nodal type bound states of schrodinger equations via invariant set and minimax methods for nonlinear schrodinger equations in the entire space we present new results on invariant sets of the gradient flows of the corresponding variational functionals the structure of the invariant sets will be built into minimax procedures to construct nodal type bound state solutions of nonlinear schrodinger type equations 2004 elsevier inc all rights reserved

multiple solutions of nonlinear schrodinger equations via flow invariance and morse theory we prove the existence of multiple bound states of the nonlinear schrodinger equation u v x u f u here the linear potential v is continuous and bounded from below and the nonlinearity f is of asymptotically linear type we show that under certain assumptions on the spectrum of the schrodinger operator v and the asymptotic behaviour of f u u the above equation has at least four non trivial solutions two of them sign changing

the union of convex polyhedra in three dimensions we show that the number of vertices edges and faces of the union of k convex polyhedra in 3 space having a total of n faces is o k3 kn log k this bound is almost tight in the worst case as there exist collections of polyhedra with omega k 3 kn alpha k union complexity we also describe a rather simple randomized incremental algorithm for computing the boundary of the union in o k3 kn log k log n expected time

invariant sets of descending flow in critical point theory with applications to nonlinear differential equations abstract properties of invariant sets of descending flow defined by a pseudogradient vector field of a functional in a banach space are studied in this way several critical points can be found by constructing different invariant sets on which the functional is bounded below under suitable conditions the existence of at least four critical points of a functional is proved each critical point being in a certain invariant set the theoretical results are applied to nonlinear elliptic boundary value problems and nonlinear systems of ordinary differential equations in variant cases at least four solutions are obtained for these equations

homoclinic orbits for a class of first order nonperiodic asymptotically quadratic hamiltonian systems with spectrum point zero in this paper we study the existence and multiplicity of homoclinic orbits for a class of first order nonperiodic hamiltonian systems by applying two recent critical point theorems for strongly indefinite functionals we give some new criteria to guarantee that hamiltonian systems with asymptotically quadratic terms and spectrum point zero have at least one and a finite number of pairs of homoclinic orbits under some adequate conditions respectively

on hamiltonian elliptic systems with periodic or non periodic potentials abstract this paper is concerned with the following hamiltonian elliptic system u b x u v x u h v x u v v b x v v x v h u x u v for x r n existence and multiplicity of solutions are obtained for the systems with periodic or non periodic potentials v via variational methods

multiple solutions for a semiclassical schrodinger equation abstract this paper is concerned with the semiclassical nonlinear schrodinger equation e 2 u v x u f x u x r n u x 0 as x where e 0 is a small parameter under certain hypotheses on v and a general spectral assumption the existence and multiplicity of semiclassical solutions are obtained for asymptotically linear nonlinearity via variational methods

sharp bounds on davenport schinzel sequences of every order one of the longest standing open problems in computational geometry is bounding the complexity of the lower envelope of n univariate functions each pair of which crosses at most s times for some fixed s this problem is known to be equivalent to bounding the length of an order s davenport schinzel sequence namely a sequence over an n letter alphabet that avoids alternating subsequences of the form a b a b with length s 2 these sequences were introduced by davenport and schinzel in 1965 to model a certain problem in differential equations and have since been applied to bound the running times of geometric algorithms data structures and the combinatorial complexity of geometric arrangements let s n be the maximum length of an order s ds sequence over n letters what is s asymptotically this question has been answered satisfactorily hart and sharir 1986 agarwal et al 1989 klazar 1999 nivasch 2010 when s is even or s 3 however since the work of agarwal et al in the mid 1980s there has been a persistent gap in our understanding of the odd orders in this work we effectively close the problem by establishing sharp bounds on davenport schinzel sequences of every order s our results reveal that contrary to one s intuition s n behaves essentially like s 1 n when s is odd this refutes conjectures by alon et al 2008 and nivasch 2010

improved bounds for the union of locally fat objects in the plane we show that for any gamma 0 the combinatorial complexity of the union of n locally gamma fat objects of constant complexity in the plane is frac n gamma 4 2 o log n for the special case of gamma fat triangles the bound improves to o n log n frac n gamma log 2 frac 1 gamma

on ground state solutions for some non autonomous schrodinger poisson systems abstract in this paper we study the schrodinger poisson system sp u u k x x u a x f u in r 3 k x u 2 in r 3 and prove the existence of ground state solutions for system sp under certain assumptions on the linear and nonlinear terms some recent results from different authors are extended

solutions of a schrodinger poisson system with combined nonlinearities abstract in this paper we are concerned with the following schrodinger poisson system u v x u u k x u q 2 u f x u x r 3 u 2 lim x x 0 x r 3 where 0 is a parameter 1 q 2 and f x u is linearly bounded in u at infinity we establish the existence and multiplicity of solutions under suitable assumptions on v k and f

fusion trees can be implemented with ac 0 instructions only addressing a problem of fredman and willard we implement fusion trees in deterministic linear space using ac o instructions only more precisely we show that a subset of 0 2 w 1 of size n

