Matrix Realization Research Articles

Refined inertias of positive and hollow positive patterns

Abstract We investigate refined inertias of positive patterns and patterns that have each off-diagonal entry positive and each diagonal entry zero, i.e., hollow positive patterns. For positive patterns, we prove that every refined inertia ( n + , n − , n z , 2 n p ) \left({n}_{+},{n}_{-},{n}_{z},2{n}_{p}) with n + ≥ 1 {n}_{+}\ge 1 can be realized. For hollow positive patterns, we prove that every refined inertia with n + ≥ 1 {n}_{+}\ge 1 and n − ≥ 2 {n}_{-}\ge 2 can be realized. To illustrate these results, we construct matrix realizations using circulant matrices and bordered circulants. For both patterns of order n n , we show that as n → ∞ n\to \infty , the fraction of possible refined inertias realized by circulants approaches 1/4 for n n odd and 3/4 for n n even.

W1+∞ and [formula omitted] algebras, and Ward identities

It was demonstrated recently that the W1+∞ algebra contains commutative subalgebras associated with all integer slope rays (including the vertical one). In this paper, we realize that every element of such a ray is associated with a generalized W˜ algebra. In particular, the simplest commutative subalgebra associated with the rational Calogero Hamiltonians is associated with the W˜ algebras studied earlier. We suggest a definition of the generalized W˜ algebra as differential operators in variables pk basing on the matrix realization of the W1+∞ algebra, and also suggest an unambiguous recursive definition, which, however, involves more elements of the W1+∞ algebra than is contained in its commutative subalgebras. The positive integer rays are associated with W˜ algebras that form sets of Ward identities for the WLZZ matrix models, while the vertical ray associated with the trigonometric Calogero-Sutherland model describes the hypergeometric τ-functions corresponding to the completed cycles.

Bisymmetric non-negative Jacobi matrix realizations

Within the symmetric inverse eigenvalue problem, the case of bisymmetric Jacobi matrices occupies a central place, since for any strictly monotone list of n real numbers there exists a unique bisymmetric Jacobi matrix realizing the list. Apart from their meaning in several issues such as physics, mechanics, statistics, to cite some of them, the families of this kind of matrices whose spectrum is known are used as models for testing the different algorithms to recover the entries of matrices from spectra data. However, the spectrum is known only for a few families of bisymmetric Jacobi matrices and the examples mainly refer to the case when the spectrum is given by a linear or quadratic function of the order and of the row index. In the first part of this paper, we join all known cases by proving a general result about bisymmetric Jacobi realizations of strictly monotone sequences that are quadratic at most. In the second part, we focus on the non-negative bisymmetric realizations, obtaining new necessary conditions for a given list to be realized by a non-negative bisymmetric Jacobi matrix. The main novelty in our techniques is considering the gaps between the eigenvalues instead of focusing on the eigenvalues themselves. In the last part of this paper, we explicitly obtain the bisymmetric realization of any list for order less or equal to 6.

On allowable properties and spectrally arbitrary sign pattern matrices

In this paper, we give a geometric construction for different allowable properties for sign pattern matrices. In Section 2, we give a construction for detecting a sign pattern matrix to be potentially nilpotent and also compute the nilpotent matrix realization for a given sign pattern matrix if it exists. In Section 3, we develop a geometric construction for potential stability. In Section 4, we establish a necessary and sufficient condition for a sign pattern matrix S to be spectrally arbitrary. For a given sign pattern matrix of order n, we prove that there exists a surface of dimension at most m for m≤n such that for every vector a1,a2,⋯,an on the same surface, there exists a matrix A∈Q(S), a qualitative class of S whose characteristic polynomial is xn−a1xn−1+⋯+(−1)nan.

