The article is devoted to different aspects of the question: ”What can be done with a matrix by a low rank perturbation?” It is proved that one can change a geometrically simple spectrum drastically by a rank 1 perturbation, but the situation is quite different if one restricts oneself to normal matrices. Also the Jordan normal form of a perturbed matrix is discussed. It is proved that with res...

The article is devoted to different aspects of the question: ”What can be done with a complex-valued matrix by a low rank perturbation?” From the works of Thompson [15] we know how the Jordan normal form can be changed by a rank k perturbation, see Theorem 2. Particulary, it follows that one can do everything with a geometrically simple spectrum by a rank 1 perturbation, see Corollary 1. But th...

The rank-width is a graph parameter related in terms of fixed functions to cliquewidth but more tractable. Clique-width has nice algorithmic properties, but no good “minor” relation is known analogous to graph minor embedding for tree-width. In this paper, we discuss the vertex-minor relation of graphs and its connection with rank-width. We prove a relationship between vertex-minors of bipartit...

‎‎‎This paper presents a remarkable formula for spectral distance of a given block normal matrix $G_{D_0} = begin{pmatrix}‎ ‎A & B \‎ ‎C & D_0‎ ‎end{pmatrix} $ to set of block normal matrix $G_{D}$ (as same as $G_{D_0}$ except block $D$ which is replaced by block $D_0$)‎, ‎in which $A in mathbb{C}^{ntimes n}$ is invertible‎, ‎$ B in mathbb{C}^{ntimes m}‎, ‎C in mathbb{C}^{mti...

In this note we illuminate and apply the equivalence of quadratic APN functions to certain subspaces of alternating bilinear forms. These subspaces can be characterized by the rank-distance of the dual subspace, or equivalently , as the subspaces of 2 F m 2 avoiding the variety of elements of rank 2. Or, in the geometric language, as subspaces external (or skew) to the line Grassmannian.

Rank-1 lattices are available in any dimension for any number of lattice points and because their generation is so efficient, they often are used in quasi-Monte Carlo methods. Applying the Fourier transform to functions sampled on rank-1 lattice points turns out to be simple and efficient if the number of lattice points is a power of two. Considering the Voronoi diagram of a rank-1 lattice as a...

The multiple root loci among univariate polynomials of degree n are indexed by partitions of n. We study these loci and their conormal varieties. The projectively dual varieties are joins of such loci where the partitions are hooks. Our emphasis lies on equations and parametrizations that are useful for Euclidean distance optimization. We compute the ED degrees for hooks. Among the dual hypersu...

Several problems in optimization and control involve a matrix of decision variables to be subject to a rank constraint. Although semidefinite programming is used as a generalpurpose tool to provide strong relaxations of such problems, finding feasible solutions mostly relies on algorithmic techniques specific to the problem at hand. We present models for expressing rank constraints using mathem...

In this paper, we consider a pseudo-random generator based on the difficulty of the syndrome decoding problem for rank metric codes. We also study the resistance of this problem against a quantum computer. Our results show that with rank metric it is possible to obtain fast PRNG with small public data, without considering additional structure for public matrices like quasi-cyclicity for Hamming...

Channel capacity in a waveguide with mode coupling is limited by a keyhole effect caused by the propagation mechanism. The coupling among the propagating modes reduces the channel rank, and at sufficient distances between the transmitter and receiver, the channel rank converges to 1. Measurements of the received power and channel capacity along a hallway and in rooms beside it, show a decrease ...