A 类量子群和 Hecke 代数的 Schur-Weyl 对偶

设Vm是量子群Uq（SLm)的标准表示,通过Hecke代数的作用,作者将Vm的张量积V^nm分解成了Uq(SLm)的不可约表示的直和,从而给出了Uq(SLm)与Hecke代数的H（q,n)Schur-Weyl对偶的完整证明,进一步得到,当q是实数时,在Hilbert空间H^n上,Uq(SL∞）和H（q,n)之间存在着Schur-Weyl对偶。     

A Duality Approach to Nonlinear Diffusion Equations

We provide new existence results for a nonlinear diffusion equation with a monotonically increasing multivalued time-dependent nonlinearity, under minimal growth and coercivity conditions. The results given in this paper prove that a generalized solution to the nonlinear equation is provided by a solution to an equivalent minimization problem for a convex functional involving the potential of the nonlinearity and its conjugate, in the case when the potential is time and space depending. If the potential is time depending only and it has a symmetry at infinity, the null minimizer in the minimization problem is found to coincide with a weak solution to the nonlinear equation.     

A Unifying Approach to Robust Convex Infinite Optimization Duality

This paper considers an uncertain convex optimization problem, posed in a locally convex decision space with an arbitrary number of uncertain constraints. To this problem, where the uncertainty only affects the constraints, we associate a robust (pessimistic) counterpart and several dual problems. The paper provides corresponding dual variational principles for the robust counterpart in terms of the closed convexity of different associated cones.     

A New Duality Approach to Solving Concave Vector Maximization Problems

We introduce a special class of monotonic functions with the help of support functions and polar sets, and use it to construct a scalarized problem and its dual for a vector optimization problem. The dual construction allows us to develop a new method for generating weak efficient solutions of a concave vector maximization problem and establish its convergence. Some numerical examples are given to illustrate the applicability of the method.     

On an Algebro-Geometric Approach to Ribaucour Transformations and Bianchi Cubes

Mathematical Notes - In this paper, we develop an algebro-geometric approach to Ribaucour transformations and Bianchi cubes of orthogonal nets. Explicit transformations of algebro-geometric data...     

关于随机对偶性的一个注记

对任意随机局部凸模（S,{x^d}d∈D）,本文证明了{x^d}d∈D可表示成关于自然的随机对偶对〈S,S＊〉的—个随机可允许结构．     

Algebraic Approach to Duality in Optimization and Applications

Set-Valued and Variational Analysis - This paper studies duality of optimization problems in a vector space without topological structure. A strong duality relation is established by means of...     

Asymptotic Properties of Fibonacci Cubes and Lucas Cubes

It is proved that the asymptotic average eccentricity and the asymptotic average degree of both Fibonacci cubes and Lucas cubes are \({(5+\sqrt{5})/10}\) and \({(5-\sqrt{5}) /5}\) , respectively. A new labeling of the leaves of Fibonacci trees is introduced and it is proved that the eccentricity of a vertex of a given Fibonacci cube is equal to the depth of the associated leaf in the corresponding Fibonacci tree. Hypercube density is also introduced and studied. The hypercube density of both Fibonacci cubes and Lucas cubes is shown to be \({(1-1/\sqrt{5})/ \rm log_ {2}\varphi}\) , where \({\varphi}\) is the golden ratio, and the Cartesian product of graphs is used to construct families of graphs with a fixed, non-zero hypercube density. It is also proved that the average ratio of the numbers of Fibonacci strings with a 0 (a 1, respectively) in a given position, where the average is taken over all positions, converges to \({\varphi^{2}}\) , and likewise for Lucas strings.     

