PDF superior Distribution of zeros of discrete and continuous polynomials from their recurrence relation.

Distribution of zeros of discrete and continuous polynomials from their recurrence relation.

Distribution of zeros of discrete and continuous polynomials from their recurrence relation.

The global behaviour of the zeros of the discrete and continuous classical orthogonal poly- nomials in both nite and asymptotic cases has received a great deal of attention from the early times [22, 27, 45] of approximation theory up to now [8, 9, 20, 23, 24, 31, 32, 33, 34, 36, 37, 41, 42, 46, 47, 48, 50]. Indeed, numerous interesting results have been found from the dif- ferent characterizations (explicit expression, weight function, recurrence relation, second order dierence or dierential equation) of the polynomial. See [16] for a survey of the published results up to 1977; more recent discoveries are collected in [49] and [31] for continuos and dis- crete polynomials, respectively. Still now, however, there are open problems which are very relevant by their own and because of its numerous applications to a great variety of classical systems [29, 35] as well as quantum-mechanical systems whose wavefunctions are governed by orthogonal polynomials in a \discrete" [2, 3, 40, 43, 44] or a \continuous" variable [5, 18, 19, 39]. In this paper the attention will be addressed to the problem of determination of the moments of the distribution density of zeros for a classical orthogonal polynomial of a given order n in both discrete and continuous cases as well as its asymptotic values (i.e, when n ! 1 ), which
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19 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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E. Buend a - The distribution of zeros of general q-polynomials.

E. Buend a - The distribution of zeros of general q-polynomials.

The method of proof used is very straightforward. It is based on an explicit formula for the moments-around-the-origin of the discrete density of zeros of a polynomial with a given degree in terms of the coecients of the three term recurrence relation [37], as described in Lemma 1 given below. This method was previously employed to normal (non-q) polynomials where recurrence coecients are given by means of a rational function of the degree [38], as well as to correspond- ing Jacobi matrices [39] encountered in quantum mechanical description of some physical systems. The paper is structured as follows. Firstly, in section 2, one introduces a general set of q- polynomials f P n ( x ) q g Nn
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29 Lee mas

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

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

Markett [37] for symmetric orthogonal polynomials, has designed a method which be- gins with the three-term recurrence relation of the involved orthogonal polynomial system to set up a partial dierence equation for the (orthogonal) polynomial, in case of connec- tion problems, or for the product of two (orthogonal) polynomials, in case of linearization problems, to be expanded; then, this equation has to be solved in terms of the initial data. Ronveaux et al. [47, 48] for classical and semiclassical orthogonal polynomials and Lewanowicz [30, 34] for classical orthogonal polynomials have proposed alternative, sim- pler techniques of the same type although they require the knowledge of not only the recurrence relation but also the dierential-dierence relation or/and the second-order dierence equation, respectively, satised by the polynomials of the orthogonal set of the expansion problem in consideration. See also [5, 11, 23, 31, 45, 46] for further description and applications of this method in the discrete case, and [19, 32] in the continuous case as well as [3, 33] for the q -discrete orthogonality. Koepf and Schmersau [25] has proposed a computer-algebra-based method which, starting from the second order dierence hyper- geometric equation, produces by symbolic means and in a recurrent way the expansion coecients of the classical discrete orthogonal hypergeometric polynomials (CDOHP) in terms of the falling factorial polynomials (already obtained analytically by Lesky [28]; see also [41], [42]) as well as the expansion coecients of its corresponding inverse problem. The combination of these two simple expansion problems allows these authors to solve the connection problems within each specic CDOHP set.
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À la carte recurrence relations for continuous and discrete hypergeometric functions

À la carte recurrence relations for continuous and discrete hypergeometric functions

The recurrence relations for the special functions (and thus for the associated wave functions) are interesting not only from the theoretical point of view (they are useful for computing the values of matrix elements of certain physical quantities see e.g. [11] and references therein) but also they can be used to numerically compute the values of the functions as well as their derivatives, as it is shown in [6] for the case of Laguerre polynomials and the associated wave functions of the harmonic oscillator and the hydrogen atom. Nevertheless, we need to point out that, although the recurrence relations seem to be more useful for the evaluation of the corresponding functions than other direct methods, one should be very careful when using them (see e.g. the nice surveys [16, 23] on numerical evaluation and the convergence problem that appears when dealing with recurrence relations, or the most recent ones [17, 18] for the case of the hypergeometric function 2 F 1 ).
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A multidimensional dynamical approach to iterative methods with memory

A multidimensional dynamical approach to iterative methods with memory

The behavior of these fixed points is reflected in the dynamical planes; it can be observed that the roots of the polynomial and also the possible attracting fixed points are located on the bisectrix z = x of the plane. The stable behavior of the origin can be seen at Figure 5b, where the basin of attraction of point (0, 0) is plotted in blue. Moreover, in Figures 5c, 5d and 5e only the basins of attraction of the solutions are found but the second one shows wider regions of stability; let us remark that this picture corresponds to the value of θ that provides higher order of convergence of the family. Finally, a highly unstable behavior can be observed in Figures 5a and 5f, where big black areas of no convergence to the roots appear and the basins of attraction of the roots are very narrow.
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Estudio de audiometría para músicos

