Abstracts of

International Conference

on Analysis and Applications

24-26 January 2010, Muscat, Oman

Contents

1 Invited Speakers 1

2 Applications of Analysis 5

3 Complex Analysis 19

4 Functional Analysis 29

5 Numerical Analysis 38

6 Topology 44

1 Invited Speakers

Difference Equations: A Gentle InvitationRaghib Abu-Saris (Walden University, USA)

Abstract

We review some of the recent developments pertinent to long termbehaviour of difference equations. In particular, we focus our attentionon the asymptotic behaviour of nonlinear difference equations, existenceof invariants and their relationship with periodic behaviour and Gausscompounding of means, and chaotic behaviour. This talk will culminatein presenting some open problems and research project.

On ϕ-normal functionsRauno Aulaskari (University of Joensuu, Finland)

Shamil Makhmutov (Sultan Qaboos University, Oman)Jouni Rattya (, )

Abstract

1

Let ϕ : [0, 1) → (0,∞) be an increasing function such that ϕ(r)(1 −r) → ∞, as r → 1−. A meromorphic function f in the unit disk belongsto the class Nϕ of ϕ-normal functions if its spherical derivative satis-fies f#(z) = O(ϕ(|z|)) as |z| → 1−. We study meromorphic ϕ-normalfunctions, and in particular an analogue of the Lohwater-Pommerenketheorem, and several equivalent characterizations for ϕ-normal functionsare established under certain regularity conditions on ϕ.

The Wijsman ConvergenceJiling Cao

(Auckland University of Technology, New Zealand)

Abstract

In 1960’s, when he studied some optimum properties of sequentialprobability ratio test, R. A. Wijsman considered a mode of convergence forsequences of closed sets such that if a sequence of proper lower semicontin-uous convex functions defined on R

n (as associated with their epigraphs)converged, then the same could be said for the induced sequence of con-jugate functions. In recognition of his contribution, this mode of conver-gence is now called the Wijsman convergence. Of course, the Wijsmanconvergence is topological, and the associated topology is called the Wi-jsman topology.

In the past 40 years, there has been a considerable effort in extendingWijsman’s results in infinite-dimensional Banach spaces, and exploringsome properties of the Wijsman topology. It turns out that the Wijs-man convergence and its topology have played important roles in theBanach Space Theory and Set-Valued Analysis. In this talk, I shall givean overview of the recent developments on the Wijsman topology. Inparticular, I shall discuss the Polishness, Baireness, the Amsterdam andother properties of this topology. Parts of my recent work with H. J. K.Junnila and A. H. Tomita will be presented, and some open questions inthis area will be posed.

Commutators of singular integralsYong Ding(Beijing Normal University, China)

Abstract

For a measurable function b on Rn , the multiplier Mb is defined by

Mbf = bf for measurable function f . Let T be the singular integraloperator with kernel K, i.e.,

Tf(x) = p.v.

Z

mathbbRn

k(x − y)f(y)dy.

Then the commutator [b, T ] formed by Mb and T is defined by

[b, T ]f(x) := (MbT − TMb)(f)(x) = p.v.

Z

Rn

k(x − y)[b(x) − b(y)]f(y)dy.

It is well known that the commutators [b, T ] play an important role inharmonic analysis and PDE. In this talk, we first recall the background

2

of the commutators for singular integral operators briefly and some earlyimportant results. Then we present some results obtained recently oncommutators of singular integral operators with rough kernels.

Foliations and non-metrisable manifoldsDavid Gauld (University of Auckland, New Zealand)

Abstract

A manifold is a topological space such that each point has an openneighbourhood homeomorphic to euclidean space R

n for some fixed n.Typically manifolds are expected to satisfy some standard topologicalconditions like Hausdorff, paracompact and connected. Using the home-omorphisms we may transfer from R

n to the manifold properties of theformer, including the differential or (piecewise-)linear structure. Safe-guards are needed to make sure that using two different homeomorphismsdoes not lead to ambiguity.

Loosely speaking a foliation on a manifold M consists of a collection(Uα, ϕα) / α ∈ A where each chart (Uα, ϕα) consists of an open subsetUα ⊂ M and a homeomorphism ϕα : Uα → R

n so that the charts col-lectively and consistently split M locally into subsets looking like parallelaffine subsets of R

n. As examples consider the flowlines of a non-singularflow on a surface or the pages of a slightly twisted soft-cover book.

Topics discussed in this talk will include.

• The connected components of the subsets of the manifold looking likeparallel affine subsets are called leaves. If the underlying manifold ismetrisable then there are uncountably many leaves, but M. Kneserand H. Kneser exhibited a foliated manifold of dimension 3 havingonly one leaf.

• Connected non-Hausdorff manifolds play an important role in de-scribing foliations of the plane R

2. I shall describe how this worksand how one may use the theory of such manifolds to find a rigidfoliation of R

2. This is joint work with Paul Gartside and SinaGreenwood.

• Another reason that a manifold is non-metrisable is that it is toolarge in some sense. With Mathieu Baillif and Alexandre Gabard Ihave been studying foliations on surfaces based on the long ray. Wehave found that in contrast with the case of R

2 there is very littlefreedom.

Supported by the Marsden Fund Council from Government funding,administered by the Royal Society of New Zealand.

Shadowing, chain transitivity, expansivity and ω-limit sets.Chris Good (University of Birmingham, UK)

Abstract

Abstract: Let X be a compact metrioc space and let f : X → X becontinuous. The ω-limit set ω(x) of a point x is the set of limit points ofsequences of the form fnk (x) for some sequence of natural numbers (nk).

3

It is well known that ω-limit sets have the property that for any open setU , the closure of the image of U is not a subset of U .

In this talk we look at conditions under which this property charac-terizes ω-limit sets. It turns out these ideas are closely related to otherwell known proeprties of dynamical systems.

Hardy spaces associated with Schrodinger operators onthe Heisenberg group

Chin-Cheng Lin (National Central University, Taiwan)

Abstract

Let L = −∆Hn +V be a Schrodinger operator on the Heisenberg groupH

n, where ∆Hn is the sub-Laplacian and the nonnegative potential V be-longs to the reverse Holder class B Q

2

and Q is the homogeneous dimension

of Hn. The Riesz transforms associated with the Schrodinger operator L

are bounded from L1(Hn) to L1,∞(Hn). The L1 integrability of the Riesztransforms associated with L characterizes a certain Hardy type space de-noted by H1

L(Hn) which is larger than the usual Hardy space H1(Hn). Wedefine H1

L(Hn) in terms of the maximal function with respect to the semi-group

˘e−sL : s > 0

¯, and give the atomic decomposition of H1

L(Hn).As an application of the atomic decomposition theorem, we prove thatH1

L(Hn) can be characterized by the Riesz transforms associated with L.

This talk is based on joint work with Heping Liu and Yu Liu.

The Governing PDE’s for nonlinear material’s science,conformal geometry and the Hilbert Smith Conjecture.

Gaven J. Martin (Massey University, New Zealand)

Abstract

The governing equations for the theories of conformal geometry andnon-linear materials science are basically the same and have been muchstudied for centuries. Despite some significant advances in the theory ofthese non-linear elliptic equations there is still much to be considered.Here we discuss a little of what is known about these equations leading toa fascinating connection with Hilbert’s fifth problem on Lie groups (fromhis famous list of 23 such problems in 1900). We will present the latestand best result toward a solution of this problem.

This talk will cover a broad sweep of modern mathematical ideas andconnections between Analysis, PDE, Geometry, Algebra and Group theorypainted with a broad brush and with few technical details.

Computational Noise in Simulation-Based OptimizationProblems

Jorge More (Argonne National Laboratory, USA)

Abstract

4

The simulation of complex scientific phenomena often requires thedetermina- tion of model parameters that optimize design criteria or thatmatch the model with the experimental data. These complex optimizationproblems arise in important scientific ap- plications (accelerator design,chemistry, nuclear physics, climate modeling, groundwater remediation,fusion, . . . ), and are often of the form

minF [s(x)] : xLxxU,

where the mapping s : Rn → R

m describes the simulation as a function ofthe parameters (or controls) x, and the mapping F : R

m → R defines theobjective in terms of the simulation. These optimization problems can beformulated in terms of the bound-constrained optimization problem

minf(x) : xLxxU,

where f : Rn → R is defined by f(x) = F [s(x)]. However, this optimiza-

tion problem is not amenable to standard gradient-related optimizationalgorithms due to the computa- tional noise and lack of smoothness in-troduced by the simulation. Computational noise also impacts sensitivityanalysis and any application that depends on the simulation. We outlinea theoretical framework for studying computational noise in simulation-based optimization problems. Our approach defines computational noisein terms of a stochastic model, but we show that this model applies todeterministic optimization problems. We present an algorithm for deter-mining the computational noise in a simulation and show that in mostcases we can reliably determine the computational noise with 8 additionalevaluations of the simulation. We also discuss the theoretical and algo-rithmic implications of computational noise. Computational experimentswith simulation-based optimization problems are used to illustrate theimplications of the theory.

2 Applications of Analysis

The boundary layer inside a rotating circular cylinderOmar A. Abdulwanis (Al-Fateh University, Libya)

Mohamed M. Arebi (Seventh of April University, Libya)

Abstract

Here we consider a boundary layer, initially of zero thickness, whichforms inside a rotating pipe, this layer increases in thickness downstreamuntil it becomes comparable to the radius of the pipe, the on coming flowcomprises uniform stremwise component plus a swirl. Since the flux offluid a cross any section is constant, as the boundary-layer thickness in-creases, the core of the flow is accelerated, and there is a correspondingfall of pressure. Far down stream from the entrance to the pipe the flowassumes the poisuille parabolic distribution, and theoretically this is at-tained asymptotically far down stream. It was shown in Pedly (1968),that a cylindrically symmetric shear flow of an incompressible fluid, such

5

as poiseuille flow in a circular pipe, is unstable to non-axisymmetric invis-cid disturbances when subjected to a rapid, almost rigid, rotation aboutits axis. Also Pedley (1969) examined the instability of rotating poiseuilleof this type. Beran & Culick (1992) studied viscous swirling flows throughpipes of constant radius and also circular pipes whit throats. They com-puted numerical solutions of both the Navier-Stokes equations and thequasi-cylindrical equations, for several values of vortex circulation. In ourwork, a study is made of the motion of an incompressible viscous fluidthrough a rotating circular cylinder, whose leading edge is perpendicu-lar to a freestream flow which also involves a swirling component togetherwith a uniform streamwise velocity component. The associated boundary-layer flow is computed using the quasi cylindrical equations.

Difference Equations: A Gentle InvitationRaghib Abu-Saris (Walden University, USA)

Abstract

We review some of the recent developments pertinent to long termbehaviour of difference equations. In particular, we focus our attentionon the asymptotic behaviour of nonlinear difference equations, existenceof invariants and their relationship with periodic behaviour and Gausscompounding of means, and chaotic behaviour. This talk will culminatein presenting some open problems and research project.

SYSTEM OF GENERALIZED H-RESOLVENTEQUATIONS WITH CORRESPONDING SYSTEM OF

GENERALIZED VARIATIONAL INCLUSIONSRAIS AHMAD (Aligarh Muslim University, India)

Abstract

The aim of this paper is to introduce a new system of generalizedH-resolvent equations in uniformly smooth Banach spaces and we alsomention the corresponding system of variational inclusions. An equiv-alence relation is established between system of generalized H-resolventequations and system of variational inclusions. We also prove the ex-istence of solutions for the system of generalized H-resolvent equationsand the convergence of iterative sequences generated by the algorithm.Our results are new and generalize many known results appeared in theliterature.

Keywords: Generalized H-resolvent equations, system, variational in-clusions, algorithm, convergence.

On the Weakly Nonlinear Development ofTollmien-Schlichting Wavetrains over compliant Surfaces

Mohamed M. Arebi (Seventh of April University, Libya)Omar A. Abdulwanis (Al-Fateh University, Libya)

Fatma Hasan Elshiek (Seventh of April University, Libya)

Abstract

6

The nonlinear development of a weakly modulated Tollmien-SchlichtingWavetrains over a compliant boundary is studied theoretically using high-Reynolds-number asymptotic methods. The carrier wave is taken to betwo-dimensional, and the envelope is assumed to be a slowly varying func-tion of time and of the streamwise and spanwise coordinates. Attentionis focused on the scaling appropriate to the so called upper branch andhigh-frequency lower branch. The dominant nonlinear effects are foundto arise in the critical layer and the surrounding diffusion layers: in theseregions nonlinear interactions influence the development of the wavetrainby producing a spanwise-dependent mean-flow distortion. The amplitudeevolution is governed by integro-differential equation involves the highestderivative with respect to spanwise position, and an additional term dueto compliant boundary is obtained.

References[1] Arebi, M. A. & Abdulwanis, O. A. 2006. Nonlinear evolution ofwaves in a two dimensional boundary layer flows over a compliant wall.CMS2006 Zarqa Private University, Jordan. .[2] Carpenter, P. W. 1990. Status of transition delay using compliantwalls. Visc. Drag Red. In Boundary layers (ed. D. M. Bushnell & J. N.Heffner) AIAA, NY. 79-113.[3] Carpenter, P. W. and Gajjar, J. S. B. 1990. A general theory fortwo- and three-dimensional wall mode instabilities in boundary layers overisotropic and anisotropic compliant walls. Theor. Comp. fluid dyn. 1,349-378.

ON STABILITY OF ABSTRACT MEASURE DELAYINTEGRO-DIFFERENTIAL EQUATIONS

S. S. Bellale (Bellale’s Maths Institute, India)

7

Abstract

In this paper existence theorem for the second order functional integro-differential equations in Banach algebras is proved under the mixed gener-alized Lipschitz and Caratheodory conditions. The existence of extremalsolutions is also proved under certain monotonicity conditions.

