7th International ISAAC Congress

Volume of Abstracts

European MathematicalSociety

InternationalMathematical Union

London MathematicalSociety

7th International ISAAC Congress—

Abstracts

Edited by M. Ruzhansky and J. Wirth. Prepared and typeset using LATEX.

Department of MathematicsImperial College London180 Queen’s GateLondon SW7 2AZ

Welcoming address

The ISAAC board, the Local Organising Committee and the Department of Mathematics at Imperial Col-lege London, are pleased to welcome you to the 7th International ISAAC Congress in London. The 7thInternational ISAAC congress continues the successful series of meetings previously held in the Delaware(USA) 1997; Fukuoka (Japan) 1999; Berlin (Germany) 2001, Toronto (Canada) 2003, Catania (Italy) 2005and Ankara (Turkey) 2007.

The success of such a series of congresses would not be possible without all the valuable contributions of allthe participants.

We acknowledge the financial support for this congress given by

the London Mathematical Society (LMS),

the International Mathematical Union (IMU),Commission on Development and Exchanges (CDE),and Developing Countries Strategy Group (DCSG),

the Engineering and Physical Sciences Research Council (EPSRC),

the Oxford Centre in Collaborational and Applied Mathematics (OCCAM),

the Oxford Centre for Nonlinear Partial Differential Equations (OxPDE),

the Bath Institute for Complex Systems (BICS),

the Imperial College London, Strategic Fund,

and the Department of Mathematics, Imperial College London.

ISAAC BoardMan Wah Wong (Toronto, Canada), President of the ISAACHeinrich Begehr (Berlin, Germany)Alain Berlinet (Montpellier, France)Bogdan Bojarski (Warsaw,Poland)Erwin Bruning (Durban, South Africa)Victor Burenkov (Padova, Italy)Okay Celebi (Istanbul, Turkey)Robert Gilbert (Newark, Delaware, USA)Anatoly Kilbas (Minsk, Belarus)Massimo Lanza de Cristoforis (Padova, Italy)Michael Reissig (Freiberg, Germany)Luigi Rodino (Torino, Italy)Michael Ruzhansky (London, UK)John Ryan (Fayetteville, Arkansas, USA)Saburou Saitoh (Aveiro, Portugal)Bert-Wolfgang Schulze (Potsdam, Germany)Joachim Toft (Vaxjo, Sweden)Yongzhi Xu (Louisville, Kentucky, USA)Masahiro Yamamoto (Tokyo, Japan)Shangyou Zhang (Newark, Delaware, USA)

Local Organising CommitteeMichael Ruzhansky (Chairman)Dan CrisanBrian DaviesJeroen LambAri Laptev (President of the European Mathematical Society)Jens WirthBoguslaw Zegarlinski

with further support by Laura Cattaneo, Federica Dragoni, Nikki Elliott, James Inglis, Vasileios Kontis andIoannis Papageorgiou as well as further student helpers.

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Abstracts

Plenary talks 15Sir John Ball : The Q-tensor theory of liquid crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Louis Boutet de Monvel : Asymptotic equivariant index of Toeplitz operators and Atiyah-Weinstein conjecture 15Brian Davies : Non-self-adjoint spectral theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Simon Donaldson : Asymptotic analysis and complex differential geometry . . . . . . . . . . . . . . . . . . . 16Carlos Kenig : The global behavior of solutions to critical nonlinear dispersive and wave equations . . . . . . 16Vakhtang Kokilashvili : Nonlinear harmonic analysis methods in boundary value problems of analytic and

harmonic functions, and PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Nicolas Lerner : Instability of the Cauchy-Kovalevskaya solution for a class of non-linear systems . . . . . . 17Paul Malliavin : Non-ergodicity of Euler deterministic fluid dynamics via stochastic analysis . . . . . . . . . 18Vladimir Maz’ya : Higher order elliptic problems in non-smooth domains . . . . . . . . . . . . . . . . . . . 18Bert-Wolfgang Schulze : Operator algebras with symbolic hierarchies on stratified spaces . . . . . . . . . . . 18Gunther Uhlmann : Visibility and Invisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Masahiro Yamamoto : Practise of industrial mathematics related with the steel manufacturing process . . . 20

Public lecture 21Pierre-Louis Lions : Analysis, Models and Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Sessions 23

I.1. Complex variables and potential theory 23Tahir Aliyev Azeroglo : Analytic functions in contour-solid problems . . . . . . . . . . . . . . . . . . . . . . 23Rauno Aulaskari : A non-α-normal function whose derivative has finite area integral of order less than 2/α 23Cristina Ballantine : Global mapping properties of rational functions . . . . . . . . . . . . . . . . . . . . . . 23Bogdan Bojarski : Beltrami equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Matteo Dalal Riva : A functional analytic approach for a singularly perturbed non-linear traction problem

in linearized elastostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Peter Dovbush : Boundary behavior of Bloch functions and normal functions . . . . . . . . . . . . . . . . . 23Anatoly Golberg : Spatial quasiconformal mappings and directional dilatations . . . . . . . . . . . . . . . . 24Dorin Ghisa : Global mapping properties of entire and meromorphic functions . . . . . . . . . . . . . . . . . 24Daniyal Israfilov : Approximation in Morrey-Smirnov classes . . . . . . . . . . . . . . . . . . . . . . . . . . 24Dmitri Karp : Two-sided bounds for the logarithmic capacity of multiple intervals . . . . . . . . . . . . . . . 24Olena Karupu : On boundary smoothness of conformal mapping . . . . . . . . . . . . . . . . . . . . . . . . 24Boris Kats : Structures of non-rectifiable curves and solvability of the jump problem . . . . . . . . . . . . . 25Gabriela Kohr : The Loewner differential equations and univalent subordination chains in several complex

variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Mirela Kohr : Boundary integral equations in the study of some porous media flow problems . . . . . . . . . 25Massimo Lanza de Cristoforis : Singular perturbation problems in potential theory: a functional analytic

approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Jamal Mamedkhanov : Classic theorems of approximation in a complex plane by rational functions . . . . 25Sergiy Plaksa : Commutative algebras of monogenic functions and biharmonic potentials . . . . . . . . . . . 26Osamu Suzuki : Fractal method for Clifford algebra and complex analysis . . . . . . . . . . . . . . . . . . . . 26Yunus Emre Yildirir : Approximation theorems in weighted Lorenz spaces . . . . . . . . . . . . . . . . . . . 26El Hassan Youssfi : Hankel operator on generalized fock spaces . . . . . . . . . . . . . . . . . . . . . . . . . 26Yuriy Zelinskiy : Continues mappings between domains of manifolds . . . . . . . . . . . . . . . . . . . . . . 26

I.2. Differential equations: Complex and functional analytic methods, applications 26Umit Aksoy : A hierarchy of polyharmonic kernel functions and the related integral operators . . . . . . . . 27Heinrich Begehr : Boundary value problems for complex partial differential equations . . . . . . . . . . . . . 27Peter Berglez : On some classes of bicomplex pseudoanalytic functions . . . . . . . . . . . . . . . . . . . . . 27Carmen Bolosteanu : Boundary value problems on Klein surfaces . . . . . . . . . . . . . . . . . . . . . . . . 27Ilya Boykov : Optimal methods for evaluation hypersingular integrals and solution of hypersingular integral

equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Okay Celebi : Complex partial differential equations with mixed-type boundary conditions . . . . . . . . . . . 28Natalia Chinchaladze : On a mathematical model of a cusped plate with big deflections . . . . . . . . . . . . 28

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Jin-Yuan Du : Mixed boundary value problem with a shift for some pair of metaanalytic function and analyticfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Grigory Giorgadze : Generalized analytic functions on Riemann surfaces . . . . . . . . . . . . . . . . . . . . 28Sonnhard Graubner : Optimization of fixed point methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Azhar Hussain : Generating functions of the Laguerre-Bernoulli polynomials involving bilateral series and

applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Alexander Kheyfits : Asymptotic behavior of subparabolic functions . . . . . . . . . . . . . . . . . . . . . . 29Giorgi Khimshiashvili : Elliptic Riemann-Hilbert problems for generalized Cauchy-Riemann systems . . . . 29Nino Manjavidze : On some qualitative issues of the elliptic systems . . . . . . . . . . . . . . . . . . . . . . 29Alip Mohammed : Poisson equation with the Robin boundary condition . . . . . . . . . . . . . . . . . . . . . 29Nusrat Rajabov : Investigation of one class of two-dimensional conjugating model and non model integral

equation with fixed super-singular kernels in connection with hyperbolic equation . . . . . . . . . . . . . 29Lutfya Rajabova : About one class of two-dimensional Volterra type integral equation with two interior

sinqular lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Roman Saks : Explicit global solutions of 3D rotating Navier-Stokes equations . . . . . . . . . . . . . . . . 30Emma Samoylova : Methods of solutions of an singular integrodifferential equation . . . . . . . . . . . . . . 30Tynysbek Sharipovich Kal’menov : A boundary condition of the volume potential . . . . . . . . . . . . . . . 30Durbudkhan Suragan : Eigenvalues and eigenfunctions of volume potential . . . . . . . . . . . . . . . . . . 30Zhaxylyk Tasmambetov : The ending solutions of Ince system with irregular features . . . . . . . . . . . . 31Ismail Taqi : Fractional integrals and hypersingular integrals in variable order Holder spaces on homogeneous

spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Yufeng Wang : On mixed boundary-value problems of polyanalytic functions . . . . . . . . . . . . . . . . . . 31Oleg N. Zhdanov : An algorithm of solving the Cauchy problem and mixed problem for the two-dimensional

system of quasi-linear hyperbolic partial differential equations . . . . . . . . . . . . . . . . . . . . . . . 32Shouguo Zhong : On solution of a kind of Riemann boundary value problem on the real axis with square roots 32Zhongxiang Zhang : Some Riemann boundary value problems in Clifford analysis . . . . . . . . . . . . . . . 32

I.3. Complex-analytical methods for applied sciences 32Vladimir Mityushev : R-linear problem and its applications to composites . . . . . . . . . . . . . . . . . . . 33Michael Porter : Application of the spectral parameter power series method to conformal mapping problems 33Sergei Rogosin : Recent results on analytic methods for 2D composite materials . . . . . . . . . . . . . . . . 33

I.4. Zeros and Gamma lines – value distributions of real and complex functions 33Grigor Barsegian : An universal value distribution: for arbitrary meromorphic function in a given domain 33Petter Branden : A generalization of the Stieltjes-Van Vleck-Bocher theorem . . . . . . . . . . . . . . . . . . 33David Cardon : A criterion for the reality of zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Marios Charalambides : New properties of a class of Jacobi and generalized Laguerre polynomials . . . . . . 34George Csordas : Meromorphic Laguerre operators and the zeros of entire functions . . . . . . . . . . . . . 34Arturo Fernandez : On the logarithmic order of meromorphic functions . . . . . . . . . . . . . . . . . . . . 34Steve Fisk : An introduction to upper (stable) polynomials in several variables . . . . . . . . . . . . . . . . . 34Paul Gauthier : Perturbations of L-functions with or without non-trivial zeros off the critical line . . . . . 34Rod Halburd : Tropical and number theoretic analogues of Nevanlinna theory . . . . . . . . . . . . . . . . . 34Aimo Hinkkanen : Growth of analytic functions in unbounded open sets . . . . . . . . . . . . . . . . . . . . 34Kazuko Kato : Zeros de la fonction holomorphe et bornee dans un polyhedre analytique de C2 . . . . . . . . 35Victor Katsnelson : Steiner and Weyl polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Anand Prakash Singh : Spiraling Baker domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Anatoliy Prykarpatsky : The algebraic Liouville integrability and the related Picard-Fuchs type equations . 35Armen Sergeev : Quantization of universal Teichmuller space: an interplay between complex analysis and

quantum field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

II.1 Clifford and quaternion analysis 36Hendrik de Bie : Clifford analysis for orthogonal, symplectic and finite reflection groups . . . . . . . . . . . 36Cinzia Bisi : Mobius transformations and Poincare distance in the quaternionic setting . . . . . . . . . . . 36Paula Cerejeiras : Wavelets invariant under reflection groups . . . . . . . . . . . . . . . . . . . . . . . . . . 36Fabrizio Colombo : Some consequences of the quaternionic functional calculus . . . . . . . . . . . . . . . . . 36Kevin Coulembier : Orthogonality of Clifford-Hermite polynomials in superspace. . . . . . . . . . . . . . . 36Sirkka-Liisa Eriksson : Recent results on hyperbolic function theory . . . . . . . . . . . . . . . . . . . . . . . 37Ming-Gang Fei : Symmetric properties of the Fourier transform in Clifford analysis setting . . . . . . . . . 37Milton Ferreira : Factorization of Mobius gyrogroups - the paravector case . . . . . . . . . . . . . . . . . . . 37Peter Franek : Higher spin analogues of the Dirac operator in two variables and its resolution . . . . . . . . 37Ghislain R. Franssens : Cauchy kernels in ultrahyperbolic Clifford analysis – Huygens cases . . . . . . . . . 37Graziano Gentili : Power series and analyticity over the quaternions . . . . . . . . . . . . . . . . . . . . . . 37Anastasia Kisil : Isomorphic action of SL(2,R) on hypercomplex numbers . . . . . . . . . . . . . . . . . . . 38Rolf Soeren Krausshar : Construction of 3D mappings on to the unit ball with the hypercomplex Szego kernel 38Lukas Krump : Explicit description of the resolution for 4 Dirac operators in dimension 6 . . . . . . . . . 38Roman Lavicka : On polynomial solutions of Moisil-Theodoresco systems in Euclidean spaces . . . . . . . . 38Matvei Libine : Quaternionic analysis, representation theory and Physics . . . . . . . . . . . . . . . . . . . 38

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Maria Elena Luna-Elizarraras : Hyperholomorphic functions in the sense of Moisil-Thodoresco and theirdifferent hyperderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Mircea Martin : Dirac and semi-Dirac pairs of differential operators . . . . . . . . . . . . . . . . . . . . . . 39

Heikki Orelma : A differential form approach to Dirac operators on surfaces . . . . . . . . . . . . . . . . . 39

Dixan Pena Pena : CK-extension and Fischer decomposition for the inframonogenic functions . . . . . . . . 39

Alessandro Perotti : A new approach to slice-regularity on real algebras . . . . . . . . . . . . . . . . . . . . . 39

Yuying Qiao : Clifford analysis with higher order kernel over unbounded domains . . . . . . . . . . . . . . . 39

Guangbin Ren : Complex Dunkl operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

John Ryan : p-Dirac equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Irene Sabadini : Duality theorems for slice hyperholomorphic functions . . . . . . . . . . . . . . . . . . . . 40

Tomas Salac : Explicit description of operators in the resolution for the Dirac operator . . . . . . . . . . . . 40

Michael Shapiro : On the relation between the Fueter operator and the Cauchy-Riemann-type operators ofClifford analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Petr Somberg : Conformally invariant boundary valued problems for spinors and families of homomorphismsof generalized Verma modules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Frank Sommen : Clifford calculus in quantum variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Vladimir Soucek : On relative BGG sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Caterina Stoppato : Regular Moebius transformations over the quaternions . . . . . . . . . . . . . . . . . . 41

Adrian Vajiac : Singularities of functions of one and several bicomplex variables . . . . . . . . . . . . . . . 41

Fabio Vlacci : Multiplicities of zeroes and poles of regular functions . . . . . . . . . . . . . . . . . . . . . . . 41

Zuzana Vlasakova : Gauss-Codazzi-Ricci equations in Riemannian, conformal, and CR geometry . . . . . . 42

Liesbet Van de Voorde : Compatibility conditions and higher spin Dirac operators . . . . . . . . . . . . . . . 42

II.2 Analytical, geometrical and numerical methods in Clifford- and Cayley-Dickson-algebras 42Swanhild Bernstein : Wavelets on spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Sebastian Bock : On special monogenic power and Laurent series expansions and applications . . . . . . . . 42

Ruth Farwell : Spin gauge models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Nelson Faustino : Further results in discrete Clifford analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Thanasis Fokas : Integrability in multidimensions, complexification and quaternions . . . . . . . . . . . . . . 43

Svetlin Georgiev : Note on the linear systems in quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Jacques Helmstetter : Minimal algorithms for Lipschitzian elements and Vahlen matrices . . . . . . . . . . 43

Jeff Hogan : Clifford-Fourier transforms and hypercomplex signal processing . . . . . . . . . . . . . . . . . . 43

Uwe Kahler : Discrete Clifford analysis by means of skew-Weyl relations . . . . . . . . . . . . . . . . . . . . 44

Vladimir Kisil : Hypercomplex analysis in the upper half-plane . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Rolf Soeren Krausshar : Formulas for reproducing kernels of solutions to polynomial Dirac equations in theannulus of the unit ball in Rn and applications to inhomogeneous Helmholtz equations . . . . . . . . . 44

Remi Leandre : The Ito transform for partial differential equations . . . . . . . . . . . . . . . . . . . . . . . 44

Dimitris Pinotsis : Quaternionic analysis and boundary value problems . . . . . . . . . . . . . . . . . . . . 44

Vitalii Shpakivskii : Integral theorems in a commutative three-dimensional harmonic algebra . . . . . . . . . 44

Wolfgang Sproßig : Initial boundary value problems with quaternionic analysis . . . . . . . . . . . . . . . . . 45

Tolksdorf, Jurgen : Real bi-graded Clifford modules, the Majorana equation and the standard model action . 45

Nelson Vieira : The regularized Schrodinger semigroup acting on tensors with values in vector bundles . . . 45

III.1. Toeplitz operators and their applications 45Cristina Camara : On the relations between the kernel of a Toeplitz operator and the solutions to some

associated Riemann-Hilbert problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Luis Castro : Convolution type operators with symmetry in exterior wedge diffraction problems . . . . . . . 45

Miroslav Englis : Berezin transform on the harmonic Fock space . . . . . . . . . . . . . . . . . . . . . . . . 46

Sergey Grudsky : Inside the eigenvalues of certain Hermitian Toeplitz band matrices . . . . . . . . . . . . . 46

Turan Gurkanlı : Toeplitz operators of M(p, q, w)(Rd) spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Oleksandr Karelin : Presentation of the kernel of a special structure matrix characteristic operator by thekernels of two operators one of them is a scalar characteristic operator . . . . . . . . . . . . . . . . . . 46

Edixon Rojas : Bounds for the kernel dimension of singular integral operators with Carleman shift . . . . . 46

Anabela Silva : Invertibility of matrix Wiener-Hopf plus Hankel operators with different Fourier symbols . . 46

Harald Upmeier : Flat Hilbert bundles and Toeplitz operators on symmetric spaces . . . . . . . . . . . . . . 47

Nikolai Vasilevski : Commutative algebras of Toeplitz operators on the unit ball . . . . . . . . . . . . . . . . 47

Kehe Zhu : Toeplitz operators on the Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

III.2. Reproducing kernels and related topics 47Belkacem Abdous : A general theory for kernel estimation of smooth functionals . . . . . . . . . . . . . . . 47

Som Datt Sharma : Weighted composition operators on some spaces of analytic functions . . . . . . . . . . 47

Keiko Fujita : Integral formulas on the boundary of some ball . . . . . . . . . . . . . . . . . . . . . . . . . . 48

John Rowland Higgins : Paley–Wiener spaces and their reproducing formulae. . . . . . . . . . . . . . . . . . 48

Darian Onchis : Irregular sampling in multiple-window spline-type spaces . . . . . . . . . . . . . . . . . . . 48

Kazuo Takemura : Free boundary value problem for (−1)M (d/dx)2M and the best constant of Sobolev inequality 48

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III.3. Modern aspects of the theory of integral transforms 48Liubov Britvina : Integral transforms related to generalized convolutions and their applications to solving

integral equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Qiuhui Chen : Bedrosian identity for Blaschke products in n-parameter cases . . . . . . . . . . . . . . . . . 49Dong Hyun Cho : Evaluation formulae for analogues of conditional analytic Feynman integrals over a

function space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Diana Dolicanin : An equation with symmetrized fractional derivatives . . . . . . . . . . . . . . . . . . . . . 49Hiroshi Fujiwara : Numerical real inversion of the Laplace transform by reproducing kernel and multiple-

precision arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Anatoly Kilbas : Method of integral transforms in the theory of fractional differential equations . . . . . . . 49Bong Jin Kim : Notes on the analytic Feynman integral over paths in abstract Wiener space . . . . . . . . . 50Sanja Konjik : On the fractional calculus of variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Anna Koroleva : Integral transforms with extended generalized Mittag-Leffler function . . . . . . . . . . . . 50Ljubica Oparnica : Systems of differential equations containing distributed order fractional derivative . . . . 50Juri M. Rappoport : Some aspects of modified Kontorovitch-Lebedev integral transforms . . . . . . . . . . . 50Semyon Yakubovich : A new class of polynomials related to the Kontorovich-Lebedev transform . . . . . . . 50

III.4. Spaces of differentiable functions of several real variables and applications 50Alexandre Almeida : Hardy spaces with generalized parameter . . . . . . . . . . . . . . . . . . . . . . . . . . 51Tsegaye Gedif Ayele : Iterated norms in Nikol’skiı-Besov type spaces with generalized smoothness . . . . . 51Ismail Aydın : Embeddings Properties of The Spaces L

p(.)w (Rn) . . . . . . . . . . . . . . . . . . . . . . . . . 51

Canay Aykol : On the boundedness of fractional B-maximal operators in the Lorentz spaces Lp,q,γ(Rn) . . . 51Oleg Besov : Spaces of functions of fractional smoothness on an irregular domain . . . . . . . . . . . . . . . 51Santiago Boza : Rearrangement transformations on general measure spaces . . . . . . . . . . . . . . . . . . 52Maria Carro : Last developments on Rubio de Francia’s extrapolation theory . . . . . . . . . . . . . . . . . . 52Gurgen Dallakyan : On transformation of coordinates invariant relative to Sobolev spaces with polyhedral

anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Ismail Ekincioglu : The boundedness of high order Riesz-Bessel transformations generated by the generalized

shift operator in weighted Lpw spaces with general weights . . . . . . . . . . . . . . . . . . . . . . . . . 52Vladimir Goldshtein : Composition Operators for Sobolev spaces of functions and differential forms . . . . . 53Vagif Guliyev : Boundedness of the fractional maximal operator and fractional integral operators in general

Morrey type spaces and some applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Mubariz Hajibayov : Weighted estimates of generalized potentials in variable exponent Lebesque spaces . . . 53Ritva Hurri-Syrjanen : Our talk is on vanishing exponential integrability for Besov functions. . . . . . . . . 53Gennady Kalyabin : New sharp estimates for function in Sobolev spaces on finite Interval . . . . . . . . . . 53Leili Kusainova : On real interpolation of weighted Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . 53Elijah Liflyand : The Fourier transform of a radial function . . . . . . . . . . . . . . . . . . . . . . . . . . 54Yagub Mammadov : Necessary and sufficient conditions for the boundedness of Riesz potential in Morrey

spaces associated with Dunkl operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Ana Moura Santos : Image normalization of Wiener-Hopf operators in diffraction problems . . . . . . . . . 54Bohumır Opic : Weighted estimates for the averaging integral operator and reverse Holder inequalities . . . 54Humberto Rafeiro : Characterization of the variable exponent Bessel potential spaces via the Poisson semigroup 55Evgeniy Radkevich : On the Maxwell problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Natasha Samko : Weighted potential operators in Morrey spaces. . . . . . . . . . . . . . . . . . . . . . . . . 55Stefan Samko : Fractional integrals and hypersingular integrals in variable order Holder spaces on homoge-

neous spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Kader Senouci : Equivalent semi-norms for Nikol’skii- Besov spaces on an interval . . . . . . . . . . . . . . 55Ayhan Serbetci : Stein-Weiss inequalities for the fractional integral operators in Carnot groups and applications 55Javier Soria : Translation-invariant bilinear operators with positive kernels . . . . . . . . . . . . . . . . . . . 55Sergey Tikhonov : Sharp inequalities for moduli of smoothness and K-functionals . . . . . . . . . . . . . . . 56Boris V. Trushin : Sobolev embedding theorems for a class of anisotropic irregular domains . . . . . . . . . 56Yusuf Zeren : Necessary and sufficient conditions for the boundedness of the Riesz potential in modified

Morrey spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

III.5. Analytic and harmonic function spaces 56Miloud Assal : Multiplier theorem in the setting of Laguerre hypergroups and applications . . . . . . . . . . 56Boo Rim Choe : Progress on finite rank Toeplitz products . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56Daniel Girela : Functions and operators in analytic Besov spaces . . . . . . . . . . . . . . . . . . . . . . . . 56Maria Jose Gonzales : Square functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56Sanjiv Gupta : Convolutions of generic orbital measures in compact symmetric spaces . . . . . . . . . . . . 57H. Turgay Kaptanoglu : Harmonic Besov spaces on the real unit ball: reproducing kernels and Bergman

projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Young Joo Lee : Sums of Toeplitz products on the Dirichlet space . . . . . . . . . . . . . . . . . . . . . . . . 57Jasbir Singh Manhas : Weighted composition operators on weighted spaces of analytic functions . . . . . . . 57Auxiliadora Marquez : Superposition operators between Qp spaces and Hardy spaces . . . . . . . . . . . . . 57Malgorzata Michalska : Bounded Toeplitz and Hankel products on Bergman space . . . . . . . . . . . . . . . 57

8

Kyesook Nam : Optimal norm estimate of the harmonic Bergman projection . . . . . . . . . . . . . . . . . 57Pekka Nieminen : Old and new on composition operators on VMOA and BMOA spaces . . . . . . . . . . . 58Maria Nowak : On Libera and Cesaro operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Jordi Pau : Integration operators on weighted Bergman spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 58Amol Sasane : Extension to an invertible matrix in Banach algebras of measures . . . . . . . . . . . . . . . 58Benoit F. Sehba : Multiplication operators on weighted BMOA spaces . . . . . . . . . . . . . . . . . . . . . . 58Pawel Sobolewski : Inequalities for Hardy spaces on the unit ball . . . . . . . . . . . . . . . . . . . . . . . . 58Mubariz Tapdıgoglu : On the Duhamel algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Jari Taskinen : Toeplitz operators on Bergman spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Luis Manuel Tovar : Hyperbolic weighted Bergman classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Dragan Vukotic : Multiplicative isometries and isometric zero-divisors . . . . . . . . . . . . . . . . . . . . . 59Zhijian Wu : Area operators on analytic function spacess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Hasi Wulan : Composition operators on BMOA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Wen Xu : Lacunary series and QK spaces on the unit ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Congli Yang : Some results on ϕ-Bloch functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Kehe Zhu : Holomorphic mean Lipschitz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Nina Zorboska : Univalently induced closed range composition operators on the Bloch-type spaces . . . . . . 59

III.6. Spectral theory 59Mikhael Agranovich : Strongly elliptic second-order systems in Lipschitz domains: surface potentials, equa-

tions at the boundary, and corresponding transmission problems. . . . . . . . . . . . . . . . . . . . . . 60Shavkat Alimov : On the spectral expansions associated with Laplace-Beltrami operator . . . . . . . . . . . . 60Victor Burenkov : Sharp spectral stability estimates for higher order elliptic operators . . . . . . . . . . . . 60Daniel Elton : Strong field asymptotics for zero modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Leander Geisinger : A universal bound for the trace of the heat kernel . . . . . . . . . . . . . . . . . . . . . 60Tigran Harutyunyan : The eigenvalues function of the family of Sturm-Liouville operators and its applications 61Jan Janas : Generalized eigenvectors of some Jacobi matrices in the critical case . . . . . . . . . . . . . . . 61Thomas Krainer : Trace expansions for elliptic cone operators . . . . . . . . . . . . . . . . . . . . . . . . . . 61Pier Domenico Lamberti : Stability estimates for eigenfunctions of elliptic operators on variable domains . . 61Oleksii Mokhonko : Spectral theory of the normal operator with the spectra on an algebraic curve . . . . . . 61Jiri Neustupa : Spectral properties of operators arising from modelling of flows around rotating bodies . . . . 62Serge Richard : New formulae for the wave operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Benedetto Silvestri : Spectral bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Alexander Strohmaier : Scattering theory for manifolds and the scattering length . . . . . . . . . . . . . . . 62Yuriy Tomilov : Spectrum and wandering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Tomio Umeda : Eigenfunctions at the threshold energies of magnetic Dirac operators . . . . . . . . . . . . . 62

IV.1. Pseudo-differential operators 62Mikhael Agranovich : Strongly elliptic second-order systems in Lipschitz domains: Dirichlet and Neumann

problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Chikh Bouzar : Generalized ultradistributions and their microlocal analysis . . . . . . . . . . . . . . . . . . 63Ernesto Buzano : Some remarks on the Sjostrand class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Viorel Catana : The heat equation for the generalized Hermite and the generalized Landau operators . . . . 63Leon Cohen : Generalization of the Weyl rule for arbitrary operators . . . . . . . . . . . . . . . . . . . . . . 63Elena Cordero : Sharp results for the STFT and localization operators . . . . . . . . . . . . . . . . . . . . . 63Yasuo Chiba : Fuchsian mild microfunctions with fractional order and their applications to hyperbolic equations 63Paulo Dattori da Silva : About Gevrey semi-global solvability of a class of complex planar vector fields with

degeneracies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Julio Delgado : Invertibility for a class of degenerate elliptic operators . . . . . . . . . . . . . . . . . . . . . 64Kenro Furutani : Heat kernel of a sub-Laplacian and Grushin type operators . . . . . . . . . . . . . . . . . . 64Lorenzo Galleani : Time-frequency analysis of stochastic differential equations . . . . . . . . . . . . . . . . . 64Gianluca Garello : Lp-microlocal regularity for pseudodifferential operators of quasi-homogeneous type . . . 64Claudia Garetto : Generalized Fourier integral operators methods for hyperbolic problems . . . . . . . . . . . 65Juan Gil : Resolvents of regular singular elliptic operators on a quantum graph . . . . . . . . . . . . . . . . 65Todor Gramchev : Hyperbolic systems of pseudodifferential equations with irregular symbols in t admitting

superlinear growth for |x| → ∞. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Bernhard Gramsch : Analytic perturbations for special Frechet operator algebras in the microlocal analysis . 65Gunther Hormann : The Cauchy problem for a paraxial wave equation with non-smooth symbols . . . . . . 65Eugenie Hunsicker : Pseudodifferential operators on locally symmetric spaces . . . . . . . . . . . . . . . . . 65Wataru Ichinose : On the continuity of the solutions with respect to the electromagnetic potentials to the

Schrodinger and the Dirac equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Chisato Iwasaki : Calculus of pseudo-differential operators and a local index of Dirac operators . . . . . . . 66Jon Johnsen : On the theory of type 1, 1-operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Yuryi Karlovych : Pseudo-differential operators with discontinuous symbols and their applications . . . . . . 66Thomas Krainer : On maximal regularity for parabolic equations on complete Riemannian manifolds . . . . 66Roberto de Leo : On the cohomological equation in the plane for regular vector fields . . . . . . . . . . . . 66

9

Yu Liu : Lp-boundedness and compactness of localization operators associated with Stockwell transform . . . 66

Jean-Andre Marti : About transport equation with irregular coefficient and data . . . . . . . . . . . . . . . . 67

Shahla Molahajloo : The Heat Kernel of τ -Twisted Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Alessandro Morando : Regularity of characteristic initial-boundary value problems for symmetrizable systems 67

David Natroshvili : Application of pseudodifferential equations in stress singularity analysis for thermo-electro-magneto-elasticity problems: a new approach for calculation of stress singularity exponents . . . 67

Alessandro Oliaro : Wigner type transforms and pseudodifferential operators . . . . . . . . . . . . . . . . . . 67

Michael Oberguggenberger : Local regularity of solutions to PDEs by asymptotic methods . . . . . . . . . . 68

Nusrat Rajabov : Modern results by theory of the three dimensional Volterra type linear integral equationswith singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Frederic Rochon : The adiabatic limit of the Chern character . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Bert-Wolfgang Schulze : Boundary value problems as edge problems . . . . . . . . . . . . . . . . . . . . . . 68

Elmar Schrohe : Noncommutative residues and projections associated to boundary value problems . . . . . 68

Jorg Seiler : On maximal regularity for mixed order systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Tatyana Shaposhnikova : Dirichlet problem for higher order elliptic systems with BMO assumptions on thecoefficients and the boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Hidetoshi Tahara : Gevrey regularities of solutions of nonlinear singular partial differential equations . . . . 69

Nenad Teofanov : Wave-front sets and SG type operators in Fourier-Lebesgue spaces . . . . . . . . . . . . . 69

Joachim Toft : Wave-front sets of Fourier Lebesgue types . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Ville Turunen : Pseudo-differential operators and symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Vladimir Vasilyev : Pseudo differential equations and boundary value problems in non-smooth domains . . 70

Andras Vasy : Diffraction at corners for the wave equation on differential forms . . . . . . . . . . . . . . . 70

Ingo Witt : Formation of singularities near Morse points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Man Wah Wong : Phases of modified Stockwell transforms and instantaneous frequencies . . . . . . . . . . 70

Hongmei Zhu : Generalized cosine transforms in image compression . . . . . . . . . . . . . . . . . . . . . . 70

IV.2. Dispersive equations 71Marcello D’Abbico : Lp–Lq estimates for hyperbolic systems . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Q-Heung Choi : Multiple solutions for non-linear parabolic systems . . . . . . . . . . . . . . . . . . . . . . 71

Ferruccio Colombini : Local sovability beyond condition ψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Daniele Del Santo : Continuous dependence for backward parabolic operators with Log-Lipschitz coefficients 71

Marcello Ebert : On the loss of regularity for a class of weakly hyperbolic operators . . . . . . . . . . . . . 72

Daoyuan Fang : Zakharov system in infinite energy space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Anahit Galstyan : Wave equation in Einstein-de Sitter spacetime . . . . . . . . . . . . . . . . . . . . . . . . 72

Vladimir Georgiev : Stability of solitary waves for Hartree type equation . . . . . . . . . . . . . . . . . . . . 72

Marina Ghisi : Hyperbolic-parabolic singular perturbations for Kirchhoff-equations . . . . . . . . . . . . . . . 72

Massimo Gobbino : Existence and uniqueness results for Kirchhoff equations in Gevrey-type spaces . . . . . 73

Torsten Herrmann : Precise loss of derivatives for evolution type models . . . . . . . . . . . . . . . . . . . . 73

Fumihiko Hirosawa : Wave equations with time dependent coefficients . . . . . . . . . . . . . . . . . . . . . 73

Tacksun Jung : Critical point theory applied to a class of systems of super-quadratic wave equations . . . . 73

Lavi Karp : On the well-posdness of the vacuum Einstein equations . . . . . . . . . . . . . . . . . . . . . . 73

Hideo Kubo : Generalized wave operator for a system of nonlinear wave equations . . . . . . . . . . . . . . 74

Tokio Matsuyama : Strichartz estimates for hyperbolic equations in an exterior domain . . . . . . . . . . . 74

Kiyoshi Mochizuki : Uniform resolvent estimates and smoothing effects for magnetic Schrodinger operators 74

Hideo Nakazawa : Decay and scattering for wave equations with dissipations in layered media . . . . . . . . 74

Tatsuo Nishitani : On the Cauchy problem for non-effectively hyperbolic operators, the Gevrey 4 well-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Rainer Picard : On the structure of the material law in a linear model of poro-elasticity . . . . . . . . . . . 74

Marco Pivetta : Backward uniqueness for the system of thermo-elastic waves with non-lipschitz continuouscoefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Michael Reissig : The log-effect for 2 by 2 hyperbolic systems . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Jun-ichi Saito : The Boussinesq equations based on the hydrostatic approximation . . . . . . . . . . . . . . . 75

Ryuichi Suzuki : Blow-up of solutions of a quasilinear parablolic equation . . . . . . . . . . . . . . . . . . . 75

Hiroshi Uesaka : Blow-up and a blow-up boundary for a semilinear wave equation with some convolutionnonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Karen Yagdjian : Fundamental solutions for hyperbolic operators with variable coefficients . . . . . . . . . . 75

Borislav Yordanov : Global existence in Sobolev spaces for a class of nonlinear Kirchhoff equations . . . . . 76

IV.3. Control and optimisation of nonlinear evolutionary systems 76Lorena Bociu : Global well-posedness and long-time behavior of solutions to a wave equation . . . . . . . . . 76

Mahdi Boukrouche : Distributed optimal controls for second kind parabolic variational inequalities . . . . . 76

Muriel Boulakia : Controllability of a fluid-structure interaction problem . . . . . . . . . . . . . . . . . . . 76

Marcello Cavalcanti : Uniform decay rate estimates for the wave equation on compact surfaces and locallydistributed damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Moez Daoulatli : Rate of decay for non-autonomous damped wave systems . . . . . . . . . . . . . . . . . . . 77

10

Valeria Domingos Cavalcanti : On qualitative aspects for the damped Korteweg-de Vries and Airy typeequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Matthias Eller : Optimal control of waves in anisotropic media via conservative boundary conditions . . . . 77Genni Fragnelli : Stability for some nonlinear damped wave equations . . . . . . . . . . . . . . . . . . . . . 77Anahit Galstyan : Global existence for the one-dimensional semilinear Tricomi-type equation . . . . . . . . 77Catherine Lebiedzik : Optimal control of a thermoelastic structural acoustic model . . . . . . . . . . . . . . 78Walter Littman : The Balayage method: Boundary control of a thermo-elastic plate . . . . . . . . . . . . . . 78Paola Loreti : Hopf-Lax type formulas and Hamilton-Jacobi equations . . . . . . . . . . . . . . . . . . . . . 78Vyacheslav Maksimov : Investigation of boundary control problems by on-line inversion technique . . . . . . 78Patrick Martinez : Null controllability properties of some degenerate parabolic equations . . . . . . . . . . . 78Maria Grazia Naso : Dissipation in contact problems: an overview and some recent results . . . . . . . . . 78Luciano Pandolfi : Heat equations with memory: a Riesz basis approach . . . . . . . . . . . . . . . . . . . . 79Michael Renardy : A note on a class of observability problems for PDEs . . . . . . . . . . . . . . . . . . . . 79Roland Schnaubelt : Invariant manifolds for parabolic problems with dynamical boundary conditions . . . . 79Ilya Shvartsman : On regularity properties of optimal control and Lagrange multipliers . . . . . . . . . . . . 79Daniela Sforza : Evolution equations with memory terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Daniel Toundykov : Stabilization of structure-acoustics interactions for a Reissner-Mindlin plate by localized

nonlinear boundary feedbacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Julie Valein : Exponential stability of the wave equation with boundary time varying delay . . . . . . . . . . 80Masahiro Yamamoto : State estimation for some parabolic systems . . . . . . . . . . . . . . . . . . . . . . . 80Jean-Paul Zolesio : Euler flow and Morphing Shape Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

IV.4. Nonlinear partial differential equations 80Piero D’Ancona : Evolution equations in nonflat waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . 80Mersaid Aripov : Investigation of solutions of one not divergent type . . . . . . . . . . . . . . . . . . . . . . 80Davide Catania : Asymptotic behavior of subparabolic functions . . . . . . . . . . . . . . . . . . . . . . . . 81Kuan-Ju Chen : On multiple solutions of concave and convex effects for nonlinear elliptic equation on RN 81Kazuyuki Doi : Nonlinear gauge invariant evolution of the plane wave . . . . . . . . . . . . . . . . . . . . . 81Mohammad Dehghan : New approach to solve linear parabolic problems via semigroup approximation . . . 81Albert Erkip : Global existence and blow-up for the nonlocal nonlinear Cauchy problem . . . . . . . . . . . . 81Marius Ghergu : Qualitative properties for reaction-diffusion systems modelling chemical reactions . . . . . 81Marco Antonio Taneco-Hernandez : Scattering in the zero-mass Lamb system . . . . . . . . . . . . . . . . . 82Soichiro Katayama : Global existence for systems of the nonlinear wave and Klein-Gordon equations in 3D 82Hideo Kubo : Global existence for nonlinear wave equations exterior to an obstacle in 2D . . . . . . . . . . 82Petr Kucera : Remark on Navier-Stokes equations with mixed boundary conditions . . . . . . . . . . . . . . 82Ut van Le : Contraction-Galerkin method for a semi-linear wave equation with a boundary-like antiperiodic

condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82Sandra Lucente : p− q systems of nonlinear Schrodinger equations . . . . . . . . . . . . . . . . . . . . . . . 83Satoshi Masaki : Semiclassical analysis for nonlinear Schrodinger equations . . . . . . . . . . . . . . . . . . 83Gianluca Mola : 3-D viscous Cahn-Hilliard equation with memory . . . . . . . . . . . . . . . . . . . . . . . 83Itir Mogultay : A symmetric error estimate for Galerkin approximations of time dependant Navier-Stokes

equations in two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83Masahito Ohta : Stability of standing waves for some systems of nonlinear Schrodinger equations with

three-wave interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83Michael Reissig : Decay rates for wave models with structural damping . . . . . . . . . . . . . . . . . . . . . 83Yoshihiro Shibata : Stability theorems in the theory of mathematical fluid mechanics . . . . . . . . . . . . . 84Laszlo Simon : On singular systems of parabolic functional equations . . . . . . . . . . . . . . . . . . . . . 84Zdenek Skalak : Survey of recent results on asymptotic energy concentration in solutions of the Navier-Stokes

equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Atanas Stefanov : Conditional stability theorems for Klein-Gordon type equations . . . . . . . . . . . . . . 84Sergio Spagnolo : A regularity result for a class of semilinear hyperbolic equations . . . . . . . . . . . . . . . 84Kamal Soltanov : On nonlinear equations, fixed-point theorems and their applications . . . . . . . . . . . . . 84Alessandro Teta : Dynamics of a quantum particle in a cloud chamber . . . . . . . . . . . . . . . . . . . . . 84Yoshihiro Ueda : Half space problem for the damped wave equation with a non-convex convection term . . . 84Nicola Visciglia : On the time-decay of solutions to a family of defocusing NLS . . . . . . . . . . . . . . . . 85Karen Yagdjian : The semilinear Klein-Gordon equation in de Sitter spacetime . . . . . . . . . . . . . . . . 85

IV.5. Asymptotic and multiscale analysis 85Natalia Babych : On the essential spectrum and singularities of solutions for Lame problem in cuspoidal

domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85Michel Bellieud : Torsion effects in elastic composites with high contrast . . . . . . . . . . . . . . . . . . . . 85Yves Capdeboscq : Enhanced resolution in structured media . . . . . . . . . . . . . . . . . . . . . . . . . . . 85Juan Casado-Diaz : Homogenization of elliptic partial differential equations with unbounded coefficients in

dimension two . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86Mikhail Cherdantsev : Two-scale Γ-convergence and its applications to homogenisation of non-linear high-

contrast problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

11

Valentina Alekseevna Golubeva : Construction of the two-parametric generalizations of the Knizhnik-Zamolodchikov equations of Bn type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Fabricio Macia : Long-time behavior for the Wigner equation and semiclassical limits in heterogeneous media 86Peter Markowich : On nonlinear dispersive equations in periodic structures: Semiclassical limits and nu-

merical schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86Karsten Matthies : Derivation of Boltzmann-type equations from hard-sphere dynamics . . . . . . . . . . . 87Bernd Schmidt : Minimizing atomic configurations of short range pair potentials in two dimensions: crys-

tallization in the Wulff shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Valery Smyshlyaev : Homogenization with partial degeneracies: analytic aspects and applications . . . . . . 87

V.1. Inverse problems 87Abdellatif El Badia : An inverse conductivity problem with a single measurement . . . . . . . . . . . . . . . 87Fabrizio Colombo : Global in time existence and uniqueness results for some integrodifferential identification

problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Mourad Choulli : Stability estimate for an inverse problem for the magnetic Schrodinger equation from the

Dirichlet-to-Neumann map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Mikko Kaasalainen : Optimal combination of data modes in inverse problems: maximum compatibility estimate 88Christian Daveau : On an inverse problem for a linear heat conduction problem . . . . . . . . . . . . . . . 88Matti Lassas : Inverse problems for wave equation and a modified time reversal method . . . . . . . . . . . . 88Koung Hee Leem : Picard condition based regularization techniques in inverse obstacle scattering . . . . . . 88William Lionheart : Limited data problems in tensor tomography . . . . . . . . . . . . . . . . . . . . . . . . 89Marco Marletta : The finite data non-selfadjoint inverse resonance problem . . . . . . . . . . . . . . . . . . 89Tsutomu Matsuura : Numerical solutions of nonlinear simultaneous equations . . . . . . . . . . . . . . . . . 89George Pelekanos : A fixed-point algorithm for determining the regularization parameter in inverse scattering 89Roland Potthast : A time domain probe method for inverse scattering problems . . . . . . . . . . . . . . . . 89Saburou Saitoh : Explicit and direct representations of the solutions of nonlinear simultaneous equations . 89Vassilios Sevroglou : Direct and inverse mixed impedance problems in linear elasticity . . . . . . . . . . . . 89Igor Trooshin : On inverse scattering for nonsymmetric operators . . . . . . . . . . . . . . . . . . . . . . . 90

V.2. Stochastic analysis 90David Applebaum : Cylindrical Levy processes in Banach space . . . . . . . . . . . . . . . . . . . . . . . . 90Vlad Bally : Integration by parts for locally smooth laws and applications to jump type diffusions . . . . . . 90Dorje Brody : Information and asset pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Michael Caruana : A (rough) pathwise approach to fully non-linear stochastic partial differential equations . 91Dan Crisan : Solving backward stochastic differential equations using cubature methods . . . . . . . . . . . . 91Ana Bela Cruzeiro : Some results on Lagrangian Navier-Stokes flows . . . . . . . . . . . . . . . . . . . . . 91Alexander Davie : A uniqueness problem for SDEs and a related estimate for transition functions . . . . . 91Mark H. A. Davis : Risk-sensitive portfolio optimization with jump-diffusion asset prices . . . . . . . . . . 91Istvan Gyongy : Accelerated numerical schemes for nonlinear filtering . . . . . . . . . . . . . . . . . . . . . 91Martin Hairer : Periodic homogenisation with an interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Lane Hughston : Wiener chaos models for interest rates and foreign exchange . . . . . . . . . . . . . . . . . 92Saul Jacka : Minimising the time to a decision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Mark Kelbert : Markov process representations for polyharmonic functions . . . . . . . . . . . . . . . . . . 92Wilfried Kendall : Networks and Poisson line patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Vassili Kolokoltsov : The Levy-Khinchine type operators with variable Lipschitz continuous coefficients and

stochastic differential equations driven by nonlinear Levy noise . . . . . . . . . . . . . . . . . . . . . . 92Thomas Kurtz : Equivalence of stochastic equations and martingale problems . . . . . . . . . . . . . . . . . 92Xue-Mei Li : Aida’s logarithmic Sobolev inequality with weights and Poincare inequalities. . . . . . . . . . . 93Terence Lyons : Evolution equations for communities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93Aleksandar Mijatovic : On the martingale property of certain local martingale . . . . . . . . . . . . . . . . . 93Khairia El-Said El-Nadi : On some stochastic dynamical systems and cancer . . . . . . . . . . . . . . . . . 93Anastasia Papavasiliou : Statistical inference for rough differential equations . . . . . . . . . . . . . . . . . 93Martijn Pistorius : First passage for stochastic volatility models . . . . . . . . . . . . . . . . . . . . . . . . . 94Boris Rozovsky : Unbiased random perturbations of Navier-Stokes equation . . . . . . . . . . . . . . . . . . 94Marta Sanz-Sole : A Poisson equation with fractional noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 94Radu Tunaru : Constructing discrete exact approximations algorithms for financial calculus from weak

convergence results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94Michael Tretyakov : Numerical methods for parabolic SPDEs based on the averaging-over-characteristics

formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94Elena Usoltseva : Consistent estimator in AFTM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

V.3. Coercivity and functional inequalities 95Franck Barthe : Remarks on non-interacting conservative spin systems . . . . . . . . . . . . . . . . . . . . . 95Sergey Bobkov : On weak forms of Poincare-type inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 95Messaoud Boulbrachene : L∞-Error estimate for variational inequalities with vanishing zero order term . . 95Federica Dragoni : Convexity along vector fields and application to equations of Monge-Ampere type . . . . 95Ivan Gentil : Φ-entropy inequalities for diffusion semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

12

Alexander Grigoryan : On positive solutions of semi-linear elliptic inequalities on manifolds . . . . . . . . 95Martin Hairer : Hypoellipticity in infinite dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Waldemar Hebisch : Logaritmic Sobolev inequality on nilpotent groups . . . . . . . . . . . . . . . . . . . . . 96Nolwen Huet : Isoperimetry for spherically symmetric log-concave probability measures . . . . . . . . . . . . 96James Inglis : Operators on the Heisenberg group with discrete spectra . . . . . . . . . . . . . . . . . . . . . 96Mikhail Neklyudov : Liggett inequality and interacting particle systems . . . . . . . . . . . . . . . . . . . . . 96Felix Otto : A new criterion for a covariance estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Ioannis Papageorgiou : The Log-Sobolev inequality for non quadratic interactions . . . . . . . . . . . . . . . 96Cyril Roberto : Isoperimetry for product probability measures . . . . . . . . . . . . . . . . . . . . . . . . . . 96

V.4. Dynamical systems 97Marco Abate : Poincare-Bendixson theorems in holomorphic dynamics . . . . . . . . . . . . . . . . . . . . 97Jose Ferreira Alves : On the liftability of absolutely continuous ergodic expanding measures. . . . . . . . . . 97Flavio Abdenur : New results on stability and genericity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Pierre Berger : Abundance of one dimensional non uniformly hyperbolic attractors for surface dynamics . . 97Svetlana Aleksandrovna Budochkina : First integrals in mechanics of infinite-dimensional systems . . . . . 97Keith Burns : Partial hyperbolicity and ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Jean-Rene Chazottes : On tilings, multidimensional subshifts of finite type and quasicrystals . . . . . . . . . 98Yi-Chiuan Chen : On topological entropy of billiard tables with small inner scatterers . . . . . . . . . . . . . 98Bau-Sen Du : On the nature of chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98Michael Field : Mixing for flows and skew extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98Jorge Freitas : Rates of mixing, large deviations and recurrence times . . . . . . . . . . . . . . . . . . . . . 98Giovanni Forni : Limiting distributions for horocycle flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98Valery Gaiko : Limit cycle problems and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98Thomas Jordan : Hausdorff dimension of Projections of McMullen-Bedford carpets . . . . . . . . . . . . . 99Jan Cees van der Meer : Fourfold 1:1 resonance, relative equilibria and moment polytopes . . . . . . . . . . 99Matthew Nicol : A dynamical Borel-Cantelli lemma for a class of non-uniformly hyperbolic systems . . . . 99Asad Niknam : Approximately inner C∗-dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 99Giovanni Panti : Dynamical systems arising in algebraic logic . . . . . . . . . . . . . . . . . . . . . . . . . . 99Chen-chang Peng : Existence of transversal homoclinic orbits for Arneodo-Coullet-Tresser map . . . . . . . 99Martin Rasmussen : Bifurcations of random diffeomorphisms with bounded noise . . . . . . . . . . . . . . . 100Felix Sadyrbaev : Bifurcations of period annuli and solutions of nonlinear boundary value problems . . . . . 100Jorg Schmeling : Large intersection properties of some invariant sets in number-theoretic dynamical systems 100Mike Todd : Thermodynamic formalism for unimodal maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 100Qiudong Wang : Dynamics of periodically perturbed homoclinic solutions . . . . . . . . . . . . . . . . . . . . 100

V.5. Functional differential and difference equations 100Jaromır Bastinec : Oscillation and non-oscillation of solutions of linear second order discrete delayed equa-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100Leonid Berezansky : New stability conditions for linear differential equations with several delays . . . . . . . 101Aleksandr Boichuk : Boundary-value problems for differential systems with a single delay . . . . . . . . . . 101Josef Diblık : Representation of solutions of linear differential and discrete systems and their controllability 101Alexander Domoshnitsky : Maximum principles and nonoscillation intervals in the theory of functional

differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101Marcia Federson : Averaging for impulsive functional differential equations: a new approach . . . . . . . . . 101Yakov Goltser : Some bifurcation problems in the theory quasilinear integro differential equations . . . . . . 102Istvan Gyori : Stability in Volterra type population model equations with delays . . . . . . . . . . . . . . . . 102Ferenc Hartung : On parameter dependence in functional differential equations with state-dependent delays 102Zeynep Kayar : Lyapunov type inequalities for nonlinear impulsive differential systems . . . . . . . . . . . . 102Conall Kelly : Evaluating the stochastic theta method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102Gabor Kiss : Delay-distribution effect on stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102Martina Langerova : Solutions of linear impulsive differential systems bounded on the entire real axis . . . . 102Malgorzata Migda : Oscillatory and asymptotic properties of solutions of higher-order difference equations

of neutral type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Abdullah Ozbekler : Principal and non-principal solutions of impulsive differential equations with applications103Mihali Pituk : Nonnegative iterations with asymptotically constant coefficients . . . . . . . . . . . . . . . . . 103Irena Rachunkova : On singular models arising in hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . 103Andrejs Reinfelds : Decoupling and simplifying of noninvertible difference equations in the neighbourhood of

invariant manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103David W. Reynolds : Precise asymptotic behaviour of solutions of Volterra equations with delay . . . . . . 103Alexandra Rodkina : On local stability of solutions of stochastic difference equations . . . . . . . . . . . . . 104Miroslava Ruzickova : Convergence of the solutions of a differential equation with two delayed terms . . . . 104Vladimir Mikhailovich Savchin : Inverse problems of the calculus of variations for functional differential

equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104Ewa Schmeidel : Existence and nonexistence of asymptotically periodic solutions of Volterra linear difference

equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

13

Andrei Shindiapin : Gene regulatory networks and delay equations . . . . . . . . . . . . . . . . . . . . . . . 104Benzion Shklyar : The moment problem approach for the zero controllability of ecolution equations . . . . . 105Svatoslav Stanek : Properties of maximal solutions of autonomous functional-differential equations with

state-dependent deviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105Stevo Stevic : Boundedness character of some classes of difference equations . . . . . . . . . . . . . . . . . . 105Milan Tvrdy : Continuous dependence of solutions of generalized ordinary differential equations on a parameter105Mehmet Unal : Lyapunov type inequalities on time scales: A survey . . . . . . . . . . . . . . . . . . . . . . 105Agacık Zafer : Interval criteria for oscillation of delay dynamic equations with mixed nonlinearities . . . . . 105

V.6. Mathematical biology 105Robert Gilbert : Cancellous bone with a random pore structure . . . . . . . . . . . . . . . . . . . . . . . . . 106Irina Alekseevna Gainova : New computer technologies for the construction and numerical analysis of math-

ematical models for molecular genetic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106Sandra Ilic : Application of the multiscale FEM in modeling the cancellous bone . . . . . . . . . . . . . . . 106Mark D. Ryser : Bone growth and destruction at the cellular level: a mathematical model . . . . . . . . . . 106

VI. Others 106Ruben Airapetyan : The relationship between Bezoutian matrix and Newton’s matrix of divided differences

and separation of zeros of interpolation polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106Hadeel Alkutubi : Bayesian shrinkage estimation of parameter exponential distribution . . . . . . . . . . . . 107Mohammed Bokhari : Interpolation beyond the interval of convergence: An extension of Erdos-Turan Theorem107Zoubir Dahmani : The ADM method and the Tanh method for solving some non linear evolutions equations 107Anvar Hasanov : Boundary-value problems for generalized axially-symmetric Helmholtz equation . . . . . . . 107Maximilian Hasler : Asymptotic extension of topological modules and algebras . . . . . . . . . . . . . . . . . 107S. Moghtada Hashemiparast : Approximation of fractional derivatives . . . . . . . . . . . . . . . . . . . . . 107Hailiza Kamarulhaili : Discrepancy estimate for uniformly distributed sequence . . . . . . . . . . . . . . . . 108Erdal Karapinar : Bounded linear operators on l-power series spaces . . . . . . . . . . . . . . . . . . . . . . 108Erkinjon Karimov : On a three-dimensional elliptic equation with singular coefficients . . . . . . . . . . . . 108Nabiullah Khan : A unified presentation of a class of generalized Humbert polynomials . . . . . . . . . . . . 108Lixia Liu : Direct estimate for modified beta operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108Eduard Marusic-Paloka : Mathematical model of an undergorund nuclear waste disposal site . . . . . . . . . 108Abdeslam Mimouni : Compact and coprime packedness and semistar operations . . . . . . . . . . . . . . . 108S. A. Mohiuddine : Characterization of some matrix classes involving (σ, λ)-convergence . . . . . . . . . . . 109Mohammad Mursaleen : Sequence spaces of invariant mean and some matrix transformations . . . . . . . . 109Ali Mussa : New convection theory for thermal plasma and NHD convection in rapidly rotating spherical

configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109Kourosh Nourouzi : Characterizations of Isometries on 2-modular spaces . . . . . . . . . . . . . . . . . . . . 109Shariefuddin Pirzada : On r-imbalances in tripartite r-digraphs . . . . . . . . . . . . . . . . . . . . . . . . . 110Hashem Parvaneh Masiha : Invariance conditions and amenability of locally compact groups . . . . . . . . . 110Zaure Rakisheva : Motion stabilisation of a solid body with fixed point . . . . . . . . . . . . . . . . . . . . . 110Lyazzat Sarybekova : A Lizorkin type theorem for Fourier series multipliers in regular systems . . . . . . . 110Pedro A. Santos : Inverse-closedness problems in the stability of sequences in Banach Algebras . . . . . . . 110Ridha Selmi : Smoothing effects for periodic NSE in critical Sobolev space . . . . . . . . . . . . . . . . . . . 110Mariana Sibiceanu : Large deviations and almost sure convergence . . . . . . . . . . . . . . . . . . . . . . . 111Tanfer Tanriverdi : The k-ε Model in Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Johnson Olaleru : The equivalence between modified Mann (with errors), Ishikawa (with errors), Noor

(with errors) and modified multi-step iterations (with errors) for non-Lipschitzian strongly successivelypseudo-contractive operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Serap Oztop : A characterization for multipliers of weighted Banach valued Lp(G)-spaces . . . . . . . . . . 111Karlyga Zhilisbaeva : Stationary motion of the dynamical symmetric satellite in the geomagnetic field . . . 111

Index 113

14

Plenary talks

The Q-tensor theory of liquid crystals

Sir John BallMathematical Institute, 24-29 St Giles’, Oxford OX1 3LB, U.K.ball@maths.ox.ac.uk

The lecture will survey what is known about the mathematics of the de Gennes Q-tensor theory for describ-ing nematic liquid crystals. This theory, despite its popularity with physicists, has been little studied bymathematicians and poses many interesting questions. In particular the lecture will describe the relation ofthe theory to other theories of liquid crystals, specifically those of Oseen-Frank and Onsager/Maier-Saupe.This is joint work with Apala Majumdar and Arghir Zarnescu.

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Sir John Ball, FRS, is Sedleian Professor of Natural Philosophy at the University of Oxford and director ofthe Oxford Centre for Nonlinear PDE. He was president of the International Mathematical Union from 2003to 2006.

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Asymptotic equivariant index of Toeplitz operators and Attiyah-Weinstein conjecture

Louis Boutet de MonvelUniversite Pierre et Marie Curie, Institut de Mathematiques de Jussieu, 4 place Jussieu, F-75252 ParisCEDEX 05, FRANCEboutet@math.jussieu.fr

The equivariant index of transversally elliptic equivariant operators was introduced by M.F. Atiyah (1974); itis a virtual trace class representation of a compact group, or equivalently the character of this representation,which is a central distribution. This does not make sense for general Toeplitz operators because the Toeplitzspace where they act is only defined up to a finite dimensional space.The asymptotic index is an avatar of this, which works for Toeplitz operators : essentially it is a virtual traceclass representation mod finite representations; equivalently its character is a singularity (distribution modC∞). It still is compatible with many natural operations, in particular the direct image by homogeneoussymplectic maps.With E. Leichtnam, X. Tang and A. Weinstein, we have used this theory to give a new natural proof of theAtiyah-Weinstein conjecture (which was proved by C. Epstein): let X, X ′ be two compact strictly pseudo-convex boundaries (of complex domains): they carry natural cooriented contact structures. If f : X → X ′

is a contact isomorphism, we define the holomorphic pushforward Tf : u 7→ S′(u f−1) where u is theboundary value of a holomorphic function, and S′ is the Szego projector, i.e. the orthogonal projector onthe subspace of boundary values of holomorphic functions in L2(X ′) (ker ∂b). It is well known that Tf isa Fredholm operator; the Weinstein conjecture proposed a topological formula for its index. A particularcase of this, proposed earlier by Atiyah, is the following: let V, V ′ be two smooth compact manifolds, andf a homogeneous symplectic isomorphism T ∗V − 0 → T ∗V ′ − 0 (equivalently a contact isomorphismbetween the cotangent spheres); then there exists an elliptic Fourier integral operator attached to f , whoseindex is given essentially by the same formula (this is a special case of the former because, if V is realanalytic, the algebra of pseudodifferential operators acting on distributions is isomorphic to the algebra ofToeplitz operators acting on holomorphic boundary values on the boundary of a small tubular neighborhoodof V in its complexification).One difficulty in this problem is that, since we are modifying the boundary CR structures (there are two ofthem), we are typically in the framework of general Toeplitz operators where the index is not well defined.Our way out was to construct a related G-elliptic operator where the index is repeated infinitely many times,but still well related geomerically to the problem, so the asymptotic index theory can be used.

15

Atiyah, M.F. Elliptic operators and compact groups. Lecture Notes in Mathematics, Vol. 401. Springer-Verlag, Berlin-New York, 1974.

Boutet de Monvel, L. Asymptotic equivariant index of Toeplitz operators, RIMS Kokyuroku Bessatsu(2008).

Boutet de Monvel, L.; Leichtnam E.; Tang, X. ; Weinstein A. Asymptotic equivariant index of Toeplitzoperators and relative index of CR structures arXiv:0808.1365v1; to appear in the Duistermaat 65volume, Progress in Math, Birkhauser.

Weinstein, A.: Some questions about the index of quantized contact transformations RIMS KokyurokuNo. 1014, pages 1-14, 1997.

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Louis Boutet de Monvel was awarded with the 2007 Medaille Emile Picard of the French Academy of Sciences.

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Non-self-adjoint spectral theory

Brian DaviesDepartment of Mathematics, King’s College London, Strand, London WC2R 2LS, U.K.e.brian.Davies@kcl.ac.uk

Over the last twenty years there has been remarkable progress in understanding the spectral behaviourof highly non-self-adjoint operators, particularly differential operators, partly as the result of numericalexperiments. The lecture will describe some of the discoveries that have been made, and theorems proved,and will contrast them with the very different spectral behaviour of self-adjoint operators. Connections withso-called pseudospectral theory, that is bounds on the norms of the resolvent operators, will be explainedand illustrated.

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Brian Davies, FRS, is Professor of Mathematics at King’s College London. In 1998 he was awarded theSenior Berwick Prize of the LMS. Brian Davies was president of the London Mathematical Society from2007 to 2009.

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Asymptotic analysis and complex differential geometry

Simon DonaldsonDepartment of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, U.K.s.donaldson@imperial.ac.uk

A long-standing problem in complex differential geometry is to find various preferred metrics on a complexmanifold. These include Kahler-Einstein, constant scalar curvature and extremal metrics. Finding suchmetrics comes down to solving highly nonlinear partial differential equations. For some manifolds solutionsdo not exist, and this is known to be related to the algebro-geometric notion of “stability”. The talk willgive an overview of this area, emphasising the role of asymptotic analysis, applied to holomorphic sectionsof high powers of a complex line bundle. This gives a bridge between the analytical problems and algebraicgeometry which is important in the general existence theory. The ideas can also be applied to constructnumerical approximations to the desired metrics.

