three positive solutions to an indefinite neumann problem a shooting method abstract we deal with the neumann boundary value problem u a t a t g u 0 0 u t 1 t 0 t u 0 u t 0 where the weight term has two positive humps separated by a negative one and g 0 1 r is a continuous function such that g 0 g 1 0 g s 0 for 0 s 1 and lim s 0 g s s 0 we prove the existence of three solutions when and are positive and sufficiently large

multiple positive solutions of superlinear elliptic problems with sign changing weight we study the existence of multiple positive solutions for a superlinear elliptic pde with a sign changing weight our approach is variational and relies on classical critical point theory on smooth manifolds a special care is paid to the localization of minimax critical points

iterative strong coupling of dimensionally heterogeneous models in this article we address decomposition strategies especially tailored to perform strong coupling of dimensionally heterogeneous models under the hypothesis that one wants to solve each submodel separately and implement the interaction between subdomains by boundary conditions alone the novel methodology takes full advantage of the small number of interface unknowns in this kind of problems existing algorithms can be viewed as variants of the natural staggered algorithm in which each domain transfers function values to the other and receives fluxes or forces and vice versa this natural algorithm is known as dirichlet to neumann in the domain decomposition literature essentially we propose a framework in which this algorithm is equivalent to applying gauss seidel iterations to a suitably defined linear or nonlinear system of equations it is then immediate to switch to other iterative solvers such as gmres or other krylov based method which we assess through numerical experiments showing the significant gain that can be achieved indeed the benefit is that an extremely flexible automatic coupling strategy can be developed which in addition leads to iterative procedures that are parameter free and rapidly converging further in linear problems they have the finite termination property copyright 2009 john wiley sons ltd

nonequilibrium time dynamics of genetic evolution biological systems are typically highly open nonequilibrium systems that are very challenging to understand from a statistical mechanics perspective while statistical treatments of evolutionary biological systems have a long and rich history examination of the time dependent nonequilibrium dynamics has been less studied in this paper we first derive a generalized master equation in the genotype space for diploid organisms incorporating the processes of selection mutation recombination and reproduction the master equation is defined in terms of continuous time and can handle an arbitrary number of gene loci and alleles and can be defined in terms of an absolute population or probabilities we examine and analytically solve several prototypical cases which illustrate the interplay of the various processes and discuss the timescales of their evolution the entropy production during the evolution towards steady state is calculated and we find that it agrees with predictions from nonequilibrium statistical mechanics where it is large when the population distribution evolves towards a more viable genotype the stability of the nonequilibrium steady state is confirmed using the glansdorff prigogine criterion

positive solutions with a complex behavior for superlinear indefinite odes on the real line abstract we show the existence of infinitely many positive solutions defined on the real line for the nonlinear scalar ode u a t a t u 3 0 where a is a periodic sign changing function and the parameter 0 is large such solutions are characterized by the fact of being either small or large in each interval of positivity of a in this way we find periodic solutions having minimal period arbitrarily large and bounded non periodic solutions exhibiting a complex behavior the proof is variational exploiting suitable natural constraints of nehari type

positive solutions of a one dimensional indefinite capillarity type problem a variational approach abstract we prove the existence and the multiplicity of positive solutions of the one dimensional capillarity type problem u 1 u 2 a x f u u 0 0 u 1 0 where a l 1 0 1 changes sign and f 0 0 is continuous and has a power like behavior at the origin and at infinity our approach is variational and relies on a regularization procedure that yields bounded variation solutions which are of class w loc 2 1 and hence satisfy the equation pointwise almost everywhere on each open interval where the weight function a has a constant sign

sensitivity analysis in kinematically incompatible models abstract a natural question that arises when working with the so called kinematically incompatible models is about the positioning of the internal artificial boundary that splits the domain of analysis into the sub domains for which the different incompatible models are established although the experience in modeling might provide an ad hoc solution to such matter it may also draw misleading conclusions here the sensitivity analysis furnishes a systematic and reliable framework to study the impact of the positioning of the artificial boundary over the solution of the problem in this sense the present work is concerned with the sensitivity analysis of the variational formulation corresponding to the kinematically incompatible models when the boundary over which the incompatibility between the models takes place is changed assuming the presence of a discontinuity in the fields over a given interface between two incompatible models this analysis allows to measure the sensitivity of a given cost functional when the location of such interface is changed the application for which this analysis is envisaged is to assess the correctness or incorrectness in the definition of the positioning of such coupling interface between dimensionally heterogeneous domains also some numerical results are provided to give numerical evidence of the usefulness of the present tool