Estudio de audiometría para músicos

According to Dr.Marshall Chasin from the hearing journal, a critical bandwidth is approximately a third of an octave, which is why one-third octave bandwidth analysis is so common in our field. One can attribute loudness for a single pure tone at 1000 Hz. Adding another at 1050 Hz will slightly increase its intensity but not its loudness. The same is true of adding a third pure tone at 1100 Hz, and this holds true as long as the sound energy is within a critical bandwidth. When adding another pure tone at 1200 Hz (exceeding the critical bandwidth at 1000 Hz, which is about 160 Hz), however, a person will subjectively report that the sound is now louder because the sound spectrum has exceeded the critical bandwidth. A bandwidth is explained as well using filters, for example a band-pass filter allows a range of frequencies within the bandwidth to pass through while stopping those outside the cut-off frequencies as it is shown Figure 2.
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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)

The research of the rst author was supported by a grant of Instituto de Cooperacion Iberoamericana (I.C.I.) of Spain. He is very greateful to Departamento de Ingeniera, Uni- versidad Carlos III de Madrid for his kind hospitality. The research of the second author was supported by Comision Interministerial de Ciencia y Tecnologa (CICYT) of Spain under grant PB 93-0228-C02-01.

14 Lee mas

Catalog Competition: Theory and Experiments

Catalog Competition: Theory and Experiments

Palfrey 2002 and Hummel, 2010). That is, we consider that two …rms compete in catalogs (locations and prices at the same time) but, compared to the standard setup in which …rms locate at a point in the unit interval we focus on a case in which …rms can locate to the western, to the central or to the eastern district of the linear city. We have to stress here, though, that despite the fact that the set of locations in our setup is …nite, the set of available prices is not. Therefore, since the strategy space of each …rm is the Cartesian product of these two sets, we have that the strategy space of each …rm is in…nite. This, along with the facts that the game is not a constant-sum game and that …rms’ payo¤ functions exhibit discontinuities, implies that existence of a unique symmetric equilibrium and, thereafter, possibility of full characterization are not guaranteed by any known theorem.
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20 Lee mas

Dispersal and extrapolation on the accuracy of temporal predictions from distribution models for the Darwin's frog

Dispersal and extrapolation on the accuracy of temporal predictions from distribution models for the Darwin's frog

are valid (Fitzpatrick and Hargrove 2009). Our results suggest that, similarly to spatial extrapolation (Heikki- nen et al. 2012), a good capability of SDMs to predict species distributions under training conditions does not guarantee equally good performance when these are transferred in time. In spite of this, environmental extrapolation seems to be a situation that often cannot be avoided when correlative SDMs are being transferred in space or time. For this reason, our findings demon- strate the importance of environmental extrapolation for temporal transference of SDMs, and is consistent with the recommendations of reporting the degree of environ- mental extrapolation both for temporal and spatial transference of SDMs (e.g., Elith et al. 2010, Zurell et al. 2012, Mesgaran et al. 2014) to prevent erroneous or imprecise predictions, or at least communicate where model predictions are reliable and where they are not. Bayesian Hierarchical models (e.g., Dynamic Range Models; Pagel and Schurr 2009) have been shown to pro- duce reliable predictions in time (using simulated data; Schurr 2012), with several advantages including the inference of spatiotemporal range dynamics in equilib- rium and non-equilibrium conditions, and the capacity of reporting the within-model uncertainty of the predic- tions. These advantages could be especially relevant in predicting range dynamics when models are transferred in time or space and extrapolated to novel environmen- tal conditions (Schurr 2012).
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Lista de Publicaciones del CIMAR (1979   2004)

Lista de Publicaciones del CIMAR (1979 2004)

007 Price, K., M. Bussing, W.A. Bussing, D. Maurer & C. Bartels 1980. Finfish survey: 83-144. In: D, Mau- rer, C. Epifanio and K. Price (Eds.). Ecological as- sessment of finfish and megabenthic invertebrates as indicators of natural and impacted habitats in the Gulf of Nicoya, Costa Rica. Progress Report of the 1979 International Sea Grant Program University of Delaware, College of Marine Studies, Newark. 008 Maurer, D., C. Epifanio, K. Price, J.A. Vargas, M.M.

16 Lee mas

Mischa Cotlar, in memoriam (1913 2007)

Mischa Cotlar, in memoriam (1913 2007)

[7] M. Cotlar and L. Recht, Rodolfo A. Ricabarra (August 1925–November 1984), in Volume in homage to Dr. Rodolfo A. Ricabarra (Spanish), i–v, Univ. Nac. del Sur, Bah´ıa Blanca, 1995. [8] M. Cotlar and C. Sadosky, The Adamjan-Arov-Kre˘ın theorem in general and regular repre- sentations of R 2 and the symplectic plane, in Toeplitz operators and related topics (Santa

11 Lee mas

On the “Favard theorem” and its extensions

On the “Favard theorem” and its extensions

claims “We have been in possession of this proof for several years. Recently Favard published an identical proof in the Comptes Rendus”. Also Natanson in his book [17, p. 167] said “This theorem was also discovered (independent of Favard) by the author (Natanson) in the year 1935 and was presented by him in a seminar led by S.N. Bernstein. He then did not publish the result since the work of Favard appeared in the meantime”. The “same” theorem was also obtained by Perron [19], Wintner [28] and Sherman [20], among others.