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Simon Donaldson, FRS, holds a Royal Society Research Professorship at Imperial College London. Hereceived a Fields Medal in 1986, was awarded with the Crafoord Prize 1994, the King Faisal InternationalPrize in 2006 and the Nemmers Prize in Mathematics in 2008. He will receive the 2009 Shaw Prize inMathematical Sciences.

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16

The global behavior of solutions to critical nonlinear dispersive and waveequations

Carlos KenigDepartment of Mathematics, University of Chicago, 5734 University Avenue, Chicago, IL 60637-1514, USAcek@math.uchicago.edu

In this lecture we will describe a method (which I call the concentration-compactness/rigidity theoremmethod) which Frank Merle and I have developed to study global well-posedness and scattering for criticalnon-linear dispersive and wave equations. Such problems are natural extensions of non-linear elliptic prob-lems which were studied earlier, for instance in the context of the Yamabe problem and of harmonic maps.We will illustrate the method with some concrete examples and also mention other applications of theseideas.

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Carlos Kenig is Louis Block Distinguished Service Professor of the University of Chicago. He was awardedthe 2008 Bocher Memorial Prize for his contributions to harmonic analysis and non-linear dispersive partialdifferential equations.

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Nonlinear harmonic analysis methods in boundary value problems ofanalytic and harmonic functions, and PDE

Vakhtang KokilashviliA. Razmadze Mathematical Institute, 1, M. Aleksidze st., 0193 Tbilisi, Georgiakokil@rmi.acnet.ge

The goal of our lecture is to present a survey of recent results in the nonlinear harmonic analysis operatortheory and their applications in the boundary value problems for harmonic and analytic functions and relatedintegral operators. We plan to discuss the above mentioned problems in the frame of Banach function spaceswith nonstandard growth condition.For the sake of presentation, we have split the talk in the following topics:• One and two-weight norm estimates for the Cauchy singular integrals on Carleson curves in variableexponent Lebesgue spaces.• The Riemann-Hilbert problem for holomorphic functions from weighted classes of the Cauchy type integralswith densities in Lp(·)(Γ) in simply connected domains with piecewise-smooth boundaries Γ.Our aim is to give a complete solvability picture; to reveal the influence on the solvability character of thegeometry of a boundary, of a weight function, and of the values of the space exponent at angular points; togive explicit formulas for solutions.• The Riemann-Hilbert-Poincare problem in the class of holomorphic functions whose mth order derivativesare representable by the Cauchy type integrals with densities from the variable exponent Lebesgue spaceswith weights. The solvability criteria are given for the problem. The study of the problem is heavily based onthe extension of I.Vekua’s integral representation of holomorphic function whose derivative is representableby the Cauchy type integral in simply connected domain with non-smooth boundary.• Baundary value problem with shift (the Hasemann BVP) for holomorphic functions in the domain witharc-chord condition. The solvability criteria and explicit formulas for solutions are established.Some part of the talk is based on joint research with V.Paatashvili.

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Vakhtang Kokilashvili is Head of the Mathematical Analysis Department of the Razmadze MathematicalInstitute. He was awarded the Razmadze Prize of the Georgian Academy of Sciences.

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Instability of the Cauchy-Kovalevskaya solution for a class of non-linearsystems

Nicolas LernerInstitut de Mathematiques de Jussieu, Universite Paris 6, 175 rue du Chevaleret, 75013 Paris, Francelerner@math.jussieu.fr

17

We prove that in any C-infinity neighborhood of an analytic Cauchy datum, there exists a smooth functionsuch that the corresponding initial value problem does not have any classical solution for a class of first-ordernon-linear systems. We use a method initiated by G. Metivier for elliptic systems based on the representationof solutions and on the FBI transform; in our case the system can be hyperbolic at initial time, but thecharacteristic roots leave the real line at positive times.

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Non-ergodicity of Euler deterministic fluid dynamics via stochastic anal-ysis

Paul MalliavinUniversite Pierre et Marie Curie, Institut de Mathematiques de Jussieu, 4 place Jussieu, F-75252 ParisCEDEX 05, FRANCE

Unitary representation associated to the motion of an incompressible fluid on the Tori. Fourier analysisof vector fields with vanishing divergence. Ergodi-city implies existence of an infinitesimal Haar measure.Randomization of Euler deterministic dynamics. Stochastic differential geometry on the group of volumepreserving diffeomorphism of the Tori. Jump process describing the evolution of the repartition of the energybetween modes. Non ergodicity of Euler equation via the transfert of energy towards micro scale.

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Paul Malliavin is famous for his contributions to stochastic analysis and stochastic differential geometry.Among other distinctions he received in 1974 the Prix Gaston Julia of the French Academy of Science andis member of the Royal Swedish Academy of Sciences.

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Higher order elliptic problems in non-smooth domainsBICS Lecture, with an introduction by Valery Smyshlyaev

Vladimir Maz’yaUniversity of Liverpool and Linkoeping Universityvlmaz@mai.liu.se

We discuss sharp regularity results for solutions of the polyharmonic equation in an arbitrary open set.The absence of information about geometry of the domain puts the question of regularity beyond the scopeof applicability of the methods devised previously, which typically rely on specific geometric assumptions.Positive results have been available only when the domain is sufficiently smooth, Lipschitz or diffeomorphicto a polyhedron.The techniques developed in the present work allow to establish the boundedness of derivatives of solutionsto the Dirichlet problem for the polyharmonic equation under no restrictions on the underlying domain andto show that the order of the derivatives is maximal.Then we introduce an appropriate notion of polyharmonic capacity which allows us to describe the precisecorrelation between the smoothness of solutions and the geometry of the domain.This is a joint work with S.Mayboroda, Perdue University.

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His honours include the prize of the Leningrad Mathematical Society 1962, Doctor honoris causa of theUniversity of Rostock 1990, Humbold Prize 1999, Corresponding Fellow of the Royal Society of Edinburgh2001, Member of Royal Swedish Academy of Sciences 2002, Verdaguer Prize of the French Academy ofSciences 2003, The Celsius Gold Medal of the Royal Society of Sciences at Uppsala 2004. He is author ofmore than 20 books and more than 430 articles.

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Operator algebras with symbolic hierarchies on stratified spaces

Bert-Wolfgang SchulzeInstitute of Mathematics, University Potsdam, Am Neuen Palais 10, Potsdam, D-14469 Germanyschulze@math.uni-potsdam.de

18

We establish operator algebras on certain categories of stratified spaces (“corner manifolds”, or “manifoldswith singularities”) that are designed to express parametrices of elliptic operators in terms of symbolic hierar-chies. Our calculus contains special cases such as (pseudo-differential) boundary value problems with/withoutthe transmission property at the boundary, and also mixed, transmission, and crack problems. The bound-aries or interfaces may be smooth or have again singularities (conical points, edges, etc.). Other examplesare equations on a smooth manifold, where the coefficients may have jumps or poles of a specific kind alongsome interfaces, smooth or singular in the above-mentioned sense, for instance, the Laplacian plus a singularinteraction potential from a many-particle system. It is typical in such problems that a concrete situation(for instance, for the Laplacian in a corner domain) may generate operator-valued amplitude functions of arelatively high generality, consisting of operator functions on configurations of lower singularity order, nowdepending on various variables and covariables along the singular lower-dimensional strata. The calculusalso contains analogues of Green functions, known from “standard” elliptic boundary value problems. Inthe singular case those refer again to all singular strata, operating on infinite cones. Moreover, when astratum is of dimension zero the operator functions globally act on compact (in general singular) bases ofsuch cones, with meromorphic dependence on a complex covariable, where non-bijectivity points (turninginto poles under inversion) contribute to the asymptotics of solutions. Ellipticity in such a scenario is definedas invertibility of such operator-valued symbols. This depends on chosen weights in the respective distribu-tion spaces. When a stratum is of dimension at least 1, this cannot be achieved in general, unless we poseextra edge conditions (analogues of boundary conditions), here of trace and potential type. The latter arepossible when an analogue of the Atiyah-Bott condition for the existence of Shapiro-Lopatinskij boundaryconditions is satisfied; otherwise another concept, namely, with global projection conditions may work (atleast for smooth boundaries or edges, cf. the well-known work of Atiyah, Patodi, Singer, and papers of manyother authors, especially, Seeley, Grubb, and also by the author, partly in joint work with J. Seiler, wherecorresponding operator algebras are established in a Toeplitz operator framework, unifying the structuresof the Shapiro-Lopatinskij and the global projection set-up). The construction of parametrices relies on theinversion of the components of the principal symbolic hierarchy, combined with algebraic operations. Thosesymbols take values in spaces of operators referring to lower singularity orders. At this point, in order toexpress parametrices within our spaces, we need the calculus as an algebra. The analysis which is doingall this is rich in detail. Many authors contributed to the pseudo-differential methods in this framework,especially, Melrose, Mendoza, Gil, Seiler, Schrohe, Witt, and Krainer.There are several monographs of theauthor, a few jointly with coauthors (Rempel, Egorov, Kapanadze, Harutyunyan) containing the basics ofthe approach, including applications, and more references. In order to keep the calculus manageable it isimportant to reduce the stuctures to a few “axiomatic” principles and then to proceed in an iterative way,beginning with the pseudo-differential calculus on a smooth manifold, and then successively building up thealgebras for conical, edge, corner, . . . , higher singularities. The focus of our talk is just a program of thatkind. We present such an iterative process to obtain operator algebras containing the desirable (“typical”)differential operators (corner-degenerate in streched coordinates), together with the parametrices of ellipticelements, where the above-mentioned examples are covered. One of the principles to make the calculusiterative is to impose a relatively simple behaviour of the growth of norms of parameter-dependent operatorswhen the parameters tend to infinity, then to make the parameter-dependence “edge-degenerate” at infinityof an infinite cone, and then to observe that this behaviour survives the step to the next floor of singularcalculus, cf. a joint article with Abed. The general structure theory is full of new challenges and “unex-pected” problems, for instance, from the point of view of index theory, or extensions to non-elliptic operators.Moreover, in concrete cases other substantial aspects remain essential, namely, to compute several data asexplicitly as possible, e.g., the index of operators on infinite cones, or the number of extra edge conditions,the right weights that depend on the individual operator, the asymptotics of solutions, including iteratedasymptotics, or the variable and branching behaviour connected with the above-mentioned poles when thosedepend on edge variables and change multipicities (cf. earlier work of Bennish, or the author, and a cycle ofpapers in progress jointly with Volpato).

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Bert-Wolfgang Schulze is author of more than 240 publications and 20 books. He received the Euler Medalof the Berlin Academy of Sciences in 1984 and is doctor honoris causa of the Vekua Institute of AppliedMathematics in Tbilisi.

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Visibility and Invisibility

Gunther UhlmannDepartment of Mathematics, C-449 Padelford Hall, Seattle, Washington 98195-4350, USA

19

gunther@math.washington.edu

We will describe the method of complex geometrical optics and its applications to find acoustic, quantum,and electromagnetic parameters of a body by making measurements at the boundary of the body. We willalso survey recent results on how to make objects invisible to acoustic, quantum and electromagnetic waves.

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Gunther Uhlmann is Walker Family Endowed Professor of Mathematics at the University of Washington.He is Fellow of the American Academy of Arts and Sciences, corresponding member of the Chilean Academyof Sciences, Fellow of the Institute of Physics and will be Clay Senior Scholar at MSRI in 2010.

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Practise of industrial mathematics related with the steel manufacturingprocessOCCAM Lecture on Applied Mathematics, with an introduction by John Ockendon

Masahiro YamamotoUniversity of Tokyo, Department of Mathematical Sciences, 3-8-1 Komaba Meguro Tokyo 153, Japanmyama@next.odn.ne.jp

We will discuss several problems given by the steel industry. Those problems have originated from realworking sites, are related for example with heat conduction processes and have been solved by the speakerand his research groups. Those problems can be modelled mathematically, on such a a theoretical basis, wehave solved them practically as well as mathematically to satisfy demands by industry for lowering costsand improving securities. For more fruitful contribution in the industrial mathematics from the side ofmathematicians, we will discuss also possible schemes.

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20

Public lecture

Analysis, Models and SimulationsOxPDE Public Lecture on Nonlinear PDE, with an introduction by Sir John Ball

Pierre-Louis LionsCollege de France, 3 rue d’Ulm, 75005 Paris, Francelions@dmi.ens.fr

In this talk, we shall first present several examples of numerical simulations of complex industrial systems.All these simulations rely upon some mathematical models involving Partial Differential Equations and weshall briefly explain the nature, the history and the role of such equations. Then, some examples showingthe importance of the mathematical analysis (i.e. understanding) of those models will be presented. Andwe shall conclude indicating a few trends and perspectives.

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Pierre-Louis Lions is the son of the famous mathematician Jacques-Louis Lions and has himself becomea renowned mathematician, making numerous important contributions to the theory of non-linear partialdifferential equations. He was awarded a Fields Medal in 1994, in particular for his work with Ron DiPernagiving the first general proof that the Boltzmann equation of the kinetic theory of gases has solutions. Otherawards Lions has received include the IBM Prize in 1987 and the Philip Morris Prize in 1991. Currently heholds the position of Chair of Partial Differential Equations and their Applications at the prestigious Collegede France in Paris.

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21

Sessions

I.1. Complex variables and potential theory

Organisers:Tahir Aliyev, Massimo Lanza de Cristoforis,Sergiy Plaksa, Promarz Tamrazov

This session is devoted to a wide range of directionsof complex analysis, potential theory, their applicationsand related topics.

—Abstracts—

Analytic functions in contour-solid problems

Tahir Aliyev AzerogloGebze Institute of Technology, Istanbul Caddesi, P.K.141, Gebze, Kocaeli, 41400 Turkeyaliyev@gyte.edu.tr

We generalize and strengthen certain contour-solid the-orems. The generalization consists in consideringfinely meromorphic functions besides holomorphic, andstrengthening is connected with taking into account ze-roes and the multivalence of functions.

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A non-α-normal function whose derivative has finitearea integral of order less than 2/α

Rauno AulaskariUniversity of Joensuu Department of Physics and Math-ematics Joensuu, Joensuu 80101 FinlandRauno.Aulaskari@joensuu.fi

LetD be the unit disk z : |z| < 1 in the complex plane.A function f , meromorphic in D, is normal, denoted byf ∈ N , if supz∈D(1 − |z|2)f#(z) < ∞, where f#(z) =|f ′(z)|/(1+ |f(z)|2). For α > 1, a meromorphic functionf is called α-normal if supz∈D(1− |z|2)αf#(z) <∞. H.Allen and C. Belna [J. Math. Soc. Japan, 24 (1972)128–132] have proved that there is an analytic functionf1, defined in D, such thatZZ

D|f ′1(z)| dxdy <∞

but f1 6∈ N . S. Yamashita [Ann. Acad. Sci. Fenn. Ser.Math. 4 (1978/1979) 293–298] sharpened this result byshowing that for another analytic function f2 which doesnot belong to N it holdsZZ

D|f ′2(z)|p dxdy <∞ (*)

for all p, 0 < p < 2. Further, H. Wulan [Progress in anal-ysis Vol. I,II, World Sci. Publ. 2003, 229–234] studiedmore the function f2 and showed that f2 6∈

S0<p<∞Q

#p

but f2 ∈T

0<p<∞M#p . We construct a class of analytic

functions fs which satisfy (*) for 0 < p < 2α

but fs 6∈ Nα

for α > 1. Further, the question if fs belongs or not toS0<p<∞M

#p is considered.

This is joint work with Shamil Makhmutov and JouniRattya.

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Global mapping properties of rational functions

Cristina BallantineDept. of Mathematics, College of the Holy Cross, 1College Street, Worcester, Massachusetts 01610 UnitedStatescballant@holycross.edu

The talk is based on a joint work with Dorin Ghisa.The main result is the following theorem. Every rationalfunction f of degree n defines a partition bC = ∪nk=1Ωkof the Riemann sphere such that the interior of everyΩk is mapped conformally by f on bC \ Lk, where Lk ispart of a cut L. The mapping extends conformally to theboundary of every Ωk except forsome points b1, b2, ..., bj ,j ≤ n, in the neighborhood of which f has one of theforms:

(i) f(z) = f(bk) + (z − bk)αkϕk(z), or

(ii) f(z) = (z − bk)−αkϕk(z),

where αk is an integer, αk ≥ 2, and ϕk is an analyticfunction with ϕk(bk) 6= 0.

Actually, (bC, f) is a branched covering Riemann surface

of bC having the branch points b1, b2, ..., bj . In the neigh-borhood of z =∞ we have:

(iii) f(z) = zαϕ(z), where α ∈ Z and ϕ is analytic withlimz→∞

ϕ(z) finite and non-zero.

If f is a polynomial, then every Ωk is bounded by arcsapproaching asymptotically rays of the form zk(t) =

tei(γ+2kπ/n), t > 0, γ ∈ R.We will present examples of color mapping visualiza-tions.

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Beltrami equations

Bogdan BojarskiIM PAN, Sniadeckich 8, Warsaw, 00-956 Polandbojarski@impan.gov.pl

In the talk will be discussed some new approaches to theBeltrami equations and operators in the complex planeand on Riemann surfaces in connections with the gen-eral theory of quasiconformal mappings and automor-phic functions.

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A functional analytic approach for a singularly per-turbed non-linear traction problem in linearized elas-tostatics

Matteo Dalal RivaUniversita’ degli Studi di Padova, Via Trieste, 63Padova, Italy/Padova/Veneto 35121, Italymdallari@math.unipd.it

We consider an application of an approach based on po-tential theory and functional analysis to analyze a non-linear traction problem of linearized elasticity in a do-main with a small hole. The results are obtained incollaboration with Professor Massimo Lanza de Cristo-foris.

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23

I.1. Complex variables and potential theory

Boundary behavior of Bloch functions and normal func-tions

Peter DovbushInstitute of Mathematics and Computer Science, 5Academy Street, Kishinev, MD-2028 Moldovapeter.dovbush@gmail.com

We give the version of the Lindelof principle which validin bounded domains in Cn with C2-smooth boundary.We also prove that if a Bloch function is bounded on aK-special curve ending at a given boundary point it isbounded on any admissible domain with vertex at thesame point.

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Spatial quasiconformal mappings and directional dilata-tions

Anatoly GolbergHolon Institut of Technology, 52 Golomb St., Holon58102 Israelgolberga@hit.ac.il

Two dilatations in Rn connected with a given directionare considered. They are spatial counterparts of the an-gular and radial dilatations in the plane. We establishnew geometric estimates of module of ring domains un-der quasiconformal mappings. These estimates are writ-ten in the terms of integrals depending on the directionaldilatations. The sharpness of the bounds is illustratedby certain examples. We also present new criteria forquasiconformal mapping to be Lipschitz or weekly Lip-schitz continuous in a prescribed point.

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Global mapping properties of entire and meromorphicfunctions

Dorin GhisaYork University, 5795 Young Street #1205, Toronto, On-tario M2M 4J3 Canadadghisa@yorku.ca

We consider that an analytic function f is the canonicalprojection of a branched Riemann surface (bC \E, f) onbC. Here E = E(f) is the set of (isolated, or non isolated)

essential singular points of f. Then bC \E consists of allregular points of f, as well as of a discrete set of poles.The main result shows that, for wide classes of integerand meromorphic functions f, disjoint unions bC \ E =∪∞n=1Ωn exist, such that the interior of every (fundamen-

tal region) Ωn is mapped conformally by f on bC \ Ln,where Ln is a part of a cut L.The global mapping properties of f concern these map-pings. If f is an infinite Blaschke product or Blaschkequotient whose set E of the cluster points of poles isa generalized Cantor subset of the unit circle, then thefundamental regions Ωn accumulate to every point of Eand f : Ωn → bC is surjective. In particular, the state-ment of the Big Picard Theorem is obvious for the pointsof E, which are non isolated essential singular points off. The same is true about a theorem of Hadamard whichsays that an entire function of fractional order assumesevery finite value infinitely many times.

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Approximation in Morrey-Smirnov classes

Daniyal IsrafilovDepartment of Mathematics, Faculty of Art and Sci-ences, Balikesir University,10145 Balikesir, Turkeymdaniyal@balikesir.edu.tr

In this talk we discuss the constructive characteriza-tion problems in the Morrey-Smirnov classes, defined onthe finite domain G with a sufficiently smooth Jordanboundary Γ. The Morrey spaces, introduced by Morreyin 1938, have been studied intensively by various authorsand together with weighted Lebesgue spaces play an im-portant role in theory of partial equations, and also inthe fluid dynamics. They also provide a large class ofexamples of mild solutions to the Navier-Stokes system.Nowadays there are sufficiently wide investigations relat-ing to the fundamental problems in these spaces, in viewof the differential equations, potential theory, maximaland singular operator theory and others.To the best of the author’s knowledge in the literaturethere are no results relating to the approximation prob-lems in the Morrey and Morrey-Smirnov classes, definedon the interval of the real line and on the sets of thecomplex plane.In the current talk we discuss the direct and inversetheorems in these spaces and obtain the constructivecharacterization of the generalized Lipschitz classes offunctions defined in the Morrey-Smirnov classes, in par-ticular.

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Two-sided bounds for the logarithmic capacity of mul-tiple intervals

Dmitri KarpInstitute of Applied Mathematics, 7 Radio Street, Vladi-vostok, 690041, Russiadimkrp@gmail.com

Potential theory on the complement of a subset of thereal axis attracts a lot of attention both in function the-ory and applied sciences (in particular in signal analy-sis). In the talk we discuss one aspect of the theory -the logarithmic capacity of closed subsets of the real line.We give simple but precise upper and lower bounds forthe logarithmic capacity of multiple intervals and a lowerbound valid also for closed sets comprising an infinitenumber of intervals. We discuss the existing methodsto compute the exact value of capacity and demonstrategraphically the results of numerical comparisons of ourestimates with exact values of capacities. The main ma-chinery behind our results are separating transforma-tion and dissymmetrization developed by V.N. Dubininand a version of the latter by K. Haliste as well as someclassical symmetrization and projection result for loga-rithmic capacity. The results presented in the talk im-prove some previous achievements by A.Yu. Solynin andK. Shiefermayr.The work reported here was done jointly withV.N. Dubinin and was supported by Far Eastern Branchof the Russian Academy of Sciences (grants 09-III-A-01-008 and 09-II-CO-01-003), Russian Basic ResearchFund (grant 08-01-00028-a) and the Presidential Grantfor Leading Scientific Schools (grant 2810.2008.1).

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24

I.1. Complex variables and potential theory

On boundary smoothness of conformal mapping

Olena KarupuNational Aviation Univesity, Kosmonavta Komarovaave. 1 Kiev, Kiev 03058 Ukrainekarupu@ukr.net

Let two simply connected domains bounded by thesmooth Jordan curves be given. Boundaries of thesedomains are characterized by the angles between thetangents to the curves and the positive real axis whichare considered as the functions of the arc length on thecurves. The estimates on the boundaries of the domainsfor the general moduli of smoothness of arbitrary or-der for the homeomorphisms between the closures of theconsidered domains, conformal in open domains are es-tablished.In partial case when moduli of smoothness of arbitraryorder for the functions characterizing boundaries of thedomains satisfy Holder condition, moduli of smoothnessof the same order for the derivatives of the functions re-alizing conformal mapping satisfy Holder condition withthe same index.

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Structures of non-rectifiable curves and solvability ofthe jump problem

Boris KatsKazan State Architecture and Civil Engineering Univer-sity, Zelenaya Street 1, Kazan, Tatarstan 420043 Russiakatsboris877@gmail.com

Let Γ be a given closed non-rectifiable curve on the com-plex plane C. We consider so-called jump problem, i.e.the boundary value problem for determination of a holo-morphic in C \ Γ function F (z) satisfying equality

F+(t)− F−(t) = g(t), t ∈ Γ,

where g(t) is a given jump. As the author have shownearlier, for non-rectifiable curve Γ this problem has a so-lution if the jump g satisfies the Holder condition withexponent exceeding half of upper metric dimension of Γ.In the present report we prove new conditions of solvabil-ity of the jump problem and new boundary properties ofits solutions in terms of decomposition of C\Γ into infi-nite sum of disjoint domains with rectifiable boundaries.In particular, the self-similarity of Γ weakens the condi-tions of solvability and improves smoothness of solutionsof the jump problem.

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The Loewner differential equations and univalent sub-ordination chains in several complex variables

Gabriela KohrFaculty of Mathematics and Computer Science, Babes-Bolyai University, 1 M. Kogalniceanu Str., Cluj-Napoca,Cluj 400084 Romaniagkohr@math.ubbcluj.ro

In this talk we present a survey with recent results con-cerning the Loewner differential equations and univalentsubordination chains on the unit ball in Cn. Various ap-plications and examples are also presented.

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Boundary integral equations in the study of someporous media flow problems

Mirela KohrFaculty of Mathematics and Computer Science, Babes-Bolyai University, 1 M. Kogalniceanu Str., Cluj-Napoca,Cluj 400084 Romaniamkohr@math.ubbcluj.ro

In this talk we present a survey of recent results concern-ing existence and uniqueness in Sobolev spaces for trans-mission problems associated with Stokes and Brinkmanequations on Lipschitz domains in Rn, by using potentialtheory and indirect boundary methods. Various appli-cations in the study of viscous incompressible flows inporous media or past porous bodies are also discussed.

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Singular perturbation problems in potential theory: afunctional analytic approach

Massimo Lanza de CristoforisDipartimento di Matematica Pura ed Applicata, Via Tri-este 63, Padova, Italy 35127, Italymldc@math.unipd.it

This talk is dedicated to the analysis of boundary valueproblems on singularly perturbed domains by an ap-proach which is alternative to those of asymptotic anal-ysis and of homogenization theory.In particular, we will consider a certain linear or non-linear boundary value problem on a domain with one orpossibly infinitely many holes, whose size is determinedby a positive parameter ε and we will consider a familyof solutions depending on ε as ε approaches 0. Then weshall represent the dependence on ε of the family of so-lutions, or of corresponding functionals of the solutionssuch as the energy integral, in terms of possibly singularat 0 but known functions of ε such as ε−1 or log ε, andin terms of possibly unknown real analytic operators.

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Classic theorems of approximation in a complex planeby rational functions

Jamal MamedkhanovZ.Khalilov, 23, Baku State University, Department oftheory of functions and functional analysis Baku, BakuAZ-1148 / 994 Azerbaijanjamalmamedkhanov@rambler.ru

Jackson’s classic theorem on closed boundary domainsrelated to approximation on the boundary of domainof functions determined on the boundary by means ofpolynomials assumes analyticity of the given function inthe considered domain. If the function is not analytic inthe considered domain, the approximation of the givenfunction on the boundary of the given domain, generallyspeaking, is not possible.Natural aggregates of approximation of function deter-mined only on the boundary of the domain are gener-alized polynomials Pn(z, 1

z), Pn(z, z) that coincide with

rational functions of the form

Rn(z) =

nXk=1

ak(zk +1

zk) = Pn(z,

1

z),

R∗n(z) =

nXk=1

bk(zk + zk) = Pn(z, z).

25

I.2. Differential equations: Complex and functional analytic methods, applications

In this case as in the case of polynomial approximationJ.Walsh problem is urgent. Namely, what necessary andsufficient conditions should satisfy a closed curve in or-der Jackson’s theorem and under some additional con-ditions the Bernstein’s theorem be true on it.

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Commutative algebras of monogenic functions and bi-harmonic potentials

Sergiy PlaksaInstitute of Mathematics of the National Academy ofSciences of Ukraine, Tereshchenkivska str., 3, Kiev-4,01601, Ukraineplaksa@imath.kiev.ua

An associative commutative two-dimensional algebra Bwith unit 1 over the field of complex numbers is bihar-monic if in B there exists a biharmonic basis e1, e2satisfying the conditions

(e21 + e2

2)2 = 0, e21 + e2

2 6= 0. (*)

I. Mel’nichenko proved that there exists the unique bi-harmonic algebra, and he constructed all biharmonicbases.Consider a biharmonic plane µ := ζ := x e1 + y e2,where x, y are real. Inasmuch as divisors of zero don’tbelong to the biharmonic plane, analytic functions in µare defined in the same way as in the complex planeand have similar properties: Cauchy’s integral theorem,Cauchy’s integral formula, Taylor’s expansion, Morera’stheorem.Differentiable functions

Φ(ζ) = U1(x, y)e1 + U2(x, y)ie1+

U3(x, y)e2 + U4(x, y)ie2 (**)

defined in µ form an commutative algebra and their com-ponents satisfy the biharmonic equation

∆2U :=

„∂4

∂x4+ 2

∂4

∂x2∂y2+

∂4

∂y4

«U(x, y) = 0 (***)

owing to equality ∆2Φ = Φ(4)(ζ) (e21 + e2

2)2 and equality(*).We proved that every solution of equation (***) in a sim-ply connected domain is the component U1 of analyticfunction (**) found explicitly. We proved also that ev-ery analytic function in the plane µ is expressed via twoanalytic functions of complex variable. We establishedan isomorphism between algebras of analytic functionsgiven in various biharmonic planes.This is joint work with S. Grishchuk.

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Fractal method for Clifford algebra and complex analy-sis

Osamu SuzukiDepartment of Computer Sciences and System Analy-sis,College of Humanities and Sciences, Nihon Univer-sity, Sakurajousui 3-25-40, Tokyo, Setagaya-ku 156-0045Japanosuzuki@cssa.chs.nihon-u.ac.jp

Wavelet expansion is introduced on a fractal boundaryand its differential and integral calculus are given. Us-ing this analysis, we construct renormalization theory of

Dirac operator on an infinite dimensional Clifford alge-bra and give a theory of hyperfunctions on the fractalboundary.

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Approximation theorems in weighted Lorentz spaces

Yunus Emre YildirirBalikesir University, Necatibey faculty of Education,Department of Mathematics Balikesir, Central 10100Turkeyyildirir@balikesir.edu.tr

In this work, we deal with the simultaneous and con-verse approximation of functions possessing derivativesof positive orders by trigonometric polynomials in theweighted Lorentz spaces with weights satisfying so calledMuckenhoupt’s Ap-condition.

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Hankel operator on generalized fock spaces

El Hassan YoussfiCentre de Mathematiques et Informatique, Universite deProvence, LAT UMR CNRS 6632, 39 Rue F.Joliot-CurieMarseille, 13453 Franceyoussfi@cmi.univ-mrs.fr

We consider Hankel operators Hf with antiholomorphicsymbol f on the generalized Fock space A2(µm)of squareintegrable functions with respect to µm, the measurewith weight e−|z|

m

,m > 0 with respect to the Lebesguemeasure in Cn. We prove that Hf is bounded if and onlyif f is a polynomial of degree at most m

2. We show that

Hf is compact if and only if f is a polynomial of degreestrictly smaller that m

2. We also establish that Hf is in

the Schatten class Sp if and only if p > 2n and f is a

polynomial of degree strictly smaller than m (p−2n)2p

.

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Continuous mappings between domains of manifolds

Yuriy ZelinskiyInstitute of Mathematics Ukr.Ac.Sci., Tereshchenkivskastr.3 Kyiv, Kyiv 01601 Ukrainezel@imath.kiev.ua

Let f be continues mapping between domains of man-ifolds, with disjoint images of the boundary and of thedomain interior, and certain degree k, then either themapping is interior in the sense of Stoilow or there is apoint possessing at least |k|+ 2 preimages. If addition-ally in the last case the map f be zerodimensional inthe domain interior, then the set of points possessing atleast |k|+ 2 preimages contains open ball.

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I.2. Differential equations: Complex andfunctional analytic methods, applications

Organisers:Heinrich Begehr, Dao-Qing Dai, Jinyuan Du

Complex analytic and functional analytic methods areused extensively to treat complex ordinary and partialdifferential equations. The main subject of the session

26

I.2. Differential equations: Complex and functional analytic methods, applications

will be higher order partial differential equations. In-tegral representations, boundary value problems, singu-lar integral equations, properties of integral transforms,polyharmonic Green, Robin, Neumann functions are re-lated. Particular subjects will be special equations asthe Vekua equation, Poisson equation, Bitsadze equa-tion, inhomogeneous biharmonic equation. Hyperana-lytic function theory as a tool for treating elliptic sys-tems in plane domains, systems in several complex vari-ables, metaanalytic function theory, Riemann-Hilbertproblem and applications e.g. for orthogonal polynomi-als might be also discussed. Ordinary complex differen-tial equations and applications in mathematical physicsare an other subject of the session.

—Abstracts—

A hierarchy of polyharmonic kernel functions and therelated integral operators

Umit AksoyAtilim University, Department of Mathematics, Kizil-casar Mahallesi, Incek, Golbasi, Ankara 06836 Turkeyuaksoy@atilim.edu.tr

Iterations of the harmonic Green, Neumann and Robinfunctions leads to hybrid polyharmonic Green functions.A hierarchy of these polyharmonic kernel functions willbe discussed. Taking these functions as a family of kernelfunctions, a class of integral operators are established.Some properties concerning the related boundary valueproblems are investigated.

———

Boundary value problems for complex partial differen-tial equations

Heinrich BegehrFreie Universitat Berlin, Mathematisches Institut, Arn-imallee 3, Berlin 14195, Germanybegehr@math.fu-berlin.de

Some particular complex partial differential equationsare investigates as e.g. the Cauchy-Riemann equationthe Beltrami equation, the polyanalytic and the poly-harmonic equation. Integral representations of Cauchy-Pompeiu type are presented. They are adjusted tocertain boundary value problems of Schwarz, Dirich-let, Neumann, Robin type. In particular some polyhar-monic Green functions are introduced which occur whenmerging different polyharmonic Green, Neumann, Robinfunctions of lower order. This process leads to new repre-sentation formulas and to certain related boundary valueproblems. Their solutions and if appropriate also solv-ability conditions are given.

———

On some classes of bicomplex pseudoanalytic functions

Peter BerglezDepartment of Mathematics, Graz University of Tech-nology, Steyrergasse 30, Graz 8010, Austriaberglez@tugraz.at

We consider certain classes of bicomplex pseudoanalyticfunctions using the commutative ring of bicomplex num-bers T ∼= ClC(1, 0) ∼= ClC(0, 1). They obey specific

bicomplex Vekua equations. We give representationsfor these functions using suitable differential operatorsacting on T-holomorphic functions as well as on otherbicomplex pseudoanalytic functions. For example thefunctions investigated here are of interest in connectionwith the complexified stationary Schrodinger equation.

———

Boundary value problems on Klein surfaces

Carmen BolosteanuUniversity “Spiru Haret” Bucharest, Faculty of Account-ing and Finance, 223 Traian Street, Campulung Muscel,Arges 115100 Romaniabolosteanuc@yahoo.com

In this paper we introduce some basic notions and solveRiemann-Hilbert problem on the Mobius strip and onthe real projective plane endowed with a dianalyticstructure. We use the symmetry in the sense of Klein,showing how symmetric conditions on the boundarytranslate into symmetric solutions, which generate so-lutions to similar problems for Klein surfaces.

———

Optimal methods for evaluation hypersingular integralsand solution of hypersingular integral equations

Ilya BoykovKrasnay Str. 40, Penza, 440026 RussiaI.V.Boykov@gmail.com

Discussed the optimal with respect to accuracy algo-rithms for evaluation of hypersingular integrals

f(s) =

1Z−1

f(t)dt

(t− s)p ,

f1(s1, s2) =

1Z−1

1Z−1

f1(t1, t2)dt1dt2(t1 − s1)p1(t2 − s2)p2

,

f2(s1, s2) =

1Z−1

1Z−1

f2(t1, t2)dt1dt2((t1 − s1)2 + (t2 − s2)2)p

,

f3(s1, s2) =

1Z−1

1Z−1

f3(t1, t2)dt1dt2((t1 − s1)2 + (t2 − s2)2)p+λ

,

where 0 < λ < 1, p, p1, p2 = 2, 3, . . . .Investigated a smooth of functions f1(s), fi(s1, s2), i =1, 2, 3. Introduced a classes of functions Ψ, Ψi, i =1, 2, 3,. Functions f(s), fi(s1, s2), i = 1, 2, 3. be-long to the classes Ψ, Ψi, i = 1, 2, 3, when functionsf(s), fi(s1, s2), i = 1, 2, 3, belong to the functionalclasses Ψ,Ψi, i = 1, 2, 3. Evaluated Babenko and Kol-mogorov widths of functional classes Ψ, Ψi, i = 1, 2, 3,and constructed a local splines for approximation func-tions from functional classes Ψ, Ψi, i = 1, 2, 3, whichare optimal algorithms for approximation conjugatefunctions f(s), fi(s1, s2), i = 1, 2, 3, from function setsΨ, Ψi, i = 1, 2, 3.Given optimal with respect to accuracy algorithms forsolution of many-dimensional weakly singular, singularand hypersingular integral equations

x(t) + λ

Z. . .

ZD

h(t, τ)x(τ)

(r(τ − t))v dτ = f(t),

27

I.2. Differential equations: Complex and functional analytic methods, applications

a(t)x(t) +

1Z−1

. . .

1Z−1

q(θ)h(t, τ)x(τ)

(r(τ − t))l dτ = f(t),

a(t)x(t) +

1Z−1

. . .

1Z−1

h(t, τ)x(τ)

(r(τ − t))(l+λ)dτ = f(t).

Here D = [−1, 1]l, 0 ≤ v < l, 0 ≤ λ ≤ 1, θ = ((τ −t)/(r(τ−t)), r(τ−t) = ((τ1−t1)2+· · ·+(τl−tl)2)1/2, t =(t1, . . . , tl), τ = (τ1, . . . , τl).Given applications singular and hypersingular integralequations in aerodynamics and electrodynamics.

———

Complex partial differential equations with mixed-typeboundary conditions

Okay CelebiYeditepe University, Department of Mathematics Kay-isdagi Caddesi, Kadikoy Istanbul, 34755 Turkeyacelebi@yeditepe.edu.tr

We consider the linear elliptic complex partial differen-tial equations of higher order with mixed-type bound-ary conditions involving combinations of Dirichlet, Neu-mann and Robin conditions. The solvability of problemsare discussed via the theory of singular integral equa-tions.

———

On a mathematical model of a cusped plate with bigdeflections

Natalia ChinchaladzeI.Vekua Institute of Applied Mathematics of TbilisiState University, 2 University St., Tbilisi 0186 Georgiachinchaladze@gmail.com

The talk deals with big deflections by the cylindricalbending of a cusped plate with the variable flexuralrigidity vanishing at the cusped edge. The setting ofboundary conditions at the plate edges depends on thegeometry of sharpening of the cusped edges. All the ad-missible classical bending boundary-value problems areformulated. Existence and uniqueness theorems for thesolutions of these boundary-value problems are proved.

———

Mixed boundary value problem with a shift for somepair of metaanalytic function and analytic function

Jin-Yuan DuSchool of Mathematics and Statistics, Wuhan Univer-sity, Wuhan, 430072 Chinajydu@whu.edu.cn

In this article, we reconsider the same mixed bound-ary value problem on the unit circumference for somepair of a metaanalytic function and an analytic func-tion given in reference (Jinyuan Du and Ying Wang.Mixed boundary value problem for some pair of meta-analytic function and analytic function. MathematicalMethods in the Applied Sciences, 31(15): 1761 - 1779,2008). By adopting appropriate function transforma-tion, we directly turn the problem into two independentboundary value problems for analytic functions. Thenwe obtain the expression of solution and the condition of

solvability for the mixed boundary value problem. Theformat of the expression of solution and the conditionof solvability here is rather dissimilar with that in ref-erence (Jinyuan Du and Ying Wang. Mixed boundaryvalue problem for some pair of metaanalytic functionand analytic function. Mathematical Methods in theApplied Sciences, 31(15): 1761 - 1779, 2008), but theequivalence for them is concretely proved in the end ofthis article.

———

Generalized analytic functions on Riemann surfaces

Grigory GiorgadzeTbilisi State University, Faculty of Exact and NaturalSciences, Chavchavadze ave. 3, Tbilisi, GE 0128 Geor-giagiorgadze@rmi.acnet.ge

The Riemann-Hilbert monodromy problem (21st Hilbertproblem) is to construct the Fuchsian system by themarked points and given non-degenerate matrices forwhich the marked points will be poles of system and themonodromy matrices coincide with given matrices. TheRiemann-Hilbert boundary value problem (the problemof linear conjugation) was posed by Riemann in the samework as one of the methods of the solution of the mon-odromy problem.Notwithstanding the fact that the problem of linear con-jugation is studied for the generalized analytic functions(vectors) and the progress achieved in research of singu-lar elliptic systems, the connection between Riemann-Hilbert problem with singular elliptic systems and prob-lem of linear conjugation for the generalized analyticfunctions was not noted until now. We consider theanalogous problems and apply the methods of algebraictopology. The analog of Fuchsian system in our ap-proach will be Carleman-Bers-Vekua system with sindu-lar points. Research object will be the space of multiple-valued solutions, the analog of the monodromy theo-rem, Chen iterated integral,Riemann-Roch theorem andRiemann-Hurwitz formula.

———

Optimization of fixed point methods

Sonnhard GraubnerKant-Gymnasium Leipzig, Scharnhorststrasse 15,Leipzig, 04275 GermanySonnhmath@aol.com

The talk deals with operator equations with Lipschitzcontinuous right-hand sides. In the case the Lipschitzcondition is only local one, fixed-point methods can beapplied only in subdomains (polydisc) of the underlyingfunction space.The polydisc is optmal if it leads restric-tion of the norm of the corresponding operator whichis as small as possible. In the talk we prove a criteronleading to a uniquely determined optimal polydisc.

———

Generating functions of the Laguerre-Bernoulli polyno-mials involving bilateral series and applications

Azhar HussainDepartment of Physics, Veer Kunwar Singh University,Ara, Bihar, 802301 Indiaalazharhussain@yahoo.co.in

28

I.2. Differential equations: Complex and functional analytic methods, applications

The object of the paper derives a new class of gener-ating functions for two variable Laguerre and Laguerre-Bernoulli polynomials involving bilateral series by ap-propriately specializing a number of known or new partlyunilateral and partly bilateral generating functions areshown to follow as applications of the main results.

———

Asymptotic behavior of subparabolic functions

Alexander KheyfitsBronx Community College, Department of Mathemat-ics and CS, 2155 University Avenue, Bronx, NY 10453United Statesakheyfits@gc.cuny.edu

For subsolutions of the heat equation, we proveanalogs of some classical results of complex analysis,for example, Phragmen-Lindelof principle and Valiron-Titchmarsh theorem.

———

Elliptic Riemann-Hilbert problems for generalizedCauchy-Riemann systems

Giorgi KhimshiashviliI.Chavchavadze ave. 32, Ilia Chavchavadze Sate Univer-sity, Tbilisi 0179 Georgiakhimsh@rmi.acnet.ge

Generalized Cauchy-Riemann systems introduced byE.Stein and G.Weiss are considered. For such systems, anatural formulation of Riemann-Hilbert problem will besuggested which gives a direct generalization of the clas-sical Rieman-Hilbert problem for holomorphic functions.Using the concept of Baum-Douglas obstruction for firstorder systems, a topological characterization of general-ized Cauchy-Riemann systems systems possessing ellip-tic Riemann-Hilbert problems will be given. Moreover,it will be explained how to verify the resulting conditionin terms of the associated Clifford algebra representa-tion, which enables us to obtain the complete list of suchsystems. For such systems, it will be also shown thatnonlinear Riemann-Hilbert problems with target man-ifolds satisfying certain conditions of transversality aredescribed by Fredholm operators in appropriate func-tional spaces.

———

On some qualitative issues of the elliptic systems

Nino ManjavidzeGeorgian Technical University, Department of Mathe-matics, Kostava str.77, Tbilisi 0175 Georgianinomanjavidze@yahoo.com

The theory of elliptic systems on the plane is the clas-sical object of investigation. We introduce and analyzethe generalized Cauchy-Lebesgue classes for some partic-ular cases of such systems. The analogs of the maximummodulus theorem are found.This is joint work with G.Akhalaia, G.Makatsaria.

———

Poisson equation with the Robin boundary condition

Alip MohammedDepartment of Mathematics and Statistics, N520 Ross

Building, 4700 Keele Street Toronto, Ontario M3J 1P3Canadaalipm@mathstat.yorku.ca

The inhomogeneous Robin/third boundary conditionwith general coefficient for the Poisson equation on theunit disc is studied in terms of holomorphic functions us-ing Fourier analysis. It is shown that against the usualexpectations this problem cannot have a unique solu-tion unless the coefficient of the first order term in theboundary condition is a constant. For the case of generalcoefficient, it is actually a problem with essential singu-larity in the domain, but still well-posed under properassumptions and the unique solution is given explicitly.

———

Investigation of one class of two-dimensional conju-gating model and non model integral equation withfixed super-singular kernels in connection with hyper-bolic equation

Nusrat RajabovTajik National University Rudaki Av. 17 Dushanbe,Dushanbe 734025 Tajikistannusrat38@mail.ru

LetD denote the rectangleD = a < x < a0, b0 < y < b.In D we consider the conjugating integral equation cor-responding to two-dimentional model integral equationwith fixed super-singular kernels, that is the integralequation

Tα,βλ,µ (V ) ≡V (x, y) +λ

(x− a)α

Z a

x

V (t, y)dt

− µ

(b− y)β

Z y

b0

V (x, s)ds

+δ

(x− a)α(b− y)β

Z a

x

dt

Z y

b0

V (t, s)ds

= g(x, y), (*)

where α = constant > 1, β = constant > 1, λ, µ, δ =constant, and g(x, y) is a given function, V (x, y) is thedesired function.To problem investigation one dimensional Volterra typeintegral equation in the case, when kernels have bound-ary and interior fixed singularity and super-singularity,investigated N. Rajabov [Volterra type integral equationwith fixed boundary and interior singular and super-singular kernels and its applications, Dushanbe, 2007,222p]. Two-dimentional and three-dimentional Volterratype integral equation investigated N. Rajabov, L. Ra-jabova [Dokl. Math., V. 71, No. 1, 2005, pp 111–114; V.409, No. 6, 2006, pp. 749–753; Math. Notes, Miskols,V. 4, No. 1 (2003), pp. 65–76].In this lecture for different values of parametrs in theintegral equation (*) existance theorems are proved forinhomogeneous equation. In case, when δ 6= −λµ, hasproved that for existence solution inhomogeneous equa-tion (*) representable in the form

V (x, y)− exp[−λωαa (x) + µωβb (y)]×

×∞Xn=1

(exp(−ωαa (x)))n(x− a)−αVn(y),

ωαa (x) = [(α− 1)(x− α)α−1]−1,

it is necessary and sufficientey infinity number solvabil-ity conditions to right part equation (*).

29

I.2. Differential equations: Complex and functional analytic methods, applications

———

About one class of two-dimensional Volterra type inte-gral equation with two interior sinqular lines

Lutfya RajabovaTajik Technical University Academician Rajabovs Av.10 Dushanbe, Dushanbe 734025 Tajikistanlutfya62@mail.ru

Let D denote the rectangle D = D1∪D2∪D3∪D4, D1 =a1 < x < a, b1 < y < b, D2 = a < x < a2, b1 <x < b, D3 = (x, y) : a1 < x < a, b < y < b2,D4 = a < x < a2, b < y < b2, Γ1 =a1 < x < a2, y = b, Γ2 = x = a, b1 < y < b2.In D we consider the two dimensional linear Volterratype Integral Equation with two interior fixed singularKernels:

u(x, y) +

Z x

a

A(t)u(t, y)

(t− a)αdt−

Z b

y

B(s)u(x, s)

(b− s)β ds

+

Z x

a

dt

(t− a)α

Z b

y

C(t, s)u(t, s)

(b− s)β ds = f(x, y), (*)

where A(x), B(y), C(x, y), f(x, y) are given functionsand a ∈ Γ1, b ∈ Γ2. In this paper the solution to (*) isconstructed in the case α = 1, β = 1. In this cases itis proved that, for certain values A(a), B(b), the homo-geneous Integral Equation (*) has an infinite number oflinearly independent solutions, and for other certain val-ues A(a), B(b) homogeneous Integral Equation (*) hasnot solution, exepting zero.The non-homogeneous Integral Equation (*) for certainvalues A(a), B(b) has always a solution and its generalsolutions contain arbitrary functions of one variable. Forother numbers A(a), B(b) the non-homogeneous IntegralEquation (*) has a unique solution.

———

Explicit global solutions of 3D rotating Navier-Stokesequations

Roman SaksMathematical Institute of RAS, 112, ChernyshevskiStreet, Ufa, 450077, Russiaromen-saks@yandex.ru

We study the Cauchy problem for 3D Navier-Stockesequations in a frame uniform rotation around the verti-cal axis with a periodical conditions by xj coordinats.Investigation is based on Fourier development knownand unknown vector function by eigen functions of thecurl and Stockes operators, using Galerkin equations.This system has very simple explicit form and its linearpart is diagonal.The familie of the explicit periodical solutions of lin-ear Stokes-Sobolev equations were found. This solutionsare also the global solutions of the nonlinear Navier-Stockes equations. New families of these equations areconstracted.For given function we have made program of calculationits Fourier coefficients. We have calculated the coeffi-cients of some Galerkin systems. There is also programmof numerical solution of its Cauchy problem.Our research have been done with the support of grantof RFFR 09-01-00349.

———

Methods of solutions of an singular integrodifferentialequation

Emma SamoylovaPeople’s Friendship University of Russia, Mikluho-Maklaya street 6, Moscow 117198 Russiaemma-samoilova@mail.ru

It is considered an integrodifferential equation with thesingular integral in the sence of Cauchy-Lebesgue prin-cipale value. Necessary and sufficiant conditions for theexistance and unicvenesse of it solution in correspondingfunctions spaces are obtained.

———

A boundary condition of the volume potential

Tynysbek Sharipovich Kal’menovInstitute of Mathematics, Computer Science, and Me-chanics, Ministry of Education and Science of Kaza-khstan, ul.Shevchenko 28., Almaty 050010 Kazakhstankalmenov.t@mail.ru

Let Ω ⊂ Rn be a finite domain with smooth boundaryS.The volume potential is considered in Ω,which

u(x) = Kf(x) =

ZΩ

εn(x− y)f(y)dy

where ε2(x − y) = − 12π

log |x − y|, εn(x − y) =1

(n−2)σn

1|x−y| , n ≥ 3 be a principal fundamental solution

of the Laplace equation namely

−4xεn(x− y) = −nXi=1

∂2(εn(x− y))

∂x2i

in Rn,σn = 2πn2

Γ(n2 )is the area of the unit sphere in Rn,and

Γ is the gamma-function. The volume potential can beapplied not only to solve problems in the theory of grav-itation but, in general, to solve a wide range of problemsin mathematical physics, in particular in electrostaticsand magnetism. So finding its boundary condition hasgreat theoretic and practical interests.The main resultof this paper is the following theorem.

Theorem. The volume potential with

−4xu(x) = f(x)

satisfies the boundary condition

1

2u(x) =

ZS

∂εn(x− y)

∂nyu(y)dS(y)

−ZS

εn(x− y)∂u(y)

∂nydS(y),

x ∈ S, for each function f ∈ L2(Ω). Inversely, if a func-tion u ∈ W 2

2 (Ω) satisfies the Poisson equation and theboundary condition then the function u is defined the vol-ume potential. Where

RS

∂εn(x−y)∂ny

u(y)dS(y) - in terms

of principal value of Cauchy.

———

Eigenvalues and eigenfunctions of volume potential

Durbudkhan SuraganInstitute of Mathematics, Computer Science, and Me-chanics, Ministry of Education and Science of Kaza-khstan, ul.Shevchenko 28., Almaty 050010 Kazakhstansuragan@list.ru

30

I.2. Differential equations: Complex and functional analytic methods, applications

Let Ω ≡ |x| < δ, x ∈ Rn, n = 2, 3 with smooth bound-ary S. We have to find eigenvalues and eigenfunctionsof the volume potential

u(x) = λ

ZΩ

εn(x− y)u(y)dy

where ε2(x − y) = − 12π

log |x − y|, εn(x − y) =1

(n−2)σn

1|x−y| , n ≥ 3 Note, we proved that the vol-

ume potential correspondingly satisfying the followingboundary condition

1

2u(x) =

ZS

∂εn(x− y)

∂nyu(y)dS(y)−

ZS

εn(x−y)∂u(y)

∂nydS(y),

x ∈ S.

Theorem. a) Let n = 2.Then eigenvalues and eigen-functions of the volume potential are represented corre-

spondingly λkj =[µ

(k)j ]2

δ2and ukj = ckjJk(µ

(k)j

rδ)eikϕ,

k = 0, 1, ..., j = 1, 2, ..., where c∗ are constants, (r, ϕ)are polar coordinates, Jk is the Bessel function of thefirst kind , µ

(k)j are roots of the following equation

Jk(µ(k)j ) +

µ(k)j

2k(Jk−1(µ

(k)j )− Jk+1(µkj )) = 0.

b) Let n = 3 . Then eigenvalues and eigenfunctionsof the volume potential are represented correspondingly

λkj =[µ

(l+ 12 )

j ]2

δ2and uljm =

cljm√rJl+ 1

2(µl+ 1

2j

rδ)Y ml (θ, ϕ),

l = 0, 1, ..., j = 1, 2, ...,m = 0,±1, ...,±l where (r, θ, ϕ)are spherical coordinates, Y mk is the spherical harmonic

function, µl+ 1

2j are roots of the following equation

Jl+ 12(µl+ 1

2j ) +

µl+ 1

2j

l + 1(Jl− 1

2(µl+ 1

2j )− Jl+ 3

2(µl+ 1

2j )) = 0.

———

The ending solutions of Ince system with irregular fea-tures

Zhaxylyk TasmambetovAktobe State University after K. Zhubanov 263, BratievZhubanov’s street, Aktobe city, 030000 Kazakhstantasmam@rambler.ru

The Ince system with irregular features:(p(0)Zxx + p(1)q(4)Zxy + p(2)Zx + q(5)Zy + p(3)Z = 0,

q(0)Zyy + p(4)q(1)Zxy + p(5)Zx + q(2)Zy + q(3)Z = 0,

where coefficients p(i) = p(i)(x) and q(i) = q(i)(y)(i = 0, 5) are polynomials of

p(i)(x) =

δiXj=πi

pijxj , q(i)(y) =

ζiXj=ξi

qijxj

type (πi, δi, ξi, ζi (i = 0, 5) - certain numbers), is studied.Let the system be collocated and let the integrabilitycondition be executable

p(0)q(0) − p(1)q(1)p(4)q(4) 6= 0.

Ince established that singular curves of this system aredefined by the coefficients in the case of second-orderprivate derivatives and in the case of certain additional

hypothesizes it has four linear-independent partial so-lutions. Ince type systems with regular and irregularfeatures were studied in work and the possibility of con-structing of normal and normal-regular solutions nearthese features were presented.The aim of this work is to establish using the conceptsof rank, antirank m and range (v+ 1) of coefficients, anessential and sufficient existence condition of the endingsolution with unknown parameters is rational functionor polynomial of two-variables.Also, there is an order of proved theorems establishingnecessary existence conditions of the ending solutions.This is joint work with A.Zh. Tasmambetova (Aktobe).

———

Fractional integrals and hypersingular integrals in vari-able order Holder spaces on homogeneous spaces

Ismail TaqiArab Open University - Kuwait Branch AlSharhabeelStreet Khaitan, 92400 Kuwaittaqiismail@yahoo.com

Mittag-Leffler [Mittag-Leffler, G. (1903). Sur la nouvellefunction Eα(X), C.R. Acad. Sci. Paris, (Ser. II). 137,554-558] introduced a function defined by an infinite se-ries

Eα(z) =

∞Xk=0

zk

Γ(αk + 1), α > 0

and investigated some of its properties. This is an entirefunction of order 1/α.Another function having similar properties to those ofMittag-Leffler functions is given by

Eα,β(z) =

∞Xk=0

zk

Γ(αk + β), α > 0, β > 0.

For β = 1, Eα,1 = Eα.Such functions arise naturally in the solution of frac-tional integral equations [Saxena, R., Mathai, A. andHaubold, H. (2002). On fractional kinetic equations,Astrophysics and Space Science, 282, 281-287] and es-pecially in the study of the fractional kinetic equation,random walks, etc.We study Mittag-Leffler type functions and derive someof their properties including integrals and recurrence re-lations. We also study fractional equations of the form

N(t)−N0 = −c 0D−1t N(t),

and its generalization, where 0D−νt is the Riemann-

Liouville operator of fractional integration.

———

On mixed boundary-value problems of polyanalyticfunctions

Yufeng WangSchool of Mathematics and Statistics, Wuhan Univer-sity, Wuhan 430072 Chinawh yfwang@hotmail.com

Recently, boundary value problems of higher-order com-plex partial differential equations have been widely in-vestigated. For example, various kinds of boundaryvalue problems of two-order complex partial differen-tial equations, including the Poisson equation and the

31

I.3. Complex-analytical methods for applied sciences

Bitsadze equation, have been systematically discussed,and the explicit expression of solution and the conditionof solvability have already been obtained. In addition,some boundary value problems of polyanalytic equation,polyharmonic equation and metaanalytic function havealso been discussed.In this paper, under the appropriate decompositionof polyanalytic functions, some mixed boundary-valueproblems of polyanalytic functions have been discussed,and the explicit expression of solution and the conditionof solvability have been obtained.

———

An algorithm of solving the Cauchy problem and mixedproblem for the two-dimensional system of quasi-linearhyperbolic partial differential equations

Oleg N. ZhdanovSiberian State Aerospace University “M.F. Reshetnyov”,Krasnoyarsk, Russiaonzhdanov@mail.ru

Let’s consider the system of homogeneous quasilinearhyperbolic partial differential equations

aij(u1, u2)∂xu

j + bij(u1, u2)∂yu

j = 0, i, j = 1, 2, (*)

where aij , bij - smooth functions in area D.There are 3 classical boundary problems for system (*):the Cauchy problem, the Riemann problem and the an-mixed problem. Earlier Cauchy and Riemann problemswere solved for some particular cases using conservationlaws. And now we have algorithm for the solution ofthe Cauchy problem of system (*) in general. Attemptsto solve the mixed prolem weren’t successful for a longtime. Our approach consists in applying to this systemnot only one conservation law, as was done in many pa-pers, but a family of such laws with functions dependingon parameters.Let’s accurately formulate the problem. Let the func-tion u be specified on the non-characteristic curve MNin the plane C, and functions u, v be specified on a char-acteristic curve crossing MN . It is important that everycharacteristic crosses the curve MN only in one pointand is not tangent to it in any point. Our aim is tofind the intersections of characteristics and the values offunctions u and v in these points.We reduce the mixed problem to the Cauchy problem.We choose a point on the curve MN and a point onthe characteristic, and we have a system of algebraicequations - corollary fact of conservation law. Using re-sultant, we obtained one equation for the value of htefunction v in the initial point. We find this value andrepeat the procedure with another points. It allows usto find the intersections of characteristics and functionvalues in these points with preassigned exactness us-ing a well-known method described in [Kiryakov P. P.,Senashov S. I., Yakhno A. N. Application of symmetriesand conservation laws to differential equations solving.Novosibirsk, 2001., p. 170]. As application we obtainedthe solution of systems, describing state of plane stressof Mises‘s plastic surroundings– a problem that is inter-esting for mechanics for more than 100 years.

———

On solution of a kind of Riemann boundary value prob-lem on the real axis with square roots

Shouguo ZhongSchool of Mathematics and Statistics, Wuhan Univer-sity, Wuhan 430072 Chinazhangzx9@sohu.com

Solution of the Riemann boundary value problem on thereal axis X with square rootsp

Ψ+(x) = G(x)p

Ψ−(x) + g(x), x ∈ X

for analytic function is considered, which was solved un-der certain assumptions on the branch points appeared.

———

Some Riemann boundary value problems in Cliffordanalysis

Zhongxiang ZhangSchool of Mathematics and Statistics, Wuhan Univer-sity, Wuhan 430072 Chinazhangzx9@126.com

In this paper, we mainly study the Rm (m > 0) Rie-mann boundary value problems for functions with valuesin a Clifford algebra C(V3,3). We firstly prove a general-ized Liouville theorem for harmonic functions and bihar-monic functions by combining the growth behaviour esti-mates with the series expansions for k-regular functions.We obtain the result under only one growth condition atinfinity by using the integral representation formulas forharmonic functions and biharmonic functions. By us-ing the Plemelj formula and the integral representationformulas, a more generalized Liouville theorem for har-monic functions and biharmonic functions is presented.Combining the Plemelj formula, the integral representa-tion formulas with the above generalized Liouville theo-rem, we prove that the Rm (m > 0) Riemann boundaryvalue problems for regular functions, harmonic functionsand biharmonic functions are solvable. The explicit so-lutions are given.

———

I.3. Complex-analytical methods for appliedsciences

Organisers:Viktor Mityushev, Sergei Rogosin

The main attention will be paid to analytic-type resultsin complex analysis, especially those which have appli-cations in Mathematical Physics, Mechanics, Chemistry,Biology, Medicine, Economics etc. Among the meth-ods under consideration are: boundary value problemsfor holomorphic and harmonic functions and their gen-eralizations, singular integral equations, potential anal-ysis, conformal mappings, functional equations, entireand meromorphic functions, elliptic and doubly peri-odic functions etc. Applications in Fluid Mechanics,Composite Materials, Porous Media, Hydro- Aero- andThermo-Dynamics, Elasticity, Elasto-Plasticity, will bethe most considered at the session.

32

I.4. Zeros and Gamma lines – value distributions of real and complex functions

—Abstracts—

R-linear problem and its applications to composites

Vladimir MityushevPodchorazych 2 Krakow, Malopolska 30-084 Polandvmityu@yahoo.com

We develop the method of functional equation to deriveanalytical approximate formulae for the effective con-ductivity tensor of the two–dimensional composites withelliptical inclusions. The sizes, the locations and the ori-entations of the ellipses can be arbitrary. The analyticalformulae contains all above geometrical parameters insymbolic form.

———

Application of the spectral parameter power seriesmethod to conformal mapping problems

Michael PorterDepartment of Mathematics, CINVESTAV-IPN, Li-bramiento Norponiente 2000, Fracc. Real de JuriquillaQueretaro, 76230 Mexicomike@math.cinvestav.mx

Many problems in conformal mapping of plane domainsare determined by the Schwarzian derivative of the map-ping, a third-order nonlinear differential operator, andit is well known that this can be rephrased in terms ofa second-order linear differential equation y′′ + φy = 0.For many mapping problems the coefficient function φ inthis equation depends on one or more real or complex pa-rameters; a typical formulation might be y′′+ qy = λry.The global aspect of a mapping problem often translatesinto boundary conditions (possibly nonlinear) on a realinterval and a spectral problem is thus presented. Weapply the recently developed spectral parameter powerseries (SPPS) method for Sturm-Liouville problems togain insight into conformal mapping problems. In par-ticular we will calculate the complete parameter spacefor conformal mappings from the disk to a symmetriccircular quadrilateral with right angles.

———

Recent results on analytic methods for 2D compositematerials

Sergei RogosinDepartment of Mathematics and Mechanics, BelarusianState University, Nezavisimosti ave, 4 Minsk, BY-220030Belarusrogosin@gmail.com

It is a survey talk on the recent analytic results for 2Dcomposite materials. Special attention will be paid toapplication of the boundary value problems for analyticfunctions, of the functional equations method and of theintegral equation method.

