fractional variational calculus and the transversality conditions this paper presents the euler lagrange equations and the transversality conditions for fractional variational problems the fractional derivatives are defined in the sense of riemann liouville and caputo the connection between the transversality conditions and the natural boundary conditions necessary to solve a fractional differential equation is examined it is demonstrated that fractional boundary conditions may be necessary even when the problem is defined in terms of the caputo derivative furthermore both fractional derivatives the riemann liouville and the caputo arise in the formulations even when the fractional variational problem is defined in terms of one fractional derivative only examples are presented to demonstrate the applications of the formulations

lyapunov exponents in disordered chaotic systems avoided crossing and level statistics the behavior of the lyapunov exponents les of a disordered system consisting of mutually coupled chaotic maps with different parameters is studied the les are demonstrated to exhibit avoided crossing and level repulsion qualitatively similar to the behavior of energy levels in quantum chaos recent results for the coupling dependence of the les of two coupled chaotic systems are used to explain the phenomenon and to derive an approximate expression for the distribution functions of le spacings the depletion of the level spacing distribution is shown to be exponentially strong at small values the results are interpreted in terms of the random matrix theory

calculus of variations on time scales with nabla derivatives abstract we prove a necessary optimality condition of the euler lagrange type for variational problems on time scales involving nabla derivatives of higher order the proof is done using a new and more general fundamental lemma of the calculus of variations on time scales

noether s theorem on time scales abstract we show that for any variational symmetry of the problem of the calculus of variations on time scales there exists a conserved quantity along the respective euler lagrange extremals

universality in chaos lyapunov spectrum and random matrix theory we propose the existence of a new universality in classical chaotic systems when the number of degrees of freedom is large the statistical property of the lyapunov spectrum is described by random matrix theory we demonstrate it by studying the finite time lyapunov exponents of the matrix model of a stringy black hole and the mass deformed models the massless limit which has a dual string theory interpretation is special in that the universal behavior can be seen already at t 0 while in other cases it sets in at late time the same pattern is demonstrated also in the product of random matrices

generalized hamilton s principle with fractional derivatives we generalize hamilton s principle with fractional derivatives in the lagrangian l t y t 0d ty t so that the function y and the order of the fractional derivative are varied in the minimization procedure we derive stationarity conditions and discuss them through several examples

progresses in the analysis of stochastic 2d cellular automata a study of asynchronous 2d minority cellular automata are often used to model systems in physics social sciences biology that are inherently asynchronous over the past 20 years studies have demonstrated that the behavior of cellular automata drastically changes under asynchronous updates still the few mathematical analyses of asynchronism focus on 1d probabilistic cellular automata either on single examples or on specific classes as for other classic dynamical systems in physics extending known methods from 1d to 2d systems is a long lasting challenging problem in this paper we address the problem of analyzing an apparently simple 2d asynchronous cellular automaton 2d minority where each cell when fired updates to the minority state of its neighborhood our simulations reveal that in spite of its simplicity the minority rule exhibits a quite complex response to asynchronism by focusing on the fully asynchronous regime we are however able to describe completely the asymptotic behavior of this dynamics as long as the initial configuration satisfies some natural constraints besides these technical results we have strong reasons to believe that our techniques relying on defining an energy function from the transition table of the automaton may be extended to the wider class of threshold automata

noether s theorems for nonshifted dynamic systems on time scales abstract conserved quantities on time scales for the nonshifted birkhoffian system and the nonshifted hamiltonian system are investigated firstly nonshifted birkhoff equation and nonshifted hamilton equation on time scales are obtained secondly noether identities and noether theorems for the nonshifted birkhoffian system and the nonshifted hamiltonian system are achieved thirdly special cases such as the classical birkhoffian system the discrete birkhoffian system the classical hamiltonian system the discrete hamiltonian system and the nonshifted lagrangian system are discussed

around probabilistic cellular automata we survey probabilistic cellular automata with approaches coming from combinatorics statistical physics and theoretical computer science each bringing a different viewpoint some of the questions studied are specific to a domain and some others are shared most notably the ergodicity problem

