Pieri-type formulas for the non-symmetric Jack polynomials

In the theory of symmetric Jack polynomials the coefficients in the expansion of the $p$th elementary symmetric function $e_p(z)$ times a Jack polynomial expressed as a series in Jack polynomials are known explicitly. Here analogues of this result for the non-symmetric Jack polynomials $E_\eta(z)$ are explored. Necessary conditions for non-zero coefficients in the expansion of $e_p(z) E_\eta(z)$ as a series in non-symmetric Jack polynomials are given. A known expansion formula for $z_i E_\eta(z)$ is rederived by an induction procedure, and this expansion is used to deduce the corresponding result for the expansion of $\prod_{j=1, \, j\ne i}^N z_j \, E_\eta(z)$, and consequently the expansion of $e_{N-1}(z) E_\eta(z)$. In the general $p$ case the coefficients for special terms in the expansion are presented.     

On ANOVA expansions and strategies for choosing the anchor point

The classic Lebesgue ANOVA expansion offers an elegant way to represent functions that depend on a high-dimensional set of parameters and it often enables a substantial reduction in the evaluation cost of such functions once the ANOVA representation is constructed. Unfortunately, the construction of the expansion itself is expensive due to the need to evaluate high-dimensional integrals. A way around this is to consider an alternative formulation, known as the anchored ANOVA expansion. This formulation requires no integrals but has an accuracy that depends sensitively on the choice of a special parameter, known as the anchor point.We present a comparative study of several strategies for the choice of this anchor point and argue that the optimal choice of this anchor point is the center point of a sparse grid quadrature. This choice induces no additional cost and, as we shall show, results in a natural truncation of the ANOVA expansion. The efficiency and accuracy is illustrated through several standard benchmarks and this choice is shown to outperform the alternatives over a range of applications.     

需求随机下房地产开发商最优进入-退出与能力扩张决策模型

在租赁市场上,房地产开发商常常需要同时决定进入-退出时机及开发能力扩张的的时机.然而这一研究在已往的房地产投资有关文献中有所忽视.鉴于此,在需求随机的条件下,通过一两阶段决策模型同时研究了房地产开发商在租赁市场的进入-退出及能力扩张问题.指出了进入、退出决策的隐式解并给出了扩张决策的阀值及扩张投资额度.研究同时得出结论:不确定性与成本的提高会增大了开发商进入-退出的决策刚性,并同时抑制了开发商的扩张投资.文章同时在行文中分析了结论的经济含义与政策含义.     

An asymptotic expansion for the tail of compound sums of Burr distributed random variables

In this paper we show that it is possible to write the Laplace transform of the Burr distribution as the sum of four series. This representation is then used to provide a complete asymptotic expansion of the tail of the compound sum of Burr distributed random variables. Furthermore it is shown that if the number of summands is fixed, this asymptotic expansion is actually a series expansion if evaluated at sufficiently large arguments.     

非线性算子方程的泰勒展式算法

本文的目的是给出一种解Ｈｉｌｂｅｒｔ空间中非线性方程的ｋ阶泰勒展式算法（ｋ１）．标准Ｇａｌｅｒｋｉｎ方法可以看作１阶泰勒展式算法，而最优非线性Ｇａｌｅｒｋｉｎ方法可视为２阶泰勒展式算法．我们应用这种算法于定常的Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程的数值逼近．在一定情景下，最优非线性Ｇａｌｅｒｋｉｎ方法提供比标准Ｇａｌｅｒｋｉｎ方法和非线性Ｇａｌｅｒｋｉｎ方法更高阶的收敛速度．     

形式Laurent级数域上交错Oppenheim展开的研究

该文介绍了形式Laurent级数域上交错Oppenheim展开的算法,得到了该展开中数字的强(弱)大数定理、中心极限定理和重对数率,并且研究了这些级数部分和的逼近的度.     

Exponentially small expansions of the Wright function on the Stokes lines

We investigate a particular aspect of the asymptotic expansion of the Wright function pΨq(z) for large |z|. In the case p?=?1, q ? 0, we establish the form of the exponentially small expansion of this function on certain rays in the z-plane (known as Stokes lines). The importance of such exponentially small terms is encountered in analytic probability theory and in the theory of generalised linear models. In addition, the transition of the Stokes multiplier connected with the subdominant exponential expansion across the Stokes lines is shown to obey the familiar error-function smoothing law expressed in terms of an appropriately scaled variable. Some numerical examples which confirm the accuracy of the expansion are given.     

