an orlicz space approach to superlinear elliptic systems in this paper we study superlinear elliptic systems in hamiltonian form using an orlicz space setting we extend the notion of critical growth to superlinear nonlinearities which do not have a polynomial growth existence of nontrivial solutions is proved for superlinear nonlinearities which are subcritical in this generalized sense

an infinite dimensional morse theory with applications in this paper we construct an infinite dimensional extraordinary cohomology theory and a morse theory corresponding to it these theories have some special properties which make them useful in the study of critical points of strongly indefinite functionals by strongly indefinite we mean a functional unbounded from below and from above on any subspace of finite codimension several applications are given to hamiltonian systems the onedimensional wave equation of vibrating string type and systems of elliptic partial differential equations 0 introduction let e be a real hilbert space with an inner product and let be a twice continuously differentiable functional denote the frechet derivative and the gradient of at x by x and x respectively where as usual x y x y y e recall that a point x0 e is said to be critical if x0 0 or equivalently if x0 0 the level c r will be called regular if 1 c contains no critical points and critical if x0 0 for some x0 1 c let a b a b be two regular levels of denote m 1 a b and consider the restriction of to m in morse theory one is interested in the local topological structure of the level sets of m near a critical point and in the relation between this local structure and the topological structure of the set m to be more specific suppose that x0 m is an isolated critical point of then one defines a sequence of critical groups of at x0 by setting cq x0 hq c u u x0 q 0 1 2 0 1 where c x0 c x e x c hq is the q th singular homology group with coefficients in some field f and u is a neighbourhood of x0 define the morse index of x0 to be the maximal dimension of a subspace of e on which the quadratic form x0 y y is negative definite one shows that if x0 is a nondegenerate critical point i e if x0 e e is invertible then cq x0 f received by the editors march 20 1995 1991 mathematics subject classification primary 58e05 secondary 34c25 35j65 35l05 55n20 58f05

population models with diffusion strong allee effect and nonlinear boundary conditions abstract we consider a population model with diffusion a strong allee effect per capita growth function and constant yield harvesting in particular we focus our study on a population living in a patch r n with n 1 that satisfies a certain nonlinear boundary condition we establish our existence results by the method of sub supersolutions

existence and multiplicity of solutions for asymptotically hamiltonian elliptic systems in rn abstract this paper is concerned with the following nonperiodic hamiltonian elliptic system u v x u h u x u v in r n v v x v h v x u v in r n u x 0 and v x 0 as x where the potential v is bounded below and h is asymptotically linear in z as z with z u v by applying a generalized linking theorem of strongly indefinite functionals we obtain solutions for the above system

population models with diffusion strong allee effect and constant yield harvesting abstract we study the steady state distribution of reaction diffusion equations with strong allee effect type growth and constant yield harvesting semipositone in heterogeneous bounded habitats assuming the exterior of the habitat is completely hostile we establish existence results for positive solutions we also establish a multiplicity result for the non harvested case we obtain our results via the method of sub super solutions

joint state and parameter estimation for a target directed nonlinear dynamic system model we present a new approach to joint state and parameter estimation for a target directed nonlinear dynamic system model with switching states the model recently proposed for representing speech dynamics is called the hidden dynamic model hdm the model parameters subject to statistical estimation consist of the target vector and the system matrix also called time constants as well as parameters characterizing the nonlinear mapping from the hidden state to the observation we implement these parameters as the weights of a three layer feedforward multilayer perceptron mlp network the new estimation approach is based on the extended kalman filter ekf and its performance is compared with the traditional expectation maximization em based approach extensive simulation results are presented using both approaches and under typical hdm speech modeling conditions the ekf based algorithm demonstrates superior convergence performance compared with the em algorithm but the former suffers from excessive computational loads when adopted for training the mlp weights in all cases the simulated model output converges to the given observation sequence however only in the case where the mlp weights or the target vector are assumed known do the time constant parameters converge to their true values we also show that the mlp weights never converge to their true values thus demonstrating the many to one mapping property of the feedforward mlp we conclude that for the system to be identifiable restrictions on the parameter space are needed

