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

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

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

of such properties in terms of the location of the mass points with respect to the support of the measure. Particular emphasis was given to measures supported in [ 1 ; 1] and satisfying some extra conditions in terms of the parameters of the three term recurrence relation which the corresponding sequence of orthogonal polynomials satisfy.

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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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On the modification of classical orthogonal polynomials: The symmetric case.

On the modification of classical orthogonal polynomials: The symmetric case.

The analysis of properties of polynomials orthogonal with respect to a perturbation of a measure via the addition of mass points was introduced by P.Nevai [23]. There the asymptotic properties of the new polynomials have been considered. In particular, he proved the dependence of such properties in terms of the location of the mass points with respect to the support of the measure. Particular emphasis was given to measures sup- ported in [ 1 ; 1] and satisfying some extra conditions in terms of the parameters of the three-term recurrence relation that the corresponding sequence of orthogonal polynomials satisfy.
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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.

A general system of q-orthogonal polynomials is dened by means of its three-term recur- rence relation. This system encompasses many of the known families of q-polynomials, among them the q-analog of the classical orthogonal polynomials. The asymptotic density of zeros of the system is shown to be a simple and compact expression of the parameters which char- acterize the asymptotic behavior of the coecients of the recurrence relation. This result is applied to specic classes of polynomials known by the names q-Hahn, q-Kravchuk, q-Racah, q-Askey & Wilson, Al Salam-Carlitz and the celebrated q-little and q-big Jacobi.
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We obtain an explicit expression for the Sobolev-type orthogonal polynomials

We obtain an explicit expression for the Sobolev-type orthogonal polynomials

polynomials have been obtained in [3]. However, the explicit form of these polynomials in the general case remains as an open question as well as the study of their zeros. We are trying in this paper to cover this lack. Moreover, some of the usual properties of classical orthogonal polynomials { sym- metry property, their representation as hypergeometric series and the second order linear dierential equation { are translated to the context of Sobolev-type ortogonality.

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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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Asymptotic properties of generalized Laguerre orthogonal polynomials

Asymptotic properties of generalized Laguerre orthogonal polynomials

L (α) n (x) = 2 − 1 π − 1/2 e x/2 ( − x) − α/2 − 1/4 n α/2 − 1/4 e 2( − nx) 1 / 2 1 + O n − 1/2 . (4) This relation holds for x in the complex plane cut along the positive real semiaxis; both ( − x) − α/2 − 1/4 and ( − x) 1/2 must be taken real and positive if x < 0 . The bound of the remainder holds uniformly in every closed domain which does not overlap the positive real semiaxis.

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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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Procesos de difusión en una dimensión y polinomios ortogonales

Procesos de difusión en una dimensión y polinomios ortogonales

We describe the general theory of diffusion processes, which contains as a particular case the solutions of stochastic differential equations. The idea of the theory is to construct explicitly the generator of the Markov process using the so-called scale function and the speed measure. We also explain how the theory of orthogonal polynomials help to study some diffusions. In addition, using the theory of diffusions, we present the Brox model, which is a process in a random environment.
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Two applications of the subnormality of the Hessenberg matrix related to general orthogonal polynomials

Two applications of the subnormality of the Hessenberg matrix related to general orthogonal polynomials

A matrix versión of this theorem, (see [5,pp. 68]), and Zhedanov has constructed, using the symmetrized Al-Salam-Carlitz polynomials, examples of orthogonal polynomials for a discrete m[r]

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INTERPOLATION OF LOW-ORDER HRTF FILTERS USING A ZERO DISPLACEMENT MEASURE

INTERPOLATION OF LOW-ORDER HRTF FILTERS USING A ZERO DISPLACEMENT MEASURE

This operation guarantees that the substituted zeros are inside the unit circle, not disturbing the minimum-phase property of the HRTF filter. The maximum error in the magnitude spectrum is less than 0.3dB when ε is selected as 0.001. The same operation can also be applied to a couple of zeros that are both negative, which will produce new negative quasi-complex zeros. The number of real zeros was decreased to 2 for all of the FIR filters. After this reduction, it is possible to represent the FIR filters as a combination of 62 complex zeros and 2 real zeros:
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The phenomenological, epistemological, and semiotic components of generalization

