on structure sensitivity of degree based topological indices one of the general requirements for any topological index ti is that similar molecules have near lying ti values which is referred to as smoothness curiously however smoothness of topological indices was until now never examined and never quantified we now propose a pertinent mathematical model for this property and undertake a comparative study of the structure sensitivity of 12 degree based topological indices by using trees

on topological indices of certain interconnection networks abstract in qsar qspr study physico chemical properties and topological indices such as randic atom bond connectivity abc and geometric arithmetic ga index are used to predict the bioactivity of chemical compounds a topological index is actually designed by transforming a chemical structure into a numeric number these topological indices correlate certain physico chemical properties like boiling point stability strain energy etc of chemical compounds graph theory has found a considerable use in this area of research the topological properties of certain networks are studied recently in 13 by hayat and imran 2014 in this paper we extend this study to interconnection networks and derive analytical closed results of general randic index r g for different values of for butterfly and benes networks we also compute first zagreb abc and ga indices for these important classes of networks moreover we construct two new classes of mesh derived networks by using some basic operations of graphs on m n mesh networks and then study certain topological indices for these classes of networks

when do nonlinear filters achieve maximal accuracy the nonlinear filter for an ergodic signal observed in white noise is said to achieve maximal accuracy if the stationary filtering error vanishes as the signal to noise ratio diverges we give a general characterization of the maximal accuracy property in terms of various systems theoretic notions when the signal state space is a finite set explicit necessary and sufficient conditions are obtained while the linear gaussian case reduces to a classic result of kwakernaak and sivan ieee trans automatic control ac 17 1972 pp 79 86

an ergodic theorem for filtering with applications to stability abstract ergodic properties of the signal filtering pair are studied for continuous time finite markov chains observed in white noise the obtained law of large numbers is applied to the stability problem of the nonlinear filter with respect to initial conditions the furstenberg khasminskii formula is derived for the top lyapunov exponent of the zakai equation and is used to estimate the stability index of the filter

quasi periodic solutions for 1d wave equation with the nonlinearity u2p 1 abstract in this paper one dimensional 1d nonlinear wave equation u t t u x x m u u 2 p 1 0 p n on the finite x interval 0 with dirichlet boundary conditions is considered it is proved that there are many 2 dimensional elliptic invariant tori and thus quasi periodic solutions for the above equation the proof is based on infinite dimensional kam theory and partial birkhoff normal form

robustness of nonlinear filters over the infinite time interval nonlinear filtering is one of the classical areas of stochastic control from the point of view of practical usefulness it is important that the filter not be too sensitive to the assumptions made on the initial distribution the transition function of the underlying signal process and the model for the observation this is particularly acute if the filter is of interest over a very long or potentially infinite time interval then the effects of small errors in the model which is used to construct the filter might accumulate to make the output useless for large time the problem of asymptotic sensitivity to the initial condition has been treated in several papers we are concerned with this as well as with the sensitivity to the signal model uniformly over the infinite time interval it is conceivable that the effects of even small errors in the model will accumulate so that the filter will eventually be useless the robustness is shown for three classes of problems for the first two cases the signal model is markov and the observations are taken in discrete time and the observation is the usual function of the signal plus noise the last class treated is a continuous time markov process with a point process observation

effects of statistical dependence on multiple testing under a hidden markov model the performance of multiple hypothesis testing is known to be affected by the statistical dependence among random variables involved the mechanisms responsible for this however are not well understood we study the effects of the dependence structure of a finite state hidden markov model hmm on the likelihood ratios critical for optimal multiple testing on the hidden states various convergence results are obtained for the likelihood ratios as the observations of the hmm form an increasing long chain analytic expansions of the first and second order derivatives are obtained for the case of binary states explicitly showing the effects of the parameters of the hmm on the likelihood ratios

invariant tori of full dimension for a nonlinear schrodinger equation abstract in this paper we consider the one dimensional nonlinear schrodinger equation i u t u x x m u f u 2 u 0 with periodic boundary conditions or dirichlet boundary conditions where f is a real analytic function in some neighborhood of the origin satisfying f 0 0 f 0 0 we prove that for each given constant potential m when the frequencies as a function of the amplitudes can be regarded as the independent parameters the equation admits a whitney smooth family of small amplitude time almost periodic solutions with all frequencies the proof is based on a birkhoff normal form reduction and an improved version of the kam theorem

