An orthogonal polynomial family that generalizes the **Racah** coefficients or 6j-symbols (so-called **Racah** **and** **q**-**Racah** **polynomials**) was introduced **in** [5]. These **polynomials** are at the top **of** the so-called Askey scheme (see, e.g., [6]) that contains all classical families **of** hypergeometric orthogonal polynomi- als. Some years later the same authors [7] introduced the celebrated Askey–Wilson **polynomials**. The important property **of** these **polynomials** is the possibility to obtain from them all known families **of** hypergeometric **polynomials** **and** **q**-**polynomials** as particular or limit cases (the review is done **in** the nice survey [6]). The main tool **of** [6, 7] was the hypergeometric **and** basic series, respectively. On the other hand, **in** [8] (see also [9]) **q**-**polynomials** were considered as the solution **of** a second-order difference equation **of** the hypergeometric type on the nonlinear lattice,

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To conclude this Section we want to remark that the same procedure can be applied to the negative discrete series **of** IR. Moreover, from the nite dierence equation **and** the dierentiation formulas (**2**), (14) **and** (16) we can obtain some new recurrence relations for the CGC's **of** the SU **q** (1 ; 1) **quantum** **algebra**.

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k = b − s − 1. Here we do this for the classical **and** alternative Hahn **and** **Racah** **polynomials** as well as for their **q**-analogs. Also we establish the connection between classical **and** alternative families. This allows us to obtain new expressions for the Clerbsch–Gordan **and** **Racah** coefficients **of** the **quantum** **algebra** U **q** (su(**2**)) **in** terms **of** various Hahn **and** **Racah** **q**-**polynomials**.

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Inspired by the appearance **of** Macfarlane’s [10] **and** Biedenharn’s [11] important construc- tions **of** **q**-analogues **of** **quantum** harmonic oscillator, this technique **of** factorization **of** difference equations was later employed **in** a number **of** publications [12]–[16] **in** order to study group theoretic properties **of** the various well-known families **of** orthogonal **polynomials**, which can be viewed as **q**-extensions **of** the classical Hermite **polynomials**. So our purpose here is to formu- late a unified approach to deriving all **of** these results, which correspond to the **q**-linear spectrum. An important aspect to observe at this point is that we shall mainly (except for the exam- ples **in** subsection 4.2) confine our attention to those families **of** **q**-**polynomials**, which satisfy discrete orthogonality relation **of** the type (5). The explanation **of** such preference is that the factorization **of** difference equations for instances **of** **q**-**polynomials** with continuous orthogonality property has been already thoroughly studied **in** [12]–[16]. Observe also that our approach still remains valid **in** the limit as **q** → 1; so classical counterparts **of** **q**-**polynomials**, which will be discussed **in** this paper, are **in** fact incorporated as appropriate limit cases. But the reader who desires to know more about the factorization **in** the cases **of** classical orthogonal **polynomials** (such as the Kravchuk, Charlier, Meixner, Meixner–Pollaczek, **and** Hahn) may be referred to [17, 18] **and** references therein.

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Different path deformations performed on the analytic solutions give rise to Theorems **2** **and** 3, where upper bounds on the difference **of** two consecutive solutions are attained (consecutive solutions **in** the sense that they are related to consecutive sectors **in** a good covering). Such bounds are related to null Gevrey **and** **q**-Gevrey asymptotic expansions **of** some positive order. As a matter **of** fact, the previous differences allow for applying a novel (( **q**, k ) ; s ) -version **of** the cohomological criteria known as a Ramis–Sibuya theorem. Such result is related to functions admitting **q**-Gevrey asymptotic expansions **of** order k **and** a Gevrey sub-level **of** order s; see Theorem 4. We also apply a **q**-**analog** **of** Ramis–Sibuya Theorem; see Theorem 5.

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For every n ∈ N consider the general linear group GL(n + 1, K ) formed by all (n + 1) × (n + 1) invertible matrices with coefficients **in** K . Let R be the Riordan group. Since every Riordan matrix is lower triangular, we can define a natural homomorphism Π n : R → GL(n + 1, K ) given by

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En el primero 26 , afirma Santo Tomás que “el acto del creyente depende de tres cosas: del intelecto [“ex intellectu”] que termina en algo uno; de la voluntad [“ex voluntate”] que determina al intelecto mediante su “imperio” [“per suum imperium”]; y de la “razón” [“ex ratione”], que inclina la voluntad. Y asigna a cada uno de estos los tres modos que encontramos en el artículo de la Summa. En cuanto que el intelecto se determina a algo uno, el acto de fe es “credere Deum”, porque el objeto de la fe es Dios considerado en sí mismo, o algo en relación a Él o causado por Él. En cuanto que el intelecto es determinado por la voluntad el acto de fe es “credere **in** Deum”, es decir “amando hacia Él tender” [“amando **in** eum tendere”] porque es propio de la voluntad amar [“est enim voluntatis amare”]. Finalmente, en cuanto que la razón inclina a la voluntad al acto de fe, este es “credere Deo”: la razón por la cual la voluntad se inclina a asentir a algo que no ve, es porque Dios lo dice, como el hombre en aquellas cosas que no ve, cree al testimonio de algún hombre bueno que ve lo que él mismo no ve.

