entropy for non additive measures in continuous domains in a recent paper we introduced a definition of f divergence for non additive measures in this paper we use this result to give a definition of entropy for non additive measures in a continuous se

semidirect products and reduction in mechanics this paper shows how to reduce a hamiltonian system on the cotangent bundle of a lie group to a hamiltonian system in the dual of the lie algebra of a semidirect product the procedure simplifies unifies and extends work of greene guillemin holm holmes kupershmidt marsden morrison ratiu sternberg and others the heavy top compressible fluids magnetohydrodynamics elasticity the maxwell vlasov equations and multifluid plasmas are presented as examples starting with lagrangian variables our method explains in a direct way why semidirect products occur so frequently in examples it also provides a framework for the systematic introduction of clebsch or canonical variables

discrete hamilton jacobi theory we develop a discrete analogue of hamilton jacobi theory in the framework of discrete hamiltonian mechanics the resulting discrete hamilton jacobi equation is discrete only in time we describe a discrete analogue of jacobi s solution and also prove a discrete version of the geometric hamilton jacobi theorem the theory applied to discrete linear hamiltonian systems yields the discrete riccati equation as a special case of the discrete hamilton jacobi equation we also apply the theory to discrete optimal control problems and recover some well known results such as the bellman equation discrete time hjb equation of dynamic programming and its relation to the costate variable in the pontryagin maximum principle this relationship between the discrete hamilton jacobi equation and bellman equation is exploited to derive a generalized form of the bellman equation that has controls at internal stages

on the f divergence for non additive measures the f divergence evaluates the dissimilarity between two probability distributions defined in terms of the radon nikodym derivative of these two probabilities the f divergence generalizes the hellinger distance and the kullback leibler divergence among other divergence functions in this paper we define an analogous function for non additive measures we discuss them for distorted lebesgue measures and give examples examples focus on the hellinger distance

explicit lie poisson integration and the euler equations we give a wide class of lie poisson systems for which explicit lie poisson integrators preserving all casimirs can be constructed the integrators are extremely simple and have widespread applicability examples are the free rigid body a moment truncation and a new fast algorithm for the sine bracket truncation of the 2d euler equations

integrators for lie poisson or dynamical systems abstract we reformulate the ge marsden 1988 lie poisson integration algorithm in terms of algebra variables alone and show how to implement it for regular quadratic lie algebras to arbitrary order and for arbitrary lie algebras to first order we also describe two generalizations of ruth s method 1983 that can be used to generate very fast algorithms

existence of strong solutions and global attractors for the suspension bridge equations abstract in this paper we show the existence of strong solutions for the suspension bridge equations furthermore the existence of strong global attractors is investigated using a new semigroup scheme

a bifurcation method for solving multiple positive solutions to the boundary value problem of the henon equation on a unit disk three algorithms based on the bifurcation theory are proposed to compute the o 2 symmetric positive solutions to the boundary value problem of the henon equation on the unit disk taking l in the henon equation as a bifurcation parameter the symmetry breaking bifurcation point on the branch of the o 2 symmetric positive solutions is found via the extended systems finally other symmetric positive solutions are computed by the branch switching method based on the lyapunov schmidt reduction

fuzzy measure and probability distributions distorted probabilities this work studies fuzzy measures and their application to data modeling we focus on the particular case when fuzzy measures are distorted probabilities we analyze their properties and introduce a new family of measures m dimensional distorted probabilities the work finishes with the application of two dimensional distorted probabilities to a modeling problem results of the application of such fuzzy measure to data modeling using choquet integral are discussed

a generalized hellinger distance for choquet integral abstract in this paper we introduce a generalized hellinger distance between two monotone measures including a derivative with respect to monotone measures then some properties of this distance are studied we also obtain a lower bound for the generalized hellinger distance finally an error bound for the generalized hellinger distance on conjugates is studied

a note on derivatives of functions with respect to fuzzy measures this paper deals with the choquet integral on the non negative real line first it gives a representation of the choquet integral of a non negative continuous and increasing function with respect to a fuzzy measure next restricting fuzzy measures to a class of distorted lebesgue measures it considers choquet integral equations in order to solve choquet integral equations a concept of the derivatives of functions with respect to fuzzy measures is introduced for distorted lebesgue measures it is shown that choquet integral equations are formulated as volterra integral equations of the first kind the differentiability of functions with respect to fuzzy measures is also discussed it further shows a relation of a choquet integral equation with the abel integral equation finally this paper introduces simple differential equations with respect to fuzzy measures and gives their solutions

