Wavelet Approximation of Distributions with Bounded Variation Derivatives

Let BV ^r denote the space of distributions f such that the distributional derivatives D ^α f with |α|≤r exist as measures of bounded variation. This paper discusses estimates for wavelet coefficients of BV ^r distributions, direct (Jackson) and inverse (Bernstein) inequalities for n-term approximation of elements of BV ^r in the L _p spaces using compactly supported wavelets. In particular, optimal rates of approximation are established. Linear approximation in similar contexts is also considered for comparison. This research was supported by the 2003–2007 Academic Grant of Prof. P. Wojtaszczyk awarded by the Foundation for Polish Science. Part of this research was supported within the HASSIP framework.     

The Upper Bounds of Arbitrary Eigenvalues for Uniformly Elliptic Operators with Higher Orders

Let Ωbelong to R^m （m≥ 2） be a bounded domain with piecewise smooth and Lipschitz boundary δΩ Let t and r be two nonnegative integers with t ≥ r ＋ 1. In this paper, we consider the variable-coefficient eigenvalue problems with uniformly elliptic differential operators on the left-hand side and （-Δ）^T on the right-hand side. Some upper bounds of the arbitrary eigenvalue are obtained, and several known results are generalized.     

On a Sharp Degree Sum Condition for Disjoint Chorded Cycles in Graphs

Let r and s be nonnegative integers, and let G be a graph of order at least 3r + 4s. In Bialostocki et al. (Discrete Math 308:5886–5890, 2008), conjectured that if the minimum degree of G is at least 2r + 3s, then G contains a collection of r + s vertex-disjoint cycles such that s of them are chorded cycles, and they showed that the conjecture is true for r = 0, s = 2 and for s = 1. In this paper, we settle this conjecture completely by proving the following stronger statement; if the minimum degree sum of two nonadjacent vertices is at least 4r + 6s−1, then G contains a collection of r + s vertex-disjoint cycles such that s of them are chorded cycles.     

Multicriteria graph problems with the MAXMIN criterion

Under study is the r-criteria problems for the r-weighted graphs (r ≥ 2). Certain kinds of subgraphs are called admissible. Solving some problem means choosing a Pareto optimal admissible subgraph from the complete set of alternatives (CSA). The main result of this paper is as follows: Suppose that a criterion denoted by MAXMIN requires maximization of the minimal first edges’ weight of the admissible subgraph and there is an effective procedure constructing the CSA for a (r − 1)-criteria problem without this MAXMIN criterion. Then the CSA for the initial r-criteria problem is created effectively.     

Differential-geometric structures on generalized Reidemeister and Bol three-webs

In this paper, we present the main results of the study of multidimensional three-websW(p, q, r) obtained by the method of external forms and moving Cartan frame. The method was developed by the Russian mathematicians S. P. Finikov, G. F. Laptev, and A. M. Vasiliev, while fundamentals of differential-geometric (p, q, r)-webs theory were described by M. A. Akivis and V. V. Goldberg. Investigation of (p, q, r)-webs, including algebraic and geometric theory aspects, has been continued in our papers, in particular, we found the structure equations of a three-web W(p, q, r), where p = λl, q = λm, and r = λ(l + m − 1). For such webs, we define the notion of a generalized Reidemeister configuration and proved that a three-web W(λl, λm, λ(l + m − 1)), on which all sufficiently small generalized Reidemeister configurations are closed, is generated by a λ-dimensional Lie group G. The structure equations of the web are connected with the Maurer–Cartan equations of the group G. We define generalized Reidemeister and Bol configurations for three-webs W(p, q, q). It is proved that a web W(p, q, q) on which generalized Reidemeister or Bol configurations are closed is generated, respectively, by the action of a local smooth q-parametric Lie group or a Bol quasigroup on a smooth p-dimensional manifold. For such webs, the structure equations are found and their differential-geometric properties are studied.     

Estimates for the Sobolev trace constant with critical exponent and applications

In this paper we find estimates for the optimal constant in the critical Sobolev trace inequality that are independent of Ω. This estimates generalized those of Adimurthi and Yadava (Comm Partial Diff Equ 16(11):1733–1760, 1991) for general p. Here p _* : =  p(N  −  1)/(N  −  p) is the critical exponent for the immersion and N is the space dimension. Then we apply our results first to prove existence of positive solutions to a nonlinear elliptic problem with a nonlinear boundary condition with critical growth on the boundary, generalizing the results of Fernández Bonder and Rossi (Bull Lond Math Soc 37:119–125, 2005). Finally, we study an optimal design problem with critical exponent.      

