Two expansion theorems by diagonal elements for the permanent and the determinant

Based on two set partitions of the symmetric group S_n expansion theorems by diagonal elements for the permanent and the determinant are derived, for both the generic commuting and noncommuting cases. They are of the same type as the well-known Laplace expansions where either fixed rows or columns of a given matrix are chosen instead of diagonal elements.     

The kernel of the bezoutian for operator Polynomials ∗

Based on two set partitions of the symmetric group S_n expansion theorems by diagonal elements for the permanent and the determinant are derived, for both the generic commuting and noncommuting cases. They are of the same type as the well-known Laplace expansions where either fixed rows or columns of a given matrix are chosen instead of diagonal elements.     

Markov processes on partitions

We introduce and study a family of Markov processes on partitions. The processes preserve the so-called z-measures on partitions previously studied in connection with harmonic analysis on the infinite symmetric group. We show that the dynamical correlation functions of these processes have determinantal structure and we explicitly compute their correlation kernels. We also compute the scaling limits of the kernels in two different regimes. The limit kernels describe the asymptotic behavior of large rows and columns of the corresponding random Young diagrams, and the behavior of the Young diagrams near the diagonal. Our results show that recently discovered analogy between random partitions arising in representation theory and spectra of random matrices extends to the associated time– dependent models.     

On the invertibility of “rectangular” bi-infinite matrices and applications in time–frequency analysis

Finite dimensional matrices with more columns than rows have no left inverses while those with more rows than columns have no right inverses. We give generalizations of these simple facts to bi–infinite matrices. Our results are then used to obtain density results for p-frames of time–frequency molecules in modulation spaces and identifiability results for operators with bandlimited Kohn–Nirenberg symbols.     

Covering and radius-covering arrays: Constructions and classification

The minimum number of rows in covering arrays (equivalently, surjective codes) and radius-covering arrays (equivalently, surjective codes with a radius) has been determined precisely only in special cases. In this paper, explicit constructions for numerous best known covering arrays (upper bounds) are found by a combination of combinatorial and computational methods. For radius-covering arrays, explicit constructions from covering codes are developed. Lower bounds are improved upon using connections to orthogonal arrays, partition matrices, and covering codes, and in specific cases by computation. Consequently for some parameter sets the minimum size of a covering array is determined precisely. For some of these, a complete classification of all inequivalent covering arrays is determined, again using computational techniques. Existence tables for up to 10 columns, up to 8 symbols, and all possible strengths are presented to report the best current lower and upper bounds, and classifications of inequivalent arrays.     

Generating functions for plane partitions of a given shape

For fixed integers α and β, planar arrays of integers of a given shape, in which the entries decrease at least by α along rows and at least by β along columns, are considered. For various classes of these (α,β)-plane partitions we compute three different kinds of generating functions. By a combinatorial method, determinantal expressions are obtained for these generating functions. In special cases these determinants may be evaluated by a simple determinant lemma. All known results concerning plane partitions of a given shape are included. Thus our approach of a given shape provides a uniform proof method and yields numerous generalizations of known results.     

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.     

The bijection between plane partitions and nonnegative integer matrices

The one-to-one correspondence between the set of plane partitions withr rows andm columns and the set of matrices of nonnegative integers with the same numbers of rows and columns has been constructed. Published in Lietuvos Matematikes Rinkinys, Vol. 35, No. 2, pp. 204–210, April–June, 1995.     

Constructing heterogeneous hash families by puncturing linear transversal designs

Constructions that use hash families to select columns from small covering arrays in order to construct larger ones can exploit heterogeneity in the numbers of symbols in the rows of the hash family. For specific distributions of numbers of symbols, the efficacy of the construction is improved by accommodating more columns in the hash family. Known constructions of such heterogeneous hash families employ finite geometries and their associated transversal designs. Using thwarts in transversal designs, specific constructions of heterogeneous hash families are developed, and some open questions are posed.     

Invariant sublattices for positive operators

There are, by now, many results which guarantee that positive operators on Banach lattices have non-trivial closed invariant sublattices. In particular, this is true for every positive compact operator. Apart from some results of a general nature, in this paper we present several examples of positive operators on Banach lattices which do not have non-trivial closed invariant sublattices. These examples include both AM-spaces and Banach lattices with an order continuous norm and which are and are not atomic. In all these cases we can ensure that the operators do possess non-trivial closed invariant subspaces.     

Springer fibers and Schubert points

Springer fibers are subvarieties of the flag variety parametrized by partitions; they are central objects of study in geometric representation theory. Schubert varieties are subvarieties of the flag variety that induce a well-known basis for the cohomology of the flag variety. This paper relates these two varieties combinatorially. We prove that the Betti numbers of the Springer fiber associated to a partition with at most three rows or two columns are equal to the Betti numbers of a specific union of Schubert varieties.     

