on frechet differentiability of lipschitz maps between banach spaces a well known open question is whether every countable collection of lipschitz functions on a banach space x with separable dual has a common point of frechet differentiability we show that the answer is positive for some infinite dimensional x previously even for collections consisting of two functions this has been known for finite dimensional x only although for one function the answer is known to be affirmative in full generality our aims are achieved by introducing a new class of null sets in banach spaces called f null sets whose definition involves both the notions of category and measure and showing that the required differentiability holds almost everywhere with respect to it we even obtain existence of frechet derivatives of lipschitz functions between certain infinite dimensional banach spaces no such results have been known previously our main result states that a lipschitz map between separable banach spaces is frechet differentiable f almost everywhere provided that it is regularly gateaux differentiable f almost everywhere and the gateaux derivatives stay within a norm separable space of operators it is easy to see that lipschitz maps of x to spaces with the radon nikodym property are gateaux differentiable f almost everywhere moreover gateaux differentiability implies regular gateaux differentiability with exception of another kind of negligible sets so called a porous sets the answer to the question is therefore positive in every space in which every r porous set is f null we show that this holds for c k with k countable compact the tsirelson space and for all subspaces of co but that it fails for hilbert spaces

uniform homeomorphisms of banach spaces and asymptotic structure we give a general result on the behavior of spreading models in banach spaces which coarse lipschitz embed into asymptotically uniformly convex spaces we use this result to study the uniqueness of the uniform structure in p sums of finite dimensional spaces for 1 p in particular we give some new examples of spaces with unique uniform structure

quantal problems with partial algebraization of the spectrum we discuss a new class of spectral problems discovered recently which occupies an intermediate position between the exactly solvable problems e g harmonic oscillator and all others the problems belonging to this class are distinguished by the fact that a part of the eigenvalues and eigenfunctions can be found algebraically but not the whole spectrum the reason explaining the existence of the quasi exactly solvable problems is a hidden dynamical symmetry present in the hamiltonian for one dimensional motion this hidden symmetry issl 2 r it is shown that other groups lead to a partial algebraization in multidimensional quantal problems in particular sl 2 r sl 2 r so 3 andsl 3 r are relevant to two dimensional motion inducing a class of quasi exactly solvable two dimensional hamiltonians typically they correspond to systems in a curved space but sometimes the curvature turns out to be zero graded algebras open the possibility of constructing quasi exactlysolvable hamiltonians acting on multicomponent wave functions for example with a non minimal superextension ofsl 2 r we get a hamiltonian describing the motion of a spinor particle

trees and branches in banach spaces an infinite dimensional notion of asymptotic structure is considered this notion is developed in terms of trees and branches on banach spaces every countably infinite countably branching tree t of a certain type on a space x is presumed to have a branch with some property it is shown that then x can be embedded into a space with an fdd e i so that all normalized sequences in x which are almost a skipped blocking of e i have that property as an application of our work we prove that if x is a separable reflexive banach space and for some 1 0 there exists a subspace of x having finite codimension which c 2 e embeds into the l p sum of finite dimensional spaces

abelian mathrm c ast subalgebras of mathrm c ast algebras of real rank zero and inductive limit mathrm c ast algebras

on asymptotically uniformly smooth banach spaces abstract the class of asymptotically uniformly smoothable banach spaces has been considered in connection with several problems of nonlinear functional analysis as the differentiability of lipschitz functions the uniform classification of banach spaces or the fixed point property the known characterizations for those spaces come from p estimates when finite dimensional block decompositions are available or via duality by means of the szlenk index in this paper we found a geometrical characterization for the existence of asymptotically uniformly smooth renorming using an ordinal type index defined for subsets of the space among the applications we prove that the modulus of asymptotic uniform smoothness is the same for all the non asymptotically uniformly smoothable banach spaces

on the embeddability of the family of countably branching trees into quasi reflexive banach spaces abstract in this note we extend to the quasi reflexive setting the result of f baudier n kalton and g lancien concerning the non embeddability of the family of countably branching trees into reflexive banach spaces whose szlenk index and szlenk index from the dual are both equal to the first infinite ordinal in particular we show that the family of countably branching trees does neither embed into the james space j p nor into its dual space j p for p 1

