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 Invitation Raghib Abu-Saris (Walden University, USA) Abstract We review some of the recent developments pertinent to long term behaviour of difference equations. In particular, we focus our attention on the asymptotic behaviour of nonlinear difference equations, existence of invariants and their relationship with periodic behaviour and Gauss compounding of means, and chaotic behaviour. This talk will culminate in presenting some open problems and research project. On ϕ-normal functions Rauno Aulaskari (University of Joensuu, Finland) Shamil Makhmutov (Sultan Qaboos University, Oman) Jouni Rättyä (, ) 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 belongs to the class N ϕ of ϕ-normal functions if its spherical derivative satisfies f # (z) = O(ϕ(|z|)) as |z| → 1− . We study meromorphic ϕ-normal functions, and in particular an analogue of the Lohwater-Pommerenke theorem, and several equivalent characterizations for ϕ-normal functions are established under certain regularity conditions on ϕ. The Wijsman Convergence Jiling Cao (Auckland University of Technology, New Zealand) Abstract In 1960’s, when he studied some optimum properties of sequential probability ratio test, R. A. Wijsman considered a mode of convergence for sequences of closed sets such that if a sequence of proper lower semicontinuous convex functions defined on Rn (as associated with their epigraphs) converged, then the same could be said for the induced sequence of conjugate functions. In recognition of his contribution, this mode of convergence is now called the Wijsman convergence. Of course, the Wijsman convergence is topological, and the associated topology is called the Wijsman topology. In the past 40 years, there has been a considerable effort in extending Wijsman’s results in infinite-dimensional Banach spaces, and exploring some properties of the Wijsman topology. It turns out that the Wijsman convergence and its topology have played important roles in the Banach Space Theory and Set-Valued Analysis. In this talk, I shall give an overview of the recent developments on the Wijsman topology. In particular, I shall discuss the Polishness, Baireness, the Amsterdam and other 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 in this area will be posed. Commutators of singular integrals Yong Ding(Beijing Normal University, China) Abstract For a measurable function b on Rn , the multiplier Mb is defined by Mb f = bf for measurable function f . Let T be the singular integral operator with kernel K, i.e., Z T f (x) = p.v. k(x − y)f (y)dy. mathbbRn Then the commutator [b, T ] formed by Mb and T is defined by Z [b, T ]f (x) := (Mb T − T Mb )(f )(x) = p.v. k(x − y)[b(x) − b(y)]f (y)dy. Rn It is well known that the commutators [b, T ] play an important role in harmonic analysis and PDE. In this talk, we first recall the background 2 of the commutators for singular integral operators briefly and some early important results. Then we present some results obtained recently on commutators of singular integral operators with rough kernels. Foliations and non-metrisable manifolds David Gauld (University of Auckland, New Zealand) Abstract A manifold is a topological space such that each point has an open neighbourhood homeomorphic to euclidean space Rn for some fixed n. Typically manifolds are expected to satisfy some standard topological conditions like Hausdorff, paracompact and connected. Using the homeomorphisms we may transfer from Rn to the manifold properties of the former, including the differential or (piecewise-)linear structure. Safeguards are needed to make sure that using two different homeomorphisms does 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 subset Uα ⊂ M and a homeomorphism ϕα : Uα → Rn so that the charts collectively and consistently split M locally into subsets looking like parallel affine subsets of Rn . As examples consider the flowlines of a non-singular flow 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 like parallel affine subsets are called leaves. If the underlying manifold is metrisable then there are uncountably many leaves, but M. Kneser and H. Kneser exhibited a foliated manifold of dimension 3 having only one leaf. • Connected non-Hausdorff manifolds play an important role in describing foliations of the plane R2 . I shall describe how this works and how one may use the theory of such manifolds to find a rigid foliation of R2 . This is joint work with Paul Gartside and Sina Greenwood. • Another reason that a manifold is non-metrisable is that it is too large in some sense. With Mathieu Baillif and Alexandre Gabard I have been studying foliations on surfaces based on the long ray. We have found that in contrast with the case of R2 there is very little freedom. 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 be continuous. The ω-limit set ω(x) of a point x is the set of limit points of sequences of the form f nk (x) for some sequence of natural numbers (nk ). 3 It is well known that ω-limit sets have the property that for any open set U , the closure of the image of U is not a subset of U . In this talk we look at conditions under which this property characterizes ω-limit sets. It turns out these ideas are closely related to other well known proeprties of dynamical systems. Hardy spaces associated with Schrödinger operators on the Heisenberg group Chin-Cheng Lin (National Central University, Taiwan) Abstract Let L = −∆Hn +V be a Schrödinger operator on the Heisenberg group Hn , where ∆Hn is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class B Q and Q is the homogeneous dimension 2 of Hn . The Riesz transforms associated with the Schrödinger operator L are bounded from L1 (Hn ) to L1,∞ (Hn ). The L1 integrability of the Riesz transforms associated with L characterizes a certain Hardy type space denoted by HL1 (Hn ) which is larger than the usual Hardy space H 1 (Hn ). We 1 define H (Hn ) in terms ˘ L−sL ¯ of the maximal function with respect to the semigroup e : s > 0 , and give the atomic decomposition of HL1 (Hn ). As an application of the atomic decomposition theorem, we prove that HL1 (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 and non-linear materials science are basically the same and have been much studied for centuries. Despite some significant advances in the theory of these non-linear elliptic equations there is still much to be considered. Here we discuss a little of what is known about these equations leading to a fascinating connection with Hilbert’s fifth problem on Lie groups (from his famous list of 23 such problems in 1900). We will present the latest and best result toward a solution of this problem. This talk will cover a broad sweep of modern mathematical ideas and connections between Analysis, PDE, Geometry, Algebra and Group theory painted with a broad brush and with few technical details. Computational Noise in Simulation-Based Optimization Problems Jorge Moré (Argonne National Laboratory, USA) Abstract 4 The simulation of complex scientific phenomena often requires the determina- tion of model parameters that optimize design criteria or that match the model with the experimental data. These complex optimization problems arise in important scientific ap- plications (accelerator design, chemistry, nuclear physics, climate modeling, groundwater remediation, fusion, . . . ), and are often of the form min{F[s(x)] : xL xxU }, where the mapping s : Rn → Rm describes the simulation as a function of the parameters (or controls) x, and the mapping F : Rm → R defines the objective in terms of the simulation. These optimization problems can be formulated in terms of the bound-constrained optimization problem min{f (x) : xL xxU }, where f : Rn → R is defined by f (x) = F [s(x)]. However, this optimization problem is not amenable to standard gradient-related optimization algorithms due to the computa- tional noise and lack of smoothness introduced by the simulation. Computational noise also impacts sensitivity analysis and any application that depends on the simulation. We outline a theoretical framework for studying computational noise in simulationbased optimization problems. Our approach defines computational noise in terms of a stochastic model, but we show that this model applies to deterministic optimization problems. We present an algorithm for determining the computational noise in a simulation and show that in most cases we can reliably determine the computational noise with 8 additional evaluations of the simulation. We also discuss the theoretical and algorithmic implications of computational noise. Computational experiments with simulation-based optimization problems are used to illustrate the implications of the theory. 2 Applications of Analysis The boundary layer inside a rotating circular cylinder Omar 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, which forms inside a rotating pipe, this layer increases in thickness downstream until it becomes comparable to the radius of the pipe, the on coming flow comprises uniform stremwise component plus a swirl. Since the flux of fluid a cross any section is constant, as the boundary-layer thickness increases, the core of the flow is accelerated, and there is a corresponding fall of pressure. Far down stream from the entrance to the pipe the flow assumes the poisuille parabolic distribution, and theoretically this is attained 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 inviscid disturbances when subjected to a rapid, almost rigid, rotation about its axis. Also Pedley (1969) examined the instability of rotating poiseuille of this type. Beran & Culick (1992) studied viscous swirling flows through pipes of constant radius and also circular pipes whit throats. They computed numerical solutions of both the Navier-Stokes equations and the quasi-cylindrical equations, for several values of vortex circulation. In our work, a study is made of the motion of an incompressible viscous fluid through a rotating circular cylinder, whose leading edge is perpendicular to a freestream flow which also involves a swirling component together with a uniform streamwise velocity component. The associated boundarylayer flow is computed using the quasi cylindrical equations. Difference Equations: A Gentle Invitation Raghib Abu-Saris (Walden University, USA) Abstract We review some of the recent developments pertinent to long term behaviour of difference equations. In particular, we focus our attention on the asymptotic behaviour of nonlinear difference equations, existence of invariants and their relationship with periodic behaviour and Gauss compounding of means, and chaotic behaviour. This talk will culminate in presenting some open problems and research project. SYSTEM OF GENERALIZED H-RESOLVENT EQUATIONS WITH CORRESPONDING SYSTEM OF GENERALIZED VARIATIONAL INCLUSIONS RAIS AHMAD (Aligarh Muslim University, India) Abstract The aim of this paper is to introduce a new system of generalized H-resolvent equations in uniformly smooth Banach spaces and we also mention the corresponding system of variational inclusions. An equivalence relation is established between system of generalized H-resolvent equations and system of variational inclusions. We also prove the existence of solutions for the system of generalized H-resolvent equations and the convergence of iterative sequences generated by the algorithm. Our results are new and generalize many known results appeared in the literature. Keywords: Generalized H-resolvent equations, system, variational inclusions, algorithm, convergence. On the Weakly Nonlinear Development of Tollmien-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-Schlichting Wavetrains over a compliant boundary is studied theoretically using highReynolds-number asymptotic methods. The carrier wave is taken to be two-dimensional, and the envelope is assumed to be a slowly varying function of time and of the streamwise and spanwise coordinates. Attention is focused on the scaling appropriate to the so called upper branch and high-frequency lower branch. The dominant nonlinear effects are found to arise in the critical layer and the surrounding diffusion layers: in these regions nonlinear interactions influence the development of the wavetrain by producing a spanwise-dependent mean-flow distortion. The amplitude evolution is governed by integro-differential equation involves the highest derivative with respect to spanwise position, and an additional term due to compliant boundary is obtained. References [1] Arebi, M. A. & Abdulwanis, O. A. 2006. Nonlinear evolution of waves 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 compliant walls. 