gibbs states of the hopfield model with extensively many patterns we consider the hopfield model withm n n patterns wheren is the number of neurons we show that if is sufficiently small and the temperature sufficiently low then there exist disjoint gibbs states for each of the stored patterns almost surely with respect to the distribution of the random patterns this solves a provlem left open in previous work the key new ingredient is a self averaging result on the free energy functional this result has considerable additional interest and some consequences are discussed a similar result for the free energy of the sherrington kirkpatrick model is also given

solutions of systems of elliptic differential equations on circular domains abstract we study global bifurcation of weak solutions of systems of elliptic differential equations considered on so 2 invariant domains we formulate sufficient conditions for the existence of unbounded continua of nontrivial solutions branching from the trivial ones as the main tool we use the degree for so 2 equivariant gradient maps defined by the second author in rybicki nonlinear anal tma 23 1 1994 83

rigidity of proper holomorphic maps between bounded symmetric domains our first main result gives assumptions guaranteeing that proper holomorphic maps between cartan type i bounded symmetric domains have simple block matrix shape answering positively a question of mok the proof is based on the second main result establishing similar phenomenon for local cr maps between arbitrary boundary components of two bounded symmetric domains of the above type since boundary components other than shilov boundaries are levi degenerate our analysis is based on their 2 nondegeneracy combining levi forms with higher order tensors

partial rigidity of degenerate cr embeddings into spheres abstract in this paper we study degenerate cr embeddings f of a strictly pseudoconvex hypersurface m c n 1 into a sphere s in a higher dimensional complex space c n 1 the degeneracy of the mapping f will be characterized in terms of the ranks of the cr second fundamental form and its covariant derivatives in 2004 the author together with x huang and d zaitsev established a rigidity result for cr embeddings f into spheres in low codimensions a key step in the proof of this result was to show that degenerate mappings are necessarily contained in a complex plane section of the target sphere partial rigidity in the 2004 paper it was shown that if the total rank d of the second fundamental form and all of its covariant derivatives is n here n is the cr dimension of m then f m is contained in a complex plane of dimension n d 1 the converse of this statement is also true as is easy to see when the total rank d exceeds n it is no longer true in general that f m is contained in a complex plane of dimension n d 1 as can be seen by examples in this paper we carry out a systematic study of degenerate cr mappings into spheres we show that when the ranks of the second fundamental form and its covariant derivatives exceed the cr dimension n then partial rigidity may still persist but there is a defect k that arises from the ranks exceeding n such that f m is only contained in a complex plane of dimension n d k 1 moreover this defect occurs in general as is illustrated by examples

unbounded sets of solutions of non cooperative elliptic systems on spheres abstract the aim of this paper is to show that any continuum of nontrivial solutions of a non cooperative system of elliptic equations on the sphere s n 1 bifurcating from the set of trivial solutions is unbounded moreover we characterize bifurcation points of this system at which the global symmetry breaking phenomenon occurs as the main tool we use the degree for so 2 invariant strongly indefinite functionals defined in 13

a reflection principle with applications to proper holomorphic mappings j l holomorphic mappings from b into b k we then apply this principle to the problem of characterizing the proper holomorphic mappings from b into b k two such mappings f and g are said to be equivalent if there exist holomorphic automorphisms b and p of b and bk respectively such that f pogo49 there are at present three significant papers on the subject the first by alexander 1 proves that i f f is a proper holomorphic mapping of b into b then f is a holomorphic automorphism of b webster i 5 proved that for n 3 i f f is a proper holomorphic mapping of b into b 1 which is c3 then f maps b into a hyperplane in r hence using alexander s result and properties of the holomorphic automorphisms f is equivalent to the mapping zl z2 z zi z2 zn o alexander s example zl z2 z2 zlz2 z 2 shows that webster s result does not hold for n 2 in a recent paper faran 2 has characterized all proper holomorphic mappings f from b 2 to b 3 that satisfy the additional smoothness condition fec3 2 he proves that such mappings are equivalent to one of the four mappings z w z w 0 z w z zw w2 z w r z 2 l zw w2