———

I.4. Zeros and Gamma lines – valuedistributions of real and complex functions

Organisers:Grigor Barsegian, George Csordas

The numbers of zeros of certain classes of meromor-phic functions are studied, particularly, in the classi-cal Nevanlinna and Ahlfors theories. Some analogousresults were obtained also for the Gamma-lines of func-tions (i.e., preimages of curves). This enlarges the valuedistribution, describes not only the numbers but also thelocations of a-points and, unexpectedly, leads to new dis-tribution type phenomena for the zeros in real analysisand real algebraic geometry. Thus we are now in a stageof formation of some methods working in both real andcomplex analysis. The zeros (a-points, fixed-point) andGamma-lines arising in complex analysis (particularlymeromorphic functions and solutions of ODE, harmonicand polynomial mappings), real analysis, real and com-plex algebraic geometry will be subject of this session.

—Abstracts—

An universal value distribution: for arbitrary meromor-phic function in a given domain

Grigor BarsegianInstitute of Mathematics of the National Academy ofSciences, 24-b Bagramian ave. Yerevan, 375019 Arme-niabarseg@instmath.sci.am

Some purely geometric results analogous to the secondfundamental theorems in the classical Nevanlinna andAhlfors theories are revealed. These analogs are valid forarbitrary analytic (meromorphic) functions in given do-mains unlike the classical results that are valid only forsome known sub classes of functions that have “equidis-tributions”. The obtained results are sharp as for func-tions in the complex plane (the classical case) as well asfor functions in a given domain.

———

A generalization of the Stieltjes-Van Vleck-Bocher the-orem

Petter BrandenDepartment of Mathematics Royal Institute of Technol-ogy Stockholm, Stockholm 100 44 Swedenpbranden@math.kth.se

A classical theorem of Stieltjes, Van Vleck and Bocherdescribes the polynomial solutions f(z), v(z) to the sec-ond order differential equation

dYj=1

(z−αj)f ′′(z)+

dXj=1

βjYi6=j

(z−αi)f ′(z)+v(z)f(z) = 0

where α1 < · · · < αd are real and β1, . . . , βd are posi-tive. B. Shapiro has recently developed a Heine-Stieltjestheory for linear differential operators of higher order.He conjectured a vast generalization of the Stieltjes–Van Vleck–Bocher theorem. We prove this conjectureand describe the intricate structure of the zeros of thesolutions.

———

A criterion for the reality of zeros

David CardonDepartment of Mathematics, Brigham Young Univer-sity, Provo, Utah 84604 United Statescardon@math.byu.edu

33

I.4. Zeros and Gamma lines – value distributions of real and complex functions

I will discuss a necessary and sufficient condition for cer-tain real entire functions to have only real zeros.

———

New properties of a class of Jacobi and generalized La-guerre polynomials

Marios CharalambidesDepartment of Business Administration, Frederick Uni-versity, 7 Yianni Frederickou street, Nicosia, Pallourio-tisa 1036 Cyprusbus.chm@fit.ac.cy

New properties of a class of Jacobi and generalized La-guerre polynomials are presented. The results give newclasses of stable polynomials and polynomials with realnegative roots. Implications of these results on the areasof geometry of polynomials and numerical analysis arealso discussed.

———

Meromorphic Laguerre operators and the zeros of en-tire functions

George CsordasDepartment of Mathematics University of Hawaii, Hon-olulu, Hawaii 96822 United Statesgeorge@math.hawaii.edu

The purpose of this lecture is to announce new re-sults pertaining to the following open problem. Char-acterize the meromorphic functions, F (x), such thatP∞k=0 F (k)akx

k/k! is a transcendental entire functionwith only real zeros (or that the zeros all lie in thehalf-plane <z < 0), whenever the entire functionP∞k=0 akx

k/k! has only real zeros. These results will beused to investigate the distribution of zeros of the Rie-mann ξ-function and the Fourier transforms of kernelsrelated to the Jacobi theta function.

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On the logarithmic order of meromorphic functions

Arturo FernandezDepartamento de Matematicas Fundamentales, Facul-tad de Ciencias, Universidad Nacional de Educacion aDistancia (UNED), Avda Senda del Rey n 9, Madrid28040, Spainafernan@mat.uned.es

The order of a meromorphic function is roughly speak-ing the exponent for which the power rλ yields thebest approximation for the growth of the characteris-tic function T (r, f). P. Chern compared for functionsof zero order the growth of T (r, f) with powers (log r)µ.This was already suggested in a more general way inthe classical literature, for instance, Nevanlinna, Valironwhere it was proposed to compare with the functionsrλ (log r) · · · (logl r)

µ.In this work we extend some conclusions by Chern forthis more general notions of order of growth.

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An introduction to upper (stable) polynomials in severalvariables

Steve FiskDepartment of Mathematics, Virginia Tech, Blacksburg,

VA 24061-0123 United Statesfisk@bowdoin.edu

Polynomials with real coefficients and all real roots havemany interesting and useful properties. This talk willintroduce an elegant generalization to polynomials withcomplex coefficients in seeveral variables. These newpolynomials are called upper (or stable) polynomials andare defined by their non-vanishing on the upper halfplane. This is recent work of J. Borcea, P. Brand en,S. Fisk, B. Shapiro, A. Sokal, and D. Wagner.

———

Perturbations of L-functions with or without non-trivialzeros off the critical line

Paul GauthierDepartement de mathematiques et de statistique, Uni-versite de Montreal, CP-6128 Centreville Montreal, Que-bec H3Y1Y8 Canadagauthier@dms.umontreal.ca

Joint with X. Xarles.There exist small perturbations of L-functions, satis-fying the appropriate functional equation, for whichthe analogue of the Riemann hypothesis fails radically.Moreover, this phenomenon is generic. Moreover, thereexist small perturbations, for which the analogue of theRiemann hypothesis holds.

———

Tropical and number theoretic analogues of Nevanlinnatheory

Rod HalburdDepartment of Mathematics, University College Lon-don, Gower Street, London WC1E 6BT United King-domR.Halburd@ucl.ac.uk

A tropical version of Nevanlinna theory is described inwhich the role of meromorphic functions is played bycontinuous piecewise linear functions of a real variablewhose one-sided derivatives are integers at every point.These functions are naturally defined on the max-plus(or tropical) semi-ring. Analogues of the Nevanlinnacharacteristic, proximity and counting functions are de-fined and versions of Nevanlinna’s first main theorem,the lemma on the logarithmic derivative and Clunie’slemma are proved. Connections with number theory,specifically Diophantine approximation, will also be dis-cussed.

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Growth of analytic functions in unbounded open sets

Aimo HinkkanenDepartment of Mathematics, University of Illinois 1409W. Green St. Urbana, IL 61801 United Statesaimo@illinois.edu

Let G be a non-empty open set in the complex plane Cwith at least two finite boundary points. Let f : G→ Cbe a continuous function that is analytic in G. Let µ(t)be a non-negative non-decreasing function defined fort ≥ 0 such that µ(2t) ≤ 2µ(t) for all t ≥ 0. Supposethat |f(z1) − f(z2)| ≤ µ(|z1 − z2|) for a fixed z1 ∈ ∂G

34

I.4. Zeros and Gamma lines – value distributions of real and complex functions

and for all z2 ∈ ∂G. Suppose that for each unboundedcomponent D of G, if any, there is a positive number qsuch that f(z) = O(|z|q) as z →∞ in D. We prove thatthen one of the following holds:

(i) For all z2 ∈ G we have |f(z1)− f(z2)| ≤ Cµ(|z1 −z2|) where C = 3456.

(ii) The set G contains a neighbourhood of infinity,so that G has exactly one unbounded component,and f has a pole at infinity.

Such problems have a long history, with contributionsmade by Hardy and Littlewood, Walsh, Sewell, Tamra-zov, Hayman, Gehring, and the speaker, among others.Earlier such a result, with an absolute constant C, hadonly been known whenG is simply or doubly connected.

———

Zeros de la fonction holomorphe et bornee dans unpolyhedre analytique de C2

Kazuko Kato2-407 takehanakinomoto-cho yamashina-ku kyoto-si,kyoto-fu 607-8083 Japankkato@vesta.ocn.ne.jp

On cherche la condition necessaire pour les zeros de lafonction holomorphe et bornee dans un polyhedre ∆ an-alytique de C2.Et, pour les zeros verifiant la condition necessaire, onconstruit la solution bornee du deuxieme problem deCousin dans ∆.

———

Steiner and Weyl polynomials

Victor KatsnelsonHerzl Departmet of Mathematics, the Weizmann Insti-tute, Rehovot, 76100 Israelvictor.katsnelson@weizmann.ac.il

We introduce certain polynomials, so-called H.Weyl andH.Minkowski polynomials, which have a geometric ori-gin. The location of roots of these polynomials is stud-ied.

———

Spiraling Baker domains

Anand Prakash SinghDepartment of Mathematics, University of Jammu,Jammu-180006, INDIAsinghanandp@rediffmail.com

Let f be a transcendental entire function. For n ∈ N,let fn denote the nth iterate of f . Fatou set F (f) of f isdefined to be the set of all points z in the complex planeC such that the family fnn≥1 forms a normal familyin some neighbourhood of z. Julia set is defined to bethe complement of Fatou set.A periodic component U of F (f) of period m is calleda Baker domain if fmn(z) → ∞ as n → ∞ for allz ∈ U . Further we define a Baker domain B as a posi-tively oriented spiraling Baker domain if there exist pos-itive continuous functions A(r), φ(r), ψ(r), of r all tend-ing to ∞ as r → ∞ such that φ(r) in non decreasing,0 < ψ(r)− φ(r) < 2π and

B ⊂ G := A(r)eiθ | φ(r) < θ < ψ(r), r > r1for some r1 > 0. Similarly we define a negatively ori-ented Baker domain. By Spiraling Baker domain, wemean either positively oriented Spiraling Baker domainor a negatively oriented spiraling Baker domain.In this paper we show the existence of Spiraling Bakerdomain and obtain several properties of these.

———

The algebraic Liouville integrability and the relatedPicard-Fuchs type equations

Anatoliy PrykarpatskyThe AGH-University of Science and Technology,Krakow, Poland, and Ivan Franko State PedagogicalUniversity, Drohobych, Lviv region, Ukraine 30 AlejaMickiewicz, N120-C Krakow, 30059 Krakow Polandpryk.anat@ua.fm

We consider a completely integrable Liouville-ArnoldHamiltonian system on a cotangent canonically symplec-tic manifold (T ∗(Rn), ω(2)), n ∈ Z+, possessing exactlyn ∈ Z+ functionally independent and Poisson commut-ing algebraic polynomial invariants Hj : T ∗(Rn) → R,j = 1, n. Due to the Liouville-Arnold theorem thisHamiltonian system can be completely integrated byquadratures in quasi-periodic functions on its integralsubmanifold when taken compact. It is equivalent tothe statement that this compact integral submanifoldis diffeomorphic to a torus Tn, that makes it possibleto integrate the system by means of searching the corre-sponding integral submanifold imbedding mapping. Thefollowing theorems are stated.

Theorem. Every completely algebraically integrableHamiltonian system admitting an algebraic submanifoldMnh ⊂ T ∗(Rn) possesses a separable canonical trans-

formation which is described by differential algebraicPicard-Fuchs type equations whose solution is a set ofsome algebraic curves

Theorem. Consider a completely integrable Hamilto-nian system on the coadjoint manifold T ∗(Rn) whose in-tegral submanifold Mn

h ⊂ T ∗(Rn) is described by Picard-Fuchs type algebraic equations. The corresponding in-tegrability embedding mapping πh : Mn

h → T ∗(Rn) isa solution of a compatibility condition subject to thedifferential-algebraic relationships on the correspondingcanonical transformations generating function.

———

Quantization of universal Teichmuller space: an inter-play between complex analysis and quantum field the-ory

Armen SergeevSteklov Mathematical Institute, Gubkina 8, Moscow,119991 Russiasergeev@mi.ras.ru

Universal Teichmuller space T is the quotient of thegroup QS(S1) of quasisymmetric homeomorphisms of S1

modulo Mobius transformations. It contains the quo-tient S of the group Diff+(S1) of diffeomorphisms of S1

modulo Mobius transformations. Both groups act natu-rally on Sobolev space H := H

1/20 (S1,R).

Quantization problem for T and S arises in string theorywhere these spaces are considered as phase manifolds.To solve the problem for a given phase space means to

35

II.1 Clifford and quaternion analysis

fix a Lie algebra of functions (observables) on it and con-struct its irreducible representation in a Hilbert (quan-tization) space.For S an algebra of observables is given by Lie algebraVect(S1) of Diff+(S1). For quantization space we take

the Fock space F (H), associated with H = H1/20 (S1,R).

Infinitesimal version of Diff+(S1)-action on H generatesan irreducible representation of Vect(S1) in F (H), yield-ing quantization of S.For T the situation is more subtle since QS(S1)-actionon T is not smooth. So there is no classical Lie al-gebra, associated to QS(S1). However, we can definea quantum Lie algebra of observables Derq(QS), gener-ated by quantum differentials, acting on F (H). Thesedifferentials arise from integral operators dqh on H withkernels, given essentially by finite-difference derivativesof h ∈ QS(S1).

———

II.1 Clifford and quaternion analysis

Organisers:Irene Sabadini, Frank Sommen

We call for contributions in the fields of theoreticalquaternionic and Clifford analysis and, more in general,hypercomplex analysis intended as the study of the func-tion theory related to the Dirac operator and systems ofpartial differential operators taking values in a Cliffordalgebra. All the topics varying from the study of mono-genic functions, its generalisations to higher spin such asthe Rarita-Schwinger system, Clifford analysis on super-space, Clifford-Radon and Fourier transforms, discreteClifford analysis to functions with values in more gen-eral non-commutative structures are welcome.

—Abstracts—

Clifford analysis for orthogonal, symplectic and finitereflection groups

Hendrik de BieDepartment of Mathematical Analysis, Ghent Univer-sity, Krijgslaan 281, 9000 Ghent (Belgium)Hendrik.DeBie@UGent.be

In recent work we have developed a theory of Cliffordanalysis in superspace. This can be seen as Clifford anal-ysis invariant under the product of the symplectic withthe orthogonal group. Other authors have recently alsostudied Clifford analysis with respect to finite reflectiongroups (using Dunkl operators).In this talk we will give a general and unified frameworkthat can be used for these different symmetries.We will also discuss some typical problems that dependon the symmetry at hand. These will include the Fischerdecomposition, the Fourier transform and the Hermitepolynomials. We also discuss related quantum systems.

———

Mobius transformations and Poincare distance in thequaternionic setting

Cinzia BisiDipartimento Matematica, Universita’ della Calabria,Cubo 30b, Ponte P.Bucci, Arcavacata di Rende Cosenza,

Calabria 87036 Italybisi@mat.unical.it

In the space H of quaternions, we investigate the natural,invariant geometry of the open, unit disc ∆H and of theopen half-space H+. These two domains are diffeomor-phic via a Cayley-type transformation. We first studythe geometrical structure of the groups of Mobius trans-formations of ∆H and H+ and identify original ways ofrepresenting them in terms of two (isomorphic) groups ofmatrices with quaternionic entries. We then define thecross-ratio of four quaternions, prove that, when real, itis invariant under the action of the Mobius transforma-tions, and use it to define the analogous of the Poincaredistances and differential metrics on ∆H and H+. As aspin-off, we directly deduce that there exists no isometrybetween the quaternionic Poincare distance of ∆H andthe Kobayashi distance inherited by ∆H as a domain ofC2, in accordance with the well known classification ofthe non compact, rank 1, symmetric spaces.

———

Wavelets invariant under reflection groups

Paula CerejeirasDepartment of Mathematics, University of Aveiro,Aveiro, P-3810-193 Portugalpceres@ua.pt

For signal reconstruction over a sphere, two main ap-proaches are used: the group-theoretical one (see,for instance, Antoine/Vandergheynst or M. Ferreira)where the authors use representations over homogeneousspaces and the one using approximate identities and sin-gular kernels (see Freeden, or Swelden). However, bothrely on the Lorentz group and, therefore, are not suit-able for signals with predefined symmetries which in-volve reflections. To overcome this problem, we considerdifferential-difference operators associated to specific fi-nite reflection groups, the so-called Dunkl operators. Inthis setting we construct spherical Dunkl wavelets basedon approximate identities and we give practical exam-ples.

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Some consequences of the quaternionic functional cal-culus

Fabrizio ColomboDipartimento di Matematica, Politecnico di Milano, viaBonardi 9 Milano, Mi 20133 Italyfabrizio.colombo@polimi.it

We show some of the most recent results on the quater-nionic functional calculus for left and right linear quater-nionic operators defined on quaternionic Banach spaces.This approach allows us to deal both with bounded andunbounded operators. In particular we use such a func-tional calculus to study the quaternionic evolution op-erator.

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Orthogonality of Clifford-Hermite polynomials in super-space.

Kevin CoulembierDepartment of Mathematical Analysis, Ghent Univer-sity, Krijgslaan 281 Ghent 9000 BelgiumCoulembier@cage.ugent.be

36

II.1 Clifford and quaternion analysis

In previous work by De Bie and Sommen, the Clifford-Hermite polynomials were generalized to superspace. Inthis talk we will construct an inner product for whichthese polynomials are orthogonal, using the Berezin in-tegral. This inner product can moreover be used forquantum mechanics in superspace, as it restores the her-miticity of the anharmonic oscillator.As an application we will also derive a Mehler formulawith O(m) × Sp(2n) symmetry. The Mehler formulagives an expansion of the kernel of the fractional Fouriertransform in terms of the super Clifford-Hermite poly-nomials. This was already known in one dimension(Hermite polynomials) and formally for O(m) (Clifford-Hermite polynomials), but the O(m) × Sp(2n) posessome extra difficulties.

———

Recent results on hyperbolic function theory

Sirkka-Liisa ErikssonDepartment of Mathematics, Tampere University ofTechnology , P.O.Box 553, Tampere 33101 Finlandsirkka-liisa.eriksson@tut.fi

The aim of this talk is to consider the hyperbolic versionof the standard Clifford analysis. The need for such amodification arises when one wants to make sure thatthe power function xm is included. The leading idea isthat the power function is the conjugate gradient of aharmonic function, defined with respect to the hyper-bolic metric of the upper half space. We present re-sults and problems concerning power series presentationof hypermonogenic functions This work is done jointlywith professor Heinz Leutwiler, University of Erlangen-Nurnberg, Department of Mathematics, Erlangen, Ger-many, email: leutwil@mi.uni-erlangen.de.

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Symmetric properties of the Fourier transform in Clif-ford analysis setting

Ming-Gang FeiDepartamento de Matematica, Universidade de Aveiro,Campus Universitario de Santiago Aveiro, Aveiro 3810-193 Portugalfei@ua.pt

In this talk we present Fueter’s Theorem for Dunkl-monogenic functions. We show that if f is a holomor-phic function in one complex variable, then for any un-derlying space Rd1 the induced function ∆

γκ+(d−1)/2h f(x)

is Dunkl-monogenic whenever γκ + (d − 1)/2 is a non-negative integer, where ∆h is Dunkl Laplacian. Tothis end Vekua-type systems for axial Dunkl-monogenicfunctions are studied.

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Factorization of Mobius gyrogroups - the paravectorcase

Milton FerreiraCampus Universitario de Santiago, Departamento deMatematica, Universidade de Aveiro, Aveiro 3810-193Portugalmferreira@ua.pt

We consider a Mobius gyrogroup on the unit ball ofthe vector space F ⊕ V, where V is a finite dimensional

vector space over the scalar field F = R or C. We willpresent the factorizations of the paravector unit ball bygyro-subgroups and subgroups, generalizing the case ofthe unit ball on Euclidean space Rn. The main differ-ences between both cases are the replacement of the Spingroup by the Spoin group and the establishment of a ge-ometric product for the paravector case, analogous tothe geometric product in the vector case.

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Higher spin analogues of the Dirac operator in two vari-ables and its resolution

Peter FranekMathematical Institute, Charles University Praha,Sokolovska 83 Prague, 8 186 75 Czech Republicfranp9am@artax.karlin.mff.cuni.cz

A resolution of the Dirac operator in two variables is wellknown and well understood. It consists of three invariantoperators (on of those of second order) expressed usingthe Dirac operators in two individual variables. We shalldiscuss higher spin analogues of such resolutions. Theyare again complexes of three invariant operators actingon functions with values in more complicated represen-tation spaces.

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Cauchy kernels in ultrahyperbolic Clifford analysis –Huygens cases

Ghislain R. FranssensBelgian Institute for Space Aeronomy, Ringlaan 3, B-1180 Brussels, Belgiumghislain.franssens@aeronomy.be

Let Rp,q ,`Rp+q, P

´, with P the canonical quadratic

form of signature (p, q). Clifford Analysis (CA) overRp,q, called Ultrahyperbolic Clifford Analysis (UCA), isa non-trivial extension of the familiar (Euclidean) CAover Rn.Essential for stating integral representation theorems inUCA is the determination of a reproducing (or Cauchy)kernel Cx0 of Rp,q, ∀p, q ∈ Z+, for the Dirac operator ∂.Any such kernel can be obtained as Cx0 = ∂gx0 , with gx0

a fundamental distribution of the ultrahyperbolic equa-tion p,qgx0 = δx0 , x0 ∈ Rp+q. The complexity of UCAis due to the fact that Cx0 is a rather complicated dis-tribution, whose form profoundly depends on the parityof p and q.Iff p and q are odd is gx0 proportional to a delta distribu-tion δ(P (x−x0)), having as support the null space of Rp,q

relative to x0, and then gx0 is said to satisfy Huygens’principle. In this talk, explicit expressions for the distri-butions gx0 and Cx0 will be presented for the Huygenscases. We will see how δ(P (x−x0)) arises as a pullback ofthe one-dimensional delta distribution δ and the matterof “regularization”, required for some of these distribu-tions, will be carefully addressed.

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Power series and analyticity over the quaternions

Graziano GentiliDipartimento di matematica ”U.Dini”, viale Morgagni67/a, 50134 Firenze, Italygentili@math.unifi.it

37

II.1 Clifford and quaternion analysis

We study power series and analyticity in the quater-nionic setting. We first consider a function f definedas the sum of a power series

Pn∈N q

nan in its domainof convergence, which is a ball B(0, R) centered at 0.At each p ∈ B(0, R), f admits expansions in terms ofappropriately defined regular power series centered atp,Pn∈N(q − p)∗nbn. The expansion holds in a ball

Σ(p,R − |p|) defined with respect to a (non-Euclidean)distance σ. We thus say that f is σ-analytic in B(0, R).Furthermore, we remark that Σ(p,R− |p|) is not alwaysan Euclidean neighborhood of p; when it is, we say thatf is quaternionic analytic at p. It turns out that f isquaternionic analytic in a neighborhoodA ofB(0, R)∩R,with A strictly contained in B(0, R) unless R is infinite.We then extend these results to the larger class ofquaternionic slice regular functions, enriching their the-ory. Indeed, slice regularity proves equivalent to σ-analyticity and slice regular functions are quaternionicanalytic only in a neighborhood of the real axis.

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Isomorphic action of SL(2,R) on hypercomplex num-bers

Anastasia KisilTriniti College Cambridge, University Cambridge, Cam-bridgeshire CB2 1TQ, United Kingdomkisilv@maths.leeds.ac.uk

We investigate the SL(2,R) invariant geodesic curveswith the associated invariant distance function inparabolic geometry. Parabolic geometry naturally oc-curs as action of SL(2,R) on dual numbers and is placedin between the elliptic and the hyperbolic geometries(which arise from the action of SL(2,R) on complex anddouble numbers). Initially we attempt to use standardmethods of finding geodesics but they lead to degener-acy in this set-up. Instead, by studying closely the tworelated hypercomplex numbers we discover a unified ap-proach to a more exotic and less obvious dual number’scase. With aid of common invariants we describe thepossible distance functions that turn out to have someunexpected, interesting properties.

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Construction of 3D mappings on to the unit ball withthe hypercomplex Szego kernel

Rolf Soeren KraussharDepartment of Mathematics, Katholieke UniversiteitLeuven, Celestijnenlaan, 200-B Leuven, Vlaams Bra-bant, 3001 BelgiumSoeren.Krausshar@wis.kuleuven.be

In this talk we present a hypercomplex generalizationof the Szego kernel method that allows us to construct3D mappings from some elementary domains of R3 ontothe unit sphere. More precisely, we consider an ap-propriately chosen line integral over the square of thehypercomplex Szego kernel. The latter one is approxi-mated numerically by the monogenic Fueter polynomialsfor rectangular domains, an L-shaped domain, circularcylinders and the double cone. In all these cases the lineintegration provides an amazingly good mapping ontothe unit sphere. We also compare the quality of resultsontained with this method with the results that wereobtained previously by using alternatively the Bergmankernel method.

This is joint work with D. Constales and D. Grob.

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Explicit description of the resolution for 4 Dirac opera-tors in dimension 6

Lukas KrumpMathematical Institute of the Charles University,Sokolovska 83, Praha 8, 186 75 Czech Republickrump@karlin.mff.cuni.cz

There are several approaches to the construction of a res-olution of several Dirac operators in higher dimensions.Among them, the Penrose transform method gives satis-fying results in both stable and unstable cases. Recentlythis method was used to determine the shape of suchresolution in many cases and the next step is an explicitdescription of operators involved. This will be shown forthe unstable case of four operators in dimension six.

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On polynomial solutions of Moisil-Theodoresco systemsin Euclidean spaces

Roman LavickaMathematical Institute, Charles University Sokolovska83 Praha 8, Praha 186 75 Czech Republiclavicka@karlin.mff.cuni.cz

Let k be a positive integer and 0 ≤ s ≤ m. Denote byPk the space of real-valued k-homogeneous polynomialsin Rm. Moreover, Λs stands for the space of s-vectorsover Rm and Psk = Pk

NR Λs. We are interested in the

following space

Hsk = P ∈ Psk ; dP = 0, d∗P = 0.

Here d and d∗ is the de Rham differential and its ad-joint, respectively. Moreover, assume that r, p and q arenon-negative integers such that p < q and r + 2q ≤ m.Putting

P(r,p,q)k =

qMj=p

Pr+2jk ,

the space

MT(r,p,q)k = P ∈ P(r,p,q)

k ; (d+ d∗)P = 0

is formed by all k-homogeneous polynomial solutions ofthe Moisil-Theodoresco system of type (r, p, q). We showthat

MT(r,p,q)k '

qMj=p

Hr+2jk ⊕

q−1Mj=p

Hr+2j+1k−1 .

Later on, the spaces Hsk are considered as SO(m)-

modules. We are interested in irreducibility, the high-est weights and dimensions of such modules. In par-ticular, we give a formula for the dimension of thespace MT

(r,p,q)k . Moreover, we decompose the kernel of

the Hodge laplacian on polynomial forms into SO(m)-modules.These results were obtained jointly with R. Delangheand V. Soucek.

———

Quaternionic analysis, representation theory andPhysics

Matvei LibineDepartment of Mathematics, Indiana University, Rawles

38

II.1 Clifford and quaternion analysis

Hall, 831 East 3rd St Bloomington, IN 47405 UnitedStatesmlibine@indiana.edu

This is a joint work with Igor Frenkel.I will describe our new developments of quaternionicanalysis using representation theory of various real formsof the conformal group as a guiding principle. These de-velopments will lead to a solution of Gelfand-Gindikinproblem. Along the way we discover striking new con-nections between quaternionic analysis and mathemat-ical physics. In particular, the Maxwell equations arerealized as the quaternionic counterpart of the Cauchyformula for the second order pole. We also find arepresentation-theoretic meaning of the polarization ofvacuum and one-loop Feynman integrals.This talk is partially based on the joint paper with IgorFrenkel, “Quaternionic analysis, representation theoryand physics”, Advances in Mathematics 218 (2008) pp1806-1877; also available at arXiv:0711.2699.

———

Hyperholomorphic functions in the sense of Moisil-Thodoresco and their different hyperderivatives

Maria Elena Luna-ElizarrarasESFM-IPN, U.P.A.L.M. Av. IPN s/n Col.LindavistaMexico City, D.F. 07338 Mexicoeluna@esfm.ipn.mx

Any Moisil-Theodoresco-hyperholomorphic function isalso Fueter-hyperholomorphic, but its hyperderivativeis always zero, so one could consider then that thesefunctions are a kind of “constants” for the Fueter oper-ator. It turns out that the skew-field of quaternions asa real linear space is wide enough, so it is possible togive another type of hyperderivatives “consistent” withthe Moisil-Theodoresco operator. In this talk we presentthese notions of different hyperderivatives and the rela-tion between them.The talk is based on a joint work with M. A. MacıasCedeno and M. Shapiro. The research was partiallysupported by CONACYT projects as well as by Insti-tuto Politecnico Nacional in the framework of COFAAand SIP programs.

———

Dirac and semi-Dirac pairs of differential operators

Mircea MartinDepartment of Mathematics, Baker University, 8th andGrove, Baldwin City, Kansas 66006 United Statesmircea.martin@bakeru.edu

The Euclidean Dirac operator Deuc,n on Rn, n ≥ 2, is adifferential operator with coefficients in the Clifford al-gebra of Rn that has the defining property D2

euc,n = −∆,where ∆ = ∆euc,n is the Laplace operator on Rn.As generalizations of this class of operators we investi-gate pairs (D,D†) of differential operators on Rn withcoefficients in a Banach algebra A, such that eitherDD† = µL∆ and D†D = µR∆, or DD†+ D†D = µ∆,where µL, µR, or µ are some elements of A. Such pairs(D,D†) are called Dirac or semi-Dirac pairs of dif-ferential operators. The typical examples of a Dirac orsemi-Dirac pair on Rn are given by D = D† = d + d∗,or D = d and D† = −d∗, where d is the operator of

exterior differentiation acting on forms on Rn, and d∗ isits formal adjoint.Our goal is to prove that any Dirac and semi-Diracpair (D,D†) has two Cauchy-Pompeiu and two Bochner-Martinelli-Koppelman type integral representation for-mulas.

———

A differential form approach to Dirac operators on sur-faces

Heikki OrelmaInstitute of Mathematics, Tampere University of Tech-nology, P.O. Box 553, FI-33101 Tampere, FinlandHeikki.Orelma@tut.fi

In this talk we consider Dirac operators on surfaces. Sur-faces are k-dimensional embedded submanifolds of Rm.Let F be a Clifford algebra-valued differential form and∂x be the Dirac operator on Rm. F is called monogenicif it is a solution of the equation

L∂xF = 0,

where L∂xF is the Lie derivative of F with respect to∂x. The aim of this talk is to show that if F and ∂xare restricted to the k-surface S we obtain a Dirac typeequation

L∂x|SF |S = 0

on S. As an application we shall consider winding num-bers.

This is joint work with Frank Sommen (Gent).

———

CK-extension and Fischer decomposition for the infra-monogenic functions

Dixan Pena PenaDepartment of Mathematics, University of Aveiro, Cam-pus Universitario de Santiago, Aveiro 3810-193, Portu-galdixanpena@ua.pt

Let ∂x denote the generalized Cauchy-Riemann opera-tor in Rm+1. In this communication, we will present arefinement of the biharmonic functions and at the sametime an extension of the monogenic functions by consid-ering the solutions of the sandwich equation ∂xf∂x = 0.In this setting a CK-extension and a Fischer decompo-sition are studied.

———

A new approach to slice-regularity on real algebras

Alessandro PerottiDept. Mathematics, Univ. of Trento, Via Sommarive14 Povo, Trento I-38100 Italyperotti@science.unitn.it

We rivisit the concept of primary functions introducedby Rinehart in the ’60’s and apply it to the theory ofslice regular functions introduced recently by Gentili,Struppa and other authors.(Joint work with Riccardo Ghiloni, Trento, Italy)

———

39

II.1 Clifford and quaternion analysis

Clifford analysis with higher order kernel over un-bounded domains

Yuying QiaoYuhua east Road 113, College of Mathematics and Infor-mation Science, Hebei Normal University, Shijiazhuang,Hebei Province 050016 ChinaQiaoyuying@hebtu.edu.cn

In this paper we talk Clifford analysis with higher or-der kernel over unbounded domains. First we derivean higher order Cauchy-Pompeiu formula for the func-tions with rth order continuous differentiability overan unbounded domain whose complementary set con-tains nonempty open set. Then we obtain higher or-der Cauchy integral formula for k-regular functions andprove Cauchy inequality. Based on the higher orderCauchy integral formula, we define higher order Cauchy-type integrals and the Plemelj formula.

———

Complex Dunkl operators

Guangbin RenDepartamento de Matematica - Universidade de Aveiro,Campus de Santiago, Aveiro 3810-193 Portugalrengb@ustc.edu.cn

Complex Dunkl operators for certain Coxeter groups areintroduced. These complex Dunkl operators have thecommutative property, which makes it possible to es-tablish the corresponding complex Dunkl analysis.

———

p-Dirac equations

John RyanDepartment of Mathematics, University of Arkansas,Fayetteville, Arkansas 72703, United Statesjryan@uark.edu

Associated to Laplacians there are first order operatorscalled Dirac operators. For instance the Dirac operatorassociated to the Laplacian in the complex plane is theCauchy-Riemann operator. In euclidean space there isthe euclidean Dirac operator Similar such operators ex-ist for Laplace-Beltrami operators on Riemannian man-ifolds.Besides the usual Laplace equation in euclidean spacethere are the non-linear p-Laplace equations. Theseequations are covariant under Mobius transformationsand are invariant when p = n. Here we shall intro-duce non-linear p-Dirac equations. We shall demon-strate their link to the p-Laplacian in euclidean spaceand demonstrate their covariance under Mobius trans-formations. Other basic properties of these equationswill be investigated. We shall extend the p-Dirac andp-Laplace equations to spin manifolds.This is joint work with Craig A. Nolder (Florida StateUniversity).

———

Duality theorems for slice hyperholomorphic functions

Irene SabadiniDipartimento di Matematica, Politecnico di Milano, ViaBonardi 9, Milano, Mi 20133 Italyirene.sabadini@polimi.it

Purpose of this talk is to provide a characterization ofthe dual of the Rn-module of slice monogenic functionson a class of compact sets in the Euclidean space Rn+1.We are able to establish a duality theorem which, sinceholomorphic functions are a very special case of slicemonogenic functions, is the generalization of the classi-cal Kothe’s theorem. The duality results are also dis-cussed in the quaternionic setting.

———

Explicit description of operators in the resolution forthe Dirac operator

Tomas SalacFaculty of Mathemtarics and Physics, Sokolovska 83,Prague 8, 18675 Czech Republicsalac@karlin.mff.cuni.cz

A study of Dirac operator D in several variables is atraditional part of Clifford analysis. A lot of effort wasspent to find an analogue of the Dolbeaut complex, i.e.a resolution starting with the operator D. The resolu-tion is composed (in the stable range) from operatorsof the first and the second order. Using representationtheory, it is possible to write down an explicit form ofthe first order operators in the resolution. It is, however,much more difficult to compute an explicit form of sec-ond order operators. In the lecture, we shall use Casimiroperators (recently introduced in study of parabolic ge-ometries) as a new tool helping to get these explicit for-mulae for the second order part of the resolution.

———

On the relation between the Fueter operator and theCauchy-Riemann-type operators of Clifford analysis.

Michael ShapiroESFM-IPN, U.P.A.L.M. Av. IPN s/n Col.LindavistaMexico City, D.F. 07338 Mexicoshapiro@esfm.ipn.mx

The Moisil-Theodoresco operator has an explicitly givenrelation with the classic Dirac operator of Clifford anal-ysis for Cl0,3. It turns out that the Fueter operator doesnot have, as one would expect, a similar relation withthe corresponding classic Cauchy-Riemann operator buta modification of the latter is necessary. The aim of thetalk is to explain all this in detail thus establishing a di-rect relation between, on one hand, what is usually calledquaternionic analysis, and, on the other hand, Cliffordanalysis.This is joint work with J. Bory-Reyes. M. Shapiro waspartially supported by CONACYT projects as well asby Instituto Politecnico Nacional in the framework ofCOFAA and SIP programs.

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Conformally invariant boundary valued problems forspinors and families of homomorphisms of generalizedVerma modules.

Petr SombergMathematical Institute of Charles University,Sokolovska 83, Prague, Karlin 180 00 Czech Repub-licsomberg@karlin.mff.cuni.cz

On a conformal manifold M with boundary ∂M there is aconstruction associating conformally invariant non-local

40

II.1 Clifford and quaternion analysis

operators to the boundary valued problems for confor-mally invariant operators on M with symbols given bypower of Laplace operator. These operators belong toone parameter families of conformally invariant oper-ators, generalizing conformal Dirichlet-to-Robin opera-tor. We will discuss generalization towards conformallyinvariant boundary valued problems for the spinor rep-resentation.

———

Clifford calculus in quantum variables

Frank SommenDepartment of Mathematics, University of Ghent, Gal-glaan 2. B-9000 Gent, Belgiumfs@cage.ugent.be

Starting from the axioms of the algebra R(S) of abstractvector variables over a set S (radial algebra):

z(xy + yx) = (xy + yx)z, x, y, z ∈ S,

together with the basic q-commutation relations for co-ordinates:

xixj = qijxjxi

we arrive at the defining relations for the q-Clifford al-gebra:

eiej + qjiejei = −2gij ,

whereby gij is the q-metric which also consists of non-commuting parameters. The partial derivatives ∂xj sat-isfy the same q-relations

∂xi∂xj = qij∂xj∂xi

together with the q-Weyl relations:

∂xixj = qjix

j∂xi + δij .

This leads to the introduction of a reciprocal Cliffordbasis ej satisfying:

ejei + qjie

iej = −2δij ,

which is linked to the original Clifford basis by relationsof the form (Einstein summation convention):

ej = gjkek.

The vector derivative (Dirac operator) is then given by∂x = ∂xj e

j and the basic rules of Clifford calculus maybe derived. On the level of radial algebra these rules arethe same as for standard Clifford analysis, which indi-cates that the q-deformation aspect is only visible whencalculations are expressed in coordinates. This raisesthe problem to define a kind of q-deformation on thelevel of abstract vector variables. This can be done bydefining the Dirac operator ∂x in a suitable way as anendomorphism on R(S). This may be done by assumingthe operator relation ∂xx = −qx∂x +m+ 2qE, wherebym is the dimension of space and E ∈ End(R(S)) is theq-Euler operator given by the operator relations

Ex− qxE = x, Ez = zE, x, z ∈ S.

However, the identities for the q-quantum lattice seemto lead (in the first approximation) to a relation of theform

∂xx+ qx∂x = m+ q(q + 1)E

whereby Ex− q2xE = x.

———

On relative BGG sequences

Vladimir SoucekSokolovska 83 Mathematical Institute, Charles Univer-sity Praha, Czech Republic 186 75 Praha Czech Republicsoucek@karlin.mff.cuni.cz

The Penrose transform is a perfect tool for a study ofgeneralised Dolbeault resolutions in the theory of sev-eral Clifford variables. An important notion used in thedefinition of the Penrose transform is the relative BGGresolution. Its construction is indicated in the book byBaston and Eastwood on the Penrose transform. They,however, deserve a better attention; their constructioncan be made more detailed using tools used for construc-tion of the classical BGG sequences.

———

Regular Moebius transformations over the quaternions

Caterina StoppatoDipartimento di Matematica “U. Dini”, Universita diFirenze, Viale Morgagni 67/A, I-50134 Firenze, Italystoppato@math.unifi.it

Let H denote the real algebra of quaternions. We presentquaternionic transformations that are included in theclass of regular quaternionic functions introduced by G.Gentili and D.C. Struppa in recent years. Regularityyields to properties that recall the complex case, al-though the diversity of the quaternionic setting intro-duces new phenomena. Specifically, the group Aut(H)of biregular functions H → H coincides with the groupof regular affine transformations (namely, q 7→ qa + bwith a, b ∈ H and a 6= 0). Moreover, inspired by theclassical quaternionic linear fractional transformations,we define the class of regular fractional transformations.This class strictly includes the set of regular injectivefunctions from bH = H∪ ∞ to itself. Finally, we studyregular Moebius transformations, which map the unitball B = q ∈ H : |q| < 1 onto itself. All regularbijections from B to itself prove to be regular Moebiustransformations.

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Singularities of functions of one and several bicomplexvariables

Adrian VajiacChapman University, Dept of Math/CS, One UniversityDrive, Orange, CA 92866 United Statesavajiac@chapman.edu

In this talk we introduce the notion of regularity forfunctions of one, as well as several bicomplex variables.Moreover, using computational algebra techniques, weprove that regular functions of one bicomplex variablehave the property that their compact singularities canbe removed.

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Multiplicities of zeroes and poles of regular functions

Fabio VlacciDepartment of Mathematics Ulisse Dini, viale Morgagni67/a FIRENZE, FI 50134 Italyvlacci@math.unifi.it

41

II.2 Analytical, geometrical and numerical methods in Clifford- and Cayley-Dickson-algebras

The aim of this talk is to give a survey on some recentresults which have been obtained for the description ofthe zero sets (and poles) of regular functions. In partic-ular we will focus our attention to define (and evaluate)a multiplicity for zeroes and poles of regular functions.

———

Gauss-Codazzi-Ricci equations in Riemannian, confor-mal, and CR geometry

Zuzana VlasakovaSokolovska 83, Faculty of Mathematics and Physics,Praha 8, 18675, Czech Republiczuzana.kasarova@email.cz

We will remind the Gauss-Codazzi-Ricci equations inRiemannian geometry, and the work of David Calder-bank with Francis Burstal and Diemer on similar equa-tions for conformal geometry. Then we introduce the CRgeometry and explain that we can do the same thing alsofor this geometry (it is a complex analogue of conformalgeometry).

———

Compatibility conditions and higher spin Dirac opera-tors

Liesbet Van de VoordeDepartment of Mathematical Analysis, Clifford Re-search Group, Galglaan 2, 9000 Gent, BelgiumLiesbet.VandeVoorde@UGent.be

In this talk, we investigate polynomial solutions for gen-eralized Rarita-Schwinger operators. We will explainthat there are two types of solutions, and we will explic-itly construct one of them using results on compatibilityconditions for systems in several Dirac operators. Thisis joint work with David Eelbode and Fred Brackx.

———

II.2 Analytical, geometrical and numericalmethods in Clifford- andCayley-Dickson-algebras

Organisers:Klaus Gurlebeck, Vladimir Kisil,Wolfgang Sproßig

The mathematical use of above mentioned algebrasreaches from hypercomplex analysis and differential ge-ometry up to corresponding numerical methods. There-fore we call especially for contributions with applicationsin gauge theories, mathematical physics, image process-ing, robotics, cosmology, engineering sciences etc.

—Abstracts—

Wavelets on spheres

Swanhild BernsteinFreiberg University of Mining and Technology, Instituteof Applied Analysis, Pruferstr. 9, D-09596 Freiberg,Germanyswanhild.bernstein@math.tu-freiberg.de

The construction of wavelets relies on translations anddilations which are perfectly given in R. On the spheretranslations can be considered as rotations but it diffi-cult to say what are dilations. For the 2-dimensionalsphere there exist two different approaches. The firstconcept defines wavelets by means of kernels of spher-ical integrals. The other approach is a purely grouptheoretical approach and defines dilations as dilations inthe tangent plane. Surprisingly both concepts coincidesfor zonal functions. We will define wavelets on the 3-dimensional sphere by means of kernels of integrals anddemonstrate that wavelets constructed according to thegroup-theoretical approach for zonal functions meet ourdefinition.Typical examples arise quite easily from the Abel-Poisson and Gauß-Weierstraß kernel.We will extend these kernels and wavelets into theClifford-algebra setting. We specifically define spheri-cal wavelets of order m.

Theorem. The elements of Ψρ, ρ > 0 are wavelets oforder m (m ≥ 0) if the following admissibility conditionsare satisfied:Z ∞

0

eΨ2ρ(k)α(ρ) dxρ = (k + 1)2, k = m+ 1, m+ 2, ...

eΨρ(k) = 0, k = 0, ...,m; ∀ρ ∈ (0, ∞)Z π

0

˛Z ∞R

Ψ(2)ρ (θ)α(ρ) dxρ

˛sin2(θ) dxθ ≤ T, ∀R ∈ (0, ∞),

(T > 0, independent of R).

Here, Ψ(2)ρ stands for Ψρ ∗Ψρ.Ψ1 (ρ = 1) is the mother

wavelet.

———

On special monogenic power and Laurent series expan-sions and applications

Sebastian BockBauhaus-University Weimar, Institute for Mathemat-ics/Physics, Coudraystraße 13B, Weimar, 99421 Ger-manysebastian.bock@uni-weimar.de

The contribution focuses on some recently developed(orthogonal) monogenic power and Laurent-series ex-pansions which are complete in the space of square inte-grable quaternion-valued functions and have as similarproperties as the respective complex series expansionsbased on the well known z-powers. Starting with theFourier series expansion we will show some structuralproperties of the series expansion with respect to theirhypercomplex derivative and primitive. These specialcharacteristics of the used orthonormal basis enable fur-ther the construction of a new Taylor type series expan-sion which can be explicitly related to the correspond-ing Fourier series analogously as in the complex one-dimensional case. We end up by showing some orthog-onality results for the exterior domain and present thecorresponding Laurent series expansion for the domainof the spherical shell.These series expansions find applications in the descrip-tion of the hypercomplex derivative as well as the mono-genic primitive of a monogenic function which are rep-resented as Fourier series, Taylor type and Laurent se-ries. In this connection some further applications arepresented.

42

II.2 Analytical, geometrical and numerical methods in Clifford- and Cayley-Dickson-algebras

———

Spin gauge models

Ruth FarwellBuckinghamshire New University, Queen AlexandraRoad, High Wycombe, Bucks HP11 2JZ United King-domruth.farwell@bucks.ac.uk

In 1999 we defined a form of spin gauge theory of particleinteractions in which both standard ’left-hand’ and new’right-hand’ interaction terms occur. In the proceed-ings of the 2005 Toulouse conference we reported thepredictions of the value of the Weinberg angle and themass of the Top quark based on a particular ’two-sided’model, and we introduced the concepts of the ’quark’and ’centroid’ representations. We also discussed newgravitational effects and the replacement of the gravitonby the ’frame field quantum’.Recently, we have studied a variety of other two-sidedmodels, and we present the predictions of another model,in which a different choice of spinor idempotent allowsus to introduce a new particle interaction term.This is joint work with Roy Chisholm (Kent).

———

Further results in discrete Clifford analysis

Nelson FaustinoDepartamento de Matematica, Campus Universitario deSantiago Aveiro, Aveiro 3810-193 Portugalnfaust@ua.pt

In this talk we will present the fundamentals of a higherdimensional discrete function theory by combining theClifford algebra setting with the umbral calculus ap-proach.Starting with the umbral version of Fischer decomposi-tion, we decompose the space of umbral homogeneouspolynomials in terms of umbral monogenic polynomials.This allows us to build up in a combinatorial way thetheory of discrete spherical monogenics as a refinementof the theory of spherical harmonics.Furthermore, the interplay between discrete Cliffordanalysis and the physical model of the discrete harmonicoscillator will be explored along this talk by means of thecanonical generators of Wigner Quantum Systems.

———

Integrability in multidimensions, complexification andquaternions

Thanasis FokasDepartment of Applied Mathematics and TheoreticalPhysics, Centre for Mathematical Sciences, WilberforceRoad Cambridge, Cambridgeshire CB3 0WA UnitedKingdomT.Fokas@damtp.cam.ac.uk

One of the most important open problems in the areaof integrable nonlinear evolution equations has been theconstruction of integrable equations in 3+1, i.e. in threespatial and one temporal dimensions. The celebratedKdV and NLS equations are integrable evolution equa-tions in 1+1; the KP and DS equations are physicallysignificant generalizations of the KdV and NLS in 2+1.Do there exist analogous equations in 3+1?

Recent progress in this direction will be reviewed. Inparticular integrable generalizations of KdV and NLS in4+2 will be presented and the question of their reduc-tion to 3+1 will be discussed. The role of quaternionsfor generalizing these results to higher dimensions willbe investigated.

———

Note on the linear systems in quaternions

Svetlin GeorgievSofia University, Faculty of Mathematics and Informat-ics, Department of Differential Equations, Blvd JamesBoucher 126, Sofia 1000 Bulgariasgg2000bg@yahoo.com

In this talk we will discuss the linear system

rXl=1

nXm=1

pslmxmqslm = As, s = 1, 2, . . . , n, (*)

where n, r ≥ 1 are given constants, pslm, qslm, As,l = 1, . . . , r, m = 1, . . . , n, s = 1, . . . , n, are given realquaternions, xm, m = 1, . . . , n, are unkown real quater-nions.Here a propose an algorithm for finding a solution tothe system (*). Also, we give necessary and sufficientcondition for the solvability of the system (*) and someexamples.

———

Minimal algorithms for Lipschitzian elements andVahlen matrices

Jacques Helmstetter15 rue de l Oisans, St-Martin d’Heres, Isere 38400 Francejacques.helmstetter@ujf-grenoble.fr

If S is a closed algebraic manifold in a vector space V ,and if d is the codimention of S in V , an algorithm thatallows us to test whether an element of V belongs to Sby means of only d numerical verifications, is called aminimal algorithm. If Cl(M, q) is the Clifford algebraderived from a quadratic module (M, q), the Lipschitzmonoid Lip(M, q) is (in most cases but not in all cases)the monoid generated in Cl(M, q) by M . From the in-variance property of Lipschitz monoids, a minimal algo-rithm can be deduced for the even and odd componentsof Lip(M, q). A minimal algorithm can also be deducedfor the two components of the monoid of Vahlen matri-ces.

———

Clifford-Fourier transforms and hypercomplex signalprocessing

Jeff HoganSchool of Mathematical and Physical Sciences, Univer-sity of Newcastle V-128, University Drive Callaghan,NSW 2308 AustraliaJeff.Hogan@newcastle.edu.au

In this talk we attempt to synthesize recent progressmade in the mathematical and electrical engineeringcommunities on topics in Clifford analysis and the pro-cessing of colour images, in particular the constructionand application of Clifford-Fourier transforms designedto treat multivector-valued signals. Emphasis will beplaced on the two-dimensional setting where the appro-priate underlying Clifford algebra is the familiar set of

43

II.2 Analytical, geometrical and numerical methods in Clifford- and Cayley-Dickson-algebras

quaternions. We’ll describe some results and problemsin the construction of discrete wavelet bases for colourimages, and the difficulties encountered in constructingClifford-Fourier kernels in dimensions 3 and higher.

———

Discrete Clifford analysis by means of skew-Weyl rela-tions

Uwe KahlerDepartment of Mathematics, University of Aveiro,Aveiro, P-3810-193 Portugalukaehler@ua.pt

Recently one can observe an increased interest in higherdimensional discrete function theories. This is not onlydriven by the numerical application of continuous meth-ods but also due to problems from combinatorics andquantum physics. While there is now a well-establishedapproach in the continuous case, by means of the so-called radial algebra (F. Sommen), unfortunately, a di-rect translation to the discrete case is problematic. Inthis talk we present an alternative approach based on arecent idea of F. Sommen of replacing the Weyl relationsby skew-Weyl relations. We will construct the basic in-gredients for discrete Clifford analysis in this contextand illustrate its applicability.

———

Hypercomplex analysis in the upper half-plane

Vladimir KisilSchool of Mathematics, Woodhouse Lane, University ofLeeds, LS2 9JT, United Kingdomkisilv@maths.leeds.ac.uk

Complex analysis seems to be the only non-trivial ana-lytic function theory in the two dimensional case. How-ever one can employ the group SL(2, R) and its repre-sentation theory in order to build elements of analyticfunctions with complex, dual and double numbers. Thisis a part of “Erlangen Programme at Large” approachin analysis.

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Formulas for reproducing kernels of solutions to poly-nomial Dirac equations in the annulus of the unit ballin Rn and applications to inhomogeneous Helmholtzequations

Rolf Soeren KraussharDepartment of Mathematics, Katholieke UniversiteitLeuven, Celestijnenlaan, 200-B Leuven, Vlaams Bra-bant, 3001 BelgiumSoeren.Krausshar@wis.kuleuven.be

Let D :=Pni=1

∂∂xi

ei be the Euclidean Dirac operator in

Rn and let P (X) = amXm + . . .+ a1X1 + a0 be a poly-

nomial with arbitrary complex coefficients. Differentialequations of the form P (D)f = 0 are called polynomialDirac equations with complex coefficients.In this talk we consider Hilbert spaces of Clifford algebravalued functions that satisfy such a polynomial Diracequation in annuli of the unit ball in Rn. We deter-mine a fully explicit formula for the associated Bergmankernel for solutions of complex polynomial Dirac equa-tions of any degree m in the annulus of radii µ and 1

where µ ∈]0, 1[. We further give explicit formulas forthe Szego kernel for solutions to polynomial Dirac equa-tions of polynomial degree m < 3 in the annulus. Asconcrete application we give an explicit representationformula for the solutions of generalized Helmholtz andKlein-Gordon type equation inside the annulus and withprescribed data at the boundary of the annulus. Thesolutions are represented in terms of integral operatorsthat involve the explicit formulas of the Bergman kernelthat we computed.

———

The Ito transform for partial differential equations

Remi LeandreInstitut de Mathematiques. Universite de Bourgogne BdAlain. Savary Dijon, Cote d’Or 21078. Franceabdessatar.khelifi@fsb.rnu.tn

We give an interpretation of the celebrated Ito formulaof stochastic analysis in various contexts where there isno convenient measure on a convenient path space.We begin by the case of a diffusion (the classical one),we study after the case of the heat-equation associatedto an operator of order four on a torus, we continue bystudying the case of the Schroedinger equation associ-ated to a big order operator on a torus, we considerafter the case on the wave equation on a torus and wefinish by studying the case of a Levy type operator asso-ciated to a big fractional power of the Laplacian on thelinear space.

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Quaternionic analysis and boundary value problems

Dimitris PinotsisDepartment of Mathematics, University of Reading,RG6 6AX, UKd.pinotsis@reading.ac.uk

First, we will review some results appearing in the the-ory of quaternions. Then, we will apply these results tosolve boundary value problems for linear elliptic equa-tions in four dimensions. Further extensions of theseresults will also be discussed.

———

Integral theorems in a commutative three-dimensionalharmonic algebra

Vitalii ShpakivskiiInstitute of Mathematics of National Academy of Sci-ences of Ukraine, Tereshchenkivska str., 3, Kiev-4,01601, Ukrainespws@zu.edu.ua

An associative commutative three-dimensional algebraA3 with unit 1 is harmonic if in A3 there exists a har-monic basis e1, e2, e3 satisfying the conditions

e21 + e2

2 + e23 = 0, e2

j 6= 0 for j = 1, 2, 3. (*)

There are three harmonic algebras exactly over the fieldof complex numbers only, and all harmonic bases areconstructed by I. Mel’nichenko.We consider a harmonic algebra A3 containing the rad-ical with basis ρ1, ρ2 and multiplication table:

ρ21 = ρ2, ρ2

2 = 0, ρ1ρ2 = 0.

44

III.1. Toeplitz operators and their applications

We proved that every locally bounded function differen-tiable in the sense of Gateaux (such a function is mono-genic)

Φ(ζ) = U1(x, y, z)e1 + U2(x, y, z)e2 + U3(x, y, z)e3

(here ζ = xe1 + ye2 + ze3 and x, y, z are real) has n-th Gateau derivative for any n. So, the componentsU1, U2, U3 satisfy the three-dimensional Laplace equa-tion

∆3U :=

„∂2

∂x2 +∂2

∂y2 +∂2

∂z2

«U(x, y, z) = 0

owing to equality ∆3Φ = Φ′′(ζ) (e21 + e2

2 + e23) and equal-

ity (*).For monogenic functions Φ(ζ) taking values in A3, weproved Cauchy’s theorems for surface integral and curvi-linear integral. We proved also an analog of Cauchy’sformula that yields Taylor’s expansion of monogenicfunction. Morera’s theorem is also established. Thus,as in the complex plane, one can give different equiva-lent definitions of monogenic functions taking values inthe algebra A3.This is joint work with S. Plaksa.

———

Initial boundary value problems with quaternionic anal-ysis

Wolfgang SproßigTU Bergakademie Freiberg, Institute of Applied Analy-sis, Pruferstr. 9, Freiberg 09599 Germanysproessig@math.tu-freiberg.de

A quaternionic operator calculus is used to find represen-tations of the solution of several initial boundary valueproblems in mathematical physics.

———

Real bi-graded Clifford modules, the Majorana equationand the standard model action

Tolksdorf, JurgenMax-Planck-Institute for Mathematics in the Sciences,Inselstraße 22, 04105 Leipzig, Germanyjuergen.tolksdorf@mis.mpg.de

The fundamental grading involution that underlies theDirac equation is provided by parity. In contrast, theMajorana equation is based on charge conjugation. To-gether, these two grading involutions form what is calleda Majorana module. On these modules there exist a nat-ural class of Dirac operators encoding the action func-tional of the Standard Model of particle physics.

———

The regularized Schrodinger semigroup acting on ten-sors with values in vector bundles

Nelson VieiraDepartamento de Matematica-Universidade de Aveiro,Campus Universitario de Santiago, P-3810-193 Aveiro,Portugalnvieira@ua.pt

In this talk we apply known techniques from semigrouptheory and Clifford analysis to the homogeneous prob-lem with initial condition of the Schrodinger equation.

To do this end, we start by express the arising tenso-rial spaces in terms of complexified Clifford algebras andwe construct a fiber bundle identification of our spaceswith appropriated vector spaces of tensors and differen-tial forms. We then establish the semi-groups for thefamily of regularized Schrodinger operators and provetheir dissipative property. We end with an applicationto the non-stationary Schroringer equation.

———

III.1. Toeplitz operators and theirapplications

Organisers:Sergei Grudsky, Nikolai Vasilevski

The idea of the session is to bring together the ex-perts actively working on Toeplitz operators acting onBergman, Fock or Hardy spaces, as well as in variousrelated areas where Toeplitz operators play an essentialrole, such as asymptotic linear algebra, quantisation, ap-proximation, singular integral and convolution type op-erators, financial mathematics, etc.We expect that the results presented, together withfruitful discussions, will serve as a snapshot of the cur-rent stage of the area, as well as for better understandingof the priority directions and themes of future develop-ments.

—Abstracts—

On the relations between the kernel of a Toeplitz op-erator and the solutions to some associated Riemann-Hilbert problems

Cristina CamaraDepartamento de Matematica, Instituto SuperiorTecnico, Av. Rovisco Pais ,Lisboa, 1049-001 Portugalcristina.camara@math.ist.utl.pt

It is possible, in many cases, to determine some solutionto a Riemann-Hilbert problem associated to TG, of theform

Gh+ = h−, h± ∈ (H±∞)n. (*)

Such a solution can provide important information onthe properties of TG. Namely, for G ∈ (L∞(R))n×n,with detG = 1 (or admitting a bounded canonical fac-torization), if h± = (h1±, h2±) are corona pairs in C±,i.e.,

infξ∈C±

(|h1±(ξ)|+ |h2±(ξ)|) > 0, (**)

it can be shown that TG is invertible.In this talk, the question of what information can beobtained, as regards the kernel of TG, from a solution to(*), is considered. Several classes of symbols are studiedwhich, if n = 2, correspond to a situation where (**) isnot, or may not, be satisfied.

———

Convolution type operators with symmetry in exteriorwedge diffraction problems

Luis CastroCampus Universitario, Department of Mathematics,University of Aveiro, Aveiro 3810-193 Portugalcastro@ua.pt

45

III.1. Toeplitz operators and their applications

We will use convolution type operators with symme-try in a Bessel potential spaces framework to anal-yse classes of problems of wave diffraction by aplane angular screen occupying an infinite 270 degreeswedge sector. The problems are subjected to differ-ent possible combinations of boundary conditions onthe faces of the wedge. Namely, under considerationthere will be boundary conditions of Dirichlet-Dirichlet,Neumann-Neumann, Neumann-Dirichlet, impedance-Dirichlet, and impedance-Neumann types. Existenceand uniqueness results are proved for all these cases inthe weak formulation. In addition, the solutions are pro-vided within the spaces in consideration, and higher reg-ularity of solutions are also obtained in a scale of Besselpotential spaces. The talk is based on a joint work withD. Kapanadze.

———

Berezin transform on the harmonic Fock space

Miroslav EnglisMathematics Institute AS CR Zitna 25, Prague 1,Prague 11567 Czech Republicenglis@math.cas.cz

The standard Berezin-Toeplitz quantization is based onthe asymptotic expansion of the Berezin transform asthe weight parameter tends to infinity. We discuss anextension of this result to the case of the harmonic Segal-Bargmann-Fock space on Cn.

———

Inside the eigenvalues of certain Hermitian Toeplitzband matrices

Sergey GrudskyDepartment of Mathematics, CINVESTAV, Av. Insti-tuto Politecnico Nacional 2508, Col. San Pedro Zaca-tenco, 07360 Mexicogrudsky@math.cinvestav.mx

While extreme eigenvalues of large Hermitian Toeplitzmatrices have been studied in detail for a long time,much less is known about individual inner eigenvalues.This paper explores the behavior of the jth eigenvalue ofan n-by-n banded Hermitian Toeplitz matrix as n goesto infinity and provides asymptotic formulas that areuniform in j for 1 ≤ j ≤ n.The real-valued generating function of the matrices is as-sumed to increase strictly from its minimum to its max-imum and then to decrease strictly back from the max-imum to the minimum, having nonzero second deriva-tives at the minimum and the maximum.

———

Toeplitz operators of M(p, q, w)(Rd) spaces

Turan GurkanlıOndokuz Mayıs University, Faculty of Arts and Sciences,Department of Mathematics, Kurupelit Samsun, 55139Turkeygurkanli@omu.edu.tr

Let g be a function in S(Rd)/0, where S(Rd) is Schwartzspace, and 1 ≤ p, q ≤ ∞. The space M(p, q, w)(Rd) de-notes the subspace of all tempered distributions f suchthat the Gabor transform Vgf of f is in the weighted

Lorentz space L(p, q, wdµ)(R2d). M(p, q, w)(Rd) is a Ba-nach space with the norm ‖f‖M(p,q,w) = ‖Vgf‖pq,w.In this paper we discussed the boundedness ofToeplitz operator on M(p, q, w)(Rd) under some as-sumptions. We also proved that the Toeplitz oper-ator Tpg(F ) of M(2, p, w1)(Rd) into M(2, p, w1)(Rd)is S2 with the Hilbert-Schmidt norm bounded by‖Tpg(F )‖S2C‖F‖(1,t) under some condition.This is joint work with Ayse Sandikci.

———

Presentation of the kernel of a special structure matrixcharacteristic operator by the kernels of two operatorsone of them is a scalar characteristic operator

Oleksandr KarelinAdvanced Research Center on Industrial Engineering,Autonomous University of the Hidalgo State, Pachuca,Hidalgo 42184 Mexicokarelin@uaeh.edu.mx

We denote the Cauchy singular integral operator alongthe upper part of the unit semicircle T+ by

(ST+ϕ)(x) =1

πi

ZT+

ϕ(τ)

τ − xdτ

and the identity operator on T+ by (IT+ϕ)(t) = ϕ(t).By operator equalities, results about the integral opera-tors with endpoint singularities are extended to matrixcharacteristic operators

DR+ = uIT+ + vST+ , DT+ ∈ [L22(T+)]

with the coefficients u, v of a special structure. The fol-lowing decomposition

kerDT+ = ker H\F kerC,

is found. Here operator C is a scalar characteristic op-erator, C ∈ [L2(T+)], operator F is invertible operator,F ∈ [L2(T+), L2

2(T+)]. Operators H and C are con-structed by an arbitrary nontrivial element of kerDT+

or by an arbitrary nontrivial element of the kernel of theassociated operator.This is joint work with Anna Tarasenko.

———

Bounds for the kernel dimension of singular integral op-erators with Carleman shift

Edixon RojasCampus Universitario, Department of Mathematics,University of Aveiro, Aveiro 3810-193 Portugaledixonr@ula.ve

Upper bounds for the kernel dimension of singular in-tegral operators with preserving-orientation Carlemanshift are obtained. This is implemented by using someestimations which are derived with the help of certainexplicit operator relations. In particular, the interplaybetween classes of operators with and without Carle-man shifts has a preponderant importance to achievethe mentioned bounds.

