nonnegativity of discrete quadratic functionals corresponding to symplectic difference systems we study the nonnegativity of quadratic functionals with separable endpoints which are related to the discrete symplectic system s in particular we characterize the nonnegativity of these functionals in terms of i the focal points of the natural conjoined basis of s and ii the solvability of an implicit riccati equation associated with s this result is closely related to the kernel condition for the natural conjoined basis of s we treat the situation when this kernel condition is possibly violated at a certain index to accomplish this goal we derive a new characterization of the set of admissible pairs sequences that does not require the validity of the above mentioned kernel condition finally we generalize our results to the variable stepsize case

coupled intervals for discrete symplectic systems in this paper we introduce strict coupled intervals for discrete symplectic systems and characterize in terms of the nonexistence of such coupled intervals the definiteness of the associated discrete quadratic functional with variable endpoints this strict coupled interval notion generalizes i the strict conjugate interval notion known for discrete variational problems with fixed right endpoint and ii the strict coupled interval notion known for the special case of linear hamiltonian systems the applicability of this theory of coupled intervals is clearly illustrated by a numerical example emanating from a nonlinear discrete control problem

discrete oscillation theorems for symplectic eigenvalue problems with general boundary conditions depending nonlinearly on spectral parameter abstract in this paper we establish new oscillation theorems for discrete symplectic eigenvalue problems with general boundary conditions we suppose that the symplectic coefficient matrix of the system and the boundary conditions are nonlinear functions of the spectral parameter and that they satisfy certain natural monotonicity assumptions in our new theory we admit possible oscillations in the coefficients of the symplectic system and the boundary conditions by incorporating their nonconstant rank with respect to the spectral parameter we also prove necessary and sufficient conditions for boundedness of the real part of spectrum of these eigenvalue problems

riccati equations for abnormal time scale quadratic functionals abstract this paper focuses on developing new riccati type conditions for an abnormal time scale symplectic system s these conditions provide characterizations of the nonnegativity with and without a certain image condition and positivity of the quadratic functionals associated with such a system the novelty of these conditions rely on the natural conjoined basis x a u a of s in which x a t is not necessarily invertible and thus the system s could be abnormal these results are new even in the special case of continuous time as are some of them in the discrete time setting

a fast algorithm for nonunitary joint diagonalization and its application to blind source separation a fast algorithm named complex valued fast frobenius diagonalization cvffdiag is proposed for seeking the nonunitary approximate joint diagonalizer of a given set of complex valued target matrices it adopts a multiplicative update to minimize the frobenius norm formulation of the approximate joint diagonalization problem at each of multiplicative iterations a strictly diagonally dominant updated matrix is obtained this scheme ensures the invertibility of the diagonalizer the cvffdiag relaxes several constraints on the target matrices and thus has much general applications furthermore the special approximation of the cost function the ingenious utilization of some structures and the adequate notation of concerned variables lead to the high computational efficiency of the proposed algorithm numerical simulations are conducted to illustrate good performances of the cvffdiag

riccati inequality and other results for discrete symplectic systems in this paper we establish several new results regarding the positivity and nonnegativity of discrete quadratic functionals f associated with discrete symplectic systems in particular we derive i the riccati inequality for the positivity of f with separated endpoints ii a characterization of the nonnegativity of f for the case of general jointly varying endpoints and iii several perturbation type inequalities regarding the nonnegativity of f with zero endpoints some of these results are new even for the special case of discrete hamiltonian systems

a residual a posteriori error estimator for the finite element solution of the helmholtz equation this paper suggests an expression of a residual based a posteriori error estimation for a galerkin finite element discretisation of the helmholtz operator applied in acoustics in the first part we summarise the finite element approach and the a priori estimates the new residual estimator is then formulated illustrated and tested on two one dimensional model problems with closed form solution

coupled points in the calculus of variations and applications to periodic problems the aim of this paper is to introduce the definition of coupled points for the problems of the calculus of variations with general boundary conditions and to develop second order necessary conditions for optimality when one of the end points is fixed our necessary conditions reduce to the known ones involving conjugate points we also apply our results to the periodic problems of the calculus of variations

coupled intervals in the discrete calculus of variations necessity and sufficiency in this work we study nonnegativity and positivity of a discrete quadratic functional with separately varying endpoints we introduce a notion of an interval coupled with 0 and hence extend the notion of conjugate interval to 0 from the case of fixed to variable endpoint s we show that the nonnegativity of the discrete quadratic functional is equivalent to each of the following conditions the nonexistence of intervals coupled with 0 the existence of a solution to riccati matrix equation and its boundary conditions natural strengthening of each of these conditions yields a characterization of the positivity of the discrete quadratic functional since the quadratic functional under consideration could be a second variation of a discrete calculus of variations problem with varying endpoints we apply our results to obtain necessary and sufficient optimality conditions for such problems this paper generalizes our recent work in r hilscher v zeidan comput math appl to appear where the right endpoint is fixed

