Diffraction of light by supersonic waves: The solution of the raman-nath equations—I

In the problem of the diffraction of light by a supersonic wave, at normal incidence of the light, the solution of the system of difference-differential equations of Raman and Nath, for the amplitudes of the diffracted light beams, is reduced to the integration of a partial differential equation. The coefficients of the Laurent expansion of the solution of the latter equation yield the expressions for the amplitudes of the diffracted light waves. The partial differential equation has been integrated for two approximations. (1) Forρ=0, the well-known results of Raman and Nath’s preliminary theory are re-established. (2) Forρ?1 a power series inρ, the terms of which are calculated as far as the third one, leads to the solution of Mertens and Berry obtained by a perturbation method.     

Diffraction of light by superposed parallel supersonic waves,one being then-th harmonic of the other. Solution of the system of difference-differential equations for the amplitudes by a series expansion method

In the problem of the diffraction of light by two parallel supersonic waves, consisting of a fundamental tone and itsn-th harmonic, the solution of the system of difference-differential equations for the amplitudes has been reduced to the integration of a partial differential equation. The expressions for the amplitudes of the diffracted light waves are obtained as the coefficients of the Laurent expansion of the solution of this partial differential equation. The latter has been integrated for two approximations:

1. Forρ = 0, the results of Murty’s elementary theory are reestablished.
2. Forρ ≤ 1, a power series inρ, the terms of which are calculated as far as the third one, leads to a new expression for the intensities of the diffracted light waves, verifying the general symmetry properties obtained by Mertens.

     

Diffraction of light by superposed parallel supersonic waves,being harmonics of the same fundamental. Solution of the system of difference-differential equations for the amplitudes

Starting from the general system of difference-differential equations for the amplitudes of the diffracted beams of light, given by Mertens, and using the method of Kuliasko, Mertens and Leroy for the diffraction of light by one supersonic wave, it is possible to reduce the solution of the system of difference-differential equations, to the solution of a partial differential equation. In this way it is possible to calculate the intensities of the ordern and ?n, as a series expansion in ρ. Here we only considered terms up to ρ². It was also possible to verify the general symmetry properties for the intensities studied by Leroy and Mertens.     

Remarks on Hilbert series of graded modules over polynomial rings

In this article we discuss a result on formal Laurent series and some of its implications for Hilbert series of finitely generated graded modules over standard-graded polynomial rings: For any integer Laurent function of polynomial type with non-negative values the associated formal Laurent series can be written as a sum of rational functions of the form ${\frac{Q_j(t)}{(1-t)^j}}$ , where the numerators are Laurent polynomials with non–negative integer coefficients. Hence any such series is the Hilbert series of some finitely generated graded module over a suitable polynomial ring ${\mathbb{F}[X_1 , \ldots , X_n]}$ . We give two further applications, namely an investigation of the maximal depth of a module with a given Hilbert series and a characterization of Laurent polynomials which may occur as numerator in the presentation of a Hilbert series as a rational function with a power of (1 ? t) as denominator.     

Constructing Solutions for the Generalized Hénon–Heiles System Through the Painlevé Test

The generalized Hénon–Heiles system is considered. New special solutions for two nonintegrable cases are obtained using the Painlevé test. The solutions have the form of the Laurent series depending on three parameters. One parameter determines the singularity-point location, and the other two parameters determine the coefficients in the Laurent series. For certain values of these two parameters, the series becomes the Laurent series for the known exact solutions. It is established that such solutions do not exist in other nonintegrable cases.     

Invariant Hyperfunction Solutions to Invariant Differential Equations on the Space of Real Symmetric Matrices

The real special linear group of degree n naturally acts on the vector space of n×n real symmetric matrices. How to determine invariant hyperfunction solutions of invariant linear differential equations with polynomial coefficients on the vector space of n×n real symmetric matrices is discussed in this paper. We prove that every invariant hyperfunction solution is expressed as a linear combination of Laurent expansion coefficients of the complex power of the determinant function with respect to the parameter of the power. Then the problem is reduced to the determination of Laurent expansion coefficients.     

