examples of attractors in scalar reaction diffusion equations on considere l equation de reaction diffusion scalaire avec des conditions aux limites de neumann u t e 2 u xx f x u 0 x 1 u x 0 t u x 1 t 0 pour une non linearite f x u u 1 u u a x ou a 0 1 0 1 est une fonction echelon on etudie la caracterisation de l attracteur quand e est petit

a fock space approach to severi degrees the classical severi degree counts the number of algebraic curves of fixed genus and class passing through points in a surface we express the severi degrees of p1 p1 as matrix elements of the exponential of a single operator ms on fock space the formalism puts severi degrees on a similar footing as the more developed study of hurwitz numbers of coverings of curves the pure genus 1 invariants of the product e p1 with e an elliptic curve are solved via an exact formula for the eigenvalues of ms to initial order the severi degrees of p2 are also determined by ms via the 1 d 1 d2 disk multiple cover formula for calabi yau threefold geometries

a sequence of order relations encoding heteroclinic connections in scalar parabolic pde abstract we address the problem of heteroclinic connections in the attractor of dissipative scalar semilinear parabolic equations type display u t u xx f x u u x 0 on a bounded interval with neumann conditions introducing a sequence of order relations we prove a new and simple criterion for the existence of heteroclinic connections using only information about nodal properties of solutions to the stationary ode problem this result also allows for a complete classification of possible attractors in terms of the permutation of the equilibria given by their order at the two boundaries of the interval

sturm attractors for quasilinear parabolic equations abstract the goal of this paper is to construct explicitly the global attractors of quasilinear parabolic equations as it was done for the semilinear case by brunovsk and fiedler 1986 8 and generalized by fiedler and rocha 1996 13 in particular we construct heteroclinic connections between hyperbolic equilibria stating necessary and sufficient conditions for heteroclinics to occur such conditions can be computed through a permutation of the equilibria lastly an example is computed yielding the well known chafee infante attractor

limit linear series basic theory in this paper we introduce techniques for handling the degeneration of linear series on smooth curves as the curves degenerate to a certain type of reducible curves curves of compact type the technically much simpler special case of 1 dimensional series was developed by beauville 2 knudsen 21 23 harris and mumford 17 in the guise of admissible covers it has proved very useful for studying the moduli space of curves the above papers and harris 16 and the simplest sorts of weierstrass points diaz 4 with our extended tools we are able to prove for example that in this paper we present the basic theory of limit linear series necessary for proving these results the results themselves will be taken up in our forthcoming papers 8 12 simpler applications not requiring the tools developed in this paper but perhaps clarified by them have already been given in our papers 5 7

on the solution of nonlinear fractional order differential equation abstract this paper investigates the uniqueness and dependence of the solution of nonlinear differential equation with fractional order by the fixed points theorem the obtained results include naturally those in open literature for some special cases and a numerical approach to the discussed problem is suggested

existence of the mild solution for fractional semilinear initial value problems abstract in this paper we will study the existence and uniqueness of mild solution for the semilinear initial value problem of non integer order u t a u t f t u t g u t s u t where 0 1 and f t u t g u t s u t is a given function

positive solutions of a system of non autonomous fractional differential equations abstract existence of positive solutions for the following system of fractional differential equations d i u i f i t u 1 u 2 u n u i 0 0 0 i 1 1 i n where d i denotes riemann liouville derivative of order i has been studied

numbers of zeros on invariant manifolds in reaction diffusion equations on considere l equation a une dimension u t u xx f x u t 0 0 x 1 avec les conditions aux limites de dirichlet u t 0 t 1 0 ou f 0 1 r r est bc 1 c k k 1

bifurcations in a parabolic equation with variable diffusion on considere le probleme aux valeurs limites u t a 2 x u x x f u 0 x 1 u x 0 t u x 1 t 0 pour 0 f u u u 3 a x 1 x et a x x 0 1 on etudie la stabilite et la bifurcation des solutions d equilibre pour les cas x 0 1 0 1 et x 0 1 2 0 1 2 et a x a 1 x pour x 1 2 1

