semilinear elliptic equations on unbounded domains on presente une nouvelle approche variationnelle a l existence des solutions pour le probleme aux valeurs limites u u r x f u x x x h 0 1 sur des domaines non bornes

multiple solutions to a linearly coupled elliptic system with critical exponents abstract we are concerned with the linearly coupled elliptic system with critical exponents u a x u u 2 2 u u p 2 u v x r n v b x v v 2 2 v v q 2 v u x r n u v h 1 r n where n 3 2 p q 2 2 n n 2 and are nonnegative constants and a x b x are positive periodic functions the existence of a positive ground state and a higher energy solution of this system is proved using variational methods additionally the asymptotic behavior of these solutions as 0 is obtained

on the fay identity for korteweg de vries tau functions and the identity for the wronskian of squared solutions of sturm liouville equation we show that the well known identity for the wronskian of squared solutions of a sturm liouville equation follows from the fay identity we also study some odd order 2n 1 order n 2 3 identities which are specific for tau functions related to the kdv hierarchy

characterizing jacobians via trisecants of the kummer variety we prove welters trisecant conjecture an indecomposable principally polarized abelian variety x is the jacobian of a curve if and only if there exists a trisecant of its kummer variety k x

multiple positive solutions for quasilinear problems with indefinite sublinear nonlinearity abstract we have established multiplicity of nontrivial solutions for the quasilinear elliptic problem p u h x u 1 g x u in u 0 in u 0 on where r n is a smooth bounded domain p 1 1 p h is a measurable function and g r r is a caratheodory function such that g x 0 0 which is asymptotically linear we also prove existence and nonexistence results when g x t k x t p 1

random matrices virasoro algebras and noncommutative kp department of mathematics brandeis university waltham mass 02254 usa email adler math brandeis edu the support of a national science foundation grant dms 9503246 is gratefully acknowledged department of mathematics kyoto university kyoto 606 japan the hospitality of the university of louvain and brandeis university is gratefully acknowledged department of mathematics universite de louvain 1348 louvain la neuve belgium and brandeis university waltham mass 02254 usa e mail vanmoerbeke geom ucl ac be and math brandeis edu the support of a national science foundation grant dms 9503246 a nato a fnrs and a francqui foundation grant is gratefully acknowledged a corrected version has appeared in the duke mathematical journal 94 379 431 1998

characterization of jacobian varieties in terms of soliton equations pour dans le demi espace superieur de siegel h g soit x c g z g z g la variete abelienne principalement polarisee correspondante et soit z z m e z gexp 2 i t mz i t m m la fonction theta de riemann pour x les deux conditions a et b pour x sont equivalentes a 1 il existe des vecteurs a 1 a 2 a 3 c g a 1 0 et une forme quadratique q t i j 1 3 q ij t i t j q ij c tel que pour c g t t exp q t t 1 a 1 t 2 a 2 t 3 a 3 donne une solution a l equation de kadomtsev petviashvili sous la forme d hirota d 1 4 3d 2 2 4d 1 d 3 0 2 le diviseur de x est irreductible b x est isomorphe a la variete jacobienne d une courbe lisse complete de genre g sur c

enumerative geometry tau functions and heisenberg virasoro algebra in this paper we establish relations between three enumerative geometry tau functions namely the kontsevich witten hurwitz and hodge tau functions the relations allow us to describe the tau functions in terms of matrix integrals virasoro constraints and kac schwarz operators all constructed operators belong to the algebra or group of symmetries of the kp hierarchy

multiple positive solutions for semilinear dirichlet problems with sign changing weight function in infinite strip domains abstract in this paper existence and multiplicity results to the following dirichlet problem u u f x u q 1 h x u p 1 in u 0 in u 0 on are established where r r n 1 is bounded smooth domain and n 2 here 1 q 2 p 2 2 2 n n 2 if n 3 2 if n 2 is a positive real parameter the function f among other conditions can possibly change sign in and the function h satisfies suitable conditions the study is based on the comparison of energy levels on nehari manifold

a criterion for jacobi varieties is a 2 1 map onto its image the kummer wirtinger variety kw x associated with x the following is well known cf 6 cf also 10 assume that x is the polarized jacobian of a smooth curve c and consider c as embedded in x by an arbitrary element of picl c let d be an effective divisor of degree 3 on c and call y the subscheme of x defined by d put s s y the abelian sum of the components of the corresponding 0 cycle of degree 3 on x then for any d ex with 2d e c s there exists a line 1 in pn such that the translated subscheme d y satisfies d y c 4 1 l in other words 1 is a trisecant or a flex of the kummer wirtinger variety and the image of d y lies in the intersection of 1 with kw x in his paper 3 r c gunning has shown how to use the above property in order to characterize jacobians gunning s result can be stated as follows

