Bulletin of Iranian Mathematical Society
Volume:37 Issue: 2, 2011

A random walk on a lattice is one of the most fundamental models in probability theory. When the random walk is inhomogenous and its inhomogeniety comes from an ergodic stationary process, the walk is called a random walk in a random environment (RWRE). The basic questions such as the law of large numbers (LLN), the central limit theorem (CLT), and the large deviation principle (LDP) are not fully understood for RWRE. Some known results in the case of LLN and LDP are reviewed. These results are closely related to the homogenization phenomenon for Hamilton-Jacobi-Bellman equations when both space and time are discretized.

Properties of the hybrid of block-pulse functions and Lagrange polynomials based on the Legendre-Gauss-type points are investigated and utilized to define the composite interpolation operator as an extension of the well-known Legendre interpolation operator. The uniqueness and interpolating properties are discussed and the corresponding differentiation matrix is also introduced. The applicability and effectiveness of the method are illustrated by two numerical experiments.

We introduce a new concept of general $G$-$eta$-monotone operator generalizing the general $(H,eta)$-monotone operator cite{arvar2, arvar1}, general $H-$ monotone operator cite{xiahuang} in Banach spaces, and also generalizing $G$-$eta$-monotone operator cite{zhang}, $(A, eta)$-monotone operator cite{verma2}, $A$-monotone operator cite{verma0}, $(H, eta)$-monotone operator cite{fanghuang}, $H$-monotone operator cite{fanghuang1, {fanghuangthompson}}, maximal $eta$-monotone operator cite{fanghuang0} and classical maximal monotone operators cite{zeid} in Hilbert spaces. We provide some examples and study some properties of general $G$-$eta$-monotone operators. Moreover, the generalized proximal mapping associated with this type of monotone operator is defined and its Lipschitz continuity is established. Finally, using Lipschitz continuity of generalized proximal mapping under some conditions a new system of variational inclusions is solved.

By means of He's homotopy perturbation method (HPM) an approximate solution of velocity eld is derived for the ow in straight pipes of non-Newtonian uid obeying the Sisko model. The nonlinear equations governing the ow in pipe are for- mulated and analyzed, using homotopy perturbation method due to He. Furthermore, the obtained solutions for velocity eld is graphically sketched and compared with Newtonian uid to show the accuracy of this work. Volume ux, average velocity and pres- sure gradient are also calculated. Results reveal that the proposed method is very e ective and simple for solving nonlinear equations like non-Newtonian uids.

We focus on the use of two stable and accurate explicit finite difference schemes in order to approximate the solution of stochastic partial differential equations of It¨o type, in particular, parabolic equations. The main properties of these deterministic difference methods, i.e., convergence, consistency, and stability, are separately developed for the stochastic cases.

We establish a relationship between general constrained pseudoconvex optimization problems and globally projected dynamical systems. A corresponding novel neural network model, which is globally convergent and stable in the sense of Lyapunov, is proposed. Both theoretical and numerical approaches are considered. Numerical simulations for three constrained nonlinear optimization problems are given to show that the numerical behaviors are in good agreement with the theoretical results.

We consider the number of zeros of the integral $I(h) = oint_{Gamma_h} omega$ of real polynomial form $omega$ of degree not greater than $n$ over a family of vanishing cycles on curves $Gamma_h:$ $y^2+3x^2-x^6=h$, where the integral is considered as a function of the parameter $h$. We prove that the number of zeros of $I(h)$, for $0 < h < 2$, is bounded above by $2[frac{n-1}{2}]+1$.

