positive periodic solutions of singular systems with a parameter abstract the existence and multiplicity of positive periodic solutions for second order non autonomous singular dynamical systems are established with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter our results provide a unified treatment for the problem and significantly improve several results in the literature the proof of our results is based on the krasnoselskii fixed point theorem in a cone

weak singularities may help periodic solutions to exist abstract in a periodically forced semilinear differential equation with a singular nonlinearity a weak force condition enables the achievement of new existence criteria through a basic application of schauder s fixed point theorem the originality of the arguments relies in that contrary to the customary situation in the available references a weak singularity facilitates the arguments of the proofs

nonlinear periodic boundary value problems with sign changing green s function abstract we study the existence and nonexistence of positive solutions of nonlinear periodic boundary value problems u a t u b t f u a e t 0 t u 0 u t u 0 u t where b 0 a 0 and the green s function g t s of the linear problem u a t u 0 a e t 0 t u 0 u t u 0 u t may change sign on 0 t 0 t

an entropy based measure for comparing distributions of complexity abstract this paper is part of a series addressing the empirical statistical distribution of the diversity of complexity within and amongst complex systems here we consider the problem of measuring the diversity of complexity in a system given its ordered range of complexity types i and their probability of occurrence p i with the understanding that larger values of i mean a higher degree of complexity to address this problem we introduce a new complexity measure called case based entropy c c a modification of the shannon wiener entropy measure h the utility of this measure is that unlike current complexity measures which focus on the macroscopic complexity of a single system c c can be used to empirically identify and measure the distribution of the diversity of complexity within and across multiple natural and human made systems as well as the diversity contribution of complexity of any part of a system relative to the total range of ordered complexity types

multiplicity of positive periodic solutions to superlinear repulsive singular equations abstract in this paper we study positive periodic solutions to the repulsive singular perturbations of the hill equations it is proved that such a perturbation problem has at least two positive periodic solutions when the anti maximum principle holds for the hill operator and the perturbation is superlinear at infinity the proof relies on a nonlinear alternative of leray schauder type and on krasnoselskii fixed point theorem on compression and expansion of cones

remarks on periodic orbits of dynamical systems with repulsive singularities abstract we study existence and multiplicity of periodic solutions with prescribed period in the case of conservative systems having several singularities of repulsive type these results are also used to prove existence and multiplicity of periodic bounce trajectories in exterior domains

existence of one signed periodic solutions of some second order differential equations via a krasnoselskii fixed point theorem this paper is devoted to study the existence of periodic solutions of the second order equation x 00 f t x where f is a caratheodory function by combining some new properties of green s function together with krasnoselskii fixed point theorem on compression and expansion of cones as applications we get new existence results for equations with jumping nonlinearities as well as equations with a repulsive or attractive singularity in this latter case our results cover equations with weak singularities and are compared with some recent results by i rachunkova m tvrdyand i vrkoc r 2002 elsevier science usa all rights reserved

on some examples and constructions of contact manifolds the first goal of this paper is to construct examples of higher dimensional contact manifolds with specific properties our main results in this direction are the existence of tight virtually overtwisted closed contact manifolds in all dimensions and the fact that every closed contact 3 manifold which is not smoothly a rational homology sphere contact embeds with trivial normal bundle inside a hypertight closed contact 5 manifold this uses known construction procedures by bourgeois on products with tori and geiges on branched covering spaces we pass from these procedures to definitions this allows to prove a uniqueness statement in the case of contact branched coverings and to study the global properties such as tightness and fillability of the results of both constructions without relying on any auxiliary choice in the procedures a second goal allowed by these definitions is to study relations between these constructions and the notions of supporting open book as introduced by giroux and of contact fiber bundle as introduced by lerman for instance we give a definition of bourgeois contact structures on flat contact fiber bundles which is local strictly includes the results of the bourgeois construction and allows to recover an isotopy class of supporting open books on the fibers this last point relies on a reinterpretation inspired by an idea by giroux of supporting open books in terms of pairs of contact vector fields

multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces abstract the existence of multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces is discussed in this paper our nonlinearity may be singular in its dependent variable and our analysis relies on a nonlinear alternative of leray schauder type and on a fixed point theorem in cones

canonical forms for symmetric skew symmetric real matrix pairs under strict equivalence and congruence a systematic development is made of the simultaneous reduction of pairs of quadratic forms over the reals one of which is skew symmetric and the other is either symmetric or skew symmetric these reductions are by strict equivalence and by congruence over the reals or over the complex numbers and essentially complete proofs are presented the proofs are based on canonical forms attributed to jordan and kronecker some closely related results which can be derived from the canonical forms of pairs of symmetric skew symmetric real forms are also included they concern simultaneously neutral subspaces hamiltonian and skew hamiltonian matrices and canonical structures of real matrices which are selfadjoint or skew adjoint in a regular skew symmetric indefinite inner product and real matrices which are skew adjoint in a regular symmetric indefinite inner product the paper is largely expository and continues the comprehensive account of the reduction of pairs of matrices started in p lancaster l rodman canonical forms for hermitian matrix pairs under strict equivalence and congruence siam rev in press

noncollision periodic solutions to some singular dynamical systems with very weak forces abstract we prove a double variational characterization of the set of all the periodic solutions of the kepler problem we provide a sufficient condition for the existence of a periodic solution with prescribed period or energy for autonomous second order differential systems with keplerian type potentials the proofs are based on linking arguments

tight contact structures with no symplectic fillings we exhibit tight contact structures on 3 manifolds that do not admit any symplectic fillings

periodic solutions of lienard equations with singular forces of repulsive type in this paper we will use the coincidence degree to give an existence result of periodic solutions for the scalar lienard equations with singular forces of repulsive type the main result of this paper clearly describes what balance conditions between the singular force at the singularity and at the infinity should be imposed if the equation has a periodic solution as an application we will improve the results of ding ye and wang for the brillouin electron beam focusing system

positive periodic solutions of singular systems of first order ordinary differential equations the existence and multiplicity of positive periodic solutions for first non autonomous singular systems are established with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter the proof of our results is based on the krasnoselskii fixed point theorem in a cone

positive periodic solutions of first order singular systems abstract we establish a complete structure for the existence of positive periodic solutions of first order non autonomous singular system u i t a i t u i t b i t f i u t i 1 n the methods employed are fixed point index theorems leray schauder degree and lower and upper solutions method

symplectic manifolds with contact type boundaries an example of a 4 dimensional symplectic manifold with disconnected boundary of contact type is constructed a collection of other results about symplectic manifolds with contact type boundaries are derived using the theory ofj holomorphic spheres in particular the following theorem of eliashberg floer mcduff is proved if a neighbourhood of the boundary of v is symplectomorphic to a neighbourhood ofs2n 1 in standard euclidean space and if vanishes on all 2 spheres inv thenv is diffeomorphic to the ballb2n

pseudoholomorphic curves in symplectizations with applications to the weinstein conjecture in dimension three l in t roduct ion 515 1 1 the weinstein conjecture 515 1 2 the main results 517 1 3 sketch of the proof 522 2 local f redho lm theory 525 2 1 similar i ty principle and consequences 525 2 2 n m as a lmost complex manifold and local fillings 529 2 3 local propert ies of ho lomorphic curves 531 3 bubbl ing off analysis and global fredholm theory 533 3 l grad ien t bounds imply ca bounds 533 3 2 bubbing off analysis of ho lomorphic planes 534 3 3 grad ien t bounds near the boundary 537 4 proof of the main results 544 4 1 set up for a global fredholm theory 544 4 2 bishop s theorem 545 4 3 proof of theorem 5 546 4 4 proof of theorems 8 and 11 547 4 5 uniqueness results for foliations 547 5 appendix 550 5 1 per turbing surfaces 550 5 2 ki l l ing crit ical points of hypersurfaces 552 5 3 adding a thin tube to a sphere 560