maximum likelihood estimation via the ecm algorithm a general framework two major reasons for the popularity of the em algorithm are that its maximum step involves only complete data maximum likelihood estimation which is often computationally simple and that its convergence is stable with each iteration increasing the likelihood when the associated complete data maximum likelihood estimation itself is complicated em is less attractive because the m step is computationally unattractive in many cases however complete data maximum likelihood estimation is relatively simple when conditional on some function of the parameters being estimated

on the existence of positive solutions of a perturbed hamiltonian system in rn using the legendre fenchel transformation and the mountain pass theorem due to ambrosetti and rabinowitz we establish an existence result for perturbations of periodic and asymptotically periodic semilinear hamiltonian systems of the type pw u u w2 x v p 1vinrn v v w1 x u q 1uinrn u x v x 0as x u 0 v 0inrn n 2 here the numbers p q 1 are below the critical hyperbola if n 3 that is they satisfy 1 p 1 1 q 1 n 2 n while no additional restrictions on p and q are required if n 2 the functions wi i 1 2 are bounded positive continuous functions

semiclassical states for nonlinear schrodinger equations with sign changing potentials we establish the existence and multiplicity of semiclassical bound states of the following nonlinear schrodinger equation e2 u v x u g x u for x rn u x 0as x where v changes sign and g is superlinear with critical or supercritical growth as u

diffusive logistic equation with constant yield harvesting and negative density dependent emigration on the boundary abstract the structure of positive steady state solutions of a diffusive logistic population model with constant yield harvesting and negative density dependent emigration on the boundary is examined in particular a class of nonlinear boundary conditions that depends both on the population density and the diffusion coefficient is used to model the effects of negative density dependent emigration on the boundary our existence results are established via the well known sub super solution method

on use of the em algorithm for penalized likelihood estimation summary the em algorithm is a popular approach to maximum likelihood estimation but has not been much used for penalized likelihood or maximum a posteriori estimation this paper discusses properties of the em algorithm in such contexts concentrating on rates of conver gence and presents an alternative that is usually more practical and converges at least as quickly the em algorithm is a general approach to maximum likelihood estimation rather than a specific algorithm dempster et al 1977 discussed the method and derived basic properties demonstrating that a variety of procedures previously developed rather informally could be unified the common strand to problems where the approach is applicable is a notion of incomplete data this includes the conventional sense of missing data but is much broader than that the em algorithm demon strates its strength in situations where some hypothetical experiment yields data from which estimation is particularly convenient and economical the incomplete data actually at hand are regarded as observable functions of these complete data the resulting algorithms while usually slow to converge are often extremely simple and remain practical in large problems where no other approaches may be feasible dempster et al 1977 briefly refer to the use of the same approach to the problem of finding the posterior mode maximum a posteriori estimate in a bayesian estima

variational learning for switching state space models we introduce a new statistical model for time series that iteratively segments data into regimes with approximately linear dynamics and learns the parameters of each of these linear regimes this model combines and generalizes two of the most widely used stochastic time series models hidden markov models and linear dynamical systems and is closely related to models that are widely used in the control and econometrics literatures it can also be derived by extending the mixture of experts neural network jacobs jordan nowlan hinton 1991 to its fully dynamical version in which both expert and gating networks are recurrent inferring the posterior probabilities of the hidden states of this model is computationally intractable and therefore the exact expectation maximization em algorithm cannot be applied however we present a variational approximation that maximizes a lower bound on the log likelihood and makes use of both the forward and backward recursions for hidden markov models and the kalman filter recursions for linear dynamical systems we tested the algorithm on artificial data sets and a natural data set of respiration force from a patient with sleep apnea the results suggest that variational approximations are a viable method for inference and learning in switching state space models

diffusive logistic equation with constant yield harvesting i steady states we consider a reaction diffusion equation which models the constant yield harvesting to a spatially heterogeneous population which satisfies a logistic growth we prove the existence uniqueness and stability of the maximal steady state solutions under certain conditions and we also classify all steady state solutions under more restricted conditions exact global bifurcation diagrams are obtained in the latter case our method is a combination of comparison arguments and bifurcation theory

on superquadratic elliptic systems in this article we study the existence of solutions for the elliptic system updelta u frac partial h partial v left u v x right text in upomega updelta v frac partial h partial v left u v x right text in upomega u 0 v 0 text on partial upomega where upomega is a bounded open subset of mathbb r n with smooth boundary partial upomega and the function h mathbb r 2 times overline upomega to mathbb r is of class c 1

