topological entanglement in polymers the brereton shah problem of two polymer loops topologically linked is studied by field theoretical methods in the limit that one polymer is allowed to fill a macroscopic volume at finite density rho for l 1 it is shown that typically for winding numbers m2 lalpha rho alpha 2 epsilon nu universality is observed with the critical exponents of the unconstrained system despite the presence of a dangerous renormalisation group instability here l is the length of the smaller polymer loop whose mean square size r2 approximately l2 nu nu 1 2 for random flight statistics or nu approximately 3 5 for the swelled chain case

unambiguous 1 uniform morphisms a morphism s is unambiguous with respect to a word a if there is no other morphism t that maps a to the same image as s in the present paper we study the question of whether for any given word there exists an unambiguous 1 uniform morphism i e a morphism that maps every letter in the word to an image of length 1

unified convergence results on a minimax algorithm for finding multiple critical points in banach spaces a minimax method for finding multiple critical points in banach spaces is successfully developed in x yao and j zhou siam j sci comput 26 2005 pp 1796 1809 by using a projected pseudogradient as a search direction since several different techniques can be used to compute a projected pseudogradient the uniform stepsize and the continuity of a search direction two key properties for the convergence results in y li and j zhou siam j sci comput 24 2002 pp 865 885 get lost in this paper instead of proving convergence results of the algorithm for each technique unified convergence results are obtained with a weaker stepsize assumption an abstract existence convergence result is also established it is independent of the algorithm and explains why function values always converge faster than their gradients do the weaker stepsize assumption is then verified for several different cases as an illustration to the new results the banach space w 0 1 p omega is considered and the conditions posed in the new results are verified for a quasi linear elliptic pde

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

cylinder buckling the mountain pass as an organizing center we revisit the classical problem of the buckling of a long thin axially compressed cylindrical shell by examining the energy landscape of the perfect cylinder we deduce an estimate of the sensitivity of the shell to imperfections key to obtaining this estimate is the existence of a mountain pass point for the system we prove the existence on bounded domains of such solutions for almost all loads and then numerically compute example mountain pass solutions numerically the mountain pass solution with lowest energy has the form of a single dimple we interpret these results and validate the lower bound against some experimental results available in the literature

a new augmented singular transform and its partial newton correction method for finding more solutions to nonvariational quasilinear elliptic pdes abstract in this paper in order to find more solutions to a nonvariational quasilinear pde a new augmented singular transform ast is developed to form a barrier surrounding previously found solutions so that an algorithm search from outside cannot pass the barrier and penetrate into the inside to reach a previously found solution thus a solution found by the algorithm must be new mathematical justifications of ast formulation are established a partial newton correction method is designed accordingly to solve the augmented problem and to satisfy a constraint in ast the new method is applied to numerically investigate bifurcation symmetry breaking phenomena to a non variational quasilinear elliptic equation through finding multiple solutions such phenomena are numerically captured and visualized for the first time and still open for theoretical verification since the formulation is general and simple it opens a door to solve other multiple solution problems

morphically primitive words in the present paper we introduce an alternative notion of the primitivity of words that unlike the standard understanding of this term is not based on the power and hence the concatenation of words but on morphisms for any alphabet s we call a word w s morphically imprimitive provided that there are a shorter word v and morphisms h h s s satisfying h v w and h w v and we say that w is morphically primitive otherwise we explain why this is a well chosen terminology we demonstrate that morphic im primitivity of words is a vital attribute in many combinatorial domains based on finite words and morphisms and we study a number of fundamental properties of the concepts under consideration

a minimax method for finding multiple critical points in banach spaces and its application to quasi linear elliptic pde a minimax method for finding multiple critical points in banach spaces is developed by using a modified pseudogradient the method is implemented successfully to solve several quasi linear elliptic boundary value problems for multiple solutions numerical solutions are presented by their profiles to illustrate the theory and method a unified convergence result of the algorithm is established in a subsequent paper x yao and j zhou a unified convergence result on a minimax method for finding multiple critical points in banach spaces preprint

