hamiltonian formulation of systems with linear velocities within riemann liouville fractional derivatives the link between the treatments of constrained systems with fractional derivatives by using both hamiltonian and lagrangian formulations is studied it is shown that both treatments for systems with linear velocities are equivalent

noether s theorem of fractional birkhoffian systems abstract in this paper we study noether type symmetry theorem to fractional birkhoffian systems with riemann liouville derivatives this theorem provides an explicit algorithmic way to compute a constant for any birkhoffian systems admitting a symmetry finally we extend our noether s theorem to fractional birkhoffian systems base on caputo or riesz derivatives

multilevel large deviations and interacting diffusions let n be a sequence of random variables with values in a topological space which satisfy the large deviation principle for eachm and eachn let m n denote the empirical measure associated withm independent copies of n as a main result we show that m n also satisfies the large deviation principle asm n we derive several representations of the associated rate function these results are then applied to empirical measure processes m n t m 1 i 1n in t 0 t t where 1n mn t is a system of weakly interacting diffusions with noise intensity 1 n this is a continuation of our previous work on the mckean vlasov limit and related hierarchical models 4 5

a variational formula for the free energy of an interacting many particle system we consider n bosons in a box in r d with volume n rho under the influence of a mutually repellent pair potential the particle density p is an element of 0 infinity is kept fixed our main result is the identification of the limiting free energy f beta p at positive temperature 1 beta in terms of an explicit variational formula for any fixed rho if beta is sufficiently small and for any fixed beta if rho is sufficiently small the thermodynamic equilibrium is described by the symmetrized trace of e beta h n where h n denotes the corresponding hamilton operator the well known feynman kac formula reformulates this trace in terms of n interacting brownian bridges due to the symmetrization the bridges are organized in an ensemble of cycles of various lengths the novelty of our approach is a description in terms of a marked poisson point process whose marks are the cycles this allows for an asymptotic analysis of the system via a large deviations analysis of the stationary empirical field the resulting variational formula ranges over random shift invariant marked point fields and optimizes the sum of the interaction and the relative entropy with respect to the reference process in our proof of the lower bound for the free energy we drop all interaction involving infinitely long cycles and their possible presence is signalled by a loss of mass of the finitely long cycles in the variational formula in the proof of the upper bound we only keep the mass on the finitely long cycles we expect that the precise relationship between these two bounds lies at the heart of bose einstein condensation and intend to analyze it further in future

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

polynomials and operator orderings it is shown that there is an exact one to one correspondence between all possible sets of polynomials both orthogonal and nonorthogonal and rules for operator orderings operator orderings that are hermitian give polynomials with definite parity most of the standard classical orthogonal polynomials are associated with operator orderings that are not particularly simple however there is a special one parameter class of hermitian operator orderings that corresponds to a class of elegant but little known orthogonal polynomials called continuous hahn polynomials

fractional radiative transfer equation within chebyshev spectral approach in this work we report the convergence of the chebyshev polynomials combined with the s n method for the steady state transport equation using the fractional derivative the procedure is based on the expansion of the angular flux in a truncated series of orthogonal polynomials that results in the transformation of the multidimensional problem into a system of fractional differential equations the convergence of this approach is studied in the context of the multidimensional discrete ordinates equations

variational problems with fractional derivatives invariance conditions and nother s theorem abstract a variational principle for lagrangian densities containing derivatives of real order is formulated and the invariance of this principle is studied in two characteristic cases necessary and sufficient conditions for an infinitesimal transformation group basic nother s identity are obtained these conditions extend the classical results valid for integer order derivatives a generalization of nother s theorem leading to conservation laws for fractional euler lagrangian equation is obtained as well results are illustrated by several concrete examples finally an approximation of a fractional euler lagrangian equation by a system of integer order equations is used for the formulation of an approximated invariance condition and corresponding conservation laws

a diffusion wave equation with two fractional derivatives of different order we analyse a diffusion wave equation with two fractional derivatives of different order on bounded and unbounded spatial domains thus our model represents a generalized telegraph equation solutions to signalling and cauchy problems in terms of a series and integral representation are given classical wave and heat conduction equations are obtained as limiting cases

mechanics with fractional derivatives lagrangian and hamiltonian mechanics can be formulated to include derivatives of fractional order f riewe phys rev 53 1890 1996 lagrangians with fractional derivatives lead directly to equations of motion with nonconservative classical forces such as friction the present work continues the development of fractional derivative mechanics by deriving a modified hamilton s principle introducing two types of canonical transformations and deriving the hamilton jacobi equation using generalized mechanics with fractional and higher order derivatives the method is illustrated with a frictional force proportional to velocity in contrast to conventional mechanics with integer order derivatives quantization of a fractional derivative hamiltonian cannot generally be achieved by the traditional replacement of momenta with coordinate derivatives instead a quantum mechanical wave equation is proposed that follows from the hamilton jacobi equation by application of the correspondence principle

