In this paper, the power spectrums ou the discrote Chrestenson transform are considered. They have the chractor of invariance for recurrent shift. Based on this charactr, the power spectrum and Q spectrum are defined. Moreover, a relation between the Q speetrum and transfarmati.on of self-correlation function is olrtainod. The correspondiiy properties of power spectrum and Q spectrum of Walsh-Hadamard transform are special causes of our results.

This paper investigates the problem of permmbations with sizes.Several counting formulae are obtained for the number of runs in a line and in a ring, respectively, with restrietion on the run-length and the number of runs. Sopne results in [1],[3] and [8] are included as special cases.

For each edge e=uv of a simple graph G, let d(e)=d(u)+d(v) where d(u) and d(v) are the degrrees of the vertices u and v respectively. The main result is as follows: Let G be a simple graph with p≥4 vertices and at least 2 edges. If G is not a path of length 3 and for each pair of edges e₀ and e₁ which have no common vertex, d(e₀)+d(e₁)>2p-2, then the line graph L(G) of G has a Hamiltonian cycle.

For the distribution family with density{f(x,θ),θ∈Θ} where f(x,θ) >0 and Θ∈Rr(r>1),the conditions for the covariance matrix of the unbiased estimators Ĝ(x) of a k-dimensional vector function G(θ) to attain the Graxnér-Rao lower bound everywhere are discussed under some regular assumption for the case k=r, k=1 and 1 < k < r respectively.

We study the prablem of three-dimensional, two-phase immiscible flow displacement and consider the influence of the gravity and the capillary pressure. The mathematical model of the problem is the initial boundary problem of systems of nonlinear degenerative parabolic equations. We put forward finite element methods of the continwous time and discuss intensively the existence and uniqueness and error estimates.

References Li and Chen and Li Guo-ying proposed and dvscussed a class of rubust estimators for dispersion matrices and principal compenents via projection pursuit. This paper shows that the estimators converge in probability to the estimands at an eaponential rate uniformly for any underlying distribution Ding in a neighlaorhood of the assumed distribution. Furthermore, the results with known location are extended to the case with unknown location.

Some mistakes in [1] are pointed out and some simple recursive formulas and explicit formnlations are given to solve the problem of counting the number of m level digital sequences of length n that satisfy some constraints.

This paper discusses a system of reaction-diffution equatnons with two diffusion Constants D₁, D₂. First, the system degenerates into a single equations as D₂ tends to infinity, The existance and stability of non-constant equilibria solutuons for the single equation are discussed. Next, the system can be considered as a perturbation for the aingle equation with D₂ large enough and the eaistance of its non-constant equilibria solutions is proved.

The random error sequence {u(t)} (t=1,2,…) in multiple regression models is assumed to be a linear process herein. Theorems on the retes of almost sure convergence of sample autocovariances of {u(t)} to their true values are proved. These rates are uniform in the log up to same order p(n),increasing with n.When {u(t)} is an i.i.d random sequence, the rate of slmost sure convergence of sample variance to its true value is also discussed.

For the factor incident graphs of shift registers with linear feedback functions a class of isomorphisms and automorphisms is constructed, by which a large class of feed-back functions of M sequences can be derived from one such function.

In this paper, we consider the problem

which arises from chemical reaction theory, where Ω is a smooth and bounded domain, r,σ and k are some positive constants.

Let K_n (1,0) denote the graph obtained by subdividing an edge of a complete graph K_n and K_n (1,m) denote the graph obtained by adding to K_n (1,0)m edges which are all incident with the subdividing vertex. It is shown that K_n (1,m) are chromatically unique when n≠m+5 and n≠m+6. The unique graph which is chromatically equivalent to K_m+5(1,m) is found. All of four graphs which are chromatically equivalent to K_m+k (1,m) are found too.

This paper deals with a one-unit repairable system with a good state and K failure states.For this system we consider apreventive maintenance policy with minimal repair at failure.When the system life distribution and all the repair timed distributions are arbitrary,using the supplementary variable method we obtain system availability and discuss the optimum policy maximizing the availability.

