kinematical symmetries of three dimensional incompressible flows abstract the motion of an incompressible fluid in lagrangian coordinates involves infinitely many symmetries generated by the left lie algebra of group of volume preserving diffeomorphisms of the three dimensional domain occupied by the fluid utilizing a 1 3 dimensional hamiltonian setting an explicit realization of this symmetry algebra is constructed recursively a dynamical connection is used to split the symmetries into reparametrization of trajectories and one parameter family of volume preserving diffeomorphisms of the fluid domain algebraic structures of symmetries and hamiltonian structures of their generators are inherited from the same construction a comparison with the properties of two dimensional flows is included

a class of label correcting methods for the k shortest paths problem in this paper we deal with the problem of finding the firstk shortest paths from a single origin node to all other nodes of a directed graph in particular we define the necessary and sufficient conditions for a set of distance label vectors on the basis of which we propose a class of methods which can be viewed as an extension of the generic label correcting method for solving the classical single origin all destinations shortest path problem the data structure used is characterized by a set ofk lists of candidate nodes and the proposed methods differ in the strategy used to select the node to be extracted at each iteration the computational results show that 1 some label correcting methods are generally much faster than the double sweep method of shier 1979 2 the most efficient node selection strategies used for solving the classical single origin all destinations shortest path problem have proved to be effective also in the case of thek shortest paths

shape invariant bounds for reaction diffusion systems with unequal diffusion coefficients abstract estimates by invariant regions or shape invariant bounds are a very useful tool in the study of reaction diffusion systems however the class of the regions available for such estimates is severely restricted if the diffusion coefficients are not equal thus many systems cannot be treated by these methods in the present article we generalize the estimates by shape invariant bounds with arbitrary smooth bounded regions to the case of unequal diffusion coefficients the efficiency of our method increases as the quotient of the largest and the smallest diffusion coefficient approaches 1 the bounds are used to prove existence and stability results

a new class of volterra differential equations for which the solutions are globally asymptotically stable on etudie le systeme de lotka volterra i u i e i j 1 m p ij u j u i 0 0 0 t ou p p ij est une matrice reelle mxm et ou i 1 2 m les constantes e i sont choisies de telle maniere qu un point stationnaire q q i existe dans le premier quadrant

on the a posteriori error analysis for equations of prescribed mean curvature we present two approaches to the a posteriori error analysis for prescribed mean curvature equations the main difference between them concerns the estimation of the residual without or with computable weights in the second case the weights are related to the eigenvalues of the underlying operator and thus provide local and computable information about the conditioning we analyze the two approaches from a theoretical viewpoint moreover we investigate and compare the performance of the derived indicators in an adaptive procedure our theoretical and practical results show that it is advantageous to estimate the residual in a weighted way

global stability of volterra models with time delay abstract easily verifiable conditions are obtained for the global stability of the positive steady state in volterra models with discrete and distributed time delays

stability of mass action reaction diffusion systems abstract results on stability of two types of chemical reactions one represented by an acyclic graph and the other as a reversible reaction have been extended to the case of reaction diffusion systems lyapunov functions are used as the major method for showing asymptotic stability of spatially homogeneous equilibria some examples are considered for illustration

on the asymptotic behavior of a class of deterministic models of cooperating species in this paper the global behavior of solutions of a class of ordinary differential equations modeling n cooperating biological species is determined the class of systems includes the classical lot

a posteriori error analysis for the mean curvature flow of graphs we study the equation describing the motion of a nonparametric surface according to its mean curvature flow this is a nonlinear nonuniformly parabolic pde that can be discretized in space via a finite element method we conduct an aposteriori error analysis of the spatial discretization and derive upper bounds on the error in terms of computable estimators based on local residual indicators the reliability of the estimators is illustrated with two numerical simulations one of which treats the case of a singular solution

environmental noise impact on regularity and extinction of population systems with infinite delay abstract populations of biological species are often subject to different types of environmental noises these noises play crucial roles and have significant impact on the evolution and biodiversity to understand the effects of different types of noises on the asymptotic properties of the populations this paper reveals the influence of inherent net birth noise and the interaction noise among stochastically perturbed population systems by considering systems with infinite delays 1 this paper discovers that a the interaction noise may suppress the potential explosion and guarantee the existence of the global solution which yields regularity of the stochastic system and b the inherent net birth noise may lead to extinction of the populations 2 this paper provides sufficient conditions that ensure the stochastic persistence stochastic boundedness and asymptotic polynomial growth of the population system 3 this paper demonstrates that these asymptotic properties are completely determined by stochastic noises and are independent of other parameters

