noether theorem for nonholonomic nonconservative mechanical systems in phase space on time scales the paper focuses on studying the noether theorem for nonholonomic nonconservative mechanical systems in phase space on time scales first the hamilton equations of nonholonomic nonconservative systems on time scales are established which is based on the lagrange equations for nonholonomic systems on time scales then based upon the quasi invariance of hamilton action of systems under the infinitesimal transformations with respect to the time and generalized coordinate on time scale the noether identity and the conserved quantity of nonholonomic nonconservative systems on time scales are obtained finally an example is presented to illustrate the application of the results

fractional variational problems depending on indefinite integrals abstract we obtain necessary optimality conditions for variational problems with a lagrangian depending on a caputo fractional derivative a fractional and an indefinite integral main results give fractional euler lagrange type equations and natural boundary conditions which provide a generalization of the previous results found in the literature isoperimetric problems problems with holonomic constraints and depending on higher order caputo derivatives as well as fractional lagrange problems are considered

generalizing the variational theory on time scales to include the delta indefinite integral we prove necessary optimality conditions of euler lagrange type for generalized problems of the calculus of variations on time scales with a lagrangian depending not only on the independent variable an unknown function and its delta derivative but also on a delta indefinite integral that depends on the unknown function such kinds of variational problems were considered by euler himself and have been recently investigated in j gregory generalizing variational theory to include the indefinite integral higher derivatives and a variety of means as cost variables methods appl anal 15 4 2008 427 435 our results not only provide a generalization to previous results but also give some other interesting optimality conditions as special cases

time scale differential integral and variational embeddings of lagrangian systems we introduce differential integral and variational delta embeddings we prove that the integral delta embedding of the euler lagrange equations and the variational delta embedding coincide on an arbitrary time scale in particular a new coherent embedding for the discrete calculus of variations that is compatible with the least action principle is obtained

picone type identities and definiteness of quadratic functionals on time scales in this paper we derive a new sufficient condition for the nonnegativity of time scale quadratic functionals associated to time scale symplectic systems to establish this result a new global picone formula is derived another proof of a special case of the result is shown to be obtained via a sturmian comparison technique furthermore we derive several new picone type identities which in particular do not impose a certain delta differentiability assumption and we survey known ones from the literature the results in this paper complete our earlier work on the definiteness of a time scale quadratic functional in terms of its corresponding time scale symplectic system

eigenvalue theory for time scale symplectic systems depending nonlinearly on spectral parameter abstract in this paper we develop the eigenvalue theory for general time scale symplectic systems in which the dependence on the spectral parameter is allowed to be nonlinear at the same time we do not impose any controllability or strict normality assumptions we prove the oscillation theorems for eigenvalue problems with dirichlet separated and jointly varying endpoints including the periodic boundary conditions we also allow the boundary conditions depending on the spectral parameter our new theory generalizes and unifies recently published results on continuous time linear hamiltonian systems and discrete symplectic systems with nonlinear dependence on and on time scale symplectic systems with linear dependence on the results of this paper are also new in the special case of linear hamiltonian systems with variable endpoints as well as for sturm liouville dynamic equations

a non differentiable noether s theorem in the framework of the nondifferentiable embedding of lagrangian systems defined by cresson and greff non dierentiable embedding of lagrangian systems and partial dierential equations preprint max plank institut fur mathematik in den naturwissenschaften leipzig 16 26 2010 we prove a noether s theorem based on the lifting of one parameter groups of diffeomorphisms

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

backward linear control systems on time scales we show how a linear control systems theory for the backward nabla differential operator on an arbitrary time scale can be obtained via caputo s duality more precisely we consider linear control systems with outputs defined with respect to the backward jump operator kalman criteria of controllability and observability as well as realisability conditions are proved

