morse theory for periodic solutions of hamiltonian systems and the maslov index in this paper we prove morse type inequalities for the contractible 1 periodic solutions of time dependent hamiltonian differential equations on those compact symplectic manifolds m for which the symplectic form and the first chern class of the tangent bundle vanish over q m the proof is based on a version of infinite dimensional morse theory which is due to floer the key point is an index theorem for the fredholm operator which plays a central role in floer homology the index formula involves the maslov index of nondegenerate contractible periodic solutions this maslov index plays the same role as the morse index of a nondegenerate critical point does in finite dimensional morse theory we shall use this connection between floer homology and maslov index to establish the existence of infinitely many periodic solutions having integer periods provided that every i periodic solution has at least one floquet multiplier which is not equal to 1

local properties of j complex curves in lipschitz continuous structures we prove the existence of primitive curves and positivity of intersections of j complex curves for lipschitz continuous almost complex structures these results are deduced from the comparison theorem for j holomorphic maps in lipschitz structures previously known for j of class mathcal c 1 lip we also give the optimal regularity of curves in lipschitz structures it occurs to be mathcal c 1 lnlip i e the first derivatives of a j complex curve for lipschitz j are log lipschitz continuous a simple example that nothing better can be achieved is given further we prove the genus formula for j complex curves and determine their principal puiseux exponents all this for lipschitz continuous j s

self dual instantons and holomorphic curves a gradient flow of a morse function on a compact riemannian manifold is said to be of morse smale type if the stable and unstable manifolds of any two critical points intersect transversally for such a morse smale gradient flow there is a chain complex generated by the critical points and graded by the morse index the boundary operator has as its x y entry the number of gradient flow lines running from x to y counted with appropriate signs whenever the difference of the morse indices is 1 the homology of this chain complex agrees with the homology of the underlying manifold m and this can be used to prove the morse inequalities cf 33 26 around 1986 floer generalized this idea to infinite dimensional variational problems in which every critical point has infinite morse index but the moduli spaces of connecting orbits form finite dimensional manifolds for every pair of critical points the dimensions of these spaces give rise to a relative morse index and the boundary operator is defined by counting connecting

hodge integrals and degenerate contributions hodge integral techniques are used to compute the degree 1 degenerate contributions of curves of arbitrary genus in the gromov witten theory of 3 folds in the calabi yau case the contributions are compared to related m theoretic calculations in the fano case the contributions suggest new integrality conditions

the principle of strictly increasing mixing character in incompatible general measurement it is known that the density operators rho of a quantal system are grouped into classes of equally mixed ones the class to which rho belongs is its so called mixing character the set of classes is known to be a lattice the ruch and mead 1976 principle of increasing mixing character in complete measurement is extended to general observables it is shown that the principle of strictly increasing mixing character holds in general incompatible measurement and some consequences are discussed a sufficient condition for strictly mixing homomorphic functionals i e for those which preserve the strictly larger relation in the lattice is obtained density operator dependent strengthening of the natural pre order in the set of general observables is investigated

real gromov witten theory in all genera and real enumerative geometry construction the first part of this work constructs positive genus real gromov witten invariants of realorientable symplectic manifolds of odd complex dimensions the second part studies the orientations on the moduli spaces of real maps used in constructing these invariants the present paper applies the results of the latter to obtain quantitative and qualitative conclusions about the invariants defined in the former after describing large collections of real orientable symplectic manifolds we show that the genus 1 real gromov witten invariants of sufficiently positive almost kahler threefolds are signed counts of real genus 1 curves only and thus provide direct lower bounds for the counts of these curves in such targets we specify real orientations on the real orientable complete intersections in projective spaces the real gromov witten invariants they determine are in a sense canonically determined by the complete intersection itself at least in most cases we also obtain equivariant localization data that computes the real invariants of projective spaces and determines the contributions from many torus fixed loci for other complete intersections our results confirm walcher s predictions for the vanishing of these invariants in certain cases and for the localization data in other cases

