GENERALIZED RECIPROCAL THEOREMS AND THEIR APPLICATIONS

Generalized reciprocal theorems of non-coupled and coupled systems, which are valid for two deformed bodies with different constitutive relations are established by generalizing the idea of Betti's reciprocal theorem. When the constitutive relations of the two deformed bodies are all alike and linear elastic the generalized reciprocal theorem of non-coupled systems just becomes Betti's. Meanwhile the generalized reciprocal theorems are applied to simulate calculations in elasticity.

Generalized S-KKM Theorems and Minimax Inequalities in FC-Spaces

The concept of FC-closed subset in FC-space without any linear structure was introduced. Then, the generalized KKM theorem is proved for FC-closed value mappings under some conditions. The FC-space in the theorem is the generalization of L-convex space and the condition of the mapping with finitely FC-closed value is weaker than that with finitely L-closed value.

Poisson Theory of Generalized Birkhoff Systems

To study the Poisson theory of the generalized Birkhoff systems, the Lie algebra and the Poisson brackets were used to establish the Poisson theorem. The generalized Poisson condition for the first integral and the generalized Poisson theorem of the generalized Birkhoff systems are obtained. An example is given to illustrate the application of the result.

Multiple homoclinics in a non-periodic Hamiltonian system

This paper concerns the existence of multiple homoclinic orbits for the second-order Hamiltonian system-L（t）z＋Wz（t,z）=0,where L∈C（R,RN2）is a symmetric matrix-valued function and W（t,z）∈C1（R×RN,R）is a nonlinear term.Since there are no periodic assumptions on L（t）and W（t,z）in t,one should overcome difficulties for the lack of compactness of the Sobolev embedding.Moreover,the nonlinearity W（t,z）is asymptotically linear in z at infinity and the system is allowed to be resonant,which is a case that has never been considered before.By virtue of some generalized mountain pass theorem,multiple homoclinic orbits are obtained.

Equivalence of crossed product of linear categories and generalized Maschke theorem

Some sufficient and necessary conditions are given for the equivalence between two crossed product actions of Hopf algebra H on the same linear category, and the Maschke theorem is generalized. Based on the result of the crossed product in the classic Hopf algebra theory, first, let A be a k-linear category and H be a Hopf algebra, and the two crossed products A#_σH and A#＇_σH are isomorphic under some conditions. Then, let A#_σH be a crossed product category for a finite dimensional and semisimple Hopf algebra H. If V is a left A#σH-module and WC V is a submodule such that W has a complement as a left A-module, then W has a complement as a A#_σH-module.

NEW PRINCIPLES OF WORK AND ENERGY AS WELLAS POWER AND ENERGY RATE FORCONTINUUM FIELD THEORIES

New principles of work and energy as well as power and energy rate with cross terms for polar and nonlocal polar continuum field theories were presented and from them all corresponding equations of motion and boundary conditions as well as complete equations of energy and energy rate with the help of generalized Piola's theorems were naturally derived in all and without any additional requirement. Finally, some new balance laws of energy and energy rate for generalized continuum mechanics were established. The new principles of work and energy as well as power and energy rate with cross terms presented in this paper are believed to be new and they have corrected the incompleteness of all existing corresponding principles and laws without cross terms in literatures of generalized continuum field theories.

EXISTENCE OF SOLUTIONS FOR NON-PERIODIC SUPERLINEAR SCHRDINGER EQUATIONS WITHOUT (AR) CONDITION

The existence of solutions is obtained for a class of the non-periodic SchrSdinger equation -△u ＋ V（x）u = f（x,u）, x E RN, by the generalized mountain pass theorem, where V is large at infinity and f is superlinear as ｜u｜→ ∞.

