A matrix G satisfying AGA=A is called the generalized inverse of matrix A, and is denoted by G=A^-. It is not necessarily unique in,general. By giving.the particular solutions of for B=AH, C=KA and for A.D nonnegative definite, we present a way to finding the generalized inverse of various kinds in the most general case. As applications, we get the formula of (A B)^-;give the unified treatment of some extremum problems of quadratic form;establish the formula of the oblique projector with given direction;prove a property of the so called random projector and present a new treatment of statistical covariawce amalysis.

In this paper we prove a functional invariance principle with a logarithmic form to obtain some invariance principles for power-of-sum process.

In this paper the author presents the multinomial expression of a characteristic equation on the torsional vibration of lumped mass systems, proves that the problem of the general symmetrical matrix characteristic value can be reduced to that of a straight torsional vibration system, and analyses the construction of the characteristic equation and the effect of the shafting parameters on the roots.

The purpose of this paper is to discuss the forms of the Euler-Lagrange equations.The polynomial-index law has been obtained:For higher derivatives of these Pquations, their order and (algbraic) degrse have to satisfy a sim ple inequality. Especially, in the two dimensional space, a second order partial differential equation, which is a Euler-Lagrange equation, must be a Monge-Ampere equation.

In this paper we discuss a fairly general model of the stochastic control problem dealing with the dynamic systems whose state processes are multivariate point processes. We derive an optimal principle applicable to the cases where the controlled probabilities need not be absolutely continuous with respect to a prior probability. The existence theorem of optimal contrel obtained in this paper improves the corresponding result of C. B. Wan and M. H. A, Davis.

Suppose that F={f:f∈Cⁿ[0,1],||f⁽ⁿ⁾||_∞≤1} and Δ_N are the equidistance partitions of interval [0, 1] s(f,x, N) expresses the interpolation spline of degree n-1 of f∈F on Δ_N, and e(f,x,N) is the pointwise error function, If the boundary conditions added on the endpoints 0 and 1 are not equal in number then the norm of e(f,x,N) is unbounded with respect to class F when N tends to infinity.

J. Abrham and A. Kotzig stated in 1980 that a Euler tour in an arbitrary Eulerian graph can be obtained from any other Euler tour of this graph by a finite number of simple transformations (K-transformations).In this paper, the author presents a new transformation T-transformation, for the directed Euler tours and shows that a directed Euler tour in an arbitrary directed Eulerian graph can be obtained from any other directed Euler tour of this graph by a finite number of T-transformations, if there are at least two directed Euler tours in the graph. A necessary and sufficient condition for the directed Eulerian graph to have a unique directed Euler tour is given.

In this paper, a cut-join method for constructing all the M-sequences with assigned degree over GF(q) is given, which is an extension of the method in[1] to any finite field. Using two different cut-joins (in finite steps) we can obtain all the M-sequences of degree n from a given one.

Two theorems for the lower bound of the bandwidth of a graph are given to generalize a theorem by Harper in 1964 and solve a computing problem of the bandwidth of K_m×K_n,which cannot be solved only by Harpers' theorem. Moreover, the concept of condense label is introduW some general theorems for the bandwidth of a graph are given, and the bandwidths of several types of graph such as P_m×P_n×P_l, are computed.

In this paper, we generalize Berman's lemma (1964) and obtain results about asymptotic distributions of the r-th largest values in nonstationary Gaussian Suquences where Leadbetter's joint work with Lindgtren and Rootzen (1978) is extended to the case of nonstationary Gaussian Sequences.

In the paper we give a simple pivotal operation and an improved reduced gradient method for the nonlinear programming problem min {f(x)|Ax=b,x≥0}. Under the assumptions that is continuously differentiable and the constraints are nondegenerate, we can prove that (ⅰ) either the iterative sequence {x^k} generated by the method leads to a K-T point after a finite number of iterations, or any cluster point of {x^k} is a K-T point. (ⅱ) if the sequence {x^k} is convergent, then the total number of pivotal operations is finite in the whole run of iterations.

In this paioer, by constructing Lyapunov's function of global stability, we solve the problems of global qualitative construction of two kinds of third order loop equations with tangent discriminator characteristic, and explain mathematically why this loop with tangent discriminator characteristic of high order never loses its locked state, thus providing a theoretical basis for such phenomena.

