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John Knopfmacher Centre for Applicable Analysis and Number TheorySchool of MathematicsSummary 2006 – 2010

The aim of the Centre, which was established as a research unit at the University of the Witwatersrand in 1992 by the late Professor John Knopfmacher, is to stimulate and pursue research within various topics belonging to pure mathematics. The mathematical subject areas described in the Centre name are an historical legacy reflecting the interests of the original membership. The research interests of the current members include topological algebra, enumerative combinatorics, mathematical logic, functional analysis, differential equations, probability theory, number theory and stochastic processes. The Centre is an officially recognized URC research unit. Each year Centre members publish papers in internationally recognized mathematics research journals, supervise higher degree students and attend local and overseas conferences in their disciplines.Membership is open to all staff in the School of Mathematics with NRF ratings or similar research recognition.The Centre members each belong to one of several different research sub-groups. These sub-groups also include PhD students and other staff in the School who are not currently members of the Centre.Each sub-group is encouraged to have regular meetings and to offer short courses given by its members or by visiting researchers.

Brief Summary of research activities during this period:The Centre members produced numerous research publications in their field of expertise, made presentations of their work at local and international conferences and supervised Masters and PhD students in their disciplines. Subjects of interest included topological algebra, enumerative combinatorics, mathematical logic, functional analysis, differential equations, probability theory, number theory and stochastic processes.

Number of Publications/Conference Presentations/higher degree awardedResearch

Type 2006 2007 2008 2009 2010DE Accredited

24 21 17 17 25

Proceedings 2 0 3 0 3Non Accredited

3 1 2 3 4

ISI Units(Publications)

13.90 15.42 12.48 13.66 11.59

NRF ratingsStaff Rating

Bau, S C (lapsed)Currie, S YKnopfmacher, A BLabuschagne, C (2006 – 2007) C

Möller, M BVan Alten, C CWatson, B BZelenyuk, (Ye) A (2006-2009), B (2010)Zelenyuk, (Yu) Y

Prizes/awards/commendations receivedProfessor A Knopfmacher was elected a Fellow of the Royal Society of South Africa in December 2010.Dr Yuliya Zelenyuk received a Friedel Sellschop Award.The research of A-Rated researcher Yevhen Zelenyuk led to papers in top international journals such as the “Proceedings of the American Mathematical Society”.Closing remarksThe Centre has served as a forum for the research mathematicians in the School of Mathematics since its inception by John Knopfmacher in 1992.

University of the Witwatersrand, JohannesburgFaculty of Science

Research Entity - Faculty of Science 2006Annual Progress Report

NB: Not to be completed by entities undergoing a five year review in the current year

Please use MS Word to fill-in the relevant fields and e-mail the completed Progress Report to the Chairperson Fazal Mahomed of the Faculty Research Committee (e-mail: [email protected]).

Name of Research Programme/Group/Unit/Institute/Centre

John Knopfmacher Centre for Applicable Analysis and Number Theory

Surname of Director Knopfmacher Initials A Title Professor

List participating University Academics

Names School % Time on EntityResearch Activities

Labuschagne, Professor C C A MathematicsMöller, Professor M H MathematicsMunagi, Dr A O MathematicsVan Alten, Dr C J MathematicsWarlimont, Honorary Professor R MathematicsWatson, Professor B A MathematicsZelenyuk, Professor Y Mathematics

Briefly state aims, objectives and strategic plan of the entity

The aim of the Centre, which was established as a research unit at the University of the Witwatersrand in 1992 by Professor John Knopfmacher, is to stimulate and pursue research within various topics belonging to pure mathematics. The mathematical subject areas described in the Centre name are an historical legacy reflecting the interests of the original membership. The research interests of the current members include topological algebra, enumerative combinatorics, mathematical logic, functional analysis, differential equations, probability theory, number theory and stochastic processes. The Centre is an officially recognized URC research unit.Membership is open to all staff in the School of Mathematics with NRF ratings or similar research recognition.The Centre members each belong to one of several different research sub-groups. These sub-groups also include PhD students and other staff in the School who are not currently members of the Centre.Each sub-group is encouraged to have regular meetings and to offer short courses given by its members or by visiting researchers.

http://www.wits.ac.za/science/number_theory/jplk.htm

Provide summaries of projects undertaken and brief descriptions of progress made against names of people involved (also list current and proposed future works)Knopfmacher, Prof A : Projects in Progress:With Margaret Archibald, statistics for the maximum of geometric samplesWith Charlotte Brennan and Stephan Wagner, Ascents of size d or more in partitionsWith Charlotte Brennan, Acents of size d or more in compositionsWith Miklos Bona, Pairs of compositions with the same number of partsWith Augustine Munagi, Successions in partitionsWith Mike Mays, Tree compositions

Labuschagne, Prof C C A :Project: Measure-free stochastic processes on Riesz spacesCo-workers: Korostenski M, Kuo W-C, Watson B AThis is an ongoing project on measure-free stochastic processes. The project has and still continues to yield a number of research papers in ISI journals (see list of publications above) and refereed conference proceedings. The project has also yielded three completed MSc’s and one completed PhD.

Project: Martingale Techniques in Banach spacesCo-worker: Cullender SThis is an ongoing project on studying the geometry of Banach spaces using measure-free martingale techniques.It has yielded a number of research papers in ISI journals (see list of publications above), one completedMSc and a PhD which will be completed in 2007.

Möller, Prof M : Due to administrative duties as Acting Head of the School of Mathematics, very little time was spent doing research with only two projects receiving some attention.Project: Negativity indices for indefinite Sturm-Liouville problemsCo-worker: Paul Binding, University of CalgaryThis is a continuation of earlier work, furthering the investigation of several functions associated with the problem, which gives information about the spectrum of the Sturm-Liouville problem. The poles of these functions are carefully analyzed and their pole orders as well as leading coefficients are compared. This provides us with upper and lower bounds for negativity indices.

Project: Differential equationsCo-workers: Igor Fedotov, Tswane University of Technology, and Michael Shatalov, CSIR.Investigation of differential equations which originate in engineering, e.g. underwater sonar. These DEs are fourth order or second order with singular highest coefficient. Thus standard methods to find the spectrum of the problem do not work, and new methods have to be devised. Some preliminary results have been obtained for particular cases.

Munagi, Dr A :Accepted Papers:“The Rademacher Conjecture and q-Partial Fractions”, The Ramanujan Journal“Combinations with Successions and Fibonacci Numbers”, Fibonacci Quarterly“Computation of q-Partial Fractions”, INTEGERS

Submitted Papers:“Pairing Conjugate Partitions by Residue Classes” “Enumerative Properties of Gap-Bounded Sequences” “Extended Set Partitions with Successions”

Projects in Progress:(with Knopfmacher A) “Successions in Integer Partitions” “Labelled Factorizations of Integers and Generalized Stirlingl Numbers”“Enumerative Proof of the Complete Compositions Theorem”“Circulating Functions and Group Characters”“Patterns in Combinations”

Van Alten, Dr C J :Project: NRF-sponsored joint SA-Polish research group in non-classical logicCo-workers: Prof Ewa Orlowska, Dr Wojciech DzikTwo joint papers were published in 2006: Relational representation theorems for lattices with negation, Lecture Notes in Computer Science.Relational representation theorems for lattices with negation (a survey), Lecture Notes in Computer Science. This project has run its course, however, the collaborators will seek to continue joint research. Project: Decidability in Non-classical Logics. This is an NRF-sponsored project started in 2006. The following paper has been accepted for publication: Axiomatizations of subreducts of n-potent residuated lattices, Algebra Universalis. I plan to undertake a number of academic visits during my half-sabbatical in 2007 to pursue research on this project.