ideals associated to deformations of singular plane curves we consider in this paper the geometry of certain loci in deformation spaces of plane curve singularities these loci are the equisingular locus es which parametrizes equisingular or topologically trivial deformations the equigeneric locus eg which parametrizes deformations of constant geometric genus and the equiclassical locus ec which parametrizes deformations of constant geometric genus and class the class of a reduced plane curve is the degree of its dual it was previously known that the tangent space to es corresponds to an ideal called the equisingular ideal and that the support of the tangent cone to eg corresponds to the conductor ideal we show that the support of the tangent cone to ec corresponds to an ideal which we call the equiclassical ideal by studying these ideals we are able to obtain information about the geometry and dimensions of es ec and eg this allows us to prove some theorems about the dimensions of families of plane curves with certain specified singularities

parabolic equations for curves on surfaces 2 intersections blow up and generalized solutions we describe a theory for parabolic equations for immersed curves on surfaces which generalizes the curve shortening or flow by mean curvature problem as well as several models in the theory of phase transitions in two dimensions a class of equations is described for which the initial value problem is well posed for rough initial data for which one can give a description of the way a smooth solution becomes singular and for which one can define generalized solutions i e solutions which are smooth except at a discrete set of times the methods which are used in this paper are more geometrical than those of part i by comparing arbitrary solutions with certain special solutions and by considering the way they intersect we derive estimates for the curvature and the tangent which allow one to study the initial value problem and the way solutions become singular

the morse smale property for a semi linear parabolic equation on etudie la dynamique d equations aux derivees partielles paraboliques semi lineaires du type suivant u t u xx f x u 0 0 u 0 t u 1 t 0 t 0 u x 0 x

on the singular problem for the scalar parabolic equation with variable diffusion we consider the scalar parabolic equation ut 2 a2 x ux x u 0 x 1 satisfying neumann boundary conditions and appropriate conditions on the non linearity the attractor a for the dynamical system generated by this equation has been widely studied in the literature and it is known that as the diffusion decreases 0 it becomes increasingly complex with an unbounded number of equilibria appearing through bifurcation hale and sakamoto have considered this singular limit in the case of x u and have shown the existence of stable solutions with transition layers when is sufficiently small and the nonlinearity satisfies an additional nondegeneracy condition however this condition is not satisfied when u and the corresponding result do not follow for the problem considered here we are concerned with the study of the shape and the number of the equilibrium solutions with lower morse indices in particular its stable equilibria

dimension theoretic characterization of maximal irreducible algebraic systems of plane nodal curves of a given order n and with a given number d of nodes introduction let k be an algebraically closed ground field of characteristic zero the following result is well known see severi 1 anhang f p 317 if en d is a complete i e maximal irreducible algebraic system defined over k of plane algebraic not necessarily irreducible curves of a given order n such that the general curve c of en d k has d nodes and no other singularities then the dimension of en d is equal to 3n p 1 where p n 1 n 2 2 d is the effective genus of c we note that 3n p 1 is equal to n n 3 2 d here n n n 3 2 is the dimension of the linear system of all plane curves of order n this is in agreement with the intuitive expectation that the requirement that a curve of order n possess d nodes in non assigned position imposes d independent algebraic non linear conditions on the curve the above result is easily proved and will be included by us in a more general theorem the proof of which is the main object of this paper theorem 2 section 3 and which deals with irreducible algebraic systems e of curves of order n in which the general curve c of e k has arbitrary singularities but is free from multiple components equivalently c is a reduced curve this theorem asserts that if p is the effective genus of c the definition of the effective genus of a reducible but reduced curve is given in section 3 formula 2 then dim e c 3n p 1 with equality if and only if c has only nodes as singularities it is easily seen see severi loc cit that if 0 c d n n 1 2 then there always exists curves of order n having d nodes the maximum

