homoclinic solutions for a class of the second order hamiltonian systems abstract we study the existence of homoclinic orbits for the second order hamiltonian system q v q t q f t where q r n and v c 1 r r n r v t q k t q w t q is t periodic in t a map k satisfies the pinching condition b 1 q 2 k t q b 2 q 2 w is superlinear at the infinity and f is sufficiently small in l 2 r r n a homoclinic orbit is obtained as a limit of 2 kt periodic solutions of a certain sequence of the second order differential equations

existence of homoclinic orbits for a class of asymptotically p linear aperiodic p laplacian systems abstract by applying a variant version of mountain pass theorem in critical point theory the existence of homoclinic solutions is obtained for the following asymptotically p linear aperiodic p laplacian system d dt u t p 2 u t k t u t w t u t 0 where p 1 t r u r n k w c 1 r r n r are not periodic in t and w is asymptotically p linear at infinity

existence of homoclinic solutions for a class of second order hamiltonian systems abstract a new result for existence of homoclinic orbits is obtained for the second order hamiltonian systems under a class of new superquadratic conditions a homoclinic orbit is obtained as a limit of solutions of a certain sequence of boundary value problems which are obtained by the minimax methods

homoclinic solutions for nonautonomous second order hamiltonian systems with a coercive potential abstract a new result for the existence of homoclinic orbits is obtained for the second order hamiltonian system u t f t u t f t where t r u r n f c 1 r r n r is t periodic with respect to t t 0 and f r r n is a continuous and bounded function this result generalizes and improves some known results in the previous literature

existence of homoclinic solutions for a class of time dependent hamiltonian systems n 1 n n 2 where q s q q g r w g c r r r and l t g c r r 1 n is a positive definite symmetric matrix for all t g r the case where l t and w t q are either periodic in t or independent of t were studied by several authors see for instance ambrosetti and w x w x bertotti 1 coti zelati ekeland and sere 4 coti zelati and rabinowitz w x w x w x 5 rabinowitz 15 17 sere 19 and their references in this kind of problem the function l t plays an important role for example if l t s ai where a 0 and i denotes the identity matrix n n w x felmer and silva 8 have showed the existence of homoclinic orbits as the limit of subharmonic solutions provided that hs is autonomous see also w x w x rabinowitz and tanaka 18 serra tarallo and terracini 20 using an w x w x approach introduced by 4 see also 19 treated the hs problem

homoclinic orbits for second order hamiltonian systems possessing superquadratic potentials received by the editors november 27 1990 1991 mathematics subject classification primary 34c99 58e99 58f99 the first author was supported by ministero p i gruppo 40 calcolo delle variazioni the second author s research was sponsored in part by the u s army research office under contract daal03 87 k 0043 the national science foundation under grant mcs 8110556 and the office of naval research under grant n00014 88 ko134 any reproduction for the purpose of the united states govemment is permitted

infinitely many homoclinic orbits for the second order hamiltonian systems with general potentials abstract in this paper we study the following nonperiodic second order hamiltonian systems u t l t u t r t u t r where l t may not be uniformly positive definite for all t r under more general conditions on r u t u we prove that the above system has infinitely many homoclinic orbits provided r is even in u

estimation and control with bounded data uncertainties the paper describes estimation and control strategies for models with bounded data uncertainties we shall refer to them as bdu estimation and bdu control methods for brevity they are based on constrained game type formulations that allow the designer to explicitly incorporate into the problem statement a priori information about bounds on the sizes of the uncertainties in this way the effect of uncertainties is not unnecessarily over emphasized beyond what is implied by the a priori bounds consequently overly conservative designs as well as overly sensitive designs are avoided a feature of these new formulations is that geometric insights and recursive techniques which are widely known and appreciated for classical quadratic cost designs can also be pursued in this new framework also algorithms for computing the optimal solutions with the same computational effort as standard least squares solutions exist thus making the new formulations attractive for practical use moreover the framework is broad enough to encompass applications across several disciplines not just estimation and control examples will be given of a quadratic control design an h control design a total least square design image restoration image separation and co channel interference cancellation a major theme in this paper is the emphasis on geometric and linear algebraic arguments which lead to useful insights about the nature of the new formulations despite the interesting results that will be discussed several issues remain open and indicate potential future developments these will be briefly discussed