———

Invertibility of matrix Wiener-Hopf plus Hankel opera-tors with different Fourier symbols

Anabela SilvaDepartamento de Matematica - Universidade de Aveiro,

46

III.2. Reproducing kernels and related topics

Campus de Santiago, Aveiro 3810-193 Portugalanabela.silva@ua.pt

Based on different kinds of auxiliary operators and corre-sponding operator relations, we will present conditionswhich characterize the invertibility of matrix Wiener-Hopf plus Hankel operators having different Fouriersymbols in the class of almost periodic elements.

———

Flat Hilbert bundles and Toeplitz operators on symmet-ric spaces

Harald UpmeierDepartment of Mathematics, University of Marburg,Hans-Meerwein-Strasse, Lahnberge Marburg, Hessen35032 Germanyupmeier@mathematik.uni-marburg.de

In generalization of the classical Fock spaces we con-struct a family of Hilbert spaces, viewed as a Hilbertbundle over a bounded symmetric domain (Cartan do-main) B, which is equivariant under a suitable, non-holomorphic, action of the holomorphic automorphismgroup G of B (a semisimple Lie group). Geometrically,these Hilbert spaces live on the so-called Matsuki dualassociated with the G-orbits in the boundary of B.We show that the Hilbert bundle carries a natural con-nection over B which is projectively flat, similar as thewell-known case for the metaplectic representation onFock space. The associated parallel transport (Bogoluy-bov transformations) is also determined. In the talk weemphasize relations to classical Fock spaces over real,complex and quaternion matrix spaces, although the ba-sic construction depends mainly on the Jordan algebraicdescription of bounded symmetric domains.

———

Commutative algebras of Toeplitz operators on the unitball

Nikolai VasilevskiDepartment of Mathematics, CINVESTAV, Av. Insti-tuto Politecnico Nacional 2508, Col. San Pedro Zaca-tenco, 07360 Mexiconvasilev@math.cinvestav.mx

All known commutative C∗-algebras generated byTopelitz operators on the unit disk are classified as fol-lows. Given a maximal commutative subgroup of bi-holomorphisms of the unit ball, the C∗-algebra gener-ated by Toeplitz operators, whose symbols are invariantunder the action of this subgroup, is commutative oneach weighted Bergman space.Surprisently there exist many other Banach algebrasgenerated by Toeplitz operators which are commuta-tive on each weighted Bergman space. These last al-gebras are non conjugated via biholomorphisms of theunit ball, non of them is a C∗-algebra, and for n = 1 allof them collapse to commutative C∗-algebra generatedby Toeplitz operators on the unit disk.

———

Toeplitz operators on the Fock space

Kehe ZhuDepartment of Mathematics and Statistics, 1400 Wash-ington Ave, SUNY Albany, New York 12222 United

Stateskzhu@math.albany.edu

We will discuss Toeplitz operators on the Fock spaceinduced by positive measures. Problems considered in-clude boundedness, compactness, and membership inthe Schatten classes.

———

III.2. Reproducing kernels and related topics

Organisers:Alain Berlinet, Saburu Saitoh

Since the first works laying its foundations as a subfieldof Complex Analysis, the theory of reproducing kernelshas proved to be a powerful tool in many fields of Pureand Applied Mathematics. The aim of this session isto gather researchers interested in theoretical as well asapplied modern problems involving this theory.

—Abstracts—

A general theory for kernel estimation of smooth func-tionals

Belkacem AbdousUniversite Laval Medecine Sociale et Preventive, Pavil-lon de l’Est, Quebec, Qc G1K 7P4 Canadabelkacem.abdous@msp.ulaval.ca

In this talk, we present a general framework for estimat-ing smooth functionals of the probability distributionfunctions, such as the density, the hazard rate function,the mean residual time, the Lorenz curve, the spectraldensity, the tail index, the quantile function and manyothers. This framework is based on maximizing a localasymptotic pseudo-likelihood associated to the empiri-cal distribution function. An explicit solution of thisproblem is obtained by means of reproducing kernelsapproach. Some asymptotic properties of the obtainedestimators are presented as well.

———

Weighted composition operators on some spaces of an-alytic functions

Som Datt SharmaDepartment of Mathematics, University of Jammu,Jammu-180006, India 66 Ashok Nagar, Canal Road,Jammu, Jammu & Kashmir 180016 Indiasomdatt jammu@yahoo.co.in

Let D be the open unit disk in the complex plane C andH(D) be the space of holomorphic functions on D. Inthis article, we give a short and selective account of re-sults known about weighted compostion operator Wψ,ϕ

defined by

Wψ,ϕf(z) = ψ(z)f(ϕ(z)), f ∈ H(D),

where ϕ is a holomorphic map of D that takes D intoitself and ψ is any holomorphic map of D. Discriptionsof weighted composition operators acting from Hardyspaces, weighted Bergman spaces, α-Bloch spaces andA−α-spaces into other spaces of holomorphic functions

47

III.3. Modern aspects of the theory of integral transforms

have been obtained by a number of authors during re-cent years. We provide a unified way of treating theseoperators.

———

Integral formulas on the boundary of some ball

Keiko FujitaFaculty of Culture and Education, Saga university, Saga840-8502keiko@cc.saga-u.ac.jp

We have been studied integral representations for holo-morphic functions and complex harmonic functions onsome balls, which we call the ”Np-balls”. One of Np-balls is the Lie ball. For holomorphic functions onthe Lie ball we know the Cauchy-Hua integral formula,whose integral is taken over the Shilov boundary of theLie ball. A generalization of the Cauchy-Hua integralformula was considered for holomorpic functions on sub-spaces of the Lie ball by M.Morimoto. Since Np-ball canbe represented by a union of these subspace, the bound-ary of the Np-ball can be represented by a union of theboundaries of the subspaces. Considering the fact, weconsider an integral representation for holomorphic func-tions on the Np-ball by an iterated integral.In this talk, we will review some integral formulas onholomorphic functions on the Np-ball and treat sometopics.

———

Paley–Wiener spaces and their reproducing formulae.

John Rowland HigginsI.H.P., 4 rue du Bary, 11250 Montclar, France.rowlandhiggins@yahoo.com

Classical Paley–Wiener space, denoted by PW, consistsof functions that are inverse Fourier transforms of thosemembers of L2(R) that are null outside [−π, π]. It iswell known that PW possesses two reproducing formu-lae; a reproducing equation and a ‘discrete’ analogue,or sampling series, and that these make a remarkable‘concrete – discrete’ comparison. It is shown that suchanalogies persist in the setting of more general Paley–Wiener spaces. ‘Operator’ versions of the reproducingequation and of the sampling series will be given thatare also comparable, but now in a slightly different way.The setting emerges from two sources, the approach tosampling theory via the reproducing kernel theory dueto S. Saitoh, and the approach via harmonic analysisof I. Kluvanek, M.M. Dodson et al. The capacity foramalgamation of these two sources has gone unnoticedhitherto.The special case of multiplier operators with respectto the Fourier transform acting on Paley–Wiener spacewill be considered. The Hilbert transform, and in two-dimensions the Riesz transforms, provide examples withpossibilities of extension to higher dimensions and tofurther classes of operators.

———

Irregular sampling in multiple-window spline-typespaces

Darian OnchisFaculty of Mathematics, University of Vienna, Nord-bergstrae 15 (Universitats Zentrum Althanstrae, UZA

4) Vienna, A-1090 Austriadarian.onchis@univie.ac.at

Irregular sampling in spline-type spaces has become avivid research area, with many contributions in the re-cent literature. We will describe efficient implementa-tions of operators related to spline-type spaces with fi-nite sets of generators on Rd, covering both the case ofregular and irregular sampling. In contrast to earlierpapers, which either treat the continuous setting usingabstract methods (i.e. continuous Fourier transforms) ordeal with the discrete case when it comes to numericalimplementations, we are discussing the problem of con-structively realizing the abstract concepts with methodsthat can be implemented on a computer, achieving asmall error of reconstruction in a certain given norm.In such a situation the trade-off between realizing in-dividual iterative steps with high precision but at highcomputational costs, versus the option of doing a largernumber of iterations has to be analyzed.Joint work with Prof. Hans Feichtinger.

———

Free boundary value problem for (−1)M (d/dx)2M andthe best constant of Sobolev inequality

Kazuo TakemuraShinei 2-11-1, Narashino, Chiba 275-8576 Japantakemura.kazuo@nihon-u.ac.jp

Green function of free boundary value problem for(−1)M (d/dx)2M is found using Whipple’s formula. ItsGreen function is constructed through so-called symmet-ric orthogonalization method under a suitable solvabilityconditions. As an application, we found the best con-stant of Sobolev inequality for M = 1, 2, 3, 4, 5 by inves-tigating an aspect of Green function as a reproducingkernel. For M ≥ 6, this is still open.

———

III.3. Modern aspects of the theory ofintegral transforms

Organisers:Anatoly Kilbas, Saburu Saitoh

—Abstracts—

Integral transforms related to generalized convolutionsand their applications to solving integral equations

Liubov BritvinaDepartment of Theoretical and Mathematical Physics,Novgorod State University, ul.St.Petersburgskaya 41,Veliky Novgorod, Novgorod region 173003 RussiaLyubov.Britvina@novsu.ru

The present research is devoted to some integral trans-forms of convolution type. The definition of polyconvo-lution, or generalized convolution, was first introducedby V.A. Kakichev in 1967. Let A1, A2 and A3 be oper-ators. The generalized convolution of function f(t) andk(t), under A1, A2, A3, with weighted function α(x),

48

III.3. Modern aspects of the theory of integral transforms

is the function h(t) denoted by“fA1

α∗ kA2

”A3

(t) for

which the following factorization property is valid:

(A3h)(x) = A3

»“fA1

α∗ kA2

”A3

–(x)

= α(x)(A1f)(x)(A2k)(x).

Here we consider the generalized convolution for integraltransforms with the Bessel functions in the kernels. Us-ing the differential properties of these convolutions weconstruct some integral transforms and find their exis-tence conditions and inverse formulas. Natural applica-tions to the corresponding class of convolution integralequations are demonstrated.

———

Bedrosian identity for Blaschke products in n-parametercases

Qiuhui ChenDepartamento de Matematica - Universidade de Aveiro,Campus de Santiago, Aveiro 3810-193 Portugalqiuhui.chen@ua.pt

We establish a necessary and sufficient condition forthe amplitude function such that a Bedrosian identityholds in the case when the phase function is determinedby the boundary value of a Blaschke product with n-parameters.

———

Evaluation formulae for analogues of conditional ana-lytic Feynman integrals over a function space

Dong Hyun ChoDepartment of Mathematics, Kyonggi University,Young-Tong-Gu Suwon, Kyonggido 443-760 South Ko-reaj94385@kyonggi.ac.kr

In this talk, we introduce two simple formulae for theconditional expectations over an analogue C[0, t] of theWiener space, the space of continuous real-valued pathson the interval [0, t]. Using these formulae, we estab-lish various formulae for analogues of the conditionalanalytic Wiener and Feynman integral of the function-als in a Banach algebra which corresponds to the Ba-nach algebra on the classical Wiener space introducedby Cameron and Storvick. Finally, we evaluate the ana-logues of the conditional analytic Wiener and Feynmanintegral for the functional

exp

Z t

0

θ(s, x(s)) dη(s)

ffwhich is defined on C[0, t] and is of interest in Feynmanintegration theories and quantum mechanics.

———

An equation with symmetrized fractional derivatives

Diana DolicaninFaculty of Technical Sciences, University of Pristina- Kosovska Mitrovica, Kneza Milosa, 28000 KosovskaMitrovica, Serbiadolicanin d@yahoo.com

We study equation

d2

dt2u (t)+

Z 1

0

±EαT u (t)φ(α)dα+ F (t, u (t)) = 0, 0 < t < T

where, ±EαT u (t) is the symmetrized Caputo fractionalderivative of u, φ(α), α ∈ (0, 1), is a positive integrablefunction or a positive compactly supported distributionwith the support in (0, 1) and F is a continuous func-tion in [0, T ] × R and locally Lipschitz continuous withrespect to the second variable.This is joint work with T. Atanackovic, S. Konjik andS. Pilipovic.

———

Numerical real inversion of the Laplace transform byreproducing kernel and multiple-precision arithmetic

Hiroshi FujiwaraKyoto University, Yoshida-Honmachi Sakyo-ku Kyoto,Kyoto 606-8501 Japanfujiwara@acs.i.kyoto-u.ac.jp

We consider the real inversion of the Laplace transform.It appears in engineering or physics, and it is ill-posedin the sense of Hadamard. We introduce some repro-ducing kernel Hilbert spaces and propose an inversionalgorithm employing Tikhonov regularization. The reg-ularized equation is well-posed, and its discretizationis expected to have the stability and convergence witha suitable norm. However, theoretical stability is notequivalent to the stability of computational processes.We propose the use of multiple-precision arithmetic toreduce the influence of rounding errors for reliable nu-merical computations. Multiple-precision arithmetic isuseful for regularization as approximation.

———

Method of integral transforms in the theory of frac-tional differential equations

Anatoly KilbasFaculty of Mathematics and Mechanics, BelarusianState University, Independence Avenue, 4 Minsk, 220030Belarusanatolykilbas@gmail.com

Our report deals with the method of integral trans-forms in investigtation of differential equations with or-dinary and partial fractional derivatives. First we givean overview of results in this field. Then we present ap-plication of one-dimensional Laplace, Mellin and Fourierintegral transforms to solution of ordinary diferentialequations with Riemann-Liouville and Caputo fracitonalderivatives. Further we give application of Laplaceand Fourier integral transforms to obtain explicit solu-tions of Cauchy-type and Cauchy problems for the two-and multi-dimensional diffusion-wave equations with theRiemann-Liouville and Caputo partial fractional deriva-tives, respectively, and indicate conditions for the ex-istence of classical solutions of these problems. Fi-nally, we use Laplace and Fourier integral transformsto deduce explicit solutions of fractional evolution equa-tions involving partial fractional derivatives of Riemann-Liouville or Caputo with respect to time and partial Li-oville fractional derivative with respect to real axis, andindicate applications.

49

III.4. Spaces of differentiable functions of several real variables and applications

We note that explicit solutions of the consideredfractional differential equations and Cauchy-type andCauchy problems for them are expressed in terms of spe-cial functions of Mittag-Leffler, Wright and the so-calledH-functions.

———

Notes on the analytic Feynman integral over paths inabstract Wiener space

Bong Jin KimDepartment of Mathematics, Daejin University,Pocheon, Kyeonggi-Do 487-711 South Koreabjkim@daejin.ac.kr

In this talk, we study some results about analytic Feyn-man integral over paths in abstract Wiener space.

———

On the fractional calculus of variations

Sanja KonjikFaculty of Agriculture, Department of Agricultural En-gineering, Trg D. Obradovica 8, Novi Sad, 21000 Serbiasanja konjik@uns.ac.rs

The purpose of this talk is to study variational principlesallowing Lagrangian density to contain derivatives of ar-bitrary real order. We derive a necessary condition forexistence of a solution to a fractional variational problemand examine invariance under the action of transforma-tion groups. As the results we obtain the Euler-Lagrangeequations, as well as infinitesimal criterion and Noether’stheorem, which in fact extend the well-known classicalresults. In addition, we also study the case when bothfunction and the order of fractional derivative are variedin the minimization procedure.

———

Integral transforms with extended generalized Mittag-Leffler function

Anna KorolevaDepartment of Mathematics and Mechanics, Nezavici-mosti ave 4, Minsk BY-220030 Belarusankakoroleva@gmail.com

Asymptotic results for integral transforms with extendedgeneralized Mittag-Leffler function in the kernel arediscussed. Inversion formulas for such transforms inweighted spaces of integrable functions are found.

———

Systems of differential equations containing distributedorder fractional derivative

Ljubica OparnicaSerbian Academy of Science and Art, Kneza Mihaila 36,Belgrade, 11000 Serbialjubicao@yahoo.com

Distributed order fractional derivatives has appeared asgeneralization of the finite sum of fractional derivativesand has wide applications in technical sciences.We define distributed order fractional derivative of dis-tribution u ∈ S ′R) supported in R+ by fomula

〈Z

suppu

φ(γ)Dγu dγ, ϕ〉 = 〈φ, 〈Dγu, ϕ〉〉, ϕ ∈ S ′,

where φ is ”weight” distribution with compact supportand, Dγ denotes the Riemman-Liouvill fractional deriva-tive of order γ.Differential equations of the formZ 2

0

φ1(γ)Dγu dγ =

Z 2

0

φ2(γ)Dγv dγ (*)

are constitutive equations for viscoelastic body.We consider (*) coupled with nonlinear ordinary differ-ential equation

D2u(t) + v = f(t, u(t)) (**)

and show existence and uniqueness of the solution tothe problem (*)–(**) with initial conditions u(0) = u0,u′(0) = v0 in classical and mild sence.

———

Some aspects of modified Kontorovitch-Lebedev inte-gral transforms

Juri M. RappoportVlasov street Building 27 apt.8 Moscow 117335jmrap@landau.ac.ru

A proof of inversion formulas of the modifiedKontorovitch-Lebedev integral transforms is developed.The Parceval equations for modified Kontorovitch-Lebedev integral transforms are proved and sufficientconditions for them are found. Some new representa-tions and properties of these transforms are justified.The inequalities which give estimations for their ker-nels - the real and imaginary parts of the modifiedBessel functions of the second kind ReK1/2+iτ (x) andImK1/2+iτ (x) for all values of the variables x and τare obtained. The applications of Kontorovitch-Lebedevtransforms to the solution of some mixed boundary valueproblems in the wedge domains are accomplished. Thesolution of the appropriate dual and singular integralequations is considered. The numerical aspects of usingof these transforms are elaborated in detail.

———

A new class of polynomials related to the Kontorovich-Lebedev transform

Semyon YakubovichDepartment of Pure Mathematics, Faculty of Science,University of Porto, Campo Alegre str. 687, Porto 4169-007 Portugalsyakubov@fc.up.pt

We consider a class of polynomials related to the kernelKiτ (x) of the Kontorovich-Lebedev transformation. Al-gebraic and differential properties are investigated andintegral representations are derived. We draw a paralleland establish a relationship with the Bernoullis and Eu-lers numbers and polynomials. Finally, as an applicationwe invert a discrete transformation with the introducedpolynomials as the kernel, basing it on a decompositionof Taylors series in terms of the Kontorovich-Lebedevoperator.

———

50

III.4. Spaces of differentiable functions of several real variables and applications

III.4. Spaces of differentiable functions ofseveral real variables and applications

Organisers:Viktor Burenkov, Stefan Samko

This session intends to cover various aspects of the the-ory of Real Variables Function Spaces (Lebesgue, Or-lich, Sobolev, Nikol’skii-Besov, Lizorkin-Triebel, Mor-rey, Campanato, and other spaces with zero or non-zero smoothness), such as imbedding properties, den-sity of nice functions, weight problems, trace prob-lems, extension theorems, duality theory etc. Vari-ous generalizations of these spaces are welcome, suchas for example Orlicz-Sobolev spaces, in particulargeneralized Lebesgue-Sobolev spaces of variable or-der, Morrey-Sobolev spaces, Muiselak-Orlich spaces andtheir Sobolev counterparts etc. Other topics: any in-equalities related to these spaces, properties of operatorsof real analysis acting in such spaces and also various ap-plications to partial differential equations and integralequations.

—Abstracts—

Hardy spaces with generalized parameter

Alexandre AlmeidaDepartment of Mathematics, University of AveiroAveiro, Aveiro 3810-193, Portugaljaralmeida@ua.pt

Hardy spaces with generalized parameter are introducedfollowing the maximal characterization approach. Asparticular cases, they include the classical Hardy spacesHp and the Hardy-Lorentz spaces Hp,q. Real interpola-tion results with function parameter are obtained. Basedon them, the behavior of some classical operators is stud-ied in this generalized setting.This talk is based on joint work with A. Caetano.

———

Iterated norms in Nikol’skiı-Besov type spaces with gen-eralized smoothness

Tsegaye Gedif AyeleDepartment of Mathematics, Addis Ababa UniversityP.O.Box 1176 Addis Ababa - Ethiopia.tsegayeg@math.aau.edu.et

In works of V.I. Burenkov iterated norms of Nikol’skiı-Besov type in spaces Blθ(· · · Blθ(Lp(Ω)) · · · ), k times it-erated, were introduced. Using these norms, it wasproved that every classical solution of the partial dif-ferential equation with constant coefficients is infinitelydifferentiable. In this paper we consider iterated normsof Nikol’skiı-Besov type in spaces B~ϕθ (· · · B~ϕθ (Lp(Ω)) · · · )with generalized smoothness ~ϕ belonging to some classof functions Φ(~σ, θ) and with the norm

‖f‖B~ϕθ (· · · B~ϕθ (Lp(Ω)) · · · )| z k

and holds the following

Theorem. Let 1 < p < ∞, 1 ≤ θ < ∞, ~σ =(σ1, σ2, · · · , σn), ~ϕ = (ϕ1, ϕ2, · · · , ϕn) ∈ Φ(~σ, θ)

such that ~ϕk = ϕk1 , ϕk2 , · · · , ϕkn and G ⊂ Rn be anopen parallelepiped with sides parallel to the coordinateplanes. Then

a. holds true the inclusion

B~ϕθ (· · · B~ϕθ (Lp(G)) · · · )| z k

→ B~ϕk

p,θ(G)

b. under the additional assumption that there existsbounded extension operator

S : B~ϕk

θ

`G´→ B~ϕ

k

θ

`Rn´,

holds true the equality of spaces

B~ϕθ (· · · B~ϕθ (Lp(G)) · · · )| z k

= B~ϕk

p,θ(G)

with equivalence of norms.

In these results: (1.) If we set ϕj(h) = hlj we get resultswhich are in works of V.I. Burenkov. (2.) In case, whenϕ(h) satisfy additional condition

∃ε > 0 :ϕ(h)

hε↑ on (0, H]

leads to increment of smoothness using iterated norms.This is joint work with Abraham N. Abebe.

———

Embeddings Properties of The Spaces Lp(.)w (Rn)

Ismail AydınSinop University, Faculty of Arts and Sciences, Depart-ment of Mathematics, 57000. Sinop-TURKIYEismaila@omu.edu.tr

We derive some of the basic properties of weighted vari-able exponent Lebesgue spaces L

p(.)w (Rn) and investi-

gate continuous embeddings Lp2(.)w2 (Rn) → L

p1(.)w1 (Rn)

with respect to variable exponents and weight functionsunder some conditions.

———

On the boundedness of fractional B-maximal operatorsin the Lorentz spaces Lp,q,γ(Rn)

Canay AykolAnkara University, Faculty of Science, Department ofMathematics, Ankara, Tandogan 06100 TurkeyCanay.Aykol@science.ankara.edu.tr

In this study, sharp rearrangement inequalities for thefractional B-maximal function Mα,γf are obtained inthe Lorentz spaces Lp,q,γ and by using these inequal-ities the boundedness conditions of the operator Mα,γ

are found. Then, the conditions for the boundedness ofthe B-maximal operator Mγ are obtained in Lp,q,γ .

———

Spaces of functions of fractional smoothness on an ir-regular domain

Oleg BesovSteklov Institute of Mathematics, Department of Func-tion Theory, 8 Gubkina Str, Moscow 119991 Russiabesov@mi.ras.ru

In 1938 S.L. Sobolev proved his well-known embeddingtheorem

Wmp (G) ⊂ Lq(G), m ∈ N, 1 < p < q <∞, (*)

51

III.4. Spaces of differentiable functions of several real variables and applications

m− n

p+n

q≥ 0, (**)

for domains G ⊂ Rn satisfying the cone condition.Relationship (**) (which determines the maximum pos-sible value of q in theorem (*)) is also a necessary con-dition for the embedding. Sobolev’s result has been ex-tended to more general domains.Definition. Given σ ≥ 1, a domain G ⊂ Rn is saidto satisfy the σ-cone condition if, for some T > 0 and0 < κ0 ≤ 1, for any x ∈ G there exists a piecewisesmooth path

γ = γx : [0, T ]→ G, γ(0) = x, |γ′| ≤ 1 a.e.,

such that

dist (γ(t),Rn \G) ≥ κ0tσ for 0 < t ≤ T.

The author established in (2001) that embedding (*) ona domain with the flexible σ-cone condition holds if

m− σ(n− 1) + 1

p+n

q≥ 0. (***)

We construct two families of function spaces Ls(m)p,θ (G)

and Bs(m)p,θ (G) of fractional smoothness s > 0 on domains

G satisfying the flexible σ-cone condition such that em-beddings

Wmp (G) ⊂ Ls(m)

q,θ (G), Wmp (G) ⊂ Bs(m)

q,θ (G),

Ls(m)p,θ (G) ⊂ Lq(G), B

s(m)p,θ (G) ⊂ Lq(G),

hold with the same loss of smoothness as in (***).

———

Rearrangement transformations on general measurespaces

Santiago BozaEPSEVG, Avda Victor Balaguer s/n. Vilanova i Geltru.08800 (SPAIN)boza@ma4.upc.edu

For a general set transformation R between two mea-sure spaces, we define the rearrangement of a measur-able function by means of the Layer’s cake formula. Westudy some functional properties of the Lorentz spacesdefined in terms of R, giving a unified approach to theclassical rearrangement, Steiner’s symmetrization, themultidimensional case, and the discrete setting of trees.

———

Last developments on Rubio de Francia’s extrapolationtheory

Maria CarroDepartment of Applied Mathematics and Analysis, Uni-versity of Barcelona, Gran Via 585, Barcelona 08007Spaincarro@ub.edu

Since in the early 80’s, J.L. Rubio de Francia devel-oped his celebrated extrapolation theorem, this theoryhas been developed in order to cover many other situa-tions such as boundedness of operator on rearrangementinvariant spaces or multilinear operator.In this talk, we shall present a new estimate on the dis-tribution function of Tf in terms of the distribution of

Mf where M is the Hardy-Littlewood maximal opera-tor and presetn several applications of it. In particular,we shall give some applications to the setting of the twoweights problem for Calderon-Zygmund operators.This is a joint work with J. Soria and R. Torres.

———

On transformation of coordinates invariant relative toSobolev spaces with polyhedral anisotropy

Gurgen DallakyanRussian-Armenian State University, Yerevan, Armeniagurg@aport.ru

Let Rn be n -dimentional euclidean space of the pointswith real coordinates, Nn

0 - the set of multi-indices.

Definition. Nonempty polyhedron ℵ ⊂ Rn with verticesin Nn

0 is said to be complete, if it has vertices at the ori-gin and at all coordinate axes of Nn

0 . Complete polyhe-dron ℵ is called completely regular (CR), if all the coor-dinates of outward normals of the noncoordinate (n−1)-dimentional faces of ℵ are positive.

Let ℵ ⊂ Rn- (CR) polyhedron with verticese0 , e1 , e2, e3, ..., eM , where the vertices ej (j =1, ..., n) lie on the j-th coordinate axe, e0 = (0, ..., 0), l = max

1≤j≤n

˛ej˛

.

For any domain Ω ⊂ Rn denote by Wℵp (Ω) (1 < p <∞)the Sobolev space with polyhedral anisotropy, i.e. thespace of functions with finite norm

‖u ‖ℵ,Ω =Xα∈ℵ

‖Dαu‖Lp(Ω).

Consider the m -dimentional manifold Γm ⊂ Ω .

Definition. The piece σ ⊂ Γm we call ℵ -regular, if forsome n -dimentional subdomain ω , σ ⊂ ω ⊂ Ω, thereexist transformation of coordinates invariant relative toWℵp (Ω), mapping σ onto σ′ ⊂ Rm .

Remark. Note, that any bounded domain always haspieces of boundary, which are not ℵ -regular.Let Γ ∈ Cs be m -dimentional manifold, where s ≥ r,

(r = max1≤j≤M

˛ej˛), i.e., Γ =

NSk=1

σk and each piece σk has

representation

xik = ψi,k(x′) , i = 1, ...,m,

where x′ ∈ Gk , Gk ∈ Rn−m , ψi,k ∈ Cs(Gk) .

Theorem. Let ℵ ⊂ Rn be (CR) polyhedron, Ω ⊂ Rn

satisfies the week condition of rectangle, Γ ∈ Cs -m di-mentional manifold ( Γ ⊂ Ω). Let σ ⊂ Γ has the repre-sentation xi = ψi(x

′) ,i = 1, ...,m ,x′ = (xm+1, ..., xn) ∈G , ψi ∈ Cs(G). Then the piece σ is ℵ -regular, if1) s ≥ r ;2) l ≥ r ;3) for all i = 1, ...,m the condition

˛ei˛

= l holds.

———

The boundedness of high order Riesz-Bessel transfor-mations generated by the generalized shift operator inweighted Lpw spaces with general weights

Ismail EkinciogluKutahya Dumlupinar University, Kutahya, Turkeyekinci69@dpu.edu.tr

In this study, the boundedness of the the high orderRiesz-Bessel transformations generated by generalized

52

III.4. Spaces of differentiable functions of several real variables and applications

shift operator in weighted Lpwv-spaces with generalweights is proved.

———

Composition Operators for Sobolev spaces of functionsand differential forms

Vladimir GoldshteinDepartment of Mathematics, Ben Gurion University ofthe Negev, P.O.Box 653, Beer Sheva, 84105 Israelvladimir@bgu.ac.il

Composition operators for Sobolev spaces with firstderivatives will be discussed. For such spaces compo-sition operators are induced by mappings with boundedmean distortion. These classes of mappings representa generalization of quasiconformal mappings. Applica-tions to embedding theorems will be described. In theframework of so-called Lq,p-cohomology similar classesof mappings play an important role for quasiisometricalan/or Lipschitz classification of complete noncompactRiemannian manifolds with bounded geometry

———

Boundedness of the fractional maximal operator andfractional integral operators in general Morrey typespaces and some applications

Vagif GuliyevF. Agayev str, 7 Rasim Mukhtarov str, 10 Baku, BakuAZ 1069 Azerbaijanvagif@guliyev.com

The theory of boundedness of fractional maximal oper-ator and fractional integral operators from one weightedLebesgue space to another one is by now well stud-ied. These results have good applications in the theoryof partial differential equations. However, in the the-ory of partial differential equations, along with weightedLebesgue spaces, general Morrey-type spaces also playan important role, but until recently there were noresults, containing necessary and sufficient conditionson the weight functions ensuring boundedness of theaforementioned operators from one general Morrey-typespace to another one (apart from the cases in which thisfollows directly from the appropriate results for weightedLebesgue spaces). The case of power-type weights waswell studied C.B. Morrey 1938, D.R. Adams 1975, F.Chiarenza and M. Frasca 1987, but for general Morrey-type spaces only sufficient conditions were known (T.Mizuhara 1991, E. Nakai 1994, V.S. Guliyev 1994).In the talk a survey of results, containing necessary andsufficient conditions for boundedness of fractional max-imal operator and fractional integral operators, will begiven, and open problems will be discussed in detail.As applications, we establish the boundedness of someSchodinger type operators on general Morrey-typespaces related to certain nonnegative potentials belong-ing to the reverse Holder class.

———

Weighted estimates of generalized potentials in variableexponent Lebesque spaces

Mubariz HajibayovInstitute of Mathematics and Mechanics of NAS of Azer-baijan, F.Agayev 9, Baku, Azerbaijan AZ1141 Azerbai-janhajibayovm@yahoo.com

For generalized potential operators with the kernela[%(x,y)]

[%(x,y)]Non bounded measure metric space (X,µ, %) with

doubling measure µ satisfying the upper growth condi-tion µB(x, r) ≤ CrN , N ∈ (0,∞), we prove weightedestimates in the case of radial type power weight w =[%(x, x0)]ν . Under some natural assumptions on a(r) interms of almost monotonicity we prove that such poten-tial operators are bounded from the weighted variableexponent Lebesgue space Lp(·)(X,w, µ) into a certain

weighted Musielak-Orlicz space LΦ(X,w1

p(x0) , µ) withthe N-function Φ(x, r) defined by the exponent p(x) andthe function a(r).

———

Our talk is on vanishing exponential integrability forBesov functions.

Ritva Hurri-SyrjanenDepartment of Mathematics and Statistics, University ofHelsinki PL 68 (Gustaf Hallstrominkatu 2 b), HelsinkiFI-00014 Finlandritva.hurri-syrjanen@helsinki.fi

Our talk is on vanishing exponential integrability forBesov functions.

———

New sharp estimates for function in Sobolev spaces onfinite Interval

Gennady KalyabinPeoples Friendship University of Russia, Miklukho-Maklaya Str 6, Moscow, 117198 Russiagennadiy.kalyabin@gmai.com

The smallest constants in new kind of Kolmogorov typeinequalities for intermediate derivatives are obtained,namely: the quantities Ar,k(x) defined as

supff (k)(x) : ‖f (r)‖L2(−1,1) ≤ 1;

f (s)(±1) = 0, s ∈ 0, . . . , r − 1,

are calculated for all natural r, k ∈ 0, . . . , r − 1 andx ∈ (−1, 1).In particular

A2r,0(x) =

(1− x2)2r−1

Γ2(r)22r−1(2r − 1).

As a Corollary it is established that for the first eigen-value of the boundary problem

(−D2)ry(x) = λy(x), y(s)(±1) = 0, s ∈ 0, . . . , r − 1,

the asymptotic formula

λ1,r ≈ 2π√

2r(2r

e)2r, r →∞,

holds.

———

On real interpolation of weighted Sobolev spaces

Leili KusainovaL.N. Gumilev Eurasian National University Astana, Mu-naitpasov 5, Akmola 010008 Kazakhstanleili2006@mail.ru

53

III.4. Spaces of differentiable functions of several real variables and applications

Let 1 < p < ∞, m ∈ N, Ω ∈ Rn an open set, and let υbe a non-negative function locally integrable in Ω. Wedenote by Wm

p (υ) the weighted Sobolev space with thefinite norm

|u;Wmp (υ)| = |∇mu;Lp(Ω)|+ |u;Lp(Ω; υ)|.

In this talk we describe Peetre interpolation spaces`Wm0p (υ0),Wm1

p (υ1

´θ,p

for weights υi, which allow in-

troducing local variable average characteristics. Here0 ≤ m1 < m0, 1 ≤ p < ∞, mip 6= n (i = 0, 1).Let d(x) be a positive bounded function in Ω such thatfor some a > 1 and all x ∈ Ω Qad(x)(x) ⊂ Ω, whereQd(x) = y ∈ Rn : |yi − xi| < d/2, i = 1, 2, ..., n. LetBsp(υ) denotes Besov space with the finite norm (s > 0)

|u;Bsp(υ)| = |u; bsp|+ |u;Lp(Ω; υ)|.

For certain class of weights υi (i = 0, 1) we have obtainedthe equality of type`

Wm0p (υ0),Wm1

p (υ1)´θ,p

= Bsp((υ∗)−sp

),

where 0 < θ < 1, s = (1− θ)m0 + θm1, υ∗ = maxi=0,1

υ∗i (x)

and

0 < υ∗i (x) = supd>0d : dmip−nυi(Qd(x)) ≤ 1 ≤ d(x).

———

The Fourier transform of a radial function

Elijah LiflyandDepartment of Mathematics Bar-Ilan University Ramat-Gan, Gush-Dan 52900 Israelliflyand@gmail.com

This talk naturally consists of two parts. In the firstone we survey the known results on representation ofthe Fourier transform of a radial function as the one-dimensional Fourier transform of a related function. Oneof such results, due to Leray, gave an impact to obtaininga series of new such formulas. We discuss those alreadyobtained in a joint work with S. Samko as well as tenta-tive formulas. Correspondingly, already obtained appli-cations are given and certain conjectures are posed.

———

Necessary and sufficient conditions for the bounded-ness of Riesz potential in Morrey spaces associated withDunkl operator

Yagub MammadovInstitute of Mathematics and Mechanics, RasimMukhtarov str. 10, Narimanov area Baku, AZ 1141Azerbaijanyagubmammadov@yahoo.com

The maximal function, fractional maximal functionand fractional integrals associated with the Dunkl op-erator were studied extensively in Lebesgue spaceson R. We study the fractional maximal function(Dunkl-type fractional maximal function) and fractionalintegrals (Dunkl-type fractional integrals) associatedwith the Dunkl operator in the Dunkl-type Morreyspace Lp,λ,α(R) and Dunkl-type Besov-Morrey spacesBspθ,λ,α(R). We obtain the necessary and sufficient con-ditions for the boundedness of Dunkl-type fractional

maximal operator Mβ and Dunkl type fractional integraloperator Iβ on the Dunkl-type Morrey spaces Lp,λ,α(R),1 ≤ p <∞.

———

Image normalization of Wiener-Hopf operators indiffraction problems

Ana Moura SantosDept. de Matematica, IST, Av. Rovisco Pais 1, Lisbon,1049-001 Portugalamoura@math.ist.utl.pt

In this work we discuss the normalization problem forWiener-Hopf Operators (WHO), which arrives in certainill-posed boundary-transmission value problems on half-planes. We first consider a wave diffraction problem by ajunction of two infinite half-planes, and different combi-nations of normal and oblique derivatives on the planes.Then a generalization for higher order derivatives fol-lows. For all studied diffraction problems, which are as-sociated with not normally solvable WHO, we solve thenormalization problem based on the image normaliza-tion technique previously developed for one half-plane.

———

Weighted estimates for the averaging integral operatorand reverse Holder inequalities

Bohumır OpicInstitute of Mathematics, AS CRZitna 2511567 Praha 1Czech Republicopic@math.cas.cz

Let 1 < p < +∞ and let v be a weight on (0,+∞)satisfying v(x)xρ is equivalent to a non-decreasing func-tion on (0,+∞) for some ρ ≥ 0. Let A be the aver-aging operator given by (Af)(x) := 1

x

R x0f(t) dt, x ∈

(0,+∞), and let Lp(v) denote the weighted Lebesguespace of all measurable functions f on (0,+∞) for which“R +∞

0|f(x)|pv(x) dx

”1/p

< +∞.

First, we prove that the following statements are equiv-alent:

(i) A is bounded on Lp(v);

(ii) A is bounded on Lp−ε(v) for some ε ∈ (0, p− 1);

(iii) A is bounded on Lp(v1+ε) for some ε > 0;

(iv) A is bounded on Lp(v(x)xε) for some ε > 0.

Moreover, if A is bounded on Lp(v), then A is boundedon Lq(v) for all q ∈ [p,+∞).Second, we show that the boundedness of the averagingoperator A on the space Lp(v) implies that, for all r > 0,

the weight v1−p′ satisfies the reverse Holder inequalityover the interval (0, r) with respect to the measure dt,while the weight v satisfies the reverse Holder inequalityover the interval (r,+∞) with respect to the measuret−p dt.Third, assuming moreover that p ≤ q < +∞ and that wis a weight on (0,+∞) such that

[w(x)x]1/q ≈ [v(x)x]1/p for all x ∈ (0,+∞),

we prove that the operator A is bounded on Lp(v) if andonly if the operator A : Lp(v)→ Lq(w) is bounded.

———

54

III.4. Spaces of differentiable functions of several real variables and applications

Characterization of the variable exponent Bessel poten-tial spaces via the Poisson semigroup

Humberto RafeiroUniversidade do Algarve, Dep. Matematica, Campus deGambelas Faro, Faro 8005-139 Portugalhrafeiro@ualg.pt

In this talk we give a characterization of the variableexponent Bessel potential space in terms of the con-vergence of the Poisson semigroup. We show thatGrunwald-Letnikov construction with respect to thePoisson semigroup coincides with the Riesz fractionaldifferentiation under some natural restrictions on the ex-ponent p(x).

———

On the Maxwell problem

Evgeniy RadkevichMathematics Department, Moscow State University,Vorobievy Gori, Moscow 119992 Russiaevrad07@gmail.com

TBA

———

Weighted potential operators in Morrey spaces.

Natasha SamkoDepartment of Mathematics, University of Algarve,Campus de Gambelas, 8005-139 Faro, Portugalnsamko@ualg.pt

We study the weighted (p, λ)-(q, λ)-boundedness ofHardy and potential operators. We show that theweighted boundedness of potential operators is reducedto the boundedness of weighted Hardy operators. In caseof power weights or oscillating weights from the Bary-Stechkin class we find conditions for weighted Hardy op-erators to be bounded in Morrey spaces.

———

Fractional integrals and hypersingular integrals in vari-able order Holder spaces on homogeneous spaces

Stefan SamkoUniversidade do Algarve Campus de Gambelas Faro, Al-garve 8005-139 Portugalssamko@ualg.pt

We consider non-standard Holder spaces Hλ(·)(X) offunctions f on a metric measure space (X, d, µ), whoseHolder exponent λ(x) is variable, depending on x ∈ X.We establish theorems on mapping properties of poten-tial operators of variable order α(x), from such a vari-able exponent Holder space with the exponent λ(x) toanother one with a ‘better’ exponent λ(x) + α(x), andsimilar mapping properties of hypersingular integrals ofvariable order α(x) from such a space into the space withthe ‘worse’ exponent λ(x)−α(x) in the case α(x) < λ(x).These theorems are derived from the Zygmund type es-timates of the local continuity modulus of potential andhypersingular operators via such modulus of their densi-ties. These estimates allow us to treat not only the caseof the spaces Hλ(·)(X), but also the generalized Holderspaces Hw(·,·)(X) of functions whose continuity modulusis dominated by a given function w(x, h), x ∈ X,h > 0.

We admit variable complex valued orders α(x), where<α(x) may vanish at a set of measure zero. To coverthis case, we consider the action of potential operatorsto weighted generalized Holder spaces with the weightα(x).

———

Equivalent semi-norms for Nikol’skii- Besov spaces onan interval

Kader SenouciUniversity City Zaaroura, 50 logts, Tiaret 14000 Algeria

kamer102000@yahoo.fr

Let 1 < p, θ 6 ∞, l > 0, k ∈ N, k > l, −∞ ≤ a 6 b 6∞. Recall thatf ∈ blp,θ(a, b), if f is measurable on (a, b)and

‖f‖blp,θ

(a,b) =

0B@b−akZ

0

“h−l‖∆k

hf‖Lp(a,b−kh)

”θ dhh

1CA1θ

is finite.

Theorem. 1 < p, θ 6 ∞, l > 0,k ∈ N, 0 < l < k,α1 ≥0, α2 ≥ k. Then for an arbitrary interval (a, b)

‖f‖blp,θ

(a,b) ∼

0BB@b−a

α1+α2Z0

“h−l‖∆k

hf‖Lp(a+α1h,b−α2h)

”θ dhh

1CCA1θ

,

where the equivalence constants are independent of a andb.

———

Stein-Weiss inequalities for the fractional integral op-erators in Carnot groups and applications

Ayhan SerbetciDepartment of Mathematics, Ankara University, Tando-gan, Ankara 06100 Turkeyserbetci@science.ankara.edu.tr

In this study we consider the fractional integral opera-tor Iα on any Carnot group G (i.e., nilpotent stratifiedLie group) in the weighted Lebesgue spaces Lp,ρ(x)β (G).We establish Stein-Weiss inequalities for Iα, and obtainnecessary and sufficient conditions on the parametersfor the boundedness of the fractional integral operatorIα from the spaces Lp,ρ(x)β (G) to Lq,ρ(x)−γ (G), and fromthe spaces L1,ρ(x)β (G) to the weak spaces WLq,ρ(x)−γ (G)by using the Stein-Weiss inequalities.In the limiting case p = Q

α−β−γ , we prove that the

modified fractional integral operator eIα is boundedfrom the space Lp,ρ(x)β (G) to the weighted BMO spaceBMOρ(x)−γ (G), where Q is the homogeneous dimensionof G.As applications of the properties of the fundamental so-lution of sub-Laplacian L on G, we prove two Sobolev-Stein embedding theorems on weighted Lebesgue andweighted Besov spaces in the Carnot group setting. Asan another application, we prove the boundedness of Iαfrom the weighted Besov spaces Bspθ,β(G) to Bsqθ,−γ(G).

———

55

III.5. Analytic and harmonic function spaces

Translation-invariant bilinear operators with positivekernels

Javier SoriaDepartment of Applied Mathematics and Analysis, Uni-versity of Barcelona, Gran Via 585, Barcelona 08007Spainsoria@ub.edu

We study the boundedness of bilinear convolutions oper-ators with positive kernels. We prove both necessary andsufficient conditions and, by means of several counterex-amples we show that near the endpoints the behavior ofpositive translation-invariant bilinear operators can bequite different than that of positive linear ones.

———

Sharp inequalities for moduli of smoothness and K-functionals

Sergey TikhonovCentre de Recerca Matematica, Facultat de Ciencies,UAB, Bellaterra, Barcelona 08193 Spainstikhonov@crm.cat

We discuss the (p − p) and (p − q) sharp inequalities(Jackson-type, Marchaud-type, Ulyanov-type, etc) formoduli of smoothness/K-functionals. Correspondingembedding theorems are studied.

———

Sobolev embedding theorems for a class of anisotropicirregular domains

Boris V. TrushinMIAN (Departament of Function Theory), ul. Gubkina,d. 8, Moscow 119991 Russiatrushinbv@gmail.com

Sufficient conditions for the embedding of a Sobolevspace in Lebesgue spaces and the space of continuousfunctions on a domain depend on the integrability andsmoothness parameters of the spaces and on the geo-metric features of the domain. In our talk, Sobolev em-bedding theorems will obtaine for a class of domainswith irregular boundary. This new class includes thewell-known classes of σ-John domains, domains with theflexible cone condition, and their anisotropic analogs.The results can be extended to weighted spaces withpower weights.

———

Necessary and sufficient conditions for the boundednessof the Riesz potential in modified Morrey spaces

Yusuf ZerenDepartment of Mathematics of Harran University, Cam-pus of Osmanbey, SanliUrfa, Region 6300 Turkeyyusufzeren@hotmail.com

We obtain necessary and sufficient conditions on theparameters for the boundedness of the fractional max-imal operator Mα, and the Riesz potential operatorIα from the modified Morrey spaces eLp,λ(Rn) to the

spaces eLq,λ(Rn), 1 < p < q < ∞, and from the

spaces eL1,λ(Rn) to the weak modified Morrey spaces

W eLq,λ(Rn), 1 < q <∞.

———

III.5. Analytic and harmonic function spaces

Organisers:Rauno Aulaskari, Turgay Kaptanoglu,Jouni Rattya

Anticipated topics are normal families, complex valuedfunction spaces and classes, function spaces and localtheory of complex differential equations, compositionoperators between function spaces, boundary behaviouretc.; Hardy, Bergman, Bloch, Besov, Lipschitz, Fock, Qpspaces of one and several holomorphic or harmonic vari-ables, Toeplitz, Hankel, composition, Volterra, multipli-cation operators, C∗ or other algebras of such operators,Toeplitz algebras, reproducing kernel Hilbert spaces ofholomorphic or harmonic functions, and other similartopics.

—Abstracts—

Multiplier theorem in the setting of Laguerre hyper-groups and applications

Miloud AssalDepartment of Mathematics, Faculty of Sciences of Biz-erte Zarzouna, Bizerte 7021, TunisiaMiloud.assal@fst.rnu.tn

In this work we study a multiplier theorem in the set-ting of Laguerre hypergroups and their applications toestimate the solution of Schrdinger equation in Hardyspaces.

———

Progress on finite rank Toeplitz products

Boo Rim ChoeDepartment of Mathematics, Korea University, Anam-dong 5 ga 1, Seongbuk-gu, Seoul 136-713 South Koreacbr@korea.ac.kr

It has been conjectured that a product of Toeplitz op-erators with function symbols, either on Hardy space orBergman space, has finite rank, then one of the factormust be the zero operator.In this talk we survey recent results towards the conjec-ture as well as related results.

———

Functions and operators in analytic Besov spaces

Daniel GirelaDepartamento de Analisis Matematico, Facultad deCiencias, Campus de Teatinos, Universidad de Malaga,Malaga 29071 Spaingirela@uma.es

In this talk we shall focus on structural and geometricproperties of the functions in analytic Besov, primarilyon the univalent functions in such spaces, and in opera-tors acting on them.

———

Square functions

Maria Jose GonzalesDepartment of Mathematics, Casem Rio San Pedro

56

III.5. Analytic and harmonic function spaces

Puerto Real, Cadiz 11560 Spainmajose.gonzalez@uca.es

We will study multiplicative versions of the usual mar-tingale square function and of the Lusin area of a har-monic function.

———

Convolutions of generic orbital measures in compactsymmetric spaces

Sanjiv GuptaDOMAS, PO BOX-36 Al-Khodh-123 Sultan QaboosUniversity Muscat, Omangupta s 63@yahoo.com

We prove that in any compact symmetric space, G/K,there is a dense set of a1, a2 ∈ G such that if µj =mK ∗ δaj ∗mk is the K-bi-invariant measure supportedon KajK, then µ1 ∗µ2 is absolutely continuous with re-spect to Haar measure on G. Moreover, the product ofdouble cosets, Ka1Ka2K, has non-empty interior in G.

———

Harmonic Besov spaces on the real unit ball: reproduc-ing kernels and Bergman projections

H. Turgay KaptanogluDepartment of Mathematics, Bilkent University,Ankara 06800, Turkeykaptan@fen.bilkent.edu.tr

Weighted Bergman spaces bpq are well-known spaces ofharmonic functions for which q > −1 and 1 ≤ p < ∞.Besov spaces, also denoted bpq , generalize them to allq ∈ R. Our Besov spaces consist of harmonic functionson the unit ball B of Rn so that their sufficiently high-order (t) derivatives are in Bergman spaces (q+pt > −1).We compute the reproducing kernels Rq(x, y) of theBesov spaces b2q with q ≤ −1. The kernels turn out to beweighted infinite sums of zonal harmonics, and also ra-dial fractional derivatives of the Poisson kernel. The newkernels give rise to generalized Bergman projections byway of Qsϕ(x) =

RB ϕ(y)Rs(x, y) (1−|x|2)s dν(y), where

s ∈ R. We prove that Qs : Lpq → bpq are bounded if andonly if q + 1 < p(s + 1). This requires new estimateson the integral growth of Bergman kernels near ∂B. Weobtain various applications of the Qs.This is joint work with Secil Gergun and A. ErsinUreyen. The work is supported by TUBITAK underResearch Project Grant 108T329.

———

Sums of Toeplitz products on the Dirichlet space

Young Joo LeeDepartment of Mathematics, Chonnam National Uni-versity, Gwangju, Yongbongdong 500-757, South Korealeeyj@chonnam.ac.kr

In this talk, we will consider a class of operators whichcontains finite sums of products of two Toeplitz opera-tors with harmonic symbols on the Dirichlet space of theunit disk.We will give characterizations of when an operators inthat class is zero or compact. Also, we solve the zeroproduct problem for products of finitely many Toeplitzoperators with harmonic symbols.

———

Weighted composition operators on weighted spaces ofanalytic functions

Jasbir Singh ManhasSultan Qaboos University, Department of Mathematics& Statistics, College of Science, P.O. Box 36, Al-KhodMuscat, Muscat 123 Omanmanhas@squ.edu.om

Let V be an arbitrary system of weights on an openconnected subset G of CN (N ≥ 1). Let HV0(G) andHVb(G) be the weighted locally convex spaces of analyticfunctions with topology defined by seminorms which areweighted analogues of the supremum norm. Let Hv0(G)and Hvb(G) be the weighted Banach spaces of analyticfunctions defined by a single weight v. In this talk be-sides presenting the characterizations of weighted com-position operators on HV0(G) ( Hv0(G) )and HVb(G)( Hvb(G) ), we shall present some results pertaining totopological structures ( e.g. component structure, Iso-lated points, compact differences ) of weighted compo-sition operators on the spaces H∞(D) and Hv0(D) (Hvb(D) ).

———

Superposition operators between Qp spaces and Hardyspaces

Auxiliadora MarquezDepartamento de Analisis Matematico, Facultad deCiencias, Campus de Teatinos, Malaga 29071 Spainauxim@uma.es

For any pair of numbers (s, p) with 0 ≤ s < ∞ and0 < p ≤ ∞ we characterize the superposition operatorswhich apply the conformally invariant Qs space into theHardy space Hp and, also, those which apply Hp intoQs.

———

Bounded Toeplitz and Hankel products on Bergmanspace

Malgorzata MichalskaInstytut Matematyki UMCS, Pl. M. Curie Sklodowskiej1, Lublin, woj. lubelskie, 20-031 Polanddaisy@hektor.umcs.lublin.pl

We improve the sufficient condition for boundedness ofproducts of Toeplitz operators TfTg on the Bergmanspace obtained by K. Stroethoff and D. Zheng in 1999.Using our result we give a short proof of the sufficientand necessary condition for the boundedness of TfT1/f

obtained also by Stroethoff and Zheng in 2002. We con-sider also the products of Hankel operators HfH

∗g .

———

Optimal norm estimate of the harmonic Bergman pro-jection

Kyesook NamDepartment of Mathematics, Hanshin University, Osan-si, Gyeonggi-do 447-791 South Koreaksnam@hs.ac.kr

On the unit ball of the Euclidean n-spaces, we give anoptimal norm estimate for one-parameter family of op-erators associated with the weighted harmonic Bergman

57

III.5. Analytic and harmonic function spaces

projections. Using this result, we obtain an optimalnorm estimate for the weighted harmonic Bergman pro-jections.This is the joint work with Boo Rim Choe and Hyung-woon Koo.

———

Old and new on composition operators on VMOA andBMOA spaces

Pekka NieminenDept. of Mathematics and Statistics, Univ of Helsinki,PO Box 68, Helsinki, 00014 Finlandpjniemin@cc.helsinki.fi

We review various compactness characterizations foranalytic composition operators acting on the spacesVMOA and BMOA, and give some new formulations.We also discuss the equivalence of weak compactnessand (norm) compactness for these operators. Joint workwith Jussi Laitila, Eero Saksman and Hans-Olav Tylli(Helsinki).

———

On Libera and Cesaro operators

Maria NowakInstytut Matematyki UMCS, Pl. M. Curie Sklodowskiej1, Lublin, woj. lubelskie, 20-031 Polandmt.nowak@poczta.umcs.lublin.pl

Let H(D) denote the class of functions holomorphic inthe unit disk D. The Cesaro operator C is defined on

H(D) by Cf(z) =P∞n=0

“1

n+1

Pnk=0 f(k)

”zn, where

f(z) =P∞n=0 f(n)zn. The Libera operator L, defined

by Lf(z) =P∞n=0

“P∞k=n

f(k)k+1

”zn, can be considered

as an extension of the conjugate operator C∗ defined onH(D) - the space of holomorphic functions defined in aneighborhood of D. We obtain results on Libera opera-tor acting on known spaces of holomorphic functions inthe unit disk. (Joint work with Miroslav Pavlovic)

———

Integration operators on weighted Bergman spaces

Jordi PauDepartament de Matematica Aplicada i Analisi, Univer-sitat de Barcelona, Gran Via de les Corts Catalanes 585,Barcelona, 08007 Spainjordi.pau@ub.edu

For an analytic function g on the unit disc, we considerthe operators

Jgf(z) =

Z z

0

f(ζ)g′(ζ)dζ.

We describe the boundedness and compactness of Jg onBergman spaces with exponential weights, answering anopen question posed by Aleman and Siskakis in 1997.

———

Extension to an invertible matrix in Banach algebras ofmeasures

Amol SasaneMathematics Department, London School of Economics,

Houghton Street London, WC2A 2AE United Kingdom

A.J.Sasane@lse.ac.uk

We will address the question of whether a left invertiblematrix with entries in certain convolution Banach alge-bras of measures supported in [0,+∞) can be completedto an invertible matrix with entries from the same Ba-nach algebra. The Banach algebras we consider arisenaturally in control theory as classes of inverse Laplacetransforms of stable transfer functions, and the relevanceof the problem of completion to an isomorphism in con-trol theory will also be explained.

———

Multiplication operators on weighted BMOA spaces

Benoit F. SehbaDepartment of Mathematics, University of Glasgow,G12 8QW, Glasgow, UKbs@maths.gla.ac.uk

We give some (test function) criteria for symbols ofbounded multiplication operators for a special famillyof weighted BMOA spaces in the unit ball.

———

Inequalities for Hardy spaces on the unit ball

Pawel SobolewskiInstytut Matematyki UMCS, Pl. M. Curie Sklodowskiej1, Lublin, woj. lubelskie, 20-031 Polandptsob@hektor.umcs.lublin.pl

In 1988 (TAMS 103(3)) D. Luecking obtained the follow-ing results for Hardy spaces Hp in the unit disk D ⊂ C.The inequalityZ

D|h(z)|p−s|h′(z)|s(1− |z|)s−1dA(z) ≤ C‖h‖pHp

holds for h ∈ Hp, p > 0 if and only if 2 ≤ s < p + 2.We obtain analogous results for the Hardy spaces on theunit ball of Cn, n ≥ 2.

———

On the Duhamel algebras

Mubariz TapdıgogluIsparta Vocational School, Suleyman Demirel Univer-sity, Dogu Campus Isparta, Cunur 32260 Turkeygarayev@fef.sdu.edu.tr

We introduce the notion of Duhamel algebra. We provethat under some natural conditions any Banach spaceof analytic functions in the unit disc D is the Duhamelalgebra and describe its all closed ideals. In particular,we improve some results of Wigley.

———

Toeplitz operators on Bergman spaces

Jari TaskinenP.O.Box 68, Department of Mathematics and StatisticsUniversity of Helsinki Helsinki, Helsinki FI-00014 Fin-landjari.taskinen@helsinki.fi

We give sufficient conditions for boundedness and com-pactness of Toeplitz operators in the Bergman spaces onthe unit disc of the complex plane. We consider both thecases 1 < p < ∞ and p = 1. The conditions concern a

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III.6. Spectral theory

kind of averages of the symbol on hypebolic rectangles.The sufficient condition is also necessary in the case ofpositive symbols, and it thus coincides with known re-sults in this case. An approach to the Fredholm proper-ties of Toeplitz operators is also given.

———

Hyperbolic weighted Bergman classes

Luis Manuel TovarDepartment of Mathematics, Esc. Sup. de Fsica y Mat.I.P.N., Edificio 9, Unidad Prof. A.L.M. Zacatenco delI.P.N., Mexico City, 07738 Mexicotovar@esfm.ipn.mx

A new class of like-hyperbolic Bergman class of analyticfunctions in the unit complex disk is introduced, whichhas several interesting properties and relationships withseveral classical weighted spaces, like Bloch, Dirichletand Qp.

———

Multiplicative isometries and isometric zero-divisors

Dragan VukoticDepartamento de Matematicas & ICMAT, Modulo C-XV, Universidad Autonoma de Madrid, Madrid, 28049Spaindragan.vukotic@uam.es

For some Banach spaces of analytic functions in the unitdisk (weighted Bergman spaces, Bloch space, Dirichlet-type spaces), we show that their isometric pointwisemultipliers are necessarily unimodular constants. As aconsequence, it follows that none of those spaces haveisometric zero-divisors. We also investigate the isomet-ric coefficient multipliers.

———

Area operators on analytic function spacess

Zhijian WuDepartment of Mathematics, The University of Al-abama, Tuscaloosa, Alabama 35487 United Stateszwu@as.ua.edu

We characterize non-negative measures µ on the unitdisk D for which the area operator Aµ is bounded orcompact on Hardy and Bergman spaces.

———

Composition operators on BMOA

Hasi WulanDepartment of Mathematics, Shantou University Shan-tou, Guangdong 515063 Chinawulan@stu.edu.cn

We give a new and simple compactness criterion for com-position operators Cϕ on BMOA and the Bloch space interms of the norms of ϕn in the respective spaces.

———

Lacunary series and QK spaces on the unit ball

Wen XuYliopistokatu 7 Metria Building (Y6) 3rd floor Joensuu,

joensuu 80100 Finlandwenxupine@gmail.com

In this paper, we give a necessary and sufficient condi-tion for a kind of lacunary series on the unit ball to bein Qp spaces for (m − 1)/m < p ≤ 1. The necessity isextended to more general QK spaces. This is a gener-alization of the result of Aulaskari, Xiao and Zhao forthat on the unit disk.

———

Some results on ϕ-Bloch functions

Congli YangYliopistokatu 7 Metria Building (Y6), Joensuu 80101Finlandcongli@cc.joensuu.fi

Let ϕ : [0, 1)→(0, ∞) be an increasing function, suchthat ϕ(r)(1−r)→∞, as r → 1−. An analytic functionf(z) in the unit disc is said to be ϕ-Bloch function ifit’s derivative satisfies |f ′(z)| = O(ϕ(|z|)) as |z| → 1−.This paper is devoted to the study of analytic ϕ-Blochfunctions. we obtain some new characterizations for ϕ-Bloch functions are established under certain regularityconditions on ϕ.

———

Holomorphic mean Lipschitz spaces

Kehe ZhuDepartment of Mathematics and Statistics, 1400 Wash-ington Ave, SUNY Albany, New York 12222 UnitedStateskzhu@math.albany.edu

I will talk about the connections between holomorphicmean Lipschitz spaces and several other classes of func-tion spaces, including Bergman spaces, Besov spaces,and Bloch type spaces. The setting is the open unit ballin Cn.

———

Univalently induced closed range composition operatorson the Bloch-type spaces

Nina ZorboskaDepartment of Mathematics, University of Manitoba,Winnipeg, Manitoba R3T 2N2 Canadazorbosk@cc.umanitoba.ca

We will show that if the closed range composition oper-ator is univalently induced, then the inducing functionhas to be a disk automorphism, whenever the underly-ing space is a Bloch-type space Bα with alpha not equalto one.The proof uses a combination of methods and resultsfrom operator theory, complex analysis and the pseudo-hyperbolic geometry on the unit disk.

———

III.6. Spectral theory

Organisers:Brian Davies, Ari Laptev, Yuri Safarov

59

III.6. Spectral theory

Anticipated topics are: Spectral theory of differentialoperators. Spectra of non-self-adjoint operators. Spec-tral asymptotics. Scattering theory. General spectraltheory and related topics.

—Abstracts—

Strongly elliptic second-order systems in Lipschitz do-mains: surface potentials, equations at the boundary,and corresponding transmission problems.

Mikhael Agranovich

magran@orc.ru

We consider a strongly elliptic second-order system in abounded Lipschitz domain Ω. For convenience, we as-sume that Ω = Ω+ lies in the standard torus T = Tn

and consider the system in the domain Ω− = T \Ω too.Assuming that the Dirichlet and Neumann problems inthe variational setting in Ω± are uniquely solvable insome spaces Hσ

p or Bσp , we describe properties of thesurface potentials. We define these operators and derivecorresponding formulas following Costabel and McLean,without using properties of the fundamental solution(but do not assume that p = 2 and that the coefficientsare smooth). Main results: boundedness of the surfacepotentials, their invertibility at the boundary (in partic-ular, of the single layer and hypersingular operators) inBesov spaces, and a description of their spectral proper-ties in these spaces (including the case p 6= 2). We alsodescribe applications to the corresponding transmissionproblems, general and spectral.

———

On the spectral expansions associated with Laplace-Beltrami operator

Shavkat AlimovVuzgorodok National University of Uzbekistan,Tashkent, 100174 Uzbekistanshavkat alimov@hotmail.com

The eigenfunction expansions associated with Laplace-Beltrami operator on n-dimensional symmetrical man-ifold Ω of rank 1 is considered. If the eigenfunctionexpansion of the piecewise smooth function, which de-pends on the geodesic distance from some point, con-verges at this point, then considered function belongsto C(n−3)/2(Ω). This result is the generalization of theresult, which was proved by M. Pinsky and W. O. Brayfor geodesic ball.