renyi entropy and complexity measure for skew gaussian distributions and related families abstract in this paper we provide the renyi entropy and complexity measure for a novel flexible class of skew gaussian distributions and their related families as a characteristic form of the skew gaussian shannon entropy we give closed expressions considering a more general class of closed skew gaussian distributions and the weighted moments estimation method in addition closed expressions of renyi entropy are presented for extended skew gaussian and truncated skew gaussian distributions finally additional inequalities for skew gaussian and extended skew gaussian renyi and shannon entropies are reported

geometric criteria for overtwistedness we establish geometric criteria to decide whether a contact manifold is over twisted starting with the original denition we rst relate the dierent overtwisted disks d 2n ot in each dimension and show that a manifold is overtwisted if the legendrian un knot is loose then we characterize overtwistedness in terms of open book decompositions and provide several applications

existence multiplicity and infinite solvability of positive solutions to a nonlinear fourth order periodic boundary value problem abstract the existence of n and infinitely many positive solutions is proved for the nonlinear fourth order periodic boundary value problem u 4 t u t u t f t u t 0 t 1 u i 0 u i 1 i 0 1 2 3 where n is an arbitrary natural number and 2 2 0 1 2 2 2 2 4 2 1 0 this kind of fourth order boundary value problems usually describes the equilibrium state of an elastic beam with periodic boundary condition the main results show that the problem may have n or infinitely many positive solutions provided the growth rates of nonlinear term f t l are appropriate on some bounded subsets of its domain

multiple positive solutions of some boundary value problems abstract we study the existence of multiple positive solutions of the equations u t u subject to linear boundary conditions we show that there are at least two positive solutions if t u is superlinear at one end point zero or infinity and sublinear at the other applications of these results are provided to yield multiple positive solutions of some elliptic boundary value problems on an annulus

polyfolds cobordisms and the strong weinstein conjecture abstract we prove the strong weinstein conjecture for closed contact manifolds that appear as the concave boundary of a symplectic cobordism admitting an essential local foliation by holomorphic spheres

local systems on the free loop space and finiteness of the hofer zehnder capacity in this article we examine under which conditions symplectic homology with local coefficients of a unit disk bundle d m vanishes for instance this is the case if the hurewicz map pi 2 m rightarrow h 2 m mathbb z is nonzero as an application we prove finiteness of the pi 1 sensitive hofer zehnder capacity of unit disk bundles in these cases we also prove uniruledness for such cotangent bundles moreover we find an obstruction to the existence of h space structures on general topological spaces formulated in terms of local systems

a relationship between the periodic and the dirichlet bvps of singular differential equations in this paper a relationship between the periodic and the dirichlet boundary value problems for second order ordinary differential equations with singularities is established this relationship may be useful in explaining the difference between the nonresonance of singular and nonsingular differential equations using this relationship we give in this paper an existence result of positive periodic solutions to singular differential equations when the singular forces satisfy some strong force condition at the singularity 0 and some linear growth condition at infinity

fredholm theory and transversality for noncompact pseudoholomorphic maps in symplectizations we study pseudoholomorphic maps from a punctured riemann surface into the symplectization of a contact manifold a fredholm theory yields the virtual dimension of the moduli spaces of such maps in terms of the euler characteristic of the riemann surface and the asymptotics data given by the periodic solutions of the reeb vector field associated to the contact form the transversality results establish the existence of additional structure for these spaces to be more precise we prove that these spaces are generically smooth manifolds and therefore their virtual dimension coincides with their actual dimension

weak and strong fillability of higher dimensional contact manifolds for contact manifolds in dimension three the notions of weak and strong symplectic fillability and tightness are all known to be inequivalent we extend these facts to higher dimensions in particular we define a natural generalization of weak fillings and prove that it is indeed weaker at least in dimension five while also being obstructed by all known manifestations of overtwistedness we also find the first examples of contact manifolds in all dimensions that are not symplectically fillable but also cannot be called overtwisted in any reasonable sense these depend on a higher dimensional analogue of giroux torsion which we define via the existence in all dimensions of exact symplectic manifolds with disconnected contact boundary