renormalization group and the melnikov problem for pde s we give a new proof of persistence of quasi periodic low dimensional elliptic tori in infinite dimensional systems the proof is based on a renormalization group iteration that was developed recently in bgk to address the standard kam problem namely persistence of invariant tori of maximal dimension in finite dimensional near integrable systems our result covers situations in which the so called normal frequencies are multiple in particular it provides a new proof of the existence of small amplitude quasi periodic solutions of nonlinear wave equations with periodic boundary conditions

invariant cantor manifolds of quasi periodic oscillations for a nonlinear schrodinger equation this paper was written while both authors were guests of the forschungsinstitut fur mathematik at the eth zurich and we thank the institute for its hospitality pleasant working atmosphere and helpful staff in particular we thank jiirgen moser for numerous stimulating discussions on the subject we also benefitted from a remark by sigurd angenent concerning the analyticity of the solutions the second author also thanks the deutsche forschungsgemeinschaft for their financial support through a heisenberg grant

newton s method and periodic solutions of nonlinear wave equations we prove the existence of periodic solutions of the nonlinear wave equation satisfying either dirichlet or periodic boundary conditions on the interval o the coefficients of the eigenfunction expansion of this equation satisfy a nonlinear functional equation using a version of newton s method we show that this equation has solutions provided the nonlinearity g x u satisfies certain generic conditions of nonresonance and genuine nonlinearity 1993 john wiley sons inc

quasi periodic solutions for quasi periodically forced nonlinear schrodinger equations with quasi periodic inhomogeneous terms abstract in this paper it is shown that there exist many small amplitude quasi periodic solutions for non autonomous quasi periodically forced in time nonlinear schrodinger equations with quasi periodic inhomogeneous terms under periodic spatial boundary conditions via kam theory

a kam theorem with applications to partial differential equations of higher dimensions the existence of lower dimensional kam tori is shown for a class of nearly integrable hamiltonian systems of infinite dimensions where the second melnikov s conditions are completely eliminated and the algebraic structure of the normal frequencies are not needed as a consequence it is proved that there exist many invariant tori and thus quasi periodic solutions for nonlinear wave equations schrodinger equations and other equations of any spatial dimensions

entropy bounds for dendrimers abstract many graph invariants have been used for the construction of entropy based measures to characterize the structure of complex networks when considering shannon entropy based graph measures there has been very little work to find their extremal values a reason for this might be the fact that shannon s entropy represents a multivariate function and all probability values are not equal to zero when considering graph entropies dehmer and kraus proved some extremal results for graph entropies which are based on information functionals and express some conjectures generated by numerical simulations to find extremal values of graph entropies dehmer and kraus discussed the extremal values of entropies for dendrimers in this paper we continue to study the extremal values of graph entropy for dendrimers which has most interesting applications in molecular structure networks and also in the pharmaceutical and biomedical area among all dendrimers with n vertices we obtain the extremal values of graph entropy based on different well known information functionals numerical experiments verifies our results

asymptotic stability of the optimal filter with respect toits initial condition consider the problem of estimation of a diffusion signal observed in additive white noise if the solution to the filtering equations initialized with an incorrect prior distribution approaches the true conditional distribution asymptotically in time then the filter is said to be asymptotically stable with respect to perturbations of the initial condition this paper presents asymptotic stability results for linear filtering problems and for signals with limiting ergodic behavior for the linear case stability of the riccati equation of kalman filtering is used to derive almost sure asymptotic stability of linear filters for possibly non gaussian initial conditions in the nonlinear case asymptotic stability in a weak convergence sense is shown for filters of signal diffusions which converge in law to an invariant distribution

almost periodic solutions for a class of linear schrodinger equations with almost periodic forcing in this paper the existence of almost periodic solutions corresponding to full dimensional invariant tori is shown for a class of linear schrodinger equations with almost periodic forcing subject to periodic boundary conditions it is proved via an improved kolmogorov arnold moser reducibility