the study of a class of rational difference equations abstract in this paper we study the global behavior of the higher order nonlinear difference equation x n 1 x n k 1 x n 2 k 1 bx n k 1 cx n 2 k 1 n 0 1 where b c and are positive k 1 2 3 and the initial conditions x 2 k 1 x 1 x 0 are positive real numbers we investigate the character of semicycles periodicity global stability and convergence to 2 k cycle solutions of mentioned equation

the qualitative behavior of solutions of a nonlinear difference equation abstract this paper is concerned with the qualitative behavior of solutions to the difference equation x n 1 p qx n k 1 x n n 0 1 2 where the initial conditions x k x 1 x0 are non negative k 1 2 3 and the parameters p q are non negative we start by establishing the periodicity the character of semicycles the global stability and the boundedness of the above mentioned equation we also present solutions that have unbounded behavior it is worth to mention that this difference equation is a special case of an open problem was introduced by m r s kulenovic and g ladas dynamics of second order rational difference equations with open problems and conjectures chapman and hall crc boca raton 2002 several computational examples are given to support our theoretical discussions the presented numerical tests represent different types of qualitative behavior of solutions to our nonlinear difference equation

on the higher order rational recursive sequence xn axn k bx n 3k the main purpose of the current paper is to prove that every positive solution of the delay difference equationx n ax n k bx n 3 k where a b 0 the initial conditions x 3 k 1 x 3 k 2 x 0 0 and k 1 2 3 converges eventually to a period k solution we also give the results of computational examples to support our theoretical discussion

on the global behavior of higher order recursive sequences our basic purpose in this paper is to establish some conditions for the global attractivity and convergence of the kth order recursive sequencex n f x n 1 x n 2 x n k n 1 2 3 we will show that if f is a real valued function with a convex basin w in r k which is differentiable with bounded differential then f has an equilibrium point our final comments are about informative examples

stability criteria for two partial difference equations abstract this paper is concerned with two classes of linear partial difference equations with constant coefficients explicit conditions are derived which are sufficient and or necessary for these equations to have stable solutions

a comparison theorem for gromov witten invariants in the symplectic category abstract we exploit the geometric approach to the virtual fundamental class due to fukaya ono and li tian to compare gromov witten invariants of a symplectic manifold and a symplectic submanifold whenever all constrained stable maps to the former are contained in the latter to first order various special cases of the comparison theorem in this paper have long been used in the algebraic category some of them have also appeared in the symplectic setting combined with the inherent flexibility of the symplectic category the main theorem leads to a confirmation of pandharipande s gopakumar vafa prediction for gw invariants of fano classes in 6 dimensional symplectic manifolds the proof of the main theorem uses deformations of the cauchy riemann equation that respect the submanifold and carleman similarity principle for solutions of perturbed cauchy riemann equations in a forthcoming paper we apply a similar approach to relative gromov witten invariants and the absolute relative correspondence in genus 0

sub poissonian and anti bunching criteria via majorization of statistics we use majorization and confidence intervals as a convenient tool to compare the uncertainty in photon number for different quantum light states to this end majorization is formulated in terms of confidence intervals as a suitable case study we apply this tool to sub and super poissonian behavior and bunching and anti bunching effects we focus on the most significant classical and nonclassical states such as glauber coherent thermal photon number and squeezed states we show that majorization provides a more complete analysis that in some relevant situations contradicts the predictions of variance

mirror symmetry for closed open and unoriented gromov witten invariants abstract in the first part of this paper we obtain mirror formulas for twisted genus 0 two point gromov witten gw invariants of projective spaces and for the genus 0 two point gw invariants of fano and calabi yau complete intersections this extends previous results for projective hypersurfaces following the same approach but we also completely describe the structure coefficients in both cases and obtain relations between these coefficients that are vital to the applications to mirror symmetry in the rest of this paper in the second and third parts of this paper we confirm walcher s mirror symmetry conjectures for the annulus and klein bottle gw invariants of calabi yau complete intersection threefolds these applications are the main results of this paper in a separate paper the genus 0 two point formulas are used to obtain mirror formulas for the genus 1 gw invariants of all calabi yau complete intersections

