lagrangian hamiltonian unified formalism for field theory the rusk skinner formalism was developed in order to give a geometrical unified formalism for describing mechanical systems it incorporates all the characteristics of lagrangian and hamiltonian descriptions of these systems including dynamical equations and solutions constraints legendre map evolution operators equivalence etc in this work we extend this unified framework to first order classical field theories and show how this description comprises the main features of the lagrangian and hamiltonian formalisms both for the regular and singular cases this formulation is a first step toward further applications in optimal control theory for partial differential equations

multivector fields and connections setting lagrangian equations in field theories the integrability of multivector fields in a differentiable manifold is studied then given a jet bundle j1e e m it is shown that integrable multivector fields in e are equivalent to integrable connections in the bundle e m that is integrable jet fields in j1e this result is applied to the particular case of multivector fields in the manifold j1e and connections in the bundle j1e m that is jet fields in the repeated jet bundle j1j1e in order to characterize integrable multivector fields and connections whose integral manifolds are canonical lifting of sections these results allow us to set the lagrangian evolution equations for first order classical field theories in three equivalent geometrical ways in a form similar to that in which the lagrangian dynamical equations of nonautonomous mechanical systems are usually given then using multivector fields we discuss several aspects of these evolution equations both for the regular and singular cases namely the existence and nonuniqueness o

a class of nonholonomic kinematic constraints in elasticity we propose a first example of a simple classical field theory with nonholonomic constraints our model is a straightforward modification of a cosserat rod based on a mechanical analogy we argue that the constraint forces should be modelled in a special way and we show how such a procedure can be naturally implemented in the framework of geometric field theory finally we derive the equations of motion and we propose a geometric integration scheme for the dynamics of a simplified model

euler lagrange and hamilton equations for non holonomic systems in field theory a generalization of the concept of a system of non holonomic constraints to fibred manifolds with n dimensional bases is considered motion equations in both lagrangian and hamiltonian settings for systems subjected to such constraints are investigated regularity conditions for the existence of a non holonomic legendre transformation and the corresponding formulae for hamiltonian and momenta are found in particular lagrangian constraints and semi holonomic constraints and simplifications arising in this case are discussed

ito type stochastic fuzzy differential equations with delay abstract in the paper we give some foundations for the studies of stochastic fuzzy delayed differential equations we prove the existence and uniqueness of solutions to such the equations to obtain our result we assume that the coefficients of the equation satisfy the lipschitz condition together with linear growth condition we estimate the distance between approximate solution and exact solution also the stability of solution with respect to the initial history is shown an application of stochastic fuzzy delayed differential equations in the modeling of population growth is indicated

on a new set valued stochastic integral with respect to semimartingales and its applications abstract we present a new approach to a concept of a set valued stochastic integral with respect to semimartingales such an integral called set valued stochastic up trajectory integral is compatible with the decomposition of the semimartingale some properties of this integral are stated we show applicability of the new integral in set valued stochastic integral equations driven by multidimensional semimartingales the uniqueness theorem is presented then we extend the notion of the set valued stochastic up trajectory integral to definition of a fuzzy stochastic up trajectory integral with respect to semimartingales a result on uniqueness of a solution to fuzzy stochastic integral equations incorporating the new fuzzy stochastic up trajectory integral driven by the multidimensional semimartingale is stated

covariant poisson brackets in geometric field theory we establish a link between the multisymplectic and the covariant phase space approach to geometric field theory by showing how to derive the symplectic form on the latter as introduced by crnkovic witten and zuckerman from the multisymplectic form the main result is that the poisson bracket associated with this symplectic structure according to the standard rules is precisely the covariant bracket due to peierls and dewitt

interconnection between set and fuzzy differential equations abstract the original formulation of fuzzy differential equations suffers from the disadvantage since the solutions increases as time increases in this paper employing the results of set differential equations obtained in v laksmikantham s leela a s vatsala setvalued hybrid differential equations and stability in terms of two measures j hybrid systems 2 2 2002 169 188 a new approach is suggested in order to capture the vagueness and the rich properties of solutions without fuzziness