Minimum number of distinct eigenvalues allowed by a sign pattern

This article initiates the study of the minimum number of distinct eigenvalues allowed by a sign pattern. Non-trivial examples of sign patterns that allow matrices with only one eigenvalue include the potentially nilpotent and the spectrally arbitrary sign patterns, in contrast to the inverse eigenvalue problem for graphs. Necessary digraph cycle conditions are developed for a sign pattern to have a matrix realization with exactly one eigenvalue. Certain manipulations of sign patterns provide new n×n patterns that preserve the minimum number of eigenvalues allowed, including some Jacobian methods. The 2×2 and the irreducible 3×3 sign patterns are classified according to the minimum number of eigenvalues allowed by the pattern.

Preparation and Study of Solid Lipid Nanoparticles Based on Curcumin, Resveratrol and Capsaicin Containing Linolenic Acid.

Linolenic acid (LNA) is the most highly consumed polyunsaturated fatty acid found in the human diet. It possesses anti-inflammatory effects and the ability to reverse skin-related disorders related to its deficiency. The purpose of this work was to encapsulate LNA in solid lipid nanoparticles (SLNs) based on curcumin, resveratrol and capsaicin for the treatment of atopic dermatitis. These compounds were first esterified with oleic acid to obtain two moonoleate and one oleate ester, then they were used for SLN matrix realization through the emulsification method. The intermediates of the esterification reaction were characterized by FT-IR and 1N-MR analysis. SLNs were characterized by dimensional analysis and encapsulation efficiency. Skin permeation studies, antioxidant and anti-inflammatory activities were evaluated. LNA was released over 24 h from nanoparticles, and resveratrol monooleate-filled SLNs exhibited a good antioxidant activity. The curcumin-based SLNs loaded or not with LNA did not induce significant cytotoxicity in NCTC 2544 and THP-1 cells. Moreover, these SLNs loaded with LNA inhibited the production of IL-6 in NCTC 2544 cells. Overall, our data demonstrate that the synthesized SLNs could represent an efficacious way to deliver LNA to skin cells and to preserve the anti-inflammatory properties of LNA for the topical adjuvant treatment of atopic dermatitis.

Universal scaling and criticality of extremes in random matrix theory.

We present a random matrix realization of a two-dimensional percolation model with the occupation probability p. We find that the behavior of the model is governed by the two first extreme eigenvalues. While the second extreme eigenvalue resides on the moving edge of the semicircle bulk distribution with an additional semicircle functionality on p, the first extreme exhibits a disjoint isolated Gaussian statistics which is responsible for the emergence of a rich finite-size scaling and criticality. Our extensive numerical simulations along with analytical arguments unravel the power-law divergences due to the coalescence of the first two extreme eigenvalues in the thermodynamic limit. We develop a scaling law that provides a universal framework in terms of a set of scaling exponents uncovering the full finite-size scaling behavior of the extreme eigenvalue's fluctuation. Our study may provide a simple practical approach to capture the criticality in complex systems and their inverse problems with a possible extension to the interacting systems.

An algebraic Monte-Carlo algorithm for the partition adjacency matrix realization problem

The graphical realization of a given degree sequence and given partition adjacency matrix simultaneously is a relevant problem in data driven modeling of networks. Here we formulate common generalizations of this problem and the Exact Matching Problem, and solve them with an algebraic Monte-Carlo algorithm that runs in polynomial time if the number of partition classes is bounded.

Connection between cut-and-join and Casimir operators

We study cut-and-join operators for spin Hurwitz partition functions. We provide explicit expressions for these operators in terms of derivatives in p-variables without straightforward matrix realization, which is yet to be found. With the help of these expressions spin cut-and-join operators can be calculated directly and algorithmically. The reason why it is possible is the connection between mentioned operators and specially chosen Casimir operators that are easy to compute. An essential part of the connection involves shifted Q-Schur functions.

