Generalized q-Hermite Polynomials

We consider two operators A and A ⁺ in a Hilbert space of functions on the exponential lattice , where 0<q<1. The operators are formal adjoints of each other and depend on a real parameter . We show how these operators lead to an essentially unique symmetric ground state ψ₀ and that A and A ⁺ are ladder operators for the sequence . The sequence (ψ _n /ψ₀) is shown to be a family of orthogonal polynomials, which we identify as symmetrized q-Laguerre polynomials. We obtain in this way a new proof of the orthogonality for these polynomials. When γ=0 the polynomials are the discrete q-Hermite polynomials of type II, studied in several papers on q-quantum mechanics. Received: 6 December 1999 / Accepted: 21 May 2001     

Loop and Path Spaces and¶Four-Dimensional BF Theories:¶Connections, Holonomies and Observables

We study the differential geometry of principal G-bundles whose base space is the space of free paths (loops) on a manifold M. In particular we consider connections defined in terms of pairs (A,B), where A is a connection for a fixed principal bundle P(M,G) and B is a 2-form on M. The relevant curvatures, parallel transports and holonomies are computed and their expressions in local coordinates are exhibited. When the 2-form B is given by the curvature of A, then the so-called non-abelian Stokes formula follows. For a generic 2-form B, we distinguish the cases when the parallel transport depends on the whole path of paths and when it depends only on the spanned surface. In particular we discuss generalizations of the non-abelian Stokes formula. We study also the invariance properties of the (trace of the) holonomy under suitable transformation groups acting on the pairs (A,B). In this way we are able to define observables for both topological and non-topological quantum field theories of the BF type. In the non-topological case, the surface terms may be relevant for the understanding of the quark-confinement problem. In the topological case the (perturbative) four-dimensional quantum BF-theory is expected to yield invariants of imbedded (or immersed) surfaces in a 4-manifold M. Received: 28 March 1998 / Accepted: 12 September 1998     

Reduction of Quantum Systems with Arbitrary First Class Constraints and Hecke Algebras

We propose a method for reduction of quantum systems with arbitrary first-class constraints. An appropriate mathematical setting for the problem is the homology of associative algebras. For every such algebra A and subalgebra B with augmentation ɛ there exists a cohomological complex which is a generalization of the BRST one. Its cohomology is an associative graded algebra Hk ^*(A,B) which we call the Hecke algebra of the triple (A,B,ɛ). It acts in the cohomology space H ^*(B,V) for every left A module V. In particular the zeroth graded component $Hk^{0}(A,B)$ acts in the space of B invariants of $V$ and provides the reduction of the quantum system. Received: 15 June 1998 / Accepted: 25 January 1999     

CoFe/TiZr多层膜材料界面结构与性能的极化中子反射研究

利用极化中子反射技术较系统地研究了CoFe/TiZr复合多层膜材料界面结构,结果表明.1)从多层膜超镜传输特性角度考虑,等厚对层不如非等厚对层膜结构好.2)等厚对层的最佳退火温度约250℃,非等厚对层退火温度低于250℃影响不明显,等厚和非等厚对层经350℃退火后膜层变化严重.3)从布拉格峰位变化看,随着退火温度的升高,等厚对层膜的厚度先变小后变大;非等厚对层与之相反. 关键词： CoFe/TiZr多层膜 退火影响 膜层结构 极化中子反射     

Solutions of convex Bethe Ansatz equations and the zeros of (basic) hypergeometric orthogonal polynomials

Via the solutions of systems of algebraic equations of Bethe Ansatz type, we arrive at bounds for the zeros of orthogonal (basic) hypergeometric polynomials belonging to the Askey–Wilson, Wilson and continuous Hahn families.

     

Microphase separation at two length scales

The possibility of microphase separation at two different length scales in monodisperse AB block copolymer melts consisting of a homopolymer A block and either a linear alternating AB copolymer block (poly(A)_m-block-poly(B-alt -A)_n) or an AB comb copolymer block poly(A)_m-block-poly(A-graft-B)_n, is investigated. An analysis of the structure factor reveals that in the parameter space of n and m three different cases can be distinguished: I) The structure factor has only one minimum corresponding to the short length scale (i.e. the characteristic length of the repeating unit of the alternating or comb block). II) The structure factor has only one minimum corresponding to the long length scale (the characteristic length of the blocks). III) Two minima are present leading to a competition between microphase separation at the short and the long length scale. Depending on the choice of n and m, one of these three possibilities will occur. Received 25 August 2000     

A Generalized Hypergeometric Function¶Satisfying Four Analytic Difference Equations¶of Askey--Wilson Type

