liouville theorem for elliptic equations with nonlinear boundary value conditions and finite morse indices abstract in this paper we study the nonexistence result for the following nonlinear elliptic equation with nonlinear boundary value condition u u p 1 u in r n u u q 1 u on r n if 1 p n 2 n 2 1 q n n 2 and p q n 2 n 2 n n 2 then we will show that this problem does not possess nontrivial bounded solution with finite morse index

using dual approximation algorithms for scheduling problems theoretical and practical results the problem of scheduling a set of n jobs on m identical machines so as to minimize the makespan time is perhaps the most well studied problem in the theory of approximation algorithms for np hard optimization problems in this paper the strongest possible type of result for this problem a polynomial approximation scheme is presented more precisely for each e an algorithm that runs in time o n e 1 e 2 and has relative error at most e is given in addition more practical algorithms for e 1 5 2 k and e 1 6 2 k which have running times o n k log n and o n km 4 log n are presented the techniques of analysis used in proving these results are extremely simple especially in comparison with the baroque weighting techniques used previously the scheme is based on a new approach to constructing approximation algorithms which is called dual approximation algorithms where the aim is to find superoptimal but infeasible solutions and the performance is measured by the degree of infeasibility allowed this notion should find wide applicability in its own right and should be considered for any optimization problem where traditional approximation algorithms have been particularly elusive

approximation techniques for average completion time scheduling we consider the problem of nonpreemptive scheduling to minimize average weighted completion time allowing for release dates parallel machines and precedence constraints recent work has led to constant factor approximations for this problem based on solving a preemptive or linear programming relaxation and then using the solution to get an ordering on the jobs we introduce several new techniques which generalize this basic paradigm we use these ideas to obtain improved approximation algorithms for one machine scheduling to minimize average completion time with release dates in the process we obtain an optimal randomized on line algorithm for the same problem that beats a lower bound for deterministic on line algorithms we consider extensions to the case of parallel machine scheduling and for this we introduce two new ideas first we show that a preemptive one machine relaxation is a powerful tool for designing parallel machine scheduling algorithms that simultaneously produce good approximations and have small running times second we show that a nongreedy rounding of the relaxation yields better approximations than a greedy one we also prove a general theorem relating the value of one machine relaxations to that of the schedules obtained for the original m machine problems this theorem applies even when there are precedence constraints on the jobs we apply this result to obtain improved approximation ratios for precedence graphs such as in trees out trees and series parallel graphs

a superquadratic indefinite elliptic system and its morse conley floer homology we study critical points of the indefinite functional f h u v int nabla u cdot nabla v h x u v mathrm d x by applying floer s homology construction to the ordinary gradient flow of the functional f on a suitable sobolev space one of our main observations is that even though this flow is well posed in both time directions and lacks any kind of smoothing property one can still obtain compactness of connecting orbit spaces and thus define the floer homology for f h

nodal sets and morse indices of solutions of super linear elliptic pdes in this paper we develop a sturm liouville type theory for the nodal sets and morse indices of solutions of super linear elliptic pdes with dirichlet boundary condition it shows that there are some relationships between analytic properties e g lp norm vanishing order of the nodal point andn 1 dimensional hausdorff measure of the nodal set of the solutions as the functions on the domain and morse indices of the solutions as the critical points of the functionalj u 12 u 2 f x u dxonh10

amplitude equations for a sub diffusive reaction diffusion system a sub diffusive reaction diffusion system with a positive definite memory operator and a nonlinear reaction term is analysed amplitude equations ginzburg landau type are derived for short wave turing and long wave hopf bifurcation points

subdiffusive master equation with space dependent anomalous exponent and structural instability we derive the fractional master equation with space dependent anomalous exponent we analyze the asymptotic behavior of the corresponding lattice model both analytically and by monte carlo simulation we show that the subdiffusive fractional equations with constant anomalous exponent in a bounded domain 0 l are not structurally stable with respect to the nonhomogeneous variations of parameter in particular the gibbs boltzmann distribution is no longer the stationary solution of the fractional fokker planck equation whatever the space variation of the exponent might be we analyze the random distribution of in space and find that in the long time limit the probability distribution is highly intermediate in space and the behavior is completely dominated by very unlikely events we show that subdiffusive fractional equations with the nonuniform random distribution of anomalous exponent is an illustration of a black swan the low probability event of the small value of the anomalous exponent that completely dominates the long time behavior of subdiffusive systems

stable and finite morse index solutions on ⁿ or on bounded domains with small diffusion in this paper we study bounded solutions of u f u on r n where n 2 and sometimes n 3 and show that for most f s the weakly stable and finite morse index solutions are quite simple we then use this to obtain a very good understanding of the stable and bounded morse index solutions of e 2 u f u on with dirichlet or neumann boundary conditions for small e

morse index of some min max critical points i application to multiplicity results on donne un resultat general sur des bornes inferieures pour des indices de morse de points critiques obtenus par des principes de min max en combinant cette information avec une inegalite semiclassique on obtient des estimations pointues sur la croissance de certaines valeurs critiques a partir desquelles on deduit de nouveaux resultats de multiplicite pour des solutions d equations elliptiques semi lineaires d ordre 2