conditional persistence in logistic models via nonlinear diffusion a nonlinear diffusion process modelling aggregative dispersal is combined with local in space population dynamics given by a logistic equation and the resulting growth dispersal model is analysed the nonlinear diffusion process models aggregation via a diffusion coefficient which is decreasing with respect to the population density at low densities this mechanism is similar to area restricted search but it is applied to conspecifics rather than prey the analysis shows that in some cases the models predict a threshold effect similar to an allee effect that is for some parameter ranges the models predict a form of conditional persistence where small populations go extinct but large populations persist this is somewhat surprising because logistic equations without diffusion or with non aggregative diffusion predict either unconditional persistence or unconditional extinction furthermore in the aggregative models the minimum patch size needed to sustain an existing population at moderate to high densities may be smaller than the minimum patch size needed for invasibility by a small population the tradeoff is that if a population is inhabiting a large patch whose size is reduced below the size needed to sustain any population then the population on the patch can be expected to experience a sudden crash rather than a steady decline

global bifurcation of steady state solutions in this paper we study the bifurcation of steady state solutions of a reaction diffusion equation in one space variable the steady state solutions satisfy the equation a f u 0 on the interval l 0 or 0 for some nonzero function c c a for the neumann problem we use entirely different techniques to prove that s is never critical for any cubic polynomial j this implies at once that the neumann problem can have at most one nonconstant solution having a given number of maxima or minima this solution is necessarily strongly nondegenerate in the sense that zero is not contained in the spectrum of the linearized operator see 2 for the dirichlet problem the situation is far more complicated and the bifurcation diagrams undergo qualitative changes depending on the positions of the roots off for example fig 1 shows the bifurcation diagram for the positive solutions for two different cubic functions of the form f u a u u b u c it is interesting to note however that in every case we study there are at most three solutions to the dirichlet problem for each

homoclinic orbits for a nonperiodic hamiltonian system abstract in this paper we prove the existence and multiplicity of homoclinic orbits for first order hamiltonian systems of the form z j h z t z where h z is asymptotically linear at and is not assumed to be periodic

homoclinic orbits for asymptotically linear hamiltonian systems we study the existence of homoclinic orbits for first order time dependent hamiltonian systems z jhz z t where h z t depends periodically on t and hz z t is asymptotically linear in z as z we also consider an asymptotically linear schrodinger equation in rn

dynamic linear models with switching abstract the problem of modeling change in a vector time series is studied using a dynamic linear model with measurement matrices that switch according to a time varying independent random process we derive filtered estimators for the usual state vectors and also for the state occupancy probabilities of the underlying nonstationary measurement process a maximum likelihood estimation procedure is given that uses a pseudo expectation maximization algorithm in the initial stages and nonlinear optimization we relate the models to those considered previously in the literature and give an application involving the tracking of multiple targets

decay symmetry and existence of solutions of semilinear elliptic systems our utmost aim was originally to establish the existence of solutions of system however in this effort we got sidetracked to consider other questions in this way we come to interesting results on asymptotic behavior of solutions on symmetry properties of such solutions and the existence of ground states

ground state solutions for superlinear elliptic systems on rn abstract this paper is concerned with the existence of the ground state solutions for the following superlinear elliptic system of hamiltonian type u v x u g x v in r n v v x v f x u in r n u x 0 and v x 0 as x where v c r n r is periodic in x 1 x 2 x n we assume that 0 lies in a gap of the spectrum v and f and g are both superlinear at 0 and infinity but they have different increasing rates at infinity by proving all cerami sequences for the energy functional are bounded existence of a ground state solution is obtained

diffusive logistic equation with non linear boundary conditions abstract we analyze the solutions of a population model with diffusion and logistic growth in particular we focus our study on a population living in a patch r n with n 1 that satisfies a certain non linear boundary condition and on its survival when constant yield harvesting is introduced we establish our existence results by the method of sub super solutions

existence and multiplicity of solutions for asymptotically linear nonperiodic hamiltonian elliptic system abstract this paper is concerned with the following nonperiodic hamiltonian elliptic system u v x u h v x u v v v x v h u x u v x r n u x v x 0 as x where z u v r n r r n 3 assuming the potential v is nonperiodic and sign changing h x z is nonperiodic in x and asymptotically quadratic in z u v existence and multiplicity of solutions are obtained via a variational approach

