Orthogonal polynomials and quadratic transformations

33 pages, no figures.-- MSC1991 code: Primary 42C05.MR#: MR1680116 (2000b:42021)Zbl#: Zbl 0936.42012Starting from a sequence $\{P_n\}_{n\geq 0}$ of monic polynomials orthogonal with respect to a linear functional ${\bf u}$, we find a linear functional ${\bf v}$ such that $\{Q_n\}_{\geq 0}$, with either $Q_{2n}(x)=P_n(T(x))$ or $Q_{2n+1}(x)=(x-a)\,P_n(T(x))$ where $T$ is a monic quadratic polynomial and a\in\C, is a sequence of monic orthogonal polynomials with respect to ${\bf v}$. In particular, we discuss the case when ${\bf u}$ and ${\bf v}$ are both positive definite linear functionals. Thus, we obtain a solution for an inverse problem which is a converse, for quadratic mappings, of one analyzed in [11].This paper was finished with financial support of a Grant from Junta Nacional de Investigação Científica e Tecnológica (JNITC) - BD976 - and Centro de Matemática da Universidade de Coimbra (CMUC) of Portugal. The work of the first author was supported by the Dirección General de Enseñanza Superior (DGES) of Spain - PB96-0120-C03-01.Publicad

Orthogonal polynomials and cubic polynomial mappings. II. The positive-definite case

10 pages, no figures.-- MSC1991 codes: Primary 42C05.MR#: MR1862232 (2002g:42029)Let $\{P_n\}_{n\geq 0}$ be a sequence of polynomials orthogonal with respect to some distribution function $\sigma$ and let $\{Q_n\}_{n\geq 0}$ be a simple set (i.e., each $Q_n$ has degree exactly $n$) of polynomials such that $Q_{3n+m}(x)=\theta_m(x)P_n(\pi_3(x))$ for all $n=0,1,2,\dots$ where $\pi_3$ is a fixed monic polynomial of degree 3 and $\theta_m$ a fixed polynomial of degree $m$ with $0\leq m\leq 2$. We give necessary and sufficient conditions in order that $\{Q_n\}_{n\geq 0}$ be a sequence of polynomials orthogonal with respect to some distribution function $\tilde\sigma$. Under these conditions, we prove that d\tilde\sigma(x)=\sum _{i=1}M_i\delta_{x_i}(x)dx+\chi_{\pi_3 -1}([\xi,\eta])}(x)\left frac{\theta_{2-m}(x)}{\theta_m(x)}\right frac{d\sigma(\pi_3(x))}{\pi_3'(x)} where $\chi_A$ means the characteristic function of the set $A$, $[\xi,\eta]$ is the support of $d\sigma$, $\theta_{2-m}$ denotes a polynomial of degree exactly $2-m$ and, if $m\geq 1$, $M_i$ is a mass located at the zero $x_i$ of $\theta_m(x)\equiv\prod _{i=1}(x-x_i),\ \delta_{x_i}(x)$ being the Dirac functional at the point $x_i$.First author (F.M.) supported by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain under grant BHA2000-0206 C04-01. Second author (J.P.) supported by Centro de Matemática da Universidade de Coimbra (CMUC) of Portugal.Publicad

Orthogonal polynomials and cubic polynomial mappings (I)

29 pages, no figures.-- MSC1991 code: Primary 42C05.MR#: MR1789676 (2001m:42047)We present characterization theorems for orthogonal polynomials obtained from a given system of orthogonal polynomials by a cubic polynomial transformation in the variable. Since such polynomials are the denominators of the approximants for the expansion in continued fractions of the x-transform of the moment sequences associated with the linear functionals with respect to which such polynomials are orthogonal, we state the explicit relation for the corresponding formal Stieltjes series. As an application, we study the eigenvalues of a tridiagonal 3-Toeplitz matrix. Finally, we deduce the second-order linear differential equation satisfied by the new family of orthogonal polynomials when the initial family satisfies such a kind of differential equation.The work of the first author was supported by Dirección General de Enseñanza Superior (DGES) of Spain under grant PB96-0120-C03-01 and INTAS program, INTAS-93-0219 Ext. The work of the second author was supported by Junta Nacional de Investigação Científica e Tecnológica (JNICT) under grant BD976 and Centro de Matemática da Universidade de Coimbra (CMUC) of Portugal.Publicad

