arXiv subjects

Harish K. Pillai

Publications and source records attributed to Harish K. Pillai.

12 recordsLinked to original sources

arXiv2023

Unified framework for Fiedler-like strong linearizations of polynomial and rational matrices

Linearization is a widely used method for solving polynomial eigenvalue problems (PEPs) and rational eigenvalue problem (REPs) in which the PEP/REP is transformed to a generalized eigenproblem and then solve this generalized eigenproblem with algorithms available in the literature. Fiedler-like pencils (Fiedler pencils (FPs), generalized Fiedler pencils (GFPs), Fiedler pencils with repetition (FPRs) and generalized Fiedler pencils with repetition (GFPRs)) are well known classes of strong linearizations. GFPs are an intriguing family of linearizations, and GF pencils are the fundamental building blocks of FPRs and GFPRs. As a result, FPRs and GFPRs have distinctive features and they provide structure-preserving linearizations for structured matrix polynomials. But GFPRs do not use the full potential of GF pencils. Indeed, not all the GFPs are FPRs or GFPRs, and vice versa. The main aim of this paper is two-fold. First, to build a unified framework for all the Fiedler-like pencils FPs, GFPs, FPRs and GFPRs. To that end, we construct a new family of strong linearizations (named as EGFPs) of a matrix polynomial $P(\lam)$ that subsumes all the Fiedler-like linearizations. A salient feature of the EGFPs family is that it allows the construction of structured preserving banded linearizations with low bandwidth for structured (symmetric, Hermitian, palindromic) matrix polynomial. Low bandwidth structured linearizations may be useful for numerical computations. Second, to utilize EGFPs directly to form a family of Rosenbrock strong linearizations of an $n \times n$ rational matrix $G(\lam)$ associated with a realization. We describe the formulas for the construction of low bandwidth linearizations for $P(\lam)$ and $G(\lam)$. We show that the eigenvectors, minimal bases/indices of $P(\lam)$ and $G(\lam)$ can be easily recovered from those of the linearizations of $P(\lam)$ and $G(\lam)$.

math.NA↗

arXiv2022

Permutation Polynomials of $\mathbb{F}_{q^2}$ : A Linear Algebraic Approach

In this paper, we present a linear algebraic approach to the study of permutation polynomials that arise from linear maps over a finite field $\mathbb{F}_{q^2}$. We study a particular class of permutation polynomials over $\mathbb{F}_{q^2}$, in the context of rank deficient and full rank linear maps over $\mathbb{F}_{q^2}$. We derive necessary and sufficient conditions under which the given class of polynomials are permutation polynomials. We further show that the number of such permutation polynomials can be easily enumerated. Only a subset of these permutation polynomials have been reported in literature earlier. It turns out that this class of permutation polynomials have compositional inverses of the same kind and we provide algorithms to evaluate the compositional inverses of most of these permutation polynomials.

math.CO↗

arXiv2019

A Classification of Permutation Polynomials through Some Linear Maps

In this paper, we propose linear maps over the space of all polynomials $f(x)$ in $\mathbb{F}_q[x]$ that map $0$ to itself, through their evaluation map. Properties of these linear maps throw up interesting connections with permutation polynomials. We study certain properties of these linear maps. We propose to classify permutation polynomials by identifying the generalized eigenspaces of these maps, where the permutation polynomials reside. As it turns out, several classes of permutation polynomials studied in literature neatly fall into classes defined using these linear maps. We characterize the shapes of permutation polynomials that appear in the various generalized eigenspaces of these linear maps. For the case of $\mathbb{F}_p$, these generalized eigenspaces provide a degree-wise distribution of polynomials (and therefore permutation polynomials) over $\mathbb{F}_p$. We show that for $\mathbb{F}_q$, it is sufficient to consider only a few of these linear maps. The intersection of the generalized eigenspaces of these linear maps contain (permutation) polynomials of certain shapes. In this context, we study a class of permutation polynomials over $\mathbb{F}_{p^2}$. We show that the permutation polynomials in this class are closed under compositional inverses. We also do some enumeration of permutation polynomials of certain shapes.

math.NT↗

arXiv2019

Yet another approach to the Algebraic Riccati Inequality

We give a rank characterization of the solution set of algebraic Riccati inequality (ARI) for both controllable and uncontrollable systems. Assuming an existence of a solution of the corresponding algebraic Riccati equation (ARE), we characterize the boundedness/unboundedness properties of solutions of ARI for controllable/uncontrollable systems without any assumption on sign controllability. As a consequence of our observations, we obtain Willems' result $K_{min}\leq K\leq K_{max}$ for an ARI in the case of controllable systems and explore some structure on the extremal solutions. We also consider the curious case of uncontrollable purely imaginary eigenvalues and the behavior of the solution set of ARI. In particular, we show that a system is controllable if and only if the set of solutions of an ARI is bounded. In addition, we study the effect of the position of eigenvalues of the system matrix in the complex plane on the behavior of the solution set of ARIs. Furthermore, we obtain a rank parametrization for solutions of ARI for controllable systems.

math.OC↗

arXiv2017

A Complete Characterization of Determinantal Quadratic Polynomials

The problem of expressing a multivariate polynomial as the determinant of a monic (definite) symmetric or Hermitian linear matrix polynomial (LMP) has drawn a huge amount of attention due to its connection with optimization problems. In this paper we provide a necessary and sufficient condition for the existence of \textit{monic Hermitian determinantal representation} as well as \textit{monic symmetric determinantal representation} of size $2$ for a given quadratic polynomial. Further we propose a method to construct such a monic determinantal representtaion (MDR) of size $2$ if it exists. It is known that a quadratic polynomial $f(\x)=\x^{T}A\x+b^{T}\x+1$ has a symmetric MDR of size $n+1$ if $A$ is \textit{negative semidefinite}. We prove that if a quadratic polynomial $f(\x)$ with $A$ which is not negative semidefinite has an MDR of size greater than $2$, then it has an MDR of size $2$ too. We also characterize quadratic polynomials which exhibit diagonal MDRs.