almost automorphic mild solutions to fractional differential equations abstract we introduce the concept of resolvent families to prove the existence of almost automorphic mild solutions to the differential equation d t u t a u t t n f t 1 2 n z considered in a banach space x where f r x is almost automorphic we also prove the existence and uniqueness of an almost automorphic mild solution of the semilinear equation d t u t a u t f t u t 1 2 assuming f t x is almost automorphic in t for each x x satisfies a global lipschitz condition and takes values on x finally we prove also the existence and uniqueness of an almost automorphic mild solution of the semilinear equation d t u t a u t f t u t u t 1 2 under analogous conditions as in the previous case

existence of pseudo almost automorphic mild solutions to stochastic fractional differential equations abstract fractional stochastic differential equations have gained considerable importance due to their application in various fields of science and engineering this paper is concerned with the square mean pseudo almost automorphic solutions for a class of fractional stochastic differential equations in a hilbert space the main objective of this paper is to establish the existence and uniqueness of square mean pseudo almost automorphic mild solutions to a linear and semilinear case of these equations a new set of sufficient conditions is obtained to achieve the required result by using the stochastic analysis theory and fixed point strategy finally an example is provided to illustrate the obtained theory

on the homotopy index for infinite dimensional semiflows absrac r in this paper we consider semiflows whose solution operator is eventually a conditional a contraction such semiflows include solutions of retarded and neutral functional differential equations of parabolic and certain other classes of partial differential equations we prove existence of nonsmooth isolating blocks and index pairs for such semiflows via the construction of special lyapunov functionals we show that index pairs enjoy all the properties needed to define the notion of a homotopy index thus generalizing earlier results of conley 2 finally using a result of maii 9 we prove that under additional smoothness assumptions on the semiflow the homotopy index is essentially a finite dimensional concept this gives a formal justification of the applicability of waiewski s principle to infinite dimensional problems several examples illustrate the theory

geometry of families of nodal curves on the blown up projective plane let p2r be the projective plane blown up at r generic points denote by e0 e1 er the strict transform of a generic straight line on p2 and the exceptional divisors of the blown up points on p2r respectively we consider the variety virr of all irreducible curves c with k nodes as the only singularities and give asymptotically nearly optimal sufficient conditions for its smoothness irreducibility and non emptiness moreover we extend our conditions for the smoothness and the irreducibility on families of reducible curves for r 9 we give the complete answer concerning the existence of nodal curves in virr

the gonality of general smooth curves with a prescribed plane nodal model let c be a smooth curve of genus g it is well known that c possesses a linear system g l if q l g 2n g 2 0 moreover if c is general and q g 2 k l and 6 kd k l 2 3 theorem 2 1 in loc cit and for each d this bound on 6 is sharp examples 4 1 and 4 2 in loc cit in this paper we consider the question for a general choice of f since the space parametrizing integral plane nodal curves with prescribed d and 6 is irreducible famous conjecture of severi proved by harris 9 the meaning of general becomes represented by a point of a particular nonempty zariski open subset of that parameter space note that qn x 3 g 0 if and only if 6 d 2 7d 18 2

moduli of nodal curves on k3 surfaces abstract we consider modular properties of nodal curves on general k3 surfaces let k p be the moduli space of primitively polarized k3 surfaces s l of genus p 3 and v p m k p be the universal severi variety of nodal irreducible curves in m l on s l k p we find conditions on p m for the existence of an irreducible component v of v p m on which the moduli map v m g with g m 2 p 1 1 has generically maximal rank differential our results which for any p leave only finitely many cases unsolved and are optimal for m 5 except for very low values of p are summarized in theorem 1 1 in the introduction