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

galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics error estimates for galerkin proper orthogonal decomposition pod methods for nonlinear parabolic systems arising in fluid dynamics are proved for the time integration the backward euler scheme is considered the asymptotic estimates involve the singular values of the pod snapshot set and the grid structure of the time discretization as well as the snapshot locations

a reduced order extrapolated crank nicolson finite spectral element method based on pod for the 2d non stationary boussinesq equations abstract in this paper we mainly utilize proper orthogonal decomposition pod to reduce the order for the coefficient vector of the classical crank nicolson finite spectral element ccnfse method of the two dimensional 2d non stationary boussinesq equations about vorticity stream functions so that the reduced order method maintains all the advantages of the ccnfse method toward this end we first establish a ccnfse format with the second order time accuracy for the two dimensional 2d non stationary boussinesq equations about vorticity stream functions and analyze the existence stability and convergence of the ccnfse solutions and then by pod we establish a reduced order extrapolated crank nicolson finite spectral element roecnfse method and analyze the existence stability and convergence of the roecnfse solutions as well as offer the flowchart for seeking roecnfse solutions finally we use two sets of numerical experiments to validate that the numerical computational consequences are accorded with the theoretical ones such that the effectiveness and feasibility of the roecnfse method are further verified both theory and method in this paper is new and completely different from the existing reduced order methods

hamilton jacobi theory over time scales and applications to linear quadratic problems abstract in this paper we first derive the verification theorem for nonlinear optimal control problems over time scales that is we show that the value function is the only solution of the hamilton jacobi equation in which the minimum is attained at an optimal feedback controller applications to the linear quadratic regulator problem lqr problem gives a feedback optimal controller form in terms of the solution of a generalized time scale riccati equation and that every optimal solution of the lqr problem must take that form a connection of the newly obtained riccati equation with the traditional one is established problems with shift in the state variable are also considered as an important tool for the latter theory we obtain a new formula for the chain rule on time scales finally the corresponding lqr problem with shift in the state variable is analyzed and the results are related to previous ones

a reduced order extrapolated finite difference iterative scheme based on pod method for 2d sobolev equation abstract in this study we devote ourselves to the reduced order extrapolated finite difference iterative roefdi modeling and analysis for the two dimensional 2d sobolev equation to this end we first establish the reduced order extrapolated finite difference iterative roefdi scheme holding sufficiently high accuracy but containing very few degrees of freedom for the 2d sobolev equation via the proper orthogonal decomposition pod technique and then we analyze the stability and convergence of the roefdi solutions finally we use the numerical experiments to verify the feasibility and effectiveness of the roefdi scheme

mixed finite element formulation and error estimates based on proper orthogonal decomposition for the nonstationary navier stokes equations in this paper proper orthogonal decomposition pod is used for model reduction of mixed finite element mfe for the nonstationary navier stokes equations and error estimates between a reference solution and the pod solution of reduced mfe formulation are derived the basic idea of this reduction technique is that ensembles of data are first compiled from transient solutions computed equation system derived with the usual mfe method for the nonstationary navier stokes equations or from physics system trajectories by drawing samples of experiments and interpolation or data assimilation and then the basis functions of the usual mfe method are substituted with the pod basis functions reconstructed by the elements of the ensemble to derive the pod reduced mfe formulation for the nonstationary navier stokes equations it is shown by considering numerical simulation results obtained for the illustrating example of cavity flows that the error between pod solution of reduced mfe formulation and the reference solution is consistent with theoretical results moreover it is also shown that this result validates the feasibility and efficiency of the pod method

finding structure with randomness probabilistic algorithms for constructing approximate matrix decompositions low rank matrix approximations such as the truncated singular value decomposition and the rank revealing qr decomposition play a central role in data analysis and scientific computing this work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low rank matrix approximation these techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets this paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions these methods use random sampling to identify a subspace that captures most of the action of a matrix the input matrix is then compressed either explicitly or implicitly to this subspace and the reduced matrix is manipulated deterministically to obtain the desired low rank factorization in many cases this approach beats its classical competitors in terms of accuracy robustness and or speed these claims are supported by extensive numerical experiments and a detailed error analysis the specific benefits of randomized techniques depend on the computational environment consider the model problem of finding the k dominant components of the singular value decomposition of an m times n matrix i for a dense input matrix randomized algorithms require bigo mn log k floating point operations flops in contrast to bigo mnk for classical algorithms ii for a sparse input matrix the flop count matches classical krylov subspace methods but the randomized approach is more robust and can easily be reorganized to exploit multiprocessor architectures iii for a matrix that is too large to fit in fast memory the randomized techniques require only a constant number of passes over the data as opposed to bigo k passes for classical algorithms in fact it is sometimes possible to perform matrix approximation with a single pass over the data

