A note on <Emphasis Type="Italic">q</Emphasis>-Bernstein polynomials

Recently, Simsek-Acikgoz [17] and Kim-Jang-Yi [9] have studied the q-extension of Bernstein polynomials. In the present paper, we suggest q-extensions of Bernstein polynomials of degree n which differ from the q-Bernstein polynomials of Simsek-Acikgoz [17] and Kim-Jang-Yi [9]. Using these q-Bernstein polynomials, we derive fermionic p-adic integral representations of several q-Bernstein-type polynomials. Finally, we investigate identities between q-Bernstein polynomials and q-Euler numbers.     

On the multiple <Emphasis Type="Italic">q</Emphasis>-Genocchi and Euler numbers

The purpose of this paper is to present a systematic study of some families of multiple q-Genocchi and Euler numbers by using the multivariate q-Volkenborn integral (= p-adic q-integral) on ℤ _p . The investigation of these q-Genocchi numbers and polynomials of higher order leads to interesting identities related to these objects. The results of the present paper cover earlier results concerning ordinary q-Genocchi numbers and polynomials. This paper is supported by Jangjeon Research Institute for Mathematical Science (JRIMS-11R-2007).     

Hypercontractivity on the <Emphasis Type="Italic">q</Emphasis>-Araki-Woods Algebras

Extending a work of Carlen and Lieb, Biane has obtained the optimal hypercontractivity of the q-Ornstein-Uhlenbeck semigroup on the q-deformation of the free group algebra. In this note, we look for an extension of this result to the type III situation, that is for the q-Araki-Woods algebras. We show that hypercontractivity from L ^p to L ² can occur if and only if the generator of the deformation is bounded.     

Exact Results for Topological Strings on Resolved <Emphasis Type="Italic">Y</Emphasis><Superscript><Emphasis Type="Italic"> p</Emphasis>,<Emphasis Type="Italic">q</Emphasis></Superscript> Singularities

We obtain exact results in α′ for open and closed A-model topological string amplitudes on a large class of toric Calabi-Yau threefolds by using their correspondence with five dimensional gauge theories. The toric Calabi-Yaus that we analyze are obtained as minimal resolution of cones over Y ^p,q manifolds and give rise via M-theory compactification to SU(p) gauge theories on . As an application we present a detailed study of the local case and compute open and closed genus zero Gromov-Witten invariants of the orbifold. We also display the modular structure of the topological wave function and give predictions for higher genus amplitudes. The mirror curve in this case is the spectral curve of the relativistic A ₁ Toda chain. Our results also indicate the existence of a wider class of relativistic integrable systems associated to generic Y ^p,q geometries.     

<Emphasis Type="Italic">p</Emphasis>-adic Superspaces and Frobenius

The notion of a p-adic superspace is introduced and used to give a transparent construction of the Frobenius map on p-adic cohomology of a smooth projective variety over (the ring of p-adic integers), as well as an alternative construction of the crystalline cohomology of a smooth projective variety over (finite field with p elements). Partly supported by NSF grant No. DMS 0505735.     

Non-vacuum Bianchi types <Emphasis Type="Italic">I</Emphasis> and <Emphasis Type="Italic">V</Emphasis> in <Emphasis Type="Italic">f</Emphasis> (<Emphasis Type="Italic">R</Emphasis>) gravity

In a recent paper (Sharif and Shamir in Class. Quantum Grav. 26:235020, 2009), we have studied the vacuum solutions of Bianchi types I and V spacetimes in the framework of metric f (R) gravity. Here we extend this work to perfect fluid solutions. For this purpose, we take stiff matter to find energy density and pressure of the universe. In particular, we find two exact solutions in each case which correspond to two models of the universe. The first solution gives a singular model while the second solution provides a non-singular model. The physical behavior of these models has been discussed using some physical quantities. Also, the function of the Ricci scalar is evaluated.     

<Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>) Cosmology from <Emphasis Type="Italic">q</Emphasis>-theory

From a macroscopic theory of the quantum vacuum in terms of conserved relativistic charges (generically denoted by q ^(a) with label a), we have obtained, in the low-energy limit, a particular type of f(R) model relevant to cosmology. The macroscopic quantum-vacuum theory allows us to distinguish between different phenomenological f(R) models on physical grounds. The text was submitted by the authors in English.     