A systematization of the saddle point method. Application to the Airy and Hankel functions

The standard saddle point method of asymptotic expansions of integrals requires to show the existence of the steepest descent paths of the phase function and the computation of the coefficients of the expansion from a function implicitly defined by solving an inversion problem. This means that the method is not systematic because the steepest descent paths depend on the phase function on hand and there is not a general and explicit formula for the coefficients of the expansion (like in Watson's Lemma for example). We propose a more systematic variant of the method in which the computation of the steepest descent paths is trivial and almost universal: it only depends on the location and the order of the saddle points of the phase function. Moreover, this variant of the method generates an asymptotic expansion given in terms of a generalized (and universal) asymptotic sequence that avoids the computation of the standard coefficients, giving an explicit and systematic formula for the expansion that may be easily implemented on a symbolic manipulation program. As an illustrative example, the well-known asymptotic expansion of the Airy function is rederived almost trivially using this method. New asymptotic expansions of the Hankel function Hn(z) for large n and z are given as non-trivial examples.     

The expansion theorem for the deviation integra

The so-called deviation integral (functional) describes the logarithmic asymptotics of the probabilities of large deviations for random walks generated by sums of random variables or vectors. Here an important role is played by the expansion theorem for the deviation integral in which, for an arbitrary function of bounded variation, the deviation integral is represented as the sum of suitable integrals of the absolutely continuous, singular, and discrete components composing this function. The expansion theorem for the deviation integral was proved by A. A. Borovkov and the author in [9] under some simplifying assumptions. In this article, we waive these assumptions and prove the expansion theorem in the general form.     

Cutting Sequences for Geodesic Flow on the Modular Surface and Continued Fractions

 This paper describes the cutting sequences of geodesic flow on the modular surface with respect to the standard fundamental domain of . The cutting sequence for a vertical geodesic is related to a one-dimensional continued fraction expansion for θ, called the one-dimensional Minkowski geodesic continued fraction (MGCF) expansion, which is associated to a parametrized family of reduced bases of a family of 2-dimensional lattices. The set of cutting sequences for all geodesics forms a two-sided shift in a symbol space which has the same set of forbidden blocks as for vertical geodesics. We show that this shift is not a sofic shift, and that it characterizes the fundamental domain ℱ up to an isometry of the hyperbolic plane . We give conversion methods between the cutting sequence for the vertical geodesic , the MGCF expansion of θ and the additive ordinary continued fraction (ACF) expansion of θ. We show that the cutting sequence and MGCF expansions can each be computed from the other by a finite automaton, and the ACF expansion of θ can be computed from the cutting sequence for the vertical geodesic θ + it by a finite automaton. However, the cutting sequence for a vertical geodesic cannot be computed from the ACF expansion by any finite automaton, but there is an algorithm to compute its first symbols when given as input the first symbols of the ACF expansion, which takes time and space . (Received 11 August 2000; in revised form 18 April 2001)     

Cutting Sequences for Geodesic Flow on the Modular Surface and Continued Fractions

 This paper describes the cutting sequences of geodesic flow on the modular surface with respect to the standard fundamental domain of . The cutting sequence for a vertical geodesic is related to a one-dimensional continued fraction expansion for θ, called the one-dimensional Minkowski geodesic continued fraction (MGCF) expansion, which is associated to a parametrized family of reduced bases of a family of 2-dimensional lattices. The set of cutting sequences for all geodesics forms a two-sided shift in a symbol space which has the same set of forbidden blocks as for vertical geodesics. We show that this shift is not a sofic shift, and that it characterizes the fundamental domain ℱ up to an isometry of the hyperbolic plane . We give conversion methods between the cutting sequence for the vertical geodesic , the MGCF expansion of θ and the additive ordinary continued fraction (ACF) expansion of θ. We show that the cutting sequence and MGCF expansions can each be computed from the other by a finite automaton, and the ACF expansion of θ can be computed from the cutting sequence for the vertical geodesic θ + it by a finite automaton. However, the cutting sequence for a vertical geodesic cannot be computed from the ACF expansion by any finite automaton, but there is an algorithm to compute its first symbols when given as input the first symbols of the ACF expansion, which takes time and space .     

Interpreting the monadic second order theory of one successor in expansions of the real line

We give sufficient conditions for a first order expansion of the real line to define the standard model of the monadic second order theory of one successor. Such an expansion does not satisfy any of the combinatorial tameness properties defined by Shelah, such as NIP or even NTP₂. We use this to deduce the first general results about definable sets in NTP₂ expansions of (R,<, +).     

A theorem concerning a product of two general classes of polynomials and the multivariableH-function

A theorem concerning a product of two general classes of polynomials and the multivariableH-function is established. Certain integrals and expansion formulae have also been derived by the application of this theorem. This general theorem yields a number of new, interesting and useful theorems, integrals and expansion formulae as its particular cases.     

