LARGE DEVIATIONS FOR MARKOV CHAINS

Donsker-Varadhan's large deviation theory is applied to Markov chains with continuous parameter and countable state space. Since the lower estimate given by Donsker and Varadhan [5; part 4] is general enough in our context, we study only the upper estimate, and make the conditions for the estimate explicit.     

LARGE DEVIATIONS FOR SYMMETRIC DIFFUSION PROCESSES

Let a(x)=(a_(ij)(x)) be a uniformly continuous, symmetric and matrix-valued function satisfying uniformly elliptic condition, p(t, x, y) be the transition density function of the diffusion process associated with the Diriehlet space (, H_0~1 (R~d)), where(u, v)=1/2 integral from n=R~d sum from i=j to d(u(x)/x_i v(x)/x_ja_(ij)(x)dx).Then by using the sharpened Arouson's estimates established by D. W. Stroock, it is shown that2t ln p(t, x, y)=-d~2(x, y).Moreover, it is proved that P_y~6 has large deviation property with rate functionI(ω)=1/2 integral from n=0 to 1<(t), α~(-1)(ω(t)),(t)>dtas s→0 and y→x, where P_y~6 denotes the diffusion measure family associated with the Dirichlet form (ε, H_0~1(R~d)).     

LARGE DEVIATIONS AND MODERATE DEVIATIONS FOR m-NEGATIVELY ASSOCIATED RANDOM VARIABLES

M-negatively associated random variables,which generalizes the classical one of negatively associated random variables and includes m-dependent sequences as its par- ticular case,are introduced and studied.Large deviation principles and moderate devi- ation upper bounds for stationary m-negatively associated random variables are proved. Kolmogorov-type and Marcinkiewicz-type strong laws of large numbers as well as the three series theorem for m-negatively associated random variables are also given.     

LARGE DEVIATIONS FOR FIELDS WITH STATIONARY INDEPENDENT INCREMENTS

ＬＡＲＧＥＤＥＶＩＡＴＩＯＮＳＦＯＲＦＩＥＬＤＳＷＩＴＨＳＴＡＴＩＯＮＡＲＹＩＮＤＥＰＥＮＤＥＮＴＩＮＣＲＥＭＥＮＴＳＧＡＯＦＵＱＩＮＧ（高付清）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＨｕｂｅｉＵｎｉｖｅｒｓｉｔｙ，Ｗｕｈａｎ４３００７２...     

LARGE DEVIATIONS FOR STATIONARY φ-MIXING SEQUENCES IN τ-TOPOLOGY

The results of Bryc on large deviations for empirical measures of stationary Φ-mixing sequences are extended. Bryc states his results in the usual weak topology on the space ofprobability measures. In this paper, under somewhat weaker assumptions than those of Bryc,the author extends Bryc's results by taking the finer topology which is generated by the integralsover bounded measurable functions.     

LARGE DEVIATIONS FOR INFINITE DIMENSIONAL AND REVERSIBLE REACTION－DIFFUSION PROCESSES

ＬＡＲＧＥＤＥＶＩＡＴＩＯＮＳＦＯＲＩＮＦＩＮＩＴＥＤＩＭＥＮＳＩＯＮＡＬＡＮＤＲＥＶＥＲＳＩＢＬＥＲＥＡＣＴＩＯＮ－ＤＩＦＦＵＳＩＯＮＰＲＯＣＥＳＳＥＳＣＨＥＮＪＩＮＷＥＮ（陈金文）（ＤｅｐｅｒｔｍｅｎｔｏｆＡｐｐｌｉｅｄＭａｔｈｅｍａｔｉｃｓ，Ｔ...     

ESTIMATES OF EIGENVALUES FOR UNIFORMLY ELLIPTIC OPERATOR OF SECOND ORDER

ＥＳＴＩＭＡＴＥＳＯＦＥＩＧＥＮＶＡＬＵＥＳＦＯＲＵＮＩＦＯＲＭＬＹＥＬＬＩＰＴＩＣＯＰＥＲＡＴＯＲＯＦＳＥＣＯＮＤＯＲＤＥＲＱＩＡＮＣＨＵＮＬＩＮ（钱椿林）ＣＨＥＮＺＵＣＨＩ（陈祖墀）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｔｎａｔｉｃｓ，Ｕｎｉｖｅｒ...     

INEQUALITIES FOR EIGENVALUES OF THE POLYHARMONIC OPERATOR Δ~p

In this paper we consider the bounds of the eigenvalues for a class of polyharmonic operators andobtain the bounds for(n 1)th eigenvalue interm of the first n eigenvalues.Those estimates do notdepend on the domain in which the problem is considered.     

A CLASS OF COLLINEAR SCALING ALGORITHMS FOR UNCONSTRAINED OPTIMIZATON

A Class of Collinear Scaling Algorithms for Unconstrained Optimization. An appealing approach to the solution of nonlinear optimization problems based on conic models of the objective function has been in troduced by Davidon (1980). It leads to a broad class of algorithms which can be considered to generalize the existing quasi-Newton methods. One particular member of this class has been deeply discussed by Sorensen (1980), who has proved some interesting theoretical properties. In this paper, we generalize Sorensen's technique to Spedicato three-parameter family of variable-metric updates. Furthermore, we point out that the collinear scaling three- parameter family is essentially equivalent to the Spedicato three-parameter family. In addition, numerical expriments have been carried out to compare some colliner scaling algorithms with a straightforward implementation of the BFGS quasi-Newton method.     