the interrelation between stochastic differential inclusions and set valued stochastic differential equations abstract in this paper we connect the well established theory of stochastic differential inclusions with a new theory of set valued stochastic differential equations solutions to the latter equations are understood as continuous mappings taking on their values in the hyperspace of nonempty bounded convex and closed subsets of the space l 2 consisting of square integrable random vectors we show that for the solution x to a set valued stochastic differential equation corresponding to a stochastic differential inclusion there exists a solution x for this inclusion that is a l 2 continuous selection of x this result enables us to draw inferences about the reachable sets of solutions for stochastic differential inclusions as well as to consider the viability problem for stochastic differential inclusions

fixed final time constrained optimal control of nonlinear systems using neural network hjb approach in this paper fixed final time constrained optimal control laws using neural networks nns to solve hamilton jacobi bellman hjb equations for general affine in the constrained nonlinear systems are proposed an nn is used to approximate the time varying cost function using the method of least squares on a predefined region the result is an nn nearly constrained feedback controller that has time varying coefficients found by a priori offline tuning convergence results are shown the results of this paper are demonstrated in two examples including a nonholonomic system

a stacking theorem approach for fuzzy differential equations abstract this paper presents new existence results for fuzzy differential equations our analysis combines the stacking theorem with new solution set results for ordinary differential equations

multivector field formulation of hamiltonian field theories equations and symmetries we state the intrinsic form of the hamiltonian equations of first order classical field theories in three equivalent geometrical ways using multivector fields jet fields and connections thus these equations are given in a form similar to that in which the hamiltonian equations of mechanics are usually given then using multivector fields we study several aspects of these equations such as the existence and non uniqueness of solutions and the integrability problem in particular these problems are analysed for the case of hamiltonian systems defined in a submanifold of the multimomentum bundle furthermore the existence of first integrals of these hamiltonian equations is considered and the relation between cartan noether symmetries and general symmetries of the system is discussed noether s theorem is also stated in this context both the classical version and its generalization to include higher order cartan noether symmetries finally the equivalence between the lagrangian and hamiltonian formalisms is also discussed

remarks on unboundedness of set valued ito stochastic integrals abstract the paper deals with set valued stochastic integrals which are defined as a random multifunctions integrable boundedness of a such integral is one of the most important features in potential applications unfortunately up to now there were no correct proofs of such property surprisingly also a negative answer to this problem is still not explained correctly hence the problem seems to be still undetermined we shall show that in general the answer is negative we shall provide several relatively simple examples both in convex and nonconvex valued case as a consequence we will show that under some restrictions on the class of selections of the integrand the integrable boundedness of set valued ito s integral is equivalent to its single valuedness

on set valued stochastic integrals and fuzzy stochastic equations in this paper we extend the notion of set valued and fuzzy stochastic integrals to semimartingale integrators we present their main properties and finally we establish the existence of solutions of a fuzzy integral stochastic equation driven by a brownian motion the approach is based on the existence of solutions of an appropriately formulated martingale problem for a system of stochastic inclusions and the theorem of negoita and ralescu

fuzzy differential equation with completely correlated parameters abstract in this paper we study fuzzy differential equations with parameters and initial conditions interactive the interactivity is given by means of the concept of completely correlated fuzzy numbers we consider the problem in two different ways the first by using a family of differential inclusions in the second the extension principle for completely correlated fuzzy numbers is employed to obtain the solution of the model we conclude that the solutions of the fuzzy differential equations obtained by these two approaches are the same the solutions are illustrated with the radioactive decay model where the initial condition and the decay rate are completely correlated fuzzy numbers we also present an extension principle for completely correlated fuzzy numbers and we show that nguyen s theorem remains valid in this environment in addition we compare the solution via extension principle of the fuzzy differential equation when the parameters are non interactive fuzzy numbers and when the parameters are completely correlated fuzzy numbers finally we study the si epidemiological model in two forms first considering that the susceptible and infected individuals are completely correlated and then assuming that the transfer rate and the initial conditions are completely correlated

lagrangian systems with constraints a geometric approach to the method of lagrange multipliers a geometric approach to the method of lagrange multipliers is presented using the framework of the tangent bundle geometry the nonholonomic systems with constraint functions linear in the velocities are studied in the first place and then and using this study of the nonholonomic mechanical systems as a previous result the holonomic systems are considered the lagrangian inverse problem is also analysed and finally the theory is illustrated with several examples