The phenomenological, epistemological, and semiotic components of generalization

The second article, “The Distributed Nature of Pattern Generalization” (Rivera, 2015), focuses on the students’ ability to generalize patterns. Rivera highlights the in- terrelation between pattern generalization and mathematical structure, and explores the cognitive and noncognitive factors that influence pattern generalization. In his analysis, he resorts to the sequence shown in Figure 2 (see above) and a variant of it— the same sequence without a dark square. He proposes a conceptual framework of fac- tors influencing pattern generalization, which includes: (a) natures and sources of generalization (something related to what I termed the ground of the generalization at the beginning of this Introduction); (b) types of structures, which include additive, multiplicative, and iterative thinking, and that are important in the construction of functional-based generalizations; (c) attention or awareness; (d) representations; and (e) context, which will favour or encumber the formulation of a general formula (in arithmetic or verbal procedural terms).
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Normalizing or not normalizing? An open question for floating-point arithmetic in embedded systems

Normalizing or not normalizing? An open question for floating-point arithmetic in embedded systems

We also test intermediate solutions, where left-shifting is performed but only for a few bits. Fig. 2 shows this new adder that we denote as A2 (A2H for HUB version). Comparing it with A1, the A2 design has a special leading zero detector, which detects up to two leading zeros at the output of the absolute value circuit. Furthermore, it has a barrel shifter that can perform a one-position right-shifting (in case of detecting overflow) and left-shifting up to 2-bit positions. This will increase the area and the delay of the critical path, but it will improve the error figures as we will see in section V. In this architecture the exponent has to be decremented when left- shifting is performed, and therefore underflow could happen. Although it is not depicted in Fig. 2, this situation is detected in the design and the result flushes to zero.
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Effects of physicochemical soil properties of five agricultural soils on herbicide soil adsorption and leaching

Effects of physicochemical soil properties of five agricultural soils on herbicide soil adsorption and leaching

amount of this herbicide molecule being present in cationic form in these soils (Table 4). In the case of the Ultisol soil, the higher CEC is de- rived from the clay content, and in the Andisol soil, it is associated with the FAH content, which is highly related to the adsorption of herbicides like MCPA (Iglesias et al., 2009) (Table 5). In general, the desorption process did not have a highly significant relationship with any pa- rameter, except for the clay content (Table 5). The contrast between the high correlation of the desorption percentages of herbicides with the soil clay content, and not with the OC content, and the high correlation between the adsorption of these herbicides and the soil OC content (es- pecially the HS content), but not the soil clay content, indicates the presence of a certain de- gree of hysteresis in the adsorption-desorption process. Therefore, the adsorption mechanism of the herbicides in these soils might differ from the desorption mechanism (Weber and Weil- ing, 1998). However, it is important to consider that the estimation of the desorption process is associated with a high degree of uncertainty because the adsorption-desorption process re- quires long periods of time to equilibrate and, in some cases, may take weeks or months to ap- proach a real balance (Ball and Roberts, 1991; Weiling and Weber, 1998). Subsequently, the times for which desorption measurements were taken in this study might have been too short to achieve a realistic balance.
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Higher order hypergeometric Lauricella function and zero asymptotics of orthogonal polynomials

Higher order hypergeometric Lauricella function and zero asymptotics of orthogonal polynomials

Our aim here is to express in terms of higher order hypergeometric Lauricella functions the corresponding asymptotic contracted measure of zeros for the sequence {P n (x)}™ =1 to be de[r]

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Layers of generality and types of generalization in pattern activities

Layers of generality and types of generalization in pattern activities

As previously described, the objectification of knowledge is a theoretical construct to account for the way in which the students engage with something in order to notice and make sense of it. By focusing on the students’ phenomenol- ogical mathematical experience, it emphasizes the subjective dimension of know- ing. But this is only half of the story. Since we are sociocultural knowers, objec- tification takes also account of the social and cultural dimensions of knowing. The concept of knowledge objectification rests indeed on the idea that class- rooms are not merely a bunch of external conditions to which the students must adapt. Classrooms are rather seen as interactive zones of mediated activities con- veying scientific, ethical, aesthetical and other culturally and historically formed values that the students objectify through reflective and active participation (Rad- ford, 2008). In these activities, embedded in cultural, historical traditions, the students relate not only to the objects of knowledge (the subject-object plane), but also to other students through face-to-face, virtual or potential communica- tive actions (the subject-subject plane or plane of social interaction).
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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.

