## Browse

### Recent Submissions

Now showing 1 - 20 of 435

Publication Interlacing of zeros of Laguerre polynomials of equal and consecutive degree(Taylor & Francis, 2021-08-03) Arvesú Carballo, Jorge; Driver, K.; Littlejohn, L. L.; Ministerio de Economía y Competitividad (España)We investigate interlacing properties of zeros of Laguerre polynomials L n ( alpha ) ( x ) and L n + 1 ( alpha + k ) ( x ) , alpha > - 1 , where n is an element of N and k is an element of { 1 , 2 } . We prove that, in general, the zeros of these polynomials interlace partially and not fully. The sharp t-interval within which the zeros of two equal degree Laguerre polynomials L n ( alpha ) ( x ) and L n ( alpha + t ) ( x ) are interlacing for every n is an element of N and each alpha > - 1 is 0 < t <= 2 , [Driver K, Muldoon ME. Sharp interval for interlacing of zeros of equal degree Laguerre polynomials. J Approx Theory, to appear.], and the sharp t-interval within which the zeros of two consecutive degree Laguerre polynomials L n ( alpha ) ( x ) and L n - 1 ( alpha + t ) ( x ) are interlacing for every n is an element of N and each alpha > - 1 is 0 <= t <= 2 , [Driver K, Muldoon ME. Common and interlacing zeros of families of Laguerre polynomials. J Approx Theory. 2015;193:89-98]. We derive conditions on n is an element of N and alpha, alpha > - 1 that determine the partial or full interlacing of the zeros of L n ( alpha ) ( x ) and the zeros of L n ( alpha + 2 + k ) ( x ) , k is an element of { 1 , 2 } . We also prove that partial interlacing holds between the zeros of L n ( alpha ) ( x ) and L n - 1 ( alpha + 2 + k ) ( x ) when k is an element of { 1 , 2 } , n is an element of N and alpha > - 1 . Numerical illustrations of interlacing and its breakdown are provided.Publication Zeros of Jacobi and ultraspherical polynomials(Springer, 2023-06) Arvesú Carballo, Jorge; Driver, K.; Littlejohn ., Lance Lee; Ministerio de Economía y Competitividad (España)Suppose {P-n((alpha,beta))(x)}(n=0)(infinity) is a sequence of Jacobi polynomials with alpha, beta > -1. We discuss special cases of a question raised by Alan Sokal at OPSFA in 2019, namely, whether the zeros of P-n((alpha,beta)) (x) and P-n+k((alpha+t,beta+s)(x)) are interlacing if s, t > 0 and k is an element of N. We consider two cases of this question for Jacobi polynomials of consecutive degree and prove that the zeros of P-n((alpha,beta)) (x) and P-n+1((alpha,beta+1)) (x), alpha > -1, beta > 0, n is an element of N, are partially, but in general not fully, interlacing depending on the values of alpha, beta and n. A similar result holds for the extent to which interlacing holds between the zeros of P-n((alpha,beta)) (x) and P-n+1((alpha+1,beta+1)) (x), alpha > -1, beta > -1. It is known that the zeros of the equal degree Jacobi polynomials P-n((alpha,beta)) (x) and P-n((alpha-t,beta+s)) (x) are interlacing for alpha - t > -1, beta > -1, 0 <= t, s <= 2. We prove that partial, but in general not full, interlacing of zeros holds between the zeros of P-n((alpha,beta)) (x) and P-n((alpha+1,beta+1)) (x), when alpha > -1, beta > -1. We provide numerical examples that confirm that the results we prove cannot be strengthened in general. The symmetric case alpha = beta = lambda - 1/2 of the Jacobi polynomials is also considered. We prove that the zeros of the ultraspherical polynomials C-n((lambda))(x) and C-n+1((lambda+1)) (x), lambda > -1/2, are partially, but in general not fully, interlacing. The interlacing of the zeros of the equal degree ultraspherical polynomials C-n((lambda)) (x) and C-n((lambda+3)) (x), lambda > -1/2, is also discussed.Publication Asymptotics of matrix valued orthogonal polynomials on [−1,1](Elsevier Inc, 2023-06-15) Deaño Cabrera, Alfredo; Kuijlaars, Arno B.J.; Román, P.We analyze the large degree asymptotic behavior of matrix valued orthogonal polynomials (MVOPs), with a weight that consists of a Jacobi scalar factor and a matrix part. Using the Riemann–Hilbert formulation for MVOPs and the Deift–Zhou method of steepest descent, we obtain asymptotic expansions