Uniqueness theorem,theorem of reciprocity,and eigenvalue problems in linear theory of porous piezoelectricity

The uniqueness theorem and the theorem of reciprocity in the linearized porous piezoelectricity are established under the assumption of positive definiteness of elastic and electric fields. General theorems in the linear theory of porous piezoelectric materials are proved for the quasi-static electric field approximation. The uniqueness theorem is also proved without using the positive definiteness of the elastic field. An eigenvalue problem associated with free vibrations of a porous piezoelectric body is studied using the abstract formulation. Some properties of operators are also proved. The problem of frequency shift due to small disturbances, based on an abstract formulation, is studied using a variational and operator approach. A perturbation analysis of a special case is also given.     

Applications of the division theorem inH s

We prove a version of the division theorem in Sobolev spaces with an estimate of the constant ass tends to infinity. We then apply it to derive spatial decay estimates for time-periodic solutions of linear wave equations in one space dimension and to prove that the space of decaying solutions is finite-dimensional. The main point is to show that some of the arguments used to analyze embedded eigenvalues of Schrödinger operators can be extended to cases where positivity arguments are not available. This has implications for nonlinear Klein-Gordon equations. A different approach, based on the proof of the stable manifold theorem, is also worked out, under slightly different assumptions.     

A general topological version of minimax theorem

ＡＧＥＮＥＲＡＬＴＯＰＯＬＯＧＩＣＡＬＶＥＲＳＩＯＮＯＦＭＩＮＩＭＡＸＴＨＥＯＲＥＭＺｈａｎｇＳｈｉｓｈｅｎｇ（张石生）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ,ＳｉｃｈｕａｎＵｎｉｖｅｒｓｉｔｙ,Ｃｈｅｎｇｄｕ６１００６４,Ｐ．Ｒ．Ｃｈｉｎ...     

Quasi-momentum theorem in Riemann-Cartan space

The geometric formulation of motion of the first-order linear homogenous scleronomous nonholonomic system subjected to active forces is studied with the nonholonomic mapping theory. The quasi-Newton law, the quasi-momentum theorem, and the second kind Lagrange equation of dynamical systems are obtained in the Riemann-Cartan configuration spaces. By the nonholonomic mapping, a Euclidean configuration space or a Riemann configuration space of a dynamical system can be mapped into a Riemann-Cartan configuration space with torsion. The differential equations of motion of the dynamical system can be obtained in its Riemann-Cartan configuration space by the quasi-Newton law or the quasi-momentum theorem. For a constrained system, the differential equations of motion in its Riemann-Cartan configuration space may be simpler than the equations in its Euclidean configuration space or its Riemann configuration space. Therefore, the nonholonomic mapping theory can solve some constrained problems, which are difficult to be solved by the traditional analytical mechanics method. Three examples are given to illustrate the effectiveness of the method.     

CONTINUOUS SELECTION THEOREMS FOR FAN-BROWDER MAPPINGS IN TOPOLOGICAL SPACES AND THEIR APPLICATIONS

The concept of Fan-Browder mappings was first introduced in topological spaces without any convex structure. Then a new continuous selection theorem was obtained for the Fan-Browder mapping with range in a topological space without any convex structure and noncompact domain. As applications, some fixed point theorems, coincidence theorems and a nonempty intersection theorem were given. Both the new concepts and results unify and extend many known results in recent literature.     

Noether's theorem for nonholonomic systems of non-Chetaev's type with unilateral constraints in event space

Ｔｈｅｓｔｕｄｙｏｆｃｏｎｓｅｒｖａｔｉｏｎｌａｗｓｏｆｄｙｎａｍｉｃａｌｓｙｓｔｅｍｉｓａｓｉｇｎｉｆｉｃａｎｔａｓｐｅｃｔｏｆｍｏｄｅｒｎａｎａｌｙｔｉｃａｌｍｅｃｈａｎｉｃｓ.Ｉｎ1918,ｕｓｉｎｇｔｈｅｉｎｖａｒｉａｎｃｅｏｆＨａｍｉｌｔｏｎａｃｔｉｏｎｕｎｄｅｒｔｈｅｉｎｆｉｎｉｔｅｓｉｍａｌｔｒａｎｓｆｏｒｍａｔｉｏｎｓ,Ｎｏｅｔｈｅｒ[1]ｓｔｕｄｉｅｄｃｏｎｓｅｒｖａｔｉｏｎｌａｗｓｏｆｍｅｃｈａｎｉｃａｌｓｙｓｔｅｍａｎｄｐｒｅｓｅｎｔｅｄｔｈｅｆａｍｏｕｓＮｏｅｔｈｅｒ’ｓｔｈｅｏｒ…     

