Linear Algebraic Properties Research Articles

On linear algebra of r-Hankel and r-Toeplitz matrices with geometric sequence

AbstractMatrix theory plays a crucial role in solving practical problems and performing computational operations. In particular, specific types of matrices and their linear algebraic properties are of paramount significance for these processes. In this paper, we study the properties of r-Hankel and r-Toeplitz matrices whose entries are geometric sequences, and then the determinants, inverse matrix, generalized inverse matrix (the Moore-Penrose inverse), and spectral norms of such matrices are obtained.

Some results for min matrices associated with Chebyshev polynomials

Abstract In the present study, inspired by the studies in the literature, we consider Min matrix and its Hadamard exponential matrix family whose elements are Chebyshev polynomials of the first kind. Afterwards, we examine their various linear algebraic properties and obtain some inequalities. Furthermore, we shed light on the results we obtained to boost the clarity of our paper with the illustrative examples. In addition to all these, we give two MATLAB-R2023a codes that compute the Min matrix and the Hadamard exponential matrix with Chebyshev polynomials of the first kind entries, as well as calculate some matrix norms.

Some results on geometric circulant matrices involving the Leonardo numbers

In this study, by the motivation of the papers in the literature, we construct a special geometric circulant matrix $Le_{r^*}$ whose entries are the Leonardo numbers. Then, we investigate some linear algebraic properties of these matrices. More specifically, we present some bounds for the spectral norm, as well as Euclidean norm of this matrix form. For this purpose, we benefit from the spectacular properties of the Leonardo numbers. Furthermore, we throw light on the obtained results with examples. In addition to all these, we give two Matlab code in order to calculate the results related norms more easily and more accurately in a short time in the computer environment.

On Harmonic Complex Balancing Numbers

In the present work, we define harmonic complex balancing numbers by considering well-known balancing numbers and inspiring harmonic numbers. Mainly, we investigate some of their basic characteristic properties such as the Binet formula and Cassini identity, etc. In addition, one type of symmetric matrix family whose entries are harmonic complex balancing numbers is constructed. Additionally, some linear algebraic properties are obtained. Furthermore, some inequalities are stated by exploiting the well-known inequalities between various matrix norms. Finally, we illustrate the results with some numerical examples.

One Type of Symmetric Matrix with Harmonic Pell Entries, Its Inversion, Permanents and Some Norms

The Pell numbers, named after the English diplomat and mathematician John Pell, are studied by many authors. At this work, by inspiring the definition harmonic numbers, we define harmonic Pell numbers. Moreover, we construct one type of symmetric matrix family whose elements are harmonic Pell numbers and its Hadamard exponential matrix. We investigate some linear algebraic properties and obtain inequalities by using matrix norms. Furthermore, some summation identities for harmonic Pell numbers are obtained. Finally, we give a MATLAB-R2016a code which writes the matrix with harmonic Pell entries and calculates some norms and bounds for the Hadamard exponential matrix.

THE USES AND ABUSES OF AN AGE-PERIOD-COHORT METHOD: ON THE LINEAR ALGEBRA AND STATISTICAL PROPERTIES OF INTRINSIC AND RELATED ESTIMATORS.

As a sophisticated and popular age-period-cohort method, the Intrinsic Estimator (IE) and related estimators have evoked intense debate in demography, sociology, epidemiology and statistics. This study aims to provide a more holistic review and critical assessment of the overall methodological significance of the IE and related estimators in age-period-cohort analysis. We derive the statistical properties of the IE from a linear algebraic perspective, provide more precise mathematical proofs relevant to the current debate, and demonstrate the essential, yet overlooked, link between the IE and classical statistical tools that have been employed by scholars for decades. This study offers guidelines for the future use of the IE and related estimators in demographic research. The exposition of the IE and related estimators may help redirect, if not settle, the logic of the debate.