estimate of morse index of cooperative elliptic systems and its application to spatial vector solitons local instability index of unstable solutions to single partial differential equations pdes by a local minimax method lmmm was established in zhou 2005 it is known that the local min orthogonal method lmom which was first proposed in zhou 2004 and then further developed in chen et al 2008 can find more general unstable solutions to both single pdes and cooperative elliptic systems this paper is to carry out instability analysis of unstable solutions by lmom to which an infinite dimensional functional space can be decomposed as a direct sum of a finite dimensional subspace and its orthogonal complement a morse index approach is developed to show that with lmom instability behavior of a solution in such infinite dimensional complement subspace can be totally determined usual instability analysis in an entire space is then reduced to analysis in its finite dimensional subspace for which a corresponding matrix decomposition is proposed to analyze a solution s instability behavior estimates of morse index are also established finally numerical examples of both 2 and 3 component cooperative systems arising in nonlinear optics are carried out for spatial vector solitons whose local instabilities are numerically confirmed by the new estimates certain important properties of the examples are also verified or presented

the speed of reaction diffusion wavefronts in nonsteady media the evolution of the speed of wavefronts for reaction diffusion equations with time varying parameters is analysed we make use of singular perturbative analysis to study the temporal evolution of the speed for pushed fronts the analogy with hamilton jacobi dynamics allows us to consider the problem for pulled fronts which is described by kolmogorov petrovskii piskunov kpp reaction kinetics both analytical studies are in good agreement with the results of numerical solutions

fast approximation algorithms for uniform machine scheduling with processing set restrictions abstract we consider the problem of nonpreemptively scheduling a set of independent jobs on a set of uniform machines where each job has a set of machines to which it can be assigned this kind of restriction is called the processing set restriction in the literature there are many kinds of processing set restrictions that have been studied in this paper we consider two kinds the inclusive processing set and the tree hierarchical processing set epstein and levin 2011 have given polynomial time approximation schemes ptas to solve both classes however the running times of their ptas are rather high in this paper we give fast approximation algorithms for both cases and show that they both have a worst case performance bound of 4 3 moreover we show that the bounds are achievable

approximation algorithms for scheduling unrelated parallel machines we consider the following scheduling problem there arem parallel machines andn independent jobs each job is to be assigned to one of the machines the processing of jobj on machinei requires timep ij the objective is to find a schedule that minimizes the makespan our main result is a polynomial algorithm which constructs a schedule that is guaranteed to be no longer than twice the optimum we also present a polynomial approximation scheme for the case that the number of machines is fixed both approximation results are corollaries of a theorem about the relationship of a class of integer programming problems and their linear programming relaxations in particular we give a polynomial method to round the fractional extreme points of the linear program to integral points that nearly satisfy the constraints in contrast to our main result we prove that no polynomial algorithm can achieve a worst case ratio less than 3 2 unlessp np we finally obtain a complexity classification for all special cases with a fixed number of processing times

partial regularity of stable solutions to the supercritical equations and its applications abstract we prove the partial regularity of stable solutions of supercritical elliptic equations as an application we prove that any smooth stable entire solution to supercritical equations with p in a suitable range is radially symmetric

systematic derivation of reaction diffusion equations with distributed delays and relations to fractional reaction diffusion equations and hyperbolic transport equations application to the theory of neolithic transition we introduce a general method for the systematic derivation of nonlinear reaction diffusion equations with distributed delays we study the interactions among different types of moving individuals atoms molecules quasiparticles biological organisms etc the motion of each species is described by the continuous time random walk theory analyzed in the literature for transport problems whereas the interactions among the species are described by a set of transformation rates which are nonlinear functions of the local concentrations of the different types of individuals we use the time interval between two jumps the transition time as an additional state variable and obtain a set of evolution equations which are local in time in order to make a connection with the transport models used in the literature we make transformations which eliminate the transition time and derive a set of nonlocal equations which are nonlinear generalizations of the so called generalized master equations the method leads under different specified conditions to various types of nonlocal transport equations including a nonlinear generalization of fractional diffusion equations hyperbolic reaction diffusion equations and delay differential reaction diffusion equations thus in the analysis of a given problem we can fit to the data the type of reaction diffusion equation and the corresponding physical and kinetic parameters the method is illustrated as a test case by the study of the neolithic transition we introduce a set of assumptions which makes it possible to describe the transition from hunting and gathering to agriculture economics by a differential delay reaction diffusion equation for the population density we derive a delay evolution equation for the rate of advance of agriculture which illustrates an application of our analysis

fronts with continuous waiting time distributions theory and application to virus infections we generalize to arbitrary waiting time distributions some results which were previously derived for discrete distributions we show that for any two waiting time distributions with the same mean delay time that with higher dispersion will lead to a faster front experimental data on the speed of virus infections in a plaque are correctly explained by the theoretical predictions using a gaussian delay time distribution which is more realistic for this system than the dirac delta distribution considered previously j fort and v mendez phys rev lett 89 178101 2002

