Test sets for generalized matrix functions

Let G = (V,E) be a biconnected graph and let C be a cycle in G. The subgraphs of G identified with the biconnected components of the contraction of C in G are called the bridges of C. Associated with the set of bridges of a cycle C is an auxilliary graphical structure G_C called a bridge graph or an overlap graph. Such auxilliary graphs have provided important insights in classical graph theory, algorithmic graph theory, and complexity theory. In this paper, we use techniques from algorithmic combinatorics and complexity theory to derive canonical forms for cycles in bridge graphs. These canonical forms clarify the relationship between cycles in bridge graphs, the structure of the underlying graph G, and lexicographic order relations on the vertices of attachment of bridges of a cycle. The first canonical form deals with the structure of induced bridge graph cycles of length greater than three. Cycles of length three in bridge graphs are studied from a different point of view, namely that of the characterization of minimal elements in certain related posets: ordered bridge three-cycles (10 minimal elements), bridge three-cycles (5 minimal elements), bridge deletion three-cycles (infinite number, 7 classes), minor order (K ₅ K _3,3), chordal bridge three-cycles (13 minimal elements), contraction poset (5 minimal elements), cycle-minor poset (infinite number, 14 classes). These results, each giving a different insight into the structure of bridge three-cycles, follow as corollaries from the characterization of the 10 minimal elements of the ordered bridge three-cycle poset. This characterization is constructive and may be regarded as an extension of the classical Kuratowski's Theorem which follows as a corollary. Algorithms are described for constructing these various minimal elements in time O(∣E∣) or O(∣V∣) depending on the case. The first canonical form gives a constructive proof of the result that a graph is nonplanar if and only if it has a cycle C whose bridge graph G_C (alternatively, skew bridge graph) has a three-cycle. An algorithm is described that constructs this three-cycle in time O(∣E∣). This is best possible.     

Positivity of matrices with generalized matrix functions

Using an elementary fact on matrices we show by a unified approach the positivity of a partitioned positive semidefinite matrix with each square block replaced by a compound matrix, an elementary symmetric function or a generalized matrix function. In addition, we present a refined version of the Thompson determinant compression theorem.     

Regular sets of matrices and applications

SupposeA ₁,...,A _s are (1, - 1) matrices of order m satisfying 1 $$A_i A_j = J, i,j \in \left\{ {1,...s} \right\}$$ 2 $$A_i^T A_j = A_j^T A_i = J, i \ne j, i,j \in \left\{ {1,...,s} \right\}$$ 3 $$\sum\limits_{i = 1}^s {(A_i A_i^T = A_i^T A_i ) = 2smI_m } $$ 4 $$JA_i = A_i J = aJ, i \in \left\{ {1,...,s} \right\}, a constant$$ Call A₁,…,A _s ,a regular s- set of matrices of order m if Eq. 1-3 are satisfied and a regular s-set of regular matrices if Eq. 4 is also satisfied, these matrices were first discovered by J. Seberry and A.L. Whiteman in “New Hadamard matrices and conference matrices obtained via Mathon’s construction”, Graphs and Combinatorics, 4(1988), 355-377. In this paper, we prove that

1. if there exist a regular s-set of order m and a regulart-set of order n there exists a regulars-set of ordermn whent =sm
2. if there exist a regular s-set of order m and a regulart-set of order n there exists a regulars-set of ordermn when 2t = sm (m is odd)
3. if there exist a regulars-set of order m and a regulart-set of ordern there exists a regular 2s-set of ordermn whent = 2sm As applications, we prove that if there exist a regulars-set of order m there exists
4. an Hadamard matrices of order4hm whenever there exists an Hadamard matrix of order4h ands =2h
5. Williamson type matrices of ordernm whenever there exists Williamson type matrices of ordern and s = 2n
6. anOD(4mp;ms₁,…,ms_u whenever anOD (4p;s₁,…,s_u)exists and s = 2p
7. a complex Hadamard matrix of order 2cm whenever there exists a complex Hadamard matrix of order 2c ands = 2c

This paper extends and improves results of Seberry and Whiteman giving new classes of Hadamard matrices, Williamson type matrices, orthogonal designs and complex Hadamard matrices.     

Regular sets in generalized topological spaces

We define and study regular and locally closed sets in generalized topological spaces.     

