A Finite Algorithm for a Class of Nonlinear Multiplicative Programs

The nonconvex problem of minimizing the product of a strictly convex quadratic function and the p-th power of a linear function over a convex polyhedron is considered. Some theoretical properties of the problem, such as the existence of minimum points and the generalized convexity of the objective function, are deepened on and a finite algorithm which solves the problem is proposed.     

Global Maximization of a Generalized Concave Multiplicative Function

This article presents a branch-and-bound algorithm for globally solving the problem (P) of maximizing a generalized concave multiplicative function over a compact convex set. Since problem (P) does not seem to have been studied previously, the algorithm is apparently the first algorithm to be proposed for solving this problem. It works by globally solving a problem (P1) equivalent to problem (P). The branch-and-bound search undertaken by the algorithm uses rectangular partitioning and takes place in a space which typically has a much smaller dimension than the space to which the decision variables of problem (P) belong. Convergence of the algorithm is shown; computational considerations and benefits for users of the algorithm are given. A sample problem is also solved.     

Multiplicative Mappings of Rings

Let R and F be arbitrary associative rings. A mapping φ of R onto F is called a multiplicative isomorphism if φ is bijective and satisfies φ（xy） = φ（x）φ（y） for all x, y ∈ R. In this short note, we establish a condition on R, in the case where R may not contain any non-zero idempotents, that assures that φ is additive, which generalizes the famous Martindale＇s result. As an application, we show that under a mild assumption every multiplicative isomorphism from the radical of a nest algebra onto an arbitrary ring is additive.     

保谱乘法映射

设X,Y为Banach空间,证明了B（X）到B（Y）的保谱乘法满射φ具有形式φ（T）＝ATA-1,其中A为X到Y上的同构．     

Multiplicative Lie Isomorphisms Between Prime Rings

Let ? be a prime ring with 1 containing a nontrivial idempotent E, and let ?′ be another prime ring. If Φ:? → ?′ is a multiplicative Lie isomorphism, then Φ(T + S) = Φ(T) + Φ(S) + Z′ _T,S for all T, S ∈ ?, where Z′ _T,S is an element in the center 𝒵′ of ?′ depending on T and S.     

乘积型模糊B-代数

引入乘积型模糊B-代数的概念,提供它们的几个例子,研究它们的一些性质.讨论模糊B-代数与乘积型模B-代数的关系,研究乘积型模糊B-代数的同态象与同态原象的性质,给出B-代数上乘积型模糊B-代数与B-代数的积代数上乘积型模糊B-代数的关系.     

Branch-and-reduce algorithm for convex programs with additional multiplicative constraints

This article presents a branch-and-reduce algorithm for globally solving for the first time a convex minimization problem (P) with p?1$p?1$ additional multiplicative constraints. In each of these p   additional constraints, the product of two convex functions is constrained to be less than or equal to a positive number. The algorithm works by globally solving a 2p$2p$-dimensional master problem (MP) equivalent to problem (P). During a typical stage k of the algorithm, a point is found that minimizes the objective function of problem (MP) over a nonconvex set F^k$F^k$ that contains the portion of the boundary of the feasible region of the problem where a global optimal solution lies. If this point is feasible in problem (MP), the algorithm terminates. Otherwise, the algorithm continues by branching and creating a new, reduced nonconvex set F^k+1$F^{k+1}$ that is a strict subset of F^k$F^k$. To implement the algorithm, all that is required is the ability to solve standard convex programming problems and to implement simple algebraic steps. Convergence properties of the algorithm are given, and results of some computational experiments are reported.     

自然数乘法分拆个数及因子个数上界估计

我们证明，若n充分大，则其乘法分拆数小于n/lnn，这几乎解决了关于自然数乘法分拆数的一个猜测,也得到了自然数因子个数的一个上界估计。     

Multiplicative convexity and its applications

This paper is concerned with some sequences based on multiplicative convexity. Our results yield a class of new inequalities between ratios and differences of means, some of which extend the known ones.     

Multiplicative Invariants of Root Lattices

We describe the multiplicative invariant algebras of the root lattices of all irreducible root systems under the action of the Weyl group. In each case, a finite system of fundamental invariants is determined and the class group of the invariant algebra is calculated. In some cases, a presentation and a Hironaka decomposition of the invariant algebra is given.     

Mathematical Programs with Vector Optimization Constraints

In this paper, we introduce mathematical programs with vector optimization constraints. For these problems, we establish two models in both the weak Pareto solution and Pareto solution setting. Some new existence results are obtained under rather weak conditions. We establish also equivalences between mathematical programs with vector optimization constraints and mathematical programs with vector variational inequality constraints.This research was partially supported by a grant from the National Science Council of the ROC. The authors thank the referees for helpful suggestions and comments.     

Multiplicative Maps With Specified (1, 1)-Entry

In this article, we will show that for many complex-valued maps f over a semigroup G of matrices, there exists a minimum k for the existence of a multiplicative map for which the (1, 1)-entry of } ( A ) is f ( A ). We obtain results on such multiplicative maps, and use them to classify all the multiplicative maps on G such that f ( A ) = ( g ( A )) for any A k G , where f and g are given complex-valued maps.     

Multiplicative Maps With Specified (1, 1)-Entry

In this article, we will show that for many complex-valued maps f over a semigroup G of matrices, there exists a minimum k for the existence of a multiplicative map for which the (1, 1)-entry of ϕ( A ) is f ( A ). We obtain results on such multiplicative maps, and use them to classify all the multiplicative maps τon G such that f ( A ) = τ( g ( A )) for any A ε G , where f and g are given complex-valued maps.     