First data on feeding ecology of Goshawk Accipiter gentilis during the breeding season in the natura 2000 site Dunas de Mira, Gândara e Gafanhas (Beira Litoral, Portugal)

This study describes the diet of the Goshwak Accipiter gentilis within littoral ecosystems dominated by pine trees. Samples were collected during the breeding season of 2001 in Dunas de Mira, Gândara e Gafanhas, located in the littoral centre of Portugal and recently included in the Portuguese Natura 2000 network (PTCON 055). Diet composition was assessed from the analysis of 89 pellets and 154 prey remains, collected between the 15th of April and the 25th of July of 2001. Birds represented 86.6% of the total prey items. The remaining prey items were coleoptera (6.5%), mammals (3.6%) and reptiles (3.3%). In terms of biomass birds comprised 90.9%, followed by mammals (7.3%), reptiles (1.7%) and coleoptera (0.05%). Results show that this raptor has a predominantly ornitophagous diet. Domestic pigeon Columba livia var is the most important prey in this area both in terms of frequency and biomass (42.2% and 49.9%, respectively). During the breeding season the diet of the Goshawk is largely connected to Man.Este trabalho pretendeu obter informação relativa à ecologia trófica do Açor Accipiter gentilis, de modo a tentar compreender o papel desempenhado por esta espécie de ave de rapina em ecossistemas litorais. O estudo decorreu durante a época de nidificação de 2001, na área do Sítio das "Dunas de Mira, Gândara e Gafanhas", recentemente incluÌdo na Rede Natura 2000 (PTCON 055). A composição do regime alimentar foi estudada com base em 89 regurgitaÁıes e 153 restos de presas, recolhidas entre 15 de Abril e 25 de Julho de 2001. As aves representaram 86,6% do total de presas consumidas, os coleópteros 6,5%, os mamíferos 3,6% e os répteis 3,3%. Em termos de biomassa, as aves continuam a ter um papel predominante representando 90,9% da biomassa consumida, seguidas pelos mamÌferos (7,3%), répteis (1,7%) e coleópteros (0,05%). Na área de estudo o Açor tem um comportamento alimentar predominantemente ornitófago, sendo o pombo doméstico Columba livia var a presa mais importante, tanto em termos numéricos como em biomassa (42,2% e 49,9%, respectivamente). Na base deste comportamento, estarão a elevada disponibilidade alimentar e uma maior vulnerabilidade à predação por parte desta espécie. Este regime alimentar do Açor durante o perÌodo reprodutor evidência uma clara dependÍncia desta ave de rapina em relação a uma espécie estreitamente ligada ao homem

WKB approximation and Krall-Type orthogonal polynomials

We give an uni ed approach to the Krall-type polynomials orthogonal with respect to a positive measure consisting of an absolutely continuous one \perturbed" by the addition of one or more delta Dirac functions. Some examples studied by di erent authors are considered from an unique point of view. Also some properties of the Krall polynomials are studied. The three-term recurrence relation is calculated explicitly, as well as some asymptotic formulas. With special emphasis will be considered the second order di erential equations that such polynomials satisfy which allows us to obtain the central moments and the WKB approximation of the distribution of zeros. Some examples coming from quadratic transformation polynomial mappings and tridiagonal periodic matrices are also studied.Comisión Interministerial de Ciencia y TecnologíaJunta Nacional de Investigacao Científica e Tecnológica de Portuga

(M, N)-coherent pairs of order (m, k) and Sobolev orthogonal polynomials

A pair of regular linear functionals (U,V)(U,V) is said to be a (M,N)(M,N)-coherent pair of order (m,k)(m,k) if their corresponding sequences of monic orthogonal polynomials...We thank the referees for the careful revision of the manuscript. Their comments and suggestions have contributed to improve substantially the presentation. The work of F. Marcellán, J. Petronilho, and N.C. Pinzón-Cortés has been supported by Dirección General de Investigación Científica y Técnica, Ministerio de Economía y Competitividad of Spain, under grants MTM2012-36732-C03-01 (FM and NCP-C) and MTM2012-36732-C03-02 (JP). The work of J. Petronilho was also supported by the Centro de Matemática da Universidade de Coimbra (CMUC), funded by the European Regional Development Fund through the program COMPETE and by the Portuguese Government through the FCT—Fundação para a Ciência e a Tecnologia under the project PEst-C/MAT/UI0324/2011, and the research project PTDC/MAT/098060/2008 (FCT)