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

vanishing of rabinowitz floer homology on negative line bundles following frauenfelder rabinowitz action functional on very negative line bundles habilitationsschrift munich munchen 2008 albers and frauenfelder bubbles and onis 2014 arxiv 1412 4360 we construct rabinowitz floer homology for negative line bundles over symplectic manifolds and prove a vanishing result ritter adv math 262 1035 1106 2014 showed that symplectic homology of these spaces does not vanish in general thus the theorem mathrm sh 0 leftrightarrow mathrm rfh 0 ritter in j topol 6 2 391 489 2013 does not extend beyond the symplectically aspherical situation we give a conjectural explanation in terms of the cieliebak frauenfelder oancea long exact sequence cieliebak et al ann sci ec norm super 4 43 6 957 1015 2010

dynamics of a higher order rational difference equation abstract our aim in this paper is to investigate the dynamics of difference equation y n 1 p qy n 1 y n k n 0 1 2 where the initial conditions y k y 1 y0 are non negative k 1 2 3 and the parameters p and q are non negative we study characteristics such as the global stability the boundedness of positive solutions and the character of semicycles of the above mentioned difference equation in particular our results solve the open problem introduced by kulenovic and ladas in their monograph m r s kulenovic g ladas dynamics of second order rational difference equations with open problems and conjectures chapman and hall crc boca raton 2002

a quantum cup length estimate for symplectic fixed points a new lower bound for the number of fixed points of hamiltonian automorphisms of closed symplectic manifolds m is established the new estimate extends the previously known estimates to the class of weakly monotone symplectic manifolds we prove for arbitrary closed symplectic manifolds with rational symplectic class that the cup length estimate holds true if the hofer energy of the hamiltonian automorphism is sufficiently small for arbitrary energy and on weakly monotone symplectic manifolds we define an analogon to the cup length based on the quantum cohomology ring of m providing a quantum cup length estimate

finite energy foliations of tight three spheres and hamiltonian dynamics surfaces of sections are a classical tool in the study of 3 dimensional dynamical systems their use goes back to the work of poincare and birkhoff in the present paper we give a natural generalization of this concept by constructing a system of transversal sections in the complement of finitely many distinguished periodic solutions such a system is established for nondegenerate reeb flows on the tight 3 sphere by means of pseudoholomorphic curves the applications cover the nondegenerate geodesic flows on t1s2 rp3 via its double covering s3 and also nondegenerate hamiltonian systems in ir4 restricted to sphere like energy surfaces of contact type

floer cohomology and geometric composition of lagrangian correspondences abstract we prove an isomorphism of floer cohomologies under geometric composition of lagrangian correspondences in exact and monotone settings

a module structure on the symplectic floer cohomology the aim of this paper is to offer an affirmative answer to the floer conjectures in 2 p 589 which states that there is a module structure on the z 2 n graded symplectic floer cohomology for monotone symplectic manifolds by constructing a z graded symplectic floer cohomology as an integral lift of the z 2 n graded symplectic floer cohomology we gain control of the holomorphic bubbling spheres this makes a module structure on the z graded floer cohomology there is a spectral sequence with e 1 given by the z graded symplectic floer cohomology such a spectral sequence converges to the z 2 n graded symplectic floer cohomology hence we induce a module structure for the z 2 n graded symplectic floer cohomology by the spectral sequence and algebraic topology methods

enumeration of genus two curves with a fixed complex structure in ℙ 2 and ℙ 3 the main concrete result of this paper is enumeration of genus two curves with complex structure fixed in p and p along the way rational curves with certain simple singularities are counted as well while the methods described can be used to count positive genus curves in some other cases the most powerful direct applications of the machinery developed are to enumeration of rational curves with a very large class of singularities in projective spaces