Comparison of approximation properties of generalized polynomials and splines

We establish that, for p ∈ [2, ∞), q = 1 or p = ∞, q ∈ [ 1, 2], the classes W _p^rof functions of many variables defined by restrictions on the L _p-norms of mixed derivatives of order r = (r ₁, r ₂, ..., r _m) are better approximated in the L _q-metric by periodic generalized splines than by generalized trigonometric polynomials. In these cases, the best approximations of the Sobolev classes of functions of one variable by trigonometric polynomials and by periodic splines coincide. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 8, pp. 1011–1020, August, 1998.     

Optimality conditions and duality for a class of nondifferentiable multi-objective fractional programming problems

A class of multi-objective fractional programming problems (MFP) are considered where the involved functions are locally Lipschitz. In order to deduce our main results, we give the definition of the generalized (F,θ,ρ,d)-convex class about the Clarke’s generalized gradient. Under the above generalized convexity assumption, necessary and sufficient conditions for optimality are given. Finally, a dual problem corresponding to (MFP) is formulated, appropriate dual theorems are proved.      

Generalized Nash equilibrium problems for lower semi-continuous strategy maps

We consider a class of generalized Ky Fan inequalities (quasi-variational inequalities) in which the involved multi-valued mapping is lower semi-continuous. We present a relaxed version of the generalized Nash equilibrium problem involving strategy maps, which are only lower semi-continuous. This relaxed version may have no exact Nash equilibrium, but we prove that it has an ε-Nash equilibrium for every ε > 0. Easy examples of such problems show no existence of exact solutions, but existence of ε-solutions for every ε > 0. We give positive answers to two questions (in the compact case) raised in a recent paper of Cubiotti and Yao.     

Zero-Sum Flows in Regular Graphs

For an undirected graph G, a zero-sum flow is an assignment of non-zero real numbers to the edges, such that the sum of the values of all edges incident with each vertex is zero. It has been conjectured that if a graph G has a zero-sum flow, then it has a zero-sum 6-flow. We prove this conjecture and Bouchet’s Conjecture for bidirected graphs are equivalent. Among other results it is shown that if G is an r-regular graph (r ≥ 3), then G has a zero-sum 7-flow. Furthermore, if r is divisible by 3, then G has a zero-sum 5-flow. We also show a graph of order n with a zero-sum flow has a zero-sum (n + 3)²-flow. Finally, the existence of k-flows for small graphs is investigated.     

Coxian representations of generalized Erlang distributions

This paper studies Coxian representations of generalized Erlang distributions. A nonlinear program is derived for computing the parameters of minimal Coxian representations of generalized Erlang distributions. The nonlinear program is also used to characterize the triangular order and the admissible region of generalized Erlang distributions. It is shown that the admissible region associated with a triangular order may not be convex. For generalized Erlang distributions of ME-order 3, a minimal Coxian representation is found explicitly. In addition, an algorithm is developed for computing a special type of ordered Coxian representations - the bivariate Coxian representation - for generalized Erlang distributions.     

L ∞ andW 1, ∞ estimates for second order generalized difference methods on two-point boundary value problems

In this paper, we consider the second order generalized difference scheme for the two-point boundary value problem and obtain optimal order error estimates inL ^∞ andW ^{1, ∞}. The results in this paper perfect the theory of the second order generalized difference method.     

Bivariate Fibonacci polynomials of order <Emphasis Type="Italic">k</Emphasis> with statistical applications

In the present article, we investigate the properties of bivariate Fibonacci polynomials of order k in terms of the generating functions. For k and ℓ (1 ≤ ℓ ≤ k − 1), the relationship between the bivariate Fibonacci polynomials of order k and the bivariate Fibonacci polynomials of order ℓ is elucidated. Lucas polynomials of order k are considered. We also reveal the relationship between Lucas polynomials of order k and Lucas polynomials of order ℓ. The present work extends several properties of Fibonacci and Lucas polynomials of order k, which will lead us a new type of geneses of these polynomials. We point out that Fibonacci and Lucas polynomials of order k are closely related to distributions of order k and show that the distributions possess properties analogous to the bivariate Fibonacci and Lucas polynomials of order k.     

Limit distributions of smallest gap and largest repeated part in integer partitions

We study two parameters in random integer partitions, namely the first gap and the last repeated part, that have been introduced by Grabner and Knopfmacher in a recent paper (Ramanujan J. 12(3):439–454, 2006). More generally, the first part that occurs at most r times and the last part that occurs at least r times are considered. For both parameters, we determine the limit distribution, which turn out to be the Rayleigh and Gumbel distributions, respectively. This also generalises the well-known result by Erdős and Lehner on the distribution of the largest part in a random integer partition. Furthermore, extensions to general Λ-partitions and results on related parameters, such as the length of the first gap, are provided.     