ON A CLASS OF NEW KKM THEOREM WITH APPLICATIONS

ＯＮＡＣＬＡＳＳＯＦＮＥＷＫＫＭＴＨＥＯＲＥＭＷＩＴＨＡＰＰＬＩＣＡＴＩＯＮＳＺｈａｎｇＳｈｉｓｈｅｎｇ（张石生），ＺｈａｎｇＸｉａｎ（张宪）（ＲｅｃｅｉｖｅｄＭａｙ１５，１９９５）Ａｂｓｔｒａｃｔ：Ｉｎｔｈｉｓｐａｐｅｒ，ａｃｌａｓｓｏｆｎｅｗＫＫ...     

FC_空间中的参数型KKM定理及其应用

The paper first proves a characteristic property in FC-spaces. By the use of the connectedness of sets, a parametric type of KKM theorem is then established in noncompact FC-spaces by introducing a linear ordered space. As a consequence, some recent results, such as noncompact minimax inequalities, saddle point theorem, and section theorem, are improved. The results generalize the corresponding results in the literatures.     

The relativistic virial theorem in plasma EOS calculations

A method is presented for accurate calculation of equation of state (EOS) for warm dense matter. The method extends an approach presented recently, based on the adjustment of the correlation energy to impose consistency between two pressure representations: the volume derivative of the free energy and the relativistic virial theorem. In this work we show that the free energy of any neutral system obeys a fundamental differential equation, which bypasses the correlation specifics and serves as a basis to enhance EOS approximations. Specifically, we start with LDA calculations and improve the results significantly using this equation with a boundary condition at the zero pressure point. The method retains the emphasis on thermal excitations, but connects to the appropriate results at low temperatures. It effectively compensates for simplifications, including the use of a spherical model to account for global solid structure effects. EOS and opacities are calculated on the same footing for low to high Z elements and in large domains of density and temperature without recourse to parametric fitting procedures. Excellent agreement is obtained with experiments. Finally the method is applied successfully to calculate EOS and opacities for mixtures. Results for C–H mixture are compared with other calculations.     

Approximate sampling theorem for bivariate continuous function

An approximate solution of the refinement equation was given by its mask, and the approximate sampling theorem for bivariate continuous function was proved by applying the approximate solution . The approximate sampling function defined uniquely by the mask of the refinement equation is the approximate solution of the equation , a piece-wise linear function , and posseses an explicit computation formula . Therefore the mask of the refinement equation is selected according to one' s requirement, so that one may controll the decay speed of the approximate sampling function .     

Generalized variational inequalities and minimax theorems

In the present paper we-discuss the variational problem of operator which maps aHausdorff topological linear space into a Riesz space.We give absolute inequality andrelative inequality.respectively.     

SYSTEM OF COINCIDENCE THEOREMS IN PRODUCT TOPOLOGICAL SPACES AND APPLICATIONS (Ⅰ)

A new notion of finite continuous topological space (in short, FC-space) with out convexity structure was introduced. A new continuous selection theorem was established in FC-spaces. By applying the continuous selection theorem, some new coincidence theorems for two families of set-valued mappings defined on product space of noncompact FC-spaces are proved under much weak assumptions. These results generalize many known results in recent literature. Some applications will be given in a follow-up paper.     

SYSTEM OF COINCIDENCE THEOREMS IN PRODUCT TOPOLOGICAL SPACES AND APPLICATIONS (Ⅰ)

A new notion of finite continuous topological space (in short, FC-space)with out convexity structure was introduced. A new continuous selection theorem was established in FC-spaces. By applying the continuous selection theorem, some new coincidence theorems for two families of set-valued mappings defined on product space of noncompact FC-spaces are proved under much weak assumptions. These results generalize many known results in recent literature. Some applications will be given in a follow-up paper.     