———

Sharp spectral stability estimates for higher order ellip-tic operators

Victor BurenkovVia Trieste 63, Padova University, Padova, 35121, Italyburenkov@math.unipd.it, burenkov@cf.ac.uk

We consider the eigenvalue problem for the operator

Hu = (−1)mX

|α|=|β|=m

Dα“Aαβ(x)Dβu

”, x ∈ Ω,

subject to homogeneous Dirichlet or Neumann boundaryconditions, where m ∈ N, Ω is a bounded open set in RN

and the coefficients Aαβ are real-valued Lipschitz con-tinuous functions satisfying Aαβ = Aβα and the uniformellipticity conditionX

|α|=|β|=m

Aαβ(x)ξαξβ ≥ θ|ξ|2

for all x ∈ Ω and for all ξα ∈ R, |α| = m, where θ > 0is the ellipticity constant. We consider open sets Ω forwhich the spectrum is discrete and can be representedby means of a non-decreasing sequence of non-negativeeigenvalues of finite multiplicity λ1[Ω] ≤ λ2[Ω] ≤ · · · ≤λn[Ω] ≤ . . . Here each eigenvalue is repeated as manytimes as its multiplicity and lim

n→∞λn[Ω] =∞ .

The aim is sharp estimates for the variation |λn[Ω1] −λn[Ω2]| of the eigenvalues corresponding to two open setsΩ1, Ω2 with continuous boundaries, described by meansof the same fixed atlas A.Three types of estimates will be under discussion: foreach n ∈ N for some cn > 0 depending only on n, A,m, θand the Lipschitz constant L of the coefficients Aαβ

|λn[Ω1]− λn[Ω2]| ≤ cndA(Ω1,Ω2),

where dA(Ω1,Ω2) is the so-called atlas distance of Ω1 toΩ2,

|λn[Ω1]− λn[Ω2]| ≤ cnω(dHP(∂Ω1, ∂Ω2)),

where dHP(∂Ω1, ∂Ω2) is the so-called lower Hausdoff-Pompeiu deviation of the boundaries ∂Ω1 and ∂Ω2 andω is the common modulus of continuity of ∂Ω1 and ∂Ω2,and, under certain regularity assumptions on ∂Ω1 and∂Ω2,

|λn[Ω1]− λn[Ω2]| ≤ cnmeas (Ω1∆Ω2) ,

where Ω1∆Ω2 is the symmetric difference of Ω1 and Ω2.Joint work with Dr P. D. Lamberti.

———

Strong field asymptotics for zero modes

Daniel EltonDepartment of Mathematics and Statistics, Fylde Col-lege, Lancaster University, Lancaster LA1 4YF UnitedKingdomd.m.elton@lancaster.ac.uk

Given a magnetic potential A one can consider the exis-tence of zero modes (or zero-energy L2 eigenfunctions)of the Weyl-Dirac operator σ.(−i∇−tA) on R3; here t isa positive parameter, with the limit t→∞ correspond-ing to the strong field (or, equivalently, semi-classical)regime. General O(t3) bounds on the number of zeromodes can be obtained. These bounds can be refinedto O(t2) asymptotics for a special class of potentials Athat are constructed from potentials on R2; a key stepinvolves localising the Aharonov-Casher theorem to ob-tain good estimates for the number of “approximate zeromodes” for two-dimensional Pauli operators.

———

A universal bound for the trace of the heat kernel

Leander GeisingerUniversitat Stuttgart, Fakultat Mathematik und Physik,IADM, Pfaffenwaldring 57, Stuttgart 70569, Germanyleander.geisinger@mathematik.uni-stuttgart.de

60

III.6. Spectral theory

We derive a unviersal upper bound for the trace of theheat kernel Z(t) =

Pk e−λkt, where (λk)k∈N denote the

eigenvalues of the Dirichlet Laplace Operator in an openset Ω ⊂ R2 with finite volume. The result improves aninequality of Kac and holds true without further assump-tions on Ω. The proof is based on improved Berezin-Li-Yau inequalities with a remainder term.

———

The eigenvalues function of the family of Sturm-Liouville operators and its applications

Tigran HarutyunyanFaculty of Math. and Mechanics of Yerevan State Uni-versity, Alek Manukyan 1, Yerevan 0049 Armeniahartigr@yahoo.co.uk

In order to study the dependence of the eigenvalues ofthe Sturm-Liouville problem on parameters, defining theboundary conditions, we introduce the concept of theeigenvalues function of the family of Sturm-Liouville op-erators.We find the necessary and sufficient conditions for somefunction (of two variables) to be the eigenvalues func-tion.Actually, we solve the direct and inverse Sturm-Liouvilleproblems. This solution particularly includes:

a) the new (more precise) asymptotic formulae for theeigenvalues and normalized constants,

b) some new uniqueness theorems in the inverse prob-lems,

c) the constructive solution of the inverse problemsin known and some new statements.

Also we introduce the concept of the eigenvalues func-tion of the family of Dirac operators and solve similarproblems for that case.

———

Generalized eigenvectors of some Jacobi matrices in thecritical case

Jan JanasSniadeckich 8 Warsaw, Warsaw 00-956 Polandnajanas@cyf-kr.edu.pl

The talk will be concerned with asymptotic behavior ofgeneralized eigenvectors of a class of Hermitian Jacobimatrices J in the critical case. The last means that thefraction qn/λn generated by the diagonal entries qn ofJ and its subdiagonal elements λn has the limit ±2. Inother word, the limit transfer matrix as n → ∞ con-tains a Jordan box (=double root in terms of Birkhoff-Adams theory). This is the situation where the asymp-totic Levinson theorem does not work and one has toelaborate more special methods for asymptotic analysis.

———

Trace expansions for elliptic cone operators

Thomas KrainerPenn State Altoona 3000 Ivyside Park Altoona, Penn-sylvania 16601 United Stateskrainer@psu.edu

I plan to report on recent joint work with Juan Gil andGerardo Mendoza on the expansion of the resolvent traceand the heat kernel for (nonselfadjoint) elliptic operatorson manifolds with conical singularities. Our approach al-lows for the treatment of elliptic operators A of generalform without simplifying assumptions on the coefficientsor the geometry near the singularities, and we achieveresults for a wide range of closed extensions of A in theappropriate metric L2-space. In particular, we obtainresults for selfadjoint and nonselfadjoint extensions ofHodge-Laplacians in the presence of warped conical sin-gularities where conventional methods that are based onseparation of variables and special functions fail.

———

Stability estimates for eigenfunctions of elliptic opera-tors on variable domains

Pier Domenico LambertiDipartimento di Matematica Pura ed Applicata, Via Tri-este, 63 Padova, Padova 35121, Italylamberti@math.unipd.it

We prove stability estimates for the variation of resol-vents and eigenfunctions of second order uniformly ellip-tic operators subject to homogeneous boundary condi-tions upon variation of the domain. We consider classesof open sets Ω parametrized by suitable bi-Lipschitzhomeomorphisms φ defined on a fixed reference do-main Ω. We obtain estimates expressed in terms of‖∇φ − I‖Lp(Ω) for finite values of p. We apply theseestimates in order to control the variation of the eigen-functions via the measure of the symmetric differenceΩ M Ω. We also discuss an application to the stabilityof the solutions to the Poisson problem.This is joint work with G. Barbatis and V.I. Burenkov.

———

Spectral theory of the normal operator with the spectraon an algebraic curve

Oleksii MokhonkoKyiv National Taras Shevchenko University, 64Volodymyrska street, 01033 Kyiv, UkraineAlexeyMohonko@univ.kiev.ua

The Jacobi (three-diagonal) structure of self-adjointmultiplication operator is well-known. BerezanskyYu.M. and Dudkin M.E. proved that similar Jacobistructure is typical not only for self-adjoint operators butalso for arbitrary unitary and even for any bounded nor-mal operators for which a cyclic vector exists. This leadsto numerous applications of these objects just in thesame way as it is for the classical Jacobi matrices, e.g.application to non-abelian difference-differential latticesgenerated by Lax equation (Golinskii L.B., MokhonkoO.A.). The following results will be presented.

1. Block Jacobi matrix of a bounded normal operatorJ acts in C1⊕C2⊕C3⊕C4⊕· · · . If one knows thatthe spectrum of J is a subset of a curve z ∈ C :p(z, z) = 0, p ∈ C[x, y] then its structure can besimplified: it acts over C1⊕C2⊕· · ·⊕Cn⊕Cn⊕· · ·(dimension stabilization phenomenon). E.g. if anormal operator is in fact the unitary one thenit acts over C1 ⊕ C2 ⊕ C2 ⊕ · · · (CMV matrixstructure) and if it is self-adjoint then it acts over`2 ' C1 ⊕ C1 ⊕ · · · .

61

IV.1. Pseudo-differential operators

2. The Direct Spectral Problem (the generalizedeigenvalue expansion theorem) and the InverseSpectral Problem will be presented for this typeof normal operators.

———

Spectral properties of operators arising from modellingof flows around rotating bodies

Jiri NeustupaMathematical Institute of the Czech Academy of Sci-ences Zitna 25 Prague 1, Czech Republic 115 67 CzechRepublicneustupa@math.cas.cz

We give a description of the spectrum of a Stokes-type oran Oseen-type operator which appears in mathematicalmodels of flows of a viscous incompressible fluid aroundrotating bodies. The special attention is paid to the es-sential spectrum. The operator is considered in an Lq

space.

———

New formulae for the wave operators

Serge RichardDepartment of Pure Mathematics and MathematicalStatistics, Wilberforce Road, Cambridge CB3 0WBUnited Kingdomsr510@cam.ac.uk

We review some new formulae recently obtained for thewave operators of various scattering systems. Differentapplications of these formulae will be presented.

———

Spectral bundles

Benedetto SilvestriDipartimento di Matematica Pura ed Applicata, Univer-sit di Padova, Via Trieste 63 Padova, 35121 Italysilvestr@math.unipd.it

In this talk I will construct certain bundles 〈M, ρ,X〉and 〈B, η,X〉 of Hausdorff locally convex spaces associ-ated to a given Banach bundle 〈E, π,X〉. Then I willpresent conditions ensuring the existence of boundedselections U ∈

Qx∈X Mx and P ∈

Qx∈X Bx both

continuous at a point x∞ ∈ X, such that U(x) is aC0−semigroup on Ex and P(x) is a spectral projectorof the infinitesimal generator of the semigroup U(x), forevery x ∈ X.

———

Scattering theory for manifolds and the scatteringlength

Alexander StrohmaierDepartment of Mathematical Sciences, LoughboroughUniversity, Loughborough, Leicestershire LE11 3TU,United Kingdoma.strohmaier@lboro.ac.uk

We define the so-called scattering length for Rieman-nian manifolds with cylindrical ends as the time delaythat waves experience when scattered in the manifold.We show that this scattering length can be estimated

by geometric quantities. For vector valued wave equa-tions our estimates depend on quite involved geometricquantities like lengths of homological systoles.

———

Spectrum and wandering

Yuriy TomilovChopina Str. 12/18 Department of Mathematics andInformatics, Nicholas Copernicus University, Torun andInstitute of Mathematics, PAN, Warsaw Torun, Torun87-100, Polandtomilov@mat.uni.torun.pl

Let T be a bounded linear operator on a Hilbert spaceH. A vector x ∈ H is called weakly wandering for T ifthere is an increasing sequence (nk) such that the vectorsTnkx are mutually orthogonal. By a well-known resultdue to Krengel, every unitary operator on H withoutpoint spectrum has a dense subset of weakly wanderingvectors.We will present several far-reaching extensions of theKrengel result. In particular, we will show that if Tis a power bounded operator on H with infinite periph-eral spectrum and with empty peripheral point spectrumthen the set of weakly wandering vectors for T is densein H. Our spectral assumptions on T are in a sense bestpossible.This is joint work with V. Muller (Prague).

———

Eigenfunctions at the threshold energies of magneticDirac operators

Tomio UmedaDepartment of Mathematical Science University of Hy-oto, Shosha 2167 Himeji, Hyogo 671-2201 Japanumeda@sci.u-hyogo.ac.jp

This talk will be devoted to investigation of the eigen-functions at the threshold energies ±m of the magneticDirac operator H = α ·

`− i∇x − A(x)

´+ mβ, where

α = (α1, α2, α3) and β are Dirac matrices and m isa positive constant. It will be considered three differentcases of the vector potential A to decay at infinity. In allthe cases, it will be shown that zero modes of the Weyl-Dirac operator σ ·

`− i∇x − A(x)

´play crucial roles in

the analysis of the eigenfunctions at the threshold of H.Here σ = (σ1, σ2, σ3) denotes Pauli matrices. It turnsout that many existing works on the Weyl-Dirac opera-tor can be utilized. Accordingly, various results on thethreshold eigenfunctions of the magnetic Dirac operatorH are obtained.This talk is based on joint work with Yoshimi Saito,University of Alabama at Birmingham, U.S.A.

———

IV.1. Pseudo-differential operators

Organisers:Luigi Rodino, Man Wah Wong

Topics related to pseudo-differential operators such asPDE, geometry, quantisation, wavelet transforms, lo-calisation operators on groups and symmetric domains,mathematical physics, signal and image processing,among others, are the embodiment of the special ses-sion.

62

IV.1. Pseudo-differential operators

—Abstracts—

Strongly elliptic second-order systems in Lipschitz do-mains: Dirichlet and Neumann problems.

Mikhael Agranovich

magran@orc.ru

This is a survey talk. We consider a strongly ellip-tic second-order system in a bounded Lipschitz domain.The coefficients have minimized smoothness. The aim ofthe talk is to describe the investigation of the Dirichletand (under natural additional assumptions) Neumannproblems in the variational setting in the spaces Hσ

p

and Bσp . The main case: the principal symbol is Her-mitian. Then we can use the Savare approach to theanalysis of the smoothness of solutions and combine itwith some tools of the interpolation theory, in particular,with Shneiberg’s results on the extrapolation of the in-vertibility of operators. The main results: conditions forthe unique solvability of the problems and some spectralresults (including the case p 6= 2) for the correspondingoperators. Applications to Neumann-to-Dirichlet oper-ators. Some results are also true for general stronglyelliptic systems. We compare this approach with thedeep approach based on the investigation of the surfacepotentials and corresponding equations at the boundary(Calderon, Jerisson, Kenig, Verchota and many othermathematicians) in terms of the non-tangential conver-gence, maximal functions, Rellich-type identities, etc.

———

Generalized ultradistributions and their microlocal anal-ysis

Chikh BouzarDepartment of Mathematics, Oran-Essenia UniversityB. P. 1925 EL MNAOUER Oran, Oran 31003 Algeriabouzar@yahoo.com

We first introduce new algebras of generalized functionscontaining ultradistributions. We then develop a mi-crolocal analysis suitable for these algebras. Finally, wegive an application through an extension of the well-known Hormander’s theorem on the wave front of theproduct of two distributions.

———

Some remarks on the Sjostrand class

Ernesto BuzanoDipartimento di Matematica, Universita di Torino, ViaCarlo Alberto 10, Torino 10123 Italyernesto.buzano@unito.it

We show that the bi-dual of the closure of C∞0 in M∞,1

is an extension of M∞,1 as a subalgebra of the algebraof bounded operators on L2.

———

The heat equation for the generalized Hermite and thegeneralized Landau operators

Viorel CatanaUniversity Politehnica of Bucharest, Splaiul Indepen-dentei 313, Bucharest 060042 Romaniacatana viorel@yahoo.co.uk

Following Wong’s point of view (Wong M.W., The heatequation for the Hermite operator on the Heisenberggroup, Hokkaido Math. Journal, vol. 34 (2005), 393-404), we give a formula for the one-parameter strongly

continuous semigroup e−tLλ

, t > 0, generated by thegeneralized Hermite operator Lλ, for a fixed λ ∈ R\0,in terms of the Weyl transforms. Then we use it to ob-tain an L2 estimate for the solution of the initial valueproblem for the heat equation governed by Lλ, in termsof the Lp norm, 1 ≤ p ≤??, of the initial data.Similar results have also been derived for the general-ized Landau operator A which was firstly introduced byM.A. De Gosson (M.A. De Gosson, Spectral Propertiesof a class of generalized Landau operators, Comm. Part.Diff. Equ., 33 (2008), 2096–2104), who has studied itsspectral properties.

———

Generalization of the Weyl rule for arbitrary operators

Leon CohenCity University-Hunter College, 695 Park. Ave, NewYork, 10471 United Statesleon.cohen@hunter.cuny.edu

The Weyl rule generally deals with two operators whosecommutator is a c-number. The generalization to arbi-trary operators is of importance and offers interestingand challenging mathematical issues. We review the ba-sic ideas, present new results and discuss the unsolvedproblems. We also show how our generalization leads tothe consideration of quasi-probability distributions forarbitrary variables. In addition to the Weyl rule we con-sider other rules of association between operators andsymbols.

———

Sharp results for the STFT and localization operators

Elena CorderoDepartimento di Matematica, Universita di Torino, viaCarlo Alberto 10 Torino, TO 10123 Italyelena.cordero@unito.it

We completely characterize the boundedness on Lp

spaces and on Wiener amalgam spaces of the short-time Fourier transform (STFT) and of a special classof pseudodifferential operators, called localization oper-ators. Precisely, a well-known STFT boundedness re-sult on Lp spaces is proved to be sharp. Then, sufficientconditions for the STFT to be bounded on the Wieneramalgam spaces W (Lp, Lq) are given and their sharp-ness is shown. Localization operators are treated sim-ilarly. Using different techniques from those employedin the literature, we relax the known sufficient bounded-ness conditions for localization operators on Lp spacesand prove the optimality of our results. More generally,we prove sufficient and necessary conditions for such op-erators to be bounded on Wiener amalgam spaces.

———

Fuchsian mild microfunctions with fractional order andtheir applications to hyperbolic equations

Yasuo Chiba1404-1, Katakura-cho Hachioji, Tokyo 1920982 Japanychiba@cs.teu.ac.jp

63

IV.1. Pseudo-differential operators

Kataoka introduced a concept of mildness in bound-ary value problems. He defined mild microfunctionswith boundary values. This theory has effective resultsin propagation of singularities of diffraction. Further-more, Oaku introduced F-mild microfunctions and ap-plied them to Fuchsian partial differential equations.Based on these theories, we introduce Fuchsian mild mi-crofunctions with fractional order. We show the proper-ties of such microfunctions and their applications to par-tial differential equations of hyperbolic type. By usinga fractional coordinate transform and a quantized Leg-endre transform, degenerate hyperbolic equations aretransformed into equations with derivatives of fractionalorder. We present a correspondence between solutionsfor the hyperbolic equations and those for the trans-formed equations.

———

About Gevrey semi-global solvability of a class of com-plex planar vector fields with degeneracies

Paulo Dattori da SilvaFaculdade de Filosofia, Ciencias e Letras de RibeiraoPreto - Departamento de Fsica e Matematica, Avenidados Bandeirantes, 3900 - Monte Alegre Ribeirao Preto,Sao Paulo 14040-901 Brazildattori@ffclrp.usp.br

Let Ωε = (−ε, ε) × S1, where ε > 0 and S1 is the unitcircle. Let

L = ∂/∂t+ (a(x) + ib(x))∂/∂x, b 6≡ 0, (*)

be a complex vector field defined on Ωε, where a and bare real-valued s-Gevrey functions on (−ε, ε), and s ≥ 1.We will assume that Σ = 0 × S1 is the characteristicset of L and that L is tangent to Σ. In particular, L is el-liptic on Ωε\Σ and (a+ib)(0) = 0. Hence, we may write(a+ ib)(x) = xna0(x) + ixmb0(x) in Ωε, with m,n ≥ 1,and a0, b0 smooth.In this talk we shall present results about Gevrey solv-ability of L, given by (*), in a neighborhood of Σ, in thefollowing sense: there exists s′ > 1 such that for any fbelonging to a subspace of finite codimension of Gs(Ωε)

there exists a solution, u ∈ Gs′, to the equation Lu = f

in a neighborhood of Σ. We will see that the interplaybetween the order of vanishing of the functions a and bat x = 0 plays a role in the Gevrey solvability. Moreover,lost of regularity occurs.This is a joint work with Adalberto P. Bergamasco(ICMC/USP) and Marcelo R. Ebert (FFCLRP/USP).

———

Invertibility for a class of degenerate elliptic operators

Julio DelgadoCra 82 Bis 49-03 Ciudad Real Cali, Valle 9999 Colombiadelgado@math.jussieu.fr

In this work we study fundamental solutions for a classof degenerate elliptic operators. The type of operatorconsidered is obtained as a sum of operators of the formD2xi + x2k

i D2xj . The invertibility for an operator of type

D2x1 +x2k

1 D2x2 on R2 is known, here we extend this result

to higher dimensions.

———

Heat kernel of a sub-Laplacian and Grushin type oper-ators

Kenro FurutaniDepartment of Mathematics, Science University ofTokyo, 2641 Yamazaki Noda, Chiba 278-8510 Japanfurutani@ma.noda.sut.ac.jp

First, I will introduce a framework of a sub-Riemannianstructure which is compatible with a submersion anddefine Grushin type operators. My purpose is to con-struct heat kernel for various Grushin type operatorsfrom known heat kernel in an explicit integral form. Asa typical example, I explain the original Grushin oper-ator and its heat kernel constucted from the heat ker-nel on three dimensional Heisenberg group. Then as ageneralization to dimension three, I define Grushin typeoperators on R3, R4 and R5 from a sub-Laplacian on the6−dimensional free nilpotent Lie group, and give theirheat kernels in terms of fiber integration.Also in the case that the submersion is a covering mapfrom the Heisenberg group to Heisenberg manifolds,I will determine the spectral zeta function for a sub-Laplacian on them in terms of Riemann zeta function.If possible, I will also show a heat kernel for a sphericalGrushin operator on S2 and CP 3 which come from asub-Laplacian on S3 or S7, respectively.

———

Time-frequency analysis of stochastic differential equa-tions

Lorenzo GalleaniPolitecnico di Torino, Corso Duca degli Abruzzi 24,Torino, TO 10129 Italygalleani@polito.it

Most of the stochastic processes used to model physicalsystems are nonstationary, and yet most of the theo-retical results on stochastic processes are related to thestationary case. We consider a nonstationary randomprocess defined as the solution of a stochastic differen-tial equation. We first transform the stochastic equa-tion to the Wigner spectrum domain, where we obtaina deterministic differential equation. Then, by applyingthe Laplace transform, we obtain the exact solution ofthe deterministic equation. Finally, we rewrite the gen-eral solution in a form which clarifies the structure ofthe nonstationary stochastic process, and which high-lights the connection to the classical results obtained byFourier analysis.

———

Lp-microlocal regularity for pseudodifferential operatorsof quasi-homogeneous type

Gianluca GarelloUniversita di Torino, Department of Mathematics, ViaCarlo Alberto 10 Torino, Torino I-107123 Italygianluca.garello@unito.it

Pseudodifferential operators whose symbols have de-cay at infinitive of quasi-homogeneous are consideredand their behavior on the wave front set of distribu-tions in weighted Zygmund-Holder spaces and weightedSobolev spaces in Lp framework is studied. Then mi-crolocal properties for solutions of linear partial differ-ential equations with coefficients in weighted Zygmund-Holder spaces are obtained.

64

IV.1. Pseudo-differential operators

———

Generalized Fourier integral operators methods for hy-perbolic problems

Claudia GarettoArbeitsbereich fur technische Mathematik, UniversitatInnsbruck Technikerstrasse 13 Innsbruck, Austria 6020Austriaclaudia@mat1.uibk.ac.at

The past decade has seen the emergence of a differential-algebraic theory of generalized functions that answereda wealth of questions on solutions to partial differentialequations involving non-smooth coefficients and stronglysingular data. In such cases, the theory of distributionsdoes not provide a general framework in which solutionsexist due to inherent constraints in dealing with nonlin-ear operations.An alternative framework is provided by the theory ofColombeau algebras of generalized functions.In this talk we solve hyperbolic equations, generated byhighly singular coefficients and data, by means of gener-alized FIO techniques developed in the Colombeau con-text. Finally, we provide a careful microlocal investiga-tion of the solution by studying the microlocal mappingproperties of these operators.

———

Resolvents of regular singular elliptic operators on aquantum graph

Juan GilPenn State Altoona, 3000 Ivyside Park, Altoona, Penn-sylvania 16601 United Statesjgil@psu.edu

We will discuss the pseudodifferential structure of theresolvent of regular singular differential operators on agraph. For second order operators, we give a simple, ex-plicit, sufficient condition for the existence of a sector ofminimal growth. In particular, we will discuss operatorswith a singular potential of Coulomb type. Our analysisis based on the theory of elliptic cone operators.

———

Hyperbolic systems of pseudodifferential equations withirregular symbols in t admitting superlinear growth for|x| → ∞.

Todor GramchevDipartimento di Matematica e Informatica, Universitadi Cagliari, via Ospedale 72, 09124 Cagliaritodor@unica.it

We consider hyperbolic systems of pseudodifferentialequations with irregular symbols with respect to thetime variable t and admitting superlinear growth for|x| → ∞. We investigate the global well-posedness of theCauchy problem for such systems in the framework ofweighted spaces which generalize the Cordes type spacesHs1,s2(Rn).

———

Analytic perturbations for special Frechet operator al-gebras in the microlocal analysis

Bernhard GramschInstitut fur Mathematik, Universitat Mainz, Staudinger-weg 9, Mainz, 55099 Germanygramsch@mathematik.uni-mainz.de

The symmetric Hoermander class of type (1, 1) (inter-esting for paradifferential operators) is included in thetheory of holomorphic Fredholm functions in connectionwith the Oka principle. This class is known to be notspectrally invariant. But commutator methods lead tothe submultiplicativity of this symmetric Frechet alge-bra. Some relations to operator algebras on singularand stratified spaces are given. Stochastic PDE lead toholomorphic operator functions on infinite dimensionaldomains in DFN - spaces with basis such as the distri-bution space S′ of Schwartz. A series of open problemsis mentioned for Frechet operator algebras connected toparameter dependent equations on singular rep. rami-fied manifolds.

———

The Cauchy problem for a paraxial wave equation withnon-smooth symbols

Gunther HormannNordbergstraße 15 Fakultat fur Mathematik Wien, WienA-1090 Austriaguenther.hoermann@univie.ac.at

We discuss evolution systems in L2 for Schroedinger-type pseudodifferential equations with non-Lipschitz co-efficients in the principal part. The underlying operatorstructure is motivated from models of paraxial approx-imations of wave propagation in geophysics. Thus, theevolution direction is a spatial coordinate (depth) withadditional pseudodifferential terms in time and low regu-larity in the lateral variables. We formulate and analyzethe Cauchy problem in distribution spaces with mixedregularity.Solutions with low regularity in the operator symbol willprovide a basis for an inverse analysis which allows toinfer the lack of lateral regularity in the medium frommeasured data.

———

Pseudodifferential operators on locally symmetricspaces

Eugenie HunsickerDepartment of Mathematical Sciences, LoughboroughUniversity, Loughborough, LE11 3TU United Kingdome.hunsicker@lboro.ac.uk

I will discuss recent work with D. Grieser of U. Olden-burg on the first stages of the construction of a pseu-dodifferential operator calculus tailored to locally sym-metric spaces and other noncompact spaces with similarstructures.

———

On the continuity of the solutions with respect to theelectromagnetic potentials to the Schrodinger and theDirac equations

Wataru IchinoseDepartment of Mathematical Science, Shinshu Univer-sity Matsumoto, Nagano 390-8621 Japanichinose@math.shinshu-u.ac.jp

The initial problem to families of the Schrodinger equa-tions and the Dirac equations with the electromagnetic

65

IV.1. Pseudo-differential operators

potentials are studied, respectively. Assume that thesolutions have the same initial data and that the elec-tromagnetic potentials converge.Then, it is proved that the solutions of Schrodinger equa-tions and the Dirac equations with the correspondingelectromagnetic potentials also converge, respectively.The proof follows from the uniqueness and the bound-edness of the solutions, and the functional method, ex.the abstract Ascoli-Arzela theorem, which will be seento be applied to nonlinear equations.

———

Calculus of pseudo-differential operators and a local in-dex of Dirac operators

Chisato IwasakiDepartment of Mathematical Sciences, Shosha 2167Himeji, Hyogo 671-2201 Japaniwasaki@sci.u-hyogo.ac.jp

I will show a method to obtain a local index of Diracoperators. This method depends on construction of thefundamental solution to the Cauchy problem for heatequations by introducing a weight for symbols of pseudo-differential operators.

———

On the theory of type 1, 1-operators

Jon JohnsenMathematics Department, Aalborg University, FredrikBajers Vej 7G, Dk-9220 Aalborg Øst, Denmarkjjohnsen@math.aau.dk

After an introduction with a brief review of celebratedcontributions on type 1, 1-operators of G. Bourdaud(1983,1988) and L. Hormander (1988–89), their resultswill be set in relation to the general definition of type1, 1-operators, which was introduced at the ISAAC 2007congress. Progress in the area will be described as timepermits.

———

Pseudo-differential operators with discontinuous sym-bols and their applications

Yuryi KarlovychUniversidad Autonoma del Estado de Morelos, Facultadde Ciencias, Av. Universidad 1001, Cuernavaca, More-los 62209 Mexicokarlovich@uaem.mx

Applying a weighted analogue of the Litllewood-Paleytheorem and the boundedness of the maximal singularintegral operator S∗ related to the Carleson-Hunt theo-rem on almost everywhere convergence on all weightedLebesgue spaces Lp(R, w), where 1 < p < ∞ andw ∈ Ap(R), we study the boundedness and compactnessof pseudo-differential operators a(x,D) with non-regularsymbols in the classes L∞(R, V (R)) and Λγ(R, Vd(R)) onthe spaces Lp(R, w). The Banach algebra L∞(R, V (R))consists of all bounded measurable V (R)-valued func-tions on R where V (R) is the Banach algebra of all func-tions on R of bounded total variation, and the Banachalgebra Λγ(R, Vd(R)) consists of all Lipschitz Vd(R)-valued functions of exponent γ ∈ (0, 1) on R whereVd(R) is the Banach algebra of all functions on R of

bounded variation on dyadic shells. For some Banachalgebras of pseudo-differential operators acting on thespace Lp(R, w) and having symbols discontinuous withrespect to spatial and dual variables, we construct a non-commutative Fredholm symbol calculi and give Fred-holm criteria and index formulas for the operators inthese algebras. Applications to algebras of generalizedsingular integral operators with shifts are considered.

———

On maximal regularity for parabolic equations on com-plete Riemannian manifolds

Thomas KrainerPenn State Altoona 3000 Ivyside Park Altoona, Penn-sylvania 16601 United Stateskrainer@psu.edu

In this talk I plan to demonstrate how recent advancesin the theory of pseudodifferential operators lead to amethod to effectively establish optimal Lp–Lq a prioriestimates for solutions to parabolic equations on certaincomplete Riemannian manifolds.The approach is based on Weis’ functional analytic char-acterization of maximal regularity in terms of the R-boundedness of the resolvent. In recent work, partly incollaboration with Robert Denk (Univ. of Constance,Germany), we have shown that the approximation ofresolvents of elliptic operators by parameter-dependentparametrices in suitable classes of pseudodifferential op-erators readily leads to the desired R-boundedness, thusto maximal regularity.In my talk I plan to survey our results and the basicunderlying principles of the method.

———

On the cohomological equation in the plane for regularvector fields

Roberto de LeoINFN, Complesso Universitario di Monserrato Monser-rato (CA), Sardegna 09042 Italyroberto.deleo@ca.infn.it

In this talk we present our recent results about the solv-ability of the equation Xf = g, where X is a vectorfield on the plane without zeros, in the cases when Xis generic and when it is Hamiltonian with respect tosome symplectic form. This work slightly generalizes arecent result of S.P. Novikov, which showed recently thata generic vector field on a compact surface, seen as a 1-st order operator on the set of smooth functions, has aninfinite-dimensional cokernel. Our study is also relatedto aspects of pseudo-differential operators on the plane.

———

Lp-boundedness and compactness of localization oper-ators associated with Stockwell transform

Yu LiuDepartment of Mathematics and Statistics, York Univer-sity, 4700 Keele St., Toronto, Ontario M3J1P3 Canadahanliu@yorku.ca

Localization operators associated with the Stockwelltransform, with respect to the filter symbol and the win-dows, are a class of operators defined on Lp(R). Under

66

IV.1. Pseudo-differential operators

suitable conditions for the symbol and the windows, thelocalization operators turn to be bounded and compact.

———

About transport equation with irregular coefficient anddata

Jean-Andre MartiCampus de Schoelcher, Laboratoire CEREGMIA, Uni-versite des Antilles et de la Guyane Schoelcher, Mar-tinique B.P. 7209-97275 Francejamarti@univ-ag.fr

We are interested in the study of the Cauchy problem fortransport equation in the formally simplified case wherethe coefficients α and β are discontinuous and even dis-tributions. For the data u0, we suppose it is a distribu-tion and even a more singular object like δpx⊗ δqy we willgive later a generalized meaning. Then the problem isformally written as

(Pform)

8<:∂

∂tu+ α⊗ 1xy

∂

∂xu+ β ⊗ 1xy

∂

∂yu = 0,

u |t=0= u0(= δpx ⊗ δqy.)

We remark that the product and the restriction writ-ten above are generally not defined in a distributionalsense. Consequently we begin in associating to (Pform)a generalized one (Pgen) well formulated in a convenient(C, E ,P) algebra A

`R3´

and recall the definition andmain properties of this generalized multiparametric fac-tor algebra.In our case, we construct such an algebra by means ofindependant regularizations involving three independantparameters and obtain

(Pgen)

8<:∂

∂tu+ F

∂

∂xu+G

∂

∂yu = 0,

u |t=0= H.

where F and G (resp.H) are the classes in A`R3´

(resp.A`R2´) of the families regularizing the coefficients

(resp. the data).First we solve (Pgen) and examine the existence of asolution. To study more pecisely its singularities, we re-fer to a generalization of the asymptotic singular spec-trum defined previously and adapted here to the three-parametric case. The so-called ”(a,D′)-singular spec-trum” of u ∈ A(R3) propose a spectral analysis of thesingularities: by means of an ”analyzing” function a wecan see where and why u is not locally (associated toa section of) D′.The localization of such singularitiesof u is always the ”D′-singular support” of u, and theasymptotic causis is described by a fiber ΣX(u) (aboveeach X = (t, x, y) ∈ R3) which is the complement in R3

+

of a conic subset of R3+.

In our case, the D′-singularities of the data propagatealong the ”regularized characteristic Γ of the problem(Pgen)” on which the fiber ΣX(u) remains constant.This joint work of V. Dvou, M. Hasler and J.-A. Martiof Universit Antilles-Guyane.

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The Heat Kernel of τ -Twisted Laplacian

Shahla MolahajlooDepartment of Mathematics and statistics, York Uni-versity 4700 Keele street, Toronto, Ontario M3J1P3

Canadasmollaha@mathstat.yorku.ca

For a family of τ -twisted Laplacians that includes theusual twisted Laplacian when τ = 1/2, we compute theheat kernel for each τ -twisted Laplacian for [0, 1].

———

Regularity of characteristic initial-boundary value prob-lems for symmetrizable systems

Alessandro MorandoDepartment of Mathematics - University of Brescia, ViaValotti, 9, I-25133, Brescia, Italymorando@ing.unibs.it

We study the initial-boundary value problem for a lin-ear Friedrichs symmetrizable system, with characteristicboundary of constant rank. We assume the existence ofthe strong L2 solution satisfying a suitable energy esti-mate, but we do not assume any structural assumptionsufficient for existence, such as the fact that the bound-ary conditions are maximally dissipative or the Kreiss-Lopatinski condition. We show that this is enough inorder to get the regularity of solutions, in the naturalframework of weighted anisotropic Sobolev spaces, pro-vided the data are sufficiently smooth.

———

Application of pseudodifferential equations in stress sin-gularity analysis for thermo-electro-magneto-elasticityproblems: a new approach for calculation of stress sin-gularity exponents

David NatroshviliGeorgian Technical University 77 M.Kostava st. Tbilisi,Tbilisi 0175 Georgianatrosh@hotmail.com

We apply the potential method and the pseudodifferen-tial equations technique to the mathematical model ofthe thermo-electro-magneto-elasticity theory. We studymixed and crack type boundary value problems. Alongwith the existence and uniqueness questions our maingoal is a detailed theoretical investigation of singularitiesof the thermo-mechanical and electro-magnetic fieldsnear the crack edges and the curves where the boundaryconditions change their type. In particular, the most im-portant question is description of the dependence of thestress singularity exponents on the material parameters.We reduce the three-dimensional mixed and cracktype boundary value problems of the thermo-electro-magneto-elasticity to the equivalent system of pseudo-differential equations which live on proper parts of theboundary of the elastic body under consideration.We show that with the help of the principal homoge-neous symbol matrices of the corresponding pseudodif-ferential operators it is possible to determine explicitlythe singularity exponents for physical fields. We givean efficient method for computation of these exponents.Moreover, we establish that these exponents essentiallydepend on the material parameters, in general.

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Wigner type transforms and pseudodifferential opera-tors

Alessandro OliaroDepartment of Mathematics, University of Torino, Via

67

IV.1. Pseudo-differential operators

Carlo Alberto, 10 Torino, TO I-10123 Italyalessandro.oliaro@unito.it

We present some modifications of the Wigner transform(Wig), suggested by the connections of Wig with pseu-dodifferential operators. We analyze some properties ofthese representations, in particular the positivity andthe behaviour with respect to the cross terms.

———

Local regularity of solutions to PDEs by asymptoticmethods

Michael OberguggenbergerUnit for Engineering Mathematics, University of Inns-bruck, A-6020 Innsbruck, Austriamichael.oberguggenberger@uibk.ac.at

In the nonlinear theory of generalized functions, alge-bras of generalized functions are commonly constructedby means of nets of smooth functions (uε)ε∈E depend-ing on one or more parameters. Typically, these nets donot converge as ε→ 0, say, but exhibit a certain asymp-totic behavior. This behavior not only determines thealgebras to which such an object belongs to, but maydescribe local regularity properties. This type of regu-larity theory has become increasingly important in ap-plications to partial- and pseudodifferential operators.This presentation is devoted to a general framework –the so-called asymptotic spectrum – for measuring theasymptotic behavior in algebras of generalized functions,using asymptotic scales and various topologies. It hasbeen developed in joint work with A. Delcroix and J.-A.Marti [Asymptotic Analysis 59(2008), 169 – 199] andforms a nonlinear alternative to the wave front set ap-proach.Various applications to propagation of singularities aswell as to regularity in Colombeau algebras and to jumpdiscontinuities in hyperbolic systems will be given.

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Modern results by theory of the three dimensionalVolterra type linear integral equations with singularity

Nusrat RajabovTajik National University Rudaki Av. 17 Dushanbe,Dushanbe 734025 Tajikistannusrat38@mail.ru

Let Ω denote the parallelepipedΩ = (x, y, z) : a < x < a0, b < y < b0, c < z < c0,D1 = (x, y) : a < x < a0, b < y < b0, z = c,D2 = (x, z) : a < x < a0, y = b, c < z < c0,D3 = (y, z) : x = a, b < y < b0, c < z < c0.In the domain Ω we consider the following integral equa-tion

Φ(x, y, z) +A

Z x

a

Φ(t, y, z)

t− a dt+B

Z b

y

Φ(x, s, z)

s− b ds

+ E

Z z

c

Φ(x, y, τ)

τ − c dτ +A1

Z x

a

dt

t− a

Z y

b

Φ(t, s, z)

s− b ds

+B1

Z x

a

dt

t− a

Z z

c

Φ(t, y, τ)

τ − c dτ

+ C1

Z y

b

ds

s− b

Z z

c

Φ(x, s, τ)

τ − c dτ

+D

Z x

a

dt

t− a

Z y

b

ds

s− b

Z z

c

Φ(t, s, τ)

τ − s dτ

= f(x, y, z), (*)

where A, B, E, A1, B1, C1, D are constants, f(x, y, z) –is a given function in Ω, Φ(x, y, z) is the desired function.Some cases equation (*) investigated N. Rajabov [Ac.of Sciences Dokl. V. 409, No 6, 2006, pp.749-753]. Inthis lecture the general solution of the integral equa-tion (*) is constructed, using the connection equation(*) with one dimensional integral equation of the type(*). In the case, when A1 = AB, B1 = AE, D = AC1,then the problem is determination general solution equa-tion (*) redused, to problems found general solution sin-gle one-dimensional integral equation and single two–dimensional integral equation of the type (*). In this ba-sis in the case when C1 = EB and A < 0, B < 0, E < 0,find general solution equation (*) by three arbitraryfunctions two variables. In the case when C1 6= EB,find the solution equation (*) by means of one arbitraryfunction two variabe and infinity number arbitrary func-tion one variabe. Select the cases, when equation (*) hasunique solution.

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The adiabatic limit of the Chern character

Frederic RochonDepartment of Mathematics, 40 St. George Street,Toronto, Ontario M5S 2E4 Canadarochon@math.toronto.edu

Certain spaces of pseudo-differential operators can beused as classifying spaces for K-theory. In this context,Bott periodicity can be realized by taking a certain adi-abatic limit. In this talk, we will indicate how natu-ral forms representing the universal Chern chararcter onthese spaces behave under such an adiabatic limit. Thisa joint work with Richard Melrose.

———

Boundary value problems as edge problems

Bert-Wolfgang SchulzeInstitute of Mathematics, University Potsdam, AmNeuen Palais 10, Potsdam, D-14469 Germanyschulze@math.uni-potsdam.de

The calculus of pseudo-differential operators on a mani-fold with edges can be established in such a way thatstandard boundary value problems (BVPs) with thetransmission property at the boundary appear as a spe-cial case (up to some simple modifications). Also thecase without the transmission property can be formu-lated as a special case of the edge calculus (as is shownin a joint paper of the author with J. Seiler, 2009). Theremarkable fact here is that the symbols of the respective(classical) pseudo-differential operators are not requiredto be of edge-degenerate form but are only smooth up tothe boundary in the usual sense. In our talk we illustratethe specific properties of that theory for the case of sym-bols with the anti-transmission property (recently sin-glet out by the author to investigate specific asymptoticsof solutions). Symbols with the transmission propertytogether with those with the anti-transmission prope rtyspan the full space of symbols that are smooth up to theboundary.

———

68

IV.1. Pseudo-differential operators

Noncommutative residues and projections associated toboundary value problems

Elmar SchroheInstitut fur Analysis, Leibniz Universitat Hannover,Welfengarten 1, 30167 Hannoverschrohe@math.uni-hannover.de

On a compact manifold X with boundary we considerthe realization B = PT of an elliptic boundary problem,consisting of a differential operator P and a differentialboundary condition T . We assume that B is parameter-elliptic in small sectors around two rays in the complexplane, say arg λ = φ and arg λ = θ. Associated to thecuts along the rays one can then define two zeta functionζφ and ζθ for B. Both extend to meromorphic functionson the plane; the origin is a regular point. We relate thedifference of the values at the origin to the associatedspectral projection Πθ,φ(B) defined by

Πθ,φu =i

2π

ZΓθ,φ

λ−1B(B − λ)−1u dλ, u ∈ dom(B),

where Γθ,φ is the contour which runs on the first rayfrom infinity to r0e

iφ for some r0 > 0, then clockwiseabout the origin on the circle of radius r0 to r0e

iθ andback to infinity along the second ray.

———

On maximal regularity for mixed order systems

Jorg SeilerSchool of Mathematics Loughborough University Lough-borough, Leicestershire LE113TU United KingdomJ.Seiler@lboro.ac.uk

I will discuss some results on maximal Lp-regularity forparabolic mixed order systems based on the so-calledH∞-calculus as well as on a calculus of Volterra pseu-dodifferential operators. This is a joint work with R.Denk and J. Saal.

———

Dirichlet problem for higher order elliptic systems withBMO assumptions on the coefficients and the boundary

Tatyana ShaposhnikovaDepartment of Mathematics, Linkoeping UniversitymLinkoeping, Ostergotland SE-58183 Swedentasha@mai.liu.se

Given a bounded Lipschitz domain, we consider theDirichlet problem with boundary data in Besov spacesfor divergence form strongly elliptic systems of arbitraryorder with bounded complex-valued coefficients.The main result gives a sharp condition on the localmean oscillation of the coefficients of the differentialoperator and the unit normal to the boundary (auto-matically satisfied if these functions belong to the spaceVMO) which guarantee that the solution operator asso-ciated with this problem is an isomorphism.

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Gevrey regularities of solutions of nonlinear singular par-tial differential equations

Hidetoshi TaharaDepartment of Mathematics, Sophia University, Kioicho

7, Chiyoda-ku Tokyo, Tokyo 102-8554 Japanh-tahara@sophia.ac.jp

In this talk, I will consider the regularity of the solutionof a nonlinear singular partial differential equation (E):

(t∂/∂t)mu = F (t, x, (t∂/∂t)j(∂/∂x)αuj+|α|≤m,j<m)

in Gevrey classes and gives a sufficient condition forthe following assertion to be valid: if F (t, x, z) withz = zj,αj+|α|≤m,j<m is in the Gevrey class Gσof order σ with respect to (t, x, z) and if u(t, x) ∈C∞([0, T ], Gσ(V )) is a solution of (E) on [0, T ]×V , thenwe have u(t, x) ∈ Gσ([0, T ]× V ).

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Wave-front sets and SG type operators in Fourier-Lebesgue spaces

Nenad TeofanovTrg D. Obradovica 4 Novi Sad, Vojvodina 21000 Serbianenad.teofanov@im.ns.ac.yu

The recent study of pseudo-differential and Fourier inte-gral operators in Fourier-Lebesgue spaces as well as theirconnection with modulation spaces in different contextsincreased the interest for such spaces, we refer to thework of Concetti-Toft, Cordero-Nikola-Rodino, Okoud-jou, Ruzhansky-Sugimoto-Tomita-Toft.In particular, we refer to several papers of Pilipovic-Teofanov-Toft for the micro-local analysis of FourierLebesgue spaces.In this lecture we study continuity properties on theFourier-Lebesgue spaces by observing the localized ver-sion of the class S0

0,0. Furthermore, we prove an ex-tension to operators whose symbols enjoy certain decaywith respect to the x variable. These operators belong tothe class of symbol global type operators, recently stud-ied by Cappiello-Gramchev-Rodino, Coriasco-Rodino,Dasgupta-Wong.... At the end, a continuity result fora class of elliptic operators is given.

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Wave-front sets of Fourier Lebesgue types

Joachim ToftDepartment of Mathematics and Systems Engineering,Vejdes plats 6,7, Vxj University, Vxj, Smland 351 95Swedenjoachim.toft@vxu.se

Roughly speaking, a wave-front set WF∗(f) of the dis-tribution f with respect to “something”, gives infor-mation where the distribution f has singularities withrespect to this “something”, as well as what directionsin these points of singularities, the singularities propa-gates. The wave-front set with respect to smoothness(by Hormander) for a distribution f is the set of pairs((x, ξ) ∈ Rd × (Rd \ 0), where ξ is the directions were fis non-smooth at x.In this talk we introduce wave-front sets WF∗(f) =WFFL

q(ω)

(f) of the distribution f with respect to

(weighted) Fourier Lebesgue spaces FLq(ω), where ω isan appropriate weight function. An advantage with suchwave-front sets is that we may examine micro-local prop-erties more close to differentiability up to a certain order,instead of complete smoothness only.Especially we show the usual property

WF∗(a(x,D)f) ⊆WF∗(f) ⊆WF∗(a(x,D)f)[

Char(a),

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IV.1. Pseudo-differential operators

when a(x,D) is an appropriate pseudo-differential oper-ator with smooth coefficients and Char(a) is the set ofcharacteristic points for a. We remark that our Char(a)is smaller than what is usual. For example, that Char(a)can be chosen to be empty when a(x,D) is hypoelliptic.In particular, WF∗(a(x,D)f) = WF∗(f).Finally we remark that one may get the “usual” wave-front set (with respect to smoothness) by consideringsequences of wave-front sets of Fourier Lebesgue types.

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Pseudo-differential operators and symmetries

Ville TurunenHelsinki University of Technology, Institute of Mathe-matics, P.O.Box 1100, FIN-02015 HUT, Finlandville.turunen@hut.fi

This work is joint with M. Ruzhansky (Imperial CollegeLondon). We study pseudo-differential equations glob-ally on compact Lie groups, without resorting to localcharts. We obtain a full global symbol and global cal-culus. This can be done by presenting functions on thegroup by Fourier series obtained from the representa-tions of the group. A pseudo-differential operator canbe presented as a convolution operator valued mappingon the group.The complete treatise can be found in the follow-ing monograph: M. Ruzhansky, V. Turunen: Pseudo-Differential Operators and Symmetries. Birkhauser2009.

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Pseudo differential equations and boundary value prob-lems in non-smooth domains

Vladimir VasilyevBryansk State University, Bezhitskaya 14, Bryansk241036, Russiavbv57@inbox.ru

One discusses a possibility for constructing theory ofboundary value problems for pseudo differential equa-tions under existence of so called wave factorization ofelliptic symbol in non-smoothness points at the bound-ary. It leads to well-posed statements of boundary valueproblems in Sobolev-Slobodetskii spaces (both old andnew) in non-smooth domains. Some of such problemsand their solvability have been described in author’s pa-pers earlier, some of them will be considered at firsttime. One suggests also to consider discrete analoguesof such equations (and boundary value problems), forwhich some preliminary results have been obtained bythe author.

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Diffraction at corners for the wave equation on differ-ential forms

Andras VasyDepartment of Mathematics, Stanford University, 450Serra Mall, CA 94305-2125, USAandras@math.stanford.edu

I will describe the propagation of smooth (C∞) andSobolev singularities for the wave equation on smoothmanifolds with corners X equipped with a Lorentzian

metric g. That is, we consider the d’Alembertian gon differential forms, and u ∈ H1

loc(X; ΛX) solvinggu = 0 with relative or absolute boundary conditions,ν ∧ u|S = 0, resp. ινu|S = 0, at all boundary hypersur-faces S, where ν = νS is the conormal of S. We showthat the appropriate wave front set WFb(u) of u is aunion of maximally extended generalized broken bichar-acteristics.I will indicate the key ideas of the proof, such as mi-crolocalization with respect to the appropriate ps.d.o.algebra, Ψb(X), and gaining b-regularity (i.e. conormalregularity) relative to H1

loc(X; ΛX) via positive commu-tator estimates.These results are analogous to those obtained by theauthor for the scalar wave equation and for the waveequation on systems with Dirichlet or Neumann bound-ary conditions. The main novelty is thus the presence ofnatural boundary conditions, which effectively make theproblem non-scalar, even ‘to leading order’, at cornersof codimension ≥ 2.

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Formation of singularities near Morse points

Ingo WittMathematical Institute, University of Gottingen, Bun-senstr. 3-5 Gottingen, 37073 Germanyiwitt@uni-math.gwdg.de

Given a Morse function f and a Riemannian metric h ona C∞ manifold M , we study solutions u to the equation

gu = 0,

where g is the wave operator associated with theLorentzian metric

g = ‖df‖2h h− ζ df ⊗ df,

and ζ > 1 a constant. As turns out, at Morse pointsinitially regular solutions u start to form singularitiesthat are then propagated as usual. These singularitiescan be described in terms of certain classes of conormaldistributions.Such models arise in quantum field theory on curvedspace-times with changing topology.

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Phases of modified Stockwell transforms and instanta-neous frequencies

Man Wah WongDepartment of Mathematics and Statistics, York Uni-versity, 4700 Keele Street Toronto, Ontario M3J 1P3Canadamwwong@mathstat.yorku.ca

Modified Stockwell transforms are introduced to includethe classical Stockwell transforms and related wavelettransforms. We begin with highlighting the subtle dif-ferences between the Stockwell transforms and the Mor-let wavelet transforms. The focus of the talk, however,is on the characteristics of the phases of modified Stock-well transforms in general and explicit formulas for in-stantaneous frequencies of signals in terms of the TT-transforms, which are closely related to the Stockwelltransforms. (This is joint work with Shahla Molaha-jloo.)

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IV.2. Dispersive equations

Generalized cosine transforms in image compression

Hongmei ZhuYork University, Department of Mathematics and Statis-tics, 4700 Keele ST, Toronto, ON M3J1P3 Canadahmzhu@yorku.ca

The generalized cosine transforms are defined by the so-called C-functions, a new family of special functions thatarise in connection with compact semi-simple Lie groupof rank 2. For groups A1 and A1xA1, the generalizedcosine transform coincide with the well-known 1D and2D cosine transform. Here, we focus on the generalizedcosine transform for group C2 and investigate its appli-cations in image compression.

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IV.2. Dispersive equations

Organisers:Michael Reissig, Fumihiko Hirosawa

The goal of the session is to discuss the state-of-the-artof qualitative properties of solutions of dispersive equa-tions. Among other things Strichartz decay estimates,Strichartz estimates, and dispersive estimates are of in-terest. The question of the influence of low regularitycoefficients on the well-posedness of the Cauchy prob-lem is another key topic.

—Abstracts—

Lp–Lq estimates for hyperbolic systems

Marcello D’AbbicoUniversita di Bari, Dipartimento di Matematica, Via EOrabona 4, Bari, BA 70125 Italydabbicco@dm.uniba.it

We establish Lp−Lq estimates for the solution of M×Msystems with bounded time dependent coefficients:

DtU =

nXj=1

Aj(t)DxjU +B(t)U , U(0, x) = U0(x) .

In the equation setting, Reissig and others obtained suchestimates by using WKB representation of the solutions.We put fγ = (t+ e3)

`log(t+ e3)

´−γ,

Tγm =

a ∈ C∞

˛|Dk

t a(t)| ≤ Ck (fγ(t))m−kff,

for some γ ∈ [0, 1]; let Aj ∈ Tγ0, B ∈ Tγ−1, and‚‚‚‚Z t

0

B(r) dr

‚‚‚‚ ≤ c1`log(t+ e3)´γ.

Theorem. Let A(t, ξ) =PAj(t)ξj. We assume that

there exists a smooth, regular matrix N(t, ξ) such thatNAN−1 is diagonal and, if ζ = (DtN +N B)N−1, then˛Z t

0

=`ζjj (r, ξ)

´dr

˛≤ c2 , j = 1, . . . ,M .

We assume that det |A(t, ξ)| ≥ c3 > 0, for t ≥ t0.Let U be the solution of the Cauchy Problem; then

‖U(t, ·)‖Lq ≤ C(n, p) (1 + t)−n−1

2 ( 1p− 1q

)+s0‖U0‖HNp,p ,

for some s0 > 0, where 1 = p−1 + q−1, 1 < p ≤ 2 andNp ≥ n(1/p − 1/q). One can take s0 = 0 if γ = 0 ands0 = ε for any ε > 0 if γ ∈ (0, 1) and C = C(n, p, ε).

This is a joint work with Sandra Lucente and GiovanniTaglialatela from University of Bari.

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Multiple solutions for non-linear parabolic systems

Q-Heung ChoiDept. of Mathematics San 68 Miryong Dong KunsanNational University , Kunsan 573-701 South Koreaqheung@inha.ac.kr

We have a concern with the existence of solutions (ξ, η)for perturbations of the parabolic system with Dirichletboundary condition

ξt = −Lξ + µg(3ξ + η)− sφ1 − h1(x, t) in Ω× (0, 2π),

ηt = −Lη + νg(3ξ + η)− sφ1 − h2(x, t) in Ω× (0, 2π).

We prove the uniqueness theorem when the nonlin-earity does not cross eigenvalues. We also investigatemultiple solutions (ξ(x, t), η(x, t)) for perturbations ofthe parabolic system with Dirichlet boundary conditionwhen the nonlinearity f ′ is bounded and f ′(−∞) <λ1, λn < (3µ+ ν)f ′(+∞) < λn+1.This is joint work with Tacksun Jung.

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Local sovability beyond condition ψ

Ferruccio ColombiniDepartment of Mathematics, University of Pisa, LargoBruno Pontecorvo 5 Pisa, PI 56127 Italycolombini@dm.unipi.it

It is well known that condition ψ (PSI) is necessary andsufficient in order to have local solvability for differen-tial (pseudo-differential) operators of principal type withcoefficients sufficiently regular.We study some cases when such conditions are not sat-isfied.These are two joint papers with Ludovico Pernazza andFrancois Treves and with Paulo Cordaro and LudovicoPernazza.

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Continuous dependence for backward parabolic opera-tors with Log-Lipschitz coefficients

Daniele Del SantoDipartimento di Matematica e Informatica, Via Valerio12/1, Trieste, 34127 Italydelsanto@units.it

We consider the following backward parabolic equation

∂tu+Xi,j

∂xi(ai,j(t, x)∂xju)

+Xj

bj(t, x)∂xju+ c(t, x)u = 0 (*)

on the strip [0, T ]× Rn 3 (t, x). We suppose that

• for all (t, x) ∈ [0, T ]× Rn and for all i, j = 1 . . . n,

ai,j(t, x) = aj,i(t, x);

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IV.2. Dispersive equations

• there exists k > 0 such that, for all (t, x, ξ) ∈[0, T ]× Rn × Rn,

k|ξ|2 ≤Xi,j

ai,j(t, x)ξiξj ≤ k−1|ξ|2;

• for all i, j = 1, . . . , n, ai,j ∈ LL([0, T ], L∞(Rn)) ∩L∞([0, T ], C2

b (Rn)) and bj , c ∈ L∞([0, T ], C2b (Rn)),

(where a ∈ LL([0, T ], L∞(Rn)) means that the functiona is Log–Lipschitz–continuous with respect to time withvalues in L∞, i.e.

supt,s∈[0,T ], 0<|t−s|≤1

‖a(t, ·)− ai,j(s, ·)‖L∞(Rn)

|t− s|(1 + | log |t− s||) ≤ ∞).

Let E := C0([0, T ], L2(Rn)) ∩ C0([0, T [, H1(Rn)) ∩C1([0, T [, L2(Rn)).Our main reslut is the following.

Theorem. For all T ′ ∈ ]0, T [ and for all D > 0 thereexist ρ′, M ′, N ′, δ′ > 0 such that if u ∈ E is a solutionof the equation (*) with supt∈[0,T ] ‖u(t, ·)‖L2 ≤ D and‖u(0, ·)‖L2 ≤ ρ′, then

supt∈[0,T ′]

‖u(t, ·)‖L2 ≤M ′e−N′| log ‖u(0,·)‖

L2 |δ′

.

(joint work with Martino Prizzi, Trieste University)

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On the loss of regularity for a class of weakly hyperbolicoperators

Marcello EbertUniversidade de Sao Paulo, Faculdade de Filosofia,Ciencias e Letras, Dept. de Fisica e Matematica,Av. dos Bandeirantes, 3900 Ribeirao Preto, Sao Paulo14040-901 Brazilebert@ffclrp.usp.br

In this work we consider the Cauchy problem

Pu = ∂2t u−λ2(t)

nXi,j=1

aij(t)∂2xixju+λ(t)

nXi=1

ci(t)∂2txiu

= f(x, t, u, ∂tu, λ′(t)∇xu), (*)

u(x, 0) = u0(x), ∂tu(x, 0) = u1(x) (**)

where P is weakly hyperbolic in a neighborhood oft = 0, that is,

the roots of p(x, t, ξ, τ) in τ are real; (***)

here p = p(x, t, ξ, τ) is the principal symbol of P . Ex-amples show that, differently of the hyperbolic case, un-der (*), (**) and (***) the solution might not exist. Inaddition to condition (***), various authors presentedsufficient conditions, usually called Levi conditions, forthe Cauchy problem to be well posed in Sobolev spaces.Those type of conditions relate p with lower order termsof P . In this work, we narrowed the bounds for the op-timal Sobolevs loss of regularity under some sharp Leviconditions.This work was done in collaboration with Rafael A. dosSantos Kapp and Jos Ruidival dos Santos Filho, bothfrom UFSCar(Brazil).

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Zakharov system in infinite energy space

Daoyuan FangZhejiang University, Hangzhou, Chinadyf@zju.edu.cn

We consider the Zakharov system in space dimensiontwo. We will show a L2-concentration result for the datawithout finite energy, when blow-up of the solution hap-pens, and a low regularity global well-posedness result.The proof uses a refined I-method originally initiated byColliander, Keel, Staffilani, Takaoka and Tao. A poly-nomial growth bound for the solution is also given.This talk is based on some joint works with Sijia Zhongand Hartmut Pecher.

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Wave equation in Einstein-de Sitter spacetime

Anahit GalstyanDepartment of Mathematics, University of Texas-PanAmerican, 1201 West University Drive, Edinburg, Texas78539 United Statesagalstyan@utpa.edu

In this talk we introduce the fundamental solutions ofthe wave equation in the Einstein-de Sitter spacetime.The last one describes the simplest non-empty expand-ing model of the universe. The covariant d’Alembert’soperator in the Einstein-de Sitter spacetime belongs tothe family of the non-Fuchsian partial differential opera-tors. In this talk we investigate initial value problem forthis equation and give the explicit representation formu-las for the solutions.The equation is strictly hyperbolic in the domain withpositive time. On the initial hypersurface its coefficientshave singularities that make difficulties in studying ofthe initial value problem. In particular, one cannot an-ticipate the well-posedness in the Cauchy problem forthe wave equation in the Einstein-de Sitter spacetime.The initial conditions must be modified to so-calledweighted initial conditions in order to adjust them tothe equation.We also show the Lp−Lq estimates for solutions. Thus,we have prepared all necessary tools in order to study thesolvability of semilinear wave equation in the Einstein-deSitter spacetime.This is a joint work with Tamotu Kinoshita (Univer-sity of Tsukuba, Japan) and Karen Yagdjian (UTPA,U.S.A.).

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Stability of solitary waves for Hartree type equation

Vladimir GeorgievDepartment of Mathematics, University of Pisa, LargoBruno Pontecorvo 5 Pisa, PI 56127 Italygeorgiev@dm.unipi.it

We prove the stability of solitary manifold associatedwith the solitary solutions of Hatree type equation withexternal Coulomb type potential.

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Hyperbolic-parabolic singular perturbations forKirchhoff-equations

Marina GhisiDepartment of Mathematics, University of Pisa, Largo

72

IV.2. Dispersive equations

Pontecorvo 5 Pisa, Pi 56127 Italyghisi@dm.unipi.it

We consider the second order Cauchy problem

εu′′+ g(t)u′+m(|A1/2u|2)Au = 0, u(0) = u0, u′(0) = u1

where ε > 0, g is a positive function, m is a non-negative C1 function, A is a self-adjoint non-negativeoperator with dense domain D(A) in a Hilbert space,and (u0, u1) ∈ D(A)×D(A1/2).We prove the global solvability of the Cauchy problemunder different conditions on the functions m and g, in-cluding the case where m(0) = 0, and the case whereg(t) tends to 0 as t tends to +infinity (weak dissipa-tion). We also consider the behavior of solutions as ttends to +infinity (decay estimates), and as ε tends to0.

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Existence and uniqueness results for Kirchhoff equa-tions in Gevrey-type spaces

Massimo GobbinoDipartimento di Matematica Applicata, via FilippoBuonarroti 1c, Pisa, PI 56127 Italym.gobbino@dma.unipi.it

We consider the second order Cauchy problem

u′′ +m(|A1/2u|2)Au = 0, u(0) = u0, u′(0) = u1,

where m : [0,+∞) → [0,+∞) is a continuous function,and A is a self-adjoint nonnegative operator with densedomain on a Hilbert space.In this conference we present three results.

• The first result is local existence for initial data insuitable spaces depending on the continuity mod-ulus of the nonlinear term m. This spaces are anatural generalization of Gevrey spaces to the ab-stract setting. We also show that solutions withless regular data may exhibit an instantaneousderivative loss.

• The second result concerns uniqueness in the casewhere the nonlinear term is not Lipschitz continu-ous.