The hypergeometric function ₂ F ₁ can be written in terms of a contour integral involving gamma functions. We generalize this (Barnes) representation by using a certain generalized gamma function as a building block. In this way we obtain a new ₂ F ₁-generalization with various symmetry features. We determine the analyticity properties of the R-function in all of its eight arguments, and show that it is a joint eigenfunction of four distinct Askey–Wilson type difference operators, two acting on v and two on . The Askey–Wilson polynomials can be obtained by a suitable discretization of v or . Received: 21 December 1998 / Accepted: 14 April 1999     

Determining bottom price-levels after a speculative peak

During a stock market peak the price of a given stock (i) jumps from an initial level p ₁(i) to a peak level p ₂(i) before falling back to a bottom level p ₃(i). The ratios A(i) = p ₂(i)/p ₁(i) and B(i)= p ₃(i)/p ₁(i) are referred to as the peak- and bottom-amplitude respectively. The paper shows that for a sample of stocks there is a linear relationship between A(i) and B(i) of the form: B=0.4A+b. In words, this means that the higher the price of a stock climbs during a bull market the better it resists during the subsequent bear market. That rule, which we call the resilience pattern, also applies to other speculative markets. It provides a useful guiding line for Monte Carlo simulations. Received 9 June 2000     

Survival and extinction in cyclic and neutral three-species systems

We study the ABC model ( A + B↦2B, B + C↦2C, C + A↦2A), and its counterpart: the three-component neutral drift model ( A + B↦2A or 2B, B + C↦2B or 2C, C + A↦2C or 2A.) In the former case, the mean-field approximation exhibits cyclic behaviour with an amplitude determined by the initial condition. When stochastic phenomena are taken into account the amplitude of oscillations will drift and eventually one and then two of the three species will become extinct. The second model remains stationary for all initial conditions in the mean-field approximation, and drifts when stochastic phenomena are considered. We analyzed the distribution of first extinction times of both models by simulations of the master equation, and from the point of view of the Fokker-Planck equation. Survival probability vs. time plots suggest an exponential decay. For the neutral model the extinction rate is inversely proportional to the system size, while the cyclic model exhibits anomalous behaviour for small system sizes. In the large system size limit the extinction times for both models will be the same. This result is compatible with the smallest eigenvalue obtained from the numerical solution of the Fokker-Planck equation. We also studied the behaviour of the probability distribution. The exponential decay is found to be robust against certain changes, such as the three reactions having different rates. Received 14 August 2002 and Received in final form 14 February 2003 / Published online: 1 April 2003 RID="a" ID="a"e-mail: ita@physics.ubc.ca     

Properties of Generalized Univariate Hypergeometric Functions

Based on Spiridonov’s analysis of elliptic generalizations of the Gauss hypergeometric function, we develop a common framework for 7-parameter families of generalized elliptic, hyperbolic and trigonometric univariate hypergeometric functions. In each case we derive the symmetries of the generalized hypergeometric function under the Weyl group of type E ₇ (elliptic, hyperbolic) and of type E ₆ (trigonometric) using the appropriate versions of the Nassrallah-Rahman beta integral, and we derive contiguous relations using fundamental addition formulas for theta and sine functions. The top level degenerations of the hyperbolic and trigonometric hypergeometric functions are identified with Ruijsenaars’ relativistic hypergeometric function and the Askey-Wilson function, respectively. We show that the degeneration process yields various new and known identities for hyperbolic and trigonometric special functions. We also describe an intimate connection between the hyperbolic and trigonometric theory, which yields an expression of the hyperbolic hypergeometric function as an explicit bilinear sum in trigonometric hypergeometric functions.     

On the Law of Addition of Random Matrices

Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices A _n and B _n rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix U _n (i.e. A _n +U _n ^* B _n U _n ) is studied in the limit of large matrix order n. Convergence in probability to a limiting nonrandom measure is established. A functional equation for the Stieltjes transform of the limiting measure in terms of limiting eigenvalue measures of A _n and B _n is obtained and studied. Received: 27 October 1999/ Accepted: 22 March 2000     

Calculated electronic structure of metallic multilayers formed by noble and transition metals

Full-potential Linearized Augmented Plane Wave calculations are performed to investigate the properties of the electronic charge of metallic multilayers formed by non-magnetic and magnetic elements (i.e. Ag, Cu and Fe). The multilayer structure is of the type A_n, A _n B _n or (AB)_n where A, B indicate Ag, Cu and Fe and n is the number of layers of the element A or B. The problem addressed by this study is the transition from the 2D behavior of the isolated monolayer to the 3D bulklike character. Therefore the calculations, carried out at paramagnetic level, illustrate the dependence of the density of states on the multilayer thickness and composition. For the three elements the main feature of the inter-layer coupling is the absence of charge intermixing and hybridization. For structures with a number of layers n? 5 the density of states bandwidth has a decrease, with respect to the bulk value, approximately proportional to the reduced coordination. At the critical thickness n = 5 and above, a noticeable difference exists between the charge in the outer layers, with reduced coordination and bandwidth, and the central layers with a bulklike density of states. Averaging between these contributions leads to the re-installment of bulklike properties. These results are in essential agreement with analytical band theories and quantum mechanical calculations for similar systems and with experiments. Received 3 October 2001     