distribution of the ratio of consecutive level spacings in random matrix ensembles we derive expressions for the probability distribution of the ratio of two consecutive level spacings for the classical ensembles of random matrices this ratio distribution was recently introduced to study spectral properties of many body problems as contrary to the standard level spacing distributions it does not depend on the local density of states our wigner like surmises are shown to be very accurate when compared to numerics and exact calculations in the large matrix size limit quantitative improvements are found through a polynomial expansion examples from a quantum many body lattice model and from zeros of the riemann zeta function are presented

the distribution of lyapunov exponents exact results for random matrices simple exact expressions are derived for all the lyapunov exponents of certainn dimensional stochastic linear dynamical systems in the case of the product of independent random matrices each of which has independent gaussian entries with mean zero and variance 1 n the exponents have an exponential distribution asn in the case of the time ordered product integral of exp n 1 2dw where the entries of then n matrixw t are independent standard wiener processes the exponents are equally spaced for fixedn and thus have a uniform distribution as n

calculus of variations on time scales weak local piecewise crd1 solutions with variable endpoints abstract a nonlinear calculus of variations problem on time scales with variable endpoints is considered the space of functions employed is that of piecewise rd continuously differentiable functions c1prd for this problem the euler lagrange equation the transversality condition and the accessory problem are derived as necessary conditions for weak local optimality assuming the coercivity of the second variation a corresponding second order sufficiency criterion is established

isotropic brownian motions over complex fields as a solvable model for may wigner stability analysis we consider matrix valued stochastic processes known as isotropic brownian motions and show that these can be solved exactly over complex fields while these processes appear in a variety of questions in mathematical physics our main motivation is their relation to a may wigner like stability analysis for which we obtain a stability phase diagram the exact results establish the full joint probability distribution of the finite time lyapunov exponents and may be used as a starting point for a more detailed analysis of the stability instability phase transition our derivations rest on an explicit formulation of a fokker planck equation for the lyapunov exponents this formulation happens to coincide with an exactly solvable class of models of the calgero sutherland type originally encountered for a model of phase coherent transport the exact solution over complex fields describes a determinantal point process of biorthogonal type similar to recent results for products of random matrices and is also closely related to hermitian matrix models with an external source

a direct approach to the construction of standard and non standard lagrangians for dissipative like dynamical systems with variable coefficients we present a direct approach to the construction of lagrangians for a large class of one dimensional dynamical systems with a simple dependence monomial or polynomial on the velocity we rederive and generalize some recent results and find lagrangian formulations which seem to be new some of the considered systems e g motions with the friction proportional to the velocity and to the square of the velocity admit infinite families of different explicit lagrangian formulations

conserved quantities for hamiltonian systems on time scales abstract conserved quantities for hamiltonian systems on time scales with nabla derivatives and delta derivatives are presented first hamilton principle on time scales with nabla derivatives is established and hamilton canonical equation with nabla derivatives is obtained second noether identity and noether theorem for hamiltonian systems with nabla derivatives are achieved third hamilton canonical equation with delta derivatives noether identity and noether theorem for hamiltonian systems with delta derivatives are gotten through duality principle on the basis of the corresponding results with nabla derivatives fourth some special cases of noether identity and noether theorem are given and finally two examples are devoted to illustrate the methods and results

noether theorem for birkhoffian systems on time scales birkhoff equations on time scales and noether theorem for birkhoffian system on time scales are studied first some necessary knowledge of calculus on time scales are reviewed second birkhoff equations on time scales are obtained third the conditions for invariance of pfaff action and conserved quantities are presented under the special infinitesimal transformations and general infinitesimal transformations respectively fourth some special cases are given and finally an example is given to illustrate the method and results