• The last result concerns the global solvability.Roughly speaking, we show that every initial da-tum in the spaces where local solutions exist is thesum of two initial data for which the solution isactually global.

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Precise loss of derivatives for evolution type models

Torsten HerrmannFaculty 1, TU Bergakademie Freiberg, Pruferstr. 9,Freiberg, 09596 Germanytorsten.herrmann@math.tu-freiberg.de

The goal of this talk is to present statements aboutwell-posedness for Cauchy problems for degenerate p-evolution equations with time-dependent coefficients.Degeneracy means that the p-evolution operators mayhave characteristics of variable multiplicity. On the onehand we are interested to apply phase space analysis to

derive results for well-posedness with a (possible) lossof regularity. On the other hand we discuss strategieshow to show optimality of the results and sharpness ofthe assumptions. Here Floquet theory and instabilityarguments form the core of our strategies. We distin-guish between optimality for the leading coefficients ofthe principal part and for coefficients of the remainingprincipal part.

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Wave equations with time dependent coefficients

Fumihiko HirosawaDepartment of Mathematics, Yamaguchi University,753-8512, Japanhirosawa@yamaguchi-u.ac.jp

The total energy of the wave equation is conserved withrespect to time if the propagation speed is a constant,but it is not true in general for time dependent propa-gation speeds. Indeed, it is considered in [F. Hirosawa,Math. Ann. 339 (2007), 819-839] that the followingproperties of the propagation speed are crucial for theestimates of the total energy: oscillating speed, differ-ence from the mean, and the smoothness in Cm cate-gory. The main purpose of our talk is to derive a benefitof a further smoothness of the propagation speed in theGevrey class for the energy estimates.

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Critical point theory applied to a class of systems ofsuper-quadratic wave equations

Tacksun JungDept. of Mathematics San 68 Miryong Dong KunsanNational University , Kunsan 573-701 South Koreatsjung@kunsan.ac.kr

We show the existence of a nontrivial solution for a classof the systems of the super-quadratic nonlinear waveequations with Dirichlet boundary conditions and pe-riodic conditions with super-quadratic nonlinear termsat infinity which have continuous derivatives. We ap-proach the variational method and use the critical pointtheory which is the Linking Theorem for the stronglyindefinite corresponding functional.This is joint work with Q-Heung Choi.

———

On the well-posdness of the vacuum Einstein equations

Lavi KarpP.O. Box 78 Karmiel, Galilee 21982 Israellavikarp@gmail.com

The Cauchy problem of the vacuum Einstein’s equationsdetermines a semi-metric gαβ of a spacetime with van-ishing Ricci curvature Rα,β and prescribe initial data.under harmonic gauge condition, the equations Rα,β = 0are transferred into a system of quasi-linear wave equa-tions which are called the reduced Einstein equations.The initial data for Einstein’s equations are a properRiemannian metric hab and a second fundamental formKab. However, these data must satisfy Einstein con-straint equations and therefore the pair (hab,Kab) can-not serve as initial data for the reduced Einstein equa-tions.Previous results in the case of asymptotically flat space-times provide a solution to the constraint equations in

73

IV.2. Dispersive equations

one type of Sobolev spaces, while initial data for theevolution equations belong to a different type of Sobolevspaces. The aim of our work is to resolve this incompati-bility and to show well-posedness of the reduced Einsteinvacuum equations in one type of Sobolev spaces.

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Generalized wave operator for a system of nonlinearwave equations

Hideo KuboGraduate School of Information Sciences, Tohoku Uni-versity 6-3-09 Aramaki-Aza-Aoba, Aoba-ku Sendai ,Miyagi 980-8579 Japankubo@math.is.tohoku.ac.jp

In this talk we discuss the asymptotic behavior of so-lutions to a system of nonlinear wave equations whosedecaying rate is actually slower than that of the free so-lutions. Desipte of that fact, we are able to constructwave operators in a generalized sense. The proof is doneby finding a nice approximation and introducing a suit-able metric (that is not a norm in fact). Moreover, thesacttering operators are defined in a generarized sence.

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Strichartz estimates for hyperbolic equations in an ex-terior domain

Tokio MatsuyamaTokai University 1117 Kitakaname Hiratsuka, Kanagawa259-1292 Japantokio@keyaki.cc.u-tokai.ac.jp

In this talk we will present Strichartz estimates forhigher oder hyperbolic equations in an exterior domainoutside a star-shaped obstacle.

———

Uniform resolvent estimates and smoothing effects formagnetic Schrodinger operators

Kiyoshi MochizukiDepartment of Mathematics, Chuo University, Kasuga,Bunnkyo 1-13-27, Tokyo 112-8551 Japanmochizuk@math.chuo-u.ac.jp

Uniform resolvent estimates for magnetic Schrodingeroperators in an exterior domain are obtained undersmallness conditions on the magnetic fields and scalarpotentials. The results are then used to obtain space-time L2-estimates for the corresponding Schrodinger,Klein-Gordon and wave equations.

———

Decay and scattering for wave equations with dissipa-tions in layered media

Hideo NakazawaChiba Institute of Technology, Shibazono 2-1-1Narashino, Chiba 275-0023 Japannakazawa.hideo@it-chiba.ac.jp

We consider wave equations with linear dissipations insome layered regions;

wtt(x, y, t)−∆w(x, y, t) + b(x, y)wt(x, y, t) = 0,

(x, y, t) ∈ RN × [0, π]× (0,∞)

with Dirichlet boundary conditions

w(x, 0, t) = w(x, π, t) = 0, (x, t) ∈ RN × (0,∞).

For long-range type of dissipations, e.g., b0(1 + |x|)−1 ≤b(x, y) ≤ b1 in RN × [0, π] for some b0, b1 > 0, the to-tal energy decays as t goes to infinity. For short-rangetype of dissipations, e.g., 0 ≤ b(x, y) ≤ b2(1 + |x|)−1−δ

in RN × [0, π] for some b2 > 0 and δ > 0, scatter-ing solution exists. Although the proof for scatteringis based on Kato’s smooth perturbation theory, the sin-gular points called thresholds in the spectrum cause todifficulty. To eliminate this, density argument usingsome approximate operators are employed. This is jointwork with Mitsuteru Kadowaki (Ehime University) andKazuo Watanabe (Gskushuin University).

———

On the Cauchy problem for non-effectively hyperbolicoperators, the Gevrey 4 well-posedness

Tatsuo NishitaniMachikaneyama-cho 1-1 Toyonaka, Osaka 560-0043Japannishitani@math.sci.osaka-u.ac.jp

The Cauchy problem for non-effectively hyperbolic oper-ators is discussed in the Gevrey classes. Our operatorsbelong to the class of non-effectively hyperbolic oper-ators with symbols vanishing of order 2 on a smoothsubmanifold of codimension 3 on which the canonicalsymplectic 2-form has a constant rank. Assuming thatthere is no null bicharacteristic issuing from a simplecharacteristic point and landing tangentialy on the dou-ble characteristic manifold, we prove that the Cauchyproblem is Gevrey s well-posed for any lower order termwhenever 1 ≤ s < 4.

———

On the structure of the material law in a linear modelof poro-elasticity

Rainer PicardInstitut fur Analysis, FB Mathematik,TU Dresden,01062 Dresden, Germanyrainer.picard@tu-dresden.de

A modification of the material law associated withthe well-known Biot system first investigated byR.E. Showalter is re-considered in the light of a newapproach to a comprehensive class of evolutionary prob-lems.The particular material law is of the form

T = (C + trace∗λ trace ∂0) E − trace∗αp

connecting the stress tensor T with strain tensor E andfluid pressure p via parameters λ, α and C as theisotropic elasticity tensor. Here ∂0 denotes the timederivative and trace the matrix trace operation withtrace∗ as its adjoint. This model is generalized toanisotropic media and well-posedness of the generalizedmodel is shown.

———

Backward uniqueness for the system of thermo-elasticwaves with non-lipschitz continuous coefficients

Marco PivettaDipartimento di Matematica e Informatica, Via Valerio

74

IV.2. Dispersive equations

12/1, Trieste, Italy 34127 Italym.pive77@gmail.com

Using the Carleman estimates developed by Koch andLasiecka [Functional analysis and evolution equations,389-403, Birkhauser, Basel, 2008] together with an ap-proximation technique in the phase space, a uniquenessresult for the backward Cauchy problem is proved for thesystem of themoelastic waves having coefficients whichare in a class of log-Lipschitz-continuous functions.

———

The log-effect for 2 by 2 hyperbolic systems

Michael ReissigFaculty 1, TU Bergakademie Freiberg, Pruferstr. 9,Freiberg, 09596 Germanyreissig@math.tu-freiberg.de

In the talk we are interested to explain how to ex-tend the Log-effect from wave equations with time-dependent coefficients to 2 by 2 strictly hyperbolic sys-tems ∂tU − A(t)∂xU = 0. From wave models we knowthat besides oscillations in the coefficients a possible in-teraction of oscillations has a strong influence on H∞

well- or ill-posedness. Moreover, the precise loss ofderivatives can be proved. In the case of systems thesituation is more complicate. Besides the effects of os-cillating entries of the matrix A = A(t) and interactionsbetween the entries of A we have to take into consid-eration the system character itself. We will prove byusing tools from phase space analysis results about H∞

well- or ill-posedness. The precise loss of regularity is ofinterest. Moreover, we discuss the question if the lossof derivatives does really appear. These considerationsbase on the interplay between the Ljapunov and energyfunctional. Finally, we discuss the cone of dependenceproperty for solutions to 2 by 2 systems.This is a joint talk with T.Kinoshita (Tsukuba).

———

The Boussinesq equations based on the hydrostatic ap-proximation

Jun-ichi SaitoMinamisenju 8-17-1 Arakawa-ku, Tokyo 116-0003 Japanj saito@acp.metro-cit.ac.jp

The Boussinesq equations is studied in the field of dy-namic meteorology. Atmospheric flow in meteorologyare described by the Boussinesq equations. Due to thefact that the aspect ratio

ε =characteristic depth

characteristic width

is very small in most geophysical domains, asymptoticmodels have been used. One of the models is the hydro-static approximation of the Boussinesq equations.We consider the Boussinesq equations in the domainswith very small aspect ratio and prove the convergencetheorem for this model.

———

Blow-up of solutions of a quasilinear parablolic equation

Ryuichi SuzukiSchool of Science and Engineering, Kokushikan Uni-versity, 4-28-1 Setagaya, Setagaya-ku Tokyo, 154-8515

Japanrsuzuki@kokushikan.ac.jp

We consider nonnegative solutions of the Cauchy prob-lem for quasilinear parabolic equations

ut = ∆um + f(u),

where m > 1 and f(ξ) is a positive function in ξ > 0 sat-isfying f(0) = 0 and a blow-up condition

R∞1

1/f(ξ) dξ <∞. We study under what conditions on f(ξ) all nontriv-ial solutions blow up.

———

Blow-up and a blow-up boundary for a semilinear waveequation with some convolution nonlinearity

Hiroshi UesakaDepartment of Mathematics, College of Science andTechnology, Nihon University, Tokyo 101-8308, Chiyo-daku Kanda Surugadai 1-8, Japanuesaka@math.cst.nihon-u.ac.jp

We consider the Cauchy problem with a convolutionnonlinearity,

(∂2t −4)u = uq(V ∗ up), in R3 × (0, T ),

u(x, 0) = f(x), ∂tu(x, 0) = g(x) in R3,(0.1)

where uq(V ∗ up)(x, t) = uq(x, t)(R

R3up(y,t)|x−y|γ dy) with

p, q > 1, 0 < γ < 3.The blow-up boundary is defined by Γ = ∂u < ∞ ∩t > 0.We can give several suitable conditions to initial data toshow that

1. the Cauchy problem has a classical positive real-valued local solution u,

2. u is monotone increasing in t for any fixed x andmoreover satisfies ∂tu ≥ |∇u|,

3. there exists a positive T (x) for any x suchthat u keeps its regularity in (0, T (x)) andlimtT (x) u(x, t) =∞ .

Then the blow-up boundary Γ exists and T (x) satisfies|T (x)− T (y)| ≤ |x− y|.

———

Fundamental solutions for hyperbolic operators withvariable coefficients

Karen YagdjianDepartment of Mathematics, University of Texas-PanAmerican, 1201 W. University Drive, Edinburg, TX78541-2999, USAyagdjian@utpa.edu

The goal of this talk is to give a survey of a new approachin the constructing of fundamental solutions for thepartial differential operators with variable coefficientsand of some resent results obtaining by that approach.This new approach appeals neither to the Fourier trans-form, nor to the Microlocal Analysis, nor to the WKB-approximation. More precisely, the new integral trans-formation is suggested which transforms the family ofthe fundamental solutions of the Cauchy problem for theoperator with the constant coefficients to the fundamen-tal solutions for the operators with variable coefficients.

75

IV.3. Control and optimisation of nonlinear evolutionary systems

The kernel of that transformation contains Gauss’s hy-pergeometric function.This approach was applied by the author and hiscoauthors, T.Kinoshita (University of Tsukuba) andA.Galstyan (University of Texas-Pan American), to in-vestigate in the unified way several equations such asthe linear and semilinear Tricomi and Tricomi-typeequations, Gellerstedt equation, the wave equation inEinstein-de Sitter spacetime, the wave and the Klein-Gordon equations in the de Sitter and anti-de Sitterspacetimes. The listed equations play important role inthe gas dynamics, elementary particle physics, quantumfield theory in the curved spaces, and cosmology. In par-ticular, for all above mentioned equations, we have ob-tained representation formulas for the initial-value prob-lem, the Lp − Lq-estimates, local and global solutionsfor the semilinear equations, blow up phenomena, self-similar solutions and number of other results.

———

Global existence in Sobolev spaces for a class of non-linear Kirchhoff equations

Borislav YordanovBorislav Yordanov, 226 Swain Ct, Belle Mead, NJ 08502-4239 United Statesyordanov@math.utk.edu

The nonlinear Kirchhoff equation

utt −m(‖∇u‖2L2)∆u = 0

is studied for initial data (u, ut)t=0 = (u0, u1) in theSobolev spaces Hs(Rn)×Hs−1(Rn) with s ≥ 2 and forsmooth perturbations m(ρ) of the Pokhozhaev functionm0(ρ) = (k1ρ + k0)−2 with k0, k1 > 0. Global exis-tence is shown when ‖u1‖L2 is large and m is close tom0 in a suitable metric. Moreover, the asymptotic be-havior of solutions is found as t → ±∞. It turns outthat the norms ‖∇u‖L2 grow like |t|, so the propagationspeeds decrease like t−2 and the waves remain trappedin bounded regions.This is joint work with Lubin Vulkov.

———

IV.3. Control and optimisation of nonlinearevolutionary systems

Organisers:Francesca Bucci, Irena Lasiecka

The session is focused on new developments in the areaof well-posedness, optimisation, and control of systemsdescribed by evolutionary partial differential equations.These include: non-linear wave and plate equations,Navier-Stokes and Euler equations, non-linear thermoe-lasticity, viscoelasticity and electromagnetism. Of par-ticular interest to the session are interacting systemsthat involve PDE’s of different type describing the dy-namics on two (or more) separate regions with a cou-pling on an interface between these regions. Particularexamples of such systemsare: structural acoustic inter-actions, fluid structure interactions, magnetostructureinteractions. These have a wide range of applicationsthat include medicine (diagnostic imaging such as MRI,ultrasound), engineering (noise reduction in an acoustic

cavities, control of turbulence), geophysics (reconstruc-tion of seismic data) and others.Recent years have witnessed rapid development of newmathematical tools in both analysis and geometry thatallow to obtain various PDE estimates of inverse type.These are enabling to establish properties such as con-trollability, reconstruction of the data, stabilisation oroptimal feedback control.

—Abstracts—

Global well-posedness and long-time behavior of solu-tions to a wave equation

Lorena BociuUniversity of Nebraska-Lincoln, Department of Mathe-matics, 203 Avery Hall Lincoln, NE 68588 United Stateslbociu2@math.unl.edu

The model under consideration is the semilinear waveequation with supercritical nonlinear sources and damp-ing terms and the aim is to discuss the wellposednessof the system on finite energy space and the long-timebehavior of solutions. A distinct feature of the equa-tion is the presence of the double interaction of sourceand damping, both in the interior of the domain andon the boundary. Moreover, the nonlinear boundarysources are driven by Neumann boundary conditions.Since Lopatinski condition fails to hold for dimensiongreater or equal than 2, the analysis of the nonlineari-ties supported on the boundary, within the frameworkof weak solutions, is a rather subtle issue and involvesstrong interaction between the source and the damping.I will provide positive answers to the questions of localexistence and uniqueness of weak solutions and moreovergive complete and sharp description of parameters cor-responding to global existence and blow-up of solutionsin finite time.I will also discuss asymptotic energy-decay rates andblow-up of solutions originating in a potential well.

———

Distributed optimal controls for second kind parabolicvariational inequalities

Mahdi BoukroucheLaMUSE Saint-Etienne University, 23 Rue Dr PaulMichelon, Saint-Etienne, 42023, FranceMahdi.Boukrouche@univ-st-etienne.fr

Let ugi be the unique solution of a second kind parabolicvariational inequality with second member gi (i = 1, 2).We establish, in the general case, the error estimatebetween u3(µ) = µug1 + (1 − µ)ug2 and u4(µ) =uµg1+(1−µ)g2 for µ ∈ [0, 1], and prove a monotony prop-erty between u3(µ) and u4(µ) using a regularizationmethod.For a given constant M > 0, and the cost functional weestablish the existence of solutions for a family of controlproblems, over the the external force g for each parame-ter h > 0. Using the monotony property between u3(µ)and u4(µ), we establish the uniqueness of the solutionfor each control problem of the above family. We provealso the convergence of the optimalcontrols and statesassociated to this family of control problems governedby a second kind parabolic variational inequalities.

———

76

IV.3. Control and optimisation of nonlinear evolutionary systems

Controllability of a fluid-structure interaction problem

Muriel Boulakia175 rue du Chevaleret, 75013 Paris, Franceboulakia@ann.jussieu.fr

We are interested by the controllability of a fluid-structure interaction problem. The fluid in governed bythe incompressible Navier-Stokes equations and a rigidstructure is immersed inside. The control acts on a fixedsubset of the fluid domain. For small initial data, weprove that this system is null controllable i.e. that thesystem can be driven at rest. This result is obtainedwith the help of a Carleman inequality proven for theadjoint linearized system.

———

Uniform decay rate estimates for the wave equation oncompact surfaces and locally distributed damping

Marcello CavalcantiDepartment of Mathematics - State University ofMaringa, Av. Colombo 5790, Maringa, PR 87020-900Brazilmmcavalcanti@uem.br

In this talk we present new contributions concerning uni-form decay rates of the energy associated with the waveequation on compact surfaces subject to a dissipationlocally distributed. We present a method that gives usa sharp result in what concerns of reducing arbitrarilythe area where the dissipative effect lies.

———

Rate of decay for non-autonomous damped wave sys-tems

Moez DaoulatliISSATS, University of Sousse (& LAMSIN) Cite Taffala(Ibn Khaldoun), Sousse 4003 Tunisiamoez.daoulatli@infcom.rnu.tn

We study the rate of decay of solutions of the wave sys-tems with time dependent nonlinear damping which islocalized on a subset of the domain. We prove that theasymptotic decay rates of the energy functional are ob-tained by solving nonlinear non-autonomous ODE.

———

On qualitative aspects for the damped Korteweg-deVries and Airy type equations

Valeria Domingos CavalcantiDepartment of Mathematics - State University ofMaringa, Av. Colombo 5790, Maringa, PR 87020-900Brazilvndcavalcanti@uem.br

This talk is concerned with the study of the dampedKorteweg-de Vries equation posed in whole real line

ut + uxxx + uux + λu = 0, in R× [0,+∞), λ > 0.

We establish two invariant subsets of H1(R) where justone of the following statements holds: (i) solutions de-cay exponentially in H1− level or (ii) solutions do notdecay to zero in H1− level as t goes to infinity.

In addition, we investigate the existence of uniform de-cay rates for both, the Airy type equation

ut + uxxx + g(u) = 0, in [0, L]× (0,+∞),

posed in a bounded interval [0, L] and supplemented bya nonlinear damping g(u). By considering suitable as-sumptions on g and on the initial data, general decayrates are proved in L2− level as well as exponential de-cay rates are established in H1−level.

———

Optimal control of waves in anisotropic media via con-servative boundary conditions

Matthias EllerDepartment of Mathematics, Georgetown University,Washington, DC 20057 United Statesmme4@georgetown.edu

An optimal boundary control problem for symmetric hy-perbolic systems is considered. The quadratic cost func-tional is of tracking type and the control acts through aconservative boundary condition. Some loss of regularityis associated with these boundary conditions. This re-sults in certain choices for the underlying function spacesin the cost functional. The loss of regularity occurs onlynear the boundary and it may be attributed to the oc-currence of surface waves. As examples we consider theanisotropic Maxwell equations as well as the anisotropicequations of elasticity.

———

Stability for some nonlinear damped wave equations

Genni FragnelliDipartimento di Ingegneria dell’Informazione, Univer-sita degli Studi di Siena, via Roma 56, c.a.p. 53100fragnelli@dii.unisi.it

We prove stability results for a large class of abstractnonlinear damped wave equations, whose prototype isthe usual wave equation8<:

utt + h(t)ut = ∆u+ f(u) in (0,+∞)× Ω,u(t, x) = 0 in (0,+∞)× ∂Ω,u(0, x) = u0(x), ut(0, x) = u1(x) x ∈ Ω,

where Ω is a bounded and smooth domain of RN , N ≥ 1,u0 ∈ H1

0 (Ω), u1 ∈ L2(Ω) and f : R→ R.At first, the damping is nonnegative, but it is allowedto be zero either on negligible sets or even in a sequenceof intervals. Then, also the case of a positive–negativedamping is treated.

———

Global existence for the one-dimensional semilinearTricomi-type equation

Anahit GalstyanDepartment of Mathematics, University of Texas-PanAmerican, 1201 W. University Dr., Edinburg 78541, TX,U.S.A.agalstyan@utpa.edu

In this talk the issue of global existence of the solutionsof the Cauchy problem for one-dimensional semilinearweakly hyperbolic equations, appearing in the boundaryvalue problems of gas dynamics is investigated. We solvethe Cauchy problem trough integral equation and givesome sufficient conditions for the existence of the global

77

IV.3. Control and optimisation of nonlinear evolutionary systems

weak solutions. The necessary condition for the exis-tence of the similarity solutions for the one-dimensionalsemilinear Tricomi-type equation will be presented aswell. Our approach is based on the fundamental solu-tion of the operator and the Lp − Lq estimates for thelinear Tricomi equation.

———

Optimal control of a thermoelastic structural acousticmodel

Catherine LebiedzikDepartment of Mathematics, Wayne State University,656 W Kirby Detroit, MI 48202 United Statesar6554@wayne.edu

We consider point control of a structural acoustic modelwith thermoelastic effects. The key feature of this pa-per is that the two-dimensional plate modeling the activewall of the acoustic chamber has clamped boundary con-ditions. For this case a new optimal regularity result hasrecently become available. Using this new result for theplate alone, we derive a sharp regularity result for theoverall coupled system of wave and thermoelastic plateequations. This allows for the study of optimal controlof the coupled system.

———

The Balayage method: Boundary control of a thermo-elastic plate

Walter LittmanUniversity of Minnesota, School of Mathematics, 206Church Street, Southeast Minneapolis, Minnesota 55455United Stateslittm001@umn.edu

We discuss the null boundary controllablity of a linearthermo-elastic plate. The method employs a smooth-ing property of the system of PDEs which allows theboundary controls to be calculated directly by solvingtwo Cauchy problems.

———

Hopf-Lax type formulas and Hamilton-Jacobi equations

Paola LoretiDipartimento di Metodi e Modelli Matematici per leScienze Applicate, via Antonio Scarpa n.16, 00161Roma, Italia.loreti@dmmm.uniroma1.it

Here we discuss Hopf-Lax type formulas related to theclass of Hamilton-Jacobi equations

ut(x, t) + αxDu(x, t) +H(Du(x, t)) = 0,

in RN × (0,+∞) with initial condition u(x, 0) = u0 inRn, with α a positive, real number.The talk is based on some joint works with A. Avantag-giati.

———

Investigation of boundary control problems by on-lineinversion technique

Vyacheslav MaksimovS.Kovalevskaya 16 Ekaterinburg, Sverdlovsk 620219

Russiamaksimov@imm.uran.ru

A number of applied studies address such fundamentalissues as

(i) reconstruction of uncertain parameters of multi-dimensional dynamical systems and

(ii) control of uncertain dynamical systems.

We discuss a technical approach intended to help solveproblems of this kind. The approach employs the on-lineinversion theory adjoining theory of closed-loop controland theory of ill-posed problems. On-line inversion al-gorithms involve artificially designed dynamical modelswhose parameters track non-observable parameters ofthe system; it is important that the tracking quality isinsensitive to perturbations in the observation channels.In combination with appropriate closed-loop regulators,on-line parameter tracking algorithms give raise to ro-bust observer-controller patterns allowing one to guidethe uncertain system close to the trajectories designedvia an optimal feedback to a complete set of observedsignals. The goal of this report is to demonstrate theessence and abilities of the on-line inversion technique;for this purpose we consider three types of problems,namely, a problem of etalon motion tracking, a prob-lem of game control, and a problem of dynamical inputidentification for a parabolic equation with the Neumannand Dirichlet boundary condition.

———

Null controllability properties of some degenerateparabolic equations

Patrick MartinezUniversite Paul Sabatier Toulouse III, Institut deMathematiques, 118 route de Narbonne, Toulouse,31062 Francemartinez@mip.ups-tlse.fr

Motivated by several problems in fluid dynamics, biol-ogy, or economics, we are interested in controllabilityproperties of parabolic equations degenerating at theboundary of the space domain. After considering theone dimensional case, this talk will mainly focus on theN-dimensional case:

ut − div(A(x)∇u) = f(x, t)χω(x), x ∈ Ω, t > 0

where ω ⊂ Ω and the matrix A(x) is definite positive forall x ∈ Ω, and but has at least one eigenvalue equal to0 for all x ∈ ∂Ω. Mainly, we assume that- the least eigenvalue of the matrix A(x) behaves asd(x, ∂Ω)α, where α ≥ 0,- the degeneracy occurs in the normal direction: whenx ∈ ∂Ω, the associated eigenvector is the unit outwardvector.When α ∈ [0, 2), we prove the null controllability via newCarleman estimates for the adjoint degenerate parabolicequation. When α ∈ [2,+∞), we prove that the prob-lem is not null controllable, using earlier results of Micu -Zuazua, Escauriaza-Seregin-Sverak related to nondegen-erate parabolic equations in unbounded domains.These results were obtained in collaboration with P.Cannarsa (Univ Tor Vergata, Roma 2), and J. Van-costenoble (Univ Toulouse 3).

———

78

IV.3. Control and optimisation of nonlinear evolutionary systems

Dissipation in contact problems: an overview and somerecent results

Maria Grazia NasoDipartimento di Matematica, Universita degli Studi diBrescia, Via Valotti, 9 Brescia, BS 25133 Italynaso@ing.unibs.it

In this talk we investigate the longtime behaviour of adynamic unilateral contact problem between two ther-moelastic beams. Under suitable mechanical and ther-mal boundary conditions the evolution problem is shownto possess an energy decaying exponentially to zero, astime goes to infinity.

———

Heat equations with memory: a Riesz basis approach

Luciano PandolfiPolitecnico di Torino, Dipartimento di Matematica, C.soDuca degli Abruzzi 24, Torino, 10129 Italyluciano.pandolfi@polito.it

In this talk we present recent results concerning a Rieszbasis approach to the heat equation with memory

θt(x, t) =

Z t

0

N(t− s) [∆θ(x, s)− q(x)θ(x, s)] ds

(x ∈ [0, π]) and square integrable initial conditions.We shall construct a special sequence zn(t) associ-ated to this equations and we shall prove that it is aRiesz sequence on a suitable interval [0, T ], using BariTheroem. These results are applied to the study of con-trol/observability problems.

———

A note on a class of observability problems for PDEs

Michael RenardyDepartment of Mathematics, Virginia Tech, Blacksburg,VA 24061-0123 United Statesrenardym@math.vt.edu

The question of observability arises naturally in the anal-ysis of control problems. If the solution of a PDE initial-boundary value problem is known to be zero in a part ofthe domain, does this guarantee it is zero everywhere?The most popular techniques to establish such resultsare based on local unique continuation results (Holm-gren’s theorem) or Carleman estimates. The lecture willdraw attention to a class of problems where the observedregion is bounded by characteristics, and local uniquecontinuation fails. Nevertheless, observability may hold.A problem of this nature arose in recent work by theauthor on control of viscoelastic flows.

———

Invariant manifolds for parabolic problems with dynam-ical boundary conditions

Roland SchnaubeltUniversity of Karlsruhe, Department of Mathematics,Kaiserstrasse 89, Karlsruhe, 76128 Germanyschnaubelt@math.uni-karlsruhe.de

We study a class of nonlinear parabolic systems de-scribed by coupled evolution equations on a domain

and on its boundary combined with a nonlinear cou-pled boundary condition. Such problems arise from freeboundary value problems as the Stefan problem withsurface tension after a suitable transformation. Besideslocal well posedness and smoothing properties, we focuson the qualitative behavior near an equilibrium. To thatpurpose we construct locally invariant manifolds and es-tablish their main properties.

———

On regularity properties of optimal control and La-grange multipliers

Ilya ShvartsmanDept. of Mathematics and Computer Science, 777 W.Harrisburg Pike Middletown, PA 17110 United Statesius13@psu.edu

In this talk we will go over classical and recent resultson regularity properties (such as continuity, Holder andLipschitz continuity) of optimal controls and Lagrangemultipliers.

———

Evolution equations with memory terms

Daniela SforzaDipartimento di Metodi e Modelli Matematici per leScienze Applicate, via Antonio Scarpa 16, Roma, 00161Italysforza@dmmm.uniroma1.it

The purpose of the talk is to show some results concern-ing control problems for integro-differential equations ofhyperbolic type.More precisely, we consider non-linear equations inHilbert spaces with integral convolution terms and as-sume the corresponding kernels to exhibit a polynomialor exponential decay. We show that the solutions havethe same decay behaviour as the kernel. Our main toolis the multipliers method and we succeed in finding suit-able multipliers which work even in the presence of in-tegral terms.Besides, we provide a reachability result for a classof linear integro-differential problems. Our strategy isfounded on the so-called Reachability Hilbert Unique-ness Method, introduced by Lagnese - Lions, whichamounts to proving Ingham type inequalities for theFourier series expansion of the solution of the adjointproblem.To conclude, we observe that our abstract results maybe used to treat some problems arising in the study ofviscoelastic systems.

———

Stabilization of structure-acoustics interactions for aReissner-Mindlin plate by localized nonlinear boundaryfeedbacks

Daniel ToundykovUniversity of Nebraska-Lincoln Department of Mathe-matics, 203 Avery Hall Lincoln, NE 68588 United Statesdtoundykov@math.unl.edu

This work addresses observability and energy decay for astructural-acoustics model comprised of a wave equationcoupled with a Reissner-Mindlin plate. Both compo-nents of the dynamics are subject to localized boundarydamping: the acoustic dissipative feedback is restrictedto the flexible boundary and only a portion of the rigid

79

IV.4. Nonlinear partial differential equations

wall; the plate is likewise damped on a segment of itsboundary.The derivation of the “coupled” stabiliza-tion/observability inequalities requires weighted energymultipliers related to the geometry of the domain, andspecial tangential trace estimates for the displacementand the filament angles of the Reissner-Mindlin platemodel. The behavior of the energy at infinity can bequantified by a solution to an explicitly constructednonlinear ODE. The nonlinearities in the feedbacks mayinclude sub- and super-linear growth at infinity, in whichcase the decay scheme presents a trade-off between theregularity of solutions and attainable uniform decayrates of the finite-energy.

———

Exponential stability of the wave equation with bound-ary time varying delay

Julie ValeinUniversite de Valenciennes et du Hainaut-CambresisLAMAV - ISTV2 - Le Mont-Houy Valenciennes, Nord-Pas de Calais 59313 Francejulie.valein@univ-valenciennes.fr

We consider the wave equation with a time-varying de-lay term in the boundary condition in a bounded domainΩ ⊂ Rn with a boundary Γ of class C2.We assume Γ = ΓD ∪ ΓN , with ΓD ∩ ΓN = ∅, ΓD 6= ∅,and we consider8>>>>>>>><>>>>>>>>:

utt(x, t)−∆u(x, t) = 0 in Ω× (0,+∞)u(x, t) = 0 on ΓD × (0,+∞)∂u∂ν

(x, t) = −µ1ut(x, t)− µ2ut(x, t− τ(t))on ΓN × (0,+∞)

u(x, 0) = u0(x) and ut(x, 0) = u1(x) in Ωut(x, t− τ(0)) = f0(x, t− τ(0))

in ΓN × (0, τ(0)),(*)

where τ(t) is the delay, µ1, µ2 > 0.We assume

0 ≤ τ(t) ≤ τ , τ ′(t) ≤ d < 1,

∀ t > 0 and τ ∈W 2,∞([0, T ]), ∀ T > 0.

Under

µ2 <√

1− dµ1,

we prove the existence and uniqueness results of (*) byusing the variable norm technique of Kato and we showthe exponential decay of an appropriate energy. Dueto the time-dependence of the delay, we can not use anobservability estimate since the system is not invariantby translation in time. Hence we introduce a Lyapunovfunctional.We extend this result to a nonlinear version of the model.This is a joint work with Serge Nicaise and Cristina Pig-notti.

———

State estimation for some parabolic systems

Masahiro YamamotoUniversity of Tokyo, Department of Mathematical Sci-ences, 3-8-1 Komaba Meguro Tokyo 153, Japanmyama@next.odn.ne.jp

For some types of parabolic systems, we consider in-equalities of Carleman’s type and prove conditional sta-bility estimates for state determination problems suchas a backward problem.

———

Euler flow and Morphing Shape Metric

Jean-Paul ZolesioCNRS-INLN nd INRIA RTE Lucioles 1361 and 2004Sophia Antipolis, France 06565 Francejean-paul.zolesio@sophia.inria.fr

We extend the so-called ”Courant metric” into a new”Tube metric” beetwen measurable sets and character-ize the geodesic as variational solution to incompress-ible Euler flow. Such geodesic can modelise topologicalchanges. We make use of a new ”Sobolev perimeter”,and Sobolev Mean curvature for the moving boundarywhich turns to be Shape differentiable under smoothtransverse perturbation. Then working with L2 speedvector fields (we don’t use any renormalization bene-fit) we succed in the existence of connecting tubes andin variational solution to the usual incompressible Eu-ler flow under surface tension associated to the Sobolevperimeter. This technic applies to several situationsin [Shape-Morphing Metric by Variational Formulationfor Incompressible Euler Flow.J. of Control and Cy-bernetics, vol 38 (2009), No. 4], [Control of MovingDomains...,. Int.Ser.Num.Math., vol. 155, 329-382,Birkhauser Verlag Basel,2007].

———

IV.4. Nonlinear partial differential equations

Organisers:Vladimir Georgiev, Tohru Ozawa

The Session intends to discuss various nonlinear partialdifferential equations in mathematical physics.Among possible arguments the following ones shall bediscussed: existence and qualitative properties of thesolutions, existence of wave operators and scattering forthese problems, stability of solitary waves and other spe-cial solutions.

—Abstracts—

Evolution equations in nonflat waveguides

Piero D’AnconaSapienza - Universita di Roma - Dipartimento di Matem-atica P. Moro, 2 Roma, RM 00185 Italydancona@mat.uniroma1.it

In a joint work with Reinhard Racke (Konstanz) weprove smoothing and Strichartz estimates for evolutionequations of Schroedinger or wave type on waveguideswhich are deformations, in a suitable sense, of flat waveg-uides of the form O×Rk, O a bounded open set in Rm.For the proof, new weighted estimates for fractional pow-ers of Schroedinger operators are required.

———

80

IV.4. Nonlinear partial differential equations

Investigation of solutions of one not divergent type

Mersaid AripovMech.,Math, National University of Uzbekistan, Univer-sitet 1, Tashkent, Tashkent 100174 Uzbekistanmirsaidaripov@mail.ru

The properties of the weak solution of problem Cauchyand the first boundary value problem for one parabolicequation of not divergent type double nonlinearity andwith lower members are investigated. The researchedequation is the best combination of forms of the equa-tion of nonlinear diffusion, fast diffusion, the equationto very fast diffusion and p-Laplace heat conductivityequation. This equation describes various processes ofnonlinear diffusion, heat conductivity, a filtration, mag-netic rheology, etc.The method of investigation of the qualitative proper-ties having physical sense weak solution on the basisof a method of a nonlinear splitting and a method ofthe standard equations is offered. Two side estima-tions of the solutions and free boundary, a conditionof existence of global solutions (including case of crit-ical value of parameter and exponent) generalizing ofknown results of H. Fujite, A.A., Samarskii, S.P. Kur-dyumov, A.P. Mikhajlov, V.A. Galaktionov, H.Vaskes,S. A. Posashkov are received. On the basis of the analy-sis of properties of solutions the numerical modeling andvisualization of solutions carried out.

———

Asymptotic behavior of subparabolic functions

Davide CataniaUniversita di Brescia - Dip. Matematica Via Valotti, n.9 Brescia, BS 25133 Italydavide.catania@ing.unibs.it

We consider an MHD-α model with regularized veloc-ity for an incompressible fluid in two space dimension.Such a model is introduced in analogy with the Navier–Stokes equation to study the turbulent behavior of flu-ids in presence of a magnetic field, since this problemis otherwise difficult to study, both analitically and nu-merically.We prove local and global existence for the relatedCauchy problem, where the velocity field is viscous,while we have not any magnetic diffusivity.

———

On multiple solutions of concave and convex effects fornonlinear elliptic equation on RN

Kuan-Ju ChenDepartment of Applied Science, Naval Academy,P.O.BOX 90175 Zuoying, Taiwan, R.O.C.kuanju@mail.cna.edu.tw

In this paper we consider the existence of multiple so-lutions of the elliptic equation on RN with concave andconvex nonlinearities.

———

Nonlinear gauge invariant evolution of the plane wave

Kazuyuki DoiGraduate School of Information Sciences, Tohoku Uni-versity, 6-3-09 Aramaki-Aza-Aoba, Aoba-ku, Sendai,

Miyagi 980-8579 Japandoi@ims.is.tohoku.ac.jp

We consider nonlinear gauge invariant evolution of theplane wave. In this talk, we deal with the power and log-arithmic type nonlinearities. Although the plane wavedoes not decay at infinity, by an elementary and simpleargument we find an extremely smooth solution whichhas an explicit expression. Additionally, we study theglobal behavior of the solution from its representation.

———

New approach to solve linear parabolic problems viasemigroup approximation

Mohammad DehghanFerdowsi University of Mashhhad Azadi Square Mash-had, Khorasan-e-Razavi 9177948974 Iranmudenayyeri@gmail.com

We consider Linear Parabolic Problems (LPPs) whosesolutions can be expressed via semigroups. Computingthe solutions of these LPPs depends on existing explicitformulas for the corresponding semigroups. However,in general explicit formulas are not available. The pro-posed approach defines a sequence of linear problemswhich are semidiscrete approximations of the consid-ered LPP. The solutions of approximant linear problemscan be expressed via corresponding semigroups whichhave explicit formulas. These solutions converge uni-formly to the solution of LPP. So the correspondingsemigroup of LPP can be approximated by semigroupswhich have explicit formulas. The approximant linearproblems are defined on the finite dimensional subspacesof the LPP solution space, via a hybrid finite-difference-projection method. The accuracy of approximations,order of convergency and their relations to the proposedhybrid method are discussed and some examples are pre-sented.

———

Global existence and blow-up for the nonlocal nonlinearCauchy problem

Albert ErkipSabanci University, Faculty of Engineering and NaturalSciences, Orhanli, Tuzla Istanbul / 3495 Turkeyalbert@sabanciuniv.edu

We study the Cauchy problem

utt = (β ∗ (u+ g (u)))xx x ∈ R, t > 0

u (x, 0) = φ (x) , ut (x, 0) = ψ (x) x ∈ R,

for a general class of nonlinear nonlocal wave equationsarising in one-dimensional nonlocal elasticity.The model involves a convolution integral operator witha general kernel function β whose Fourier transform isnonnegative. We show that some well-known examplesof nonlinear wave equations, such as Boussinesq-typeequations, follow from the present model for suitablechoices of the kernel function.We establish global existence of solutions of the modelassuming enough smoothness on the initial data togetherwith some positivity conditions on the nonlinear term.Furthermore, conditions for finite time blow-up are pro-vided.

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81

IV.4. Nonlinear partial differential equations

Qualitative properties for reaction-diffusion systemsmodelling chemical reactions

Marius GherguSchool of Mathematical Sciences University CollegeDublin Belfield, , Dublin 4 Dublin Irelandmarius.ghergu@ucd.ie

In 1952 the British mathematician Alan M. Turing pub-lished the foundation of reaction-diffusion theory formorphogenesis, the development of form and shape in bi-ological systems. Since then, many Turing-type modelsdescribed by coupled reaction-diffusion equations havebeen proposed for generating patterns in both organicand inorganic systems.In this talk we present a qualitative study for reaction-diffusion systems of the type

ut − d1∆u = a+ bu+ f(u)v in Ω× (0,∞),

vt − d2∆v = c+ du− f(u)v in Ω× (0,∞),

u(x, 0) = u0(x), v(x, 0) = v0(x) on Ω,

∂u

∂ν(x, t) =

∂u

∂ν(x, t) = 0 on ∂Ω× (0,∞).

Here Ω ⊂ RN (N ≥ 1) is a bounded domain,a, b, c, d, d1, d2 ∈ R, u0, v0 ∈ C(Ω) are non-negative andf ∈ C[0,∞) ∩ C1(0,∞) is a non-negative and nonde-creasing such that f(0) = 0 and f > 0 in (0,∞).The system encompasses two well known chemical mod-els: the Brusselator and the Schnackenberg modelswhich are a rich source of varied spatio-temporal pat-terns.We present several existence and stability results. A par-ticular attention is paid to the associated steady-statesystem where the crucial role played by the diffusion co-efficients d1, d2 and the behavior of the nonlinearity f isemphasized.The proofs rely on a-priori estimates combined with an-alytical and topological methods.

———

Scattering in the zero-mass Lamb system

Marco Antonio Taneco-HernandezInstituto de Fısica y Matematicas, Universidad Michoa-cana de San Nicolas de Hidalgo, Edificio C-3, CiudadUniversitaria Av., Francisco J. Mujica s/n, Colonia Fe-licitas del Rio Morelia, Michoacan 58040, Mexicomath23@ifm.umich.mx

We consider nonlinear conservative Lamb system, whichis the wave equation coupled with a particle of zero mass:

u(x, t) = u′′(x, t),

F (y(t)) + u′(0+, t)− u′(0−, t), y(t) = u(0, t),

with x ∈ R \ 0, t ∈ R. Here u := ∂u∂t

, u′ := ∂u∂x

and so on. The solutions u(x, t) take the values in Rdwith d ≥ 1 and F := −∇V with V : Rd → R is a po-tential force field. For the first time we establish longtime asymptotics in global energy norm for all finite en-ergy solutions. Namely, under some Ginzburg-Landautype conditions to V , each solution from some functionalspace decays to a sum of a stationary state, outgoingwave and the rest which tends to zero in global energynorm as t→ +∞.

The outgoing wave is a solution to the free wave equationwith some asymptotics states as initial data. We intro-duce corresponding nonlinear scattering operator, andobtain a necessary condition for the asymptotic states.Also we prove the asymptotics completeness in the Lambsystem.This is joint work with A.E. Merzon and A.I. Komech.

———

Global existence for systems of the nonlinear wave andKlein-Gordon equations in 3D

Soichiro KatayamaDepartment of Mathematics, Wakayama University, 930Sakaedani Wakayama, Wakayama 640-8510 Japankatayama@center.wakayama-u.ac.jp

We consider the Cauchy problem for coupled systems ofthe nonlinear wave and Klein-Gordon equations in threespace dimensions. We present a sufficient condition forglobal existence of small amplitude solutions to such sys-tems.Our condition is much weaker than the strong null con-dition for this kind of coupled system, and our resultis a natural extension of the global existence theoremfor the nonlinear wave equations under the null con-dition, as well as that for the Klein-Gordon equationswith quadratic nonlinearities. Our result is applicableto a certain kind of model equation in physics, suchas the Klein-Gordon-Dirac equations, the Klein-Gordon-Zakharov equations, and the Dirac-Proca equations.

———

Global existence for nonlinear wave equations exteriorto an obstacle in 2D

Hideo KuboGraduate School of Information Sciences, Tohoku Uni-versity 6-3-09 Aramaki-Aza-Aoba, Aoba-ku Sendai ,Miyagi 980-8579 Japankubo@math.is.tohoku.ac.jp

In this talk we discuss the global existence for the exte-rior problem of nonlinear wave equations in two space di-mensions. The obstacle is assumed to be a star-shaped,so that the decay of the local energy is available. Themain difficulty compared with the three space dimen-sional case is the weaker decay of solutions in 2D, aswell as the lack of the sharp Huygens principle. How-ever, we are able to show the global existence for smallinitial data, provided the nonlinearity is of the cubic or-der and fulfills the so-called null condition.

———

Remark on Navier-Stokes equations with mixed bound-ary conditions

Petr KuceraCzech Technical University, Fac. of Civil Engineering,Dept. of Math., Thakurova 7, Prague 166 29 Czech Re-publickucera@mat.fsv.cvut.cz

We solve a system of the Navier-Stokes equations for in-compressible heat conducting fluid with mixed bound-ary conditions (of the Dirichlet or non-Dirichlet type ondifferent parts of the boundary). We suppose that theviscosity of the fluid depends on temperarure.

———

82

IV.4. Nonlinear partial differential equations

Contraction-Galerkin method for a semi-linear waveequation with a boundary-like antiperiodic condition

Ut van LeDepartment of Mathematical Sciences, P.O. Box 3000,Oulu FI-90014 Finlandut.van.le@oulu.fi

We consider the unique solvability of initial-boundaryvalue problems of semi-linear wave equations with space-time dependent coefficients and special mixed non-homogeneous boundary values which make the so-calledboundary-like antiperiodic condition. The procedure inthis project is the combination of the Galerkin methodand a contraction.

———

p− q systems of nonlinear Schrodinger equations

Sandra LucenteDipartimento di Matematica, Via Orabona 4, Bari70124, Italylucente@dm.uniba.it

In a joint work with L. Fanelli and E. Montefusco, weconsider coupled nonlinear Schrodinger equations

iut + ∆u±N1(u, v) = 0,ivt + ∆v ±N2(u, v) = 0,

with suitable semilinear terms N1(u, v) and N2(u, v)having polynomial growth. We investigate on local andglobal existence critical exponents and describe the cor-responding solutions.

———

Semiclassical analysis for nonlinear Schrodinger equa-tions

Satoshi Masaki6-3-09 Aza-aoba Aramki Aoba-ku Sendai, Miyagi 980-8579 Japanmasaki@ims.is.tohoku.ac.jp

We consider the semiclassical limit of the nonlinearSchrodinger equations. We approximate the solution bya function of phase-amplitude form, called WKB analy-sis. We mainly treat the nonlocal nonlinearites.

———

3-D viscous Cahn-Hilliard equation with memory

Gianluca MolaUniversita di Milano, Dipartimento di Matamatica, viaSaldini 50 Milano, MI 20133 Italygianluca.mola@unimi.it

We deal with the memory relaxation of the viscousCahn-Hilliard equation in 3-D, covering the well–knownhyperbolic version of the model. We study the longtermdynamic of the system in dependence of the scaling pa-rameter of the memory kernel ε and of the viscosity co-efficient δ. In particular we construct a family of expo-nential attractors which is robust as both ε and δ go tozero, provided that ε is linearly controlled by δ.

———

A symmetric error estimate for Galerkin approximationsof time dependant Navier-Stokes equations in two di-mensions

Itir MogultayDepartment of Mathematics, Yeditepe University, 26Agustos Yerlesimi Kayisdagi Caddesi Kayisdagi Istan-bul, 81120 Turkeyitir.mogultay@yeditepe.edu.tr

A symmetric error estimate for Galerkin approximationof solutions of the Navier-Stokes equations in two spacedimensions plus time is given. The finite dimensionalfunction spaces are taken to be divergence free, and timeis left continuous. The estimate is similar to known re-sults for scalar parabolic equations. An application ofthe result is given for mixed method formulations. Ashort discussion of examples is included. Finally, thereare some remarks about a partial expansion to threespace dimensions.Note: This is a joint work with Prof. Todd F. Dupontat the University of Chicago.

———

Stability of standing waves for some systems of nonlin-ear Schrodinger equations with three-wave interactions

Masahito OhtaDepartment of Mathematics, Saitama University, 255Shimo-Ohkubo, Saitama, 338-8570 Japanmohta@mail.saitama-u.ac.jp

We discuss orbital stability and instability of severaltypes of standing waves for some three-component sys-tems of nonlinear Schrodinger equations.

———

Decay rates for wave models with structural damping

Michael ReissigFaculty 1, TU Bergakademie Freiberg, Pruferstr. 9,Freiberg, 09596 Germanyreissig@math.tu-freiberg.de

In this talk, we will present results on the behaviorof higher order energies of solutions to the followingCauchy problem for a wave model with structural damp-ing:

utt −∆u+ b(t)(−∆)σut = 0,

u(0, x) = u0(x), ut(0, x) = u1(x),

σ ∈ (0, 1], b(t) = µ(1 + t)δ, µ > 0, δ ∈ [−1, 1].

We are interested in the influence of the structural dis-sipation (between external and visco-elastic damping)b(t)(−∆)σut on L2 − L2 estimates.Our main goal is to study under which conditions do wehave a parabolic effect for the solutions, that is, the decayrates depend on the order of energy.In the talk we will explain how hyperbolic or ellipticWKB analysis comes in. The main tools are a correctdivision of the extended phase space into zones, diago-nalization procedures, construction of fundamental so-lutions and a gluing procedure. Some open problemscomplete the talk.This is joint work with Xiaojun Lu (Hangzhou).

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83

IV.4. Nonlinear partial differential equations

Stability theorems in the theory of mathematical fluidmechanics

Yoshihiro ShibataDepartment of Mathematics, Waseda University,Ohkubo 3-4-1 Shinjuku-ku Tokyo, Tokyo 169-8555Japanyshibata@waseda.jp

I would like to talk about some stability theorem of sta-tionary solutions of incompressible fluid flow with initialdisturbance.

———

On singular systems of parabolic functional equations

Laszlo SimonPazmany P. setany 1/C, L. Eotvos University, Instituteof Mathematics Budapest, Hungary H-1117 Hungarysimonl@ludens.elte.hu

We shall consider initial-boundary value problems fora system consisting of a quasilinear parabolic functionalequation and an ordinary differential equation with func-tional terms. The parabolic equation may contain thegradient with respect to the space variable of the un-known function in the ODE. It will be proved globalexistence of weak solutions, by using the theory of mono-tone type operators and Schauder’s fixed point theo-rem. Such problems are motivated by models describ-ing reaction-mineralogy-porosity changes in porous me-dia and polymer diffusion.

———

Survey of recent results on asymptotic energy concen-tration in solutions of the Navier-Stokes equations

Zdenek SkalakThakurova 7, Czech Technical University, Prague, 16629Czech Republicskalak@mat.fsv.cvut.cz

We present some recent results on asymptotic energyconcentration in solutions of the Navier-Stokes equa-tions. For example, if w is such a solution satisfying thestrong energy inequality then there exists a ≥ 0 suchthat

limt→∞

||Eλw(t)||/||w(t)|| = 1

for every λ > a, where Eλ;λ ≥ 0 denotes the resolu-tion of identity of the Stokes operator.

———

Conditional stability theorems for Klein-Gordon typeequations

Atanas Stefanov1460, Jayhawk Blvd., Department of Mathematics, Uni-versity of Kansas, Lawrence, KS 66049, USAstefanov@math.ku.edu

We consider unstable ground state solutions of the Klein-Gordon equation with various power nonlinearities. Themain result is a fairly precise construction of a stablemanifold in a close vicinity of the ground state. In par-ticular, we provide an asymptotic formula for the asymp-totic phase, an estimate of the rate of the convergence

towards the stable manifold etc. Several outstandingopen questions will be discussed as well.

———

A regularity result for a class of semilinear hyperbolicequations

Sergio SpagnoloDepartment of Mathematics, University of Pisa, LargoPontecorvo 5 Pisa, 56127 Italyspagnolo@dm.unipi.it

We first recall a former result of global wellposedness inC-infinity (resp., in each Gevrey class) for a special kindof homogeneous, linear hyperbolic equations with an-alytic (resp., C-infinity) coefficients depending only ontime. Then, we add to these equations an analytic semi-linear term, and we prove that the resulting equationsenjoy the following regularity property; each solutionwhich is real analytic at the initial time together withall its time derivatives, remains analytic as long as it isbounded in C-infinity (resp., in some Gevrey class).

———

On nonlinear equations, fixed-point theorems and theirapplications

Kamal SoltanovDepartment of Mathematics, Faculty of Sciences,Hacettepe University, Beytepe Campus Ankara,Cankaya TR-06532 Turkeysoltanov@hacettepe.edu.tr

In this work we investigated some class of the nonlinearoperators and a nonlinear equations with such type op-erators in a Banach spaces. Here we obtained some newresults on the solvability of the nonlinear equations, andalso a fixed-point theorems for continuous mappings.With use of the obtained here results we studied variousboundary value problems (BVP) (and mixed problems)for the different nonlinear differential equations.

———

Dynamics of a quantum particle in a cloud chamber

Alessandro TetaDipartimento di matematica pura e applicata, Univer-sita’ di L’Aquila via Vetoio - loc. Coppito L’Aquila,Abruzzo 67100 Italyteta@univaq.it

We consider the Schroedinger equation for a system com-posed by a particle (the α-particle) interacting with twoother particles (the atoms) subject to attractive poten-tials centered in a1, a2 ∈ R3. At time zero the α-particleis described by a diverging spherical wave centered in theorigin and the atoms are in their ground state. The aimis to show that, under suitable assumptions on the phys-ical parameters of the system and up to second order inperturbation theory, the probability that both atoms areionized is negligible unless a2 lies on the line joining theorigin with a1.The work (in collaboration with G. Dell’Antonio and R.Figari) is a fully time-dependent version of the originalanalysis performed by Mott in 1929.

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84

IV.5. Asymptotic and multiscale analysis

Half space problem for the damped wave equation witha non-convex convection term

Yoshihiro UedaGraduate School of Sciences, Tohoku University 6-3Aramaki-Aza-Aoba, Aoba-ku Sendai, Miyagi 980-8578Japanueda@math.tohoku.ac.jp

We consider the initial-boundary value problem fordamped wave equations with a nonlinear convectionterm in the half space. In the case where the flux isconvex, it had already known that the solution tends tothe corresponding stationary wave. In this talk, we showthat even for a quite wide class of flux functions whichare not necessarily convex, such the stationary wave isasymptotically stable. The proof is given by a technicalweighted energy method.

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On the time-decay of solutions to a family of defocusingNLS

Nicola ViscigliaDipartimento di Matematica, Universita di Pisa, Via F.Buonarroti 2, Pisa 56127 Italyviscigli@dm.unipi.it

Let u(t, x) be any solution to the defocusing NLS withpure power nonlinearity u|u|α, where 0 < α < 4

n−2, and

with initial condition u(0, x) ∈ H1(Rn). Then the Lp

norm of u(t, x) goes to zero as t → ∞ provided that2 < p < 2n

n−2. In particular we extend previous result

due to Ginibre and Velo who have shown the propertyabove under the extra assumption 4

n< α < 4

n−2.

———

The semilinear Klein-Gordon equation in de Sitterspacetime

Karen YagdjianDepartment of Mathematics, University of Texas-PanAmerican, 1201 W. University Drive, Edinburg, TX78541-2999, USAyagdjian@utpa.edu

In this talk we present the blow-up phenomena forthe solutions of the semilinear Klein-Gordon equationgφ − m2φ = −|φ|p with the small mass m ≤ n/2 inde Sitter spacetime with the metric g. We prove that forevery p > 1 large energy solutions blow up, while for thesmall energy solutions we give a borderline p = p(m,n)for the global in time existence. The consideration isbased on the representation formulas for the solutionof the Cauchy problem and on some generalizations ofKato’s lemma.

———

IV.5. Asymptotic and multiscale analysis

Organisers:Ilia Kamotski, Valery Smyshlyaev

BICS Mini-SymposiumThe minisymposium will focus on fundamental analyt-ical issues associated with differential equations (linear

and nonlinear, partial or ordinary) with a small param-eter and/or multiple scales, and relevant applications.This includes singularly perturbed problems, problemsin thin domain or with singular boundaries, homoge-nization. The applications may include propagation andlocalization of waves, blow-up phenomena, metamateri-als, etc. The relevant analytic issues are convergenceand relevant functional spaces, compactness and propa-gation of oscillations, asymptotic expansions with errorbounds, etc.

—Abstracts—

On the essential spectrum and singularities of solutionsfor Lame problem in cuspoidal domain

Natalia BabychDepartment of Mathematical Sciences, University ofBath, Bath, BA2 7AY United Kingdomn.babych@bath.ac.uk

Within a Lame problem of linear elasticity, we investi-gate singularities of solutions in the vicinity of an out-ward cusp at the boundary. In case of a sharp cusp(the Holder constant is less or equal 1

2), we describe the

essential spectrum that consists of a certain real ray ac-cessing +∞. We analyse all possible local singularitiesof solutions and construct radiation conditions definingsuitable spaces that guarantee a Fredholm type solv-ability for the problem. We demonstrate that the sharpoutward cusp at the boundary is somewhat similar toinfinity for unbounded domains.This is joint work with Dr. I. Kamotski.

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Torsion effects in elastic composites with high contrast

Michel BellieudUniversite de Perpignan, 52 avenue Paul Alduy, Perpig-nan, 66860 Francebellieud@univ-perp.fr

In the context of linearized elasticity, we analyze asε → 0 a vibration problem for a two-phase mediumwhereby an ε-periodic set of ”stiff” elastic fibers of elas-tic moduli of the order 1 is embedded in a ”soft” elasticmatrix of elastic moduli of the order ε2. We show thattorsional vibrations take place at an infinitesimal scale.

———

Enhanced resolution in structured media

Yves CapdeboscqOxPDE Centre for Nonlinear Partial Differential Equa-tions, University of Oxford, Mathematical Institute, Ox-ford, OX1 3LB United Kingdomyves.capdeboscq@gmail.com

In this talk, we show that it is possible to achieve a reso-lution enhancement in detecting a target inclusion if it issurrounded by an appropriate structured medium. Thiswork is motivated by the advances in physics concerningthe so-called super resolution, or resolution beyond thediffraction limit. We first revisit the notion of resolu-tion and focal spot, and then show that in a structuredmedium, the resolution is conditioned by effective pa-rameters.This is a joint work with Habib Ammari & Eric Bon-netier

85

IV.5. Asymptotic and multiscale analysis

———

Homogenization of elliptic partial differential equationswith unbounded coefficients in dimension two

Juan Casado-DiazDpto. de Ecuaciones Diferenciales y Analisis Numerico,Facultad de Matematicas, C. Tarfia s/n Sevilla, Sevilla41012 Spainjcasadod@us.es

This is work in collaboration with Marc Briane, wherewe study the asymptotic behaviour of a given sequence ofdiffusion energies in L2(Ω) for a bounded open subset Ωof R2. The corresponding diffusion matrices are assumedto be coercive but any upper bound is considered. Weprove that, contrary to the three dimension (or greater),the Γ-limit of any convergent subsequence of Fn is stilla diffusion energy. We also provide an explicit represen-tation formula of the Γ-limit when its domains containsthe regular functions with compact support in Ω. Theseresults are based on the uniform convergence satisfied bysome minimizers of the equicoercive sequence Fn, whichis specific to the dimension two.

———

Two-scale Γ-convergence and its applications to ho-mogenisation of non-linear high-contrast problems

Mikhail CherdantsevSchool of Mathematics, Cardiff University, SenghennyddRoad, Cardiff, CF24 4AG United Kingdomcherdantsevm@cardiff.ac.uk

It is a resent results of Bouchitte, Felbacq, Zhikov andothers that passing to the limit in high-contrast ellipticPDEs may lead to non-classical effects, which are due tothe two-scale nature of the limit problem. These haveso far been studied in the linear setting, or under the as-sumption of convexity of the stored energy function. Itseems of practical interest however to investigate the ef-fect of high-contrast in the general non-linear case, suchas of finite elasticity.With this aim in mind, we develop a new tool tostudy non-linear high-contrast problems, which may bethought of as a “hybrid” of the classical Γ-convergence(De Giorgi, Dal Maso, Braides) and two-scale conver-gence (Allaire, Briane, Zhikov). We demonstrate theneed for such a tool by showing that in the high-contrastcase the minimising sequences may be non-compact inLp space and the corresponding minima may not con-verge to the minimum of the usual Γ-limit. We prove acompactness principle for high-contrast functionals withrespect to the two-scale Γ-convergence, which in partic-ular implies convergence of their minima. We brieflydiscuss possible applications of this new technique inthe mechanics of composites. (This is a joint work withK.D. Cherednichenko.)

———

Construction of the two-parametric generalizations ofthe Knizhnik-Zamolodchikov equations of Bn type

Valentina Alekseevna GolubevaSteklov Mathematical Institute, Gubkina 8, Moscow119991 Russiagoloubeva@yahoo.com

The Knizhnik-Zamolodchikov equation associated withthe root system Bn is investigated. This root systemhas two orbits with respect Weyl group. By this reasonKZ equation naturally contains two parameters. Sin-gular locus of this equation consists from hyperplanesxi − xj = 0, xi + xj = 0, xk = 0, i, j, k = 1, 2, . . . , n,x = (x1, . . . , xn) ∈ Cn. The following inverse problem ofRiemann-Hilbert type is considered: given a representa-tion of a fundamental group of complement to the singu-lar locus in Cn to the orthogonal group of odd order. Todefine the coefficients of the two-parametric differentialKZ equation as elements of tensor power of universal en-velopping algebra for odd orthogonal Lie algebra. One-parametric case was investigated by A.Leibman. Forcoefficients were used Casimir elements of second order.In two-parametric case the coefficients are defined byusing the families of Casimir elements of higher orderdescribed by A. Molev. For construction of these ele-ments are used Capelli operators which permit to de-scribe the centre of corresponding universal envelopingalgebra. The invariants used in explicit form of coeffi-cients for the case o(5) are expressed by means of Pfaf-fian for matrix defined using the root system.

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Long-time behavior for the Wigner equation and semi-classical limits in heterogeneous media

Fabricio MaciaUniversidad Politecnica de Madrid, DEBIN ETSINavales, Avda. Arco de la Victoria, Madrid 28040 Spainfabricio.macia@upm.es

We study the semiclassical limit for a class of linearSchrodinger equations in an heterogeneous medium (forinstance, a Riemannian manifold) at time scales tendingto infinity as the characteristic frequencies of the initialdata tend to zero. We are interested, in particular, indealing with time scales larger than the Eherenfest time,for which the high frequency behavior is completelycharacterized by classical mechanics via Egorov’s the-orem. Our analysis is performed by studying the high-frequency behavior of Wigner functions corresponding tosolutions to the Schrodinger equation at very long times.We give a complete characterization of their structurefor systems arising as the quantization of a completelyintegrable classical Hamiltonian flow. In particular, weprove that in such systems the asymptotic behavior ofWigner functions for times larger than Ehrenfest’s mightno longer be determined by the classical flow. This is dueto effects caused by resonances, that have to be studiedvia a new object, the resonant Wigner distribution.