optimal order a posteriori error estimates for a class of runge kutta and galerkin methods we derive a posteriori error estimates which exhibit optimal global order for a class of time stepping methods of any order that include runge kutta collocation rk c methods and the continuous galerkin cg method for linear and nonlinear stiff odes and parabolic pdes the key ingredients in deriving these bounds are appropriate one degree higher continuous reconstructions of the approximate solutions and pointwise error representations the reconstructions are based on rather general orthogonality properties and lead to upper and lower bounds for the error regardless of the time step they do not hinge on asymptotics

coarse graining schemes for stochastic lattice systems with short and long range interactions we develop coarse graining schemes for stochastic many particle microscopic models with competing short and long range interactions on a dimensional lattice we focus on the coarse graining of equilibrium gibbs states and by using cluster expansions we analyze the corresponding renormalization group map we quantify the approximation properties of the coarse grained terms arising from different types of interactions and present a hierarchy of correction terms we derive semi analytical numerical coarse graining schemes that are accompanied by a posteriori error estimates for lattice systems with short and long range interactions

elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems we derive a posteriori error estimates for fully discrete approximations to solutions of linear parabolic equations the space discretization uses finite element spaces that are allowed to change in time our main tool is an appropriate adaptation of the elliptic reconstruction technique introduced by makridakis and nochetto we derive novel a posteriori estimates for the norms of l 0 t l2 and the higher order spaces l 0 t h1 and h1 0 t l2 with optimal orders of convergence

stochastic lotka volterra population dynamics with infinite delay we discover in this paper that when environmental noise is strongly dependent on the population size this noise may suppress the population explosion in a finite time and guarantee the global positive solution when the environmental noise is weakly dependent on the population size the conditions that guarantee the global positive solution are independent of this environmental noise we also discuss the pth moment boundedness stochastic ultimate boundedness and moment average boundedness in time under two classes of conditions these properties are natural requirements from the biological point of view

convergence conditions for some classes of nonlinear systems abstract systems of differential equations with nonlinearities of a sector type and almost periodic perturbations are studied with the aid of the lyapunov direct method conditions are found under which the considered systems admit globally asymptotically stable almost periodic solutions moreover it is shown that the proposed approach permits us to derive new convergence conditions for some models of neural networks and generalized lotka volterra models of population dynamics an example is presented to demonstrate the effectiveness of the obtained results

parametric fem for geometric biomembranes we consider geometric biomembranes governed by an l 2 gradient flow for bending energy subject to area and volume constraints helfrich model we give a concise derivation of a novel vector formulation based on shape differential calculus and corresponding discretization via parametric fem using quadratic isoparametric elements and a semi implicit euler method we document the performance of the new parametric fem with a number of simulations leading to dumbbell red blood cell and toroidal equilibrium shapes while exhibiting large deformations

stability analysis of a fractional order diffused prey predator model with prey refuges abstract in this paper a fractional order diffused prey predator model with prey refuges is considered the existence uniqueness non negativity and boundedness of the solutions for the model are proved moreover some sufficient conditions are given to ensure the existence and uniform asymptotic stability of the equilibrium point of the studied system finally numerical simulations are presented in order to support the theoretical results

finite element discretization of the navier stokes equations with a free capillary surface the instationary navier stokes equations with a free capillary boundary are considered in 2 and 3 space dimensions a stable finite element discretization is presented the key idea is the treatment of the curvature terms by a variational formulation in the context of a discontinuous in time space time element discretization stability in weak energy norms can be proved numerical examples in 2 and 3 space dimensions are given

a new approach to front propagation problems theory and applications in this paper we present a new definition for the global in time propagation motion of fronts hypersurfaces boundaries with a prescribed normal velocity past the first time they develop singularities we show that if this propagation satisfies a geometric maximum principle inclusion avoidance type property then the normal velocity must depend only on the position of the front and its normal direction and principal curvatures this new approach which is more geometric and as it turns out equivalent to the level set method is then used to develop a very general and simple method to rigorously validate the appearance of moving interfaces at the asymptotic limit of general evolving systems like interacting particles and reaction diffusion equations we finally present a number of new asymptotic results among them are the asymptotics of i reaction diffusion equations with rapidly oscillating coefficients ii fully nonlinear nonlocal integral differential equations and iii stochastic ising models with long range anisotropic interactions and general spin flip dynamics

