<p>Shiv Nadar University is a comprehensive, multidisciplinary, research-focused, and student-centric University offering a full range of academic programs at the undergraduate, postgraduate and doctoral level. With state-of-the-art infrastructure, the University comprises of academic wings, sophisticated labs, international standard sports facilities, amphitheaters, auditorium, conference rooms and smart classrooms. The University&rsquo;s goal is to become internationally recognized for the quality of its research and creative endeavors and their applicability to improving quality of life, generating new insights and expanding the boundaries of human knowledge creativity. Committed to excellence in teaching, research and service, the University aims to serve the higher education needs of India and the world beyond.</p>

Shiv Nadar University

Let PG ([0, ∞)) and P′G([0, ∞)) be the sets of positive semidefinite and positive definite matrices of order n, respectively, with nonnegative entries, where some positions of zero entries are restricted by a simple graph G with n vertices. It is proved that for a connected simple graph G of order n ≥ 3, the set of powers preserving positive semidefiniteness on PG ([0, ∞)) is precisely the same as the set of powers preserving positive definiteness on P′G([0, ∞)). In particular, this provides an explicit combinatorial description of the critical exponent for positive definiteness, for all chordal graphs. Using chain sequences, it is proved that the Hadamard powers preserving the positive (semi) definiteness of every tridiagonal matrix with nonnegative entries are precisely r ≥ 1. The infinite divisibility of tridiagonal matrices is studied. The same results are proved for a special family of pentadiagonal matrices. © 2022, International Linear Algebra Society. All rights reserved.

World University root node

Mathematics

(SoNS) School of Natural Sciences

Electronic Journal of Linear Algebra

POSITIVITY OF HADAMARD POWERS OF A FEW BAND MATRICES

Let GL3(Zn) denote the set of 3 × 3 invertible matrices with entries from Zn, the ring of integers modulo n. We study the distribution of a matrix function called the permanent, restricting its domain to GL3(Zn). Given x ∈ Z, we count the number of elements in the set G3(n, x) = {M ∈ GL3(Zn) | perm(M) ≡ x (mod n)}. © 2022, Colgate University. All rights reserved.

Integers

PERMANENTS OF 3 × 3 INVERTIBLE MATRICES MODULO N

In this article, we prove various properties of the Calkin-Wilf tree. We also see how the Minkowski question mark function acts on the Calkin-Wilf tree and its diagonals. © 2020, Indian Academy of Sciences.

Resonance

Calkin-Wilf Tree

It is shown that for positive real numbers 0&lt;λ1&lt;…&lt;λn, [Formula presented], where β(⋅,⋅) denotes the beta function, is infinitely divisible and totally positive. For [Formula presented], the Cholesky decomposition and successive elementary bidiagonal decomposition are computed. Let w(n) be the nth Bell number. It is proved that [w(i+j)] is a totally positive matrix but is infinitely divisible only upto order 4. It is also shown that the symmetrized Stirling matrices are totally positive. © 2020 Elsevier Inc.

Fulltext

Linear Algebra and Its Applications

Positivity properties of some special matrices

In this article, we consider the polynomials of the form f(x) = a0+ a1x + a2X2+ ... + anxn∈ Z[x], where |a0| = |a1| + ... + |a0| and |ao| is a prime. We show that these polynomials have a cyclotomic factor whenever reducible. As a consequence, we give a simple procedure for checking the irreducibility of trinomials of this form and separability criterion for certain quadrinomials. © 2020 Societatea de Stiinte Matematice din Romania. All rights reserved.

Bulletin Mathematique de la Societe des Sciences Mathematiques de Roumanie

An irreducibility criterion for polynomials over integers

A nonnegative matrix A is primitive if for some positive integer m all entries in Am are positive. The smallest such m is called the exponent of A and written exp(A). For the class of primitive companion matrices X, we find exp(A) for certain A ∈ X. We find certain values of m for which there is an n×n primitive companion matrix (for given n) with exponent m. We also propose open problems for further research. © 2019 Rocky Mountain Mathematics Consortium.

Rocky Mountain Journal of Mathematics

Exponents of primitive companion matrices

A cyclotomic polynomial φk(x) is an essential cyclotomic factor of f (x) ∈ Z[x] if φk(x) ∥ f (x) and every prime divisor of k is less than or equal to the number of terms of f: We show that if a monic polynomial with coefficients from {-1, 0, 1} has a cyclotomic factor, then it has an essential cyclotomic factor. We use this result to prove a conjecture posed by Mercer ['Newman polynomials, reducibility, and roots on the unit circle', Integers 12(4) (2012), 503-519]. © 2019 Australian Mathematical Publishing Association Inc.

