Class Of Matrices Research Articles

Numerical semigroups from rational matrices I: power-integral matrices and nilpotent representations

Our aim in this paper is to initiate the study of exponent semigroups for rational matrices. We prove that every numerical semigroup is the exponent semigroup of some rational matrix. We also obtain lower bounds on the size of such matrices and discuss the related class of power-integral matrices.

The green approach of chitosan/Fe2O3/ZnO-nanocomposite synthesis with an evaluation of its biological activities

Biopolymers embedded with nanoparticles of metal oxides (MOs) demonstrate a wide range of bio-functions. Chitosan-incorporated MOs are an interesting class of support matrices for enhancing the biological function, compared to other support matrices. Therefore, the importance of this study lies in exploiting chitosan as a carrier not of one metal as in previous studies, but of two metals in the form of a nanocomposite to carry out several biological functions. The coprecipitation approach was employed to synthesize chitosan/Fe2O3/ZnO-nanocomposite in the present research. The characterization of chitosan/Fe2O3/ZnO-nanocomposite was performed to find out the morphology and dispersion properties of chitosan/Fe2O3/ZnO-nanocomposite. The X-ray diffraction (XRD) investigation revealed that these were crystalline. Fourier transforms infrared (FTIR) spectrum bands were viewed at 400/cm and 900/cm, due to the stretching vibration of Fe and Zn oxygen bond. TEM showed that chitosan/Fe2O3/ZnO-nanocomposite was of 20–95 nm in size. chitosan/Fe2O3/ZnO-nanocomposite exhibited inhibitory potential against Staphylococcus aureus, Bacillus subtilis, Escherichia coli, and Candida albicans with inhibition zones of 25 ± 0.1, 28 ± 0.2, 27 ± 0.1, and 27 ± 0.2 mm, respectively while didn’t inhibited Aspergillus niger. MIC value of nanocomposite was 15.62 ± 0.33 µg/mL for C. albicans, B. subtilis and E. coli, while it was 62.50 ± 0.66 µg/mL for Pseudomonas aeruginosa. Ranged values of nanocomposite MBC (15.62 ± 0.33 to 125 ± 1 µg/mL) were attributed to all tested bacteria. Different concentrations of chitosan/Fe2O3/ZnO-nanocomposite MBC (25, 50, and 75%) reflected anti-biofilm activity against E. coli (85.0, 93.2, and 96.0%), B. subtilis (84.88, 92.21, and 96.99%), S. aureus 81.64, 90.52, and 94.64%) and P. aurogenosa (90.11, 94.43, and 98.24%), respectively. The differences in the levels of antimicrobial activities may depend on the type of examined microbes. Antioxidant activity of chitosan/Fe2O3/ZnO-nanocomposite was recorded with excellent IC50 values of 16.06 and 32.6 µg/mL using DPPH and ABTS scavenging, respectively. Wound heal by chitosan/Fe2O3/ZnO-nanocomposite was achieved with 100% compared to the untreated cells (76.75% of wound closer). The cytotoxicity outcomes showed that the IC50 of the chitosan/Fe2O3/ZnO-nanocomposite was 564.32 ± 1.46 µg/mL normal WI-38 cells. Based on the achieved findings, the chitosan/Fe2O3/ZnO-nanocomposite is a very promising agent for perform pharmacological activities.

Some remarks on permanental dominance conjecture

In this paper we provide an identity between the determinant and other generalized matrix functions, and give a criterion for positive semi-definite matrices to satisfy the permanental dominance conjecture. As a consequence, infinitely many classes of positive semi-definite matrices satisfying the conjecture (does not depend on groups or characters) are provided by generating from any positive semi-definite matrix having no zero in the first column.