homoclinic orbits for a class of hamiltonian systems consider the second order hamiltonian system where q n and v c 1 n is t periodic in t suppose v q t 0 0 0 is a local maximum for v t and v t x x under these and some additional technical assumptions we prove that hs has a homoclinic orbit q emanating from 0 the orbit q is obtained as the limit as k of 2 kt periodic solutions i e subharmonics q k of hs the subharmonics q k are obtained in turn via the mountain pass theorem

homoclinic solutions for a class of second order hamiltonian systems abstract some new results for the existence of homoclinic orbits are obtained for the second order hamiltonian systems u t v t u t f t where t r u r n and v c 1 r r n r v t x k t x w t x is t periodic with respect to t t 0 and f r r n is a continuous and bounded function our assumptions on k and f are rather loose

homoclinics for asymptotically quadratic and superquadratic hamiltonian systems abstract in this paper we find new conditions which are different from those used in previous related studies to ensure the existence of infinitely many homoclinic orbits for the second order hamiltonian systems of the form q v q t q here we assume that v t q depends periodically on t and assume on q that v t q is asymptotically quadratic at q 0 and is as q either asymptotically quadratic or superquadratic as well as the new conditions

a subgradient solution to structured robust least squares problems a standard and established method for solving a least squares problem in the presence of a structured uncertainty is to assemble and solve a semidefinite programming sdp equivalent problem when the problem s dimensions are high the solution of the structured robust least squares rls problem via sdp becomes an expensive task in a computational complexity sense we propose a subgradient based solution that utilizes the minmax structure of the problem this algorithm is justified by danskin s minmax theorem and enjoys the well known convergence properties of the subgradient method the complexity of the new scheme is analyzed and its efficiency is verified by simulations of a robust equalization design

regularized constrained total least squares image restoration in this paper the problem of restoring an image distorted by a linear space invariant lsi point spread function psf that is not exactly known is formulated as the solution of a perturbed set of linear equations the regularized constrained total least squares rctls method is used to solve this set of equations using the diagonalization properties of the discrete fourier transform dft for circulant matrices the rctls estimate is computed in the dft domain this significantly reduces the computational cost of this approach and makes its implementation possible even for large images an error analysis of the rctls estimate based on the mean squared error mse criterion is performed to verify its superiority over the constrained total least squares ctls estimate numerical experiments for different errors in the psf are performed to test the rctls estimator objective and visual comparisons are presented with the linear minimum mean squared error lmmse and the regularized least squares rls estimator our experiments show that the rctls estimator reduces significantly ringing artifacts around edges as compared to the two other approaches

on remarkable properties of invariance superalgebras for the harmonic oscillator starting from the orthosymplectic superalgebra osp 3 4 the authors rigorously explain the correspondence between two superalgebras as invariance superalgebras of the one dimensional quantum harmonic oscillator i e osp 2 2 sh 2 2 and osp 3 2 the study of the corresponding root systems as well as their properties with respect to their even or odd characters gives them remarkable properties for the enclosed subsuperalgebras of osp 3 4 such properties and correspondences can be extended to the n dimensional context

the oscillator model for the lie superalgebra sh 2 2 and charlier polynomials we investigate an algebraic model for the quantum oscillator based upon the lie superalgebra sh 2 2 known as the heisenberg weyl superalgebra or the algebra of supersymmetric quantum mechanics and its fock representation the model offers some freedom in the choice of a position and a momentum operator leading to a free model parameter using the technique of jacobi matrices we determine the spectrum of the position operator and show that its wavefunctions are related to charlier polynomials cn with parameter 2 some properties of these wavefunctions are discussed as well as some other properties of the current oscillator model

methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation the application of regularization to ill conditioned problems necessitates the choice of a regularization parameter which trades fidelity to the data with smoothness of the solution the value of the regularization parameter depends on the variance of the noise in the data the problem of choosing the regularization parameter and estimating the noise variance in image restoration is examined an error analysis based on an objective mean square error mse criterion is used to motivate regularization two approaches for choosing the regularization parameter and estimating the noise variance are proposed the proposed and existing methods are compared and their relationship to linear minimum mean square error filtering is examined experiments are presented that verify the theoretical results