a posteriori error analysis for higher order dissipative methods for evolution problems we prove a posteriori error estimates for time discretizations by the discontinuous galerkin method dg q and the corresponding implicit runge kutta radau method irk r q of arbitrary order q 0 for both linear and nonlinear evolution problems of the form u prime mathfrak f u f with 2 angle bounded operator mathfrak f the key ingredient is a novel higher order reconstruction widehat u of the discrete solution u which restores continuity and leads to the differential equation widehat u prime pi mathfrak f u f for a suitable interpolation operator and piecewise polynomial approximation f of f we discuss applications to linear pde such as the convection diffusion equation 1 2 and the wave equation formally and nonlinear pde corresponding to subgradient operators 1 such as the p laplacian as well as lipschitz operators 1 2 we also derive conditional a posteriori error estimates for the time dependent minimal surface problem

a posteriori error estimates for the crank nicolson method for parabolic equations we derive optimal order a posteriori error estimates for time discretizations by both the crank nicolson and the crank nicolson galerkin methods for linear and nonlinear parabolic equations we examine both smooth and rough initial data our basic tool for deriving a posteriori estimates are second order crank nicolson reconstructions of the piecewise linear approximate solutions these functions satisfy two fundamental properties i they are explicitly computable and thus their difference to the numerical solution is controlled a posteriori and ii they lead to optimal order residuals as well as to appropriate pointwise representations of the error equation of the same form as the underlying evolution equation the resulting estimators are shown to be of optimal order by deriving upper and lower bounds for them depending only on the discretization parameters and the data of our problem as a consequence we provide alternative proofs for known a priori rates of convergence for the crank nicolson method

unified approach fuzzy graph problems we present a taxonomy of fuzzy graphs that treats fuzziness in vertex existence edge existence edge connectivity and edge weight within that framework we formulate some standard graph theoretic problems shortest paths and minimum cut for fuzzy graphs using a unified approach distinguished by its uniform application of guiding principles such as the construction of membership grades via the ranking of fuzzy numbers the preservation of membership grade normalization and the collapsing of fuzzy sets of graphs into fuzzy graphs finally we provide algorithmic solutions to these problems with examples

the lyapunov functionals for delay lotka volterra type models by means of lyapunov functionals we have succeeded in establishing the global attractivity of the positive equilibrium of n species lotka volterra systems modeled by equations of the pure delay type with both finite and infinite delays involved as a corollary we show that if the delay system is dissipative and the coefficient matrix is vl stable then the global attractivity of the unique positive equilibrium is maintained provided the delays are small estimates on the allowable sizes of delays are indicated applications to 2 species prey predator models and n species food chains and competition models are included

low and high frequency oscillations in three dimensional food chain systems the dynamic behavior of a three trophic level food chain prey predator superpredator is analyzed in this paper prey is logistic while predator and superpredator have functional response of the holling type moreover the trophic levels are assumed to be characterized by increasing and quite diversified time responses using a singular perturbation approach explicit conditions for the persistence of the three populations are derived and the structure of the corresponding attractors is noted as well as the nature of transients the analysis shows that the system can have low frequency cycles due to the interactions between predator and superpredator nevertheless the interactions between prey and predator can give rise to high frequency oscillations which can arise during the transients toward the attractor or during the low frequency cycle this periodic burst of high frequency oscillations develops in particular when the two predators are fairly efficient these results note in a very concise w

graph theory based approach for stability analysis of stochastic coupled systems with levy noise on networks in this paper a novel class of stochastic coupled systems with levy noise on networks scslnns is presented both white noise and levy noise are considered in the networks by exploiting graph theory and lyapunov stability theory criteria ensuring p th moment exponential stability and stability in probability of these scslnns are established respectively these principles are closely related to the topology of the network and the perturbation intensity of white noise and levy noise moreover to verify the theoretical results stochastic coupled oscillators with levy noise on a network and stochastic volterra predator prey system with levy noise are performed finally a numerical example about oscillators network is provided to illustrate the feasibility of our analytical results

