Strong invariance principle for a counterbalanced random walk

We study a counterbalanced random walkS_(n)=X_(1)+…+X_(n),which is a discrete time non-Markovian process andX_(n) are given recursively as follows.For n≥2,X_(n) is a new independent sample from some fixed law̸=0 with a fixed probability p,andX_(n)=−X_(v(n))with probability 1−p,where v(n)is a uniform random variable on{1;…;n−1}.We apply martingale method to obtain a strong invariance principle forS_(n).

A Supplement to the Invariance Principle of the Speed of Light and the Quantum Theory

Richard Feynman once said, “I think it is safe to say that no one understands Quantum Mechanics”. The well-known article on the Einstein-Podolsky-Rosen (EPR) paradox brought forth further doubts on the interpretation of quantum theory. Einstein’s doubt on quantum theory is a doubleedged sword: experimental verification of quantum theory would contradict the hypothesis that speed of light is finite. It has been almost a century since the creation of quantum theory and special relativity, and the relevant doubts brought forward remain unresolved. We posit that the existence of discontinuity points and quantum wormholes would imply superluminal phenomenon or infinite speed of light, which provides for an important supplement to the invariance principle of the speed of light and superluminal phenomena. This can potentially resolve the inconsistency between special relativity and quantum theory.

Should the Galilean Invariance be Kept as a Guiding Principle for Selecting the Interaction Potentials?

Galilean invariance is a nonrelativistic principle,which should not be kept as a guid-ing principle in discriminating the interaction potential terms derived from field theory.

Controlling Chaos in Neuron Based on Lasalle Invariance Principle

A new control law is proposed to asymptotically stabilize the chaotic neuron system based on LaSalleinvariant principle.The control technique does not require analytical knowledge of the system dynamics and operateswithout an explicit knowledge of the desired steady-state position.The well-known modified Hodgkin-Huxley (MHH)and Hindmarsh-Rose (HR) model neurons are taken as examples to verify the implementation of our method.Simulationresults show the proposed control law is effective.The outcome of this study is significant since it is helpful to understandthe learning process of a human brain towards the information processing,memory and abnormal discharge of the brainneurons.

An Invariance Principle for ρ^--Mixing Random Sequences

在这份报纸，我们建立一个不变性原则为 -- 在某时刻 condition.The 结果下面混合随机的 se quences 改进并且延长吴(2003 ) 的相关结果。

STRONG COMPARISON PRINCIPLES FOR SOME NONLINEAR DEGENERATE ELLIPTIC EQUATIONS

In this paper, we obtain the strong comparison principle and Hopf Lemma for locally Lipschitz viscosity solutions to a class of nonlinear degenerate elliptic operators of the form △2ψ ＋ L（x,△ ψ）, including the conformal hessian operator.

STUDY OF THE EQUIVALENT THEOREM OF GENERALIZED VARIATIONAL PRINCIPLES IN ELASTICITY

The relations of all generalized variational principles in elasticity are studied by employing the invariance theorem of field theory. The infinitesimal scale transformation in field theory was employed to investigate the equivalent theorem. Among the results found particularly interesting are those related to that all generalized variational principles in elasticity are equal to each other. Also studied result is that only two variables are independent in the functional and the stress-strain relation is the variational constraint condition for all generalized variational principles in elasticity. This work has proven again the conclusion of Prof. Chien Wei-zang.

FORM INVARIANCE AND NOETHER SYMMETRICAL CONSERVED QUANTITY OF RELATIVISTIC BIRKHOFFIAN SYSTEMS

A form invariance of the relativistic Birkhoffian system is studied, and the conserved quantities of the system are obtained. Under the infinitesimal transformation of groups, the definition and criteria of the form invariance of the system were given. In view of the invariance of relativistic Pfaff_Birkhoff_ D'Alembert principle under the infinitesimal transformation of groups, the theory of Noether symmetries of the relativistic Birkhoffian system were constructed. The relation between the form invariance and the Noether symmetry is studied, and the results show that the form invariance can also lead to the Noether symmetrical conserved quantity of the relativistic Birkhoffian system under certain conditions.