for the MVOPs as the degree tends to infinity, in different regions of the complex plane (outside the interval of orthogonality, on the interval away from the endpoints and in neighborhoods of the endpoints), as well as for the matrix coefficients in the three-term recurrence relation for these MVOPs. The asymptotic analysis follows the work of Kuijlaars, McLaughlin, Van Assche and Vanlessen on scalar Jacobi-type orthogonal polynomials, but it also requires several different factorizations of the matrix part of the weight, in terms of eigenvalues/eigenvectors and using a matrix Szegő function. We illustrate the results with two main examples, MVOPs of Jacobi and Gegenbauer type, coming from group theory.Publication The Cauchy Exponential of Linear Functionals on the Linear Space of Polynomials(MDPI, 2023-04-02) Marcellán Español, Francisco José; Sfaxi, Ridha; Comunidad de Madrid; Universidad Carlos III de Madrid; Agencia Estatal de Investigación (España); Ministerio de Ciencia e Innovación (España)In this paper, we introduce the notion of the Cauchy exponential of a linear functional on the linear space of polynomials in one variable with real or complex coefficients using a functional equation by using the so-called moment equation. It seems that this notion hides several properties and results. Our purpose is to explore some of these properties and to compute the Cauchy exponential of some special linear functionals. Finally, a new characterization of the positive-definiteness of a linear functional is given.Publication Generalized mixed type Bernoulli-Gegenbauer polynomials(University of Kragujevac. Faculty of Science, 2023-04) Quintana, YamiletThe generalized mixed type Bernoulli-Gegenbauer polynomials of order (infinite) > 1/2 are special polynomials obtained by use of the generating function method. These polynomials represent an interesting mixture between two classes of special functions, namely generalized Bernoulli polynomials and Gegenbauer polynomials. The main purpose of this paper is to discuss some of their algebraic and analytic properties.Publication Differential properties of Jacobi-Sobolev polynomials and electrostatic interpretation(MDPI, 2023-08-06) Pijeira Cabrera, Héctor Esteban; Quintero Roba, Javier Alejandro; Toribio Milane, JuanWe study the sequence of monic polynomials {S-n}n >= 0, orthogonal with respect to the JacobiSobolev inner product < f,g > s = integral(1)(-1) f (x)g(x) d mu(alpha,beta)(x) + Sigma (N)(dj)(j=1) lambda(j,k),f(k) (c(j))g((k))(cj), where N, d(j) is an element of Z(+), lambda(j,k) >= 0, d mu(alpha,beta)(x) = (1-x)(alpha)(1 + x)beta (dx), alpha, beta > -1, and c(j) is an element of R backslash(-1, 1). A connection formula that relates the Sobolev polynomials Sn with the Jacobi polynomials is provided, as well as the ladder differential operators for the sequence {S-n}(n >= 0) and a second-order differential equation with a polynomial coefficient that they satisfied. We give sufficient conditions under which the zeros of a wide class of Jacobi-Sobolev polynomials can be interpreted as the solution of an electrostatic equilibrium problem of n unit charges moving in the presence of a logarithmic potential. Several examples are presented to illustrate this interpretation.Publication Some relations between the Riemann zeta function and the generalized Bernoulli polynomials of level m(DergiPark Akademik, 2019-12-26) Quintana, Yamilet; Torres Guzmán, HéctorThe main purpose of this paper is to show some relations between the Riemann zeta function and the generalized Bernoulli polynomials of level m. Our approach is based on the use of Fourier expansions for the periodic generalized Bernoulli functions of level m, as well as quadrature formulae of Euler-Maclaurin type. Some illustrative examples involving such relations are also given.Publication The uniform Roe algebra of an inverse semigroup(Elsevier, 2021-07-01) Lledó Macau, Fernando; Martinez, Diego; Ministerio de Economía y Competitividad (España)Given a discrete and countable inverse semigroup S one can study, in