coexistence and survival in conservative lotka volterra networks analyzing coexistence and survival scenarios of lotka volterra lv networks in which the total biomass is conserved is of vital importance for the characterization of long term dynamics of ecological communities here we introduce a classification scheme for coexistence scenarios in these conservative lv models and quantify the extinction process by employing the pfaffian of the network s interaction matrix we illustrate our findings on global stability properties for general systems of four and five species and find a generalized scaling law for the extinction time

a general monotone scheme for elliptic systems with applications to ecological models we consider weakly coupled elliptic systems of the type with each f i being either an increasing or a decreasing function of each u j assuming the existence of coupled super and subsolutions we prove the existence of solutions and provide a constructive iteration scheme to approximate the solutions we then apply our results to study the steady states of two species interaction in the volterra lotka model with diffusion

global stability for large systems of volterra lotka type integrodifferential population delay equations on considere l equation integrodifferentielle non lineaire x i t x i t e i j 1 n p ij x j t j 1 n o r s ij k ij x j t d i 1 n ou 1 i j n e i p ij s ij sont des constantes on etudie la stabilite globale de la solution q 1 q n par rapport aux solutions positives

dynamics of the attractor in the lotka volterra equations abstract we show that for stable dissipative lotka volterra systems the dynamics on the attractor are hamiltonian and we argue that complex dynamics can occur

adaptive finite element methods for parabolic problems i a linear model problem this paper is the first part in a series of papers on adaptive finite element methods for parabolic problems in this paper an adaptive algorithm is presented and analyzed for choosing the space a

analysis for time discrete approximations of blow up solutions of semilinear parabolic equations we prove a posteriori error estimates for time discrete approximations for semilinear parabolic equations with solutions that might blow up in finite time in particular we consider the backward euler and the crank nicolson methods the main tools that are used in the analysis are the reconstruction technique and energy methods combined with appropriate fixed point arguments the final estimates we derive are conditional and lead to error control near the blow up time

algorithms for computing motion by mean curvature we propose a finite element algorithm for computing the motion of a surface moving by mean curvature the algorithm uses the level set formulation so that changes in topology of the surface can be accommodated stability is deduced by showing that the discrete solutions satisfy both l infty and w 1 1 bounds existence of discrete solutions and connections with brakke flows are established some numerical examples and application to related problems such as the phase field equations are also presented

hamiltonian structures for the n dimensional lotka volterra equations under what algebraic conditions n dimensional lotka volterra equations are hamiltonian for a suitable poisson structure is investigated herein

asymptotic behaviour of comparable skew product semiflows with applications the almost 1 cover lifting property of omega limit sets is established for nonmonotone skew product semiflows which are comparable to uniformly stable eventually strongly monotone skew product semiflows these results are then applied to study the asymptotic behavior of solutions to the nonmonotone comparable systems of odes reaction diffusion systems differential systems with time delays and semilinear parabolic equations

analysis of a stochastic logistic model with diffusion stochastic logistic model with diffusion and levy jumps is proposed critical value between stability and extinction for species in each patch is obtained critical value has a close relationship with levy jumps numerical simulations are introduced to validate the results taking both white noise and levy jump noise into account a stochastic logistic model with diffusion is proposed and considered under some simple assumptions the almost complete parameters analysis of the model is carried out in each case it is shown that the population in each patch is either stable in time average or extinct depending on the parameters of the model especially depending on the intensity of the levy jump noise some simulation figures are introduced to validate the theoretical results

global stability problem for coupled systems of differential equations on networks the global stability problem of equilibria is investigated for coupled systems of differential equations on networks using results from graph theory we develop a systematic approach that allows one to construct global lyapunov functions for large scale coupled systems from building blocks of individual vertex systems the approach is applied to several classes of coupled systems in engineering ecology and epidemiology and is shown to improve existing results

adaptivity and blow up detection for nonlinear evolution problems this work is concerned with the development of a space time adaptive numerical method based on a rigorous a posteriori error bound for a semilinear convection diffusion problem which may exhibit blow up in finite time more specifically a posteriori error bounds are derived in the l infty l 2 l 2 h 1 type norm for a first order in time implicit explicit interior penalty discontinuous galerkin in space discretization of the problem although the theory presented is directly applicable to the case of conforming finite element approximations in space the choice of the discretization in time is made based on a careful analysis of adaptive time stepping methods for odes that exhibit finite time blow up the new adaptive algorithm is shown to accurately estimate the blow up time of a number of problems including one which exhibits regional blow up

