Manuel Bogoya

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• PhD
• Professor (Associate) at University of Valle

We study a family of non-Hermitian tetradiagonal Toeplitz matrices having a limiting set consisting of one analytic arc only. We derive individual asymptotic expansions for all eigenvalues as the matrix size grows to infinity. Additionally, we provide specific expansions for the extreme eigenvalues, which are those approaching the endpoints of the...

Under appropriate technical assumptions, the simple-loop theory allows to derive various types of asymptotic expansions for the eigenvalues of Toeplitz matrices () generated by a function. Unfortunately, such a theory is not available in the preconditioning setting, that is for matrices of the form −1 () () with , real-valued, nonnnegative and not...

It is known that the generating function of a sequence of Toeplitz matrices may not describe the asymptotic distribution of the eigenvalues of the considered matrix sequence in the non-Hermitian setting. In a recent work, under the assumption that the eigenvalues are real, admitting an asymptotic expansion whose first term is the distribution func...

8 The eigenvalues of Toeplitz matrices Tn(f) with a real-valued generating function f , satisfying some 9 conditions and tracing out a simple loop over the interval [−π, π], are known to admit an asymptotic 10 expansion with the form 11 λj(Tn(f)) = f (σj,n) + c1(σj,n)h + c2(σj,n)h 2 + O(h 3), where h = 1/(n + 1), σj,n = πjh, and c k are some bounde...

The present paper is a survey of some of the authors’ results on the asymptotic behavior of individual eigenvalues and eigenvectors of sequences of Toeplitz matrices when their size tends to infinity. The symbols of the matrices are supposed to have power singularities and are special cases of so-called Fisher–Hartwig symbols.

In this paper we consider a family of tetradiagonal (= four non-zero diagonals) Toeplitz matrices with a limiting set consisting in one analytic arc only and obtain individual asymptotic expansions for all the eigenvalues, as the matrix size goes to infinity. Additionally, we provide specific expansions for the extreme eigenvalues which are the eig...

The present work is devoted to the construction of an asymptotic expansion for the eigenvalues of a Toeplitz matrix Tn(a)$$ {T}_n(a) $$ as n$$ n $$ goes to infinity, with a continuous and real‐valued symbol a$$ a $$ having a power singularity of degree γ$$ \gamma $$ with 1<γ<2$$ 1<\gamma <2 $$, at one point. The resulting matrix is dense and its en...

The present work is devoted to the construction of an asymptotic expansion for the eigenvalues of a Toeplitz matrix Tn(a) as n goes to infinity, with a continuous and real-valued symbol a having a power singularity of degree γ with 1 < γ < 2, at one point. The resulting matrix is dense and its entries decrease slowly to zero when by going the main...

The present work is devoted to the eigenvalue asymptotic expansion of the Toeplitz matrix Tn(a) whose generating function a is complex valued and has a power singularity at one point. As a consequence, Tn(a) is non-Hermitian and we know that the eigenvalue computation is a non-trivial task in the non-Hermitian setting for large sizes. We follow the...

It is known that the generating function f of a sequence of Toeplitz matrices {T n (f)} n may not describe the asymptotic distribution of the eigen-values of T n (f) in the non-Hermitian setting. In a recent paper, we assumed the following working hypothesis: if the eigenvalues of T n (f) are real for all n,

The analysis of the spectral features of a Toeplitz matrix-sequence {Tn(f)}n∈N, generated by the function f∈L1([−π,π]), real-valued almost everywhere (a.e.), has been provided in great detail in the last century, as well as the study of the conditioning, when f is nonnegative a.e. Here we consider a novel type of problem arising in the numerical ap...

The eigenvalues of Toeplitz matrices Tn(f) with a real-valued generating function f , satisfying some conditions and tracing out a simple loop over the interval [−π, π], are known to admit an asymptotic expansion with the form λj(Tn(f)) = f (σj,n) + c1(σj,n)h + c2(σj,n)h 2 + O(h 3), where h = 1 n+1 , σj,n = πjh, and ck are some bounded coefficients...

Under appropriate technical assumptions, the simple-loop theory allows to derive various types of asymptotic expansions for the eigenvalues of Toeplitz matrices generated by a function f. Independently and under the milder hypothesis that f is even and monotone over [0,π], matrix-less algorithms have been developed for the fast eigenvalue computati...