Watson, Prof B A :Project: Stochastic process on Riesz spacesCullender S, Kuo W-C, Labuschagne CCA, Watson BA, “Measure-free martingales with applications to classical martingales”, in: Soft Methods for Integrated Uncertainty Modelling, Springer, Berlin, 2006. Advances in Soft Computing, Springer Verlag, 2006, pp.121-128.Kuo W-C, Labuschagne CCA, Watson BA, “Convergence of Riesz Space Martingales”, Indagationes Mathematicae, 17 (2006), 271-283.Kuo W-C, Labuschagne CCA, Watson BA, “A Zero-One Law for Riesz Space and Fuzzy Processes”, IFSA2005, Beijing, Springer Verlag, 2005, pp.393-397.Kuo W-C, Labuschagne CCA, Watson BA, “Conditional Expectations on Riesz spaces”, J. Mathematical Analysis and Applications, 303 (2005), 509-521.Kuo W-C, Labuschagne CCA, Watson BA, “Ergodic Theory and the Law of Large Numbers on Riesz Spaces”, J. Mathematical Analysis and Applications, (to appear).Kuo W-C, Labuschagne, CCA, Watson BA, “Amarts on Riesz spaces”, (submitted).Korostenski, M, Labuschagne, CCA, Watson, BA, “Reverse martingales in Riesz spaces”, (submitted).Watson, BA, “A Radon-Nikodym property in Riesz spaces with a conditional expectation”, (submitted).Project: Non-standard boundary value problemsBinding PA, Browne PJ, Watson BA, “Sturm-Liouville problems with reducible boundary conditions”, Proceedings of the Edinburgh Mathematical Society, 49 (2006), 593-608.Binding PA, Browne PJ, Watson BA, “Factorization of Sturm-Liouville problems and the Darboux-Crum transformation”, (in preparation).Binding PA, Watson BA, “Operator commutation in Pontryagin spaces”, (in preparation).Project: Boundary value problems on graphs Currie S, Watson BA, “Dirichlet-Neumann Bracketing for Boundary Value problems on Graphs”, Electronic Journal of Differential Equations, 2005 No. 93 (2005), 1-11.Currie S, Watson BA, “Eigenvalue Asymptotics for Differential Operators on Graphs”, Journal of Computational and Applied Mathematics, 182 (2005), 13-31.Currie S, Watson BA, “Green’s functions and regularized traces of Sturm-Liouville operators on graphs”, Proceedings of the Edinburgh Mathematical Society, (to appear).Currie S, Watson BA, “Boundary value problems on graphs – M-matrix asymptotics”, (submitted).Currie S, Watson BA, “Inverse nodal problems on graphs”, (in preparation).Project: Multiparameter boundary value problems Faierman M, Möller, Watson BA, “Completeness theorems for a non-standard two-parameter eigenvalue problem”, (submitted).

Zelenyuk, Prof Y : Project: Topological groupsThis is an ongoing project on studying topological groups with extremal properties like irresolvability, extremal disconnectedness, nodec and so on. An important partial case of the Arhangel’skii problem (raised in 1967) is solved in the paper.:Zelenyuk, Y, “On extremally disconnected topological groups”, Topology and its Applications, 153 (2006), 2382-2385.

List of Publications (which appeared in print, sorted by DoE categories):Category 3: Published Conference Proceedings

1. Cullender SF, Kuo W-C, Labuschagne CCA and Watson BA, “Measure-Free Martingales with Application to Classical Martingales”, Soft Methods for Integrated Uncertainty Modelling, pp.121-128, Springer-Verlag, 2006.

2. Denk R, Möller M and Tretter C, “The Spectrum of the Multiplication Operator Associated with a Family of Operators in a Banach Space”, Operator Theory in Krein Spaces and Nonlinear Eigenvalue Problems, pp.103-116, Birkhäuser, 2006.

Category 4 : DE Accredited Journals1. Andrews GE, Knopfmacher A and Zimmermann B, “On the number of distinct multinomial coefficients”,

Journal of Number Theory, 118 (2006), 15-30.2. Archibald M and Knopfmacher A, “Samples of geometric random variables with multiplicity constraints”,

Discrete Mathematics and Theoretical Computer Science, AG (2006), 437-440.3. Archibald M, Knopfmacher A and Prodinger H, “The number of distinct values in a geometrically distributed

sample”, European Journal of Combinatorics, 27 (2006), 1059-1081.4. Binding PA, Browne, PJ and Watson BA, “Sturm-Liouville problems with reducible boundary conditions”,

Proceedings of the Edinburgh Mathematical Society, 49 (2006), 593-608.5. Brennan C and Knopfmacher A, “The first ascent of size d or more in compositions”, Discrete Mathematics

and Theoretical Computer Science, AG (2006) 261-270.6. Christea LL and Prodinger H, “Moments of distributions related to digital expansions'', Journal of

Mathematical Analysis and Applications, 315 (2006), 606-625.7. Cullender SF and Labuschagne CCA, “A description of norm-convergent martingales on vector-valued

-spaces'', Journal of Mathematical Analysis and Applications, 323 (2006), 119-130.8. Driver K and Johnston SJ, “An integral representation of some hypergeometric functions'', Electronic

Transactions on Numerical Analysis, 25 (2006), 115-120.9. Driver KA, Prodinger H, Schneider C and Weideman JAC, “Padé approximations to the logarithm II:

Identities, recurrences, and symbolic computation'', The Ramanujan Journal, 11 No. 2 (2006), 139-158.10. Driver KA, Prodinger H, Schneider C and Weideman JAC, “Padé approximations to the logarithm III:

Alternative methods and additional results'', The Ramanujan Journal, 12 No. 3 (2006), 299-314.11. Dzik W, Orlowska E and van Alten C, “Relational Representation Theorems for General Lattices with

Negations'', Lecture Notes in Computer Science, 4136 (2006), 162-176.12. Grabner PJ and Knopfmacher A, “Analysis of some new partition statistics”, The Ramanujan Journal, 12

No. 3 (2006), 439-454.13. Hassi S, Möller M and de Snoo H, “A class of Nevanlinna functions related to singular Sturm-Liouville

problems'', Proceedings of the American Mathematical Society, 134 (10) (2006), 2885-2893.14. Knopfmacher A and Prodinger H, “The first descent in samples of geometric random variables and

permutations'', Discrete Mathematics and Theoretical Computer Science, 8 (2006), 215-234.15. Koné S and Möller M, “Mixed Order Systems of Ordinary Linear Differential Equations”, Rocky Mountain

Journal of Mathematics, 36 No. 3 (2006), 957-980.16. Kuo W-C, Labuschagne CCA and Watson BA, “Convergence of Riesz space martingales'', Indagationes

Mathematicae - New Series, 17 (2006), 271-283.17. Labuschagne CCA, “A Dodds-Fremlin Property for Asplund and Radon-Nikodým Operators'', Positivity, 10

(2006), 391407.18. Labuschagne CCA and Pinchuck AL, “The set of fuzzy points of a fuzzy vector lattice is not a vector

lattice'', Fuzzy Sets and Systems, 157 (2006), 2783-2785.19. Möller M and Pivovarchik V, “Spectral Properties of a Fourth Order Differential Equation'', Zeitschrift für

Analysis und ihre Anwendungen, 25 (2006), 341-366.20. Warlimont R, “Complete compositions of a natural number”, Quaestiones Mathematicae, 29 (2) (2006),

229-238.21. Zelenyuk Y, “On extremally disconnected topological groups”, Topology and its Applications, 153 (2006),

2382-2385.Category 4a : Non-Accredited Journals

1. Van Alten CJ, “On Varieties of Biresiduation Algebras”, Studia Logica, 83 No. 1-3 (2006), 425-445.2. Knopfmacher A and Mays M, “Dirichlet generating functions and factorization identities”, Congressus

Numerantium, 173 (2005), 117-127.3. Knopfmacher A and May M, “Ordered and Unordered Factorizations of Integers”, The Mathematica Journal, 10

(2006), 72-89.

List Researchers from other centres who visited and contributed to research:Professor Paul Binding, University of Calgary, CanadaProfessor Miklos Bona, Florida State UniversityDr Wojciech Dzik, Silesian University, PolandProfessor Mike Mays, West Virginia UniversityProfessor Ewa Orlowska, National Institute of Telecommunications, Warsaw, PolandDr Stephan Wagner, Technical University of Graz

Details of MSc and PHD students who completed their degrees in the one year period

Student No: Name Degree Registration

Month & Year of first

registration

Month and Year of

completionSupervisor(S)

0108851W Hawke, P MSc Feb 2005 Feb 2006 Labuschagne, C C A9700502J Kuo, W-C PhD July 2004 Aug 2006 Labuschagne, C C A9206749J Rusznyak, D PhD Hartney, J & Watson, B

Tabulate details of current MSc and PhD students



registration

Month & Year of completion Supervisor(s)

0201319N Botha, G MSc (f/t) Zelenyuk, Y9900444T Emmett, R MSc Watson, B9700100P Moletsane, B MSc (f/t) Möller, M

0420408P Brooks, H PhD (p/t) Mar 2004 Möller, M9711450V Cullender, S PhD (f/t) Feb 2005 May 2007 Labuschagne, C C A0618369F Pinchuck, A PhD (p/t) Feb 2006 Unknown Labuschagne, C C A0114294H Matlala, A M PhD Watson, B8902505T Mavhungu, S N PhD (p/t) Mar 2001 March

2007Knopfmacher, A

9713739J Mdhluli, D S PhD (f/t) Möller, M

List of external/internal research grants received during the one year periodKnopfmacher, A : NRF: Unlocking the Future, 2002-2006

Möller, M: NRF Grant: Unlocking the Future, 2002-2006Van Alten, C J : NRF: Unlocking the Future, Ref. FA2005030900033, duration: 2006-2008.Title: Decidability in Non-Classical Logics.

All Centre members also received John Knopfmacher Centre Funding based on their ISI output in the previousfew years.

