Narrow Results

A new membrane finite element with four nodes that is based on the strain approach using Airy’s function is developed for the resolution of plane linear elasticity problems in general. The adoption of linear variations for the normal components of the strain and constant shear strain improves the accuracy of the elements. However, any singularity is eliminated by the use of a local reference frame oriented in an optimal manner. From numerical examples of a beam subjected to bending and shearing, we conclude that exact solutions can be obtained and they demonstrate the accuracy of the proposed procedure.

The classical Airy function has been generalized by Kontsevich to a function of a matrix argument, which is an integral over the space of skew-hermitian matrices of a unitary-invariant exponential kernel. In this paper, the Kontsevich integral is further generalized to integrals over the Lie algebra of an arbitrary connected compact Lie group, using exponential kernels invariant under the group. The (real) polynomial defining this kernel is said to have the Airy property if the integral defines a function of moderate growth. A very general sufficient criterion for a polynomial to have the Airy property is given. It is shown that an invariant polynomial on the Lie algebra has the Airy property if its restriction to a Cartan subalgebra has the Airy property. This result is used to evaluate these invariant integrals completely and explicitly on the hermitian matrices, obtaining formulae that contain those of Kontsevich as special cases.

We study the behavior of the fermion propagator in an external time-dependent potential in 0 + 1 dimension. We show that, when the potential has up to quadratic terms in time, the propagator can be expressed in terms of generalized Airy functions (or standard Airy functions depending on the exact time dependence). We study various properties of these new generalized functions which reduce to the standard Airy functions in a particular limit.

In this paper, we obtain an asymptotic expansion for the Gauss hypergeometric function ₂F₁(a + λ, b + 2λ; c; -z), as |λ| → ∞. The expansion holds for fixed values of a, b, and c, and is uniformly valid for z in the domain |ph z| < π.

In this paper, we develop the Riemann–Hilbert approach to study the global asymptotics of discrete orthogonal polynomials with infinite nodes. We illustrate our method by concentrating on the Charlier polynomials . We first construct a Riemann–Hilbert problem Y associated with these polynomials and then establish some technical results to transform Y into a continuous Riemann–Hilbert problem so that the steepest descent method of Deift and Zhou ([8]) can be applied. Finally, we produce three Airy-type asymptotic expansions for in three different but overlapping regions whose union is the entire complex z-plane. When z is real, our results agree with the ones given in the literature. Although our approach is similar to that used by Baik, Kriecherbauer, McLaughlin and Miller ([3]), there are crucial differences in the details. For instance, our expansions hold in much bigger regions. Our results are completely new, and one of them answers a question raised in Bo and Wong ([4]). Asymptotic formulas are also derived for large and small zeros of the Charlier polynomials.

We study the uniform asymptotics of a system of polynomials orthogonal on [-1, 1] with weight function w(x) = exp{-1/(1 - x²)^μ}, 0 < μ < 1/2, via the Riemann–Hilbert approach. These polynomials belong to the Szegö class. In some earlier literature involving Szegö class weights, Bessel-type parametrices at the endpoints ±1 are used to study the uniform large degree asymptotics. Yet in the present investigation, we show that the original endpoints ±1 of the orthogonal interval are to be shifted to the MRS numbers ±β_n, depending on the polynomial degree n and serving as turning points. The parametrices at ±β_n are constructed in shrinking neighborhoods of size 1 - β_n, in terms of the Airy function. The polynomials exhibit a singular behavior as compared with the classical orthogonal polynomials, in aspects such as the location of the extreme zeros, and the approximation away from the orthogonal interval. The singular behavior resembles that of the typical non-Szegö class polynomials, cf. the Pollaczek polynomials. Asymptotic approximations are obtained in overlapping regions which cover the whole complex plane. Several large-n asymptotic formulas for π_n(1), i.e. the value of the nth monic polynomial at 1, and for the leading and recurrence coefficients, are also derived.

In this paper, we study the asymptotics of the Hahn polynomials Q_n(x; α, β, N) as the degree n grows to infinity, when the parameters α and β are fixed and the ratio of n/N = c is a constant in the interval (0, 1). Uniform asymptotic formulas in terms of Airy functions and elementary functions are obtained for z in three overlapping regions, which together cover the whole complex plane. Our method is based on a modified version of the Riemann–Hilbert approach introduced by Deift and Zhou.

Asymptotic formulas are derived for the Stieltjes–Wigert polynomials S_n(z; q) in the complex plane as the degree n grows to infinity. One formula holds in any disc centered at the origin, and the other holds outside any smaller disc centered at the origin; the two regions together cover the whole plane. In each region, the q-Airy function A_q(z) is used as the approximant. For real x > 1/4, a limiting relation is also established between the q-Airy function A_q(x) and the ordinary Airy function Ai(x) as q → 1.

In this paper, we develop the Riemann–Hilbert method to study the asymptotics of discrete orthogonal polynomials on infinite nodes with an accumulation point. To illustrate our method, we consider the Tricomi–Carlitz polynomials $f_n^{(\alpha )}(z)$ where $\alpha$ is a positive parameter. Uniform Plancherel–Rotach type asymptotic formulas are obtained in the entire complex plane including a neighborhood of the origin, and our results agree with the ones obtained earlier in [W. M. Y. Goh and J. Wimp, On the asymptotics of the Tricomi–Carlitz polynomials and their zero distribution. I, SIAM J. Math. Anal.25 (1994) 420–428] and in [K. F. Lee and R. Wong, Uniform asymptotic expansions of the Tricomi–Carlitz polynomials, Proc. Amer. Math. Soc.138 (2010) 2513–2519].