existence and uniqueness for solutions to fuzzy stochastic differential equations driven by local martingales under the non lipschitzian condition abstract this paper makes a research into a class of fuzzy stochastic differential equations fsdes driven by a continuous local martingale under the non lipschitzian condition such equations can be useful in modelling of hybrid systems where the phenomena are subjected to two kinds of uncertainties randomness and fuzziness simultaneously the solutions of fsdes are the fuzzy stochastic processes and their uniqueness is considered to be in a strong sense thus the existence and uniqueness of solutions to fsdes under the non lipschitzian condition is first proven and the continuity of solutions to fsdes with respect to the initial data or the coefficients of the equations is investigated

a geometrical framework for the study of non holonomic lagrangian systems a geometrical framework is presented for the treatment of a class of dynamical systems which are modelled by a system of second order differential equations coupled with first order equations linear in the derivatives such problems in particular make their appearance in the study of lagrangian systems with non holonomic constraints among other things we discuss the concepts of symmetry and adjoint symmetry for such systems and identify for that purpose an appropriate notion of dynamical covariant derivative and jacobi endomorphism the intrinsic tools which are being developed further allow a direct geometrical construction of the dynamics of non holonomic systems the vertically rolling disc is chosen as an illustrative example for the newly proposed formalism and results

fuzzy set valued stochastic lebesgue integral this paper studies lebesgue integral of a fuzzy closed set valued stochastic process with respect to the time t firstly a progressively measurable fuzzy closed set valued stochastic process is discussed and an almost everywhere problem in the former aumann type lebesgue integral of the level set process is pointed out secondly a new definition of the lebesgue integral by decomposable closure is given focusing on aumann representation theorem representation theorem and property of convexity it is proved that the fuzzy closed set valued stochastic lebesgue integral is a fuzzy closed set valued stochastic process which is widely used in the fuzzy world with randomness finally the fuzzy closed set valued stochastic lebesgue integral in l p space is studied especially on an almost everywhere problem

a constrained backpropagation approach for the adaptive solution of partial differential equations this paper presents a constrained backpropagation cprop methodology for solving nonlinear elliptic and parabolic partial differential equations pdes adaptively subject to changes in the pde parameters or external forcing unlike existing methods based on penalty functions or lagrange multipliers cprop solves the constrained optimization problem associated with training a neural network to approximate the pde solution by means of direct elimination as a result cprop reduces the dimensionality of the optimization problem while satisfying the equality constraints associated with the boundary and initial conditions exactly at every iteration of the algorithm the effectiveness of this method is demonstrated through several examples including nonlinear elliptic and parabolic pdes with changing parameters and nonhomogeneous terms

geometry of multisymplectic hamiltonian first order field theories in the jet bundle description of field theories multisymplectic models in particular there are several choices for the multimomentum bundle where the covariant hamiltonian formalism takes place as a consequence several proposals for this formalism can be stated and on each one of them the differentiable structures needed for setting the formalism are obtained in different ways in this work we make an accurate study of some of these hamiltonian formalisms showing their equivalence in particular the geometrical structures canonical or not needed for the hamiltonian formalism are introduced and compared and the derivation of hamiltonian field equations from the corresponding variational principle is shown in detail furthermore the hamiltonian formalism of systems described by lagrangians is performed both for the hyper regular and almost regular cases finally the role of connections in the construction of hamiltonian field theories is clarified

artificial neural network method for solution of boundary value problems with exact satisfaction of arbitrary boundary conditions a method for solving boundary value problems bvps is introduced using artificial neural networks anns for irregular domain boundaries with mixed dirichlet neumann boundary conditions bcs the approximate ann solution automatically satisfies bcs at all stages of training including before training commences this method is simpler than other ann methods for solving bvps due to its unconstrained nature and because automatic satisfaction of dirichlet bcs provides a good starting approximate solution for significant portions of the domain automatic satisfaction of bcs is accomplished by the introduction of an innovative length factor several examples of bvp solution are presented for both linear and nonlinear differential equations in two and three dimensions error norms in the approximate solution on the order of 10 4 to 10 5 are reported for all example problems

on the geometry of non holonomic lagrangian systems we present a geometric framework for non holonomic lagrangian systems in terms of distributions on the configuration manifold if the constrained system is regular an almost product structure on the phase space of velocities is constructed such that the constrained dynamics is obtained by projecting the free dynamics if the constrained system is singular we develop a constraint algorithm which is very similar to that developed by dirac and bergmann and later globalized by gotay and nester special attention to the case of constrained systems given by connections is paid in particular we extend the results of koiller for caplygin systems an application to the so called non holonomic geometry is given