Soliton Perturbation Theory for Phi-Four Model andNonlinear Klein-Gordon EquationsAnjan Biswas (Delaware State University, USA)

Ryan Sassaman (Delaware State University, USA)

Abstract

This talk is on the study of adiabatic variation of the soliton velocity,in presence of perturbation terms, of the phi-four model and the nonlinearKlein-Gordon equations. There are four types of models of the nonlinearKlein-Gordon equation, that will be talked about. The soliton perturba-tion theory is utilized to carry out this investigation.

Parabolic problems with discontinuous nonlinearities andnonlocal conditions

Abdelkader Boucherif (KFUPM, Saudi Arabia)

Abstract

In this talk I shall present a study of a class of parabolic problems withdiscontinuous nonlinearities and subjected to nonlocal conditions. Morespecifically, I consider the following problem

ut + Lu + F (x, t, u), (t, t) ∈ Ω × (0, T ]

u(x, t) = 0, (x, t) ∈ Ω × (0, T ]

u(x, 0) =

Z T

0

φ(t)u(x, t)dt, x ∈ Ω (1)

where Ω is an open bounded domain in RN , N ≥ 2 with a smoothboundary, and L is a strongly elliptic operator, F : Ω× (0, T )×R → R issuch that F (., ., u) is measurable and F (x, t, .) is of bounded variations onevery compact interval in R, and φ is a nonnegative continuous functionon [0, T ].

Objective: provide sufficient conditions on L, F , φ that will guaranteethe existence of at least one solution.

A Mathematical Approach on Field Equations with ACase for Higher Dimensional Cosmological Models

R K Dubey (A P S University, India)Abhijeet Mitra (Govt. Science P G College, India) and

Bijendra Kumar Singh (Govt. Science P G College, India)

Abstract

8

A mathematical solution to Einstein’s field equations with a perfectfluid source, with variable constant G and Cosmological constant Λ forFRW space-time in higher dimensions is gravitational obtained and casestudy has also been done where the values of ρ(t), G(t), Λ(t), T (t), q(t)and dH(t) has been obtained and their nature is also analysised.

Reducing the composition of mappings in Torus basedCryptosystems

G.Geetha (Sathyabama University, India)Abdul Hameed (Ibra College of Technology, Oman)

Abstract

Encryption, a primary method to protect valuable electronic infor-mation ensures secrecy or confidentiality of information transmittedacross an insecure communication channel.

In 1976 Whitefield Diffie and Martin Hellman introduced the idea ofpublic key cryptography. The security of the Diffie-Hellman protocol relieson the presumed intractability of the Discrete-log problem. The discretelogarithm problem (DLP) is computationally feasible in finite fields ofsmall size, so in order for the Diffie-Hellman protocol to be secure, thebit-size of the field, n log q must be large. Since the Diffie-Hellman pro-tocol transmits finite fields elements, field sizes large enough to ensurecryptographic security also increase the cost of bandwidth significantly.

Cryptosystems that transmit as little data as possible are valuable.But, smaller the amount of information communicated, the easier it is foran attacker to compromise system security. Torus-based cryptosystemsimprove on conventional cryptosystems by representing some elements oflarge finite fields compactly, and therefore they transmit fewer bits.

At Crypto 2004, van Dijk and Woodruff introduced a new way ofusing the Algebraic Tori Tn in cryptography, and obtained an asymptot-ically optimal n/φ(n) savings in bandwidth and storage for a number ofcryptographic applications. However, the computational requirements ofcompression and decompression in their scheme were impractical, and itwas left open to reduce them to a practical level. Marten van Dijk,Robert Granger et al., suggested a new method that compressed or-ders of magnitude faster than the original, while also speeding up thedecompression and improving on the compression factor (by a constantterm)

Brouwer, Pellikaan and Verheul showed that compression can beachieved by going up to an extension field. They conjectured that onecan attain a compression ratio of n/φ(n), where n is the degree of theextension. For n = 2, the LUC cryptosystem already achieved thesesavings. For n = 6, Brouwer et al. described a system that was laterimproved upon by Lenstra and Verheul, resulting in the XTR publickey cryptosystem.

The problem here is to reduce a composition of mappings which willresult in better percentage of compression and decompression process intorus based cryptosystems.

It is to be noted that there is an isomorphism between two algebraicstructures G and iG . We can as well produce an isomorphism between G

9

and its quotient structure. This in turn identifies an isomorphism betweenthe Quotient structure and G. Thus the complexity is reduced to n/φ(n).This is evident from the commutative diagram of the isomorphisms. Thiscan be extended to any algebraic structure in algebraic varieties. In thispaper, Weil restriction on the extension of the finite field is applied, thusreducing the composition which will result in better percentage of com-pression and decompression process. Also Weil restriction on the abelianvariety is used to study the compression and decompression problem inhigh dimensional tori. Our first application is ElGamal encryption witha small message domain.

We obtained an additional 25% compression over CEILIDH even forthe encryption of a single message. Given a fixed finite separable extensionof a field, one can attach in a functorial way to every algebraic varietyassociated with the natural transformation on it. This association helpsus to attack major problems on cryptography. In this paper, the reductionof time and space complexity of compression and decompression processin encoding is elaborated.

References:Bosch, S. Lutkebohmert W.,Raynaud, M, Neron models, Springer Ver-

lag, Berlin 1980.Bosma. W, Hutton.J and Verheul.E.R. Looking beyond XTR,. Asi-

acrypt02, LNCS 2501, 4663.Brouwer A.E, Pellikaan.R and Verheul.E.R, Doing More with Fewer

Bits, Asiacrypt99, LNCS 1716, 321-332.Diffie.W and Hellman M.E, New directions in cryptography, IEEE

trans.Infom. Theory, vol. IT-22(1976), pp 644-654Itoh. T and Tsujii.S, A Fast Algorithm for Computing Multiplicative

Inverses in GF(2m) Using Normal Bases, In Info. and Comp., 78(3),171177, 1988.

Klyachko.A.A, On the Rationality of Tori with Cyclic Splitting Field(Russian) In Arithmetic and Geometry of Varieties, Kuybyshev Univ.Press, 7378, 1988.

Lennon.M.J.J. and Smith.P.J, LUC: A New Public Key System, InIFIP TC11 Ninth International Conference on Information Security IFIP/Sec,103117, 1993.

Lenstra. A.K. and Verheul.E.R. The XTR Public Key System, Crypto00,LNCS 1880, 119.

Lenstra. A.K. and Verheul.E.R. An Overview of the XTR PublicKey System. In Public-Key Cryptography and Computational NumberTheory, Verlages Walter de Gruyter, 151180, 2001.

Marten van Dijk and Woodruff D, Asymptotically Optimal Commu-nication for Torus-Based Cryptography,Crypto04, LNCS 3152, 157178.

Marten van Dijk, Robert Granger, Dan Page, Karl Rubin, Alice Sil-verbert, Martijn Stam and David Woodruff, Practical Cryptography inHigh Dimensional Tori, In: Advances in Cryptology (Eurocrypt 2005),pages 234-250. Springer Verlag LNCS 3494, May 2005.

Rubin K and Alice Silverberg, Torus-Based Cryptography, Crypto03,LNCS 2729, 349365.

Thiyagarajan M, Geetha G, Computational requirement in high di-mensional tori, In the Proc. of International conference on Mathematics

10

and computer science ICMCS 2007, Chennai, pp 367-369VoskresenskiV.E, Algebraic Groups and Their Birational Invariants,

Translations of Mathematical Monographs 179, American MathematicalSociety, 1998.

STRONGLY NONLINEAR AND NONLLOCALELLIPTIC THIRD BOUNDARY VALUE PROBLEMWITH NONLINEAR INTEGRAL CONDITION ON

THE BOUNDARYHASSAN I. M, (Alfateh University, Libya)

Abstract

The aim of this paper is to prove existence of solutions of second orderpartial differential equations with nonlocal boundary conditions i.e. weconsider the problem:(1)

P|α|≤1(−1)α∂α[fα (id, u, u′)] + g (id, u) = F in Ω

(2) ∂∗vu := h1(x, u(x)) + h2(x, u(φ(x)) +

R∂Ω

h3(x, t, u(Ψ(t)))dσt

Where Ω ⊂ Rn is (bounded or unbounded) domain, α = (α1, α2, · · · , αn)

is multindex, |α| =nX

j=1

αj αj ≥ 0 and ∂α = ∂α1

1 · ∂α2

2 . . . ∂αnn , ∂∗

vu :=

X

|α|=1

[fα (id, u, u′)]vα vα denote the coordinates of the normal unite vec-

tor on ∂Ω, φ, Ψ are C1-diffeomorphism in a neighbourhood of ∂Ω suchThat Φ(∂Ω) ⊂ Ω, Ψ(∂Ω) ⊂ Ω, ∂Ω is bounded and belong to C1. Theterms fα(x, ξ) and h2 are required to have polynomial growth with re-spect to the second variable, while in the terms g(x, u(x)), h1(x, u(x)) nosuch growth is imposed, but it supposed that g, h1 satisfy the sign condi-tions g(x, η)η ≥ 0, h1(x, η)η ≤ 0. Linear elliptic equations with nonlocalboundary conditions have been considered by carleman and then by someauthers see ,e.g [3] , [7] and [9] . nonlinear elliptic equatios with non-local boundary conditions have been studied in [2],[10] and [11] .similarproblem with out inteqral term have been considered in [2] and [4] . Theweak solution of (1),(2) will be defined as follows: Assuming that u is aclassical soluation of (1),(2) by using Gauss- ostrogradskij theorem andby as integral trans formation we obtain:

(3)X

|α|≤1

Z

Ω

[fα(x, u, ∂u)]ϑ−

Z

∂Ω

h1(x, u(x))ϑ(x)dσx−

Z

S

eh2(x, u(x))ϑ(φ−1(x))dσx

−

Z∂Ω

Z

Γ

eh3(x, τ, u(τ ))dστ

ffϑ(x)dσx+

Z

Ω

g(x, u(x))ϑ(x)dx =

Z

Ω

Fϑdx =

〈G, ϑ〉∀ϑ ∈ C1(Ω) with bounded support . Thus weak solution of (1) , (2) willbe defined by (3).

Solving mathematical programs with fuzzy equilibriumconstraints

Cheng-Feng Hu (I-Shou University, Taiwan)Fung-Bao Liu (, )

11

Abstract

This work deals with the mathematical programs with fuzzy equilib-rium constraints. It shows that solving the fuzzy MPEC is equivalent tosolving a fuzzy complementarity constrained optimization problem. Byusing the tolerance approach, we show that the fuzzy complementarityconstrained optimization problem can be converted to a regular nonlin-ear programming problem. A new smoothing approach based on entropicregularization is developed for solving the resulting optimization problem.Numerical examples are also included to illustrate the solution procedure.

On properties of geodesic η-preinvex functionsI. Ahmad (King Fahd University, Saudi Arabia)Akhlad Iqbal (Aligarh Muslim University, India)Shahid Ali (Aligarh Muslim University, India)

Abstract

The present paper deals with the properties of geodesic η-preinvexfunctions and their relationships with η-invex functions and strictly geodesicη-preinvex functions. The geodesic η-pre-pseudo-invex and geodesic η-pre-quasi-invex functions on the geodesic invex set are introduced and someof their properties are discussed.

Keywords: Geodesic invex set; Geodesic η-preinvex function; Geodesic η-pre-pseudo-invex function; Geodesic η-pre-quasi-invex function; Rieman-nian manifold.

On a nonlinear nonlocal in time parabolic equationMokhtar Kirane (University of La Rochelle, France)

Abstract

We present, for a nonlocal in time hyperbolic equation, not only theFujita exponent separating global existing solutions from blowing-up ones,but also, the blowing-up profil of a blowing-up solution. Furthemore,necessary conditions on the initial data for the local existence or globalexistence will be put in evidence.

Study of a mathematical model in a Banach spaceRajiv Kumar (Birla Institute of Technology and Science, India)Padma Murali(Birla Institute of Technology and Science, India)

Abstract

In the paper, a new mathematical model of bacterial culture in achemostat is studied in a Banach space .In view of the above, we studya general nonlinear age dependent population dynamics model which is ageneralization of the chemostat model. We prove the existence, unique-ness the semigroup property and the continuous dependence of the solu-tions on the initial data for the general model. Then, we show that thechemo stat model is a particular case of the general model and hence con-clude the wellposedness of the chemostat model. Key words: Partial in-tegrodifferential equation, Banach space, contraction semigroup, Chemo-stat, age dependent model

12

Eigenvalue Approach to Three Dimensional CoupledThermoelasticity in a Rotating Transversely Isotropic

MediumA. Lahiri (Jadavpur University, India)

Abstract

The theory of coupled thermoelasticity in three dimensions is em-ployed to determine the distribution of temperature and stresses in aninfinite medium having an instantaneous point heat source at the originin a rotating medium. Laplace transform along with the double Fouriertransforms have been applied in the basic equations of coupled thermoe-lasticity and finally the resulting equations are written in the form of avector-matrix differential equation which is then solved by eigenvalue ap-proach. The inversion of the Laplace transform solution is carried outby applying Bellman method and computations have been done by usingMATLAB software. Numerical computations of temperature and stresseshave been made in space time domain and presented graphically.

n-Orthogonality in n-normed spacesH. Mazaheri (Yazd University, Iran)

Abstract

In 1965, Gahler introduced the idea of 2-normed spaces by a publi-cation in Math. Nachr. entitled Linear 2-normietre Raume. In 1989,by another publication in Math. Nachr. entitled n-inner product spaces,Misiak developed the idea of n-inner product spaces. Since then manyothers have studied these concepts and obtained various results. In thisarticle we introduce the idea of constructing n-best approximation andn-orthogonality with elements in an n-norm space.

References

[1] Y. J. Cho, M. Matic, J. E. Pecaric, On Grams determinant in 2-innerproduct spaces, J. Korean Math. Soc., 38 (6) (2001) 1125-1156.

[2] Z. Lewandowska, Linear operators on generalized 2-normed spaces,Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 42(90) (1999)] no. 4,353-368.

[3] Z. Lewandowska, Generalized 2-normed spaces, Supskie Space Matemayczno Fizyczne 1 (2001), 33-40.