discrete versions of some classical integrable systems and factorization of matrix polynomials discrete versions of several classical integrable systems are investigated such as a discrete analogue of the higher dimensional force free spinning top euler arnold equations the heisenberg chain with classical spins and a new discrete system on the stiefel manifold the integrability is shown with the help of a lax pair representation which is found via a factorization of certain matrix polynomials the complete description of the dynamics is given in terms of abelian functions the flow becomes linear on a prym variety corresponding to a spectral curve the approach is also applied to the billiard problem in the interior of ann dimensional ellipsoid

equivariant symplectic difference schemes and generating functions abstract we show that all the known results on the preservation of first integrals by a symplectic difference scheme can be unified under the equivalence theorem of feng ge in particular when the equivalence theorem is applied to the first type of generating function on a lie group we obtain the lie poisson integrator

the jacobiator of nonholonomic systems and the geometry of reduced nonholonomic brackets in this paper we consider the hamiltonian formulation of nonholonomic systems with symmetries and study several aspects of the geometry of their reduced almost poisson brackets including the integrability of their characteristic distributions our starting point is establishing global formulas for the nonholonomic jacobiators before and after reduction which are used to clarify the relationship between reduced nonholonomic brackets and twisted poisson structures for certain types of symmetries generalizing the chaplygin case we obtain genuine poisson structures on the reduced spaces and analyze situations in which the reduced nonholonomic brackets arise by applying a gauge transformation to these poisson structures we illustrate our results with mechanical examples and in particular show how to recover several well known facts in the special case of chaplygin symmetries

analysis and computation for a class of semilinear elliptic boundary value problems in this paper with the help of super solutions and sub solutions we set up a general framework and get a positive threshold l for solution existence and non existence of a class of semilinear elliptic dirichlet boundary value problems moreover a result on multiplicity is obtained when l is large enough we also give a numerical method to solve and visualize the positive solutions of the problem theoretical results are illustrated by numerical simulation

a refined holder s inequality for choquet expectation by cauchy schwarz s inequality abstract in this paper we first introduce a refined holder s inequality in choquet calculus then by this inequality we show that cauchy schwarz inequality and holder s inequality for choquet expectation are equivalent when the monotone measure is submodular or the integrands are comonotonic thus closing the series of papers in this literature note that it is obvious that holder s inequality implies cauchy schwarz s inequality but the converse is an interesting subject this paper focuses on this subject

existence of strong solutions and global attractors for the coupled suspension bridge equations abstract in this paper we show the existence of the strong solutions for the coupled suspension bridge equations furthermore existence of the strong global attractors is investigated using a new semigroup scheme since the solutions of the coupled equation have no higher regularity and the semigroup associated with the solutions is not continuous in the strong hilbert space the results are new and appear to be optimal

a modified kullback leibler divergence for non additive measures based on choquet integral abstract the kullback leibler divergence is a very important concept in statistics and probability which helps us in many problems of information systems the main property of the kullback leibler divergence is non negativity the study of kullback leibler divergence for choquet integral has been led to a challenging problem torra 2017 1 introduced the definition of the kullback leibler divergence for two non additive measures including choquet integral and derivative with respect to non additive measures this paper focuses on the non negativity of kullback leibler divergence for two non additive measures and shows that it is not always non negative in general this reason permits us to introduce new versions of kullback leibler divergence and f divergence for choquet integral which are always non negative we also focus on the lower bounds of f divergence for non additive measures and improve some previous results appearing in the literature

finite dimensional lie poisson approximations to vlasov poisson equations many of the basic equations of conservative continuum mechanics euler vlasov poisson vlasov maxwell mhd etc are hamiltonian systems with respect to lie poisson brackets on dual spaces of infinite dimensional lie algebras the development of lie poisson integrators for finite dimensional lie poisson systems has shown that they are superior in the numerical simulations of these systems especially with regard to long term phenomena this paper shows how to truncate one of these systems the vlasov poisson equation of plasma physics to a finite dimensional lie poisson system this requires replacing the functions on single particle phase space with their poisson bracket lie algebra structure by a finite dimensional lie algebra replacing the densities by their moments of order up to k about a fixed reference point corresponds to replacing the functions by their taylor expansions up to order k unfortunately these truncated taylor expansions do not form a lie algebra since the functions which vanish through order k do not form an ideal under poisson bracket geometrically this corresponds to the fact that canonical transformations which fix the reference point do not form a normal subgroup introducing the location of a reference point in phase space as an extra variable and truncating with respect to this moving point turns out to decouple the location from shape coordinates of a lump of density in phase space as far as the poisson bracket is concerned one can then replace the shape coordinates by a finite number of moments the central result of the paper is a construction of the decoupling map described above in the general context of the decomposition of a lie group as a product of subgroups the main theorem is first proved by the general theory of poisson reduction then by explicit calculation and lastly by showing that the poisson isomorphism follows from the lift of a natural groupoid isomorphism the groupoid aspect of the theory also provides natural poisson maps useful in the application of ruth type integration techniques which do not seem easily derivable from the general theory of poisson reduction 1994 john wiley sons inc