sanov results for glauber spin glass dynamics in this paper we prove a sanov result i e a large deviation principle ldp for the distribution of the empirical measure for the annealed glauber dynamics of the sherrington kirkpatrick spin glass without restrictions on time or temperature we prove a full ldp for the asymmetric dynamics and the crucial upper large deviations bound for the symmetric dynamics in the symmetric model a new order parameter arises corresponding to the response function in sozi83 in the asymmetric case we show that the corresponding rate function has a unique minimum given as the solution of a self consistent equation the key argument used in the proofs is a general result for mixing of what is known as large deviation systems lds with measures obeying an independent ldp

symmetric langevin spin glass dynamics we study the asymptotic behavior of symmetric spin glass dynamics in the sherrington kirkpatrick model as proposed by sompolinsky zippelius we prove that the averaged law of the empirical measure on the path space of these dynamics satisfies a large deviation upper bound in the high temperature regime we study the rate function which governs this large deviation upper bound and prove that it achieves its minimum value at a unique probability measure q which is not markovian we deduce an averaged and a quenched law of large numbers we then study the evolution of the gibbs measure of a spin glass under sompolinsky zippelius dynamics we also prove a large deviation upper bound for the law of the empirical measure and describe the asymptotic behavior of a spin on path space under this dynamic in the high temperature regime

proper holomorphic mappings from the two ball to the three ball we prove that a proper mapping of the two ball in cn into the three ball which is c2 on the closed two ball is equivalent to one of four normalized polynomial mappings this improves the known result of faran the proof is basic using taylor expansions

a reflection principle for proper holomorphic mappings of strongly pseudoconvex domains and applications this result is proved in 5 the techniques of 12 and 5 involve detailed computat ion of differential invariants on the boundary another paper philosophically related to the above but involving more elementary techniques is 1 in which proper holomorphic maps of b n to b are shown to be automorphisms more recently this result has been proved on strongly pseudoconvex domains in 2 and in 4 in the paper 3 which is the starting point for the present work completely elementary means are used to obtain a reflection principle for suitable maps of b to b k this in turn leads to a proof of the result of 12 assuming only c 2 smoothness to the boundary further detailed information is obtained about the maps even when k n 1 the elementary techniques of 3 may be adapted to use on strictly

multiple eigenvalue bifurcation for fredholm operators on fait le point sur les resultats de bifurcation locale et globale et on considere la stabilite des branches de bifurcation

rigidity of cr maps between shilov boundaries of bounded symmetric domains our goal is to establish what seems to be the first rigidity result for cr embeddings between shilov boundaries of bounded symmetric domains of higher rank the result states that any such cr embedding is the standard linear embedding up to cr automorphisms our basic assumption extends precisely the well known optimal bound for the rank one case there are no other restrictions on the ranks in particular the difficult case when the target rank is larger than the source rank is also allowed

initial monomial invariants of holomorphic maps we study a new biholomorphic invariant of holomorphic maps between domains in different dimensions based on generic initial ideals we start with the standard generic monomial ideals to find invariants for rational maps of spheres and hyperquadrics giving a readily computable invariant in this important case for example the generic initial monomials distinguish all four inequivalent rational proper maps from the two to the three dimensional ball next we associate to each subspace x subset mathcal o u a generic initial monomial subspace which is invariant under biholomorphic transformations and multiplication by nonzero functions the generic initial monomial subspace is a biholomorphic invariant for holomorphic maps if the target automorphism is linear fractional as in the case of automorphisms of spheres or hyperquadrics

the sherrington kirkpatrick model of spin glasses and stochastic calculus the high temperature case we study the fluctuations of free energy energy and entropy in the high temperature regime for the sherrington kirkpatrick model of spin glasses we introduce here a new dynamical method with the help of brownian motions and continuous martingales indexed by the square root of the inverse temperature as parameter thus formulating the thermodynamic formalism in terms of random processes the well established technique of stochastic calculus leads us naturally to prove that these fluctuations are simple gaussian processes with independent increments a generalization of a result proved by aizenman lebowitz and ruelle 1

on the c version of the reflection principle for mappings between cr manifolds we prove a smooth version of the classical schwarz reflection principle for cr mappings between an abstract cr manifold m and a generic cr manifold embedded in euclidean complex space as a consequence of our results we settle a conjecture of x huang formulated in 1994

