measures for oscillatory sequences abstract the classical concept of oscillation is not sufficient to fully describe the oscillatory behavior of a real sequence by introducing the concepts of frequency measures of subsets of integers and investigating their properties we are able to strengthen the original concept of an oscillatory sequence and thus new oscillation criteria for difference equations can be derived these criteria differ from the classical ones in that they are described by means of the measures of the level sets of the involved parameter sequences a brief description of the theory of frequency measures for multiple sequences is also given

oscillations of nonlinear partial difference systems abstract this paper studies the following two dimensional nonlinear partial difference systems t 1 2 x mn b mn g y mn 0 t 1 2 y mn a mn f x mn 0 where m n n0 0 1 2 t 1 2 1 2 i t 1 2 1 2 i 1ymn ym 1 n ymn 2ymn ym n 1 ymn iymn ymn 1ymn ym 1 n ymn 2ymn ym n 1 ymn amn and bmn are real sequences m n n0 and f g r r are continuous with uf u 0 and ug u 0 for all u 0 a solution xmn ymn of the system is oscillatory if both components are oscillatory some sufficient conditions for all solutions of this system to be oscillatory are derived

data driven operator inference for nonintrusive projection based model reduction abstract this work presents a nonintrusive projection based model reduction approach for full models based on time dependent partial differential equations projection based model reduction constructs the operators of a reduced model by projecting the equations of the full model onto a reduced space traditionally this projection is intrusive which means that the full model operators are required either explicitly in an assembled form or implicitly through a routine that returns the action of the operators on a given vector however in many situations the full model is given as a black box that computes trajectories of the full model states and outputs for given initial conditions and inputs but does not provide the full model operators our nonintrusive operator inference approach infers approximations of the reduced operators from the initial conditions inputs trajectories of the states and outputs of the full model without requiring the full model operators our operator inference is applicable to full models that are linear in the state or have a low order polynomial nonlinear term the inferred operators are the solution of a least squares problem and converge with sufficient state trajectory data in the frobenius norm to the reduced operators that would be obtained via an intrusive projection of the full model operators our numerical results demonstrate operator inference on a linear climate model and on a tubular reactor model with a polynomial nonlinear term of third order

oscillation criteria of first order linear difference equations with delay argument abstract this paper presents new sufficient conditions for the oscillation of all proper solutions of the first order linear difference equation with delay argument u k p k u k 0 k n where u k u k 1 u k p n r n n and lim k k examples illustrating the results are given it is to be pointed out that this is the first paper dealing with the oscillatory behaviour of the equation in the case of a general delay argument k

oscillation for difference equations with continuous variable consider the delay difference equation y t y t p t h y t t t 0 where 0 0 and p c r r f c r r h c r r some oscillation criteria for this equation are obtained

frequency measures on time scales with applications the concept of frequency measures of subsets of a time scale is introduced and the relevant properties are discussed then frequent oscillation is defined to strengthen the classical concept of oscillation applications are shown by deriving oscillation criteria for first order dynamic equations on time scales

multiple time scale analysis of interacting diffusions we consider the questions how can the long term behavior of large systems of interacting components be described in terms of infinite systems on what time scale does the infinite system give a qualitatively correct description and what happens at large resp critical time scales lety n t be a solution y i n t i n n of the system of stochastic differential equations w i t are i i d brownian motions dy i n t left frac 1 2n 1 sum limits j n n y j n t y i n t right dt sqrt 2g y i n t dw i t in the mckean vlasov limit n we obtain the infinite independent system dy i infty t e y i infty t y i infty t dt sqrt 2g y i infty t dw i t i in z this infinite system has a one parameter set of invariant measures v theta mathop otimes limits x in z gamma theta with the equilibrium measure of dx t theta x t dt sqrt 2g x t dw t letq s be the transition kernel of the diffusion with generator u g x left frac partial partial x right 2 with u g x int g y gamma x dy then one main result is that asn mathcal l y n s 2n 1 rightarrow int q s theta d theta v theta theta e y 0 this provides a new example of a phenomenon also exhibited by the voter model and branching random walk in particular we are also able to modify our model by adding the termcn 1 a y i n t dt to obtain the first example in which the analog ofq s converges to an honest equilibrium instead of absorption in traps as in all models previously studied in the literature finally we discuss a hierarchical model with two levels from the point of view discussed above but now in two fast time scales