———

On nonlinear dispersive equations in periodic structures:Semiclassical limits and numerical schemes

Peter MarkowichDAMTP, University of Cambridge, Wilberforce Road,Cambridge, CB3 0WA United KingdomP.A.Markowich@damtp.cam.ac.uk

We discuss (nonlinear) dispersive equations, such asthe Schrdinger equation, the Gross-Pitaevskii equationmod- eling Bose-Einstein condensation, the Maxwell-Dirac system and semilinear wave equations. Semiclas-sical limits are analysed using WKB and Wigner tech-

86

V.1. Inverse problems

niques, in particular for periodic structures, and connec-tions to classical homogenisation problems for Hamilton-Jacobi equations and hyperbolic conservation laws areestablished. We present a new numerical technique forsuch PDE problems, based on Bloch decomposition, andshow applications in semiconductor modelling, Bose-Einstein condensation and Anderson localisation for ran-dom wave equations.

———

Derivation of Boltzmann-type equations from hard-sphere dynamics

Karsten MatthiesDepartment of Mathematical Sciences, University ofBath, Bath, BA2 7AY United Kingdomk.matthies@bath.ac.uk

The derivation of the continuum models from determin-istic atomistic descriptions is a longstanding and fun-damental challenge. In particular the emergence of irre-versible macroscopic evolution from reversible determin-istic microscopic evolution is still not fully understood.We study a classic system: N balls that interact witheach other via a hard-core potential and show rigorouslythat in the case of kinetic annihilation (particles annihi-late each other upon collision) the asymptotic behavioras N tends to infinity is correctly described by the Boltz-mann equation without gain-term for non-concentratedinitial distributions. The mean-field description fails,when there are concentrations in the space or the veloc-ity coordinates.This is joint work with Florian Theil.

———

Minimizing atomic configurations of short range pairpotentials in two dimensions: crystallization in theWulff shape

Bernd SchmidtZentrum Mathematik, TU Muenchen, Boltzmannstr. 3,Garching b. Muenchen, 85747 Germanyschmidt@ma.tum.de

We investigate ground state configurations of atomic sys-tems in two dimensions as the number of atoms tendsto infinity for suitable pair interaction models. Suit-ably rescaled, these configurations are shown to crystal-lize on a triangular lattice and to converge to a macro-scopic Wulff shape which is obtained from an anisotropicsurface energy induced by the microscopic atomic lat-tice. Moreover, sharp estimates on the microscopic fluc-tuations about the limiting Wullf shape are obtained.(Joined work with Y. Au Yeung and G. Friesecke.)

———

Homogenization with partial degeneracies: analytic as-pects and applications

Valery SmyshlyaevDepartment of Mathematical Sciences, University ofBath, Claverton Down Bath, BA2 7AY United King-domvps@maths.bath.ac.uk

We consider homogenization problems for a generic classof (scalar or vector) operators with ”partial” degeneracy

in the (tensorial) coefficients. The employed tools arethose of ”non-classical” (high contrast type) homogeni-sation. This leads to interesting effect physically, for ex-ample allowing ”directional localisation”, with no wavepropagation in certain directions, and mathematicallyallows treating form a unified perspective ”classical”,high-contrast homogenizations and intermediate cases.We discuss some related analytic issues, including theneed to develop appropriate versions of two-scale conver-gence and of the theory of compensated compactness.

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V.1. Inverse problems

Organisers:Yaroslav Kurylev, Masahiro Yamamoto

Inverse problems is a multidisciplinary subject having itsfirm origin in application of mathematics to such prob-lems as search for oil, gas and other mineral resources,medical imaging, process monitoring in micro-biological,chemical and other industries, non-destructive testing ofmaterials, to mention just few. Its mathematical un-derpinning stretches from discrete mathematics, to ge-ometry, to computational methods with, however, theprincipal background being in analysis. In particular,the use of analytic methods makes it possible to addresssuch issues of IP as their strongly non-linear nature andsevere ill-posdnesss.In recent years, these relations have made it possible tosolve a number of long-standing inverse problems, in-cluding those with data on a part of the boundary, withsignificantly reduced requirements on regularity and thenumber of measurements, etc. These were based on theadvancing and employing such topics in analysis as Car-leman estimates for PDE’s, harmonic and quasiconfor-mal analysis, global and geometric analysis, microlocalcalculus and stochastic/probabilistic methods. In thissection we intend to represent those progress by invitingthe leading people in the area to give relevant talks.

—Abstracts—

An inverse conductivity problem with a single measure-ment

Abdellatif El BadiaLMAC, University of Compiegne, Compiegne, Oise60200 Franceabdellatif.elbadia@utc.fr

We revisit in this paper the inverse boundary valueproblem of Calderon for a coated domain, where theconductivity is constant in each subdomain. This ge-ometric distribution of conductivity corresponds to thewell accepted model of heads in ElectroEncephaloGra-phy (EEG). For instance, the inmost interior domainis occupied by the brain, and it is surrounded by theskull and the scalp. The so-called spherical model, wherethese regions are concentric spherical layers, is also fre-quently used. We show for this distribution of conduc-tivity that the inverse problem is completely solved withonly one suitably chosen Cauchy data, instead of thewhole Dirichlet-to-Neumann operator. The criterion ofchoice for these Cauchy data is completely set up in

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V.1. Inverse problems

the spherical model, using spherical harmonics. Also,a stability result is established. As for the numericalmethod to compute the conductivity, we propose a leastsquare procedure with a Kohn-Vogelius functional, anda boundary integral method for the direct problem.This is joint work with T. Ha-Duong.

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Global in time existence and uniqueness results for someintegrodifferential identification problems

Fabrizio ColomboDipartimento di Matematica, Politecnico di Milano, viaBonardi 9 Milano, Mi 20133 Italyfabrizio.colombo@polimi.it

We show some results on the identification of memorykernels in some nonlinear equations such as the heatequation with memory, the strongly damped wave equa-tion with memory, the beam equation with memory anda peculiar model in the theory of combustion.An additional restriction on the state variable is givento determine both the state variable and the memorykernels.We prove global in time uniqueness results and for suit-able nonlinearities we prove existence and uniquenessresults for the solution of the identification problems as-sociated to the models mentioned above.

———

Stability estimate for an inverse problem for themagnetic Schrodinger equation from the Dirichlet-to-Neumann map

Mourad ChoulliDepartment of Mathematics, Metz University, Ile duSaulcy Metz, Lorraine 57000 Francechoulli@univ-metz.fr

In this talk we consider the problem of stability esti-mate of the inverse problem of determining the mag-netic field entering the magnetic Schrodinger equationin a bounded smooth domain of Rn with input Dirich-let data, from measured Neumann boundary observa-tions. This information is enclosed in the dynamicalDirichlet-to-Neumann map associated to the solutionsof the magnetic Schrodinger equation. We prove in di-mension n ≥ 2 that the knowledge of the Dirichlet-to-Neumann map for the magnetic Schrodinger equa-tion measured on the boundary determines uniquely themagnetic field and we prove a Holder-type stability indetermining the magnetic field induced by the magneticpotential.

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Optimal combination of data modes in inverse prob-lems: maximum compatibility estimate

Mikko KaasalainenDepartment of Mathematics and Statistics, PO Box 68,Helsinki, FI-00014 Finlandmikko.kaasalainen@helsinki.fi

We present an optimal strategy for weighting variousdata modes in inverse problems. The solution, maxi-mum compatibility estimate, corresponds to the maxi-mum likelihood estimate of the single-mode case (with,

e.g., regularization functions included). We illustratethe method by showing that one can reconstruct a bodywith sparse data of the boundary curves (profiles) andvolumes (brightnesses) of its generalized projections.

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On an inverse problem for a linear heat conductionproblem

Christian DaveauCNRS (UMR 8088) and Department of Mathematics,University of Cergy-Pontoise, 2 avenue Adolphe Chau-vin, 95302 Cergy-Pontoise Cedex, France.Christian.Daveau@u-cergy.fr

In this talk, a boundary integral method is used tosolve an inverse linear heat conduction problem in two-dimensional bounded domain. An inverse problem ofmeasuring the heat flux from partial (on a part of theboundary) dynamic boundary measurements is consid-ered.This talk presents joint work with A. Khelifi.

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Inverse problems for wave equation and a modified timereversal method

Matti LassasDepartment of Mathematics and Statistics, P.O. Box68 (Gustaf Hallstromin katu 2b), Helsinki, Universityof Helsinki 00014 FinlandMatti.Lassas@helsinki.fi

A novel method to solve inverse problems for the waveequation is introduced. Suppose that we can send wavesfrom the boundary into an unknown body with spatiallyvarying wave speed c(x). Using a combination of theboundary control method and an iterative time reversalscheme, we show how to focus waves near a point x0

inside the medium and simultaneously recover c(x0) ifthe wave speed is isotropic. In the anisotropic case wecan reconstruct the wave speed up to a change of coordi-nates. These results are obtained in collaboration withKenrick Bingham, Yaroslav Kurylev, and Samuli Silta-nen. Also, we will disucss how the energy of a wave canbe focused near a single point in an unknown medium.These results are done in collaboration with Matias Dahland Anna Kirpichnikova.

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Picard condition based regularization techniques in in-verse obstacle scattering

Koung Hee LeemDept. of Mathematics & Statistics, Southern IllinoisUniversity, Edwardsville, IL 62026 United Stateskleem@siue.edu

The problem of determining the shape of an obstaclefrom far-field measurements is considered. It is wellknown that linear sampling methods have been widelyused for shape reconstructions obtained via the singularsystem of an ill conditioned discretized far-field opera-tor. For our reconstructions we assume that the far fielddata are noisy and we present two novel regularizationmethods that are based on the Picard Condition and donot require a priori knowledge of the noise level. Both

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V.1. Inverse problems

approaches yield results comparable to the ones obtainedvia the L-curve method and the discrepancy principle.

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Limited data problems in tensor tomography

William LionheartSchool of Mathematic, University of Manchester, OxfordRd, Manchester, M13 9PL United Kingdombill.lionheart@manchester.ac.uk

n photoelastic tomography one seeks to recover a tracefree symmetric second rank tensor from its truncatedtransverse ray transform. We present constructiveuniqueness results in the case where realistic subsets ofdata are known and numerical reconstruction methods.This is joint work with V Sharafutdinov and D Szotten.

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The finite data non-selfadjoint inverse resonance prob-lem

Marco MarlettaCardiff School of Mathematics Senghennydd RoadCardiff, Wales CF24 4AG United KingdomMarlettaM@cardiff.ac.uk

We consider Schrodinger operators on [0,∞) with com-pactly supported, possibly complex-valued potentials inL1[0,∞). It is known (at least in the case of a real-valued potential) that the location of eigenvalues andresonances determines the potential uniquely. From thephysical point of view one expects that large resonancesare increasingly insignicant for the reconstruction of thepotential from the data. We prove the validity of thisstatement, i.e., we show conditional stability for nitedata. As a by-product we also obtain a uniqueness resultfor the inverse resonance problem for complex-valued po-tentials.This is joint work with S. Naboko, S. Shterenberg andR, Weikard.

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Numerical solutions of nonlinear simultaneous equa-tions

Tsutomu MatsuuraGraduate School of Engineering, Gunma University 1-5-1 Tenjintyo Kiryu, Gunma 376-8515 Japanmatsuura@gunma-u.ac.jp

In this paper we shall give practical and numerical rep-resentations of inverse mappings of 2-dimensional map-pings (of the solutions of 2-nonlinear simultaneous equa-tions) and show their numerical experiments by usingcomputers. We derive a concrete formula from a verygeneral idea for the representation of the inverse func-tion

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A fixed-point algorithm for determining the regulariza-tion parameter in inverse scattering

George PelekanosDept. of Mathematics & Statistics, Southern IllinoisUniversity, Edwardsville, IL 62026 United Statesgpeleka@siue.edu

The factorization method is a fast inversion techniquefor visualizing the profile of a scatterer from measure-ments of the far-field pattern. The mathematical basisof this method is given by the far-field equation, whichis a Fredholm integral equation of the first kind in whichthe data function is a known analytic function and theintegral kernel is the measured (and therefore noisy) farfield pattern. We present a Tikhonov parameter choiceapproach based on a fast fixed-point method developedby Bazan. The method determines a Tikhonov parame-ter associated with a point near the corner of the L-curvein log-log scale and it works well even for cases wherethe L-curve exhibits more than one convex corner. Theperformance of the method is evaluated by comparingour reconstructions with those obtained via the L-curvemethod.

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A time domain probe method for inverse scatteringproblems

Roland PotthastDepartment of Mathematics, University of Reading,Whiteknights, PO Box 220, Berkshire, RG6 6AX, UK

r.w.e.potthast@reading.ac.uk

The goal of the talk is to discuss the development ofprobe methods for inverse scattering problems in thetime-domain. We will study wave scattering by three-dimensional rough surface problems. Both the math-ematics of these problems as well as the algorithmicalsolution of direct and inverse problems and the numer-ical analysis of algorithms provide a sincere challengessince the methods developed for bounded objects can-not be directly translated into the setting of unboundedscatterers. We survey recent results on the direct andinverse problems by Burkard, Chandler-Wilde, Heine-meyer, Lindner and the speaker. With the multi-sectionmethod we present a numerical scheme for which con-vergence both for direct and inverse scattering (usinga multi-section Kirsch-Kress approach) can be shown.The time-domain probe method is then formulated anddiscussed. Convergence for the reconstruction of sur-faces can be shown and numerical examples are pre-sented.

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Explicit and direct representations of the solutions ofnonlinear simultaneous equations

Saburou SaitohDepartment of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugalsaburou.saitoh@gmail.com

We shall present our recent results with Dr. MasatoYamada on practical, numerical and explicit represen-tations of inverse mappings of n-dimensional mappings(of the solutions of n-nonlinear simultaneous equations)and show their numerical experiments by using comput-ers. We derive those concrete formulas from very generalideas for the representation of the inverse functions.

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V.2. Stochastic analysis

Direct and inverse mixed impedance problems in linearelasticity

Vassilios SevroglouUniversity of Piraeus, Department of Statistics and In-surance Science, 80 Karaoli & Dimitriou Str., Piraeus,Athens 18534, Greecebsevro@unipi.gr

Direct and inverse scattering problems with mixedboundary conditions in linear elasticity are considered.We formulate the direct scattering problem for a par-tially coated obstacle as well as the mathematical settingfor the inverse one. Uniqueness theorems are presentedand an inversion algorithm for the determination of thescattering obstacle is established. In particular, a lin-ear integral equation due to the linear sampling methodwhich arises from an application of the reciprocity gapfunctional and the fundamental solution, connected withthe appoximate solution of the inverse problem, is inves-tigated. Finally, a discussion about the validity of ourmethod for mixed boundary value problems in elasticscattering theory is presented.

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On inverse scattering for nonsymmetric operators

Igor TrooshinInstitute of Problems of Precise Mechanics and Control,Russian Academy of Sciences, Rabochaya 24, Saratov,410028 Russiatrooshin@jcom.home.ne.jp

We consider a nonsymmetric operator AP inL2(0,∞)2. defined by differential expression

(APu)(x) = Bu′(x) + P (x)u(x), 0 < x <∞where

B =

„0 11 0

«, P (x) =

„p11(x) p12(x)p21(x) p22(x)

«,

with the domain

D = u(x) =

„u1(x)u2(x)

«∈ H1(R+)2; u1(0) = hu2(0).

An inverse problem of reconstruction of complex-valuedcoefficients pij(x) from the scattering data of operatorAP is investigated..

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V.2. Stochastic analysis

Organisers:Dan Crisan, Terence Lyons

Stochastic analysis aims to provide mathematical toolsto describe and model high-dimensional random systemsthat arise in the study of stochastic differential equationsand stochastic partial differential equations, infinite di-mensional stochastic geometry, random media and inter-acting particle systems, super-processes, stochastic fil-tering, mathematical finance, etc. It has emerged asa core area of late 20th century mathematics and iscurrently undergoing quite rapid scientific development.The section will provide a forum for researchers workingon the different aspects of stochastic analysis to presenttheir findings, and to interact with people working inthe wider area of analysis.

—Abstracts—

Cylindrical Levy processes in Banach space

David ApplebaumDept of Probability and Statistics, Hicks BuildingHounsfield Road, University of Sheffield Sheffield, York-shire S3 7RH United Kingdomd.applebaum@sheffield.ac.uk

Cylindrical probability measures are finitely additivemeasures on Banach spaces that have sigma-additiveprojections to Euclidean spaces of all dimensions. Theyare naturally associated to notions of weak random vari-able and hence weak processes which may be good can-didates to be the driving noise in stochastic evolutionequations. In this talk I’ll focus on cylindrical Levy pro-cesses. These have (weak) Levy-Ito decompositions andan associated Levy-Khintchine formula. If the processis weakly square integrable, its covariance operator canbe used to construct a reproducing kernel Hilbert spacein which the process has a decomposition as an infiniteseries built from a sequence of uncorrelated bona fideone-dimensional Levy processes. This series is used todefine cylindrical stochastic integrals from which cylin-drical Ornstein-Uhlenbeck processes can be constructed.(Based on joint work with Markus Reidel, Manchester)

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Integration by parts for locally smooth laws and appli-cations to jump type diffusions

Vlad BallyUniversite Marne-le-Vallee, Boulevard Descartes CiteDescartes - Champs-sur-Marne Marne-la-Vallee, 77454Francebally@univ-mlv.fr

Our main result concerns the regularity of the law ofsolutions of jump type stochastic equations with discon-tinuous coefficients. Since the coefficients are discontin-uous the solution is not in the domain of the Malliavindifferential operators and so the usual Malliavin calculuson the Poisson space does not work.The main tool in our approach is an integration by partsformula for finte dimensional random variables whichhave a locally smooth law. This is an abstract versionof the Malliavin calculus for simple functionals. Thenwe approximate a general functional (the solution of theequation in our case) by a sequence of simple functionalsand we use the integration by parts formula for them.But the weights which appear in the integration by partsformula blow up. Nevertheless, using a balance argu-ment we are able to estimate the Fourier transform ofthe functional and to conclude that the law has a smoothdensity with respect to the Lebesgue measure.

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Information and asset pricing

Dorje BrodyDepartment of Mathematics, Imperial College London,180 Queen’s Gate, London SW7 2AZ, UKd.brody@imperial.ac.uk

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V.2. Stochastic analysis

A framework for asset price dynamics is introduced inwhich the concept of noisy information about future cashflows is used to derive the corresponding price processes.In this framework an asset is defined by its cash-flowstructure. With each cash flow we associate a mar-ket information process, the values of which we assumeare accessible to market participants. Each informationprocess consists of a sum of two terms; one containstrue information (the signal) about the value of the as-sociated market factor, and the other represents noise.The market filtration is assumed to be that generatedby the aggregate of the independent information pro-cesses. The price of an asset is given by the expectationof the discounted cash flows in the risk neutral measure,conditional on the information provided by the marketfiltration. The work is done in corroboration with L.P.Hughston, A. Macrina, as well as others including M.Davis and R. Friedman.

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A (rough) pathwise approach to fully non-linearstochastic partial differential equations

Michael CaruanaDepartment of Pure Mathematics & MathematicalStatistics, University of Cambridge, Wilberforce RoadCambridge, Cambridgeshire CB3 0WB United Kingdomm.caruana@statslab.cam.ac.uk

In a series of papers, P. L. Lions and P. Souganidis pro-posed a pathwise theory for fully non-linear stochasticpartial differential equations. We present a (partial) ex-tension towards certain spatial dependence in the noiseterm. The main novelty is the use of rough path the-ory in this context. This joint work with P. Friz and H.Oberhauser.

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Solving backward stochastic differential equations usingcubature methods

Dan CrisanDepartment of Mathematics, Imperial College London,180 Queen’s Gate, London SW7 2AZ, United Kingdomd.crisan@imperial.ac.uk

In the last decade, a new class of numerical methodsfor approximating weak solutions of SDEs have beenintroduced by Kusuoka, Lyons, Ninomiya and Victoir.These methods are based on the work of Kusuoka andStroock who established refined gradient upper boundsfor the associated semigroup using Malliavin Calculustechiniques. In this talk, I will present an application ofthese methods to the numerical solution ofBackward SDEs and some applications to option pricing.The talk is based on joint work with K. Manolarakis.

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Some results on Lagrangian Navier-Stokes flows

Ana Bela CruzeiroDep. Mathematics, IST and GFMUL, Av. Prof. GamaPinto Lisboa, 2 1649-003 Portugalabcruz@math.ist.utl.pt

We consider stochastic Lagrangian trajectories associ-ated to the Navier-Stokes equation and some of its prop-erties, namely stability.

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A uniqueness problem for SDEs and a related estimatefor transition functions

Alexander DavieSchool of Mathematics, University of Edinburgh, King’sBuildings, Mayfield Road, Edinburgh, EH9 3JZ UnitedKingdoma.davie@ed.ac.uk

Existence and uniqueness theorems for (vector) stochas-tic differential equations dx = a(t, x)dt + b(t, x)dW areusually formulated at the level of stochastic processes.This talk will consider instead a uniqueness question foran individual driving Brownian path W , when the equa-tion is interpreted using rough path theory, and in thiscontext a uniqueness theorem can be proved (for a.e. W )if b has suitable regularity and a is bounded Borel. Theproof depends on an estimate for the transition functionof an associated diffusion process.

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Risk-sensitive portfolio optimization with jump-diffusion asset prices

Mark H. A. DavisDepartment of Mathematics, Imperial College, LondonSW7 2AZmark.davis@imperial.ac.uk

This paper considers a portfolio optimization problemin which asset prices are represented by SDEs driven byBrownian motion and a Poisson random measure, withdrifts that are functions of an auxiliary diffusion ‘fac-tor’ process. The criterion, following earlier work byBielecki, Pliska, Nagai and others, is risk-sensitive opti-mization (equivalent to maximizing the expected growthrate subject to a constraint on variance.) By using achange of measure technique introduced by Kuroda andNagai we show that the problem reduces to solving acertain stochastic control problem in the factor process,which has no jumps. The main result of the paper is thatthe Hamilton-Jacobi-Bellman equation for this problemhas a classical (C1,2) solution. The proof uses Bellman’s‘policy improvement’ method together with results onlinear parabolic PDEs due to Layzhenskaya et al. Thisis joint work with Sebastien Lleo.

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Accelerated numerical schemes for nonlinear filtering

Istvan GyongySchool of Mathematics, Edinburgh University, MayfieldRoad, Edinburgh, EH9 3JZ United KingdomI.Gyongy@ed.ac.uk

Accelerated numerical schemes for deterministic andstochastic PDEs are discussed. In particular, acceler-ated finite difference schemes for stochastic PDEs arepresented and sufficient conditions are given under whichthe convergence of finite difference approximations canbe accelerated to any given order of convergence byRichardson’s method.The results are applied to numerical solutions of nonlin-ear filtering problems. The talk is based on joint workwith N.V. Krylov.

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V.2. Stochastic analysis

Periodic homogenisation with an interface

Martin HairerCourant Institute, 251 Mercer Street, New York, NY10012 United Statesmhairer@cims.nyu.edu

It is well-known that, under a diffusive space-time rescal-ing, a typical diffusion process with periodic coefficientsconverges in law to a Brownian motion with a certaineffective diffusion tensor. The twist on this old prob-lem considered in this talk is that we allow the presenceof an interface that partitions the space into two half-spaces. We then consider a diffusion whose coefficientsare periodic in each half-space (with possibly differentperiodic structures), with a smooth transition of orderone around the interface.In this talk, we will show that while the long-time large-scale limit of such a process converges to the expectedhomogenised Brownian motion on either side of the in-terface, additional local time terms can appear on the in-terface. We will provide a complete description of theseadditional terms and outline the main ideas appearingin the proofs.

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Wiener chaos models for interest rates and foreign ex-change

Lane HughstonDepartment of Mathematics, Imperial College London,London, SW7 2BZ United Kingdomlane.hughston@imperial.ac.uk

We consider the general problem of modeling thearbitrage-free dynamics of the nominal interest-rateterm structure in the case when the discount-bond sys-tem is driven by Brownian motion. We show that underrather general assumptions the pricing kernel for such asystem is given by the conditional variance of a randomvariable that admits a Wiener chaos expansion. Theresulting interest rate models can be classified in a hi-erarchical fashion according to the degree of the chaosexpansion. A number of specific models are constructedaccording to this scheme, with extensions to foreign ex-change and other asset classes.

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Minimising the time to a decision

Saul JackaDepartment of Statistics, Zeeman Building Universityof Warwick, Gibbet Hill Road Coventry, West MidlandsCV4 7AL United KingdomS.D.Jacka@warwick.ac.uk

We consider a stochastic control problem motivated bya conjecture of Peres. The problem is to choose (at eachtime point) which of three absorbing BMs to run, insuch a way as to minimise the time until at least twohave been absorbed at either zero or one.

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Markov process representations for polyharmonic func-tions

Mark KelbertDepartment of Mathematics, Swansea University, Sin-gleton Park Swansea, Wales SA2 8PP United Kingdomm.kelbert@swansea.ac.uk

We present the higher-order Feynman-Kac formula forsolution of equation (∆ + V )mu = 0 in a domainD. Probabilistic representation implies some a prioribounds on the growth of solutions when the domain Dextends to Rd. The estimations of moments of randomtime to reach a high level for Bessel processes are used toestablish a weak dependence of polyharmonic functionsof some boundary values.

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Networks and Poisson line patterns

Wilfried KendallDepartment of Statistics, University of Warwick, Coven-try, West Midlands CV4 7AL United Kingdomw.s.kendall@warwick.ac.uk

How best to join up n nodes in a planar network? Thecomplete planar graph provides short connections at theexpense of large network length. The Steiner tree min-imizes total network length at the expense of generat-ing potentially long connections. It turns out that aconstruction based on Poisson line processes does a re-markably good job of solving the frustrated optimizationproblem based on the competing criteria of reducing to-tal network length versus providing connections whichare short in an average sense. An associated randomgraph presents intriguing questions of stochastic anal-ysis motivated by this application: what is the typicalbehaviour of a geodesic? and what can one say abouttraffic flow in the graph? Answers involve considerationof exponential functionals of Brownian motion and otherLevy processes, and a curious limiting object based ona highly improper Poisson line process.(Joint work with David Aldous.)

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The Levy-Khinchine type operators with variable Lips-chitz continuous coefficients and stochastic differentialequations driven by nonlinear Levy noise

Vassili KolokoltsovDepartment Statistics, University of Warwick, Coventry,West Midlands CV4 7AL United Kingdomv.kolokoltsov@warwick.ac.uk

Theory of stochastic integrals and SDEs driven by dis-tribution dependent nonlinear Levy noise is developedyielding an effective construction of linear and nonlin-ear Markov semigroups and the corresponding processeswith a given pseudo-differential (pre)generator. It isshown that a conditionally positive integro-differentialoperator (of the Levy-Khintchine type) with variablecoefficients (diffusion, drift and Levy measure) depend-ing Lipschitz continuously on its parameters (positionand/or its distribution) generates a linear or nonlinearMarkov semigroup, where the measures are metricizedby the Wasserstein-Kantorovich metrics. This is a non-trivial but natural extension to general Markov processesof a long known fact for ordinary diffusions.

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Equivalence of stochastic equations and martingaleproblems

Thomas KurtzDepartment of Mathematics/UW-Madison, 480 Lincoln

92

V.2. Stochastic analysis

Drive, Madison, WI 53706 United Stateskurtz@math.wisc.edu

The fact that the solution of a martingale problem for adiffusion process gives a weak solution of the correspond-ing Ito equation is well-known since the original work ofStroock and Varadhan. The result is typically provedby constructing the driving Brownian motion from thesolution of the martingale problem and perhaps an aux-iliary Brownian motion. This constructive approach ismuch more challenging for more general Markov pro-cesses where one would be required to construct a Pois-son random measure from the sample paths of the solu-tion of the martingale problem. A soft approach to thisequivalence will be given which begins with a joint mar-tingale problem for the solution of the desired stochasticequation and the driving processes and applies a Markovmapping theorem to show that any solution of the origi-nal martingale problem corresponds to a solution of thejoint martingale problem.

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Aida’s logarithmic Sobolev inequality with weights andPoincare inequalities.

Xue-Mei LiDepartment of Mathematics, University of Warwick,Coventry, CV4 7AL United Kingdomxuemei@xuemei.org

Although a Logarithmic Sobolev inequality holds for theBrownian bridge measure on the Wiener space and forthe Brownian motion measure on the path space over acompact manifolds, it may not hold on a general loopspace. As noted big L. Gross Poincare inequalities donot hold on the Lie groups. A. Eberle gave an exam-ple of a compact simply connected Riemannian mani-fold on which the Poincare inequality does not hold forthe Brownian bridge measure. For the Brownian bridgemeasure a positive result was obtained by Aida for theHyperbolic space H where he obtained a weak form Log-arithmic Sobolev inequality with a weight function. Weshow that Aidas type weak logarithmic Sobolev inequal-ity leads to a weak logarithmic sobolev inequality usingthe non-homogeneous H1 norm together with an L∞

norm. We also show that there is a precise passage fromweak Logarith-mic Sobolev inequality to weak Poincareinequality. As a corollary we obtain a Poincare inequal-ity for the Brownian bridge measure on loop spaces overthe hyperbolic space where the Bismut tangent space isdened using the Levi-Civita connection. This is jointwork with Chenand Wu.

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Evolution equations for communities

Terence LyonsMathematical Institute, 24-29 St Giles, Oxford, Oxford-shire OX1 3LB United KingdomTerry.Lyons@maths.ox.ac.uk

There are many situations where one would like to modelthe evolution of a large community, where each memberhas their own preferences. The members of the commu-nity evolves in ways which depend on the behaviour ofthe ensemble of other community members as well astheir own preferences.

The theory of rough paths allows one to build rigorousmathematical models for the evolution of these commu-nities, even in the contiuum context.This is joint work with Thomas Cass.

———

On the martingale property of certain local martingale

Aleksandar MijatovicDepartment of Mathematics, Imperial College London,180 Queen’s Gate London, England SW7 United King-doma.mijatovic@imperila.ac.uk

The stochastic exponential

Zt = expMt −M0 − (1/2)〈M,M〉tof a continuous local martingale M is itself a continuouslocal martingale. We present a necessary and sufficientcondition for the process Z to be a true martingale andto be a uniformly integrable martingale in the case whereMt =

R t0b(Yu) dWu and Y is a one-dimensional diffusion

driven by a Brownian motion W . These conditions aredeterministic and expressed only in terms of the func-tion b and the drift and diffusion coefficients of Y . Asan application of the result we describe a deterministicnecessary and sufficient condition for the existence offinancial bubbles in the diffusion based models.

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On some stochastic dynamical systems and cancer

Khairia El-Said El-NadiDepartment of Mathematics, Faculty of Science, Alexan-dria University, Alexandria Egyptkhairia el said@hotmail.com

Different models of tumor growth are considered. Somemathematical methods are developed to analyze the dy-namics of mutations enabling cells in cancer patients tometastize.The mathematical models consist of some stochastic dy-namical systems describing tumor cells and immune ef-fectors. It is also considered a method to contrast theideal outcomes of some treatments. The results of theconsidered model predict continuous under which somesuitable treatment can be successful in returning an ag-gressive tumor to its passive, non-immune evading state.The principle goal of this paper is to find ways to treat-ment the cancer tumor before they can reach an ad-vanced stage development.

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Statistical inference for rough differential equations

Anastasia PapavasiliouUniversity of Warwick, Coventry, CV4 7AJ, UK.a.papavasiliou@warwick.ac.uk

Our goal is to estimate unknown parameters in the poly-nomial vector field of a differential equation driven byrough paths. We assume that we know the expected sig-nature of the drivers and we observe several independentcopies of the signature of the response. By approximat-ing the theoretical expected signature of the response bythat of the Picard iterations and its empirical signatureby Monte-Carlo, we end up with a polynomial systemwhose solution is the “expected signature matching es-timator”. We prove its consistency and asymptotic nor-mality.

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V.2. Stochastic analysis

At the second part of the talk, we will discuss in moredetail the computational challenges this approach poses.More precisely, the degree of the polynomials grow as qr,where q is the degree of the polynomial vector field andr the Picard iteration. This makes the computation ofthe polynomial computationally very demanding. Wewill suggest ways for improving the efficiency by utiliz-ing a different definition for the product. Finally, we willdiscuss applications to multiscale modelling.

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First passage for stochastic volatility models

Martijn PistoriusSouth Kensington Campus, Imperial College, Depart-ment of Mathematics, London SW7 2AZ, UKmartijn.pistorius@imperial.ac.uk

Barrier options are financial contracts that are acti-vated or de-activated when the underlying price pro-cess crosses a specific level; they are among the mostwidely traded of exotic contracts. The pay-offs of bar-rier options are path dependent and their valuation re-quires the specification of the first-hitting-time distribu-tion. In this talk, we present a new approach to obtainfirst passage probabilities for stochastic volatility mod-els (i.e. diffusions whose coefficients are functions ofa one-dimensional diffusion). We illustrate the resultsby calculating the values and Greeks of barrier options,and compare the outcomes with Monte Carlo simula-tion results. The talk is based on joint work with MarcJeannin.

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Unbiased random perturbations of Navier-Stokes equa-tion

Boris RozovskyDivision of Applied Mathematics, 182 George Street,Providence, Rhode Island 02912 United Statesrozovsky@dam.brown.edu

A random perturbation of a deterministic Navier-Stokesequation is considered in the form of an Stochastic PDEwith Wick product in the nonlinear term. The equationis solved in the space of generalized stochastic processesusing the Cameron-Martin version of the Wiener chaosexpansion. The generalized solution is obtained as an in-verse of solutions to corresponding quantized equations.An interesting feature of this type of perturbation is thatit preserves the mean dynamics: the expectation of thesolution of the perturbed equation solves the underlyingdeterministic Navier-Stokes equation. From the standpoint of a statistician it means that the perturbed modelis unbiased.The talk is based on a joint work with R. Mikulevicius.

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A Poisson equation with fractional noise

Marta Sanz-SoleFacultat de Matematiques, Universitat de Barcelona,Gran Via de les Corts Catalanes 585, Barcelona 08028,Spainmarta.sanz@ub.edu

We consider a stochastic elliptic SPDE on a bounded do-main driven by a fractional Brownian field with Hurstparameter H = (H1, . . . , Hk) ∈ [ 1

2, 1[k.

Firstly we give a meaning to the stochastic convolu-tion derived from the Green kernel. Using monotonicitymethods, we prove existence and uniqueness of solution,along with regularity of the sample paths. Finally, for agiven lattice scheme, we prove convergence to the solu-tion of the SPDE.

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Constructing discrete exact approximations algorithmsfor financial calculus from weak convergence results

Radu TunaruCass Business School, Faculty of Finance, 106 BunhillRow, London EC1Y 8TZ United KingdomR.Tunaru@city.ac.uk

In financial calculus the calculation of moments in gen-eral and the expectation in particular is extremely im-portant. This can be difficult in a multi-dimensionalset-up and Monte Carlo methods are not always satis-factory. In this paper we show how to construct exactdiscrete approximations schemes that can be used for awide range of financial mathematics problems. The al-gorithms we present are derived from well-known weakconvergence results. Theoretical results are adapted towork for unbounded payoffs. In addition we show howto circumvent the problem caused by the singularity ofthe covariance matrix that appears with the CLT forthe multinomial distribution. The approximation gridsdeveloped here are shown to be dense in the set of realnumbers, for the one-dimensional case. The results areproved for the geometric Brownian set-up but it ca beeasily adapted to other frameworks.

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Numerical methods for parabolic SPDEs based on theaveraging-over-characteristics formula

Michael TretyakovDepartment of Mathematics, University of Leicester, Le-icester LE1 7RH United KingdomM.Tretyakov@le.ac.uk

The method of characteristics (the averaging over thecharacteristic formula) and the weak-sense numerical in-tegration of ordinary stochastic differential equations areused to propose numerical methods for stochastic par-tial differential equations (SPDEs). Their orders of con-vergence in the mean-square sense and in the sense ofalmost sure convergence are obtained. The developedapproach is supported by numerical experiments. Thetalk is based on a joint work with G.N. Milstein (Eka-terinburg, Russia).

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Consistent estimator in AFTM

Elena UsoltsevaVasilkovskaya str. 94, room 823 Kiev, 03022 Ukraineelena usolceva@ukr.net

We consider a sample of n nonnegative i.i.d. randomvariables Ti, i = 1, n from Accelerated Failure TimeModel. This model (AFTM) is described in general casein the following way

log Ti = β0 + βTX + ψεi, ψ > 0,

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V.3. Coercivity and functional inequalities

where εi are i.i.d. with zero mean. A measurement er-ror is present in our model therefore instead of Xi weobserve the surrogate data Wi = Xi+Ui, where Ui forma centered i.i.d. sequence with finite second moments.Some lifetimes Ti may be censored, in that only a lowerbound for the lifetime is recorded. The distribution of Tidepends on unknown parameter ν, which is estimated.In general AFTM, the adjusted Quasi-Likelihood equa-tion leads to estimating equation:

nXi=1

„1Xi

«“Ti − eβ0+βTXXi

”= 0.

Under the present of measurement error, the adjustedunbiased Corrected Score estimating function for messylifetime was constructed. In this report, based on thetheory of estimating equation, it is proved that this esti-mating function yields a consistent estimator under non-singular correlation matrix of regressor and some condi-tions on censoring distribution.

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V.3. Coercivity and functional inequalities

Organisers:Dominique Bakry, Boguslav Zegarlinski

The session intend to present an overview of a recentprogress in coercive and functional isoperimetric inqual-ities and their applications to study long time behaviourof (sub-)elliptic problems, variety of probabilistic prob-lems, analysis on groups as well as other related areas.

—Abstracts—

Remarks on non-interacting conservative spin systems

Franck BartheInstitut de Mathematiques de Toulouse, Universite PaulSabatier, Toulouse, cedex 9 31062 Francebarthe@math.univ-toulouse.fr

We compute precise estimates for the spectral gap andlog-Sobolev constants in the special particular case ofgamma distributions. This turns out to be related tothe Kannan-Lovasz-Simonovits conjecture for simplicesand Lp balls (joint work with Pawel Wolff).

———

On weak forms of Poincare-type inequalities

Sergey Bobkov127 Vincent Hall, 206 Church St.S.E. Minneapolis, Min-nesota 55455 United Statesbobkov@math.umn.edu

We will discuss weak forms of Poincare-type inequali-ties for probability measures on Euclidean and abstractmetric spaces.

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L∞-Error estimate for variational inequalities with van-ishing zero order term

Messaoud BoulbracheneSultan Qaboos University, Department of Mathematics

and Statistics, Al Khod, Muscat 123 Omanboulbrac@squ.edu.om

In this paper, a new method for the finite element ap-proximation of variational inequalities (VI) with vanish-ing zero order term is introduced and analyzed. Errorestimate in the maximum norm is derived and a basiciterative scheme for the computation of the discrete so-lution is also provided.

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Convexity along vector fields and application to equa-tions of Monge-Ampere type

Federica DragoniDepartment of Mathematics, Imperial College London,180 Queen’s Gate, London SW7 2AZ, UKf.dragoni@imperial.ac.uk

We introduce and study a new notion of convexity andsemiconvexity along vector fields. We give in particularcharacterizations in terms of inequalities (in the viscositysense) for the matrix of second derivatives with respectto the fields. Our notion of convexity is equivalent tothe horiontal convexity in Carnot groups but holds for afar more general class of vector fields. As an application,we get comparison principles for subelliptic equations ofMonge-Ampere type, extending a recent result of Bardiand Mannucci for Carnot groups to more general vec-tor fields. The key point is that convex functions alongvector fields satisfy a-priori gradient bounds similar tothose satisfied by euclidean convex funtions.This is joint work with Martino Bardi.

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Φ-entropy inequalities for diffusion semigroups

Ivan GentilCEREMADE, Universite Paris-Dauphine, Pl. du Mal

De L. De Tassigny. 75016 PARIS, FRANCE.gentil@ceremade.dauphine.fr

We obtain and study new Φ-entropy inequalities for dif-fusion semigroups, with Poincare or logarithmic Sobolevinequalities as particular cases. From this study we de-rive the asymptotic behaviour of a large class of linearFokker-Plank type equations under simple conditions,widely extending previous results. The Γ2 criterion ofD. Bakry and M. Emery [Diffusions hypercontractives.Seminaire de probabilites, XIX, 1983/84, Lecture Notesin Math., 1123, 177–206, (1985)] appears as a main toolin the analysis, in local or integral forms.The presentation is based on [Bolley, F.; Gentil, I. Phi-entropy inequalities for diffusion semigroups Preprint,(2008)].

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On positive solutions of semi-linear elliptic inequalitieson manifolds

Alexander GrigoryanDepartment of Mathematics, University of Bielefeld,Bielefeld, D-33501 Germanygrigor@math.uni-bielefeld.de

We consider elliptic inequalities of the type ∆u+uσ ≤ 0on geodesically complete Riemannian manifolds and pro-vide sharp sufficient conditions in terms of capacities andvolumes for the non-existence of positive solutions. Jointwork with V.A.Kondratiev.

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V.3. Coercivity and functional inequalities

———

Hypoellipticity in infinite dimensions

Martin HairerCourant Institute, 251 Mercer Street, New York, NY10012 United Statesm.hairer@warwick.ac.uk

One of Hormander’s legacies is to provide a con-structive and essentially sharp criterion for checkingwhether a second-order differential operator is hypoel-liptic: Hormander’s bracket condition. This result hasfound numerous applications, one of them being thestudy of regularising properties for Markov semigroupsarising from diffusion processes.It is natural to try to extend these results to an infinite-dimensional setting, for example in the context of thestudy of Markov semigroups arising from stochastic par-tial differential equations (SPDEs). We will present atheory that allows to obtain such results in the settingof semilinear parabolic SPDEs with multilinear nonlin-earities. Applications of these results to linear responsetheory and ergodic theory will also be discussed.

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Logaritmic Sobolev inequality on nilpotent groups

Waldemar HebischMathematical Institute, University of Wroc law,Pl. Grunwaldzki 2/4, 50-384 Wroc law, Polandhebisch@math.uni.wroc.pl

We discuss recent results about logaritmic Sobolev (LS)inequality and related coercive inequalities on nilpo-tent groups. Our main result is LS inequality for mea-sures of form Z−1 exp(−βd2)dλ, where d is Carnot-Caratheodory distance on H-type group. We will alsomention related results for heat kernel measure and moregeneral groups.

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Isoperimetry for spherically symmetric log-concaveprobability measures

Nolwen HuetInstitut de Mathematiques de Toulouse, Universite Paul-Sabatier (Toulouse III), 31062 Toulouse Cedex 9, Francenolwen.huet@math.univ-toulouse.fr

We prove an isoperimetric inequality for probabil-ity measures µ on Rn with density proportional toexp(−φ(λ|x|)), where |x| is the Euclidean norm on Rnand φ is a non-decreasing convex function. It appliesin particular when φ(x) = xα with α ≥ 1. Under mildassumptions on φ, the inequality is dimension-free if λis chosen such that the covariance of µ is the identity.

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Operators on the Heisenberg group with discrete spec-tra

James InglisDepartment of Mathematics, Imperial College London,180 Queen’s Gate, London SW7 2AZ, UKjames.inglis06@imperial.ac.uk

We show that a certain class of hypoelliptic operatorson the Heisenberg group have discrete spectra, using a

spectral representation of the Heisenberg Laplacian. Wethen extend the result to other operators related to theheat kernel using functional inequalities.

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Liggett inequality and interacting particle systems

Mikhail NeklyudovDepartment of Mathematics, University of York, YorkYO10 5DD, UKmn505@york.ac.uk

In this talk we discuss one class of interacting parti-cle systems which correspond to degenerate generator.In this case Poincare inequality doesn’t hold even in aweak sense. Instead it is possible to show that Liggettinequality holds in certain sense.

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A new criterion for a covariance estimate

Felix OttoInstitute for Applied Mathematics, University of Bonn,Endenicher Allee 60 Bonn, 53115 Germanyotto@iam.uni-bonn.de

We present a simple criterion for a covariance estimatefor a spin system with continuous spin space which isformulated in terms of the single-site spectral gap andthe interaction between the sites. It is optimal in caseof Gaussians. A typical application is the exponentialdecay of correlations in case of weak interactions.This is joint work with Georg Menz.

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The Log-Sobolev inequality for non quadratic interac-tions

Ioannis PapageorgiouDepartment of Mathematics, Imperial College London,180 Queen’s Gate, London SW7 2AZ, UKioannis.papageorgiou05@imperial.ac.uk

We are interested on the Logarithmic Sobolev q inequal-ity for unbounded spin systems on the Lattice with in-teractions that are non quadradic. We present criteri-ons that allow to extend the inequality from the onedimesional measure to the infinite dimensional Gibbsmeasure, for variables on the real line as well as on theHeisenberg group.

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Isoperimetry for product probability measures

Cyril RobertoUniversit de Marne la Valle, LAMA 5bv DescartesChamps sur Marne, Marne la Valle 77454 Francecyril.roberto@univ-mlv.fr

In this talk we shall give a short overview on the isoperi-metric problem for product probability measures.An isoperimetric inequality is a lower bound on theboundary measure of sets in terms of their measure.Finding the optimal sets (of given measure and of min-imal boundary measure) is very difficult, and the onlyhope is to estimate the isoperimetric function. This iswell understood on the line (Bobkov) and for the prod-uct of standard Gaussian measures (Sudakov-Tsirel’son,Borell). We shall start by recalling those known results.

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V.4. Dynamical systems

Then, we shall explain how functional inequalities canbe used to get dimension free isoperimetric inequalitiesfor measures between exponential and Gaussian. Also,using the transport of mass technique we shall derviveisoperimetric inequalities (depending on the dimension)for measures with tails larger than exponential.

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V.4. Dynamical systems

Organisers:Jeroen Lamb, Stefan Luzzatto

Dynamical Systems aims to provide the mathematicaltools to describe and model deterministic systems thatarise in the study of ordinary Differential Equations andin the iteration of maps on smooth manifolds. A varietyof ideas and techniques from Analysis and other areascome together to provide existence and classification re-sults regarding the dynamical properties of systems fromgeometrical, topological, probabilistic points of view.This section will provide a forum for high level re-searchers working mainly in Bifurcation Theory and Er-godic Theory to present their recent research and to dis-cuss open problems and technical issues.

—Abstracts—

Poincare-Bendixson theorems in holomorphic dynamics

Marco AbateDipartimento di Matematica, Universita di Pisa, LargoPontecorvo 5, Pisa 56127 Italyabate@dm.unipi.it

I shall present a recent Poincare-Bendixson theorem de-scribing recurrence properties for geodesics of a mero-morphic connections on the complex projective line.Then I shall explain how to use this theorem to studythe dynamics of homogeneous vector fields in C2, andwhy this approach provides tools for studying the localdynamics of holomorphic maps tangent to the identity.(Joint work with F. Tovena.)

———

On the liftability of absolutely continuous ergodic ex-panding measures.

Jose Ferreira AlvesDepartamento de Matemtica Pura, Rua do Campo Ale-gre 687, 4169-007 Porto, Portugaljfalves@fc.up.pt

We consider maps on a compact manifold of arbitrarydimension possibly admitting critical points, discontinu-ities or singularities. Under some mild nondegeneracyassumptions we show that the map admits an inducedGibbs-Markov map with integrable inducing times if andonly if it has an ergodic invariant probability measurewhich is absolutely continuous with respect to the Rie-mannian volume and has all Lyapunov exponents posi-tive.

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New results on stability and genericity

Flavio AbdenurRua Marques de Sao Vicente 225, Departamento deMatematica, PUC-Rio Rio de Janeiro, Rio de JaneiroCEP 22453-900 Brazilfabdenur@yahoo.com

I will briefly explain some very recent results on stabil-ity and genericity of different types of discrete dynamicalsystems.In short:

1. there exist non-hyperbolic diffeomorphisms whichare ’weakly’ structurally stable in a very naturalsense (joint w/ L. J. Diaz and E. Pujals)

2. generic partially hyperbolic transitive diffeomor-phisms are robustly transitive/robustly mixing(joint w/ S. Crovisier)

3. generic continuous maps have highly weird ergodicproperties (joint w/ M. Andersson)

I will discuss some or all of these three topics, as time al-lows and interest arises. I might also discuss some otherrelated results, as whim intervenes.

———

Abundance of one dimensional non uniformly hyperbolicattractors for surface dynamics

Pierre BergerMathematisches Forschungsinstitut Oberwolfach,Schwarzwaldstr. 9-11 (Lorenzenhof), Oberwolfach-Walke, 77709 Germanypierre.berger@normalesup.org

We present a (new) proof of the existence of a non uni-formly hyperbolic attractor for a positive set of param-eters a in the family of endomorphisms:

(x, y) 7→ (x2 + a+ y, 0) +B(x, y),

where B is any fixed C2 small function. For B = 0, thisis the Jackoson theorem. For B = b · (0, x), we get theBenedicts-Carleson theorem for the Hnon map.The proof is done thanks to analytical and probabilis-tic tools of (B-C) in the geometric and combinatorialformalism of Yoccoz puzzles generalized in a very al-gebraic way (pseudo-semi-group). These theorems arenotably generalized to the C2-case and to the endomor-phisms. The theorem is an answer to question of Pesin-Yurchenko reaction-diffusion EDPs in applied mathe-matics.

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First integrals in mechanics of infinite-dimensional sys-tems

Svetlana Aleksandrovna BudochkinaMiklukho-Maklaya Str. 6, Moscow, 117198 Russiasbudotchkina@yandex.ru

Equation of motion represented in the operator form isconsidered. Formulas for finding some first integrals ofthe equation of motion are given.

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Partial hyperbolicity and ergodicity

Keith BurnsMathematics Department, Northwestern University,

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V.4. Dynamical systems

Evanston, IL 60201 United Statesburns@math.northwestern.edu

I will survey recent results which extend Hopf’s methodfor proving ergodicity to a large class of partially hyper-bolic diffeomorphisms.

———

On tilings, multidimensional subshifts of finite type andquasicrystals

Jean-Rene ChazottesCPHT, Ecole Polytechnique Palaiseau, Cedex 91128Francejeanrene@cpht.polytechnique.fr

Subshifts of finite type (SFT) in dimension one (Z-actions) are well known objects used in the symbolicdynamics of hyperbolic dynamical systems. In dimen-sion greater than one (Zd-action, d ≥ 2), they are veryrich and complicated objects for which few results areknown. We will see how SFTs arise naturally as sup-ports of ground states in lattice models in StatisticalPhysics and how they are connected to toy-models ofquasicrystals. We will also see how tiling dynamical sys-tems can be used to derive results on multidimensionalSFTs.

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On topological entropy of billiard tables with small innerscatterers

Yi-Chiuan ChenInstitute of Mathematics, Academia Sinica, 128Academia Road, Section 2 Nankang, Taipei 11529 Tai-wanYCChen@math.sinica.edu.tw

An approach to studying the topological entropy of aclass of billiard systems will be presented. In this class,any billiard table consists of strictly convex domain inthe plane and strictly convex inner scatterers. Usingthe concept of anti-integrable limit, we show that a bil-liard system in this class generically admits a set ofnon-degenerate anti-integrable orbits which correspondsbijectively to a topological Markov chain of arbitrarilylarge topological entropy. Consequently, we prove thetopological entropy of the first return map to the scat-terers can be made arbitrarily large provided the innerscatterers are sufficiently small.

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On the nature of chaos

Bau-Sen DuInstitute of Mathematics, Academia Sinica 128, Sec. 2,Ian-Jiou-Yuan Rd., NanKang, Taipei 11529 Taiwandubs@math.sinica.edu.tw

Based on a very special property of the shift map(Theorem 1), we believe that chaos should involve notonly nearby points can diverge apart but also far-away points can get close to each other. Therefore,we propose to call a continuous map f from an in-finite compact metric space (X, d) to itself chaotic ifthere exists a positive number δ such that for anypoint x and any nonempty open set V (not necessar-ily an open neighborhood of x) in X there is a point

y in V such that lim supn→∞ d(fn(x), fn(y)) ≥ δ andlim infn→∞ d(fn(x), fn(y)) = 0.

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Mixing for flows and skew extensions

Michael FieldDepartment of Mathematics, University of Houston,4800 Calhoun Houston, Texas TX 77204-3008 UnitedStatesmikefield@gmail.com

Problems and results on rates of mixing and exponentialestimates.

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Rates of mixing, large deviations and recurrence times

Jorge FreitasDep Matematica Pura, Fac Ciencias, Univ Porto, Ruado Campo Alegre, 687 Porto, 4169-007 Portugaljmfreita@fc.up.pt

It is very well known that one can derive rates for the de-cay of correlations of stationary stochastic processes aris-ing from dynamical systems admitting a Young tower.These rates depend on the volume decay of the tail setof the inducing times.In this work we exploit the connection between decayof correlations of certain classes of observables and largedeviations estimates of stochastic processes generated bythe system. We also show the relation between the largedeviations of the potential corresponding to the loga-rithm of the derivative and the volume decay of the tailset of hyperbolic times (the set of points that resist topresent hyperbolic behavior in short time range).Based on these considerations we obtain a converse ofL. S. Young’s result, namely, if we have a system witha certain rate of decay of correlations then the systemadmits a Young tower with the same type of rate forthe volume of the tail of inducing times. Moreover, wecan show how to obtain an estimate for the large devia-tions of a whole class of observable functions, when weonly have an estimate for the large deviations for thelogarithm of the derivative.

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Limiting distributions for horocycle flows

Giovanni ForniDepartment of Mathematics, University of Maryland,College Park, MD 20742 United Statesgforni@math.umd.edu

We give results on the existence and non-existence oflimiting distributions for horocycle flows on the unit tan-gent bundle of hyperbolic (negative constant curvature)surfaces. This is joint work with A. Bufetov.

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Limit cycle problems and applications

Valery GaikoBelarusian State University of Informatics and Radio-electronics, L. Beda Str. 6-4 Minsk, 220040 Belarusvalery.gaiko@yahoo.com

We establish the global qualitative analysis of planarpolynomial dynamical systems and suggest a new geo-metric approach to solving Hilbert’s Sixteenth Problem

98

V.4. Dynamical systems

on the maximum number and relative position of theirlimit cycles in two special cases of such systems. First,using geometric properties of four field rotation param-eters of a new canonical system, we present the proofof our earlier conjecture that the maximum number oflimit cycles in a quadratic system is equal to four andtheir only possible distribution is (3 :1). Then, by meansof the same geometric approach, we solve the Problemfor Lienard’s polynomial system (in this special case, itis considered as Smale’s Thirteenth Problem). Besides,generalizing the obtained results, we present the solutionof Hilbert’s Sixteenth Problem on the maximum num-ber of limit cycles surrounding a singular point for anarbitrary polynomial system and, applying the Wintner–Perko termination principle for multiple limit cycles, wedevelop an alternative approach to solving the Prob-lem. By means of this approach we complete also theglobal qualitative analysis of a generalized Lienard cubicsystem, a neural network cubic system, a Lienard-typepiecewise linear system and a quartic dynamical systemwhich models the population dynamics in ecological sys-tems.

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Hausdorff dimension of Projections of McMullen-Bedford carpets

Thomas JordanDepartment of Mathematics, The University of Bristol,University Walk Clifton, Bristol BS8 1TW United King-domThomas.Jordan@bristol.ac.uk

Joint work with Andrew Ferguson and Pablo Shmerkin.Marstand’s Projection Theorem states that If E ⊂ R2

has Hausdorff dimension less than 1 then orthogonal pro-jections in almost all directions preserve this dimension.For general sets very little is known about exactly whichdirections preserve the dimension. We show that if E isa type of self-affine set investigated by Bedford and Mc-Mullen then orthogonal projections in all directions in(0, π/2) preserve the dimension. This is an extensionof a result on products of Cantors sets by Peres andShmerkin.

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Fourfold 1:1 resonance, relative equilibria and momentpolytopes

Jan Cees van der MeerDept. of Mathematics and Computer Science, Eind-hoven University of Technology, Den Dolech 2, Eind-hoven, N.B. 5612 AZ Netherlandsj.c.v.d.meer@tue.nl

A uniparametric 4-DOF family of perturbed Hamilto-nian oscillators in 1:1:1:1 resonance, with two additionalrotational symmetries, is studied. These systems gener-alize several models of perturbed Keplerian systems. Af-ter normalization the truncated normal form is reducedin stages to a one-degree-of-freedom system. In this re-duction process moment polytopes turn up describingpart of the relative equilibria for such systems.Joint work with S. Ferrer, G. Diaz, J. Egea, J.A. Vera.

———

A dynamical Borel-Cantelli lemma for a class of non-uniformly hyperbolic systems

Matthew NicolDepartment of Mathematics University of HoustonHouston, Texas 77204-3008 United Statesnicol@math.uh.edu

We establish a dynamical Borel-Cantelli lemma forshrinking balls for certain classes of non-uniformly hy-perbolic dynamical systems. As an application we es-tablish results on almost sure behavior of extremes forthese classes of dynamical systems.This work is joint with Chinmaya Gupta and WilliamOtt (both University of Surrey).

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Approximately inner C∗-dynamical systems

Asad NiknamDepartement of Mathematics, Ferdowsi University ofMashad, Vakilahbad Boulvar Mashad, Khorahsan 1159-91775, Irandassamankin@yahoo.co.uk

In quantum statistical mechanics one often describe aphysical system in terms of a C∗-algebra A. The dy-namics or time evoluton of the systems is given in termsof one parameter group of ∗-automophisims on A. Westudy such C∗-dynamical systems. We prove that un-der some restriction the dynamic is approximately in-ner. Moreover we construct a dynamical system which isnot approximatelly inner and therefore without groundstate.

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Dynamical systems arising in algebraic logic

Giovanni PantiDepartment of Mathematics, via delle Scienze, 208Udine, UD 33100 Italypanti@dimi.uniud.it

Algebraic logic studies the algebras associated to cer-tain logical systems. Standard examples are booleanalgebras, MV-algebras, Heyting algebras, associated toclassical logic, many-valued logic, intuitionistic logic, re-spectively. Typically, these algebras have dual spectralspaces, and can be represented as algebras of functionson the spectrum: automorphisms of the algebras corre-spond then to dynamical systems on the dual.We survey here the structure of the relevant dynamicalsystems, the results that have been obtained and theirsignificance, the open problems and directions for fur-ther research.

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Existence of transversal homoclinic orbits for Arneodo-Coullet-Tresser map

Chen-chang PengDepartment of Applied Mathematics, National ChiayiUniversity No.300 Syuefu Rd. Chiayi City, Taiwan60004 Taiwanccpeng@mail.ncyu.edu.tw

In this talk, first we study difference equations

xk+n = F (xk+n−1, · · · , xk, b1xk−1, · · · , bmxk−m)

99

V.5. Functional differential and difference equations

as C1-perturbation of the equation:

xk+n = f(xk+n−1, · · · , xk) ≡ F (xk+n−1, · · · , xk, 0, · · · , 0).

We prove that if f has a snapback repeller then F hasa transversal homoclinic orbit for all |bi| < ε for someε > 0. Second, we study a class of two-dimensional maps(or called Mira map) and prove that there exist snapbackrepellers for the map near its anti-integrable limits. Fi-nally, combining the above two results, we establish theexistence of transversal homoclinic orbits in family ofArneodo-Coullet-Tresser map near singularities.

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Bifurcations of random diffeomorphisms with boundednoise

Martin RasmussenDepartment of Mathematics, Imperial College, LondonSW7 2AZ, United Kingdomm.rasmussen@imperial.ac.uk

We discuss iterates of random diffeomorphisms withidentically distributed and bounded noise. In this con-text, minimal forward invariant sets play an importantrole, since they support stationary measures, and whenthe noise is interpreted as external control, minimal for-ward invariant sets coincide with invariant control sets.Discontinuous bifurcations of minimal forward invariantsets are analysed, and a numerical method to approx-imate these sets is presented. The results are appliedto study a bifurcation of the randomly perturbed Henonmap. This talk is based on joint work with Jeroen Lamb(Imperial College) and Christian Rodrigues (Universityof Aberdeen).

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Bifurcations of period annuli and solutions of nonlinearboundary value problems

Felix SadyrbaevInstitute of Mathematics and Computer Science, Rainisboul. 29 Riga, Latvia LV-1459 Latviafelix@latnet.lv

Differential equations of the type x′′ + λf(x) = 0 areconsidered, where f(x) are polynomials. First bifurca-tions of period annuli (continua of periodic solutions)are studied under the change of coefficients of f(x). Sec-ondly, bifurcations of solutions to the Dirichlet problemx(a) = 0, x(b) = 0 are investigated under the change ofλ.

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Large intersection properties of some invariant sets innumber-theoretic dynamical systems

Jorg SchmelingCenter of Mathematical Sciences, LTH, Box 118, Slveg-atan 18 Lund, 22100 Swedenjoerg@maths.lth.se

In this talk we consider sets of real numbers that have agiven approximation property by rationals with denom-inators gn. We prove that these sets have large inter-section properties and are winning in a modified (α, β)game or belong to Falconers s-class. This result will

be generalized to non-linear and also non-integer expan-sions of a real number. This talk is based on joined workwith T. Persson and D. Farm.

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Thermodynamic formalism for unimodal maps

Mike ToddDepartamento de Matematica Pura, Rua do Campo Ale-gre, 687 Porto, 4169-007 Portugalmtodd@fc.up.pt

Notions from thermodynamic formalism such as pres-sure, equilibrium states and large deviations can give arich qualitative description of a dynamical system. Re-cently there has been a lot of activity in the developmentof thermodynamic formalism applied to non-uniformlyhyperbolic dynamical systems. These systems have beenshown to exhibit a wide variety of phenomena, most in-terestingly critical phenomena such as phase transitions.In this talk I will give a fairly complete description ofthe possible behaviour of the class of unimodal intervalmaps, including the relation between phase transitionsand the existence of a natural measure for the system.

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Dynamics of periodically perturbed homoclinic solu-tions

Qiudong WangDepartment of Mathematics, University of Arizona,Tucson, Arizona 85721 United Statesdwang@math.arizona.edu

We study the dynamics of homoclinic tangles in periodi-cally perturbed second order equations. Let µ be the sizeof the perturbation and Λµ be the homoclinic tangles.We prove that (i) for infinitely many µ, Λµ contain noth-ing else but a horseshoe of infinitely many branches; (ii)for infinitely many µ, Λµ contain nothing else but onesink and one horseshoe of infinitely many branches; and(iii) there are positive measure set of µ so that Λµ admitsstrange attractors with Sinai-Ruelle-Bowen measure.

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V.5. Functional differential and differenceequations

Organisers:Leonid Berezansky, Josef Diblık, Agacık Zafer

Scope of the session: Qualitative theory of functionaldifferential and difference equations: stability, bound-edness, oscillation, asymptotic behaviour, positive solu-tions, dynamic equations on time scales, applications topopulation dynamics.

—Abstracts—

Oscillation and non-oscillation of solutions of linear sec-ond order discrete delayed equations

Jaromır BastinecDepartment of Mathematics, The Faculty of Electri-cal Engineering and Communication, Brno University

100

V.5. Functional differential and difference equations

of Technology, Technicka 8, 616 00 Brno, Czech Repub-licbastinec@feec.vutbr.cz

The phenomenon of the existence of a positive solution ofdifference equations is often encountered when analysingmathematical models describing various processes. Thisis a motivation for an intensive study of the conditionsfor the existence of positive solutions of difference equa-tions. Such analysis is related to an investigation of thecase of all solutions being oscillating. In the talk, con-ditions for the existence of a positive solution are givenfor a class of linear delayed discrete equations

∆x(n) = −p(n)x(n− 1)

where n ∈ Z∞a := a, a + 1, . . . , a ∈ N is fixed,∆x(n) = x(n + 1) − x(n), p : Z∞a → (0,∞). For thesame class of equations, also conditions are given for allthe solutions being oscillating. The results obtained in-dicate sharp sufficient conditions for the existence of apositive solution or for the case of all solutions being os-cillating. The investigation was supported by the grant201/07/0145 of the Czech Grant Agency (Prague) andby the Councils of Czech Government MSM 0021630529and by MSM 00216 30503.This is joint work with Josef Diblik.