multiple solutions for semilinear elliptic systems with non homogeneous boundary conditions since the early seventies many authors have widely investigated existence and multiplicity of solutions for semilinear elliptic problems with dirichlet boundary conditions especially by means of variational methods see 22 and references therein in particular if is a real l2 function on a bounded domain rn p 2 and pi 2 if n 3 here 2 2n n 2 the following model problem 6u u p 2u in

turing instability in sub diffusive reaction diffusion systems linear stability characteristics of an anomalous sub diffusive activator inhibitor system are investigated through a mesoscopic model where anomaly ensues by a memory integro differential operator with an equal anomaly exponent for all species it is shown that monotonic instability known from normal diffusion persists in the anomalous system thereby allowing turing pattern occurrence presence of anomaly stabilizes the system by diminution of the range of diffusion coefficients ratio involving instability in addition the maximal growth rate is diminished proving the existence of an absolutely stable anomalous system

multidelayed random walks theory and application to the neolithic transition in europe we present a model in which particles or individuals of a biological population disperse with a rest time between consecutive motions or migrations which may take several possible values from a discrete set particles or individuals may also react or reproduce we derive a new equation for the effective rest time stackrel ifmmode tilde else fi t of the random walk application to the neolithic transition in europe makes it possible to derive more realistic theoretical values for its wavefront speed than those following from the single delayed framework presented previously j fort and v m endez phys rev lett 82 867 1999 the new results are consistent with the archaeological observations of this important historical process

tight bounds for online class constrained packing we consider class constrained packing problems in which we are given a set of bins each having a capacity v and c compartments and n items of m different classes and the same unit size we need to fill the bins with items subject to capacity constraints such that items of different classes are placed in separate compartments thus each bin can contain items of at most c distinct classes we consider two optimization goals in the class constrained bin packing problem ccbp our goal is to pack all the items in a minimal number of bins in the class constrained multiple knapsack problem ccmk we wish to maximize the total number of items packed in m bins for m 1 the ccbp and ccmk problems model fundamental resource allocation problems in computer and manufacturing systems both are known to be strongly np hard in this paper we derive tight bounds for the online variants of these problems we first present a lower bound of 1 on the competitive ratio of any deterministic algorithm for the online ccbp where 0 1 depends on v c m and n we show that this ratio is achieved by the algorithm first fit we then consider the temporary ccbp in which items may be packed for a bounded time interval that is unknown in advance we obtain a lower bound of v c on the competitive ratio of any deterministic or randomized algorithm we show that this ratio is achieved by all any fit algorithms finally tight bounds are derived for the online ccmk and the temporary ccmk problems

connecting the dots semi analytical and random walk numerical solutions of the diffusion reaction equation with stochastic initial conditions we study a system with bimolecular irreversible kinetic reaction a b a where the underlying transport of reactants is governed by diffusion and the local reaction term is given by the law of mass action we consider the case where the initial concentrations are given in terms of an average and a white noise perturbation our goal is to solve the diffusion reaction equation which governs the system and we tackle it with both analytical and numerical approaches to obtain an analytical solution we develop the equations of moments and solve them approximately to obtain a numerical solution we develop a grid less monte carlo particle tracking approach where diffusion is modeled by a random walk of the particles and reaction is modeled by annihilation of particles the probability of annihilation is derived analytically from the particles co location probability we rigorously derive the relationship between the initial number of particles in the system and the amplitude of white noise represented by that number this enables us to compare the particle simulations and the approximate analytical solution and offer an explanation of the late time discrepancies

subordinated stochastic processes with aged operational time in this paper subordinated stochastic processes are considered where the renewal process acting as the operational time it is assumed that the observation of the process begins at a certain time after the start of the renewal process a recurrence formula was derived for calculating the multipoint probability density functions of the aged renewal process two point correlation functions for certain subordinated stochastic processes particularly for the generalized ornstein uhlenbeck process were calculated a model of relaxation in a disordered medium with traps and obstacles is proposed

on a nonlinear elliptic eigenvalue problem abstract let n be the number of the solutions of the equation u f u in u 0 on where is a bounded domain in r n n 2 with smooth boundary under suitable conditions on f we proved that n as

scheduling parallel machines with inclusive processing set restrictions and job release times we consider the problem of scheduling a set of jobs with different release times on parallel machines so as to minimize the makespan of the schedule the machines have the same processing speed but each job is compatible with only a subset of those machines the machines can be linearly ordered such that a higher indexed machine can process all those jobs that a lower indexed machine can process we present an efficient algorithm for this problem with a worst case performance ratio of 2 we also develop a polynomial time approximation scheme ptas for the problem as well as a fully polynomial time approximation scheme fptas for the case in which the number of machines is fixed

topological results on a certain class of functionals and application abstract we study in this paper problems of the type u u p 1 u x bounded r n u 0 i where x is given and where p 1 n 2 n 2 p 1 if n 2 our main result is that i has an infinite number of solutions for a residual set of in h 1 in particular for many n n there exists an open and dense subset of in h 1 such that i has n distinct solutions for such an this result is to be related to the conjecture developed in 1 of the existence of an infinite number of solutions to i the proof relies on a general characterization of level sets for a certain class of functionals when there are no critical value in a large enough interval in addition to the study of problem i we apply this characterization to give another proof using e g brouwer s fixed point theorem for some classical results about even functionals and saddle points

