stability of runge kutta methods in the numerical solution of equation u t au t a 0 u t this paper is concerned with the stability analysis of the runge kutta methods for the equation u t au t a0u t the stability regions for the runge kutta methods are determined the conditions that the analytic stability region is contained in the numerical stability region are obtained and some numerical experiments are given

stability of solutions of nonlinear neutral differential equations with piecewise constant delay and their discretizations this paper is devoted to stability analysis of solutions of nonlinear neutral differential equations with piecewise constant delay these results form the basis for obtaining insight into the analogous properties of numerical solutions generated by bn f stable runge kutta methods it is showed theoretically and numerically that two classes of runge kutta methods considered in literature are contractive and asymptotically stable numerical experiments also demonstrate that a class of runge kutta methods can preserve their original convergence order for odes but not the other class of runge kutta methods

bifurcation of stable equilibria under nonlinear flux boundary condition with null average weight abstract the bifurcation and stability structures of equilibria of a parabolic problem motivated by a population genetics model are completely described in a previous work the authors considered weights having nonzero average and drew the bifurcation and stability diagrams of equilibria herein that study is completed by considering weights with null average over the boundary of the domain and other approach is employed to overcome a degeneration which occurs in such case for each value of the parameter we prove existence of an unique nonconstant equilibrium solution it belongs to a globally parametrized bifurcation curve constructed using a lyapunov schmidt type reduction combined with the morse lemma that solution is showed to be a global minimizer of the corresponding energy functional which attracts all nontrivial semi orbits the behavior of its trace when the parameter is large is established and complete diagrams containing all bifurcation and stability information are also provided

nonuniqueness of an indefinite nonlinear diffusion problem in population genetics abstract we study the following neumann problem in one dimension space arising from population genetics u t d u x x h x u 2 1 u in 1 1 0 0 u 1 in 1 1 0 u 1 t u 1 t 0 in 0 where h changes sign in 1 1 and d is a positive parameter lou and nagylaki 2002 6 conjectured that if 1 1 h x d x 0 then this problem has at most one nontrivial steady state i e a time independent solution which is not identically equal to zero or one nakashima 2018 15 proved this uniqueness under some additional conditions on h x unexpectedly in this paper we discover 3 nontrivial steady states for some h x satisfying 1 1 h x d x 0 moreover bi stable phenomenon occurs in this scenario one with two layers is stable two with one layer each are ordered with the smaller one being stable and the larger one being unstable

variational principles for second order quasi linear scalar equations abstract the problem of constructing variational principles for a given second order quasi linear partial differential equation is considered in particular we address the problem of finding a first order function f whose product with the given differential operator is the euler lagrange operator derived from some lagrangian two sets of equations for such a function f are obtained necessary and sufficient conditions for the integration of the first set are established in general and these lead to a considerable simplification of the second set in certain special cases such as the case when the operator is elliptic the problem is completely solved the utility of our results is illustrated by a variety of examples

stability of runge kutta methods for the alternately advanced and retarded differential equations with piecewise continuous arguments this paper deals with the numerical properties of runge kutta methods for the solution of u t au t a 0u t 12 it is shown that the runge kutta method can preserve the convergence order the necessary and sufficient conditions under which the analytical stability region is contained in the numerical stability region are obtained it is interesting that the q methods with 0 q 12 are asymptotically stable some numerical experiments are given

an integro pde model from population genetics abstract we investigate an integro partial differential equation that models the evolution of the frequencies for two alleles at a single locus under the joint action of migration selection and partial panmixia i e global random mating we extend previous analyses t nagylaki clines with partial panmixia theor popul biol 81 2012 45 68 on the maintenance of both alleles from conservative to arbitrary migration and prove the uniqueness and global asymptotic stability of the nontrivial equilibrium for conservative migration we show that increasing the rate of panmixia makes it harder to maintain the allele with the smaller average fitness in the population in terms of the selection function we estimate the dependence on the panmictic rate of the minimal value of the selection intensity for the persistence of the allele with the smaller average fitness we also show that at least in an average sense increasing panmixia flattens the cline

asymptotic stability of runge kutta methods for nonlinear differential equations with piecewise continuous arguments this paper deals with the asymptotic stability of numerical solutions for differential equations with piecewise continuous arguments epcas the necessary and sufficient condition is given for non confluent runge kutta methods to preserve the stability of nonlinear scalar epcas as for systems we prove that some algebraically stable methods can preserve the asymptotic stability