University of the Witwatersrand, Johannesburg

Faculty of Science









Labuschagne, Professor C C A MathematicsMöller, Professor M H MathematicsMunagi, Dr A O MathematicsShonhiwa, Dr T MathematicsVan Alten, Dr C J MathematicsWarlimont, Honorary Professor R MathematicsWatson, Professor B A MathematicsZelenyuk, Professor Y Mathematics

Briefly state aims, objectives and strategic plan of the entity The aim of the Centre, which was established as a research unit at the University of the Witwatersrand in 1992 by Professor John Knopfmacher, is to stimulate and pursue research within various topics belonging to pure mathematics. The mathematical subject areas described in the Centre name are an historical legacy reflecting the interests of the original membership. The research interests of the current members include topological algebra, enumerative combinatorics, mathematical logic, functional analysis, differential equations, probability theory, number theory and stochastic processes. The Centre is an officially recognized URC research unit.Membership is open to all staff in the School of Mathematics with NRF ratings or similar research recognition.The Centre members each belong to one of several different research sub-groups. These sub-groups also include PhD students and other staff in the School who are not currently members of the Centre.Each sub-group is encouraged to have regular meetings and to offer short courses given by its members or by visiting researchers.

Provide summaries of projects undertaken and brief descriptions of progress made against names of people involved (also list current and proposed future works)


Knopfmacher, Prof A : Projects in Progress:

1. A Knopfmacher and N Robbins,”On multi-color partitions with distinct parts'', accepted by Ars Combinatorica. 2. C Brennan, A Knopfmacher and S Wagner, “On the number of ascents of a given size in integer partitions'', accepted

by Combinatorics Probability and Computing. 3. A Knopfmacher and A. Munagi, “Permutations with interval cycles'', accepted by Ars Combinatorica. 4. M Bona and A Knopfmacher, “On The Probability that Certain Compositions Have The Same Number Of Parts'',

accepted by Annals of Combinatorics.

Labuschagne, Prof C C A :

Project: Measure-free stochastic processes on Riesz spacesCo-workers: Korostenski M, Kuo W-C, Watson B AThis is an ongoing project on measure-free stochastic processes. The project has and still continues to yield a number of research papers in ISI journals (see list of publications above) and refereed conference proceedings.

Project: Martingale Techniques in Banach spacesCo-worker: Cullender SThis is an ongoing project on studying the geometry of Banach spaces using measure-free martingale techniques.It has yielded a number of research papers in ISI journals (see list of publications above). These projects yielded 2 completed PhD’s in 2007.

Möller, Prof M :

Due to administrative duties as Head of the School of Mathematics, very little time was spent doing research with only one project receiving some attention.

Project: Negativity indices for indefinite Sturm-Liouville problemsCo-worker: Paul Binding, University of CalgaryThis is a continuation of earlier work, furthering the investigation of several functions associated with the problem, which gives information about the spectrum of the Sturm-Liouville problem. The poles of these functions are carefully analyzed and their pole orders as well as leading coefficients are compared. This provides us with upper and lower bounds for negativity indices. The paper has been submitted in 2007.

Munagi, Dr A :

Projects: “Labeled Factorization of Integers”. (nearly completed).“Recurrence relation and Plane Compositions” (with T. Mansour, A. Knopfmacher). (nearly completed)“Enumeration of Smooth Words” (with A. Knopfmacher, T. Mansour, H. Prodinger). (nearly completed)“Enumeration of Words with Successions” (with S. Wagner, A. Knopfmacher). (in progress)

Accepted Papers:A. Knopfmacher and A. O. Munagi, “ Permutations with interval cycles”, Ars Combinatoria.N. Y. Lee, T. Mansour and A.O. Munagi, “Plane trees and reduction rules”, Utilitas Math.A. O. Munagi, A. Knopfmacher, “Successions in Integer Partitions”, Ramanujan Journal.A. O. Munagi, “Alternating Subsets and Successions”, Ars Combinatoria.A. O. Munagi, “Alternating Subsets and Permutations”, Rocky Mountain Journal of Math.T. Mansour and A. O. Munagi, “Enumeration of partitions by rises, levels and descents”, Proceedings of the 5 th Conference on Permutation Patterns, Scotland, UK, 2007.A.O. Munagi, “Generalized alternating subsets with permutations”, Proceedings of the 13th International Conference on Fibonacci Numbers and their Applications, Patras, Greece, 2008.

Submitted Papers:“On the Partitions of a Number into Arithmetic Progressions” (with T. Shonhiwa).“Avoiding permutation patterns of type (2,1) in compositions” (with Silvia Heubach and T. Mansour).“Enumeration of partitions by long rises, levels and descents” (with T. Mansour).

Shonhiwa, Dr T:

Accepted Papers:(i) ``Generalized LCM Product Functions”, Utilitas Mathematica.(ii) `Àrithmetical Function Characterizations and Identities Induced Through Equivalence Relations”, Notes on Number Theory and Discrete Mathematics.

Submitted Papers:(i) `Òn Compositeness of Multicompositions”, Quaestiones Mathematicae.(ii) `Òn the Partition of a Number into Arithmetic Sequences”, Journal of Integer Sequences. (With Munagi)

Projects in Progress:(i) Characterizations of Additive Functions

Van Alten, Dr C J :Project: Decidability in Non-classical Logics. Submitted paper: “Structural completeness in substructural logics”, by Olsen, J., Raftery, J. and Van Alten, C J.

In Preparation: Completion and finite embeddability property for residuated ordered algebras, Van Alten, C J.

Watson, Prof B A :1. Binding, Browne, Watson – Study the spectral behaviour of non-standard boundary value problems, in particular boundary value problems in which the boundary conditions depend on the eigenparameter. Have completed manuscripts on: (a) the factorization of the operators arising from such problems;(b) the spectral behaviour of a certain class of quadratic pencil;(c) a two-parameter inverse nodal problem.

2. Currie, Watson – Study forward and inverse problems four boundary value problems on graphs: manuscript has been submitted in which we have solved the inverse spectral problems where the spectral data is either the M-matrix or spectrum and norming constants.

3. Korostenski, Kuo, Labuschagne, Watson – Develop an order based (measure-free) approach to stochastic processes. Have submitted a manuscript on the convergence of reverse martingales (this provides a key step in the consideration of continuous time processes); have submitted and accepted a paper on a measure-free Radon-Nikodym theorem (an essential step in the consideration of Markov processes in a measure-free setting); manuscript on a measure-free stochastic integral should be completed by the end of 2008.

Zelenyuk, Prof Y : Project: Topological GroupsThis is an ongoing project on studying various questions related to topological groups. Several important problems have been solved. In particular, it was shown that (a) every infinite group admits a nondiscrete 0-dimensional Hausdorff topology with continuous translations and inversion, and (b) every countable topological group containing a discrete subset with exactly one accumulation point admits a local homomorphism onto the natural numbers. These and other results have been published in 5 papers, one of them jointly with Yuliya Zelenyuk.

List of Publications (which appeared in print, sorted by DoE categories):Category 4 : DE Accredited Journals

1. Archibald, M. and Knopfmacher“The first position of the maximum in samples of geometric random variables”DMTCS, AH (2007) 269-278.

2. Cullender, S.F. and Labuschagne, C.C.A. "Unconditional martingale difference sequences in Banach spaces", Journal of Mathematical Analysis and Applications, 326 (2007), 1291-1309.

3. Cullender, S.F. and Labuschagne, C.C.A.“A note on the -norm of Chaney-Schaefer”,Quaestiones Mathematicae, 30 (2007), 152-158.

4. Cullender, S.F. and Labuschagne, C.C.A. "Unconditional Schauder decompositions and stopping times in the Lebesgue-Bochner spaces", Journal of Mathematical Analysis and Applications, 336 (2007), 849-864.

5. Currie, S. and Watson, B.A.“Inverse nodal problems for Sturm-Liouville equations on graphs”,Inverse Problems, 23 (2007), 2029-2040.

6. Driver, K. and Jordaan, K."Separation theorems for the zeros of certain hypergeometric polynomials", Journal of Computational and Applied Mathematics, 199 (2007), 48-55.

7. Faierman, M. and Möller, M.“Eigenvalue asymptotics for an elliptic boundary problem”,Proceedings of the Royal Society of Edinburgh, 137A (2007), 281-302.

8. Knopfmacher, A. and Prodinger, H.“The Last Descent in Samples of Geometric Random Variables and Permutations”,Ars Combinatoria, 83 (2007), 365-379.

9. Knopfmacher, A., R.F. Tichy, S. Wagner and Ziegler, V. "Graphs, partitions and Fibonacci numbers", Discrete Applied Mathematics, 155 (2007), 1175-1344.

10. Knopfmacher, A. and Warlimont, R.“Gaps in integer partitions”,Utilitas Mathematica, 71 (2006), 257-267.

11. Korostenski, M. and Labuschagne, C.C.A. "Lax proper maps of locales", Journal of Pure and Applied Algebra, 208 (2007), 655-664.

12. Kuo, W-C., Labuschagne, C.C.A. and Watson, B.A. "Ergoic theory and the Strong Law of Large Numbers on Riesz spaces", Journal of Mathematical Analysis and Applications, 325 (2007), 422-437.