comparison theorems and oscillation criteria for difference equations abstract first we establish the equivalence of the oscillation of the delay difference equation x n p n x n k 0 and the second order difference equation 2 y n 1 2 k 1 k k k 1 p n k k k 1 k 1 y n 0 where pn is a sequence of nonnegative real numbers and k is a positive integer next we obtain some sharp conditions for oscillations and nonoscillations of the first equation

coalescing random walks and voter model consensus times on the torus in mathbb z d voter models on z d and on the torus of site n in z d are considered the second ones get trapped with probability 1 as t tends to infinity at all 0 s or all 1 s the asymptotic growth of these trapping times are examined as n tends to infinity this is done with limit thewems for coalescing random walk systems on the torus of side n in z d

oscillation of second order nonlinear difference equation with continuous variable in this paper we are mainly concerned with the second order nonlinear difference equation with continuous variable here by using the iterated integral transformations generalized riccati transformations and integrating factors we give some oscillatory criteria for this equation

asymptotic stability for a linear difference equation with variable delay we shall obtain sufficient conditions for the uniform stability and the global asymptotic stability of the linear difference equation with variable delay xn 1 xn pnxn kn 0 n n where pn is a sequence of nonnegative real numbers kn is a sequence of nonnegative integers and there exists a nonnegative integer k such that kn k for n n

a note on oscillation of second order nonlinear difference equation with continuous variable abstract in this paper by improving the proofs of some theorems in j math anal appl 255 2001 349 357 we obtain some new oscillation criteria for the second order nonlinear difference equation with continuous variable

oscillation of partial difference equations with variable coefficients for the partial difference equations am 1 n am n 1 am n pm nam k n 1 0 and am 1 n am n 1 am n i 1upi m n fi am ki n li 0 we shall obtain sufficient conditions for the oscillation of all solutions

invariant measures of critical spatial branching processes in high dimensions we consider two critical spatial branching processes on mathbb r d critical branching brownian motion and the critical dawson watanabe process a basic feature of these processes is that their ergodic behavior is highly dimension dependent it is known that in low dimensions d leq 2 the only invariant measure is delta 0 the unit point mass on the empty state in high dimensions d geq 3 there is a family nu theta theta epsilon 0 infty of extremal invariant measures the measures nu theta are translation invariant and indexed by spatial intensity we prove here for d geq 3 that all invariant measures are convex combinations of these measures

oscillations of delay difference equations with oscillating coefficients abstract in this paper we establish sufficient conditions for the oscillation of all solutions of the delay difference equation y n 1 y n p n y n k 0 n 0 1 2 where pn is a sequence of real numbers and k is a positive integer an open problem in ladas in international conference on differential equations stability and control dekker new york 1990 is solved

different clustering regimes in systems of hierarchically interacting diffusions we study a system of interacting diffusions dx t a x t x t dt g x t dw t indexed by the hierarchical group as a genealogical two genotype model where x t denotes the frequency of say type a with hierarchically determined degrees of relationship between colonies in the case of short interaction range it is known that the system clusters that is locally one genotype dies out we focus on the description of the different regimes of cluster growth which is shown to depend on the interaction kernel a via its recurrent potential kernel one of these regimes will be further investigated by mean field methods for general interaction range we shall also relate the behavior of large finite systems indexed by finite subsets n of to that of the infinite system on the way we will establish relations between hitting times of random walks and their potentials