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 for two- and three-dimensional wall mode instabilities in boundary layers over isotropic and anisotropic compliant walls. Theor. Comp. fluid dyn. 1, 349-378. ON STABILITY OF ABSTRACT MEASURE DELAY INTEGRO-DIFFERENTIAL EQUATIONS S. S. Bellale (Bellale’s Maths Institute, India) 7 Abstract In this paper existence theorem for the second order functional integrodifferential equations in Banach algebras is proved under the mixed generalized Lipschitz and Caratheodory conditions. The existence of extremal solutions is also proved under certain monotonicity conditions. Soliton Perturbation Theory for Phi-Four Model and Nonlinear Klein-Gordon Equations Anjan 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 nonlinear Klein-Gordon equations. There are four types of models of the nonlinear Klein-Gordon equation, that will be talked about. The soliton perturbation theory is utilized to carry out this investigation. Parabolic problems with discontinuous nonlinearities and nonlocal conditions Abdelkader Boucherif (KFUPM, Saudi Arabia) Abstract In this talk I shall present a study of a class of parabolic problems with discontinuous nonlinearities and subjected to nonlocal conditions. More specifically, 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 φ(t)u(x, t)dt, x∈Ω (1) 0 where Ω is an open bounded domain in RN , N ≥ 2 with a smooth boundary, and L is a strongly elliptic operator, F : Ω × (0, T ) × R → R is such that F (., ., u) is measurable and F (x, t, .) is of bounded variations on every compact interval in R, and φ is a nonnegative continuous function on [0, T ]. Objective: provide sufficient conditions on L, F , φ that will guarantee the existence of at least one solution. A Mathematical Approach on Field Equations with A Case 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 perfect fluid source, with variable constant G and Cosmological constant Λ for FRW space-time in higher dimensions is gravitational obtained and case study 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 based Cryptosystems G.Geetha (Sathyabama University, India) Abdul Hameed (Ibra College of Technology, Oman) Abstract Encryption, a primary method to protect valuable electronic information ensures secrecy or confidentiality of information transmitted across an insecure communication channel. In 1976 Whitefield Diffie and Martin Hellman introduced the idea of public key cryptography. The security of the Diffie-Hellman protocol relies on the presumed intractability of the Discrete-log problem. The discrete logarithm problem (DLP) is computationally feasible in finite fields of small size, so in order for the Diffie-Hellman protocol to be secure, the bit-size of the field, n log q must be large. Since the Diffie-Hellman protocol transmits finite fields elements, field sizes large enough to ensure cryptographic 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 for an attacker to compromise system security. Torus-based cryptosystems improve on conventional cryptosystems by representing some elements of large finite fields compactly, and therefore they transmit fewer bits. At Crypto 2004, van Dijk and Woodruff introduced a new way of using the Algebraic Tori Tn in cryptography, and obtained an asymptotically optimal n/φ(n) savings in bandwidth and storage for a number of cryptographic applications. However, the computational requirements of compression and decompression in their scheme were impractical, and it was left open to reduce them to a practical level. Marten van Dijk, Robert Granger et al., suggested a new method that compressed orders of magnitude faster than the original, while also speeding up the decompression and improving on the compression factor (by a constant term) Brouwer, Pellikaan and Verheul showed that compression can be achieved by going up to an extension field. They conjectured that one can attain a compression ratio of n/φ(n), where n is the degree of the extension. For n = 2, the LUC cryptosystem already achieved these savings. For n = 6, Brouwer et al. described a system that was later improved upon by Lenstra and Verheul, resulting in the XTR public key cryptosystem. The problem here is to reduce a composition of mappings which will result in better percentage of compression and decompression process in torus based cryptosystems. It is to be noted that there is an isomorphism between two algebraic structures G and iḠ . We can as well produce an isomorphism between G 9 and its quotient structure. This in turn identifies an isomorphism between the Quotient structure and Ḡ. Thus the complexity is reduced to n/φ(n). This is evident from the commutative diagram of the isomorphisms. This can be extended to any algebraic structure in algebraic varieties. In this paper, Weil restriction on the extension of the finite field is applied, thus reducing the composition which will result in better percentage of compression and decompression process. Also Weil restriction on the abelian variety is used to study the compression and decompression problem in high dimensional tori. Our first application is ElGamal encryption with a small message domain. We obtained an additional 25% compression over CEILIDH even for the encryption of a single message. Given a fixed finite separable extension of a field, one can attach in a functorial way to every algebraic variety associated with the natural transformation on it. This association helps us to attack major problems on cryptography. In this paper, the reduction of time and space complexity of compression and decompression process in encoding is elaborated. References: Bosch, S. Lutkebohmert W.,Raynaud, M, Neron models, Springer Verlag, Berlin 1980. Bosma. W, Hutton.J and Verheul.E.R. Looking beyond XTR,. Asiacrypt02, 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 644-654 Itoh. 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, In IFIP 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 Public Key System. In Public-Key Cryptography and Computational Number Theory, Verlages Walter de Gruyter, 151180, 2001. Marten van Dijk and Woodruff D, Asymptotically Optimal Communication for Torus-Based Cryptography,Crypto04, LNCS 3152, 157178. Marten van Dijk, Robert Granger, Dan Page, Karl Rubin, Alice Silverbert, Martijn Stam and David Woodruff, Practical Cryptography in High 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 dimensional tori, In the Proc. of International conference on Mathematics 10 and computer science ICMCS 2007, Chennai, pp 367-369 VoskresenskiV.E, Algebraic Groups and Their Birational Invariants, Translations of Mathematical Monographs 179, American Mathematical Society, 1998. STRONGLY NONLINEAR AND NONLLOCAL ELLIPTIC THIRD BOUNDARY VALUE PROBLEM WITH NONLINEAR INTEGRAL CONDITION ON THE BOUNDARY HASSAN I. M, (Alfateh University, Libya) Abstract The aim of this paper is to prove existence of solutions of second order partial differential equations with nonlocal boundary conditions i.e. we consider P the problem: (1) |α|≤1 (−1)α ∂ α [fα ◦ (id, u, u′ )] + g ◦ (id, u) = F in Ω R (2) ∂v∗ u := h1 (x, u(x)) + h2 (x, u(φ(x)) + ∂Ω h3 (x, t, u(Ψ(t)))dσt Where Ω ⊂ Rn is (bounded or unbounded) domain, α = (α1 , α2 , · · · , αn ) n X is multindex, |α| = αj αj ≥ 0 and ∂ α = ∂1α1 · ∂2α2 . . . ∂nαn , ∂v∗ u := X j=1 [fα ◦ (id, u, u′ )]vα vα denote the coordinates of the normal unite vec- |α|=1 tor on ∂Ω, φ, Ψ are C 1 -diffeomorphism in a neighbourhood of ∂Ω such That Φ(∂Ω) ⊂ Ω̄, Ψ(∂Ω) ⊂ Ω̄, ∂Ω is bounded and belong to C 1 . The terms fα (x, ξ) and h2 are required to have polynomial growth with respect to the second variable, while in the terms g(x, u(x)), h1 (x, u(x)) no such growth is imposed, but it supposed that g, h1 satisfy the sign conditions g(x, η)η ≥ 0, h1 (x, η)η ≤ 0. Linear elliptic equations with nonlocal boundary conditions have been considered by carleman and then by some authers see ,e.g [3] , [7] and [9] . nonlinear elliptic equatios with nonlocal boundary conditions have been studied in [2],[10] and [11] .similar problem with out inteqral term have been considered in [2] and [4] . The weak solution of (1),(2) will be defined as follows: Assuming that u is a classical soluation of (1),(2) by using Gauss- ostrogradskij theorem and by as integral trans formation Z we obtain: Z X Z [fα (x, u, ∂u)]ϑ− h2 (x, u(x))ϑ(φ−1 (x))dσx h1 (x, u(x))ϑ(x)dσx− e (3) Ω ∂Ω S ﬀ Z Z Z|α|≤1 Z e F ϑdx = h3 (x, τ, u(τ ))dστ ϑ(x)dσx+ g(x, u(x))ϑ(x)dx = − ∂Ω Ω Γ Ω hG, ϑi ∀ϑ ∈ C 1 (Ω̄) with bounded support . Thus weak solution of (1) , (2) will be defined by (3). Solving mathematical programs with fuzzy equilibrium constraints Cheng-Feng Hu (I-Shou University, Taiwan) Fung-Bao Liu (, ) 11 Abstract This work deals with the mathematical programs with fuzzy equilibrium constraints. It shows that solving the fuzzy MPEC is equivalent to solving a fuzzy complementarity constrained optimization problem. By using the tolerance approach, we show that the fuzzy complementarity constrained optimization problem can be converted to a regular nonlinear programming problem. A new smoothing approach based on entropic regularization is developed for solving the resulting optimization problem. Numerical examples are also included to illustrate the solution procedure. On properties of geodesic η-preinvex functions I. 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-preinvex functions and their relationships with η-invex functions and strictly geodesic η-preinvex functions. The geodesic η-pre-pseudo-invex and geodesic η-prequasi-invex functions on the geodesic invex set are introduced and some of their properties are discussed. Keywords: Geodesic invex set; Geodesic η-preinvex function; Geodesic ηpre-pseudo-invex function; Geodesic η-pre-quasi-invex function; Riemannian manifold. On a nonlinear nonlocal in time parabolic equation Mokhtar Kirane (University of La Rochelle, France) Abstract We present, for a nonlocal in time hyperbolic equation, not only the Fujita 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 global existence will be put in evidence. Study of a mathematical model in a Banach space Rajiv 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 a chemostat is studied in a Banach space .In view of the above, we study a general nonlinear age dependent population dynamics model which is a generalization of the chemostat model. We prove the existence, uniqueness the semigroup property and the continuous dependence of the solutions on the initial data for the general model. Then, we show that the chemo stat model is a particular case of the general model and hence conclude the wellposedness of the chemostat model. Key words: Partial integrodifferential equation, Banach space, contraction semigroup, Chemostat, age dependent model 12 Eigenvalue Approach to Three Dimensional Coupled Thermoelasticity in a Rotating Transversely Isotropic Medium A. Lahiri (Jadavpur University, India) Abstract The theory of coupled thermoelasticity in three dimensions is employed to determine the distribution of temperature and stresses in an infinite medium having an instantaneous point heat source at the origin in a rotating medium. Laplace transform along with the double Fourier transforms have been applied in the basic equations of coupled thermoelasticity and finally the resulting equations are written in the form of a vector-matrix differential equation which is then solved by eigenvalue approach. The inversion of the Laplace transform solution is carried out by applying Bellman method and computations have been done by using MATLAB software. Numerical computations of temperature and stresses have been made in space time domain and presented graphically. n-Orthogonality in n-normed spaces H. Mazaheri (Yazd University, Iran) Abstract In 1965, Gähler introduced the idea of 2-normed spaces by a publication 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 many others have studied these concepts and obtained various results. In this article we introduce the idea of constructing n-best approximation and n-orthogonality with elements in an n-norm space. References [1] Y. J. Cho, M. Matic, J. E. Pecaric, On Grams determinant in 2-inner product 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 Matema yczno Fizyczne 1 (2001), 33-40. A singular Gierer-Meinhardt system of elliptic equations in 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 considerable attention in the last decade. They arise in the study of biological pattern formation by auto and cross catalysis in