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Namita Behera

Publications and source records attributed to Namita Behera.

15 recordsLinked to original sources
arXiv2026

Arithmetical Structures on Ladder Graphs

In this paper, we investigate arithmetical structures on Cartesian product graphs, particularly, ladder graph of the form P2\square Pm and grid graph of the form Pn \square Pm. An arithmetical structure on a finite and connected graph G is a pair (d, r) of positive integer vectors such that r is primitive (the gcd of its entries is 1) and (diag(d) - A)r = 0, where A is the adjacency matrix of G. Arithmetical structures have been widely studied for basic graph families such as paths and cycles. Extending these ideas to graph products, we first analyze the ladder graph P2 \square Pm, deriving structural properties and identifying patterns in the corresponding arithmetical configurations. We then generalize these results to the grid graph Pn \square Pm, where increased complexity arises due to higher-dimensional interactions. Our work provides new insights into the behavior, characterization, and enumeration of arithmetical structures on grid-like graphs, contributing to the broader understanding of Laplacian based invariants and their combinatorial properties.

math.CO↗
arXiv2026

Structured Linearizations of Structured Rational Matrices

Numerical computations involving rational matrices often benefit from preserving underlying matrix structures such as symmetry, Hermitian properties, or sparsity that reflect physical, geometric, or algebraic characteristics of the system. Maintaining such structures enhances stability, accuracy, and efficiency. Linearization, a technique that reformulates rational matrix problems as generalized eigenvalue problems (GEPs) of larger matrices, is widely used but does not automatically retain structure. In this chapter, we focus on structured linearizations, which preserve both the spectral information of the original rational matrix and its intrinsic structural properties. To achieve this, we present the construction of a family of linearizations called generalized Fiedler pencils with repetition (GFPR), which we prove to be valid linearizations for rational matrices. Moreover, we demonstrate that the GFPR family serves as a versatile framework for generating structured linearizations, specifically symmetric, skew-symmetric, T-even, and T-odd linearizations, provided the original rational matrix exhibits the corresponding structure. These structured linearizations facilitate the use of specialized, structure-preserving algorithms, reduce numerical errors, and yield physically meaningful solutions in application

math.RA↗
arXiv2025

Arithmetical Structures On Fan Graphs

In this paper, we study the arithmetical structures on Fan Graphs Fn. Let G be a finite and connected graph. An arithmetical structure on G is a pair (d, r) of positive integer vectors such that r is primitive (the greatest common divisor of its coefficients is 1) and (diag(d)-A)r = 0, where A represents the adjacency matrix of G. This work explores the combinatorial properties of the arithmetical structures associated with Fn. Further, we discuss the arrow-star graph, a structure derived from the fan graph, along with its properties. Additionally, we investigate the critical group linked to each such structure on Fn.

math.CO↗
arXiv2024

Arithmetical Structures on Wheel Graphs

An arithmetical structure on a finite and connected graph G is a pair (d, r) of positive integer vectors such that r is primitive (the gcd of its entries is 1) and (diag(d) - A)r = 0, where A is the adjacency matrix of G. In this article, we investigate arithmetical structures on the wheel graphs.

math.CO↗
arXiv2024

Vector Spaces of Linearizations for Multivariable State-Space Systems

Consider a multivariable state space system and associated transfer function G(λ). The aim of this paper is to define and analyze two vector spaces of matrix pencils associated with the matrix G(λ) and show that almost all of these pencils are linearizations of G(λ). We also construct symmetric/Hermitian linearizations of G(λ) when G(λ) is regular and symmetric/Hermitian.

math.OC↗
arXiv2024

Cohomology and deformation of compatible Hom-Leibniz algebras

In this paper, we consider compatible Hom-Leibniz algebra where the Hom map twists the operations in the compatible system. We consider a suitably graded Lie algebra whose Maurer-Cartan elements characterize the structure of compatible Hom-Leibniz algebras. Using this, we study cohomology, infinitesimal deformations, the Nijenhuis operator, and their relation for compatible Hom-Leibniz algebras. Finally we see the cohomology of compatible Hom-Leibniz algebra with coefficients in an arbitrary representation.