on the uniqueness and stability of an inverse problem in photo acoustic tomography this article deals with the uniqueness and stability of the solution of a problem of optimal control related to the photo acoustic tomography process we prove stability results of the optimal solution with respect to the source and to the observation data and we compute the corresponding derivatives

a method for construction of the matrix solvable models we give a method for construction of the solvable models the method is based on an idea of turbiner s to transform the hamiltonian into a simple solvable form this method gives a possibility of constructing some matrix models we apply this method to the models of the calogero type and obtain solvable models of the three particle system with the matrix 2 2 potential which was constructed by polychronakos and a new model with the matrix 3 3 potential

numerical radius attaining compact linear operators abstract we show that there are compact linear operators on banach spaces which cannot be approximated by numerical radius attaining operators

a compact null set containing a differentiability point of every lipschitz function we prove that in a euclidean space of dimension at least two there exists a compact set of lebesgue measure zero such that any real valued lipschitz function defined on the space is differentiable at some point in the set such a set is constructed explicitly

a n type dunkl operators and new spin calogero sutherland models a new family of an type dunkl operators preserving a polynomial subspace of finite dimension is constructed using a general quadratic combination of these operators and the usual dunkl operators several new families of exactly and quasi exactly solvable quantum spin calogero sutherland models are obtained these include in particular three families of quasi exactly solvable elliptic spin hamiltonians

a universal differentiability set in banach spaces with separable dual abstract we show that any non zero banach space with a separable dual contains a totally disconnected closed and bounded subset s of hausdorff dimension 1 such that every lipschitz function on the space is frechet differentiable somewhere in s

lipschitz quotients from metric trees and from banach spaces containing l1 abstract if x is any separable banach space containing l1 then there is a lipschitz quotient map from x onto any separable banach space y

the uniform structure of banach spaces we explore the existence of uniformly continuous sections for quotient maps using this approach we are able to give a number of new examples in the theory of the uniform structure of banach spaces we show for example that there are two non isomorphic separable mathcal l 1 subspaces of l1 which are uniformly homeomorphic we also prove the existence of two coarsely homeomorphic banach spaces i e with lipschitz isomorphic nets which are not uniformly homeomorphic answering a question of johnson lindenstrauss and schechtman we construct a closed subspace of l1 whose unit ball is not an absolute uniform retract answering a question of the author

infinite dimensional generalized jacobian properties and calculus rules the extension to infinite dimensional domains of clarke s generalized jacobian is the focus of this paper first a generalization of a fabian preiss theorem to the infinite dimensional setting is obtained as a consequence a new formula relating the clarke s generalized jacobians corresponding to finite dimensional spaces k l with k l is established furthermore in the infinite dimensional case basic properties pertaining the generalized jacobian are developed and then an identification of this set valued map is produced applications of these results in the form of chain rules including sum and product rules and a computational formula for continuous selections are derived

infinite games banach space geometry and the eikonal equation we study a class of infinite games which turns out to be closely related to banach space geometry using one of these games we construct a bounded differentiable almost everywhere solution of the eikonal equation u 1 onr d w ithd 2

lenses and asymptotic midpoint uniform convexity abstract we introduce a new isometric geometric property namely asymptotic midpoint uniform convexity and investigate its connection to similar asymptotic properties