ergodicity of filtering process by vanishing discount approach a new approach to study ergodicity of filtering processes is presented it is based on the vanishing discount approach to discounted functional of filtering process we show that limit superior of the cesaro averages of the functionals is the same for all initial conditions from which the uniqueness of invariant measures of filtering processes follows the approach is based on certain assumption for which we provide a sufficient condition using concavity arguments in addition we show the existence of solutions to the poisson equation corresponding to filtering process with concave functional the assumptions are then extended to the controlled case and using similar concave arguments we obtain the existence of solutions to the bellman equation corresponding to partially observed average cost per unit time problem

quasi periodic solutions of two dimensional schrodinger equations with quasi periodic forcing abstract this work focuses on two dimensional 2 d quasi periodically forced nonlinear schrodinger equations this means studying i u t u u e t u u 2 u 0 0 x t 2 t r with periodic boundary conditions where e is a small positive parameter t is a real analytic quasi periodic function in t with frequency vector 1 2 m it is shown that under suitable hypothesis on t there are many quasi periodic solutions for the above equation via kam theory

almost periodic solutions of an almost periodically forced wave equation abstract this work focuses on a wave equation with almost periodic forcing t x subject to dirichlet boundary conditions for the equation we obtain almost periodic solutions corresponding to full dimensional invariant tori of an associated infinite dimensional dynamical system the proof is based on the reducibility via an improved kam theory

quasi periodic solutions of nonlinear wave equations with quasi periodic forcing abstract this work focuses on one dimensional 1d quasi periodically forced nonlinear wave equations this means studying u t t u x x u e t h u 0 0 with dirichlet boundary conditions where e is a small positive parameter t is a real analytic quasi periodic function in t with frequency vector 1 2 m and the nonlinearity h is a real analytic odd function of the form h u 1 u 2 r 1 u 2 r 1 k r 1 2 k 1 u 2 k 1 1 2 r 1 0 r n it is shown that under a suitable hypothesis on t and h there are many quasi periodic solutions for the above equation via kam theory

periodic and quasi periodic solutions of nonlinear wave equations via kam theory in this paper the nonlinear wave equation u u u xx v x u x t varepsilon u 3 x t 0 is studied it is shown that for a large class of potentials v x one can use kam methods to construct periodic and quasi periodic solutions in time for this equation

degenerate lower dimensional tori in hamiltonian systems abstract we study the persistence of lower dimensional tori in hamiltonian systems of the form h x y z y 1 2 z m z e p x y z where x y z t n r n r 2 m e is a small parameter and m can be singular we show under a weak melnikov nonresonant condition and certain singularity removing conditions on the perturbation that the majority of unperturbed n tori can still survive from the small perturbation as an application we will consider the persistence of invariant tori on certain resonant surfaces of a nearly integrable properly degenerate hamiltonian system for which neither the kolmogorov nor the g nondegenerate condition is satisfied

the stability of conditional markov processes and markov chains in random environments we consider a discrete time hidden markov model where the signal is a stationary markov chain when conditioned on the observations the signal is a markov chain in a random environment under the conditional measure it is shown that this conditional signal is weakly ergodic when the signal is ergodic and the observations are nondegenerate this permits a delicate exchange of the intersection and supremum of fields which is key for the stability of the nonlinear filter and partially resolves a long standing gap in the proof of a result of kunita j multivariate anal 1 1971 365 393 a similar result is obtained also in the continuous time setting the proofs are based on an ergodic theorem for markov chains in random environments in a general state space

kam for reversible derivative wave equations we prove the existence of cantor families of small amplitude analytic linearly stable quasi periodic solutions of reversible derivative wave equations

discrete time filtering for linear systems with non gaussian initial conditions asymptotic behavior of the difference between the mmse and lmse estimates the authors consider the one step prediction problem for discrete time linear systems in correlated plant and observation gaussian white noises with nongaussian initial conditions they investigate the large time asymptotics of epsilon sub t the expected squared difference between the mmse and lmse or kalman estimates of the state of time t given past observations they characterize the limit of their error sequence epsilon sub t t 0 1 and obtain some related rates of convergence a complete analysis is provided for the scalar case the discussion is based on explicit representations for the mmse and lmse estimates recently obtained by the authors which display the dependence of these quantities on the initial distribution