Spectral properties of certain tridiagonal matrices

We study spectral properties of irreducible tridiagonal k−Toeplitz matrices and certain matrices which arise as perturbations of them.Dirección General de Enseñanza SuperiorJunta de Andalucí

On linearly related sequences of difference derivatives of discrete orthogonal polynomials

Proceedings of: OrthoQuad 2014. Puerto de la Cruz, Tenerife, Spain. January 20–24, 2014Let ν be either ω∈C∖{0} or q∈C∖{0,1} , and let Dν be the corresponding difference operator defined in the usual way either by Dωp(x)=p(x+ω)−p(x)ω or Dqp(x)=p(qx)−p(x)(q−1)x . Let U and V be two moment regular linear functionals and let {Pn(x)}n≥0 and {Qn(x)}n≥0 be their corresponding orthogonal polynomial sequences (OPS). We discuss an inverse problem in the theory of discrete orthogonal polynomials involving the two OPS {Pn(x)}n≥0 and {Qn(x)}n≥0 assuming that their difference derivatives Dν of higher orders m and k (resp.) are connected by a linear algebraic structure relation such as ∑Mi=0ai,nDmνPn+m−i(x)=∑Ni=0bi,nDkνQn+k−i(x),n≥0, Turn MathJax off where M,N,m,k∈N∪{0} , aM,n≠0 for n≥M , bN,n≠0 for n≥N , and ai,n=bi,n=0 for i>n . Under certain conditions, we prove that U and V are related by a rational factor (in the ν− distributional sense). Moreover, when m≠k then both U and V are Dν -semiclassical functionals. This leads us to the concept of (M,N) - Dν -coherent pair of order (m,k) extending to the discrete case several previous works. As an application we consider the OPS with respect to the following Sobolev-type inner product ⟨p(x),r(x)⟩λ,ν=⟨U,p(x)r(x)⟩+λ⟨V,(Dmνp)(x)(Dmνr)(x)⟩,λ>0, Turn MathJax off assuming that U and V (which, eventually, may be represented by discrete measures supported either on a uniform lattice if ν=ω , or on a q -lattice if ν=q ) constitute a (M,N) - Dν -coherent pair of order m (that is, an (M,N) - Dν -coherent pair of order (m,0) ), m∈N being fixed.We are grateful to Prof. Francisco Marcellán for his valuable comments and remarks that helped us to improve the paper. This work was supported by Dirección General de Investigación, Desarrollo e Innovación, Ministerio de Economía y Competitividad of Spain, under grants MTM2012-36732-C03 (RAN, NCP-C, JP), Junta de Andalucía (Spain) under grants FQM262, FQM-7276, and P09-FQM-4643 (RAN), FEDER funds (RAN

Dependents admitted to LTMU and ICCT: a comparative study on health condition evolution

Communication included in symposium: National Network of Integrated Continuous Care: evolution of the dependents health profile. (3rd World Congress of Health Research) Symposium coordinator: Fernando Petronilho, Escola Superior de Enfermagem da Universidade do Minho, [email protected]: The National Network of Integrated Continuous Care (NNICC) integrates a set of internment types and home support. The main objective is the provision of care for dependents and the training of family members to take care. Objective: Compare the health condition evolution among dependents admitted on the Long Term and Maintenance Units (LTMU) and the Integrated Continuous Care Teams (ICCT). Material and Methods: Exploratory study of quantitative profile with an inter-subject and intra-subject design. A sample of 466 cases: 241 admitted in 3 LTMU and 225 in 4 ICCT of Minho, Portugal. Was applied the form “Profile of dependents integrated in NNICC providers" in admission (A) and clinical discharge (D), for 1 year (between 2013 and 2014). Results: Dependents on the ICCT, are older, have longer dependency time and greater autonomy reconstruction potential; fewer deaths and fewer exacerbations. In LTMU there is a higher % of: pressure ulcers, joint stiffness and ineffective coughing. In an inter-subject evaluation, dependents on the ICCT are less dependent on self-care, in A [t (464) = - 5.573, p <.001], and D [t (232) = - 4.303, p <.001] ; lower bodily processes commitment either in A [t (464) = - 2.567], p = .011] and D [t (233) = - 2.009, p = .046]. In an intra-subject evaluation, on ICCT there are a higher number of cases with positive development between admission and discharge, on the dependency level and bodily processes commitment. Conclusion: The dependents accompanied by ICCT have better health condition and positive evolution compared to those admitted to LTMU