the equivalence problem and rigidity for hypersurfaces embedded into hyperquadrics we consider the class of levi nondegenerate hypersurfaces m in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i that admit a local cr transversal embedding near a point p e m into a standard nondegenerate hyperquadric in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i with codimension k n n small compared to the cr dimension n of m we show that for hypersurfaces in this class there is a normal form which is closely related to the embedding such that any local equivalence between two hypersurfaces in normal form must be an automorphism of the associated tangent hyperquadric we also show that if the signature of m and that of the standard hyperquadric in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i are the same then the embedding is rigid in the sense that any other embedding must be the original embedding composed with an automorphism of the quadric

holomorphic mappings between hyperquadrics with small signature difference in this paper we study holomorphic mappings sending a hyperquadric of signature ell in bbb c n into a hyperquadric of signature ell in bbb c n we show theorem 1 1 that if the signature difference ell ell is not too large then the mapping can be normalized by automorphisms of the target hyperquadric to a particularly simple form and in particular the image of the mapping is contained in a complex plane of a dimension that depends only on ell and ell and not on the target dimension n we also prove a hopf lemma type result theorem 1 3 for such mappings

local rigidity of discrete groups acting on complex hyperbolic space the superrigidity theorem of margulis see zimmer 17 classifies finite dimensional representations of lattices in semi simple lie groups of real rank strictly larger than 1 it is a fundamental problem to obtain the classification of finite dimensional representations of lattices in rank i semi simple lie groups that this problem will be considerably harder than the previous one is suggested by the existence of continuous families of inequivalent representations or in other words the existence of non trivial deformations in johnson millson 6 and kourouniotis 8 such deformations were constructed for certain representations of lattices in so n 1 based on a construction of thurston called bending the deformation space of the representation of a lattice f so n 1 obtained by restricting an inclusion so n 1 so n l 1 to f is of particular interest if n 2 the space of infinitesimal deformations is hi f r 1 where f acts on r 1 by the restriction of the standard action of so n 1 the space of infinitesimal deformations is non zero for the standard arithmetic examples and the main point of the papers cited above was to establish that some of these infinitesimal deformations are integrable in 6 it is also shown that some are not in this paper we study the complex analogue of the above example we let f be a cocompact torsion free lattice in su n 1 and consider the deformation space of the representation of f obtained by restricting an inclusion su n 1 su n i 1 if n l the space of infinitesimal deformations is h f r h f c 1 in the first summand f acts trivially and the infinitesimal deformations are tangent to the obvious deformations obtained by deforming f in u n 1 by a curve of homomorphisms into the center of u n 1 observe that the above inclusion factors as su n 1 u n 1 su n i 1 in the second summand f acts by the restriction of the standard action of su n 1 this summand is non zero for the standard arithmetic examples

regularity of mappings into classical domains we study the regularity and algebraicity for mappings into classical domains among other things we establish everywhere regularity and algebraicity results for c 2 cr maps into the set of smooth boundary points of a classical domain where the codimension can be arbitrarily large

separation of global solution branches of elliptic systems with symmetry via nodal properties the idea that nodal properties are invariant along global solution branches of bifurcation problems goes back to the work of kolodner l and wolkowsky 2 31 in the context of nonlinear sturm liouville eigenvalue problems in that setting one can count the number of zeros of the solution a real valued function of a single real variable for the same class of problems it was shown later by crandall and rabinowitz 4 that this node count is inherited from the associated eigenfunction of the linearized problem about the trivial solution and that the invariance of nodal properties implies separation and unboundedness of global bifurcating solution branches it is natural to seek generalizations of these ideas to higher dimensions e g to problems involving more than one independent variable second order elliptic partial differential equations and or more than one dependent variable systems of second order differential equations these lead to immediate difficulties for elliptic partial differential equations we are confronted with the task of characterizing the set of zeros of functions of two or more variables in recent papers 5 6 we have shown how this problem simplifies considerably for equations with symmetries specifically in that setting we have demonstrated that the entire nodal set comprising onlyfixed hyper planes is precisely inherited from the eigenfunction of the linearization and is invariant along global branches of solutions possessing certain prescribed symmetries the proof requires groupinvariance arguments combined with subtle maximum principles in this work we take up the more challenging problem of systems of second order elliptic equations in the presence of symmetry to illustrate the difficulties involved we first discuss the case of a system of second order sturm liouville problems of the form