approximation and limit results for nonlinear filters over an infinite time interval this paper is concerned with approximations to nonlinear filtering problems that are of interest over a very long time interval since the optimal filter can rarely be constructed one needs to compute with numerically feasible approximations the signal model can be a jump diffusion or just a process that is approximated by a jump diffusion the observation noise can be either white or of wide bandwidth the observations can be taken in either discrete or continuous time the cost of interest is the pathwise error per unit time over a long time interval it is shown under quite reasonable conditions on the approximating filter and the signal and noise processes that as time bandwidth process and filter approximation etc go to their limit in any way at all the limit of the pathwise average costs per unit time is just what one would get if the approximating processes were replaced by their ideal values and the optimal filter were used analogous results are obtained with appropriate scaling if the observations are taken in discrete time and the sampling interval also goes to zero for these cases the approximating filter is a numerical approximation to the optimal filter for the presumed limit signal observation noise problem

lyapunov exponents for finite state nonlinear filtering consider the wonham optimal filtering problem for a finite state ergodic markov process in both discrete and continuous time and let sigma be the noise intensity for the observation we examine the sensitivity of the solution with respect to the filter s initial conditions in terms of the gap between the first two lyapunov exponents of the zakai equation for the unnormalized conditional probability this gap is studied in the limit as sigma to 0 by techniques involving considerations of nonlinear filtering and the stochastic feynman kac formula conditions are given for the limit to be either negative or infty asymptotic bounds are derived in the latter case

markov property and ergodicity of the nonlinear filter in this paper we first prove under quite general conditions that the nonlinear filter and the pair signal filter are feller markov processes the state space of the signal is allowed to be nonlocally compact and the observation function h can be unbounded our proofs in contrast to those of kunita j multivariate anal 1 1971 pp 365 393 spatial stochastic processes birkhauser 1991 pp 233 256 and stettner stochastic differential equations springer verlag 1989 pp 279 292 do not depend upon the uniqueness of the solutions to the filtering equations we then obtain conditions for existence and uniqueness of invariant measures for the nonlinear filter and the pair process these results extend those of kunita and stettner which hold for locally compact state space and bounded h to our general framework finally we show that the recent results of ocone and pardoux siam j control optim 34 1996 pp 226 243 on asymptotic stability of the nonlinear filter which use the kunita stettner setup hold for the general situation considered in this paper

long time behavior for some dynamical noise free nonlinear filtering problems a possible approach to studying the accuracy of optimal nonlinear filtering is to look at the conditional law on an arbitrary neighborhood of the current unknown state as the time goes to infinity this paper is devoted to a restricted class of systems with a deterministic state equation and under additional assumptions results concerning the concentration of the conditional law are shown as is how estimating the current state could be different from estimating the initial condition the assumptions on the system could seem quite restrictive however they concern only the observability of the system without any reference to ergodicity as in previous works

periodic solutions for completely resonant nonlinear wave equations with dirichlet boundary conditions we consider the nonlinear string equation with dirichlet boundary conditions utt uxx u with u u3 o u5 odd and analytic 0 and we construct small amplitude periodic solutions with frequency for a large lebesgue measure set of close to 1 this extends previous results where only a zero measure set of frequencies could be treated the ones for which no small divisors appear the proof is based on combining the lyapunov schmidt decomposition which leads to two separate sets of equations dealing with the resonant and non resonant fourier components respectively the q and the p equations with resummation techniques of divergent powers series allowing us to control the small divisors problem the main difficulty with respect to the nonlinear wave equations utt uxx mu u m 0 is that not only the p equation but also the q equation is infinite dimensional

degenerate kam theory for partial differential equations abstract this paper deals with degenerate kam theory for lower dimensional elliptic tori of infinite dimensional hamiltonian systems depending on one parameter only we assume that the linear frequencies are analytic functions of the parameter satisfy a weak non degeneracy condition of russmann type and an asymptotic behavior an application to nonlinear wave equations is given

