PDF superior On the properties for modifications of classical orthogonal polynomials of discrete variables. (revised version October 1996)

On the properties for modifications of classical orthogonal polynomials of discrete variables. (revised version October 1996)

On the properties for modifications of classical orthogonal polynomials of discrete variables. (revised version October 1996)

The structure of the paper is as follows. In Section 2, we provide the basic properties of the classical orthogonal polynomials of discrete variable which will be needed, as well as the main data for the Meixner, Kravchuk and Charlier polynomials. In Section 3 we deduce expressions of the generalized Meixner, Kravchuk and Charlier polynomials and its rst dierence derivatives, as well as their representation as hypergeometric functions in the direction raised by Askey. In Section 4, we nd the second order dierence equation which these generalized polynomials satisfy. In Section 5, from the three term recurrence relation (TTRR) of the classical orthogonal polynomials we nd the TTRR which satisfy the perturbed ones. In Section 6, from the relation of the perturbed polynomials P An (x) as a linear combination of the classical ones, we nd the tridiagonal matrices associated with the perturbed monic orthogonal polinomial sequence (PMOPS) f P An (x) g
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Some Extension of the Bessel Type Orthogonal Polynomials.

Some Extension of the Bessel Type Orthogonal Polynomials.

The structure of the paper is as follows. In Section 2 we list some of the main properties of the classical Bessel polynomials which will be used later on. In Section 3 we dene the generalized polynomials and nd some of their properties. In Section 4 we obtain the representation of the generalized Bessel polynomials in terms of the hypergeometric functions. In Section 5 we obtain an asymptotic formula for these polynomials and in Section 6 we establish their quasi-orthogonality. Finally, in Sections 7 and 8 we obtain the three-term recurrence relation that such polynomials satisfy as well as the corresponding Jacobi matrices.
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On characterizations of classical polynomials

On characterizations of classical polynomials

It can be shown (see e.g., [3,8,25]) that the only families satisfying the above definition are the Hermite, Laguerre, Jacobi, and Bessel polynomials. Nevertheless there are other properties characterizing such families and that can be used to define the classical OPS. The oldest one is the so called Hahn characterization—unless this was firstly observed and proved for the Jacobi, Laguerre, and Hermite polynomials by Sonin in 1887. In [12], Hahn proved the following, Theorem 1.2 (Sonin–Hahn [12,19]). A given sequence of orthogonal polynomials (P n ) n , is a classical sequence if
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18 Lee mas

Matrix moment perturbations and the inverse Szegő matrix transformation

Matrix moment perturbations and the inverse Szegő matrix transformation

The term moment problem was used for the first time in T. J. Stieltjes’ clas- sic memoir [32] (published posthumously between 1894 and 1895) dedicated to the study of continued fractions. The moment problem is a question in classical analysis that has produced a rich theory in applied and pure mathematics. This problem is beautifully connected to the theory of orthogonal polynomials, spectral representation of operators, matrix factorization problems, probability, statistics, prediction of stochastic processes, polynomial optimization, inverse problems in financial mathematics and function theory, among many other areas. In the ma- trix case, M. Krein was the first to consider this problem in [21], and later on some density questions related to the matrix moment problem were addressed in [14, 15, 24, 25]. Recently, the theory of the matrix moment problem is used in [10] for the analysis of random matrix-valued measures. Since the matrix moment problem is closely related to the theory of matrix orthogonal polynomials, M. Krein was the first to consider these polynomials in [22]. Later, several researchers have made contributions to this theory until today. In the last 30 years, several known properties of orthogonal polynomials in the scalar case have been extended to the matrix case, such as algebraic aspects related to their zeros, recurrence relations, Favard type theorems, and Christoffel–Darboux formulas, among many others.
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DUAL PROPERTIES OF ORTHOGONAL POLYNOMIALS OF DISCRETE VARIABLES ASSOCIATED WITH THE QUANTUM ALGEBRA Uq (su(2))

DUAL PROPERTIES OF ORTHOGONAL POLYNOMIALS OF DISCRETE VARIABLES ASSOCIATED WITH THE QUANTUM ALGEBRA Uq (su(2))