d angelo conjecture in the third gap interval we show that the d angelo conjecture holds in the third gap interval more precisely we prove that the degree of any rational proper holomorphic map from mathbb b n to mathbb b 4n 6 with n ge 7 is not more than 3

on the regularity of cr mappings between cr manifolds of hypersurface type we prove smooth and analytic versions of the classical schwarz reflection principle for transversal cr mappings between two levi nondegenerate cr manifolds of hypersurface type

global bifurcations of solutions of elliptic differential equations abstract the aim of this article is to prove global bifurcation theorems for s 1 equivariant potential operators of the form compact perturbation of identity as an application we prove that components of the set of nontrivial solutions of system formula which bifurcate from the set of trivial solutions are unbounded in suitably chosen space the important point to note here is that our results apply only to systems with an even number of equations as the main tool we use degree theory for s 1 equivariant potential operators constructed by the author in 16 and developed in 17

rigidity of proper holomorphic mappings between nonequidimensional bounded symmetric domains we prove that any proper holomorphic mapping from d p p 1 i to d p p i p geq 3 is necessarily a totally geodesic isometric embedding with respect to their bergman metrics and therefore is the standard linear embedding up to their automorphisms the proof of the main result in this paper will be achieved by a differential geometric study of a special class of complex geodesic curves on the bounded symmetric domains with respect to their bergman metrics

rigidity and holomorphic segre transversality for holomorphic segre maps let mathcal h n and mathcal h mathcal n denote the complexifications of heisenberg hypersurfaces in mathbf c n and mathbf c n respectively we show that non degenerate holomorphic segre mappings from mathcal h n into mathcal h n with n leq2n 2 possess a partial rigidity property as an application we prove that the holomorphic segre non transversality for a holomorphic segre map from mathcal h n into mathcal h n with n leq2n 2 propagates along segre varieties we also give an example showing that this propagation property of holomorphic segre transversality fails when n 2n 2

bifurcation and symmetry breaking of solutions of systems of elliptic differential equations in this article we study global bifurcations of weak solutions of the following variational system of elliptic differential equations u au u u in u 0on we prove sufficient conditions for the existence of unbounded continua of nontrivial solutions emanating from the trivial ones and necessary conditions for the existence of bounded continua moreover we describe a symmetry breaking phenomenon that occurs on that continua

symmetries in cr complexity theory abstract we introduce the hermitian invariant group f of a proper rational map f between the unit ball in complex euclidean space and a generalized ball in a space of typically higher dimension we use properties of the groups to define the crucial new concepts of essential map and the source rank of a map we prove that every finite subgroup of the source automorphism group is the hermitian invariant group of some rational proper map between balls we prove that f is non compact if and only if f is a totally geodesic embedding we show that f contains an n torus if and only if f is equivalent to a monomial map we show that f contains a maximal compact subgroup if and only if f is equivalent to the juxtaposition of tensor powers we also establish a monotonicity result the group after intersecting with the unitary group does not decrease when a tensor product operation is applied to a polynomial proper map we give a necessary condition for f when the target is a generalized ball to contain automorphisms that move the origin

on the third gap for proper holomorphic maps between balls let f be a proper rational map from the complex ball mathbb b n into mathbb b n with n 7 and 3n 1 le n le 4n 7 then f is equivalent to a map g 0 dots 0 where g is a proper holomorphic map from mathbb b n into mathbb b 3n

large deviations for langevin spin glass dynamics we study the asymptotic behaviour of asymmetrical spin glass dynamics in a sherrington kirkpatrick model as proposed by sompolinsky zippelius we prove that the annealed law of the empirical measure on path space of these dynamics satisfy a large deviation principle in the high temperature regime we study the rate function of this large deviation principle and prove that it achieves its minimum value at a unique probability measureq which is not markovian we deduce that the quenched law of the empirical measure converges to q extending then the preceeding results to replicated dynamics we investigate the quenched behavior of a single spin we get quenched convergence toq in the case of a symmetric initial law and even potential for the free spin

rational proper holomorphic maps from b n into b 2 n in this paper we will investigate rational proper holomorphic maps from b n into b2 n when n 4