24 Lee mas

Recurrence relations for discrete hypergeometric functions.

Recurrence relations for discrete hypergeometric functions.

Concluding remarks. In this paper we present a simple, unified and con- structive approach for finding linear recurrence relations for the difference hypergeometric-type functions, i.e., solutions of the hypergeometric differ- ence equation (2.1), and apply the general results to some discrete mod- els (e.g. discrete oscillators). Furthermore, the method described here is valuable for new situations, such as higher order recurrence relations and ladder-type relations for the classical discrete orthogonal polynomials.

22 Lee mas

On the distribution of gymnosperm genera and their areas of endemism and cladistic biogeography

On the distribution of gymnosperm genera and their areas of endemism and cladistic biogeography

Abstract. A distributional analysis of 81 gymnosperm genera was undertaken. On the basis of the congruence in the distribution of these genera, nine areas of endemism were recognised. Many of these areas also represent areas of endemism for other plant and animal taxa. South-western China and New Caledonia are particularly interesting from the viewpoint of gymnosperm diversity and endemism. The suggested areas of endemism agree in part with some floristic regions previously proposed. The congruence between the areas of endemism suggested and postulated Pleistocene refuges and panbiogeographic nodes is discussed. A cladistic biogeographic analysis was carried out and a general area cladogram obtained by strict consensus shows two major components, one Gondwanic and the other almost Laurasian. This cladogram was compared with previous studies and the similarities and differences among relationship areas are discussed.
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12 Lee mas

Numerical Implementation of Gradient Algorithms

Numerical Implementation of Gradient Algorithms

where the apostrophe 0 denotes matrix transpose. We emphasize that the con- struction of a practical optimization algorithm from Hopfield networks stems from the dynamical properties [2]: since a Lyapunov function is decreasing over time, thus the evolution of the states of the network heads towards a stable equilibrium, where the Lyapunov function presents a—possibly local—minimum. Therefore, an optimization problem where the target function has the same struc- ture of the Lyapunov function given by Equation (6) is solved by matching the target and the Lyapunov function, so that the weights and biases are obtained. Finally, the network is “constructed”, which usually means that some numerical method is used to integrate Equation (4) until the states reach an equilibrium, which provides the minimum. Therefore, using a numerical method that pre- serves the dynamical properties of the continuous ODE under discretization is a crucial step, since a solution is only obtained by an algorithm that mimics the correct dynamical behaviour.
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10 Lee mas

On weighted average interpolation with cardinal splines

On weighted average interpolation with cardinal splines

After this paper was flnished, the authors knew that the same result had been obtained independently and at the same time by Ponnaian and Shanmugam in [8]. Although both papers use Theorem 3, the way those authors follow to study the zeros of the G function is different from ours. Ponnaian and Shanmugam base their arguments on the study that they carry out about the roots of the exponential Euler splines. In particular, they need to obtain a recursion relation for those exponential splines. Moreover, they consider four different cases on the degree d. On the contrary, our arguments may be more direct, since we apply already known properties of the Euler-Frobenius polynomials and only have to consider even and odd degree cases separately.
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10 Lee mas

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

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

In other words, these polynomials are obtained via the addition of delta Dirac measures to a positive denite weight function [11, 14]. A special emphasis was given to the case when is the classical \discrete" weight function corresponding to the classical \discrete" [12] polynomials of Hahn [3], Meixner [1, 2, 6], Kravchuk [2] and Charlier [2, 7] for which the coecients , and in (1.2) are polynomials such that and do not depend of n , is a constant, degree( ) 2 and degree( ) 1. In all these cases, when one or
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9 Lee mas

A generalization of the classical Laguerre polynomials: Asymptotic Properties and Zeros.

A generalization of the classical Laguerre polynomials: Asymptotic Properties and Zeros.

The rst approach to such a kind of modications of a moment linear functional is [3]. In particular, necessary and sucient conditions for the existence of a sequence of polynomials orthogonal with respect to such a linear functional are obtained. Furthermore, an extensive study for the new orthogonal polynomials was performed when the initial functional is semi- classical.

18 Lee mas

On the Krall-type discrete polynomials

On the Krall-type discrete polynomials

Special cases of quasi-definite linear functionals are the classical ones (those of Ja- cobi, Laguerre, Hermite, and Bessel). In the last years perturbations of the func- tional via the addition of Dirac delta functions —the so-called Krall-type orthog- onal polynomials— have been extensively studied (see e.g. [6, 7, 16, 20, 21, 22, 26] and references therein), i.e., the linear functional

18 Lee mas

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