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addition **of** one or more delta Dirac functions. Some examples studied by dierent 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 dierential 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.

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In 1940, H. L. Krall [14] obtained three new classes **of** **polynomials** **orthogonal** with respect to measures which are not absolutely continuous with respect to the Lebesgue measure. In fact, his study is related to an extension **of** the very well known characterization **of** classical **orthogonal** **polynomials** by S. Bochner. This kind **of** measures was not considered in [22].

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We consider the modications **of** the monic Hermite and Gegenbauer **polynomials** via the addition **of** one point mass at the origin. Some properties **of** the resulting **polynomials** are studied: three-term recurrence relation, dierential equation, ratio asymptotics, hypergeometric representation as well as, for large n , the behaviour **of** their **zeros**.

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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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The hypergeometric **polynomials** in a continous or a discrete variable, whose canonical forms are the so-called classical **orthogonal** polynomial systems, are objects which naturally appear in a broad range **of** physical and mathematical elds from quantum mechanics, the theory **of** vibrating strings and the theory **of** group representations to numerical analysis and the theory **of** Sturm-Liouville dierential and dierence equations. Often, they are encoun- tered in the form **of** a three term recurrence relation (TTRR) which connects a polynomial **of** a given order with the polynomial **of** the contiguous orders. This relation can be directly found, in particular, by use **of** Lanczos-type methods, tight-binding models or the appli- cation **of** the conventional discretisation procedures to a given dierential operator. Here the distribution **of** **zeros** and its asymptotic limit, characterized by means **of** its moments around the origin, are found for the continuous classical (Hermite, Laguerre, Jacobi, Bessel) **polynomials** and for the discrete classical (Charlier, Meixner, Kravchuk, Hahn) **polynomials** by means **of** a general procedure which (i) only requires the three-term recurrence relation and (ii) avoids the often high-brow subleties **of** the potential theoretic considerations used in some recent approaches. The moments are given in an explicit manner which, at times, allows us to recognize the analytical form **of** the corresponding distribution.

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In the last decade an increasing interest on the so called q-**orthogonal** **polynomials** (or basic **orthogonal** **polynomials**) is observed ( for a review see [1], [2] and [3]). The reason is not only **of** purely intrinsic nature but also because **of** the so many applications in several areas **of** Math- ematics ( e.g., continued fractions, eulerian series, theta functions, elliptic functions,...; see for instance [4] and [5]) and Physics ( e.g., angular momentum [6] and [7] and its q-analog [8]-[11], q-Shrodinger equation [12] and q-harmonic oscillators [13]-[19]). Moreover, it is well known the connection between the representation theory **of** quantum algebras (Clebsch-Gordan coecients, 3j and 6j symbols) and the q-**orthogonal** **polynomials**, (see [20], [21] (Vol. III), [22], [23], [24] ), and the important role that these q-algebras play in physical applications (see for instance [26]-[31] and references therein).

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Special important examples appear when is a classical linear functional and the mass points are located at the ends **of** the interval or **orthogonal**- ity. In this case we have studied the corresponding sequences **of** **orthogonal** **polynomials** in several papers: For the Laguerre linear functional see [2, 3], for the Bessel linear functional see [6], for the Jacobi linear functional see [7] and for the Hermite case see [1]. In particular the quasi-definiteness **of** e , relative asymptotics, the representation as a hypergeometric function, and the location **of** their **zeros** have been obtained.

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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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After studying some examples that the literature provides us with, one may realize that, even thought it is generic to assume the perturbing matrix polynomial W ( x ) to have a nonsingular leading coefﬁcient, many examples do have a singular matrix as its leading coefﬁcient. This situation is a special feature **of** the matrix case setting since in the scalar case, having a singular leading term would mean that this coefﬁcient is just zero (affecting, **of** course, to the degree **of** the polynomial). For this reason, when dealing with this kind **of** matrix **polynomials** talking about their degree should make no sense. The effect that this fact has on our reasoning is that since deg [ det W ( x )] ≤ Np the information encoded in the **zeros** (and corresponding adapted **polynomials**) **of** det W (x ) is no longer enough to make the matrices kN **of** the needed size. Therefore, there will be no way

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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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Let us point out here that the theory **of** **orthogonal** **polynomials** on the non-uniform lattices is based not on the Pearson equation and on the hypergeometric-type di<erence equation **of** the non-uniform lattices as it is shown in papers [7,26,28] and obviously it is possible to derive many properties **of** the q-classical **polynomials** from this di<erence hypergeometric equation. Our purpose is not to show howfrom the di<erence equation many properties can be obtained, but to showthat some **of** them characterize the q-classical **polynomials**, i.e., the main aim is the proof **of** several characterizations **of** these q-families as well as the explicit computations **of** the corresponding coeMcients in a uni1ed way. Some **of** these results on characterizations (e.g. the Al-Salam-Chihara or Marcell+an et al. characterization for classical **polynomials**) are completely new as far as we know.