multibump nodal solutions for an indefinite superlinear elliptic problem abstract we define some nehari type constraints using an orthogonal decomposition of the sobolev space h 0 1 and prove the existence of multibump nodal solutions for an indefinite superlinear elliptic problem

solvable biological evolution models with general fitness functions and multiple mutations in parallel mutation selection scheme in a recent paper phys rev e 69 046121 2004 we used the suzuki trottere formalism to study a quasispecies biological evolution model in a parallel mutation selection scheme with a single peak fitness function and a point mutation in the present paper we extend such a study to evolution models with more general fitness functions or multiple mutations in the parallel mutation selection scheme we give some analytical equations to define the error thresholds for some general cases of mean field like or symmetric mutation schemes and fitness functions we derive some equations for the dynamics in the case of a point mutation and polynomial fitness functions we derive exact dynamics for two point mutations asymmetric mutations and the four value spin model with a single peak fitness function the same method is applied for the model with a royal road fitness function we derive the steady state distribution for the single peak fitness function

indefinite weight nonlinear problems with neumann boundary conditions abstract we present a multiplicity result of positive solutions for the neumann problem associated with a second order nonlinear differential equation of the following form u a t g u 0 where the weight function a t has indefinite sign the only assumption we make for the nonlinear term g u is that its primitive g u presents some oscillations at infinity expressed by the condition involving lim g u u 2 0 lim g u u 2 as an application we obtain multiple radially symmetric solutions for neumann problems associated with u a x g u 0

multiple positive solutions to elliptic boundary blow up problems abstract we prove the existence of multiple positive radial solutions to the sign indefinite elliptic boundary blow up problem u a x a x g u 0 x 1 u x x 1 where g is a function superlinear at zero and at infinity a and a are the positive negative part respectively of a sign changing function a and 0 is a large parameter in particular we show how the number of solutions is affected by the nodal behavior of the weight function a the proof is based on a careful shooting type argument for the equivalent singular ode problem as a further application of this technique the existence of multiple positive radial homoclinic solutions to u a x a x g u 0 x r n is also considered

existence and uniqueness of solutions of nonlinear neumann problems soit d r n un domaine connexe borne lisse et on suppose qu en chaque point x d la normale exterieure n est bien definie soit a une fonction arbitraire continue de holder dans d on etudie les solutions positives du probleme de neumann non lineaire u a x g u 0 dans d u n 0 sur d ou g r r est une fonction de classe c r c 1 r 0 1

topological sensitivity analysis the so called topological derivative concept has been seen as a powerful framework to obtain the optimal topology for several engineering problems this derivative characterizes the sensitivity of the problem when a small hole is created at each point of the domain however the greatest limitation of this methodology is that when a hole is created it is impossible to build a homeomorphic map between the domains in study because they have not the same topology therefore some specific mathematical framework should be developed in order to obtain the derivatives this work proposes an alternative way to compute the topological derivative based on the shape sensitivity analysis concepts the main feature of this methodology is that all the mathematical procedure already developed in the context of shape sensitivity analysis may be used in the calculus of the topological derivative this idea leads to a more simple and constructive formulation than the ones found in the literature further to point out the straightforward use of the proposed methodology it is applied for solving some design problems in steady state heat conduction

four state quantum chain as a model of sequence evolution a variety of selection mutation models for dna or rna sequences well known in molecular evolution can be translated into a model of coupled ising quantum chains this correspondence is used to investigate the genetic variability and error threshold behaviour in dependence of possible fitness landscapes in contrast to the two state models treated hitherto the model explicitly takes the four state nature of the nucleotide alphabet into account and allows for the distinction of mutation rates for the different base substitutions as given by standard mutation schemes of molecular phylogeny as a consequence of this refined treatment new phase diagrams for the error threshold behaviour are obtained with appearance of a novel phase in which the nucleotide ordering of the wildtype sequence is only partially conserved explicit analytical and numerical results are presented for evolution dynamics and equilibrium behaviour in a number of accessible situations such as quadratic fitness landscapes and the kimura 2 parameter mutation scheme