a reduced finite volume element formulation and numerical simulations based on pod for parabolic problems a proper orthogonal decomposition pod method is applied to a usual finite volume element fve formulation for parabolic equations such that it is reduced to a pod fve formulation with lower dimensions and high enough accuracy the error estimates between the reduced pod fve solution and the usual fve solution are analyzed it is shown by numerical examples that the results of numerical computation are consistent with theoretical conclusions moreover it is also shown that the reduced pod fve formulation based on pod method is both feasible and highly efficient

some errors estimates for the box method we define and analyze several variants of the box method for discretizing elliptic boundary value problems in the plane our estimates show the error to be comparable to a standard galerkin finite element method using piecewise linear polynomials

the conjugate point condition for smooth control sets abstract in this paper we present for optimal control problems with smooth control constraints second order necessary conditions stated in terms of conjugate points and a riccati type equation we show explicitly how to construct a set of feasible directions and how to derive the corresponding accessory problem

hamiltonian systems on time scales linear and nonlinear hamiltonian systems are studied on time scales t we unify symplectic flow properties of discrete and continuous hamiltonian systems a chain rule which unifies discrete and continuous settings is presented for our so called alpha derivatives on generalized time scales this chain rule allows transformation of linear hamiltonian systems on time scales under simultaneous change of independent and dependent variables thus extending the change of dependent variables recently obtained by dosl and hilscher we also give the legendre transformation for nonlinear euler lagrange equations on time scales to hamiltonian systems on time scales

a reduced order extrapolation central difference scheme based on pod for two dimensional fourth order hyperbolic equations the reduced order extrapolating fd schemes are just developed by luo s research team since 2013 see 47 52 and a very new technique the reduced order extrapolation central difference scheme based on pod for two dimensional fourth order hyperbolic equations is first built by us we provide theoretical analysis and examples to show the advantage of our method this paper is concerned with establishing the reduced order extrapolation central difference roecd scheme based on proper orthogonal decomposition pod for two dimensional 2d fourth order hyperbolic equations for this purpose we first develop the classical central difference cd scheme for the 2d fourth order hyperbolic equations and analyze its stability and convergence then by making use of the pod method we build the roecd scheme with fewer degrees of freedom and sufficiently high accuracy and furnish the error estimates of the roecd solutions and the algorithm procedure for solving the roecd scheme finally we employ some numerical examples to confirm the correctness of theoretical conclusions this implies that roecd scheme is feasible and efficient for seeking the numerical solutions of the 2d fourth order hyperbolic equations

oscillation criteria for second order matrix dynamic equations on a time scale we obtain oscillation criteria for a second order self adjoint matrix differential equation on a measure chain in terms of the eigenvalues of the coefficient matrices and the graininess function we illustrate our results with some nontrivial examples

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

leitmann s direct method of optimization for absolute extrema of certain problems of the calculus of variations on time scales abstract the fundamental problem of the calculus of variations on time scales concerns the minimization of a delta integral over all trajectories satisfying given boundary conditions this includes the discrete time the quantum and the continuous classical calculus of variations as particular cases in this note we follow leitmann s direct method to give explicit solutions for some concrete optimal control problems on an arbitrary time scale

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

existence results for dynamic sturm liouville boundary value problems via variational methods abstract several conditions ensuring existence of solutions of a dynamic sturm liouville boundary value problem are derived variational methods are utilized in the proofs an example illustrating the main results is given

oscillation theorems and rayleigh principle for linear hamiltonian and symplectic systems with general boundary conditions the aim of this paper is to establish the oscillation theorems rayleigh principle and coercivity results for linear hamiltonian and symplectic systems with general boundary conditions i e for the case of separated and jointly varying endpoints and with no controllability normality and strong observability assumptions our method is to consider the time interval as a time scale and apply suitable time scales techniques to reduce the problem with separated endpoints into a problem with dirichlet boundary conditions and the problem with jointly varying endpoints into a problem with separated endpoints these more general results on time scales then provide new results for the continuous time linear hamiltonian systems as well as for the discrete symplectic systems this paper also solves an open problem of deriving the oscillation theorem for problems with periodic boundary conditions furthermore the present work demonstrates the utility and power of the analysis on time scales in obtaining new results especially in the classical continuous and discrete time theories

discrete optimal control the accessory problem and necessary optimality conditions abstract in this work we derive the accessory problem for a general nonlinear discrete optimal control problem we obtain necessary conditions for optimality nonnegativity of the second variation as the second variation is a discrete quadratic functional we proceed by deriving necessary conditions for the nonnegativity of such functionals a certain controllability or normality assumption is required

