Puzzles,tableaux, and mosaics

We define mosaics, which are naturally in bijection with Knutson-Tao puzzles. We define an operation on mosaics, which shows they are also in bijection with Littlewood-Richardson skew-tableaux. Another consequence of this construction is that we obtain bijective proofs of commutativity and associativity for the ring structures defined either of these objects. In particular, we obtain a new, easy proof of the Littlewood-Richardson rule. Finally we discuss how our operation is related to other known constructions, particularly jeu de taquin.     

Jeu de taquin and a monodromy problem for Wronskians of polynomials

The Wronskian associates to d linearly independent polynomials of degree at most n, a non-zero polynomial of degree at most d(n−d). This can be viewed as giving a flat, finite morphism from the Grassmannian Gr(d,n) to projective space of the same dimension. In this paper, we study the monodromy groupoid of this map. When the roots of the Wronskian are real, we show that the monodromy is combinatorially encoded by Schützenberger's jeu de taquin; hence we obtain new geometric interpretations and proofs of a number of results from jeu de taquin theory, including the Littlewood-Richardson rule.     

Schur positivity of skew Schur function differences and applications to ribbons and Schubert classes

Some new relations on skew Schur function differences are established both combinatorially using Schützenberger’s jeu de taquin, and algebraically using Jacobi-Trudi determinants. These relations lead to the conclusion that certain differences of skew Schur functions are Schur positive. Applying these results to a basis of symmetric functions involving ribbon Schur functions confirms the validity of a Schur positivity conjecture due to McNamara. A further application reveals that certain differences of products of Schubert classes are Schubert positive. For Manfred Schocker 1970–2006. S.J. van Willigenburg was supported in part by the National Sciences and Engineering Research Council of Canada.     

Growth diagrams for the Schubert multiplication

We present a partial generalization of the classical Littlewood-Richardson rule (in its version based on Schützenberger's jeu de taquin) to Schubert calculus on flag varieties. More precisely, we describe certain structure constants expressing the product of a Schubert and a Schur polynomial. We use a generalization of Fomin's growth diagrams (for chains in Young's lattice of partitions) to chains of permutations in the so-called k-Bruhat order. Our work is based on the recent thesis of Beligan, in which he generalizes the classical plactic structure on words to chains in certain intervals in k-Bruhat order. Potential applications of our work include the generalization of the S₃-symmetric Littlewood-Richardson rule due to Thomas and Yong, which is based on Fomin's growth diagrams.     

Standard tableaux and Klyachko's theorem on Lie representations

We show that for all but two partitions λ of n>6 there exists a standard tableau of shape λ with major index coprime to n. In conjunction with a deep result of Kra?kiewicz and Weyman this provides a new purely combinatorial proof of Klyachko's famous theorem on Lie representations of the general linear group.     

Enumeration formulas for standard Young tableaux of nearly hollow rectangular shapes

This paper considers the enumeration problem of a generalization of standard Young tableau (SYT) of truncated shape. Let $\lambda ?\mu |\left\{(i_0,j_0)\right\}$ be the SYT of shape $\lambda$ truncated by $\mu$ whose upper left cell is $(i_0,j_0)$, where $\lambda$ and $\mu$ are partitions of integers. The summation representation of the number of SYT of the truncated shape $(n+k+2,{(n+2)}^{m+1})?\left(n^m\right)|\left\{(2,2)\right\}$ is derived. Consequently, three closed formulas for SYT of hollow shapes are obtained, including the cases of (i). $m=n=1$, (ii). $k=0$, and (iii). $k=1,m=n$. Finally, an open problem is posed.     

A comment on the jäger-kačur linearization scheme for strongly nonlinear parabolic equations

The aim of this paper is to demonstrate how the variational equations from [11] can be formulated and solved in some abstract Banach spaces without any a priori construction of special linearization schemes. This should be useful e.g. in the analysis of heat conduction problems and modelling of flows in porous media.     

A note on quantum products of Schubert classes in a Grassmannian

Given two Schubert classes σ_λ and σ_μ in the quantum cohomology of a Grassmannian, we construct a partition ν, depending on λ and μ, such that σ_ν appears with coefficient 1 in the lowest (or highest) degree part of the quantum product σ_λ⋆σ_μ. To do this, we show that for any two partitions λ and μ, contained in a k × (n − k) rectangle and such that the 180^∘-rotation of one does not overlap the other, there is a third partition ν, also contained in the rectangle, such that the Littlewood-Richardson number c _λμ ^ν is 1.     

On Young's Orthogonal Form and the Characters of the Alternating Group

A combinatorial method of determining the characters of the alternating group is presented. We use matrix representations, due to Thrall, that are closely related to Young's orthogonal form of representations of the symmetric group. The characters are computed directly from matrix entries of these representations and entries of the character table of the symmetric group.     

Pseudoparabolic equations with convection

The existence of solutions of pseudoparabolic equations withconvection by using discretization along characteristics isshown. The uniqueness of the solution of a pseudoparabolic equationis proved for a linear elliptic part and for a space dimensionN 4.     

The use of linear approximation scheme for solving the Stefan problem

This paper deals with the linear approximation scheme to approximate a singular parabolic problem: the two-phase Stefan problem on a domain consisting of two components with imperfect contact. The results of some numerical experiments and comparisons are presented. The method was used to determine the temperature of steel in the process of continuous casting.     

Bi-cycle extendable through a given set in balanced bipartite graphs

Let G=(X,Y;E) be a balanced bipartite graph of order 2n. The path-cover numberpc(H) of a graph H is the minimum number of vertex-disjoint paths that use up all the vertices of H. S⊆V(G) is called a balanced set of G if |S∩X|=|S∩Y|. In this paper, we will give some sufficient conditions for a balanced bipartite graph G satisfying that for every balanced set S, there is a bi-cycle of every length from |S|+2pc(〈S〉) up to 2n through S.     

如何利用单纯形表上的信息

本文分析了求解线性规划的基本方法－－单纯形法所使用的单纯形表，将表中所提供的信息分为直接信息和间接信息两类，论述了如何充分利用这些信息的方法。例如如何由最终表求原问题、如何利用表中的数据互相推演和校正等。这是一篇教学经验的总结，对初学者可能有一定的帮助。