THE TRANSFORMATION Φ AND GENERAL PYTHAGORAS’THEOREM ON THE LINEAR MANIFOLD

The primary aim of this paper is to describe the transformation function Φ^[1]. to extend the conclusion of Pythagoras’ theorem for "squared length" of any triangle and to study some geometrical significance of its application     

THE RECIPROCAL THEOREM AND LINEAR ELASTIC VARIATIONAL PRINCIPLES

In this paper classical linear elastic variational principles are systematically derivedfrom the reciprocal theorem and mixed variational principles of variations of boundaryconditions are given.     

THE TRANSFORMATION FUNCTION Φ AND THE CONDITION NEEDED FOR KUR SPACE HAVING THE FIXED POINT

In the last several years some progress has been made in the study of the properties ofthe extent of Banach space:In 1979 for example.when Suillivan discussed a relatedcharacterization of real L~P(X)space.he used uniform behavior of all two-dimensionalsubspace and defined this concept of a KUR space.In 1980 Huff used the concept of a NUCspace when he discussed the property of generalizing uniform convexity which was definedin terms of sequence.And in 1980 Yu Xin-tai stated,certainly and proved that the KURspace is equal to the NUC space.However,the following quite interesting questions raised respectively by Suillivan andHuff merit attention:Does every super-reflexive space have the fixed point property?The purpose of this paper is to study the characterization of transformationfunction and relationships between transformation function and the two questions above.     

THE TRANSFORMATION FUNCTION Φ AND THE CONDITION NEEDED FOR KUR SPACE HAVING THE FIXED POINT

In the last several years some progress has been made in the study of the properties of the extent of Banach space: In 1979 for example, when Suillivan discussed a related characterization of real L^p (X) space, he used uniform behavior of all two-dimensional subspace and defined this concept of a KUR space. In 1980 Huff used the concept of a NUC space when he discussed the property of generalizing uniform convexity which was defined in terms of sequence. And in 1980 Yu Xin-tai stated certainly and proved that the KUR space is equal to the NUC space^[1]. However, the following quite interesting questions raised respectively by Suillivan and Huff merit attention: Does every super-reflexive space have the fixed point property?^[2] The purpose of this paper is to study the characterization of transformation function^[4] and relationships between transformation function and the two questions above.     

ON THE MODIFIED CASTIGLIANO’S THEOREM

This paper gives the modified Castigliano’s theorem, which is more convenient and more extensive for applications than the classical Castigliano’s theorem.The modifications to the classical Castigliano ’s theorem are in following two respects, the first respect is that the expression of the partial derivative with respect to the concentrated load p of the complementary energy density in the clasical Castigliano ’s theorem is replaced by the expressions with teh products of the external loads and influence functions, the modification brings us greatest simplicity and greatest convenience for calculations under various loads; the second respect is that the expressions with the products of the inhomogeneous boundary displacements and the influence functions are introduced into the classical Castigliano’s theorem, the modification provides the theoretical fundamental for solving the problems of various boundary conditions.We show also the method of how to apply the modified Castigliano’s theorem to solve the problems of the surface structural mechanics.Finally, as a calculated example of the applications of the modified Castigliano’s theorem we solved the equation of the deflection surface of the rectangular plate with two adjacent edges built-in and other two adjacent edges free.     

MINIMAX THEOREM AND SADDLE POINT THEOREM WITHOUT LINEAR STRUCTURE

I.IntroductionFanKyinequalityisaveryusefultoolinnoolinearanalysiswhichasserts:LetXbeanonemptycompactconvexsetinatopologicallinearspaceandWbeareal-valuedfunctiondefinedonXxX.Assumethat(i)9(x,y)islowersemicontinuousinac;(n)9(x,y)isquasiconcaveiny.ThenthereisboaXsuchthatsHPg(xo'Y)$sZPp(x,x).SinceFanKyprovedtheinequalityin1972,variousg...,.li..tlon,havebeen;v..byseveralauthors(see[l-3]andreferencestherein).Thepreviousgeneralizationsallremainedtheconditions(i)-'9(x,y)beinglowersemicontinuous…     

ON THE GENERAL EQUATION AND THE GENERAL SOLUTION IN PROBLEMS FOR PLASTODYNAMICS WITH RIGID-PLASTIC MATERIAL

This work is the continuation of the discussion of refs.[1-2].We discuss thedynamics problems of ideal rigid—plastic material in the flow theory of plasticity in thispaper.From introduction of the theory of functions of complex variable under Dirac-Paulirepresentation we can obtain a group of the so-called“general equations”(i.e.have twoscalar equations)expressed by the stream function and the theoretical ratio.In this paperwe also testify that the equation of evolution for time in plastodynamics problema is neitherdissipative nor disperive,and the eigen-equation in plastodynamics problems is a stationarySchr(?)dinger equation,in which we take partial tensor of stress-increment as eigenfunctionsand take theoretical ratio as eigenvalues.Thus,we turn nonlinear plastodynamics problemsinto the solution of linear stationary Schr(?)dinger equation,and from this we can obtain thegeneral solution of plastodynamics problems with rigid-plastic material.     

THE ERECTING EGG IN WATER AND LAGRANGE’S THEOREM

鸡蛋在液体中的平衡稳定性问题可作为应用拉格朗日定理判断保守系统稳定性的有趣例子. 分析表明,当液体比重超过某个临界值时,鸡蛋可在液体中实现竖直状态的稳定平衡. 讨论鸡蛋平衡位置数目与稳定性随液体比重变化的静态分岔问题.     

