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Formulae of type Ankeny-Artin-Chowla for class numbers of general cyclic sextic fields

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Chinese Science Bulletin 1998年第10期43卷 824-826页

作者： LIU Tong department of applied mathematics, tsinghua university, beijing 100084, china department of applied mathematics Tsinghua University Beijing China

Let K 6 be a real cyclic sextic number fields, and K 2, K 3 be its quadratic and cubic subfields. Let h(L) denote the ideal class number of field L. Seven congruences for h -=h(K 6)/h(K 2)h(K 3) are obtained. In particular, when conductor f\-6 of K 6 is prime p, then Ch -≡B p-16B 5(p-1)6 (mod p), where C is an explicitly given constant, and B n is the Bernoulli number. These results for real cyclic sextic fields are an extension of results for quadratic and cyclic quartic fields obtained by Ankeny_Artin_Chowla, Kiselev, Carlitz, Lu Hongwen, Zhang Xianke from 1948 to 1988.

关键词： real cyclic sextic number fields class number fundamental relative unit.

RECOGNITION AND CLASSIFICATION FOR O(n)-EQUIVARIANT BIFURCATIONS WITH O(n)-CODIMENSION LESS THAN 5

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Chinese Annals of mathematics,Series B 1998年第4期19卷 391-400页

作者： WANG XIAOFENG (department of applied mathematics, tsinghua university, beijing 100084, china.)TANG YUN (department of applied mathematics, tsinghua university, beijing 100084, china.)WANG DUO (School of Mathematical Science, Peking universityl Beliing 10 Tsing Hua Univ Dept Appl Math Beijing 100084 Peoples R China Peking Univ Sch Math Sci Beijing 100080 Peoples R China

Bifurcation problems equivariant under the standard action of the orthogonal group O(n) up to O(n)-codimension 4 are classified into 19 classes. For each class the normal form and one universal unfolding are calculate... 详细信息

关键词： O(n)-equivariant bifurcation Normal form Universal unfolding Recognition problem

Centralizers and Iterate Radicals of Morse-Smale Diffeomorphisms of the Circle

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Acta Mathematica Sinica,English Series 1995年第1期11卷 1-11页

作者： Zhang Meirong department of applied mathematics tsinghua university beijing,100084 china department of applied mathematics Tsinghua University Beijing People's Republic of China

In this paper,we will use the embedding flows in[1],[2]to give a complete descrip- tion of the smooth centralizers and iterate radicals of all C^r(r≥2)Morse-Smale diffeomorphisms of the circle S^*** a result,we prove... 详细信息

关键词： Centralizers and Iterate Radicals of Morse-Smale Diffeomorphisms of the Circle TDFs

A Remark on the Symplectic Blow-up in Dimension 4

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Acta Mathematica Sinica,English Series 1996年第4期12卷 379-384页

作者： Ma Renyi department of applied mathematics tsinghua university beijing, 100084 china department of applied mathematics Tsinghua University Beijing China

In this note, we prove that the symplectic blow-up or blow-down in the dimension 4 is rigid, i.e. the symplectic area of the divisor does not exceed the symplectic radius of the neighbourhood on which we do the blow-u... 详细信息

关键词： J-Holomorphic disc Contact manifold Symplectic blow-up

THE EXISTENCE OF RADIAL LIMITS OF ANALYTIC FUNCTIONS IN BANACH SPACES

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Chinese Annals of mathematics,Series B 2001年第4期22卷 513-518页

作者： BU SHANGQUAN department of applied mathematics, tsinghua university, beijing 100084, china. E-mail: sbu@*** department of applied mathematics Tsinghua University Beijing 100084 China

Let X be a complex Banach space without the analytic Radon-Nikodym property. The author shows that G = {f∈H∞(D,X) there exists∈＞ 0, such that for almost allθ∈[0, 2], lim sup f(reiθ) - f(seiθ)≥∈ } is a dense open subset of H∞(D, X). It is also shown that for every open subset B of T, there exists F∈H∞ (D, X), such that F has boundary values everywhere on Bc and F has radial limits nowhere on B. When A is a measurable subset of T with positive measure, there exists f∈H∞ (D, X), such that f has nontangential limits almost everywhere on Ac and f has radial limits almost nowhere on A.