math.RA↗
arXiv2023

Fiedler Linearizations of Rectangular Rational Matrix Functions

Linearization is a standard approach in the computation of eigenvalues, eigenvectors and invariant subspaces of matrix polynomials and rational matrix value functions. An important source of linearizations are the so called Fiedler linearizations, which are generalizations of the classical companion forms. In this paper the concept of Fiedler linearization is extended from square regular to rectangular rational matrix valued functions. The approach is applied to Rosenbrock functions arising in mathematical system theory.

math.CT↗
arXiv2023

Cohomology and deformations of compatible Leibniz algebras

In this paper we study a cohomology theory of compatible Leibniz algebra. We construct a graded Lie algebra whose Maurer-Cartan elements characterize the structure of compatible Leibniz algebras. Using this, we study cohomology, infinitisimal deformations, Nijenhuis operator and their relation for compatible leibniz algebras. Finally using cohomology of compatible Leibniz algebra with coefficients in an arbitrary representation we study the abelian extensions of compatible Leibniz algebra.

math.RA↗
arXiv2022

Fiedler Linearizations of Multivariable State-Space System and its Associated System Matrix

Linearization is a standard method in the computation of eigenvalues and eigenvectors of matrix polynomials. In the last decade a variety of linearization methods have been developed in order to deal with algebraic structures and in order to construct efficient numerical methods. An important source of linearizations for matrix polynomials are the so called Fiedler pencils, which are generalizations of the Frobenius companion form and these linearizations have been extended to regular rational matrix function which is the transfer function of LTI State-space system in [1, 6]. We consider a multivariable state-space system and its associated system matrix S(λ). We introduce Fiedler pencils of S(λ) and describe an algorithm for their construction. We show that Fiedler pencils are linearizations of the system matrix S(λ).

math.NA↗
arXiv2021

Solution Method For Higher Order System

Consider a higher order state space system and the aim of this paper is to linearize the system preserving system characteristics. That is, linearization preserving system characteristics(e.g, controllability, observability, various zeros and transfer function) for analysis of higher order systems gives the solution for higher order system. We study recovery of zero directions of higher order state space system from those of the linearizations. That is, the zero directions of the transfer functions associated to higher order state space system are recovered from the eigenvectors of the Fiedler pencils without performing any arithmetic operations

math.OC↗
arXiv2021

Backward Error of Matrix Rational Function

We consider a minimal realization of a rational matrix functions. We perturb the polynomial part and one of the constant matrices from the realization part. We derive explicit computable expressions of backward errors of approximate eigenvalue of rational matrix function. We also determine minimal perturbations for which approximate eigenvalue are exact eigenvalue of the perturbed matrix rational functions.

math.NA↗
arXiv2020

Equivariant one-parameter deformations of Lie triple systems

In this article, we introduce equivariant formal deformation theory of Lie triple systems. We introduce an equivariant deformation cohomology of Lie triple systems and using this we study the equivariant formal deformation theory of Lie triple systems.

math.RA↗
arXiv2015

Linearizations for Rosenbrock system polynomials and rational matrix functions

Our aim in this paper is two-fold: First, for computing zeros of a linear time-invariant (LTI) system $Σ$ in {\em state-space form}, we introduce a "trimmed structured linearization", which we refer to as {\em Rosenbrock linearization}, of the Rosenbrock system polynomial $\mathcal{S}(\lam)$ associated with $Σ.$ We also introduce Fiedler-like matrices for $\mathcal{S}(\lam)$ and describe constructions of Fiedler-like pencils for $\mathcal{S}(\lam).$ We show that the Fiedler-like pencils of $\mathcal{S}(\lam)$ are Rosenbrock linearizations of the system polynomial $\mathcal{S}(\lam).$ Second, with a view to developing a direct method for solving rational eigenproblems, we introduce "linearization" of a rational matrix function. We describe a state-space framework for converting a rational matrix function $G(\lam)$ to an "equivalent" matrix pencil $\mathbb{L}(\lam)$ of smallest dimension such that $G(\lam)$ and $\mathbb{L}(\lam)$ have the same "eigenstructure" and we refer to such a pencil $\mathbb{L}(\lam)$ as a "linearization" of $G(\lam).$ Indeed, by treating $G(\lam)$ as the transfer function of an LTI system $Σ_G$ in state-space form via state-space realization, we show that the Fiedler-like pencils of the Rosenbrock system polynomial associated with $Σ_G$ are "linearizations" of $G(\lam)$ when the system $Σ_G$ is both controllable and observable.

math.NA↗