the reduced genus 1 gromov witten invariants of calabi yau hypersurfaces 0 introduction 691 0 1 mirror symmetry predictions for a quintic threefold 691 0 2 computing gw invariants of hypersurfaces 693 0 3 mirror symmetry formulas for projective cy hypersurfaces 695 1 equivariant cohomology and stable maps 698 1 1 definitions and notation 698 1 2 setup for localization computation on m1 1 p d 701 1 3 contributions from fixed loci i 704 1 4 contributions from fixed loci ii 708 2 localization computations 712 2 1 regularizable power series in rational functions 712 2 2 regularizability of gw generating functions 714 2 3 proofs of propositions 1 1 and 1 2 717 3 algebraic computations 721 3 1 linear independence in symmetric rational functions 722 3 2 the genus 0 generating functions 723 3 3 proof of theorem 3 725 appendix a some combinatorics 730 appendix b comparison of mirror symmetry formulations 733 acknowledgments 736 references 736

entropic measures of joint uncertainty effects of lack of majorization we compute renyi entropies for the statistics of a noisy simultaneous observation of two complementary observables in two dimensional quantum systems the relative amount of uncertainty between two states depends on the uncertainty measure used these results are not reproduced by a more standard duality relation we show that these behaviors are consistent with the lack of majorization relation between the corresponding statistics

morse theory for lagrangian intersections soit p une variete symplectique compacte et soit l p une sous variete lagrangienne avec 2 p l 0 pour un diffeomorphisme exact de p avec la propriete que l coupe l transversalement on demontre une inegalite de morse reliant l ensemble l l a la cohomologie de l

lower bounds for enumerative counts of positive genus real curves abstract we transform the positive genus real gromov witten invariants of many real orientable symplectic threefolds into signed counts of curves these integer invariants provide lower bounds for counts of real curves of a given genus that pass through conjugate pairs of constraints we conclude with some implications and related conjectures for hodge integrals

quasi implicit and two level explicit finite difference procedures for solving the one dimensional advection equation a variety of explicit and implicit algorithms has been studied dealing with the solution of the one dimensional advection equation these schemes are based on the weighted finite difference approximations this solution approach proves to be very effective this procedure is carried out in many disciplines including hydrology oceanography and meteorology the main idea behind the finite difference methods for obtaining the solution of a given partial differential equation is to approximate the derivatives appearing in the equation by a set of values of the function at a selected number of points the most usual way to generate these approximations is through the use of taylor series the basis of analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach it is worth noting that from the truncation error of the modified equivalent equation it is possible to eliminate the dominant error terms associated with the finite difference equations that contain weights thus leading to more accurate techniques quasi implicit and two level explicit finite difference techniques are considered in this article to approximately solve the one dimensional advection problem stability of the techniques are investigated using the von neumann stability analysis the results of a numerical experiment are presented and the accuracy and central processor unit cpu time needed are discussed and compared numerical tests show that the new third order or fourth order methods are more accurate than either of the second order techniques

the birkhoff lewis fixed point theorem and a conjecture of v i arnold abstract the following conjecture of v i arnold is proved every measure preserving diffeomorphism of the torus t2 which is homologeous to the identity and which leaves the center of mass invariant possesses at least 3 fixed points the proof of this global fixed point theorem does not make use of the generating function technique the theorem is a consequence of the statement that a hamiltonian vector field on a torus t2n which depends periodically on time possesses at least 2n 1 forced oscillations these periodic solutions are are found using the classical variational principle by means of two qualitative statements for general flows a second conjecture of v i arnold proved concerns a birkhoff lewis type fixed point theorem for symplectic maps additional keywords periodic functions variational principles equations reprints author

reduced genus one gromov witten invariants in a previous paper we described a natural closed subset m 0 1 k x a j of the moduli space m1 k x a j of stable genus one j holomorphic maps into a symplectic manifold x in this paper we generalize the definition of the main component to moduli spaces of perturbed in a restricted way j holomorphic maps this generalization implies that m 0 1 k x a j just like m1 k x a j carries a virtual fundamental class and can be used to define symplectic invariants these truly genus one invariants constitute part of the standard genus one gromovwitten invariants which arise from the entire moduli space m1 k x a j the new invariants are more geometric and can be used to compute the genus one gw invariants of complete intersections as shown in a separate paper

on the arnold conjecture for weakly monotone symplectic manifolds we show the arnold conjecture concerning symplectic fixed points in the case that the symplectic manifold is weakly monotone and all the fixed points are non degenerate in particular the conjecture is true in dimension 2 4 6 if all the fixed points are non degenerate