on nonlinear oscillations in a suspension bridge system in this paper we study nonlinear oscillations in a suspension bridge system governed by two coupled nonlinear partial differential equations by applying the leray schauder degree theory it is proved that the suspension bridge system has at least two solutions one is a near equilibrium oscillation and the other is a large amplitude oscillation

boundary element monotone iteration scheme for semilinear elliptic partial differential equations the monotone iteration scheme is a constructive method for solving a wide class of semilinear elliptic boundary value problems with the availability of a supersolution and a subsolution the iterates converge monotonically to one or two solutions of the nonlinear pde however the rates of such monotone convergence cannot be determined in general in addition when the monotone iteration scheme is implemented numerically through the boundary element method error estimates have not been analyzed in earlier studies in this paper we formulate a working assumption to obtain an exponentially fast rate of convergence this allows a margin 6 for the numerical implementation of boundary elements within the range of monotone convergence we then interrelate several approximate solutions and use the aubin nitsche lemma and the triangle inequalities to derive error estimates for the galerkin boundary element iterates with respect to the h r 0 r 2 sobolev space norms such estimates are of optimal order furthermore as a peculiarity we show that for the nonlinearities that are of separable type higher than optimal order error estimates can be obtained with respect to the mesh parameter h several examples of semilinear elliptic partial differential equations featuring different situations of existence nonexistence uniqueness multiplicity and stability are discussed computed and the graphics of their numerical solutions are illustrated

fourth order symplectic integration abstract in this paper we present an explicit fourth order method for the integration of hamilton s equations this method preserves the property that the time evolution of such a system yields a canonical transformation from the initial conditions to the final state that is the integration step is an explicit symplectic map although the result is first derived for a specific type of hamiltonian it is shown to be quite general in particular the results can be applied to any lie group

on the geometry of the hamilton jacobi equation and generating functions in this paper we develop a geometric version of the hamilton jacobi equation in the poisson setting specifically we geometrize what is usually called a complete solution of the hamilton jacobi equation we use some well known results about symplectic groupoids in particular cotangent groupoids as a keystone for the construction of our framework our methodology follows the ambitious program proposed by weinstein in mechanics day waterloo on 1992 volume 7 of fields institute communications american mathematical society providence 1996 in order to develop geometric formulations of the dynamical behavior of lagrangian and hamiltonian systems on lie algebroids and lie groupoids this procedure allows us to take symmetries into account and as a by product we recover results from channell and scovel phys d 50 1 80 88 1991 ge indiana univ math j 39 3 859 876 1990 ge and marsden phys lett a 133 3 134 139 1988 but even in these situations our approach is new a theory of generating functions for the poisson structures considered here is also developed following the same pattern solving a longstanding problem of the area how to obtain a generating function for the identity transformation and the nearby poisson automorphisms of poisson manifolds a direct application of our results gives the construction of a family of poisson integrators that is integrators that conserve the underlying poisson geometry these integrators are implemented in the paper in benchmark problems some conclusions current and future directions of research are shown at the end of the paper

search extension method for multiple solutions of a nonlinear problem abstract based on the eigensystem j j of multiple solutions of nonlinear problem u u 3 0 min u 0 on are approximated a new method called the search extension method sem composed of three steps is proposed numerical examples in a square and the unit circle are presented to show the efficiency of our method

choquet functional and it s applications in information theory abstract in this paper we first focus on a general functional satisfying only monotonicity positive homogeneity and translation invariance axioms which includes several well known concepts such as choquet integral natural risk statistics and owa risk statistics then a general version of jensen s inequality for this functional is studied as an application we also present some important results to f divergence for non additive measures finally the lower bounds of variation and hellinger distances between two monotone measures are given

entropy of discrete choquet capacities we introduce a measure of entropy for any discrete choquet capacity and we interpret it in the setting of aggregation by the choquet integral