morse indices of solutions for super linear elliptic pde s abstract in this paper we establish l estimates for solutions of some general superlinear and subcritical elliptic equations via the morse index our results generalize and improve the previous works bahri and lions 1992 yang 1998

speed of reaction diffusion fronts in spatially heterogeneous media the front speed problem for nonuniform reaction rate and diffusion coefficient is studied by using singular perturbation analysis the geometric approach of hamilton jacobi dynamics and the local speed approach exact and perturbed expressions for the front speed are obtained in the limit of large times for linear and fractal heterogeneities the analytic results have been compared with numerical results exhibiting a good agreement finally we reach a general expression for the speed of the front in the case of smooth and weak heterogeneities

pattern formation on networks with reactions a continuous time random walk approach we derive the generalized master equation for reaction diffusion on networks from an underlying stochastic process the continuous time random walk ctrw the nontrivial incorporation of the reaction process into the ctrw is achieved by splitting the derivation into two stages the reactions are treated as birth death processes and the first stage of the derivation is at the single particle level taking into account the death process while the second stage considers an ensemble of these particles including the birth process using this model we have investigated different types of pattern formation across the vertices on a range of networks importantly the ctrw defines the laplacian operator on the network in a non ad hoc manner and the pattern formation depends on the structure of this laplacian here we focus attention on ctrws with exponential waiting times for two cases one in which the rate parameter is constant for all vertices and the other where the rate parameter is proportional to the vertex degree this results in nonsymmetric and symmetric ctrw laplacians respectively in the case of symmetric laplacians pattern formation follows from the turing instability however in nonsymmetric laplacians pattern formation may be possible with or without a turing instability

kinetic equations for reaction subdiffusion systems derivation and stability analysis we derive general kinetic equations for reacting and subdiffusing entities based on a nonlinear continuous time random walk formalism proposed by vlad and ross phys rev e 66 061908 2002 reaction and diffusion processes are separable in a typical reaction diffusion system and their combined influence on the evolution of the density of a species is a simple sum our derivation shows that this is no longer true for subdiffusive entities undergoing reactions the strong memory effects in the transport process i e the non markovian nature of subdiffusion results in a nontrivial combination of reactions and spatial dispersal which we discuss in detail we carry out a linear stability analysis of the derived reaction subdiffusion system to understand the effects of memory on pattern formation we find that the turing instability persists in the subdiffusive system however the memory modifies the turing threshold and the characteristics of the band of unstable modes close to this threshold

solutions of superlinear elliptic equations and their morse indices we consider here solutions of semilinear second order elliptic equations with superlinear nonlinearities and we present some relationships between their morse indices and some qualitative properties in particular we show that for uo subcritical nonlinearities bounds on solutions are equivalent to bounds on their morse indices

exact multiplicity of solutions to superlinear and sublinear problems f2 lim u f u u f 0 lim u f u u f 0 for the de5niteness we assume f f and when f f we use f to represent it we will consider f being either superlinear or sublinear f is said to be superlinear if f u u is decreasing in 0 and is increasing in 0 and f is said to be sublinear if f u u is increasing in 0 and is decreasing in 0 the semilinear equation 1 1 with f satisfying f1 f2 has been studied extensively since early 1970s several di9erent approaches like variational methods morse

approximation algorithms for scheduling jobs with release times and arbitrary sizes on batch machines with non identical capacities abstract we consider the problem of scheduling jobs with release times and arbitrary sizes on batch machines with non identical capacities to minimize makespan a job can only be assigned to a machine whose capacity is not less than the size of the job a machine can process a group of jobs as a batch simultaneously as long as the total size of the jobs in this batch does not exceed the capacity of the machine the processing time of a batch is equal to the longest processing time of all the jobs in the batch for the case of equal release times we obtain a fast 5 approximation algorithm as well as an algorithm with approximation ratio arbitrarily close to 2 for the case of unequal release times and a fixed number of different batch capacities we obtain an algorithm with approximation ratio arbitrarily close to 2 we also study the problem of scheduling jobs with equal release times arbitrary sizes inclusive processing set restrictions and incompatible families on batch machines with identical capacities to minimize makespan we obtain a 8 3 approximation algorithm as well as an algorithm with approximation ratio arbitrarily close to 2 if only a single family is considered a 2 approximation algorithm is presented it is likely that these results cannot be too substantially improved upon since all of the problems studied cannot be approximated to within a ratio smaller than 2 unless p np

existence of radial solutions with prescribed number of zeros for elliptic equations and their morse index abstract in this paper we consider radially symmetric solutions of the nonlinear dirichlet problem u f x u 0 in where is a ball in r n n 3 and f satisfies some appropriate assumptions we prove existence of radially symmetric solutions with k prescribed number of zeros moreover when f x u k x u p 1 u using the uniqueness result due to tanaka 2008 21 we verify that these solutions are non degenerate and we prove that their radial morse index is exactly k