‎In this paper‎, ‎we present a novel approach for image selective smoothing by the evolution of two paired nonlinear‎ ‎partial differential equations‎. ‎The distribution coefficient in de-noising equation controls the speed of distribution‎, ‎and is‎ ‎determined by the edge-strength function‎. ‎In the previous works‎, ‎the edge-strength function depends on isotropic‎ ‎smoothing of the image‎, ‎which results in failing to preserve corners and junctions‎, ‎and may also result in failing to resolve‎ ‎small features that are closely grouped together‎. ‎The proposed approach obtains the edge-strength function by solving a‎ ‎nonlinear distribution equation governed by the norm of the image gradient‎. ‎This edge strength function is then introduced‎ ‎into a well-studied anisotropic distribution model to yield a regularized distribution coefficient for image smoothing‎. ‎An explicit‎ ‎numerical scheme is employed to efficiently solve these two paired equations‎. ‎Compared with the existing methods‎, ‎the‎ ‎proposed approach has the advantages of more detailed preservation and implementational simplicity‎. ‎Experimental results‎ ‎on the synthesis and real images confirm the validity of the proposed approach‎.

In this survey, we give an overview over some aspects of the set of tilting objects in an $m-$cluster category, with focus on those properties which are valid for all $m geq 1$. We focus on the following three combinatorial aspects: modeling the set of tilting objects using arcs in certain polygons, the generalized assicahedra of Fomin and Reading, and colored quiver mutation.

Let $A$ be a nite dimensional $k-$algebra and $R$ be a locally bounded category such that $R rightarrow R/G = A$ is a Galois covering de ned by the action of a torsion-free group of automorphisms of $R$. Following [30], we provide criteria on the convex subcategories of a strongly simply connected category R in order to be a cycle- nite category and describe the module category of $A$. We provide criteria for $A$ to be of polynomial growth.

These are notes from introductory survey lectures given at the Institute for Studies in Theoretical Physics and Mathematics (IPM), Teheran, in 2008 and 2010. We present the definition and the fundamental properties of Fomin-Zelevinsky’s cluster algebras. Then, we introduce quiver representations and show how they can be used to construct cluster variables, which are the canonical generators of cluster algebras. From quiver representations, we proceed to the cluster category, which yields a complete categorification of the cluster algebra and its combinatorial underpinnings.

This paper is a slightly revised version of an introduction into singularity theory corresponding to a series of lectures given at the ``Advanced School and Conference on homological and geometrical methods in representation theory'' at the International Centre for Theoretical Physics (ICTP), Miramare - Trieste, Italy, 11-29 January 2010. We show how to associate to a triple of positive integers $(p_1,p_2,p_3)$ a two dimensional isolated graded singularity by an elementary procedure that works over any field $k$ (with no assumptions on characteristic, algebraic closedness or cardinality). This assignment starts from the triangle singularity $x_1^{p_1}+x_2^{p_2}+x_3^{p_3}$ and when applied to the Platonic (or Dynkin) triples, it produces the famous list of A-D-E singularities. As another particular case, the procedure yields Arnold's famous strange duality list consisting of the 14 exceptional unimodular singularities (and an infinite sequence of further singularities not treated here in detail). As we are going to show, weighted projective lines and various triangulated categories attached to them play a key role in the study of the triangle and associated singularities.

Let $X$ be a sufficiently nice scheme. We survey some recent progress on dualizing complexes. It turns out that a complex in $kinj X$ is dualizing if and only if tensor product with it induces an equivalence of categories from Murfet's new category $kmpr X$ to the category $kinj X$. In these terms, it becomes interesting to wonder how to glue such equivalences.

Since 2005 a new powerful invariant of an algebra has emerged using the earlier work of Horvath, Hethelyi, Kulshammer and Murray. The authors studied Morita invariance of a sequence of ideals of the center of a nite dimensional algebra over a eld of nite characteristic. It was shown that the sequence of ideals is actually a derived invariant, and most recently a slightly modi ed version of it is an invariant under stable equivalences of Morita type. The invariant was used in various contexts to distinguish derived and stable equivalence classes of pairs of algebras in very subtle situations. Generalisations to non symmetric algebras and to higher Hochschild (co-)homology were given. This article surveys the results and gives some of the constructions in more details.