A singular Gierer-Meinhardt system of elliptic equationsin mathbbRN

Abdelkrim MOUSSAOUI (A. Mira BEJAIA University, Algeria)Brahim KHODJA (ANNABA University, ALGERIA)Saadia TAS (A. Mira BEJAIA University, Algeria)

Abstract

13

The nonlinear Gierer-Meinhardt systems have received considerableattention in the last decade. They arise in the study of biological patternformation by auto and cross catalysis in biochemical processes. Gierer-Meinhardt equations model the interaction between activators and in-hibitors. If the activators, u (x, t), have a source distribution ρ (x) andthe inhibitors, v (x, t), have a source distribution ρ′ (x), and assumingthat both act proportionally to some powers of u and v in the sourceterm, the general model proposed by Gierer-Meinhardt may be written as

8<:

ut = d1∆u − αu + cρ uk

vq + ρ0ρ in Ω × [0, T ]

vt = d2∆v − βv + c′ρ′ ur

vs in Ω × [0, T ] ,

(2)

under Neumann boundary conditions. Here Ω is bounded domain of RN

(N ≥ 1), d1, d2 are diffusion coefficients with d1 ≪ d2 and α, β, c, c′ andρ0 are positive constants. The exponents k, q, r, s ≥ 0 satisfy the relationqr > (k − 1) (s + 1).

In this paper, we are interested in examining the existence and unique-ness of solutions of elliptic system (2) in the stationary case, where Ω =R

N (N ≥ 3), ρ (x) = 0, the exponent k = 0 while the nonnegative param-eters α and β are both in L∞

loc

`R

N´

and satisfy

α (x) ≥ α0, β (x) ≥ β0 for |x| ≥ R for some a0, β0, R > 0. (3)

In other words, we are concerned with the following problem:8>>>>>>>><>>>>>>>>:

−∆u + α (x)u = h1 (x) 1vq in R

N

−∆v + β (x) v = h2 (x) ur

vs in RN

u (x) , v (x) −→ 0 as |x| −→ ∞

u, v > 0 in RN ,

(4)

where hi ∈ L2 ∩ Lθi , hi 6= 0, i = 1, 2 and θ1 = 21+q

, θ2 = 21+s

. We assumethat the exponents q, r > 0, s ∈ ]0, 1[ such that r − s ≤ 1 and h1, h2 arenonnegative L∞

loc

`R

N´

functions. Our demonstration strategy will be toshow – by applying Schauder’s fixed point theorem – that the regularizedsystem of (4) has a solution uε, vε ∈ W 2,p

loc

`R

N´, 1 < p < ∞, and then

derive a solution of (4) by passing to the limit.

References

[1] C. O Alves, J. V. Goncales and L. A. Maia, Singular nonlinear ellipticequations in R

N , Abstract and Applied Analysis 03 (1998), 411-423.

[2] Y. S. Choi and P. J. McKenna, A singular Gierer-Meinhardt systemof elliptic equations, Ann. Inst. H. Poincare, Anal. Non Lineaire 17(2000), 503-522.

[3] Y. S. Choi and P. J. McKenna, A singular Gierer-Meinhardt systemof elliptic equations: the classical case, Nonlinear Anal. 55 (2003),521-541.

14

[4] M. Del Pino, M. Kowalczyk and X. Chen, The Gierer-Meinhardtsystem: the breaking of homoclinics and multi-bump ground states,Commun. Contemp. Math. 3 (2001), 419-439.

[5] M. Del Pino, M. Kowalczyk and J. Wei, Multi-bump ground statesof the Gierer-Meinhardt system in R

2, Ann. Inst. H. Poincare, Anal.Non Lineaire 20 (2003), 53-85.

[6] M. Ghergu and V. Radulescu, On a class of Gierer-Meinhardt systemsarising in morphogenesis, C. R.Acad.Sci. Paris, Ser.I 344 (2007).

[7] A. Gierer and H. Meinhardt, A theory of biological pattern formation,Kybernetik 12 (1972), 30-39.

[8] E. H. Kim, A class of singular Gierer-Meinhardt systems of ellipticboundary value problems, Nonlinear Anal. 59 (2004), 305-318.

[9] H. Meinhardt and A. Gierer, Generation and regeneration of sequenceof structures during morphogenesis, J. Theoret. Biol. 85 (1980), 429-450.

[10] W. M. Ni and I. Takagi, On the shape of least-energy solutions to asemilinear Neumann problem, Comm. Pure Appl. Math. 44 (1991),819-851.

[11] W. M. Ni and I. Takagi, Locating the peaks of least-energy solutionsto a semilinear Neumann problem, Duke Math. J. 70 (1993), 247-281.

[12] W. M. Ni and J. Wei, On positive solutions concentrating on spheresfor the Gierer-Meinhardt system, J. Differential Equations 221(2006), 158-189.

[13] J. Wei, On the interior spike layer solutions for some singular pertur-bation problems, Proc. Roy. Soc. Edinburgh section A 128 (1998),849-874.

[14] J. Wei and M. Winter, Spikes for the Gierer-Meinhardt system intwo dimensions: the strong coupling case, J. Differential Equations178 (2004), 478-518.

[15] J. Wei and M. Winter, Existence and stability analysis of asymmetricfor the Gierer-Meinhardt system, J. Math. Pures Appl. 83 (2004),433-476.

On the control of solutions of a coupled system ofnonlinear viscoelastic equations

Salim A. Messaoudi (King Fahd University of Petroleum and Minerals, SaudiArabia)

Muhammad Islam Mustafa (Prince Sultan University, Saudi Arabia)

Abstract

Due to its importance and application in material science, the study ofviscoelasticity has attracted a great deal of attention. Engineers, Mathe-maticians and Scientists have considered several models and used differentkinds of damping mechanisms to control and obtain stability of the sys-tems governing the motion of these materials. The issue of the rate of

15

decay of these motions was always an important issue while studying thestability and decay of solutions of these systems. In most literature, therelaxation functions were considered to be of either polynomial or expo-nential decay only. Lately, few works allowed a wider a class of decayingrelaxation functions. In this paper we consider a system, of two nonlinearcoupled viscoelastic equations, which describes the interaction betweentwo different fields arising in viscoelasticity. For a wider class of relax-ation functions, we establish a general stability result for this system, fromwhich the usual polynomial or exponential decay are only special cases.Keywords: nonlinear, general decay, relaxation function, stability, vis-coelastic material.

A Brief History of the Development of Analysis Topics inthe Twentieth Century

Abdul-Majid Nusayr (Jordan University of Science and Technology, Jordan)

Abstract

Analysis is one of the oldest topics that received much of the mathe-maticians attention since the seventeenth century. It had been refined andwidened by time. Its rigor was well established in the nineteenth century.Also, so many topics were invented or discovered within the ever expand-ing tent of analysis. In spite of the great strides that were taken, thatdid not mean that mathematical analysis was closed for more expansionand depth. As the human genius and invention know no boundaries orpermanent halts, then analysis continued to expand and got more depthin the twentieth century. This article throws some light on the develop-ment of different topics, old and new in analysis, in the twentieth century.However, this is not a survey of this development. That is a great andimportant project not yet dealt with as far as I know. It needs hundredsof pages, and should be written by a group of experts in the different fieldsof analysis. The writer will stick to a brief historical development withoutgoing into details or covering every important aspect in any field. In termsof titles, the writer will cover the following fields: real numbers, founda-tion of arithmetic, functions, ordinary differential equations, calculus ofvariations, functional analysis and potential theory.

HAMILTON-JACOBI FORMULATIONS FORSYSTEMS IN TERMS OF RIESZ FRACTIONAL

DERIVATIVESEqab M. RABEI (AL al-Bayt University, Jordan)

IBRAHIM M. RAWASHDEH (AL al-Bayt University, Jordan)

Abstract

The traditional calculus of variations is extended to be applicable forsystems containing fractional derivatives. This paper presents fractionalLagrangian and fractional Hamiltonian for systems containing Riesz frac-tional derivatives (RFDs). The Hamilton’s Equations of motion are de-fined in terms of (RFDs). Besides, The Hamilton-Jacobi formulations forthese systems are developed. An illustrative example for harmonic oscil-lator has been discussed. The classical results are recovered for integerorder derivative.

16

Eigenvalues Calculation of Sturm-Liouville Problem inBessel’s Equation

Ayad Muftah Shahoot (Mergib University, Libya)

Abstract

The aim of the study is to calculate eigenvalues of Sturm-Liouvilleproblem emerged from solving third boundary problem, applied on heatflow through hollow cylindrical bodies. The encountered difficulty is man-ifested in the calculation of the primary eigenvalues of Bessel’s equation.This study is a continuation of a previous study to solve the same prob-lem using alterative roots property of Bessel’s equation. The early effortsbased on trial and error process and guessing to determine eigenvalueswere time consuming. Changing the cylinder thickness requires furthertrail and approaching numerical solutions is even more problematic asguesses can only be made based on visual observations of the graphicalrepresentation of Bessel’s function roots. This study suggests representingBessel’s functions by Taylor’s series. A comparison of the results showshigh consistency on order of 10−8.

Harvesting in Discrete Population Models: OpenProblems and Conjectures

Ziyad M. Al-Sharawi (Sultan Qaboos University, Oman)

Abstract

In this talk, we discuss several harvesting strategies on discrete pop-ulation models. Since the theory of difference equations is not well-established as in differential equations, investigating the dynamics of dis-crete models in general, and under the effect of harvesting in particular isa challenging task. We consider the Beverton-Holt and Pielou’s models,and give several unanswered questions and conjectures.

Remark: This talk is about ongoing collaborative work with R. Abu-Saris and M. Rhouma.

REMARKS ON A NONQUADRATIC HAMILTONIANELLIPTIC SYSTEMS

SHILGBA Leonard Karshima (American University of Nigeria, Nigeria)

Abstract

In this paper, we study the problem:

−∆u = δu + γ|v|β−2v +∂H

∂v(x, u, v)

−∆v = δv + λ|u|α−2u +∂H

∂u(x, u, v)

in a smooth bounded domain Ω ∈ RN , N ≥ 3, subject to Dirichlet bound-

ary conditions. The tool we have used to prove existence of solutions isthe Rabinowitz linking theorem, Theorem 5.29 in Rabonowitz’s Minimax

methods in critical point theory with applications to differential equations,CBMS, 65 (1986). This problem has been studied by D.G. Figueiredoand C. A. Magalhaes in their paper, On Nonquadratic Hamiltonian Sys-tems, Adv. in Diff. Eq.1, 5 (1996), 881-898, with a restriction to whenγ, λ > 0. We have extended our study to the case when γ, λ < 0. Fur-thermore, we have made some important remarks, which further elucidatethe subject. For compactness, we have used an Alama-Tarantello type ofgrowth restriction on the Hamiltonian H.

Keywords: Nonquadratic, Hamiltonian, elliptic.

On an abstract second order fractional differentialequation

Nasser-eddine Tatar (King Fahd University of Petroleum and Minerals, SaudiArabia)

Abstract

Of concern is an abstract general second order differential equation in-volving different derivatives of non-integer order in the nonlinearity. Thisproblem covers many of the well-known problems involving integer or-der derivatives. Some of the problems which arise in applications will bediscussed. We establish some existence and uniqueness theorems for dif-ferent types of nonlinearities. Moreover, we introduce nonlocal conditionsof fractional type. The key tools are the cosine families and some fixedpoint theorems. Cosine families are the equivalent of semigroups for thecase of a differential equation of first order.

Fractal Image CompressionF. Ghezelbash (Islamic Azad University, Iran)A. Tavakoli (Islamic Azad University, Iran)

Abstract

Michael Barnsley led development of fractal compression in 1987, andholds several patents on the technology. The most widely known practi-cal fractal compression algorithm was invented by Barnsley and AlanSloan. Barnsleys graduate student Arnaud Jacquin implemented the firstautomatic algorithm in software in 1992. All methods are based on thefractal transform using iterated function systems. Michael Barnsley andAlan Sloan formed Iter- ated Systems Inc.in 1987 which was granted over20 additional patents related to fractal compression. A major break-through for Iterated Systems Inc. was the automatic fractal transformprocess which eliminated the need for human intervention during com-pression as was the case in early experimentation with fractal compres-sion technology. In 1992 Iterated Systems Inc. received a 2.1 milliongovernment grant to develop a prototype digital image storage and de-compression chip using fractal transform image compression technology.To do fractal compression, the image is divided into sub-blocks.then foreach block, the most similar block if found in a half size version of the

18

image and stored. this is done for each block. then during decompression,the opposite is done iteratively to recover the original image. We applyAffine transforms to compression and decompression and introduce a fastalgorithm for converges. in fact, in this method we represent images as afixed point of a contractive iterated function system.

On the qualitative behaviors of solutions to theLinard-type equations with adeviating argument

Cemil Tunc (Yuzuncu Yl University, Turkey)

Abstract

In this paper, we establish some new results related to the stability,boundedness and existence of periodic solutions of some Linard type equa-tions with a deviating argument. We use the Lyapunovs second method toprove our results, and examples are given to illustrate the importance ofthe results. Our results include and improve some recent papers publishedin literature.

References

[1] Liu, B. and Huang, L. Boundedness of solutions for a class of re-tarded Linard equation. J. Math. Anal. Appl. 286 (2003), no. 2, 422-434.[2] Liu, B. and Huang, L. Boundedness of solutions for a class of Linardequations with a deviating argument. Appl. Math. Lett. 21 (2008), no.2, 109-112.[3] Liu, C. J. and Xu, S. L. Boundedness of solutions of Linard equations.J. Qingdao Univ. Nat. Sci. Ed. 11 (1998), no. 3, 12-16.[4] Tun, C., Some stability and boundedness results to nonlinear differen-tial equations of Linard type with finite delay. J. Comput. Anal. Appl.11(4), (2009), 711-727.[5] Tun, C., Asymptotic stable and bounded solutions of a kind of non-linear differential equations with variable delay. Functional DifferentialEquations, 2009, (accepted for publication).[6] Tun, C., On the existence of periodic solutions to nonlinear third or-der ordinary differential equations with delay, Journal of ComputationalAnalysis and Applications, (2009), (accepted for publication).[7] Tun, C. and Tun, E. On the asymptotic behavior of solutions of cer-tain second-order differential equations. J. Franklin Inst., Engineeringand Applied Mathematics 344 (5), (2007), 391-398.

