Tenth order mock theta functions in Ramanujan's lost notebook (IV)

Ramanujan's lost notebook contains many results on mock theta functions. In particular, the lost notebook contains eight identities for tenth order mock theta functions. Previously the author proved the first six of Ramanujan's tenth order mock theta function identities. It is the purpose of this paper to prove the seventh and eighth identities of Ramanujan's tenth order mock theta function identities which are expressed by mock theta functions and a definite integral. L. J. Mordell's transformation formula for the definite integral plays a key role in the proofs of these identities. Also, the properties of modular forms are used for the proofs of theta function identities.

     

Some Theorems on the Rogers-Ramanujan Continued Fraction and Associated Theta Function Identities in Ramanujan's Lost Notebook

In his lost notebook, Ramanujan recorded several modular equations of degree 5 related to the Rogers-Ramanujan continued fraction R(q). We prove several of these identities and give factorizations of some of them in this paper.The parameter k = R(q) R₂(q₂) introduced by Ramanujan in his second notebook has not been recognized for its usefulness. In this work, we demonstrate how beautifully the parameter k works, as we prove several identities involving k stated by Ramanujan in the lost notebook.     

一类Sparre Andersen风险模型的Gerber-Shiu函数

本文考虑了一类相邻两次索赔的时间间隔服从Erlang($n$)和Erlang($m$)的混合分布的Sparre Andersen风险模型.主要目的是研究Gerber-Shiu函数$\phi_\delta(u)$,首先证明了$\phi_\delta(u)$满足一个高阶的积分微分方程,然后讨论了广义Lundberg方程根的性质,在此基础上导出了$\phi_\delta(u)$的拉普拉斯变换并且证明了$\phi_\delta(u)$满足一个更新方程,最后给出了一个例子.     

Eisenstein Series in Ramanujan's Lost Notebook

In his lost notebook, Ramanujan stated without proofs several beautifulidentities for the three classsical Eisenstein series (in Ramanujan's notation) P(q), Q(q), and R(q). The identities are given in terms of certain quotients of Dedekind eta-functions called Hauptmoduls. These identities were first proved by S. Raghavan and S.S. Rangachari, but their proofs used the theory of modular forms, with which Ramanujan was likely unfamiliar. In this paper we prove all these identities by using classical methods which would have been well known to Ramanujan. In fact, all our proofs use only results from Ramanujan's notebooks.     

Implicitly enriched Galerkin methods for numerical solutions of fourth‐order partial differential equations containing singularities

Highlights are the following:

• For any integer , we construct ‐continuous partition of unity (PU) functions with flat‐top from B‐spline functions to have numerical solutions of fourth‐order equations with singularities. B‐spline functions are modified to satisfy clamped boundary conditions.
• To handle singularity arising in fourth‐order elliptic differential equations, these modified B‐spline functions are enriched either by introducing enrichment basis functions implicitly through particular geometric mappings or by adding singular basis functions explicitly.
• To show the effectiveness of the proposed implicit enrichment methods (mapping method), the accuracy, the number of degrees of freedom (DOF), and matrix condition numbers are computed and compared in the h‐refinement, the p‐refinement, and the k‐refinement of the approximation space of B‐spline basis functions.

Using Partition of unity (PU) functions with flat‐top, B‐spline functions are modified to satisfy boundary conditions of the fourth‐order equations. Since the standard isogeometric analysis (IGA) as well as the conventional FEM have limitations in handling fourth‐order differential equations containing singularities, we consider two enrichment methods (explicit and implicit) in the framework of the p‐, the k, and the h‐refinements of IGA. We demonstrate that both enrichment methods yield good approximate solutions, but explicit enrichment methods give large (almost singular) matrix condition numbers and face integrating singular functions. Because of these limitations of external enrichment methods, we extensively investigate implicit enrichment methods (mapping methods) that virtually convert fourth‐order elliptic problems with singularities to problems with no influence of the singularities. Effectiveness of the proposed mapping method extensively tested to one‐dimensional fourth‐order equation with singularities. The implicit enrichment (mapping) method is extended to the two‐dimensional cases and test it to fourth‐order partial differential equations on cracked domains.     