A detailed study of this family was done in [16] and their main characteristics are given in Table 2 of [16]. Let us now study the duality properties of the q-Racah polynomials. First of all, notice that all the characteristics of these polynomials transform into the corresponding ones by replacing the q-numbers [m] with the standard ones m and the q-Gamma functions e Γ q (x), with the classical ones Γ(x). Therefore, it is reasonable to expect that all the results in Sec. 3 can be extended to this case just replacing the standard numbers and functions by their symmetric q-analogs. We will show only the details for the first case, since the other three are equivalent and we will include only the final result.
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28 Lee mas

A characterization of the classical orthogonal discrete and q-polynomials

A characterization of the classical orthogonal discrete and q-polynomials

The classical orthogonal polynomials are very interesting mathematical objects that have attracted the attention not only of mathematicians since their appearance at the end of the XVIII century connected with some physical problems. They are used in several branches of mathematical and physical sciences and they have a lot of useful properties: they satisfy a three-term recurrence relation (TTRR), they are the solution of a second order linear differential (or difference) equation, their derivatives (or finite differences) also constitute an orthogonal family, their generating functions can be given explicitly, among others (for a recent review see e.g. [1]). Among such properties, a fundamental role is played by the so-called characterization theorems, i.e., such properties that completely define and characterize the classical polynomials. Obviously not every property characterize the classical polynomials and as an example we can use the TTRR. It is well-known that, under certain conditions—by the so-called Favard Theorem (for a review see [7])—, the TTRR characterizes the orthogonal polynomials (OP) but there exist families of OP that satisfy a TTRR but not a linear differential equation with polynomial coefficients, or a Rodrigues-type formula, etc. In this paper we will complete the works [3,10] proving a new characterization for the classical discrete [3,6] and the q-classical [4,10] polynomials. For the continuous case see [8,9].
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The modification of classical Hahn polynomials of a discrete variable.(revised version October 1996)

The modification of classical Hahn polynomials of a discrete variable.(revised version October 1996)

as a modication of the rst ones troughtout the addition of two mass points. All the formulas for the classical Hahn polynomials can be found in a lot of books ( see for instance the excellent monograph Orthogonal Polynomials in Discrete Variables by A.F. Nikiforov, S. K. Suslov, V. B. Uvarov [16], Chapter 2.)

14 Lee mas

The limit relations between generalized orthogonal polynomials.(revised version October 1996)

The limit relations between generalized orthogonal polynomials.(revised version October 1996)

In this Section we will study limit relations involving the modications of the Jacobi and Laguerre polynomials as well as the modications of the classical polynomials of discrete variables. In some way we will obtain an analogue of the Askey-scheme of hypergeometric polynomials (for a review see [18]). Results are predictible but we have found nothing of this kind in the literature. Anyway, we want to remark that the main dierence with respect to the classical case is the fact, as we will show below, that the point masses change.
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22 Lee mas

On the q-polynomials: A distributional study.

On the q-polynomials: A distributional study.

In this work we will present a di<erent approach: It can be considered a pure algebraic approach and constitutes an alternative to the two previous ones, and, in some sense is the continuation of the Hahn’s work [16]. Furthermore, we will prove here that the q-classical polynomials are characterized by several relations, analogue to the ones satis1ed by the classical “continuous” (Jacobi, Bessel, Laguerre, Hermite) and “discrete” (Hahn, Meixner, Kravchuk and Charlier) orthogonal polynomials [1,13,21,22] and references therein. Besides, our point of viewis very di<erent from the previous ones based on the basic hypergeometric series and the di<erence equation, respectively. In fact we start with the distributional equation that the q-moment functionals satisfy and we will prove all the other characterizations using basically the algebraic theory developed by Maroni [23]. So, somehow, this paper is the natural continuation of the study started in [22,13] for the “continuous” and “discreteorthogonal polynomials, respectively. Another advantage of this approach is the uni1ed and simple treatment of the q-polynomials where all the information is obtained from the coeMcients of the polynomials and of the distributional or Pearson equation (compare it with the method by the American school [20] or the Russian ones [29]).
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40 Lee mas

Second order difference equations for certain families of "discrete" polynomials.

Second order difference equations for certain families of "discrete" polynomials.

In this paper we will consider two algorithms which allow us to obtain second order linear dierence equations for certain families of polynomials. The corresponding algorithms can be implemented in any computer algebra system in order to obtain explicit expressions of the coecients of the dierence equations.