data driven parametrized model reduction in the loewner framework we introduce a new method for constructing reduced order models that accurately approximate the frequency domain behavior of large scale parametrized linear dynamical systems using multivariate rational interpolation we compute reduced order models that match transfer function measurements of the large scale system the main tools are a new barycentric formula and loewner matrices formed directly from measurements this data driven approach introduces a new degree of freedom for parametrized model reduction the ability to choose separate reduced orders for each parameter more precisely each reduced order is determined by computing the rank of appropriate loewner matrices moreover we show how to control the pointwise approximation error through a new formula involving the barycentric form and the smallest singular value of a loewner matrix finally we also give new state space realizations for multivariate rational functions and conclude with several numerical examples showcasing the effectivene

finite and infinite systems of interacting diffusions we study the problem of relating the long time behavior of finite and infinite systems of locally interacting components we consider in detail a class of lincarly interacting diffusionsx t x i t i d in the regime where there is a one parameter family of nontrivial invariant measures for these systems there are naturally defined corresponding finite systems x n t left x i n t i in lambda n right with lambda n n n d cap mathbb z d our main result gives a comparison between the laws ofx t n andx n t n for timest n asn the comparison involves certain mixtures of the invariant measures for the infinite system

on the long term behavior of some finite particle systems we consider the problem of comparing large finite and infinite systems with locally interacting components and present a general comparison scheme for the case when the infinite system is nonergodic we show that this scheme holds for some specific models one of these is critical branching random walk onz d let t denote this system and let denote a finite version of t defined on the torus n n d z d ford 3 we prove that for stationary shift ergodic initial measures with density that ift n andt n 2n 1 d s 0 asn then v 0 is the set of extremal invariant measures for the infinite system t andq s is the transition function of feller s branching diffusion we prove several extensions and refinements of this result the other systems we consider are the voter model and the contact process

algorithms for model reduction of large dynamical systems three algorithms for the model reduction of large scale continuous time time invariant linear dynamical systems with a sparse or structured transition matrix and a small number of inputs and outputs are described they rely on low rank approximations to the controllability and observability gramians which can efficiently be computed by adi based iterative low rank methods the first two model reduction methods are closely related to the well known square root method and schur method which are balanced truncation techniques the third method is a heuristic balancing free technique the performance of the model reduction algorithms is studied in numerical experiments

the asymptotic and oscillatory behavior of neutral difference inclusions in this paper the asymptotic and oscillatory properties of solutions to the first order neutral difference inclusion of the email protected x n r nx n t p nf x n s 1 x n s 2 x n s s n 0 1 2 are discussed where f r s 2 r 1 is a compact convex upper semicontinuous multifunction some examples are given to illustrate main theorems

an oscillation criterion for a delay difference equation by using the method of yu zhang and qian in j s yu b g zhang x z qian oscillation of delay difference equations with oscillating coefficients j math anal appl 177 1993 432 444 we prove another oscillation criterion for all solutions of the following linear delay difference equation x n 1 x n p nx n k 0 n 0 1 2 where p n is a sequence of real numbers and k is a positive integer

oscillation criteria for first order nonlinear delay difference equations in this paper two interesting oscillation criteria are obtained for all solutions of the nonlinear delay difference equation of the formx n 1 x n p nf x n l 1 x n l 2 x n l m 0 n 0 1 2 the results extend some well known results in the literature and two examples are given to demonstrate the advantage of our results

an oscillation criterion for linear difference equations with oscilling coefficients abstract relatively few oscillation criteria have been established for delay difference equations of the form yn 1 yn qnyn 0 where qn is an oscillatory sequence in this note we make use of the convexity property of the function x ln x to obtain such a criterion which complements some of the existing results even when qn is nonnegative

boundedness and asymptotic behavior results for nonlinear difference equations with positive and negative coefficients this paper is concerned with the nonlinear delay difference equations with positive and negative coefficients x n 1 x n p n f x n k q n f x n l 0 n n 0 sufficient conditions are obtained under which every solution of eq is bounded and tends to a constant as n

