Upper triangular operator matrices with the single-valued extension property

If A=(Aij)_1?i,j?n∈B(X) is an upper triangular Banach space operator such that AiiAij=AijAjj for all 1?i?j?n, then A has SVEP or satisfies (Dunford's) condition (C) or (Bishop's) property (β) or (the decomposition) property (δ) if and only if Aii, 1?i?n, has the corresponding property.     

A graph theoretic upper bound on the permanent of a nonnegative integer matrix. I

Let A be a fully indecomposable n×n matrix with nonnegative integer entries. Then the permanent of A is bounded above by 1+min{Π(c_i?1), Π(r_i?1)}, where c_i and r_i are the column and row sums of A. The inequality results from a bound on the number of disjoint cycle unions in an associated multigraph. This bound can improve via contractions.     

Fast canonization of circular strings

A circular string A = (a₁,…,a_n) is a string that has n equivalent linear representations A_i = a_i,…,a_n,a₁,…,a_i?1 for i = 1,…,n. The a_i's are assumed to be well ordered. We say that A_i < A_j if the word a_i…a_na₁…a_i?1 precedes the word a_j…a_na₁…a_j?1 in the lexicographic order, A_i ? A_j if either A_i < A_j or A_i = A_j. A_i0 is a minimal representation of A if A_i0 ? A_i for all 1 ≤ i ≤ n. The index i₀ is called a minimal starting point (m.s.p.). In this paper we discuss the problem of finding m.s.p. of a given circular string. Our algorithm finds, in fact, all the m.s.p.'s of a given circular string A of length n by using at most $\text{n + ?}\text{d}\text{2}\text{?}$ comparisons. The number d denotes the difference between two successive m.s.p.'s (see Lemma 1.1) and is equal to n if A has a unique m.s.p. Our algorithm improves the result of 3n comparisons given by K. S. Booth. Only constant auxiliary storage is used.     

Symposium on combinatorial mathematics and optimal design

Let A₁,A₂,…,A_n be finite sets such that A_i?A_j for all i≠j. Let F be an intersecting family consisting of sets contained in some A_i, i=1,2,…,n. Chvátal conjectured that among the largest intersecting families, there is always a star. In this paper, we obtain another proof of a result of Schönheim: If A₁∩A₂∩?∩A_n≠?, then the conjecture is true. We also prove that if A_i∩A_jA_k = ? for all i≠j≠k≠i or if the independent system satisfies a hereditary tree structure, then the conjecture is also true.     

On separating systems whose elements are sets of at most k elements

A system A₁,…,A_m of subsets of X?{1,…,n} is called a separating system if for any two distinct elements of X there is a set A_i (1?i?m) that contains exactly one of the two elements. We investigate separating systems where each set A_i has at most k elements and we are looking for minimal separating systems, that means separating systems with the least number of subsets. We call this least number m(n,k). Katona has proved good bounds on m(n,k) but his proof is very complicated. We give a shorter and easier proof. In doing so we slightly improve the upper bound of Katona.     

A continuous family of partition statistics equidistributed with length

This article investigates a remarkable generalization of the generating function that enumerates partitions by area and number of parts. This generating function is given by the infinite product _∏i?11/(1−tqi). We give uncountably many new combinatorial interpretations of this infinite product involving partition statistics that arose originally in the context of Hilbert schemes. We construct explicit bijections proving that all of these statistics are equidistributed with the length statistic on partitions of n. Our bijections employ various combinatorial constructions involving cylindrical lattice paths, Eulerian tours on directed multigraphs, and oriented trees.     

Simultaneous block diagonalization of two real symmetric matrices

The simultaneous diagonalization of two real symmetric (r.s.) matrices has long been of interest. This subject is generalized here to the following problem (this question was raised by Dr. Olga Taussky-Todd, my thesis advisor at the California Institute of Technology): What is the first simultaneous block diagonal structure of a nonsingular pair of r.s. matrices ? For example, given a nonsingular pair of r.s. matrices S and T, which simultaneous block diagonalizations X′SX = diag(A₁, , A_k), X′TX = diag(B₁,, B_k) with dim A_i = dim B_i and X nonsingular are possible for 1 ? k ? n; and how well defined is a simultaneous block diagonalization for which k, the number of blocks, is maximal? Here a pair of r.s. matrices S and T is called nonsingular if S is nonsingular.If the number of blocks k is maximal, then one can speak of the finest simultaneous block diagonalization of S and T, since then the sizes of the blocks A_i are uniquely determined (up to permutations) by any set of generators of the pencil P(S, T) = {aS + bT|a, tb ε R} via the real Jordan normal form of S^?1T. The proof uses the canonical pair form theorem for nonsingular pairs of r.s. matrices. The maximal number k and the block sizes dim A_i are also determined by the factorization over C of ? (λ, μ) = det(λS + μT) for λ, μ ε R.     

The characteristic polynomial of a principal subpencil of a Hermitian matrix pencil

Given n-square Hermitian matrices A,B, let A_i,B_i denote the principal (n?1)- square submatrices of A,B, respectively, obtained by deleting row i and column i. Let μ, λ be independent indeterminates. The first main result of this paper is the characterization (for fixed i) of the polynomials representable as det(μA_i?λB_i) in terms of the polynomial det(μA?λB) and the elementary divisors, minimal indices, and inertial signatures of the pencil μA?λB. This result contains, as a special case, the classical interlacing relationship governing the eigenvalues of a principal sub- matrix of a Hermitian matrix. The second main result is the determination of the number of different values of i to which the characterization just described can be simultaneously applied.     