discrete uncertainty relations generalized uncertainty relations based upon fourier transforms of both discrete and continuous functions are briefly reviewed we extend these results in order to establish discrete versions of the angular momentum uncertainty relations based upon su 2 transformations possible applications in quantum signal processing and computations are briefly discussed

complete equivalence of the gibbs ensembles for one dimensional markov systems we show the equivalence of the gibbs ensembles at the level of measures for one dimensional markov systems with arbitrary boundary conditions that is the limit of the microcanonical gibbs ensemble is a gibbs measure with an interaction depending on the microcanonical constraint in fact the usual microcanonical condition is replaced by the sharper constraint that all type frequencies of neighboring spins including the boundary spins are fixed when conditioning on a set of different frequencies of neighboring spins compatible with physical quantities like energy density we get the usual microcanonical ensemble we show that the limit is a gibbs measure for a nearest neighbor potential depending on the pair measure which maximizes the entropy on the given set of pair measures for this we show the large deviation property of the pair empirical measure for arbitrary boundary conditions we establish analogous results for finite range potentials

entropy minimization and schrodinger processes in infinite dimensions schrodinger processes are defined as mixtures of brownian bridges which preserve the markov property in finite dimensions they can be characterized as h transforms in the sense of doob for some space time harmonic function h of brownian motion and also as solutions to a large deviation problem introduced by schrodinger which involves minimization of relative entropy with given marginals as a basic case study in infinite dimensions we investigate these different aspects for schrodinger processes of infinite dimensional brownian motion the results and examples concerning entropy minimization with given marginals are of independent interest

wright functions as scale invariant solutions of the diffusion wave equation the time fractional diffusion wave equation is obtained from the classical diffusion or wave equation by replacing the first or second order time derivative by a fractional derivative of order 0 2 using the similarity method and the method of the laplace transform it is shown that the scale invariant solutions of the mixed problem of signalling type for the time fractional diffusion wave equation are given in terms of the wright function in the case 0 1 and in terms of the generalized wright function in the case 1 2 the reduced equation for the scale invariant solutions is given in terms of the caputo type modification of the

a counterexample to a frederico torres fractional noether type theorem abstract in this note we present a counterexample to theorem 24 in g s f frederico and d f m torres 2010 3

some aspects of fractional diffusion equations of single and distributed order abstract the time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first order time derivative with a fractional derivative of order 0 1 the fundamental solution for the cauchy problem is interpreted as a probability density of a self similar non markovian stochastic process related to a phenomenon of sub diffusion the variance grows in time sub linearly a further generalization is obtained by considering a continuous or discrete distribution of fractional time derivatives of order less than one then the fundamental solution is still a probability density of a non markovian process that however is no longer self similar but exhibits a corresponding distribution of time scales

a fractional calculus of variations for multiple integrals with application to vibrating string we introduce a fractional theory of the calculus of variations for multiple integrals our approach uses the recent notions of riemann liouville fractional derivatives and integrals in the sense of jumarie the main results provide fractional versions of the theorems of green and gauss fractional euler lagrange equations and fractional natural boundary conditions as an application we discuss the fractional equation of motion of a vibrating string

convergence of the grunwald letnikov scheme for time fractional diffusion using bivariate generating functions we prove convergence of the grunwald letnikov difference scheme for the fractional diffusion equation in one space dimension with and without central linear drift in the fourier laplace domain as the space and time steps tend to zero in a well scaled way this implies convergence in distribution weak convergence of the discrete solution towards the probability of sojourn of a diffusing particle the difference schemes allow also interpretation as discrete random walks for fractional diffusion with central linear drift we show that in the fourier laplace domain the limiting ordinary differential equation coincides with that for the solution of the corresponding diffusion equation

time fractional 2 1 dimensional wu zhang system dispersion analysis similarity reductions conservation laws and exact solutions abstract this paper examines the time fractional 2 1 dimensional wu zhang system describing dispersive long waves the dispersion relation of waves has been obtained from linear analysis and used to find the phase and group velocity the dispersion relation shows an anomalous dispersion of waves lie symmetries and the corresponding similarity reductions are carried out using riemann liouville fractional derivative and the order of the system is reduced to lower dimension furthermore the system is analysed for nonlinear self adjointness and the conservation laws are obtained associated with infinitesimal symmetries by applying new conservation theorem the adjoint variables are used to convert conservation laws into original dependent variables the exponential rational function method is employed to find exact solutions revealing kink and bell type profiles the effect of fractional order on the wave profile of solutions is discussed graphically