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Beyond the cutoff power (p 1), the shape **of** g ∞ MI depends only on the Raman characteristics **of** the transmission medium. From a significant number **of** numerical simulations, we verified that this is satisfied for p ≥ 5. Figure 3 shows the gain profile g MI for p = 10 (a), **and** the dependence **of** Ω MI with p for several

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Moreover, inasmuch as measuring a classical bit does not disturb its state, a measurement **of** a qubit destroys its coherence **and** irrevocably disturbs the superposition state. Then, how can the state **of** a qubit be measured? i.e., how can we assess the probability for the system to collapse to one state or the other? The procedure used to perform measurements on a qubit is simple to a certain extent. An analogy with a coin helps better understand the nature **of** the qubit. This coin has two possible static states, head or tails that could be represented: |Hi , |T i. Now image this coin is falling through the air. **In** a certain moment it could be interpreted that it is **in** a superposition state **of** both basis states: |H + Ti = α |Hi + β |T i . We can not measure exactly the probability **of** the coin to fall on each side, **and** when it falls, the superposition state collapses to only one **of** the basis states: head or tails, i.e, the coherence is irrevocably disturbed. Now, if the event can be replayed from the first standpoint an empirical measurement **of** the number **of** times it lays on tails can be obtained. Then, using the classical definition **of** probability: P(Tails) = 1 - P(Heads) = tails

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Semiconductor microcavities are one **of** the most suitable structures to study light-matter interaction. **In** the strong coupling regime excitons **and** photons form mixed states, named cavity polaritons. Our activities on polariton dynamics **and** its spin properties started more than then 10 years ago with the discovery **of** a strong influence **of** exciton-cavity detuning on the spin relaxation **of** polaritons, [10] **and** the demonstration **of** the feasibility to control the polarization **of** the non-linear, stimulated emission [11] (with samples from E. Mendez at Stony Brook **and** R. André at Université Joseph Fourier, respectively). Figure 4 shows the polarization-resolved time evolution **of** the

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tramural da origen a complejos ventriculares de diferentes morfologías en las derivaciones corres- pondientes: W, qRS, qrS, qrSr’, rSRS’. Si ellos se inician con ondas **Q**, la duración de éstas no apa- rece prolongada de manera significativa. Los po- tenciales de las fibras de Purkinje no resultan afec- tados, pese al discreto retardo del proceso de activación y al empastamiento de los complejos QRS. Dichos potenciales pueden explicarse por la colisión de dos frentes de activación: uno su- bendocárdico y otro subepicárdico. Este hecho sería la causa de una ligera aberrancia de los com- plejos ventriculares, semejante a la producida por el bloqueo periférico izquierdo, y capaz de ocul- tar la verdadera extensión de la zona de infarto. 9-11

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this separation result, it might be simpler to combine Theorem 2.1 **and** the inner faithfulness criterion for representations **of** pointed Hopf algebras **in** [4] (Theorem 4.1). This last theorem is as follows: a pointed Hopf **algebra** H is inner linear if **and** only if there exists a finite-dimensional representation π : H −→ A such that for any group-like g ∈ Gr(H), the restriction map π |P g, 1 (H) : P g,1 (H ) −→ A is injective. A

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H (µ) 2m (x; **q**) **and** **of** an odd degree H (µ) 2n+1 (x; **q**), m, n = 0, 1, **2**, . . ., are evidently orthogonal to each other. Consequently, it suffices to prove only those cases **in** (3.1), when degrees **of** **polynomials** m **and** n are either simultaneously even or odd. Let us consider first the former case. From (2.7) **and** (2.3) it follows that

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?t 1,00 Hora Peri?do de Tr?nsito Caudal de Entrada Caudal de Salida (I?+I???)/2 (Q?+Q???)/2 ?t [(I?+I???)/2 (Q?+Q???)/2 ] S??? = ?t [(I?+I???)/2 (Q?+Q???)/2 ] + S? j (Horas) I (m3/hora) Q (m3/hora) (m[.]

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Paso 3: el valor de la hipotenusa es **2**, luego con la ayuda de un compás podemos representar en la recta el valor **2** la siguiente manera. Con el compás se toma la dimensión de la hipotenusa, que en este caso es **2** , con como centro el cero. Luego se traza un arco de circunferencia y el punto de corte con la recta numérica será el valor de raíz de **2** (longitud desde el punto cero al punto P).

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[18] F. A. Gr¨ unbaum, L. Haine, **and** E. Horozov, On the Krall-Hermite **and** Krall-Bessel **polynomials**, Internat. Math. Res. Not. 19 (1997), 953-966. [19] L. Haine, The Bochner-Krall problem: some new perspectives. **In** Special Functions 2000: Current Perspective **and** Future Directions, J. Bustoz et al. (Eds.) NATO ASI Series, Dordrecht, Kluwer (2002),141-178. [20] L. Haine **and** P. Iliev, Askey-Wilson type functions, with bound states,

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On the other hand the complementary microfabrication technology is the lithography on glass. The purpose **of** lithography is to define the regions or patterns on the substrate where the components or circuits are going to be made. There are three primary exposure methods: contact, proximity, **and** projection. With these methods **and** using high intensity UV exposure systems we can get resolutions **of** about 1μm. Better resolutions are achieved with expensive projections systems using shorter UV wavelengths, however if sub-micron resolution is needed at a low cost, other techniques are wanted. This is the case for patterning very small circuits, as directional couplers with very sharp edges, at a standard laboratory. The most interesting technique which satisfies both requirements **of** resolution **and** cost for craft production is the laser beam lithography which has been widely developed **in** our group [13]. It allows us to get resolutions close to the most accurate techniques with an equipment affordable for most laboratories. As an illustrative example we show **in** Fig. 1 the result **of** a laser beam lithography corresponding to a 1 to 4 power splitter where the waveguide widths become lower than 1μm.

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