fast approximation algorithms for job scheduling with processing set restrictions we consider the problem of scheduling n independent jobs on m parallel machines where the machines differ in their functionality but not in their processing speeds each job has a restricted set of machines to which it can be assigned called its processing set preemption is not allowed our goal is to minimize the makespan of the schedule we study two variants of this problem 1 the case of tree hierarchical processing set and 2 the case of nested processing set we first give a fast algorithm for the case of tree hierarchical processing set with a worst case bound of 4 3 which is better than the best known algorithm whose worst case bound is 2 we then give a more complicated algorithm for the case of nested processing set with a worst case bound of 5 3 which is better than the best known algorithm whose worst case bound is 7 4 in both cases we will give examples achieving the worst case bounds

anomalous diffusion with linear reaction dynamics from continuous time random walks to fractional reaction diffusion equations we have revisited the problem of anomalously diffusing species modeled at the mesoscopic level using continuous time random walks to include linear reaction dynamics if a constant proportion of walkers are added or removed instantaneously at the start of each step then the long time asymptotic limit yields a fractional reaction diffusion equation with a fractional order temporal derivative operating on both the standard diffusion term and a linear reaction kinetics term if the walkers are added or removed at a constant per capita rate during the waiting time between steps then the long time asymptotic limit has a standard linear reaction kinetics term but a fractional order temporal derivative operating on a nonstandard diffusion term results from the above two models are compared with a phenomenological model with standard linear reaction kinetics and a fractional order temporal derivative operating on a standard diffusion term we have also developed further extensions of the ctrw model to include more general reaction dynamics

traveling waves in a reaction diffusion system diffusion with finite velocity and kolmogorov petrovskii piskunov kinetics an asymptotic method is presented for the analysis of the traveling waves in the one dimensional reaction diffusion system with the diffusion with a finite velocity and kolmogorov petrovskii piskunov kinetics the analysis makes use of the path integral approach scaling procedure and the singular perturbation techniques involving the large deviations theory for the poisson random walk the exact formula for the position and speed of reaction front is derived it is found that the reaction front dynamics is formally associated with the relativistic hamiltonian lagrangian mechanics

an explicit positivity preserving finite difference scheme for the classical fisher kolmogorov petrovsky piscounov equation abstract in this work we introduce a family of explicit finite difference schemes to approximate positive solutions of the classical fisher kolmogorov petrovsky piscounov equation from population dynamics a neumann stability analysis reveals that our methods are indeed stable under certain choices of the model and numerical parameters and numerical comparisons with analytical solutions are carried out in some detail

a bounded linear integrator for some diffusive nonlinear time dependent partial differential equations we propose a numerical method to approximate the solutions of generalized forms of two bi dimensional models of mathematical physics namely the burgers fisher and the burgers huxley equations in one dimensional form the literature in the area gives account of the existence of analytical solutions for both models in the form of traveling wave fronts bounded within an interval i of the real numbers motivated by this fact we propose a finite difference methodology that guarantees that under certain analytical conditions on the model and computer parameters estimates within i will evolve discretely into new estimates which are likewise bounded within i additionally we establish the preservation in the discrete domain of the skew symmetry of the solutions of the models under study our computational implementation of the method confirms numerically that the properties of positivity and boundedness are preserved under the analytical constraints derived theoretically our simulations show a good agreement between the analytical solutions derived in the present work and the corresponding numerical approximations

anomalous subdiffusion with multispecies linear reaction dynamics we have introduced a set of coupled fractional reaction diffusion equations to model a multispecies system undergoing anomalous subdiffusion with linear reaction dynamics the model equations are derived from a mesoscopic continuous time random walk formulation of anomalously diffusing species with linear mean field reaction kinetics the effect of reactions is manifest in reaction modified spatiotemporal diffusion operators as well as in additive mean field reaction terms one consequence of the nonseparability of reaction and subdiffusion terms is that the governing evolution equation for the concentration of one particular species may include both reactive and diffusive contributions from other species the general solution is derived for the multispecies system and some particular special cases involving both irreversible and reversible reaction dynamics are analyzed in detail we have carried out monte carlo simulations corresponding to these special cases and we find excellent agreement with theory

solutions of fractional laplacian equations and their morse indices abstract in this paper we study the solutions of the following fractional laplacian equation s u f x u in u 0 in c under some assumptions on the nonlinearity we will show that the boundedness of solutions is equivalent to the finiteness of their morse indices we extend the result of bahri and lions 1992 1 from the laplacian equations to the fractional laplacian equations

on a conditionally stable nonlinear method to approximate some monotone and bounded solutions of a generalized population model in this work we design a numerical method to approximate some solutions of a generalization of a nonlinear diffusion reaction model which appears in the context of population dynamics the existence of traveling wave solutions for the equation under consideration is a well known fact some of such solutions are positive bounded from above and monotone in both space and time motivated by these facts we propose an explicit nonlinear finite difference methodology to approximate consistently the solutions of the model under investigation in the linear regime the method is consistent of first order in time and of second order in space under certain flexible parameter conditions the method is capable of preserving the positivity the boundedness and the spatial and the temporal monotonicity of the traveling wave solutions moreover we establish analytically and numerically that the nonlinear method is conditionally stable a computational implementation of our technique shows that the method preserves in practice the mathematical features of interest of the exact solutions considered