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

time scale symplectic systems without normality abstract we present a theory of the definiteness nonnegativity and positivity of a quadratic functional f over a bounded time scale the results are given in terms of a time scale symplectic system s which is a time scale linear system that generalizes and unifies the linear hamiltonian differential system and discrete symplectic system the novelty of this paper resides in removing the assumption of normality in the characterization of the positivity of f and in establishing equivalent conditions for the nonnegativity of f without any normality assumption to reach this goal a new notion of generalized focal points for conjoined bases x u of s is introduced results on the piecewise constant kernel of x t are obtained and various picone type identities are derived under the piecewise constant kernel condition the results of this paper generalize and unify recent ones in each of the discrete and continuous time setting and constitute a keystone for further development in this theory

error estimate procedure in the finite element method and applications statically and kinematically admissible fields are explicitly derived from the finite element solution of the primal form of linear models the error on constitutive law for these fields yields an expression of the finite element error moreover the contribution of each element to this error allows to implement an automatic mesh refinement procedure leading to a uniform distribution of a given accuracy

recessive solutions for nonoscillatory discrete symplectic systems abstract in this paper we introduce a new concept of a recessive solution for discrete symplectic systems which does not require any eventual controllability assumption we prove that the existence of a recessive solution is equivalent to the nonoscillation of the system and that recessive solutions can have any rank between explicitly given lower and upper bounds the smallest rank corresponds to the minimal recessive solution which is unique up to a right nonsingular multiple while the largest rank yields the traditional maximal recessive solution we also present a method for constructing some but not all recessive solutions having a block diagonal structure from systems in lower dimension our results are new even for special discrete symplectic systems such as for even order sturm liouville difference equations and linear hamiltonian difference systems

linear hamiltonian difference systems disconjugacy and jacobi type conditions abstract we consider a linear hamiltonian difference system for the so called singular case so that discrete sturm liouville equations of higher order are included in our theory we introduce the concepts of focal points for matrix valued and generalized zeros for vector valued solutions of the system and define disconjugacy for linear hamiltonian difference systems we prove a reid roundabout theorem which gives conditions equivalent to positive definiteness of a certain discrete quadratic functional among them the strengthened jacobi s condition and a condition on a certain riccati difference equation the key to this theorem is a discrete version of picone s identity furthermore for the sake of generalization of our theorem we introduce controllability for linear hamiltonian difference systems and prove a reid roundabout theorem for a more general functional and more general boundary conditions

oscillation theorems for discrete symplectic systems with nonlinear dependence in spectral parameter abstract in this paper we open a new direction in the study of discrete symplectic systems and sturm liouville difference equations by introducing nonlinear dependence in the spectral parameter we develop the notions of finite eigenvalues and finite eigenfunctions and their multiplicities and prove the corresponding oscillation theorem for dirichlet boundary conditions the present theory generalizes several known results for discrete symplectic systems which depend linearly on the spectral parameter our results are new even for special discrete symplectic systems namely for sturm liouville difference equations symmetric three term recurrence equations and linear hamiltonian difference systems

a remark on focal points of recessive solutions of discrete symplectic systems abstract we investigate nonoscillatory and controllable symplectic difference systems we show that the recessive solution of such a system at has the same number of focal points counting multiplicities as the recessive solution at

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

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

on a weyl titchmarsh theory for discrete symplectic systems on a half line recently bohner and sun introduced basic elements of a weyl titchmarsh theory into the study of discrete symplectic systems we extend this development through the introduction of weyl titchmarsh functions together with a preliminary study of their properties a limit point criterion is described and characterized green s function for the half line is introduced as a limit of such functions in the regular case and half line solutions obtained are seen to satisfy lambda dependent boundary conditions at infinity

disconjugacy for linear hamiltonian difference systems abstract in this paper we study the linear hamiltonian difference system y t b t y t 1 c t z t z t a t y t 1 b t z t where a t c t are hermitian matrices with i b t and c t invertible i is identity disconjugacy criteria analogous to those for linear hamiltonian differential systems are obtained by a discrete riccati method

applications of time scale symplectic systems without normality recently the authors obtained new characterizations of the positivity and nonnegativity of a time scale quadratic functional f with separable endpoints related to a time scale symplectic system s in these results the assumption of normality is absent in this paper we present applications of such results namely without assuming normality we derive sturmian comparison theorems results for general jointly varying endpoints and characterizations of the positivity of f via the corresponding time scale riccati equation a certain perturbed quadratic functional and a time scale riccati inequality these results generalize and unify many recent as well as classical ones

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