a combinatorial analysis of severi degrees abstract based on results by brugalle and mikhalkin fomin and mikhalkin give formulas for computing classical severi degrees n d using long edge graphs in 2012 block colley and kennedy considered the logarithmic version of a special function associated to long edge graphs which appeared in fomin mikhalkin s formula and conjectured it to be linear they have since proved their conjecture at the same time motivated by their conjecture we consider a special multivariate function associated to long edge graphs that generalizes their function the main result of this paper is that the multivariate function we define is always linear a special case of our result gives an independent proof of block colley kennedy s conjecture the first application of our linearity result is that by applying it to classical severi degrees we recover quadraticity of q d and a bound for the threshold of polynomiality of n d next in separate joint work with osserman we apply the linearity result to a special family of toric surfaces and obtain universal polynomial results having connections to the gottsche yau zaslow formula as a result we provide combinatorial formulas for the two unidentified power series b 1 q and b 2 q appearing in the gottsche yau zaslow formula the proof of our linearity result is completely combinatorial we define graphs which generalize long edge graphs and a closely related family of combinatorial objects we call n words by introducing height functions and a concept of irreducibility we describe ways to decompose certain families of n words into irreducible words which leads to the desired results

universal polynomials for severi degrees of toric surfaces abstract the severi variety parameterizes plane curves of degree d with nodes its degree is called the severi degree for large enough d the severi degrees coincide with the gromov witten invariants of c p 2 fomin and mikhalkin 2010 10 proved the 1995 conjecture that for fixed severi degrees are eventually polynomial in d in this paper we study the severi varieties corresponding to a large family of toric surfaces we prove the analogous result that the severi degrees are eventually polynomial as a function of the multidegree more surprisingly we show that the severi degrees are also eventually polynomial as a function of the surface we illustrate our theorems by explicitly computing for a small number of nodes the severi degree of any large enough hirzebruch surface and of a singular surface our strategy is to use tropical geometry to express severi degrees in terms of brugalle and mikhalkin s floor diagrams and study those combinatorial objects in detail an important ingredient in the proof is the polynomiality of the discrete volume of a variable facet unimodular polytope

the geometry of the moduli space of curves of genus 23 the problem of describing the birational geometry of the moduli space mg of complex curves of genus g has a long history severi already knew in 1915 thatmg is unirational for g 10 cf sev see also ac1 for a modern proof in the same paper severi conjectured thatmg is unirational for all genera g then for a long period this problem seemed intractable mumford writes in mu p 51 whether more mg s g 11 are unirational or not is a very interesting problem but one which looks very hard too especially if g is quite large the breakthrough came in the eighties when eisenbud harris and mumford proved that mg is of general type as soon as g 24 and that the kodaira dimension of m23 is 1 see hm eh3 we note that mg is rational for g 6 see dol for problems concerning the rationality of various moduli spaces severi s proof of the unirationality of mg for small g was based on representing a general curve of genus g as a plane curve of degree d with nodes this is possible when d 2g 3 2 when the number of nodes is small i e d 1 d 2 6 the dominant map from the variety of plane curves of degree d and genus g to mg yields a rational parametrization of the moduli space the two conditions involving d and can be satisfied only when g 10 so severi s argument cannot be extended for other genera however using much more subtle ideas chang ran and sernesi proved the unirationality of mg for g 11 12 13 see cr1 se1 while for g 15 16 they proved that the kodaira dimension is see cr2 4 the remaining cases g 14 and 17 g 23 are still quite mysterious harris and morrison conjectured in hmo that mg is uniruled precisely when g 23 all these facts indicate that m23 is a very interesting transition case our main result is the following

the zero set of a solution of a parabolic equation on etudie l ensemble nul d une solution u t x de l equation u t a x t u xx b x t u x c x t u sous des hypotheses tres generales sur les coefficients a b et c

existence of solutions for neutral functional differential evolution equations with nonlocal conditions abstract in this paper by using fractional power of operators and sadovskii s fixed point theorem we study the existence of mild and strong solutions of semilinear neutral functional differential evolution equations with nonlocal conditions the results we obtained are a generalization and continuation of the recent results on this issue in the end an example is given to show the application of our results