Entropy-variation with resistance in a quantized RLC circuit derived by the generalized Hellmann-Feynman theorem

By virtue of the generalized Hellmann-Feynman theorem for the ensemble average, we obtain the internal energy and average energy consumed by the resistance R in a quantized resistance-inductance-capacitance （RLC） electric circuit. We also calculate the entropy-variation with R. The relation between entropy and R is also derived. By the use of figures we indeed see that the entropy increases with the increment of R.

Generalized Hellmann-Feynman Theorem and Its Applications

The Hellmann-Feynman （H-F） theorem is generalized from stationary state to dynamical state. The generalized H-F theorem promotes molecular dynamics to go beyond adiabatic approximation and clears confusion in the Ehrenfest dynamics.

GENERALIZED BOCHNER'S THEOREM FOR RADIAL FUNCTION

A radial function can be expressed by its generator through The positive definite of the function plays an important rote in the radial basis interpolation. We can naturally use Bochner's Theorem to check if is positive definite. This requires however a n-dhnensiotial Fourier transformation and it is not very easy to calculate. Furthermore in a lot of cases we will use for spaces of various dimensions too, then for every fixed n we need do the Fourier transformation once to check if the function is positive definite in the n-di-mensional space. The completely monotone function:, which is discussed in [4] is positive definite for arbitrary space dimensions. With this technique tve can very easily characterize the positive definite, of a radial function through its generator. Unfortunately there is only a very small subset of radial function which is completely monotone. Thus this criterion excluded a lot of interesting functions such as compactly supported radial function, whcih are very useful in application. Can we find some conditions (as the completely monotone function) only for the \-dimen simial Fourier transform of the generator epto characterize a radial function 9, which is positivedefinite in n-dimensional (fixed n) spacel In this paper we defined a kind of incompletelymonotone function of order a, for a= 0,,1/2 ,1,3/2,(we denote the function class by ICM) ,in this sence a normal positvie function is in ICM a positive monotone decreasing function is inICM and a positive monotone decreasing and convex function is in ICM2- Based on this definition we get a generalized Bochner's Theorem for radial function-. If dimensional Fouriertransform of the generator of a radial function can be written as , then corre-spending radial function (x) is positive definite as a n-variate function iff F is an incomplete-ly monotone function of order a= (n- 1 )/2 (or simply In this way we have characterized the positive definite of the radial function as a n-vari-ate function through its generator in the sense of the Bocher's Theorem.

Connectedness of Cone-efficient Solution Set for Cone-quasiconvex Multiobjective Programming in Hausdorff Topological Vector Spaces

This paper deals with the connectedness of the cone-efficient solution set for vector optimization in locally convex Hausdorff topological vector spaces. The connectedness of the cone-efficient solution set is proved for multiobjective programming defined by a continuous one-to-one cone-quasiconvex mapping on a compact convex set of alternatives. During the proof, the generalized saddle theorem plays a key role.

CHARACTERISTICS OF SUBDIFFERENTIALS OF FUNCTIONS

In the presem paper, some important characteristics of Fenchel-, Frechet-,Hademard-, and Gateaux-Subdifferentials are showed up, and properties of functions, especially. convexity of functions, are described by subdifferentials.

NEW PRINCIPLES OF POWER AND ENERGY RATE OF INCREMENTAL RATE TYPE FOR GENERALIZED CONTINUUM FIELD THEORIES

The aim of this paper is to establish new principles of power and energy rate of incremental type in generalized continuum mechanics BY combining new principles of virtual velocity and virtual angular velocity as well as of virtual stress anti virtual couple stress with c ross terms of incremental rate type a new principle of power anti energy rate of incremental rate type with cross terms for micropolar continuum field theories is presented and from it all corresponding equations of motion and boundary conditions as well as power and energy rate equations of incremental rate type for micropolar and nonlocal micropolar continua with the help of generalized Piola's theorems in all and without any additional requirement are derived. Complete results for micromorphic continua could be similarly derived. The derived results in the present paper are believed to be new. They could be used to establish corresponding finite element methods of incremental rate type for generalized continuum mechanics.