differential difference operators associated to reflection groups there is a theory of spherical harmonics for measures invariant under a finite reflection group the measures are products of powers of linear functions whose zero sets are the mirrors of the reflections in the group times the rotation invariant measure on the unit sphere in rn a commutative set of differential difference operators each homogeneous of degree 1 is the analogue of the set of first order partial derivatives in the ordinary theory of spherical harmonics in the case of r2 and dihedral groups there are analogues of the cauchy riemann equations which apply to gegenbauer and jacobi polynomial expansions the analysis of orthogonality structures for polynomials in several variables is a problem of vast dimensions this paper is part of an ongoing program to establish a workable theory for one particular class the underlying structure is based on finite coxeter groups these are finite groups acting on euclidean space generated by reflections in the zero sets of a collection of linear functions the roots the weight functions for the orthogonality are products of powers of these linear functions restricted to the surface of the unit sphere in addition the weight function is required to be invariant under the action of the group the resulting theory has strong similarities to the theory of spherical harmonics this was established in previous papers of the author 3 4 5 most notably a homogeneous polynomial is orthogonal to all polynomials of lower degree if and only if it is annihilated by a certain second order differential difference operator ordinary partial differentiation acts as an endomorphism on ordinary harmonic functions the use of such operators and their adjoints leads to recurrence formulas and orthogonal decompositions for harmonic polynomials in this paper we construct a commutative set of first order differentialdifference operators associated to the second order operator previously mentioned received by the editors june 1 1987 1980 mathematics subject classification 1985 revision primary 33a45 33a65 20h15 secondary 20c30 42c10 51f15

differentiability of lipschitz functions on banach spaces abstract the main result of this note says that if the norm of a banach space e is differentiable frechet gateaux or in some intermediate sense away from the origin then every locally lipschitz function defined on an open subset g of e is differentiable in the same sense at every point of some subset of g dense in g as the main corollary we prove that every lipschitz function on an asplund space is frechet differentiable on a dense set moreover in all these results the mean value theorem for the points of differentiability which is equivalent to the possibility of computing the clarke subdifferential from the points of differentiability is shown to hold

a topological and geometric approach to fixed points results for sum of operators and applications abstract in this paper we establish a fixed point result of krasnoselskii type for the sum a b where a and b are continuous maps acting on locally convex spaces our results extend previous ones we apply such results to obtain strong solutions for some quasi linear elliptic equations with lack of compactness we also provide an application to the existence and regularity theory of solutions to a nonlinear integral equation modeled in a banach space in the last section we develop a sequentially weak continuity result for a class of operators acting on vector valued lebesgue spaces such a result is used together with a geometric condition as the main tool to provide an existence theory for nonlinear integral equations in l p e

asymptotic structure lp estimates of sequences and compactness of multilinear mappings abstract we relate the moduli of asymptotic uniform smoothness and convexity of a banach space with the existence of upper and lower l p estimates of sequences in the space to this end we introduce two properties which are related to the m p property defined by kalton and werner in this way we obtain a connection between the moduli of asymptotic uniform smoothness and convexity and compactness or weak sequential continuity of multilinear mappings finally we give some applications to the existence of analytic and asymptotically flat norms on a banach space

szlenk indices and uniform homeomorphisms we prove some rather precise renorming theorems for banach spaces with szlenk index wo we use these theorems to show the invariance of certain quantitative szlenk type indices under uniform homeomorphisms

compact spaces of szlenk index ω abstract the szlenk index has found many applications in the isomorphic theory of banach spaces its definition is based in some kind of interplay between a weak topology and the norm metric with not much care on the linear structure there is no obstacle to consider the notion of szlenk index in more general settings in this paper we study the compact spaces of szlenk index at most with respect to an associated metric we include new applications to banach spaces of the these methods where the estimations of the growth speed of the finite szlenk indices play a fundamental role

the coarse lipschitz geometry of ell_p oplus ell_q we show that for 1 p with p 2 the space l p 0 1 is not uniformly homeomorphic to ell p oplus ell 2 we also show that if 1 p 2 q the space ell p oplus ell q has unique uniform structure answering a question of johnson lindenstrauss and schechtman geom funct anal 6 430 470 1996

a universal reflexive space for the class of uniformly convex banach spaces we show that there exists a separable reflexive banach space into which every separable uniformly convex banach space isomorphically embeds this solves problems raised by j bourgain b in 1980 and by w b johnson in 1977 jo we also give intrinsic characterizations of separable reflexive banach spaces which embed into a reflexive space with a block q hilbertian and or a block p besselian finite dimensional decomposition