A further look into combinatorial orthogonality

Strongly quadrangular matrices have been introduced in the study of the combinatorial properties of unitary matrices. It is known that if a (0, 1)-matrix supports a unitary then it is strongly quadrangular. However, the converse is not necessarily true. In this paper, we fully classify strongly quadrangular matrices up to degree 5. We prove that the smallest strongly quadrangular matrices which do not support unitaries have exactly degree 5. Further, we isolate two submatrices not allowing a (0, 1)-matrix to support unitaries.Comment: 11 pages, some typos are corrected. To appear in The Electronic journal of Linear Algebr

Structure theorem of square complex orthogonal design

Square COD (complex orthogonal design) with size $[n, n, k]$ is an $n \times n$ matrix $\mathcal{O}_z$, where each entry is a complex linear combination of $z_i$ and their conjugations $z_i^*$, $i=1,\ldots, k$, such that $\mathcal{O}_z^H \mathcal{O}_z = (|z_1|^2 + \ldots + |z_k|^2)I_n$. Closely following the work of Hottinen and Tirkkonen, which proved an upper bound of $k/n$ by making a crucial observation between square COD and group representation, we prove the structure theorem of square COD

Transfer Matrix Analysis of the Unidirectional Grating-Assisted Codirectional Coupler

The unidirectional grating-assisted codirectional coupler (U-GACC) has recently been proposed. This unique structure permits irreversible coupling between orthogonal waveguide eigenmodes by means of simultaneous modulation of both the real and imaginary parts of the refractive index in the coupling region. Analysis of the U-GACC has until now relied on coupled mode theory, which can be restrictive in its application as a design tool. We analyze the U-GACC by the transfer matrix method, which demonstrates in a simple fashion why the device operates in a unidirectional manner. In addition, we show that for all practical designs, there is a limit to the minimum cross talk between outputs, a phenomenon that has not been previously identified

The history of degenerate (bipartite) extremal graph problems

This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version of our survey presented in Erdos 100. In this version 2 only a citation was complete

A quadratically convergent parallel Jacobi-process for diagonal dominant matrices with nondistinct eigenvalues

Matrices;Eigenvalues;mathematics