a reduced order extrapolation technique for solution coefficient vectors in the mixed finite element method for the 2d nonlinear rosenau equation abstract this study focuses mainly on a reduced order method for the coefficient vectors in the crank nicolson mixed finite element cnmfe solutions for the two dimensional nonlinear rosenau equation we first provide a cnmfe method for the rosenau equation and give relative theoretical results as well as rewrite the cnmfe method in matrix form we then produce two sets of proper orthogonal decomposition pod bases by means of the first few coefficient vectors of the cnmfe solutions and set up a reduced order extrapolating cnmfe roecnmfe model with the pod bases as well as discuss the existence and stability as well as errors of the roecnmfe solutions by means of the matrix tool resulting in the theoretical analysis becoming very succinct lastly we present some numerical experiments to validate the correctness of the theoretical results finding that the superiority of the roecnmfe method is further verified

convergence of finite volume schemes for poisson s equation on nonuniform meshes an error analysis of a family of cell center finite volume schemes for poisson s equation on cartesian product nonuniform meshes in two dimensions is presented optimal order error estimates are derived in the discrete h 1 norm under minimum smoothness requirements on the exact solution and without any additional assumption on the regularity of the mesh on quasi uniform meshes analogous estimates are obtained in the maximum norm

higher order hahn s quantum variational calculus abstract we prove a necessary optimality condition of euler lagrange type for quantum variational problems involving hahn s derivatives of higher order

low rank approximation and regression in input sparsity time we design a new distribution over m n matrices s so that for any fixed n d matrix a of rank r with probability at least 9 10 psaxp2 e 1 e paxp2 simultaneously for all x rd here m is bounded by a polynomial in re 1 and the parameter e 0 1 such a matrix s is called a subspace embedding furthermore sa can be computed in o nnz a time where nnz a is the number of nonzero entries of a this improves over all previous subspace embeddings for which computing sa required at least ndlog d time we call these s sparse embedding matrices using our sparse embedding matrices we obtain the fastest known algorithms for overconstrained least squares regression low rank approximation approximating all leverage scores and ep regression more specifically let b be an n 1 vector e 0 a small enough value and integers k p g 1 our results include the following regression the regression problem is to find d 1 vector x for which pax bpp l 1 e min xpax bpp for the euclidean case p e 2 we obtain an algorithm running in o nnz a o d3e 2 time and another in o nnz a log 1 e o d3 log 1 e time here o f e f log o 1 f for p 1 more generally we obtain an algorithm running in o nnz a log n o r e 1 c time for a fixed c low rank approximation we give an algorithm to obtain a rank k matrix k such that pa kpf 1 e p a akpf where ak is the best rank k approximation to a that is ak is the output of principal components analysis produced by a truncated singular value decomposition useful for latent semantic indexing and many other statistical problems our algorithm runs in o nnz a o nk2e 4 k3e 5 time leverage scores we give an algorithm to estimate the leverage scores of a up to a constant factor in o nnz a log n o r3 time

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

galerkin proper orthogonal decomposition methods for parabolic problems in this work error estimates for galerkin proper orthogonal decomposition pod methods for linear and certain non linear parabolic systems are proved the resulting error bounds depend on the number of pod basis functions and on the time discretization numerical examples are included

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

weak maximum principle and accessory problem for control problems on time scales abstract in this paper we derive the first and second variations for a nonlinear time scale optimal control problem with control and state endpoints equality constraints using the first variation a first order necessary condition for weak local optimality is obtained under the form of a weak maximum principle generalizing the dubois reymond lemma to the optimal control setting and time scales a second order necessary condition in terms of the accessory problem is derived by using the nonnegativity of the second variation at all admissible directions the control problem is studied under a controllability assumption and with or without the shift in the state variable these two forms of the problem are shown to be equivalent

first order conditions for generalized variational problems over time scales in this paper we derive the weak pontryagin principle for generalized optimal control problems over time scales three types of problems are considered namely i the problems involving the values of the state endpoints in the lagrangian and the dynamics ii the problems with an integral of the state in the lagrangian and the dynamics and iii the isoperimetric problems as special cases we obtain the first order optimality conditions for the corresponding calculus of variations problems which have been of a recent interest in the literature our method is based on transforming the generalized optimal control or calculus of variations problem into a traditional optimal control problem on time scales to which the known weak pontryagin principle can be applied