existence of many positive solutions of semilinear elliptic equations on annulus abstract we study the existence of many nonradial positive solutions in an annulus of rn it is an improvement of results in c v coffman a nonlinear boundary value problem with many positive solutions j differential equations 54 1984 429 437 an annulus is invariant under many group actions and these group actions act naturaly on the functional and leave the functional invariant therefore when we look for critical points of the functional we can restrict the functional to the manifold consisting of all fixed points under the group action we are going to use different group actions to obtain critical points and furthermore we can distinguish them

a bifurcation method for solving multiple positive solutions to the boundary value problem of the henon equation on a unit disk three algorithms based on the bifurcation theory are proposed to compute the o 2 symmetric positive solutions to the boundary value problem of the henon equation on the unit disk taking l in the henon equation as a bifurcation parameter the symmetry breaking bifurcation point on the branch of the o 2 symmetric positive solutions is found via the extended systems finally other symmetric positive solutions are computed by the branch switching method based on the lyapunov schmidt reduction

numerical methods for computing nonlinear eigenpairs part i iso homogeneous cases with a rayleigh quotient formulation a local minimax method is developed to solve a class of iso homogeneous nonlinear eigenpair problems for multiple solutions in banach spaces following their instability order the algorithm is implemented to compute weighted eigenpairs of the p laplacian numerical eigenfunctions are illustrated by their graphics several interesting phenomena have been observed and are open for further investigation mathematical analysis related to convergence and instability order of computed eigenfunctions etc is also presented

two classes of ljusternik schnirelman minimax algorithms and an application for finding multiple negative energy solutions of a class of p laplacian equations abstract in this paper two classes of ls minimax algorithms are presented they are applied to numerically find multiple negative energy solutions of the p laplacian equation p u u r 1 u u q 1 u x r l u 0 x where is an open bounded domain 0 r p 1 q p 1 0 and p is the sobolev exponent and mathematical justification and global convergence result for them are established by combining ls minimax algorithm with the finite element method it is verified that as element size goes to zero numerical solutions of p laplacian equation captured by ls minimax algorithm converge to solutions of p laplacian equation two ls minimax algorithms developed in 1 are two special algorithms in these two classes of algorithms

a local min orthogonal method for finding multiple saddle points abstract the purpose of this paper is twofold the first is to remove a possible ill posedness related to a local minimax method developed in siam j sci comput 23 2001 840 865 siam j sci comput 24 2002 840 865 and the second is to provide a local characterization for nonminimax type saddle points to do so a local l selection is defined and a necessary and sufficient condition for a saddle point is established which leads to a min orthogonal method those results exceed the scope of a minimax principle the most popular approach in critical point theory an example is given to illustrate the new theory with this local characterization the local minimax method in siam j sci comput 23 2001 840 865 siam j sci comput 24 2002 840 865 is generalized to a local min orthogonal method for finding multiple saddle points in a subsequent paper this approach is applied to define a modified pseudo gradient flow of a functional for finding multiple saddle points in banach spaces

ground state solutions for some indefinite variational problems abstract we consider the nonlinear stationary schrodinger equation u v x u f x u in r n here f is a superlinear subcritical nonlinearity and we mainly study the case where both v and f are periodic in x and 0 belongs to a spectral gap of v inspired by previous work of li et al 2006 11 and pankov 2005 13 we develop an approach to find ground state solutions i e nontrivial solutions with least possible energy the approach is based on a direct and simple reduction of the indefinite variational problem to a definite one and gives rise to a new minimax characterization of the corresponding critical value our method works for merely continuous nonlinearities f which are allowed to have weaker asymptotic growth than usually assumed for odd f we obtain infinitely many geometrically distinct solutions the approach also yields new existence and multiplicity results for the dirichlet problem for the same type of equations in a bounded domain

analysis and computation for a class of semilinear elliptic boundary value problems in this paper with the help of super solutions and sub solutions we set up a general framework and get a positive threshold l for solution existence and non existence of a class of semilinear elliptic dirichlet boundary value problems moreover a result on multiplicity is obtained when l is large enough we also give a numerical method to solve and visualize the positive solutions of the problem theoretical results are illustrated by numerical simulation