invariant tori for 1d quintic nonlinear wave equation abstract in this paper one dimensional 1d nonlinear wave equation u t t u x x m u u 5 0 on the finite x interval 0 with dirichlet boundary conditions is considered it is proved that for any integer b 2 there are many b dimensional elliptic invariant tori and thus quasi periodic solutions for the above equation this is an extension of the previous work 6 by the same authors where b 2 the proof is based on infinite dimensional kam theory and partial birkhoff normal form

quasi periodic solutions of hamiltonian perturbations of 2d linear schrodinger equations the general problem discussed here is the persistency of quasi periodic solutions of linear or integrable equations after hamiltonian perturbation this subject is closely related to the well known kam theory of invariant tori in smooth dynamical systems the main source of new difficulties is the fact that one considers here finite dimensional tori in an infinite dimensional phase space for instance already in the study of time periodic solutions the fact that the phase space is infinite dimensional leads to small divisor problems now the standard kam technology may be reworked in a form applicable to treat persistency problems in space dimension 1d in the case of dirichlet boundary conditions say where in particular the normal frequencies of the problem are well separated results along this line were obtained by s kuksin ki and c e wayne w among other works we mentioned ki as the main expository reference for the kam approach considering pde s in id with periodic boundary conditions and especially in space dimension 1 a significant new problem arises due to the presence of clusters of normal frequencies

absence of diffusion in the anderson tight binding model for large disorder or low energy we prove that the green s function of the anderson tight binding hamiltonian decays exponentially fast at long distances on v with probability 1 we must assume that either the disorder is large or the energy is sufficiently low our proof is based on perturbation theory about an infinite sequence of block hamiltonians and is related to kam methods

perturbations of lower dimensional tori for hamiltonian systems abstract this paper provides a generalized kolmogorov arnold moser theorem for lower dimensional tori in hamiltonian systems which applies to multiple normal frequency case the proof is based on newton s iteration method and a generalized version of small divisor conditions

on zagreb indices zagreb polynomials of some nanostar dendrimers dendrimers are highly branched organic macromolecules with successive layers or generations of branch units surrounding a central core ashrafi and mirzargar 2008 3 chen et al 2014 7 klajnert and bryszewska 2001 26 yamamoto et al 2002 36 these are key molecules in nanotechnology and can be put to good use topological indices are numbers associated with molecular graphs for the purpose of allowing quantitative structure activity property toxicity relationships these topological indices correlate certain physico chemical properties like boiling point stability strain energy etc of chemical compounds in this paper we determine hyper zagreb index first multiple zagreb index second multiple zagreb index and zagreb polynomials for some nanostar dendrimers

periodic solutions of nonlinear wave equations on obtient des conditions suffisantes pour l existence de solutions doublement periodiques de l equation d onde non lineaire u ts f t s u u t t s u t s u t s t ou f est une fonction continue donnee dans r 3 et f est t periodique en t et s

on quasi periodic solutions for a generalized boussinesq equation abstract in this paper a one dimensional generalized boussinesq equation u t t u x x u 3 u x x x x 0 with hinged boundary conditions is considered it is proved that the above equation admits small amplitude quasi periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional hamiltonian system the proof is based on an infinite dimensional kam theorem and the birkhoff normal form

on filtering of markov chains in strong noise the filtering problem for finite state markov chains is revisited in the low signal to noise regime we give a description of conditional measure concentration around the invariant distribution of the signal and derive asymptotic expressions for the performance indices of the minimum mean square error mmse and minimum a posteriori probability map filtering estimates

observability and nonlinear filtering this paper develops a connection between the asymptotic stability of nonlinear filters and a notion of observability we consider a general class of hidden markov models in continuous time with compact signal state space and call such a model observable if no two initial measures of the signal process give rise to the same law of the observation process we demonstrate that observability implies stability of the filter i e the filtered estimates become insensitive to the initial measure at large times for the special case where the signal is a finite state markov process and the observations are of the white noise type a complete necessary and sufficient characterization of filter stability is obtained in terms of a slightly weaker detectability condition in addition to observability the role of controllability is explored finally the results are partially extended to non compact signal state spaces

kam for the derivative nonlinear schrodinger equation with periodic boundary conditions abstract this paper is concerned with the derivative nonlinear schrodinger equation with periodic boundary conditions i u t u x x i f u 2 u x 0 x t r 2 z where f is real analytic in some neighborhood of the origin in c f 0 0 and f 0 0 we show the above equation possesses cantor families of smooth quasi periodic solutions of small amplitude the proof is based on an infinite dimensional kam theorem for unbounded perturbation vector fields