EIGENVALUES ON THE BOUNDARY OF CERTAIN INCLUSION FOR THE SPECTRUM OF A MATRIX

We introduce four types of special eigenvalues which lie on the boundary of certain inclusion regions for the spectrum of a complex square matrix, i.e. , R_r(G_c)-,O(a)-,B_r(B_c)-. and OB(a)- eigenvalues. Then we characterize these eigenvalues and their corresponding eigenvectors for irreducible matrices, Finally we give some new sufficient conditions for an irreducible complex matrix to be nonsingular.     

FRANCIS QL ALGORITHM FOR FINDING THE EIGENVALUES OF ANTI-SYMMETRIC MATRICES

In this paper,we demonstrate that the double-shift QL algorithm for an irreducible anti-symmetric iridiagonal matrix with the shifts being two eigenvalues of the 2×2 matrix in the left upper corner of this matrix is convergent and the convergence rale of Ms kind of algorithm is generally cubic.     

LARGE DEVIATION FOR EMPIRICAL FIELD OF A SYMMETRIC MEASURE

This paper discusses the large deviation for the empirical field of a symmetricmeasure.The lower bound of the large deviation is obtained by extending the classcal Shannon-McMillan theorem. The upper bound is established by means of Legendre transformation and the minimax theorem.     

LARGE TIME STEP GENERALIZATIONS OF GLIMM''''S SCHEME FOR SYSTEMS OF CONSERVATION LAWS

Two kinds of generalizations of Glimm's scheme for Courant numbers larger than1/2 is introduced. For one kind of the generalizations, referred to as L. T. S. Glimm's scheme(I), it is shown that for any fixed (but arbitrary large) Courant number if a sequence ofapproximate solutions converges to a limit u as the mesh is refined, then u is a werk solutionto the system of conservation laws/for almost choise of random sequence. Furthermore it is,obtained that for scalar equations and systems of conservation laws the family ofapproximate solutions contains convergent subsequnce. For another kind of generalizations with any fixed (but arbitrary large) Courantnumber, referred to as L. T. S. Glhnm's scheme (Ⅱ), it is proved that the family ofapproximate solutions to the system of isothermal gas dynamics equations contains aconvergent subsequence provided the total varigtion of the initial data is bounded.     

ASYMPTOTIC BEHAVIOUR OF EIGENVALUES FOR THE DISCONTINUOUS BOUNDARY-VALUE PROBLEM WITH FUNCTIONAL-TRANSMISSION CONDITIONS

In this study, the boundary-value problem with eigenvalue parameter generated by the differential equation with discontinuous coefficients and boundary conditions which contains not only endpoints of the considered interval, but also point of discontinuity and linear functionals is investigated. So, the problem is not pure boundary-value. The authors single out a class of linear functionals and find simple algebraic conditions on coefficients, which garantee the existence of innnit number eigenvalues. Also the asymptotic formulas for eigenvalues are found.     

ENTROPY CONSISTENCY OF LARGE TIME STEP SCHEMES FOR ISENTROPIC EQUATIONS OF GAS DYNAMICS

In this paper, We show for isentropic equations of gas dynamics with adiabatic exponent γ=3 that approximations of weak solutions generated by large time step Godunov's scheme or Glimm's scheme give entropy solution in the limit if Courant number is less than or equal to 1.     

LARGE TIME EXISTENCE AND BEHAVIOR OF SOLUTION FOR SYSTEMS OF NONLINEAR REACTION-DIFFUSION EQUATIONS

This paper deals with the existence of solution for the systems u_i-K△u=F(t, x, u, ▽u) with the initial boundary value condition, where F (t, x, y, z) grows in y, z with arbitrary power and goes to infinity as t→∞. Under suitable assumptions on the nonlinear term F and the initial value condition, we will obtain a unique global solution and an asymptotic estimate.     

THE ERDOS-RENYI LAWS OF LARGE NUMBERS FOR NON-IDENTICALLY DISTRIBUTED RANDOM VARIABLES

The Erdos-Renyi law of large numbers (1970) is the first important result forasymptotic behaviours of increments of partial sams of a sequence of random variableswith apan [ClogN]. Some generalizations have been done sinoe then, such as conver-gence rate of the limit, some results when order of span being either higher or lowerthan log N. But all these results are only obtained in the case of i. i. d. random variables.This paper aims at the generalization of these results to the ease when random variablesare independent, but not necessarily identically distributed. To this end Chernoff Theoremis generalized to the corlesponding case at first.     

THE UPPER BOUND ESTIMATE OF LARGE DEVIATION FOR INHOMOGENEOUS MARKOV PROCESSES

In this paper, the Feynmann-Kac formula is extended to the inhomogeneous case. Using the extended formula, the author studies the upper bound estimate of large deviation for some class of inhomogenous Markov processes. The results in the paper can beconsidered as a generalization of the homogeneous ones.     

EQUIVALENCE OF COMPLETE CONVERGENCE AND LAW OF LARGE NUMBERS FOR B-VALUED RANDOM ELEMENTS＇

Under some conditions on probability, this note discusses the equivalence between the complete convergence and the law of large number forB- valued independent random elements. The results of [10] become a simple corollary of the results here. At the same time, the author uses them to investigate the equivalence of strong and weak law of large numbers, and there exists an example to show that the conditions on probability are weaker. Project supported by the National Natural Science Foundation of China.     

LARGE DEVIATION FOR THE EMPIRICAL CORRELATION COEFFICIENT OF TWO GAUSSIAN RANDOM VARIABLES

In this article,the author obtains the large deviation principles for the empir- ical correlation coefficient of two Gaussian random variables X and Y.Especially,when considering two independent Gaussian random variables X,Y with the means EX. EY (both known),wherein the author gives two kinds of different proofs and gets the same results.