包括激发和衰减的粘弹性Ⅱ型破裂过程的研究

用非线性Rayleigh阻尼公式描述初始破裂时有激发而加速,至一定高速时有衰减而止裂。视介质为匀质各向同性的Voigt线性粘弹性体,用小参数摄动法把滑开型(Ⅱ型)破裂定义的非线性偏微分方程组线性化,得出各次逼近解所定义的线性方程组,再用动坐标表示的广义Fourier级数把问题简化为非齐次的Mathieu方程,用WKBJ法给出问题在稳定区域的渐近解。     

Non-commutative integrability, paths and quasi-determinants

In previous work, we showed that the solution of certain systems of discrete integrable equations, notably Q and T-systems, is given in terms of partition functions of positively weighted paths, thereby proving the positive Laurent phenomenon of Fomin and Zelevinsky for these cases. This method of solution is amenable to generalization to non-commutative weighted paths. Under certain circumstances, these describe solutions of discrete evolution equations in non-commutative variables: Examples are the corresponding quantum cluster algebras (Berenstein and Zelevinsky (2005) [3]), the Kontsevich evolution (Di Francesco and Kedem (2010) [10]) and the T-systems themselves (Di Francesco and Kedem (2009) [8]). In this paper, we formulate certain non-commutative integrable evolutions by considering paths with non-commutative weights, together with an evolution of the weights that reduces to cluster algebra mutations in the commutative limit. The general weights are expressed as Laurent monomials of quasi-determinants of path partition functions, allowing for a non-commutative version of the positive Laurent phenomenon. We apply this construction to the known systems, and obtain Laurent positivity results for their solutions in terms of initial data.     

A generalization of the Laurent series

A system of functions, biorthogonal to the D-basis introduced by M. K. Fage for a linear differential operator D with analytic coefficients, is constructed. Series in the combination of the D-basis with this system generalize Laurent series in the same way as series in the D-basis generalize Taylor series.Translated from Matematicheskie Zametki, Vol. 9, No. 5, pp. 521–531, May, 1971.     

Laurent-Padé approximants to four kinds of Chebyshev polynomial expansions. Part I. Maehly type approximants

Laurent Padé-Chebyshev rational approximants,A _m (z,z ⁻¹)/B _n (z, z ⁻¹), whose Laurent series expansions match that of a given functionf(z,z ⁻¹) up to as high a degree inz, z ⁻¹ as possible, were introduced for first kind Chebyshev polynomials by Clenshaw and Lord [2] and, using Laurent series, by Gragg and Johnson [4]. Further real and complex extensions, based mainly on trigonometric expansions, were discussed by Chisholm and Common [1]. All of these methods require knowledge of Chebyshev coefficients off up to degreem+n. Earlier, Maehly [5] introduced Padé approximants of the same form, which matched expansions betweenf(z,z ⁻¹)B _n (z, z ⁻¹)). The derivation was relatively simple but required knowledge of Chebyshev coefficients off up to degreem+2n. In the present paper, Padé-Chebyshev approximants are developed not only to first, but also to second, third and fourth kind Chebyshev polynomial series, based throughout on Laurent series representations of the Maehly type. The procedures for developing the Padé-Chebyshev coefficients are similar to that for a traditional Padé approximant based on power series [8] but with essential modifications. By equating series coefficients and combining equations appropriately, a linear system of equations is successfully developed into two sub-systems, one for determining the denominator coefficients only and one for explicitly defining the numerator coefficients in terms of the denominator coefficients. In all cases, a type (m, n) Padé-Chebyshev approximant, of degreem in the numerator andn in the denominator, is matched to the Chebyshev series up to terms of degreem+n, based on knowledge of the Chebyshev coefficients up to degreem+2n. Numerical tests are carried out on all four Padé-Chebyshev approximants, and results are outstanding, with some formidable improvements being achieved over partial sums of Laurent-Chebyshev series on a variety of functions. In part II of this paper [7] Padé-Chebyshev approximants of Clenshaw-Lord type will be developed for the four kinds of Chebyshev series and compared with those of the Maehly type.     

Generalizations of the image conjecture and the Mathieu conjecture

We first propose a generalization of the image conjecture Zhao (submitted for publication) [31] for the commuting differential operators related with classical orthogonal polynomials. We then show that the non-trivial case of this generalized image conjecture is equivalent to a variation of the Mathieu conjecture Mathieu (1997) [21] from integrals of G-finite functions over reductive Lie groups G to integrals of polynomials over open subsets of Rⁿ with any positive measures. Via this equivalence, the generalized image conjecture can also be viewed as a natural variation of the Duistermaat and van der Kallen theorem Duistermaat and van der Kallen (1998) [14] on Laurent polynomials with no constant terms. To put all the conjectures above in a common setting, we introduce what we call the Mathieu subspaces of associative algebras. We also discuss some examples of Mathieu subspaces from other sources and derive some general results on this newly introduced notion.     

The McKay–Thompson series of Mathieu Moonshine modulo two

In this note, we describe the parity of the coefficients of the McKay–Thompson series of Mathieu moonshine. As an application, we prove a conjecture of Cheng, Duncan, and Harvey stated in connection with umbral moonshine for the case of Mathieu moonshine.     