on two variable rational interpolation abstract the goal of this contribution is to investigate interpolation of two variable rational functions the tool is the two variable loewner matrix which is an extension of its single variable counterpart the main property of the loewner matrix is that its rank encodes the information concerning minimal complexity interpolants both polynomial and generalized state space descriptor realizations of interpolants are presented examples illustrate how two variable rational functions can be recovered from appropriate measurements

oscillation criteria for delay and difference equations with non monotone arguments consider the first order linear delay differential equation 1 x t p t x t t 0 t t 0 where p email protected c t 0 r t t t 0 and lim t email protected t and the discrete analogue difference equation 1 dx n p n x t n 0 n 0 1 2 where d denotes the forward difference operator dx n x n 1 x n p n is a sequence of nonnegative real numbers and t n is a sequence of integers such that t n 0 and lim n email protected n the state of the art on the oscillation of all solutions to these equations are established especially in the case of non monotone arguments examples illustrating the results are given

oscillation theorems for second order nonlinear partial difference equations 479 abstract the oscillation of the second order nonlinear partial difference equation t 1 2 c mn 1 y mn i 1 s a i m n f i y m 1 n 1 y mn 0 is investigated some sufficient conditions for the oscillation of all solutions of the above equation are obtained

a framework for the solution of the generalized realization problem in this paper we present a novel way of constructing generalized state space representations e a b c d of interpolants matching tangential interpolation data the loewner and shifted loewner matrices are the key tools in this approach

oscillation of a class of partial difference equations with unbounded delay abstract this paper is concerned with the partial difference equation where and n z are nondecreasing n and n are strictly less than n for n n lim n n lim n n a m n b m n and p m n are three real double sequences some oscillation criteria for this equation are obtained

new oscillation criteria for delay difference equations abstract this paper is concerned with the oscillation of all solutions of the delay difference equations and where pn and pi n are sequences of nonnegative real numbers an k and ki are positive integers new oscillation criteria of the forms for equation and for equation are obtained where lim infn 1 k n 1i n k pi and c is the the best possible function of in some sense

nonexistence of positive solution of a class of nonlinear difference equation sufficient conditions are derived for the nonexistence of positive solution of a class of nonlinear difference equations the conditions are based on nonexistence of positive characteristic roots of the characteristic equation of the nonlinear difference equation

interpolatory projection methods for parameterized model reduction we provide a unifying projection based framework for structure preserving interpolatory model reduction of parameterized linear dynamical systems i e systems having a structured dependence on parameters that we wish to retain in the reduced order model the parameter dependence may be linear or nonlinear and is retained in the reduced order model moreover we are able to give conditions under which the gradient and hessian of the system response with respect to the system parameters is matched in the reduced order model we provide a systematic approach built on established interpolatory mathcal h 2 optimal model reduction methods that will produce parameterized reduced order models having high fidelity throughout a parameter range of interest for single input single output systems with parameters in the input output maps we provide reduced order models that are optimal with respect to an mathcal h 2 otimes mathcal l 2 joint error measure the capabilities of these approaches are illustrated by several numerical examples from technical applications

a machine learning minimal residual ml mres framework for goal oriented finite element discretizations abstract we introduce the concept of machine learning minimal residual ml mres finite element discretizations of partial differential equations pdes which resolve quantities of interest with striking accuracy regardless of the underlying mesh size the methods are obtained within a machine learning framework during which the parameters defining the method are tuned against available training data in particular we use a provably stable parametric petrov galerkin method that is equivalent to a minimal residual formulation using a weighted norm while the trial space is a standard finite element space the test space has parameters that are tuned in an off line stage finding the optimal test space therefore amounts to obtaining a goal oriented discretization that is completely tailored towards the quantity of interest we use an artificial neural network to define the parametric family of test spaces using numerical examples for the laplacian and advection equation in one and two dimensions we demonstrate that the ml mres finite element method has superior approximation of quantities of interest even on very coarse meshes