biochemical processes. GiererMeinhardt equations model the interaction between activators and inhibitors. If the activators, u (x, t), have a source distribution ρ (x) and the inhibitors, v (x, t), have a source distribution ρ′ (x), and assuming that both act proportionally to some powers of u and v in the source term, the general model proposed by Gierer-Meinhardt may be written as 8 k < ut = d1 ∆u − αu + cρ uvq + ρ0 ρ in Ω × [0, T ] (2) : r in Ω × [0, T ] , vt = d2 ∆v − βv + c′ ρ′ uvs 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′ and ρ0 are positive constants. The exponents k, q, r, s ≥ 0 satisfy the relation qr > (k − 1) (s + 1). In this paper, we are interested in examining the existence and uniqueness of solutions of elliptic system (2) in the stationary case, where Ω = RN (N ≥ 3), ρ (x) = 0, the exponent ` N ´ k = 0 while the nonnegative parameters α and β are both in L∞ and satisfy loc R α (x) ≥ α0 , β (x) ≥ β0 for |x| ≥ R for some a0 , β0 , R > 0. In other words, we are concerned with the following problem: 8 −∆u + α (x) u = h1 (x) v1q in RN > > > > > > r > > < −∆v + β (x) v = h2 (x) us in RN v > > > u (x) , v (x) −→ 0 as |x| −→ ∞ > > > > > : u, v > 0 in RN , (3) (4) 2 2 where hi ∈ L2 ∩ Lθi , hi 6= 0, i = 1, 2 and θ1 = 1+q , θ2 = 1+s . We assume that the exponents q, r > 0, s ∈ ]0, 1[ such that r − s ≤ 1 and h1 , h2 are ` N´ nonnegative L∞ functions. Our demonstration strategy will be to loc R show – by applying Schauder’s fixed point theorem ` N ´ – that the regularized 2,p system of (4) has a solution uε , vε ∈ Wloc R , 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 elliptic equations in RN , Abstract and Applied Analysis 03 (1998), 411-423. [2] Y. S. Choi and P. J. McKenna, A singular Gierer-Meinhardt system of elliptic equations, Ann. Inst. H. Poincaré, Anal. Non Linéaire 17 (2000), 503-522. [3] Y. S. Choi and P. J. McKenna, A singular Gierer-Meinhardt system of elliptic equations: the classical case, Nonlinear Anal. 55 (2003), 521-541. 14 [4] M. Del Pino, M. Kowalczyk and X. Chen, The Gierer-Meinhardt system: 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 states of the Gierer-Meinhardt system in R2 , Ann. Inst. H. Poincaré, Anal. Non Linéaire 20 (2003), 53-85. [6] M. Ghergu and V. Radulescu, On a class of Gierer-Meinhardt systems arising 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 elliptic boundary value problems, Nonlinear Anal. 59 (2004), 305-318. [9] H. Meinhardt and A. Gierer, Generation and regeneration of sequence of structures during morphogenesis, J. Theoret. Biol. 85 (1980), 429450. [10] W. M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math. 44 (1991), 819-851. [11] W. M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J. 70 (1993), 247-281. [12] W. M. Ni and J. Wei, On positive solutions concentrating on spheres for the Gierer-Meinhardt system, J. Differential Equations 221 (2006), 158-189. [13] J. Wei, On the interior spike layer solutions for some singular perturbation problems, Proc. Roy. Soc. Edinburgh section A 128 (1998), 849-874. [14] J. Wei and M. Winter, Spikes for the Gierer-Meinhardt system in two dimensions: the strong coupling case, J. Differential Equations 178 (2004), 478-518. [15] J. Wei and M. Winter, Existence and stability analysis of asymmetric for the Gierer-Meinhardt system, J. Math. Pures Appl. 83 (2004), 433-476. On the control of solutions of a coupled system of nonlinear viscoelastic equations Salim A. Messaoudi (King Fahd University of Petroleum and Minerals, Saudi Arabia) Muhammad Islam Mustafa (Prince Sultan University, Saudi Arabia) Abstract Due to its importance and application in material science, the study of viscoelasticity has attracted a great deal of attention. Engineers, Mathematicians and Scientists have considered several models and used different kinds of damping mechanisms to control and obtain stability of the systems governing the motion of these materials. The issue of the rate of 15 decay of these motions was always an important issue while studying the stability and decay of solutions of these systems. In most literature, the relaxation functions were considered to be of either polynomial or exponential decay only. Lately, few works allowed a wider a class of decaying relaxation functions. In this paper we consider a system, of two nonlinear coupled viscoelastic equations, which describes the interaction between two different fields arising in viscoelasticity. For a wider class of relaxation functions, we establish a general stability result for this system, from which the usual polynomial or exponential decay are only special cases. Keywords: nonlinear, general decay, relaxation function, stability, viscoelastic material. A Brief History of the Development of Analysis Topics in the 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 mathematicians attention since the seventeenth century. It had been refined and widened by time. Its rigor was well established in the nineteenth century. Also, so many topics were invented or discovered within the ever expanding tent of analysis. In spite of the great strides that were taken, that did not mean that mathematical analysis was closed for more expansion and depth. As the human genius and invention know no boundaries or permanent halts, then analysis continued to expand and got more depth in the twentieth century. This article throws some light on the development 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 and important project not yet dealt with as far as I know. It needs hundreds of pages, and should be written by a group of experts in the different fields of analysis. The writer will stick to a brief historical development without going into details or covering every important aspect in any field. In terms of titles, the writer will cover the following fields: real numbers, foundation of arithmetic, functions, ordinary differential equations, calculus of variations, functional analysis and potential theory. HAMILTON-JACOBI FORMULATIONS FOR SYSTEMS IN TERMS OF RIESZ FRACTIONAL DERIVATIVES Eqab 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 for systems containing fractional derivatives. This paper presents fractional Lagrangian and fractional Hamiltonian for systems containing Riesz fractional derivatives (RFDs). The Hamilton’s Equations of motion are defined in terms of (RFDs). Besides, The Hamilton-Jacobi formulations for these systems are developed. An illustrative example for harmonic oscillator has been discussed. The classical results are recovered for integer order derivative. 16 Eigenvalues Calculation of Sturm-Liouville Problem in Bessel’s Equation Ayad Muftah Shahoot (Mergib University, Libya) Abstract The aim of the study is to calculate eigenvalues of Sturm-Liouville problem emerged from solving third boundary problem, applied on heat flow through hollow cylindrical bodies. The encountered difficulty is manifested in the calculation of the primary eigenvalues of Bessel’s equation. This study is a continuation of a previous study to solve the same problem using alterative roots property of Bessel’s equation. The early efforts based on trial and error process and guessing to determine eigenvalues were time consuming. Changing the cylinder thickness requires further trail and approaching numerical solutions is even more problematic as guesses can only be made based on visual observations of the graphical representation of Bessel’s function roots. This study suggests representing Bessel’s functions by Taylor’s series. A comparison of the results shows high consistency on order of 10−8 . Harvesting in Discrete Population Models: Open Problems and Conjectures Ziyad M. Al-Sharawi (Sultan Qaboos University, Oman) Abstract In this talk, we discuss several harvesting strategies on discrete population models. Since the theory of difference equations is not wellestablished as in differential equations, investigating the dynamics of discrete models in general, and under the effect of harvesting in particular is a 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. AbuSaris and M. Rhouma. REMARKS ON A NONQUADRATIC HAMILTONIAN ELLIPTIC SYSTEMS SHILGBA Leonard Karshima (American University of Nigeria, Nigeria) Abstract In this paper, we study the problem: ∂H (x, u, v) ∂v ∂H (x, u, v) −∆v = δv + λ|u|α−2 u + ∂u −∆u = δu + γ|v|β−2 v + in a smooth bounded domain Ω ∈ RN , N ≥ 3, subject to Dirichlet boundary conditions. The tool we have used to prove existence of solutions is the 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. Figueiredo and C. A. Magalhaes in their paper, On Nonquadratic Hamiltonian Systems, Adv. in Diff. Eq.1, 5 (1996), 881-898, with a restriction to when γ, λ > 0. We have extended our study to the case when γ, λ < 0. Furthermore, we have made some important remarks, which further elucidate the subject. For compactness, we have used an Alama-Tarantello type of growth restriction on the Hamiltonian H. Keywords: Nonquadratic, Hamiltonian, elliptic. On an abstract second order fractional differential equation Nasser-eddine Tatar (King Fahd University of Petroleum and Minerals, Saudi Arabia) Abstract Of concern is an abstract general second order differential equation involving different derivatives of non-integer order in the nonlinearity. This problem covers many of the well-known problems involving integer order derivatives. Some of the problems which arise in applications will be discussed. We establish some existence and uniqueness theorems for different types of nonlinearities. Moreover, we introduce nonlocal conditions of fractional type. The key tools are the cosine families and some fixed point theorems. Cosine families are the equivalent of semigroups for the case of a differential equation of first order. Fractal Image Compression F. Ghezelbash (Islamic Azad University, Iran) A. Tavakoli (Islamic Azad University, Iran) Abstract Michael Barnsley led development of fractal compression in 1987, and holds several patents on the technology. The most widely known practical fractal compression algorithm was invented by Barnsley and Alan Sloan. Barnsleys graduate student Arnaud Jacquin implemented the first automatic algorithm in software in 1992. All methods are based on the fractal transform using iterated function systems. Michael Barnsley and Alan Sloan formed Iter- ated Systems Inc.in 1987 which was granted over 20 additional patents related to fractal compression. A major breakthrough for Iterated Systems Inc. was the automatic fractal transform process which eliminated the need for human intervention during compression as was the case in early experimentation with fractal compression technology. In 1992 Iterated Systems Inc. received a 2.1 million government grant to develop a prototype digital image storage and decompression chip using fractal transform image compression technology. To do fractal compression, the image is divided into sub-blocks.then for each 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 apply Affine transforms to compression and decompression and introduce a fast algorithm for converges. in fact, in this method we represent images as a fixed point of a contractive iterated function system. On the qualitative behaviors of solutions to the Linard-type equations with adeviating argument Cemil Tunç (Yüzüncü 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 equations with a deviating argument. We use the Lyapunovs second method to prove our results, and examples are given to illustrate the importance of the results. Our results include and improve some recent papers published in literature. References [1] Liu, B. and Huang, L. Boundedness of solutions for a class of retarded 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 Linard equations 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 differential 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 nonlinear differential equations with variable delay. Functional Differential Equations, 2009, (accepted for publication). [6] Tun, C., On the existence of periodic solutions to nonlinear third order ordinary differential equations with delay, Journal of Computational Analysis and Applications, (2009), (accepted for publication). [7] Tun, C. and Tun, E. On the asymptotic behavior of solutions of certain second-order differential equations. J. Franklin Inst., Engineering and Applied Mathematics 344 (5), (2007), 391-398. 3 Complex Analysis Order of Magnitude of Coefficients of Clifford Polynomials in Simple Bases M. 