witten s complex and infinite dimensional morse theory we investigate the relation between the trajectories of a finite dimensional gradient flow connecting two critical points and the cohomology of the surrounding space the results are applied to an infinite dimensional problem involving the symplect c action function

the oscillatory character of the recursive sequence xn 1 α βxn k 1a bxn 2k 1 abstract the aim of this work is to generalize techniques of difference equations in order to apply them to a wider class we start by establishing the boundedness the oscillatory character and the global stability of the equation x n 1 x n k 1 a bx n 2 k 1 n 0 1 2 with positive parameters k 1 2 and nonnegative initial conditions

quivers floer cohomology and braid group actions we consider the derived categories of modules over a certain family a m of graded rings and floer cohomology of lagrangian intersections in the symplectic manifolds which are the milnor fibres of simple singularities of type a m we show that each of these two rather different objects encodes the topology of curves on an m 1 punctured disc we prove that the braid group b m 1 acts faithfully on the derived category of a m modules and that it injects into the symplectic mapping class group of milnor fibers the philosophy behind our results is as follows using floer cohomology one should be able to associate to the milnor fibre a triangulated category its construction has not been carried out in detail this triangulated category should contain a full subcategory which is equivalent up to a slight difference in the grading to the derived category of a m modules the full embedding would connect the two occurrences of the braid group thus explaining the similarity between them

a combinatorial description of knot floer homology given a grid presentation of a knot or link k in the three sphere we describe a heegaard diagram for the knot complement in which the heegaard surface is a torus and all elementary domains are squares using this diagram we obtain a purely combinatorial description of the knot floer homology of k

transversality in elliptic morse theory for the symplectic action our goal in this paper is to settle some transversality question for the perturbed nonlinear cauchy riemann equations on the cylinder these results play a central role in the denition of symplectic floer homology and hence in the proof of the arnold conjecture there is currently no other reference to these transversality results in the open literature our approach does not require aronszajn s theorem instead we derive the unique continuation theorem from a generalization of the carleman similarity principle

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

study of a system of non linear difference equations arising in a deterministic model for hiv infection a deterministic model for hiv infection in the presence of combination therapy is introduced the model is then approximated by an implicit finite difference method a system of difference equations to solve the ivp system we initially purpose the discrete analogue of the continuous model of hiv infection then the boundedness the local and the global asymptotic stability of the equilibria of the system of difference equations the bistability phenomenon and the bifurcation phenomenon are also investigated in addition we will show that the associated approximational method is very robust since the equilibria and the stability conditions are independent on the time step finally numerical simulations of the system are presented when the system has meaningless solutions in view point of biology or our theory is incapable

dynamics of a rational difference equation using both theoretical and computational approaches this paper is concerned with the qualitative behavior of solutions to the difference equationx n 1 x n px n qx n k n 0 1 2 where the initial conditions x k x 1 x 0 are non negative email protected 1 2 3 and the parameters p q are non negative our concentration is on invariant intervals the character of semicycles the global stability and the boundedness of the above mentioned equation it is worth to mention that this difference equation was an open problem introduced by kulenovic and ladas in m r s kulenovic and g ladas dynamics of second order rational difference equations with open problems and conjectures chapman and hall crc boca raton 2002 our final comments are about informative examples

homological mirror symmetry for calabi yau hypersurfaces in projective space we prove homological mirror symmetry for a smooth d dimensional calabi yau hypersurface in projective space for any d 3 for example d 3 is the quintic threefold the main techniques involved in the proof are the construction of an immersed lagrangian sphere in the d dimensional pair of pants the introduction of the relative fukaya category and an understanding of its grading structure a description of the behaviour of this category with respect to branched covers via an orbifold fukaya cate gory a morse bott model for the relative fukaya category that allows one to make explicit computations and the introduction of certain graded categories of matrix factorizations mirror to the relative fukaya category