analogy to the group case, its geometric aspects. In particular, we can equip S with a natural metric, given by the path metric in the disjoint union of its Schützenberger graphs. This graph, which we denote by ΛS, inherits much of the structure of S. In this article we compare the C*-algebra RS, generated by the left regular representation of S on 2(S) and ∞(S), with the uniform Roe algebra over the metric space, namely C∗u(ΛS). This yields a characterization of when RS = C∗u(ΛS), which generalizes finite generation of S. We have termed this by admitting a finite labeling (or being FL), since it holds when ΛS can be labeled in a finitary manner. The graph ΛS, and the FL condition, also allow to analyze large scale properties of ΛS and relate them with C*-properties of the uniform Roe algebra. In particular, we show that domain measurability of S (a notion generalizing Day’s definition of amenability of a semigroup, cf., [6]) is a quasi-isometric invariant of ΛS. Moreover, we characterize property A of ΛS (or of its components) in terms of the nuclearity and exactness of the corresponding C*-algebras. We also treat the special classes of F-inverse and E-unitary inverse semigroups from this large scale point of view.Publication Stability of the volume growth rate under quasi-isometries(Springer Nature, 2020-01) Granados, Ana; Pestana Galván, Domingo de Guzmán; Portilla, Ana; Rodríguez García, José Manuel; Tourís, Eva; Ministerio de Economía y Competitividad (España); Agencia Estatal de Investigación (España)Kanai proved powerful results on the stability under quasi-isometries of numerous global properties (including the volume growth rate) between non-bordered Riemannian manifolds of bounded geometry. Since his work focuses more on the generality of the spaces considered than on the two-dimensional geometry, Kanai's hypotheses are not usually satisfied in the context of Riemann surfaces endowed with the Poincaré metric. In this work we try to fill that gap and prove the stability of the volume growth rate by quasi-isometries, under hypotheses that many bordered or non-bordered Riemann surfaces (and even Riemannian surfaces with pinched negative curvature) satisfy. In order to get our results, it is shown that many bordered Riemannian surfaces with pinched negative curvature are bilipschitz equivalent to bordered surfaces with constant negative curvature.Publication Direct and inverse results for multipoint Hermite-Padé approximants(Springer Nature, 2019-06) Bosuwan, Nattapong; López Lagomasino, Guillermo; Zaldivar Gerpe, Yanely; Ministerio de Economía y Competitividad (España)Given a system of functions f = ( f1,..., fd ) analytic on a neighborhood of some compact subset E of the complex plane with simply connected complement in the extended complex plane, we give necessary and sufficient conditions for the convergence with geometric rate of the common denominators of row sequences of multipoint Hermite–Padé approximants under a general extremal condition on the table of interpolation points. The exact rate of convergence of these denominators is provided and the rate of convergence of the simultaneous approximants is estimated. These results allow us to detect the location of the poles of the system of functions which are in some sense closest to E.Publication Direct and inverse results on row sequences of simultaneous Pade-Faber approximants(Springer Nature, 2019-04) Bosuwan, Nattapong; López Lagomasino, Guillermo; Ministerio de Economía y Competitividad (España)Given a vector function F=(F1,...,Fd), analytic on a neighborhood of some compact subset E of the complex plane with simply connected complement, we define a sequence of vector rational functions with common denominator in terms of the expansions of the components Fk,k=1,...,d, with respect to the sequence of Faber polynomials associated with E. Such sequences of vector rational functions are analogous to row sequences of type II Hermite-Pade approximation. We give necessary and sufficient conditions for the convergence with geometric rate of