泰勒公式及泰勒级数之妙用

泰勒公式及泰勒级数是非常重要的数学工具,除了读者熟知的应用方面外,在其他问题的解决中也有妙用.举例介绍了应用泰勒公式及泰勒级数解决判断级数的敛散性、证明与积分有关的不等式等问题.     

Extremal integral representations

Let X be a closed bounded convex subset with the Radon-Nikodym property of a Banach space. For tight Borel probability measures μ, v on X, define μ ? v iff there is a dilation T on X such that T(μ) = v. Then, for every x?X, there is a measure μ on X which is maximal in the partial order ? and which has barycenter x. If X is separable, then μ(ex X) = 1 for all maximal measures μ. In general, a maximal measure need not be “on” ex X in this strong sense. If X is weakly compact, then a maximal measure is “on” ex X in the looser sense that μ(B) = 1 for all weak Baire sets B ? ex X.     

The spiral minimal surfaces and their Legendre and Weierstrass representations

A class of spiral minimal surfaces in E³ is constructed using a symmetry reduction. The reduction leads to a cubic-nonlinear ODE whose phase portrait is described using an auxiliary Riccati's equation and the Warzewski topological principle for its solutions. The new surfaces are invariant with respect to the composition of rotation and dilation. The solutions are obtained in parametric form through the Legendre and the Weierstrass representations, and also their asymptotic behaviour is described.     

The nonparametric integral of the Calculus of Variations as a Weierstrass integral: Existence and representation

The definition and properties of an abstract and very general nonparametric integral of the Calculus of Variations is presented. In harmony with the Lewy-McShane approach, the nonparametric integral ∝ f, for set functions ? taking their values in a Banach space E, is defined in terms of its associated parametric integral. For the latter use is made of the abstract parametric integral proposed by Cesari in Rⁿ and then extended to Banach spaces by Breckenridge, Warner, and the authors. A condition (c) is shown to be relevant for the existence of the integral, and is preserved by the nonlinear operation f. Also, for f nonnegative, a Tonelli-type theorem is proved in the sense that the so defined Weierstrass integral ∝ f is always larger than or equal to the corresponding Lebesgue integral, and equality holds if and only if absolute continuity conditions hold. In the proof a suitable martingale is associated and a convergence theorem for martingales is applied. Applications to the calculus of variations will follow.     

关于Taylor级数与Fourier级数的几点注记

Taylor级数与Fourier级数是两类非常重要的函数项级数,二者在发展与应用背景、展开条件、收敛性和展开的唯一性等方面不尽相同,本文对此作了一些总结与探讨。     

Smooth Universal Taylor Series

Let W í \Bbb C\Omega \subseteq {\Bbb C} be a simply connected domain in \Bbb C{\Bbb C} , such that {￥} è[ \Bbb C \[`(W)]]\{\infty\} \cup [ {\Bbb C} \setminus \bar{\Omega}] is connected. If g is holomorphic in Ω and every derivative of g extends continuously on [`(W)]\bar{\Omega} , then we write g ∈ A^∞ (Ω). For g ∈ A^∞ (Ω) and z ? [`(W)]\zeta \in \bar{\Omega} we denote S_N (g,z)(z) = ?^N_l=0\fracg^(l) (z)l ! (z-z)^lS_N (g,\zeta )(z)= \sum^{N}_{l=0}\frac{g^{(l)} (\zeta )}{l !} (z-\zeta )^l . We prove the existence of a function f ∈ A^∞(Ω), such that the following hold:

Smooth Universal Taylor Series

Let be a simply connected domain in , such that is connected. If g is holomorphic in Ω and every derivative of g extends continuously on , then we write g ∈ A^∞ (Ω). For g ∈ A^∞ (Ω) and we denote . We prove the existence of a function f ∈ A^∞(Ω), such that the following hold:

关于ζ函数的积分表示

主要研究了ζ函数的积分表示形式;通过解析数论的研究方法,利用黎曼ζ函数方程,给出了关于赫尔维茨ζ函数的埃尔米特公式,利用埃尔米特公式得出关于Γ函数的比内第二表达式,通过ζ函数得出Γ函数一些性质.     

应用多元泰勒级数展开式计算电多极矩

对矩形排列且正负相间的4个点电荷,应用泰勒级数的三维展开式,可发现其电势由电多极矩所激发的各级电势叠加组成,从而得出其各级近似.     

Regularization and integral representations of Hermite processes

It is known that Hermite processes have a finite-time interval representation. For fractional Brownian motion, the representation has been well known and plays a fundamental role in developing stochastic calculus for the process. For the Rosenblatt process, the finite-time interval representation was originally established by using cumulants. The representation was extended to general Hermite processes through the convergence of suitable partial sum processes. We provide here an alternative and different proof for the finite-time interval representation of Hermite processes. The approach is based on regularization of Hermite processes and the fractional Gaussian noises underlying them, and does not use cumulants nor convergence of partial sums.     

Eigenspaces of the Laplacian; integral representations and irreducibility

An integral representation is given for eigenfunctions of the Laplacian on a noncompact two-point homogeneous space. For the flat case an irreducibility criterion for the eigenspace representations is proved, complementing an earlier result [6a, p. 143] for the nonflat cases.     

Additive representations of non-additive measures and the choquet integral

This paper studies some new properties of set functions (and, in particular, “non-additive probabilities” or “capacities”) and the Choquet integral with respect to such functions, in the case of a finite domain. We use an isomorphism between non-additive measures on the original space (of states of the world) and additive ones on a larger space (of events), and embed the space of real-valued functions on the former in the corresponding space on the latter. This embedding gives rise to the following results:

the Choquet integral with respect to any totally monotone capacity is an average over minima of the integrand;

the Choquet integral with respect to any capacity is the difference between minima of regular integrals over sets of additive measures;

under fairly general conditions one may define a “Radon-Nikodym derivative” of one capacity with respect to another;

the “optimistic” pseudo-Bayesian update of a non-additive measure follows from the Bayesian update of the corresponding additive measure on the larger space.

We also discuss the interpretation of these results and the new light they shed on the theory of expected utility maximization with respect to non-additive measures.     

A method of generating integral representations

Conclusion The objective of this paper was to present a unified method of generating integral representations in special function theory using the idea of simultaneous separability of ²U+U=0 in orthogonal curvilinear coordinate systems. The main results of this paper are formulas (8), (9) and (10) with formula (11) as a special case of (10).This paper contains results submitted by W. W.Turner as a dissertation for the degree of Doctor of Philosophy at Michigan State University.     

On integral representations of linear forms

Starting from a given *-algebra, we consider integral representations of positive linear forms on the hermitian spectrum of the algebra, providing necessary and sufficient conditions theorem. This specializes to previous results of R. S. Bucy—G. Maltese and G. Maltese for Banach *-algebras, and M. Fragoulopoulou for Imc *-algebras.