PDF superior A q-ANALOG OF RACAH POLYNOMIALS AND q-ALGEBRA SUq (2) IN QUANTUM OPTICS

A q-ANALOG OF RACAH POLYNOMIALS AND q-ALGEBRA SUq (2) IN QUANTUM OPTICS

A q-ANALOG OF RACAH POLYNOMIALS AND q-ALGEBRA SUq (2) IN QUANTUM OPTICS

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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The dual Hahn q-Polynomials in the lattice

The dual Hahn q-Polynomials in the lattice

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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DUAL PROPERTIES OF ORTHOGONAL POLYNOMIALS OF DISCRETE VARIABLES ASSOCIATED WITH THE QUANTUM ALGEBRA Uq (su(2))

DUAL PROPERTIES OF ORTHOGONAL POLYNOMIALS OF DISCRETE VARIABLES ASSOCIATED WITH THE QUANTUM ALGEBRA Uq (su(2))

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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Factorization of the hypergeometric-type difference equation on the non-uniform lattices: dynamical algebra

Factorization of the hypergeometric-type difference equation on the non-uniform lattices: dynamical algebra

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

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

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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Research lines of the Quantum and Atom Optics Group at the Universitat Aut•noma de Barcelona

Research lines of the Quantum and Atom Optics Group at the Universitat Aut•noma de Barcelona

Quantum registers with single-site addressing of about hundred quantum bits [2] and cluster entangled states of thousands of atoms [3] have been reported, respectively, in 2D optical microtrap arrays and 3D optical lattices. The loading of quantum gases into 3D optical lattices achieving the Mott insulator regime for both bosons and fermions has been experimentally demonstrated [4,5], reaching one of the main goals for QIP with neutral atoms. In this context, we developed a set of coherent and robust tools [6,7], termed three-level atom optics (TLAO) techniques, to adiabatically transport neutral atoms between the two extreme traps of a triple- well potential. We have very recently extended the TLAO techniques to the coherent manipulation of the external degrees of freedom of defects [8], i.e., holes, in single occupancy dipole trap arrays, see Fig. 1, and investigated their potential application to the initialization of defect-free trap domains, the building-up of single atom transistors, and even for QIP with the hole itself being the quantum bit carrier. By means of the Hubbard formalism, we extended the previous results for a single hole in a triple well potential to a single hole in an arbitrarily long trap array. We are presently addressing the application of optimal control methods to the TLAO techniques in order to achieve a high fidelity fast transport with realistic potentials.
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On a q analog of a singularly perturbed problem of irregular type with two complex time variables

On a q analog of a singularly perturbed problem of irregular type with two complex time variables

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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Struggling with Riordan involutions formula

Struggling with Riordan involutions formula

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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Credere in Deum: una lectura de II II, q  2, a  2

Credere in Deum: una lectura de II II, q 2, a 2

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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Simple method for estimating the fractional Raman contribution

Simple method for estimating the fractional Raman contribution

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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TítuloQuantum modeling of uncertainty in classical rule based systems

TítuloQuantum modeling of uncertainty in classical rule based systems

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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Quantum and non-linear optics with semiconductor nanostructures

Quantum and non-linear optics with semiconductor nanostructures

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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Infartos miocrdicos con onda Q y sin onda Q

Infartos miocrdicos con onda Q y sin onda Q

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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Examples of inner linear Hopf algebras

Examples of inner linear Hopf algebras

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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A q-extension of the generalized Hermite polynomials with the continuous orthogonality property on R

A q-extension of the generalized Hermite polynomials with the continuous orthogonality property on R

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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ANÁLISIS DEL TRÁNSITO DE CRECIENTES MEDIANTE EL MÉTODO DE MUSKINGUM EN EL RÍO NEGRO QUE PASA POR EL DEPARTAMENTO DE CUNDINAMARCA

ANÁLISIS DEL TRÁNSITO DE CRECIENTES MEDIANTE EL MÉTODO DE MUSKINGUM EN EL RÍO NEGRO QUE PASA POR EL DEPARTAMENTO DE CUNDINAMARCA

?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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Guía-taller  N°2 Mat 8° -2020-1P Q-Q'.pd

Guía-taller N°2 Mat 8° -2020-1P Q-Q'.pd

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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On the Krall-type polynomials on q-quadratic lattices

On the Krall-type polynomials on q-quadratic lattices

[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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Quantum integrated optics: Theory and applications

Quantum integrated optics: Theory and applications

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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