algebraic properties and topological properties of the quotient space of fuzzy numbers based on mares equivalence relation abstract in this paper we obtain some algebraic properties and topological properties of the quotient space of fuzzy numbers with respect to the equivalence relation defined by mares every fuzzy number has only one mares core and equivalent fuzzy numbers have the same mares core in addition equivalence classes of fuzzy numbers can be only expressed as the sum of its mares core and the set of symmetric fuzzy numbers which shows the notable difference between the equivalence classes of fuzzy numbers and the cosets of the normal subgroup in a group based on these results we introduce a new concept of convergence under which the quotient space is complete as an application of the main results it is shown that if we identify every fuzzy number with the corresponding equivalence class there would be more differentiable fuzzy functions than what is found in the literature

the nonholonomic variational principle a variational principle for mechanical systems and fields subject to nonholonomic constraints is found providing chetaev reduced equations as equations for extremals investigating nonholonomic variations of the chetaev type and their properties we develop foundations of the calculus of variations on constraint manifolds modelled as fibred submanifolds in jet bundles this setting is appropriate to study general first order nonlinear nonitegrable constraints that locally are given by a system of first order ordinary or partial differential equations we obtain an invariant constrained first variation formula and constrained euler lagrange equations both in intrinsic and coordinate forms and show that the equations are the same as chetaev equations without lagrange multipliers introduced recently by other methods we pay attention to two possible settings first when the constrained system arises from an unconstrained lagrangian system defined in a neighbourhood of the constraint and second more generally when an internal constrained system on the constraint manifold is given in the latter case a corresponding unconstrained system need not be a lagrangian nor even exist we also study in detail an important particular case nonholonomic constraints that can be alternatively modelled by means of co distributions in the total space of the fibred manifold in nonholonomic mechanics this happens whenever constraints affine in velocities are considered it becomes clear that and why if the distribution is completely integrable the constraints are semiholonomic the principle of virtual displacements holds and can be used to obtain the constrained first variational formula by a more or less standard procedure traditionally used when unconstrained or holonomic systems are concerned if however the constraint is nonintegrable no significant simplifications are available among others some properties of nonholonomic systems are clarified that without a deeper insight seem rather mysterious

smooth function approximation using neural networks an algebraic approach for representing multidimensional nonlinear functions by feedforward neural networks is presented in this paper the approach is implemented for the approximation of smooth batch data containing the function s input output and possibly gradient information the training set is associated to the network adjustable parameters by nonlinear weight equations the cascade structure of these equations reveals that they can be treated as sets of linear systems hence the training process and the network approximation properties can be investigated via linear algebra four algorithms are developed to achieve exact or approximate matching of input output and or gradient based training sets their application to the design of forward and feedback neurocontrollers shows that algebraic training is characterized by faster execution speeds and better generalization properties than contemporary optimization techniques

viability theory and fuzzy differential equations new results are presented which guarantee the existence of viable solutions to differential inclusions we show that a viable solution exists if i a tangency condition is satisfied and ii there exists a maximal solution to an appropriate differential equation in addition we discuss the topological structure of the solution set these results will then be used with the stacking theorem to establish new existence criteria for fuzzy differential equations if there exists a maximal solution for an appropriate set of differential equations

stochastic choice of basis functions in adaptive function approximation and the functional link net a theoretical justification for the random vector version of the functional link rvfl net is presented in this paper based on a general approach to adaptive function approximation the approach consists of formulating a limit integral representation of the function to be approximated and subsequently evaluating that integral with the monte carlo method two main results are 1 the rvfl is a universal approximator for continuous functions on bounded finite dimensional sets and 2 the rvfl is an efficient universal approximator with the rate of approximation error convergence to zero of order o c spl radic n where n is number of basis functions and with c independent of n similar results are also obtained for neural nets with hidden nodes implemented as products of univariate functions or radial basis functions some possible ways of enhancing the accuracy of multivariate function approximations are discussed

k cosymplectic manifolds and lagrangian field theories a geometrical description of classical field theories of first order is given the underlying k cosymplectic structure permits to derive the corresponding field equations