3 Complex Analysis

Order of Magnitude of Coefficients of CliffordPolynomials in Simple BasesM. Abul-Ez(Sohag University, Egypt)

D. Constales(Ghent University, Belgium)M. Zayed(King Khalid University, Saudi Arabia)

Abstract

19

n this paper we investigate the relation between the order of magnitudeof coefficients of Clifford polynomials and the convergence properties ofthe simple bases to which these Clifford polynomials belong. Two typesof restrictions on the coefficients are here considered; the first restrictionleads to the effectiveness of the base in closed balls of sufficiently largeradii, while the other implies the boundedness of the order of the base

On ϕ-normal functionsRauno Aulaskari (University of Joensuu, Finland), Shamil Makhmutov

(Sultan Qaboos University, Oman) and Jouni Rattya (, )

Abstract

Let ϕ : [0, 1) → (0,∞) be an increasing function such that ϕ(r)(1 −r) → ∞, as r → 1−. A meromorphic function f in the unit disk belongsto the class Nϕ of ϕ-normal functions if its spherical derivative satis-fies f#(z) = O(ϕ(|z|)) as |z| → 1−. We study meromorphic ϕ-normalfunctions, and in particular an analogue of the Lohwater-Pommerenketheorem, and several equivalent characterizations for ϕ-normal functionsare established under certain regularity conditions on ϕ.

Convex null sequence technique for analytic and univalentmappings of the unit disk

K. O. BABALOLA (University of Ilorin, Nigeria)

Abstract

In this paper we employ a technique based on the convex null proper-ties of certain infinite sequences to study various classes of analytic andunivalent functions in the open unit disk. The technique simplifies manyproblems of the theory of geometric functions and our results generalizeand extend many earlier ones.

Composition operators on normal functionsMahmoud Ali Bakhit (University of Alazhar, Egypt)

Abstract

In this paper we gave a function-theoretic characterization of compo-sition operators sending (meromorphic) normal classes to their Mobiusinvariant subclasses. The boundedness and compactness of compositionoperator Cφ from normal classes to the class Q#

K are investigated. Thecompactness of the composition operator from normal classes to the classQ#

α are completely characterized.

AAK-theory for generalized Hankel operatorsMarcus Carlsson (Purdue University, USA)

Abstract

20

An algorithm from 2004 by G. Beylkin and L. Monzon, aimed at ap-proximating a function by a sum of few exponentials, is shown to be aconsequence of the Adamyan, Arov and Krein theorem on Hankel oper-ators. The algorithm does however work in greater generality than theAAK-theory supports, which has necessitated further research in this area.We present a number of new theorems and open problems, in particular wewill give an improvement of an AAK-type theorem by Treil and Volberg,concerning Hankel operators on weighted spaces.

The escaping set for quasiregular mappingsAlastair Fletcher (University of Warwick, UK)

Abstract

Complex analysis has in recent years been used to great effect in thefield of complex dynamics. The escaping set plays a central role in complexdynamics and, in this talk, we will see some ideas that carry over tothe higher dimensional world of quasiregular dynamics. In particular, ifa quasiregular mapping is of polynomial type and the degree is largerthan the distortion, then the boundary of its escaping set is a perfect set(compare with the Julia set).

This talk is based on joint work with Dan Nicks (Nottingham).

Local ABC theorems for analytic functionsKonstantin Dyakonov (University of Barselona, Spain)

Abstract

The classical abc theorem for polynomials (often called Mason’s theo-rem) deals with nontrivial polynomial solutions to the equation a+ b = c.It provides a lower bound for the number of distinct zeros of the poly-nomial abc in terms of deg a, deg b, and deg c. We prove some abc typetheorems for general analytic functions on a (reasonable) simply connecteddomain. Mason’s theorem is then derived as a corollary.

A Non-discrete Convergence Group of QuasiconformalMappings

Jianhua Gong (United Arab Emirates University, UAE)

Abstract

Convergence groups are introduced by Gehring and Martin in [4], andhave found wide application in geometric group theory and low-dimensiontopology and geometry [1, 2, 3, 10]. Quasiconformal mappings are almosteverywhere differentiable mappings on a subdomain of the extended Eu-clidean space. They map infinitesimal circles to infinitesimal ellipses whichhas the property that the ratio of the major to minor axes is uniformlybounded from above [5,6,7].

Groups here are topological groups of homeomorphisms, each of whichis a K−quasiconformal mapping, under composition with respect to thecompact–open topology which is equivalent to the topology induced fromlocally uniformly convergence. A subgroup G is called discrete if theinduced topology coincides with the discrete topology [8, 9,10].

We will a non-discrete convergence group of quasiconformal mappingsin this talk.

21

References

[1] . H. Bowditch, A topological characterization of hyperbolic groups,J. Amer. Math. Soc. 11(1998) 643-667.

[2] . Casson and D. Jungreis, Convergence groups and Seifert fibered3-manifolds, Invent. Math. 118 (1994) 441-456.

[3] . Gabai, Convergence groups are Fuchsian, Ann. of Math. (2) 136(1992) 447-510.

[4] . W. Gehring and G. Martin, Discrete quasiconformal groups I, Proc.London Math. Soc. (3) 55 (1987) 331-358.

[5] . Gong and G. Martin, Aspects of quasiconformal homogeneity, NewZealand J. Math., accepted.

[6] . Gong, Quasiconformal homogeneities, Verlag Dr. Muller (VDM),German, 2008.

[7] . Gong, Non-elementary K-quasiconformal groups are Lie groups,Acta Mathematica Academiae Paedagogicae Nyıregyhaziensis, 24(2008), 367–371.

[8] . Hinkkanen and G. Martin, Limit functions for convergence groupsand uniformly quasiregular maps, J. London Math. Soc. (3) 73 (2006)716-726.

[9] . Hinkkanen and G. Martin, Abelian non-discrete convergence groupsin the plane, Ann.Acad. Sci. Fenn. Math. 19 (1994) 205-246.

[10] . Tukia, Convergence groups and Gromov’s metric hyperbolic spaces,New Zealand J. Math. 23 (1994) 157-187.

Weighted spaces of monogenic functionsKlaus Gurlebeck (Bauhaus-Universitat, Germany)

Abstract

Monogenic functions are defined in domains of Euclidean spaces ofarbitrary dimensions as null solutions of a generalized Cauchy-Riemannoperator or of the Dirac operator. A popular example are solutions ofthe Riesz system in R3. For such functions different representations inform of Taylor or Fourier expansions are known. We introduce in the talka recently developed representation based on an orthogonal Appell basisof monogenic polynomials. This representation is used to characterizefunctions belonging to weighted Dirichlet spaces, Qp-spaces and Bergmanspaces by means of the behaviour of their Taylor or Fourier coefficients,respectively.

Boundary behavior of special classes of functionsMubariz Tapdygoglu Karaev (Suleyman Demirel University, Turkey)

Abstract

We give a concrete example of a diagonal operator acting in the Hardyspace H2 (D) for which the Berezin symbol has radial limits at no point of

22

the boundary ∂D. We use the Berezin symbol technique in the discussionof several old problems from the classical book of I.I. Privalov related withthe Taylor coefficients and boundary behavior of analytic functions. In

particular, we give in terms of Taylor coefficientsn

bf (n)o

and Berezin

symbols necessary and sufficient conditions ensuring existence of radialboundary values of the functions f (z) =

Pn≥0

bf (n) znfrom the classeslpA (D) , 0 < p ≤ ∞. Some other questions are also discussed.

On rationality of the generating function of the solution ofa two-dimensional difference equation

Alexander P. Lyapin (, Country)

The work was supported in part by the Russian Foundation of Basic Research(RFBR) and by the National Natural Science Foundation of China (NSFC) inthe framework of the bilateral project ¡¡Complex Analysis and its applications¿¿(project No. 08-01-92208 GFEN).

Abstract

In this note we give a formula for the generating function of the solu-tion of a two-dimensional difference equation under the assumption thatthe generating function of the initial data is known. We also state thenecessary and sufficient condition for rationality of the generating func-tion.

Let C = α, where α = (α1, . . . , αn), be a finite subset of the positiveoctant Z

n+ of the integer lattice Z

n and let m = (m1, m2, . . . , mn) ∈ C.Moreover for all α ∈ C the condition

α1 6 m1, . . . , αn 6 mn (∗)

be fulfilled. The problem Cauchy is to find the solution f(x, y) of thedifference equation (we use a multidimensional notation)

X

α∈C

cαf(x + α) = 0, (1)

which coincides with the some given function ϕ : Xm → C on the setXm = Z

n+ \ (m + Z

n+).

Let J = (j1, ..., jn), where jk ∈ 0, 1, k = 1, ..., n, is an ordered setof zeros and ones. With every such set J we associate the face ΓJ of then-dimensional integer parallelepiped Πm = x ∈ Z

n : 0 6 xk 6 mk, k =1, ..., n as follows:

ΓJ = x ∈ Πm : xk = mk, if jk = 1, and xk < mk, if jk = 0.

The generating function

Φ(z) =X

x∈Xm

ϕ(x)

zx+1

of the initial data of the difference equation (1) can be represented as thesum

Φ(z) =X

J

ΦJ (z), where ΦJ (z) =X

τ∈ΓJ

Φτ,J (z), Φτ,J (z) =X

y>0

ϕ(τ + Jy)

zτ+Jy+I.

23

Theorem. The generating function F (z) =P

x∈Zn+

ϕ(x)z−x−1 of the

solution of the difference equation (1) is

F (z)P (z) =X

J

X

τ∈ΓJ

Φτ,J (z)Pτ (z), where Pτ (z) =X

α6m

ατ

cαzα

and P (z) =P

α∈C

cαzα is the characteristic polynomial of the difference

equation (1).Corollary. The generating function F (z, w) of the solution of the

difference equation (1) is rational if and only if the generating functionΦ(z, w) of the initial data is rational.

Real-Part Estimates for Solutions ofthe Riesz System in R

3

K. Gurlebeck (Bauhaus-Universitat, Germany)J. Morais (Bauhaus-Universitat, Germany)

Abstract

The study of estimates for analytic functions and their derivatives isrich from a purely mathematical point of view and provides an indispens-able and powerful tool for solving a wide range of problems. Concreteapplications arise, for instance, in complex dynamics, boundary valueproblems and partial differential equations theory or other fields of physicsand engineering. The estimates in matter, known as ”real part theorems”,attracted the attention of many complex analysts since the classical as-sertion namely Hadamard’s real part theorem (1892). Excellent contribu-tions to this subject have been made, in particular, by Hadamard, Landau,Wiman, Jensen, Koebe, Borel, Riesz, Littlewood, Titchmarsh, Rajagopal,Elkins, Holland, Hayman, Levin and Kresin and Maz’ya. Being such aclassical object of analysis it became recently more and more importantto perform an analogous study in higher dimensions and/or for other par-tial differential equations. One way to generalize complex function theoryto higher dimensional spaces is offered by the Riemann approach whichconsiders (monogenic) functions with values in a Clifford algebra thatsatisfy generalized Cauchy-Riemann or Dirac systems. This approach isnowadays called Clifford analysis.

In view of many applications in physics and engineering, in this lecturewe aim to generalize Hadamard’s real part theorem and invariant forms ofBorel-Caratheodory’s theorem to the three-dimensional Euclidean spacein the framework of quaternionic analysis.

Movable discrete solitonsV.Yu.Novokshenov (Ufa Science Center RAS, Country)

Abstract

We study the phase dynamics of a chain of autonomous, self-sustained,dispersively coupled oscillators. In the quasicontinuum limit the basic

24

discrete model reduces to a Kortevegde Vries-like equation, but with anonlinear dispersion. The system supports “compactons” – solitary waveswith a compact support and kink - antikink pairs that propagate with aunique speed, but may assume an arbitrary width. We demonstrate thatlattice solitary waves, though not exactly compact, have tails which decayat a superexponential rate. They are robust and collide nearly elasticallyand together with wave sources are the building blocks of the dynamicsthat emerges from typical initial conditions. In finite lattices, after a longtime, the dynamics becomes chaotic. Numerical studies of the complexGinzburgLandau lattice show that the non-dispersive coupling causes adamping and deceleration, or growth and acceleration, of compactons.Asimple perturbation method is applied to study these effects.

Weighted Paley-Wiener SpacesPhilippe Poulin (UAE University, UAE)

Abstract

In an explorative paper published in 2002, Professors Kristian Seip andYurii Lyubarskii (NTNU, Norway) investigated a family of de Brangesspaces subject to a natural axiom: in these spaces, the norm of a func-tion is comparable with its L2-norm against ‖kx‖

−2dx, where kx is thereproducing kernel at x. Doing so, they invented the notion of weightedPaley-Wiener space.

From a deep study of the Hermite-Biehler functions associated withweighted Paley-Wiener spaces, they deduced many structural results, whichI revised during my postdoctoral fellowship under their kind supervision.In this talk I will discuss these revised results, with emphasis on an auxil-iary class of de Branges spaces introduced in the same work, which appearsto be more general.

On the Libera OperatorMaria Nowak (Affiliation, Country)

Miroslav Pavlovic (Affiliation, Country)

Abstract

Let H(D) denote the class of all functions holomorphic in the unitdisk D of the complex plane. The Libera operator, L, defined on thespace H(D) by Lg(z) = 1

1−z

R 1

zg(ζ) dζ is the conjugate of the Cesaro op-

erator acting on H(D). We show that the operator L can be continuouslyextended to some known subspaces of H(D). In particular, we prove,using the duality arguments, that L maps boundedly the Bloch spaceinto BMOA. We also obtain some results on the best approximation bypolynomials in Hardy and Bergman spaces.