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 magnitude of coefficients of Clifford polynomials and the convergence properties of the simple bases to which these Clifford polynomials belong. Two types of restrictions on the coefficients are here considered; the first restriction leads to the effectiveness of the base in closed balls of sufficiently large radii, while the other implies the boundedness of the order of the base On ϕ-normal functions Rauno Aulaskari (University of Joensuu, Finland), Shamil Makhmutov (Sultan Qaboos University, Oman) and Jouni Rättyä (, ) 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 belongs to the class N ϕ of ϕ-normal functions if its spherical derivative satisfies f # (z) = O(ϕ(|z|)) as |z| → 1− . We study meromorphic ϕ-normal functions, and in particular an analogue of the Lohwater-Pommerenke theorem, and several equivalent characterizations for ϕ-normal functions are established under certain regularity conditions on ϕ. Convex null sequence technique for analytic and univalent mappings of the unit disk K. O. BABALOLA (University of Ilorin, Nigeria) Abstract In this paper we employ a technique based on the convex null properties of certain infinite sequences to study various classes of analytic and univalent functions in the open unit disk. The technique simplifies many problems of the theory of geometric functions and our results generalize and extend many earlier ones. Composition operators on normal functions Mahmoud Ali Bakhit (University of Alazhar, Egypt) Abstract In this paper we gave a function-theoretic characterization of composition operators sending (meromorphic) normal classes to their Möbius invariant subclasses. The boundedness and compactness of composition operator Cφ from normal classes to the class Q# K are investigated. The compactness of the composition operator from normal classes to the class Q# α are completely characterized. AAK-theory for generalized Hankel operators Marcus Carlsson (Purdue University, USA) Abstract 20 An algorithm from 2004 by G. Beylkin and L. Monzon, aimed at approximating a function by a sum of few exponentials, is shown to be a consequence of the Adamyan, Arov and Krein theorem on Hankel operators. The algorithm does however work in greater generality than the AAK-theory supports, which has necessitated further research in this area. We present a number of new theorems and open problems, in particular we will give an improvement of an AAK-type theorem by Treil and Volberg, concerning Hankel operators on weighted spaces. The escaping set for quasiregular mappings Alastair Fletcher (University of Warwick, UK) Abstract Complex analysis has in recent years been used to great effect in the field of complex dynamics. The escaping set plays a central role in complex dynamics and, in this talk, we will see some ideas that carry over to the higher dimensional world of quasiregular dynamics. In particular, if a quasiregular mapping is of polynomial type and the degree is larger than 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 functions Konstantin Dyakonov (University of Barselona, Spain) Abstract The classical abc theorem for polynomials (often called Mason’s theorem) deals with nontrivial polynomial solutions to the equation a + b = c. It provides a lower bound for the number of distinct zeros of the polynomial abc in terms of deg a, deg b, and deg c. We prove some abc type theorems for general analytic functions on a (reasonable) simply connected domain. Mason’s theorem is then derived as a corollary. A Non-discrete Convergence Group of Quasiconformal Mappings Jianhua Gong (United Arab Emirates University, UAE) Abstract Convergence groups are introduced by Gehring and Martin in [4], and have found wide application in geometric group theory and low-dimension topology and geometry [1, 2, 3, 10]. Quasiconformal mappings are almost everywhere differentiable mappings on a subdomain of the extended Euclidean space. They map infinitesimal circles to infinitesimal ellipses which has the property that the ratio of the major to minor axes is uniformly bounded from above [5,6,7]. Groups here are topological groups of homeomorphisms, each of which is a K−quasiconformal mapping, under composition with respect to the compact–open topology which is equivalent to the topology induced from locally uniformly convergence. A subgroup G is called discrete if the induced topology coincides with the discrete topology [8, 9,10]. We will a non-discrete convergence group of quasiconformal mappings in 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 fibered 3-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, New Zealand 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 Nyı́regyháziensis, 24 (2008), 367–371. [8] . Hinkkanen and G. Martin, Limit functions for convergence groups and uniformly quasiregular maps, J. London Math. Soc. (3) 73 (2006) 716-726. [9] . Hinkkanen and G. Martin, Abelian non-discrete convergence groups in 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 functions Klaus Gürlebeck (Bauhaus-Universität, Germany) Abstract Monogenic functions are defined in domains of Euclidean spaces of arbitrary dimensions as null solutions of a generalized Cauchy-Riemann operator or of the Dirac operator. A popular example are solutions of the Riesz system in R3 . For such functions different representations in form of Taylor or Fourier expansions are known. We introduce in the talk a recently developed representation based on an orthogonal Appell basis of monogenic polynomials. This representation is used to characterize functions belonging to weighted Dirichlet spaces, Qp -spaces and Bergman spaces by means of the behaviour of their Taylor or Fourier coefficients, respectively. Boundary behavior of special classes of functions Mubariz Tapdygoglu Karaev (Suleyman Demirel University, Turkey) Abstract We give a concrete example of a diagonal operator acting in the Hardy space H 2 (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 discussion of several old problems from the classical book of I.I. Privalov related with the Taylor coefficients and boundary behavior of analytic n o functions. In particular, we give in terms of Taylor coefficients fb(n) and Berezin symbols necessary and sufficient conditions Pensuring existence of radial boundary values of the functions f (z) = n≥0 fb(n) z n from the classes p lA (D) , 0 < p ≤ ∞. Some other questions are also discussed. On rationality of the generating function of the solution of a 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) in the 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 solution of a two-dimensional difference equation under the assumption that the generating function of the initial data is known. We also state the necessary and sufficient condition for rationality of the generating function. Let C = {α}, where α = (α1 , . . . , αn ), be a finite subset of the positive n octant Zn + of the integer lattice Z 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 the difference equation (we use a multidimensional notation) X cα f (x + α) = 0, (1) α∈C which coincides with the some given function ϕ : Xm → C on the set n Xm = Zn + \ (m + Z+ ). Let J = (j1 , ..., jn ), where jk ∈ {0, 1}, k = 1, ..., n, is an ordered set of zeros and ones. With every such set J we associate the face ΓJ of the n-dimensional integer parallelepiped Πm = {x ∈ Zn : 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 ϕ(x) z x+1 x∈X m of the initial data of the difference equation (1) can be represented as the sum X X X ϕ(τ + Jy) Φ(z) = ΦJ (z), where ΦJ (z) = Φτ,J (z), Φτ,J (z) = . z τ +J y+I J τ ∈Γ J 23 y>0 Theorem. The generating function F (z) = P ϕ(x)z −x−1 of the x∈Zn + solution of the difference equation (1) is X X X F (z)P (z) = Φτ,J (z)Pτ (z), where Pτ (z) = cα z α J and P (z) = P τ ∈ΓJ α6m α τ cα z α is the characteristic polynomial of the difference α∈C 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 function Φ(z, w) of the initial data is rational. Real-Part Estimates for Solutions of the Riesz System in R3 K. Gürlebeck (Bauhaus-Universität, Germany) J. Morais (Bauhaus-Universität, Germany) Abstract The study of estimates for analytic functions and their derivatives is rich from a purely mathematical point of view and provides an indispensable and powerful tool for solving a wide range of problems. Concrete applications arise, for instance, in complex dynamics, boundary value problems and partial differential equations theory or other fields of physics and engineering. The estimates in matter, known as ”real part theorems”, attracted the attention of many complex analysts since the classical assertion namely Hadamard’s real part theorem (1892). Excellent contributions 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 a classical object of analysis it became recently more and more important to perform an analogous study in higher dimensions and/or for other partial differential equations. One way to generalize complex function theory to higher dimensional spaces is offered by the Riemann approach which considers (monogenic) functions with values in a Clifford algebra that satisfy generalized Cauchy-Riemann or Dirac systems. This approach is nowadays called Clifford analysis. In view of many applications in physics and engineering, in this lecture we aim to generalize Hadamard’s real part theorem and invariant forms of Borel-Carathéodory’s theorem to the three-dimensional Euclidean space in the framework of quaternionic analysis. Movable discrete solitons V.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 a nonlinear dispersion. The system supports “compactons” – solitary waves with a compact support and kink - antikink pairs that propagate with a unique speed, but may assume an arbitrary width. We demonstrate that lattice solitary waves, though not exactly compact, have tails which decay at a superexponential rate. They are robust and collide nearly elastically and together with wave sources are the building blocks of the dynamics that emerges from typical initial conditions. In finite lattices, after a long time, the dynamics becomes chaotic. Numerical studies of the complex GinzburgLandau lattice show that the non-dispersive coupling causes a damping and deceleration, or growth and acceleration, of compactons.A simple perturbation method is applied to study these effects. Weighted Paley-Wiener Spaces Philippe Poulin (UAE University, UAE) Abstract In an explorative paper published in 2002, Professors Kristian Seip and Yurii Lyubarskii (NTNU, Norway) investigated a family of de Branges spaces subject to a natural axiom: in these spaces, the norm of a function is comparable with its L2 -norm against kkx k−2 dx, where kx is the reproducing kernel at x. Doing so, they invented the notion of weighted Paley-Wiener space. From a deep study of the Hermite-Biehler functions associated with weighted Paley-Wiener spaces, they deduced many structural results, which I revised during my postdoctoral fellowship under their kind supervision. In this talk I will discuss these revised results, with emphasis on an auxiliary class of de Branges spaces introduced in the same work, which appears to be more general. On the Libera Operator Maria Nowak (Affiliation, Country) Miroslav Pavlović (Affiliation, Country) Abstract Let H(D) denote the class of all functions holomorphic in the unit disk D of the complex plane. R 1 The Libera operator, L, defined on the 1 space H(D) by Lg(z) = 1−z g(ζ) dζ is the conjugate of the Cesàro opz erator acting on H(D). We show that the operator L can be continuously extended to some known subspaces of H(D). In particular, we prove, using the duality arguments, that L maps boundedly the Bloch space into BMOA. We also obtain some results on the best approximation by polynomials in Hardy and Bergman spaces. EXPANSIONS FOR GENERALISED HYPERGEOMETRIC FUNCTIONS P . A . Padmanabham (Pondicherry Engineering College, Country) Abstract 25 Motivated by the fact that Generalized Hypergeometric Functions like Appell function, Saran functions Lauricella function etc.. have many applications in physical and quantum chemical problems, we drive three summations formulas for a three variable Hypergeometric function. Special cases of the main results are also discussed. Linear differential equations in the unit disc: growth and zeros of solutions Jouni Rättyä (University of Joensuu, Finland) Abstract We discuss the growth and the oscillation of solutions of complex linear differential 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 necessary conditions for the coefficients such that (i) all solutions are of order of growth at most α ∈ [0, ∞), or (ii) the exponent of convergence of all solutions is at most α ∈ [0, ∞). Qp classes in Rn Lino F. Reséndis O. (Universidad Autónoma 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 ZZ |f ′ (z)|2 g(z, a)p dx dy < ∞ sup a∈D D where g(z, a) is defined by ˛ ˛ ˛ 1 − az ˛ ˛, g(z, a) = ln ˛˛ a−z ˛ that is, the Green’s function of the unit disk with logarithmic singularity at a ∈ D, after Aulaskari, Xiao and Zhao considered 0 < p < ∞. In 1999, K. Güerlebeck, Kähler, Tovar and Shapiro defined Qp -spaces in the 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 Operators Alejandro Rodrı́guez-Martı́nez (Zayed University, Country) Abstract In this talk we present an example of the combination of finite dimensional matrices along with approximation methods to the study of the spectrum of certain Composition Operators. In particular, we prove that increasing composition symbols produce non-negative real spectrum and 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 composition Volterra operators. The present work is part of the author’s PhD thesis that was directed by A. Montes-Rodrı́guez and S.A. Shkarin. Some Properties of Certain Integral Operators Pravati Sahoo (Banaras Hindu University, India) Abstract The object of the talk is to discuss some interesting properties of the integral operator Z (p + 1)α z “ z ”α−1 P α f (z) = f (t) dt, for α > 0, p ∈ N = {1, 2, . . .