the common denominators of the sequence of vector rational functions so constructed. The exact rate of convergence of these denominators is provided and the rate of convergence of the approximants is estimated. It is shown that the common denominators of the approximants detect the poles of the system of functions closest to E and their order.Publication Domination on hyperbolic graphs(Elsevier, 2020-11) Reyes Guillermo, Rosalío; Rodríguez García, José Manuel; Sigarreta Almira, José María; Villeta, María; Ministerio de Economía y Competitividad (España); Agencia Estatal de Investigación (España)If k ≥ 1 and G = (V, E) is a finite connected graph, S ⊆ V is said a distance k-dominating set if every vertex v ∈ V is within distance k from some vertex of S. The distance k-domination number γ kw (G) is the minimum cardinality among all distance k-dominating sets of G. A set S ⊆ V is a total dominating set if every vertex v ∈ V satisfies δS (v) ≥ 1 and the total domination number, denoted by γt(G), is the minimum cardinality among all total dominating sets of G. The study of hyperbolic graphs is an interesting topic since the hyperbolicity of any geodesic metric space is equivalent to the hyperbolicity of a graph related to it. In this paper we obtain relationships between the hyperbolicity constant δ(G) and some domination parameters of a graph G. The results in this work are inequalities, such as γkw(G) ≥ 2δ(G)/(2k + 1) and δ(G) ≤ γt(G)/2 + 3.Publication Computational and analytical studies of the Randic index in Erdös-Rényi models(Elsevier, 2020-07-15) Martínez-Martínez, C. T.; Méndez-Bermúdez, J. A.; Rodríguez García, José Manuel; Sigarreta Almira, José María; Ministerio de Economía y Competitividad (España); Agencia Estatal de Investigación (España)In this work we perform computational and analytical studies of the Randic´ index R(G) in Erdös–Rényi models G(n, p) characterized by n vertices connected independently with probability p ∈ (0, 1). First, from a detailed scaling analysis, we show that R(G) = {R(G)}/(n/2) scales with the product ξ ≈ np, so we can define three regimes: a regime of mostly isolated vertices when ξ < 0.01 (R(G) ≈ 0), a transition regime for 0.01 < ξ < 10 (where 0 < R(G) < n/2), and a regime of almost complete graphs for ξ > 10 (R(G) ≈ n/2). Then, motivated by the scaling of R(G), we analytically (i) obtain new relations connecting R(G) with other topological indices and characterize graphs which are extremal with respect to the relations obtained and (ii) apply these results in order to obtain inequalities on R(G) for graphs in Erdös–Rényi models.Publication A look at generalized degenerate Bernoulli and Euler matrices(MDPI, 2023-06-02) Hernández, Juan; Peralta, Dionisio; Quintana, Yamilet; Comunidad de Madrid; Universidad Carlos III de Madrid; Ministerio de Ciencia e Innovación (España); Agencia Estatal de Investigación (España)In this paper, we consider the generalized degenerate Bernoulli/Euler polynomial matrices and study some algebraic properties for them. In particular, we focus our attention on some matrix-inversion formulae involving these matrices. Furthermore, we provide analytic properties for the so-called generalized degenerate Pascal matrix of the first kind, and some factorizations for the generalized degenerate Euler polynomial matrix.Publication Higher-order recurrence relations, Sobolev-type inner products and matrix factorizations(Springer, 2023-01) Hermoso, Carlos; Huertas, Edmundo J.; Lastra, Alberto; Marcellán Español, Francisco José; Comunidad de Madrid; Agencia Estatal de Investigación (España)It is well known that Sobolev-type orthogonal polynomials with respect to measures supported on the real line satisfy higher-order recurrence relations and these can be expressed as a (2N + 1)-banded symmetric semi-infinite matrix. In this paper, we state the connection between these (2N + 1)-banded matrices and the Jacobi matrices associated with the three-term recurrence relation satisfied by the standard sequence of orthonormal polynomials with respect to the 2-iterated