Index Coding: Rank-Invariant Extensions

An index coding (IC) problem consisting of a server and multiple receivers with different side-information and demand sets can be equivalently represented using a fitting matrix. A scalar linear index code to a given IC problem is a matrix representing the transmitted linear combinations of the message symbols. The length of an index code is then the number of transmissions (or equivalently, the number of rows in the index code). An IC problem ${\cal I}_{ext}$ is called an extension of another IC problem ${\cal I}$ if the fitting matrix of ${\cal I}$ is a submatrix of the fitting matrix of ${\cal I}_{ext}$. We first present a straightforward $m$\textit{-order} extension ${\cal I}_{ext}$ of an IC problem ${\cal I}$ for which an index code is obtained by concatenating $m$ copies of an index code of ${\cal I}$. The length of the codes is the same for both ${\cal I}$ and ${\cal I}_{ext}$, and if the index code for ${\cal I}$ has optimal length then so does the extended code for ${\cal I}_{ext}$. More generally, an extended IC problem of ${\cal I}$ having the same optimal length as ${\cal I}$ is said to be a \textit{rank-invariant} extension of ${\cal I}$. We then focus on $2$-order rank-invariant extensions of ${\cal I}$, and present constructions of such extensions based on involutory permutation matrices

Distributed LQR-based Suboptimal Control for Coupled Linear Systems

A well-established distributed LQR method for decoupled systems is extended to the dynamically coupled case where the open-loop systems are dynamically dependent. First, a fully centralized controller is designed which is subsequently substituted by a distributed state-feedback gain with sparse structure. The control scheme is obtained by optimizing an LQR performance index with a tuning parameter utilized to emphasize/de-emphasize relative state difference between interconnected systems. Overall stability is guaranteed via a simple test applied to a convex combination of Hurwitz matrices, the validity of which leads to stable global operation for a class of interconnection schemes. It is also shown that the suboptimality of the method can be assessed by measuring a certain distance between two positive definite matrices which can be obtained by solving two Lyapunov equations

Spectra of quadratic vector fields on C2\mathbb{C}^2: The missing relation

Consider a quadratic vector field on $\mathbb{C}^2$ having an invariant line at infinity and isolated singularities only. We define the extended spectra of singularities to be the collection of the spectra of the linearization matrices of each of the singular points over the affine part, together with all the characteristic numbers (i.e. Camacho-Sad indices) at infinity. This collection consists of 11 complex numbers, and is invariant under affine equivalence of vector fields. In this paper we describe all polynomial relations among these numbers. There are 5 independent polynomial relations; four of them follow from the Euler-Jacobi, the Baum-Bott and the Camacho-Sad index theorems, and are well known. The fifth relation was, until now, completely unknown. We provide an explicit formula for the missing 5th relation, discuss it's meaning and prove that it cannot be formulated as an index theorem.Comment: 16 pages, 1 appendix. Note that the title has changed from the previous versio

The connection matrix theory for semiflows on (not necessarily locally compact) metric spaces

AbstractThe index theory of Rybakowski for isolated invariant sets and attractor-repeller pairs in the setting of a semiflow on a not necessarily locally compact metric space is extended to include a connection matrix theory for Morse decompositions. Partially ordered Morse decompositions and attractor semifiltrations of invariant sets are defined and shown to be equivalent. The definition and proof of existence of index filtrations for an ordered Morse decomposition is provided. Via the index filtration, the homology index braid and the connection matrices of the Morse decomposition are defined

The pseudo core inverse of a companion matrix

The notion of core inverse was introduced by Baksalary and Trenkler for a complex matrix of index 1. Recently, the notion of pseudo core inverse extended the notion of core inverse to an element of an arbitrary index in *-rings; meanwhile, it characterized the core-EP inverse introduced by Manjunatha Prasad and Mohana for complex matrices, in terms of three equations. Many works have been done on classical generalized inverses of companion matrices and Toeplitz matrices. In this paper, we discuss existence criteria and formulae of the pseudo core inverse of a companion matrix over a *-ring. A {1, 3}-inverse of a Toeplitz matrix plays an important role in that process.- This research is supported by the National Natural Science Foundation of China (No.11771076, No.11471186), the Scientific Innovation Research of College Graduates in Jiangsu Province (No.KYZZ16_0112)