}, log zΓ(α) 0 t which was introduced and studied by Jung, Kim and Srivastava [J. Math Anal Appl., 176(1993), 138-147], for the class of all analytic functions f (z) P n of the form f (z) = z + ∞ n=p+1 an z , for z ∈ ∆ = {z ∈ C : |z| < 1}. 2000 AMS Subject Classification: 30C45 Keywords: Analytic functions; Differential subordination; Convex functions; Hadamard product. References [1] I.B. Jung, Y.C, Kim and H.M. Srivastava: The Hardy space of analytic functions associated with certain one-parameter families of integral operators, J. Math Anal Appl., 176(1993), 138-147. [2] Y. Komatu: On a one-parameter additive family of operators defined 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, Heidelberg and New York, 1980. [3] J.L. Liu: Notes on Jung-Kim-Srivastava integral operator, J. Math Anal Appl., 294(2004), 96-103. [4] S.S. Miller and P.T. Mocanu: On some classes of first order differential subordination, Michigan Math. J., 32(1985), 185-195. [5] H. M. Srivastava and S. Owa: A certain one parameter additive family 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 Analysis: An Introduction to the General Theory of Infinite Processes and Analytic Functions; with an Account of the Principal Transcdental Functions, 4th editin, Cambridge University Press, Cambridge, 1927. 27 ON FRACTIONAL CALCULUS OPERATOR FOR ANALYTIC FUNCTIONS Khalifa Al-Shaqsi (Ministry of Education, Oman) Abstract For analytic function f (z) = z + a2 z 2 + · · · in the open unit disk U = {z : |z| < 1}, a new fractional operator Ωλγ is defined for any real numbers λ, γ by Ωλγ f (z) = ´ Γ(2 + γ(λ − 1) − λ) λ−γ(λ−1) λ ` γ(λ−1) z Dz z f (z) , Γ(2 + γ(λ − 1)) where Dzλ f (z) is the fractional derivative of order λ. Several basic properties for this operator are obtained. A new class of analytic functions defined by using Ωλγ are introduced and investigated some properties belong to this class. On Complex Dynamical Systems with large girth indicator, related Chaos and applications to Quantum Computing V. A. Ustimenko (University of Maria Curie Sklodowska, Poland) Abstract n Let Vn = C be the family of n-dimensional vector spaces over the field of complex numbers C. We refer to the tuple t = (t1 , t2 , . . . , tk ) of nonzero elements ti ∈ C such that ti 6= −ti+1 as irreducible tuple of length k ≥ 1. We say that the family of nonlinear polynomial bijective maps Ft n = (x1 , x2 , xn ) : C n → C n , t ∈ C , t 6= 0, n ≥ 1 of kind x1 → 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 if the inverse map for Ft n is F−t n . If all coefficients of polynomials fi , i + 1, . . . , n are integers we use term symmetric arithmetical dynamical system depending on time. We refer to the triple (n, x, t) where x ∈ C n , t is irreducible tuple of length s as n-dimensional cycle of length s if for the composition F (t1 , t2 , . . . , ts )n of Ft1 n , 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 Ft1 n , 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 Ft n 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 least 2/3. The map F = F n (t1 , t2 , . . . , ts ) of complex arithmetical dynamical system 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 a subring of C with unity (Z, Q, Gaussian integers, Gaussian rational numbers and etc) then F maps K n into itself for t ∈ K s . So the computation of restriction of F onto K n can be implemented on Quantum Computer or other probabilistic machine. In the talk we will discuss studies of mixing properties of such maps, complexity of computation, use for different tasks of Coding Theory and Cryptography (it is partially reflected in [3]). References [1] V. A. Ustimenko, Linguistic Dynamical Systems, Graphs of Large Girth and Cryptography, Journal of Mathematical Sciences, Springer, vol.140, N3 (2007) pp. 412-434. [2] T. Shaska, V. Ustimenko, On the homogeneous algebraic graphs of large girth and their applications, Linear Algebra and its Applications Article, Volume 430, Issue 7, 1 April 2009, Special Issue in Honor of Thomas J. Laffey. [3] V. A. Ustimenko, On the cryptographical properties of extremal algebraic graphs, in ”Algebraic Aspects of Digital Communications”, IOS Press, NATO Science for Peace and Security Series - D: Information and Communication Security, Volume 24, July 2009. 4 Functional Analysis EXACT CONSTANTS FOR BEST APPROXIMATION ON THE GROUP SU(2) E. AGHDASSI (University of Tabriz, Iran) Abstract In the present paper we study the properties of the least upper bounds of the best approximation by algebric polynomials in metrics L1 and L∞ for classes of convolutions defined on the group SU (2). The exact constants for best approximation by trigonometric polynomials in L∞ (−π, π) is studied by many authors. Finally in this paper we proved that for group SU (2) analog of the Favard-Akhiezer-Krein theorem dose not hold. C ∗ -algebras associated to low dimensional Leibniz algebras Massoud Amini (Tarbiat Modares University, Iran) Abstract 29 We consider the loop space associated to the enveloping Lie-algebra of Leibniz algebras and show that they carry a natural hypergroup structure. We construct the associated universal C ∗ -algebra for low dimensional Leibniz algebras and compare our construction with the construction of C ∗ -algebras of Hecke pairs. We give a classification scheme for dimensions three to five. AN EXTENSION OF THE FUGLEDE -PUTNAM THEOREM Aı̈ssa NASLI BAKIR (Chlef University, Algeria) Salah MECHERI (Tébessa University, Algeria.) Abstract The familiar Fuglede-Putnam theorem is as follows : For two normal operators A and B on a complex separable Hilbert space H, and for some operator X on H, the equation AX = XB implies A∗ X = XB ∗ . Recently in [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 reduces A. We’ll also generalize the Fuglede-Putnam theorem to another class of nonnormal operators. Key words and phrases. (p, k)-quasihyponormal operators, dominant operators, theorem of Fuglede-Putnam. AMS Classification. Primary 47B47, 47A30, 47B20, Secondary 47B10. REFERENCES 1. 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 paranormality, 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 theorem, Georgian Mathematical Journal, 16(2009), N3, 427-433. PROPER EXTENSIONS OF A CERTAIN CLASS OF CARLEMAN OPERATORS Berrabah Bendoukha() Abstract In this paper, we describe all proper (in particular, self-adjoint and dissipative) extensions of a certain type of Carleman operators, defined in Hilbert space L2 (Ω, µ). For such operators, we use the concept of bound- ary triplet to give the general form of Weil functions and proper extensions. Spectral properties of these extensions are also investigated. N A CLASS OF n-NORMED SEQUENCES RELATED TO lp -SPACE 30 Binod Chandra Tripathy (Institute of Advanced Study in Science and Technology, 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 from different aspects. Let n ∈ N and X be a real vector space. A real valued function k ., ., ., . . . kn on X n satisfying the following four properties: (1) k z1 , z2 , ., . . . zn kn = 0 if and only if z1, z2,zn are linearly dependent; (2) k z1 , z2 , ., . . . zn kn is invariant under permutation; (3) k z1 , z2 , ., . . . azn kn = |a| k z1 , z2 , ., . . . , zn−1 , zn kn , for any α ∈ R; (4) k z1 , z2 , ., . . . zn−1 , x+y kn =k z1 , z2 , ., . . . zn−1 , x kn + k z1 , z2 , ., . . . zn−1 , y kn ; is called an n-norm on X and the pair (X, k ., ., ., . . . kn ) is called an n-normed space. Sargent [J London Math. Soc, 35,1960:161-171] introduced the sequence space m(φ) and studied some properties of this space. Later on it was studied from the sequence space point of view and some matrix classes were characterized with one member as m(φ) by many others. In this paper we introduced the following sequence space: (m(φ), k ., ., ., . . . kn ) = (xk ) w :k (xk ) kn,m(φ) = sup s1,σ 1 k z1 , z2 , ., . . . zn−1 , xk kn < . φskσ We investigate some topological and algebraic properties of this space and obtain some inclusion results. Another Characterization of Semiclassical d−Orthogonal Polynomials Ammar BOUKHEMIS (University of Annaba, Algeria) Abstract We give a new characterization of d−orthogonal polynomials sequences whose derivatives sequences is strictly d−quasi-orthogonal. Also, an integral representation of the forms of orthogonality is established. —————————————————————————————– AMS Classification: [2000] , 42C05, 33C47, 33C50. Keys words: Orthogonal Polynomials, Semiclassical, Integral representation. References [1] A. I. APTEKAREV, F. MARCELLÁN and I. ROCHA. Semiclassical multiple 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 and differential equations. International Journal of Mathematics and Mathematical Sciences, Vol. 2006 (2006), Article ID 12640, 25 pages [3] A. BOUKHEMIS. On the classical 2−orthogonal polynomials sequences of Sheffer-Meixner type. Cubo A Math. J. 7 (2) (2005), 39-55. [4] A. BOUKHEMIS. A study of a sequence of classical orthogonal polynomials of dimension 2. J. Appro. Theory 90 (3) (1997), 435-454. [5] K. DOUAK and P. MARONI. Les polynômes orthogonaux ”classiques” de dimension 2. Analysis 12 (1992), 71-107. [6] P. MARONI. Prolégomene à l’étude des polynômes orthogonaux semiclassiques. annali di Mat. Pura ed Appl 149 (4) (1987), 165-184. [7] W. VAN ASSCHE and E. COUSSEMENT. Some classical multiple orthogonal polynomials. J. Comput. Appli. 127(2001), 317-347. Asymptotic Unconditionality in Banach Spaces Simon Cowell (UAE University, UAE) Abstract We show that a separable real Banach space embeds almost isometrically in a space Y with a shrinking 1-unconditional basis if and only if limn→∞ kx∗ + x∗n k = limn→∞ kx∗ − x∗n k whenever x∗ ∈ X ∗ , (x∗n )∞ n=1 is a weak∗ -null sequence and both limits exist. If X is reflexive then Y can be assumed reflexive. These results provide the isometric counterparts of recent work of Johnson and Zheng. FUSION INTEGRAL AND FUSION FRAME M.H.FAROUGHI (Azad University of Shbestar) Abstract In the present paper we shall introduce an operator-valued integral over a Hilbert space. We also, shall prove that it is additive (finite or infinite) 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 of fusion frame. Frames and Riesz bases of irregular translates Sigrid Bettina Heineken (University of Vienna, Austria) Abstract A sequence {fk }k∈Z is a frame for a separable HilbertPspace H if there exist positive constants A and B that satisfy Akf k2 ≤ k∈Z |hf, fk i|2 ≤ Bkf k2 for all f ∈ H. In this work we study frames and Riesz bases obtained by translating a function along an irregular set of points. Let P WE be the space of 32 a function in L2 (R) whose Fourier transform is supported in E. We find conditions for the familiy of irregular translates of a function to be a frame for P WE . For this, we investigate systems of the form {heλ }λ∈Λ where h ∈ L2 (E) and {eλ }λ∈Λ are complex exponentials in L2 (E), and also a certain Gramian function. These results have possible applications in image and digital data transmission, speech coding, general signal processing and geophysics. This is joint work with Peter Balazs from the Acoustic Research Institute, 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 operator T ∈ L(X, Y ) attains its norm if there is an element x0 in the unit sphere of X such that kT (x0 )k = kT k. We denote the set of norm attaining operators from X to Y by N A(X, Y ). The general question (sometimes called the Bishop-Phelps problem), is whether or not N A(X, Y ) is dense in L(X, Y ). Given an arbitrary measure µ, we show that the set of norm attaining operators from L1 (µ) into C(K) is dense in the space of all bounded linear operators L(L1 (µ), C(K)). CONDITIONAL TYPE AND WEIGHTED COMPOSITION OPERATORS BETWEEN Lp SPACES M. R. JABBARZADEH (University of Tabriz, Iran) Abstract In this paper Lambert multipliers acting between Lp spaces are characterized by using some properties of conditional expectation operator. Also, we discuss measure theoretic characterizations for some weak hyponormal classes of weighted composition operators on L2 such as, pquasihyponormal, 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 finite dimensional subspaces N of C(Q) for which there is a projection P from C(Q) onto N with kP k = 1. It is shown that if Q is a compact Hausdorff space, and N is an n-dimensional subspaces of C(Q) then there is a projection P : C(Q) → N with kP k = 1 iff N has a basis {f1 , f2 , . . . , fn } that P satisfies the property that kfi k = 1 for each i = 1, 2, . . . , n, and k n i=1 |fi |k = 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} that satisfies the property that kfi k = 1 for each i = 1, 2, . . . , n, and for each f ∈ N , kf k = max{|µi (f )|i = 1, 2, . . . , n}. This paper was published in Mathematica Japonica Vol. 37. 