On the Eventual Exponential Positivity of Some Tree Sign Patterns

An n×n matrix A is called eventually exponentially positive (EEP) if etA=∑k=0∞tkAkk!>0 for all t≥t0, where t0≥0. A matrix whose entries belong to the set {+,−,0} is called a sign pattern. An n×n sign pattern A is called potentially eventually exponentially positive (PEEP) if there exists some real matrix realization A of A that is EEP. Characterizing the PEEP sign patterns is a longstanding open problem. In this article, A is called minimally potentially eventually exponentially positive (MPEEP), if A is PEEP and no proper subpattern of A is PEEP. Some preliminary results about MPEEP sign patterns and PEEP sign patterns are established. All MPEEP sign patterns of orders n≤3 are identified. For the n×n tridiagonal sign patterns Tn, we show that there exists exactly one MPEEP tridiagonal sign pattern Tno. Consequently, we classify all PEEP tridiagonal sign patterns as the superpatterns of Tno. We also classify all PEEP star sign patterns Sn and double star sign patterns DS(n,m) by identifying all the MPEEP star sign patterns and the MPEEP double star sign patterns, respectively.

A Coalgebraic Structure on Bisexual Populations

We introduce a coalgebra structure representing the backwards inheritance in genetic systems with finitely many male and female genetic types, and study its main algebraic properties. We endow these coalgebras with a 3-way array (matrix) realization, showing that when the comultiplication structure constants have probabilistic significance as inheritance probabilities, the associated 3-way arrays are stochastic. Besides of providing a graphical representation of these (known as bisexual) populations, we propose a bivariate Markov model to study their backwards evolution.

Distance Matrix of a Class of Completely Positive Graphs: Determinant and Inverse

AbstractA real symmetric matrix A is said to be completely positive if it can be written as BBt for some (not necessarily square) nonnegative matrix B. A simple graph G is called a completely positive graph if every matrix realization of G that is both nonnegative and positive semidefinite is a completely positive matrix. Our aim in this manuscript is to compute the determinant and inverse (when it exists) of the distance matrix of a class of completely positive graphs. We compute a matrix 𝒭 such that the inverse of the distance matrix of a class of completely positive graphs is expressed a linear combination of the Laplacian matrix, a rank one matrix of all ones and 𝒭. This expression is similar to the existing result for trees. We also bring out interesting spectral properties of some of the principal submatrices of 𝒭.

A Space-Frame Periodic Table Representation System Testing Relativity in Nucleosynthesis of the Elements

The geometric Lie algebra SU(3) isotropic vector matrix realization of the periodic table reported at PIRT 2017 has now been broken up to a disjoint-set modular R3×SO(3) building kit, exactly sufficing to stage the Big Bang and ensuing nucleosynthesis events: First the ultrashort radiation/plasma inflation of the Big Bang phase transition moment with release of photons, neutrinos and module precursors, which in next seconds recombine to the Protium proton/electron compound and its neutron conversion to continue separately or in fusion of the two get on to Deuterium and from there to Tritium, Helium in isotope and α form, and traces of Lithium and possibly Beryllium. That is, literally the whole primordial start-gas delivered within a few minutes to billion-year wait for sufficiently energetic perturbations with itself for astrophysical/cosmogenic/experimental nucleosynthesis of the full periodic table and likewise replicable by systematic space-filling assembly/disassembly of the here disclosed neutrino and photon lattice vector, β particle, α wave-packet and neutron building bricks, providing clues also on isotope/neutron excess, shell/subshell, spectroscopy, and chemical bond structural make-up and disposition. Furthermore, the absolute trigonometric sharpness of the nucleosynthesis phase transition burst and expansion is reciprocal to the absolute speed of light and hence a specific test and verification of the relativity theory.

Bethe Vectors for Orthogonal Integrable Models

We consider quantum integrable models associated with $\mathfrak{so}_3$ algebra. We describe Bethe vectors of these models in terms of the current generators of the $\mathcal{D}Y(\mathfrak{so}_3)$ algebra. To implement this approach we use isomorphism between $R$-matrix and Drinfeld current realizations of the Yangians and their doubles for classical types $B$, $C$, and $D$ series algebras. Using these results we derive the actions of the monodromy matrix elements on off-shell Bethe vectors. We show that these action formulas lead to recursions for off-shell Bethe vectors and Bethe equations for on-shell Bethe vectors. The action formulas can also be used for calculating the scalar products in the models associated with $\mathfrak{so}_3$ algebra.