standard and non standard lagrangians for dissipative dynamical systems with variable coefficients dynamical systems described by equations of motion with the first order time derivative dissipative terms of even and odd powers and coefficients varying either in time or in space are considered methods to obtain standard and non standard lagrangians are presented and used to identify classes of equations of motion that admit a lagrangian description it is shown that there are two general classes of equations that have standard lagrangians and one special class of equations that can only be derived from non standard lagrangians in addition each general class has a subset of equations with non standard lagrangians conditions required for the existence of standard and non standard lagrangians are derived and a relationship between these two types of lagrangians is introduced by obtaining lagrangians for several dynamical systems and some basic equations of mathematical physics it is demonstrated that the presented methods can be applied to a broad range of physical problems

the hamilton formalism with fractional derivatives recently the traditional calculus of variations has been extended to be applicable for systems containing fractional derivatives in this paper the passage from the lagrangian containing fractional derivatives to the hamiltonian is achieved the hamilton s equations of motion are obtained in a similar manner to the usual mechanics in addition the classical fields with fractional derivatives are investigated using hamiltonian formalism two discrete problems and one continuous are considered to demonstrate the application of the formalism the results are obtained to be in exact agreement with agrawal s formalism

on green s functions and positive solutions for boundary value problems on time scales in this paper we offer a form of self adjoint differential equations on time scales so that the associated green s function is found symmetric in the usual sense for this purpose together with the delta derivative we employ the nabla derivative as well we introduce the concepts of lebesgue delta and nabla integrals on time scales next sign properties of the green s function are investigated and existence results for positive solutions of nonlinear boundary value problems are established upper and lower bounds for these positive solutions also are given

nondeterministic density classification with diffusive probabilistic cellular automata we present a probabilistic cellular automaton ca with two absorbing states which performs classification of binary strings in a nondeterministic sense in a system evolving under this ca rule empty sites become occupied with a probability proportional to the number of occupied sites in the neighborhood while occupied sites become empty with a probability proportional to the number of empty sites in the neighborhood the probability that all sites become eventually occupied is equal to the density of occupied sites in the initial string

solving the density classification problem with a large diffusion and small amplification cellular automaton abstract one of the most studied inverse problems in cellular automata cas is the density classification problem it consists in finding a ca such that given any initial configuration of 0s and 1s it converges to the all 1 fixed point configuration if the fraction of 1s is greater than the critical density 1 2 and it converges to the all 0 fixed point configuration otherwise in this paper we propose an original approach to solve this problem by designing a ca inspired by two mechanisms that are ubiquitous in nature diffusion and nonlinear sigmoidal response this ca which is different from the classical ones because it has many states has a success ratio of 100 and works for any system size any dimension and any critical density

dynamics symmetries of hamiltonian system on time scales in this paper the dynamics symmetries of hamiltonian system on time scales are studied we study the symmetries and quantities based on the calculation of variation and lie transformation group particular focus lies in the noether symmetry leads to the noether conserved quantity and the lie symmetry leads to the noether conserved quantity if the infinitesimal transformations satisfy the structure equation as the new application of result at end of the article we give a simple example of noether symmetry and lie symmetry on time scales

fractional actionlike variational problems fractional actionlike variational problems have recently gained importance in studying dynamics of nonconservative systems in this note we address multidimensional fractional actionlike problems of the calculus of variations

solution of the density classification problem with two cellular automata rules recently land and belew phys rev lett 74 5148 1995 have shown that no one dimensional two state cellular automaton which classifies binary strings according to their densities of 1 s and 0 s can be constructed we show that a pair of elementary rules namely the traffic rule 184 and the majority rule 232 performs the task perfectly this solution employs the second order phase transition between the freely moving phase and the jammed phase occurring in rule 184 we present exact calculations of the order parameter in this transition using the method of preimage counting

a formulation of noether s theorem for fractional problems of the calculus of variations abstract fractional or non integer differentiation is an important concept both from theoretical and applicational points of view the study of problems of the calculus of variations with fractional derivatives is a rather recent subject the main result being the fractional necessary optimality condition of euler lagrange obtained in 2002 here we use the notion of euler lagrange fractional extremal to prove a noether type theorem for that we propose a generalization of the classical concept of conservation law introducing an appropriate fractional operator