关于任意随机变量序列泛函的强极限定理

本文在k是固定的正整数，{fn}是R^k 1上的Borel可测函数列时，得到了任意随机变量序列{Xrn≥0}的泛函{fn(Xn-k,…,Xn)}的强极限定理，它是Chung的关于独立随机变量序列的强大数律的推广，作为推论，得到了k重非齐次马尔科夫链的一类强极限定理．     

On different geometric formulations of Lagrangian formalism

We consider two geometric formulations of Lagrangian formalism on fibred manifolds: Krupka's theory of finite order variational sequences, and Vinogradov's infinite order variational sequence associated with the -spectral sequence. On one hand, we show that the direct limit of Krupka's variational bicomplex is a new infinite order variational bicomplex which yields a new infinite order variational sequence. On the other hand, by means of Vinogradov's -spectral sequence, we provide a new finite order variational sequence whose direct limit turns out to be the Vinogradov's infinite order variational sequence. Finally, we provide an equivalence of the two finite order and infinite order variational sequences modulo the space of Euler-Lagrange morphisms.     

Variationally trivial Lagrangians and locally variational differential equations of arbitrary order

We continue our investigation of the Lagrangian formalism on jet bundle extensions using Fock space methods. We are able to provide the most general form of a variationally trivial Lagrangian of arbitrary order and we also give a generic expression for the most general locally variational differential equation. As anticipated in the literature, these expressions involve some special combinations of the highest order derivatives, called hyper-Jacobians.     

On Weitzenböck curvature operators

The Weitzenböck curvature operators are the curvature terms of order zero that appear in the well‐known classical Weitzenböck formula. In this paper, we use the formalism of double forms to prove a simple formula for these operators and to study their geometric properties.     

Statistical Inferences for Termination of Markov Type Random Search Algorithms

In this paper, we consider the application of order statistics to establish the optimality in stochastic and heuristic optimization algorithms. A method for estimating the minimum value with an associated confidence interval is developed using the formalism of the theory of order statistics for i.i.d. variables; we examine it by computer simulation. We build a method for the estimation of confidence intervals of the minimum value using order statistics, implemented for optimality testing and stopping in Markov type random search algorithms. The efficiency of this approach is discussed, using the results of application to stochastic approximation and simulated annealing.     

Modeling Discrete Event Systems With Faults Using a Rules-based Modeling Formalism

Obtaining accurate models of systems which are prone to failures and breakdowns is a difficult task. In this paper we present a methodology which makes the task of modeling failure prone discrete event systems (DESs) considerably less cumbersome, less error prone, and more user-friendly. The task of obtaining commonly used automata models for DESs is non-trivial for most practical systems, owing to the fact that the number of states in the commonly used automata models is exponential in the number of signals and faults. In contrast a model of a discrete event system, in the rules based modeling formalism proposed by the co-authors of this paper, is of size polynomial in the number of signals and faults. In order to model failures, we augment the signals set of the rules based formalism to include binary valued fault signals, the values representing either a non-faulty or a faulty state of a certain failure type. Addition of new fault signals requires introduction of new rules for the added fault signal events, and also modification of the existing rules for non-fault events. The rules based modeling formalism is further extended to model real-time systems, and we apply it to model delay-faults of the system as well. The model of a failure prone DES in the rules based can automatically be converted into an equivalent (timed)-automaton model for a failure analysis in the automaton model framework.     

Critical behavior of a monoaxial chiral helimagnet

We analyze the critical behavior of magnetically ordered phases appearing in a monoaxial chiral helimagnet in a weak external magnetic field. Using the formalism of the equations of state in the critical region, we determine the temperature dependence of the order parameters for the conical phase and the soliton-lattice phase. We calculated the critical exponents and show that they coincide with those in the three-dimensional Heisenberg model.     

Studying liquid-crystal systems with impurities using matrix-formalism methods

We use the matrix formalism to investigate a liquid-crystal system containing impurities. As an example, we calculate correlation functions for fluctuations of the director orientation and of the order parameter, which characterizes an isotropic subsystem. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 2, pp. 323–327, May, 2006.     

On the solvability of a linear homogeneous functional-differential equation of point type

We consider the Cauchy problem for a linear homogeneous functional-differential equation of point type on the real line. For the case of a one-dimensional equation, we obtain sufficient conditions for the existence and uniqueness of a solution with a prescribed order of growth. The spectral properties of the operator generated by the right-hand side of such an equation are studied in detail. The study relies upon a formalism based on the group properties of such equations.     

Nonlinear and non-separable multiscale representations based on Lipschitz perturbation

In this Note, we present a new formalism for nonlinear and non-separable multiscale representations. The new formalism we propose brings about similarities between existing nonlinear multiscale representations and also allows us to alleviate the classical hypotheses made to prove the convergence of the multiscale representations.     

