Required nonzero patterns for nonsingular sign regular matrices

An n×n real matrix is called sign regular if, for each k(1?k?n), all its minors of order k have the same nonstrict sign. The zero entries which can appear in a nonsingular sign regular matrix depend on its signature because the signature can imply that certain entries are necessarily nonzero. The patterns for the required nonzero entries of nonsingular sign regular matrices are analyzed.     

Potentially nilpotent and spectrally arbitrary even cycle sign patterns

An n × n sign pattern S_n is potentially nilpotent if there is a real matrix having sign pattern S_n and characteristic polynomial xⁿ. A new family of sign patterns C_n with a cycle of every even length is introduced and shown to be potentially nilpotent by explicitly determining the entries of a nilpotent matrix with sign pattern C_n. These nilpotent matrices are used together with a Jacobian argument to show that C_n is spectrally arbitrary, i.e., there is a real matrix having sign pattern C_n and characteristic polynomial for any real μ_i. Some results and a conjecture on minimality of these spectrally arbitrary sign patterns are given.     

Symmetric sign patterns with maximal inertias

The inertia of an n by n symmetric sign pattern is called maximal when it is not a proper subset of the inertia of another symmetric sign pattern of order n. In this note we classify all the maximal inertias for symmetric sign patterns of order n, and identify symmetric sign patterns with maximal inertias by using a rank-one perturbation.     

Dn,r is not potentially nilpotent for n ⩾ 4r − 2

An n × n sign pattern A is said to be potentially nilpotent if there exists a nilpotent real matrix B with the same sign pattern as A. Let D_n,r be an n × n sign pattern with 2 ≤ r ≤ n such that the superdiagonal and the (n, n) entries are positive, the (i, 1) (i = 1,..., r) and (i, i ? r + 1) (i = r + 1,..., n) entries are negative, and zeros elsewhere. We prove that for r ≥ 3 and n ≥ 4r ? 2, the sign pattern D_n,r is not potentially nilpotent, and so not spectrally arbitrary.     

Self-complementary totally symmetric plane partitions

A totally symmetric plane partition of size n is a plane partition whose three-dimensional Ferrers graph is contained in the box X_n = [1, n] × [1, n] × [1, n] and which is mapped to itself under all permutations of the coordinate axes. The complement of the Ferrers graph of such a plane partition (that is, the set of lattice points in the box X_n that do not belong to the Ferrers graph) is again totally symmetric when viewed from the vantage point of the vertex (n + 1, n + 1, n + 1). A totally symmetric plane partition is self-complementary if it is congruent (in the geometrical sense) to its complement. This cannot occur unless n = 2m is even. In this paper we give several conjectures and a few theorems concerning self-complementary totally symmetric plane partitions. In particular we describe evidence which indicates a close relationship with m by m alternating sign matrices. In an earlier paper we described the close connection between m by m alternating sign matrices and descending plane partitions with no parts exceeding m. We are thus left with three classes of objects which are all apparently interrelated. There remain many unsolved problems, the simplest of which is to prove that any two of the objects have the same cardinality.     

Spectrally arbitrary ray patterns

An n×n ray pattern A is said to be spectrally arbitrary if for every monic nth degree polynomial f(x) with coefficients from C, there is a matrix in the pattern class of A such that its characteristic polynomial is f(x). In this article the authors extend the nilpotent-Jacobi method for sign patterns to ray patterns, establishing a means to show that an irreducible ray pattern and all its superpatterns are spectrally arbitrary. They use this method to establish that a particular family of n×n irreducible ray patterns with exactly 3n nonzeros is spectrally arbitrary. They then show that every n×n irreducible, spectrally arbitrary ray pattern has at least 3n-1 nonzeros.     

On the primality of 2h·3n+1

We consider the primality test of Williams and Zarnke for rational integers of the form 2h·3ⁿ+1. We give an algebraic proof of the test, and we resolve a sign ambiguity. We also show that the conditions of the original test can be relaxed, especially if h is divisible by a power of 2.     

Multidimensional analog of the two-dimensional perron effect of sign change of characteristic exponents for infinitely differentiable differential systems

We obtain a general n-dimensional analog of the two-dimensional (partial) Perron effect of sign change of all arbitrarily prescribed negative characteristic exponents of an n-dimensional differential system of the linear approximation with infinitely differentiable bounded coefficients to the positive sign for the characteristic exponents of all nontrivial solutions of a nonlinear n-dimensional differential system with infinitely differentiable perturbations of arbitrary order m > 1 of smallness in a neighborhood of the origin and growth outside it. These positive exponents take n values distributed over n arbitrarily prescribed disjoint intervals and are realized on solutions issuing from nested subspaces R ¹ ? R ² ? ... ? R ⁿ .     

± sign pattern matrices that allow orthogonality

A sign pattern A is a ± sign pattern if A has no zero entries. A allows orthogonality if there exists a real orthogonal matrix B whose sign pattern equals A. Some sufficient conditions are given for a sign pattern matrix to allow orthogonality, and a complete characterization is given for ± sign patterns with n − 1 ⩽ N₋(A) ⩽ n + 1 to allow orthogonality.     

Linear relations of refined enumerations of alternating sign matrices

In recent papers we have studied refined enumerations of alternating sign matrices with respect to a fixed set of top and bottom rows. The present paper is a first step towards extending these considerations to alternating sign matrices where in addition some left and right columns are fixed. The main result is a simple linear relation between the number of n×n alternating sign matrices where the top row as well as the left and the right column is fixed and the number of n×n alternating sign matrices where the two top rows and the bottom row are fixed. This may be seen as a first indication for the fact that the refined enumerations of alternating sign matrices with respect to a fixed set of top and bottom rows as well as left and right columns can possibly be reduced to the refined enumerations where only some top and bottom rows are fixed. For the latter numbers we provide a system of linear equations that conjecturally determines them uniquely.     