Discovery of time-inconsecutive co-movement patterns of foreign currencies using an evolutionary biclustering method

This paper proposes an evolutionary biclustering algorithm to discover inconsecutive co-movement patterns of different foreign exchange rates. The rows/columns of a bicluster (i.e. a submatrix with a subset of rows and a subset of columns) are not necessarily consecutive. A typical bicluster with constant values on rows and/or columns is represented as a hyperplane in a high-dimensional space and the coefficients of the hyperplane are determined using a genetic algorithm. A detected bicluster demonstrates the co-moving behaviors of a subset of currencies in inconsecutive time periods, indicating that the currencies moved in different manners in some specific time periods. In our experiments, we relate these patterns to the geographically close economic connections and find out the correspondence between the nominal exchange rates and the economic conditions. The findings are useful as a guide for investing foreign currencies.     

Reconstructing finite group actions and characters from subgroup information

A finite group G is said to be action reconstructible if, for any action of G on a finite set, the numbers of orbits under restriction to each subgroup always give enough information to reconstruct the action up to equivalence. G is character reconstructible if, given any matrix representation of G, the mean value of the character on each subgroup always gives enough information to reconstruct the character. The conjugacy matrix of G is the matrix whose (ij) entry is the number of elements of the jth conjugacy class belonging to a typical subgroup of the ith subgroup conjugacy class. It is shown that G is action reconstructible if and only if the rows of this matrix are linearly independent (which is in turn true if and only if G is cyclic), and is character reconstructible if and only if the columns are linearly independent (which is true if and only if any two elements of G which generate conjugate cyclic subgroups are themselves conjugate).     

Convex sets of nonsingular and P:-Matrices

We show that the set r(A,B) (resp. c(A,B) of square matrices whose rows (resp. columns) are the independent convex combinations of the rows (resp.columns) of real matrices A and B consists entirely of nonsingular matrices if and only if BA^-1(resp. B^-1A) is a P-matrix. This imrpoves a theorem on P-Matrices proven in [2] and [3], in the context of interval nonsingularity. We also show that every real P-matrix admits a representation BA^-1 with the above property. These reseults are only partially true for complex P-matrices. Based on them we obtain a characterizaiton of complex P-matrices in terms of block partitions.     

A Distributive Lattice Structure Connecting Dyck Paths,Noncrossing Partitions and 312-avoiding Permutations

In [Ferrari, L. and Pinzani, R.: Lattices of lattice paths. J. Stat. Plan. Inference 135 (2005), 77–92] a natural order on Dyck paths of any fixed length inducing a distributive lattice structure is defined. We transfer this order to noncrossing partitions along a well-known bijection [Simion, R.: Noncrossing partitions. Discrete Math. 217 (2000), 367–409], thus showing that noncrossing partitions can be endowed with a distributive lattice structure having some combinatorial relevance. Finally we prove that our lattices are isomorphic to the posets of 312-avoiding permutations with the order induced by the strong Bruhat order of the symmetric group.     

关于一类矩阵的一个不变量

本文讨论满足关联方程的实数矩阵A,得到关于A的行(列)和的一个不变量。     

Matrices with prescribed rows and invariant factors

In this paper, we solve the problem of the existence of an n × n matrix over an arbitrary field when its invariant polynomials and either some rows or columns are prescribed. The solution is given in terms of invariant factor inequalities and of majorization inequalities involving controllability indices and the degrees of the invariant polynomials.     

Matrices with prescribed invariant factors and off-diagonal submatrices

We solve the problem of the existence of a matrix with prescribed invariant factors and an off-diagonal submatrix (i.e., a submatrix without entries from the main diagonal and such that the number of rows plus the number of columns is the size of the given matrix). We will use some ideas from Control theory.     

The Smale horseshoes in a class of non-invertible maps in with two-direction instability

We study the existence of Smale horseshoes of new type and the uniformly hyperbolic invariant sets for a class of non-invertible maps in three-dimensional Euclidean spaces with the dimension of instability equal to two. Parameter regions are given, for which the map has a horseshoe and a uniformly hyperbolic invariant set on which the map is topologically conjugate to the two-sided fullshift on four symbols.     

Syndetically proximal pairs

For continuous self-maps of compact metric spaces, we study the syndetically proximal relation, and in particular we identify certain sufficient conditions for the syndetically proximal cell of each point to be small. We show that any interval map f with positive topological entropy has a syndetically scrambled Cantor set, and an uncountable syndetically scrambled set invariant under some power of f. In the process of proving this, we improve a classical result about interval maps and establish that if f is an interval map with positive topological entropy and m?2, then there is n∈N such that the one-sided full shift on m symbols is topologically conjugate to a subsystem of fn² (the classical result gives only semi-conjugacy).