———

New stability conditions for linear differential equationswith several delays

Leonid BerezanskyDepartment of Mathematics, Ben-Gurion University ofthe Negev, P.O. Box 653, Beer Sheva, Negev 84105 Is-raelbrznsky@math.bgu.ac.il

New explicit conditions of asymptotic and exponen-tial stability are obtained for the general scalar nonau-tonomous linear delay differential equation with measur-able delays and coefficients. These results are comparedto known stability tests.

———

Boundary-value problems for differential systems witha single delay

Aleksandr BoichukFaculty of Science, Zilina University, Zilina, 01 026 Slo-vakiaboichuk@imath.kiev.ua

Conditions are derived of the existence of solutions of lin-ear Fredholm’s boundary-value problems for systems ofordinary differential equations with constant coefficientsand a single delay. Utilizing a delayed matrix exponen-tial and a method of pseudo-inverse by Moore-Penrosematrices led to an explicit and analytical form of a cri-terion for the existence of solutions in a relevant spaceand, moreover, to the construction of a family of linearlyindependent solutions of such problems in a general casewith the number of boundary conditions (defined by alinear vector functional) not coinciding with the numberof unknowns of a differential system with a single delay.This work was supported by the grant 1/0771/08 of theGrant Agency of Slovak Republic (VEGA) and by the

project APVV-0700-07 of Slovak Research and Develop-ment Agency.This is joint work with J. Diblık, D. Khu-sainov, M. Ruzickova.

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Representation of solutions of linear differential and dis-crete systems and their controllability

Josef DiblıkBrno University of Technology, Brno, Czech Republic,Kiev State University, Kiev, Ukrainediblik.j@fce.vutbr.cz

We study discrete controlled systems

∆x(k) = Bx(k −m) + bu(k),

where m ≥ 1 is a fixed integer, k ∈ Z∞0 , Zqs := s, s +1, . . . , q, B is a constant n × n matrix, x : Z∞−m → Rn

is unknown solution, b ∈ Rn is given nonzero vector andu : Z∞0 → R is input scalar function. Moreover, we con-sider the system of delayed linear differential equationsof second order

y′′(t) + Ω2y(t− τ) = bu(t)

and an initial problem y(t) = ϕ(t), y′(t) = ϕ′(t),t ∈ [−τ, 0] where τ > 0 and ϕ : [−τ, 0]→ Rn is twice dif-ferentiable. Special matrix functions are defined: the de-layed matrix sine and the delayed matrix cosine. Thesematrix functions are applied to obtain explicit formulasfor the solution of the initial problem and a controlla-bility criterion. The investigation was supported by thegrant 201/08/0469 of the Czech Grant Agency (Prague),by the Councils of Czech Government MSM 0021630519and MSM 00216 30503 and by the project M/34-2008 ofUkrainian Ministry of Education.This is joint work with Denys Khusainov, BlankaMoravkova.

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Maximum principles and nonoscillation intervals in thetheory of functional differential equations

Alexander DomoshnitskyAriel University Center, Department of Mathematicsand Computer Science, Ariel, 44837 Israeladom@ariel.ac.il

Many classical topics in the theory of functional differ-ential equations, such as nonoscillation, differential in-equalities and stability, were historically studied withoutany connection between them. As a result, assertionsassociated with maximum principles for such equationsin contrast with the cases of ordinary and even partialdifferential equations do not add so much in problemsof existence and uniqueness of solutions to boundaryvalue problems and stability for functional differentialequations. One of the goals of this talk is to present aconcept of the maximum principles for functional differ-ential equations. New results on existence and unique-ness of solutions of boundary value problems are pro-posed. Assertions about positivity og Green’s functionsare formulated. Tests of the exponential stability areobtained on the basis of nonoscillation and positivity ofthe Cauchy function.

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V.5. Functional differential and difference equations

Averaging for impulsive functional differential equa-tions: a new approach

Marcia FedersonAv. Trabalhador Sao-carlense 400, CP 668, Sao Carlos,SP 13560-970 Brazilfederson@icmc.usp.br

We consider a large class of functional differential equa-tions subject to impulse effects and state an averagingresult by means of the techniques of the theory of gen-eralized ordinary differential equations introduced by J.Kurzweil.

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Some bifurcation problems in the theory quasilinear in-tegro differential equations

Yakov GoltserDepartment of Computer Sciences and Mathematics,Ariel University Center of Samaria, Ariel, 44837 IsserNatanzon, 27/7, Pisgat Zeev, 97877 Jerusalem, Israeljakovgoltser@014.net

Our goal is to study parametrical perturbed nonlin-ear quasiperiodic systems of differential and integro-differential equations.Study bifurcation problems sim-ilarly Hopf bifurcation, Bogdanov-Takkens bifurcationand bifurcation of invariant torus,based on the normalform theory and the truncated method for countable sys-tems of ordinary differential equations.

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Stability in Volterra type population model equationswith delays

Istvan GyoriEgyetem u. 10 Department of Mathematics, Universityof Pannonia Veszprem, Veszprem County H-8200 Hun-garygyori@almos.vein.hu

In this talk some delay dependent and delay indepen-dent stability conditions will be given for differentialequations arising in population dynamics. The proofsare based on the construction of a Lyapunov functionaland some monotone techniques for nonautonomous sys-tems. At the end of the talk we shall formulate someopen problems and conjectures.

———

On parameter dependence in functional differentialequations with state-dependent delays

Ferenc HartungUniversity of Pannonia Egyetem str 10 Veszprem, H-8200 Hungaryhartung.ferenc@uni-pannon.hu

In this talk we study smooth dependence on parametersof solutions of several classes of functional differentialequations with state-dependent delays. As an applica-tion of our results, we discuss the parameter estimationproblems for FDEs with state-dependent delays using aquasilinearization method.

———

Lyapunov type inequalities for nonlinear impulsive dif-ferential systems

Zeynep KayarMiddle East Technical University, Department of Math-ematics, Ankara, Cankaya 06531, Turkeyzkayar@metu.edu.tr

We obtain Lyapunov-type inequalities for systems ofnonlinear impulsive differential equations. In particular,these sytems contain the Emden-Fowler-type systemsand half linear systems in the special cases. In addi-tion, as an application we make use of these inequalitiesto derive some boundedness and disconjugacy criteriaand sufficient conditions for the asymptotic behaviourof solutions.

———

Evaluating the stochastic theta method

Conall KellyDepartment of Mathematics, University of the West In-dies, Mona Kingston, Sn.Andrew 7, Jamaicaconall.kelly@uwimona.edu.jm

When a numerical method is applied to a differentialequation, the result is a difference equation. Ideally thedynamics of the difference equation should reflect thoseof the original as closely as possible, but in general thiscan be difficult to check. It is therefore useful to performa linear stability analysis: applying the method of inter-est to a linear test equation possessed of an equilibriumsolution with known stability properties, and determin-ing the asymptotic properties of the resultant differenceequation for comparison.We examine the issues that arise for this kind of analy-sis in the context of stochastic differential equations, andreview the relevant literature. These issues have yet tobe adequately addressed. We propose a new approachand demonstrate its usage for the class of θ-Maruyamamethods with constant step-size.

———

Delay-distribution effect on stability

Gabor KissDepartment of Engineering Mathematics, University ofBristol Queen’s Building Bristol, South West EnglandBS8 1TR United Kingdomgabor.kiss@bristol.ac.uk

We consider the effect of delay distribution on retartedfunctional differential equations with one delay. Morespecifically, we study the effect of delay distribution onthe stability of solutions of first- and second-order equa-tions by comparing the stability regions of the respectiveequation with a single delay with that of the equationwith distributed delays.

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Solutions of linear impulsive differential systemsbounded on the entire real axis

Martina LangerovaDept. of Mathematics, Faculty of Science, University ofZilina, Univerzitna 1, 010 26 Zilina, Slovakiamartina.langerova@fpv.uniza.sk

We consider the problem of existence and structure ofsolutions bounded on the entire real axis of the linear

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V.5. Functional differential and difference equations

differential system with impulsive action at fixed pointsof time

x = A(t)x+ f(t), t 6= τi,

∆x˛t=τi

= ai, t, τi ∈ R, i ∈ Z, ai ∈ Rn.

Under the assumption that the corresponding homoge-neous system is exponentially dichotomous on the semi-axes R+ and R− and by using the results of the well-known Palmer lemma and the theory of pseudoinversematrices we establish necessary and sufficient condi-tions for the indicated problem. Co-authors: OleksandrBoichuk, Jaroslava Skorıkova.This research was supported by the Grants 1/0771/08,1/0090/09 of the Grant Agency of Slovak Republic(VEGA) and APVV 0700-07.

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Oscillatory and asymptotic properties of solutions ofhigher-order difference equations of neutral type

Malgorzata MigdaInstitute of Mathematics, Poznan University of Technol-ogy, ul. Piotrowo 3A, Poznan, 60-965 Polandmalgorzata.migda@put.poznan.pl

We consider higher-order linear difference equationswith delayed and advanced terms

∆m(xn − pxn−τ ) = qnxn−σ + hnxn+η

where p is a nonnegative number, τ, σ, η are positive in-tegers and (qn), (hn) are sequences of nonnegative realnumbers.We give sufficient conditions under which all nonoscil-latory solutions of the delayed part of the equation areunbounded and under which all nonoscillatory solutionsof the advanced part tend to zero as n→∞.We establish also sufficient conditions for the oscillationof all solutions of the full equation.

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Principal and non-principal solutions of impulsive differ-ential equations with applications

Abdullah OzbeklerAtılım University, Department of Mathematics,Kızılcasar Koyu, Incek Golbası, Ankara 06836 Turkeyaozbekler@gmail.com

In this work we first prove a theorem on the existenceof principal and nonprincipal solutions for second orderdifferential equations having fixed moments of impulseactions. Next, by means of nonprincipal solution we givenew oscillation criteria for related impulsive differentialequations. Examples are provided with numerical simu-lations to illustrate the importance of the study.

———

Nonnegative iterations with asymptotically constantcoefficients

Mihali PitukEgyetem u. 10 Department of Mathematics, Universityof Pannonia Veszprem, Veszprem County H-8200 Hun-garypitukm@almos.uni-pannon.hu

We shall discuss some results on the asymptotic be-haviour of the nonnegative solutions of systems of lineardifference equations with asymptotically constant coef-ficients. The main result describes the relationship be-tween the nonnegative solutions of the perturbed systemand the positive eigenvalues and the corresponding non-negative eigenvectors of the limiting system. The proofsare based on Pringsheim’s Theorem and the ExtendedLiouville Theorem from complex analysis.

———

On singular models arising in hydrodynamics

Irena RachunkovaPalacky University, Fakulty of Science, Dept. of Math-ematics, Tomkova 40, Olomouc, 77900 Czech Republicrachunko@inf.upol.cz

We investigate models arising in hydrodynamics. Thesemodels have the form of the singular second order dif-ferential equation

(p(t)u′(t))′ = p(t)f(u(t))

on the half-line. Here f is locally Lipsichtz on R andchanges its sign and p iscontinuous on [0,∞) and p(0) = 0. A discrete formula-tion of this equation is investigated as well. We are inter-ested in strictly increasing solutions and homoclinic solu-tions and provide conditions for p and f which guaranteethe existence of such solutions. In particular cases a ho-moclinic solution determines an increasing mass densityin centrally symmetric gas bubbles which are surroundedby an external liquid.

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Decoupling and simplifying of noninvertible differenceequations in the neighbourhood of invariant manifold

Andrejs ReinfeldsUniversity of Latvia, Institute of Mathematics and Com-puter Science; Raina bulvaris 29, LV-1459, Rıga, Latvia

reinf@latnet.lv

In Banach space X×E the system of difference equations

x(t+ 1) = g(x(t)) +G(x(t), p(t)),p(t+ 1) = A(x(t))p(t) + Φ(x(t), p(t))

(*)

is considered. Sufficient conditions under which there isan local Lipschitzian invariant manifold u : X → E areobtained. Using this result we find sufficient conditionsof partial decoupling and simplifying of the system ofnoninvertible difference equations (*).

———

Precise asymptotic behaviour of solutions of Volterraequations with delay

David W. ReynoldsSchool of Mathematical Sciences, Dublin City Univer-sity, Dublin 9, Ireland.david.reynolds@dcu.ie

This talk considers the rates at which solutions ofVolterra equations with delay converge to asymptoticequilibria. It is found that these convergence rates de-pend delicately on prescribed data. The results are es-tablished using admissibility techniques. This work ismotivated by logistic equations with infinite delay.

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V.5. Functional differential and difference equations

———

On local stability of solutions of stochastic differenceequations

Alexandra RodkinaDepartment of Mathematics, University of the West In-dies, Mona Kingston, Sn.Andrew 7, Jamaicaalechkajm@yahoo.com

We present results on the local stability of solutions ofa stochastic difference equation with polynomial coeffi-cients. Two cases are considered: when stochastic per-turbation are a state-independent and asymptoticallyfading and when stochastic perturbation are a state-dependent.

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Convergence of the solutions of a differential equationwith two delayed terms

Miroslava RuzickovaFaculty of science, University of Zilina, Slovak Republicmiroslava.ruzickova@fpv.uniza.sk

In this contribution we deal with asymptotic behavior ofsolutions to a linear homogeneous differential equationcontaining two discrete delays

y(t) = β(t)[y(t− δ)− y(t− τ)] (*)

for t→∞. We assume δ, τ ∈ R+ := (0,+∞), τ > δ, β :I−1 → R+ is a continuous function, I−1 := [t0 − τ,∞),t0 ∈ R. Denote I := [t0,∞) and the symbol “ ˙ ” de-notes (at least) the right-hand derivative. Similarly, ifnecessary, the value of a function at a point of I−1 isunderstood (at least) as value of the corresponding limitfrom the right.The main results concern the asymptotic convergence ofall solutions of Eq. (*). Especially we deal with so calledcritical case with respect to the function β. When thefunction β is the constant function than this critical caseis represented with the value β := (τ − σ)−1. The proofof results is, except other, based on comparison of solu-tions of Eq. (*) with solutions of an auxiliary inequalitywhich formally copies Eq. (*).This research was supported the Grant No 1/0090/09of the Grant Agency of Slovak Republic (VEGA), bythe project APVV-0700-07 of Slovak Research and De-velopment Agency and by the Slovak-Ukrainian projectSK-UA-0028-07 (Ukrainian-Slovak project M/34 MOHUkraine 27.03.2008).This is joint work with Josef Diblık.

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Inverse problems of the calculus of variations for func-tional differential equations

Vladimir Mikhailovich SavchinPeoples Friendship University of Russia, Mikluxo-Maklaya street 6, Moscow, 117198 Russiavsavchin@yandex.ru

The problem of existense of solutions of inverse prob-lems of the calculus of variations for partial differencialdifference operators is investigated. Necessary and suf-ficient conditions of potentiality for such operators areobtained. Methods of construction of variational multi-plies are suggested.

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Existence and nonexistence of asymptotically periodicsolutions of Volterra linear difference equations

Ewa SchmeidelInstytute of Mathematics, ul. Piotrowo 3A, Poznan,Wielkopolska 60-965 Polandewa.schmeidel@put.poznan.pl

In this talk we investigate Volterra difference equationof the form

x(n+ 1) = a(n) + b(n)x(n) +

nXi=0

K(n, i)x(i)

where n ∈ N = 0, 1, 2, . . . , a, b, x : N → R andK : N × N → R, the special case of this equation isVolterra difference equation of convolution type

x(n+ 1) = Ax(n) +

nXi=0

K(n− i)x(i).

This equation may be considered as a discrete analogueof famous Volterra integrodifferential equation

x′(t) = Ax(t) +

Z t

0

b(t− s)x(s)ds.

Such equation has been widely used as a mathematicalmodel in population dynamics. Both discrete equationsrepresents a system in which the future state x(n + 1)does not depend only on the present state x(n) but alsoon all past states x(n−1), x(n−2), . . . , x(0). These sys-tem are sometimes called hereditary. Given the initialcondition x(0) = x0, one can easy generate the solu-tion x(n, x0). Sufficient conditions for the existence ofasymptotically periodic solutions of Volterra differenceequation are presented. In addition we present sufficientconditions for non-existence of an asymptotically peri-odic solution satisfying some auxiliary conditions. Theresults are illustrated by examples.

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Gene regulatory networks and delay equations

Andrei ShindiapinEduardo Mondlane University, Maputo, Mozambiqueandrei.olga@tvcabo.co.mz

Gene regulatory networks consist of differential equa-tions with smooth but steep nonlinearities (”sigmoids”).As the number of genes may be rather large, any theoret-ical or computer-based analysis of such networks can becomplicated. That is why a simplified approach basedon replacing sigmoids with step functions is widely used.However, this leads to some mathematical challenges, asfor instance analysis of stationary points belonging tothe discontinuity set of the system (thresholds) cannotbe done directly. Additional problems occur if one triesto incorporate time delays into the network. The delayeffects naturally arise from the time required to com-plete transcription, translation and diffusion to the placeof action of a protein. We offer an algorithm of localiz-ing stationary points in the presence of delays as well asstability analysis around such points. This algorithm iscombined with a method to study delay systems by re-placing them with an equivalent system of ordinary dif-ferential equations, commonly known as the linear chaintrick. However, a direct application of this ”trick” is not

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V.6. Mathematical biology

possible in our case, so that we suggest a modificationof it based on the general framework of representing de-lay equations as ordinary differential equations using theintegral transforms”.This is joint work with Arcady Ponosov.

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The moment problem approach for the zero controlla-bility of ecolution equations

Benzion Shklyar52 Golomb St., P.O.B. 305, Dept. of Appl. Math, HolonInstitute of Technology, Holon, 58102 Israelshk b@hit.ac.il

The exact controllability to the origin for linear evolu-tion control equation is considered.The problem is inves-tigated by its transformation to infinite linear momentproblem.Controllability conditions for linear evolution controlequations have been obtained. The obtained results areapplied to the zero controllability for partial differentialand functional differential equations.

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Properties of maximal solutions of autonomousfunctional-differential equations with state-dependentdeviations

Svatoslav StanekPalacky University, Fakulty of Science, Dept. of Math-ematics, Tomkova 40, Olomouc, 77900 Czech Republicstanek@inf.upol.cz

Equations of the type x′′+x(t−kx)) = 0 are considered.Here k is a positive parameter. It is described (i) the setof all periodic solutions x satisfying x′ < 1/k and (ii) theset of all maximal solutions x (that is, solutions whichhave no extension) satisfying x′ ≥ 1/k.

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Boundedness character of some classes of differenceequations

Stevo StevicMathematical Institute of the Serbian Academy of Sci-ences, Knez Mihailova 36/III, Beograd, 11000 Serbiasstevic@ptt.rs

Some results on the boundedness character of the pos-itive solutions of the following two classes of differenceequations

xn+1 = A+xpn

xqn−1xrn−2

, n ∈ N0;

xn+1 = max

A,

xpnxqn−1x

rn−2

ff, n ∈ N0,

where the parameters A, p, q and r are positive numbers,are presented.

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Continuous dependence of solutions of generalized or-dinary differential equations on a parameter

Milan TvrdyInstitute of Mathematics, Academy of Sciences of theCzech Republic, Zitna 25, Praha 1, CZ 115 67 Czech

Republictvrdy@math.cas.cz

This contribution deals with systems of linear general-ized linear differential equations of the form

x(t) = ex+

Z t

a

d[A(s)]x(s)+g(t)−g(a), t ∈ [a, b], (*)

where −∞ < a < b < ∞, A is an n × n-complex ma-trix valued function, g is an n-complex vector valuedfunction, A has a bounded variation on [a, b] and g isregulated on [a, b]. The integrals are understood in theKurzweil-Stieltjes sense.Our aim is to present some new results on continuousdependence of solutions to linear generalized differentialequations (*) on parameters and initial data.

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Lyapunov type inequalities on time scales: A survey

Mehmet UnalBahcesehir University, Cıragan Caddesi, OsmanpasaMektebi Sokak No. 4–6, Besiktas, Istanbul 34353 Turkeymunal@bahcesehir.edu.tr

We survey Lyapunov type inequalities for linear andnonlinear dynamic equations on time scales. The in-equalities contain the well-known classical Lyapunov in-equalities as special cases. We also give some applica-tions to illustrate the importance of such inequalities.

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Interval criteria for oscillation of delay dynamic equa-tions with mixed nonlinearities

Agacık ZaferDepartment of Mathematics Middle East Technical Uni-versity Cankaya, Ankara 06531 Turkeyzafer@metu.edu.tr

We obtain interval oscillation criteria for second-orderforced delay dynamic equations with mixed nonlineari-ties on an arbitrary time scale T. All results are neweven for T = R and T = Z. Analogous results for re-lated advance type equations are also given, as well asextended delay and advance equations. The theory canbe applied to second order delay dynamic equations re-gardless of the choice of delta (∆) or nabla (∇) deriva-tives.

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V.6. Mathematical biology

Organisers:Robert Gilbert

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VI. Others

—Abstracts—

Cancellous bone with a random pore structure

Robert GilbertDepartment of Mathematics, University of Delaware,317 Ewing Hall, Newark, DE 19716 United Statesgilbert@math.udel.edu

We continue the study of acoustic wave propagation foran elastic medium that is randomly fissured. Moreover,the fissures are assumed to be statistically homogeneous.Although the underlying stochastic process does not nec-essarily have to be ergodic, we assume for simplicity ofexposition that it is. This allows us to obtain an ex-plicit and computationaly easier auxillary problem in aRepresentative Elementary Volume. In a later work weintend to study the more general case.This is joint work with Ana Vasilic.

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New computer technologies for the construction andnumerical analysis of mathematical models for molecu-lar genetic systems

Irina Alekseevna GainovaSobolev Institute of Mathematics, Siberian Branch ofthe Russian Academy of Sciences Acad. Koptyug av-enue, 4 Novosibirsk, Novosibirsk region 630090 Russiagajnova@math.nsc.ru

We have created an integrative computer system, whichincludes three program modules (Institute of Cytol-ogy and Genetics, SB RAS): GeneNet, MGSgenera-tor, MGSmodeller, and the software package STEP+(Sobolev Institute of Mathematics, SB RAS). The sys-tem is used to construct and numerically analyze mod-els describing dynamics of the molecular genetic sys-tems (MGS) functioning in pro- and eukaryotes. Us-ing module GeneNet we can reconstruct structure func-tional organization of gene networks. We use MGSgen-erator as an intermediate module in generation of math-ematical models based on gene networks reconstructedin GeneNet. Moreover, in the module MGSgeneratorwe represent obtained mathematical models in the inputformat of STEP+. Module MGSmodeller contains toolsfor the gene network models to be developed and nu-merically analyzed. Package STEP+ is intended for thenumerical analysis of mathematical models representedby autonomous systems ODEs. We have tested our inte-grated system on the MGS model for intracellular auxinmetabolism in a plant cell.This work has been partially supported by the SiberianBranch of the Russian Academy of Sciences (Interdisci-plinary integration project Post-genomic bioinformatics:computer analysis and modeling of the molecular geneticsystems, No. 119).

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Application of the multiscale FEM in modeling the can-cellous bone

Sandra IlicInstitute of Mechanics, Ruhr-University of Bochum,Bochum, 44780 Germanysandra.ilic@rub.de

Due to the presence of the fluid and solid phase, themodeling of cancellous bone represents a complex, exten-sive task where the dynamic investigation and viscosityeffects must be taken into consideration. The already es-tablished approach for the investigation of this materialtype is Biots method, originally developed for simulat-ing saturated porous materials. In this contribution wepresent the homogenization multiscale FEM as an alter-native to Biots method. The motivation for this choice isdecreasing the extent of the necessary laboratory inves-tigations. According to the multiscale FEM, the bone isunderstood as the homogenized medium whose effectivematerial parameters are obtained by the analysis of anappropriate representative volume element (RVE). Thisis also the main topic of the presentation: a compari-son of the effective values obtained by studying differenttypes of RVEs where the particular attention is paid tothe numerical values for Youngs modulus and attenu-ation coefficient. The distinction between the modelspertains to the geometry of the solid frame of the RVE,the type of the applied elements as well as the type ofthe coupling conditions on the interface of the phases.

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Bone growth and destruction at the cellular level: amathematical model

Mark D. RyserMcGill University, W. Burnside Hall, Room 1005, 805Sherbrooke Street, Montral, Quebec H3A 2K6 Canadaryser@math.mcgill.ca

The process of bone destruction and subsequent growthis continually occurring in healthy bone tissue. Thisprocess is referred to as ’remodeling’ and plays a keyrole in many pathologies such as osteoporosis and os-teoarthritis. We describe remodeling at the cellular leveland discuss the cells and biochemical pathways involved.We then develop a mathematical model for remodeling,consisting of a system of coupled nonlinear PDEs. Wediscuss how physiological parameters may be obtainedthrough scaling of the equations and we comment ontheir mathematical properties. Numerical experimentsvalidating the model will be presented. This is jointwork with Nilima Nigam (SFU) and Svetlana Komarova(McGill).

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VI. Others

Organisers:local organising committee

—Abstracts—

The relationship between Bezoutian matrix and New-ton’s matrix of divided differences and separation ofzeros of interpolation polynomials

Ruben AirapetyanKettering University, 1700 W Third Ave. Flint, Michi-gan 48504, United Statesrhayrape@kettering.edu

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VI. Others

Let x1, . . . , xn be real numbers, Pn(x) = an(x −x1) · · · (x−xn). Denote byDng the matrix of generalizeddivided differences of function g in Newton’s interpola-tion formula with nodes x1, . . . , xn and by Gn(x) theNewton’s interpolation polynomial of function g. De-note by B = B(Pn(x), Gn(x)) the Bezoutian matrix ofPn and Gn. The relationship between the correspond-ing principal minors of the matrices Dng and B countedfrom the left lower corner is establish. Then, it followsthat if these principal minors of the matrix of divideddifferences are positive or have alternating signs thenthe roots of the interpolation polynomial are real andseparated by the nodes of interpolation.

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Bayesian shrinkage estimation of parameter exponen-tial distribution

Hadeel AlkutubiB-23-1 , The Heritage, JLN SB, Dagang Mines ResortCity, Seri Kembangan, Serdang, 43300 Malaysiahadeelkutubi@live.com

In this paper, we would like to test the best estima-tor (smallest MSE and MPE) of shrinkage estimator ofparameter exponential distribution . To do this , we de-rived this estimators depend on Bayesian method withJeffreys prior information and square error loss func-tion . To compared between estimators we used MSEand MPE with respect of simulation study. We foundthe shrinkage estimator between Bayes estimators underdifferent loss function is the best estimator.

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Interpolation beyond the interval of convergence: Anextension of Erdos-Turan Theorem

Mohammed BokhariDepartment of Mathematics & Statistics, King FahdUniversity of Petroleum & Minerals, Dhahran, SaudiArabiambokhari@kfupm.edu.sa

An elegant result due to Erdos and Turan states thatthe sequence of Lagrange interpolants to a given contin-uous function f at the zeros of orthogonal polynomialsover a closed interval converges to f in the mean squaresense. We introduce certain sequences of polynomialswhich preserve both interpolation as well as convergenceproperties of Erdos-Teran Theorem. In addition, theyinterpolate f at a finite number of pre-assigned pointslying outside the underlying open interval. We shall in-troduce a method to construct the suggested polynomi-als and also investigate their properties. Computationalaspects will also be discussed.

———

The ADM method and the Tanh method for solvingsome non linear evolutions equations

Zoubir DahmaniDepartment of Mathematics, Faculty of Sciences, Uni-versity of Mostaganem Les HLM, 21 street les HLMmostaganem, mostaganem 27000 Algeriazzdahmani@yahoo.fr

In our work, we use the ADM method for solving somenonlinear evolution equations with time and space frac-tional derivative. Then we use the Extended Tanhmethod to formally derive traveling wave solutions forsome evolution equations. The obtained solutions in-clude, also, kink soltuions.

———

Boundary-value problems for generalized axially-symmetric Helmholtz equation

Anvar Hasanov34 Durmon yoli, Tashkent branch of the Russian StateUniversity of oil and gas named after Gubkin, Tashkent,Tashkent 100125, Uzbekistananvarhasanov@yahoo.com

In this talk several main boundary-value problems suchthe Dirichlet, Neumann problem and other problems willbe considered. The unique solvability of afore-mentionedproblems will be proved.

———

Asymptotic extension of topological modules and alge-bras

Maximilian HaslerLaboratoire AOC, Universit Antilles-Guyane, B.P. 7209,campus de Schoelcher, Schoelcher, Martinique 97275Francemhasler@martinique.univ-ag.fr

Given a topological R-module or algebra E and anasymptotic scale M ⊂ RΛ, we exhibit a natural M -extended topology on the sequence space EΛ, and definethe M -extension of E as the Hausdorff space associatedwith the subspace of nets for which multiplication is con-tinuous with respect to this topology.Commonly used spaces of generalized functions are ob-tained as special cases, but this new approach applies inmany different situations. It also allows the iteration ofthe construction, which is not possible with previouslyexisting theories.We use only the topology, i.e. neighbourhoods of zero,but not its explicit definition in terms of seminorms, in-ductive or projective limits etc., which is particularlyconvenient in non-metrizable spaces.Many ideas commonly used in the context of generalizedfunctions (functoriality, association, sheaf structure, al-gebraic analysis, . . . ) can be applied to a large extent.Reasoning on a category-theoretic level allows to estab-lish several results so far only known for particular cases,for the whole class of such spaces.

———

Approximation of fractional derivatives

S. Moghtada HashemiparastMathematics and Statistics, Jolfa Ave, Seyed Khandan,Tehran 193953358 Iranhashemiparast@yahoo.com

Series represantations are presented to approximate thefractional derivatives which have extensive applicationin ordinary,partial difrential equations and specilly thestable probability distributions.The convergence of the

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series are considered and are applied to solving the equa-tions,finally toillustrate the accuracy of the apprpxima-tions examples are solved.

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Discrepancy estimate for uniformly distributed se-quence

Hailiza KamarulhailiSchool of Mathematical Sciences Universiti SainsMalaysia Minden, Penang 11800 Malaysiahailiza@cs.usm.my

A general metrical result of discrepancy estimate relatedto uniform distribution of a sequence is proved . Thiswork extends result of R.C. Baker where the sequencecan be assumed to be real. The lighter version of thistheorem will also be discussed in this talk.

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Bounded linear operators on l-power series spaces

Erdal KarapinarATILIM University, Department of Mathematics, Kizil-casar Koyu, INCEK ANKARA, 06836 Turkeyerdalkarapinar@gmail.com

Let A be the class of Banach space ` of scalar sequenceswith a norm ‖ · ‖` such that

(i) a = (ai) ∈ l∞, x = (ξi) ∈ ` ⇒ ax = (aiξi) ∈`, ‖ax‖` ≤ ‖a‖l∞‖x‖`,

(ii) ‖ei‖` = 1, ∀i ∈ N where ei = (δij)j∈N.

For a given ` ∈ A and a Kothe matrix A, we de-fine `-Kothe space K`(ai,n) as a Frechet space of allscalar sequences x = (ξi) such that (ξiai,n) ∈ ` foreach n, endowed with the topology of Fr echet space,determined by the canonical system of norms ‖x‖n =‖ (ξiai,n) ‖`, n ∈ N.We write (E,F ) ∈ B, if every continuous linear mapfrom E to F is bounded. In 1983, D.Vogt has character-ized those Frechet spaces E for which (E,Kl∞(A)) ∈ Bholds.This gives also a characterization of (E,Kc0(A)) ∈ B.We extend this results and prove that

Theorem. For Frechet space E and ` ∈ A,(E,Kl1(A)) ∈ B ⇒ (E,K`(A)) ∈ B ⇒ (E,Kl∞(A)) ∈B.

Theorem. For Frechet space F and ` ∈ A,(K`(A), F ) ∈ B ⇒ (Kl1(A), F ) ∈ B.

———

On a three-dimensional elliptic equation with singularcoefficients

Erkinjon KarimovDurmon yuli street 29, Akademgorodok Tashkent,Tashkent 100125 Uzbekistanerkinjon@gmail.com

In this talk some questions such as finding fundamentalsolutions, investigations of main boundary-value prob-lems for an equation

uxx + uyy + uzz +2α

xux +

2β

yuy +

2ζ

zuz

will be discussed. Here α, β, zeta are constants, more-over 0 < 2α, 2β, ζ < 1.

———

A unified presentation of a class of generalized Hum-bert polynomials

Nabiullah KhanDepartment of Applied Mathematics, Z.H. College ofEngineering and Technology, Aligarh Muslim University,Aligarh 202002 Indianabi khan1@rediffmail.com

The principal object of this paper is to present a naturalfurther step toward the unified presentation of a class ofHumbert’s polynomials which generalizes the wellknownclass of Gegenbauer, Humbert, Legendre, Tchebycheff,Pincherle, Horadam, Dave, Kinnsy, Sinha, Shreshtha,Horadam-Pethe, Djordjevie, Gould, Milovanovic andDjordjevic, Pathan and Khan polynomials and many notso wellknown polynomials. We shall give some basic re-lations involving the generalized Humbert polynomialsand then take up several generating functions, hyperge-ometric representations and expansions in series of somerelatively more familier polynomials of Legendre, Gegen-bauer, Rice, Hermite, Jacobi, Laguerre Fasenmyer Sis-ter M. Celine, Bateman, Rainville and Khandekar. Wealso show that our results provide useful extensions ofknown results of Dilcher, Horadam, Sinha, Shreshtha,Milovanovic-Djordjevic, Pathan and Khan.

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Direct estimate for modified beta operators

Lixia LiuYuhua east Road 113, College of Mathematics and Infor-mation Science, Hebei Normal University, Shijiazhuang,Hebei Province 050016 Chinaliulixia@hebtu.edu.cn

Use the Ditzian modulous of smoothness ω2ϕλ(f, t), (0 ≤

λ ≤ 1), to study the pointwise direct results for modifiedBeta operators, which extend the approximation resultfor Beta operators.

———

Mathematical model of an undergorund nuclear wastedisposal site

Eduard Marusic-PalokaDepartment of Mathematics, University of Zagreb, Bi-jenicka 30, Zagreb, 10000 Croatiaemarusic@math.hr

The goal of our research is to find an accurate model fornumerical simulations of the nuclear waste disposal site.The purpose of such model is to perform safety analy-sis of the site and find out its possible impact on thebiosphere. Due to the large dimension of the site andvery long lifetime of radioelements, realistic experimentsare not possible. Thus, predictions based on numericalsimulations are all we have.Starting from the microscopic model given by thereaction-diffusion-convection equation, using the asymp-totic analysis and homogenization, we derive a macro-scopic model and discuss ity accuracy.

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VI. Others

Compact and coprime packedness and semistar opera-tions

Abdeslam MimouniDepartment of Mathematics and Statistics King FahdUniversity of Petoleum and Minerals Dhahran, Estern31261 Saudi Arabiaamimouni@kfupm.edu.sa

In this talk we will present new developments on thestudy of compact and coprime packedness of an inte-gral domain with respect to a star operation of finitecharacter. Let R be an integral domain with quotientfield K and let ∗ be a star operation of finite type onR. A ∗-ideal I is said to be ∗-compaclty (respectively∗-coprimely) packed if whenever I ⊆

Sα∈Ω Pα, where

Pαα∈Ω is a family of ∗-prime ideals of R, I is actuallycontained in Pα (resp. (I + Pα)∗ ( R) for some α ∈ Ω;and R is said to be ∗-compactly (resp. ∗-coprimely)packed if every ∗-ideal of R is ∗-compactly (resp. ∗-coprimely) packed. In the particular case where ∗ = dis the trivial operation, we obtain the so-called com-pactly and coprimely packed domains. Our objectivesis to study some ring-theoretic aspects of these notionsin different classes of integral domains, paying particu-lar attention to the the t-operation as the largest andwell-known operation.

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Characterization of some matrix classes involving (σ, λ)-convergence

S. A. MohiuddineDepartment of Mathematics Aligarh Muslim UniversityAligarh, Uttar Pradesh 202002 Indiamohiuddine@gmail.com

Let σ be a one-to-one mapping from the set N of naturalnumbers into itself. A continuous linear functional ϕ onthe space `∞ of bounded single sequences is said to bean invariant mean or σ-mean if and only if (i) ϕ(x) ≥ 0if x ≥ 0 (i.e. xk ≥ 0 for all k); (ii) ϕ(e) = 1, wheree = (1, 1, 1, · · · ); (iii) ϕ(x) = ϕ((xσ(k))) for all x ∈ `∞.Let λ = (λn) be a non-decreasing sequence of positivenumbers tending to ∞ such that

λn+1 ≤ λn + 1, λ1 = 0.

In this paper, first we define (σ, λ)-convergence and showthat V λσ is a Banach space with ‖x‖ = supm,n |tmn(x)|,where V λσ is the set of all (σ, λ)-convergent sequencesx = (xk). We also define and characterize (σ, λ)-conservative, (σ, λ)-regular and (σ, λ)-coercive matrices.Further, we characterize the class (`1, V

λσ ), where `1 is

the space of all absolutely convergent series.

———

Sequence spaces of invariant mean and some matrixtransformations

Mohammad MursaleenDepartment of Mathematics Aligarh Muslim UniversityAligarh, UP 202002 Indiamursaleenm@gmail.com

Let σ be a one-to-one mapping from the set N of naturalnumbers into itself. A continuous linear functional φ onthe space `∞ is said to be an invariant mean or a σ-mean

if and only if (i) φ(x) ≥ 0 when the sequence x = (xk)has xk ≥ 0 for all k, (ii) φ(e) = 1, where e = (1, 1, 1, · · · ),and (iii) φ(x) = φ((xσ(k))) for all x ∈ `∞. Throughoutthis paper we consider the mapping σ which has no fi-nite orbits, that is, σp(k) 6= k for all integer k ≥ 0 andp ≥ 1, where σp(k) denotes the pth iterate of σ at k.Note that, a σ-mean extends the limit functional on thespace c in the sense that φ(x) = limx for all x ∈ c.In this paper we define a new sequence space V∞σ (λ)which is related to the concept of σ-mean and the se-quence λ = (λn) described as above and characterizethe matrix classes (`∞, V

∞σ (λ)) and (`1, V

∞σ (λ)).

Let λ = (λn) be a non-decreasing sequence of positivenumbers tending to∞ such that λn+1 ≤ λn+ 1, λ1 = 0.Then we define the following sequence space and showthat it is a BK-space:

V∞σ (λ) := x ∈ `∞ : supm,n|τmn(x)| ≤ ∞,

whereτmn(x) = (1/(λm))

Xj∈`m

xσj(n).

———

New convection theory for thermal plasma and NHDconvection in rapidly rotating spherical configurations

Ali MussaKing Abdulaziz City for Science and Technology Build-ing # 2 King Abdullah Bin Abdulaziz Street Riyadh,Riyadh 6086/11442 Saudi Arabiaalmussa@kacst.edu.sa

We extend Jones-Soward-Mussa (JSM) theory (2000):“analytic and computational solution for E → 0 andPr/E →∞”. We also make use of Zhang (2001) ansatzfor: “E 1 arbitrary but fixed and 0 ≤ Pr < ∞”the so-called enhanced Nearly Geostrophic Inertial Wave(NGIW) approach. Such extension represented as a con-struction of a new MHD plasma convection and magne-toconvection force theory. The flow field confinement inthe study assumed to be in spherical geometry config-uration and our investigation is made under the basisof magnetic balance and scaling theory. Furthermore,strong inertial turbulence can be achieved in presence ofhigh Reynolds number so strong forces govern the flowfields have to be sufficiently understood. Indeed, strongrotation and strong magnetic field for the flow field in-side the spherical rotating geometry; take into consid-eration the effect of the anticipated vigorous convectionand magnetoconvection in the flow field confinement.

———

Characterizations of Isometries on 2-modular spaces

Kourosh NourouziDepartment of Mathematics, K.N. Toosi University ofTechnology Tehran, Tehran 16315-1618 Irannourouzi@kntu.ac.ir

Let X be a real vector space of dimension greater thanone. A real valued function ρ(·, ·) on X2 satisfying thefollowing properties is called a 2-modular on X, for allx, y, z ∈ X:

1. ρ(x, y) = 0 if and only if x, y are linearly depen-dent,

2. ρ(x, y) = ρ(y, x),

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VI. Others

3. ρ(−x, y) = ρ(x, y),

4. ρ(x, αy + βz) ≤ ρ(x, y) + ρ(x, z), for any

nonnegative real numbers α, β with α+ β = 1.In this talk, we discuss on the characterization of isome-tries defined on 2-modular spaces.

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On r-imbalances in tripartite r-digraphs

Shariefuddin PirzadaKing Fahd University of Petroleum and Minerals,Dhahran, 31261 Saudi Arabiasdpirzada@yahoo.co.in

A tripartite r-digraph(r ≥ 1) is an orientation of a tri-partite multigraph that is without loops and containsatmost r edges between any pair of vertices from dis-tinct parts. For any vertex x in a tripartite r-digraphD(U, V,W ), let d+

x and d−x denote the outdegree andindegree respectively of x. Define aui = d+

ui − d−ui ,bvj = d+

vj −d−vj and cwk = d+

wk−d−wk as the r-imbalances

of the vertices ui in U , vj in V and wk in W respectively.In this paper, we characterize r-imbalances in tripartiter-digraphs and obtain some results.

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Invariance conditions and amenability of locally com-pact groups

Hashem Parvaneh MasihaDepartment of Mathematics, Faculty of Science, K. N.Toosi University of Technology. No. 41, Kavian St.,Seyyed Khandan Bridge (N.), Shariati Ave., Tehran,Tehran 16315-1613 Iranmasiha@kntu.ac.ir

Adler and Hamilton showed that a semigroup S is leftamenable if and only if it satisfies the following invari-ance condition. For any subsets A1, A2, · · · , Ak of S andany s1, s2, · · · , sk ∈ S, there exists a nonempty finitesubset E of S such that n(s−1

i Ai ∩ E) = n(Ai ∩ E), fori = 1, 2, · · · , k, where s−1A = t ∈ S : st ∈ A and n(A)is the number of elements in A. In this talk, we shallprove an analogous result for locally compact groups.More precisely, we show that amenability of a locallycompact group G is equivalent to: For any λ-measurablesubsets A1, A2, · · · , Ak of G, any g1, g2, · · · , gk ∈ Gand any ε > 0, there exists a compact subset K of Gsuch that |λ(g−1

i Ai ∩ K) − λ(Ai ∩ K)| < ελ(K), fori = 1, 2, · · · , k, where λ(A) denotes the left Haar mea-sure of A. In this paper, we suppose that G be a locallycompact group and λ a fixed left Haar measure onG. Welet X = K ⊂ G : K is compact and λ(K) > 0. Forf ∈ L∞(G), we define f(K) = 1

λ(K)

RKfdλ, K ∈ X,

then f : X → R is well defined.

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Motion stabilisation of a solid body with fixed point

Zaure RakishevaAl-Faraby Kazak National University, Almaty, Kazak-stanzaure ra@mail.ru

The problem of a solid body dynamics in the centralNewton field of forces is considered. It is generalization

of one of the major classical problems of theoretical me-chanics, dynamics of a solid body with one fixed pointin a gravity field. Motion of a solid body with one fixedpoint is described by the well-known system of Eulerand Poisson equations. It is known the general solutionexists if one considers two first terms of force functionexpansion into a series. By original change of variablesthe system is reduced to the normal form with the firstintegral of norm type. The solution of this system isconsidered as the non-perturbed motion and it is inves-tigated on stability. The procedure is offered for ob-taining of asymptotically steady motion in general case.The controlling force nature was defined. This methodwas applied to three cases with special restrictions onthe bodys inertia moments, so-called generalized clas-sical cases of Euler, Lagrange and Kovalevskaya. Thenumerical solution for the problem in Euler case wasconstructed.

———

A Lizorkin type theorem for Fourier series multipliers inregular systems

Lyazzat SarybekovaMunaitpasov 7, Astana, 010010 Kazakhstanlsarybekova@yandex.ru

A new Fourier series multiplier theorem of Lizorkin typeis proved for the case 1 < q < p <∞. The result is givenfor a general regular system and, in particular, for thetrigonometrical system it implies an analogy of the orig-inal Lizorkin theorem.

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Inverse-closedness problems in the stability of se-quences in Banach Algebras

Pedro A. SantosDepartamento de Matematica, Instituto SuperiorTecnico, Av Rovisco Pais, Lisboa, 1049-001 Portugalpasantos@math.ist.utl.pt

We are concerned with the applicability of the finite sec-tions method to operators belonging to the closed subal-gebra of L(Lp(R)), 1 < p < ∞, generated by operatorsof multiplication by piecewise continuous functions inR and operators of convolution by piecewise continuousFourier multipliers.The usual technique is to introduce a larger algebra ofsequences, which contains the special sequences we areinterested and the usual operator algebra generated bythe operators of multiplication and convolution. Thereis a direct relationship between the applicability of thefinite section method for a given operator and invertibil-ity of the corresponding sequence in this algebra.But, contrarily to the C∗ case and Banach analogue forToeplitz operators, in our case several inverse-closednessproblems must be solved.

———

Smoothing effects for periodic NSE in critical Sobolevspace

Ridha SelmiDepartment of Mathematics, Faculty of Sciences ofGabes, 6072, TUNISIAridha.selmi@isi.rnu.tn

We prove smoothing effects for 3D incompressible NavierStokes Equation for initial data belonging to critical

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VI. Others

Sobolev Space H12 (T3).

Asymptotic behavior of the global solution when thetime goes to +∞ is studied.

———

Large deviations and almost sure convergence

Mariana SibiceanuGh.Mihoc-C.Iacob Institute of Mathematical Statisticsand Applied Mathematics, Romanian Academy, Calea13 Septembrie Nr.13, Sector 5 Bucharest, 050711 Ro-maniamarianasibiceanu@yahoo.com

In our setup, the Large Deviation Principle for a se-quence P (n) of probabilities on a separable Banachspace E, with a convex good rate function I is assumed,also the existence of the finite limits g(w) of the associ-ated logarithmic moment generating function.We establish precise upper and lower bounds of the val-ues that P (n) assigns almost sure in the weak and strongtopology of E, respectively, determined by the amountsof the canonical dual product on E′ × N , N being thenullifying set of the rate function.Also, we reveal the significance of the derivative of thefunction g(tw) of real t for the almost sure convergence,in the situation when g is Gateaux differentiable on(E′, t(E′, E)).

———

The k-ε Model in Turbulence

Tanfer TanriverdiHarran University, Faculty of Arts and Sciences, Depart-ment of Mathematics, Sanlurfa 63300, Turkeyttanriverdi@harran.edu.tr

We prove analytically the existence of self-similar solu-tions for the k-ε model arising in the evolution of turbu-lent bursts by employing the topological shooting tech-nique where α > β with the some other conditions.The first author was supported by the Scientific andTechnological Research Council of Turkey (TUBITAK).He is also thankful to the Oxford Center for NonlinearPDE, and to the Mathematical Institute of the Univer-sity of Oxford, for the hospitality they offered him duringhis visit.This is joint work with Bryce McLeod (Oxford).

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The equivalence between modified Mann (with errors),Ishikawa (with errors), Noor (with errors) and modifiedmulti-step iterations (with errors) for non-Lipschitzianstrongly successively pseudo-contractive operators

Johnson OlaleruMathematics Department, University of Lagos, Univer-sity of Lagos Road, Yaba, Lagos, Nigeriajolaleru@unilag.edu.ng

In this paper, the equivalence of the convergence be-tween modified Mann(with errors), Ishikawa(with er-rors), Noor(with errors) and modified multistep iter-ation(with errors) is proved for generalized stronglysuccessively pseudocontractive mapping without Lips-chitzian assumption. Our results generalise and improve

on the recent results of Z.Y.Huang [Equivalence theo-rems of the convergence between Ishikawa and Mann it-erations with errors for generalized strongly successivelypseudocontractive mappings without Lipschitzian as-sumptions, J.Math.Anal.Appl. 329(2007),935-947], Z.Y.Huang, F.W. Bu, M.A. Noor [On the equivalence of theconvergence criteria between modified Mann-Ishikawaand multistep iteration with errors for strongly pseudo-contractive operators, Appl. Math. Compt. 181(2006),641-647], B.E.Rhoades, S.M.Soltuz [The equivalence be-tween the convergences of Ishikawa and Mann itera-tion for an asymptotically non-expansive in the inter-mediate sense and strongly successively pseudocontrac-tive maps, J. Math. Anal. Appl. 289(2004), 266-278]and B.E.Rhoades, S.M.Soltuz [The equivalence betweenMann-Ishikawa iterations and multi-step iteration, Non-linear Anal.58(2004),219-228] among others.

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A characterization for multipliers of weighted Banachvalued Lp(G)-spaces

Serap OztopIstanbul University, Faculty of Sciences, Istanbul,Vezneciler 34134 Turkeyoztops@istanbul.edu.tr

Let G be a locally compact group, 1 < p < ∞. Theaim of this paper is to characterize the multipliers ofthe weighted Banach valued intersection Lp(G) spacesas the space of multipliers of a certain Banach algebra.

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Stationary motion of the dynamical symmetric satellitein the geomagnetic field

Karlyga ZhilisbaevaAl-Faraby Kazak National University, Almaty, Kazak-stanzhilisbaeva@mail.ru

Stationary solutions of the system of the satellite’s mo-tion equations are of special interest for the various prob-lems of space researches, and first of all for the satellite’smagnetic stabilization.In the paper stationary motions of the equatorial magne-tized dynamically symmetric satellite round the centreof mass on a circular orbit are considered. Strong mag-nets are placed on the satellite’s board. Perturbationsare taken into account, caused by insignificant deviationof a satellite’s axis of dynamic symmetry and by mag-netization of its cover.The equations of the satellite’s perturbed motion in Eu-ler’s canonical variables are obtained. Conditions of sta-tionary motion existence are defined, necessary and suf-ficient conditions of their stability are found with takinginto account of small perturbations.

———

111

Index

Abate, M., 97D’Abbico, M., 71Abdenur, F., 97Abdous, B., 47Agranovich, M., 60, 63Airapetyan, R., 106Aksoy, U., 27Alimov, S., 60Aliyev, T., 23Alkutubi, H., 107Almeida, A., 51Alves, J.F., 97D’Ancona, P., 80Applebaum, D., 90Aripov, M., 81Assal, M., 56Aulaskari, R., 23, 56Aydin, I., 51Aykol, C., 51

Babych, N., 85El Badia, A., 87Bakry, D., 95Ball, J., 15Ballantine, C., 23Bally, V., 90Barsegian, G., 33Barthe, F., 95Bastinec, J., 101Begehr, H., 3, 26, 27Bellieud, M., 85Berezansky, L., 100, 101Berger, P., 97Berglez, P., 27Berlinet, A., 3, 47Bernstein, S., 42Besov, O., 51de Bie, H., 36Bisi, C., 36Bobkov, S., 95Bociu, L., 76Bock, S., 42Boichuk, A., 101Bojarski, B., 3, 23Bokhari, M., 107Bolosteanu, C., 27Boukrouche, M., 76Boulakia, M., 77Boulbrachene, M., 95Boutet de Monvel, L., 15Bouzar, C., 63Boykov, I., 27Boza, S., 52Branden, P., 33Britvina, L., 48Brody, D., 90Bruning, E., 3Bucci, F., 76

Budochkina, S.A., 97Burenkov, V., 3, 51, 60Burns, K., 98Buzano, E., 63

Camara, C., 45Capdeboscq, Y., 85Cardon, D., 33Carro, M., 52Caruana, M., 91Casado-Diaz, J., 86Castro, L., 45Catana, V., 63Catania, D., 81Cattaneo, L., 3Cavalcanti, M., 77Celebi, O., 3, 28Cerejeiras, P., 36Charalambides, M., 34Chazottes, J.-R., 98Chen Kuan-Ju, 81Chen Qiuhui, 49Chen Yi-Chiuan, 98Cherdantsev, M., 86Chiba, Y., 63Chinchaladze, N., 28Cho, D.H., 49Choe, B.R., 56Choi, Q-H., 71Choulli, M., 88Cohen, L., 63Colombini, F., 71Colombo, F., 36, 88Cordero, E., 63Coulembier, K, 37Crisan, D., 3, 90, 91Cruzeira, A.B., 91Csordas, G., 33, 34

Dahmani, Z., 107Dai Daoquin, 26Dalla Riva, M., 23Dallakyan, G., 52Daoulatli, M., 77Datt Sharma, S., 47Dattori da Silva, P., 64Daveau, C., 88Davie, A., 91Davies, B., 3, 16, 59Davis, M., 91Dehgan, M., 81Del Santo, D., 71Delgado, J., 64Diblik, J., 100, 101Doi, K., 81Dolicanin, D., 49Domingos Cavalcanti, V., 77Domoshnitzky, A., 101Donaldson, S., 16

Dovbush, P., 24Dragoni, F., 3, 95Du Bau-Sen, 98Du Jinyuan, 26, 28

Ebert, M., 72Ekincioglu, I., 52El-Nadi, K., 93Eller, M., 77Elliott, N., 3Elton, D., 60Englis, M., 46Eriksson, S.-L., 37Erkip, A., 81

Fang Daoyuan, 72Farwell, R., 43Faustino, N., 43Federson, M., 102Fei, M.-G., 37Fernandez, A., 34Ferreira, M., 37Field, M., 98Fisk, S., 34Fokas, T., 43Forni, G., 98Fragnelli, G., 77Franek, P., 37Franssens, G.R., 37Freitas, J., 98Fujita, K., 48Fujiwara, H., 49Furutani, K., 64

Gaiko, V., 98Gainova, I.A., 106Galleani, L., 64Galstyan, A., 72, 77Garello, G., 64Garetto, C., 65Gauthier, P., 34Gedif Ayele, T., 51Geisinger, L., 60Gentil, I., 95Gentili, G., 37Georgiev, S., 43Georgiev, V., 72, 80Ghergu, M., 82Ghisa, D., 24Ghisi, M., 73Gil, J., 65Gilbert, R., 3, 105, 106Giorgadze, G., 28Girela, D., 56Gobbino, M., 73Golberg, A., 24Goldshtein, V., 53Goltser, Y., 102Golubeva, V.A., 86

113

Index

Gonzalez, M.J., 57Gramchev, T., 65Gramsch, B., 65Graubner, S., 28Grigoryan, A., 95Grudsky, S., 45, 46Guliyev, V., 53Gupta, S., 57Gurlebeck, K., 42Gurkanlı, A.T., 46Gyongy, I., 91Gyori, I., 102

Hairer, M., 92, 96Hajibayov, M., 53Halburd, R., 34Hartung, F., 102Harutyunyan, T., 61Hasanov, A., 107Hashemiparast, S.M., 107Hasler, M., 107Hebisch, W., 96Helmstetter, J., 43Herrmann, T., 73Higgins, J.R., 48Hinkkanen, A., 34Hirosawa, F., 71, 73Hogan, J., 43Hormann, G., 65Huet, N., 96Hughston, L., 92Hunsicker, E., 65Hurri-Syrvanen, H., 53Hussain, A., 28

Ichinose, W., 65Ilic, S., 106Inglis, J., 3, 96Israfilov, D., 24Iwasaki, C., 66

Jacka, S., 92Janas, J., 61Johnson, J., 66Jordan, T., 99Jung, T., 73

Kaasalainen, M., 88Kahler, U., 44Kalmenov, T.S., 30Kalyabin, G., 53Kamarulhaili, H., 108Kamotski, I., 85Kaptanoglu, T., 56, 57Karapinar, E., 108Karelin, O., 46Karimov, E., 108Karlovych, Y., 66Karp, D., 24Karp, L., 73Karupu, O., 25Katayama, S., 82Kato, K., 35Kats, B., 25Katsnelson, V., 35Kayar, Z., 102Kelbert, M., 92

Kelly, C., 102Kendall, W., 92Kenig, C., 17Khan, N., 108Kheyfits, A., 29Khimshiashvili, G., 29Kilbas, A., 3, 48, 49Kim, B.J., 50Kisil, A., 38Kisil, V., 42, 44Kiss, G., 102Kohr, G., 25Kohr, M., 25Kokilashvili, V., 17Kolokoltsov, V., 92Konjik, S., 50Kontis, V., 3Koroleva, A., 50Krainer, T., 61, 66Krausshar, R.S., 38, 44Krump, L., 38Kubo, H., 74, 82Kucera, P., 82Kurtz, T., 93Kurylev, Y., 87Kusainova, L., 54

Lamb, J., 3, 97Lamberti, P., 61Langerova, M., 102Lanza de Cristoforis, M., 3, 23,

25Laptev, A., 3, 59Lasiecka, I., 76Lassas, M., 88Lavicka, R., 38Le, U., 83Leandre, R., 44Lebiedzik, C., 78Lee, Y.L., 57Leem, K.H., 88de Leo, R., 66Lerner, N., 17Li Xue-Mei, 93Libine, M., 39Liflyand, E., 54Lionheart, W., 89Lions, P.-L., 21Littman, W., 78Liu Lixia, 108Liu Yu, 66Loreti, P., 78Lucente, S., 83Luna-Elizarraras, M.E., 39Luzzatto, S., 97Lyons, T., 90, 93

Macia, F., 86Maksimov, V., 78Malliavin, P, 18Mamedkhanov, J., 25Mammadov, Y., 54Manhas, J.S., 57Manjavidze, N., 29Markowich, P., 86Marletta, M., 89

Marquez, A., 57Marti, J.-A., 67Martin, M., 39Martinez, P., 78Marusic-Paloka, E., 108Masaki, S., 83Matsuura, T., 89Matsuyama, T., 74Matthies, K., 87Maz’ya, V., 18van der Meer, J. C., 99Michalska, M., 57Migda, M., 103Mijatovic, A., 93Mimoiuni, A., 109Mityushev, V., 32, 33Mochizuki, K., 74Mogultay, I., 83Mohammed, A., 29Mohiuddine, S.A., 109Mokhonko, A., 61Mola, G., 83Molahajloo, S., 67Morando, A., 67Moura Santos, A., 54Mursaleen, M., 109Mussa, A., 109

Nakazawa, H., 74Nam, K., 57Naso, M.G., 79Natroshvili, D., 67Neklyudov, M., 96Neustupa, J., 62Nicol, M., 99Nieminen, P., 58Niknam, A., 99Nishitani, T., 74Nourouzi, K., 109Nowak, M., 58

Oberguggenberger, A., 68Ockendon, J., 20Ohta, M., 83Olaleru, J., 111Oliaro, A., 68Onchis, D., 48Oparnica, L., 50Opic, B., 54Orelma, H., 39Otto, F., 96Ozawa, T., 80Ozbekler, A., 103Oztop, S., 111

Pandolfi, L., 79Panti, G., 99Papageorgiou, I., 3, 96Papavasiliou, A., 93Parvaneh Masoha, H., 110Pau, J., 58Pelekanos, G., 89Pena Pena, D., 39Peng, C., 99Perotti, A., 39Picard, R., 74Pinotsis, D., 44

114

Index

Pirzada, S., 110Pistorius, M., 94Pituk, M., 103Pivetta, M., 75Plaksa, S., 23, 26Porter, M., 33Potthast, R., 89Prakash Sing, A., 35Prykarpatsky, A., 35

Quiao Yuying, 40

Rattya, J., 56Rachunkova, I., 103Radkevich, E., 55Rafeiro, H., 55Rajabov, N., 29, 68Rajabova, L., 30Rakisheva, Z., 110Rappoport, J., 50Rasmussen, M., 100Reinfelds, A., 103Reissig, M., 3, 71, 75, 83Ren Guangbin, 40Renardy, M., 79Reynolds, D., 103Richard, S., 62Roberto, C., 96Rochon, F., 68Rodino, L., 3, 62Rodkina, A., 104Rogosin, S., 32, 33Rojas, E., 46Rozovsky, B., 94Ruzickova , M., 104Ruzhansky, M., 3Ryan, J., 40Ryan, M., 3Ryser, M.D., 106

Sabadini, I., 36, 40Sadyrbaev, F., 100Safarov, Y., 59Saito, J., 75Saitoh, S., 3, 47, 48, 89Saks, R., 30Salac, T., 40Samko, N., 55Samko, S., 51, 55Samoylova, E., 30Santos, P.A., 110Sanz-Sole, M., 94Sarybekova, L., 110Sasane, A., 58Savchin, V.M., 104Schmeidel, E., 104Schmeling, J., 100Schmidt, B., 87Schnaubelt, R., 79

Schrohe, E., 69Schulze, B.-W., 3, 18, 68Sehba, B.F., 58Seiler, J., 69Selmi, R., 110Senouci, K., 55Serbetci, A., 55Sergeev, A., 35Sevroglou, V., 90Sforza, D., 79Shapiro, M., 40Shaposhnikova, T., 69Shibata, Y., 84Shindiapin, A., 104Shklyar, B., 105Shpakivskii, V., 44Shvartsman, I., 79Sibiceanu, M., 111Silva, A., 47Silvestri, B., 62Simon, L., 84Skalak, Z., 84Smyshlyaev, V., 85, 87Sobolewski, P., 58Soltanov, K., 84Somberg, P., 40Sommen, F., 36, 41Soria, J., 56Soucek, V., 41Spagnolo, S., 84Sproßig, W., 42, 45Stanek, S., 105Stefanov, A., 84Stevic, S., 105Stoppato, C., 41Strohmaier, A., 62Suragan, D., 30Suzuki, O., 26Suzuki, R., 75

Tahara, H., 69Takemura, K., 48Tamrazov, P., 23Taneco-Hernandez, M.A., 82Tanriverdi, T., 111Tapdıgoglu, M., 58Taqi, I., 31Taskinen, J., 58Tasmambetov, Zh., 31Teofanov, N., 69Teta, A., 84Tikhonov, S., 56Todd, M., 100Toft, J., 3, 69Tolksdorf, J., 45Tomilov, Y., 62Toundykov, D., 79Tovar, L.M., 59

Tretyakov, M., 94Trooshin, I., 90Trushin, B.V., 56Tunaru, R., 94Turunen, V., 70Tvrdy, M., 105

Ueda, Y., 85Uesaka, H., 75Uhlmann, G., 20Umeda, T., 62Unal, M., 105Upmeier, H., 47Usoltseva, E., 94

Vajiac, A., 41Valein, J., 80Vasilevski, N., 45, 47Vasilyev, V., 70Vasy, A., 70Vieira, N., 45Visciglia, N., 85Vlacci, F., 42Vlasakova, Z., 42van de Voorde, L., 42Vukotic, D., 59

Wang Qiudong, 100Wang Yufeng, 31Wirth, J., 3Witt, I., 70Wong, M.W., 3, 62, 70Wu Zhijian, 59Wulan Hasi, 59

Xu Wen, 59Xu Yongzhi, 3

Yagdjian, K., 75, 85Yakubovich, S., 50Yamamoto, M., 3, 20, 80, 87Yang Congli, 59Yildirir, Y.E., 26Yordanov, B., 76Youssfi, E.H., 26

Zafer, A., 100, 105Zegarlinski, B., 3, 95Zelinskiy, Y., 26Zeren, Y., 56Zhang Shangyou, 3Zhang Zhongxiang, 32Zhdanov, O.N., 32Zhilisbaeva. K., 111Zhong Shouguo, 32Zhu Hongmei, 71Zhu Kehe, 47, 59Zolesio, J.-P., 80Zorboska, N., 59

115