13. Labuschagne, C.C.A."Preduals and Nuclear Operators Associated with Bounded, -Convex,

-Concave and Positive -Summing Operators",Canadian Journal of Mathematics, 59 (3) (2007), 614-637.

14. Labuschagne, C.C.A. and van Alten, C.J.“On the variety of Riesz spaces”,Indagationes Mathematicae, N.S., 18 (1) (2007), 61-68.

15. Van Alten, C.J.“Implicational subreducts of n-potent commutative residuated lattices”,Algebra Universalis, 57 (2007), 47-62.

16. Zelenyuk, Yevhen"On topologizing groups", Journal of Group Theory, 10 (2007), 235-244.

17. Zelenyuk, Yevhen“Almost maximal spaces”,Topology and its Applications, 154 (2007), 339-357.

18. Zelenyuk, Yevhen“On the Ultrafilter Semigroup of an Abelian Topological Group”,Semigroup Forum, 74 (2007), 467-472.

19. Zelenyuk, Yevhen“Local homomorphisms of topological groups”,Journal of the Australian Mathematical Society, 83 (2007), 1-14.

20. Zelenyuk, Y. and Zelenyuk, Yuliya“Transformations and Colorings of Groups”,Canadian Mathematical Bulletin, 50 (4) (2007), 632-636.

Category 4A : Non-Accredited Journals1. Munagi, A.O.

“Computation of q-partial fractions”,Integers: Electronic Journal of Combinatorial Number Theory, 7 (2007), #A25, 21pp.

List Researchers from other centres who visited and contributed to research:Professor Paul Binding, University of Calgary, CanadaProfessor Paul Browne, University of Saskatchewan, CanadaDr Alexander Burstein, Howard University, Washington DC, USA, visited in August and presented a short course, “Patten Avoidance in Words and Permutations”.Dr R Hryniv, Institute for Applied Problems in Mechanics and Mathematics, Lviv, UkraineDr R Hryniv, Lviv National University, UkraineDr Manuel Lladser, University of Colorado, visited in November and presented a course on Multivariate Asymptotics.Dr Stephan Wagner, University of Stellenbosch, visited in January and gave a course on Tree Enumeration.

Further short-term visitors to the Centre who presented seminars:Alan Beardon, Cambridge University, U.K.G Bluman, University of British Columbia, CanadaS Damelin, Georgia Southern University, U.S.A.Elliot Lieb, Princeton University, U.S.A.Ian Short , National University of IrelandWieslaw Zelazko, Polish Academy of Sciences, Warsaw, Poland




registration

Month and Year of


9711450V Cullender, S PhD Feb 2005 May 2007 Labuschagne, C C A8902505T Mavhungu, S N PhD Mar 2001 June 2007 Knopfmacher A & Prodinger H9700502J Kuo, W-C PhD July 2004 2007 Labuschagne, C C A




registration


0201319N Botha, G MSc (f/t) Zelenyuk, Y

9900444T Emmet, R J MSc June 2008 Watson, B A9700100P Moletsane, B MSc (f/t) Möller, M

0314117G Zinsou, B MSc (f/t) Jan 2007 Möller, M

0420408P Brooks, H PhD (p/t) Mar 2004 Möller, M0618369F Pinchuck, A PhD (p/t) Feb 2006 Unknown Labuschagne, C C A

0609043F Nthebe, M J PhD (f/t) Mar 2006 Möller, M0114294H Matlala, A M PhD 2008 Watson, B A

List of external/internal research grants received during the one year periodKnopfmacher, A : NRF: Unlocking the Future grant and B2 ratingMöller, M: Unlocking the FutureMunagi, A: New and Developing Staff Grant from the Faculty of Science Research Committee – 2008 Van Alten, C J : NRF: Unlocking the Future, Ref. FA2005030900033, duration: 2006-2008.Title: Decidability in Non-Classical Logics.

All Centre members also received John Knopfmacher Centre Funding based on their ISI output in the previousfew years.

Research Entity - Faculty of Science 2008

Annual Progress Report








Möller, Professor M H MathematicsMunagi, Dr A O MathematicsShonhiwa, Dr T MathematicsVan Alten, Dr C J MathematicsWarlimont, Honorary Professor R MathematicsWatson, Professor B A MathematicsZelenyuk, Professor Y Mathematics

Briefly state aims, objectives and strategic plan of the entityThe aim of the Centre, which was established as a research unit at the University of the Witwatersrand in 1992 by Professor John Knopfmacher, is to stimulate and pursue research within various topics belonging to pure mathematics. The mathematical subject areas described in the Centre name are an historical legacy reflecting the interests of the original membership. The research interests of the current members include topological algebra, enumerative combinatorics, mathematical logic, functional analysis, differential equations, probability theory, number theory and stochastic processes. The Centre is an officially recognized URC research unit.

Membership is open to all staff in the School of Mathematics with NRF ratings or similar research recognition.

The Centre members each belong to one of several different research sub-groups. These sub-groups also include PhD students and other staff in the School who are not currently members of the Centre.

Each sub-group is encouraged to have regular meetings and to offer short courses given by its members or by visiting researchers.



Knopfmacher, Prof A

Projects in Progress:1. “The distribution of ascents of size d or more in compositions”, DMTCS (2009) (with C. Brennan)2. “The d-th position of the maximum in samples of geometric random variables”, Statistics and Prob. Letters

(2009) (with M. Archibald)3. “Successions in partitions of integers”, The Ramanujan Journal (2009) (with A. Munagi)4. “Permutations with interval cycles, Ars Combinatoria (with A. Munagi)5. “On the probability that certain compositions have the same number of parts”, Annals of Combinatorics (with M. Bona)6. “Plane compositions”, J. Diff. Eq. Appl. (with T. Mansour & A. Munagi)7. “Smooth words”, submitted (with T. Mansour, A. Munagi & H. Prodinger)8. “Record statistics in a random composition”, submitted (with T. Mansour)9. “Record statistics in integer compositions”, DMTCS, FPSAC09 Proceedings (with T. Mansour)

Möller, Prof M

Due to administrative duties as Head of the School of Mathematics, very little time was spent doing research. Three papers are in preparation as a result of student supervision.

Projects: 1. “On maximal domains for C-convex functions and convex extensions” (with T.M.J. Nthebe) (nearly completed)2. “Spectra of multiplication operators in Sobolev spaces” (with H. Brooks) (nearly completed)3. “Self-adjoint fourth order differential operators with eigenvalue parameter dependent boundary conditions”

(with B. Zinsou) (in progress)

Accepted Papers:“Negativity indices for definite and indefinite Sturm-Liouville problems”, Math. Nachrichten (with P. Binding)

Munagi, Dr A

Projects:1. Staircase compositions and staircase words (with A. Knopfmacher & T. Mansour)2. Factorization of excedent sequences3. A generalization of the Legendre-Stirling numbers4. Classification of set partitions by residue classes

Accepted Papers:1. “Permutations with interval cycles”, Ars Combinatoria (with A. Knopfmacher)2. “Enumeration of partitions by rises, levels and descents”, Proceedings of the 5th Conference on Permutation

Patterns, Scotland, UK, 2007 (with T. Mansour)3. “Generalized alternating subsets with permutations”, Proceedings of the 13th International Conference on

Fibonacci Numbers and their Applications, Patras, Greece, 20084. “Plane trees and reduction rules”, Utilitas Math. (with N.Y. Lee & T. Mansour)5. “Alternating subsets and successions”, Ars Combinatoria6. “Alternating subsets and permutations”, Rocky Mountain Journal of Math.7. “Recurrence relation with two indices and plane compositions”, Journal of Difference Equations and

Applications (with A. Knopfmacher & T. Mansour)8. “Labeled factorization of integers”, Electronic Journal of Combinatorics

Submitted Papers:1. “Avoiding permutation patterns of type (2,1) in compositions” (with S. Heubach & T. Mansour)2. “Staircase words and Chebyshev polynomials” (with A. Knopfmacher, T. Mansour & H. Prodinger)3. “Block-connected set partitions” (with T. Mansour)4. “Combinatorics of integer partitions in arithmetic progression”

Shonhiwa, Dr T

Accepted Papers:1. “Generalized LCM Product Functions”, Utilitas Mathematica.2. “On Compositeness of Multicompositions”, Quaestiones Mathematicae.

Projects in Progress:1. Characterizations of Additive Functions.2. Characterizations of Completely Multiplicative Functions.

Van Alten, Dr C J

Regarding my NRF project on “Decidability in Non-classical Logics”, a paper entitled “Completion and finite embeddability property for residuated ordered algebras” was completed and accepted for publication in Algebra Universalis. This paper proves decidability via the finite embeddability property for a large class of algebras related to non-classical logics. A second paper on this topic entitled “Preservation theorems for MTL-algebras” was submitted for publication in March 2009. The results were presented at the conference: Logic, Algebra and Truth Degrees, in Siena, Italy, 8-11 September, 2008.