analysis of the cell centred finite volume method for the diffusion equation abstract this paper presents a rigorous theoretical analysis of the cell centred finite volume method for poisson s equation we review the traditional taylor series expansion technique which suggests that the cell centred method is inconsistent on nonuniform grids which is not confirmed by numerical experiments we then present an analysis of the method which confirms that the solution error does indeed reduce as the cell size is reduced this is supported by numerical calculations

a reduced order extrapolation algorithm based on sfve method and pod technique for non stationary stokes equations in this paper a reduced order extrapolation algorithm roea based on proper orthogonal decomposition pod technique and classical stabilized finite volume element sfve method for non stationary stokes equations is established the error estimates between the roea solutions and the classical sfve solutions and the implementation for solving roea are provided some numerical examples are used to verify that the results of numerical computation are consistent with theoretical conclusions moreover it is shown that roea based on sfve formulation and pod method is feasible and efficient for solving non stationary stokes equations

isogeometric analysis and proper orthogonal decomposition for parabolic problems we investigate the combination of isogeometric analysis iga and proper orthogonal decomposition pod based on the galerkin method for model order reduction of linear parabolic partial differential equations for the proposed fully discrete scheme the associated numerical error features three components due to spatial discretization by iga time discretization with the theta scheme and eigenvalue truncation by pod first we prove a priori error estimates of the spatial iga semi discrete scheme then we show stability and prove a priori error estimates of the space time discrete scheme and the fully discrete iga theta pod galerkin scheme numerical tests are provided to show efficiency and accuracy of nurbs based iga for model order reduction in comparison with standard finite element based pod techniques

nonlinear integro differential equations and optimal control problems on time scales this paper is mainly concerned with a class of optimal control problems of a system governed by the nonlinear integro differential equation on time scales using the contraction mapping principle the existence and uniqueness of a weak solution of the nonlinear integro differential equation on time scales are proved discussing l t 1 strong weak lower semicontinuity of integral functional on time scales we give sufficient conditions for the existence of optimal control by the virtue of the integration by parts formula on time scales the necessary conditions of optimality are derived

on the accuracy of the finite volume element method for diffusion equations on composite grids this paper develops the finite volume element fve method which is similar to the so called control volume finite element method but tailored to composite grid applications fve is described for the general two dimensional diffusion equation in divergence form o h 3 2 discretization error estimates are then developed for the case of triangular elements these results are improved to o h 2 using a special modification involving rectilinear elements at the grid interface

structural conditions for projection cost preservation via randomized matrix multiplication abstract projection cost preservation is a low rank approximation guarantee which ensures that the cost of any rank k projection can be preserved using a smaller sketch of the original data matrix we present a general structural result outlining four sufficient conditions to achieve projection cost preservation these conditions can be satisfied using tools from the randomized linear algebra literature

a necessary and sufficient condition for oscillation of the sturm liouville dynamic equation on time scales in this paper we investigate oscillatory properties of the second order sturm liouville dynamic equation on a time scale r t x c t x 0 using the so called trigonometric transformation we establish a necessary and sufficient condition for oscillation of this condition is then used to derive an explicit oscillation criterion for this equation

a self contained automated methodology for optimal flow control this paper describes a self contained automated methodology for active flow control which couples the time dependent navier stokes system with an adjoint navier stokes system and optimality conditions from which optimal states i e unsteady flow fields and controls e g actuators may be determined the problem of boundary layer instability suppression through wave cancellation is used as the intital validation case to test the methodology here the objective of control is to match the stress vector along a portion of the boundary to a steady base flow control is effected through the injection or suction of fluid through a single orifice on the boundary the results demonstrate that instability suppression can be achieved without any a priori knowledge of the flow unsteadiness such as frequencies instability type etc the present methodology has been extended to three dimensions and may potentially be applied to separation control relaminarization and turbulence control applications using one to many sensors and actuators

dynamic equations on time scales a survey the study of dynamic equations on time scales which goes back to its founder stefan hilger 1988 is an area of mathematics that has recently received a lot of attention it has been created in order to unify the study of differential and difference equations in this paper we give an introduction to the time scales calculus we also present various properties of the exponential function on an arbitrary time scale and use it to solve linear dynamic equations of first order several examples and applications among them an insect population model are considered we then use the exponential function to define hyperbolic and trigonometric functions and use those to solve linear dynamic equations of second order with constant coefficients finally we consider self adjoint equations and more generally so called symplectic systems and present several results on the positivity of quadratic functionals