Global Optimization of 0-1 Hyperbolic Programs

We develop eight different mixed-integer convex programming reformulations of 0-1 hyperbolic programs. We obtain analytical results on the relative tightness of these formulations and propose a branch and bound algorithm for 0-1 hyperbolic programs. The main feature of the algorithm is that it reformulates the problem at every node of the search tree. We demonstrate that this algorithm has a superior convergence behavior than directly solving the relaxation derived at the root node. The algorithm is used to solve a discrete p-choice facility location problem for locating ten restaurants in the city of Edmonton.The research was supported in part by NSF awards DMII 95-02722 and BES 98-73586 to NVS.     

On the Multiplicative Quasi-Adequate Transversals of an Abundant Semigroup

In this article, we explore the multiplicative quasi-adequate transversals of an abundant semigroup. Let S be an abundant semigroup with multiplicative quasi-adequate transversals. Then the product of any two multiplicative quasi-adequate transversals is also a multiplicative quasi-adequate transversal. Moreover, all multiplicative quasi-adequate transversals of S form a rectangular band. Let S°and S ^? be multiplicative quasi-adequate transversals of S, and let δ°(δ^?) be the δ-relation on S°(S ^?). Then there exists a bijection ? from S°/δ°onto S ^?/δ^?. In particular, if δ°and δ^? are congruences, then the bijection ? is an isomorphism.     

A Geometric Approach to Global Optimization

In this paper we consider the problem of optimizing a piecewise-linear objective function over a non-convex domain. In particular we do not allow the solution to lie in the interior of a prespecified region R. We discuss the geometrical properties of this problems and present algorithms based on combinatorial arguments. In addition we show how we can construct quite complicated shaped sets R while maintaining the combinatorial properties.     

基于乘性模糊互补判断矩阵的FAHP

根据乘性一致性的定义,从人的主观判断与其权重的函数关系出发,获得了乘性模糊互补判断矩阵的元素表达式,论证了利用乘性模糊互补判断矩阵求取因素相对权重的可行性,最后给出了一种基于乘性一致模糊判断矩阵的排序方法.从而使得乘性模糊互补判断矩阵的应用的理论基础更加坚实.     

Multiplicative spectrum of ultrametric Banach algebras of continuous functions

Let K be an ultrametric complete field and let E be an ultrametric space. Let A be the Banach K-algebra of bounded continuous functions from E to K and let B be the Banach K-algebra of bounded uniformly continuous functions from E to K. Maximal ideals and continuous multiplicative semi-norms on A (resp. on B) are studied by defining relations of stickiness and contiguousness on ultrafilters that are equivalence relations. So, the maximal spectrum of A (resp. of B) is in bijection with the set of equivalence classes with respect to stickiness (resp. to contiguousness). Every prime ideal of A or B is included in a unique maximal ideal and every prime closed ideal of A (resp. of B) is a maximal ideal, hence every continuous multiplicative semi-norms on A (resp. on B) has a kernel that is a maximal ideal. If K is locally compact, every maximal ideal of A (resp. of B) is of codimension 1. Every maximal ideal of A or B is the kernel of a unique continuous multiplicative semi-norm and every continuous multiplicative semi-norm is defined as the limit along an ultrafilter on E. Consequently, on A as on B the set of continuous multiplicative semi-norms defined by points of E is dense in the whole set of all continuous multiplicative semi-norms. Ultrafilters show bijections between the set of continuous multiplicative semi-norms of A, Max(A) and the Banaschewski compactification of E which is homeomorphic to the topological space of continuous multiplicative semi-norms. The Shilov boundary of A (resp. B) is equal to the whole set of continuous multiplicative semi-norms.     

On Multiplicative Invariants of Finite Reflection Groups

This article focuses on two recent results on multiplicative invariants of finite reflection groups: Lorenz (2001 Lorenz , M. ( 2001 ). Multiplicative invariants and semigroup algebras . Alg. and Rep. Theory 4 : 293 – 304 . [Google Scholar]) showed that such invariants are affine normal semigroup algebras, and Reichstein (2003 Reichstein , Z. ( 2003 ). SAGBI bases in rings of multiplicative invariants . Commentarii Math. Helvetici 78 ( 1 ): 185 – 202 .[Web of Science ®] , [Google Scholar]) proved that the invariants have a finite SAGBI basis. Reichstein (2003 Reichstein , Z. ( 2003 ). SAGBI bases in rings of multiplicative invariants . Commentarii Math. Helvetici 78 ( 1 ): 185 – 202 .[Web of Science ®] , [Google Scholar]) also showed that, conversely, if the multiplicative invariant algebra of a finite group G has a SAGBI basis, then G acts as a reflection group. There is no obvious connection between these two results. We will show that multiplicative invariants of finite reflection groups have a certain embedding property that implies both results simultaneously.     

Multiplicative Jordan Decomposition in Group Rings of 3-Groups,II

In this paper, we complete the classification of those finite 3-groups G whose integral group rings have the multiplicative Jordan decomposition property. If G is abelian, then it is clear that ?[G] satisfies the multiplicative Jordan decomposition (MJD). In the nonabelian case, we show that ?[G] satisfies MJD if and only if G is one of the two nonabelian groups of order 3³ = 27.