lagrangian submanifolds and dynamics on lie algebroids in some previous papers a geometric description of lagrangian mechanics on lie algebroids has been developed in this topical review we give a hamiltonian description of mechanics on lie algebroids in addition we introduce the notion of a lagrangian submanifold of a symplectic lie algebroid and we prove that the lagrangian hamiltonian dynamics on lie algebroids may be described in terms of lagrangian submanifolds of symplectic lie algebroids the lagrangian hamiltonian formalism on lie algebroids permits us to deal with lagrangian hamiltonian functions not defined necessarily on tangent cotangent bundles thus we may apply our results to the projection of lagrangian hamiltonian functions which are invariant under the action of a symmetry lie group as a consequence we obtain that lagrange poincare hamilton poincare equations are the euler lagrange hamilton equations associated with the corresponding atiyah algebroid moreover we prove that lagrange poincare hamilton poincare equations are the local equations defining certain lagrangian submanifolds of symplectic atiyah algebroids

finite mode analogs of 2d ideal hydrodynamics coadjoint orbits and local canonical structure abstract the algebraic finite mode hydrodynamic type systems that have o n integrals of motion for o n n modes and are intrinsically connected with two dimensional ideal fluid flows are studied special emphasis is given on the structure of their phase space and the problem of finding proper dynamical variables for its description in connection with appropriate problems for the original hydrodynamic equations various possibilities for embedding in the space of canonical variables are demonstrated on the simplest example of the su 3 model giving the finite dimensional analogs of clebsch variables

a mountain pass method for the numerical solution of semilinear wave equations it is well known that periodic solutions of semilinear wave equations can be obtained as critical points of related functionals in the situation that we studied there is usually an obvious solution obtained as a solution of linear problem we formulate a dual variational problem in such a way that the obvious solution is a local minimum we then find additional non obvious solutions via a numerical mountain pass algorithm based on the theorems of ambrosetti rabinowitz and ekeland numerical results are presented

on the f divergence for discrete non additive measures abstract in this paper we study the definition of the f divergence and the hellinger distance for non additive measures in the discrete case as these measures are based on the derivatives of the measures we consider the problem of defining the radon nikodym derivative of a non additive measure while radon nikodym derivatives for additive measures exist for absolutely continuous measures this is not the case in the non additive case in this paper we will define set directional and upper lower and interval derivatives we will also define when two measures have the same sign these definitions will be used to introduce alternative definitions of the f divergence all extending the classical definition to non additive measures

numerical mountain pass periodic solutions of a nonlinear spring equation abstract in this paper we show how the introduction of a nonlinear term in the classic spring model can produce dramatic results we compute a large amplitude solution which is drastically different from the known linear small amplitude solution a dual variational formulation is given recasting the problem as one in which saddle points correspond to solutions of the differential equation our computations are based on the numerical mountain pass algorithm developed by choi and mckenna which was inspired by the theorems of ambrosetti rabinowitz and ekeland

the momentum map in poisson geometry every action on a poisson manifold by poisson diffeomorphisms lifts to a hamiltonian action on its symplectic groupoid which has a canonically defined momentum map we study various properties of this momentum map as well as its use in reduction

random attractors for the coupled suspension bridge equations with white noises the paper is devoted to the investigation of the existence of a compact random attractor for the random dynamical system generated by the coupled suspension bridge equations with white noises

random attractors for the extensible suspension bridge equation with white noise this paper is devoted to the dynamical behavior of a stochastic extensible suspension bridge equation with white noise for a deterministic suspension bridge equation there are many classical results such as existence and uniqueness of a solution and long term behavior of solutions to the best of our knowledge the existence of attractors for the suspension bridge equation with white noise was not yet considered we intend to investigate these problems we first obtain the dissipativeness of a solution in higher energy spaces h 3 h 2 h 0 1 this implies that the random dynamical system generated by the equation has a random attractor in h 2 h 0 1 l 2 which is a tempered random set in the space in h 3 h 2 h 0 1

reduced dynamics and lagrangian submanifolds of symplectic manifolds in this paper we will see that the symplectic creed by weinstein everything is a lagrangian submanifold also holds for hamilton poincare and lagrange poincare reduction in fact we show that solutions of the hamilton poincare equations and of the lagrange poincare equations are in one to one correspondence with distinguished curves in a lagrangian submanifold of a symplectic manifold for this purpose we will combine the concept of a tulczyjew triple with marsden weinstein symplectic reduction

a new class of symplectic integration schemes based on generating functions we present a new family of one step symplectic integration schemes for hamiltonian systems of the general form dot y j 1 nabla h y t such a class of methods contains as particular cases the methods of miesbach and pesch numer math 61 501 521 1992 and also the family of symplectic runge kutta methods as in the case of the methods introduced in miesbach and pesch numer math 61 501 521 1992 the new integration methods are constructed by defining a generating function which automatically determines a symplectic map the resulting methods are implicit and require the evaluation of the gradient of the hamiltonian function as well as the hessian times a vector