33 An Iterative Algorithm for a System of Implicit Variational Inclusions involving H(·, ·)-Mixed Accretive Mappings Kaleem Raza Kazmi (A.M.U. Aligarh, India) Abstract In this paper, we introduce a new class of H(·, ·)-mixed accretive mappings in Banach space. We give the notion of resolvent mapping for the H(·, ·)-mixed accretive mapping, an extension of resolvent mapping given in [Juhe Sun, Liwei Zhang, Xiantao Xiao; An algorithm based on resolvent operators for solving variational inequalities in Hilbert spaces, Nonlinear Analysis: Theory, Methods & Applications, 69(10) (2008), 3344-3357], and prove its existence and Lipschitz continuity. Further, we consider the following 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 nonlinear mappings and let Ci : E1 → 2E1 , Di : E2 → 2E2 , Mi : Ei → 2Ei be setvalued 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 implicit Wiener-Hopf equations using the concept of resolvent mappings for the H(·, ·)-mixed accretive mapping. Using this equivalence, we propose a new iterative algorithm for the system of implicit variational inclusions. Furthermore, we prove the existence of solution of the system of implicit variational inclusions and discuss the convergence analysis of the iterative algorithm. We also discuss some further extensions of the work presented in this paper. The theorems presented in this paper can be viewed as significant generalizations of many known and important results under Hilbert spaces as well as Banach spaces setting in recent literature. AMS Subject Classifications: 47H04, 49J40 Keywords: System of implicit variational inclusions, H(·, ·)-Mixed accretive mapping, Mixed Lipschitz mapping, Resolvent mapping, System of implicit Wiener-Hopf equations, Iterative algorithm, Convergence analysis. 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 of extended general variational inequality problems involving nonlinear operators in real Hilbert spaces. Using the projection operator technique, it is given that the system of extended general variational inequality problems are equivalent to the nonlinear projection equations. This alternative equivalent formulation is used to discuss the existence of a solution and suggest some general algorithms. Further we shall discuss the convergence analysis of iterative sequence generated by general algorithms. Since this system includes the system of variational inequalities, variational inequalities and related optimization problems. My results can be viewed as a refinement and improvement of previously know results for recent work in variational inequalities. Note that in the talk, we shall also discuss some new iterative method to solve such kind of problem by using some different approach and interrelate my problem to some physical problem of nonlinear analysis. An iterative process for common fixed points of two finite families of nonself nonexpansive mappings Safeer Hussain Khan (Qatar University, Qatar) Abstract In this talk, we introduce an iterative process for approximating common fixed points of two finite families of nonself nonexpansive mappings in Banach spaces. Our process contains both explicit and implicit Mann iterative processes and some other processes for nonself mappings. We prove some weak and strong convergence theorems for this iterative process. Our results generalize and improve some results in contemporary literature. Condition for maxima and minima between two points Chandan Kumar (Delhi College of Engineering, India) Abstract In the present paper, we are going to find out and study the true relationship between the curve and the point of intersection of any two tangent drawn on the curve having product of their slope negative, also the condition for which maxima and minima exists between two points. since their is a general misconception that when the product of slope of two tangent is negative then their exist atleast one point in between them whose slope is 0(zero). In this paper we have find the condition when it will satisfy this result and when not, and also derived the other possible results, for any smooth curve which is continuous and differentiable in its domain. keywords: Continuity, Differentiability, Tangents, Slopes. AMS subject classifications: 11G20, 14H50, 53A04 35 Norm for sums of two basic elementary operators Farida Lombarkia (University of Batna, Algeria) Abstract In this talk we shall be concerned to estimate the norm of the elementary operator MA1 ,B1 + MA2 ,B2 , where A1 , A2 , B1 , B2 are bounded linear operators on a normed space E, and MA1 ,B1 is the basic elementary operator defined on B(E) by MA1 ,B1 (X) = A1 XB1 , we also give necessary and sufficient conditions on the operators A1 , A2 , B1 , B2 under wich MA1 ,B1 + MA2 ,B2 attains its optimal value kA1 kkB1 k + kA2 kkB2 k. This is a jointe work with Ahmed Bachir . Composition operator on Besov algebra spaces Madani Moussai (University of MSila, Algeria) Abstract We will study the boundedness of the composition operator Tf (g) := s f ◦ g on Besov s spaces Bp,q (R). In case 1 ≤ p, q < ∞, and „ « 1 p 1 [s] ≥ 2, s − [s] ≤ (1/p), min 1 + , + − 2 < s − [s], p q p s we will prove the following: the operator Tf takes Bp,q (R) to itself if and s only if f (0) = 0 and f belongs locally to Bp,q (R). From Certainty to Uncertainty: In the Context of Fixed Point Theory and Applications-II P. P. Murthy (Guru Ghasidas University, India) Abstract Non-Fuzzy Sets ( Crisp )we mean those sets with mathematical logic i.e. the value of the membership function defined is 0 or 1 and Fuzzy Sets ( Uncertain Sets ), we mean the membership values lies between [0, 1]. The pioneer concept of Fuzzy Sets and Fuzzy Logic introduced initially by Prof. Lofti Zadeh ( Fuzzy sets, Information and Control, 8 (1965), 338-353) of University of California, Berkeley, USA in 1965. In this talk, I shall try to give some results of Fuzzy Sets and Fuzzy Logic and newly developing space Cone Metric Space ( Huang Long-Guang and Zhang Xian, Cone Metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl.332(2007), 1468 - 1476 ) in the context of Fixed Point Theory and Application. In this space we shall discuss the impact of the concepts of compatible and non-compatilbe maps for obtaining common fixed points. R-Transforms for polynomial expansion and root extraction Wajdi M. Ratemi (Alfateh University, Libya) 36 Abstract Special mathematical formulation has been introduced for polynomial expansion, Ratemi (1996), Ratemi and Eshabo(1998). It is named as the Guelph expansion, Ratemi and Abdulla (2009). n Y i=1 (ω + λi ) = n X ω n−1 Rn−k = n X ω n−k k=0 k=0 X λ . . .k . . . λ. T= n k ( ) Ratemi and Eshabo (1998), Ratemi (1998), and Aboanber (2003) have used such expansion for studies of the inhour equation and for the studies of point reactor kinetics solutions of positive and negative step insertion of reactivities in nuclear reactors. In this paper, further explorations are made for the Guelph expansion. The coefficients of such polynomial expansion have been found to be related to the roots of the polynomial in concern. Each coefficient is found by summing up the number of possible k-combinations (single, dual, triple, etc..) of the roots of the nth degree polynomial. Such ` ´ number of combinations is determined by the Tripoli index T = nk , where n represents the degree of the polynomial, and k represents the number of combinations of roots. It turns out that Tripoli index generates the entries of Pascal triangle, Hence it is here in this paper, it is reported for the first time the explanation of the entries of the Pascal triangle, that is; each entry (T ) of the Pascal triangle represents the number of k-combinations of the roots of the nth degree polynomial. Also, it is presented in this study what is called the R-transforms which find the coefficients of the polynomial expansion for given polynomial roots or finding the roots of a polynomial given the coefficients of its expansion. References: Aboanber, A.E.(2003), Analytical Solution of the Point Kinetics Equation by Exponential Mode Analysis, Progress in Nuclear Energy, Vol.42, No.2, pp.179-197. Aboanber A.E. ( 2003), An efficient analytical form for the periodreactivity 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 Inhour Equation 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 Reactor Kinetics Equation, Proceedings of the 4th Arab Conference for Peaceful Uses of Atomic Energy, Tunis, Tunisia, 16-2-/11/1998. Ratemi Wajdi M., and Hussein Abdullah (2009), Guelph Expansion: A Special Mathematical Formulation for Polynomial Expansion, accepted for presentation in the International Conference of Mathematical Sciences, 04-10 August, 2009, stanbul, Turkey Key Words: Guelph Expansion, Tripoli Index, Polynomial Expansion, Root extraction , R-transforms, Pascal triangle, Inhour Equation, Point Reactor Kinetics. 37 Common fixed point theorems for left reversible semigroups on located distance spaces Ahmad H. Soliman(Al-Azhar University, Egypt) Abstract In the present paper we introduce a new type of generalized metric spaces so called a located distance space where self-distances are zero. For example, each metric spaces are located distance spaces. The purpose of this paper is to establishe fixed point theorems for left reversible semigroup of selfmaps. Our results extend relevant fixed point theorems of Y. Y. Huang and C. C. Hong [ Common fixed point theorems for semigroups on metric spaces, Internat. J. Math. and Math. Sci.(1999) 377-386]. 5 Numerical Analysis ′ The ( GG )−expansion method for solving some evolution equations Q. Ebadi (University of tabriz, Iran) ′ Keywords: ( GG )−expansion method,travelling wave solution, FitzHugh-Nagumo equation Abstract ′ ( GG )−expansion method is used to find traveling wave solution of The FitzHugh-Nagumo and Newell-whitehead equations. The traveling wave solutions are expressed by the hyperbolic, trigonometric and rational functions. It is shown that the proposed method is direct and effective. The solutions are compared with solutions of the tanh-Coth method. On Sixtic Lacunary Spline Solutions of Fourth Order Initial Value Problem Karwan Hama Faraj Jwamer (University of Sulaimani, Iraq) Abstract Many initial value problems that arise in the real life situations defy analytical solution; hence numerical techniques are desirable to find the solution of such equations. New numerical methods which are comparatively better than the existing ones in terms of deficiency, accuracy, convergence and computational cost are always needed. In this paper, an approximation solution with spline (0, 1, 4) functions of degree six and deficiency four is derived for solving fourth order initial value problems, with prescribed nonlinear endpoint conditions. Under suitable assumptions, the existences and uniqueness of the spline (0, 1, 4) function are proved; also the upper bounds of errors are obtained. Other purpose of this construction is to solve the fourth order differential equations. The convergence analysis and the stability of the approximation solution are investigated and compared with the exact solution to demonstrate the prescribed lacunary spline (0, 1, 4) function interpolation. 