diagonal riccati stability and applications abstract we consider the question of diagonal riccati stability for a pair of real matrices a b a necessary and sufficient condition for diagonal riccati stability is derived and applications of this to two distinct cases are presented we also describe some motivations for this question arising in the theory of generalised lotka volterra systems

analysis of autonomous lotka volterra competition systems with random perturbation abstract this paper discusses a randomized n species lotka volterra competition system we show that this system is stable in time average under certain conditions furthermore there is a stationary distribution of this system if extra conditions are satisfied also we give the extinction condition of this system finally numerical simulations are carried out to support our results

adaptive finite elements for a linear parabolic problem a posteriori error estimates for the heat equation in two space dimensions are presented a classical discretization is used euler backward in time and continuous piecewise linear triangular finite elements in space the error is bounded above and below by an explicit error estimator based on the residual numerical results are presented for uniform triangulations and constant time steps the quality of our error estimator is discussed an adaptive algorithm is then proposed successive delaunay triangulations are generated so that the estimated relative error is close to a preset tolerance again numerical results demonstrate the efficiency of our approach c 1998 elsevier science s a all rights reserved

fibonacci heaps and their uses in improved network optimization algorithms in this paper we develop a new data structure for implementing heaps priority queues our structure fibonacci heaps abbreviated f heaps extends the binomial queues proposed by vuillemin and studied further by brown f heaps support arbitrary deletion from an n item heap in o log n amortized time and all other standard heap operations in o 1 amortized time using f heaps we are able to obtain improved running times for several network optimization algorithms in particular we obtain the following worst case bounds where n is the number of vertices and m the number of edges in the problem graph o n log n m for the single source shortest path problem with nonnegative edge lengths improved from o m log m n 2 n o n 2 log n nm for the all pairs shortest path problem improved from o nm log m n 2 n o n 2 log n nm for the assignment problem weighted bipartite matching improved from o nm log m n 2 n o m m n for the minimum spanning tree problem improved from o m log log m n 2 n where m n min i u log i n m n note that m n log n if m n of these results the improved bound for minimum spanning trees is the most striking although all the results give asymptotic improvements for graphs of appropriate densities

stability properties of a general class of nonlinear dynamical systems we establish sufficient conditions for the boundedness of the trajectories and the stability of the fixed points in a class of general nonlinear systems the so called quasi polynomial vector fields with the help of a natural embedding of such systems in a family of generalized lotka volterra lv equations a purely algebraic procedure is developed to determine such conditions we apply our method to obtain new results for lv systems by a reparametrization in time variable and to study general nonlinear vector fields originally far from the lv format

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

adaptive finite element methods for parabolic problems iv nonlinear problems we extend our program on adaptive finite element methods for parabolic problems to a class of nonlinear scalar problems we prove a posteriori error estimates design corresponding adaptive algorithms and present some numerical results

a k shortest path algorithm for adaptive routing in communications networks an efficient and flexible algorithm is presented for finding a k shortest loopless path with distinct initial links from one node to each other node low order polynomial bounds are established for the worst case time complexity of the algorithm showing it to offer a substantial improvement over applying known algorithms to the problem the algorithm can incorporate various extensions including the ability to handle an algebraic objective which enhance its applicability to diverse network models in addition the k shortest path formulation and algorithm are proposed as a basis for network survivability measures where path length bounds exist

nonlinear neural networks for solving the shortest path problem in this paper a new nonlinear neural network for solving the shortest path problem is presented the our nonlinear neural networks are able to generate optimal solutions to the shortest path problem the performance of the our neural networks are demonstrated by means of illustrative example the our neural networks are shown to be capable of generating the shortest path and suitable for electronic implementation

rank of stably dissipative graphs abstract for the class of stably dissipative lotka volterra systems we prove that the rank of its defining matrix which is the dimension of the associated invariant foliation is completely determined by the system s graph

the shortest path problem on a fuzzy time dependent network in this study we introduce a fuzzy time dependent network ftdn and analyze its shortest path problem the ftdn is a network in which travel times are represented as fuzzy sets and are also time dependent under these circumstances the shortest path problem on the ftdn is far more complex in comparison with the shortest path problem on the existing networks to highlight the complexity we show that on the ftdn standard shortest path algorithms e g the well known dijkstra algorithm are not able to come up with solutions subsequently we construct a suitable method which is suitable to deal with the shortest problem a fuzzy programming model is presented for finding the shortest path on the ftdn the proposed model is handled through the techniques which combine mechanisms of fuzzy simulation and genetic optimization in this particular setting fuzzy simulation is exploited to estimate the value of uncertain functions which do not exist in the general networks the proposed model is evaluated with the use of numerical experimentation a comparative analysis demonstrates that the proposed model leads to the shortest path while standard algorithms are not capable of finding the path when dealing with the shortest path problem on the ftdn