In a related direction, I started a joint project with Dr W Conradie from the University of Johannesburg on the finite embeddability property for modal algebras, which also arise in connection with non-classical logics. From July 2008, my PhD student, Wilmari Morton, joined the project. A first paper is nearly complete and should be submitted in the first half of 2009.

My interest in non-classical logic encompasses fuzzy logics and in this regard, I attended the 9th International Conference on Intelligent Technologies, in Samui, Thailand, 7-9 October 2008. My paper, which was joint work with C Labuschagne of Wits University, was published in the proceedings of the conference.

Watson, Prof B A

1. Binding, Browne, Watson – Study the spectral behaviour of non-standard boundary value problems, in particular boundary value problems in which the boundary conditions depend on the eigenparameter. Have completed manuscripts on: (a) the factorization of the operators arising from such problems;(b) the spectral behaviour of a certain class of quadratic pencil;(c) a two-parameter inverse nodal problem.

2. Currie, Watson – Study forward and inverse problems four boundary value problems on graphs: manuscript has been submitted in which we have solved the inverse spectral problems where the spectral data is either the M-matrix or spectrum and norming constants.

3. Korostenski, Kuo, Labuschagne, Watson – Develop an order based (measure-free) approach to stochastic processes. Have submitted a manuscript on the convergence of reverse martingales (this provides a key step in the consideration of continuous time processes); have submitted and accepted a paper on a measure-free Radon-Nikodym theorem (an essential step in the consideration of Markov processes in a measure-free setting); manuscript on a measure-free stochastic integral should be completed by the end of 2008.

Zelenyuk, Prof Y

Project: Topological Groups

This is an ongoing project on studying various questions related to topological groups. Several important problems have been solved. In particular, (a) it is shown that every infinite Abelian group is absolutely resolvable, and (b) some new topological groups with important extremal properties are constructed. These and other results have been published in 4 papers.

List of Publications (which appeared in print, sorted by DoE categories):

Category 3 : Published Conference Proceedings

1. Currie, S. and Watson, B.A.“Boundary estimates for solutions of non-homogeneous boundary value problems on graphs”,Proceedings of the 12th WSEAS International Conference on APPLIED MATHEMATICS, Cairo, Egypt, 29 – 31 December 2007, pp.37-42.

2. Labuschagne, C.C.A. and van Alten, C.J.“On the MacNeille Completion of MTL-chains”,Proceedings of the 9th International Conference on Intelligent Technologies (InTech’08),Samui, Thailand, 7-9 October 2008, pp.71-75.

3. Zelenyuk, Y.“Partitions of Vector Spaces over Finite Fields”,Algebraic Geometry and Its Applications, Proceedings of the First SAGA Conference, 7-11 May 2007, Papette, France. World Scientific, pp. 505-511.

Category 4 : DE Accredited Journals

1. Brennan, C., Knopfmacher, A. and Wagner, S.“The Distribution of Ascents of Size or More in Partitions of ”,Combinatorics, Probability and Computing, 17 (2008), 495-509.

2. Brown, P.J. and Watson, B.A.“Oscillation Theory for a Quadratic Eigenvalue Problem”,Quaestiones Mathematicae, 31, No. 4 (2008), 345-357.

3. Currie, S. and Watson, B.A.“Green’s functions and regularized traces of Sturm-Liouville operators on graphs”,Proceedings of the Edinburgh Mathematical Society, 51 (2008), 315-335.

4. Currie, S. and Watson, B.A.“ -matrix asymptotics for Sturm-Liouville problems on graphs”,Journal of Computational and Applied Mathematics, 218 (2008), 568-578.

5. De, D. and Paul, R.K.“Universally Image Partition Regularity”,The Electronic Journal of Combinatorics, 15 (2008), #R141, 7pp.

6. Faierman, M., Möller, M. and Watson, B.A.“Completeness Theorems for a Non-Standard Two-Parameter Eigenvalue Problem”,Integral Equations and Operatory Theory, 60 (2008), 37-52.

7. Knopfmacher, A. and Robbins, N.“On multi-color partitions with distinct parts”,Ars Combinatoria, 89 (2008), 401-419.

8. Kuo, W.-C., Labuschagne, C.C.A. and Watson, B.A.“Amarts on Riesz Spaces”,Acta Mathematica Sinica, English Series, 24 (2) (2008), 329-342.

9. Möller, M. and Szafraniec, F.H.“Adjoints and formal adjoints of matrices of unbounded operators”,Proceedings of the American Mathematical Society, 136, No. 6 (2008), 2165-2176.

10. Munagi, A.O.“Extended set partitions with successions”,European Journal of Combinatorics, 29 (2008), 1298-1308.

11. Munagi, A.O.

“Pairing conjugate partitions by residue classes”,Discrete Mathematics, 308 (2008), 2492-2501.

12. Munagi, A.O.“The Rademacher conjecture and -partial fractions”,Ramanujan Journal, 15 (2008), 339-347.

13. Munagi, A.O.“Combinations with successions and Fibonacci numbers”,Fibonacci Quarterly, 45 (2) (2007), 104-114.

14. Olson, J.S., Raftery, J.G. and van Alten, C.J.“Structural Completeness in Substructural Logics”,Logic Journal of IGPL, 16 No. 5 (2008), 453-495.

15. Zelenyuk, Y.“Finite groups in Stone-Cech compactifications”,Bulletin of the London Mathematical Society, 40 (2008), 337-346.

16. Zelenyuk, Y.“Partitions and sums with inverses in Abelian groups”,Journal of Combinatorial Theory, Series A, 115 (2008), 331-339.

17. Zelenyuk, Y.“Topologies on groups determined by discrete subsets”,Topology and its Applications, 155 (2008), 1332-1339.

Category 4A : Non-Accredited Journals

1. Gould, H.W. and Shonhiwa, T.“A catalog of interesting Dirichlet series”,Missouri Journal of Mathematical Sciences Articles, 20 Issue #1 (2008) 17 pages.

2. Munagi, A.O. and Shonhiwa, T.“On the Partitions of a Number into Arithmetic Progressions”,Journal of Integer Sequences, 11 (2008), Article 08.5.4.

List Researchers from other centres who visited and contributed to research:

Dr Margaret Archiblad, UCTProfessor Paul Binding, University of Calgary, CanadaProfessor Paul Browne, University of Saskatchewan, CanadaProfessor Toufik Mansour, Haifa University, IsraelProfessor Mike Mays, West Virginia UniversityDr Stephan Wagner, Stellenbosch University




registration

Month and Year of


9900444T Emmet, R J MSc June 2008 Watson, B A0114294H Matlala, A M PhD 2008 Watson, B ATabulate details of current MSc and PhD students



registration


0201319N Botha, G MSc (f/t) Zelenyuk, Y9900444T Emmet, R J MSc June 2008 Watson, B A9700100P Moletsane, B MSc (f/t) Möller, M0305498E Toloane, E MSc Jan 2008 van Alten, C J0314117G Zinsou, B MSc (f/t) Jan 2007 Möller, M

0420408P Brooks, H PhD (p/t) Mar 2004 Möller, M0609043F Nthebe, T M J PhD (f/t) Mar 2006 Möller, M0114294H Matlala, A M PhD 2008 Watson, B A0616713N Morton, W. PhD Jul 2008 van Alten, C J

List of external/internal research grants received during the one year periodKnopfmacher, A: Unlocking the future grant

Munagi, A: New and Developing Researchers, Faculty of Science Research Committee

Van Alten, C J: Unlocking the Future, reference FA2005030900033Watson, B A: NRF: Unlocking the Future grant and B3 rating

All Centre members also received John Knopfmacher Centre Funding based on their ISI output in the previous few years.

Research Entity - Faculty of Science 2009

Annual Progress Report








Möller, Professor M H MathematicsMunagi, Dr A O MathematicsShonhiwa, Dr T MathematicsVan Alten, Dr C J Maths/Comp. ScienceWarlimont, Honorary Professor R MathematicsWatson, Professor B A MathematicsZelenyuk, Professor Y Mathematics







Knopfmacher, Prof A

Accepted Papers:M. Archibald, A. Knopfmacher and T. Mansour, “Compositions and samples of geometric random variables with constrained multiplicities”, accepted by FPSAC 2010.

Submitted Papers:1. Knopfmacher and T. Mansour, “Record statistics in a random composition”, submitted.2. M. Archibald, A. Knopfmacher and T. Mansour, “Forbidden multiplicities in compositions and samples of

geometric random variables”, submitted. 3. M. Archibald, A. Knopfmacher and T. Mansour, “Variation statistics on compositions”, submitted. 4. A. Knopfmacher, A. Munagi and S. Wagner “Successions in words and compositions”. ?5. A . Knopfmacher and M. Mays and A. Munagi, “Inversions in compositions of integers”, submitted. 6. P. Grabner, A. Knopfmacher and S. Wagner, “A general asymptotic scheme for moments of partition statistics”,

submitted. 7. Knopfmacher and N. Robbins, “Some Properties of Cyclic Compositions”, submitted.