A Event Occurrence Rules based Compact Modeling Formalism for a Class of Discrete Event Systems

The analysis, failure diagnosis and control of discrete event systems (DESs) requires an accurate model of the system. In this paper we present a methodology which makes the task of modeling DESs considerably less cumbersome, less error prone, and more user-friendly than it usually is. In doing so we simplify the modeling formalism of [4, 5], proposed for obtaining valid models of complex discrete event systems, by eliminating ‘precedence relations’, and capturing them as part of the ‘event occurrence rules’. Under the new modeling formalism the size of the system model is polynomial in the number of signals; whereas the number of states in the commonly used automata models is exponential in the number of signals. We present automated techniques for deriving an automaton model from the model in the proposed formalism. We illustrate the modeling formalism using examples drawn from manufacturing and process control systems.     

A charged composite particle in a constant electric field

We consider the motion of a composite charged particle in a constant electric field. Using the billiard formalism, we establish exact laws of motion for such a particle with a small number of internal degrees of freedom and propose using a generalized Schwarz principle to straighten trajectories in the field presence. Within the billiard formalism, we obtain regimes of motion of a composite particle with two internal degrees of freedom in a constant field.     

The Hessian and Jacobi morphisms for higher order calculus of variations

We formulate higher order variations of a Lagrangian in the geometric framework of jet prolongations of fibered manifolds. Our formalism applies to Lagrangians which depend on an arbitrary number of independent and dependent variables, together with higher order derivatives. In particular, we show that the second variation is equal (up to horizontal differentials) to the vertical differential of the Euler-Lagrange morphism which turns out to be self-adjoint along solutions of the Euler-Lagrange equations. These two objects, respectively, generalize in an invariant way the Hessian morphism and the Jacobi morphism (which is then self-adjoint along critical sections) of a given Lagrangian to the case of higher order Lagrangians. Some examples of classical Lagrangians are provided to illustrate our method.     

Integration of positive constructible functions against Euler characteristic and dimension

Following recent work of R. Cluckers and F. Loeser [Fonctions constructible et intégration motivic I, C. R. Math. Acad. Sci. Paris 339 (2004) 411-416] on motivic integration, we develop a direct image formalism for positive constructible functions in the globally subanalytic context. This formalism is generalized to arbitrary first-order logic models and is illustrated by several examples on the p-adics, on the Presburger structure and on o-minimal expansions of groups. Furthermore, within this formalism, we define the Radon transform and prove the corresponding inversion formula.     

Fourier transforms on Cantor sets: A study in non-Diophantine arithmetic and calculus

Fractals equipped with intrinsic arithmetic lead to a natural definition of differentiation, integration, and complex structure. Applying the formalism to the problem of a Fourier transform on fractals we show that the resulting transform has all the required basic properties. As an example we discuss a sawtooth signal on the ternary middle-third Cantor set. The formalism works also for fractals that are not self-similar.     

Factorization iterative methods,M-operators andH-operators

Summary We set up here a general formalism for describing factorization iterative methods of the first order and we use it to review various methods that have been proposed in the literature; next we introduce the notions ofM- andH-operators which generalize those of block-M- and block-H-matrices; finally we discuss the properties of factorization iterative methods in relation with characteristic properties ofM- andH-operators.     

The effect of time fractality on the transition coefficients: Historical Stern–Gerlach experiment revisited

In this article, the influence of time discreteness on the transition coefficients is investigated within the framework of time fractional development of quantum systems which has been developed recently by the present authors [22]. In this formalism, fractional mathematics which is a powerful tool to study the non-Markovian and non-Gaussian properties of physical processes is used in order to obtain time fractional evolution operator and transition probability. They are given in terms of Mittag–Leffler function which plays an important role in the mathematical structure as well as the physical interpretation of the phenomena under investigation. In order to place the presented formalism on a concrete basis, historical Stern–Gerlach experiment has been revisited with the purpose of studying transition coefficients which have a non-Markovian feature. The effect of the time fractionalization has been clearly illustrated in the figures via fractional derivative order α.     

Bigravity in the Kuchař Hamiltonian formalism: The general case

We develop the Hamiltonian formalism of bigravity and bimetric theories for the general form of the interaction potential of two metrics. When studying the role of lapse and shift functions in theories with two metrics, we naturally use the Kucha? formalism in which these functions are independent of the choice of the space-time coordinate system. We find conditions on the potential necessary and sufficient for the existence of four first-class constraints. These constraints realize a well-known hypersurface deformation algebra in the framework of the formalism of Dirac brackets constructed on the base of all second-class constraints. Fixing one of the metrics, we obtain a bimetric theory not containing first-class constraints. Conserved quantities corresponding to symmetries of the background metric can then be expressed ultralocally in terms of the metric interaction potential.