<Emphasis Type="Italic">q</Emphasis>-Exponentials on quantum spaces

We present explicit formulae for q-exponentials on quantum spaces which could be of particular importance in physics, i.e. the q-deformed Minkowski space and the q-deformed Euclidean space with two, three or four dimensions. Furthermore, these formulae can be viewed as 2-, 3- or 4-dimensional analogues of the well-known q-exponential function.Received: 21 January 2004, Revised: 19 May 2004, Published online: 7 September 2004     

Static anisotropic solutions in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">T</Emphasis>) theory

In previous work, we undertook to study static and anisotropic content in f(T) theory and obtained new spherically symmetric solutions considering a constant torsion and some particular conditions for the pressure. In this paper, still in the framework of f(T) theory, new spherically symmetric solutions are obtained, first considering the general case of an isotropic fluid and later the anisotropic content case in which the generalized conditions for the matter content are considered such that the energy density, the radial and tangential pressures depend on the algebraic f(T) and its derivative f _T (T). Moreover, we obtain the algebraic function f(T) through the reconstruction method for two cases and also study a polytropic model for the stellar structure.     

Identities involving values of Bernstein, <Emphasis Type="Italic">q</Emphasis>-Bernoulli,and <Emphasis Type="Italic">q</Emphasis>-Euler polynomials

In this paper, we give relations involving values of q-Bernoulli, q-Euler, and Bernstein polynomials. Using these relations, we obtain some interesting identities on the q-Bernoulli, q-Euler, and Bernstein polynomials.     

The <Emphasis Type="Italic">q</Emphasis>-boson-fermion realizations of the quantum superalgebra <Emphasis Type="Italic">U</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>(<Emphasis Type="Italic">osp</Emphasis>(1/2))

We show that our construction of realizations for algebras and quantum algebras can be generalized to quantum superalgebras too. We studyan example of quantum superalgebra U _q (osp(1/2)) and give the boson-fermion realization with respect to one pair of q-boson operators and one pair of fermions.     

Multiple p-adic L-function

An analytic function interpolating the multiple generalized Bernoulli numbers attached to a primitive Dirichlet character X at negative integers in the complex plane is studied. The multiple p-adic L-function is constructed as the p-adic analog of the above function. Finally, the values of the partial derivative of this multiple p-adic L-function at s = 0 are given. Dedicated to the memory of Boris Moiseevich Levitan     

<Emphasis Type="Italic">q</Emphasis>-Integration on quantum spaces

In this article we present explicit formulae for q-integration on quantum spaces which could be of particular importance in physics, i.e., q-deformed Minkowski space and q-deformed Euclidean space in three or four dimensions. Furthermore, our formulae can be regarded as a generalization of Jacksons q-integral to three or four dimensions and provide a new possibility for an integration over the whole space being invariant under translations and rotations.Received: 9 September 2003, Published online: 26 November 2003     

<Emphasis Type="Italic">q</Emphasis>-plane-wave solutions of <Emphasis Type="Italic">q</Emphasis>-Weyl gravity

The solutions of the q-deformed equations of quantum conformal Weyl gravity in terms of q-deformed plane waves are given. The text was submitted by the authors in English.     

Tsallis <Emphasis Type="Italic">p</Emphasis>,<Emphasis Type="Italic">q</Emphasis>-deformed Touchard polynomials and Stirling numbers

In this paper, we develop and investigate a new two-parametrized deformation of the Touchard polynomials, based on the definition of the NEXT q-exponential function of Tsallis. We obtain new generalizations of the Stirling numbers of the second kind and of the binomial coefficients and represent two new statistics for the set partitions.     