关键词： Analytic Radon-Nikodym property Radial limits and vector-valued Hardy space

Computational intelligence:From mathematical point of view

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Chinese Science Bulletin 1999年第10期44卷 865-880页

作者： QIAN Minping and GONG Guangludepartment of Probability and Statistics, Peking university, beijing 100871, china department of applied mathematics, tsinghua university, beijing 100084, china Deportment of Probability and Statistics Peking University Beijing China department of applied mathematics Tsinghua University Beijing China

A simple but illustrative survey is given on various approaches of computational intelligence with their features, applications and the mathematical tools involved, among which the simulated annealing, neural networks, genetic and evolutionary programming, self-organizing learning and adapting algorithms, hidden Markov models are recommended intensively. The common mathematical features of various computational intelligence algorithms are ***, two common principles of concessive strategies implicated in many computational intelligence algorithms are discussed.

关键词： computational intelligence simulated annealing neural network genetic algorithm evolutionary program-ming self-organizing learning self-adapting algorithm hidden Markov model concession strategy.

Uniqueness Theorem of Solutions for Stochastic Differential Equation in the Plane

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Acta Mathematica Sinica,English Series 1998年第4期14卷 495-500,201+502-506页

作者： Liang Zongxia, department of applied mathematics, tsinghua university beijing 100084, china department of applied mathematics Tsinghua University Beijing 100084 China

Let M = {Mz, z∈R+2} be a continuous square integrable martingale and A = {Az, z∈ R+2} be a continuous adapted increasing process. Consider the following stochastic partial differential equations in the plane: dXz=α（z, Xz）dM2+β（z,Xz）dAz, z∈R+2, Xz=Zz, z∈R+2, where R+2=[0,+∞)×[0,+∞) and R+2 is its boundary, Z is a continuous stochastic process on R+2. We establish a new theorem on the pathwise uniqueness of solutions for the equation under a weaker condition than the Lipschitz one. The result concerning the one-parameter analogue of the problem we consider here is immediate （see [1, Theorem 3.2]）. Unfortunately, the situation is much more complicated for two-parameter process and we believe that our result is the first one of its kind and is interesting in itself. We have proved the existence theorem for the equation in.

关键词： Two-parameter S. D. E., Two-parameter martingale ITO’s formula Pathwise uniqueness Gronwall’s-Bellman lemma

GLOBAL PHASE-PORTRAITS OF PLANEHOMOGENEOUS POIYNOMIAL VECTOR FIELDSAND STABILITY OF THE ORIGIN

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Systems Science and Mathematical Sciences 1997年第1期10卷 33-40页

作者： YANG Xiaojing (department of applied mathematics, tsinghua university, beijing 100084, china)

In this paper, the phase-portraits of plane homogeneous polynomial vector fields are studied and some formulae for the calculation of the number of different global phase-portraits of plane homogeneous polynomial vector fields are given. Some necessary and sufficient conditions for the global stability of these vector fields are derived.

关键词： Plane homogeneous polynomial vector fields phase-portraits stability

Relative Extreme Values and Extremum-Times of Jump Process:Upward Type

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Acta Mathematica Sinica,English Series 1997年第3期13卷 347-356页

作者： Ge Yubo (department of applied mathematics,tsinghua university,beijing,100084,china) department of applied mathematics Tsinghua University

Relative extreme values are defined by the supremum and minimum of a general jump process before its first time quitting from some state set,and relative extremum-times are defined by the first times reaching relative extreme *** main objective of this paper is to find out the exact distributions and moments of them as the maximum of the set is up or equal to the process initial *** especial cases,these results are applied to a general birth-death process and generalized birth-death processes.

关键词： Strong Markovian property Stopping time Distribution and moment