lie symmetry analysis explicit solutions and conservation laws for the space time fractional nonlinear evolution equations abstract this paper studies the symmetry analysis explicit solutions convergence analysis and conservation laws cls for two different space time fractional nonlinear evolution equations with riemann liouville rl derivative the governing equations are reduced to nonlinear ordinary differential equation ode of fractional order using their lie point symmetries in the reduced equations the derivative is in erdelyi kober ek sense power series technique is applied to derive an explicit solutions for the reduced fractional odes the convergence of the obtained power series solutions is also presented moreover the new conservation theorem and the generalization of the noether operators are developed to construct the nonlocal cls for the equations some interesting figures for the obtained explicit solutions are presented

noether symmetries and conserved quantities for fractional forced birkhoffian systems abstract in this paper noether symmetries and conserved quantities for fractional forced birkhoffian systems are studied firstly a new fractional pfaff birkhoff variational principle with riemann liouville derivatives and generalized force is established and from which the fractional forced birkhoff equations are derived secondly the definitions and the criteria of the invariance of a functional under an e parameter group of infinitesimal transformations are given finally the fractional noether theorems for forced birkhoffian systems are obtained and an illustrative example is presented

variational problems with fractional derivatives euler lagrange equations we generalize the fractional variational problem by allowing the possibility that the lower bound in the fractional derivative does not coincide with the lower bound of the integral that is minimized also for the standard case when these two bounds coincide we derive a new form of euler lagrange equations we use approximations for fractional derivatives in the lagrangian and obtain the euler lagrange equations which approximate the initial euler lagrange equations in a weak sense

extending bauer s corollary to fractional derivatives we comment on the method of dreisigmeyer and young j phys a math gen 368297 to model nonconservative systems with fractional derivatives it was previously hoped that using fractional derivatives in an action would allow us to derive a single retarded equation of motion using a variational principle it is proven that under certain reasonable assumptions the method of dreisigmeyer and young fails

dynamical difference equations abstract dynamical difference equations are motivated and developed conservation and covariance laws are proved illustrative examples are described and discussed newtonian equations are stressed but extensions to relativity and quantum mechanics are indicated the physical problem of convergence to the limit as the time step goes to zero is shown to reveal an area in which quantum mechanics and relativity are in disagreement

lie symmetry analysis conservation laws and exact solutions of the time fractional generalized hirota satsuma coupled kdv system abstract in the present paper lie point symmetries of the time fractional generalized hirota satsuma coupled kdv hs ckdv system based on the riemann liouville derivative are obtained using the derived lie point symmetries we obtain similarity reductions and conservation laws of the considered system finally some analytic solutions are furnished by means of the invariant subspace method in the caputo sense

large deviations for many brownian bridges with symmetrised initial terminal condition consider a large system of n brownian motions in mathbb r d with some non degenerate initial measure on some fixed time interval 0 with symmetrised initial terminal condition that is for any i the terminal location of the i th motion is affixed to the initial point of the i th motion where is a uniformly distributed random permutation of 1 n such systems play an important role in quantum physics in the description of boson systems at positive temperature 1 in this paper we describe the large n behaviour of the empirical path measure the mean of the dirac measures in the n paths and of the mean of the normalised occupation measures of the n motions in terms of large deviations principles the rate functions are given as variational formulas involving certain entropies and fenchel legendre transforms consequences are drawn for asymptotic independence statements and laws of large numbers in the special case related to quantum physics our rate function for the occupation measures turns out to be equal to the well known donsker varadhan rate function for the occupation measures of one motion in the limit of diverging time this enables us to prove a simple formula for the large n asymptotic of the symmetrised trace of rm e beta mathcal h n where mathcal h n is an n particle hamilton operator in a trap

some properties of the k dimensional lyness map this paper is devoted to study some properties of the k dimensional lyness map f x1 xk x2 xk a ki 2xi x1 our main result presents a rational vector field that gives a lie symmetry for f this vector field is used for k 5 to give information about the nature of the invariant sets under f when k is odd we also present a new as far as we know first integral for ff which allows us to deduce in a very simple way several properties of the dynamical system generated by f in particular for this case we prove that except on a given codimension one algebraic set none of the positive initial conditions can be a periodic point of odd period