partial differential equations with differential constraints a geometric setting for constrained exterior differential systems on fibered manifolds with n dimensional bases is proposed constraints given as submanifolds of jet bundles locally defined by systems of first order partial differential equations are shown to carry a natural geometric structure called the canonical distribution systems of second order partial differential equations subjected to differential constraints are modeled as exterior differential systems defined on constraint submanifolds as an important particular case lagrangian systems subjected to first order differential constraints are considered different kinds of constraints are introduced and investigated lagrangian constraints constraints adapted to the fibered structure constraints arising from a co distribution semi holonomic constraints holonomic constraints

a tulczyjew triple for classical fields the geometrical structure known as the tulczyjew triple has proved to be very useful in describing mechanical systems even those with singular lagrangians or subject to constraints starting from basic concepts of the variational calculus we construct the tulczyjew triple for first order field theory the important feature of our approach is that we do not postulate ad hoc the ingredients of the theory but obtain them as unavoidable consequences of the variational calculus this picture of field theory is covariant and complete containing not only the lagrangian formalism and euler lagrange equations but also the phase space the phase dynamics and the hamiltonian formalism since the configuration space turns out to be an affine bundle we have to use affine geometry in particular the notion of the affine duality in our formulation the two maps and which constitute the tulczyjew triple are morphisms of double structures of affine vector bundles we also discuss the legendre transformation i e the transition between the lagrangian and the hamiltonian formulation of the first order field theory

revisiting fuzzy differential equations abstract we revisit the original formulation of fuzzy differential equations fdes and provide examples to show that the theory of fdes is richer than the corresponding theory of ordinary differential equations odes and therefore need not be rejected the alternative formulation in terms of differential inclusions does not always have advantage over odes as claimed we also discuss a stability result via lyapunov functions in this setup

neural network methods for boundary value problems with irregular boundaries partial differential equations pdes with boundary conditions dirichlet or neumann defined on boundaries with simple geometry have been successfully treated using sigmoidal multilayer perceptrons in previous works the article deals with the case of complex boundary geometry where the boundary is determined by a number of points that belong to it and are closely located so as to offer a reasonable representation two networks are employed a multilayer perceptron and a radial basis function network the later is used to account for the exact satisfaction of the boundary conditions the method has been successfully tested on two dimensional and three dimensional pdes and has yielded accurate results

covariant hamilton equations for field theory we study the relations between the equations of first order lagrangian field theory on fibre bundles and the covariant hamilton equations on the finite dimensional polysymplectic phase space of covariant hamiltonian field theory if a lagrangian is hyperregular these equations are equivalent a degenerate lagrangian requires a set of associated hamiltonian forms in order to exhaust all solutions of the euler lagrange equations the case of quadratic degenerate lagrangians is studied in detail

hamiltonian time dependent mechanics the usual formulation of time dependent mechanics implies a given splitting y r m of an event space y this splitting however is broken by any time dependent transformation including transformations between inertial frames the goal is the frame covariant formulation of time dependent mechanics on a bundle y r whose fibration y m is not fixed its phase space is the vertical cotangent bundle v y provided with the canonical 3 form and the corresponding canonical poisson structure an event space of relativistic mechanics is a manifold z whose fibration z r is not fixed

on set valued stochastic integrals in an m type 2 banach space in a separable banach space for set valued martingale several equivalent conditions based on the measurable selections are discussed and then in an m type 2 banach space at first we define single valued stochastic integral by the differential of a real valued brownian motion after that extend it to set valued case we prove that the set valued stochastic integral becomes a set valued submartingale which is different from single valued case and obtain the castaing representation theorem for the set valued stochastic integral which is applicable for set valued stochastic differential equations

the constraint algorithm for time dependent lagrangians the aim of this paper is to develop a constraint algorithm for time dependent lagrangian systems which permit us to solve the motion equations this algorithm extends the gotay and nester algorithm for autonomous lagrangians which is in fact a particular case to do this the almost stable tangent geometry of the evolution space and the notion of cosymplectic structure are used

set valued stochastic integral equations driven by martingales abstract we consider a notion of set valued stochastic lebesgue stieltjes trajectory integral and a notion of set valued stochastic trajectory integral with respect to martingale then we use these integrals in a formulation of set valued stochastic integral equations the existence and uniqueness of the solution to such the equations is proven as a generalization of set valued case results we consider the fuzzy stochastic trajectory integrals and investigate the fuzzy stochastic integral equations driven by bounded variation processes and martingales