We characterize all the multiple orthogonal three-fold symmetric polynomial sequences whose sequence of derivatives is also multiple orthogonal. Such a property is commonly called the Hahn property and it is an extension of the concept of classical polynomials to the context of multiple orthogonality. The emphasis is on the polynomials whose indices lie on the step line, also known as $2$-orthogonal polynomials. We explain the relation of the asymptotic behavior of the recurrence coefficients to that of the largest zero (in absolute value) of the polynomial set. We provide a full characterization of the Hahn-classical orthogonality measures supported on a $3$-star in the complex plane containing all the zeros of the polynomials. There are essentially three distinct families, one of them $2$-orthogonal with respect to two confluent functions of the second kind. This paper complements earlier research of Douak and Maroni.

We derive uniform and non-uniform asymptotics of the Charlier polynomials by using difference equation methods alone. The Charlier polynomials are special in that they do not fit into the framework of the turning point theory, despite the fact that they are crucial in the Askey scheme. In this paper, asymptotic approximations are obtained, respectively, in the outside region, an intermediate region, and near the turning points. In particular, we obtain uniform asymptotic approximation at a pair of coalescing turning points with the aid of a local transformation. We also give a uniform approximation at the origin by applying the method of dominant balance and several matching techniques.

The spectrum of masses of the colorless states of DW Hamiltonian operator of quantum SU(2) Yang–Mills field theory on ${ℝ}^D$ obtained via the precanonical quantization is shown to be purely discrete and bounded from below. The scale of the mass gap is estimated to be of the order of magnitude of the scale of the ultra-violet parameter $\varkappa$ which is introduced by precanonical quantization on dimensional grounds.

This paper deals with products and ratios of average characteristic polynomials for unitary ensembles. We prove universality at the soft edge of the limiting eigenvalues’ density, and write the universal limit in function of the Kontsevich matrix model (“matrix Airy function”, as originally named by Kontsevich). For the case of the hard edge, universality is already known. We show that also in this case the universal limit can be expressed as a matrix integral (“matrix Bessel function”) known in the literature as generalized Kontsevich matrix model.

This article provides a mini-introduction to three-dimensional field theories with rigid (p, q) anti-de Sitter supersymmetry.

We have studied the fermi gas system associated to the ABJM matrix model, aiming at understanding some hidden structures of M-theory. In trying to reduce the number of integrations in computing the partition functions Z_k(N), we have found an identity relating two density matrices of the opposite parities. Using this relation, we have computed exact values of the partition functions for k = 1 up to N = 9.

Two asymptotic expansions are obtained for the Laguerre polynomial for large n and fixed α > −1. These expansions are uniformly valid in two overlapping intervals covering the entire x-axis. The leading terms of both agree with the two asymptotic formulas given by Erdélyi who used the theory of differential equations. Our approach is based on two integral representations for the Laguerre polynomials. The phase function of one of these integrals has two coalescing saddle points, and to this one the cubic transformation introduced by Chester, Friedman, and Ursell is applied. The phase function of the other integral also has two coalescing saddle points, but in addition it has a simple pole. Moreover, the saddle points coalesce onto this pole. In this case a rational transformation is used, which mimics the singular behavior of the phase function. In both cases explicit expressions are given for the remainders associated with the asymptotic expansions.

In this paper, we develop the Riemann–Hilbert approach to study the global asymptotics of discrete orthogonal polynomials with infinite nodes. We illustrate our method by concentrating on the Charlier polynomials . We first construct a Riemann–Hilbert problem Y associated with these polynomials and then establish some technical results to transform Y into a continuous Riemann–Hilbert problem so that the steepest descent method of Deift and Zhou can be applied. Finally, we produce three Airy-type asymptotic expansions for in three different but overlapping regions whose union is the entire complex z-plane. When z is real, our results agree with the ones given in the literature. Although our approach is similar to that used by Baik, Kriecherbauer, McLaughlin and Miller, there are crucial differences in the details. For instance, our expansions hold in much bigger regions. Our results are completely new, and one of them answers a question raised in Bo and Wong. Asymptotic formulas are also derived for large and small zeros of the Charlier polynomials.

Using the steepest descent method for oscillatory Riemann–Hilbert problems introduced by Deift and Zhou [Ann. Math.137 (1993), 295–368], we derive asymptotic formulas for the Meixner polynomials in two regions of the complex plane separated by the boundary of a rectangle. The asymptotic formula on the boundary of the rectangle is obtained by taking limits from either inside or outside. Our results agree with the ones obtained earlier for z on the positive real line by using the steepest descent method for integrals [Constr. Approx.14 (1998), 113–150].

In this paper, we study the asymptotics of the Hahn polynomials Q_n(x; α, β, N) as the degree n grows to infinity, when the parameters α and β are fixed and the ratio of n/N = c is a constant in the interval (0, 1). Uniform asymptotic formulas in terms of Airy functions and elementary functions are obtained for z in three overlapping regions, which together cover the whole complex plane. Our method is based on a modified version of the Riemann–Hilbert approach introduced by Deift and Zhou.

Asymptotic formulas are derived for the Stieltjes–Wigert polynomials S_n(z;q) in the complex plane as the degree n grows to infinity. One formula holds in any disc centered at the origin, and the other holds outside any smaller disc centered at the origin; the two regions together cover the whole plane. In each region, the q-Airy function A_q(z) is used as the approximant. For real x > 1/4, a limiting relation is also established between the q-Airy function A_q(x) and the ordinary Airy function Ai(x) as q → 1.