A numerical method for the computation of Faber polynomials for starlike domains

We describe a simple numerical process for computing approximationsto Faber polynomials for starlike domains. This process is basedon using the Theodorsen integral equation method for computingthe Laurent series coefficients of the associated exterior conformalmapping, and then determining the corresponding Faber polynomialsby means of the well-known recurrence relation which is availablefor this purpose.     

两个垂直圆柱在水波中的水动力相互作用

基于线性势流理论研究了两个垂直圆柱在水波中的水动力相互作用．两个圆柱中的一个固定在底部，另一个铰接在底部且可以在入射波方向以小振幅振动．本文研究了绕射波和辐射波，运用加法定理得到了每个圆柱表面速度势的简单的解析表达式，用级数形式显式表示了圆柱上的波浪激励力和力矩及振动圆柱的附加质量和辐射阻尼系数．级数的系数由代数方程组的解决定．给出了一些数值例子以说明诸如间距、圆柱的相对大小、入射角等各种参数对一阶力、定常二阶力、附加质量和辐射阻尼系数以及振动圆柱的响应等的影响．     

弹性波在平面多连通域中的绕射与动应力集中

本文用复变函数论方法研究了弹性波在平面多连通域中的绕射问题,给出了这一问题解的完备逼近序列及边备条件的一般表示。问题归结为无穷代数方程组的求解,使用电子计算机可直接求得解答。特别是,对弱耦合问题,本文提出了渐近求解方法并且使用这个方法详细地讨论了P波对圆孔群的绕射问题。基于绕射波场的解,文中给出了任意形状空腔动应力集中系数的一般算式。     

The coefficients of the Laurent series expansions of real analytic functions

Letf be a real analytic function of a real variable such that 0 is an isolated (possibly essential) singularity off. In the existing literature the coefficients of the Laurent series expansion off around 0 are obtained by applying Cauchy's integral formula to the analytic continuation off on the complex plane. Here by means of a conformal mapping we derive a formula which determines the Laurent coefficients off solely in terms of the values off and the derivatives off at a real point of analyticity off. Using a more complicated mapping, we similarly determine the coefficients of the Laurent expansion off around 0 where now 0 is a singularity off which is not necessarily isolated.     

On formal Laurent series

Several kinds of formal Laurent series have been introduced with some restrictions so far. This paper systematically sets up a natural definition and structure of formal Laurent series without those restrictions, including introducing a multiplication between formal Laurent series. This paper also provides some results on the algebraic structure of the space of formal Laurent series, denoted by \mathbbL\mathbb{L}. By means of the results of the generalized composition of formal power series, we define a composition of a Laurent series with a formal power series and provide a necessary and sufficient condition for the existence of such compositions. The calculus about formal Laurent series is also introduced.     

Riordan arrays associated with Laurent series and generalized Sheffer-type groups

A relationship between a pair of Laurent series and Riordan arrays is formulated. In addition, a type of generalized Sheffer groups is defined by using Riordan arrays with respect to power series with non-zero coefficients. The isomorphism between a generalized Sheffer group and the group of the Riordan arrays associated with Laurent series is established. Furthermore, Appell, associated, Bell, and hitting-time subgroups of the groups are defined and discussed. A relationship between the generalized Sheffer groups with respect to different type of power series is presented. The equivalence of the defined Riordan array pairs and generalized Stirling number pairs is given. A type of inverse relations of various series is constructed by using pairs of Riordan arrays. Finally, several applications involving various arrays, polynomial sequences, special formulas and identities are also presented as illustrative examples.     

Deviation Matrix,Laurent Series and Blackwell Optimality in Countable State Markov Decision Processes

This paper presents a recurrent condition on Markov decision processes with a countable state space and bounded rewards. The condition is sufficient for the existence of a Blackwell optimal stationary policy, having the Laurent series expansion with continuous coefficients. It is so relaxed that the Markov chain corresponding to a stationary policy may have countably many periodic recurrent classes. Our method finds the deviation matrix in an explicit form.     

Singularity Structure of Third-Order Dynamical Systems. I

A general third-order dynamical system with polynomial right-hand sides of finite degrees in the dependent variables is analyzed to unravel the singularity structure of its solutions about a movable singular point. To that end, the system is first transformed to a second-order Briot–Bouquet system and a third auxiliary equation via a transformation, similar to one used earlier by R. A. Smith in 1973–1974 for a general second-order dynamical system. This transformation imposes some constraints on the coefficients appearing in the general third-order system. The known results for the second-order Briot–Bouquet system are used to explicitly write out Laurent or psi-series solutions of the general third-order system about a movable singularity. The convergence of the relevant series solutions in a deleted neighborhood of the singularity is ensured. The theory developed here is illustrated with the help of the May–Leonard system.