regularity aspects and hamiltonization of non holonomic systems we discuss various aspects of the transition from lagrangian to hamiltonian equations for systems with general nonlinear non holonomic constraints the emphasis is first on constructing the reduced dynamics on the constraint submanifold and then trying to start a hamiltonization procedure from there we prove a theorem concerning the regularity which is required to obtain a unique second order dynamics on the constraint submanifold and we show that the same condition allows the transition to a hamiltonian picture throughout the analysis different degrees of generality are discussed

on the structure of solutions for fuzzy initial value problem in this paper by means of banach fixed point theorem we establish the existence and uniqueness of solutions for set differential equations the continuous dependence of solutions on initial values and the structural stability of solutions on the basis of these results we define level sets of solutions to fuzzy differential equations by the solutions of set differential equations and then we obtain the corresponding results for the large solutions of the fuzzy initial value problem fivp finally discuss the relationship between small solutions and large solutions of fivp

the polysymplectic hamiltonian formalism in field theory and calculus of variations i the local case on presente un formalisme hamiltonien geometrique invariant pour des problemes variationnels a integrale multiple et des theories de champ le formalisme est base sur la notion de forme polysymplectique qui est une generalisation a valeur vectorielle des formes symplectiques on etudie des equations hamiltoniennes des transformations canoniques des systemes de lagrange des symetries des applications en pulsion de theorie du champ et une classification des systemes de theorie du champ g homogenes sur une generalisation des orbites coadjointes

non standard connections in classical mechanics in the jet bundle description of first order classical field theories there are some elements such as the lagrangian energy and the construction of the hamiltonian formalism which require the prior choice of a connection bearing these facts in mind we analyse the situation in the jet bundle description of time dependent classical mechanics we prove that this connection dependence also occurs in this case although it is usually hidden by the use of the natural connection given by the trivial bundle structure of the phase spaces under consideration however we also prove that this dependence is dynamically irrelevant except where the dynamical variation of the energy is concerned in addition the relationship between first integrals and connections is shown for a sufficiently large class of lagrangians

chetaev versus vakonomic prescriptions in constrained field theories with parametrized variational calculus starting from a characterization of admissible chetaev and vakonomic variations in a field theory with constraints we show how the so called parametrized variational calculus can help to derive the vakonomic and the nonholonomic field equations we present an example in field theory where the nonholonomic method proved to be unphysical

jensen and chebyshev inequalities for pseudo integrals of set valued functions set valued functions are an important mathematical notion and play a crucial role in several practical areas at the same time pseudo analysis as a background allows extension of some classical mathematical notions to the forms that are highly applicable in some nonstandard situations this paper focuses on pseudo integration of set valued functions which is generalization of aumann s research and corresponding extensions of the jensen and chebyshev integral inequalities to the set valued case since the integral inequalities in question are widely used in various aspects of mathematics the main motivation for the presented research lies in the possibility of expanding the applicability of these inequalities by combining the properties of set valued functions with pseudo analysis

the gunther s formalism in classical field theory momentum map and reduction the polysymplectic formalism in local field theory developed by gunther j diff geom 25 23 1987 is revised a new approach and new results on momentum maps and reduction are given

mechanical systems with nonholonomic constraints a geometric setting for the theory of first order mechanical systems subject to general nonholonomic constraints is presented mechanical systems under consideration are not supposed to be lagrangian systems and the constraints are not supposed to be of a special form in the velocities as e g affine or linear a mechanical system is characterized by a certain equivalence class of 2 forms on the first jet prolongation of a fibered manifold the nonholonomic constraints are defined to be a submanifold of the first jet prolongation it is shown that this submanifold is canonically endowed with a distribution this distribution resp its vertical subdistribution has the meaning of generalized possible resp virtual displacements the concept of a constraint force is defined and a geometric version of the principle of virtual work is proposed from the principle of virtual work a formula for a workless constraint force is obtained a mechanical system subject nonholonomic constraints is modeled as a deformation of the original unconstrained system a direct characterization of a constrained system by means of a class of 2 forms along the canonical distribution is given and constrained equations of motion in an intrinsic form are found a geometric definition of regularity for systems under nonholonomic constraints is provided in particular the case of lagrangian systems is discussed also systems subject to holonomic constraints and nonholonomic constraints affine in the velocities are investigated within the range of the general scheme