A simple recursive formula is given to solve the problem of counting the number of m-level dieital sequences of leneth n. that satisfy some constraints.

Perturbation of eigenvalues and the eigenfunction may be applied to deteraiining the natural frequencies and modes of vibration of a non-uniform beam. Mathematically this is to investigate the eigenvalue problem
.

In this paper, the seasonal adjustment of time series is discussed from the viewpoint of numerical approximation. The variance analysis periodic extrapolated method is improved and generalized into the successive average method. Some relative properties of this new method for seavsonal adjustment of time series are studied, and particularly, a convergence theorem is proved.

In this paper,we present a recursive algorithm for autocorrelation coefficients of M-sequences.We will prove that

where c_τ is the τ^th autocorrelation coefficient, α∈f₀ is a minimal term of f₀, and δ=0 or ±1.

In [1], a finite pivoting process was introduced, by which the convergence ofthe reduced gradient algorithm under certain conditions has been proved. In thispaper, we give two simplified versions of the process, which have all the propertiesstated in [1].

Generalization is made to define the B-spline at general knots as the map, under Fourier transform, of generalized g_k(u) in[1].It is pointed out that there is a map between trigonometric function e_k^int and the B-spline function. Some integral relations of B-splines are also derived.

A new general reduced gradient method for problems with nonlinear constraintsis proposed and used in finding the decreasing direction. Its global convergence isalso proved.

In this paper we discuss the existence of limit cycles of the Liénard equationand prove five theorems without the hypothesis G(±∞)=+∞. Then some resultsin [1-4] are extended. Theorems 6 and 7 are remarks about Filippov's andDragilev's theorems^[5,7].

following theorems are proved:Theorem 1. Let p and q be any two points except the endpoints on a plane curve Γ∈C², and let L_p and L_q be tangents of Γ at p and q respectively.Γ is convex if and only if it satisfies the condition that if q∈L_p,then L_p=L_q.Theorem 2: In a 3-dimensional Euclidean space, a connected surface π∈C² is convex if and only if it satisfies the condition that for any two points p,q∈π\∂π,if q∈T_p, then T_p=T_q,where T_p and T_q are tangent planes of π at p and q respectively.

In this paper, we discuss the problem of determining the orders of AR(k) and ARMA(p, q) models of time series on the basis of the Bayesian estimate theory. A general prior distribution for the order and a general family of prior distribution for the parameters are proposed. With respect to a particular loss-function, the criterion for the order of AR(k), denoted by η₁(k), and the approximate criterion for theorder of ARMA (p, q) are given. The consistency of the order K estimated by using η₁(k)is proved.Finally, the simulation comparisons between η₁(k), AIC(k) (Akaike,1976) and φ(k) (Hannan,1979) are made. The results show that η₁(k) is superior to AIC(k).

The present paper proves the existence and uniqueness (in some given sense) of the LDL^T decomposition of real symmetric non-negative definite matrices, where L is a unit lower triangular matrix with real elements and D is a diagonal matrix with real elements. The proof is made in a constructive way. By taking advantage of this decomposition, a criterion for the consistency of the linear equation with such a coefficient matrix and its whole solution set (or the least-squares solution if it is inconsistent) are obtained. Since it involves no row or column permutation, the process may be combined with any sparse technique on the computer, and hence is of practical importance in treating the large scale sparse matrices derived from such problems as the structure design by finite elements methods. Finally, the stability of such a decomposition is discussed and a backward error analysis is given.

In this paper, the trigonometrical association scheme and its design are expanded from the review of set theory to tht case of miltiple associative class.The explicit formulas of their corresponding parameters are obtained.

It is impossible to solve linear equations for symmetric semi-positive definite matrix if we apply the factorization or elimination algorithm. [1] presented a corrected elimination algorithm and gave a solution. However, this solution is not correct for structural nodal displacement in the finite element method. This paper presents a technique so as to yield a correct solution for structural nodal displacement.

Let X₁,…,X_n be a sample from a normal distribution N_p(μ,∑),where μ∈R^p and ∑ is a positive definite matrix, both μ and ∑ being unknown. In this paper it is shown that xor the loss function tr (∑^-1·d-l)² the best affine invariant estimator of the covariance matrix ∑ is inadmissible and an improved estimator is explicitly construeted. The results of [1, 2] are generalized to the multiparameter case. However, simultaneous estimation of several parameters is more difficult than that of a single parameter. Consequently, some properties of the coefficient of the best affine invariant estimator are discussed and a relavent proposition on the quadratic form is proved.