Infinite mixture models for operational modal analysis: An automated and principled approach

The development of a fully automated system identifier without the need for human intervention, is a key step for real-time vibration-based Structural Health Monitoring (SHM). In this paper a novel approach based on the Dirichlet Process Gaussian Mixture Model (DP-GMM) is developed in order to analyze the stabilization diagram. The aim is to separate the true physical modes from the mathematically spurious modes in a fully automated manner, whilst eliminating the need for any manually specified parameters or thresholds. The parametric Covariance Driven Stochastic Subspace Identification (SSI-Cov) is adopted to estimate the modal parameters, and consequently establish the initial stabilization diagram. From there, the use of a two-stage algorithm involving a DP-GMM is proposed to non-parametrically perform an automated cleaning of the stabilization diagram. The contributions of the paper are five-fold: (1) A probabilistic approach based on a DP-GMM is proposed to analyze the stabilization diagram. To the best knowledge of the authors, this study presents one of the first attempts of DP-GMM for full automation of Operational Modal Analysis (OMA). The method is validated using the field test data from a large-scale operating cable-stayed bridge, which has two closely-spaced modes around 3 Hz. Not only are these two complicated scenarios consistently identified, but the performance of the method with respect to the problem of missing modes is compared against a reference method based on the conventional multi-stage clustering technique used in OMA, wherein superior performance of the proposed method is demonstrated. (2) The method does not require specification of any threshold or parameter at any stage of the algorithm for cleaning the stabilization diagram, making the approach a potential for robust and fully automated modal identification. (3) Compared to many conventional multi-stage clustering techniques, the proposed approach is computationally efficient as intelligent updates are made to the model using multiple linear algebra properties. (4) New feature vectors are developed which are justified using a combination of mathematical rigor, visual understanding, and engineering intuition. (5) Due to the probabilistic nature of the method, the identification results are accompanied with uncertainty bounds. Several mathematical proofs are presented to explain the observed behavior of the uncertainty bounds.

Canards Existence in the Hindmarsh–Rose model

In two previous papers we have proposed a new method for proving the existence of “canard solutions” on one hand for three and four-dimensional singularly perturbed systems with only onefastvariable and, on the other hand for four-dimensional singularly perturbed systems with twofastvariables [J.M. Ginoux and J. Llibre,Qual. Theory Dyn. Syst.15(2016) 381–431; J.M. Ginoux and J. Llibre,Qual. Theory Dyn. Syst.15(2015) 342010]. The aim of this work is to extend this method which improves the classical ones used till now to the case of three-dimensional singularly perturbed systems with twofastvariables. This method enables to state a unique generic condition for the existence of “canard solutions” for such three-dimensional singularly perturbed systems which is based on the stability offolded singularities(pseudo singular pointsin this case) of thenormalized slow dynamicsdeduced from a well-known property of linear algebra. Applications of this method to a famous neuronal bursting model enables to show the existence of “canard solutions” in the Hindmarsh-Rose model.

Boolean Functions: Degree and Support

In this paper we establish some properties about Boolean functions that allow us to relate their degree and their support. These properties allow us to compute the degree of a Boolean function without having to calculate its algebraic normal form. Furthermore, we introduce some linear algebra properties that allow us to obtain the degree of a Boolean function from the dimension of a linear or affine subspace. Finally we derive some algorithms and compute the average time to obtain the degree of some Boolean functions from its support.

On Zweier generalized difference ideal convergent sequences in a locally convex space defined by Musielak-Orlicz function

Let $\mathbf{M}=(M_k)$ be a Musielak-Orlicz function. In this article, we introduce a new class of ideal convergent sequence spaces defined by Musielak-Orlicz function, using an infinite matrix, and a generalized difference matrix operator $B_{(i)}^{p}$ in locally convex spaces. We investigate some linear topological structures and algebraic properties of these spaces. We obtainsome relations related to these sequence spaces.