multiple bound states of nonlinear schrodinger systems this paper is concerned with existence of bound states for schrodinger systems which have appeared as several models from mathematical physics we establish multiplicity results of bound states for both small and large interactions this is done by different approaches depending upon the sizes of the interaction parameters in the systems for small interactions we give a new approach to deal with multiple bound states the novelty of our approach lies in establishing a certain type of invariant sets of the associated gradient flows for large interactions we use a minimax procedure to distinguish solutions by analyzing their morse indices

on the sign of the mountain pass solution we consider a class of superlinear elliptic problems on a bounded domain in r n i t is known that besides the zero solution such superlinear problems possess at least one positive and one negative solution of mountain pass type as well as a sign changing solution moreover the least energy solutions ground states are among those one sign solutions we study the eect of the nonlinearity on the sign of such ground states and the corresponding role played by the interior geometry of and the geometry of 2001 elsevier science ltd all rights reserved

integer programming with a fixed number of variables it is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable the proof depends on methods from geometry of numbers

on line bin stretching we are given a sequence of items that can be packed into m unit size bins in the classical bin packing problem we fix the size of the bins and try to pack the items in the minimum number of such bins in contrast in the bin stretching problem we fix the number of bins and try to pack the items while stretching the size of the bins as least as possible we present two on line algorithms for the bin stretching problem that guarantee a stretching factor of 5 3 for any number m of bins we then combine the two algorithms and design an algorithm whose stretching factor is 1 625 for any m the analysis for the performance of this algorithm is tight the best lower bound for any algorithm is 4 3 for any m 2 we note that the bin stretching problem is also equivalent to the classical scheduling load balancing problem in which the value of the makespan maximum load is known in advance copyright 2001 elsevier science b v

improved approximation schemes for scheduling unrelated parallel machines we consider the problem of schedulingn independent jobs onm unrelated parallel machines where each job has to be processed by exactly one machine processing jobj on machinei requiresp ijtime units and the objective is to minimize the makespan i e the maximum job completion time focusing on the case whenm is fixed we present for both preemptive and nonpreemptive variants of the problem fully polynomial approximation schemes whose running times depend only linearly onn we also study an extension of the problem where processing jobj on machinei incurs a cost ofc ij and thus there are two optimization criteria makespan and cost we show that for any fixedm there is a fully polynomial approximation scheme that given valuest andc computes for any fixed e 0 a schedule in0 n time with makespan at most 1 e t and cost at most 1 e c if there exists a schedule of makespant and costc

morse s oscillation theorem for a degenerate case abstract we consider the eigenvalue problem for a selfadjoint system of linear ordinary differential equations with general mixed boundary conditions which allow a combination of terms involving boundary values from the left and right end points we obtain a precise formula for the morse index of the problem which is valid even in the degenerate case

online hierarchical scheduling an approach using mathematical programming this paper considers an online hierarchical scheduling problem on parallel identical machines we are given a set of m machines and a sequence of jobs each machine has a different hierarchy and each job also has a hierarchy associated with it a job can be assigned to a machine only if its hierarchy is no less than that of the machine the objective is to minimize the makespan i e the maximum load of all machines two models are studied in the paper for the fractional model we present an improved algorithm and lower bounds both the algorithm and the lower bound are based on solutions of mathematical programming for any given m our algorithm is optimal by numerical calculation for the integral model we present both a general algorithm for any m and an improved algorithm with better competitive ratios of 2 333 and 2 610 for m 4 and 5 respectively

remarks on the resonant elliptic problem we consider the semilinear elliptic equation u f x u with the dirichlet boundary value for resonant nonlinearity at infinity using the morse theory and perturbation method some existence results of solution and multiple solutions are proved

fronts from integrodifference equations and persistence effects on the neolithic transition we extend a previous model of the neolithic transition in europe j fort and v mendez phys rev lett 82 867 1999 by taking two effects into account i we do not use the diffusion approximation which corresponds to second order taylor expansions and ii we take proper care of the fact that parents do not migrate away from their children we refer to this as a time order effect in the sense that it implies that children grow up with their parents before they become adults and can survive and migrate we also derive a time ordered second order equation which we call the sequential reaction diffusion equation and use it to show that effect ii is the most important one and that both of them should in general be taken into account to derive accurate results as an example we consider the neolithic transition the model predictions agree with the observed front speed and the corrections relative to previous models are important up to 70

periodic solutions of second order superquadratic hamiltonian systems with potential changing sign ii abstract we consider the periodic solutions of the second order hamiltonian system x x h t v x with v being positive and superquadratic at infinity h being c 1 2 periodic sign changing and all zeros being simple some existence and multiplicity results of periodic solutions are given

bifurcation curves of a diffusive logistic equation with harvesting orthogonal to the first eigenfunction abstract we study the global bifurcation curves of a diffusive logistic equation when harvesting is orthogonal to the first eigenfunction of the laplacian for values of the linear growth up to 2 examining in detail their behavior as the linear growth rate crosses the first two eigenvalues we observe some new behavior with regard to earlier works concerning this equation namely the bifurcation curves suffer a transformation at 1 they are compact above 1 there are precisely two families of degenerate solutions with morse index equal to zero and the whole set of solutions below 2 is not a two dimensional manifold