Deriving Internal Energy by Virtue of Generalized Feynman-Hellmann Theorem for Mixed States

We show how to directly use the generalized Feynman-Hellmann theorem, which is suitable for mixed state ensemble average, to derive the internal energy of Hamiltonian systems. A concrete example, which is a two coupled harminic oscillators, is used for elucidating our approach.

Thermal vacuum state for the two-coupled-oscillator model at finite temperature: Derivation and application

Following the spirit of thermo field dynamics initiated by Takahashi and Umezawa, we employ the technique of integration within an ordered product of operators to derive the thermal vacuum state （TVS） for the Hamiltonian H of the two-coupled-oscillator model. The ensemble averages of the system are derived conveniently by using the TVS. In addition, the entropy for this system is discussed based on the relation between the generalized Hellmann-Feynman theorem and the entroy variation in the context of the TVS.

Optimal Robust Control for Unstable Delay System

Proportional-Integral-Derivative control system has been widely used in industrial applications.For uncertain and unstable systems,tuning controller parameters to satisfy the process requirements is very challenging.In general,the whole system’s performance strongly depends on the controller’s efficiency and hence the tuning process plays a key role in the system’s response.This paper presents a robust optimal Proportional-Integral-Derivative controller design methodology for the control of unstable delay system with parametric uncertainty using a combination of Kharitonov theorem and genetic algorithm optimization based approaches.In this study,the Generalized Kharitonov Theorem(GKT)for quasi-polynomials is employed for the purpose of designing a robust controller that can simultaneously stabilize a given unstable second-order interval plant family with time delay.Using a constructive procedure based on the Hermite-Biehler theorem,we obtain all the Proportional-Integral-Derivative gains that stabilize the uncertain and unstable second-order delay system.Genetic Algorithms(GAs)are utilized to optimize the three parameters of the PID controllers and the three parameters of the system which provide the best control that makes the system robust stable under uncertainties.Specifically,the method uses genetic algorithms to determine the optimum parameters by minimizing the integral of time-weighted absolute error ITAE,the Integral-Square-Error ISE,the integral of absolute error IAE and the integral of time-weighted Square-Error ITSE.The validity and relatively effortless application of presented theoretical concepts are demonstrated through a computation and simulation example.

Generalized Virial Theorem for Mixed State When Hamiltonians Include Coordinate-Momentum Couplings

The generalized Virial theorem for mixed state, derived from the generalized Hellmann Feynman theorem, only applies to Hamiltonians in which potential of coordinates is separate from momentum energy term. In this paper we discuss Virial theorem for mixed state for some Hamiltonians with coordinate-momentum couplings in order to know their contributions to internal energy.

Biased Random Number Generator Based on Bell's Theorem

We propose a biased random number generation protocol whose randomness is based on the violation of the Clauser Home inequality. Non-maximally entangled state is used to maximize the Bell violation. Due to the rotational asymmetry of the quantum state, the ratio of Os to ls varies with the measurement bases. The experimental partners can then use their measurement outcomes to generate the biased random bit string. The bias of their bit string can be adjusted by altering their choices of measurement bases. When this protocol is implemented in a device-independent way, we show that the bias of the bit string can still be ensured under the collective attack.

HYPERCYCLICITY, SUPERCYCLICITY AND GENERALIZED RAKOEVIC'S PROPERTY

A Banach space operator satisfies generalized RakoSevi5＇s property （gw） if the complement of its upper semi B-Weyl spectrum in its approximate point spectrum is the set of eigenvalues of T which are isolated in the spectrum of T. In this note, we characterize hypecyclic and supercyclic operators satisfying the property （gw）.

A Generalized Form of Whitney's Theorem

Whitney＇s theorem is a famous theorem in the local singularity theory. In this paper, as an application of Malgrange preparation theorem, a generalized form of Whitney＇s theorem will be derived.