EXPANSIONS FOR GENERALISEDHYPERGEOMETRIC FUNCTIONS

P . A . Padmanabham (Pondicherry Engineering College, Country)

Abstract

25

Motivated by the fact that Generalized Hypergeometric Functions likeAppell function, Saran functions Lauricella function etc.. have many ap-plications in physical and quantum chemical problems, we drive threesummations formulas for a three variable Hypergeometric function. Spe-cial cases of the main results are also discussed.

Linear differential equations in the unit disc: growth andzeros of solutions

Jouni Rattya (University of Joensuu, Finland)

Abstract

We discuss the growth and the oscillation of solutions of complex lineardifferential equation

f (k) + Ak−1(z)f (k−1) + · · · + A1(z)f ′ + A0(z)f = 0

with analytic coefficients in the unit disc. We give sufficient and necessaryconditions for the coefficients such that (i) all solutions are of order ofgrowth at most α ∈ [0,∞), or (ii) the exponent of convergence of allsolutions is at most α ∈ [0,∞).

Qp classes in Rn

Lino F. Resendis O. (Universidad Autonoma Metropolitana, Country)

Abstract

R. Aulaskari and P. Lappan introduced in 1994 the Qp-spaces, 1 ≤p < ∞ as the set of analytic funtions f : D → C such that

supa∈D

ZZ

D|f ′(z)|2g(z, a)p dx dy < ∞

where g(z, a) is defined by

g(z, a) = ln

˛˛1 − az

a − z

˛˛ ,

that is, the Green’s function of the unit disk with logarithmic singularityat a ∈ D, after Aulaskari, Xiao and Zhao considered 0 < p < ∞.In 1999, K. Guerlebeck, Kahler, Tovar and Shapiro defined Qp-spaces inthe quaternionic case and later Cnops and Delanghe in Clifford Algebras.We present here a generalization that includes all the previous cases.

Totally Positive Matrices and Integral OperatorsAlejandro Rodrıguez-Martınez (Zayed University, Country)

Abstract

In this talk we present an example of the combination of finite di-mensional matrices along with approximation methods to the study ofthe spectrum of certain Composition Operators. In particular, we provethat increasing composition symbols produce non-negative real spectrumand we find a sharp estimate of the trace of these operators. We also

26

present upper and lower sharp bounds for the spectral radius of composi-tion Volterra operators.

The present work is part of the author’s PhD thesis that was directedby A. Montes-Rodrıguez and S.A. Shkarin.

Some Properties of Certain Integral OperatorsPravati Sahoo (Banaras Hindu University, India)

Abstract

The object of the talk is to discuss some interesting properties of theintegral operator

Pαf(z) =(p + 1)α

zΓ(α)

Z z

0

“log

z

t

”α−1

f(t) dt, for α > 0, p ∈ N = 1, 2, . . .,

which was introduced and studied by Jung, Kim and Srivastava [J. MathAnal Appl., 176(1993), 138-147], for the class of all analytic functions f(z)of the form f(z) = z +

P∞n=p+1 anzn, for z ∈ ∆ = z ∈ C : |z| < 1.

2000 AMS Subject Classification: 30C45Keywords: Analytic functions; Differential subordination; Convex func-tions; Hadamard product.

References

[1] I.B. Jung, Y.C, Kim and H.M. Srivastava: The Hardy space ofanalytic functions associated with certain one-parameter families ofintegral operators, J. Math Anal Appl., 176(1993), 138-147.

[2] Y. Komatu: On a one-parameter additive family of operators de-fined on analytic functions regular in the unit disk, Bull. Fac. Sci.Enrgy. Chuo Uni., 22 (1979), 1-22; see also Analytic Functions(J.Lawrynowicz, Editor), pp. 292-300, Springer-Verlag Berlin, Heidel-berg and New York, 1980.

[3] J.L. Liu: Notes on Jung-Kim-Srivastava integral operator, J. MathAnal Appl., 294(2004), 96-103.

[4] S.S. Miller and P.T. Mocanu: On some classes of first orderdifferential subordination, Michigan Math. J., 32(1985), 185-195.

[5] H. M. Srivastava and S. Owa: A certain one parameter additivefamily of operators defined on analytic functions, J. Math Anal.Appl., 118(1986), 80-87.

[6] E.T.Whittaker and G.N. Watson: A course on Modern Analy-sis: An Introduction to the General Theory of Infinite Processes andAnalytic Functions; with an Account of the Principal Transcden-tal Functions, 4th editin, Cambridge University Press, Cambridge,1927.

27

ON FRACTIONAL CALCULUS OPERATOR FORANALYTIC FUNCTIONS

Khalifa Al-Shaqsi (Ministry of Education, Oman)

Abstract

For analytic function f(z) = z + a2z2 + · · · in the open unit disk

U = z : |z| < 1, a new fractional operator Ωλγ is defined for any real

numbers λ, γ by

Ωλγf(z) =

Γ(2 + γ(λ − 1) − λ)

Γ(2 + γ(λ − 1))zλ−γ(λ−1)Dλ

z

`zγ(λ−1)f(z)

´,

where Dλz f(z) is the fractional derivative of order λ. Several basic prop-

erties for this operator are obtained. A new class of analytic functionsdefined by using Ωλ

γ are introduced and investigated some properties be-long to this class.

On Complex Dynamical Systems with large girthindicator, related Chaos and applications to Quantum

ComputingV. A. Ustimenko (University of Maria Curie Sklodowska, Poland)

Abstract

Let Vn = Cn be the family of n-dimensional vector spaces over thefield of complex numbers C. We refer to the tuple t = (t1, t2, . . . , tk) ofnonzero elements ti ∈ C such that ti 6= −ti+1 as irreducible tuple of lengthk ≥ 1.

We say that the family of nonlinear polynomial bijective maps Ftn =

(x1, x2, xn) : Cn → Cn, t ∈ C , t 6= 0, n ≥ 1 of kindx1 → f1(t, x1, x2, . . . , xn), x2 → f2(t, x1, x2, . . . , Xn), . . . , xt → fn(t, x1, x2, . . . , xn)

form a symmetric complex dynamical system depending on ”time” t ifthe inverse map for Ft

n is F−tn. If all coefficients of polynomials fi,

i + 1, . . . , n are integers we use term symmetric arithmetical dynamicalsystem depending on time.

We refer to the triple (n, x, t) where x ∈ Cn, t is irreducible tupleof length s as n-dimensional cycle of length s if for the compositionF (t1, t2, . . . , ts)

n of Ft1n, Ft2

n, . . . , Fts

n we have F n(t1, t2, . . . , ts)(x) = x.The minimal length of n-dimensional cycle in dynamical system is the

girth indicator gin(n).We say that the symmetric complex dynamical system with large girth

indicator if there is a positive real constant c such that gin(n) ≥ cn.It is easy to see that in case of such a dynamical system for each value

of n and each irreducible tuple (t1, t2, . . . , ts) of length s ≤ cn and each(x1, x2, . . . , xn) the composition F (t1, t2, . . . , ts)

n of Ft1n, Ft2

n, . . . , Fts

n

has no fixed points.The existence of such a dynamical systems is proven in [1] via explicit

constructions.We refer to the maximal constant c with the above property for the

family Ftn as the speed of growth of girth indicator.

28

We can deduce from the results of [2] the following statement.THEOREM 1: The speed of growth of gin for the symmetric complex

dynamical system with large girth indicator is bounded by 2.THEOREM 2: There is a symmetric arithmetical dynamical system

with the large girth indicator with the speed of growth for gin at least2/3.

The map F = F n(t1, t2, . . . , ts) of complex arithmetical dynamicalsystem can be used as encryption map for potentially infinite string x =(x1, x2, . . . , xn) with the password t = (t1, t2, . . . , ts). Notice that if K is asubring of C with unity (Z, Q, Gaussian integers, Gaussian rational num-bers and etc) then F maps Kn into itself for t ∈ Ks. So the computationof restriction of F onto Kn can be implemented on Quantum Computeror other probabilistic machine. In the talk we will discuss studies of mix-ing properties of such maps, complexity of computation, use for differenttasks of Coding Theory and Cryptography (it is partially reflected in [3]).

References

[1] V. A. Ustimenko, Linguistic Dynamical Systems, Graphs of LargeGirth and Cryptography, Journal of Mathematical Sciences, Springer,vol.140, N3 (2007) pp. 412-434.

[2] T. Shaska, V. Ustimenko, On the homogeneous algebraic graphs oflarge girth and their applications, Linear Algebra and its ApplicationsArticle, Volume 430, Issue 7, 1 April 2009, Special Issue in Honor ofThomas J. Laffey.

[3] V. A. Ustimenko, On the cryptographical properties of extremal alge-braic graphs, in ”Algebraic Aspects of Digital Communications”, IOSPress, NATO Science for Peace and Security Series - D: Informationand Communication Security, Volume 24, July 2009.

4 Functional Analysis

EXACT CONSTANTS FOR BEST APPROXIMATIONON THE GROUP SU(2)

E. AGHDASSI (University of Tabriz, Iran)

Abstract

In the present paper we study the properties of the least upper boundsof the best approximation by algebric polynomials in metrics L1 and L∞

for classes of convolutions defined on the group SU(2). The exact con-stants for best approximation by trigonometric polynomials in L∞(−π, π)is studied by many authors. Finally in this paper we proved that for groupSU(2) analog of the Favard-Akhiezer-Krein theorem dose not hold.

C∗-algebras associated to low dimensional Leibniz algebrasMassoud Amini (Tarbiat Modares University, Iran)

Abstract

29

We consider the loop space associated to the enveloping Lie-algebraof Leibniz algebras and show that they carry a natural hypergroup struc-ture. We construct the associated universal C∗-algebra for low dimen-sional Leibniz algebras and compare our construction with the construc-tion of C∗-algebras of Hecke pairs. We give a classification scheme fordimensions three to five.

AN EXTENSION OF THE FUGLEDE -PUTNAMTHEOREM

Aıssa NASLI BAKIR (Chlef University, Algeria)Salah MECHERI (Tebessa University, Algeria.)

Abstract

The familiar Fuglede-Putnam theorem is as follows : For two normaloperators A and B on a complex separable Hilbert space H , and for someoperator X on H , the equation AX = XB implies A∗X = XB∗. Recentlyin [4], authors proved that this result remains true for an injective (p, k)-quasihyponormal operator A∗ and a dominant operator B∗. In this paper,we’ll show that injectivity can be replaced by the condition kerA reducesA. We’ll also generalize the Fuglede-Putnam theorem to another class ofnonnormal operators.

Key words and phrases. (p, k)-quasihyponormal operators, dominantoperators, theorem of Fuglede-Putnam.

AMS Classification. Primary 47B47, 47A30, 47B20, Secondary 47B10.

REFERENCES1. A. Aluthge, On p-hyponormal operators for 0¡p¡1, Integ. Equa. Oper.Theo., 13(1990), 307-315.2. A. Aluthge, D. Wang, An operator inequality which implies paranor-mality, Math. Ineq. Appl, 2(1999), 113-119.3. T. Ando, Operators with a norm condition, Acta. Sci. Math. (Szeged),33(1972), 169-178.4. A. Nasli Bakir, S. Mecheri, Another version of Fuglede-Putnam theo-rem, Georgian Mathematical Journal, 16(2009), N3, 427-433.

PROPER EXTENSIONS OF A CERTAIN CLASS OFCARLEMAN OPERATORS

Berrabah Bendoukha()

Abstract

In this paper, we describe all proper (in particular, self-adjoint anddissipative) extensions of a certain type of Carleman operators, definedin Hilbert space L2(Ω, µ). For such operators, we use the concept ofbound- ary triplet to give the general form of Weil functions and properextensions. Spectral properties of these extensions are also investigated.

N A CLASS OF n-NORMED SEQUENCES RELATEDTO lp-SPACE

30

Binod Chandra Tripathy (Institute of Advanced Study in Science andTechnology, India)

Stuti Borgohain (Institute of Advanced Study in Science and Technology,India)

Abstract

The notion of n-normed space was studied at the initial stage by Gahler[Math. Nachr., 28(1964), 1-43] , Misiak , Gunawan and many others fromdifferent aspects.

Let n ∈ N and X be a real vector space. A real valued function‖ ., ., ., . . . ‖n on Xn satisfying the following four properties:

(1) ‖ z1, z2, ., . . . zn ‖n= 0 if and only if z1, z2,zn are linearly dependent;

(2) ‖ z1, z2, ., . . . zn ‖n is invariant under permutation;

(3) ‖ z1, z2, ., . . . azn ‖n= |a| ‖ z1, z2, ., . . . , zn−1, zn ‖n, for any α ∈ R;

(4) ‖ z1, z2, ., . . . zn−1, x+y ‖n=‖ z1, z2, ., . . . zn−1, x ‖n + ‖ z1, z2, ., . . . zn−1, y ‖n;is called an n-norm on X and the pair (X, ‖ ., ., ., . . . ‖n) is called ann-normed space.

Sargent [J London Math. Soc, 35,1960:161-171] introduced the se-quence space m(φ) and studied some properties of this space. Later onit was studied from the sequence space point of view and some matrixclasses were characterized with one member as m(φ) by many others.

In this paper we introduced the following sequence space:

(m(φ), ‖ ., ., ., . . . ‖n) = (xk) w :‖ (xk) ‖n,m(φ)= sups1,σ

1

φskσ

‖ z1, z2, ., . . . zn−1, xk ‖n< .

We investigate some topological and algebraic properties of this space andobtain some inclusion results.

Another Characterization of Semiclassical d−OrthogonalPolynomials

Ammar BOUKHEMIS (University of Annaba, Algeria)

Abstract

We give a new characterization of d−orthogonal polynomials sequenceswhose derivatives sequences is strictly d−quasi-orthogonal. Also, an inte-gral representation of the forms of orthogonality is established.