oscillations of delay difference equations consider the delay difference equation n 1 n pn n k 0 n 0 1 2 where pn is a sequence of real numbers and k is a positive integer new oscillation criteria for this equation are established an extension of the results to difference equations with several delays is also presented

model reduction for large scale systems with high dimensional parametric input space a model constrained adaptive sampling methodology is proposed for the reduction of large scale systems with high dimensional parametric input spaces our model reduction method uses a reduced basis approach which requires the computation of high fidelity solutions at a number of sample points throughout the parametric input space a key challenge that must be addressed in the optimization control and probabilistic settings is the need for the reduced models to capture variation over this parametric input space which for many applications will be of high dimension we pose the task of determining appropriate sample points as a pde constrained optimization problem which is implemented using an efficient adaptive algorithm that scales well to systems with a large number of parameters the methodology is demonstrated using examples with parametric input spaces of dimension 11 and 21 which describe thermal analysis and design of a heat conduction fin and compared with statistically based sampling methods for these examples the model constrained adaptive sampling leads to reduced models that for a given basis size have error several orders of magnitude smaller than that obtained using the other methods

diffusive clustering in an infinite system of hierarchically interacting diffusions we study a countable system of interacting diffusions on the interval 0 1 indexed by a hierarchical group a particular choice of the interaction guaranties we are in the diffusive clustering regime this means clusters of components with values either close to 0 or close to 1 grow on various different scales however single components oscillate infinitely often between values close to 0 and close to 1 in such a way that they spend fraction one of their time together and close to the boundary the processes in the whole class considered and starting with a shift ergodic initial law have the same qualitative properties universality

some oscillation criteria for difference equations abstract consider the difference equation x n 1 x n p n x n k 0 where p n is a sequence of nonnegative real numbers and k is a positive integer it is proved that all solutions oscillate if there exists some positive integer l such that lim sup n i 0 k pn i i 0 k j 1 k pn i j m 0 l 1 i 0 k j 0 m 1 pn jk 1 1 or lim sup n i 1 k pn i i 1 k j 1 k pn i j m 0 l 1 i 0 k j 0 m 1 pn jk 1 1

a further result on the oscillation of delay difference equations in this paper we are concerned with the delay difference equations of the form yn 1 yn pnyn k 0 n 0 1 2 where pn 0 and k is a positive integer we prove by using a new technique that n 0 i nn kpi ln i nn k pi 1 sign i nn k pi i n 1n k pi ln i n 1n k pi 1 sign i n 1n k pi guarantees that all solutions of equation oscillate which improves many previous well known results in particular our theorems also fit the case where n 1i n kpi kk 1 k 1 k 1 in addition we present a nonoscillation sufficient condition for equation

asymptotic behavior results for nonlinear neutral delay difference equations abstract this paper is concerned with the nonlinear neutral delay difference equation x n c n x n m p n f x n k 0 n n n 0 where is the forward difference operator defined by x n x n 1 x n c n is a sequence of real numbers p n is a positive sequence f c r r m and k are positive integers n 0 is a nonnegative integer and n n 0 n 0 n 0 1 n 0 2 sufficient conditions are obtained under which every solution of equation is bounded and tends to a constant as n our results improve and extend some known results one example is given to illustrate our results

oscillation of nonlinear delay difference equations in this paper some new oscillation criteria are obtained for all solutions of the nonlinear delay difference equation of the form xn 1 xn pnf xn k 0 n 0 1 2 our results also improve some well known results in the literature even when applying them to the linear delay difference equation

a necessary and sufficient condition for the oscillation of first order linear difference equations with several delay arguments abstract in this article we discuss the oscillation of all solutions of a first order linear delay difference equation with several delay arguments and obtain some new oscillation criteria under more relaxed conditions in particular we establish a necessary and sufficient condition only based on the coefficient functions and delay arguments when the coefficient functions are non negative examples and tables are given to illustrate our results