gidas ni nirenberg results for finite difference equations estimates of approximate symmetry are positive solutions of finite difference boundary value problems hu f u in h u 0 on h as symmetric as the domain to answer this question we first show by examples that almost arbitrary non symmetric solutions can be constructed this is in striking difference to the continuous case where by the famous gidas ni nirenberg theorem b gidas wei ming ni l nirenberg symmetry and related problems via the maximum principle comm math phys 68 1979 209 243 positive solutions inherit the symmetry of the underlying domain then we prove approximate symmetry theorems for solutions on equidistantly meshed n dimensional cubes explicit estimates depending on the data are given which show that the solutions become more symmetric as the discretization gets finer the quality of the estimates depends on whether or not f 0 0 the one dimensional case stands out in two ways the proofs are elementary and the estimates for the defect of symmetry are o h compared to o 1 log h in the higher dimensional case

on a conjecture about finite fixed points of morphisms a conjecture of m billaud is given a word w if for each letter x occurring in w the word obtained by erasing all the occurrences of x in w is a fixed point of a nontrivial morphism fx then w is also a fixed point of a non trivial morphism we prove that this conjecture is equivalent to a similar one on sets of words using this equivalence we solve these conjectures in the particular case where each morphism fx has only one expansive letter

a symmetry theorem and applications to nonlinear partial differential equations on donne des applications d un nouveau theoreme sur les proprietes de symetrie des solutions des equations differentielles non lineaires de la forme lu f x u dans u 0 ou u n 0 ou est une region bornee ou non bornee de r n

ljusternik schnirelman theory in partially ordered hilbert spaces we present several variants of ljusternik schnirelman type theorems in partially ordered hilbert spaces which assert the locations of the critical points constructed by the minimax method in terms of the order structures these results are applied to nonlinear dirichlet boundary value problems to obtain the multiplicity of sign changing solutions

the computation of one parameter families of bifurcating elastic surfaces the problem of constructing the middle surface of a deformed elastic shell from its first and second fundamental forms hat a alpha beta and hat b alpha beta is considered the undeformed shell is a spherical cap of radius r and thickness h with an angular width 2 theta 0 where 0 theta 0 pi 2 the cap is subjected to a constant uniform load lambda and is simply supported at its edge the authors seek to compute the one parameter families of buckled states that branch from the unbuckled state of the shell this is accomplished in two steps first a finite element method is used to solve the governing shell equations a pair of fourth order nonlinear partial differential equations pdes a solution of this system is a curvature potential w a stress potential f and the load lambda using liapunov schmidt reduction it can be shown that solutions possessing a variety of symmetries bifurcate from the unbuckled state of the shell in the work presented here these

saddle critical point analysis and computation a local minimax method is developed for multiple saddle point analysis and computation in this paper this method is also applied to devise a numerical algorithm for solving local minimax problems in control theory

a local minimax characterization for computing multiple nonsmooth saddle critical points this paper is concerned with characterizations of nonsmooth saddle critical points for numerical algorithm design most characterizations for nonsmooth saddle critical points in the literature focus on existence issue and are converted to solve global minimax problems thus they are not helpful for numerical algorithm design inspired by the results on computational theory and methods for finding multiple smooth saddle critical points in 14 15 19 21 23 a local minimax characterization for multiple nonsmooth saddle critical points in either a hilbert space or a reflexive banach space is established in this paper to provide a mathematical justification for numerical algorithm design a local minimax algorithm for computing multiple nonsmooth saddle critical points is presented by its flow chart

a minimax method for finding multiple critical points and its applications to semilinear pdes most minimax theorems in critical point theory require one to solve a two level global optimization problem and therefore are not for algorithm implementation the objective of this research is to develop numerical algorithms and corresponding mathematical theory for finding multiple saddle points in a stable way in this paper inspired by the numerical works of choi mckenna and ding costa chen and the idea to define a solution submanifold some local minimax theorems are established which require us to solve only a two level local optimization problem based on the local theory a new local numerical minimax method for finding multiple saddle points is developed the local theory is applied and the numerical method is implemented successfully to solve a class of semilinear elliptic boundary value problems for multiple solutions on some nonconvex non star shaped and multiconnected domains numerical solutions are illustrated by their graphics for visualization in a subsequent paper y li and j zhou convergence results of a minimax method for finding critical points in review we establish some convergence results for the algorithm

search extension method for multiple solutions of a nonlinear problem abstract based on the eigensystem j j of multiple solutions of nonlinear problem u u 3 0 min u 0 on are approximated a new method called the search extension method sem composed of three steps is proposed numerical examples in a square and the unit circle are presented to show the efficiency of our method