In this paper the theory of random operators on a special class of probabilistic metric space, E-space,is considered. As an application, we study the existence utilize the results obtained to study the existence and uniqueness of random solutions of nonlinear random operator equatibns and of the system of nonlinear random operator equations.

In this paper we establish a kind of expansion formula for functions having Lip-schitz oantinuous first order derivatives and derive sufficient optimality condition for these functions. Then the result ie applied to mathematical programming and stochastic programing to get sufficient optimality comditions.

A fast algorithm of polynomial regression is established in this paper as a result of improvement of Householder's solution for overdetermined equations while keeping its highua.ocuracy characteristics. The algorithm can detemmine the order of a polynomial model acid give the least squares estimate of the model paramieters. If the order of the model is n the computational time is only equal to that of the solution of an nth order overdetermined equation, much less than the existing methods.

This paper studies a method of nearest neighbour discrimination by randomization. It is shown that the estimates of conditional error probability converge to a constant with exponential bound.

In this paper, vve construct some algorithms for optimisation by families of point-to-set maps, some of which are improvements of the algorithms introduced in [2] and [3].We show that under weakened hypotheses the algorithms have better convergence properties. If there is only one member in the family of point-to-set maps, some extensions of Zangwill's theorem are special cases of the algorithms. As an example, the aanvergences of the variable metric algorithm and Catuahy stqepest ascent algorithm with inexact line search are liven.

Linear regression and correlation coefficient are generally defined on condition that second moments are finite. This conditiou is not natural for indicating the linear relationship between two random variables. In this paper, we consider a new form of linear predictor and comrelstion coefficient, in order to get rid of that condition. It is also shown tthat the linear predictor and the correlation coefficient considered are coincident with usual linear regression and correlation coefficient provided the distribution in question is normal.

This paper deals with the boundedness of the solutions of the following equations
(r(t)y')+h(t,y,y')+a(t)f(y)+b(t)g(y)=0
and (r(t)y')'+a(t)y+b(t)f(y(t-τ(t)))=p(t).By using the Bellman-Cronwall integral inequality as a main tool, some boundedness theorems are obtained.

The Darbouz transformation's method is used to give the lutions of the sine-Gordon equation with external field.

In this paper we give two direct representations of the of matriia Riccati differential equations, which are closely related to quadratic optimal control, differential game and oprtimal filtering. The first expression is abtainerd in terms of an arbitrary-steady solution. The second is obtained via decomposition of the Hamilton matrix and by meanu of the resolvent kernel of a matrix volterra linear integral equation.

In this paper, we discuss Hamilton systems having two degrees of freedom with guadratic and cubic nonlinearities, which exist widely in the field of nonlimear oscillation. By use of the Melnikov method, we find the conditionns that result in chaos in the systems. It is illustrated that the cractic phenomenon is an extremely universal onlinear phenomenon in the free osciillation system with multi-degrees of freedom.

Let (X_i, Y_i),i=1,…,n,be R⁴×R-valued i.i.d, samples tarken from the distribution of (X,Y).Denote the regression function by m (x) =E (Y|X=x). In this paper, we oonsider the kernel estimators m_n(x) of m(x) based on (X₁, Y₁),…,(X_n, Y_n), which were introduced by Watson [1] and Nadaraya [2],but here the window size h_n is supposed to be a sequence of positive nandbm variables which depend on x and samples X₁,…,X_n. We prove the uniform weak convergence of m (x). This case has also been studied by Schuster and Yakowitz [3],who assumed that d=1 and h_n is a sequence of positive constants.

In this paper, a problem of exact null oontronllability for a class of discrete infinite limentional linear system with control energy constraint is discussed. A necessary and sufficient condition for this controllability is obtained with the method of the eigenvector development of the linear operator.

