Scalarization of Henig Proper Efficient Points in a Normed Space

In a general normed space equipped with the order induced by a closed convex cone with a base, using a family of continuous monotone Minkowski functionals and a family of continuous norms, we obtain scalar characterizations of Henig proper efficient points of a general set and a bounded set, respectively. Moreover, we give a scalar characterization of a superefficient point of a set in a normed space equipped with the order induced by a closed convex cone with a bounded base.     

Staircase Parameterization in Dynamical Selection

This paper deals with a variant of a dynamical selection scheme introduced by Attouch and Cominetti for ill-posed convex minimization which combines approximation with the steepest descent method by mean of a suitable parameterization of the approximation parameter as a function of the time. This variant applies to a general inclusion with a maximal monotone operator by mean of a staircase parameterization. A discrete analogue is also considered. Applications to selecting a particular zero of a maximal monotone operator or a particular fixed point of a nonexpansive mapping via regularization techniques are presented. Finally, the alternative use of well-posedness by perturbations is discussed.     

A pair of orthogonal frames

We start with a characterization of a pair of frames to be orthogonal in a shift-invariant space and then give a simple construction of a pair of orthogonal shift-invariant frames. This is applied to obtain a construction of a pair of Gabor orthogonal frames as an example. This is also developed further to obtain constructions of a pair of orthogonal wavelet frames.     

Spectrum of a homogeneous graph

We describe the spectrum of the Laplacian for a homogeneous graph acted on by a discrete group. This follows from a more general result which describes the spectrum of a convolution operator on a homogeneous space of a locally compact group. We also prove a version of Harnack inequality for a Schrödinger operator on an invariant homogeneous graph.     

Two questions on semi-clean group rings

It was proved in [4] that every group ring of a torsion abelian group over a commutative local ring is a semi-clean ring. It was asked in [4] whether every group ring of a torsion abelian group over a commutative clean ring is a semi-clean ring and whether every group ring of a torsion abelian group over a commutative semi-clean ring is a semi-clean ring. In this paper, we give a positive answer to question 1 and a negative answer to question 2.     

Partially isometric dilations of noncommuting -tuples of operators

Given a row contraction of operators on a Hilbert space and a family of projections on the space that stabilizes the operators, we show there is a unique minimal joint dilation to a row contraction of partial isometries that satisfy natural relations. For a fixed row contraction the set of all dilations forms a partially ordered set with a largest and smallest element. A key technical device in our analysis is a connection with directed graphs. We use a Wold decomposition for partial isometries to describe the models for these dilations, and we discuss how the basic properties of a dilation depend on the row contraction.

     

On arithmetic Macaulayfication of Noetherian rings

The Rees algebra is the homogeneous coordinate ring of a blowing-up. The present paper gives a necessary and sufficient condition for a Noetherian local ring to have a Cohen-Macaulay Rees algebra: A Noetherian local ring has a Cohen-Macaulay Rees algebra if and only if it is unmixed and all the formal fibers of it are Cohen-Macaulay. As a consequence of it, we characterize a homomorphic image of a Cohen-Macaulay local ring. For non-local rings, this paper gives only a sufficient condition. By using it, however, we obtain the affirmative answer to Sharp's conjecture. That is, a Noetherian ring having a dualizing complex is a homomorphic image of a finite-dimensional Gorenstein ring.

     

A generalized duality and applications

The aim of this paper is to present a nonconvex duality with a zero gap and its connection with convex duality. Since a convex program can be regarded as a particular case of convex maximization over a convex set, a nonconvex duality can be regarded as a generalization of convex duality. The generalized duality can be obtained on the basis of convex duality and minimax theorems. The duality with a zero gap can be extended to a more general nonconvex problems such as a quasiconvex maximization over a general nonconvex set or a general minimization over the complement of a convex set. Several applications are given.On leave from the Institute of Mathematics, Hanoi, Vietnam.     

覆盖数不超过3的图上渡河问题

1000多年前,英国著名学者Alcuin曾提出一个古老的渡河问题,即狼、羊和卷心菜的渡河问题。2006年,Prisner把该问题推广到任意的冲突图上,考虑了一类情况更一般的渡河运输问题。所谓冲突图是指一个图G=(V,E),这里V代表某些物品的集合,V中的两个点有边连结当且仅当这两个点是冲突的,即在无人监管的情况下不允许留在一起的点。图G=(V,E)的一个可行运输方案是指在保证不发生任何冲突的前提下,把V的点所代表的物品全部摆渡到河对岸的一个运输方案。图G的Alcuin数定义为它存在可行运输方案时所需船的最小容量。本文讨论了覆盖数不超过3的连通图的Alcuin数,给出了该类图Alcuin数的完全刻画。     

A Barrier Function Method for the Nonconvex Quadratic Programming Problem with Box Constraints

In this paper a barrier function method is proposed for approximating a solution of the nonconvex quadratic programming problem with box constraints. The method attempts to produce a solution of good quality by following a path as the barrier parameter decreases from a sufficiently large positive number. For a given value of the barrier parameter, the method searches for a minimum point of the barrier function in a descent direction, which has a desired property that the box constraints are always satisfied automatically if the step length is a number between zero and one. When all the diagonal entries of the objective function are negative, the method converges to at least a local minimum point of the problem if it yields a local minimum point of the barrier function for a sequence of decreasing values of the barrier parameter with zero limit. Numerical results show that the method always generates a global or near global minimum point as the barrier parameter decreases at a sufficiently slow pace.     

