Finite Energy Solutions of Maxwell''s Equations and Constructive Hodge Decompositions on Nonsmooth Riemannian Manifolds

We consider two basic potential theoretic problems in Riemannian manifolds: Hodge decompositions and Maxwell's equations. Here we are concerned with smoothness and integrability assumptions. In the context of L^p forms in Lipschitz domains, we show that both are well posed provided that 2−<p<2+, for some >0, depending on the domain. Our approach is constructive (in the sense that we produce integral representation formulas for the solutions) and emphasizes the intimate connections between the two problems at hand. Applications to other related PDEs, such as boundary problems for the Hodge Dirac operator, are also presented.     

On <Emphasis Type="Italic">G</Emphasis>-invex multiobjective programming. Part II. Duality

This paper represents the second part of a study concerning the so-called G-multiobjective programming. A new approach to duality in differentiable vector optimization problems is presented. The techniques used are based on the results established in the paper: On G-invex multiobjective programming. Part I. Optimality by T.Antczak. In this work, we use a generalization of convexity, namely G-invexity, to prove new duality results for nonlinear differentiable multiobjective programming problems. For such vector optimization problems, a number of new vector duality problems is introduced. The so-called G-Mond–Weir, G-Wolfe and G-mixed dual vector problems to the primal one are defined. Furthermore, various so-called G-duality theorems are proved between the considered differentiable multiobjective programming problem and its nonconvex vector G-dual problems. Some previous duality results for differentiable multiobjective programming problems turn out to be special cases of the results described in the paper.     

Approximation characteristics for diagonal operators in different computational settings

First we study several extremal problems on minimax, and prove that they are equivalent. Then we connect this result with the exact values of some approximation characteristics of diagonal operators in different settings, such as the best n-term approximation, the linear average and stochastic n-widths, and the Kolmogorov and linear n-widths. Most of these exact values were known before, but in terms of equivalence of these extremal problems, we present a unified approach to give them a direct proof.     

A Polynomial Factorization Approach for the Discrete Time GIX/>G/1/K Queue

This paper proposes a polynomial factorization approach for queue length distribution of discrete time GI ^X /G/1 and GI _X /G/1/K queues. They are analyzed by using a two-component state model at the arrival and departure instants of customers. The equilibrium state-transition equations of state probabilities are solved by a polynomial factorization method. Finally, the queue length distributions are then obtained as linear combinations of geometric series, whose parameters are evaluated from roots of a characteristic polynomial.     

Feature selection for high-dimensional data

This paper focuses on feature selection for problems dealing with high-dimensional data. We discuss the benefits of adopting a regularized approach with L ₁ or L ₁–L ₂ penalties in two different applications—microarray data analysis in computational biology and object detection in computer vision. We describe general algorithmic aspects as well as architecture issues specific to the two domains. The very promising results obtained show how the proposed approach can be useful in quite different fields of application.     

A classification of some regular p-groups and its applications

In this paper we classify regular p-groups with type invariants (e/it, 1, 1, 1) for e⩾2 and (1, 1, 1, 1, 1). As a by-product, we give a new approach to the classification of groups of order p ⁵, p⩾5 a prime.     

Constructions for Octonion and Exceptional Jordan Algebras

In this note we reverse theusual process of constructing the Lie algebras of types G ₂and F ₄ as algebras of derivations of the splitoctonions or the exceptional Jordan algebra and instead beginwith their Dynkin diagrams and then construct the algebras togetherwith an action of the Lie algebras and associated Chevalley groups.This is shown to be a variation on a general construction ofall standard modules for simple Lie algebras and it is well suitedfor use in computational algebra systems. All the structure constantswhich occur are integral and hence the construction specialisesto all fields, without restriction on the characteristic, avoidingthe usual problems with characteristics 2 and 3.     