cattaneo type subdiffusion reaction equation subdiffusion in a system in which mobile particles a can chemically react with static particles b according to the rule a b b is considered within a persistent random walk model this model which assumes a correlation between successive steps of particles provides hyperbolic cattaneo normal diffusion or fractional subdiffusion equations starting with the difference equation which describes a persistent random walk in a system with chemical reactions using the generating function method and the continuous time random walk formalism we will derive the cattaneo type subdiffusion differential equation with fractional time derivatives in which the chemical reactions mentioned above are taken into account we will also find its solution over a long time limit based on the obtained results we will find the cattaneo type subdiffusion reaction equation in the case in which mobile particles of species a and b can chemically react according to a more complicated rule

global a priori bounds for weak solutions of quasilinear elliptic systems with nonlinear boundary condition abstract in this paper we study quasilinear elliptic systems with nonlinear boundary condition with fully coupled perturbations even on the boundary under very general assumptions our main result says that each weak solution of such systems belongs to l l the proof is based on moser s iteration scheme the results presented here can also be applied to elliptic systems with homogeneous dirichlet boundary condition

on line load balancing in a hierarchical server topology in a hierarchical server environment jobs are to be assigned in an on line fashion to a collection of servers which form a hierarchy of capability each job requests a specific server meeting its needs but the system is free to assign it either to that server or to any other server higher in the hierarchy each job carries a certain load which it imparts to the server it is assigned to the goal is to find a competitive assignment in which the maximum total load on a server is minimized we consider the linear hierarchy in which the servers are totally ordered in terms of their capabilities we investigate several variants of the problem in the unweighted as opposed to weighted problem all jobs have unit weight in the fractional as opposed to integral model a job may be assigned to several servers each receiving some fraction of its weight finally temporary as opposed to permanent jobs may depart after being active for some finite duration of time we show an optimal e competitive algorithm for the unweighted integral permanent model the same algorithm is e 1 competitive in the weighted case its fractional version is e competitive even if temporary jobs are allowed for the integral model with temporary jobs we show an algorithm which is 4 competitive in the unweighted case and 5 competitive in the weighted case we show a lower bound of e for the unweighted case both integral and fractional this bound is valid even with respect to randomized algorithms we also show a lower bound of 3 for the unweighted integral model when temporary jobs are allowed we generalize the problem and consider hierarchies in which the servers form a tree in the tree hierarchy any job assignable to a node is also assignable to the node s ancestors we show a deterministic algorithm which is 4 competitive in the unweighted case and 5 competitive in the weighted case where only permanent jobs are allowed randomizing this algorithm improves its competitiveness to e and e 1 respectively we also show an omega sqrt n lower bound when temporary jobs are allowed

aging in mortal superdiffusive levy walkers a growing body of literature examines the effects of superdiffusive subballistic movement premeasurement aging or time lag on observations arising from single particle tracking a neglected aspect is the finite lifetime of these levy walkers be they proteins cells or larger structures we examine the effects of aging on the motility of mortal walkers and discuss the means by which permanent stopping of walkers may be categorized as arising from natural death or experimental artifacts such as low photostability or radiation damage this is done by comparison of the walkers mean squared displacement msd with the front velocity of propagation of a group of walkers which is found to be invariant under time lags for any running time distribution of a mortal random walker the msd is tempered by the stopping rate this provides a physical interpretation for truncated heavy tailed diffusion processes and serves as a tool by which to better classify the underlying running time distributions of random walkers tempering of aged msds raises the issue of misinterpreting superdiffusive motion which appears brownian or subdiffusive over certain time scales

parallel machine scheduling with nested processing set restrictions abstract we consider the problem of scheduling a set of n independent jobs on m parallel machines where each job can only be scheduled on a subset of machines called its processing set the machines are linearly ordered and the processing set of job j is given by two machine indexes a j and b j i e job j can only be scheduled on machines a j a j 1 b j two distinct processing sets are either nested or disjoint preemption is not allowed our goal is to minimize the makespan it is known that the problem is strongly np hard and that there is a list type algorithm with a worst case bound of 2 1 m in this paper we give an improved algorithm with a worst case bound of 7 4 for two and three machines the algorithm gives a better worst case bound of 5 4 and 3 2 respectively

partial regularity of finite morse index solutions to the lane emden equation we prove regularity and partial regularity results for finite morse index solutions u h1 lp to the lane emden equation u u p 1u in

on theory of pulse transmission in a nerve fibre the ray method is used to analyse pulse propagation in a nerve fibre the initial equations are considered to be hyperbolic i e with a finite velocity of propagation and the corresponding evolution equation is obtained the evolution equation that describes the dynamical side of the process is an equation of the first order being the natural result of applying the ray method this approach avoids the paradox of infinite velocity arising from the initial parabolic system used in the hodgkin huxley model but does not make the final governing equation more complicated the biochemical side of the process is described by means of the hodgkin huxley or fitzhugh nagumo models the steady state solution if it exists leads to an equation of the lienard type and the correspondence between the equations of the catastrophe theory and the usual nerve pulse equations is established

