We verify the skew weighing matrix conjecture for orders 2t.7, t ~ 3 a positive integer, by showing that orthogonal (1, k) exist for all t k = 0, 1, .... , 2.7 1 in order 2t.7 We discuss the construction of orthogonal designs using circulant matrices. In particular we construct designs in orders 20 and 28. The weighing matrix conjecture is verified for order 60. Disciplines Physical Sciences an...

We study the range S(A) := {xT Ay : x, y are orthonormal in Rn}, where A is an n×n complex skew symmetric matrix. It is a compact convex set. Power inequality s(A) ≤ s(A), k ∈ N, for the radius s(A) := maxξ∈S(A) |ξ| is proved. When n = 3, 4, 5, 6, relations between S(A) and the classical numerical range and the k-numerical range are given. Axiomatic characterization of S(A) is given. Sharp poin...

We establish several discrepancy and isoperimetric inequalities for directed graphs by considering the associated random walk. We show that various isoperimetric parameters, as measured by the stationary distribution of the random walks, including the Cheeger constant and discrepancy, are related to the singular values of the normalized probability matrix and the normalized Laplacian. Further, ...

Given a graph G, let G be an oriented graph of G with the orientation σ and skewadjacency matrix S(G). Then the spectrum of S(G) consisting of all the eigenvalues of S(G) is called the skew-spectrum of G, denoted by Sp(G). The skew energy of the oriented graph G, denoted by ES(G), is defined as the sum of the norms of all the eigenvalues of S(G). In this paper, we give orientations of the Krone...

We introduce a function on sequences, which we call the Pfaffian transform, using the Pfaffian of a skew-symmetric matrix. We establish several basic properties of the Pfaffian transform, and we use the transfer matrix method to show that the set of sequences with rational generating functions is closed under the Pfaffian transform. We conclude by computing the Pfaffian transform of a variety o...

Using the skew-Hopf pairing, we obtain ℛ-matrix for two-parameter quantum algebra Uv,t. We further construct a strict monoidal functor

Tutte associates a V by V skew-symmetric matrix T , having indeterminate entries, with a graph G=(V,E). This matrix, called the Tutte matrix, has rank exactly twice the size of a maximum cardinality matching of G. Thus, to find the size of a maximum matching it suffices to compute the rank of T . We consider the more general problem of computing the rank of T +K where K is a real V by V skew-sy...

• In this paper, we introduce a generalized skew logistic distribution that contains the usual skew logistic distribution as a special case. Several mathematical properties of the distribution are discussed like the cumulative distribution function and moments. Furthermore, estimation using the method of maximum likelihood and the Fisher information matrix are investigated. Two real data applic...

In this paper we give formulas for performing row reduction of a matrix of Ore polynomials in a fraction-free way. The reductions can be used for finding the rank and left nullspace of such matrices. When specialized to matrices of skew polynomials our reduction can be used for computing a weak Popov form of such matrices and for computing a GCRD and an LCLM of skew polynomials or matrices of s...

It is fairly well known that rotation in three dimensions can be expressed as a quadratic in a skew symmetric matrix via the Euler-Rodrigues formula. A generalized Euler-Rodrigues polynomial of degree 2n in a skew symmetric generating matrix is derived for the rotation matrix of tensors of order n. The Euler-Rodrigues formula for rigid body rotation is recovered by n = 1. A Cayley form of the n...