In this paper, a general m,ethad of bcnerating spanning trees in the faotor incident graph of shift register is proposed. Several classes of feedback functions of M stequences are contructed directly and feedback functions of M sequences are derived from these functions. In addition, among the incident graphs, a class of homomorphism is defined, which can be used to construct M sequences.

In this paper, shrinkage estimators and ridge regression estimators of parameters of a linear model are considered when sample comes from elliptically contoured distributions, and a class of shrinkage estimators and regression estimators, which are better than the least square estimator under general quadratic loss, are given. Similar results are also discussed when the loss function is a nondecreasing concave function of quadratic loss.

Let Y be a random n-vector with mean Xβ and covariance matrix σ²V, where X is a known n×p matrix and V is a known Symmetric nonnegative definite matrix;β∈R^p and σ²>0 are parameters. Let Sβ be linearly estimable. The necessary and sufficient aanditions for a linear estimator of Sβ to be admissble among linear estimators under the quadratic loss function and matrix loss function are obtained respectively.

On the basis of the dependent central limit theorems, the asymptotic biehavior of quadratic forms of series for martingale differences is discussed and compared with the results in [3].The asymptotic distribution of covariance and spectral density is discussed as well by applying the asymptotic behavior to time series analysis. Moreover, some results in [2] one modified.

In this paper, we generalize the, well-known LaSall's theorem for posiive limit set of n-th order nonautonomous systems. We omit some restrictibns on the right hand oontral terms. Therefore, our theorems are much more convenient and more fiesible in applications.

Richard has studied impulse control of a model. In this paper we eztend his model and obtain a similar result. Far the eztended model, the policy consists of stapping times, impulse variables and acts. The state graceasses and coat functions of the new model are more general.

Infusing water into oil reservoir by high-pressure pump to squeeze the oil residual out of oil wells is a problem of two-phass immiscible flow displacement. The mathematical model of tlbe problem is the initial boundary problem of systems of partial differentidal equations. We suet up some computation schemes of the discrete operator method, and stnbdy approgimation and convergence theory. Computation of the model problem yielded aatisfactory results.

This paper gives a separation theorem of convex cone on an ordered vector space:Theorem.Let X be a linear space, Y an ordered vector space, and C⊂X×Y a convex cone.

In this paper same general results on the relations between nonsingular perturbations and generalized inverses of matrices are abtained, and several applications are given.

Let {x_ij,i=1},j=1,…,c,be c independent randcm series, each construeted by iid r.v.s. And let Φ(x_ij,i=1,…,m_i;j=1,…c)be a measurable function in R^m. Then Φ will be a symmetric function of variables x_ij,…x_mij,j=1,…,c, when other variables are fixed. P. K. Sen introduced the so-called generalized U-statistics U_R in 1972 and proved its convergence almost surely with stronger conditions. Here,the author establishes its convergence almost surely and illustrates the difference between the generalized U-statistics and the sums of iid r.v.'s under weaker conditions than Sen's.

We define a kind of homomorphisms between some factor incident graphs of feedback shift registers. Using these homomorphisms we construct a large number of M sequences.

In this paper, the construction of CCR's and PCR's factor-incident graphs is approached in a different way. A simple and convenient method for finding the quadruple-lines and the loops in Γ_x0⁽ⁿ⁾ and the double-lines in Γ_x0⁽ⁿ⁾,is proposed. Moreover, the properties on the arrangement of the binary digits of the multiple-lines are explored, and the relation between the loops and the quadruple-lines in Γ_x0⁽ⁿ⁾ is expounded. In addition, an error in counting the quadruple-lines of Γ_x0⁽ⁿ⁾ in [3] is corrected.

This paper discusses the existence of the limit cycle of the following system:

in which g(x)=mx³十2nx²+x, F(x, y)=ax²+2bxy十cy²+2dx+2ey+f. Unlike[1],[2],this paper discusses not only the case that F(x, y)=0 is an ellipse, but also the cases: F(x,y)=0 is parallel lines; a parabola; a hyperbolic type curve. The sufficient conditions of the limit cycle of system (1) is given.