green s function and stability of a linear partial difference scheme abstract this paper is concerned with a linear partial difference equation which arises from discretizing a heat and a first order wave equations a green s function for this equation is derived various representation theorems monotonicity and oscillatory properties of the green s functions are obtained then by means of the representation theorems we derive several stability criteria for arbitrary solutions of this equation

equivariant split generation and mirror symmetry of special isogenous tori abstract we prove a version of equivariant split generation of fukaya category when a symplectic manifold admits a free action of a finite group g combining this with some generalizations of seidel s algebraic frameworks from 35 we obtain new cases of homological mirror symmetry for some symplectic tori with non split symplectic forms which we call special isogenous tori this extends the work of abouzaid smith 2 we also show that derived fukaya categories are complete invariants of special isogenous tori

propagation in hamiltonian dynamics and relative symplectic homology the main result asserts the existence of noncontractible periodic orbits for compactly supported time dependent hamiltonian systems on the unit cotangent bundle of the torus or of a negatively curved manifold whenever the generating hamiltonian is sufficiently large over the zero section the proof is based on floer homology and on the notion of a relative symplectic capacity applications include results about propagation properties of sequential hamiltonian systems periodic orbits on hypersurfaces hamiltonian circle actions and smooth lagrangian skeletons in stein manifolds

bounds for solutions of a six point partial difference scheme abstract this paper is concerned with a class of partial difference equations which arise from discretising the one dimensional heat and the first order wave equations explicit bounds are found for their solutions and subexponential solutions are considered using the green s function we derive a stability criteria for arbitrary solutions of this equation

determination of nodes in numerical integration rules using difference equation in this paper the difference equations will be employed to produce a polynomial which its roots are the nodes of gauss quadrature rules in this procedure we obtain a system of nonlinear equations according to the concepts of precision degree of integration and the moments of order k hence the resulted polynomials obtained by difference equations have their roots as the nodes finally some examples are given to show the application of the procedure

monodromy of monomially admissible fukaya seidel categories mirror to toric varieties abstract mirror symmetry for a toric variety involves laurent polynomials whose symplectic topology is related to the algebraic geometry of the toric variety we show that there is a monodromy action on the monomially admissible fukaya seidel categories of these laurent polynomials as the arguments of their coefficients vary that corresponds under homological mirror symmetry to tensoring by a line bundle naturally associated to the monomials whose coefficients are rotated in the process we introduce the monomially admissible fukaya seidel category as a new interpretation of the fukaya seidel category of a laurent polynomial on c n which has other potential applications and give evidence of homological mirror symmetry for non compact toric varieties

the intrinsic normal cone let x be an algebraic stack in the sense of deligne mumford we construct a purely 0 dimensional algebraic stack over x in the sense of artin the intrinsic normal cone frak c x the notion of perfect obstruction theory for x is introduced and it is shown how to construct given a perfect obstruction theory for x a pure dimensional virtual fundamental class in the chow group of x we then prove some properties of such classes both in the absolute and in the relative context via a deformation theory interpretation of obstruction theories we prove that several kinds of moduli spaces carry a natural obstruction theory and sometimes a perfect one

punctured holomorphic curves and lagrangian embeddings we use a neck stretching argument for holomorphic curves to produce symplectic disks of small area and maslov class with boundary on lagrangian submanifolds of nonpositive curvature applications include the proof of audin s conjecture on the maslov class of lagrangian tori in linear symplectic space the construction of a new symplectic capacity obstructions to lagrangian embeddings into uniruled symplectic manifolds a quantitative version of arnold s chord conjecture and estimates on the size of weinstein neighbourhoods the main technical ingredient is transversality for the relevant moduli spaces of punctured holomorphic curves with tangency conditions

on the genus one gromov witten invariants of complete intersections we state and prove a long elusive relation between genus one gromov witten of a complete intersection and twisted gromov witten invariants of the ambient projective space as shown in a previous paper certain naturally arising cones of holomorphic vector bundle sections over the main component m 0 1 k p d of the moduli space of stable genus one holomorphic maps into p have a well defined euler class in this paper we extend this result to moduli spaces of perturbed in a restricted way j holomorphic maps this extension is used to show that these cones are the correct genus one analogues of the vector bundles relating genus zero gromov witten invariants of a complete intersection to those of the ambient projective space a relationship for highergenus invariants is conjectured as well