restricted ambiguity of erasing morphisms a morphism h is called ambiguous for a string s if there is another morphism that maps s to the same image as h otherwise it is called unambiguous in this paper we examine some fundamental problems on the ambiguity of erasing morphisms we provide a detailed analysis of so called ambiguity partitions and our main result uses this concept to characterise those strings that have a morphism of strongly restricted ambiguity furthermore we demonstrate that there are strings for which the set of unambiguous morphisms depending on the size of the target alphabet of these morphisms is empty finite or infinite finally we show that the problem of the existence of unambiguous erasing morphisms is equivalent to some basic decision problems for nonerasing multi pattern languages

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

random walks on multiconnected manifolds and conformal field theory we propose a simple geometrical method which enables us to link the topological properties of a random walk on the double punctured plane and the conformal field theory characterized by the central charge c 2 and the conformal dimension delta 1 8 we discuss briefly the connection between the topological invariants obtained from the conformal methods and the algebraic alexander invariants for the simplest non trivial braid b3

monotone explicit iterations of the finite element approximations for the nonlinear boundary value problem in this paper we consider monotone explicit iterations of the finite element schemes for the nonlinear equations associated with the boundary value problem u bu 2 based on piecewise linear polynomials and the lumping operator these iterations construct the monotonically decreasing and increasing sequences and convergence proofs are given finally we present some numerical examples verifying the effectiveness of the theory

level set methods for finding critical points of mountain pass type abstract computing mountain passes is a standard way of finding critical points we describe a numerical method for finding critical points that is convergent in the nonsmooth case and locally superlinearly convergent in the smooth finite dimensional case we apply these techniques to describe a strategy for addressing the wilkinson problem of calculating the distance from a matrix to a closest matrix with repeated eigenvalues finally we relate critical points of mountain pass type to nonsmooth and metric critical point theory

a local minimax newton method for finding multiple saddle points with symmetries in this paper a local minimax newton method is developed to solve for multiple saddle points the local minimax method siam j sci comput 23 2001 pp 840 865 is used to locate an initial guess and a version of the generalized newton method is used to speed up convergence when a problem possesses a symmetry the local minimax method is invariant to the symmetry thus the symmetry can be used to greatly enhance the efficiency and stability of the local minimax method but such an invariance is sensitive to numerical error and the haar projection has been used to enforce the symmetry siam j numer anal submitted in this paper we prove that the newton method is invariant to symmetries and that such an invariance is insensitive to numerical error when a symmetric degeneracy takes place it is proved that the newton direction can be easily solved in an invariant subspace thus the newton method can be used not only to speed up convergence but also to avoid using the haar projection if the symmetric degeneracy is removable by a discretization finally numerical examples are presented to illustrate the theory

an augmented singular transform and its partial newton method for finding new solutions to find new solutions through the information provided by previously found solution s this paper is first to develop a new singular transform to change the local basin barrier structure of the original problem in an infinite dimensional space and then to propose an augmented partial newton method to solve the transformed problem on the solution set mathematical justifications of the new formulation and method are established details on algorithm implementation are presented examples for two very different variational pde problems are carried out to illustrate the method their numerical results are displayed by profile contour plots and error convergence data the formulation and method developed in this paper can also be applied to solve multiple fixed point problems

multiple nodal solutions for elliptic equations abstract we assert the locations of critical points constructed for the c 1 functional by the minimax method in terms of the order structures these results are applied to nonlinear dirichlet boundary value problems to obtain the multiplicity of nodal sign changing solutions

constrained mountain pass algorithm for the numerical solution of semilinear elliptic problems a new numerical algorithm for solving semilinear elliptic problems is presented a variational formulation is used and critical points of a c1 functional subject to a constraint given by a level set of another c1 functional or an intersection of such level sets of finitely many functionals are sought first constrained local minima are looked for then constrained mountain pass points the approach is based on the deformation lemma and the mountain pass theorem in a constrained setting several examples are given showing new numerical solutions in various applications