Directionally-Unbiased Unitary Optical Devices in Discrete-Time Quantum Walks

The optical beam splitter is a widely-used device in photonics-based quantum information processing. Specifically, linear optical networks demand large numbers of beam splitters for unitary matrix realization. This requirement comes from the beam splitter property that a photon cannot go back out of the input ports, which we call “directionally-biased”. Because of this property, higher dimensional information processing tasks suffer from rapid device resource growth when beam splitters are used in a feed-forward manner. Directionally-unbiased linear-optical devices have been introduced recently to eliminate the directional bias, greatly reducing the numbers of required beam splitters when implementing complicated tasks. Analysis of some originally directional optical devices and basic principles of their conversion into directionally-unbiased systems form the base of this paper. Photonic quantum walk implementations are investigated as a main application of the use of directionally-unbiased systems. Several quantum walk procedures executed on graph networks constructed using directionally-unbiased nodes are discussed. A significant savings in hardware and other required resources when compared with traditional directionally-biased beam-splitter-based optical networks is demonstrated.

Dual Numbers and Operational Umbral Methods

Dual numbers and their higher-order version are important tools for numerical computations, and in particular for finite difference calculus. Based on the relevant algebraic rules and matrix realizations of dual numbers, we present a novel point of view, embedding dual numbers within a formalism reminiscent of operational umbral calculus.

Non-sparse Companion Matrices

Given a polynomial $p(z)$, a companion matrix can be thought of as a simple template for placing the coefficients of $p(z)$ in a matrix such that the characteristic polynomial is $p(z)$. The Frobenius companion and the more recently-discovered Fiedler companion matrices are examples. Both the Frobenius and Fiedler companion matrices have the maximum possible number of zero entries, and in that sense are sparse. In this paper, companion matrices are explored that are not sparse. Some constructions of non-sparse companion matrices are provided, and properties that all companion matrices must exhibit are given. For example, it is shown that every companion matrix realization is non-derogatory. Bounds on the minimum number of zeros that must appear in a companion matrix, are also given.

Leibniz Algebras Associated with Representations of Euclidean Lie Algebra

In the present paper we describe Leibniz algebras with three-dimensional Euclidean Lie algebra $\mathfrak {e}(2)$ as its liezation. Moreover, it is assumed that the ideal generated by the squares of elements of an algebra (denoted by I) as a right $\mathfrak {e}(2)$-module is associated to representations of $\mathfrak {e}(2)$ in $\mathfrak {sl}_{2}({\mathbb {C}})\oplus \mathfrak {sl}_{2}({\mathbb {C}}), \mathfrak {sl}_{3}({\mathbb {C}})$ and $\mathfrak {sp}_{4}(\mathbb {C})$. Furthermore, we present the classification of Leibniz algebras with general Euclidean Lie algebra ${\mathfrak {e(n)}}$ as its liezation I being an (n + 1)-dimensional right ${\mathfrak {e(n)}}$-module defined by transformations of matrix realization of $\mathfrak {e(n)}$. Finally, we extend the notion of a Fock module over Heisenberg Lie algebra to the case of Diamond Lie algebra $\mathfrak {D}_{k}$ and describe the structure of Leibniz algebras with corresponding Lie algebra $\mathfrak {D}_{k}$ and with the ideal I considered as a Fock $\mathfrak {D}_{k}$-module.

Matrix Bailey Lemma and the Star-Triangle Relation

We compare previously found finite-dimensional matrix and integral operator realizations of the Bailey lemma employing univariate elliptic hypergeometric functions. With the help of residue calculus we explicitly show how the integral Bailey lemma can be reduced to its matrix version. As a consequence, we demonstrate that the matrix Bailey lemma can be interpreted as a star-triangle relation, or as a Coxeter relation for a permutation group.

Increasing the Accuracy of Solving Discrete Ill-Posed Problems by the Random Projection Method

To solve discrete ill-posed problems by the random projection method, the error bias and variance that arise from averaging over the random matrix realizations are studied. An estimate for the input vector is obtained that makes possible to significantly improve the accuracy of solving such problems using the random projection method.