a kam theorem for hyperbolic type degenerate lower dimensional tori in hamiltonian systems a kam theorem for degenerate lower dimensional tori in nearly integrable hamiltonian systems is given in this paper for the non degenerate cases both hyperbolic and elliptic the kam theorem has been well established by many authors 8 9 11 13 14 17

analytic solutions of nonlinear elliptic equations on rectangular tori abstract in this paper we consider the nonlinear elliptic equations on rectangular tori using methods in the study of kam theory and anderson localization we prove that these equations admit many analytic solutions

invariant tori for nearly integrable hamiltonian systems with degeneracy does not fall into a hyperplanethrough the origin the condition 1 3 is the sharpest one we have not seenits proof yet in this paper we will give a nondegeneracy condition by meansof the derivatives of h p which is equivalent to the condition 1 3 in analyticcase and under this nondegeneracy condition we obtain the russmann s results for analytic case furthermore the arguments of this paper are available to thenonanalytic case since our nondegeneracy conditions only involve the nite orderderivatives of h p main resultstheorem a suppose that h h p f p q is analytic in t

exponential stability for nonlinear filtering of diffusion processes in a noncompact domain the optimal nonlinear filtering problem for a diffusion process in a noncompact domain observed in white noise is considered it is assumed that the process is ergodic the diffusion coefficient is constant and the observation is linear using known bounds on the conditional density it is shown that when the observation noise is sufficiently small the filter is exponentially stable and that the decay rate of the total variation distance between differently initialized filtering processes tends to infinity as the noise intensity approaches zero

a nekhoroshev type theorem for the nonlinear wave equation abstract we prove a nekhoroshev type theorem for the nonlinear wave equation nlw 0 1 u t t u x x m u f u under dirichlet boundary conditions where f is an analytic function more precisely we prove the subexponential long time stability result of the origin for equation 0 1 by using birkhoff normal form technique for infinite dimensional hamiltonian systems and the so called tame property of the nonlinearity in gevrey or analytic space which is inspired by the work 6 where the authors proved the polynomial long time stability result in sobolev space for a class of hamiltonian pdes

invariant cantor manifolds of quasi periodic solutions for the derivative nonlinear schrodinger equation abstract this paper is concerned with the derivative nonlinear schrodinger equation with periodic boundary conditions i u t u x x i f x u u x 0 x t r 2 z where f is an analytic function of the form f x u u u 2 u f 4 x u u 0 r and f 4 x u u denotes terms of order at least four in u u we show the above equation possesses cantor families of smooth quasi periodic solutions of small amplitude the proof is based on an infinite dimensional kam theorem for unbounded perturbation vector fields

periodic solutions for a class of nonlinear partial differential equations in higher dimension we prove the existence of periodic solutions in a class of nonlinear partial differential equations including the nonlinear schrodinger equation the nonlinear wave equation and the nonlinear beam equation in higher dimension our result covers cases of completely resonant equations where the bifurcation equation is infinite dimensional such as the nonlinear schrodinger equation with zero mass for which solutions which at leading order are wave packets are shown to exist

an infinite dimensional kam theorem and its application to the two dimensional cubic schrodinger equation abstract we prove an infinite dimensional kam theorem as an application we use the theorem to study the two dimensional nonlinear schrodinger equation i u t u u 2 u 0 t r x t 2 with periodic boundary conditions we obtain for the equation a whitney smooth family of small amplitude quasi periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamical system

quasi periodic solutions for 1d wave equation with higher order nonlinearity abstract in this paper one dimensional 1d nonlinear wave equation u t t u x x m u u 5 0 on the finite x interval 0 with dirichlet boundary conditions is considered it is proved that there are many 2 dimensional elliptic invariant tori and thus quasi periodic solutions for the above equation the proof is based on infinite dimensional kam theory and partial birkhoff normal form