A wide class of heavy-tailed distributions and its applications

Let F(x) be a distribution function supported on [0, ∞) with an equilibrium distribution function F _e (x). In this paper we pay special attention to the hazard rate function r _e (x) of F _e (x), which is also called the equilibrium hazard rate (E.H.R.) of F(x). By the asymptotic behavior of r _e (x) we give a criterion to identify F(x) to be heavy-tailed or light-tailed. Moreover, we introduce two subclasses of heavy-tailed distributions, i.e., ℳ and ℳ*, where ℳ contains almost all the most important heavy-tailed distributions in the literature. Some further discussions on the closure properties of ℳ and ℳ* under convolution are given, showing that both of them are ideal heavy-tailed subclasses. In the paper we also study the model of independent difference ξ = Z − θ, where Z and θ are two independent and non-negative random variables. We give intimate relationships of the tail distributions of ξ and Z, as well as relationships of tails of their corresponding equilibrium distributions. As applications, we apply the properties of class ℳ to risk theory. In the final, some miscellaneous problems and examples are laid, showing the complexity of characterizations on heavy-tailed distributions by means of r _e (x).      

Relaxed Derivatives and Extremality Conditions in Optimal Control

In previous work dating back to the early 1970’s F.H. Clarke and the author had independently derived necessary conditions for minimum including a maximum principle for optimal control problems defined by ordinary differential equations in which the right hand side f(t,·, r) and functions defining side conditions are Lipschitz continuous in their dependence on the state variable. Our results, though not the methods, were similar in the formulation of the maximum principle in which the nonexisting derivative f _v (t, v, σ) was replaced by an unknown element of Clarke’s generalized Jacobian but differed in handling some side conditions. In the present paper we exhibit a maximum principle in which the dual variables and the related functions are limits of appropriate subsequences of computable sequences.     

Remarks on Gagliardo–Nirenberg type inequality with critical Sobolev space and BMO

We consider the generalized Gagliardo–Nirenberg inequality in in the homogeneous Sobolev space with the critical differential order s = n/r, which describes the embedding such as for all q with p ≦ q < ∞, where 1 < p < ∞ and 1 < r < ∞. We establish the optimal growth rate as q → ∞ of this embedding constant. In particular, we realize the limiting end-point r = ∞ as the space of BMO in such a way that with the constant C _n depending only on n. As an application, we make it clear that the well known John–Nirenberg inequality is a consequence of our estimate. Furthermore, it is clarified that the L ^∞-bound is established by means of the BMO-norm and the logarithm of the -norm with s > n/r, which may be regarded as a generalization of the Brezis–Gallouet–Wainger inequality.     

Using Lagrangians of Hypergraphs to Find Non-Jumping Numbers (I)

Let r ≥ 2 be an integer. A real number is a jump for r if for any and any integer m, m ≥ r, any r-uniform graph with vertices and density at least contains a subgraph with m vertices and density at least α + c, where c = c(α) does not depend on or m. It follows from a result of Erdős, Stone, and Simonovits that every is a jump for r = 2. Erdőos asked whether the same is true for r ≥ 3. Frankl and R?dl gave a negative answer by showing an infinite sequence of non-jumping numbers for r ≥ 3. However, there are a lot of unknowns on determining whether a number is a jump for r ≥ 3. In this paper, we first find two infinite sequences of non-jumping numbers for r = 4, then we extend one of the results to every r ≥ 4. Our approach is still based on the approach developed by Frankl and R?dl. Received November 30, 2005     

The radial curvature of an end that makes eigenvalues vanish in the essential spectrum I

The concern of this paper is to clarify a relationship between the curvatures at infinity and the spectral structure of the Laplacian. In particular, this paper discusses the question of whether there is an eigenvalue of the Laplacian embedded in the essential spectrum or not. The borderline-behavior of the radial curvatures for this problem will be determined: we will assume that the radial curvature K _rad. of an end converges to a constant −1 at infinity with the decay order K _rad. + 1 = o(r ⁻¹) and prove the absence of eigenvalues embedded in the essential spectrum. Furthermore, in order to show that this decay order K _rad. + 1 = o(r ⁻¹) is sharp, we will construct a manifold with the radial curvature decay K _rad. + 1 = O(r ⁻¹) and with an eigenvalue \frac(n-1)²4+1{\frac{(n-1)^2}{4}+1} embedded in the essential spectrum [ \frac(n-1)²4, ￥){[ \frac{(n-1)^2}{4}, \infty)} of the Laplacian.