Coloured Permutations Containing and Avoiding Certain Patterns

Let be the set of all coloured permutations on the symbols 1, 2, . . . , n with colours 1, 2, . . . , r, which is the analogous of the symmetric group when r = 1, and the hyperoctahedral group when r = 2. Let be a subset of d colours; we define to be the set of all coloured permutations . We prove that the number of -avoiding coloured permutations in . We then prove that for any , the number of coloured permutations in which avoid all patterns in except for and contain exactly once equals . Finally, for any , this number equals . These results generalize recent results due to Mansour, Mansour and West, and Simion.AMS Subject Classification: 05A05, 05A15.     

Asymptotic behavior of solutions of A 2nth order nonlinear differential equation

In this paper we prove two results. The first is an extension of the result of G. D. Jones [4[:Every nontrivial solution for

An Application of U(g)-bimodules to Representation Theory of Affine Lie Algebras

Let be the affine Lie algebra associated to the simple finite-dimensional Lie algebra . We consider the tensor product of the loop -module associated to the irreducible finite-dimensional -module V() and the irreducible highest weight -module L _k,. Then L _k, can be viewed as an irreducible module for the vertex operator algebra M _k,0. Let A(L _k,) be the corresponding -bimodule. We prove that if the -module is zero, then the -module is irreducible. As an example, we apply this result on integrable representations for affine Lie algebras.     

Book Notices

Given the minimization problem of a real-valued function let A be any algorithm of type with that converges to a local minimum . In this note, new assumptions on f(x) under which A converges linearly to x* are established. These include the ones introduced in the literature which involve the uniform convexity of f(x).     

On a General Projection Algorithm for Variational Inequalities

Let H be a real Hilbert space with norm and inner product denoted by and . Let K be a nonempty closed convex set of H, and let f be a linear continuous functional on H. Let A, T, g be nonlinear operators from H into itself, and let be a point-to-set mapping. We deal with the problem of finding uK such that g(u)K(u) and the following relation is satisfied: , where >0 is a constant, which is called a general strong quasi-variational inequality. We give a general and unified iterative algorithm for finding the approximate solution to this problem by exploiting the projection method, and prove the existence of the solution to this problem and the convergence of the iterative sequence generated by this algorithm.     

Global convergence result for conjugate gradient methods

Conjugate gradient optimization algorithms depend on the search directions,

Decomposition Method for a Class of Monotone Variational Inequality Problems

In the solution of the monotone variational inequality problem VI(, F), with

An Intersection Theorem on an Unbounded Set and Its Application to the Fair Allocation Problem

We prove the following theorem. Let m and n be any positive integers with mn, and let be a subset of the n-dimensional Euclidean space ⁿ . For each i=1, . . . , m, there is a class of subsets M _i ^j of ^Tn . Assume that for each i=1, . . . , m, that M _i ^j is nonempty and closed for all i, j, and that there exists a real number B(i, j) such that and its jth component _{xjB(i, j)} imply . Then, there exists a partition of {1, . . . , n} such that for all i and We prove this theorem based upon a generalization of a well-known theorem of Birkhoff and von Neumann. Moreover, we apply this theorem to the fair allocation problem of indivisible objects with money and obtain an existence theorem.     

On a Singular Two-Point Boundary Value Problem for the Nonlinear mth-Order Differential Equation with Deviating Arguments

Sufficient conditions of solvability and unique solvability of the boundary value problem are established, where are measurable functions and the vector function is measurable in the first and continuous in the last kmn arguments; moreover, this function may have nonintegrable singularities with respect to the first argument.     

Solvability Theory and Projection Methods for a Class of Singular Variational Inequalities: Elastostatic Unilateral Contact Applications

The mathematical modeling of engineering structures containing members capable of transmitting only certain type of stresses or subjected to noninterpenetration conditions along their boundaries leads generally to variational inequalities of the form , where C is a closed convex set of (kinematically admissible set), (loading strain vector), and (stiffness matrix). If rigid body displacements and rotations cannot be excluded from these applications, then the resulting matrix M is singular and serious mathematical difficulties occur. The aim of this paper is to discuss the existence and the numerical computation of the solutions of problem (P) for the class of cocoercive matrices. Our theoretical results are applied to two concrete engineering problems: the unilateral cantilever problem and the elastic stamp problem.     