a note on exponential stability of the nonlinear filter for denumerable markov chains abstract asymptotic stability of the optimal filter with respect to its initial conditions is investigated in this paper under the assumption that the observation function is one to one and the observation noise is sufficiently small it is shown that exponential stability of the nonlinear filter holds for a large class of denumerable markov chains including all finite markov chains throughout this paper ergodicity of the signal process is not assumed

construction of quasi periodic solutions of delay differential equations via kam techniques abstract this work focuses on the existence of quasi periodic solutions for linear autonomous delay differential equation under quasi periodic time dependent perturbation near an elliptic hyperbolic equilibrium point using the time 1 map of the solution operator newton iteration scheme space splitting and kam techniques it is shown that under appropriate hypothesis there exist quasi periodic solutions with the same frequencies as the perturbation for most parameters we show that if the delay differential equation is analytic we obtain analytic parameterizations of the solutions

kam tori for 1d nonlinear wave equations with periodic boundary conditions in this paper one dimensional 1d nonlinear wave equations with periodic boundary conditions are considered v is a periodic smooth or analytic function and the nonlinearity f is an analytic function vanishing together with its derivative at u 0 it is proved that for most potentials v x the above equation admits small amplitude periodic or quasi periodic solutions corresponding to finite dimensional invariant tori for an associated infinite dimensional dynamical system the proof is based on an infinite dimensional kam theorem which allows for multiple normal frequencies

asymptotic stability of the wonham filter ergodic and nonergodic signals the stability problem of the wonham filter with respect to initial conditions is addressed the case of ergodic signals is revisited in view of a gap in the classic work of h kunita 1971 we give new bounds for the exponential stability rates which do not depend on the observations in the nonergodic case the stability is implied by identifiability conditions formulated explicitly in terms of the transition intensities matrix and the observation structure

exponential stability of discrete time filters for bounded observation noise this paper proves exponential asymptotic stability of discrete time filters for the estimation of solutions to stochastic difference equations when the observation noise is bounded no assumption is made on the ergodicity of the signal the proof uses the hilbert projective metric introduced into filter stability analysis by atar and zeitouni 1 2 it is shown that when the signal noise is sufficiently regular boundedness of the observation noise implies that the filter update operation is on average a strict contraction with respect to the hilbert metric asymptotic stability then follows

lagrange stability for duffing type equations abstract in this paper the duffing type equation d 2 x dt 2 x 2 n 1 p 2 n t x 2 n p 1 t x p 0 t 0 is studied where the p j s are 1 periodic it is shown that all solutions of this equation are bounded provided that the p j t 0 j n are of bounded variation in 0 1 and that the derivatives of p j t n j 2 n are lipschitzian it is also shown that there exist p j s being discontinuous everywhere such that all solutions of the equation are bounded this implies that the continuity of p j s is not necessary for the boundedness of solutions of the equation

quasi periodic solutions of a non autonomous wave equations with quasi periodic forcing abstract in this paper we prove existence of small amplitude quasi periodic solutions for a non autonomous quasi periodically forced nonlinear wave equations with periodic spatial boundary conditions via kam theory

on discrete time ergodic filters with wrong initial data for a class of non uniformly ergodic markov chains x n satisfying exponential or polynomial beta mixing under observations y n subject to an iid noise with a positive density it is shown that wrong initial data is forgotten in the mean total variation topology with a certain exponential or polynomial rate

invariant tori for a derivative nonlinear schrodinger equation with quasi periodic forcing this paper is concerned with a one dimensional derivative nonlinear schrodinger equation with quasi periodic forcing under periodic boundary conditions iut uxx ig t f u2 u x 0 x t r 2 z where g t is real analytic and quasi periodic on t with frequency vector 1 2 m f is real analytic in some neighborhood of the origin in f 0 0 and f 0 0 we show that the above equation admits cantor families of smooth quasi periodic solutions of small amplitude the proof is based on an abstract infinite dimensional kolmogorov arnold moser theorem for unbounded perturbation vector fields and partial birkhoff normal form

time periodic solutions to the one dimensional nonlinear wave equation this paper is concerned with the time periodic solutions to the one dimensional nonlinear wave equation with either variable or constant coefficients by adjusting the basis of l2 function space we can circumvent the difficulties caused by u 0 and obtain the existence of a weak periodic solution which was posed as an open problem by baubu and pavel in trans am math soc 349 2035 2048 1997 finally an application to the forced sine gordon equation is presented to illustrate the utility of this technique