the n component kp hierarchy and representation theory it is the aim of the present article to give all formulations of the n component kp hierarchy and clarify connections between them the generalization to the n component kp hierarchy is important because it contains many of the most popular systems of soliton equations like the davey stewartson system for n 2 the two dimensional toda lattice for n 2 the n wave system for n 3 and the darboux egoroff system it also allows us to construct natural generalizations to the davey stewartson and toda lattice systems of course the inclusion of all these systems in the n component kp hierarchy allows us to construct their solutions by making use of vertex operators

on a schrodinger kirchhoff type equation involving the p x laplacian abstract in this paper we deal with a schrodinger type equation involving a nonlocal kirchhoff type coefficient and depending on two real parameters working within the framework of variable exponent spaces and using the variational approach we obtain several results of existence and multiplicity of solutions depending on the range of the parameters

cohomological structure in soliton equations and jacobian varieties on etudie la structure des orbites du systeme dynamique definie par la hierarchie kp totale on montre que chaque orbite est localement isomorphe a un certain groupe de cohomologie associe a une algebre commutative le systeme dynamique kp restreint a chaque orbite determine un systeme dynamique de mouvements lineaires sur lui par rapport a la structure lineaire du groupe de cohomologie

a lax representation for the vertex operator and the central extension integrable hierarchies viewed as isospectral deformations of an operatorl may admit symmetries they are time dependent vector fields transversal to and commuting with the hierarchy and forming an algebra in this work the commutation relations for the symmetries are shown to be based on a non commutative lie algebra splitting theorem the symmetries viewed as vector fields onl are expressed in terms of a lax pair

multiple solutions for p kirchhoff equations in rn abstract in this paper we prove the existence of multiple ground state solutions for the nonhomogeneous p kirchhoff elliptic equation 0 1 m r n u p v x u p d x p u v x u p 2 u f x u g x in r n where v x c r n and v x as x the nonlinear function f x u is continuous and satisfies some conditions the solutions will be obtained by the mountain pass theorem ekeland s variational principle and krasnoselskii s genus theory in struwe 2000 1

analytic bilinear approach to integrable hierarchies ii multicomponent kp and 2d toda lattice hierarchies an analytic bilinear approach for the construction and study of integrable hierarchies is discussed generalized multicomponent kp and 2d toda lattice hierarchies are considered this approach allows us to represent generalized hierarchies of integrable equations in a condensed form of finite functional equations a generalized hierarchy incorporates basic hierarchy modified hierarchy singularity manifold equation hierarchy and corresponding linear problems different levels of generalized hierarchy are connected via invariants of combescure symmetry transformation the resolution of functional equations also leads to the function and addition formulas to it

a bayesian determination of threshold for identifying differentially expressed genes in microarray experiments the original definitions of false discovery rate fdr and false non discovery rate fnr can be understood as the frequentist risks of false rejections and false non rejections respectively conditional on the unknown parameter while the bayesian posterior fdr and posterior fnr are conditioned on the data from a bayesian point of view it seems natural to take into account the uncertainties in both the parameter and the data in this spirit we propose averaging out the frequentist risks of false rejections and false non rejections with respect to some prior distribution of the parameters to obtain the average fdr afdr and average fnr afnr respectively a linear combination of the afdr and afnr called the average bayes error rate aber is considered as an overall risk some useful formulas for the afdr afnr and aber are developed for normal samples with hierarchical mixture priors the idea of finding threshold values by minimizing the aber or controlling the afdr is illustrated using a gene expression data set simulation studies show that the proposed approaches are more powerful and robust than the widely used fdr method

solutions to semilinear elliptic problems with combined nonlinearities abstract we are concerned with the following nonlinear dirichlet problem u h x u q f x u 0 u h 0 1 0 where is a bounded smooth domain in rn n 1 and h x l f x s is asymptotically linear with respect to s at infinity by a variant version of mountain pass theorem we prove that there exist at least two nonnegative solutions under suitable assumptions on f x s our methods also work for the cases where f x s is superlinear or linear in s

multiple positive solutions for dirichlet problems involving concave and convex nonlinearities abstract in this paper we study the multiplicity of positive solutions for the following dirichlet problems involving concave and convex nonlinearities u u f x u q 2 u h x u p 2 u in a u h 0 1 a where 1 q 2 p 2 2 2 n n 2 if n 3 2 if n 2 0 a r is an infinite strip in r n is a bounded domain in r n 1 and f x h x satisfy suitable conditions

periodic multiphase solutions of the kadomtsev petviashvili equation n phase solutions of the kadomtsev petviashvili kp equation 1970 that are periodic in space variables x and y were obtained and effectively investigated using the schottky uniformisation of which a short description is given many wave patterns are represented graphically as contour plots and as isometric projections for different parameter values of two three and four phase solutions of the kp equation