oscillation of nonlinear delay difference equations abstract in this paper two interesting oscillation criteria are obtained for all solutions of the nonlinear delay difference equations of the form yn 1 yn pnf yn k 0 n 0 1 2 some applications are given to demonstrate the advantage of results obtained in this paper our results also improve the existing results in the literature even when applying them to the linear delay difference equation

necessary and sufficient condition for oscillation of higher order nonlinear delay difference equations this paper is concerned with the oscillation of solutions of higher order nonlinear delay difference equations with forcing terms of the form nx t f t x t x t h t t i 0 1 where is the forward difference operator defined by x t x t 1 x t and mx t m 1x t m 1 a necessary and sufficient condition is established under which every solution x t is oscillatory when n is even and is either oscillatory or strongly monotone when n is odd a sufficient condition for the oscillation of solutions of neutral type delay difference equations is also obtained

oscillation of delay difference equation we introduce some new techniques to analyze generalized characteristic equations to obtain several infinite sum sharp conditions for oscillation of the following delay difference equation xn 1 xn i 1mpi n xn ki 0 n 0 1 2 where pi n 0 and ki are positive integers

oscillation of delay difference equations with several delays abstract in this paper we obtain some new oscillation criteria for the difference equation with several delays x n 1 x n i 1 m p i n x n k i 0 n 0 1 2 which improve the existing results in the literature the mistakes in comput math appl 44 2002 817 821 are pointed out

frequent oscillatory criteria forpartial difference equations with several delays this paper is concerned with partial difference equations with several delays of theform aam 1 n bam n 1 dam n i 1rpi m n am i n i 0 m n 0 1 2 where a b and d are three positive real constants i i and are positive integers and pi m n are real double sequences i 1 2 some new frequent oscillation criteria for this equation are derived

new results for oscillation of delay difference equations in this paper we consider the delay difference equation xn 1 xn pnxn k 0 n 0 1 2 where pn is a sequence of nonnegative real numbers and k is a positive integer some new results for the oscillation of this equation are obtained our theorems improve all known results in the literature

data driven pod galerkin reduced order model for turbulent flows abstract in this work we present a reduced order model which is specifically designed to deal with turbulent flows in a finite volume setting the method used to build the reduced order model is based on the idea of merging combining projection based techniques with data driven reduction strategies in particular the work presents a mixed strategy that exploits a data driven reduction method to approximate the eddy viscosity solution manifold and a classical pod galerkin projection approach for the velocity and the pressure fields respectively the newly proposed reduced order model has been validated on benchmark test cases in both steady and unsteady settings with reynolds up to r e o 10 5

diffusive clustering in the two dimensional voter model on donne des resultats de simulation et une enquete concernant les resultats connus sur le modele du votant dans trois cas distincts trois ou plusieurs dimensions une dimension et deux dimensions

the semicycles of solutions of delay difference equations in this paper we study the semicycles of oscillatory solutions of the delay difference equation yn 1 yn pnyn k 0 where pn is is a sequence of nonnegative real numbers and k is a positive integer upper bound of numbers of terms of semicycles are determined

oscillation results for difference equations with several oscillating coefficients this paper presents a new sufficient condition for the oscillation of all solutions of difference equations with several deviating arguments and oscillating coefficients corresponding difference equations of both retarded and advanced type are studied examples illustrating the results are also given

semicycles of solutions of nonlinear difference equations with several delays abstract in this paper we study the semicycles of oscillatory solutions of nonlinear difference equation with several delays where pi n is a real sequence with pi n 0 for all large n l mi kij are positive integers ij i 1 2 l j 1 2 mi are nonnegative real numbers with mij 1 ij 1 i 1 2 l upper bound of numbers of terms of semicycles are determined