An extremal problem among subsets of a set

We determine the maximum size of a family of subsets in {1, 2,…, n} with the property that if A₁, A₂, A₃,… are any members of the family with $\text{∩A}{}_{\mathrm{i}}\text{= ?}$, then ∪A_i = {1, 2,…, n}.     

A combinatorial problem involving graphs and matrices

In this paper we discuss a combinatorial problem involving graphs and matrices. Our problem is a matrix analogue of the classical problem of finding a system of distinct representatives (transversal) of a family of sets and relates closely to an extremal problem involving 1-factors and a long standing conjecture in the dimension theory of partially ordered sets. For an integer n ?1, let n denote the n element set {1,2,3,…, n}. Then let A be a k×t matrix. We say that A satisfies property P(n, k) when the following condition is satisfied: For every k-taple (x₁,x₂,…,x_k?n^k there exist k distinct integers j₁,j₂,…,j_k so that x_i= a_ii for i= 1,2,…,k. The minimum value of t for which there exists a k × t matrix A satisfying property P(n,k) is denoted by f(n,k). For each k?1 and n sufficiently large, we give an explicit formula for f(n, k): for each n?1 and k sufficiently large, we use probabilistic methods to provide inequalities for f(n,k).     

Extremal Systems with Upper-Bounded Odd Intersections

In this paper we determine the maximum cardinality m of a family A= {A ₁,..., A _m} of subsets of a set of n elements with the property that for each A _i there are at most s A _j such that |A _i ∩ A _j | is odd (even). A tight upper bound is given in the case s < c(2 ^n,2/n). In the remaining cases we give an asymptotically tight upper bound. As an application we give a tight upper-bound for the cardinality of a family with even multi-intersection. Both results generalize a result of Berlekamp and Graver.     

Some extremal problems for strictly totally positive matrices

For a strictly totally positive M × N matrix A we show that the ratio $\text{∥A}\text{x}\text{∥}{}_{\mathrm{p}}\text{∥}\text{x}\text{∥}{}_{\mathrm{p}}$ has exactly R = min{ M, N} nonzero critical values for each fixed p? (1, ∞). Letting λ_i denote the ith critical value, and xⁱ an associated critical vector, we show that λ₁ > … > λ_R > 0 and xⁱ (unique up to multiplication by a constant) has exactly i ? 1 sign changes. These critical values are generalizations to l^p of the s-numbers of A and satisfy many of the same extremal properties enjoyed by the s-numbers, but with respect to the l^p norm.     

On the unconditional subsequence property

We show that a construction of Johnson, Maurey and Schechtman leads to the existence of a weakly null sequence (fi) in ?₂(∑Lpi), where pi↓1, so that for all ε>0 and 1<q?2, every subsequence of (fi) admits a block basis (1+ε)-equivalent to the Haar basis for Lq. We give an example of a reflexive Banach space having the unconditional subsequence property but not uniformly so.     

Some graft transformations and its applications on the distance spectral radius of a graph

Let D(G)=(d_i,j)_n×n denote the distance matrix of a connected graph G with order n, where d_ij is equal to the distance between v_i and v_j in G. The largest eigenvalue of D(G) is called the distance spectral radius of graph G, denoted by ?(G). In this paper, some graft transformations that decrease or increase ?(G) are given. With them, for the graphs with both order n and k pendant vertices, the extremal graphs with the minimum distance spectral radius are completely characterized; the extremal graph with the maximum distance spectral radius is shown to be a dumbbell graph (obtained by attaching some pendant edges to each pendant vertex of a path respectively) when 2≤k≤n−2; for k=1,2,3,n−1, the extremal graphs with the maximum distance spectral radius are completely characterized.     

On a problem of Katona on minimal completely separating systems with restrictions

Let S be a set of n elements, and k a fixed positive integer $\text{<}\text{1}\text{2}\text{n}$. Katona's problem is to determine the smallest integer m for which there exists a family $\text{A}$ = {A₁, …, A_m} of subsets of S with the following property: |_i| ? k (i = 1, …, m), and for any ordered pair x_i, x_i ∈ S (i ≠ j) there is A₁ ∈ $\text{A}$ such that x_i ∈ A₁, x_j ? A₁. It is given in this note that $\text{m = ?}\text{2n}\text{k}\text{?}\text{if}\text{1}\text{2}\text{k}{}^2\text{? 2}$.     

Spaces of matrices of equal rank

Let M_m,n(F) denote the space of all mXn matrices over the algebraically closed field F. A subspace of M_m,n(F), all of whose nonzero elements have rank k, is said to be essentially decomposable if there exist nonsingular mXn matrices U and V respectively such that for any element A, UAV has the form

$\text{UAV=}\text{A}{}_1\text{A}{}_2\text{A}{}_3\text{0}$

where A₁ is iX(k–i) for some i?k. Theorem: If $\text{K}$ is a space of rank k matrices, then either $\text{K}$ is essentially decomposable or dim $\text{K}$?k+1. An example shows that the above bound on non-essentially-decomposable spaces of rank k matrices is sharp whenever n?2k–1.     

Multigraphs with quasiweak odd cycles

In this paper we define a class of multigraphs with chromatic index equal to the maximum degree d. They are characterized by a property of their elementary odd cycles. It is shown that these graphs are panchromatic (i.e., they have a good k-coloring for any k). In the partially ordered set of color-feasible sequences of these graphs, all maximal sequences have at most d + 1 terms.