—————————————————————————————–

AMS Classification: [2000] , 42C05, 33C47, 33C50.Keys words: Orthogonal Polynomials, Semiclassical, Integral representa-

tion.

References

[1] A. I. APTEKAREV, F. MARCELLAN and I. ROCHA. Semiclassical multi-ple orthogonal polynomials and the properties of Jacobi-Bessel polynomials.J. Approx. Theory 90 (1) (1997), 117-146.

31

[2] A. BOUKHEMIS et E. ZEROUKI, Classical 2-orthogonal polynomials anddifferential equations. International Journal of Mathematics and Mathemat-ical Sciences, Vol. 2006 (2006), Article ID 12640, 25 pages

[3] A. BOUKHEMIS. On the classical 2−orthogonal polynomials sequences ofSheffer-Meixner type. Cubo A Math. J. 7 (2) (2005), 39-55.

[4] A. BOUKHEMIS. A study of a sequence of classical orthogonal polynomialsof dimension 2. J. Appro. Theory 90 (3) (1997), 435-454.

[5] K. DOUAK and P. MARONI. Les polynomes orthogonaux ”classiques” dedimension 2. Analysis 12 (1992), 71-107.

[6] P. MARONI. Prolegomene a l’etude des polynomes orthogonaux semi-classiques. annali di Mat. Pura ed Appl 149 (4) (1987), 165-184.

[7] W. VAN ASSCHE and E. COUSSEMENT. Some classical multiple orthog-onal polynomials. J. Comput. Appli. 127(2001), 317-347.

Asymptotic Unconditionality in Banach SpacesSimon Cowell (UAE University, UAE)

Abstract

We show that a separable real Banach space embeds almost isomet-rically in a space Y with a shrinking 1-unconditional basis if and only iflimn→∞ ‖x∗ + x∗

n‖ = limn→∞ ‖x∗ − x∗n‖ whenever x∗ ∈ X∗, (x∗

n)∞n=1 is aweak∗-null sequence and both limits exist. If X is reflexive then Y canbe assumed reflexive. These results provide the isometric counterparts ofrecent work of Johnson and Zheng.

FUSION INTEGRAL AND FUSION FRAMEM.H.FAROUGHI (Azad University of Shbestar)

Abstract

In the present paper we shall introduce an operator-valued integralover a Hilbert space. We also, shall prove that it is additive (finite orinfinite) with no more condition. We show that it has retrieval property,and fusion integral has a useful connection with the Lebesgue integral.Also, by means of the fusion integral we shall generalized the concept offusion frame.

Frames and Riesz bases of irregular translatesSigrid Bettina Heineken (University of Vienna, Austria)

Abstract

A sequence fkk∈Z is a frame for a separable Hilbert space H if thereexist positive constants A and B that satisfy A‖f‖2 ≤

Pk∈Z |〈f, fk〉|

2 ≤B‖f‖2 for all f ∈ H.

In this work we study frames and Riesz bases obtained by translatinga function along an irregular set of points. Let PWE be the space of

32

a function in L2(R) whose Fourier transform is supported in E. We findconditions for the familiy of irregular translates of a function to be a framefor PWE. For this, we investigate systems of the form heλλ∈Λ whereh ∈ L2(E) and eλλ∈Λ are complex exponentials in L2(E), and also acertain Gramian function.

These results have possible applications in image and digital datatransmission, speech coding, general signal processing and geophysics.

This is joint work with Peter Balazs from the Acoustic Research In-stitute, Vienna.

Norm Attaining Operators From L1(µ) Into C(K)Yousef Hasan (Hebron University, Palestine)

Abstract

Given two Banach spaces, X and Y , a bounded and linear operatorT ∈ L(X, Y ) attains its norm if there is an element x0 in the unit sphereof X such that ‖T (x0)‖ = ‖T‖. We denote the set of norm attainingoperators from X to Y by NA(X, Y ). The general question (sometimescalled the Bishop-Phelps problem), is whether or not NA(X, Y ) is densein L(X, Y ). Given an arbitrary measure µ, we show that the set of normattaining operators from L1(µ) into C(K) is dense in the space of allbounded linear operators L(L1(µ), C(K)).

CONDITIONAL TYPE AND WEIGHTEDCOMPOSITION OPERATORS BETWEEN Lp SPACES

M. R. JABBARZADEH (University of Tabriz, Iran)

Abstract

In this paper Lambert multipliers acting between Lp spaces are char-acterized by using some properties of conditional expectation operator.Also, we discuss measure theoretic characterizations for some weak hy-ponormal classes of weighted composition operators on L2 such as, p-quasihyponormal, p-paranormal and p-hyponormal.

On norm one complemented subspaces of C(Q)Aref Kamal (Sultan Qaboos University, Oman)

Abstract

n this talk the author gives a complete characterization for those finitedimensional subspaces N of C(Q) for which there is a projection P fromC(Q) onto N with ‖P‖ = 1. It is shown that if Q is a compact Haus-dorff space, and N is an n-dimensional subspaces of C(Q) then there is aprojection P : C(Q) → N with ‖P‖ = 1 iff N has a basis f1, f2, . . . , fnthat satisfies the property that ‖fi‖ = 1 for each i = 1, 2, . . . , n, and‖

Pni=1 |fi|‖ = 1. It is shown also that this property is equivalent to the

property that N∗, the dual space of N , has a basis µ1, µ2, . . . , µn thatsatisfies the property that ‖fi‖ = 1 for each i = 1, 2, . . . , n, and for eachf ∈ N , ‖f‖ = max|µi(f)|i = 1, 2, . . . , n. This paper was published inMathematica Japonica Vol. 37.

33

An Iterative Algorithm for a System of ImplicitVariational Inclusions involving H(·, ·)-Mixed Accretive

MappingsKaleem Raza Kazmi (A.M.U. Aligarh, India)

Abstract

In this paper, we introduce a new class of H(·, ·)-mixed accretive map-pings in Banach space. We give the notion of resolvent mapping for theH(·, ·)-mixed accretive mapping, an extension of resolvent mapping givenin [Juhe Sun, Liwei Zhang, Xiantao Xiao; An algorithm based on resolventoperators for solving variational inequalities in Hilbert spaces, NonlinearAnalysis: Theory, Methods & Applications, 69(10) (2008), 3344-3357],and prove its existence and Lipschitz continuity. Further, we consider thefollowing system of implicit variational inclusions in Banach spaces:

For each i = 1, 2, let Ei a be real Banach space; let gi, hi : Ei → Ei,Ai : E1 → E1, Bi : E2 → E2, Fi, Gi : E1 × E2 → Ei be nonlinearmappings and let Ci : E1 → 2E1 , Di : E2 → 2E2 , Mi : Ei → 2Ei be set-valued mappings such that (gi − hi)(xi) ∈ domain Mi(·) for all xi ∈ Ei.The system of implicit variational inclusions is to:

Find (x1, x2) ∈ E1 × E2, ui1 ∈ Ci(x1), ui2 ∈ Di(x2) such that

Fi(Ai(x1), Bi(x2)) − Gi(ui1, ui2) + Mi(gi(xi) − hi(xi)) ∋ θi,

where θi is a zero vector of Ei.

We show the equivalence of this system with a system of implicitWiener-Hopf equations using the concept of resolvent mappings for theH(·, ·)-mixed accretive mapping. Using this equivalence, we propose anew iterative algorithm for the system of implicit variational inclusions.Furthermore, we prove the existence of solution of the system of implicitvariational inclusions and discuss the convergence analysis of the iterativealgorithm. We also discuss some further extensions of the work presentedin this paper. The theorems presented in this paper can be viewed assignificant generalizations of many known and important results underHilbert spaces as well as Banach spaces setting in recent literature.

AMS Subject Classifications: 47H04, 49J40

Keywords: System of implicit variational inclusions, H(·, ·)-Mixed accre-tive mapping, Mixed Lipschitz mapping, Resolvent mapping, System ofimplicit Wiener-Hopf equations, Iterative algorithm, Convergence analy-sis.

Variational Inequalities and its Applications(System of extended general variational inequality

problems)Faizan Ahmad Khan (A.M.U. Aligarh, India)

34

Abstract

In this talk, we shall introduce and consider, a new class of system ofextended general variational inequality problems involving nonlinear op-erators in real Hilbert spaces. Using the projection operator technique, itis given that the system of extended general variational inequality prob-lems are equivalent to the nonlinear projection equations. This alternativeequivalent formulation is used to discuss the existence of a solution andsuggest some general algorithms. Further we shall discuss the convergenceanalysis of iterative sequence generated by general algorithms. Since thissystem includes the system of variational inequalities, variational inequal-ities and related optimization problems. My results can be viewed as arefinement and improvement of previously know results for recent work invariational inequalities.

Note that in the talk, we shall also discuss some new iterative methodto solve such kind of problem by using some different approach and inter-relate my problem to some physical problem of nonlinear analysis.

An iterative process for common fixed points of two finitefamilies of nonself nonexpansive mappings

Safeer Hussain Khan (Qatar University, Qatar)

Abstract

In this talk, we introduce an iterative process for approximating com-mon fixed points of two finite families of nonself nonexpansive mappingsin Banach spaces. Our process contains both explicit and implicit Manniterative processes and some other processes for nonself mappings. Weprove some weak and strong convergence theorems for this iterative pro-cess. Our results generalize and improve some results in contemporaryliterature.

Condition for maxima and minima between two pointsChandan Kumar (Delhi College of Engineering, India)

Abstract

In the present paper, we are going to find out and study the truerelationship between the curve and the point of intersection of any twotangent drawn on the curve having product of their slope negative, alsothe condition for which maxima and minima exists between two points.since their is a general misconception that when the product of slope oftwo tangent is negative then their exist atleast one point in between themwhose slope is 0(zero). In this paper we have find the condition when itwill satisfy this result and when not, and also derived the other possibleresults, for any smooth curve which is continuous and differentiable in itsdomain.

keywords: Continuity, Differentiability, Tangents, Slopes.

AMS subject classifications: 11G20, 14H50, 53A04

35

Norm for sums of two basic elementary operatorsFarida Lombarkia (University of Batna, Algeria)

Abstract

In this talk we shall be concerned to estimate the norm of the ele-mentary operator MA1,B1

+ MA2,B2, where A1, A2, B1, B2are bounded

linear operators on a normed space E, and MA1,B1is the basic elemen-

tary operator defined on B(E) by MA1,B1(X) = A1XB1, we also give

necessary and sufficient conditions on the operators A1, A2, B1, B2 underwich MA1,B1

+ MA2,B2attains its optimal value ‖A1‖‖B1‖ + ‖A2‖‖B2‖.

This is a jointe work with Ahmed Bachir .

Composition operator on Besov algebra spacesMadani Moussai (University of MSila, Algeria)

Abstract

We will study the boundedness of the composition operator Tf (g) :=f g on Besov s spaces Bs

p,q(R). In case 1 ≤ p, q < ∞, and

[s] ≥ 2, s − [s] ≤ (1/p), min

„1 +

1

p,p

q

«+

1

p− 2 < s − [s],

we will prove the following: the operator Tf takes Bsp,q(R) to itself if and

only if f(0) = 0 and f belongs locally to Bsp,q(R).

From Certainty to Uncertainty: In the Context of FixedPoint Theory and Applications-II

P. P. Murthy (Guru Ghasidas University, India)

Abstract

Non-Fuzzy Sets ( Crisp )we mean those sets with mathematical logici.e. the value of the membership function defined is 0 or 1 and FuzzySets ( Uncertain Sets ), we mean the membership values lies between[0, 1]. The pioneer concept of Fuzzy Sets and Fuzzy Logic introducedinitially by Prof. Lofti Zadeh ( Fuzzy sets, Information and Control, 8(1965), 338-353) of University of California, Berkeley, USA in 1965. Inthis talk, I shall try to give some results of Fuzzy Sets and Fuzzy Logic andnewly developing space Cone Metric Space ( Huang Long-Guang andZhang Xian, Cone Metric spaces and fixed point theorems of contractivemappings, J. Math. Anal. Appl.332(2007), 1468 - 1476 ) in the context ofFixed Point Theory and Application. In this space we shall discussthe impact of the concepts of compatible and non-compatilbe maps forobtaining common fixed points.

R-Transforms for polynomial expansion and rootextraction

Wajdi M. Ratemi (Alfateh University, Libya)

36

Abstract

Special mathematical formulation has been introduced for polynomialexpansion, Ratemi (1996), Ratemi and Eshabo(1998). It is named as theGuelph expansion, Ratemi and Abdulla (2009).

nY

i=1

(ω + λi) =nX

k=0

ωn−1Rn−k =nX

k=0

ωn−kX

T=(nk)

λ . . .k . . . λ.

Ratemi and Eshabo (1998), Ratemi (1998), and Aboanber (2003) haveused such expansion for studies of the inhour equation and for the studiesof point reactor kinetics solutions of positive and negative step insertionof reactivities in nuclear reactors.

In this paper, further explorations are made for the Guelph expan-sion. The coefficients of such polynomial expansion have been found tobe related to the roots of the polynomial in concern. Each coefficientis found by summing up the number of possible k-combinations (single,dual, triple, etc..) of the roots of the nth degree polynomial. Such num-ber of combinations is determined by the Tripoli index T =

`nk

´, where

n represents the degree of the polynomial, and k represents the numberof combinations of roots. It turns out that Tripoli index generates theentries of Pascal triangle, Hence it is here in this paper, it is reported forthe first time the explanation of the entries of the Pascal triangle, that is;each entry (T ) of the Pascal triangle represents the number ofk-combinations of the roots of the nth degree polynomial. Also,it is presented in this study what is called the R-transforms which findthe coefficients of the polynomial expansion for given polynomial roots orfinding the roots of a polynomial given the coefficients of its expansion.

References:

Aboanber, A.E.(2003), Analytical Solution of the Point KineticsEquation by Exponential Mode Analysis, Progress in Nuclear En-ergy, Vol.42, No.2, pp.179-197.

Aboanber A.E. ( 2003), An efficient analytical form for the period-reactivity relation of beryllium and heavy-water moderated reactors,Nuclear Engineering and Design 224, 279-292.