computer aided proof for the global stability of lotka volterra systems abstract by modifying the mechanical method of determining lasalle s invariant sets for lotka volterra chain systems 1 2 and repeatedly using wu s characteristic set method 3 4 it is proved that for a class of lotka volterra loop systems the locally asymptotically stable positive equilibrium point must be globally stable

a posteriori error estimates and maximal regularity for approximations of fully nonlinear parabolic problems in banach spaces a posteriori error estimates are provided for discretizations in time of abstract nonlinear parabolic problems u f u by the backward euler method in the maximal regularity framework of banach spaces the estimates are of conditional type i e are valid under assumptions on the approximate solution and the proofs are based on appropriate fixed point arguments

global stability of a multi group sei model in this paper a multi group sei model with the bilinear incidence and variable susceptibility and infectivity is analyzed the susceptible individuals are divided into n groups according to their susceptibility and infected individuals are divided into m groups according to infectivities each group of infected individuals has an exposed period the basic reproduction number r 0 is obtained and globally asymptotic stability of disease free equilibrium is proved if r 0 1 the disease free equilibrium is unstable and the endemic equilibrium exists and is unique which is globally asymptotically stable finally some numerical examples are given to illustrate the above analysis

decay to uniform states in food webs weakly coupled parabolic systems describing populations undergoing diffusion and interaction dynamics are discussed the systems are of lotka volterra type and model complex food webs in which there can be several trophic levels involving predator prey competition commensal and amensal interactions sufficient conditions on the interaction terms are given which imply the existence and global asymptotic stability of a unique equilibrium point with all species coexisting the theory of contracting rectangles is the main tool used in proving the stability result

orbital compactness and asymptotic behaviour of nonlinear parabolic systems with functionals abstract in recent years reaction diffusion systems have become widely used as models in biology chemistry and population dynamics a major point of interest is the long time behavior of the solutions for systems governed by ordinary differential equations the asymptotic behavior is usually investigated using liapunov functionals in conjunction with an invariance principle the purpose of this paper is to extend these methods to a general class of distributed systems that admit possible hysteresis effects in the reaction mechanism keywords nonlinear functions

qualitative stability of linear systems abstract this paper provides a finitely computable graph theoretic answer to the following question concerning linear dynamical systems when given only the signs of entries or 0 in a real square matrix a can one be certain that all positive trajectories of the system ax are bounded matrices having such sign patterns are called sign quasistable with bounded replaced by convergent to the origin the matrices are called sign stable and were fully described in earlier papers however when a s digraph has several strong components so that the system is actually a hierarchy of subsystems and when some of those subsystems fail to be sign stable the recognition of sign quasistability is a very delicate matter by means of certain graph color tests it is possible to identify the system variables that are capable for some choice of matrix entry magnitudes and initial conditions of emitting nonzero constant

a posteriori error estimates for variable time step discretizations of nonlinear evolution equations we study the backward euler method with variable time steps for abstract evolution equations in hilbert spaces exploiting convexity of the underlying potential or the angle bounded condition thereby assuming no further regularity we derive novel a posteriori estimates of the discretization error in terms of computable quantities related to the amount of energy dissipation or monotonicity residual these estimators depend solely on the discrete solution and data and impose no constraints between consecutive time steps we also prove that they converge to zero with an optimal rate with respect to the regularity of the solution we apply the abstract results to a number of concrete strongly nonlinear problems of parabolic type with degenerate or singular character 2000 john wiley sons inc

solution of the stability problem for a class of generalized volterra prey predator systems where ei pij and qj are real constants ui are real valued continuous functions of t for t 0 and zii dui dt both equations are required for i 1 2 m condition lb is here considered to be a restriction on the vector e ei rather than on the matrix p pii as is more usual the hypothesis 0 0 qi 0 is appropriate in applications but as far as the mathematical development is concerned it could be replaced by qiui 0 0 a discussion of historical background can be found in the papers 4 5 to which the present paper is a sequel suffice it to say here that the chief landmarks are volterra s book 6 la salle s theorem 1 21 and krikorian s analysis of the case n 3 3 references 4 5 provide useful