Robust sparse recovery with sparse Bernoulli matrices via expanders

Sparse binary matrices are of great interest in the field of sparse recovery, nonnegative compressed sensing, statistics in networks, and theoretical computer science. This class of matrices makes it possible to perform signal recovery with lower storage costs and faster decoding algorithms. In particular, Bernoulli (p) matrices formed by independent identically distributed (i.i.d.) Bernoulli (p) random variables are of practical relevance in the context of noise-blind recovery in nonnegative compressed sensing.In this work, we investigate the robust nullspace property of Bernoulli (p) matrices. Previous results in the literature establish that such matrices can accurately recover n-dimensional s-sparse vectors with m=O(sc(p)log⁡ens) measurements, where c(p)≤p is a constant dependent only on the parameter p. These results suggest that in the sparse regime, as p approaches zero, the (sparse) Bernoulli (p) matrix requires significantly more measurements than the minimal necessary, as achieved by standard isotropic subgaussian designs. However, we show that this is not the case.Our main result characterizes, for a wide range of sparsity levels s, the smallest p for which sparse recovery can be achieved with the minimal number of measurements. We also provide matching lower bounds to establish the optimality of our results and explore connections with the theory of invertibility of discrete random matrices and integer compressed sensing.

Characterizations of some new classes of four-dimensional matrices and dual spaces on the double spaces of Cesàro means

The main purpose in this study is to investigate some topological and algebraic properties of the absolutely double series spaces |C_{1,1}|_{k} defined by combining the first order Cesàro means with the concept of absolute summability k≥1. Beside this, we determine the α-dual of the space |C_{1,1}|₁ and the β(bp)- and γ-duals of the spaces |C_{1,1}|_{k} for k≥1. Finally, we characterize some new four-dimensional matrix classes (|C_{1,1}|_{k},υ), (|C_{1,1}|₁,υ), (|C_{1,1}|₁,L_{k}), (|C_{1,1}|_{k},L_{u}), (L_{u},|C_{1,1}|_{k}) and (L_{k},|C_{1,1}|₁), where υ∈{M_{u},C_{bp}} for 1≤k

Eigenstructure Perturbations for a Class of Hamiltonian Matrices and Solutions of Related Riccati Inequalities

Eigenstructure Perturbations for a Class of Hamiltonian Matrices and Solutions of Related Riccati Inequalities

The distribution of polynomials in monotone-independent elements

Building on the work of Arizmendi and Celestino (2021), we derive the ∗-distributions of polynomials in monotone independent and infinitesimally monotone independent elements. For non-zero complex numbers [Formula: see text] and [Formula: see text], we derive explicitly the ∗-distribution of [Formula: see text] whenever [Formula: see text] and [Formula: see text] are monotone or infinitesimally monotone independent elements. This encompasses both cases of the commutator and anti-commutator. This approach can be pushed to study more general polynomials. As applications, we derive the limiting distribution with respect to the partial trace of polynomials in a certain class of random matrices.

Structured random matrices and cyclic cumulants: A free probability approach

In this paper, we introduce a new class of large structured random matrices characterized by four fundamental properties which we discuss. We prove that this class is stable under matrix-valued and pointwise nonlinear operations. We then formulate an efficient method, based on an extremization problem, for computing the spectrum of subblocks of such large structured random matrices. We present different proofs — combinatorial or algebraic — of the validity of this method, which all have some connection with free probability. We illustrate this method with well-known examples of unstructured matrices, including Haar randomly rotated matrices, as well as with the example of structured random matrices arising in the quantum symmetric simple exclusion process.

Group theoretical classification of SIC-POVMs

The Symmetric Informationally Complete Positive Operator-Valued Measures (SIC-POVMs) are known to exist in all dimensions ⩽ 151 and many larger dimensions as high as 39604 . All known solutions with the exception of the Hoggar solutions are covariant with respect to the Weyl-Heisenberg group and in the case of dimension 3, it has been proven that all SIC-POVMs are Weyl-Heisenberg group covariant. In this work, we explore this by SIC-POVMs in dimensions 4–7 without the assumption of group covariance. We introduce two functions with which SIC-POVM Gram matrices can be generated without the group covariance constraint, and analytically show that the SIC-POVM Gram matrices exist on critical points of the surfaces defined by the two functions on a subspace of Hermitian matrices. In dimensions 4–7, all known SIC-POVM Gram matrices lie in disjoint continuous sets of solution. Thus, we define an equivalent class that relates Gram matrices based on the trivial symmetries of the two functions. In dimensions 4–7, we generated {1.7×106,1.1×105,169,50} Gram matrices, respectively. For each of the Gram matrices, we generate the symmetry group of their respective disjoint sets. In all cases, the symmetry group was isomorphic to a subgroup of the Clifford group containing the Weyl-Heisenberg group matrices and the order-3 unitaries. In dimensions 4–6, All constructed solutions belonged to a single equivalence class of Gram matrices whereas in dimension 7, we find two distinct families of SIC-POVMs. In dimensions 4 and 5, the absence of new classes of SIC-POVM strongly suggests that the functions do not have a non-trivial symmetry. Furthermore, in all the dimensions tested, the only equivalence classes found correspond to the distinct stabilizers of fiducial vectors of each dimension.