Ratemi W.M,(1996), Unpublished work.

Ratemi W.M., and Eshabo A.E.,(1998), New form of the InhourEquation and its Universal ABC-Values for Different Reactor Types,Ann.Nucl.Energy, Vol.25, No.6,pp.377-386.

Ratemi W.M., and Eshabo A.E.,(1998), New form of the Point Re-actor Kinetics Equation, Proceedings of the 4th Arab Conference forPeaceful Uses of Atomic Energy, Tunis, Tunisia, 16-2-/11/1998.

Ratemi Wajdi M., and Hussein Abdullah (2009), Guelph Expan-sion: A Special Mathematical Formulation for Polynomial Expan-sion, accepted for presentation in the International Conference ofMathematical Sciences, 04-10 August, 2009, stanbul, Turkey

Key Words: Guelph Expansion, Tripoli Index, Polynomial Expansion, Rootextraction , R-transforms, Pascal triangle, Inhour Equation, Point Reactor Ki-netics.

37

Common fixed point theorems for left reversiblesemigroups on located distance spaces

Ahmad H. Soliman(Al-Azhar University, Egypt)

Abstract

In the present paper we introduce a new type of generalized metricspaces so called a located distance space where self-distances are zero. Forexample, each metric spaces are located distance spaces. The purpose ofthis paper is to establishe fixed point theorems for left reversible semigroupof selfmaps. Our results extend relevant fixed point theorems of Y. Y.Huang and C. C. Hong [ Common fixed point theorems for semigroups onmetric spaces, Internat. J. Math. and Math. Sci.(1999) 377-386].

5 Numerical Analysis

The (G′

G)−expansion method for solving some evolution

equationsQ. Ebadi (University of tabriz, Iran)

Keywords: (G′

G)−expansion method,travelling wave solution, FitzHugh-Nagumo

equation

Abstract

The (G′

G)−expansion method is used to find traveling wave solution of

FitzHugh-Nagumo and Newell-whitehead equations. The traveling wavesolutions are expressed by the hyperbolic, trigonometric and rational func-tions. It is shown that the proposed method is direct and effective. Thesolutions are compared with solutions of the tanh-Coth method.

On Sixtic Lacunary Spline Solutions of Fourth OrderInitial Value Problem

Karwan Hama Faraj Jwamer (University of Sulaimani, Iraq)

Abstract

Many initial value problems that arise in the real life situations defyanalytical solution; hence numerical techniques are desirable to find thesolution of such equations. New numerical methods which are compara-tively better than the existing ones in terms of deficiency, accuracy, con-vergence and computational cost are always needed. In this paper, anapproximation solution with spline (0, 1, 4) functions of degree six anddeficiency four is derived for solving fourth order initial value problems,with prescribed nonlinear endpoint conditions. Under suitable assump-tions, the existences and uniqueness of the spline (0, 1, 4) function areproved; also the upper bounds of errors are obtained. Other purpose ofthis construction is to solve the fourth order differential equations. Theconvergence analysis and the stability of the approximation solution areinvestigated and compared with the exact solution to demonstrate theprescribed lacunary spline (0, 1, 4) function interpolation.

38

Exact solutions for the coupled Higgs equation and theMaccari system by using (G′/G)-expansion method

H. Kheiri, (University of Tabriz, Iran)A. Jabbari

Abstract

In this paper, the“

G′

G

”-expansion method is used to seek more gen-

eral exact solutions of the coupled Higgs equation and the Maccari system.As a result, hyperbolic function solutions, trigonometric function solutionsand rational function solutions with free parameters are obtained. Whenthe parameters are taken as special values the solitary wave solutions arealso derived from the travelling wave solutions. It is shown that the pro-posed method is more powerful and more general.PACS: 02.30.Jr; 05.45.Yv

Keywords: (G′/G)-expansion method; coupled Higgs equation; Maccarisystem; travelling wave solutions.

Numerical solution of algebraic equation fuzzy equationswith crisp variable by secant method

Seied mahmoud khorasany kiasari (Islamic Azad University, Iran)

Abstract

n this paper we introduce an algebraic fuzzy equation of degree withfuzzy coefficients and crisp variable, and we present an iterative method tofind the real roots of such equations, numerically. We present an algorithmto generate a sequence that can be converged to the root of an algebraicfuzzy equation.

Keywords: Nearest approximation; Fuzzy numbers; Fuzzy polyno-mial; Algebraic fuzzy equation

The Identification of Pollution Sources Using a PLSLinear Regression Technique

Abdelmalek Kouadri (University of Boumerdes, Algeria)Abdallah Namoune (University of Manchester, UK)

Abstract

A large number of industrial problems can be described using a regres-sion model obtained from the available data. The objective is to describethe relationships between the input variables X and observed output vari-ables Y in the lack of a theoretical model. The problem is that the numberof variables is very important in relation to the number of observations.The PLS Regression (Partial Least Squares Regression) is a data anal-ysis method used specifically to investigate this type of problems. ThePLS Regression is an extension of the multiple linear regression model. Inits simplest form, a linear model specifies the linear relationship betweenone or more dependent variables, the responses Y and a set of predictorvariables X. In many data analysis problems, the estimation of linear

39

relationship between two variables is adequate to describe and estimatethe observed data and consequently make good forecasts for further anal-ysis. The multiple regression model has been extended in many ways toadapt its form to complex data analysis problems. Hence, it is used asa basis for many multivariable methods such as: the Principle Compo-nent Regression (PCR). The PLS Regression is a recent technique whichcombines the characteristics of the principle component analysis (PCA)and multiple regression. It is particularly useful when a set of dependentvariables needs to be predicted from a large set of explanatory variables(predictors) that can be highly correlated with each other.

When there are only few predictors that are not significantly collinearbut have relationships with the expected responses, multiple linear re-gression is the best way to analyse the data. However, if one of theseconditions is not satisfied, then multiple linear regression becomes inef-fective and inappropriate.

The PLS is also used to build predictive models when the factors arenumerous and very collinear. Note that this method focuses on the predic-tion of a response and not necessarily on the identification of a relationshipbetween variables. This means that the PLS is not appropriate for distin-guishing variables with a small influence on the response, but when thegoal is exclusively the prediction there is no need to limit these variables.

Among the most obvious advantages we retain that the PLS methodmakes it possible to combine the prediction with any study of a structureenclosed in latent variables X and Y . Thus, the method requires less thanPCR components to give a good prediction. In this work, this methodenables modelling different sets of classes with proposed scenarios charac-teristics under different situations of pollutants concentrations in a river.Based on the concentration of pollution components as input variablesand reprocessing plants which ordure dump in a river as response vari-ables, we can reconstruct a relationship between these two variable blocs.The input variables are collected from a number r of sensors which aremounted longitudinally and with an appropriate distance between eachone in the river edge. These variables represent the concentration of thelead, nitrates, oxide and dioxide carbon, and oxide sulfate in each zone andin different coordinates of the river plan. Each response variable noted Pi(i = 1tok) indicates the reprocessing plant i. A number of N scenariosare disposed which represent a sufficient possible case. This number canbe at least equal to 2k − 1. To get more significant results, it is requiredto repeat each 2k − 1 for a sufficient number of experiments in order tocover all the pollutant component concentration. It is very importantto note that the PLS discriminate analysis can be used with loses data.Using all principal plans, the separation between the different plants isalmost perfect. Indeed, the two first PLS components improve enoughthe disjointing between k observed clusters. These clusters are regroupedin the Hotelling elliptic of the principle component plan at a significantlevel equals to 95

To validate the PLS regression discriminate analysis and test its abil-ity to indicate which plant or plants ordure dump in the river, severalnumerical simulations are performed. The obtained results confirm theeffectiveness and accuracy of the proposed technique in identifying the

40

sources of pollution.This research demonstrates two important points:

1. It is possible to work with less and lose number of data and animportant number of variables and to obtain operational results;

2. The PLS regression technique enables identification of existing struc-tures from the data.

Furthermore, we aim to confirm that the PLS regression discriminateanalysis represents an important and useful tool in identifying the pollu-tion sources. It offers an improved new technique for accurately disjoint-ing sources in a multi-pollutants diversion case. The Hotelling elliptic isused to quantify the devices measurements error, noises and parametricuncertainties which have gained a robust and accurate identification ofpollution sources. The interest of pollution sources identification aims tohelp the authorities penalise the plant which ordures dump in a river andstop it. Consequently, it has repercussions on the environment protection,specifically on the underground water nap.

Numerical solution of Hallen’s integral equation by usingcardinal Legendre functions

Mehrdad Lakestani (University of Tabriz, Iran)

Abstract

An approximate method is proposed for solving Hallen’s integral equa-tion based on cardinal Legendre functions. Properties of these functionsare first presented, these properties are then utilized to reduce the com-putation of Hallen’s integral equation to some algebraic equations. Themethod is computationally attractive, and applications are demonstratedthrough an illustrative example.

Keywords: Cardinal Legenre functions, Hallen integral equation, col-location method.

Global Convergence of the quasi Newton BFGS algorithmwith new nonmontone line search technique

Ivan Subhi Latif (University of Salahaddin, Iraq)

Abstract

The BFGS method is the most effective of the quasi-Newton algorithmfor solving unconstrained optimization problem. In this work we developa new nonmonotone line search of quasi-Newton algorithm for minimizingfunction having Lipschitz continuous partial derivatives,The nonmonotoneline search can guarantee the global convergence of the original quasi-Newton BFGS algorithm. Numerical experiments on sixteenth well-Knowtest functions with various dimensions generally encouraging results showthat the new algorithm line search is available and efficient in practicalcomputation by comparing with other same algorithm in many situations.

Keywords: Unconstrained Optimization, BFGS update, DescentCondition ,Nonmonotone Line Searches.

41

Solving a system of linear two-dimensional Fredholmintegral Equations of the Second Kind by using Weighted

Residual MethodsRostam K. Saeed (Salahaddin University, Iraq)

Mehdi Hassan Mahmud (, )

Abstract

This paper focuses on obtaining an approximation solution for solvinga system of two dimensional linear Fredholm integral equations of the2nd kind by using four different types of weighted residual methods. Twoexamples are solved to show the validity of the prescribed methods bydepending on the least square errors (L.S.E) and running time (R.T).

Keywords: Weighted residual methods, two-dimensional Fredholmintegral equation

Dual variational methods for the Liouville ProblemHocine SISSAOUI (University of Annaba, Algeria)

Abstract

The Liouville problem falls into the large class of quasilinear ellipticproblems and therefore has been studied for some time. For earlier nu-merical work, we mention Arthurs [1], Noor and Whiteman [2], Christie& al.[3] among others.

For our study of the Liouville problem we make use of the dual varia-tional principles derived from Moreau’s Two Cones Theorem [4] togetherwith the finite element method. The originality of the present work is theuse of the dual variational principles derived from Moreau’s Two ConesTheorem [4] which results in an inequality constraint for the field variableof the dual principle. These methods lead to mathematical programmingproblems which can be solved by optimization methods.

Keywords: Liouville problem, Duality, Finite element method .AMS subject classifications (2000): Primary 35P70, Secondary 35J65 .

References

[1] A. M. Arthurs,Complementary Variational Problems, 2nd edition,OUP, 1980

[2] Noor, M.A. & J. R. Whiteman, Error Bounds for FE Solutions ofMildly Nonlinear Elliptic BVP’s, Num. Math. 26 (1976), pp 106-116.

[3] Christie, I. & al., Product Approximation for Nonlinear Problemsin the FEM, Dept. of Maths. Report N. A. /42, July 1980, Universityof Dundee, U.K.

[4] W.D. Collins An Extension of the Method of the Hypercircle to Lin-ear Operator Problems with Unilateral Constraints, Proceedings of theRoyal Society of Edin. 85 A (1980), pp 173-193

42

On the stability analysis of weighted average finitedifference methods for fractional wave equations

N. H. Sweilam (Cairo University, Egypt)

Abstract

In this article, numerical study for the fractional wave equations is in-troduced by using a class of finite difference methods. These methods arean extension of the weighted average methods for ordinary (non-fractional)wave equations. The stability analysis of the proposed methods is givenby a recently proposed procedure akin to the standard von Neumann sta-bility analysis. A simple and accurate stability criterion valid for differentdiscretization schemes of the fractional derivative, arbitrary weight factor,and arbitrary order of the fractional derivative, is found and checked nu-merically. For comparison propose, a test example is given and comparedour results against the exact solutions.

Restarted Adomians decomposition method for quadraticRiccati differential equation

A. R. Vahidi (Islamic Azad University, Iran)

Abstract

In this paper, restarted Adomians decomposition method is proposedto solve the well-known quadratic Riccati differential equation. Compar-ison are made between Adomians decomposition method and restartedAdomians decomposition method using to exact solution. The resultsreveal that the restarted Adomians decomposition method is of higheraccuracy than Adomians decomposition method.

Keywords: Adomians decomposition method; Restarted Adomianmethod ; Riccati differen- tial equation.

One-Pass Nested QR Updating Algorithm for TheSolution of Linear Least squares Problems

Muhammad Yousaf (University of Essex, UK)Dr.Abdellah Salhi (,)

Abstract

Systems of linear equations are often difficult to solve because of theirshear size, ill-conditioning, loss of rank and so on. By concentratingon particular parts of the problem matrix, the problem may be madeamenable to solution. Here, we consider solving linear systems in thesense of least squares, i.e. minx||Ax− b||2, where A ∈ Rm×n and b ∈ Rm.The suggested approach relies on repeated updating of the QR factor of asub-matrix of A. The original matrix is first reduced by removing a blockof rows. It is then reduced by removing a block of columns. The right-hand side is reduced accordingly. The QR factorization is calculated forthe small subproblem. The block of columns is then appended and theQR factor is updated and the right hand side updated accordingly. Thisintermediary system is solved and the solution is updated after appendingthe block of rows.