a gradient algorithm locally equivalent to the em algorithm in many problems of maximum likelihood estimation it is impossible to carry out either the e step or the m step of the em algorithm the present paper introduces a gradient algorithm that is closely related to the em algorithm this em gradient algorithm approximately solves the m step of the em algorithm by one iteration of newton s method since newton s method converges quickly the local properties of the em gradient algorithm are almost identical with those of the em algorithm any strict local maximum point of the observed likelihood locally attracts the em and em gradient algorithm at the same rate of convergence and near the maximum point the em gradient algorithm always produces an increase in the likelihood with proper modification the em gradient algorithm also exhibits global convergence properties that are similar to those of the em algorithm our proof of global convergence applies and improves existing theory for the em algorithm these theoretical points are reinforced by a discussion of three realistic examples illustrating how the em gradient algorithm can succeed where the em algorithm is intractable

on a periodic schrodinger equation with nonlocal superlinear part we consider the choquard pekar equation open image in new window and focus on the case of periodic potential v for a large class of even functions w we show existence and multiplicity of solutions essentially the conditions are that 0 is not in the spectrum of the linear part v and that w does not change sign our results carry over to more general nonlinear terms in arbitrary space dimension n 2

existence and multiplicity of semiclassical solutions for asymptotically hamiltonian elliptic systems abstract this paper is concerned with the following nonperiodic hamiltonian elliptic system e 2 u v x u h u x u v in r n e 2 v v x v h v x u v in r n u x 0 and v x 0 as x where e 0 is a small parameter and the potential v is bounded below and h is asymptotically linear in z as z with z u v by applying a generalized linking theorem of strongly indefinite functionals we obtain the existence of multiple semiclassical solutions for the above system

existence and multiplicity of solutions for a non periodic schrodinger equation abstract this paper is concerned with the non periodic schrodinger equation u v x u f x u x r n u x 0 as x under a general spectral assumption the existence and multiplicity of solutions are obtained for asymptotically linear nonlinearity via variational methods

multiple solutions for asymptotically quadratic and superquadratic elliptic system of hamiltonian type this paper is concerned with the following nonperiodic hamiltonian elliptic system u v x u h v x u v x r n v v x v h u x u v x r n u x 0 v x 0 as x where z u v r n r i r n 3 and the potential v x is nonperiodic and sign changing by applying a generalized linking theorem for strongly indefinite functionals we establish the existence of multiple solutions for asymptotically quadratic nonlinearity as well as the existence of infinitely many solutions for superquadratic nonlinearity

existence of solutions for nonperiodic superquadratic hamiltonian elliptic systems abstract this paper is concerned with the following nonperiodic hamiltonian elliptic system u v x u r v x u v in r n v v x v r u x u v in r n u x 0 and v x 0 as x where r x z is superquadratic in z as z with z u v by applying a critical point theorem for strongly indefinite functionals we prove the existence of solution for the above system

differential systems with strongly indefinite variational structure abstract the euler lagrange equations of lagrangians with a strongly indefinite quadratic part are studied by means of the direct min max method of benci and rabinowitz the simplest case is a system of two coupled semilinear poisson equations using an analytic framework of a suitable family of products of fractional sobolev spaces existence results are obtained for all nonlinearities with subcritical growth rates

strongly indefinite functionals and multiple solutions of elliptic systems we study existence and multiplicity of solutions of the elliptic system left begin array 20 l updelta u h u x u v hfill text in upomega hfill updelta v h v x u v hfill text in upomega quad u x v x 0 quad text on partial upomega hfill end array right where upomega subset mathbb r n n ge 3 is a smooth bounded domain and h in mathcal c 1 overline upomega times mathbb r 2 mathbb r we assume that the nonlinear term h x u v sim left u right p left v right q r x u v text with mathop lim limits left u v right to infty frac r x u v left u right p left v right q 0 where p in 1 2 2 2n n 2 text and q in 1 infty so some supercritical systems are included nontrivial solutions are obtained when h x u v is even in u v we show that the system possesses a sequence of solutions associated with a sequence of positive energies resp negative energies going toward infinity resp zero if p 2 resp p 2 all results are proved using variational methods some new critical point theorems for strongly indefinite functionals are proved