在空间W_0~(1，x)L~(P(x))(Q)及其共轭空间中的若干收敛性定理

本文证明了在空间W_0~(1,x)L~(p(x))(Q)及其共轭空间中三个收敛性定理.在附加较弱条件的情况下,给出了几乎处处收敛蕴含弱收敛,弱收敛蕴含强收敛.     

Uniqueness Theorems for Periodic (in Mean) Functions on Quaternion Hyperbolic Space

New uniqueness theorems for periodic (in mean) functions on quaternion hyperbolic space are obtained.     

Antipodal Theorems for Sets in Rn Bounded by a Finite Number of Spheres

This is a continuation of paper in Adv. Appl. Math. 22 (1999), 219–226, on an antipodal theorem for sets Dⁿ in Rⁿ bounded by a finite number of spheres. Here this theorem is first applied to set-valued mappings from Dⁿ to the boundary of an (n + 1)-cube or a d- dimensional octahedron. Next, the antipodal theorem is reformulated in terms of real continuous functions on Dⁿ, together with applications to the classical theorems of Borsuk–Ulam and Lusternik–Schnirelmann–Borsuk.     

On a class of analytic functions related to conic domains and associated with Carlson-Shaffer operator

Making use of the Carlson-Shaffer convolution operator, we introduce and study a new class of analytic functions related to conic domains. The main object of this paper is to investigat inclusion relations, coefficient bound for this class. We also show that this class is closed under convolution with a convex function. Some applications are also discussed.     

Model for spontaneous frequency sweeping of an Alfvén wave in a toroidal plasma

We study the frequency chirping signals arising from spontaneously excited toroidial Alfvén eigenmode (TAE) waves that are being driven by an inverted energetic particle distribution whose free energy is tapped from the generic particle/wave resonance interaction. Initially a wave is excited inside the Alfvén gap with a frequency determined from the linear tip model of Rosenbluth, Berk and Van dam (RBV) [1]. Hole/clumps structures are formed and are observed to chirp towards lower energy states. We find that the chirping signals from clump enter the Alfvén continuum which eventually produce more rapid chirping signals. The accuracy of the adiabatic approximation for the mode evolution is tested and verified by demonstrating that a WKB-like decomposition of the time response for the field phase and amplitude agree with the data. Plots of the phase space structure correlate well with the chirping dependent shape of the separatrix structure. A novel aspect of the simulation is that it performed close to the wave frame of the phase space structure, which enables the numerical time step to remain the same during the simulation, independent of the rest frame frequency.     

Some analytical and numerical investigation of a family of fractional-order Helmholtz equations in two space dimensions

In this article, we aim at solving a family of two-dimensional fractional-order Helmholtz equations by using the Laplace-Adomian Decomposition Method (LADM). The fractional-order derivatives, which we use in this investigation, follows the Liouville-Caputo definition. Our results based upon the LADM are obtained in series form that helps us in analyzing the analytical solutions of the fractional-order Helmholtz equations considered here. For illustration and verification of the analytical procedure using the LADM, several numerical examples and graphical representations are presented for the analytical solution of the fractional-order Helmholtz equations. The mathematical analytic procedure, which we have used here, has shown that the LADM is a fairly accurate and computable method for the solution of problems involving fractional-order Helmholtz equations in two dimensions. In an analogous manner, one can apply the LADM for finding the analytical solution of other classes of fractional-order partial differential equations.     

A meshless method for numerical solution of a linear hyperbolic equation with variable coefficients in two space dimensions

A meshless method is proposed for the numerical solution of the two space dimensional linear hyperbolic equation subject to appropriate initial and Dirichlet boundary conditions. The new developed scheme uses collocation points and approximates the solution employing thin plate splines radial basis functions. Numerical results are obtained for various cases involving variable, singular and constant coefficients, and are compared with analytical solutions to confirm the good accuracy of the presented scheme. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

基于EMD方法的混沌信号中周期分量的提取

提出一种从Duffing振子产生的混沌信号中提取谐波分量的方法．依据任何信号由不同的固有简单振动模态组成的概念,利用经验模式分解（EMD）方法,将混沌信号分离为不同的内在模态函数(IMF),并在特定参数下从中分解出单一频率成分的谐波信号,从而成功地将混沌信号和谐波分量分离．仿真实验都表明该方法非常有效．     