double integral calculus of variations on time scales we consider a version of the double integral calculus of variations on time scales which includes as special cases the classical two variable calculus of variations and the discrete two variable calculus of variations necessary and sufficient conditions for a local extremum are established among them an analogue of the euler lagrange equation

symplectic difference systems variable stepsize discretization and discrete quadratic functionals discrete quadratic functionals with variable endpoints for variable stepsize symplectic difference systems are considered a comprehensive study is presented for characterizing the positivity of such functionals in terms of conjugate intervals conjoined bases and implicit and explicit riccati equations with various forms of boundary conditions moreover necessary conditions for the nonnegativity of these functionals are obtained in terms of the above notions furthermore we show that a variable stepsize discretization of a continuous time nonlinear control problem leads to a discrete linear quadratic problem and a hamiltonian difference system which are special cases of their symplectic counterparts

a reduced order crank nicolson finite volume element formulation based on pod method for parabolic equations in this paper a proper orthogonal decomposition pod method is used to deal with a classical crank nicolson finite volume element cnfve method for two dimensional parabolic equations a reduced order cnfve formulation with lower dimensions and sufficiently high accuracy based on pod technique is established the error estimates between reduced order cnfve solutions based on the pod method and classical cnfve solutions are provided and the extrapolation algorithm for solving reduced order cnfve formulation is implemented some numerical examples show that the results of numerical computation are consistent with previous theoretical conclusions moreover it is shown that the reduced order cnfve formulation based on pod method is feasible and efficient for solving two dimensional parabolic equations

a critical point approach for a second order dynamic sturm liouville boundary value problem with p laplacian abstract in this paper we give conditions guaranteeing the existence of at least three solutions for a second order dynamic sturm liouville boundary value problem involving two parameters in the proofs of the results we utilize critical point theory and variational methods in addition an example is given in order to illustrate our results

faster least squares approximation least squares approximation is a technique to find an approximate solution to a system of linear equations that has no exact solution in a typical setting one lets n be the number of constraints and d be the number of variables with n gg d then existing exact methods find a solution vector in o nd 2 time we present two randomized algorithms that provide accurate relative error approximations to the optimal value and the solution vector of a least squares approximation problem more rapidly than existing exact algorithms both of our algorithms preprocess the data with the randomized hadamard transform one then uniformly randomly samples constraints and solves the smaller problem on those constraints and the other performs a sparse random projection and solves the smaller problem on those projected coordinates in both cases solving the smaller problem provides relative error approximations and if n is sufficiently larger than d the approximate solution can be computed in o nd ln d time

fast monte carlo algorithms for matrices i approximating matrix multiplication motivated by applications in which the data may be formulated as a matrix we consider algorithms for several common linear algebra problems these algorithms make more efficient use of computational resources such as the computation time random access memory ram and the number of passes over the data than do previously known algorithms for these problems in this paper we devise two algorithms for the matrix multiplication problem suppose a and b which are m times n and n times p respectively are the two input matrices in our main algorithm we perform c independent trials where in each trial we randomly sample an element of 1 2 ldots n with an appropriate probability distribution cal p on 1 2 ldots n we form an m times c matrix c consisting of the sampled columns of a each scaled appropriately and we form a c times n matrix r using the corresponding rows of b again scaled appropriately the choice of cal p and the column and row scaling are crucial features of the algorithm when these are chosen judiciously we show that cr is a good approximation to ab more precisely we show that left ab cr right f o left a right f left b right f sqrt c where cdot f denotes the frobenius norm i e a 2 f sum i j a ij 2 this algorithm can be implemented without storing the matrices a and b in ram provided it can make two passes over the matrices stored in external memory and use o c m n p additional ram to construct c and r we then present a second matrix multiplication algorithm which is similar in spirit to our main algorithm in addition we present a model the pass efficient model in which the efficiency of these and other approximate matrix algorithms may be studied and which we argue is well suited to many applications involving massive data sets in this model the scarce computational resources are the number of passes over the data and the additional space and time required by the algorithm the input matrices may be presented in any order of the entries and not just row or column order as is the case in many applications where e g the data has been written in by multiple agents in addition the input matrices may be presented in a sparse representation where only the nonzero entries are written