Vortex identification from local properties of the vorticity field

A number of systematic procedures for the identification of vortices/coherent structures have been developed as a way to address their possible kinematical and dynamical roles in structural formulations of turbulence. It has been broadly acknowledged, however, that vortex detection algorithms, usually based on linear-algebraic properties of the velocity gradient tensor, can be plagued with severe shortcomings and may become, in practical terms, dependent on the choice of subjective threshold parameters in their implementations. In two-dimensions, a large class of standard vortex identification prescriptions turn out to be equivalent to the “swirling strength criterion” (λci-criterion), which is critically revisited in this work. We classify the instances where the accuracy of the λci-criterion is affected by nonlinear superposition effects and propose an alternative vortex detection scheme based on the local curvature properties of the vorticity graph (x,y,ω)—the “vorticity curvature criterion” (λω-criterion)—which improves over the results obtained with the λci-criterion in controlled Monte Carlo tests. A particularly problematic issue, given its importance in wall-bounded flows, is the eventual inadequacy of the λci-criterion for many-vortex configurations in the presence of strong background shear. We show that the λω-criterion is able to cope with these cases as well, if a subtraction of the mean velocity field background is performed, in the spirit of the Reynolds decomposition procedure. A realistic comparative study for vortex identification is then carried out for a direct numerical simulation of a turbulent channel flow, including a three-dimensional extension of the λω-criterion. In contrast to the λci-criterion, the λω-criterion indicates in a consistent way the existence of small scale isotropic turbulent fluctuations in the logarithmic layer, in consonance with long-standing assumptions commonly taken in turbulent boundary layer phenomenology.

ENSEÑANZA DEL ÁLGEBRA DE VECTORES CON ENFOQUE POR COMPETENCIAS A IMPLEMENTARSE EN FÍSICA DE EDUCACIÓN SECUNDARIA

Esta investigación ha caracterizado el proceso de enseñanza del Álgebra de Vectores implementada en Física desde la perspectiva de la Educación Matemática en la enseñanza de Educación Secundaria, sus características y dificultades en la enseñanza. El enfoque es cuantitativo con un diseño descriptivo. La muestra la conformaron 97 estudiantes del décimo grado y 4 docentes de las asignaturas de Física y Matemática, del Instituto Nacional Autónomo "Rubén Darío", del municipio de Nueva Guinea, a los cuales se les aplicaron test, encuesta, entrevista y guía de observación. En el análisis se ha utilizado el Sistema Estadístico para la Ciencias Sociales (SPSS), en los resultados de relevancia se puede mencionar el uso inadecuado de las propiedades de Álgebra de Vectores durante los procedimientos aplicados a definiciones y conceptos físicos por los estudiantes; asimismo, la búsqueda de los docentes para contrarrestar las debilidades en el aprendizaje del álgebra y la unificación de la herramienta Matemática de manera significativa, en el programa de estudio de Física de Educación Secundaria.

Canards Existence in Memristor’s Circuits

The aim of this work is to propose an alternative method for determining the condition of existence of "canard solutions" for three and four-dimensional singularly perturbed systems with only one fast variable in the folded saddle case. This method enables to state a unique generic condition for the existence of "canard solutions" for such three and four-dimensional singularly perturbed systems which is based on the stability of folded singularities of the normalized slow dynamics deduced from a well-known property of linear algebra. This unique generic condition is perfectly identical to that provided in previous works. Application of this method to the famous three and four-dimensional memristor canonical Chua's circuits for which the classical piecewise-linear characteristic curve has been replaced by a smooth cubic nonlinear function according to the least squares method enables to show the existence of "canard solutions" in such Memristor Based Chaotic Circuits.

Algebraic properties of word equations

The question about maximal size of independent system of word equations is one of the most striking problems in combinatorics on words. Recently, Aleksi Saarela has introduced a new approach to the problem that is based on linear-algebraic properties of polynomials encoding the equations and their solutions. In this paper we develop further this approach and take into account other algebraic properties of polynomials, namely their factorization. This, in particular, allows to improve the bound for the number of independent equations with maximal rank from quadratic to linear.