a mountain pass method for the numerical solution of semilinear wave equations it is well known that periodic solutions of semilinear wave equations can be obtained as critical points of related functionals in the situation that we studied there is usually an obvious solution obtained as a solution of linear problem we formulate a dual variational problem in such a way that the obvious solution is a local minimum we then find additional non obvious solutions via a numerical mountain pass algorithm based on the theorems of ambrosetti rabinowitz and ekeland numerical results are presented

weakly unambiguous morphisms a nonerasing morphism s is said to be weakly unambiguous with respect to a word s if s is the only nonerasing morphism that can map s to s s i e there does not exist any other nonerasing morphism t satisfying t s s s in the present paper we wish to characterise those words with respect to which there exists such a morphism this question is nontrivial if we consider so called length increasing morphisms which map a word to an image that is strictly longer than the word our main result is a compact characterisation that holds for all morphisms with ternary or larger target alphabets we also comprehensively describe those words that have a weakly unambiguous length increasing morphism with a unary target alphabet but we have to leave the problem open for binary alphabets where we can merely give some non characteristic conditions

rearrangement inequalities in the discrete setting and some applications abstract we prove extended hardy littlewood riesz and polya szeg inequalities in the discrete case we also establish cases of equality in these inqualities and give some applications of our result

a mountain pass algorithm with projector in this paper we present an enhanced version of the minimax algorithm of chen ni and zhou that offers the additional guarantee that the solution found is a fix point of a projector on a cone positivity negativity and monotonicity can be expressed in this way the convergence of the algorithm is proved by means of a computational deformation lemma instead of the usual deformation lemma used in the calculus of variations

an efficient and stable method for computing multiple saddle points with symmetries in this paper an efficient and stable numerical algorithm for computing multiple saddle points with symmetries is developed by modifying the local minimax method established in y li and j zhou siam j sci comput 23 2001 pp 840 865 y li and j zhou siam j sci comput 24 2002 pp 840 865 first an invariant space is defined in a more general sense and a principle of invariant criticality is proved for the generalization then the orthogonal projection to the invariant space is used to preserve the invariance and to reduce computational error across iterations simple averaging formulas are used for the orthogonal projections numerical computations of examples with various symmetries of which some can and others cannot be characterized by a compact group of linear isomorphisms are carried out to confirm the theory and to illustrate applications the mathematical features of various problems demonstrated in these examples fall into two categories nodal solutions of saddle point type with large morse indices and nonradial positive solutions via symmetry breaking in radially symmetric elliptic problems the new numerical algorithm generates these rather unstable solutions in an efficient and stable way the existence of many unstable solutions and their behavior found in this paper remain to be investigated

a computer assisted existence and multiplicity proof for travelling waves in a nonlinearly supported beam abstract for a nonlinear beam equation with exponential nonlinearity we prove existence of at least 36 travelling wave solutions for the specific wave speed c 1 3 this complements the result in smets van den berg homoclinic solutions for swift hohenberg and suspension bridge type equations j differential equations 184 2002 78 96 stating that for almost all c 0 2 there exists at least one solution our proof makes heavy use of computer assistance starting from numerical approximations we use a fixed point argument to prove existence of solutions close to the computed approximations

convergence results of a local minimax method for finding multiple critical points in y li and j zhou siam j sci comput 23 2001 pp 840 865 a new local minimax method that characterizes a saddle point as a solution to a local minimax problem was established based on the local characterization a numerical minimax algorithm was designed for finding multiple saddle points numerical computations of many examples in semilinear elliptic pde were successfully carried out to solve for multiple solutions one of the important issues which remains unsolved is the convergence of the numerical minimax method in this paper first step 5 in the algorithm is modified with the design of a new stepsize rule that is easier to implement practically and with which convergence results of the numerical minimax method are established for isolated and nonisolated critical points the convergence results show that the algorithm indeed exceeds the scope of a minimax principle in the last section numerical multiple solutions to the henon equation and a sublinear elliptic equation subject to zero dirichlet boundary condition are presented to show their numerical convergence and profiles

stability and convergence of a finite element method for a semilinear elliptical problem with small viscosity abstract in this paper we propose and analyze a new finite element method for a semilinear elliptical problem with small viscosity firstly we prove the existence and uniqueness of solution for its variational problem then we obtain the stability and convergence of the corresponding finite element galerkin approximation furthermore we derive the optimal error estimates in the l 2 and h 1 norms for the finite element approximations respectively finally several numerical experiments are provided to confirm the above theoretical results