Christoffel transformation of the measure.Publication Mixed type Hermite-Padé approximation inspired by the Degasperis-Procesi equation(Elsevier Inc., 2019-06-20) López Lagomasino, Guillermo; Medina, Peralta S.; Szmigielski, J.; Ministerio de Economía, Industria y Competitividad (España)In this work we present new results on the convergence of diagonal sequences of certain mixed type Hermite-Padé approximants of a Nikishin system. The study is motivated by a mixed Hermite-Padé approximation scheme used in the construction of solutions of a Degasperis-Procesi peakon problem and germane to the analysis of the inverse spectral problem for the discrete cubic string.Publication Mean Sombor index(Shahin Digital Publisher, 2022) Méndez-Bermúdez, J. A.; Aguilar-Sánchez, R.; Molina, Edil D.; Rodríguez García, José Manuel; Comunidad de Madrid; Ministerio de Ciencia e Innovación (España)We introduce a degree–based variable topological index inspired on the power (or generalized) mean. We name this new index as the mean Sombor index: SOα(G) = P uv∈E(G) [(d α u + d α v ) /2]1/α. Here, uv denotes the edge of the graph G connecting the vertices u and v, du is the degree of the vertex u, and α ∈ R\{0}. We also consider the limit cases mSOα→0(G) and SOα→±∞(G). Indeed, for given values of α, the mean Sombor index is related to well-known opological indices such as the inverse sum indeg index, the reciprocal Randic index, the first Zagreb index, the Stolarsky–Puebla index and several ´Sombor indices. Moreover, through a quantitative structure property relationship (QSPR) analysis we show that mSOα(G) correlates well with several physicochemical properties of octane isomers. Some mathematical properties of the mean Sombor index as well as bounds and new relationships with known topological indices are also discussed.Publication Extremal problems on the general Sombor index of a graph(AIMS Press, 2022) Hernandez, Juan C.; Rodríguez García, José Manuel; Rosario, O.; Sigarreta Almira, José María; Comunidad de Madrid; Ministerio de Ciencia e Innovación (España)In this work we obtain new lower and upper optimal bounds of general Sombor indices. Specifically, we get inequalities for these indices relating them with other indices: the first Zagreb index, the forgotten index and the first variable Zagreb index. Finally, we solve some extremal problems for general Sombor indices.Publication Sequentially ordered Sobolev inner product and Laguerre-Sobolev polynomials(MDPI, 2023-04-02) Díaz González, Abel; Hernández, Juan; Pijeira Cabrera, Héctor EstebanWe study the sequence of polynomials {Sn}n≥0 that are orthogonal with respect to the general discrete Sobolev-type inner product ⟨f,g⟩s=∫f(x)g(x)dμ(x)+∑Nj=1∑djk=0λj,kf(k)(cj)g(k)(cj), where μ is a finite Borel measure whose support supp(μ) is an infinite set of the real line, λj,k≥0 , and the mass points ci , i=1,…,N are real values outside the interior of the convex hull of supp(μ) (ci∈R\Ch(supp(μ))∘) . Under some restriction of order in the discrete part of ⟨⋅,⋅⟩s , we prove that Sn has at least n−d∗ zeros on Ch(supp(μ))∘ , being d∗ the number of terms in the discrete part of ⟨⋅,⋅⟩s . Finally, we obtain the outer relative asymptotic for {Sn} in the case that the measure μ is the classical Laguerre measure, and for each mass point, only one order derivative appears in the discrete part of ⟨⋅,⋅⟩s.Publication On the generalized ABC index of graphs(MATCH, 2022) Das, Kinkar Chandra; Rodríguez García, José Manuel; Sigarreta Almira, José María; Ministerio de Ciencia e Innovación (España)The atom-bond connectivity and the generalized atom-bond connectivity indices have shown to be useful in the QSPR/QSAR researches. In particular, the atom- bond connectivity index has been applied to study the stability of alkanes and the strain energy of cycloalkanes. In this paper we obtain some bounds on these indices in terms of graph parameters. To obtain these bounds we use the mathematical tools from analysis. Some of these bounds for ABCalpha improve, when alpha = 1/2, known results on the ABC index.