<Emphasis Type="Italic">B</Emphasis> and <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">s</Emphasis></Subscript> decay constants from moments of finite energy sum rules in QCD

We use an appropriate combination of moments of finite energy sum rules in QCD in order to compute the B _q -meson decay constants f _B and . We perform the calculation using a two-loop computation of the imaginary part of the pseudoscalar two point function in terms of the running bottom quark mass. The results are stable against the so-called QCD duality threshold, and they are in agreement with the estimates obtained from Borel transform QCD sum rules and lattice computations.Received: 28 July 2004, Revised: 10 September 2004, Published online: 23 November 2004PACS: 12.38.Bx, 12.38.LgSupported by MCYT-FEDER under contract FPA2002-00612.     

Perturbation foundation of <Emphasis Type="Italic">q</Emphasis>-deformed dynamics

In the q-deformed theory the perturbation approach can be expressed in terms of two pairs of undeformed position and momentum operators. There are two configuration spaces. Correspondingly there are two q-perturbation Hamiltonians; one originates from the perturbation expansion of the potential in one configuration space, the other one originates from the perturbation expansion of the kinetic energy in another configuration space. In order to establish a general foundation of the q-perturbation theory, two perturbation equivalence theorems are proved. The first is Equivalence Theorem I: Perturbation expressions of the q-deformed uncertainty relations calculated by two pairs of undeformed operators are the same, and the two q-deformed uncertainty relations undercut Heisenberg's minimal one in the same style. The general Equivalence Theorem II is: for any potential (regular or singular) the expectation values of two q-perturbation Hamiltonians in the eigenstates of the undeformed Hamiltonian are equivalent to all orders of the perturbation expansion. As an example of singular potentials the perturbation energy spectra of the q-deformed Coulomb potential are studied. Received: 6 September 2002 / Revised version: 21 October 2002 / Published online: 14 April 2003 RID="a" ID="a" e-mail: jzzhang@physik.uni-kl.de, jzzhang@ecust.edu.cn     

Polynomial Realization of <Emphasis Type="Italic">s</Emphasis>l<Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>(2) and Fusion Rules at Exceptional Values of <Emphasis Type="Italic">q</Emphasis>

Representations of the sℓ_q(2) algebra are constructed in the space of polynomials of real (complex) variable for q^N=1. The spin addition rule based on eigenvalues of Casimir operator is illustrated on few simplest cases and conjecture for general case is formulated.     

Lifting <Emphasis Type="Italic">q</Emphasis>-difference operators for Askey-Wilson polynomials and their weight function

We determine an explicit form of a q-difference operator that transforms the continuous q-Hermite polynomials H _n (x|q) of Rogers into the Askey-Wilson polynomials p _n (x; a, b, c, d|q) on the top level in the Askey q-scheme. This operator represents a special convolution-type product of four one-parameter q-difference operators of the form ɛ _q (c _q D _q ) (where c _q are some constants), defined as Exton’s q-exponential function ɛ _q (z) in terms of the Askey-Wilson divided q-difference operator D _q . We also determine another q-difference operator that lifts the orthogonality weight function for the continuous q-Hermite polynomialsH _n (x|q) up to the weight function, associated with the Askey-Wilson polynomials p _n (x; a, b, c, d|q).     

Fitting Cosmological Data to the Function <Emphasis Type="Italic">q</Emphasis>(<Emphasis Type="Italic">z</Emphasis>) from GR Theory: Modified Chaplygin Gas

In the Friedmann cosmology, the deceleration of the expansion q plays a fundamental role. We derive the deceleration as a function of redshift q(z) in two scenarios: ΛCDM model and modified Chaplygin gas (MCG) model. The function for the MCG model is then fitted to the cosmological data in order to obtain the cosmological parameters that minimize χ ². We use the Fisher matrix to construct the covariance matrix of our parameters and reconstruct the q(z) function. We use Supernovae Ia, WMAP5, and BAO measurements to obtain the observational constraints. We determined the present acceleration as q ₀ = − 0.65 ±0.19 for the MCG model using the Union2 dataset of SNeIa, BAO, and CMB and q ₀ = − 0.67 ±0.17 for the Constitution dataset, BAO and CMB. The transition redshift from deceleration to acceleration was found to be around 0.80 for both datasets. We have also determined the dark energy parameter for the MCG model: Ω _X0 = 0.81 ±0.03 for the Union2 dataset and Ω _X0 = 0.83 ±0.03 using the Constitution dataset.