9 Lee mas

A multidimensional dynamical approach to iterative methods with memory

A multidimensional dynamical approach to iterative methods with memory

In this paper, we have adapted some tools of the dynamical analysis of multivariate real discrete problems to analyze the stability of the fixed points of iterative methods with memory on quadratic polynomials. As far as we know, this kind of analysis has not been performed before on iterative methods with memory for solving nonlinear equations. From the well-known Secant method as initial inspiration, we have designed new methods with mem- ory from Steffensen’ or Traub’s schemes (in the last case we have previously transformed it in a derivative-free method), as well as a parametric family of iterative procedures of third- and fourth-order of convergence. Our statements, based on consistent discrete dynamics results and also on Feigenbaum diagrams of the family, allow us to select the most stable elements of the class and to find those that present convergence to other points different from the solution of our problem or even chaotic behavior.
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22 Lee mas

Generating Function: Multiple Orthogonal Polynomials

Generating Function: Multiple Orthogonal Polynomials

There is a well known equivalence between determining multiple orthogonal polynomials and Hermite-Pad´ e approximants of types I and II, see [9, 10, 11]. To see how to generate the multiple orthogonal polynomials of types I and II, and the associated polynomials of the second kind, see [2, 3, 4, 8, 10, 11].

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Psychometric properties of the Spanish version of the Collective Efficacy Questionnaire for Sports

Psychometric properties of the Spanish version of the Collective Efficacy Questionnaire for Sports

The CEQS is defi ned by fi ve factors (effort, ability, preparation, persistence and unity) each consisting of four items making a total twenty items. The items are written in a clear brief way stating, «I can do» thus refl ecting the judgement ability in accordance with the recommendations by Bandura (2006). The Collective Effi cacy Questionnaire for Sports consists of an 11- point scale (0 - 10) which scores answers from «No confi dence at all» to «Absolute confi dence». The initial instructions refl ect the confi dence of the team’s capability when faced with the situation of competing in the near future («Grade to what extent your team believes in its abilities when faced with an imminent match or competition…»). These initial instructions are written in present tense considering that effi cacy is a changing construction not a characteristic.
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Psychometric properties of the leisure time satisfaction scale in family caregivers

Psychometric properties of the leisure time satisfaction scale in family caregivers

lacking (Stevens et al., 2004). One of the most important limitations of the existing scales revolves around their conceptualization of leisure, which focuses on the frequency of participation in leisure activities rather than on the meaning and satisfaction that they lend. This is particularly the case of the Victoria Longitudinal Study Activity Questionnaire, a Likert-type scale-ranging from 0 (never) to 8 (daily) - which, in the adaptation by Jopp and Hertzog (2010), comprised 57 items relating to eleven different sorts of activities (physical, crafts, games, TV, social-private, social-public, religious, developmental, experiential, technology and travel). In spite of its acceptable psychometric properties, the Victoria Longitudinal Study Activity Questionnaire only provides information on the type of leisure activity and the amount of time that the person spends doing it. Taking into account that leisure is a subjective experience (Iso-Ahola, 1980; Neulinger, 1981) which can involve different meanings, and the fact that the farther away the leisure activity is from the routine or the amount of time spent doing it are precisely the meanings that make it a potentially positive infl uence (Carbonneau, Martineau, Andre, & Dawson, 2011), it could be posited that this instrument is limited from a conceptual perspective. According to Wakui, Saito, Agree, and Kai (2012), while accepting that different activities could bring about different and specifi c effects or benefi ts, we must not overlook the different meanings that caregivers attribute to those leisure activities.
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7 Lee mas

A Rakhmanov-like theorem for orthogonal polynomials on Jordan arcs in the complex plane

A Rakhmanov-like theorem for orthogonal polynomials on Jordan arcs in the complex plane

In this paper, we consider regular Borel measures µ defined on subsets of the complex plane which are Jordan arcs, or connected finite union of Jordan arcs, and we show how the support of µ is determined by the entries of the Hessenberg matrix D associated with µ. The Hessenberg matrix is the natural generalization of the tridiagonal Jacobi matrix to the complex plane and, in the particular case of measures with support the unit circle T, the Hessenberg matrix is a Toeplitz matrix.
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11 Lee mas