oscillations of delay difference equations in a critical state consider the first order delay difference equation xn i 1m pi n xn ki 0 n 0 1 2 in the case where lim infn pi n pi 0 ki 0 i 1 2 a necessary and sufficient condition for the oscillation of all solutions of the above equation is established in the critical state that the corresponding limiting equation xn i 1m pixn ki 0 n 0 1 2 admits a nonoscillatory solution it is to be pointed out that there is no result in this critical state for a delay difference equation with more than one delay argument

an estimate of numbers of terms of semicycles of delay difference equations in this paper we study the semicycles of oscillatory solutions of the delay difference equation yn 1 yn pnyn k 0 where pn is a sequence of nonnegative real numbers and k is a positive integer upper bound of numbers of terms of semicycles are determined in the case when our results improve and complement known results in literature

oscillatory and asymptotic behavior of positive periodic solutions of nonlinear discrete model exhibiting the allee effect in this paper we shall consider the discrete nonlinear delay population model with allee effectx n 1 x n exp a n b n x p email protected c n x q email protected where a n b n and c n are positive sequences of period w and p and q are positive integers we will establish some sufficient conditions for the oscillation of all positive solutions about its positive periodic solution x n and prove that every nonoscillatory solution converges to x n monotonically as n

persistence of critical multitype particle and measure branching processes we consider a class of systems of particles ofk types inr d undergoing spatial diffusion and critical multitype branching where the diffusions the particle lifetimes and the branching laws depend on the types we prove persistence criteria for such systems and for their corresponding high density limits known as multitype dawson watanabe processes the main tool is a representation of the palm distributions for a general class of inhomogeneous critical branching particle systems constructed by means of a backward tree

oscillation of parabolic neutral delay difference equations with several positive and negative coefficients we obtain some oscillation criteria for the parabolic neutral delay difference equations with several positive and negative coefficients

oscillation for even order delay difference equations with unstable type in this paper we first prove the even odd delay difference equation with unstable type d mx email protected i 1 np i n x n k i n 0 1 2 has always a positive solution which tends infinity as n then we obtain some oscillation criteria for all bounded solutions of eq

nonexistence of unbounded nonoscillatory solutions of partial difference equations abstract we develop criteria for the nonexistence of eventually positive negative and nondecreasing nonincreasing solutions of the partial difference equation m n y m n p m n y m k n l q m n y m k n l and m n y m n i 1 p i m n y m k i n l i i 1 q i m n y m k i n l i several examples which dwell upon the importance of our results are also included

oscillations of difference equations with several delays abstract consider the first order delay difference equation where lim infn pi n pi 0 ki 0 i 1 2 sufficient conditions for the oscillation of all solutions of the above equation are established in the case when the corresponding limiting equation admits nonoscillatory solutions oscillation criteria for the nonlinear difference equation are also derived as applications

multifidelity importance sampling estimating statistics of model outputs with the monte carlo method often requires a large number of model evaluations this leads to long runtimes if the model is expensive to evaluate importance sampling is one approach that can lead to a reduction in the number of model evaluations importance sampling uses a biasing distribution to sample the model more efficiently but generating such a biasing distribution can be difficult and usually also requires model evaluations a different strategy to speed up monte carlo sampling is to replace the computationally expensive high fidelity model with a computationally cheap surrogate model however because the surrogate model outputs are only approximations of the high fidelity model outputs the estimate obtained using a surrogate model is in general biased with respect to the estimate obtained using the high fidelity model we introduce a multifidelity importance sampling mfis method which combines evaluations of both the high fidelity and a surrogate model it uses a surrogate model to facilitate the construction of the biasing distribution but relies on a small number of evaluations of the high fidelity model to derive an unbiased estimate of the statistics of interest we prove that the mfis estimate is unbiased even in the absence of accuracy guarantees on the surrogate model itself the mfis method can be used with any type of surrogate model such as projection based reduced order models and data fit models furthermore the mfis method is applicable to black box models i e where only inputs and the corresponding outputs of the high fidelity and the surrogate model are available but not the details of the models themselves we demonstrate on nonlinear and time dependent problems that our mfis method achieves speedups of up to several orders of magnitude compared to monte carlo with importance sampling that uses the high fidelity model only