Maximum Principles for Normal Matrices

Ky Fan maximum principle is a well-known observation about traces of certain hermitian matrices. In this note, we derive a powerful extension of this claim. The extension is achieved in three ways. First, traces are replaced with norms of diagonal matrices, and any unitarily invariant norm can be used. Second, hermitian matrices are replaced by normal matrices, so the rule applies to a larger class of matrices. Third, diagonal entries can be replaced with eigenvalues and singular values. It is shown that the new maximum principle is closely related to the problem of approximating one matrix by another matrix of a lower rank.

A Formulation of Quantum Theory Based on Two Physical Principles

We demonstrate two points: 1) the formalism of quantum mechanics can be understood simply as a structure for the expression of the physical notion that not all observables can have values simultaneously;2) the specific uncertainty relations can be derived (rigorously) by combination of the invariance principle with a general uncertainty relation based only on the existence of unspecified pairs of conjugate observables. For this purpose, we present a formulation of quantum mechanics based strictly on the invariance principle and a “weak” statement of the uncertainty principle that asserts only the existence of incompatible (conjugate) observables without specifying which observables are incompatible. We go on to show that the invariance principle can be used to develop the equations of motion of the theory, including the Klein-Gordon and Schrodinger equations.

A New Version of Special Relativity Absorbed the Uncertainty Principle: Its Content as Well as Application and Experimental Test

Based on the space spherical symmetry of 3-dimensional and the translational symmetry of time and the uncertainty principle, a 4-dimensional space-time cylinder model of quarks and leptons is established. With this model, equations of the special relativity can be extended more perfectly, thereby achieving a unity of the special relativity and quantum mechanics in deeper level. New equations can not only interpret issues explained by old equations but also solve several important pending problems. For example, a formula to strictly calculate the coefficient ξ of Lorentz invariance violation (LIV) is derived, to above 4 × 1019 eV UHECR protons the calculated |ξ| -30, although there is the LIV effect it is too weak to change the GZK cutoff, which is consistent with observations of HiRes and Auger;Also, a relation formula between the Hubble constant and several basic constants is derived, thus theoretically calculated H0 = 70.937 km·s-1·Mpc-1, which is well consistent with the final observation result of HST Key Project. In addition, an unusual effect predicted by new equations can be experimentally tested in the electron storage ring;a preliminary experiment result has hinted its signs of existence.

An invariance principle for negatively associated random variables

A finite collection of random variables, X1,…, Xn （n≥2）, is said to be negatively associated （NA） if any two coordinatewise nondecreasing （or nonincreasing） functions f1 and f2 on Rn, such that （?）j=fj（X1,…,Xn） have a finite variance for j=1, 2 and Cov（（?）1,（?）2）≤0; an infinite collection is said to be NA if every finite subcollection is NA. It is relative to a lot of practical problems, such as reliability theory, percolation models and multivariate statistical analysis. Matula （1992） studied the strong laws of large numbers for an NA sequence. Recent-

Donsker’s Invariance Principle Under the Sub-linear Expectation with an Application to Chung’s Law of the Iterated Logarithm

We prove a new Donsker’s invariance principle for independent and identically distributed random variables under the sub-linear expectation.As applications,the small deviations and Chung’s law of the iterated logarithm are obtained.

Lyapunov stability and generalized invariance principle for nonconvex differential inclusions

This paper studies the system stability problems of a class of nonconvex differential inclusions. At first, a basic stability result is obtained by virtue of locally Lipschitz continuous Lyapunov functions. Moreover, a generalized invariance principle and related attraction conditions are proposed and proved to overcome the technical difficulties due to the absence of convexity. In the technical analysis, a novel set-valued derivative is proposed to deal with nonsmooth systems and nonsmooth Lyapunov functions. Additionally, the obtained results are consistent with the existing ones in the case of convex differential inclusions with regular Lyapunov functions. Finally, illustrative examples are given to show the effectiveness of the methods.