Silver Cubes

An n × n matrix A is said to be silver if, for i = 1,2,...,n, each symbol in {1,2,...,2n − 1} appears either in the ith row or the ith column of A. The 38th International Mathematical Olympiad asked whether a silver matrix exists with n = 1997. More generally, a silver cube is a triple (K _n ^d , I, c) where I is a maximum independent set in a Cartesian power of the complete graph K _n , and is a vertex colouring where, for v ∈ I, the closed neighbourhood N[v] sees every colour. Silver cubes are related to codes, dominating sets, and those with n a prime power are also related to finite geometry. We present here algebraic constructions, small examples, and a product construction. The nonexistence of silver cubes for d = 2 and some values of n, is proved using bounds from coding theory. Luis A. Goddyn: This research was supported by a Canada NSERC Discovery Grant. Ebadollah S. Mahmoodian: Partially supported by the institutes CECM and IRMACS and the departments of Mathematics and Computing Science at Simon Fraser University. Grateful thanks are extended here and also to the Institute for Advanced Studies in Basic Sciences, Iran for support in the final stages of this paper.     

A Remark on Two Duality Relations

We remark that an easy combination of two known results yields a positive answer, up to log(n) terms, to a duality conjecture that goes back to Pietsch. In particular, we show that for any two symmetric convex bodies K, T in , denoting by N(K, T) the minimal number of translates of T needed to cover K, one has:

Schur-Weyl duality and the heat kernel measure on the unitary group

We investigate a relation between the Brownian motion on the unitary group and the most natural random walk on the symmetric group, based on Schur-Weyl duality. We use this relation to establish a convergent power series expansion for the expectation of a product of traces of powers of a random unitary matrix under the heat kernel measure. This expectation turns out to be the generating series of certain paths in the Cayley graph of the symmetric group. Using our expansion, we recover asymptotic results of Xu, Biane and Voiculescu. We give an interpretation of our main expansion in terms of random ramified coverings of a disk.     

On Combinatorial Cubes

A sequence of positive integers with positive lower density contains a Hilbert (or combinatorial) cube size c log log n up to n. We prove this bound is sharp for some thinner sequence.     

Randomized Simplex Algorithms on Klee-Minty Cubes

The analysis of two most natural randomized pivot rules on the Klee-Minty cubes leads to (nearly) quadratic lower bounds for the complexity of linear programming with random pivots. Thus we disprove two bounds (for the expected running time of the random-edge simplex algorithm on Klee-Minty cubes) conjectured in the literature. At the same time, we establish quadratic upper bounds for the expected length of a path for a simplex algorithm with random pivots on the classes of linear programs under investigation. In contrast to this, we find that the average length of an increasing path in a Klee-Minty cube is exponential when all paths are taken with equal probability. Received: September 2, 1996     

Cubes and orientability

We define and study a new class of matroids: cubic matroids. Cubic matroids include, as a particular case, all affine cubes over an arbitrary field. There is only one known orientable cubic matroid: the real affine cube. The main results establish as an invariant of orientable cubic matroids the structure of the subset of acyclic orientations with LV-face lattice isomorphic to the face lattice of the real cube or, equivalently, with the same signed circuits of length 4 as the real cube.     

Unified Nonlinear Lagrangian Approach to Duality and Optimal Paths

In the context of an inequality constrained optimization problem, we present a unified nonlinear Lagrangian dual scheme and establish necessary and sufficient conditions for the zero duality gap property. From these results, we derive necessary and sufficient conditions for four classes of zero duality gap properties and establish the equivalence among them. Finally, we obtain the convergence of an optimal path for the unified scheme and present a sufficient condition for the finite termination of the optimal path. This research was partially supported by the Research Grants Council of Hong Kong Grant PolyU 5250/03E, the National Natural Science Foundation of China Grants 10471159 and 10571106, NCET, and the Natural Science Foundation of Chongqing     

A Coherent Homotopy Category of 2-track Commutative Cubes

We consider a category \({\mathcal H}^{\ominus \otimes}\) (the homotopy category of homotopy squares) whose objects are homotopy commutative squares of spaces and whose morphisms are cubical diagrams subject to a coherent homotopy relation. The main result characterises the isomorphisms of \({\mathcal H}^{\ominus \otimes}\) to be the cube morphisms whose forward arrows are homotopy equivalences. As a first application of the new category we give a direct 2-track theoretic definition of the quaternary Toda bracket operation.