38 Exact solutions for the coupled Higgs equation and the Maccari system by using (G′ /G)-expansion method H. Kheiri, (University of Tabriz, Iran) A. Jabbari “ G′ G ” Abstract -expansion method is used to seek more genIn this paper, the eral exact solutions of the coupled Higgs equation and the Maccari system. As a result, hyperbolic function solutions, trigonometric function solutions and rational function solutions with free parameters are obtained. When the parameters are taken as special values the solitary wave solutions are also derived from the travelling wave solutions. It is shown that the proposed method is more powerful and more general. PACS: 02.30.Jr; 05.45.Yv Keywords: (G′ /G)-expansion method; coupled Higgs equation; Maccari system; travelling wave solutions. Numerical solution of algebraic equation fuzzy equations with 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 with fuzzy coefficients and crisp variable, and we present an iterative method to find the real roots of such equations, numerically. We present an algorithm to generate a sequence that can be converged to the root of an algebraic fuzzy equation. Keywords: Nearest approximation; Fuzzy numbers; Fuzzy polynomial; Algebraic fuzzy equation The Identification of Pollution Sources Using a PLS Linear 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 regression model obtained from the available data. The objective is to describe the relationships between the input variables X and observed output variables Y in the lack of a theoretical model. The problem is that the number of variables is very important in relation to the number of observations. The PLS Regression (Partial Least Squares Regression) is a data analysis method used specifically to investigate this type of problems. The PLS Regression is an extension of the multiple linear regression model. In its simplest form, a linear model specifies the linear relationship between one or more dependent variables, the responses Y and a set of predictor variables X. In many data analysis problems, the estimation of linear 39 relationship between two variables is adequate to describe and estimate the observed data and consequently make good forecasts for further analysis. The multiple regression model has been extended in many ways to adapt its form to complex data analysis problems. Hence, it is used as a basis for many multivariable methods such as: the Principle Component Regression (PCR). The PLS Regression is a recent technique which combines the characteristics of the principle component analysis (PCA) and multiple regression. It is particularly useful when a set of dependent variables 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 collinear but have relationships with the expected responses, multiple linear regression is the best way to analyse the data. However, if one of these conditions is not satisfied, then multiple linear regression becomes ineffective and inappropriate. The PLS is also used to build predictive models when the factors are numerous and very collinear. Note that this method focuses on the prediction of a response and not necessarily on the identification of a relationship between variables. This means that the PLS is not appropriate for distinguishing variables with a small influence on the response, but when the goal is exclusively the prediction there is no need to limit these variables. Among the most obvious advantages we retain that the PLS method makes it possible to combine the prediction with any study of a structure enclosed in latent variables X and Y . Thus, the method requires less than PCR components to give a good prediction. In this work, this method enables modelling different sets of classes with proposed scenarios characteristics under different situations of pollutants concentrations in a river. Based on the concentration of pollution components as input variables and reprocessing plants which ordure dump in a river as response variables, we can reconstruct a relationship between these two variable blocs. The input variables are collected from a number r of sensors which are mounted longitudinally and with an appropriate distance between each one in the river edge. These variables represent the concentration of the lead, nitrates, oxide and dioxide carbon, and oxide sulfate in each zone and in different coordinates of the river plan. Each response variable noted Pi (i = 1tok) indicates the reprocessing plant i. A number of N scenarios are disposed which represent a sufficient possible case. This number can be at least equal to 2k − 1. To get more significant results, it is required to repeat each 2k − 1 for a sufficient number of experiments in order to cover all the pollutant component concentration. It is very important to note that the PLS discriminate analysis can be used with loses data. Using all principal plans, the separation between the different plants is almost perfect. Indeed, the two first PLS components improve enough the disjointing between k observed clusters. These clusters are regrouped in the Hotelling elliptic of the principle component plan at a significant level equals to 95 To validate the PLS regression discriminate analysis and test its ability to indicate which plant or plants ordure dump in the river, several numerical simulations are performed. The obtained results confirm the effectiveness 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 an important number of variables and to obtain operational results; 2. The PLS regression technique enables identification of existing structures from the data. Furthermore, we aim to confirm that the PLS regression discriminate analysis represents an important and useful tool in identifying the pollution sources. It offers an improved new technique for accurately disjointing sources in a multi-pollutants diversion case. The Hotelling elliptic is used to quantify the devices measurements error, noises and parametric uncertainties which have gained a robust and accurate identification of pollution sources. The interest of pollution sources identification aims to help the authorities penalise the plant which ordures dump in a river and stop it. Consequently, it has repercussions on the environment protection, specifically on the underground water nap. Numerical solution of Hallen’s integral equation by using cardinal Legendre functions Mehrdad Lakestani (University of Tabriz, Iran) Abstract An approximate method is proposed for solving Hallen’s integral equation based on cardinal Legendre functions. Properties of these functions are first presented, these properties are then utilized to reduce the computation of Hallen’s integral equation to some algebraic equations. The method is computationally attractive, and applications are demonstrated through an illustrative example. Keywords: Cardinal Legenre functions, Hallen integral equation, collocation method. Global Convergence of the quasi Newton BFGS algorithm with new nonmontone line search technique Ivan Subhi Latif (University of Salahaddin, Iraq) Abstract The BFGS method is the most effective of the quasi-Newton algorithm for solving unconstrained optimization problem. In this work we develop a new nonmonotone line search of quasi-Newton algorithm for minimizing function having Lipschitz continuous partial derivatives,The nonmonotone line search can guarantee the global convergence of the original quasiNewton BFGS algorithm. Numerical experiments on sixteenth well-Know test functions with various dimensions generally encouraging results show that the new algorithm line search is available and efficient in practical computation by comparing with other same algorithm in many situations. Keywords: Unconstrained Optimization, BFGS update, Descent Condition ,Nonmonotone Line Searches. 41 Solving a system of linear two-dimensional Fredholm integral Equations of the Second Kind by using Weighted Residual Methods Rostam K. Saeed (Salahaddin University, Iraq) Mehdi Hassan Mahmud (, ) Abstract This paper focuses on obtaining an approximation solution for solving a system of two dimensional linear Fredholm integral equations of the 2nd kind by using four different types of weighted residual methods. Two examples are solved to show the validity of the prescribed methods by depending on the least square errors (L.S.E) and running time (R.T). Keywords: Weighted residual methods, two-dimensional Fredholm integral equation Dual variational methods for the Liouville Problem Hocine SISSAOUI (University of Annaba, Algeria) Abstract The Liouville problem falls into the large class of quasilinear elliptic problems and therefore has been studied for some time. For earlier numerical 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 variational principles derived from Moreau’s Two Cones Theorem [4] together with the finite element method. The originality of the present work is the use of the dual variational principles derived from Moreau’s Two Cones Theorem [4] which results in an inequality constraint for the field variable of the dual principle. These methods lead to mathematical programming problems 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 of Mildly Nonlinear Elliptic BVP’s, Num. Math. 26 (1976), pp 106-116. [3] Christie, I. & al., Product Approximation for Nonlinear Problems in the FEM, Dept. of Maths. Report N. A. /42, July 1980, University of Dundee, U.K. [4] W.D. Collins An Extension of the Method of the Hypercircle to Linear Operator Problems with Unilateral Constraints, Proceedings of the Royal Society of Edin. 85 A (1980), pp 173-193 42 On the stability analysis of weighted average finite difference methods for fractional wave equations N. H. Sweilam (Cairo University, Egypt) Abstract In this article, numerical study for the fractional wave equations is introduced by using a class of finite difference methods. These methods are an extension of the weighted average methods for ordinary (non-fractional) wave equations. The stability analysis of the proposed methods is given by a recently proposed procedure akin to the standard von Neumann stability analysis. A simple and accurate stability criterion valid for different discretization schemes of the fractional derivative, arbitrary weight factor, and arbitrary order of the fractional derivative, is found and checked numerically. For comparison propose, a test example is given and compared our results against the exact solutions. Restarted Adomians decomposition method for quadratic Riccati differential equation A. R. Vahidi (Islamic Azad University, Iran) Abstract In this paper, restarted Adomians decomposition method is proposed to solve the well-known quadratic Riccati differential equation. Comparison are made between Adomians decomposition method and restarted Adomians decomposition method using to exact solution. The results reveal that the restarted Adomians decomposition method is of higher accuracy than Adomians decomposition method. Keywords: Adomians decomposition method; Restarted Adomian method ; Riccati differen- tial equation. One-Pass Nested QR Updating Algorithm for The Solution 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 their shear size, ill-conditioning, loss of rank and so on. By concentrating on particular parts of the problem matrix, the problem may be made amenable to solution. Here, we consider solving linear systems in the sense 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 a sub-matrix of A. The original matrix is first reduced by removing a block of rows. It is then reduced by removing a block of columns. The righthand side is reduced accordingly. The QR factorization is calculated for the small subproblem. The block of columns is then appended and the QR factor is updated and the right hand side updated accordingly. This intermediary system is solved and the solution is updated after appending the block of rows. 43 6 Topology On the categories of lattice-valued pretopological convergence groups and pseudotopological convergence groups T. 