On Macbeath-Singerman symmetries of Belyi surfaces with PSL(2,p) as a group of automorphisms

The famous theorem of Belyi states that the compact Riemann surface X can be defined over the number field if and only if X can be uniformized by a finite index subgroup Γ of a Fuchsian triangle group Λ. As a result such surfaces are now called Belyi surfaces. The groups PSL(2,q),q=p ⁿ are known to act as the groups of automorphisms on such surfaces. Certain aspects of such actions have been extensively studied in the literature. In this paper, we deal with symmetries. Singerman showed, using acertain result of Macbeath, that such surfaces admit a symmetry which we shall call in this paper the Macbeath-Singerman symmetry. A classical theorem by Harnack states that the set of fixed points of a symmetry of a Riemann surface X of genus g consists of k disjoint Jordan curves called ovals for some k ranging between 0 and g+1. In this paper we show that given an odd prime p, a Macbetah-Singerman symmetry of Belyi surface with PSL(2,p) as a group of automorphisms has at most     

Controlling the space with preassigned responses

It is shown that large classes of control systems, which include certain systems of the typex+A(t)x=B(t)u, can be handled in such a way that the control functionsu(t) are actually associated with responsesx(t) that belong to known families of functions. In particular, it is possible, for a variety of perturbationsB(t)u and operatorsA(t) with convex domains, to have responses that are line segments joining the origin to the reachable states.The present approach establishes the fact that a vast number of results from functional analysis concerning ranges of operators can be effectively applied to the general theory of control. It is also rather significant that the present theory does not necessarily require the solvability of the associated Cauchy problem.The operatorsB(t)u do not have to be invertible inu. However, it is shown that continuous controlsu(t) can be obtained for a variety of problems whenB ^–1(t)u exists and is continuous int.     

关于L_p球的几个新不等式和性质(英文)

本文研究了L_p球的相关问题.利用对偶混合体积、球面Radon变换和Fourier变换的方法,获得了关于L_p球的几个新不等式和性质,其中一个不等式与著名的最大切片猜想有关.     

The expected value models on Sugeno measure space

Uncertain programming is a theoretical tool to handle optimization problems under uncertain environment. The research reported so far is mainly concerned with probability, possibility, or credibility measure spaces. Up to now, uncertain programming realized in Sugeno measure space has not been investigated. The first type of uncertain programming considered in this study and referred to as an expected value model optimizes a given expected objective function subject to some expected constraints. We start with a concept of the Sugeno measure space. We revisit some main properties of the Sugeno measure and elaborate on the g_λ random variable and its characterization. Furthermore, the laws of the large numbers are discussed based on this space. In the sequel we introduce a Sugeno expected value model (SEVM). In order to construct an approximate solution to the complex SEVM, the ideas of a Sugeno random number generation and a Sugeno simulation are presented along with a hybrid approach.     

Approximations for the <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">GI</Emphasis>/<Emphasis Type="Italic">N</Emphasis>+<Emphasis Type="Italic">GI</Emphasis> type call center

In this paper, we propose approximations to compute the steady-state performance measures of the M/GI/N+GI queue receiving Poisson arrivals with N identical servers, and general service and abandonment-time distributions. The approximations are based on scaling a single server M/GI/1+GI queue. For problems involving deterministic and exponential abandon times distributions, we suggest a practical way to compute the waiting time distributions and their moments using the Laplace transform of the workload density function. Our first contribution is numerically computing the workload density function in the M/GI/1+GI queue when the abandon times follow general distributions different from the deterministic and exponential distributions. Then we compute the waiting time distributions and their moments. Next, we scale-up the M/GI/1+GI queue giving rise to our approximations to capture the behavior of the multi-server system. We conduct extensive numerical experiments to test the speed and performance of the approximations, which prove the accuracy of their predictions.      

Linear best approximation using a class ofk-majorl p norms

Consideration is given to problems of linear best approximation using a variant of the usuall _p norms referred to ask-majorl _p norms, for the cases when 1<p<. The underlying problem is the minimization of a piecewise smooth function. Best approximations are characterized, and a descent algorithm is developed.     