robust solutions to a general class of approximation problems the solution of robust counterparts of optimization problems with uncertain data is currently attracting much interest in particular this has been considered in the context of approximation using total least squares here we consider the analogue of this in the errors in variables context where attention is focused on the particular errors which arise in the individual independent variable values in data fitting problems in addition to consideration of problems which are linear in the free parameters some suggestions are made for the treatment of nonlinear problems the emphasis throughout is on providing methods which are computationally tractable and the results parallel and extend earlier results on uncertain linear approximation problems

image reconstruction and restoration overview of common estimation structures and problems developments in the theory of image reconstruction and restoration over the past 20 or 30 years are outlined particular attention is paid to common estimation structures and to practical problems not properly solved yet the problem of image reconstruction and restoration is first formulated some of the current regularization approaches used to solve the problem are then described the concepts of a priori information and compound criterion are introduced a bayesian interpretation of the regularization techniques is given which clarifies the role of the tuning parameters and indicates how they could be estimated the practical aspects of computing the solution first when the hyperparameters are known and second when they must be estimated are then considered conclusions are drawn and points that still need to be investigated are outlined

on the heisenberg and orthosymplectic superalgebras of the harmonic oscillator the new invariance superalgebra osp 2 2n sh 2 2n for an n dimensional quantum harmonic oscillator is proposed and the corresponding supersymmetric hamiltonian is constructed this superalgebra is related to the structure osp 3 2n recently studied by englefield j phys a 21 1309 1988 by interchanging the even and odd characters of the fundamental generators belonging to the heisenberg superalgebra sh 2 2n the connection between both structures is established and its consequences exploited the specific cases n 1 2 3 are discussed and compared with recent contributions some general conclusions are also summarized

homoclinic solutions for a class of nonperiodic and noneven second order hamiltonian systems abstract the existence of homoclinic solutions is obtained for second order hamiltonian systems u t l t u t w t u t f t as the limit of the solutions of a sequence of nil boundary value problems which are obtained by the mountain pass theorem when l t and w t x are neither periodic nor even with respect to t

homoclinic solutions for nonautonomous second order hamiltonian systems with a coercive potential abstract a new existence result of homoclinic orbits is obtained for the second order hamiltonian systems u t f t u t f t where f t x is periodic with respect to t this result generalizes some known results in the literature

symmetries and supersymmetries of the quantum harmonic oscillator the supersymmetric version of the one dimensional harmonic oscillator is studied by taking into account its conformal properties the largest superalgebra of symmetries and supersymmetries is derived as osp 2 2 square operator sh 1 the semidirect sum of osp 2 2 and the heisenberg superalgebra through a one to one correspondence between the nonrelativistic free case and the harmonic oscillator description the authors deduce the expected supersymmetries of the schrodinger equation the above structure appears as the largest spectrum generating superalgebra of the harmonic oscillator and its representation within an energy basis is given the physical three dimensional case is also considered when the maximal set of super symmetries is required and this case is compared with recent work

superalgebras and the supersymmetric harmonic oscillator the bound states of the d dimensional supersymmetric oscillator with a bosonic and fermionic component are shown to span an irreducible representation of osp 3 2d which contains the supersymmetric hamiltonian the representation branches in two infinite dimensional irreducible representations of osp 2 2d and osp 1 2d subalgebras the one dimensional case is used to exhibit s 2 subalgebras with representations both finite and infinite dimensional which are reducible but indecomposable the supersymmetry associated with the radial equation of the d dimensional oscillator is considered and states having a fixed angular momentum in each sector provide an irreducible representation space for an osp 2 2 superalgebra and osp 1 2 subalgebra

a framework for state space estimation with uncertain models develops a framework for state space estimation when the parameters of the underlying linear model are subject to uncertainties compared with existing robust filters the proposed filters perform regularization rather than deregularization it is shown that under certain stabilizability and detectability conditions the steady state filters are stable and that for quadratically stable models the filters guarantee a bounded error variance moreover the resulting filter structures are similar to various time and measurement update prediction and information forms of the kalman filter albeit ones that operate on corrected parameters rather than on the given nominal parameters simulation results and comparisons with spl hscr sub spl infin guaranteed cost and set valued state estimation filters are provided