Canards Existence in FitzHugh-Nagumo and Hodgkin-Huxley Neuronal Models

In a previous paper we have proposed a new method for proving the existence of “canard solutions” for three- and four-dimensional singularly perturbed systems with only onefastvariable which improves the methods used until now. The aim of this work is to extend this method to the case of four-dimensional singularly perturbed systems with twoslowand twofastvariables. This method enables stating a unique generic condition for the existence of “canard solutions” for such four-dimensional singularly perturbed systems which is based on the stability offolded singularities(pseudo singular pointsin this case) of thenormalized slow dynamicsdeduced from a well-known property of linear algebra. This unique generic condition is identical to that provided in previous works. Application of this method to the famous coupled FitzHugh-Nagumo equations and to the Hodgkin-Huxley model enables showing the existence of “canard solutions” in such systems.

STRONGLY ALMOST IDEAL CONVERGENT SEQUENCES IN A LOCALLY CONVEX SPACE DEFINED BY MUSIELAK-ORLICZ FUNCTION

In this article, we introduce a new class of ideal convergent sequence spaces using an innite matrix, Musielak-Orlicz function and a new generalized dierence matrix in locally convex spaces. We investigate some linear topological structures and algebraic properties of these spaces. We also give some relations related to these sequence spaces.

Algebraic properties of robust Padé approximants

For a recent new numerical method for computing so-called robust Padé approximants through SVD techniques, the authors gave numerical evidence that such approximants are insensitive to perturbations in the data and do not have so-called spurious poles, that is, poles with close-by zero or poles with small residuals. A black box procedure for eliminating spurious poles would have a major impact on the convergence theory of Padé approximants since it is known that convergence in capacity plus the absence of poles in some domain D implies locally uniform convergence in D.In the present paper we provide a proof for forward stability (or robustness) and show the absence of spurious poles for the subclass of so-called well-conditioned Padé approximants. We also give a numerical example of some robust Padé approximant which has spurious poles and discuss related questions. It turns out that it is not sufficient to discuss only linear algebra properties of the underlying rectangular Toeplitz matrix, since in our results other matrices like Sylvester matrices also occur. These types of matrices have been used before in numerical greatest common divisor computations.

Diagonalization of Matrices of Statistics

In the work of Varchenko, Zagier, Thibon, Renteln, Reiner, Saliola, Welker, linear algebraic properties of the multiplication map on the group algebra of the group algebra element are studied, which is the sum over all permutations weighted by q inv , q maj , inv. Here q is a variable, and inv and maj are the classical statistics inversion and major index. We define a multinomial descent statistic des X and a multinomial inversion statistic inv X . These new defined statistics are the multinomial expressions of the classical statistics descent des and inversion. We determine the spectrum and the multiplicity of each element of the spectrum of the analogously defined multiplication map on the group algebra for both des X and inv X . As corollaries, we deduce the spectrum and the multiplicity of each element of the spectrum of the defined multiplication map on the group algebra for the statistics des, maj, and inv.

On generalized A- difference strongly summable sequence spaces defined by ideal convergence on a real n-normed space

In this article, we shall introduce a new class of ideal convergent (briefly I-convergent) sequence spaces using, infinite matrix, an Orlicz function and difference operator defined on n-normed spaces. We study these spaces for some linear topological structures and algebraic properties. We also give some relations related to these sequence spaces.Mathematics Subject Classification 2010: 40A05; 40B50; 46A19; 46A45.

The GUS-property of second-order cone linear complementarity problems

The globally uniquely solvable (GUS) property of the linear transformation of the linear complementarity problems over symmetric cones has been studied recently by Gowda et al. via the approach of Euclidean Jordan algebra. In this paper, we contribute a new approach to characterizing the GUS property of the linear transformation of the second-order cone linear complementarity problems (SOCLCP) via some basic linear algebra properties of the involved matrix of SOCLCP. Some more concrete and checkable sufficient and necessary conditions for the GUS property are thus derived.