On discrete inequalities for some classes of sequences

Abstract For a given sequence a = ( a 1 , … , a n ) ∈ R n a=\left({a}_{1},\ldots ,{a}_{n})\in {{\mathbb{R}}}^{n} , our aim is to obtain an estimate of E n ≔ a 1 + a n 2 − 1 n ∑ i = 1 n a i {E}_{n}:= \left|\hspace{-0.33em},\frac{{a}_{1}+{a}_{n}}{2}-\frac{1}{n}{\sum }_{i=1}^{n}{a}_{i},\hspace{-0.33em}\right| . Several classes of sequences are studied. For each class, an estimate of E n {E}_{n} is obtained. We also introduce the class of convex matrices, which is a discrete version of the class of convex functions on the coordinates. For this set of matrices, new discrete Hermite-Hadamard-type inequalities are proved. Our obtained results are extensions of known results from the continuous case to the discrete case.

Extending EP matrices by means of recent generalized inverses

It is well known that a square complex matrix is called EP if it commutes with its Moore–Penrose inverse. In this paper, new classes of matrices which extend this concept are characterized. For that, we consider commutative equalities given by matrices of arbitrary index and generalized inverses recently investigated in the literature. More specifically, these classes are characterized by expressions of type AmX=XAm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A^mX=XA^m$$\\end{document}, where X is an outer inverse of a given complex square matrix A and m is an arbitrary positive integer. The relationships between the different classes of matrices are also analyzed. Finally, a picture presents an overview of the overall studied classes.

Universal hard-edge statistics of non-Hermitian random matrices

Random matrix theory is a powerful tool for understanding spectral correlations inherent in quantum chaotic systems. Despite diverse applications of non-Hermitian random matrix theory, the role of symmetry remains to be fully established. Here, we comprehensively investigate the impact of symmetry on the level statistics around the spectral origin—hard-edge statistics—and expand the classification of spectral statistics to encompass all the 38 symmetry classes of non-Hermitian random matrices. Within this classification, we discern 28 symmetry classes characterized by distinct hard-edge statistics from the level statistics in the bulk of spectra, which are further categorized into two groups, namely, the Altland-Zirnbauer0 classification and beyond. We introduce and elucidate quantitative measures capturing the universal hard-edge statistics for all the symmetry classes. Furthermore, through extensive numerical calculations, we study various open quantum systems in different symmetry classes, including quadratic and many-body Lindbladians, as well as non-Hermitian Hamiltonians. We show that these systems manifest the same hard-edge statistics as random matrices and that their ensemble-average spectral distributions around the origin exhibit emergent symmetry conforming to the random-matrix behavior. Our results establish a comprehensive understanding of non-Hermitian random matrix theory and are useful in detecting quantum chaos or its absence in open quantum systems. Published by the American Physical Society 2024

Cohen-Macaulay type of orders, generators and ideal classes

In this paper we study the (Cohen-Macaulay) type of orders over Dedekind domains in étale algebras. We provide a bound for the type, and give formulas to compute it. We relate the type of the overorders of a given order to the size of minimal generating sets of its fractional ideals, generalizing known results for Gorenstein and Bass orders. Finally, we give a classification of the ideal classes with multiplicator ring of type 2, with applications to the computations of the conjugacy classes of integral matrices and the isomorphism classes of abelian varieties over finite fields.