INVARIANT FORM AND INTEGRAL INVARIANTS ON KAHLER MANIFOLD

The important notions and results of the integral invariants of Poincare and Cartan-Poincare and the relationship between integral invariant and invariant form established first by E. Cartan in the classical mechanics are generalized to Hamilton mechanics on Kahler manifold, by the theory of modern geometry and advanced calculus, to get the corresponding wider and deeper results.     

ON THE MODIFIED CASTIGLIANO"S THEOREM

This paper gives the modified Castigliano's theorem, which is more convenient and more extensive for applications than the classical Castigliano's theorem. The modifications to the classical Castigliano's theorem are in following two respects, the first respect is that the expression of the partial derivative with respect to the concentrated load p of the complementary energy density in the clasical Castigliano's theorem is replaced by the expressions with teh products of the external loads and influence functions, the modification brings us greatest simplicity and greatest convenience for calculations under various loads; the second respect is that the expressions with the products of the inhomogeneous boundary displacements and the influence functions are introduced into the classical Castigliano's theorem, the modification provides the theoretical fundamental for solving the problems of various boundary conditions. We show also the method of how to apply the modified Castigliano's theorem to solve th     

THE THEOREM OF THE STABILITY OF LINEAR NONAUTONOMOUS SYSTEMS UNDER THE FREQUENTLY-ACTING PERTURBATION

In the paper a theorem on the stability of linear nonautonomous system under thefrequently-acting perturbation has been given and proved on the basis of Malkin＇sTheorem.     

ON THE DECOMPOSITION OF COMPLEX VECTOR SPACES AND THE JORDAN CANONICAL FORM OF COMPLEX LINEAR TRANSFORMATIONS

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GENERAL RANDOM FIXED POINT THEOREM AND ITS APPLICAT1ONS

In this paper, we obtain a general random fixed point theorem which generalizes the main results of Engle and Bocsan. The usefulness of this theorem seems to lie in the fact that unlike many random fixed point theorems obtained by using special methods can be proved by using out general theorem (Theorem 1 and Corollaries 1 and 2). Finally, we indicate some possible applications of our results to nonlinear random integral and differential equations.     

ON THE GENERAL EQUTIONS,DOUBLE HORMONIC EQUATION AND ELGEN-EQUATION IN THE PROBLEMS OF IDEAL PLASTICITY

In this paper the outcome of axisymmetric problems of ideal plasticity in paper[39],[19]and[37]is directly extended to the three-dimensional problems of ideal plasticity,andget at the general equation in it.The problem of plane strain for material of ideal rigid-plasticity can be solved by putting into double hormonic equation by famous Pauli matricesof quantum electrodynamics different from the method in paper[7].We lead to the eigenequation in the problems of ideal plasticity,taking partial tenson of stress-increment aseigenfunctions,and we are to transform from nonlinear equations into linear equation inthis paper.     

ON THE GENERAL SOLUTION OF CYLINDRICAL SHELL EQUATIONS

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ON UNIFORM CONVERGENCE OF LINEAR OPERATORS ON THE PROBABILISTIC NORMED SPACE

In this paper,we introduced the notion of uniform convergence of the linear operatorson the probabilistic normed space,and the notion of probabilistic distance between theoperators,which describes the above convergence completely.In terms of these notions,weobtained the essential features of the continuity of operators,and of the uniformconvergence of operator sequences,and we also obtained the closure of continuity andcomplete continuity under the operation of the limit of uniform convergence.     

A COMMENT ON THE PROOF OF FERMAT'S LAST THEOREM

I.IntroductionItiswell-knownthatFermat'sLastTheoremll'2'31:"TheequationX" yn=Z"(l.1)hasnosolutioninpositiveintegersx,.1',:whicharealldifferentfrom0foranarbitrarynumbern>2"isahardmathematicalproblem,proposedbyFrenchmathematicianPierredeFermatinabout1637.Sincethenmorethan300yearspast,manyfamousmathematicianshavemadegreatcontributionsintheresearchesanddevelopmentsinthenumbertheoryofalgebraaswellasothermathematicalbranchestolookouttheproofoftheproblematthemoment,buttilefinalproofhasonlybeenobt…     

THE LARGE DEFLECTION ON PROBLEM OF CIRCULAR PLATE ON ELASTIC FOUNDATION AND IN CONJUNCTION WITH LINEAR ELASTIC STRUCTURE

This paper deals with the axisymmetrical deformation of the circular plate in largedeflection,which is on elastic foundation and in conjunction with a certain linear elasticstructure.The governing integral equations are established by the method of mixedboundary condition I and the simplified form is given.The pertrubation method is used toobtain the solutions and an example of the composite structure made up of a circular plateand a cylindrical shell is presented.     

ON A GENERALIZATION OF BERTRAND’S THEOREM

Bertrand’s theorem for the determination of the applied forces to a holonomicsystem from one of its first integrals,is extended to nonholonomic systems.Someinteresting applications of this new result are also given.     

ON GENERAL FORM OF NAVIER-STOKES EQUATIONS AND IMPLICIT FACTORED SCHEME

A general weak conservative form of Navier-Stokes equations expressed with respect to non-orthogonal curvilinear coordinates and with primitive variables was obtained by using tensor analysis technique, where the contravariant and covariant velocity components were employed. Compared with the current coordinate transformation method, the established equations are concise and forthright, and they are more convenient to be used for solving problems in body-fitted curvitinear coordinate system. An implicit factored scheme for solving the equations is presented with detailed discussions in this paper. For n-dimensional flow the algorithm requires n-steps and for each step only a block tridiagonal matrix equation needs to be solved. It avoids inverting the matrix for large systems of equations and enhances the speed of arithmetic. In this study, the Beam-Warming's implicit factored schceme is extended and developed in non-orthogonal curvilinear coordinate system.