combined effects of concave and convex nonlinearities in some elliptic problems abstract this paper deals with a class of semilinear elliptic dirichlet boundary value problems where the combined effects of a sublinear and a superlinear term allow us to establish some existence and multiplicity results

multiple positive solutions for a class of concave convex elliptic problems in rn involving sign changing weight in this paper we study the multiplicity of positive solutions for the following concave convex elliptic equation u u f x uq 1 g x up 1in rn u 0in rn u h1 rn where 1 q 2 p 2 2 2nn 2 if n 3 2 if n 1 2 and the parameters 0 we assume that f x f x f x is sign changing and g x a x b x where the functions f a and b satisfy suitable conditions

the riemann liouville integral and continuous kp hierarchy extension of the standard construction of the kadomtsev petviashvili kp hierarchy by the use of riemann liouville integral is given in consequence we obtain the new classes of integer as well as fractional graded kp hierarchies which are further investigated the fractional calculus leads to the new generalization of the w algebra

supertori are algebraic curves super riemann surfaces of genus 1 with arbitrary spin structures are shown to be the sets of zeroes of certain polynomial equations in projective superspace we conjecture that the same is true for arbitrary genus properties of superelliptic functions and super theta functions are discussed the boundary of the genus 1 super moduli space is determined

the kp equation with quasiperiodic initial data abstract the kadomtsev petviashvili kp equation is known to admit exact quasiperiodic solutions that can be written in terms of riemann theta functions with a finite number of phases in each solution in this paper we propose a method to solve the initial value problem for the kp equation for initial data taken from this class of quasiperiodic functions

periodic solutions of the kdv equation in this paper we construct a large family of special solutions of the kdv equation which are periodic in x and almost periodic in t these solutions lie on n dimensional tori very likely they are dense among all solutions the special solutions are characterized variationally they minimize fn u subject to the constraints fj u aj j 1 n 1 here fj denote the remarkable sequence of conserved functionals discovered by kruskal and zabusky the above minimum problem was originally suggested by them in exploring the manifold of solutions of this minimum problem we make essential use of gardner s discovery that these functionals are in involution with respect to a suitable poisson bracket gardner greene kruskal and miura have shown that the eigenvalues of the schrodinger operator are conserved functionals if the potential is a function of t and satisfies the kdv equation in section 6 a new set of conserved quantities is constructed which serve as a link between the eigenvalues of the schrodinger operator and the fj another result in section 6 is a slight sharpening of an earlier result of the author and j moser for the special solutions constructed above all but 2n 1 eigenvalues of the schrodinger operator are double the simplest class of special solutions n 1 are cnoidal waves in an appendix m hyman describes the results of computing numerically the next simplest case n 2 these calculations show that the shape of these solutions recurs exactly after a finite time in a shifted position the theory verifies this fact

the kadomtsev petviashvili equations and fundamental string theory in this paper the infinite sequence of non linear partial differential equations known as the kadomtsev petviashvili equations is described in simple terms and possible applications to a fundamental description of interacting strings are addressed lines of research likely to prove useful in formulating a description of non perturbative string configurations are indicated

geometrical optics in nonlinear media and integrable equations it is shown that the geometrical optics limit of the maxwell equations for certain nonlinear media with slow variation along one axis and particular dependence of the dielectric constant on frequency and fields gives rise to the dispersionless veselov novikov equation for the refractive index it is demonstrated that the dispersionless veselov novikov hierarchy is amenable to the quasiclassical dressing method a connection is noted between the geometrical optics phenomena under consideration and the quasiconformal mappings on the plane

zero pole interpolation for matrix meromorphic functions on a compact riemann surface and a matrix fay trisecant identity this paper presents a new approach to constructing a meromorphic bundle map between flat vector bundles over a compact riemann surface having a prescribed weil divisor i e having prescribed zeros and poles with directional as well as multiplicity information included in the vector case this new formalism unifies the earlier approach of ball clancey in the setting of trivial bundles over an abstract riemann surface with an earlier approach of the authors where the riemann surface was assumed to be the normalizing riemann surface for an algebraic curve embedded in c 2 with determinantal representation and the vector bundles were assumed to be presented as the kernels of linear matrix pencils the main tool is a version of the cauchy kernel appropriate for flat vector bundles over the riemann surface our formula for the interpolating bundle map in the special case of a single zero and a single pole can be viewed as a generalization of the fay trisecant identity from the usual line bundle case to the vector bundle case in terms of cauchy kernels in particular we obtain a new proof of the fay trisecant identity