Möller, Prof M

Projects: 1. “On maximal domains for C-convex functions and convex extensions” (with T.M.J. Nthebe) 2. “Spectra of multiplication operators in Sobolev spaces” (with H. Brooks) 3. “Self-adjoint fourth order differential operators with eigenvalue parameter dependent boundary conditions”

(with B. Zinsou) (in progress) 4. “Quadraticd operator polynomials and applications

Accepted Papers:“Negativity indices for definite and indefinite Sturm-Liouville problems”, Math. Nachrichten (with P. Binding) “On maximal domains for convex functions and convex extensions”, Journal of Convex Analysis (with M.J. Nthebe)

Published Papers:“Spectra of multiplication operators in Sobolev spaces”, Result. Math. 55 (2009), 281-293 (with H. Brooks)

Submitted Papers:Self-adjoint fourth order differential operators with eigenvalue parameter dependent boundary conditions (with B. Zinsou)

Conferences attended: Boundary Relations, Leiden (The Netherlands), 14 – 18 December 2009 Presented a talk on: Self-adjoint fourth order differential operators with eigenvalue parameter dependent boundary conditions

Munagi, Dr A

Accepted Papers:1. T. Mansour, A.O. Munagi, Enumeration of gap-bounded Set Partitions, Automata,

Languages and Combinatorics.2. T. Mansour, A.O. Munagi, Alternating subsets modulo m, Rocky Mountain Journal of Math.3. Knopfmacher, T. Mansour, A. O. Munagi, Recurrence relation with two indices and plane compositions, Journal

of Difference Equations and Applications

Projects in progress:5. Factorization of excedent sequences6. Generalized Steinhaus sequences.7. IMO permutations and residue classes (with F. Luca).8. Arithmetic permutations of finite sets (with T. Mansour).9. Successions in words and compositions (with S. Wagner, A. Knopfmacher)10. Inversions in Compositions of Integers (with A. Knopfmacher, M. E. Mays).

Shonhiwa, Dr T (Passed away on 24 October 2009)

Accepted Paper: On Relatively Prime Sets Counting Functions, accepted by Electronic Journal of Combinatorial Number Theory.

Van Alten, Dr C J

Accepted papers:1. C.J. van Alten, Completion and finite embeddability property for residuated ordered algebras, to appear in

Algebra Universalis.2. C.J. van Alten, Preservation theorems for MTL-chains, to appear in the Logic Journal of the IGPL.

Papers in progress:1. W. Conradie, W. Morton, C.J. van Alten, An algebraic view of filtrations in modal logic.2. W. Morton, C.J. van Alten, Modalities on MTL-chains.

Watson, Prof B A

Current Projects:1. An abstract Darboux-Crum transformation in Pontryagin spaces. (Participants: Binding & Watson)2. Transformations within classes of boundary value problems – classes of second order boundary value

problems consistent with the symplectic nature of the boundary forms. (Participants: Kara & Watson)3. The diffusion operators on graphs – spectral asymptotics, oscillation theory, completeness of eigenfunction

expansions. (Participants: Currie & Watson)4. The Fucik spectrum for boundary value problems on graphs. (Participants: Currie & Watson)5. Spectral asymptotics for general indefinite Sturm-Liouville problems. (Participants: Love & Watson)6. Spectral asymptotics for indefinite p-Laplacian operators. (Participants: Binding, Brown & Watson)7. Indefinite Sturm-Liouville operators on graphs, the nature of the spectrum, eigenvalue asymptotics and

completeness. (Participants: Currie & Watson)8. Markov processes, spectral measures and Riesz spaces. (Participants: Vardy & Watson)9. Local martingales and their order structure. (Participants: Labuschagne, Vardy & Watson)

Zelenyuk, Prof Y

During this year I published 5 papers, two of them jointly. In these papers, several important problems in topological groups and semigroups have been solved. In particular, it is shown that (a) every countable irresolvable topological group contains an open Boolean subgroup, and (b) for every infinite abelian group G of cardinality k, the semigroup beta G contains exp(exp(k)) minimal right ideals.



1. Archibald, M. and Knopfmacher, A.“The average position of the d th maximum in a sample of geometric random variables”,Statistics & Probability Letters, 79 No. 7 (2009), 864-872.

2. Binding, P.A. and Watson, B.A.“An inverse nodal problem for two-parameter Sturm-Liouville systems”,Inverse Problems, 25 (2009), 085005 (19pp).

3. Brennan, C.A. and Knopfmacher, A.“The distribution of ascents of size d or more in compositions”,Discrete Mathematics & Theoretical Computer Science, 11 No. 1 (2009), 1-10.

4. Brooks, H. and Möller, M.

“Spectra of multiplication operators in Sobolev spaces”,Results in Mathematics, 55 (2009), 281-293.

5. Currie, S. and Watson, B.A.“The M-matrix inverse problem for the Sturm-Liouville equation on graphs”,Proceedings of the Royal Society of Edinburgh, 139A (2009), 775-796.

6. De, D. And Hindman, N.“Image partition regularity near zero”,Discrete Mathematics, 309 (2009), 3219-3232.

7. Knopfmacher, A. and Munagi, A.O.“Successions in integer partitions”,Ramanujan Journal, 18 No. 3 (2009), 239-255.

8. Munagi, A.O.“Labeled factorization of integers”,Electronic Journal of Combinatorics, 16 (1) (2009), Research Paper 50, 17 pp.

9. Shonhiwa, T.“A note on some characterizations of arithmetic functions”Rocky Mountain Journal of Mathematics, 39 (2) (2009), 645-651.

10. Shonhiwa, T.“On compositeness in multicompositions”,Quaestiones Mathematicae, 32 (2009), 35-45.

11. Shonhiwa, T.“Generalized LCM-product function”,Utilitas Mathematica, 80 (2009), 161-170.

12. Shuungula, O., Zelenyuk, Yevhen and Zelenyuk, Yuliya“The closure of the smallest ideal of an ultrafilter semigroup”,Semigroup Forum, 79 (2009), 531-539.

13. Watson, B.A.“An Andô-Douglas type theorem in Riesz spaces with a conditional expectation”,Positivity, 13 (2009), 543-558.

14. Zelenyuk, Yevhen“Regular homeomorphisms of finite order on countable spaces”,Canadian Journal of Mathematics, 61 (3) (2009), 708-720.

15. Zelenyuk, Yevhen“The number of minimal right ideals of βG”,Proceedings of the American Mathematical Society, 137 (7) (2009), 2483-2488.

16. Zelenyuk, Yevhen“The smallest ideal of S is not closed”,Topology and its Applications, 156 (2009), 2670-2673.

17. Zelenyuk, Yevhen and Zelenyuk, Yuliya.“Free groups in the smallest ideal of βG”,Semigroup Forum, 78 No. 2 (2009), 360-367.


1. De, D., Hindman, N. and Strauss, D.“Sets central with respect to certain subsemigroups of ”,Topology Proceedings, 33 (2009), 55-79.

2. Heubach, S., Mansour, T. and Munagi, A.O.“Avoiding permutation patterns of type (2,1) in compositions”,Journal of Analytic Combinatorics, Issue 4, Article 3 (2009), 9 pp.

3. Mansour, T, and Munagi, A.O.“Enumeration of partitions by long rises, levels and descents”,J. Integer Seq., 12 (2009), #09.1.8, 17 pp.

List Researchers from other centres who visited and contributed to research:Dr. Margaret Archibald, UCTProfessor Paul Binding, University of Calgary, CanadaProfessor Toufik Mansour, Haifa University

Professor Mike Mays, West Virginia University




registration

Month and Year of


0314117G Zinsou, B MSc Jan 2007 April 2009 Möller, M




registration


0201319N Botha, G MSc (f/t) Zelenyuk, Y9700100P Moletsane, B MSc (f/t) Möller, M0505606N Roth, T MSc Jan 2009 Watson, B A0305498E Toloane, E MSc Jan 2008 van Alten, C J0502393Y Vardy, J MSc Jan 2009 Watson, B A0314117G Zinsou, B MSc (f/t) Jan 2007 April 2009 Möller, M

0420408P Brooks, H PhD (p/t) Mar 2004 Möller, M0609043F Nthebe, T M J PhD (f/t) Mar 2006 Möller, M0616713N Morton, W PhD Jul 2008 van Alten, C J343099 Shuungula, O PhD Jul 2008 Zelenyuk, Y0508019R Toko, W PhD Feb 2009 Zelenyuk, Y

List of external/internal research grants received during the one year periodKnopfmacher, A: NRF Incentive Funding for B2 rating and Unlocking the future grant.Möller, M: NRF grant : Incentive Funding for Rated Researchers GUN 69659Watson, B A: NRF Unlocking the future grant and B3 rating.Zelenyuk, Y: NRF Unlocking the future grant FA2007041200005.