bianalytic maps between free spectrahedra linear matrix inequalities lmis i d sum j 1 g a jx j sum j 1 g a j x j succeq 0 play a role in many areas of applications the set of solutions of an lmi is a spectrahedron lmis in dimension free matrix variables model most problems in linear systems engineering and their solution sets are called free spectrahedra free spectrahedra are exactly the free semialgebraic convex sets this paper studies free analytic maps between free spectrahedra and under certain generically valid irreducibility assumptions classifies all those that are bianalytic the foundation of such maps turns out to be a very small class of birational maps we call convexotonic the convexotonic maps in g variables sit in correspondence with g dimensional algebras if two bounded free spectrahedra mathcal d a and mathcal d b meeting our irreducibility assumptions are free bianalytic with map denoted p then p must after possibly an affine linear transform extend to a convexotonic map corresponding to a g dimensional algebra spanned by u i a 1 ldots u i a g for some unitary u furthermore b and ua are unitarily equivalent the article also establishes a positivstellensatz for free analytic functions whose real part is positive semidefinite on a free spectrahedron and proves a representation for a free analytic map from mathcal d a to mathcal d b not necessarily bianalytic another result shows that a function analytic on any radial expansion of a free spectrahedron is approximable by polynomials uniformly on the spectrahedron these theorems are needed for classifying free bianalytic maps

extending proper holomorphic mappings of positive codimension in this paper we obtain results on holomorphic continuation of proper holomorphic mappings between pseudoconvex domains with real analytic boundaries in complex spaces of different dimensions equivalently we obtain results concerning the analyticity of cauchy riemann mappings between real analytic pseudoconvex hypersurfaces in complex spaces of different dimensions to begin with we recall the corresponding results for mappings of equidimensional domains let d and d be bounded pseudoconvex domains with smooth boundaries in c if the boundaries of d and d are strictly pseudoconvex or more generally of finite type in the sense of d angelo 15 then every proper holomorphic map of d onto d extends smoothly t o according to the results of bell and catlin 7 8 and diederich and forn ess 19 for mappings between strictly pseudoconvex domains this was proved by fefferman 26 and nirenberg webster and yang 37 if the boundaries of the pseudoconvex domains d d c c are real analytic then every proper holomorphic mapping of d onto d extends holomorphically to a neighborhood of 3 according to baouendi and rothschild 2 and diederich and forn ess 21 this reflection principle was first discovered by lewy 34 and pin6uk 38 for mappings between strictly pseudoconvex domains in the case of biholomorphic mappings between weakly pseudoconvex domains with realanalytic boundaries the result follows from the work of baouendi jacobowitz and treves 5 results in this direction were obtained in recent years by several authors see the papers 4 6 18 22 33 47 49 in this paper we are treating the case when the domains d and d have different dimensions to be specific we assume that d e c and d c c n are bounded pseudoconvex domains with real analytic boundaries and n n l in this situation a proper holomorphic map f d d need not be regular at the boundary for instance there exist proper holomorphic maps of balls of different dimensions

super rigidity for cr embeddings of real hypersurfaces into hyperquadrics we consider holomorphic mappings sending a given levi nondegenerate pseudoconcave hypersurface m in into a nondegenerate hyperquadric of the same signature in and show that if m is sufficiently close to a hyperquadric in a certain sense then any two such mappings differ only by an automorphism of the hyperquadric

maps from the two ball to the three ball in 4 webster proved the following theorem let b z e c 7 2 3 is a proper holomorphic map which is c a up to the boundary of b then f is linear fractional the purpose of this paper is to extend this result to the case n 2 however the theorem is false when n 2 f need not be linear fractional so instead of trying to prove this false theorem we shall instead try to make a list of maps f bz b a which are c 3 up to the boundary of course given such an f we can get another such by composing f with an automorphism of b 2 and an automorphism of b a so we make the

uniqueness of certain polynomials constant on a line abstract we study a question with connections to linear algebra real algebraic geometry combinatorics and complex analysis let p x y be a polynomial of degree d with n positive coefficients and no negative coefficients such that p 1 when x y 1 a sharp estimate d 2 n 3 is known in this paper we study the p for which equality holds we prove some new results about the form of these sharp polynomials using these new results and using two independent computational methods we give a complete classification of these polynomials up to d 17 the question is motivated by the problem of classification of cr maps between spheres in different dimensions