multiple periodic solutions for suspension bridge systems abstract this paper studies the existence of multiple periodic oscillations of the suspension bridge system governed by the coupled nonlinear beam and wave equations by using the degree theory it is proved that the system has at least three periodic solutions

pseudospectral methods for computing the multiple solutions of the lane emden equation based on the liapunov schmidt reduction and symmetry breaking bifurcation theory we compute and visualize multiple solutions of the lane emden equation on a square and a disc using legendre and fourier legendre pseudospectral methods starting from the nontrivial solution branches of the corresponding nonlinear bifurcation problem we obtain multiple solutions of lane emden equation with various symmetries numerically numerical results demonstrate the effectiveness of these approaches

existence of global attractors for the coupled system of suspension bridge equations a mathematical model of the suspension bridge describes the vibration of the road bed in the vertical plain and that of the main cable we show the existence of an absorbing set for the solution of the problem furthermore the global attractors of the coupled system of suspension bridge are studied by a new semigroup approach

choquet calculus using operational matrices abstract in generalized probability theory choquet calculus on a real line including choquet integral equations and differential equations with respect to non additive measures has been recently discussed by sugeno torra and narukawa these studies were based on laplace transformation as a basic tool however in many problems it is not possible to provide laplace transformation on the other hand in many problems differential equations with respect to non additive measures often do not have exact solutions these reasons permit us to propose an alternative approach for numerical choquet calculus based on operational matrices which also readily gives the approximate numerical solution of choquet integral equations and differential equations with respect to non additive measures finally we present some illustrative examples for numerical choquet calculus based on operational matrices which the laplace transformation does not seem to be applicable in these cases

newton s method s basin of attraction for sign changing solutions of concave and convex nonlinearities abstract in this paper we will illustrate and visualize the chaotic behavior for solving the sign changing solutions of concave convex nonlinearities we propose contractor iteration algorithm and bifurcation method to improve the computation speed and avoid local convergence of newton raphson method by this method sign changing numerical solutions will be computed and visualized in various domains

periodic solutions of a nonlinear suspension bridge equation with damping and nonconstant load abstract the paper is concerned with the existence of periodic solutions for the lazer mckenna suspension bridge equation with damping and nonconstant load by using the lyapunov schmidt reduction methods the author discuss the relationship between the sign changing solutions and the source terms the result answers partly the open problem in lazer and mckenna siam rev 32 1990 537 578

lie algebroid structures on a class of affine bundles we introduce the notion of a lie algebroid structure on an affine bundle whose base manifold is fibered over r it is argued that this is the framework which one needs for coming to a time dependent generalization of the theory of lagrangian systems on lie algebroids an extensive discussion is given of a way one can think of forms acting on sections of the affine bundle it is further shown that the affine lie algebroid structure gives rise to a coboundary operator on such forms the concept of admissible curves and dynamical systems whose integral curves are admissible brings an associated affine bundle into the picture on which one can define in a natural way a prolongation of the original affine lie algebroid structure

bifurcation method for computing the multiple positive solutions to p henon equation in this paper we proposed the algorithms to solving the d 4 symmetric positive solutions to the boundary value problem of p henon equation taking r in p henon equation as a bifurcation parameter the symmetry breaking bifurcation point on the branch of the d 4 symmetric positive solutions is found via the extended systems based on the liapunov schmidt reduction other symmetric positive solutions are computed by the branch switching method

gauge transformations twisted poisson brackets and hamiltonization of nonholonomic systems in this paper we study the problem of hamiltonization of nonholonomic systems from a geometric point of view we use gauge transformations by 2 forms in the sense of severa and weinstein in progr theoret phys suppl 144 145 154 2001 to construct different almost poisson structures describing the same nonholonomic system in the presence of symmetries we observe that these almost poisson structures although gauge related may have fundamentally different properties after reduction and that brackets that hamiltonize the problem may be found within this family we illustrate this framework with the example of rigid bodies with generalized rolling constraints including the chaplygin sphere rolling problem we also see through these examples how twisted poisson brackets appear naturally in nonholonomic mechanics

a high linking algorithm for sign changing solutions of semilinear elliptic equations where n is a bounded domain with regular boundary the operator can be replaced by a more general second order uniformly elliptical operator in divergence form the solutions we are interested in nding are of minimax type and sign changing nonlinear elliptic equations of type 1 1 arise naturally in physics engineering and mathematical biology it is known that many such equations have multiple solutions their study has attracted the attention of many pure and applied mathematicians during the past two decades let us assume that f satis es the following regularity and growth conditions a1 f x t is locally lipschitz continuous in