Inequalities and identities for generalized matrix functions

Let χ be a character on the symmetric group S_n, and let A = (a_ij) be an n-by-n matrix. The function d_χ(A) = Σ_σ?Snχ(σ)Πⁿ_{t = 1}a_tσ(t) is called a generalized matrix function. If χ is an irreducible character, then d_χ is called an immanent. For example, if χ is the alternating character, then d_χ is the determinant, and if χ ≡ 1, then d_χ is called the permanent (denoted per). Suppose that A is positive semidefinite Hermitian. We prove that the inequality $\text{(1/χ(}\text{id}\text{))d}{}_{\mathrm{\chi }}\text{(A) ?}\text{per}\text{A}$ holds for a variety of characters χ including the irreducible ones corresponding to the partitions (n ? 1,1) and (n ? 2,1,1) of n. The main technique used to prove these inequalities is to express the immanents as sums of products of principal subpermanents. These expressions for the immanents come from analogous expressions for Schur polynomials by means of a correspondence of D.E. Littlewood.     

Box-total dual integrality,box-integrality,and equimodular matrices

Mathematical Programming - Box-totally dual integral (box-TDI) polyhedra are polyhedra described by systems which yield strong min-max relations. We characterize them in several ways, involving the...     

A quadratic-phase integral operator for sets of generalized integrable functions

In this paper, we aim to discuss the classical theory of the quadratic-phase integral operator on sets of integrable Boehmians. We provide delta sequences and derive convolution theorems by using certain convolution products of weight functions of exponential type. Meanwhile, we make a free use of the delta sequences and the convolution theorem to derive the prerequisite axioms, which essentially establish the Boehmian spaces of the generalized quadratic-phase integral operator. Further, we nominate two continuous embeddings between the integrable set of functions and the integrable set of Boehmians. Furthermore, we introduce the definition and the properties of the generalized quadratic-phase integral operator and obtain an inversion formula in the class of Boehmians.     

Strongly internal sets and generalized smooth functions

Based on a refinement of the notion of internal sets in Colombeau's theory, so-called strongly internal sets, we introduce the space of generalized smooth functions, a maximal extension of Colombeau generalized functions. Generalized smooth functions as morphisms between sets of generalized points form a sub-category of the category of topological spaces. In particular, they can be composed unrestrictedly.     

Multiplicative properties of generalized matrix functions

We consider scalar-valued matrix functions for n×n matrices A=(a_ij) defined by Where G is a subgroup of S_n the group of permutations on n letters, and χ is a linear character of G. Two such functions are the permanent and the determinant. A function (1) is multiplicative on a semigroup S of n×n matrices if d(AB)=d(A)d(B) AB∈S.

With mild restrictions on the underlying scalar ring we show that every element of a semigroup containing the diagonal matrices on which (1) is multiplicative can have at most one nonzero diagonal(i.e., diagonal with all nonzero entries)and conversely, provided that χ is the principal character(χ≡1).     

On generalized cluster sets of functions and multifunctions

This paper introduces and studies generalized cluster sets (g-cluster sets) of functions and multifunctions on GTS, which unifies the existing notions of cluster sets, θ-cluster sets, δ-cluster sets, S-cluster sets, s-cluster sets, p-cluster sets and many more. Several properties of the functions and multifunctions as well as their range and domain spaces are observed via degeneracies of their g-cluster sets. Characterizations of g-cluster sets through filterbases and grills on a typical class of GTS’s are also obtained. Moreover, μ-compactness of a GTS is characterized through g-cluster sets of multifunctions.     

A conjecture involving generalized matrix functions

This paper resolves the following conjecture of R. Merris: Let d^G_λ be the generalized matrix function determined by a subgroup G of the symmetric group S_m and an irreducible complex character λ of G. If A, B, and A?B are m-square positive semidefinite hermitian m-square matrices and d^G_λ(A)=d^G_λ(B)≠0, then A=B.     

Generalized continuous functions defined by generalized open sets on generalized topological spaces

We introduce generalized continuous functions defined by generalized open (= g-α-open, g-semi-open, g-preopen, g-β-open) sets in generalized topological spaces which are generalized (g, g′)-continuous functions. We investigate characterizations and relationships among such functions.     

The general coupled matrix equations over generalized bisymmetric matrices

In the present paper, by extending the idea of conjugate gradient (CG) method, we construct an iterative method to solve the general coupled matrix equations