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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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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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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.)

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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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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 “**discrete**” **orthogonal** **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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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.

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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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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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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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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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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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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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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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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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