Bulletin of the Australian Mathematical Society

Cyclotomic factors of borwein polynomials

In this article, we compile the work done by various mathematicians on the topic of the fixed divisor of a polynomial. This article explains most of the results concisely and is intended to be an exhaustive survey. We present the results on fixed divisors in various algebraic settings as well as the applications of fixed divisors to various algebraic and number theoretic problems. The work is presented in an orderly fashion so as to start from the simplest case of Z, progressively leading up to the case of Dedekind domains. We also ask a few open questions according to their context, which may give impetus to the reader to work further in this direction. We describe various bounds for fixed divisors as well as the connection of fixed divisors with different notions in the ring of integervalued polynomials. Finally, we suggest how the generalization of the ring of integer-valued polynomials in the case of the ring of n × n matrices over Z (or a Dedekind domain) could lead to the generalization of fixed divisors in that setting. © 2019, Institut Camille Jordan. All rights reserved.

Electrical Engineering

(SoE) School of Engineering

Confluentes Mathematici

A survey on fixed divisors

In this work, we studied various properties of arithmetic function ˜ϕ, where ˜ϕ(n) = |{m ∈ N|1 ≤ m ≤ n, (m, n) = 1, m is not a prime}|. © Indian Mathematical Society, 2019.

Mathematics Student

Number of non-primes in the set of units modulon

This paper proposes a method for gesture classification based on Graph Fourier transform (GFT) coefficients. GFT coefficients are the projection of image pixel block onto the eigenvectors of a Laplacian matrix. This Laplacian matrix is generated from undirected graph, representing a spatial connectedness between each pixel within an image block. This work proposes a method for generating an undirected graph by using edge information of the image. Edge information of the image is obtained by average sum of absolute difference between the current pixel and its neighboring pixels by using an appropriate threshold. The resulting GFT based feature vector is formed by concatenating GFT coefficients of each block. The resultant feature vector is applied to linear Support Vector Machine (SVM) classifier to predict the gesture class. For NTU and Massey hand gesture datasets, threshold value 30 gives maximum prediction accuracy. We compare the results of the proposed GFT based descriptor approach with Karhunen-Loeve transform (K-LT) and Discrete Cosine transform (DCT) based descriptors on three different gesture datasets: NTU, Cambridge and Massey. Simulation results show that the proposed GFT based descriptor gives a comparable results with Karhunen-Loeve transform (K-LT) and Discrete Cosine transform (DCT) based descriptors for gesture classification. © 2017 IEEE.

2017 4th International Conference on Image Information Processing, ICIIP 2017

Graph fourier transform based descriptor for gesture classification

Let K be a finite extension of a characteristic zero field F. We say that a pair of n × n matrices (A,B) over F represents K if K ≅ F[A]/〈B〉, where F[A] denotes the subalgebra of Mn(F) containing A and 〈B〉 is an ideal in F[A], generated by B. In particular, A is said to represent the field K if there exists an irreducible polynomial q(x) ∈ F[x] which divides the minimal polynomial of A and K ≅ F[A]/〈q(A)〉. In this paper, we identify the smallest order circulant matrix representation for any subfield of a cyclotomic field. Furthermore, if p is a prime and K is a subfield of the p-th cyclotomic field, then we obtain a zero-one circulant matrix A of size p × p such that (A, J) represents K, where J is the matrix with all entries 1. In case, the integer n has at most two distinct prime factors, we find the smallest order 0, 1-companion matrix that represents the n-th cyclotomic field. We also find bounds on the size of such companion matrices when n has more than two prime factors. © 2013 The Indian National Science Academy.

Indian Journal of Pure and Applied Mathematics

Representation of cyclotomic fields and their subfields

For a fixed positive integer n, let Wn be the permutation matrix corresponding to permutation. In this article, it is shown that a symmetric matrix with rational entries is circulant if, and only if, it lies in the subalgebra of Q[x]/ (xn-1) generated by Wn+Wn -1. On the basis of this, the singularity of graphs on n-vertices is characterized algebraically. This characterization is then extended to characterize the singularity of directed circulant graphs. The kth power matrix Wnk+Wn -k defines a circulant graph Cnk. The results above are then applied to characterize its singularity, and that of its complement graph. The digraph Cr,s,t is defined as that whose adjacency matrix is circulant circ(a), where a is a vector with the first r-components equal to 1, and the next s, t and n - (r + s + t) components equal to zero, one, and zero respectively. The singularity of this digraph (graph), under certain conditions, is also shown to depend algebraically upon these parameters. A slight generalization of these graphs are also studied.