We classify small partial geometric designs (PGDs) by spectral characteristics of their concurrence matrices. It is well known that the matrix a PGD can have at most three distinct eigenvalues, all which are nonnegative integers. The contains useful information on incidence structure design. An ordinary 2- ( v , k λ ) $(v,k,\lambda )$ design has single $\lambda $ and its circulant, geometry two...

We prove that there is no circulant Hermitian complex Hadamard matrix of order n > 4.

by a quasi-permutation matrix we mean a square matrix over the complex field c with non-negative integral trace. thus every permutation matrix over c is a quasipermutation matrix. for a given finite group g, let p(g) denote the minimal degree of a faithful permutation representation of g (or of a faithful representation of g by permutation matrices), let q(g) denote the minimal degree of a fait...

"A 4-dimensional Riemannian manifold equipped with an additional tensor structure, whose fourth power is the identity, considered. This structure has a circulant matrix respect to some basis, i.e. circulant, and it acts as isometry metric. The product associated considered studied. Conditions for metric, which imply that belongs each of basic classes Staikova-Gribachev's classi cation, are obta...

We study continuous-time quantum walks on graphs which generalize the hypercube. The only known family of graphs whose quantum walk instantaneously mixes to uniform is the Hamming graphs with small arities. We show that quantum uniform mixing on the hypercube is robust under the addition of perfect matchings but not much else. Our specific results include: • The graph obtained by augmenting the...

We consider the number of spanning trees in circulant graphs of βn vertices with generators depending linearly on n. The matrix tree theorem gives a closed formula of βn factors; while we derive a formula of β−1 factors. The spanning tree entropy of these graphs is then compared to the one of fixed generated circulant graphs.

In present paper, we investigate 4 problems. Firstly, it is known that, a matrix is MDS if and only if all sub-matrices of this matrix of degree from 1 to n are full rank. In this paper, we propose a theorem that an orthogonal matrix is MDS if and only if all sub-matrices of this orthogonal matrix of degree from 1 to bn2 c are full rank. With this theorem, calculation of constructing orthogonal...

In the second part of this paper we study condition numbers with respect to componentwise perturbations in the input data for linear systems and for matrix inversion, and the distance to the nearest singular matrix. The structures under investigation are linear structures, namely symmetric, persymmetric, skewsymmetric, symmetric Toeplitz, general Toeplitz, circulant, Hankel, and persymmetric Ha...

We obtain closed formulas, in terms of Littlewood-Richardson coefficients, for the canonical basis elements of the Fock space representation of Uv(ŝle) which are labelled by partitions having ‘locally small’ e-quotients and arbitrary e-cores. We further show that upon evaluation at v = 1, this gives the corresponding decomposition numbers of the q-Schur algebra in characteristic l (where q is a...