All Centre members also received John Knopfmacher Centre Funding based on their ISI output in the previous few years.







List participating University AcademicsNames School % Time on Entity

Research ActivitiesBau, Professor S MathematicsBrennan, Dr C MathematicsCurrie, Dr S MathematicsMöller, Professor M H MathematicsMunagi, Dr A O MathematicsVan Alten, Dr C J Computer ScienceWarlimont, Honorary Professor R MathematicsWatson, Professor B A MathematicsZelenyuk, Professor Yevhen MathematicsZelenyuk, Dr Yuliya Mathematics







Bau, Prof S

Accepted paper:S. Bau, “Balanced binary mappings on graphs”, Bulletin of the Institute of Combinatorics and its Applications.

Submitted papers:1. S. Bau and M. Gao, “Balanced binary mappings on direct product to two paths”.2. S. Bau and J. Lin, “Balanced binary mappings on direct products of two cycles”.3. S. Bau, “Reductions and minors of graphs”.4. S. Bau, “Reductions of 3-connected graphs with minimum degree at least four.

Ongoing research:Toeplitz graphsMetric dimensionsReductions and minors of graphs

Conference attended:The 53rd Annual Congress of the South African Mathematical Society, University of Pretoria, November 3 – 5, 2010.

Brennan, Dr C

Published paper:1. Brennan, C. and Mavhungu, S., “Peaks and valleys in Motzkin paths”,

Quaestiones Mathematicae, 33 (2010), 171-188.

Accepted paper:C. Brennan, Ascents of size less than d in samples of geometric variables: mean, variance and distribution, to appear in Ars Combinatoria.

Submitted:1. C. Brennan, Toufik Mansour and Eunice Mphako-Banda, Tutte polynomials of fans and wheels, QM, 20102. C. Brennan and S. Mavhungu (University of Venda, South Africa), Visits to level r by Dyck paths,

Journal of Statistical Inference, 2010.3. C. Brennan and A. Knopfmacher, Descent variation in samples of geometric samples, Discrete Mathematics

and Theoretical Computer Science, 2010.4. C. Brennan, A. Knopfmacher and T. Mansour, Separation of the maxima in samples of geometric random

variables, Applicable analysis and Discrete Mathematics.

In preparation:1. C. Brennan, A. Knopfmacher and T. Mansour, The distance between the first and the last maximum in samples

of geometric variables, 2010.2. C. Brennan, A. Knopfmacher and T. Mansour, Height of Dyck paths with horizontal steps of length k, 2010.3. A. Blecher and C. Brennan, Number of blocks with maximal strings of ascents of at most d in compositions of n,

2010.4. A. Blecher and C. Brennan, Number of blocks with smooth maximal strings in compositions of n, 2010.

Conferences attended and paper presented:1. South African mathematical Society, 53rd Annual Congress, Pretoria, 2010.

Visits to level r by Dyck paths.2. 7th International Conference on Lattice Path Combinatorics and Application, Siena, Ital, 2010.

Vistis to level r by dyck paths.

Currie, Dr S

Published papers: S. Currie, A. Love, “Transformations of difference equations I”, Advances in Difference equations, article ID 947058S. Currie, A. Love, “Transformations of difference equations II”, Advances in Difference equations, article ID 623508

Accepted papers:S. Currie, B. A. Watson, “Quadratic Infinite boundary value problems on graphs”, Operators and Matrices.S. Currie, A. Love, “Hierarchies of difference boundary value problems”, Boundary Value Problems.

Submitted papers:S. Currie, B. A. Watson, “Refutal of Remark 2 in V. Yurko, Inverse spectral problems for differential operators on arbitrary compact graphs”, submitted.S. Currie, B. A. Watson, “Quadratic eigenvalue problems on graphs”, submitted.

Ongoing projects:Scattering problems on the infinite line (with B.A. Watson and M. Nowaczyk)The Fucik spectrum for boundary value problems on graphs. (with B.A. Watson)An application of hierarchies of difference boundary value problems (with A. Love)

Conferences attended and presented at:Functions and Operators, June 2010, Krakow, PolandIWOTA, July 2010, Berlin, Germany

Knopfmacher, Prof A

Accepted Papers:1. A.Knopfmacher, T. Mansour and A. Munagi, ``Recurrence Relation with two indices and plane compositions'',

Journal of Difference Equations and Applications, 17 (1) (2011), 115-127.

2. M. Archibald and A. Knopfmacher, `` The largest missing value in a composition of an integer'', Discrete Mathematics, 311 (2011), 723-731.

3. A. Knopfmacher, A. Munagi and S. Wagner, ``Successions in words and compositions'', Annals of Combinatorics.

4. S. Heubach, A. Knopfmacher, M. Mays and A. Munagi,” Inversions in compositions of integers”, Quaestiones Mathematicae.

5. A . Knopfmacher and F. Luca, ``On prime-perfect numbers'', Int. J. Number Theory.

6. A. Knopfmacher and A. Munagi, ``Permutations with interval cycles'', Ars Combinatorica.

Submitted Papers:

1. A. Knopfmacher and T. Mansour,``Record statistics in a random composition''.

2. M. Archibald, A. Knopfmacher and T. Mansour, ``Variation statistics on compositions''.

3. P. Grabner, A. Knopfmacher and S. Wagner, ``A general asymptotic scheme for moments of partition statistics''.

4. A. Blecher, A. Knopfmacher and A. Munagi, ``Durfee Square Areas and Associated Partition Identities''.

5. A. Knopfmacher and N. Robbins, ``Some Properties of Dihedral Compositions''.

6. A. Knopfmacher, T. Mansour and A. Munagi, ``Smooth compositions and smooth words''.

Möller, Prof M

Projects: 1. “Convex epigraphs” (with T.M.J. Nthebe).

2. “Self-adjoint fourth order differential operators with eigenvalue parameter dependent boundary conditions”

(with B. Zinsou) (in progress).

3. “Nonstandard Sturm-Liouville problems” (with I. Fedotov).

Published Papers:Möller, Manfred; Nthebe, Thabang M. J. “On maximal domains for C-convex functions and convex extensions”. J. Convex Anal. 17 (2010), no. 2, 651–658.

Binding, Paul; Möller, Manfred. “Negativity indices for definite and indefinite Sturm-Liouville problems”. Math. Nachr. 283 (2010), 180–192.

Accepted Papers:“Self-adjoint fourth order differential operators with eigenvalue parameter dependent boundary conditions”, Quaestiones Math. (with B. Zinsou).

Submitted Papers:“Spectral asymptotics of self-adjoint fourth order differential operators with eigenvalue parameter dependent boundary conditions” (with B. Zinsou).

Conferences attended: Systems and Operators, Groningen (The Netherlands), 14 – 17 December 2010 Presented a talk on: Spectra of multiplication operators in Sobolev spaces.

Munagi, Dr A

Accepted Papers:1. A. Knopfmacher and A. O. Munagi, Permutations with interval cycles, Ars Combinatoria.2. A. O. Munagi, Alternating subsets and successions, Ars Combinatoria.3. T. Mansour, A.O. Munagi, Alternating subsets modulo m, Rocky Mountain Journal of Math.4. A. Knopfmacher, A. O. Munagi and S. Wagner, Successions in words and compositions, Annals of Combinatorics.5. S. Heubach, A. Knopfmacher, M. E. Mays, A. O. Munagi, Inversions in compositions of integers, Quaestiones Mathematicae.6. A. O. Munagi, Constructive enumeration of bit strings, Proceedings of 14th Fibonacci Conference, 2010.

Submitted Papers:1. (with A. Blecher, A. Knopfmacher), Durfee Square Areas and associated partition identities 2. (with A. Knopfmacher, T. Mansour), Smooth compositions and smooth words.3. (with T. Mansour, M. Shattuck), Recurrence relation and two-dimensional set partitions.4. Homonymic classes of partitions of finite sets.

Projects in progress:1. “Enumeration of parity-alternating permutations”.2. “Generalized enumeration of alternating subsets”.3. “Inversions in set partitions” (with A. Knopfmacher, M. Mays).4. “d-Inversions in words and permutations” (with A. Knopfmacher).5. “Number of smallest parts in compositions” (with A. Knopfmacher). 6. “Enumeration of set partitions by arithmetic progressions”.

Shonhiwa, Dr T (Passed away on 24 October 2009)

Published Paper:1. Shonhiwa, T.

“On relatively prime sets counting functions”,Integers, 10 (2010), 465-476.

Watson, Prof B A

Current Projects:

10. Transformations within classes of boundary value problems – classes of second order boundary value problems consistent with the symplectic nature of the boundary forms. (Participants: Kara & Watson)

11. The diffusion operators on graphs – spectral asymptotics, oscillation theory, completeness of eigenfunction expansions. (Participants: Currie & Watson) : submitted for publication.