In the present article we consider a type of matrices stemming in the context of the numerical approximation of distributed order fractional differential equations (FDEs). From one side they could look standard, since they are real, symmetric and positive definite. On the other hand they cause specific difficulties which prevent the successful use...

A R T I C L E I N F O Keywords: Toeplitz matrix spectra preconditioned matrix asymptotic expansion A B S T R A C T Under appropriate technical assumptions, the simple-loop theory allows to deduce various types of asymptotic expansions for the eigenvalues of Toeplitz matrices () generated by a function , unfortunately, such a theory is not available...

In previous works Bohemian matrices have attracted the attention of several researchers for their rich combinatorial structure and they have been studied intensively, from several points of view, including height, determinants, characteristic polynomials, normality, and stability. Here we consider a selected number of examples of upper Hessenberg a...

The eigenvalues of Toeplitz matrices $T_{n}(f)$ with a real-valued symbol $f$, satisfying some conditions and tracing out a simple loop over the interval $[-\pi,\pi]$, are known to admit an asymptotic expansion with the form \[ \la_{j}(T_{n}(f))=f(d_{j,n})+c_{1}(d_{j,n})h+c_{2}(d_{j,n})h^{2}+O(h^{3}), \] where $h=\frac{1}{n+1}$, $d_{j,n}=\pi j h$,...

In the present note we consider a type of matrices stemming in the context of the numericalapproximation of distributed order fractional differential equations (FDEs): from one sidethey could look standard, since they are, real, symmetric and positive definite. On the otherhand they present specific difficulties which prevent the successful use of...

The analysis of the spectral features of a Toeplitz matrix-sequence Tn(f) n∈N , generated by a symbol f ∈ L 1 ([−π, π]), real-valued almost everywhere (a.e.), has been provided in great detail in the last century, as well as the study of the conditioning, when f is nonnegative a.e. Here we consider a novel type of problem arising in the numerical a...

In previous works Bohemian matrices have attracted the attention of several researchers for their rich combinatorial structure and they have been studied intensively, from several viewpoints, including height, determinants, characteristic polynomials, normality, and stability. Here we consider a selected number of examples of upper Hessenberg and T...

Low-density polyethylene (LDPE) sheets (3.0 ± 0.1 cm) received sequential treatment, first by the action of direct-current low-pressure plasma (DC-LPP) with a 100% oxygen partial pressure, 3.0 × 10⁻² mbar pressure, 600 V DC tension, 5.6 cm distance, 6-min treatment. Then, sheets were submitted to TiO2 photocatalysis at UV radiation at 254 nm (TiO2/...

Different types of structures and substrates are used for urban extensive green roofs. However, there is not enough information about the performance of these structures and substrates for growing edible plants in tropical climate conditions. This study evaluates the best combination of three different modular extensive green roof structures and tw...

Multi-objective optimization problems (MOPs) naturally arise in many applications. Since for such problems one can expect an entire set of optimal solutions, a common task in set based multi-objective optimization is to compute N solutions along the Pareto set/front of a given MOP. In this work, we propose and discuss the set based Newton methods f...

A brief but comprehensive review of the averaged Hausdorff distances that have recently been introduced as quality indicators in multi-objective optimization problems (MOPs) is presented. First, we introduce all the necessary preliminaries, definitions, and known properties of these distances in order to provide a stat-of-the-art overview of their...

In a previous work we studied the asymptotic behavior of individual inner eigenvalues of the n-by-n truncations of a certain family of infinite Hessenberg Toeplitz matrices as n goes to infinity. In the present work we deal with the extreme eigenvalues. The generating function of the Toeplitz matrices is supposed to be of the form \( a(t)\,= \, \fr...

The Hausdorff distance is a widely used tool to measure the distance between different sets. For the approximation of certain objects via stochastic search algorithms this distance is, however, of limited use as it punishes single outliers. As a remedy in the context of evolutionary multi-objective optimization (EMO), the averaged Hausdorff distanc...

Low-density polyethylene (LDPE) waste generates an environmental impact. To achieve the most suitable option for their degradation, it is necessary to implement chemical, physical and biological treatments, as well as combining procedures. Best treatment was prognosticated by Plackett-Burman Experimental Design (PB), evaluating five factors with tw...

The averaged Hausdorff distance ?p is an inframetric which has been recently used in evolutionary multiobjective optimization (EMO). In this paper we introduce a new two-parameter performance indicator ?p,q which generalizes ?p as well as the standard Hausdorff distance. For p, q 1 the indicator ?p,q (that we call the (p, q)-averaged distance) turn...