quasi periodic solutions of completely resonant nonlinear wave equations it is shown that there are many elliptic invariant tori and thus quasi periodic solutions for the completely resonant nonlinear wave equation subject to periodic boundary conditions via kam theory

asymptotic ergodicity of the process of conditional law in some problem of non linear filtering abstract we consider a sample of a trajectory of an ergodic process m x through a stationary noisy observation process we study the asymptotic ergodicity of the phase process

computation of topological indices of certain networks abstract there are certain types of topological indices such as degree based topological indices distance based topological indices and counting related topological indices etc among degree based topological indices the so called atom bond connectivity abc geometric arithmetic ga are of vital importance these topological indices correlate certain physico chemical properties such as boiling point stability and strain energy etc of chemical compounds in this paper we compute abc 4 and ga 5 indices for certain networks like silicate chain silicate hexagonal oxide and honeycomb networks the atom bond connectivity index abc index and geometric arithmetic index ga index for oxide and chain silicate networks are also computed in this paper

note on two generalizations of the randic index for a given graph g the well known randic index of g introduced by milan randic in 1975 is defined as r g u v e g d u d v 1 2 where the sum is taken over all edges uv and du denotes the degrees of u bollobas and erdos generalized this index by replacing 1 2 with any real number which is called the general randic index dvo ak et al introduced a modified version of randic index r g u v e g max d u d v 1 based on this recently knor et al introduced two generalizations r g u v e g min d u d v and r g u v e g max d u d v for any real number clearly the former is a lower bound for the general randic index and the latter is its upper bound knor et al studied extremal values of r g and r g and concluded some open problems in this paper we consider the open problems and give some comments and results some results for chemical trees are obtained

approximation and limit results for nonlinear filters over an infinite time interval part ii random sampling algorithms the paper is concerned with approximations to nonlinear filtering problems that are of interest over a very long time interval since the optimal filter can rarely be constructed one needs to compute with numerically feasible approximations the signal model can be a jump diffusion reflected or not the observations can be taken either in discrete or continuous time the cost of interest is the pathwise error per unit time over a long time interval in a previous paper of the authors a budhiraja and h j kushner siam j control optim 37 1999 pp 1946 1979 it was shown under quite reasonable conditions on the approximating filter and on the signal and noise processes that as time bandwidth process and filter approximation etc go to their limit in any way at all the limit of the pathwise average costs per unit time is just what one would get if the approximating processes were replaced by their ideal values and the optimal filter was used when suitable approximating filters cannot be readily constructed due to excessive computational requirements or to problems associated with a high signal dimension approximations based on random sampling methods or perhaps combinations of sampling and analytical methods become attractive and are the subject of a great deal of attention the work of the previous paper is extended to a wide class of such algorithms under quite broad conditions covering virtually all the cases considered to date it is shown that the pathwise average errors converge to the same limit that would be obtained if the optimal filter was used as time goes to infinity and the approximation parameter goes to its limit in any way at all all the extensions e g wide bandwidth observation or system driving noise in a budhiraja and h j kushner siam j control optim 37 1999 pp 1946 1979 hold for our random sampling algorithms as well

periodic solutions for one dimensional wave equation with bounded nonlinearity abstract this paper is concerned with the periodic solutions for the one dimensional nonlinear wave equation with either constant or variable coefficients the constant coefficient model corresponds to the classical wave equation while the variable coefficient model arises from the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media for finding the periodic solutions of variable coefficient wave equation it is usually required that the coefficient u x satisfies ess inf u x 0 with u x 1 2 u u 1 4 u u 2 which actually excludes the classical constant coefficient model for the case u x 0 it is indicated to remain an open problem by barbu and pavel 1997 6 in this work for the periods having the form t 2 p 1 q p q are positive integers and some types of boundary value conditions we find some fundamental properties for the wave operator with either constant or variable coefficients based on these properties we obtain the existence of periodic solutions when the nonlinearity is monotone and bounded such nonlinearity may cross multiple eigenvalues of the corresponding wave operator in particular we do not require the condition ess inf u x 0