Further properties of accretive matrices

To better understand the algebra \(\mathcal{M}_n\) of all \(n\times n\) complex matrices, we explore the class of accretive matrices. This class has received renowned attention in recent years due to its role in complementing those results known for positive definite matrices. Among many results, we present order-preserving results, Choi–Davis-type inequalities, mean-convex inequalities, sub-multiplicative results for the real part, and new bounds of the absolute value of accretive matrices. These results will be compared with the existing literature. In the end, we quickly pass through related entropy results for accretive matrices.

Generalized unistochastic matrices

We study a class of bistochastic matrices generalizing unistochastic matrices. Given a complex bipartite unitary operator, we construct a bistochastic matrix having as entries the normalized squared Frobenius norm of the blocks. We show that the closure of the set of generalized unistochastic matrices is the whole Birkhoff polytope. We characterize the points on the edges of the Birkhoff polytope that belong to a given level of our family of sets, proving that the different (non-convex) levels have a rich inclusion structure. We also study the corresponding generalization of orthostochastic matrices. Finally, we introduce and study the natural probability measures induced on our sets by the Haar measure of the unitary group. These probability measures interpolate between the natural measure on the set of unistochastic matrices and the Dirac measure supported on the van der Waerden matrix.

Scattering matrices of particle ensembles analytically decomposed into pure Mueller matrices

Scattering matrices of particle ensembles are analytically decomposed into sums of pure Mueller matrices. The ensembles are assumed to have equal numbers of particles and their mirror particles, both in random orientation. In the general case, there are four pure Mueller matrices in the decomposition. In the present spectral decomposition, of these four matrices, there is a single matrix that qualifies as a scattering matrix, whereas the remaining three matrices represent other classes of pure matrices. For ensembles of spherical particles, there are two pure Mueller matrices in the decomposition. Again, there is a single matrix qualifying as a scattering matrix. The analytical decomposition unveils the explicit dependencies of the pure Mueller matrices on the ensemble-averaged scattering matrix. Applications are identified in scattering by single particles and by random media of particles.

An algorithm based on quantum phase estimation for the identification of patterns

The quantum phase estimation algorithm has been utilized by a variety of quantum algorithms, most notably Shor’s algorithm, to obtain information regarding the period of a function that is appropriately encoded into a unitary operator. In many cases, it is desired to estimate whether a specific state exhibits a certain pattern quickly. In this paper, we exhibit a methodology based on the QPE algorithm to identify certain patterns. In particular, starting from a properly encoded state, we demonstrate how to construct unitary operators whose eigenvectors correspond to states with proper diagonals. QPE will then output an eigenphase equal to zero with certainty for these states, thereby identifying this class of matrices. For partial matches, our algorithm, based on the tolerance threshold used, will show areas of high similarity, and it will outperform classical ones when the number of mismatches defined by the tolerance is significantly low when compared to the size of the diagonal.

Regularization, early-stopping and dreaming: A Hopfield-like setup to address generalization and overfitting

In this work we approach attractor neural networks from a machine learning perspective: we look for optimal network parameters by applying a gradient descent over a regularized loss function. Within this framework, the optimal neuron-interaction matrices turn out to be a class of matrices which correspond to Hebbian kernels revised by a reiterated unlearning protocol. Remarkably, the extent of such unlearning is proved to be related to the regularization hyperparameter of the loss function and to the training time. Thus, we can design strategies to avoid overfitting that are formulated in terms of regularization and early-stopping tuning. The generalization capabilities of these attractor networks are also investigated: analytical results are obtained for random synthetic datasets, next, the emerging picture is corroborated by numerical experiments that highlight the existence of several regimes (i.e., overfitting, failure and success) as the dataset parameters are varied.

Preserver Problems on Infinite Divisibility and Separability

This paper is committed in characterizing the preserving maps for the class of separable ma- trices and a subclass of infinitely divisible matrices which are also separable. For this, an association between the classes of separable matrices and infinitely divisible matrices is established. Also, several properties and attributes of the above mentioned classes are stated and proved.

A matricial view of the Collatz conjecture

It is shown that the nilpotency of submatrices of a certain class of adjacency matrices is equivalent to the Collatz conjecture. Our main result extends the previous work of Alves et al. and clarifies a conjecture made by Cardon and Tuckfield.