k symplectic structures the basic properties of k symplectic structures are introduced and developed in analogy with the well known symplectic differential geometry examples of such structures related to k symplectic lie algebras are given

fuzzy differential equations abstract this paper deals with fuzzy set valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in rn we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation

non holonomic lagrangian and hamiltonian mechanics an intrinsic approach a geometrical approach to lagrangian and hamiltonian non holonomic dynamics is proposed the construction relies on a revisitation of the poincare cartan 1 form leading to the introduction of the concepts of lagrangian and hamiltonian pairs and to the implementation of a non holonomic legendre map the relationship with the standard extrinsic approach is outlined a unified canonical framework joining both lagrangian and hamiltonian aspects is proposed

extended legendre transformation approach to the time dependent hamiltonian formalism the hamiltonian formalism for time dependent systems is developed using vector fields defined in the extended cotangent bundle t q r the extended hamiltonian h is obtained using the extended legendre transformation associated to an extended lagrangian l previously defined in t q r and then the techniques of symplectic mechanics are employed in t q r to study the properties of the extended hamiltonian system finally this approach is related with the multisymplectic formalism and other extended approaches

on connections between stochastic differential inclusions and set valued stochastic differential equations driven by semimartingales abstract in the paper we study properties of solutions to stochastic differential inclusions and set valued stochastic differential equations with respect to semimartingale integrators we present new connections between their solutions in particular we show that attainable sets of solutions to stochastic inclusions are subsets of values of multivalued solutions of certain set valued stochastic equations we also show that every solution to stochastic inclusion is a continuous selection of a multivalued solution of an associated set valued stochastic equation the results obtained in the paper generalize results dealing with this topic known both in deterministic and stochastic cases

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

stochastic integrals of set valued processes and fuzzy processes we define the stochastic integrals of a set valued process and a fuzzy process with respect to a cylindrical brownian motion on a hilbert space we also give their properties which are useful for the study of fuzzy stochastic differential equations and stochastic differential inclusions with a set valued diffusion term

detection of cracks using neural networks and computational mechanics an inverse analysis method is proposed to simulate the a scan ultrasonic nondestructive testing by means of back propagation neural networks and computational mechanics both direct problem and inverse problem are considered in this study in the direct problem the frequency responses of a cracked medium subjected to an impact loading are calculated by the computational mechanics combining the finite element method with the boundary integral equation the transient responses are obtained using fast fourier transform in the inverse problem the back propagation neural networks are trained by the characteristic parameters extracted from the various surface responses obtained from the direct problem these surface responses carry a great deal of information about the structure of the medium with or without cracks the trained neural networks are then utilized for the classification and identification of the crack in the medium to determine the type location and length of the crack

set valued and fuzzy stochastic differential equations driven by semimartingales abstract in the paper we present set valued and fuzzy stochastic integrals with respect to semimartingale integrators as well as their main properties then we study the existence of solutions to set valued and fuzzy set valued stochastic differential equations driven by semimartingales the stability of solutions is also established

differentiable selections of multifunctions and their applications abstract in the paper we present some selection properties of differentiable multifunctions next we introduce the definition of a set valued stratonovich stochastic integral finally using selection properties of differentiable multifunctions we discuss the existence of weak solutions to stochastic inclusions with respect to such an integral

jet methods in nonholonomic mechanics classical nonholonomic mechanical systems are studied whose evolution space is a fiber bundle m r the framework is that of jet spaces in which the geometrical meaning of the theory emerges clearly for systems with linear constraints a new interpretation is given of appell s equations as first order differential equations associated with a suitable vector field the d alembert principle is formulated in an appropriate way to be generalized to systems with nonlinear constraints the equivalence between the equations of motion arising from this generalization the ones set up on gauss principle of least constraint and hertz s principle of least curvature is established

extended hamiltonian systems in multisymplectic field theories we consider hamiltonian systems in first order multisymplectic field theories we review the properties of hamiltonian systems in the so called restricted multimomentum bundle including the variational principle which leads to the hamiltonian field equations in an analogous way to how these systems are defined in the so called extended symplectic formulation of nonautonomous mechanics we introduce hamiltonian systems in the extended multimomentum bundle the geometric properties of these systems are studied the hamiltonian equations are analyzed using integrable multivector fields the corresponding variational principle is also stated and the relation between the extended and the restricted hamiltonian systems is established all these properties are also adapted to certain kinds of submanifolds of the multimomentum bundles in order to cover the case of almost regular field theories

