Modeling Optimal Effective Bandwidth of Page-Mode Memory for Stream-Oriented Computations

Memory system performance models have traditionally assumed that individual modules are insensitive to the sequence cf access requests. For current memory components, this assumption is not correct. Furthermore, performance models generally assume a stochastic sequence of reference. For stream-oriented computations, such models of reference are not suitable. This paper derives the optimal effective memory bandwidth achieved by stream-oriented computations at a single module of such memory components.     

Power Series Kernels

We introduce a class of analytic positive definite multivariate kernels which includes infinite dot product kernels as sometimes used in machine learning, certain new nonlinearly factorizable kernels, and a kernel which is closely related to the Gaussian. Each such kernel reproduces in a certain “native” Hilbert space of multivariate analytic functions. If functions from this space are interpolated in scattered locations by translates of the kernel, we prove spectral convergence rates of the interpolants and all derivatives. By truncation of the power series of the kernel-based interpolants, we constructively generalize the classical Bernstein theorem concerning polynomial approximation of analytic functions to the multivariate case. An application to machine learning algorithms is presented.      

Teaching Power Series

<正> Power series is an important topic in calculus. Many well-known elementary functions can be expanded into power series. Furthermore, the derivative of a power series is equal to the sum of its termwise derivatives. The proof involves the use of uniform convergence and is often omitted in a calculus course. In this note, we shall demonstrate how the subject can be     

Logarithmic-Exponential Power Series

We use generalized power series to construct algebraically anonstandard model of the theory of the real field with exponentiation.This model enables us to show the undefinability of the zetafunction and certain non-elementary and improper integrals.We also use this model to answer a question of Hardy by showingthat the compositional inverse to the function (log x) (loglog x) is not asymptotic as x+ to a composition of semialgebraicfunctions, log and exp.     

不等式证明的幂级数方法

以部分国内外数学竞赛题为例说明幂级数方法在证明不等式方面的应用.     

Encoding Algebraic Power Series

The division algorithm for ideals of algebraic power series satisfying Hironaka’s box condition is shown to be finite when expressed suitably in terms of the defining polynomial codes of the series. In particular, the codes of the reduced standard basis of the ideal can be constructed effectively.     

Three Power Series Techniques

The techniques and concepts we present are flags of regularschemes and their persistence under blow-up, the Gauss–Bruhatdecomposition of the group of formal automorphisms of affinespace, and coordinate-free initial ideals. All three are usedto construct and study invariants for resolution of singularities.2000 Mathematics Subject Classification 14B05, 14E15, 32S05,32S10, 32S45.     

两种基本函数项级数系数公式注记

分析部分高等数学教材在推导幂级数和傅里叶级数的系数公式时存在的不足,并从函数逼近论的角度,就此问题提出改进思想和方法.     

关于富里埃及数和幂级数的蔡查罗平均

Let f(x) be a periodic function of period 2π,|f(x)|and |f(x)|^p integrable on [0,2π].It is said to be f∈H_X¹,if (∫₀^2π|f(x+h)-f(x)|^pdx)^1/p≤|h|.It is said to be f∈H_X²,if (∫₀^2π|f(x+h)+f(x-h)-2f(x)|^pdx)^1/p≤2|h|.     

用傅里叶级数求自然数幂和

为寻求自然数幂和公式新方法,借助傅里叶级数这一解析工具,通过把幂函数xr在[0,n]上表示为傅里叶余弦级数,经过整点赋值求和,得到了自然数幂和的一个无穷级数表达式.运用此表达式进一步建立了自然数幂和问题与zeta函数之间的联系.     

PP-Rings of Generalized Power Series

Abstract As a generalization of power series rings, Ribenboim introduced the notion of the rings of generalized power series. Let R be a commutative ring, and (S, ≤) a strictly totally ordered monoid. We prove that (1) the ring [[R ^S,≤]] of generalized power series is a PP-ring if and only if R is a PP-ring and every S-indexed subset C of B(R) (the set of all idempotents of R) has a least upper bound in B(R) and (2) if (S, ≤) also satisfies the condition that 0 ≤s for any s∈S, then the ring [[R ^S,≤ ]] is weakly PP if and only if R is weakly PP. Research supported by National Natural Science Foundation of China, 19501007, and Natural Science Foundation of Gansu, ZQ-96-01     

Semirings of Formal Power Series

Semirings of formal power series over semirings, in particular, over k-semifields, are studied. It is also shown that the semiring of formal power series S[[x]] over a k-semifield S becomes a local semiring. Moreover, the Jacobson radical of S[x]] over a k-semifield S is described.AMS Subject Classification (1991): Primary 16Y60     

Power Series Methods and Multisummability

We recall the notion of a power series summability method and show, using results of W. Balser and A. Beck , that existence of a power series method which is stronger than multisummability would be equivalent to existence of an entire function with certain particular properties. We then prove that such entire functions cannot exist. In contrast to this negative result, we show that an iteration of power series methods is indeed stronger than multisummability.     

关于幂级数求和的教学实例分析

通过几组教学实例,阐述如何通过积分、求导、拆项等方法将复杂的幂级数转化为简单的基本型幂级数和等比级数以求其和函数.     

Restricted Universality of Power Series

We prove the existence of a power series having radius of convergence0, whose partial sums have universal approximation propertieson any compact set with connected complement that is containedin a finite union of circles centred at 0 and having rationalradii, but do not have such properties on any compact set withnonempty interior. This relates to a theorem of A. I. Seleznev.2000 Mathematics Subject Classification 30B30 (primary), 30B10,30E60 (secondary).     

What is the Power Series

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Generalized Inverse Power Series Modules

In this article, we introduce a construction called the generalized inverse power series module M[[S ^?1]] over a monoid ring R[S] with coefficients in an R-module M and exponents in a commutative monoid S. This construction is a generalization of the R[x]-modules which were discussed by S. Park in [12-14 Park , S. ( 2001 ). The general structure of inverse polynomial modules . Czechoslovak Math. J. 126 ( 2 ): 343 – 349 . Park , S. , Cho , E. ( 2004 ). Injective and projective properties of R[x]-modules . Czechoslovak Math. J. 129 ( 3 ): 573 – 578 . Park , S. , Cho , E. ( 2005 ). Purity of polynomial modules and inverse polynomial modules . Bull. Korean Math. Soc. 42 ( 3 ): 609 – 616 . ]. The injectivity and injective precovers of the generalized inverse power series module are considered. We also show that N is a pure submodule of M if and only if N[S] is a pure submodule of the monoid module M[S].     

幂级数在解方程中的应用

通过实例说明幂级数在解某些差分方程、函数方程及积分方程中的应用.     

广义幂级数环的PP性质

作为幂级数环的推广,Ｒｉｂｅｎｂｏｉｍ引入了广义幂级数环的概念．设Ｒ是有单位元的交换环,（Ｊ,≤）是严格全序半群．本文中我们证明了如下结果：（１）广义幂级数环 ［［Ｒｓ］］是ＰＰ－环当且仅当Ｒ是ＰＰ－环且Ｂ（Ｒ）的任意 Ｓ－可标子集Ｃ在Ｂ（Ｒ）中有最小上界;（２）如果对任意ｓ∈Ｓ都有０≤ｓ,则［［Ｒｓ,≤］］是弱ＰＰ－环当且仅当Ｒ是弱ＰＰ－环．我们还给出了一个例子说明交换的弱ＰＰ－环可以不是ＰＰ－环．