computation of three dimensional dendrites with finite elements starting from an initial seed crystal inside an undercooled liquid the solid phase begins to grow rapidly and develops unstable growth patterns some growth directions are preferred because of anisotropic parameters in the physical model this results in the development of dendrites the physical model includes the heat equation for both the liquid and solid phases the gibbs thomson law couples velocity and curvature of the interface and the temperature we describe a numerical method that enables us to compute dendritic growth of crystals in two and three space dimensions the method consists of two coupled finite element algorithms the first one solves the heat equation the other operates on a discretization of the free boundary and computes the evolution of this moving interface the two methods work with totally independent grids by using timedependent locally refined and coarsened adaptive meshes in both methods we are able to reach a spatial resolution necessary to compute dendritic growth in two and three space dimensions

a posteriori error estimates for nonlinear problems l r o t p ω error estimates for finite element discretizations of parabolic equations using the abstract framework of 9 we analyze a residual a posteriori error estimator for space time finite element discretizations of quasilinear parabolic pdes the estimator gives global upper and local lower bounds on the error of the numerical solution the finite element discretizations in particular cover the so called 0 scheme which includes the implicit and explicit euler methods and the crank nicholson scheme

eigenvalues stability and color tests abstract in some cases the qualitative nature of flows in a dynamical system signs of matrix entries is sufficient to insure the stability of the system coloring graphs seems to be an essential part of efficient tests for such qualitative stability

stability analysis for a class of switched nonlinear systems this paper addresses the stability analysis of a class of switched nonlinear systems the switched systems have uncertain nonlinear functions constrained in a sector set which are called admissible sector nonlinearities a sufficient condition in terms of linear inequalities is presented to guarantee the existence of a common lyapunov function and thereby to ensure that the switched system is stable for an arbitrary switching signal and any admissible sector nonlinearities a constructive algorithm based on the modified gaussian elimination procedure is given to find the solutions of the linear inequalities the obtained results are applied to a population model with switchings of parameter values and the conditions of ultimate boundedness of its solutions are investigated another example of an automatic control system is considered to demonstrate the effectiveness of the proposed approaches

degenerate lyapunov functionals of a well known prey predator model with discrete delays it is commonly believed that as far as stabilities are concerned small delays are negligible in some modelling processes however to have an affirmative answerfor this common belief is still an open problem for many nonlinear equations in this paper the classical lotka volterra prey predator equation with discrete delays is considered and by using degenerate lyapunov functionals method an affirmative answer to this open problem on both local and global stabilities of the prey predator delay equations is given it is shown that degenerate lyapunov functional method is a powerful tool for studying the stability of such nonlinear delay systems a detailed and explicit procedure of constructing such functionals is provided furthermore some explicit estimates on the allowable sizes of the delays are obtained

compactness results for time dependent parabolic systems k 1 2 n where u u 8 is a function of t x e g 24 ad at q c r is a bounded domain with smooth boundary lx ak are uniformly elliptic operators and bk linear boundary operators we say that u is a regular solution of l if the function u together with its derivatives appearing in 1 is continuous in g such that eqs 1 are identically satisfied it is the object of this paper to show that under suitable conditions on a b and g the orbit t u a t b o of a bounded regular solution of 1 is relatively compact in c2 fl w results of this type are essential for the study of the asymptotic behavior of solutions u of 1 see e g 6 sect 4 61 or 17 p 3 551 we will deduce our theorems from more general results valid for abstract nonlinear evolution equations of the form

a posteriori error estimation and adaptivity for degenerate parabolic problems two explicit error representation formulas are derived for degenerate parabolic pdes which are based on evaluating a parabolic residual in negative norms the resulting upper bounds are valid for any numerical method and rely on regularity properties of solutions of a dual parabolic problem in nondivergence form with vanishing diffusion coefficient they are applied to a practical space time discretization consisting of c 0 piecewise linear finite elements over highly graded unstructured meshes and backward finite differences with varying time steps two rigorous a posteriori error estimates are derived for this scheme and used in designing an efficient adaptive algorithm which equidistributes space and time discretization errors via refinement coarsening a simulation finally compares the behavior of the rigorous a posteriori error estimators with a heuristic approach and hints at the potentials and reliability of the proposed method