Conserved quantities and adiabatic invariants of fractional Birkhoffian system of Herglotz type

In order to further study the dynamical behavior of nonconservative systems,we study the conserved quantities and the adiabatic invariants of fractional Brikhoffian systems with four kinds of different fractional derivatives based on Herglotz differential variational principle.Firstly,the conserved quantities of Herglotz type for the fractional Brikhoffian systems based on Riemann-Liouville derivatives and their existence conditions are established by using the fractional Pfaff-Birkhoff-d Alembert principle of Herglotz type.Secondly,the effects of small perturbations on fractional Birkhoffian systems are studied,the conditions for the existence of adiabatic invariants for the Birkhoffian systems of Herglotz type based on Riemann-Liouville derivatives are established,and the adiabatic invariants of Herglotz type are obtained.Thirdly,the conserved quantities and adiabatic invariants for the fractional Birkhoffian systems of Herglotz type under other three kinds of fractional derivatives are established,namely Caputo derivative,Riesz-Riemann-Liouville derivative and Riesz-Caputo derivative.Finally,an example is given to illustrate the application of the results.

Stability of switched nonlinear systems via extensions of LaSalle's invariance principle

This paper studies the extension of LaSalle＇s invariance principle for switched nonlinear systems. Unlike most existing results in which each switching mode in the system needs to be asymptotically stable, this paper allows the switching modes to be only stable. Under certain ergodicity assumptions of the switching signals, two extensions of LaSalle＇s invariance principle for global asymptotic stability of switched nonlinear systems are obtained using the method of common joint Lyapunov function.

The Invariance Principle for p-■Chain

There are two parts in this paper. In the first part we construct the Markov chain in random environment（MCRE）, the skew product Markov chain and p-θ^→ chain from a random transition matrix and a two-dimensional probability distribution, and in the second part we prove that the invarianee principle for p-θ^→ chain, a more complex non-homogeneous Markov chain, is true under some reasonable conditions. This result is more powerful.

On a Conjecture of an Invariance Principle for Sequences of Associated Random Variables

A finite collection of random variables, X1} —, Xm, is said to be associated if any two coordinatewise nondecreasing functions /i and /a on Rm such that Jt = fj(Xi} , Xm~) has finite variance for j = l, 2, Oov(/i,fa)>0; an infinite collection is said to be associated if every finite subcollection is. associated. Thus the concept of "association" is introduced as dependence in probability (Esary, Proschan and Walkup (1967)). It is relative to a lot of practical models, such as the percolation models, the Ising models of statistical mechanics. Therefore, some people are

The Proof That There Are No Invariabilities of Lorentz Transformations in the Interaction Theories of Micro-Particle Physics

It is proved in this paper that there are at least five situations in the interaction theories of microparticle physics that the Lorentz transformations have no invariabilities. 1) In the formula to calculate transition probabilities in particle physics, the so-called invariability factor of phase space d3p/E is not invariable actually under the Lorentz transformations. Only in one-dimensional motion with uy = uz = 0, it is invariable. 2) The propagation function of spinor field in quantum theory of field has no invariability of Lorentz Transformation actually. What appears in the transformation is the sum of Lorentz factors aμνaλμ ≠ δνλ when ν, λ = 1, 4, rather than aμνaλμ = δνλ. But in the current calculation, we take aμνaλμ = δνλ. The confusion of subscript’s position leads to wrong result. 3) Though the motion equations of quantum fields and the interaction Hamiltonian are unchanged under the Lorentz transformation, the motion equation of perturbation which is used to calculate the transition probability in the interaction representation has no invariability. 4) The interactions between bound state’s particles have no Lorentz invariability. In fact, the principle of relativity has no approximation if it holds. 5) The calculation methods of high order perturbations normalization processes in quantum theory of fields violate the invariability of Lorentz transformation. The conclusions above are effective for strong, weak and electromagnetic interactions and so on. Therefore, the principle of relativity does not hold in the micro-particle’s interactions. On the other hand, the invariability principle of light’s speed is still effective. So the formulas of special relativity still hold, but we should consider them with absolute significances.

The best rates in strong invariance principle

Let {Xn} be a sequence of i.i.d.r. v. s with mean 0 and variance 1, Sn = ∑i=1nXi- Suppose H(x)】0 (x≥0) is a non-decreasing continuous function such that for some γ】0 and x0】0, x-2-γ(x)(x≥x0) is non-decreasing and x -1logH(x) (x≥x0) is non-increasing. If x-1 logH(x)→0 (x→∞), then Sn - W(n)=o (invH(n)) a.s. (n → ∞) holds if and only if EH(t|X1|)【∞ for all t】0.