Variational formulations for scattering in a three‐dimensional acoustic waveguide

Variational formulations for direct time‐harmonic scattering problems in a three‐dimensional waveguide are formulated and analyzed. We prove that the operators defined by the corresponding forms satisfy a Gårding inequality in adequately chosen spaces of test and trial functions and depend analytically on the wavenumber except at the modal numbers of the waveguide. It is also shown that these operators are strictly coercive if the wavenumber is small enough. It follows that these scattering problems are uniquely solvable except possibly for an infinite series of exceptional values of the wavenumber with no finite accumulation point. Furthermore, two geometric conditions for an obstacle are given, under which uniqueness of solution always holds in the case of a Dirichlet problem. Copyright © 2007 John Wiley & Sons, Ltd.     

Quenching rate for a nonlocal problem arising in the micro-electro mechanical system

In this paper, we study the quenching rate of the solution for a nonlocal parabolic problem which arises in the study of the micro-electro mechanical system. This question is equivalent to the stabilization of the solution to the transformed problem in self-similar variables. First, some a priori estimates are provided. In order to construct a Lyapunov function, due to the lack of time monotonicity property, we then derive some very useful and challenging estimates by a delicate analysis. Finally, with this Lyapunov function, we prove that the quenching rate is self-similar which is the same as the problem without the nonlocal term, except the constant limit depends on the solution itself.     

Boundary behavior of monotone Sobolev functions on John domains in a metric space

This article deals with weighted boundary limits of monotone Sobolev functions on bounded s-John domains in a metric space.     

A continuation and existence result for a boundary value problem on an unbounded domain arising for the electrical potential in a cylindrical double layer

Using a continuation theorem for contractions, the existence, uniqueness, and an approximation method for a class of nonlinear boundary value problems on an unbounded interval is obtained. The results apply in particular to the problem in the title.     

Global Solution Approach for a Nonconvex MINLP Problem in Product Portfolio Optimization

The rigorous and efficient determination of the global solution of a nonconvex MINLP problem arising from product portfolio optimization introduced by Kallrath (2003) is addressed. The objective of the optimization problem is to determine the optimal number and capacity of reactors satisfying the demand and leading to a minimal total cost. Based on the model developed by Kallrath (2003), an improved formulation is proposed, which consists of a concave objective function and linear constraints with binary and continuous variables. A variety of techniques are developed to tighten the model and accelerate the convergence to the optimal solution. A customized branch and bound approach that exploits the special mathematical structure is proposed to solve the model to global optimality. Computational results for two case studies are presented. In both case studies, the global solutions are obtained and proved optimal very efficiently in contrast to available commercial MINLP solvers.     

基于聚类分析和Cobb-Douglas函数的我国人才经济贡献率测算

建立人才环境综合评价指标体系,对指标体系进行因子分析,根据各个主因子得分,通过聚类将中国(除港澳台以外的)31个省(市、自治区)划分为若干类别;对于每一类别,分别构建物质资本、人才资本、一般人力资本等到经济产出之间的改进柯布道格拉斯生产函数模型,得到不同类别地区人才对经济增长的贡献率,进而计算全国人才对经济增长贡献率的总体水平.研究表明:2001-2010年期间,31个地区按照人才环境可分为3类.第一类地区包括北京、上海、天津,人才环境发展水平最高,人才经济贡献率为27.13%;第二类地区包括广东、浙江等东部沿海6个省份,人才环境发展水平居中,人才经济贡献率为24.91%;第三类地区为其余22各省份,人才环境发展水平偏低,人才经济贡献率为18.78%.全国人才经济贡献率总体水平为22.68%.     

Covering of nonlinear maps on a cone in neighborhoods of irregular points

Inverse function theorems for smooth nonlinear maps defined on convex cones in Banach spaces in a neighborhood of an irregular point are considered. The corresponding covering theorem is proved. The proofs are based on a Banach open mapping theorem for convex cones in Banach spaces, which is also proved in the paper. Sufficient conditions for tangency to the zero set of a nonlinear map without a priori regularity assumptions are obtained.Translated from Matematicheskie Zametki, vol. 77, no. 4, 2005, pp. 483–497.Original Russian Text Copyright © 2005 by A. V. Arutyunov.This revised version was published online in April 2005 with a corrected issue number.