functional representation of the ablowitz ladik hierarchy the ablowitz ladik hierarchy alh is considered in the framework of the inverse scattering approach after establishing the structure of solutions of the auxiliary linear problems the alh which has been originally introduced as an infinite system of differential difference equations is presented as a finite system of difference functional equations the representation obtained when rewritten in terms of hirota s bilinear formalism is used to demonstrate relations between the alh and some other integrable systems the kadomtsev petviashvili hierarchy in particular

an inflectionary tangent to the kummer variety and the jacobian condition in this paper we deal with a degenerate version of the trisecant conjecture let x theta be an indecomposable principally polarized abelian variety and let theta be a symmetric representative of the polarization we shall denote by theta a non zero section of the corresponding sheaf the linear system vert 2 theta vert is base point free and it is independent of the choice of theta the image of the morphism k x rightarrow vert 2 theta vert associated with the base point free linear system vert 2 theta vert is a projective variety which is called the kummer variety of x theta welters conjectured that the existence of one trisecant line to the kummer variety characterizes the jacobians it is well known that the kummer variety of a jacobian has a rich geometry in terms of trisecants and flexes we prove that if there exists an inflectionary tangent l to the kummer variety associated with x theta then x theta is a jacobian provided that there are no set theoretical d invariant components of the scheme d theta theta cap d theta 0 where d is an invariant vector field on x associated to l

the geometry of the super kp flows a supersymmetric generalization of the krichever map is used to construct algebro geometric solutions to the various super kadomtsev petviashvili skp hierarchies the geometric data required consist of a suitable algebraic supercurve of genusg generallynot a super riemann surface with a distinguished point and local coordinates z there and a generic line bundle of degreeg 1 with a local trivialization near the point the resulting solutions to the manin radul skp system describe coupled deformations of the line bundle and the supercurve itself in contrast to the ordinary kp system which deforms line bundles but not curves two new skp systems are introduced an integrable jacobian system whose solutions describe genuine jacobian flows deforming the bundle but not the curve and a nonintegrable maximal system describing independent deformations of bundle and curve the kac van de leur skp system describes the same deformations as the maximal system but in a different parametrization

remarks on sublinear elliptic problems on unbounded cylinders m l where v s o r o r is a bounded smooth domain m l g 0 m w n m q l g 1 v s o if m s 0 and v s r if l s 0 f v 0 a r is a caratheodory function and r v a r is continuous r g 0 there is by now an extensive literature on this kind of problem for w x w x w x w x instance amick and toland 1 stuart 2 esteban 3 burton 4 bona w x w x bose and turner 5 and berestycki and lions 20 where different conditions on r x and f x s as well as several techniques were considered this list is of course far from complete and we would also refer to w x w x schindler 6 costa and miyagaki 7 and their references physical motiva

functional representations of integrable hierarchies we consider a general framework for integrable hierarchies in lax form and derive certain universal equations from which functional representations of particular hierarchies such as kp discrete kp mkp akns i e formulations in terms of functional equations are systematically and quite easily obtained the formalism genuinely applies to hierarchies where the dependent variables live in a noncommutative typically matrix algebra the obtained functional representations can be understood as noncommutative analogues of fay identities for the kp hierarchy

menelaus theorem clifford configurations and inversive geometry of the schwarzian kp hierarchy it is shown that the integrable discrete schwarzian kp dskp equation which constitutes an algebraic superposition formula associated with for instance the schwarzian kp hierarchy the classical darboux transformation and quasi conformal mappings encapsulates nothing but a fundamental theorem of ancient greek geometry thus it is demonstrated that the connection with menelaus theorem and more generally clifford configurations renders the dskp equation a natural object of inversive geometry on the plane the geometric and algebraic integrability of dskp lattices and their reductions to lattices of menelaus darboux schwarzian kdv schwarzian boussinesq and schramm type are discussed the dskp and discrete schwarzian boussinesq equations are shown to represent discretizations of families of quasi conformal mappings

tetrahedra octahedra and cubo octahedra integrable geometry of multi ratios geometric and algebraic aspects of multi ratios m2n are investigated in detail connections with menelaus theorem clifford configurations and maxwell s reciprocal quadrangles are utilized to associate the multi ratios m4 m8 and m8 with tetrahedra octahedra and cubo octahedra respectively integrable maps defined on face centred fcc lattices and irregular lattices composed of the face centres of simple cubic lattices are constructed and related to the discrete kp and bkp equations and the integrable discrete darboux system governing conjugate lattices an interpretation in terms of integrable irregular lattices of slopes on the plane is also given

equations of the moduli of pointed curves in the infinite grassmannian the main result of this paper is the explicit computa tion of the equations defining the moduli space of triples c p where c is an integral and complete algebraic curve p a smooth rational point anda certain isomorphism this is achieved by introducing algebraically infinite grassmannians tau and baker ahkiezer functions and by proving an addition formula for tau functions