high order numerical method for the nonlinear helmholtz equation with material discontinuities in one space dimension the nonlinear helmholtz equation nlh models the propagation of electromagnetic waves in kerr media and describes a range of important phenomena in nonlinear optics and in other areas in our previous work we developed a fourth order method for its numerical solution that involved an iterative solver based on freezing the nonlinearity the method enabled a direct simulation of nonlinear self focusing in the nonparaxial regime and a quantitative prediction of backscattering however our simulations showed that there is a threshold value for the magnitude of the nonlinearity above which the iterations diverge in this study we numerically solve the one dimensional nlh using a newton type nonlinear solver because the kerr nonlinearity contains absolute values of the field the nlh has to be recast as a system of two real equations in order to apply newton s method our numerical simulations show that newton s method converges rapidly and in contradistinction with the iterations based on freezing the nonlinearity enables computations for very high levels of nonlinearity in addition we introduce a novel compact finite volume fourth order discretization for the nlh with material discontinuities our computations corroborate the design fourth order convergence of the method the one dimensional results of the current paper create a foundation for the analysis of multidimensional problems in the future

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

very weak solutions with boundary singularities for semilinear elliptic dirichlet problems in domains with conical corners let rn be a bounded lipschitz domain with a cone like corner at 0 we prove existence of at least two positive unbounded very weak solutions of the problem u up in u 0 on which have a singularity at 0 for any p slightly bigger that the generalized brezis turner exponent p on an example of a planar polygonal domain the actual size of the p interval on which the existence result holds is computed the solutions are found variationally as perturbations of explicitly constructed singular solutions in cones this approach also makes it possible to find numerical approximations of the two very weak solutions on following a gradient flow of a suitable functional and using the mountain pass algorithm two dimensional examples are presented

a new branch of mountain pass solutions for the choreographical 3 body problem we prove the existence of a new branch of solutions of mountain pass type for the periodic 3 body problem with choreographical constraint at first we describe the variational structure of the action functional associated to the choreographical three body problem in mathbb r 3 in the second part using a bisection algorithm we provide a numerical non rigorous solution of mountain pass type for this problem in a rotating frame with angular velocity 1 5 the last step consists in the rigorous computer assisted proof of the existence of a full branch of solutions for the problem starting from the mountain pass solution detected numerically

on finding multiple solutions to a singularly perturbed neumann problem in this paper in order to numerically solve for multiple positive solutions to a singularly perturbed neumann boundary value problem in mathematical biology and other applications a local minimax method is modified with new local mesh refinement and other strategies algorithm convergence and other related properties are verified motivated by the numerical algorithm and convinced by the numerical results a morse index approach is used to identify the morse index of the root solution u 1 varepsilon 1 at any perturbation value its bifurcation points and then the critical perturbation value many interesting numerical solutions are computed for the first time and displayed with their contours and mesh profiles to illustrate the theory and method

topological properties of a two dimensional polymer chain in the lattice of obstacles a new method of calculating the topological properties of a polymer chain in the lattice of obstacles on the plane is considered in detail it is shown that by applying the method of conformal transformations it is possible to reduce the problem of calculation of the partition function of the polymer chain in a given topological state with respect to the lattice of obstacles to a much more simple problem of the calculation of the green function of a free random walk without any topological restrictions on a riemann surface

pseudospectral methods for computing the multiple solutions of the lane emden equation based on the liapunov schmidt reduction and symmetry breaking bifurcation theory we compute and visualize multiple solutions of the lane emden equation on a square and a disc using legendre and fourier legendre pseudospectral methods starting from the nontrivial solution branches of the corresponding nonlinear bifurcation problem we obtain multiple solutions of lane emden equation with various symmetries numerically numerical results demonstrate the effectiveness of these approaches

analytical and numerical results for the fucik spectrum of the laplacian nontrivial solutions of u u vu 0 in rn with zero dirichlet condition on are studied the collection of the pairs v is the so called fucik spectrum f a new variational formulation for parts of f is presented and analyzed based on this formulation a minimization algorithm for the computation of a part of f is developed alternatively an approach using an implicit function argument is discussed analytically and via newton s method also numerically by combining the variational minimization method with newton s method a new bifurcation result is first observed numerically and then proved rigorously by replacing the variational minimization method by the mountain pass algorithm higher curves in f are found numerically several numerical results are discussed including a further example of the previously recorded phenomenon of crossing of fucik curves originating from different eigenvalues