nonparametric neyman scott problems telescoping product methods summary in this paper we consider some generalizations of the neyman scott problem of estimating a common variance in the presence of infinitely many mean nuisance parameters for example suppose observations are independent and stratified such that they have common distribution of the form f x lk within the kth stratum if both f and the location parameters glk are unknown then estimates for f obtained by centring the samples in each stratum and constructing an estimate directly from the pooled centred samples are generally inconsistent unless the stratum sizes go to infinity similarly nuisance parameters can arise from an exponential tilt of a common unknown distribution analogous estimators for the distribution produced by tilting the data and pooling the strata will be inconsistent for similar reasons to the location parameter problem in this paper we concentrate on the case where the stratum sizes are fixed and propose a method for the estimation of f using telescoping products partially multiplicative functions are introduced as a tool for the construction of counterexamples to the consistent estimation of f

characterization of distributions from maximal invariant statistics g w brown yu v prokhorov l bondesson and others have developed characterization results for maximal invariant statistics such results show that it is possible to construct tests to distinguish between types of distributions in this paper these results are proved for a large number of cases that arise in particular in spatial statistics and directional statistics the class of n collected distributions n 3 is introduced it is proved that if a distribution is n collected then a sample of size n has sufficient information to distinguish the type of the distribution in other words the distribution of the maximal invariant characterizes type

a semilinear parabolic system for migration and selection in population genetics abstract the semilinear parabolic system that describes the evolution of the gene frequencies in the diffusion approximation for migration and selection at a multiallelic locus is investigated the population occupies a finite habitat of arbitrary dimensionality and shape the selection coefficients depend on position the drift and diffusion coefficients may do so for two alleles the scalar case the global analysis of d henry 1981 geometric theory of semilinear parabolic equations lecture notes in mathematics vol 840 springer verlag berlin is extended from homogeneous isotropic migration corresponding to the laplacian to arbitrary migration corresponding to an arbitrary elliptic operator for multiple alleles sufficient conditions are given for the global loss of an allele that is nowhere the fittest in the case of no dominance with at least one change in the direction of selection sufficient conditions are established for global convergence to a stable equilibrium with all the intermediate alleles absent and one or two extreme alleles present sufficient conditions on the migration pattern for casting the elliptic operator into variational form are proved in this case the above results become more explicit

stability of θ methods for advanced differential equations with piecewise continuous arguments abstract this paper deals with the stability analysis of numerical methods for the solution of advanced differential equations with piecewise continuous arguments we focus on the behaviour of the one leg method and the linear method in the solution of the equation x t ax t a 0 x t a 1 x t 1 with real a a 0 a 1 and designates the greatest integer function the stability regions of two methods are determined the conditions under which the analytic stability region is contained in the numerical stability region are obtained and some numerical experiments are given

the improved linear multistep methods for differential equations with piecewise continuous arguments abstract this paper deals with the convergence of the linear multistep methods for the equation x t ax t a 0 x t numerical experiments demonstrate that the 2 step adams bashforth method is only of order p 0 when applied to the given equation an improved linear multistep methods is constructed it is proved that these methods preserve their original convergence order for ordinary differential equations odes and some numerical experiments are given

multivariate aspects of shape theory sis and thus provide a comprehensive view of shape distributions when landmark coordinates are gaussian distributed this work allows the statistical analysis of shape to be carried out using standard techniques of multivariate analysis the paper includes some new results in all dimensions and a general gaussian approximation to the size and shape distribution we also discuss some inference problems and give a numerical example

multiple existence of indefinite nonlinear diffusion problem in population genetics abstract we study the following neumann problem in one dimension u t d u x x h x u 2 1 u in 1 1 0 0 u 1 in 1 1 0 u 1 t u 1 t 0 in 0 where h changes sign in 1 1 and is symmetric i e h x h x and d is a positive small parameter in 9 we showed that this equation has a nontrivial steady state which is unique under the condition 1 1 h x d x 0 on the other hand in this paper we will prove that this equation has at least 8 nontrivial steady states for some h x satisfying 1 1 h x d x 0 this is a counterexample to a conjecture by lou and nagylaki 2