Sign regular matrices and Neville elimination

A pivoting strategy of O(n) operations for the Neville elimination of n × n nonsingular sign regular matrices is introduced. Among other nice properties, it is proved that it preserves sign regularity. It is also shown its relationship with scaled partial pivoting strategies for Neville elimination.     

Factorization of alternating sums of Virasoro characters

G. Andrews proved that if n is a prime number then the coefficients ak and ak+n of the product _∞(q,q)/_∞(qn,qn)=k_∑akqk have the same sign, see [G. Andrews, On a conjecture of Peter Borwein, J. Symbolic Comput. 20 (1995) 487-501]. We generalize this result in several directions. Our results are based on the observation that many products can be written as alternating sums of characters of Virasoro modules.     

Construction and nonnegativity of sign idempotent sign pattern matrices

In this paper, we modify Eschenbach’s algorithm for constructing sign idempotent sign pattern matrices so that it correctly constructs all of them. We find distinct classes of sign idempotent sign pattern matrices that are signature similar to an entrywise nonnegative sign pattern matrix. Additionally, if for a sign idempotent sign pattern matrix A there exists a signature matrix S such that SAS is nonnegative, we prove such S is unique up to multiplication by -1 if the signed digraph D(A) is not disconnected.     

Defining the Determinant-like Function for m by n Matrices Using the Exterior Algebra

In this paper, we present a function called the Determinant-Like Function that generalises the determinant function to m by n matrices using the Clifford algebra C?(V, 0). By definition, this generalisation satisfies the property that the exterior product of its column vectors has a magnitude that equals its determinant-like function if it has more rows than columns. For matrices which have more columns than rows, we make use of another property that the exterior product of its rows has a magnitude that equals the absolute value of its determinant-like function if it has more columns than rows. Defining the sign of this determinant-like function remains an open question.     

A Sharp Inequality for a Trigonometric Sum

We prove that the inequality $$-\frac{1}{2}\leq {\sum\limits_{k=1}^{n}} \left( \frac{{\rm cos}(2kx)}{2k - 1}+\frac{{\rm sin}((2k - 1)x)}{2k} \right)$$ holds for all natural numbers n and real numbers x with ${x \in [0, \pi]}$ . The sign of equality is valid if and only if n =  1 and x =  π /2.     

Equal signs additive sequences in Banach spaces

A sequence (e_n) spanning a Banach space E is called ESA or equal signs additive if the norm of a linear combination of the e_i's does not change when adjacent coefficients of equal sign are combined. Call the sequence (e_n) regular if neither E nor its dual contain an isomorphic copy of c₀. It is shown that a regular ESA sequence is a boundedly complete and 1-shrinking basis for its span which is thus quasi-reflexive. It is further possible to replace a regular ESA norm by an equivalent ESA norm rendering E isometrically isomorphic to its second dual. A sequence (e_n) is called IS or invariant under spreading if the norm of a linear combination of the e_i's does not change when the mutual distances of the terms in the sequence (but not their relative positions) change. We give a simple construction of an unconditional norm for an IS sequence, hence, in particular, for an ESA sequence. Also, an inverse construction is obtained: We prove that each unconditional IS basis gives rise to an ESA basis by means of an inversion formula; to nonequivalent IS unconditional bases correspond nonequivalent ESA bases. It follows that nonisomorphic ESA bases are plentiful.     

Congruence of Hermitian matrices by Hermitian matrices

Two Hermitian matrices A,B∈M_n(C) are said to be Hermitian-congruent if there exists a nonsingular Hermitian matrix C∈M_n(C) such that B=CAC. In this paper, we give necessary and sufficient conditions for two nonsingular simultaneously unitarily diagonalizable Hermitian matrices A and B to be Hermitian-congruent. Moreover, when A and B are Hermitian-congruent, we describe the possible inertias of the Hermitian matrices C that carry the congruence. We also give necessary and sufficient conditions for any 2-by-2 nonsingular Hermitian matrices to be Hermitian-congruent. In both of the studied cases, we show that if A and B are real and Hermitian-congruent, then they are congruent by a real symmetric matrix. Finally we note that if A and B are 2-by-2 nonsingular real symmetric matrices having the same sign pattern, then there is always a real symmetric matrix C satisfying B=CAC. Moreover, if both matrices are positive, then C can be picked with arbitrary inertia.     

The minimum upper bound on the first ambiguous power of an irreducible, nonpowerful ray or sign pattern

Let A be an n×n irreducible ray or sign pattern matrix. If A is a sign pattern, it is shown that either A is powerful or else A^k has an ambiguous entry for some , and further, sign patterns based on the Wielandt graph show that this bound is the best possible. If A is a ray pattern, partial results for the same bound are given.     

The Number of Negative Entries in a Sign Pattern Allowing Orthogonality

A ± sign pattern is a matrix whose entries are in the set {+,–}. An n×n ± sign pattern A allows orthogonality if there exists a real orthogonal matrix B in the qualitative class of A. In this paper, we prove that for n3 there is an n×n ± sign pattern A allowing orthogonality with exactly k negative entries if and only if n–1kn²–n+1.Research supported by Shanxi Natural Science Foundation 20011006, 20041010Final version received: October 22, 2003     

The number of equivalence classes of symmetric sign patterns

This paper shows that the number of sign patterns of totally non-zero symmetric n-by-n matrices, up to conjugation by permutation and signature matrices and negation, is equal to the number of unlabelled graphs on n vertices.