Genus fields of Kummer extensions of rational function fields

In this paper we obtain the genus field of a general Kummer extension of a global rational function field. We study first the case of a general Kummer extension of degree a power of a prime. Then we prove that the genus field of a composite of two abelian extensions of a global rational function field with relatively prime degrees is equal to the composite of their respective genus fields. Our main result, the genus of a general Kummer extension of a global rational function field, is a direct consequence of this fact.     

Geometry of the Space of Phylogenetic Trees

We consider a continuous space which models the set of all phylogenetic trees having a fixed set of leaves. This space has a natural metric of nonpositive curvature, giving a way of measuring distance between phylogenetic trees and providing some procedures for averaging or combining several trees whose leaves are identical. This geometry also shows which trees appear within a fixed distance of a given tree and enables construction of convex hulls of a set of trees. This geometric model of tree space provides a setting in which questions that have been posed by biologists and statisticians over the last decade can be approached in a systematic fashion. For example, it provides a justification for disregarding portions of a collection of trees that agree, thus simplifying the space in which comparisons are to be made.     

DC programming techniques for solving a class of nonlinear bilevel programs

We propose a method for finding a global solution of a class of nonlinear bilevel programs, in which the objective function in the first level is a DC function, and the second level consists of finding a Karush-Kuhn-Tucker point of a quadratic programming problem. This method is a combination of the local algorithm DCA in DC programming with a branch and bound scheme well known in discrete and global optimization. Computational results on a class of quadratic bilevel programs are reported.     

Existence of a solution for two phase flow in porous media: The case that the porosity depends on the pressure

In this paper we prove the existence of a solution of a coupled system involving a two phase incompressible flow in the ground and the mechanical deformation of the porous medium where the porosity is a function of the global pressure. The model is strongly coupled and involves a nonlinear degenerate parabolic equation. In order to show the existence of a weak solution, we consider a sequence of related uniformly parabolic problems and apply the Schauder fixed point theorem to show that they possess a classical solution. We then prove the relative compactness of sequences of solutions by means of the Fréchet-Kolmogorov theorem; this yields the convergence of a subsequence to a weak solution of the parabolic system.     

An algebraic index theorem for orbifolds

Using the concept of a twisted trace density on a cyclic groupoid, a trace is constructed on a formal deformation quantization of a symplectic orbifold. An algebraic index theorem for orbifolds follows as a consequence of a local Riemann-Roch theorem for such densities. In the case of a reduced orbifold, this proves a conjecture by Fedosov, Schulze, and Tarkhanov. Finally, it is shown how the Kawasaki index theorem for elliptic operators on orbifolds follows from this algebraic index theorem.     

On the Nature of Buffering

We state a theorem on the instability of self-similar cycles and tori of a certain type in a system which is a quasinormal form of a boundary-value problem for a nonlinear wave equation in the square.     

Kripke submodels and universal sentences

We define two notions for intuitionistic predicate logic: that of a submodel of a Kripke model, and that of a universal sentence. We then prove a corresponding preservation theorem. If a Kripke model is viewed as a functor from a small category to the category of all classical models with (homo)morphisms between them, then we define a submodel of a Kripke model to be a restriction of the original Kripke model to a subcategory of its domain, where every node in the subcategory is mapped to a classical submodel of the corresponding classical model in the range of the original Kripke model. We call a sentence universal if it is built inductively from atoms (including ? and ⊥) using ∧, ∨, ?, and →, with the restriction that antecedents of → must be atomic. We prove that an intuitionistic theory is axiomatized by universal sentences if and only if it is preserved under Kripke submodels. We also prove the following analogue of a classical model‐consistency theorem: The universal fragment of a theory Γ is contained in the universal fragment of a theory Δ if and only if every rooted Kripke model of Δ is strongly equivalent to a submodel of a rooted Kripke model of Γ. Our notions of Kripke submodel and universal sentence are natural in the sense that in the presence of the rule of excluded middle, they collapse to the classical notions of submodel and universal sentence. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear Chebyshev approximation to set-valued functions

In this paper, we give a characterization of best Chebyshev approximation to set-valued functions from a family of continuous functions with the weak betweeness property. As a consequence, we obtain a characterization of Kolmogorov type for best simultaneous approximation to an infinity set of functions. We introduce the concept of a set-sun and give a characterization of it. In addition, we prove a property of Amir–Ziegler type for a family of real functions and we get a characterization of best simultaneous approximation to two functions