on the existence of positive solutions for semilinear elliptic equations in the annulus abstract we study the existence of positive radial solutions of u g x u 0 in annuli with dirichlet dirichlet neumann boundary conditions we prove that the problems have positive radial solutions on any annulus if is sublinear at 0 and

a statistical measure of complexity with nonextensive entropy abstract a statistical measure of complexity utilising the concept of entropy or information is proposed our way in this study is to use a nonextensive entropy instead of an extensive additive shannon entropy in the definition but can be characterised as a difference between the qth order renyi entropy and the second one furthermore we devise a conditional joint and mutual complexity measure as a coherent possibility the behavior of the measure for the logistic map shows that it is more sensitive to nonextensivity at the transition point ac 3 8284 than any other values when 0

positive solutions to boundary value problems with nonlinear boundary conditions abstract in this paper we consider the boundary value problem y t f t y t subject to the boundary conditions y a y and y 2 b 0 in this setting c rd a 2 b t r r is a continuous functional which represents a nonlinear nonlocal boundary condition by imposing sufficient structure on and the nonlinearity f we deduce the existence of at least one positive solution to this problem the novelty in our setting lies in the fact that may be strictly nonpositive for some y 0 our results are achieved by appealing to the krasnosel ski fixed point theorem we conclude with several examples illustrating our results and the generalizations that they afford

global structure of positive solutions for superlinear second order periodic boundary value problems in this paper we consider the nonlinear eigenvalue problems gather u lambda h t f u 0 quad 0 0 for i 1 ldots m 2 with sum i 1 m 2 alpha i eta i 0 for t 0 in 0 1 f in c 0 infty 0 infty and f s 0 for s 0 and f 0 infty where f 0 lim s rightarrow 0 f s s we investigate the global structure of positive solutions by using the nonlinear krein rutman theorem

on the number of positive solutions of nonlinear systems abstract we prove that appropriate combinations of superlinearity and sublinearity of f u with respect to at zero and infinity guarantee the existence multiplicity and nonexistence of positive solutions to boundary value problems for the n dimensional system u h t f u 0 0 is defined by u u 1 u n where u u 1 u n and covers the two important cases u u and u u p 2 u p 1 our methods employ fixed point theorems in a cone

strongly fillable contact manifolds and j holomorphic foliations we prove that every strong symplectic filling of a planar contact manifold admits a symplectic lefschetz fibration over the disk and every strong filling of t 3 similarly admits a lefschetz fibration over the annulus it follows that strongly fillable planar contact structures are also stein fillable and all strong fillings of t 3 are equivalent up to symplectic deformation and blowup these constructions result from a compactness theorem for punctured j holomorphic curves that foliate a convex symplectic manifold we use it also to show that the compact v supported symplectomorphism group on t t 2 is contractible and to define an obstruction to strong fillability that yields a non gauge theoretic proof of gay s recent nonfillability result g for contact manifolds with positive giroux torsion

on the existence of collisionless equivariant minimizers for the classical n body problem we show that the minimization of the lagrangian action functional on suitable classes of symmetric loops yields collisionless periodic orbits of the n body problem provided that some simple conditions on the symmetry group are satisfied more precisely we give a fairly general condition on symmetry groups g of the loop space for the n body problem with potential of homogeneous degree with 0 which ensures that the restriction of the lagrangian action mathcal a to the space g of g equivariant loops is coercive and its minimizers are collisionless without any strong force assumption in proving that local minima of g are free of collisions we develop an averaging technique based on marchal s idea of replacing some of the point masses with suitable shapes see 10 as an application several new orbits can be found with some appropriate choice of g furthermore the result can be used to give a simplified and unitary proof of the existence of many already known minimizing periodic orbits

quantitative darboux theorems in contact geometry this paper begins the study of relations between riemannian geometry and contact topology on 2n 1 manifolds and continues this study on 3 manifolds specifically we provide a lower bound for the radius of a geodesic ball in a contact 2n 1 manifold m that can be embedded in the standard contact structure on r that is on the size of a darboux ball the bound is established with respect to a riemannian metric compatible with an associated contact form for in dimension three it further leads us to an estimate of the size for a standard neighborhood of a closed reeb orbit the main tools are classical comparison theorems in riemannian geometry in the same context we also use holomorphic curves techniques to provide a lower bound for the radius of a ps tight ball