generalizations of noether s theorem in classical mechanics in this paper a review is presented of various approaches to the generalization of the version of noether s theorem which is presented in most textbooks on classical mechanics its motivation is the controversy still persisting around the possible scope of a noether type theorem allowing for velocity dependent transformations our analysis is centered around the one factor common to all known treatments namely the structure of the related first integral we first discuss the most general framework in which a function of the above mentioned structure constitutes a first integral of a given lagrangian system and show that one cannot really talk about an interrelationship between symmetries and first integrals there we then compare different proposed generalizations of noether s theorem by describing the nature of the restrictions which characterize them when they are situated within the broadest framework we prove a seemingly new equivalence result between the two main approaches that of invaria

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

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

a reduced fve formulation based on pod method and error analysis for two dimensional viscoelastic problem abstract proper orthogonal decomposition pod method has been successfully used in the reduced order modeling of complex systems in this paper we extend the applications of pod method i e combine a classical finite volume element fve method with pod method to establish a reduced fve formulation with lower dimensions and sufficiently high accuracy for two dimensional viscoelastic problem with real practical applied background and analyze the errors between the reduced pod fve solution and the classical fve solution so as to provide scientific theoretic basis for service applications some numerical examples illustrate the fact that the results of numerical computation are consistent with theoretical conclusions moreover it is also shown that the reduced fve formulation based on pod method is feasible and efficient for solving two dimensional viscoelastic problem

new pod error expressions error bounds and asymptotic results for reduced order models of parabolic pdes the derivations of existing error bounds for reduced order models of time varying partial differential equations pdes constructed using proper orthogonal decomposition pod have relied on bounding the error between the pod data and various pod projections of that data furthermore the asymptotic behavior of the model reduction error bounds depends on the asymptotic behavior of the pod data approximation error bounds we consider time varying data taking values in two different hilbert spaces h and v with v subset h and prove exact expressions for the pod data approximation errors considering four different pod projections and the two different hilbert space error norms furthermore the exact error expressions can be computed using only the pod eigenvalues and modes and we prove the errors converge to zero as the number of pod modes increases we consider the pod error estimation approaches of kunisch and volkwein siam j numer anal 40 2002 pp 492 515 and chapelle gariah an

sturm liouville eigenvalue problems on time scales for sturm liouville eigenvalue problems on time scales with separated boundary conditions we give an oscillation theorem and establish rayleigh s principle our results not only unifly the corresponding theories for differential and difference equations but are also new in the discrete case

on existence of optimal control governed by a class of the first order linear dynamic systems on time scales in this paper by using the approximation of classical solution we introduce the definition of solution and prove the existence and uniqueness of solutions of the first order linear dynamic systems on time scales existence of lagrange optimal control problem governed by the first order linear dynamic systems on time scales is also presented for illustration some examples of optimal control problems on time scales are also discussed

a pod reduced order finite difference time domain extrapolating scheme for the 2d maxwell equations in a lossy medium abstract this paper is concerned with establishing a reduced order finite difference time domain fdtd extrapolating scheme based on proper orthogonal decomposition pod method for the two dimensional 2d maxwell equations in a lossy medium for this purpose we first review the classical fdtd scheme and associated results for the 2d maxwell equations in a lossy medium and then we formulate three sets of pod bases by means of the solutions obtained from the classical fdtd scheme on the very short time interval establish the pod reduced order fdtd extrapolating scheme and furnish the error estimates among the pod reduced order fdtd solutions and the classical fdtd solutions as well as the accuracy solution for the 2d maxwell equations in a lossy medium finally we enumerate two numerical examples to confirm that the pod reduced order fdtd extrapolating scheme is feasible and efficient for simulating the phenomena for the electric and magnetic fields in a lossy medium

nonshifted calculus of variations on time scales with differentiable σ abstract in calculus of variations on general time scales an euler lagrange equation of integral form is usually derived in order to characterize the critical points of nonshifted lagrangian functionals see e g ferreira et al 2011 13 in this paper we prove that the differentiability of the forward jump operator is a sharp assumption on the time scale in order to differentiate this integral euler lagrange equation this procedure leads to an euler lagrange equation of differential form furthermore from this differential form we prove a noether type theorem providing an explicit constant of motion for euler lagrange equations admitting a symmetry