hamiltonian multivector fields and poisson forms in multisymplectic field theory we present a general classification of hamiltonian multivector fields and of poisson forms on the extended multiphase space appearing in the geometric formulation of first order classical field theories this is a prerequisite for computing explicit expressions for the poisson bracket between two poisson forms

an alternative approach to the cartan form in lagrangian field theories an operator analogous to the almost tangent structure on a tangent bundle is defined on jet bundles of a fibred manifold this operator is used to construct a cartan form the construction is unique for first order lagrangians and is also unique when restricted to higher order mechanics

existence and interrelation between set and fuzzy differential equations abstract an existence result is proved for set differential equation when the function involved is upper semicontinuous and the result is very general then the connection between the solutions of fuzzy differential equation and the set differential equation that is generated from it is studied

set valued assessments of solutions to stochastic differential equations with random set parameters abstract we consider stochastic differential equations depending on parameters whose uncertainty is modeled by random compact sets several approaches are discussed how to construct set valued stochastic processes from the solutions the induced lower and upper probabilities are compared to a set valued set function and a set of probability measures constructed from the distributions of the solutions and the selections of the random set

representation theorems set valued and fuzzy set valued ito integral in this paper we shall present representation theorems of set valued martingales and set valued processes of finite variation with continuous time we shall also obtain a representation theorem of a predictable set valued stochastic process we shall give a new definition of ito integral of a set valued stochastic process with respect to a brownian motion based on the work e j jung j h kim on set valued stochastic integrals stochastic anal appl 21 2 2003 401 418 we shall also discuss some properties of set valued ito integral especially the presentation theorem of set valued ito integral finally we extend some of above results to the fuzzy set valued case

non holonomic lagrangian systems in jet manifolds a geometrical setting in terms of jet manifolds is developed for time dependent non holonomic lagrangian systems an almost product structure on the evolution space is constructed in such a way that the constrained dynamics is obtained by projection of the free dynamics a constrained poincare cartan 2 form is defined if the non holonomic system is singular a constraint algorithm is constructed special attention is devoted to caplygin systems and a reduction theorem is proved

constraints in hamiltonian time dependent mechanics the key point of the study of constraints in hamiltonian time dependent mechanics lies in the fact that a poisson structure does not provide dynamic equations and a poisson bracket of constraints with a hamiltonian is ill defined we describe hamiltonian dynamics in terms of hamiltonian forms and connections on the vertical cotangent bundle v q r seen as a momentum phase space a poisson bracket v on v q is induced by the canonical poisson bracket t on the cotangent bundle t q with v an algebra of first and second class time dependent constraints is described but we use the pull back of the evolution equation onto t x and the bracket t in order to extend the constraint algorithm to time dependent constraints the case of lagrangian constraints of a degenerate almost regular lagrangian is studied in detail one can assign to this lagrangian l a set of hamiltonian forms which are not necessarily degenerate such that any solution of the corresponding hamilton equations which lives in the lag

artificial neural networks for solving ordinary and partial differential equations we present a method to solve initial and boundary value problems using artificial neural networks a trial solution of the differential equation is written as a sum of two parts the first part satisfies the initial boundary conditions and contains no adjustable parameters the second part is constructed so as not to affect the initial boundary conditions this part involves a feedforward neural network containing adjustable parameters the weights hence by construction the initial boundary conditions are satisfied and the network is trained to satisfy the differential equation the applicability of this approach ranges from single ordinary differential equations ode to systems of coupled ode and also to partial differential equations pde in this article we illustrate the method by solving a variety of model problems and present comparisons with solutions obtained using the galerkin finite element method for several cases of partial differential equations with the advent of neuroprocessors and digital signal processors the method becomes particularly interesting due to the expected essential gains in the execution speed

hamiltonian systems on k cosymplectic manifolds the hamiltonian framework on symplectic and cosymplectic manifolds is extended in order to consider classical field theories to do this the notion of k cosymplectic manifold is introduced and a suitable hamiltonian formalism is developed so that the field equations for scalar and vector hamiltonian functions are derived