existence of nonnegative and nonpositive solutions for second order periodic boundary value problems we are interested in nonnegative and nonpositive solutions of the boundary value problem u f t u u 0 u 1 u 0 u 1 where f fulfils the caratheodory conditions on 0 1 r we generalize the results reached by m n nkashama j santanilla and l sanchez and present estimates for solutions in addition we apply our existence theorems to periodic boundary value problems for nonlinear duffing equations whose right hand sides have a repulsive or attractive singularity at the origin we extend or generalize existence results by a c lazer and s solimini and other authors moreover we get some multiplicity results and in the case of a repulsive singularity we also admit a weak singularity in constrast to the previous papers on this subject our proofs are based on the method of lower and upper functions and topological degree arguments and the results are tested on examples

contact homology and virtual fundamental cycles we give a construction of contact homology in the sense of eliashberg givental hofer specifically we construct coherent virtual fundamental cycles on the relevant compactified moduli spaces of pseudo holomorphic curves the aim of this work is to provide a rigorous construction of contact homology an invariant of contact manifolds and exact symplectic cobordisms due to eliashberg givental hofer eli98 egh00 the contact homology of y is defined in terms of pseudo holomorphic curve counts in the sense of gromov gro85 in the symplectization r y unfortunately the moduli spaces of such curves often suffer from a severe lack of transversality and for this reason it has until now remained an important open problem to define the relevant curve counts in full generality we will use the framework developed in par15 to construct coherent virtual fundamental cycles on the moduli spaces in question thus giving rise to the desired curve counts our methods are quite general and should apply equally well to many other moduli spaces of interest previously a number of important special cases of contact homology and closely related invariants have been constructed rigorously using generic and or automatic transversality techniques cylindrical contact homology of some three manifolds was constructed by bao

the topology of stein fillable manifolds in high dimensions i we continue our study of contact structures on manifolds of dimension at least five using complex surgery theory we show that in each dimension 2q 1 3 there are maximal almost contact manifolds to which there is a stein cobordism from any other 2q 1 dimensional contact manifold we show that the product m x s 2 admits a weakly fillable contact structure provided m admits a weak symplectic filling w omega with omega pi 2 m 0 we also study the connection between stein fillability and connected sums we give examples of almost contact manifolds for which the connected sum is stein fillable while the components are not concerning obstructions to stein fillings we show that the 8k 1 dimensional sphere has an almost contact structure which is not stein fillable once k 1 as a consequence we deduce that any highly connected almost contact 8k 1 manifold with k 1 admits an almost contact structure which is not stein fillable

multiple closed orbits of fixed energy forn body type problems with gravitational potentials abstract using the equivariant ljusternik schnirelmann theory and the estimates of the upper bound and lower bound of the critical values for the variational functional we obtain results in general concerning multiple geometrically distinct periodic solutions of fixed energy for a class of planarn body type problems

periodic solutions of second order non autonomous singular dynamical systems abstract in this paper we establish two different existence results of positive periodic solutions for second order non autonomous singular dynamical systems the first one is based on a nonlinear alternative principle of leray schauder and the result is applicable to the case of a strong singularity as well as the case of a weak singularity the second one is based on schauder s fixed point theorem and the result sheds some new light on problems with weak singularities and proves that in some situations weak singularities may help create periodic solutions recent results in the literature are generalized and significantly improved

periodic solutions to some problems of n body type we prove the existence of at least one t periodic solution to a dynamical system of the type m i ddot u i sum limits j 1 j ne i n triangledown v ij u i u j text t 1 where the potentials v ij are t periodic in t and singular at the origin u i e r k i 1 n and k 3 we also provide estimates on the h 1 norm of this solution the proofs are based on a variant of the ljusternik schnirelman method the results here generalize to the n body problem some results obtained by bahri rabinowitz on the 3 body problem in 6