-Optimality conditions for composed convex optimization problems

The aim of the present paper is to provide a formula for the -subdifferential of f+gh different from the ones which can be found in the existent literature. Further we equivalently characterize this formula by using a so-called closedness type regularity condition expressed by means of the epigraphs of the conjugates of the functions involved. Even more, using the -subdifferential formula we are able to derive necessary and sufficient conditions for the -optimal solutions of composed convex optimization problems.     

Distribution results for lattices in SL(2,ℚ p )

In this paper we study the ergodic properties of the linear action of lattices Γ of SL(2,ℚ_p) on ℚ_p × ℚ_p and distribution results for orbits of Γ. Following Serre, one can define a “geodesic flow” for an associated tree (actually associated to GL(2,ℚ_p)). The approach we use is based on an extension of this approach to “frame flows” which are a natural compact group extension of the geodesic flow.     

Long vectors for quasi-Newton updates

This work concerns the derivation of formulae for updating quasi-Newton matrices used in algorithms for computing approximate minima of smooth unconstrained functions. The paper concentrates strictly on the techniques used to derive update formulae. It demonstrates a technique in which problems of finding matrices in ℝ ^{n ×n} of minimum Frobenius norm are converted to equivalent problems, using vector representations in ℝ ⁿ² and ℝ ^n(n+1)/2 of these matrices, and then solvingl ₂-minimization problems. These problems are more directly dealt with, and indeed, the paper demonstrates how this technique may be used to handle weighted sparse updates.     

G-pre-invex functions in mathematical programming

In the present paper, we introduce the concept of G-pre-invex functions with respect to η defined on an invex set with respect to η. These function unify the concepts of nondifferentiable convexity, pre-invexity and r-pre-invexity. Furthermore, relationships of G-pre-invex functions to various introduced earlier pre-invexity concepts are also discussed. Some (geometric) properties of this class of functions are also derived. Finally, optimality results are established for optimization problems under appropriate G-pre-invexity conditions.     

The generalized Boardman homomorphisms

This paper provides universal upper bounds for the exponent of the kernel and of the cokernel of the classical Boardman homomorphism b ⁿ : π ⁿ (X)→H ⁿ (H;ℤ), from the cohomotopy groups to the ordinary integral cohomology groups of a spectrum X, and of its various generalizations π ⁿ (X)→E ⁿ (X), F ⁿ (X)→(E∧F) ⁿ (X), F ⁿ (X)→H ⁿ (X;π ₀ F) and F ⁿ (X)→H ^n+t (X;π _t F) for other cohomology theories E ^*(−) and F ^*(−). These upper bounds do not depend on X and are given in terms of the exponents of the stable homotopy groups of spheres and, for the last three homomorphisms, in terms of the order of the Postnikov invariants of the spectrum F.     

Semicontinuity of Solution Mappings of Parametric Generalized Vector Equilibrium Problems

In this paper, we study the ε-generalized vector equilibrium problem (ε-GVEP) and the ε-extended vector equilibrium problem (ε-EVEP), which can be regarded as approximate problems to the generalized vector equilibrium problems (GVEP). Existence results for ε-GVEP and ε-EVEP are established. We investigate also the continuity of the solution mappings of ε-GVEP and ε-EVEP. In particular, two results concerning the lower semicontinuity of the solution mappings of ε-GVEP and ε-EVEP are presented. This research was partially supported by Grant NSC 95-2811-M-110-010.     

An Assmus–Mattson-Type Approach for Identifying 3-Designs from Linear Codes over Z 4

The complete weight enumerator of the Delsarte–Goethals code over Z ₄ is derived and an Assmus–Mattson-type approach at identifying t-designs in linear codes over Z ₄ is presented. The Assmus–Mattson-type approach, in conjunction with the complete weight enumerator are together used to show that the codewords of constant Hamming weight in both the Goethals code over Z ₄ as well as the Delsarte–Goethals code over Z ₄ yield 3-designs, possibly with repeated blocks.