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Firstly, we will consider the case when we add a point mass at x = 0. This case corresponds to the Laguerre, Charlier, Meixner and Kravchuk **polynomials**. Later on, we will consider the Jacobi and Hahn **polynomials** which involve two point masses at the ends **of** the interval **of** orthogonality. The reason **of** such a choice **of** the point in which we will add our posi- tive mass will be clear from formulas (39) and (41) from below, because in such formulas appears the value **of** the kernel **polynomials** K n (x;y) and they have a very simple analyti-

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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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Head-related transfer function (HRTF) defines the spectral shaping **of** the sound signal on its way from a source location in the free field to the external ear. HRTFs are generally measured at discrete positions on a spherical grid equidistant from centre **of** the human head (or mannequin) whose measurements are being taken [1]. These measurements provide a set **of** HRTFs at discrete values **of** azimuth and elevation. The HRTFs provide a frequency dependent function **of** interaural intensity difference (IID) and a constant interaural time delay (ITD). The HRTF data obtained from these measurements are used for designing low-order digital filters to be made available for use in real-time audio spatialisation applications. A number **of** different approaches regarding the HRTF filter design process have been proposed [2][3][4].

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In this work we show that, in the case **of** regular measures µ whose support is a Jordan arc or a connected union **of** Jordan arcs in the complex plane C, the limits **of** the values at the diagonals **of** the Hessenberg matrix D **of** µ, supposing those limits exist, determine the terms **of** the coefficients **of** the series expansion **of** the Riemann map φ(z) (see [20]) which applies conformally the exterior **of** the unit disk in the exterior **of** the support **of** the measure. As a consequence, the support **of** µ can be determined just knowing the limits **of** the values at the diagonals **of** its Hessenberg matrix D.

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In Chapter III, we propose space-time-frequency codes for two transmit antennas over frequency selective fading channels. First, the channel model and design criteria for which a STF code guarantees multipath diversity and high coding gain over a MIMO-OFDM system are discussed. Furthermore, in the first part **of** the Chapter, we propose a STF code called Extended Super-**Orthogonal** Space-Time-Frequency Trellis Code (Ex-SOSTFTC). Here, we consider the concept **of** rotated constellations and we avoid parallel transitions in the trellis structure. Then, we extend the Super-**Orthogonal** Space-Time Trellis Codes (SOSTTCs) originally designed for the frequency-flat fading case, to the frequency-selective fading channel. Afterwards, decoding **of** extended-STF trellis codes is shown. In the second part **of** the chapter, we propose a coding scheme called Quasi-**Orthogonal** Space-Time-Frequency Trellis Codes (QOSTFTCs), where we systematically combine a Quasi-**Orthogonal** Space-Time Block Code (QOSTBC) with a trellis code, operating over a frequency selective fading channel. In addition, in order to provide the maximum coding gain for the QOSTFTCs, we will describe a systematic method to do set partitioning. Then, we derive a decoding process for the QOSTFTCs, and then the diversity **of** QOSTFTCs is discussed. Finally, performance simulation results **of** the proposed Ex-SOSTFTC and QOSTFTCs will be presented and examined. We show that our designs are able to exploit the full diversity gains available in the MIMO-OFDM channel, and can achieve high coding gain, good performance, and low decoding complexity with a low number **of** states in the trellis. Both proposals get full symbol rate (one symbol per frequency tone per time slot). We show with analysis and numerical simulations that our designs **of** STF codes outperform the best existing space-time-frequency trellis codes in the literature.

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By offspring selection, the best children are chosen and become the parents **of** the next generation. Typically, parent selection in ES is performed randomly with no regard to fitness; survival in ESs simply saves the µ best individuals, which is only based on the relative ordering **of** their fitness values. Basically, there are two selection strategies for ESs:

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4.- Formulas for Change of basis of chromatic polynomials. 4.1 Change from null basis to complete basis and inversely. 4.2 Change from null basis to tree basis and inversely.. ü ANALE[r]

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After the measurement **of** the machining forces, the acquired data was processed to convert from the original signal to a sin- gle force value, in order to facilitate the comparison between different machining conditions. Therefore, the machining for- ce signal was filtered using MatLab software, and a portion **of** it was further selected to obtain an average value. The se- lection criteria considered avoiding instability zones, utilizing only the data from the central millimeter, from the total 5 mm in radial depth **of** cut (i.e. data from 2 mm after the start **of** the cut and 2 mm before the end **of** the cut was discarded). Once the points were selected and averaged, a final average force va- lue was obtained among the four (4) experimental replications. The final average values from the dynamometer force compo- nents are reported in Figures 5 and 6. The cutting force compo- nent, F c , corresponds to the z dynamometer direction.

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