characterization of solutions having finite morse index for some nonlinear pde with supercritical growth abstract in this paper we study solutions of the equation u u p in r n where n 2 n 2 p p c n see 3 for the definition of p c n we first classify all solutions not necessarily bounded having finite morse index for r n or r n with dirichlet boundary condition when is a regular bounded domain of r n for a family of solutions with dirichlet boundary condition u n we prove that u n l is bounded if and only if the morse index sequence i u n of u n is bounded finally we prove the same result for other nonlinearities f x u which have similar growth respectively at

reaction diffusion waves of advance in the transition to agricultural economics in a previous paper j fort and v mendez phys rev lett 82 867 1999 the possible importance of higher order terms in a human population wave of advance has been studied however only a few such terms were considered here we develop a theory including all higher order terms results are in good agreement with the experimental evidence involving the expansion of agriculture in europe

nontrivial solutions of operator equations and morse indices of critical points of min max type on montre que si les hypotheses du theoreme de point selle sont satisfaites si dim y 2 et si f 1 c contient seulement des points critiques non degeneres alors f 1 c contient au moins un point critique u 0 tel que 2 indice de morse u 0 dim y

traveling waves with dispersive variability and time delay we first determine an approximate traveling wave profile for the cook model j murray mathematical biology i an introduction springer new york 2002 pp 471 478 for the case in which the number of dispersers is small relative to the number of nondispersers the results are consistent with the previous linearized wavefront analysis that predicts counterintuitively that relatively few dispersers can drive the population expansion wave with a wavespeed not too different from that for the case of a single dispersing population as described by the fisher equation the method of solution differs from that used in the latter case since here the dimensionless wavespeed is close to unity we next generalize the cook model to include time delay effects while the cook model like the fisher equation does not adequately describe the wave of advance during the neolithic transition in europe we show that the generalized cook model provides a close agreement with the historical record

liouville theorem for robin boundary value problems and finite morse indices in this paper we study the nonexistence result of the following nonlinear elliptic equation with robin boundary condition u u p 1uin r n u cu 0on r n under some assumptions on c and p we will show that this problem does not possess nontrivial bounded solution with finite morse index

instability analysis of saddle points by a local minimax method the objective of this work is to develop some tools for local instability analysis of multiple critical points which can be computationally carried out the morse index can be used to measure local instability of a nondegenerate saddle point however it is very expensive to compute numerically and is ineffective for degenerate critical points a local weak linking index can also be defined to measure local instability of a degenerate saddle point but it is still too difficult to compute in this paper a local instability index called a local minimax index is defined by using a local minimax method this new instability index is known beforehand and can help in finding a saddle point numerically relations between the local minimax index and other local instability indices are established those relations also provide ways to numerically compute the morse local linking indices in particular the local minimax index can be used to define a local instability index of a saddle point relative to a reference trivial critical point even in a banach space while others failed to do so

solutions of the mixed boundary problem and their morse indices abstract in this paper we study the solutions of the following mixed boundary problem u f x u in u 0 on 1 u 0 on 2 under some assumptions on the nonlinearity we will show that the boundedness of solutions is equivalent to the finiteness of their morse indices we extend the result of a bahri p l lions solutions of superlinear elliptic equations and their morse indices commun pure appl anal 45 1992 1205 1215 from the dirichlet boundary condition to the mixed boundary condition

superlinear indefinite elliptic problems and pohoz aev type identities abstract we prove the existence of a nontrivial solution for a nonlinear elliptic problem u u a x g u with dirichlet boundary condition on a bounded domain where g is superlinear both at zero and at infinity a x changes sign and 0

nontrivial solutions of schrodinger equations with indefinite nonlinearities we prove the existence of nontrivial solutions for the schrodinger equation delta u v x u a gamma x f u in r n where f is superlinear and suberitical at zero and infinity respectively v is periodic and a x chances sign c 2007 elsevier inc all rights reserved

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

on line load balancing the setup for our problem consists of n servers that must complete a set of tasks each task can be handled only by a subset of the servers requires a different level of service and once assigned cannot be reassigned we make the natural assumption that the level of service is known at arrival time but that the duration of service is not the on line load balancing problem is to assign each task to an appropriate server in such a way that the maximum load on the servers is minimized in this paper we derive matching upper and lower bounds for the competitive ratio of the on line greedy algorithm for this problem namely 3n 23 2 1 o 1 and derive a lower bound n12 for any other deterministic or randomized on line algorithm

generalized linear boltzmann equation describing non classical particle transport and related asymptotic solutions for small mean free paths abstract in classical kinetic models a particle free path distribution is exponential but this is more likely to be an exception than a rule in this paper we derive a generalized linear boltzmann equation glbe for a general free path distribution in the framework of alt s model in the case that the free path distribution has at least first and second finite moments we construct an asymptotic solution to the initial value problem for the glbe for small mean free paths in the special case of the one speed transport problem the asymptotic solution results in a diffusion approximation to the glbe