multiplicity of positive periodic solutions in the superlinear indefinite case via coincidence degree abstract we study the periodic boundary value problem associated with the second order nonlinear differential equation u c u a t a t g u 0 where g u has superlinear growth at zero and at infinity a t is a periodic sign changing weight c r and 0 is a real parameter our model includes for c 0 the so called nonlinear hill s equation we prove the existence of 2 m 1 positive solutions when a t has m positive humps separated by m negative ones in a periodicity interval and is sufficiently large thus giving a complete solution to a problem raised by g j butler in 1976 the proof is based on mawhin s coincidence degree defined in open possibly unbounded sets and applies also to neumann boundary conditions our method also provides a topological approach to detect subharmonic solutions

pairs of positive periodic solutions of second order nonlinear equations with indefinite weight abstract we study the problem of the existence and multiplicity of positive periodic solutions to the scalar ode u a t g u 0 0 where g x is a positive function on r superlinear at zero and sublinear at infinity and a t is a t periodic and sign indefinite weight with negative mean value we first show the nonexistence of solutions for some classes of nonlinearities g x when is small then using critical point theory we prove the existence of at least two positive t periodic solutions for large some examples are also provided

evolution models with base substitutions insertions deletions and selection the evolution model with parallel mutation selection scheme is solved for the case when selection is accompanied by base substitutions insertions and deletions the fitness is assumed to be either a single peak function i e having one finite discontinuity or a smooth function of the hamming distance from the reference sequence the mean fitness is calculated exactly in large genome limit in the case of insertions and deletions the evolution characteristics depend on the choice of reference sequence

the effect of varying coefficients on the dynamics of a class of superlinear indefinite reaction diffusion equations abstract in this paper we analyze how the dynamics of a class of superlinear indefinite reaction diffusion equations varies as the nodal behavior of a coefficient changes to perform this analysis we use both theoretical and numerical tools the analysis aids the numerical study and the numerical study confirms and completes the analysis the numerics in addition provides us with some further results for which at first glance analytical tools are not available yet our main analytical result shows that the problem possesses a unique positive solution which is linearly asymptotically stable if the trivial state is linearly unstable and the model admits some positive solution this result is a relevant feature for superlinear indefinite problems since our numerical computations show how these models can have an arbitrarily large number of positive solutions if the trivial state is unstable

a note on a superlinear indefinite neumann problem with multiple positive solutions abstract we prove the existence of three positive solutions for the neumann problem associated to u a t u 1 0 assuming that a t has two positive humps and 0 t a t d t is large enough actually the result holds true for a more general class of superlinear nonlinearities

high multiplicity of positive solutions for superlinear indefinite problems with homogeneous neumann boundary conditions abstract we prove that a class of superlinear indefinite problems with homogeneous neumann boundary conditions admits an arbitrarily high number of positive solutions provided that the parameters of the problem are adequately chosen the sign changing weight in front of the nonlinearity is taken to be piecewise constant which allows us to perform a sharp phase plane analysis firstly to study the sets of points reached at the end of the regions where the weight is negative and then to connect such sets through the flow in the positive part moreover we study how the number of solutions depends on the amplitude of the region in which the weight is positive using the latter as the main bifurcation parameter and constructing the corresponding global bifurcation diagrams

multiple positive solutions for a superlinear problem a topological approach we study the multiplicity of positive solutions for a two point boundary value problem associated to the nonlinear second order equation u f x u 0 we allow x f x s to change its sign in order to cover the case of scalar equations with indefinite weight roughly speaking our main assumptions require that f x s s is below 1 as s 0 and above 1 as s in particular we can deal with the situation in which f x s has a superlinear growth at zero and at infinity we propose a new approach based on topological degree which provides the multiplicity of solutions applications are given for u a x g u 0 where we prove the existence of 2n 1 positive solutions when a x has n positive humps and a x is sufficiently large

varying bifurcation diagrams of positive solutions for a class of indefinite superlinear boundary value problems in this work we analyze the existence multiplicity and stability of positive solutions for a class of indefinite superlinear elliptic boundary value problems the main contribution of this paper consists in the change of mind inherent to the fact of adding the superlinear amplitude e as an unfolding parameter this change of mind allows us to unify many previous results obtained separately in the literature it helps us to realize the global structure of the set of positive steady states and provides us with a great variety of new general results our techniques can be applied to much more general equations and systems