Levi Classes Generated by Nilpotent Groups

Let be a class of all groups G for which the normal closure (x) ^G of every element x belongs to a class . is a Levi class generated by . Let and ₀ be classes of finitely generated nilpotent groups and of torsion-free, finitely generated, nilpotent groups, respectively. We prove that and , and so and . It is shown that quasivarieties and are closed under free products, and that each contains at most one maximal proper subquasivariety. It is also proved that is closed under free products if so is .     

Mathematical Properties of Optimization Problems Defined by Positively Homogeneous Functions

We consider the nonlinear programming problem

On the Summing Property of the Sobolev Embedding Operators

We prove that the Sobolev embedding operator S _d,k,p : , where 1/s=1/p-k/d , is (v,1) -absolutely summing for appropriate v > 1 . The result is optimal for s 2 .     

The Maximum Size Of Graphs With A Unique<Emphasis Type="Italic">k</Emphasis>- Factor

For suitable positive integers n and k let m(n, k) denote the maximum number of edges in a graph of order n which has a unique k-factor. In 1964, Hetyei and in 1984, Hendry proved for even n and , respectively. Recently, Johann confirmed the following conjectures of Hendry: for and kn even and for n = 2kq, where q is a positive integer. In this paper we prove for and kn even, and we determine m(n, 3).     

The Paley--Wiener Theorem for the Generalized Radon Transform on the Plane

We consider the problem of reconstructing a function on the disk from its integrals over curves close to straight lines, i.e., the inversion problem for the generalized Radon transform. Necessary and sufficient conditions on the range of the generalized Radon transform are obtained for functions supported in a smaller disk under the additional condition that the curves that do not meet coincide with the corresponding straight lines.     

Convergence Analysis of Some Methods for Minimizing a Nonsmooth Convex Function

In this paper, we analyze a class of methods for minimizing a proper lower semicontinuous extended-valued convex function . Instead of the original objective function f, we employ a convex approximation f _{k + 1} at the kth iteration. Some global convergence rate estimates are obtained. We illustrate our approach by proposing (i) a new family of proximal point algorithms which possesses the global convergence rate estimate even it the iteration points are calculated approximately, where are the proximal parameters, and (ii) a variant proximal bundle method. Applications to stochastic programs are discussed.     

Beckman-Quarles type theorems for mappings from $$ \mathbb{R}^n $$ to $$ \mathbb{C}^n $$

Summary. Let We say that preserves the distance d 0 if for each implies Let A _n denote the set of all positive numbers d such that any map that preserves unit distance preserves also distance d. Let D _n denote the set of all positive numbers d with the property: if and then there exists a finite set S _xy with such that any map that preserves unit distance preserves also the distance between x and y. Obviously, We prove: (1) (2) for n 2 D _n is a dense subset of (2) implies that each mapping f from to (n 2) preserving unit distance preserves all distances, if f is continuous with respect to the product topologies on and      

Generalized Harish-Chandra Modules: A New Direction in the Structure Theory of Representations

Let be a reductive Lie algebra over C. We say that a -module M is a generalized Harish-Chandra module if, for some subalgebra , M is locally -finite and has finite -multiplicities. We believe that the problem of classifying all irreducible generalized Harish-Chandra modules could be tractable. In this paper, we review the recent success with the case when is a Cartan subalgebra. We also review the recent determination of which reductive in subalgebras are essential to a classification. Finally, we present in detail the emerging picture for the case when is a principal 3-dimensional subalgebra.     

Behavior of Automorphic L-Functions at the Center of the Critical Strip

Let be the Hecke eigenbasis of the space of -cusp forms of weight 2. Let p be a prime. Let be the Hecke L-series of form . The following statements are proved:

Strictly Cyclic Algebra of Operators Acting on Banach Spaces H p(β)

Let be a sequence of positive numbers and 1 p< . We consider the space H ^p() of all power series such that . We investigate strict cyclicity of the weakly closed algebra generated by the operator of multiplication by zacting on H ^p(), and determine the maximal ideal space, the dual space and the reflexivity of the algebra . We also give a necessary condition for a composition operator to be bounded on H ^p() when is strictly cyclic.