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Deformed generalization of the semiclassical entropy

Deformed generalization of the semiclassical entropy

for problems with more than a few degrees of freedom. Another great advantage of the semiclassical approximation lies in that it facilitates an intuitive understanding of the underlying physics, which is usu- ally hidden in blind numerical solutions of the Schr¨odinger equation. Although semiclassical mechanics is as old as the quantum theory itself, the field is continuously evolving. There still exist many open problems in the mathematical aspects of the approximation as well as in the quest for new effective ways to apply the approximation to various physical systems (see, for instance, [1, 2] and references therein). In a different vein, applications of the so-called q-calculus to statistical mechanics have accrued in- creasing interest lately [3]. This q-calculus [4] has its origin in the q-deformed harmonic oscillator theory, which, in turn, is based on the construction of a SU q(2) algebra of q-deformed commutation or anti-commutation relations between creation and annihilation operators [5–7]. The above mentioned applications also employ “deformed information measures” (DIM) that have been applied to different scientific disciplines (see, for example, [3, 8, 9] and references therein). DIMs were introduced long ago in the cybernetic-information communities by Harvda-Charvat [10] and Vadja [11] in 1967-68, being rediscovered by Daroczy in 1970 [12] with several echoes mostly in the field of image processing. For a historic summary and the pertinent references see Ref. [13]. In astronomy, physics, economics, biology, etc., these deformed information measures are often called q-entropies since 1988 [9].
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A new look at the Feynman ‘hodograph’ approach to the Kepler first law

A new look at the Feynman ‘hodograph’ approach to the Kepler first law

For an historical view of this question we refer to a paper by Derbes [ 2 ] which also gives a very complete discussion of the problem in the language of classical Euclidean geometry, including the contribution to this very problem of outstanding fi gures such as Maxwell [ 4 ] . The historical constructions are extended in this paper even to parabolic orbits ( see also the paper [ 5 ]) . The hodograph circular character for the Kepler problem is closely related to the exis- tence of a speci fi cally Keplerian constant of motion which which is an exceptional property of the central potential with radial dependence 1 r . From a purely historic viewpoint, this vector can be traced back to the beginning of of the 18th century, with Hermann and Bernoulli ( see two notes by Goldstein [ 6, 7 ]) , and was later rediscovered independently several times. The connection with the circular character of the hodograph seems to be due to Hamilton [ 1 ] ; from a modern viewpoint all these distinguished properties are linked to the superintegrability of the Kepler problem ( for a moderately advanced discussion, see [ 8 ]) .
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The distributed nature of pattern generalization

The distributed nature of pattern generalization

In the context of figural PG, Radford’s initial layer of structural generaliza- tion is factual, that is, it is “a generalization of numerical actions in the form of an operational scheme that remains bound to the numerical level, nevertheless allowing the students to virtually tackle any particular case successfully” (Rad- ford, 2001b, pp. 82-83). So, for example, one group of Grade 8 students in his study noticed that since the first two stages in pattern 1 of Figure 16 seem to fol- low the sense “it’s always the next… 1 + 2, 2 + 3,” that allowed them to impose the factual structure of “25 plus 26” in the case of Stage 25 of the pattern. Here the multiplicative dimension pertains to the two growing composite parts corre- sponding to the top and bottom rows of circles (versus the additive strategy of counting-all in which case circles are counted one by one and from stage to stage). Factual generalizations are often accompanied by the use of adverbs such as “the next” or “always,” including the effects of rhythm of an utterance and movement (e.g., a pointing gesture). While perhaps necessary in the beginning stage of generalizing, unfortunately, factual generalizations remain context- bound and numerical and often draw on shared “implicit agreements and mutual comprehension” (Radford, 2001b, p. 83) among those who construct them in so- cial activity.
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