continuous hahn polynomials and the heisenberg algebra continuous hahn polynomials have surfaced in a number of somewhat obscure physical applications for example they have emerged in the description of two photon processes in hydrogen hard hexagon statistical mechanical models and clebsch gordan expansions for unitary representations of the lorentz group so 3 1 in this paper it is shown that there is a simple and elegant way to construct these polynomials using the heisenberg algebra

the hahn and meixner polynomials of an imaginary argument and some of their applications the hahn and meixner polynomials belonging to the classical orthogonal polynomials of a discrete variable are analytically continued in the complex plane both in variable and parameter this leads to the origination of two systems of real polynomials orthogonal with respect to a continuous measure the meixner polynomials of an imaginary argument obtained in this manner turned out to be known in the literature as the pollaczek polynomials the orthogonality relation for the hahn polynomials with respect to a continuous measure is apparently new a close connection between the hahn polynomials of an imaginary argument and representations of the lorentz group so 3 1 is considered

underdetermined systems of partial differential equations this paper examines underdetermined systems of partial differential equations in which the independent variables may be classical c numbers or even quantum operators one can view an underdetermined system as expressing the kinematic constraints on a set of dynamical variables that generate a lie algebra the arbitrariness in the general solution reflects the freedom to specify the dynamics of such a system

quantum discrete dubrovin equations the discrete equations of motion for the quantum mappings of kdv type are given in terms of the sklyanin variables which are also known as quantum separated variables both temporal discrete time evolutions and spatial along the lattice at a constant time level evolutions are considered in the classical limit the temporal equations reduce to the classical discrete dubrovin equations as given in a previous publication nijhoff f w 2000 chaos solitons fractals 11 19 28 the reconstruction of the original dynamical variables in terms of the sklyanin variables is also achieved

phase uniqueness for the mallows measure on permutations for a positive number q the mallows measure on the symmetric group is the probability measure on sn such that pn q is proportional to q to the power inv where inv equals the number of inversions inv equals the number of pairs i j one may consider this as a mean field model from statistical mechanics the weak large deviation principle may replace the gibbs variational principle for characterizing equilibrium measures in this sense we prove the absence of phase transition i e phase uniqueness

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

large deviations for exchangeable random vectors satisfies a large deviation principle with rate function a x inf a6 x o e s u 1 where s 4 is the support of the mixing measure 4 if x1 x2 is a sequence of i i d random vectors xn the corresponding sequence of sample means and pown po x1 then pon is exponentially continuous if the classical rate function ar v is jointly lower semicontinuous and a uniform integrability condition introduced by de acosta is satisfied these results are applied in section 4 to derive a large deviation theory for exchangeable random variables the resulting rate functions are typically nonconvex if the parameter space e is not compact then examples can be constructed where a full large deviation principle is not satisfied because the upper bound fails for a noncompact set

expansion formula for fractional derivatives in variational problems abstract we modify the expansion formula introduced in t m atanackovic b stankovic an expansion formula for fractional derivatives and its applications fract calc appl anal 7 3 2004 365 378 for the left riemann liouville fractional derivative in order to apply it to various problems involving fractional derivatives as a result we obtain a new form of the fractional integration by parts formula with the benefit of a useful approximation for the right riemann liouville fractional derivative and derive a consequence of the fractional integral inequality 0 t y 0 d t y d t 0 further we use this expansion formula to transform fractional optimization minimization of a functional involving fractional derivatives to the standard constrained optimization problem it is shown that when the number of terms in the approximation tends to infinity solutions to the euler lagrange equations of the transformed problem converge in a weak sense to solutions of the original fractional euler lagrange equations an illustrative example is treated numerically

fractional variational calculus in terms of riesz fractional derivatives this paper presents extensions of traditional calculus of variations for systems containing riesz fractional derivatives rfds specifically we present generalized euler lagrange equations and the transversality conditions for fractional variational problems fvps defined in terms of rfds we consider two problems a simple fvp and an fvp of lagrange results of the first problem are extended to problems containing multiple fractional derivatives functions and parameters and to unspecified boundary conditions for the second problem we present lagrange type multiplier rules for both problems we develop the euler lagrange type necessary conditions which must be satisfied for the given functional to be extremum problems are considered to demonstrate applications of the formulations explicitly we introduce fractional momenta fractional hamiltonian fractional hamilton equations of motion fractional field theory and fractional optimal control the formulations presented and the resulting equations are similar to the formulations for fvps given in agrawal 2002 j math anal appl 272 368 2006 j phys a math gen 39 10375 and to those that appear in the field of classical calculus of variations these formulations are simple and can be extended to other problems in the field of fractional calculus of variations