12. The Fucik spectrum for boundary value problems on graphs. (Participants: Currie & Watson)13. Spectral asymptotics for general indefinite Sturm-Liouville problems. (Participants: Love & Watson)14. Spectral asymptotics for indefinite p-Laplacian operators. (Participants: Binding, Brown & Watson)

One part of 3 completed and accepted for publication.15. Indefinite Sturm-Liouville operators on graphs, the nature of the spectrum, eigenvalue asymptotics and

completeness. (Participants: Currie & Watson) : Accepted for publication.16. Markov processes, spectral measures and Riesz spaces. (Participants: Vardy & Watson)

Accepted for publication.17. Local martingales and their order structure. (Participants: Labuschagne, Vardy & Watson)18. Indefinite singular p-Laplacian operators. (Binding, Browne & Watson)19. The structure of eigencurves for p-Laplace operators. (Binding, Browne & Watson)20. Scattering and inverse scattering problems with transfer conditions. (Currie, Novaczyk & Watson)21. 2-parameter eigenvalue problems for the Dirac operator. (Currie, Dziwa & Watson)22. Variational approaches to the Dirac eigenvalue problem. (Currie, Roth & Watson)23. Dirac operators on graphs.24. Dirac operators with eigenparametric dependent boundary value problems.

Zelenyuk, Prof Yevhen

During this year I published 2 papers, one of them jointly. In these papers, several important problems in topological algebra and set-theoretic topology have been solved. In particular, it is shown that for every infinite cardinal k, there is a homogeneous regular maximal space of dispersion character k. This is the complete answer to the 1950 question of M. Katetov.

Zelenyuk, Dr Yuliya

During this year I published 3 papers, two of them jointly. In these papers, several questions in combinatorial group theory and topological algebra have been solved. In particular, (a) a general picture of asymptotic behaviour of some important Ramsey functions for compact Abelian groups was obtained, and (b) the number of semiprincipal closed ideals of the semigroup beta S was counted.

Received a Friedel Sellschop Award.


Category 3 : Published Conference Proceedings

1. Korostenski, M., Labuschagne, C.C.A. and Watson, B.A.“Reverse martingales in Riesz spaces”,Proceedings of the 18th International Workshop on Operator Theory and Applications (IWOTA 2007): Operator Theory: Advances and Applications, 195 (2010), 213-230.

2. Mansour, T. and Munagi, A.O.“Enumeration of partitions by rises, levels and descents”. The Permutation Patterns 2007 Conference, 11-15 June 2007. London Mathematical Society Lecture Note Series, 376,

3. Munagi, A.O.Generalized alternating subsets with permutations”,Congressus Numerantium, 201 (2010), 45-54.


5. Archibald, M., Knopfmacher, A. and Mansour, T.“Compositions and samples of geometric random variables with constrained multiplicities”,Discrete Mathematics and Theoretical Computer Science (DMTCS), AN (2010), 449-460.

6. Binding, P.A., Browne, P.J. and Watson, B.A.“Darboux transformations and the factorization of generalized Sturm-Liouville problems”,Proceedings of the Royal Society of Edinburgh, 140A, (2010) 1-29.

7. Binding, P. and Möller, M.“Negativity indices for definite and indefinite Sturm-Liouville problems”,Mathematische Nachrichten, 283 (2) (2010), 180-192.

8. Bona, M. and Knopfmacher, A.“On the probability that certain compositions have the same number of parts”,Annals of Combinatorics, 14 (2010), 291-306.

9. Brennan, C. and Mavhungu, S.“Peaks and valleys in Motzkin paths”,Quaestiones Mathematicae, 33 (2010), 171-188.

10. Currie, S. and Love, A.D.“Transformations of difference equations I”,Advances in Difference Equations, (2010), ID 947058, 22 pages.

11. Currie, S. and Love, A.D.“Transformations of difference equations II”,Advances in Difference Equations, (2010), ID 623508, 23 pages.

12. Knopfmacher, A. and Mansour, T.“Record statistics in integer compositions”,Discrete Mathematics and Theoretical Computer Science (DMTCS) AK, (2009), 527-536.

13. Knopfmacher, A., Mansour, T, Munagi, A. and Prodinger, H.“Staircase words and Chebyshev polynomials”,Applicable Analysis and Discrete Mathematics, 4 (2010), 81-95.

14. Knopfmacher, A., Mansour, T. and Wagner, S.“Records in set partitions”,Electronic Journal of Combinatorics, 17 (1) (2010), R109, 14 pages.

15. Knopfmacher, A. and Robbins, N.“Some properties of cyclic compositions”,The Fibonacci Quarterly, 48 (3) (2010), 249-255.

16. Korostenski, M. and Zelenyuk, Yu.“Ramsey functions for symmetric subsets in compact Abelian groups”,Quaestiones Mathematicae, 33 (2010), 161-169.

17. Labuschange, C.C.A. and Watson, B.A.“Discrete stochastic integration in Riesz spaces”,Positivity, 14 (2010), 859-875.

18. Li, N.Y., Mansour, T. and Munagi, A.O.“Plane trees and reduction rules”,Utilitas Mathematica, 81 (2010), 215-230.

19. Mansour, T. and Munagi, A.O.“Block-connected set partitions”,European Journal of Combinatorics, 31 (2010), 887-902.

20. Möller, M. and Nthebe, T.M.J.“On maximal domains for C-convex functions and convex extensions”,Journal of Convex Analysis, 17 (2) (2010), 651-658.

21. Munagi, A.O.“Alternating subsets and permutations”,Rocky Mountain Journal of Mathematics, 40 (1) (2010), 1965-1977.

22. Ross, D. and Zelenyuk, Ye.

“Topological groups and -points”,

Topology and Its Applications, 157 , Issue16 (2010), 2467-2475.

23. Toko, W. and Zelenyuk, Yu.“Semiprincipal closed ideals of S”,Proceedings of the American Mathematical Society, 138 (6) (2010), 2217-2220.

24. Versfeld, D.J.J., Vinck, A.J.H., Ridley, J.N. and Ferreira, H.C.“Constructing coset codes with optimal same-symbol weight for detecting narrowband interference in M-FSK systems”,IEEE Transactions on Information Theory, 56 (12) December, 2010, 6347-6353.

25. Zelenyuk, Ye.“Regular idempotents in S”,Transactions of the American Mathematical Society, 362 (6) (2010), 3183-3201.


2. Munagi, A.O.“Combinatorics of integer partitions in arithmetic progression”,Integers: Electronic Journal of Combinatorial Number Theory, 10 (2010), 73-82.

3. Shonhiwa, T.“On relatively prime sets counting functions”,Integers, 10 (2010), 465-476.

4. Zelenyuk, Yu.“The number of symmetric colorings of the quaternion group”,Symmetry, 2 (2010), 69-75.

List Researchers from other centres who visited and contributed to research:Dr. Margaret Archibald, UCTProfessor Paul Binding, University of Calgary, CanadaProfessor Toufik Mansour, Haifa University

Professor V Pivovarchik, Odessa, Ukraine Professor Neville Robbins, San Francisco State UniversityProfessor Stephan Wagner, University of Stellenbosch




registration

Month and Year of


0201319N Botha, G MSc April 2010 Zelenyuk, YeTabulate details of current MSc and PhD students



registration


0201319N Botha, G MSc (f/t) April 2010 Zelenyuk, Ye474653 S, Dziwa MSc (p/t) Jan 2010 Watson, B.A and Currie, S0203880R Kobo, S MSc Zelenyuk, Ye0605418D Kyriazis, E MSc Zelenyuk, Yu9700100P Moletsane, B MSc (p/t) Möller, M0505606N Roth, T MSc Jan 2009 Watson, B A0305498E Toloane, E MSc Jan 2008 van Alten, C J0502393Y Vardy, J PhD Jan 2009 Watson, B A

0420408P Brooks, H PhD (p/t) Mar 2004 2011 Möller, M0609043F Nthebe, T M J PhD (p/t) Mar 2006 Möller, M0616713N Morton, W PhD Jul 2008 van Alten, C J343099 Shuungula, O PhD Jul 2008 Zelenyuk, Ye & Zelenyuk, Yu0508019R Toko, W PhD Feb 2009 Zelenyuk, Ye & Zelenyuk, Yu

List of external/internal research grants received during the one year period

Currie, S: NRF incentive funding for Y2 rating, NRF Thuthuka grant, NRF Unlocking the future grant (with Watson and Kara)Knopfmacher, A: NRF Incentive Funding for B2 rating and Unlocking the future grant.Möller, M: NRF grant : Incentive Funding for Rated Researchers GUN 69659Watson, B A: NRF Unlocking the future grant and B3 rating. Incentive Funding for Rated Researchers.Zelenyuk, Yevhen: NRF Unlocking the future grant FA2007041200005.Zelenyuk, Yuliya: Incentive Funding for Rated Researchers IFR2008041600015

All Centre members also received School of Mathematics Funding based on their ISI output in the previous few years.

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