unique continuation and absence of positive eigenvalues for schrodinger operators on considere l operateur de laplace sur r n et une fonction v x sur un sous ensemble connexe ouvert de r n on montre que si n 3 une propriete de prolongement unique est vraie pour v l loc n 2 r n dans l espace de sobolev h loc 2 q r n avec q 2n n 2

morse index and multiple solutions for the asymptotically linear schrodinger type equation abstract we study the nonlinear problem u v x u f x u x r n where the schrodinger type operator v is semibounded and has a finite number of eigenvalues below the infimum of the essential spectrum we classify the linear schrodinger equation prove the semi continuity and monotonicity of the index function and obtain some new conditions on the existence and multiplicity for generalized asymptotically linear schrodinger equations based on an application of the classical ambrosetti rabinowitz symmetric mountain pass theorem

on a class of asymptotically linear elliptic problems in rn under these conditions p is an asymptotically linear problem when this equation is considered in a bounded domain 0 r with say the dirichlet boundary condition there is a large literature on existence and multiplicity results of particular interest is then the case of resonance where s and s is the asymptotic linearization of the problem in other words s d s l 0 l 0 is the operator given by

geometric packing under nonuniform constraints we study the problem of discrete geometric packing here given weighted regions say in the plane and points with capacities one has to pick a maximum weight subset of the regions such that no point is covered more than its capacity we provide a general framework and an algorithm for approximating the optimal solution for packing in hypergraphs arising out of such geometric settings using this framework we get a flotilla of results on this problem and also on its dual where one wants to pick a maximum weight subset of the points when the regions have capacities for example for the case of fat triangles of similar size we show an o 1 approximation and prove that no ptas is possible

on the asymptotic distribution of the eigenvalue branches of the schrodinger operator h λw in a spectral gap of h on considere des operateurs de schrodinger h v dans l espace de hilbert l 2 r avec un potentiel v satisfaisant 1 vel r on suppose que l intervalle a b est contenu dans l ensemble resolvant de h on obtient des estimations asymptotiques sur le nombre des branches de valeurs propes dans a b des familles d operateurs h w 0 quand ou w est un potentiel a courte portee non negatif

the fucik spectrum of schrodinger operator and the existence of four solutions of schrodinger equations with jumping nonlinearities abstract this paper contains the existence of four solutions of schrodinger equations with jumping nonlinearities the proof procedure is supported by a lot of new results initially a consequence is rendered as a minimax principle on h 1 r n which allows us to achieve the feasibility verification of the ps condition furthermore the constructions of minimal and maximal curves of fucik spectrum in q l see the introduction for the definition of q l warrant an intensive investigation that we encounter some thorny problems is largely due to the absence of compact embedding and the appearance of essential spectrum based on a nontrivial argument we can compute critical groups of homogeneous functional at zero if a b is free of fucik spectrum and a b q l this together with convexity and concavity offers a detailed description of the two curves by a series of sophisticated tricks additionally we present a new version of morse theory in view of the fact that classical version doesn t work directly for weak smooth functional on h 1 r n finally we prove a weak maximum principle for r n which serves as a tool to get a critical point in positive and negative cone respectively and also compute critical groups of critical points of mountain pass type with the help of above preparations we attain the ultimate aim by morse inequalities and various exact homology sequences

three dimensional weak visibility complexity and applications abstract in this paper we study the complexity of 3d weak visibility we obtain an o n 8 time and n 6 space algorithm to compute the weakly visible region of a triangle f from another triangle g among general scenes which are a set of n disjoint triangles we also consider the cases when the scenes are rectilinear objects and polyhedral terrains we show that in these special situations the weakly visible regions can be computed much faster in o n 6 time and o n 4 space with these results we obtain the first known polynomial time algorithm to decide whether or not a simple polyhedron is weakly internally or externally visible

the existence of a positive solution to asymptotically linear scalar field equations which is important in using the mountain pass theorem the aim of this paper is to discuss how to use the mountain pass theorem to show the existence of a positive solution to the present problem when we lose the above condition furthermore if f x u f u we also prove that the above problem has a ground state by using the articial constraint method

on the union of fat tetrahedra in three dimensions we show that the combinatorial complexity of the union of n fat tetrahedra in 3 space i e tetrahedra all of whose solid angles are at least some fixed constant of arbitrary sizes is o n2 e for any e 0 the bound is almost tight in the worst case thus almost settling a conjecture of pach et al 2003 our result extends in a significant way the result of pach et al 2003 for the restricted case of nearly congruent cubes the analysis uses cuttings combined with the dobkin kirkpatrick hierarchical decomposition of convex polytopes in order to partition space into subcells so that on average the overwhelming majority of the tetrahedra intersecting a subcell behave as fat dihedral wedges in as an immediate corollary we obtain that the combinatorial complexity of the union of n cubes in r3 having arbitrary side lengths is o n2 e for any e 0 again significantly extending the result of pach et al 2003 finally our analysis can easily be extended to yield a nearly quadratic bound on the complexity of the union of arbitrarily oriented fat triangular prisms whose cross sections have arbitrary sizes in r3