math.OC↗

arXiv2015

Novel representation of discrete n-D autonomous systems

In this paper we address the problem of representing solutions of a system of scalar linear partial difference equations akin to state space equations of 1-D systems theory. We first obtain a representation formula for a special class of autonomous systems. Then we show every autonomous system can be converted into the special ones by a coordinate transformation on n-d integer grid. Using this conversion we provide representation formula for general autonomous systems. The representation formula we present can be viewed as multidimensional flow operators acting on initial conditions. These initial conditions are required to satisfy certain compatibility conditions. We give a full description of the set of allowable initial conditions. In our search for a general representation formula, one algebraic result plays a very crucial role. In this result we show that every quotient ring of the n-variable Laurent polynomial ring can be made a finitely generated faithful module over another Laurent polynomial ring of smaller dimension by doing a suitable change of coordinates. We call this result a discrete version of Noether's Normalization Lemma.

math.AP↗

arXiv2014

Necessary condition on Lyapunov functions corresponding to the globally asymptotically stable equilibrium point

It is well known that, the existence of a Lyapunov function is a sufficient condition for stability, asymptotic stability, or global asymptotic stability of an equilibrium point of an autonomous system $\dot{\mathbf{x}} = f(\mathbf{x})$. In variants of Lyapunov theorems, the condition for a Lyapunov candidate $V$ (continuously differentiable and positive definite function) to be a Lyapunov function is that its time derivative along system trajectories must be negative semi-definite or negative definite. Numerically checking positive definiteness of $V$ is very difficult; checking negative definiteness of $\dot{V}(\cdot)=\langle \nabla V(\cdot), f(\cdot) \rangle$ is even more difficult, because it involves dynamics of the system. We give a necessary condition independent of the system dynamics, for every Lyapunov function corresponding to the globally asymptotically stable equilibrium point of $\dot{\mathbf{x}} = f(\mathbf{x})$. This necessary condition is numerically easier to check than checking positive definiteness of a function. Therefore, it can be used as a first level test to check whether a given continuously differentiable function is a Lyapunov function candidate or not. We also propose a method, which we call a generalized steepest descent method, to check this condition numerically. Generalized steepest descent method can be used for ruling out Lyapunov candidates corresponding to the globally asymptotically stable equilibrium point of $\dot{\mathbf{x}} = f(\mathbf{x})$. It can also be used as a heuristic to check the local positive definiteness of a function, which is a necessary condition for a Lyapunov function corresponding to a stable and/or asymptotically stable equilibrium point of an autonomous system.

math.DS↗

arXiv2012

On Multisequences and their extensions

In this paper we deal with the dimension of multisequences and related properties. For a given multisequence W and an m tuple of positive integers R, we define the R extension of W. Further we count the number of multisequences W whose R extensions have maximum dimension and give an algorithm to derive such multisequences. We then go on to use this theory to count the number of Linear Feedback Shift Register(LFSR) configurations with multi input multi output delay blocks for any given primitive characteristic polynomial and also to design such LFSRs. Further, we use the result on multisequences to count the number of Hankel matrices of any given dimension.

cs.CR↗

arXiv2011

Higher weights of Grassmann codes in terms of properties of Schubert unions

We describe the higher weights of the Grassmann codes $G(2,m)$ over finite fields ${\mathbb F}_q$ in terms of properties of Schubert unions, and in each case we determine the weight as the minimum of two explicit polynomial expressions in $q$.

math.AG↗

arXiv2011

Analysis of Stable Periodic Orbits in 1-D Linear Piecewise Smooth Maps

By varying a parameter of a one-dimensional piecewise smooth map, stable periodic orbits are observed. In this paper, complete analytic characterization of these stable periodic orbits is obtained. An interesting relationship between the cardinality of orbits and their period is established. It is proved that for any $n$, there exist $ϕ(n)$ distinct admissible patterns of cardinality $n$. An algorithm to obtain these distinct admissible patterns is outlined. Additionally, a novel algorithm to find the range of parameter for which the orbit exists is proposed.

math.DS↗

arXiv2008

Subclose Families, Threshold Graphs, and the Weight Hierarchy of Grassmann and Schubert Codes

We discuss the problem of determining the complete weight hierarchy of linear error correcting codes associated to Grassmann varieties and, more generally, to Schubert varieties in Grassmannians. The problem is partially solved in the case of Grassmann codes, and one of the solutions uses the combinatorial notion of a closed family. We propose a generalization of this to what is called a subclose family. A number of properties of subclose families are proved, and its connection with the notion of threshold graphs and graphs with maximum sum of squares of vertex degrees is outlined.

math.CO↗

arXiv2007

Decomposable Subspaces, Linear Sections of Grassmann Varieties, and Higher Weights of Grassmann Codes

Given a homogeneous component of an exterior algebra, we characterize those subspaces in which every nonzero element is decomposable. In geometric terms, this corresponds to characterizing the projective linear subvarieties of the Grassmann variety with its Plucker embedding. When the base field is finite, we consider the more general question of determining the maximum number of points on sections of Grassmannians by linear subvarieties of a fixed (co)dimension. This corresponds to a known open problem of determining the complete weight hierarchy of linear error correcting codes associated to Grassmann varieties. We recover most of the known results as well as prove some new results. In the process we obtain, and utilize, a simple generalization of the Griesmer-Wei bound for arbitrary linear codes.

math.AG↗