newton s method s basin of attraction for sign changing solutions of concave and convex nonlinearities abstract in this paper we will illustrate and visualize the chaotic behavior for solving the sign changing solutions of concave convex nonlinearities we propose contractor iteration algorithm and bifurcation method to improve the computation speed and avoid local convergence of newton raphson method by this method sign changing numerical solutions will be computed and visualized in various domains

boundary element monotone iteration scheme for semilinear elliptic partial differential equations the monotone iteration scheme is a constructive method for solving a wide class of semilinear elliptic boundary value problems with the availability of a supersolution and a subsolution the iterates converge monotonically to one or two solutions of the nonlinear pde however the rates of such monotone convergence cannot be determined in general in addition when the monotone iteration scheme is implemented numerically through the boundary element method error estimates have not been analyzed in earlier studies in this paper we formulate a working assumption to obtain an exponentially fast rate of convergence this allows a margin 6 for the numerical implementation of boundary elements within the range of monotone convergence we then interrelate several approximate solutions and use the aubin nitsche lemma and the triangle inequalities to derive error estimates for the galerkin boundary element iterates with respect to the h r 0 r 2 sobolev space norms such estimates are of optimal order furthermore as a peculiarity we show that for the nonlinearities that are of separable type higher than optimal order error estimates can be obtained with respect to the mesh parameter h several examples of semilinear elliptic partial differential equations featuring different situations of existence nonexistence uniqueness multiplicity and stability are discussed computed and the graphics of their numerical solutions are illustrated

bifurcation method for computing the multiple positive solutions to p henon equation in this paper we proposed the algorithms to solving the d 4 symmetric positive solutions to the boundary value problem of p henon equation taking r in p henon equation as a bifurcation parameter the symmetry breaking bifurcation point on the branch of the d 4 symmetric positive solutions is found via the extended systems based on the liapunov schmidt reduction other symmetric positive solutions are computed by the branch switching method

efficient gluing of numerical continuation and a multiple solution method for elliptic pdes numerical continuation calculations for ordinary differential equations odes are by now an established tool for bifurcation analysis in dynamical systems theory as well as across almost all natural and engineering sciences although several excellent standard software packages are available for odes there are for good reasons no standard numerical continuation toolboxes available for partial differential equations pdes which cover a broad range of different classes of pdes automatically a natural ach to this problem is to look for efficient gluing computation approaches with independent components developed by researchers in numerical analysis dynamical systems scientific computing and mathematical modeling in this paper we shall study several elliptic pdes lane emden fowler lane emden fowler with microscopic force caginalp via the numerical continuation software pde2path and develop a gluing component to determine a set of starting solutions for the continuation by exploiting the variational structures of the pdes in particular we solve the initialization problem of numerical continuation for pdes via a minimax algorithm to find multiple unstable solution furthermore for the caginalp system we illustrate the efficient gluing link of pde2path to the underlying mesh generation and the fem matlab pdetoolbox even though the approach works efficiently due to the high level programming language and without developing any new algorithms we still obtain interesting bifurcation diagrams and directly applicable conclusions about the three elliptic pdes we study in particular with respect to symmetry breaking in particular we show for a modified lane emden fowler equation with an asymmetric microscopic force how a fully connected bifurcation diagram splits up into c shaped isolas on which localized pattern deformation appears towards two different regimes we conclude with a section on future software development issues that would be helpful to be addressed to simplify interfaces to allow for more efficient time saving gluing computation for dynamical systems analysis of pdes in the near future

a high linking algorithm for sign changing solutions of semilinear elliptic equations where n is a bounded domain with regular boundary the operator can be replaced by a more general second order uniformly elliptical operator in divergence form the solutions we are interested in nding are of minimax type and sign changing nonlinear elliptic equations of type 1 1 arise naturally in physics engineering and mathematical biology it is known that many such equations have multiple solutions their study has attracted the attention of many pure and applied mathematicians during the past two decades let us assume that f satis es the following regularity and growth conditions a1 f x t is locally lipschitz continuous in