dynamics of xn 1 xn 2k 1xn 2k 1 αxn 2l abstract we investigate the global stability of all positive solutions of the difference equation x n 1 x n 2 k 1 x n 2 k 1 x n 2 l where is a positive real number k 1 2 l 0 1 and the initial conditions are positive real numbers when x 1 1 is globally asymptotically stable also when 1 we show that this equation possesses solutions that converge to the two cycle 0 1 0 1

symplectic homology of some brieskorn manifolds this paper consists of two parts in the first part we use symplectic homology to distinguish the contact structures on the brieskorn manifolds varsigma 2 ell 2 2 2 which contact homology cannot distinguish this answers a question from kwon and van koert brieskorn manifolds in contact topology preprint 2013 arxiv 1310 0343 in the second part we prove the existence of infinitely many exotic but homotopically trivial exotic contact structures on s 7 distinguished by the mean euler characteristic of s 1 equivariant symplectic homology apart from various connected sum constructions these contact structures can be taken from the brieskorn manifolds varsigma 78k 1 13 6 3 3 we end with some considerations about extending this result to higher dimensions

symplectic fixed points and holomorphic spheres letp be a symplectic manifold whose symplectic form integrated over the spheres inp is proportional to its first chern class on the loop space ofp we consider the variational theory of the symplectic action function perturbed by a hamiltonian term in particular we associate to each isolated invariant set of its gradient flow an abelian group with a cyclic grading it is shown to have properties similar to the homology of the conley index in locally compact spaces as an application we show that if the fixed point set of an exact diffeomorphism onp is nondegenerate then it satisfies the morse inequalities onp we also discuss fixed point estimates for general exact diffeomorphisms

on the recursive sequence xn 1 α βxn k 1 γxn 2k 1bxn k 1 cxn 2k 1 abstract our aim in this paper is to investigate the global asymptotic stability of all positive solutions of the higher order nonlinear difference equation x n 1 x n k 1 x n 2 k 1 bx n k 1 cx n 2 k 1 n 0 1 2 where b c and are positive k 1 2 3 and the initial conditions x 2 k 1 x 1 x 0 are positive real numbers we show that the unique positive equilibrium of the equation is globally asymptotically stable and has some basins that depend on certain conditions posed on the coefficients our concentration is on invariant intervals the character of semicycles and the boundedness of the above mentioned equation our final comments are about informative examples

the characteristics of a higher order rational difference equation in this paper we study the global behavior of the higher order nonlinear difference equationx n email protected email protected n k email protected n 2 k 1a bx n k 1 n 0 1 where a b and a b c are positive email protected 1 2 3 and the initial conditions x 2 k 1 x 1 x 0 are positive real numbers we investigate the character of semicycles periodicity global stability and boundedness of above mentioned equation

global stability of yn 1 p qyn ryn k1 yn in this paper we investigate the global stability of difference equation y n 1 p qy n ry n k1 y n n 0 1 2 where the initial conditions y k y 1 y 0 are non negative email protected 1 2 3 and the parameters p q and r are non negative we show that when rq 1 and k is odd then this equation possesses unbounded solutions

the kunneth formula in floer homology for manifolds with restricted contact type boundary we prove the kunneth formula in floer co homology for manifolds with restricted contact type boundary we use viterbo s definition of floer homology involving the symplectic completion by adding a positive cone over the boundary the kunneth formula implies the vanishing of floer co homology for subcritical stein manifolds other applications include the weinstein conjecture in certain product manifolds obstructions to exact lagrangian embeddings existence of holomorphic curves with lagrangian boundary condition as well as symplectic capacities

the symplectic arc algebra is formal this is the author accepted manuscript the final version is available from duke university press via http dx doi org 10 1215 00127094 3449459