some results on connecting orbits for a class of hamiltonian systems abstract the existence of various kinds of connecting orbits is established for a certain hamiltonian system as well as its time dependent analogue for the autonomous case our main assumption is that v has a global maximum e g at x o and we find a various kinds of orbits terminating at o for the time dependent case v has a local but not global maximum at x o and we find a homoclinic orbit emanating from and terminating at o keywords periodic solution

existence and multiplicity of homoclinic orbits for the second order hamiltonian systems in this paper we study the following nonperiodic second order hamiltonian system u t l t u t ur t u t r where l t may not be uniformly positive definite for all t r and r t u is asymptotically quadratic in u as u by applying a generalized linking theorem in bartsch and ding 2006 8 we obtain the existence and multiplicity of homoclinic orbits for the above system

infinitely many homoclinic solutions for second order hamiltonian systems abstract in this paper we study the existence of infinitely many homoclinic solutions for second order hamiltonian systems u l t u w u t u 0 t r where l t is unnecessarily positive definite for all t r and w t u is of subquadratic growth as u

homoclinic solutions for ordinary p laplacian systems with a coercive potential abstract a result for the existence of homoclinic orbits is obtained for p laplacian systems d d t u t p 2 u t f t u t f t where p 1 u r n f c 1 r r n r is t periodic with respect to t and f r r n is a continuous and bounded function such that f t x f t 0 b x and r f t 1 d t for some b 0 and 1

on chains of orthosymplectic lie superalgebras and the n dimensional quantum harmonic oscillator triplets of embedded orthosymplectic lie superalgebras are singled out and analyzed in terms of their respective even and odd root systems the corresponding chains of embeddings are considered for arbitrary integers m and n for both m n and m n it is shown that the second member inside each triplet is always in a one to one correspondence with the semidirect sum of the third member and the associated heisenberg superalgebra that is characterized by the same entries m and n for arbitrary m n invariance superalgebras of the n dimensional harmonic oscillator are recovered and their one to one correspondence associated with a recent character reversal phenomenon linking these invariance superstructures is rigorously explained specific and general cases as well as conclusions are presented

on generalized coherent states with maximal symmetry for the harmonic oscillator generalized coherent states for the one dimensional harmonic oscillator with maximal symmetry i e admitting the semidirect sum so 2 1 h 2 as the largest invariance lie algebra pointed out by niederer are constructed the normalization of such states as well as their completeness property are determined and discussed they are also analyzed in the subcontexts of the so 2 1 algebra and of the heisenberg h 2 algebra general considerations on heisenberg relations on minimal dispersions and on quantum mechanical entropies are presented in connection with the uncertainty principle

parameter estimation with multiple sources and levels of uncertainties least squares designs are sensitive to errors in the data which can be due to several factors including the approximation of complex models by simpler ones the presence of unavoidable experimental errors when collecting data or even due to unknown or unmodeled effects we formulate a new design criterion that treats multiple sources of uncertainties in the data with possibly varied degrees of intensity we show that the solution has a regularized form with one regularization parameter for each source of uncertainty the parameters turn out to be model dependent and can be determined optimally as the nonnegative roots of certain coupled equations applications in array signal processing and image processing are considered

finite and infinite dimensional representations of the orthosymplectic superalgebra osp 3 2 the shift operator technique is used to give a complete analysis of all finite and infinite dimensional irreducible representations of the orthosymplectic superalgebra osp 3 2 for all cases the star or grade star conditions for the algebra are investigated only two finite dimensional representations are grade star representations if the representation space is required to be a graded hilbert space when the even part is so 3 sp 2 r su 2 su 1 1 an infinite class of infinite dimensional star representations is found one of them can be realized in terms of two valued functions of a complex variable this representation reduces to the sum of two metaplectic representations of sp 2 we show that it is precisely this metaplectic representation for osp 3 2 which gives the spin energy eigenstates for the one dimensional harmonic oscillator with spin 1 2 states