discrete time quantum mechanics abstract this paper summarizes a research program that has been underway for a decade the objective is to find a fast and accurate scheme for solving quantum problems which does not involve a monte carlo algorithm we use an alternative strategy based on the method of finite elements we are able to formulate fully consistent quantum mechanical systems directly on a lattice in terms of operator difference equations one advantage of this discretized formulation of quantum mechanics is that the ambiguities associated with operator ordering are eliminated furthermore the scheme provides an easy way in which to obtain the energy levels of the theory numerically a generalized version of this discretization scheme can be applied to quantum field theory problems the difficulties normally associated with fermion doubling are eliminated also one can incorporate local gauge invariance in the finite element formulation results for some field theory models are summarized in particular we review the calculation of the anomaly in two dimensional quantum electrodynamics the schwinger model finally we discuss nonabelian gauge theories

continuous hahn polynomials a slightly more general orthogonality relation for the hahn polynomials of a continuous variable than the recent one given by atakishiyev and suslov 1985 is given here

large deviations for trapped interacting brownian particles and paths are defined in terms of transformed path measures on finite time intervals under a trap hamiltonian and two respective pair interaction hamiltonians the first pair interaction exhibits a particle repellency while the second one imposes a path repellency we analyze both models in the limit of diverging time with fixed number n of brownian motions in particular we prove large deviations principles for the normalized occupation measures the minimizers of the rate functions are related to a certain associated operator the hamilton operator for a system of n interacting trapped particles more precisely in the particle repellency model the minimizer is its ground state and in the pathrepellency model the minimizers are its ground product states in the case of path repellency we also discuss the case of a dirac type interaction which is rigorously defined in terms of brownian intersection local times we prove a large deviation result for a discrete variant of the model this study is a contribution to the search for a mathematical formulation of the quantum system of n trapped interacting bosons as a model for bose einstein condensation motivated by the success of the famous 1995 experiments recently lieb et al described the large n behavior of the ground state in terms of the well known gross pitaevskii formula involving the scattering length of the pair potential we prove that the large n behavior of the ground product states is also described by the gross pitaevskii formula however with the scattering length of the pair potential replaced by its integral

nonconservative lagrangian mechanics a generalized function approach we reexamine the problem of having nonconservative equations of motion arise from the use of a variational principle in particular a formalism is developed that allows the inclusion of fractional derivatives this is done within the lagrangian framework by treating the action as a volterra series it is then possible to derive two equations of motion one of these is an advanced equation and the other is retarded

nonconservative lagrangian and hamiltonian mechanics traditional lagrangian and hamiltonian mechanics cannot be used with nonconservative forces such as friction a method is proposed that uses a lagrangian containing derivatives of fractional order a direct calculation gives an euler lagrange equation of motion for nonconservative forces conjugate momenta are defined and hamilton s equations are derived using generalized classical mechanics with fractional and higher order derivatives the method is applied to the case of a classical frictional force proportional to velocity textcopyright 1996 the american physical society

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

brownian intersection local times upper tail asymptotics and thick points we equip the intersection of p independent brownian paths in r d d 2 with the natural measure l defined by projecting the intersection local time measure via one of the brownian motions onto the set of intersection points given a bounded domain u c r d we show that as a the probability of the event l u a decays with an exponential rate of a 1 p where is described in terms of a variational problem in the important special case when u is the unit ball in r 3 and p 2 we characterize in terms of an ordinary differential equation we apply our results to the problem of finding the hausdorff dimension spectrum for the thick points of the intersection of two independent brownian paths in r 3

a numerical scheme for two dimensional optimal control problems with memory effect a new formulation for multi dimensional fractional optimal control problems is presented in this article the fractional derivatives which are coming from the formulation of the problem are defined in the riemann liouville sense some terminal conditions are imposed on the state and control variables whose dimensions need not be the same a numerical scheme is described by using the grunwald letnikov definition to approximate the riemann liouville fractional derivatives the set of fractional differential equations which are obtained after the discretization of the time domain are solved within the grunwald letnikov approximation to obtain the state and the control variable numerically a two dimensional fractional optimal control problem is studied as an example to demonstrate the performance of the scheme

on a nonlinear distributed order fractional differential equation abstract we study the existence and the uniqueness of mild and classical solutions for a class of equations of the form y 2 t 0 2 y t d f y t such equations arise in distributed derivatives models of viscoelasticity and system identification theory we also formulate a variational principle for a more general equation based on a method of doubling of variables for such equations