Electron transport in low-dimensional systems: optoelectronic device simulations

Electron transport in low-dimensional systems: optoelectronic device simulations

Concerning dynamical models, NEGFF has been used to study the different transport material properties, such as electron conductance [33, 34] or thermoelectric characteris- tics [35]. Generally, NEGFF is used in combination with Tight Binding (TB) approach or Density Functional Theory (DFT) in order to describe from first principles the elec- trical transport. However, the computational effort demanded by NEGFF computa- tions for systems with a large number of atoms exceeds the capabilities of the current high-computing facilities being unfeasible to simulate realistic devices. Thus, several approximations have to be done like decreasing the system size; considering only one or two QDs; a simplified description of the energy level spectra of the QDs or assuming constant transitions rates [36, 37, 38, 39, 40]. Although some extra implementations have been included in NEGFF, like the potential due to the self-charge [41, 42], nobody has done a fully quantum transport study in an extended arbitrary array of QDs using this framework since this approach is usually unfeasible to implement for large systems being large QD arrays a computational challenge.
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Psychometric properties of the Spanish version of the Passion Scale

Psychometric properties of the Spanish version of the Passion Scale

Additionally, as expected, both HP and OP are positively associated with measures of valuing and passion, as in past research (Marsh et al., 2013; Vallerand et al., 2003). Surprisingly, time was only signifi cantly correlated with OP, whilst liking was only correlated with HP, instead of being both correlated with both criteria (Marsh et al., 2013; Vallerand et al., 2003, Study 1). These relationships may be better understood when we consider the type of activity separately. These results are consistent with Chamarro et al.’s (2011) fi ndings, which show that HP increases with hours of practice. It may be related to the high number of dancers involved in our sport and physical activity sample. In contrast, for videogamers, high involvement is not related to HP (Fuster et al., 2014). Also, liking was predicted by HP but not by OP. This is in line with Marsh et al. (2013), OP being associated with engagement but less with loving the activity. Nevertheless, and from the point of view of validity, the present results showed that while OP and HP were related to the various criteria, these associations varied somewhat for liking and time criteria. Future research is needed to replicate these relationship patterns with other samples.
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Modified Clebsch-Gordan-type expansions for products of discrete hypergeometric polynomials

Modified Clebsch-Gordan-type expansions for products of discrete hypergeometric polynomials

Up to now, to the best of our information, there only exists a formal, recurrent way [11] to evaluate linearization coecients. Its application to the simplest case, i.e., to the expansion of the products of two Charlier polynomials in series of Charlier polynomials of the same type, leads to a six-term recursive relation for the corresponding linearization coecients which has not yet been possible to be solved, even not at a hypergeometric level by symbolic means (Petrovsek algorithm [44]). Also , Dunkl [14] for Hahn polyno- mials and Askey and Gasper [10] for Kravchuk polynomials have been able to calculate explicitly the expansion coecients of the Clebsch-Gordan-type or conventional lineariza- tion problems (i.e., those problems which involve polynomials of the same system). They are collected in [53].
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27 Lee mas

On the “Favard theorem” and its extensions

On the “Favard theorem” and its extensions

The structure of the paper is as follows. In Section 2 we present a survey of results surrounding the Favard Theorem when a sequence of polynomials satises a linear relation like (1.1). In particular, we show that the interlacing property for the zeros of two consecutive polynomials gives basic information about the preceding ones in the sequence of polynomials.

24 Lee mas

TítuloEstudio del comportamiento de antioxidantes naturales adicionados a poliolefinas en aplicaciones industriales

TítuloEstudio del comportamiento de antioxidantes naturales adicionados a poliolefinas en aplicaciones industriales

el material que va a ser sometido a una serie de tratamientos previos como, por ejemplo, tratamientos térmicos de esterilización, retenga en un primer momento esos compuestos aditivados para que la liberación no se produzca en los primeros días o incluso horas, ya que para alargar la vida útil del alimento resulta más interesante que esa cesión sea lenta y progresiva. Por lo tanto, se busca una inmovilización del agente activo lo que implica una menor liberación inicial, lo que a su vez provoca que su actividad antioxidante dure un mayor tiempo. Para lograr este fin, en el último artículo recogido en esta memoria (“Art. 8. Interaction and release of catechin from anhidride maleic grafted polypropylene films”), se aborda la inmovilización de catequina en la matriz de polipropileno a través de la incorporación de polipropileno modificado superficialmente con anhídrido maleico.
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