A zona tectônica Teixeira Terra Nova - ZTTTN: fundamentos da geologia regional e isotópica

A sigmoidal-shape linear tectonic zone occurs from Teixeira (Paraíba, to the north-northeast)- to Terra Nova (Pernambuco, to the south-southwest) as a line diagonal to the"Transversal Zone", which is the central tectonic portion of the Borborema Province, thus dividing this province into two different litho-structural and tectonic domains. To the north and northwest occur the Early Neoproterozoic Riacho Gravatá (RG) and the Neoproterozoic Piancó-Alto Brígida (PAB) terranes, both of them characterized by supracrustal rocks with low-grade of regional metamorphism. To the south-southwest occur the Alto Pajeú terrane (TAP, Early Neoproterozoic, orthognaisses and mesozonal supracrustal rocks) and the domain of its basement, the Alto Moxotó terrane (TAM, Paleoproterozoic in age), with high grade metamorphic rocks. This proposed"tectonic zone" is mostly composed by orthogneisses (from TAP and its Paleoproterozoic basement) pierced by an almost continuous series of intrusive granitic and syenitic rocks (plutons and stocks) of the Ediacaran period (ages between 630 and 570 Ma). During the evolutionary history of this linear portion of the plateau of the Borborema province there are some (direct and indirect) records of polycyclic geologic events of the Paleoproterozoic, Early Neoproterozoic, Ediacaran, Cambrian and Cenozoic era (a sedimentary veneer, evolution of a supergenic crust and regional uplift) with interesting implications for the regional geology, geography and social-economic development. In this paper the emphasis will be given to the isotopic data (Rb-Sr, Sm-Nd, mostly) along this special zone and the domains that it separates, as well its role on the regional tectonic framework. For this purpose, we take into consideration an expressive amount of available geological and isotopic data, obtained along the last decade (most of them not published) from different sources. Additionally, some first interpretative hypotheses will be postulated, where this tectonic zone (ZTTTN) would be formed as by-product of three parallel Neoproterozoic subduction zones.Uma zona tectônica linear, de forma sigmoidal, ocorre de Teixeira (Paraíba, ao norte-nordeste) a Terra Nova (Pernambuco, ao sul-sudoeste), com disposição diagonal ao domínio da Zona Transversal - ZT, que é a porção central da Província Borborema, de forma a subdividir a ZT em dois domínios lito-estruturais e geotectônicos distintos. Para norte e noroeste ficam os terrenos do Riacho Gravatá - RG (Eoneoproterozóico) e Piancó-Alto Brígida - PAB (Neoproterozóico), caracterizados por baixo grau de metamorfismo. Para o sul e sudeste fica o terreno Alto Pajeú (TAP, Eoneoproterozóico, ortognaisses e rochas supracrustais mesozonais) e seu embasamento, terreno Alto Moxotó - TAM (Paleoproterozóico), de grau médio a elevado de metamorfismo. Esta zona linear ou"alto" constituído de ortognaisses (do TAP e do embasamento paleoproterozóico que aflora esparsamente) é perlongado e marcado por uma série quase contínua de rochas intrusivas, plutões e stocks graníticos variados e sieníticos do tempo Ediacarano (idades entre 630 e 570 Ma, principalmente). Na história evolutiva desta porção linear particularmente soerguida do Planalto da Borborema há o registro (direto e/ou indireto) de eventos geológicos policíclicos, do Paleoproterozóico, do Eoneoproterozóico, do Ediacarano, do Eocambriano e do Cenozóico (sedimentação correlativa, elaboração de crostas e soerguimento regional), com implicações interessantes na geografia e sócio-economia regional. No presente trabalho será dada ênfase aos dados isotópicos (métodos Rb-Sr, Sm-Nd sobretudo) desta zona diferenciada, e nos contextos por ela separados, assim como no seu papel dentro do contexto tectônico regional, usufruindo para isto de expressivo acervo de dados geológicos e isotópicos, obtidos na última década (grande parte do qual ainda inédita). Além disso, hipóteses interpretativas serão postuladas para esta linha axial tectônica que subdivide a Zona Transversal em domínios geológicos distintos, considerando-a como conseqüência e subproduto de três processos paralelos de subducção no Neoproterozóico