43

6 Topology

On the categories of lattice-valued pretopologicalconvergence groups and pseudotopological convergence

groupsT. M. G. Ahsanullah and Fawzi Al-Thukair (King Saud University, Saudi

Arabia )

Abstract

Starting with a frame L, we introduced in [On the category of fixedbasis frame valued topological groups, Fuzzy Sets and Syst. 159(2008),25292551] the notion of stratified L-valued neighborhood topological group,and by dropping the so-called kernel condition, we brought into light in[Frame valued stratified generalized convergence groups, Quaest. Math.31(2008), 279302] a category SL-NeighGrp having stratified L-valuedneighborhood groups as ob- jects and continuous group homomorphismsas morphisms, isomorphic to the category SL-PrTopConvGrp whoseobjects are stratified L-valued pretopologi- cal convergence groups andmorphisms are continuous group homomorphisms, also called category ofstratified L-valued principal convergence groups, SL-PConvGrp. Themotives behind this article are, in the first place, to introduce variousweaker forms of these structures and compare them. Secondly, we in-troduce a notion which we believe to be new, the notion of stratifiedL-valued pseudotopological convergence group, also called stratified L-Choquet conver- gence groups - a generalization of classical Choquet con-vergence structure [G. Jager, Subcategories of lattice-valued convergencespaces, Fuzzy Sets and Syst. 156(2005), 124], and we seek relationshipbetween stratified L-valued pretopo- logical convergence groups and strat-ified L-valued pseudotopological convergence groups, both in frame valuedcase as well as Boolean valued case. Finally, we shade light on the conceptof lattice-valued categories of stratified lattice-valued convergence groupsfrom the perspective of the preceding objects and morphisms in an at-tempt to measure the topologiness of a lattice-valued convergence groupor to measure the degree of continuity of certain algebraic operations fora lattice- valued convergence space or topological space. Keywords: L-topology, L-neighborhood topological system, stratified L-neighborhoodsystem, stratified L-pretopological space, stratified L-pseudotopologicalspace, stratified L-convergence structure of continuous convergence, func-tion space, group, category, L-category.

AMS Subject Classification (2000): 54A40, 54A20, 54H11, 18B30.

Almost Hermitian manifold with flat Bochner tensorHabeeb M. Abood (University of Basrah, Iraq)

Abstract

Many researchers investigated the flat Bochner tensor on some kindsof almost Hermitian manifold. In the present paper the author studiesthis tensor on general class almost Hermitian manifold by using a newmethodology which is called an G-structure space. Thus this study gener-alize the results which are found out by those researchers.It is proved that,

44

if M is an almost Hermitian manifold of class R1 with flat Bochner tensor,then either M is 2-dimensional flat Ricci manifold or n-dimensional flatscalar curvature tensor manifold. As well, it is proved that, if M is an al-most Hermitian manifold with flat Bochner tensor, then M is a manifoldof class R3 if and only if M is a linear complex manifold. Later on, equiv-alently of classes R2 and R3 is investigated. Finally we prove that, if Mis flat manifold with flat Bochner tensor, then M is an Einstein manifoldwith a cosmological constant.

Mathematics Subject Classification: 53C55, 53B35

Topological representation of T-fuzzy finite distributivelattices by means of T-fuzzy Priestley spaces

Abdelaziz AMROUNE (Msila University, Algeria)

Abstract

In this paper, we extend some results obtained by Priestley to therepresentation of distributive lattices more precisely we give a represen-tation theory of fuzzy distributive lattices in finite case. The notion offuzzy relations was introduced by Zadeh. In this seminal paper he intro-duced the concept of a fuzzy relation, defined the notion of similarity asa generalization of the notion of equivalence and defined the concept offuzzy ordering. The most obvious idea to define fuzzy ordering is, naturalto demand three straightforward generalizations of the classical axiomsreflexivity, antisymmetry, and transitivity. In Zadehs we extend some re-sults obtained by Priestley to the representation of distributive latticespaper only the minimum t-norm was considered. In this paper the moregeneral definition admitting an arbitrary t-norm. By using a previousresult of Priestly this extension is obtained in a very simple and naturalway.

Coincidence and common fixed points of nonlinearcontractions in PM spaces

Javid Ali* and M. Imdad (*IIT Kanpur, India. Aligarh Muslim University,India)

Abstract

In this paper, we prove some existence results on coincidence and com-mon fixed points of two pairs of self mappings without continuity underrelatively weaker commutativity requirement in Menger PM spaces. Ourresults generalize many known results in Menger as well as metric spaces.Some related results are also derived besides furnishing illustrative exam-ples.

COMMON FIXED POINT FOR GENERALIZED SETVALUED CONTRACTIONS SATISFYING AN

IMPLICIT RELATION IN PARTIALLY ORDEREDMETRIC SPACES

ISMAT BEG (Lahore University of Management Sciences, Pakistan)

45

Abstract

Let (X, d,) be a partially ordered metric space. Let F , G be two setvalued mappings on X. We obtained sufficient conditions for existence ofcommon fixed point of F , G satisfying an implicit relation in X.

Keywords and Phrases: Fixed point; partially ordered metric space; setvalued mapping; implicit relation.2000 Mathematics Subject Classification: 47H10; 47H04; 47H07.

Presic Type Extension and Generalisation of BanachContraction Principle

Reny George (St. Thomas College, India)M.S Khan (Sultan Qaboos University, Oman)

Abstract

Let (X, d) be a metric space, k a positive integer, T : Xk → X,f : X → X be mappings. In this paper we have investigated under whatconditions the mappings f and T will have a common fixed point. Ourresults extends and generalises the results of [3], [4], [5] and [6].References[1] Lj.B.Ciric, A generalisation of Banach Contraction Principle, Proc.Amer. Math. Soc. 45 (1974), 267-273.[2] Lj.B.Ciric, A generalisation of Caristis fixed point theorem, Math.Pannonica 3/2 (1992), 52-57.[3] Lj.B.Ciric and S. B. Presic, On Presic type generalisation of BanachContraction map- ping Principle, Acta. Math. Univ. Com. LXXVI(2)(2007), 143-147.[4] B. C. Dhage ,Generalisation of Banach Contraction Principle, J.IndianMath. Acad ., 9(1987),75-86.[5] S. B. Presic,Sur la convergence des suites , Comptes Rendus de lAcad.des Sci. de Paris 260 (1965), 3828-3830.[6] S. B. Presic,Sur une classe dinequations aux differences finite et surla convergence de certain es suites, Pub.de. lInst. Math. Belgrade 5(19)(1965), 75-78.

WEAK AND STRONG FORMS OFSIMPLY-CONTINUOUS FUNCTIONS

Radhi I. M. Ali (University of Baghdad, )Jalal Hatem Hussein (University of Baghdad)

Abstract

Abstract. In this paper we introduce three classes of functions calledSimply-continuous, Strong simply-continuous and Weak simply-continuousfunctions as generalization of continuous function. We obtain their charac-terizations, their basic properties and their relationships with other formsof generalization continuous functions between topological spaces.

Some fixed point theorems on ordered uniform spacesIshak ALTUN and Mohammad IMDAD* (Kirikkale University, Turkey;

*Aligarh Muslim University, India)

46

Abstract

In this paper, we introduce an order relation on uniform spaces andutilize this relation to prove some fixed point theorems for single andmulti valued mappings in ordered uniform spaces. Some related resultsalso discussed.

Grand Antiprism and QuaternionsMehmet Koca (Sultan Qaboos University, Oman )

Abstract

Abstract Vertices of the 4-dimensional semi-regular polytope, the grandantiprism and its symmetry group of order 400 are represented in terms ofquaternions with unit norm. It follows from the icosian representation ofthe root system which decomposes into two copies of the root system of.The symmetry of the grand antiprism is a maximal subgroup of the Cox-eter group It is the group which is constructed in terms of 20 quaternionicroots of the Coxeter diagram. The root system of represented by the bi-nary icosahedral group I of order 120, constitutes the regular 4D polytope600-cell. When its 20 quaternionic vertices corresponding to the roots ofthe diagram are removed from the vertices of the 600-cell the remaining100 quaternions constitute the vertices of the grand antiprism. We give adetailed analysis of the construction of the cells of the grand antiprism interms of quaternions. The dual polytope of the grand antiprism has beenalso constructed.

A new Proof of Schlafli’s FormulaTimothy Marshall (American University of Sharjah, UAE)

Abstract

In contrast to the situation in Euclidean space, the shape and size of apolyhedron in the unit n-sphere is completely determined by its dihedralan- gles (the angles between faces of codimension one). In particular thevolume of such a polyhedron is a function of these angles. The mostfamiliar case is n = 2, where we have the well known formula for the areaof a spherical k-gon

Area =X

angles − (k − 2)π (5)

In higher dimensions it is not possible to express volume as such an elemen-tary function of angles, but we have instead Schlaflis equation,

d(Volume) =1

n − 1

X(Vol. codimension 2 face) d(angle), (6)

the sum being taken over all faces of codimension 2 and their correspond-ing dihedral angles. We present a formulation of the Schlali formula thatavoids geometric terms, but instead takes the form of an identity betweentwo integrals over simplices of powers of quadratic forms. We outline howthis identity can be proved using Stokess theorem.

47

Hyperspaces and ManifoldsAbdul Mohamad (Sultan Qaboos University, Oman)

Abstract

Given a topological space X, let 2X be the family of all nonemptyclosed subsets of X. In this talk, we shall consider three well-knowntopologies on 2X , namely, the finite topology (also called the Vietoristopology), the Fell topology and the locally finite topology.

The main purpose of this talk is to consider when these three hyper-space topologies are metrizable. Also we shall explore the metrizability ofa manifold in terms of properties of its hyperspaces with various topolo-gies.

This is a joint research work with J. Cao.

Petrov Classification of Lorentzian Manifolds with StaticCylindrically Symmetric Metrics

Name(Sultan Qaboos University, Oman)

Abstract

The Lorentzian manifolds with cylindrically symmetric static metricsare classified for their Petrov types. A theorem is proved that such man-ifolds cannot be of petrov type II or III. Differential constraints are ob-tained which are satisfied by all metrics of Petrov type D. A list of allmetrics of type N is also given.

Common idempotents in compact left topological leftsemirings

Denis I. Saveliev (Moscow State University of Russian Academy of Sciences,Russia)

Abstract

A classical result of Ellis, which became crucial for applications ofRamsey theory in number theory, algebra, topological dynamics, and er-godic theory, is that any compact left topological semigroup has an idem-potent. It follows that any compact left topological left semiring has anadditive idempotent as well as a multiplicative one. We show that it has,moreover, a common, i.e., additive and multi- plicative simultaneously,idempotent. As an application, we partially answer a question related toalgebraic properties of the Stone-Cech compacti cation of natural num-bers. Finally, we notice that similar arguments establish the existence ofcommon idempotents in far more general structures than left semirings.

On the crossings in minimal chartsTeruo Nagase and Akiko Shima

(Tokai University, Japan)

Abstract

48

S. Kamada introduced charts for studying embedded closed surfaces in4-space, called surface links. Charts are oriented labeled graphs in a diskwith three kinds of vertices called black vertices (of degree 1), crossings(of degree 4), and white vertices (of degree 6). Kamada also introducedC-moves which are local modifications of charts in a disk. A C-movebetween two charts induces an ambient isotopy between the surface linkscorresponding two charts.

A surface in 4-space is called a ribbon surface if it is the boundary ofan immersed handle body with singularities which are mutually disjointdisks such that the preimage of each disk is a union of a proper disk of thedomain and a disk in the interior of the domain, a handle body. In thewords of charts, a ribbon surface is the closure of a surface braid whichcorresponds to a ribbon chart where a ribbon chart can be modified to achart without white vertices by C-moves.

In our talk, we introduce that any chart with at most one crossing isa ribbon chart. Moreover we introduce theorems for charts with two orthree crossings.

Graphs arising from Rings-ICherian Thomas (Higher College of Technology, Oman)

Abstract

In this paper we define the graph Ω(R) arising from a commutativering R. The definition is then extended to the ring Zn. Some of the char-acteristic properties of the graph Ω(Zp) are obtained where p is prime. Weprove that Ω(Zn) is complete if and only if n is prime. The automorphismgroup of the graph Ω(Zpn) is obtained. We also prove that Ω(Zn) is asplit graph if n is the power of a prime. Then we define the graph Ω∗(R)and prove that Ω∗(R) is complete if and only if R ∼= Z2 × Z2. We alsofind the automorphism group of the graph Zn defined by I Beck when nis an odd prime and also when n = pq, where p and q are distinct primes

Vague Normed Linear SpaceD.R. Prince Williams (Ministry of Higher Education, Sultanate of Oman. )

Abstract

A vague set (or in short VS) A in the universe of discourse U is char-acterized by two membership functions given by:

(VS1) a truth membership function tA : U → [0, 1],and

(VS2) a false membership function fA : U → [0, 1],

where tA(u) is a lower bound of the grade of membership of u derived fromthe evidence for u, and fA(u) is a lower bound on the negation of u derivedfrom the evidence against u, and tA(u)+fA(u)1. Thus the grade of mem-bership of u in the vague set A is bounded by a subinterval [tA(u), 1fA(u)]of [0, 1]. This indicates that if the actual grade of membership is µ(u),then tA(u)µ(u)1fA(u). The vague set A is written as

A = 〈u, [tA(u), fA(u)]〉 : u ∈ U,

49

where the interval [tA(u), 1fA(u)] is called the “vague value” of u in A andis denoted by VA(u). In this paper ,we applied vague set theory appliedin normed linear spaces and introduce a notion so called , vague normedlinear spaces ( in short VNLS ) and have studied their related properties.

On PL-absolute total curvature of surface-knotsTsukasa Yashiro (Sultan Qaboos University, Oman)

Abstract

The PL-absolute total curvature of an oriented m-manifold embeddedin n-space is defined as half of the average number of critical points withrespect to unit vectors in n-space. In this talk we give a lower bound ofthe PL-absolute total curvature of a special family of oriented surfaces ofgenus g embedded in 4-dimensional space.

50