Integral representations for the Lagrange polynomials,Shively’s pseudo-Laguerre polynomials,and the generalized Bessel polynomials

Motivated essentially by their potential for applications in the mathematical, physical, and statistical sciences, the object of this paper is to investigate several general families of hypergeometric polynomials and their associated multiple integral representations. By suitably specializing the main results presented here, the corresponding integral representations are derived for familiar simpler classes of hypergeometric polynomials such as (for example) the Lagrange polynomials, Shively’s pseudo-Laguerre polynomials, and generalized Bessel polynomials. Each of the integral representations derived in this paper may be also viewed as a linearization relationship for the product of two different members of the associated family of hypergeometric polynomials.     

<Emphasis Type="Italic">SO</Emphasis>(3,1)-Invariant Approach to Dipole Radiation

The aim of the present work is to revisit the theory of the dipole radiation, within an SO(3,1)-gauge invariant formulation, by solving the Maxwell equations. Thus, we obtain the two interconnected components, A _B , B=1,2, of the vector potential A, in terms of Hankel and Legendre polynomials. Finally, for the pure dipole-like radiation, the observables, regarded as phasors, the Umov–Poynting vector components and the well-known Larmor formula for the effective radiated power are explicitly derived.     

Some Functions that Generalize the Askey–Wilson Polynomials

We determine all biinfinite tridiagonal matrices for which some family of eigenfunctions are also eigenfunctions of a second order q-difference operator. The solution is described in terms of an arbitrary solution of a q-analogue of Gauss hypergeometric equation depending on five free parameters and extends the four dimensional family of solutions given by the Askey-Wilson polynomials. There is some evidence that this bispectral problem, for an arbitrary order q-difference operator, is intimately related with some q-deformation of the Toda lattice hierarchy and its Virasoro symmetries. When tridiagonal matrices are replaced by the Schroedinger operator, and q= 1, this statement holds with Toda replaced by KdV. In this context, this paper determines the analogs of the Bessel and Airy potentials. Received: 7 May 1996/Accepted: 30 August 1996     

Ll2超导体的Tc表达式

通过对L1₂超导体的研究,我们得出了这类超导体临界温度的表达式T_c=(15.9T_BV(B)G_A)/(M^1/2V(L1₂)G_B)。其中T_B是B组元的超导转变温度,V(B)是B组元的原子体积,V(L1₂)为L1₂化合物的平均原子体积,M是L1₂化合物的平均原子量,G_A,G_B分别为A组元、B组元的Gordy电负性值。 关键词：     

General formulae for $f_1 \to f_2 \gamma$

At one-loop level the decay , where f₁ and f₂ are two spin-1/2 particles with the same electric charge, is mediated by a boson B and a spin-1/2 fermion F. The boson B may have either spin - interacting with the fermions through the Dirac matrices 1 and - or spin 1 - with V+A and V-A couplings to the fermions. I give general formulae for the one-loop electroweak amplitude of in all these cases. Received: 24 February 2003 / Revised version: 26 March 2003 / Published online: 2 June 2003     

Formulas for An- and Bn-solutions of WDVV equations

The simplest non-trivial solutions of WDVV equations are A_n-and B_n-potentials, which describe metrics of Saito on spaces of versal deformation of A_n-and B_n-singularities. These are some polynomials, which were known for n≤4. In this paper, we find the potentials for all A_n-and B_n-singularities. We give a recurrence formula for coefficients of KP and n-KdV hierarchy.     

On the Absorption of antibaryons in nuclei

Antiproton (ˉp) and antilambda (ˉΛ) production has been measured for minimum bias in p+A collisions and central A₁+ A₂ collisions at the CERN-SPS by the collaborations NA35/49 and NA44. The measurements are extrapolated from rapidity distributions to absolute minimum bias cross sections. It is shown that the ˉp cross sections divided by A₁· A₂ follow an exponential trend as a function of a characteristic length obtained from a Glauber type absorption model, while the ˉΛ cross sections divided by A₁· A₂ are constant. The exponential trend also holds for ˉp production at the lower energies of the Brookhaven AGS. A discussion of the physics interpretation of the established trends in terms of an effective absorption cross section is presented. Received: 15 July 1999 / Revised version: 10 November 1999     

Investigation of the hyperfine structure of Ta I lines (V)

We report on further investigations of the hyperfine structure of spectral lines of the neutral tantalum atom. Besides determination of the hyperfine constants A and B of 21 levels we report on the discovery of 9 up to now unknown fine structure levels for which we could determine their energy position, parity, angular momentum and the constants A and B. For a large number of up to now unclassified lines the combinations could be identified. Received 27 June 2000 and Received in final form 23 August 2000