Analysis of the asymptotic behaviour of the spectral characteristics of Toeplitz matrices as the dimension of the matrix tends to infinity has a history of over 100 years. For instance, quite a number of versions of Szegő's theorem on the asymptotic behaviour of eigenvalues and of the so-called strong Szegő theorem on the asymptotic behaviour of th...

Исследование асимптотического поведения спектральных характеристик тeплицевых матриц, когда размерность матрицы стремится к бесконечности, имеет более чем столетнюю историю. Например, хорошо известны многочисленные варианты теоремы Сeгe об асимптотическом распределении собственных чисел и так называемой сильной теоремы Сегe об асимптотическом повед...

Colombia applies two mandatory National State tests every year. The first, known as Saber 11, is applied to students who finish the high school cycle, whereas the second, called Saber Pro, is applied to students who finish the higher education cycle. The result obtained by a student on the Saber 11 exam along with his/her gender, and socioeconomic...

In a sequence of previous works with Albrecht Böttcher, we established higher-order uniform individual asymptotic formulas for the eigenvalues and eigenvectors of large Hermitian Toeplitz matrices generated by symbols satisfying the so-called simple-loop condition, which means that the symbol has only two intervals of monotonicity, its first deriva...

The paper is devoted to the structure and the asymptotics of the eigenvector matrix of Hermitian Toeplitz matrices with moderately smooth symbols which trace out a simple loop on the real line. The results extend existing results on banded Toeplitz matrices to full Toeplitz matrices with temperate decay of the entries in the first row and column. W...

We present conditions that allow us to pass from the convergence of probability measures in distribution to the uniform convergence of the associated quantile functions. Under these conditions, one can in particular pass from the asymptotic distribution of collections of real numbers, such as the eigenvalues of a family of $n$-by-$n$ matrices as $n...

The collective behavior of the singular values of large Toeplitz matrices is described by the Avram–Parter theorem. In the case of Hermitian matrices, the Avram–Parter theorem is equivalent to Szegő’s theorem on the eigenvalues. The Avram–Parter theorem in conjunction with an improvement made by Trench implies estimates in the mean between the sing...

The paper presents higher-order asymptotic formulas for the eigenvalues of large Hermitian Toeplitz matrices with moderately smooth symbols which trace out a simple loop on the real line. The formulas are established not only for the extreme eigenvalues, but also for the inner eigenvalues. The results extend and make more precise existing results,...

The concept of academic added-value can be associated with a variation in the cognitive development of students who complete an educational cycle at a given institution belonging to a reference universe, related to the tendency shown by all students in the aforementioned universe. Certain knowledge at the beginning and end of the cycle must be eval...

The paper is concerned with finite Hermitian Toeplitz matrices whose entries in the first row grow like a polynomial. Such matrices cannot be viewed as truncations of an infinite Toeplitz matrix which is generated by an integrable function or a nice measure. The main results describe the first-order asymptotics of the extreme eigenvalues as the mat...

The paper is devoted to the eigenvectors of Hessenberg Toeplitz matrices whose symbol has a power singularity. We describe the structure of the eigenvectors and prove an asymptotic formula which can be used to compute individual eigenvectors effectively. The symbols of our matrices are special Fisher–Hartwig symbols, and the theorem of this paper c...

We study the asymptotic behavior of individual eigenvalues of the n-by-n truncations of certain infinite Hessenberg Toeplitz matrices as n goes to infinity. The generating function of the Toeplitz matrices is supposed to be of the form a(t) = t −1 (1 − t) α f (t) (t ∈ T), where α is a positive real number but not an integer and f is a smooth functi...

In a recent paper, we established asymptotic formulas for the eigenvalues of the $n\times n$ truncations of certain infinite Hessenberg Toeplitz matrices as $n$ goes to infinity. The symbol of the Toeplitz matrices was of the form $a(t)=t^{-1}(1-t)^{\alpha}f(t)$ ($t\in{\mathbb T}$), where $\alpha$ is a positive real number but not an integer and $f...

The axiom of choice says that for any collection of sets (or for any set of sets) X, exists a function f such that f(x) ∈ x for all non empty x ∈ X, i.e. f takes an element in each set of the collection X, such function is called a choice function, it is customary to weak the axiom of choice by putting some extra condition for the set X such that:...