invariants of plane algebraic curves via representations of the braid groups recently new remarkable polynomial invariants of knots in 3 sphere were introduced using some representations of the braid groups which were shown to be effective in distinguishing knots j fyhlmo the purpose of this note is to introduce using representation of the braid groups an invariant of continuous equisingular families of plane algebraic curves a pleasant feature is that this invariant is defined for any representation of the braid group no coherence of representations for different number of strings is required though for many representations this invariant is trivial in the case of the reduced burau representation this invariant essentially coincides with the alexander polynomial of plane curves introduced in l 1 this fact is the counterpart of the well known relation between the alexander polynomial of closed braids and the reduced burau representation the definition of this invariant depends on braid monodromy associated with the curve m in particular we obtain a direct not involving calculations of the fundamental group though computationally cumbersome method of computation of the alexander polynomial via braid monodromy in the next section we shall recall the background on continuous equisingular families of plane algebraic curves and their braid monodromies in the sect 3 we introduce the invariant and in the last section we consider the case of reduced burau representations

irreducibility of equisingular families of curves in 1985 joe harris proved the long standing claim of severi that equisingular families of plane nodal curves are irreducible whenever they are nonempty for families with more complicated singularities this is no longer true given a divisor d on a smooth projective surface it thus makes sense to look for conditions which ensure that the family v irr d s 1 s r of irreducible curves in the linear system d l with precisely r singular points of types s 1 s r is irreducible considering different surfaces including general surfaces in p 3 c and products of curves we produce a sufficient condition of the type r i 1 deg x s i 2 d k 2 where is some constant and x s i some zero dimensional scheme associated to the singularity type our results carry the same asymptotics as the best known results in this direction in the plane case even though the coefficient is worse for most of the surfaces considered these are the only known results in that direction

on the variety of plane curves of degree with nodes and cusps let pn be the projective space which parametrizes all plane curves of degree d and v d k the subvariety of pn consisting of all reduced and irreducible plane curves of degree d with 3 nodes and k cusps as their only singularities in this paper we prove that v d k is irreducible if k 3 except possibly when k 3 and d 5 or 6

on universal severi varieties of low genus k3 surfaces we prove the irreducibility of universal severi varieties parametrizing irreducible reduced nodal hyperplane sections of primitive k3 surfaces of genus g with 3 g 11 g 10

on almost automorphic mild solutions for fractional semilinear initial value problems abstract this paper investigates almost automorphic mild solutions of the fractional semilinear equation d x t a x t f t x t 0 1 considered in a banach space x where a is a linear operator of sectorial type 0 some sufficient conditions are given for the existence uniqueness and uniform stability of almost automorphic mild solutions to this semilinear equation

rational curves on k3 surfaces we show that projective k3 surfaces with odd picard rank contain infinitely many rational curves our proof extends the bogomolov hassett tschinkel approach i e uses moduli spaces of stable maps and reduction to positive characteristic

families of nodal curves on projective threefolds and their regularity via postulation of nodes the main purpose of this paper is to introduce a new approach to study families of nodal curves on projective threefolds precisely given x a smooth projective threefold e a rank two vector bundle on x l a very ample line bundle on x and k 0 0 integers and denoted by v e l k the subscheme of p h 0 e l k parametrizing global sections of e l k whose zero loci are irreducible and nodal curves on x we present a new cohomological description of the tangent space t s v e l k at a point s 2 v e l k this description enable us to determine effective and uniform upper bounds for which are linear polynomials in k such that the family v e lk is smooth and of the expected dimension regular for short the almost sharpness of our bounds is shown by some interesting examples furthermore when x is assumed to be a fano or a calaby yau threefold we study in detail the regularity property of a point s 2 v e lk related to the postulation of the nodes of its zero locus c v s x roughly speaking when the nodes of c are assumed to be in general position either on x or on an irreducible divisor of x having at worst log terminal singularities or to lie on a l c i and subcanonical curve in x we find upper bounds onwhich are respectively cubic quadratic and linear polynomials in k ensuring the regularity of v e l k at s finally when x p 3 we also discuss some interesting geometric properties of the curves given by sections parametrized by v e ox k