on the floer homology of cotangent bundles this paper concerns floer homology for periodic orbits and for a lagrangian intersection problem on the cotangent bundle t m of a compact orientable manifold m the first result is a new l estimate for the solutions of the floer equation which allows us to deal with a larger and more natural class of hamiltonians the second and main result is a new construction of the isomorphism between the floer homology and the singular homology of the free loop space of m in the periodic case or of the based loop space of m in the lagrangian intersection problem the idea for the construction of such an isomorphism is to consider a hamiltonian that is the legendre transform of a lagrangian on t m and to construct an isomorphism between the floer complex and the morse complex of the classical lagrangian action functional on the space of w1 2 free or based loops on m 2005 wiley periodicals inc

oscillation and asymptotic behavior of a class of higher order nonlinear recursive sequences abstract we investigate the global attractivity the periodic behavior and the invariant interval with the semicycles character of solutions of the equation x n 1 x n 2 k 1 bx n k 1 cx n 2 k 1 n 0 1 2 for all admissible non negative values of the parameters b c and the initial conditions x 2 k 1 x 1 x 0 we show that the solutions exhibit a periodic attitude depending on the parameters of the family of the above mentioned difference equation in comparison with each other

framed bordism and lagrangian embeddings of exotic spheres in dimensions congruent to 1 modulo 4 we prove that the cotangent bundle of an exotic sphere which does not bound a parallelisable manifold is not symplectomorphic to the cotangent bundle of the standard sphere more precisely we prove that such an exotic sphere cannot embed as a lagrangian in the cotangent bundle of the standard sphere the main ingredients of the construction are 1 the fact that the graph of the hopf bration embeds the standard sphere and hence any lagrangian which embeds in its cotangent bundle as a displaceable lagrangian in the product a symplectic vector space of the appropriate dimension with its complex projective space and 2 a moduli space of solutions to a perturbed cauchyriemann equation introduced by gromov

on the structure of certain natural cones over moduli spaces of genus one holomorphic maps we show that certain naturally arising cones over the main component of a moduli space of j0 holomorphic maps into pn have a well defined euler class we also prove that this is the case if the standard complex structure j0 on pn is replaced by a nearby almost complex structure j the genus zero analogue of the cone considered in this paper is a vector bundle the genus zero gromov witten invariant of a projective complete intersection can be viewed as the euler class of such a vector bundle as shown in a separate paper this is also the case for the genus one part of the genus one gw invariant the remaining part is a multiple of the genus zero gw invariant

global stability of a higher order rational recursive sequence abstract in this paper we study the qualitative behavior of solutions of the class of delay difference equation x n 1 x n k 1 x n 2 k 1 a bx n k 1 n 0 1 2 where the initial conditions x 2 k 1 x 1 x 0 are positive k 1 2 and the parameters a b are positive our concentration is on invariant intervals and the global stability of the above mentioned equation we obtain sufficient conditions for the global attractivity of all positive solutions about the zero and positive equilibrium points with basins that depend on specific conditions posed on the coefficients furthermore the oscillation and the character of semicycles about the positive equilibrium are thoroughly discussed it is also illustrated that for some special case of parameters the solution will be either a period 2 solution or will converge to the equilibrium point or will have unbounded solutions

on the recursive sequence xn 1 α βxn k 1a bxn k 1 cxn 2k 1 abstract in this paper we study the global asymptotic stability invariant intervals the character of semicycles and the boundedness of all positive solutions of the higher order nonlinear difference equation x n 1 x n k 1 a bx n k 1 cx n 2 k 1 n 0 1 where a b c and are positive k 1 2 3 and the initial conditions x 2k 1 x 1 x0 are positive real numbers we show that the unique positive equilibrium of the equation is globally asymptotically stable finally we present some informative examples

dynamics of a non linear difference equation abstract our aim in this paper is to investigate the dynamics of difference equation y n 1 y n py n k y n q n 0 1 2 where the initial conditions y k y 1 y0 are non negative k 1 2 3 and the parameters p and q are non negative we study characteristics such as periodicity invariant interval the character of semicycles the global stability and the boundedness of positive solutions of the above mentioned difference equation in particular our results solve the open problem introduced by kulenovic and ladas in their monograph dynamics of second order rational difference equations with open problems and conjectures chapman and hall crc boca raton 2002