the uniqueness of an indefinite nonlinear diffusion problem in population genetics part ii abstract we study the following neumann problem which models the complete dominance case of population genetics of two alleles u t d u g x u 2 1 u in 0 1 0 0 u 1 in 0 1 0 u 0 t u 1 t 0 in 0 where g changes sign in 0 1 it is known that this equation has a nontrivial steady state u d for d sufficiently small 5 it has been conjectured by nagylaki and lou 2 that u d is a unique nontrivial steady state if g x d x 0 this was proved in 6 if g changes sign only once in this paper under additional condition on g x we treat the case when g has multiple zeros

graph intersection and uniqueness results for some nonlinear elliptic problems when cp 0 f is continuously differentiable and concave and g is non increasing in 24 no assumption is made about the sign of g so the nonlinearity may be both concave and convex in zj on different parts of the x domain here prime denotes differentiation with respect to x and u is a function of x such problems when g takes both positive and negative values arise in population genetics see l 51 for the biological background material actually the problems which arise in population genetics tend to be partial differential equations of the form au g lxl u f u 0 in rn u o in rn lim u x 0 1x1 cc see 6 7 8 however tertikas has shown in many cases that all

evolution of a semilinear parabolic system for migration and selection without dominance the semilinear parabolic system that describes the evolution of the gene frequencies in the diffusion approximation for migration and selection at a multiallelic locus without dominance is investigated the population occupies a finite habitat of arbitrary dimensionality and shape i e a bounded open domain in r d the selection coefficients depend on position the drift and diffusion coefficients may depend on position the primary focus of this paper is the dependence of the evolution of the gene frequencies on the strength of selection relative to that of migration it is proved that if migration is sufficiently strong i e is sufficiently small and the migration operator is in divergence form then the allele with the greatest spatially averaged selection coefficient is ultimately fixed the stability of each vertex i e an equilibrium with exactly one allele present is completely specified the stability of each edge equilibrium i e one with exactly two alleles present is fully described when either i migration is sufficiently weak i e is sufficiently large or ii the equilibrium has just appeared as increases the existence of unexpected complex phenomena is established even if there are only three alleles and migration is homogeneous and isotropic corresponding to the laplacian i as increases arbitrarily many changes of stability of the edge equilibria and corresponding appearance of an internal equilibrium can occur and ii the conditions for protection or loss of an allele can both depend nonmonotonically on neither of these phenomena can occur in the diallelic case

the uniqueness of indefinite nonlinear diffusion problem in population genetics part i we study the following neumann problem in one dimension ut du g x u2 1 u in 0 1 0 0 u 1in 0 1 0 u 0 t u 1 t 0in 0 g changes sign in 0 1 this equation models the complete dominance case in population genetics of two alleles it is known that this equation has a nontrivial steady state ud for d sufficiently small we show that the steady state ud is linearly stable moreover under the condition 01g x dx 0 we show that ud is a unique nontrivial steady state a conjecture of nagylaki and lou in one dimensional case has been largely resolved

multi layered stationary solutions for a spatially inhomogeneous allen cahn equation abstract we consider stationary solutions of a spatially inhomogeneous allen cahn type nonlinear diffusion equation in one space dimension the equation involves a small parameter e and its nonlinearity has the form h x 2 f u where h x represents the spatial inhomogeneity and f u is derived from a double well potential with equal well depth when e is very small stationary solutions develop transition layers we first show that those transition layers can appear only near the local minimum and local maximum points of the coefficient h x and that at most a single layer can appear near each local minimum point of h x we then discuss the stability of layered stationary solutions and prove that the morse index of a solution coincides with the total number of its layers that appear near the local maximum points of h x we also show the existence of stationary solutions having clustering layers at the local maximum points of h x

stability of the euler maclaurin methods for neutral differential equations with piecewise continuous arguments abstract this paper deals with the stability analysis of the euler maclaurin methods for neutral differential equations with piecewise continuous arguments u t au t i 0 n a i u i t the stability regions of the euler maclaurin methods are determined the conditions under which the analytic stability region is contained in the numerical stability region are obtained and some numerical experiments are given

stability analysis of runge kutta methods for systems u t lu t mu t this paper deals with the stability of runge kutta methods applied to the complex linear system u t lu t mu t the condition under which the numerical solution is asymptotically stable is presented which is stronger than a stability and weaker than a f stability furthermore in the case of 2 norm and l being a real symmetric matrix by using email protected approximation and order star theory it is proved that for a stable runge kutta methods suppose whose stability function is given by the r s email protected approximation to e x the numerical solution is asymptotically stable if and only if r is even