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 fixed basis 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-valued neighborhood groups as ob- jects and continuous group homomorphisms as morphisms, isomorphic to the category SL-PrTopConvGrp whose objects are stratified L-valued pretopologi- cal convergence groups and morphisms are continuous group homomorphisms, also called category of stratified L-valued principal convergence groups, SL-PConvGrp. The motives behind this article are, in the first place, to introduce various weaker forms of these structures and compare them. Secondly, we introduce a notion which we believe to be new, the notion of stratified L-valued pseudotopological convergence group, also called stratified LChoquet conver- gence groups - a generalization of classical Choquet convergence structure [G. Jäger, Subcategories of lattice-valued convergence spaces, Fuzzy Sets and Syst. 156(2005), 124], and we seek relationship between stratified L-valued pretopo- logical convergence groups and stratified L-valued pseudotopological convergence groups, both in frame valued case as well as Boolean valued case. Finally, we shade light on the concept of lattice-valued categories of stratified lattice-valued convergence groups from the perspective of the preceding objects and morphisms in an attempt to measure the topologiness of a lattice-valued convergence group or to measure the degree of continuity of certain algebraic operations for a lattice- valued convergence space or topological space. Keywords: Ltopology, L-neighborhood topological system, stratified L-neighborhood system, stratified L-pretopological space, stratified L-pseudotopological space, stratified L-convergence structure of continuous convergence, function space, group, category, L-category. AMS Subject Classification (2000): 54A40, 54A20, 54H11, 18B30. Almost Hermitian manifold with flat Bochner tensor Habeeb M. Abood (University of Basrah, Iraq) Abstract Many researchers investigated the flat Bochner tensor on some kinds of almost Hermitian manifold. In the present paper the author studies this tensor on general class almost Hermitian manifold by using a new methodology which is called an G-structure space. Thus this study generalize 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 flat scalar curvature tensor manifold. As well, it is proved that, if M is an almost Hermitian manifold with flat Bochner tensor, then M is a manifold of class R3 if and only if M is a linear complex manifold. Later on, equivalently of classes R2 and R3 is investigated. Finally we prove that, if M is flat manifold with flat Bochner tensor, then M is an Einstein manifold with a cosmological constant. Mathematics Subject Classification: 53C55, 53B35 Topological representation of T-fuzzy finite distributive lattices by means of T-fuzzy Priestley spaces Abdelaziz AMROUNE (Msila University, Algeria) Abstract In this paper, we extend some results obtained by Priestley to the representation of distributive lattices more precisely we give a representation theory of fuzzy distributive lattices in finite case. The notion of fuzzy relations was introduced by Zadeh. In this seminal paper he introduced the concept of a fuzzy relation, defined the notion of similarity as a generalization of the notion of equivalence and defined the concept of fuzzy ordering. The most obvious idea to define fuzzy ordering is, natural to demand three straightforward generalizations of the classical axioms reflexivity, antisymmetry, and transitivity. In Zadehs we extend some results obtained by Priestley to the representation of distributive lattices paper only the minimum t-norm was considered. In this paper the more general definition admitting an arbitrary t-norm. By using a previous result of Priestly this extension is obtained in a very simple and natural way. Coincidence and common fixed points of nonlinear contractions 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 common fixed points of two pairs of self mappings without continuity under relatively weaker commutativity requirement in Menger PM spaces. Our results generalize many known results in Menger as well as metric spaces. Some related results are also derived besides furnishing illustrative examples. COMMON FIXED POINT FOR GENERALIZED SET VALUED CONTRACTIONS SATISFYING AN IMPLICIT RELATION IN PARTIALLY ORDERED METRIC 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 set valued mappings on X. We obtained sufficient conditions for existence of common fixed point of F , G satisfying an implicit relation in X. Keywords and Phrases: Fixed point; partially ordered metric space; set valued mapping; implicit relation. 2000 Mathematics Subject Classification: 47H10; 47H04; 47H07. Presic Type Extension and Generalisation of Banach Contraction 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 : X k → X, f : X → X be mappings. In this paper we have investigated under what conditions the mappings f and T will have a common fixed point. Our results 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 Banach Contraction map- ping Principle, Acta. Math. Univ. Com. LXXVI(2) (2007), 143-147. [4] B. C. Dhage ,Generalisation of Banach Contraction Principle, J.Indian Math. 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 sur la convergence de certain es suites, Pub.de. lInst. Math. Belgrade 5(19) (1965), 75-78. WEAK AND STRONG FORMS OF SIMPLY-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 called Simply-continuous, Strong simply-continuous and Weak simply-continuous functions as generalization of continuous function. We obtain their characterizations, their basic properties and their relationships with other forms of generalization continuous functions between topological spaces. Some fixed point theorems on ordered uniform spaces Ishak ALTUN and Mohammad IMDAD* (Kirikkale University, Turkey; *Aligarh Muslim University, India) 46 Abstract In this paper, we introduce an order relation on uniform spaces and utilize this relation to prove some fixed point theorems for single and multi valued mappings in ordered uniform spaces. Some related results also discussed. Grand Antiprism and Quaternions Mehmet Koca (Sultan Qaboos University, Oman ) Abstract Abstract Vertices of the 4-dimensional semi-regular polytope, the grand antiprism and its symmetry group of order 400 are represented in terms of quaternions with unit norm. It follows from the icosian representation of the root system which decomposes into two copies of the root system of. The symmetry of the grand antiprism is a maximal subgroup of the Coxeter group It is the group which is constructed in terms of 20 quaternionic roots of the Coxeter diagram. The root system of represented by the binary icosahedral group I of order 120, constitutes the regular 4D polytope 600-cell. When its 20 quaternionic vertices corresponding to the roots of the diagram are removed from the vertices of the 600-cell the remaining 100 quaternions constitute the vertices of the grand antiprism. We give a detailed analysis of the construction of the cells of the grand antiprism in terms of quaternions. The dual polytope of the grand antiprism has been also constructed. A new Proof of Schläfli’s Formula Timothy Marshall (American University of Sharjah, UAE) Abstract In contrast to the situation in Euclidean space, the shape and size of a polyhedron in the unit n-sphere is completely determined by its dihedral an- gles (the angles between faces of codimension one). In particular the volume of such a polyhedron is a function of these angles. The most familiar case is n = 2, where we have the well known formula for the area of a spherical k-gon X Area = angles − (k − 2)π (5) In higher dimensions it is not possible to express volume as such an elementary function of angles, but we have instead Schläflis equation, d(Volume) = 1 X (Vol. codimension 2 face) d(angle), n−1 (6) the sum being taken over all faces of codimension 2 and their corresponding dihedral angles. We present a formulation of the Schläli formula that avoids geometric terms, but instead takes the form of an identity between two integrals over simplices of powers of quadratic forms. We outline how this identity can be proved using Stokess theorem. 47 Hyperspaces and Manifolds Abdul Mohamad (Sultan Qaboos University, Oman) Abstract Given a topological space X, let 2X be the family of all nonempty closed subsets of X. In this talk, we shall consider three well-known topologies on 2X , namely, the finite topology (also called the Vietoris topology), the Fell topology and the locally finite topology. The main purpose of this talk is to consider when these three hyperspace topologies are metrizable. Also we shall explore the metrizability of a manifold in terms of properties of its hyperspaces with various topologies. This is a joint research work with J. Cao. Petrov Classification of Lorentzian Manifolds with Static Cylindrically Symmetric Metrics Name(Sultan Qaboos University, Oman) Abstract The Lorentzian manifolds with cylindrically symmetric static metrics are classified for their Petrov types. A theorem is proved that such manifolds cannot be of petrov type II or III. Differential constraints are obtained which are satisfied by all metrics of Petrov type D. A list of all metrics of type N is also given. Common idempotents in compact left topological left semirings Denis I. Saveliev (Moscow State University of Russian Academy of Sciences, Russia) Abstract A classical result of Ellis, which became crucial for applications of Ramsey theory in number theory, algebra, topological dynamics, and ergodic theory, is that any compact left topological semigroup has an idempotent. It follows that any compact left topological left semiring has an additive 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 to algebraic properties of the Stone-C̆ech compacti cation of natural numbers. Finally, we notice that similar arguments establish the existence of common idempotents in far more general structures than left semirings. On the crossings in minimal charts Teruo Nagase and Akiko Shima (Tokai University, Japan) Abstract 48 S. Kamada introduced charts for studying embedded closed surfaces in 4-space, called surface links. Charts are oriented labeled graphs in a disk with three kinds of vertices called black vertices (of degree 1), crossings (of degree 4), and white vertices (of degree 6). Kamada also introduced C-moves which are local modifications of charts in a disk. A C-move between two charts induces an ambient isotopy between the surface links corresponding two charts. A surface in 4-space is called a ribbon surface if it is the boundary of an immersed handle body with singularities which are mutually disjoint disks such that the preimage of each disk is a union of a proper disk of the domain and a disk in the interior of the domain, a handle body. In the words of charts, a ribbon surface is the closure of a surface braid which corresponds to a ribbon chart where a ribbon chart can be modified to a chart without white vertices by C-moves. In our talk, we introduce that any chart with at most one crossing is a ribbon chart. Moreover we introduce theorems for charts with two or three crossings. Graphs arising from Rings-I Cherian Thomas (Higher College of Technology, Oman) Abstract In this paper we define the graph Ω(R) arising from a commutative ring R. The definition is then extended to the ring Zn . Some of the characteristic properties of the graph Ω(Zp ) are obtained where p is prime. We prove that Ω(Zn ) is complete if and only if n is prime. The automorphism group of the graph Ω(Zpn ) is obtained. We also prove that Ω(Zn ) is a split 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 also find the automorphism group of the graph Zn defined by I Beck when n is an odd prime and also when n = pq, where p and q are distinct primes Vague Normed Linear Space D.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 characterized 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 from the evidence for u, and fA (u) is a lower bound on the negation of u derived from the evidence against u, and tA (u) + fA (u)1. Thus the grade of membership 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 = {hu, [tA (u), fA (u)]i : u ∈ U }, 49 where the interval [tA (u), 1fA (u)] is called the “vague value” of u in A and is denoted by VA (u). In this paper ,we applied vague set theory applied in normed linear spaces and introduce a notion so called , vague normed linear spaces ( in short VNLS ) and have studied their related properties. On P L-absolute total curvature of surface-knots Tsukasa Yashiro (Sultan Qaboos University, Oman) Abstract The P L-absolute total curvature of an oriented m-manifold embedded in n-space is defined as half of the average number of critical points with respect to unit vectors in n-space. In this talk we give a lower bound of the P L-absolute total curvature of a special family of oriented surfaces of genus g embedded in 4-dimensional space. 50

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# Abstracts of International Conference on Analysis and Applications Contents