In this paper, using Galilean transformation, scale transformations, generalized Lorentz transformation and Lie transformation, we obtain the relations between the auto-Backlund and the invertible Backlund transformations;the former are with differeut parameters for the Korteweg-de Vries equation, which is considered as a continuum limit of the Toda equation, second modified Korteweg-de Vries equation and Kadomtsev-Petviashvili equation, and the latter are for two generalized sine-Gordon equations.

The accelerated quadrature method is applied to the calculation of potentials and fields produced by toroidal current, and the numerical solutions show good agreement with physical analysis. When:is sufficiently large simultaneous use of accelerated quadrature and Filon quadrature methods is suggested.

A method of fitting data points with piecewise least square is provided for thecomputer aided geometric design. It contains fitting of Bezier curves, fitting ofBezier surfaces and constrained fitting of surfaces. This method has been put into usein the design system for automobile surfaces.

In this paper, a fitted autoregrcssion is used to construct the estimates of spectral functions of a class of linear processes. The invariance principle for the estimates is proved under quite general conditions.

Some properties of uniquely 3-edge-colourable graphs are observed in this paper. Furthermore, a family of counterexamples of a conjecture given by Greenwell and Kronk is constructed.

The question of selecting a specified number of representative points form a normal population with as much information of the population as possible arises in ulanv situations. To obtain these points, it is necessary to solve two systems of equations and to study their properties. As requirement a generalized Mills'ratio is defined and its basic properties are studied. A computational procedure is gimp and tabled of representative points and corresponding probabilities are listed for m≤31 where m is the specified number points.

In [1],Li Weixuan discussed the flexible analysis of structures from the view point of mathematics. He gave the precise mathematical definitions of the changeable, instantaneously changeable and unchangeable structures and obtained an algebraic computational method to decide whether a structure is unchangeable. In this paper we go further into the theory about the insi;antaneous changeability of systems of function equations. By this theory we make a finer classification of the instantaneous changeability of structures. We also give some theorems by which the instantaneously changeable order of a structure may be decided. Dloreover, we define the concepts of the generating forest and the quasi loop basis of a graph in order to simplify some computations in [1].

This paper proves that, in using Kuhn's root-finding algorithm, the cost of computing all n roots of a complex polynomial of degree n grows no faster than n² log(n/ε),where ε>0 is the accuracy demand for resulted roots and is small enough.

We discuss the equations

The model was suggested by H. Degn to account for the qualitative features of the respiratory precess in a bacterial culture.

In this paper the uniform convergence rate of strong consistency of kernel estimate f_n (x) of the density function f (x) is considered. Under mild conditions on f (x) and kernel function K(x),we obtain the order of convergence rate of |f_n (x)-f(x).With proper conditions this order may reach O((log n/n)^2/3) a.s.

In this paper, a branch-constructing algorithm for extreme vertices of the mixture convex polyhedron with lower and upper bounds is given. By use of pseudocomponent transform. it can be transformed in to a mixture convex polyhedron with upper bounds only.A branch-constructing algorithm for the polyhedron is given. On the basis of the algorithm, a modified CONSIM algorithm for the mixture convex polyhedron with multicomponent constraints is given.

In this note, we construct two classes of PBIBD by means of number theory.

Form the viewpoint of modern bifurcation theory, the buckling state of an elastic slender rod with an internal constraint and under a compressive axial thrust is discussed in this paper. The governing equations of the problem are (1.1)-(1.4).If there are no internal constraints, the linearization of the nonlinear problem about the trivial solution is self-conjugate. In this case, a detailed discussion was given by E. L. Reiss in [2].If there is an internal constraint, the linearization will become non-self-conjugate. In the paper, at first we calculate the eigenvalues and the eigenfunctions of tlm linearization problem. Then, by means of elliptical integration, the non-linear problem is transformed into a equivalent system of the transcendental equations. It is proved that all of the eagenvalues of the linearization are the bifurcation points, namely, critical thrusts. The asymtotic formulae of the small solutions of the nonlinear problems near the critical thrusts are obta incd. Finally, by means of the global branching theorem and the nodal structures of the solutions, we examine the global properties of the branching solutions.