Expanding competence sets for the consumer decision problem

The framework of competence set analysis provides a new approach to complement the existing models for the consumer decision problem. Since it is essential for the firm to improve the competence set of its product or service to fully address the consumer's truly needed benefits, the effective expansion of competence sets plays an important role in marketing reality. The previous studies regarding competence set expansion have thrown light on the tree expansion processes, but forest learning is more suitable for the acquisition of product benefits than tree learning. Thus, this study looses the assumption of tree learning and conducts forest learning to design an effective expansion program. In addition, this study explores a more general problem involving intermediate attributes, compound benefits, and experiential effects. An algorithm is also provided for effective expansion of competence sets in consumer decision analysis.     

On the Fourier transform of SO(d)-finite measures on the unit sphere

The paper presents a simple new approach to the problem of computing Fourier transforms of SO(d)-finite measures on the unit sphere in the euclidean space. Representing such measures as restrictions of homogeneous polynomials we use the canonical decomposition of homogeneous polynomials together with the plane wave expansion to derive a formula expressing such transforms under two forms, one of which was established previously by F. J. Gonzalez Vieli. We showthat equivalence of these two forms is related to a certain multi-step recurrence relation for Bessel functions, which encompasses several classical identities satisfied by Bessel functions. We show it leads further to a certain periodicity relation for the Hankel transform, related to the Bochner- Coifman periodicity relation for the Fourier transform. The purported novelty of this approach rests on the systematic use of the detailed form of the canonical decomposition of homogeneous polynomials, which replaces the more traditional approach based on integral identities related to the Funk-Hecke theorem. In fact, in the companion paper the present authors were able to deduce this way a fairly general expansion theorem for zonal functions, which includes the plane wave expansion used here as a special case.Received: 7 May 2004; revised: 11 October 2004     

The exact solution of a non-linear boundary-value problem of the theory of waves on the surface of a liquid of finite depth

A non-linear boundary-value problem of the theory of waves on the surface of a heavy ideal incompressible liquid, which arises as a result of the expansion of the required functions in amplitude, taking quadratic terms into account, is investigated. A solution is constructed, on the one hand, suitable for describing long waves, and on the other, matched to the Stokes expansion (i.e., with the expansion in amplitude of the first order of infinitesimals). A function is sought which conformally maps a strip into the plane of the complex potential in the flow region. An exact solution is obtained for this problem, defined by fairly simply formulae. This solution, in the limit of long and short waves, gives linear sinusoidal waves and cnoidal waves respectively.     

Payne-Whitham型宏观交通流模型波动特性

宏观交通流模型将交通流比拟成流体流,通过整体变量如交通流量、平均车速以及交通密度来研究其整体性质,得到了越来越多的肯定.文章采用波前展开的方法,研究Payne-Whitham型宏观交通流模型描述扰动沿交通流波动的特性,同时给出了相应的稳定性条件.最后利用Padé逼近法进行数值仿真,得到的结果与理论分析相一致.     

An Infinite Asymptotic Expansion for the Extreme Zeros of the Pollaczek Polynomials

In this paper, we first establish an integral expression for the Pollaczek polynomials P_n ( x ; a , b ) from a generating function. By applying a canonical transformation to the integral and carrying out a detailed analysis of the integrand, we derive a uniform asymptotic expansion for P_n (cosθ; a , b ) in terms of the Airy function and its derivative, in descending powers of n . The uniformity is in an interval next to the turning point , with M being a constant. The coefficients of the expansion are analytic functions of a parameter that depends only on t where , and not on the large parameter n . From the expansion of the polynomials we obtain an asymptotic expansion in powers of n ^−1/3 for the largest zeros. As a special case, a four-term approximation is provided for comparison and illustration. The method used in this paper seems to be applicable to more general situations.     

Hecke-Siegel's pull-back formula for the Epstein zeta function with a harmonic polynomial

In this paper, we discuss the generalization of the Hecke's integration formula for the Epstein zeta functions. We treat the Epstein zeta function as an Eisenstein series come from a degenerate principal series. For the Epstein zeta function of degree two, Siegel considered the Hecke's formula as the constant term of a certain Fourier expansion of the Epstein zeta function and obtained the other Fourier coefficients as the Dedekind zeta functions with Grössencharacters of a real quadratic field. We generalize this Siegel's Fourier expansion to more general Eisenstein series with harmonic polynomials. Then we obtain the Dedekind zeta functions with Grössencharacters for arbitrary number fields.     

Expansion of vectors by powers of a matrix

In this paper, we investigate the problem of expansion of any d-dimensional vector in powers of a dilation matrix M, where a dilation matrix is an integer matrix of size d × d with all modules of its eigenvalues more than one. We consider this expansion as a multidimensional system of numeration, where we take the matrix as the base of the system of numeration and a special set of vectors as the set of digits. We give a constructive method of expansion of integer vectors in powers of a dilation matrix and prove the existence of expansion for any real vector. Bibliography: 4 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 355, 2008, pp. 199–218.