nonexistence and existence of positive solutions for difference equations with unbounded delay consider the delay difference equation xn 1 xn pnx n 0 0 1 2 where n z n n for n n and limn n pn r some existence results of positive solutions and oscillation criteria for this equation are obtained

oscillation in nonlinear difference equations abstract in this paper we shall discuss oscillatory behavior of the solutions of difference equations including the self adjoint second order linear equation and the discrete version of the nonlinear wave equation our work is to give sufficient conditions such that every nontrivial solution of the equations oscillates

permanence oscillation and attractivity of the discrete hematopoiesis model with variable coefficients abstract in this paper we consider the nonlinear delay difference equation with periodic coefficients p n 1 p n n p n n 1 p n m and demonstrate that solutions are permanent the existence of a positive periodic solution is proven and sufficient conditions for the attractivity of this solution are obtained oscillation and nonoscillation about the positive periodic solution is considered

oscillation criteria for difference equations with unbounded delay consider the delay difference equation n 1 n pn n 0 n 0 1 2 where n z is nondecreasing n n for n n and limn n pn is a nonnegative sequence some oscillation criteria for this equation are obtained

a survey of projection based model reduction methods for parametric dynamical systems numerical simulation of large scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena however the inherent large scale nature of the models often leads to unmanageable demands on computational resources model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate yet accurately represent the original large scale system behavior model reduction of linear nonparametric dynamical systems has reached a considerable level of maturity as reflected by several survey papers and books however parametric model reduction has emerged only more recently as an important and vibrant research area with several recent advances making a survey paper timely thus this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods parametric model reduction targets the broad class of problems for wh

multiple scale analysis of clusters in spatial branching models in this paper we will investigate the long time behaviour of critical branching brownian motion and finite variance super brownian motion the so called dawson watanabe process on r d these processes are known to be persistent if d 3 that is there exist non trivial equilibrium measures if d 2 they cluster i e the processes converge to the 0 configuration while the surviving mass piles up in so called clusters we study the spatial profile of the clusters in the critical dimension d 2 via multiple space scale analysis we will also investigate the long time behaviour of these models restricted to finite boxes in d 2 on the way we develop coupling and comparison methods for spatial branching models

sufficient conditions for the oscillation of linear delay difference equations with oscillating coefficients abstract in this paper sufficient conditions for the oscillation of all solutions of the delay difference equation xn 1 xn pnxn k 0 n 0 1 2 are established where the coefficient pn itself may be allowed to be oscillatory we also give an example to demonstrate the advantage of our results

bounded oscillation criteria for second order nonlinear delay difference equations of unstable type this paper deals with some sufficient conditions for the bounded oscillation criteria for the second order nonlinear delay difference equation d 2x email protected i 1mp ix n k i f n x n l 1 x n l s n 0 1 2 in addition we generalize and improve the existing results and then give some examples of applications

oscillation criteria for a class of neutral difference equations with continuous variable in this paper we are mainly concerned with oscillatory behaviour of solutions for a class of second order nonlinear neutral difference equations with continuous variable using an integral transformation the riccati transformation and iteration some oscillation criteria are obtained

frequent oscillatory solutions of a nonlinear partial difference equation this paper is concerned with a class of nonlinear delay partial difference equations with variable coefficients which may change sign by making use of frequency measures some new oscillatory criteria are established

new oscillation criteria for delay difference equations abstract this paper considers the delay difference equation x n 1 x n p n x n k 0 n 0 1 2 where pn is a sequence of nonnegative real numbers and k is a positive integer some new oscillation criteria for this equation are obtained our theorems improve all known results in the literature