existence and multiplicity of solutions for asymptotically linear nonperiodic hamiltonian elliptic system abstract this paper is concerned with the following nonperiodic hamiltonian elliptic system u v x u h v x u v v v x v h u x u v x r n u x v x 0 as x where z u v r n r r n 3 assuming the potential v is nonperiodic and sign changing h x z is nonperiodic in x and asymptotically quadratic in z u v existence and multiplicity of solutions are obtained via a variational approach

sign changing solutions of a class of nonlocal quasilinear elliptic boundary value problems abstract this paper is concerned with nonlocal kirchhoff s equation a b u 2 u f x u in u 0 on via variational methods and invariant sets of descent flow we obtain existence of signed and sign changing solutions with asymptotically 3 linear bounded nonlinearity

existence and multiplicity of solutions for a non periodic schrodinger equation abstract this paper is concerned with the non periodic schrodinger equation u v x u f x u x r n u x 0 as x under a general spectral assumption the existence and multiplicity of solutions are obtained for asymptotically linear nonlinearity via variational methods

multiple solutions for a schrodinger type equation with an asymptotically linear term abstract we study the nonlinear schrodinger type equation u g x 1 u f u on the whole space r n the nonlinearity f is assumed to be asymptotically linear and g x 0 has a potential well using results from critical point theory we show how the behavior of f near zero and at infinity affects the number of solutions

on the existence of eigenvalues of the schrodinger operatorh λw in a gap of σ h the authors prove a number of results asserting the existence of eigenvalues of the schrodinger operatorh w in a gap of h whereh v andv andw are bounded the existence of these eigenvalues is an important element in the theory of the colour of crystals the basic theorems are proved in v stronger results forv 1 are also presented

semialgebraic range reporting and emptiness searching with applications in a typical range emptiness searching resp reporting problem we are given a set p of n points in mathbb r d and we wish to preprocess it into a data structure that supports efficient range emptiness resp reporting queries in which we specify a range sigma which in general is a semialgebraic set in mathbb r d of constant description complexity and we wish to determine whether p cap sigma emptyset or to report all the points in p cap sigma range emptiness searching and reporting arise in many applications and have been treated by matousek comput geom theory appl 2 1992 pp 169 186 in the special case where the ranges are half spaces bounded by hyperplanes as shown in matousek s work the two problems are closely related and they have solutions for the case of half spaces with similar performance bounds in this paper we extend the analysis to general semialgebraic ranges and show how to adapt matousek s technique without the need to linearize the ranges into a higher dimensional space this yields more efficient solutions to several useful problems and we demonstrate the new technique in four applications with the following results i an algorithm for ray shooting amid balls in mathbb r 3 which uses o n storage and o n preprocessing we use the notation o n gamma to mean an upper bound of the form c varepsilon n gamma varepsilon which holds for any varepsilon 0 where c varepsilon is a constant that depends on varepsilon and answers a query in o n 2 3 time improving the previous bound of o n 3 4 ii an algorithm that preprocesses in o n time a set p of n points in mathbb r 3 into a data structure with o n storage so that for any query line ell or for that matter any simply shaped convex set the point of p farthest from ell can be computed in o n 1 2 time this in turn yields an algorithm that computes the largest area triangle spanned by p in time o n 26 11 as well as nontrivial algorithms for computing the largest perimeter or largest height triangle spanned by p iii an algorithm that preprocesses in o n time a set p of n points in mathbb r 2 into a data structure with o n storage so that for any query alpha fat triangle delta we can determine in o 1 time whether delta cap p is empty alternatively we can report in o 1 o k time the points of delta cap p where k delta cap p iv an algorithm that preprocesses in o n time a set p of n points in mathbb r 2 into a data structure with o n storage so that given any query semidisk c or a circular cap larger than a semidisk we can determine in o 1 time whether c cap p is empty or report the k points in c cap p in o 1 o k time adapting the recent techniques of b aronov and s har peled siam j comput 38 2008 pp 899 921 b aronov s har peled and m sharir on approximate halfspace range counting and relative epsilon approximations in proceedings of the 23rd acm symposium comput geom 2007 pp 327 336 b aronov and m sharir siam j comput 39 2010 pp 2704 2725 we can turn our solutions into efficient algorithms for approximate range counting with small relative error for the cases mentioned above our technique is closely related to the notions of nearest or farthest neighbor generalized voronoi diagrams and of the union or intersection of geometric objects where sharper bounds on the combinatorial complexity of decompositions of complements of these structures yield faster range emptiness searching or reporting algorithms

compactness results for schrodinger equations with asymptotically linear terms abstract we study the nonlinear problem u v x f x u x r n lim x u x 0 where the schrodinger operator v is semibounded and the nonlinearity f is linearly bounded we establish compactness of palais smale sequences and cerami sequences for the associated energy functional under general spectral theoretic assumptions applying these results we obtain existence of three nontrivial solutions if the energy functional has a mountain pass geometry

fat triangles determine linearly many holes the authors show that for every fixed delta 0 the following holds if f is a union of n triangles all of whose angles are at least delta then the complement of f has o n connected components and the boundary of f consists of o n log log n straight segments where the constants of proportionality depend on delta this latter complexity becomes linear if all triangles are of roughly the same size or if they are all infinite wedges