A comment on the proof of Fermat's last theorem

In this paper, some conmments on the proof of Fermat’s last theorem are proposed.The main resuilt is thai the proof proposed by Wong Chiahe is only part of proof for fermat’s last theorem. That is to sqy,the proof is not all-full proof to Fermat’s last theorem.     

TRANSFER OPEN OR CLOSED SETVALUED MAPPING AND GENERALIZ－ATION OF H－KKM THEOREM WITH APPLICATIONS

ＴＲＡＮＳＦＥＲＯＰＥＮＯＲＣＬＯＳＥＤＳＥＴＶＡＬＵＥＤＭＡＰＰＩＮＧＡＮＤＧＥＮＥＲＡＬＩＺ－ＡＴＩＯＮＯＦＨ－ＫＫＭＴＨＥＯＲＥＭＷＩＴＨＡＰＰＬＩＣＡＴＩＯＮＳＬｉＢｉｎｇ－ｙｏｕ（李秉友）;ＳｕＪｉａ－ｂａｏ（苏家宝）（Ｄｅｐａｒｔｍｅｎｔ...     

Revisiting umbra-Lagrangian-Hamiltonian mechanics: Its variational foundation and extension of Noether's theorem and Poincare-Cartan integral

This paper revisits an extension of the Lagrangian-Hamiltonian mechanics that incorporates dissipative and non-potential fields, and non-integrable constraints in a compact form, such that one may obtain invariants of motion or possible invariant trajectories through an extension of Noether's theorem. A new concept of umbra-time has been introduced for this extension. This leads to a new form of equation, which is termed as the umbra-Lagrange's equation. The underlying variational principle, which is based on a recursive minimization of functionals, is presented. The introduction of the concept of umbra-time extends the classical manifold over which the system evolves. An extension of the Lagrangian-Hamiltonian mechanics over vector fields in this extended space has been presented. The idea of umbra time is then carried forward to propose the basic concept of umbra-Hamiltonian, which is used along with the extended Noether's theorem to provide an insight into the dynamics of systems with symmetries. Gauge functions for umbra-Lagrangian are also introduced. Extension of the Poincare-Cartan integral for the umbra-Lagrangian theory is also proposed, and its implications have been discussed. Several examples are presented to illustrate all these concepts.     

CONSTRAINED MULTIOBJECTIVE GAMES IN LOCALLY CONVEX H-SPACES

IntroductionTheconceptofanequilibriumpointforann_persongamewasfirstintroducedbyNash[1,2 ],whoestablishedtheexistenceoftheequilibriumpointundercertainassumptions.Sincethen ,theNashequilibriumproblemforn_persongameshavebeenintensivelystudiedandextendedbyanumberofauthorsundervariousassumptionsanddifferentdirections.Theconstrainedgameswithfiniteorinfiniteplayersisanimportantgeneralizationofn_persongameswhichhavebeenwidelystudiedbymanyauthors.Recently ,muchattentionhasbeenfocusedonthegameproblemsw…     

On uniqueness, existence and objectivity of S-R decomposition theorem

I.IntroductionThedecompositionofadeformationgradientfunctionFisafundamentalproblemforfinitedeformationtheory.Greenstraintheoryandpolardecompositiontheorywereemployedbymostscientists,However,thesetheorieshaveallsomeconsiderabledrawbacks.Inl979,ChenZhidadev…     

Note on the virial theorem

Poincaré's formalism is used to develop a variant of the usual virial theorem in which the time average of the equation of motion of a certain function is expressed in terms of the generalized Poisson brackets. Recommended by Prof. Mei Fengxiang     

An elastodynamic reciprocity theorem for linear, viscoelastic media

Summary A reciprocity theorem for impulsive disturbances in a linear, viscoelastic medium is derived. Apart from the condition that the medium be linear and viscoelastic of the Boltzmann type, no restrictions as to the properties of the medium are imposed. Hence, the reciprocity theorem is applicable to disturbances in inhomogeneous and anisotropic media.As an illustration of its usefulness the reciprocity theorem is used to derive an integral representation of the Kirchhoff type for the displacement vector in a linear, viscoelastic medium.For reference, also the Laplace transform version of the obtained results is given.