Homology decompositions for classifying spaces of compact Lie groups

Let be a prime number and be a compact Lie group. A homology decomposition for the classifying space is a way of building up to mod homology as a homotopy colimit of classifying spaces of subgroups of . In this paper we develop techniques for constructing such homology decompositions. Jackowski, McClure and Oliver (Homotopy classification of self-maps of BG via -actions, Ann. of Math. 135 (1992), 183-270) construct a homology decomposition of by classifying spaces of -stubborn subgroups of . Their decomposition is based on the existence of a finite-dimensional mod acyclic --complex with restricted set of orbit types. We apply our techniques to give a parallel proof of the -stubborn decomposition of which does not use this geometric construction.

     

On syzygies of abelian varieties

In this paper we prove the following result: Let be a complex torus and a normally generated line bundle on ; then, for every , the line bundle satisfies Property of Green-Lazarsfeld.

     

Geometry of Banach spaces having shrinking approximations of the identity

Let and let be a compact set of scalars. We introduce property of Banach spaces by the requirement that whenever is a bounded net converging weak to in and . Using with 1$">, we characterize the existence of certain shrinking approximations of the identity (in particular, those related to -, -, and -ideals of compact or approximable operators). We also show that the existence of these approximations of the identity is separably determined.

     

Principal curvatures of isoparametric hypersurfaces in

Let be an isoparametric hypersurface in , and the inverse image of under the Hopf map. By using the relationship between the eigenvalues of the shape operators of and , we prove that is homogeneous if and only if either or is constant, where is the number of distinct principal curvatures of and is the number of non-horizontal eigenspaces of the shape operator on .

     

Partial differential equations with matricial coefficients and generalized translation operators

Let be the Bessel operator with matricial coefficients defined on by

where is a diagonal matrix and let be an matrix-valued function. In this work, we prove that there exists an isomorphism on the space of even , -valued functions which transmutes and . This allows us to define generalized translation operators and to develop harmonic analysis associated with . By use of the Riemann method, we provide an integral representation and we deduce more precise information on these operators.

     

Infinite convolution products and refinable distributions on Lie groups

Sufficient conditions for the convergence in distribution of an infinite convolution product of measures on a connected Lie group with respect to left invariant Haar measure are derived. These conditions are used to construct distributions that satisfy where is a refinement operator constructed from a measure and a dilation automorphism . The existence of implies is nilpotent and simply connected and the exponential map is an analytic homeomorphism. Furthermore, there exists a unique minimal compact subset such that for any open set containing and for any distribution on with compact support, there exists an integer such that implies If is supported on an -invariant uniform subgroup then is related, by an intertwining operator, to a transition operator on Necessary and sufficient conditions for to converge to , and for the -translates of to be orthogonal or to form a Riesz basis, are characterized in terms of the spectrum of the restriction of to functions supported on

     

A reduced Tits quadratic form and tameness of three-partite subamalgams of tiled orders

Let be a complete discrete valuation domain with the unique maximal ideal . We suppose that is an algebra over an algebraically closed field and . Subamalgam -suborders of a tiled -order are studied in the paper by means of the integral Tits quadratic form . A criterion for a subamalgam -order to be of tame lattice type is given in terms of the Tits quadratic form and a forbidden list of minor -suborders of presented in the tables.

     

Mean convergence of orthogonal Fourier series of modified functions

We construct orthonormal systems (ONS) which are uniformly bounded, complete, and made up of continuous functions such that some continuous and even some arbitrarily smooth functions cannot be modified so that the Fourier series of the new function converges in the -metric for any 2. $"> We prove also that if is a uniformly bounded ONS which is complete in all the spaces , then there exists a rearrangement of the natural numbers such that the system has the strong -property for all 2$">; that is, for every and for every and 0 $">there exists a function which coincides with except on a set of measure less than and whose Fourier series with respect to the system converges in      

A new result on the Pompeiu problem

A nonempty bounded open set () is said to have the Pompeiu property if and only if the only continuous function on for which the integral of over is zero for all rigid motions of is . We consider a nonempty bounded open set with Lipschitz boundary and we assume that the complement of is connected. We show that the failure of the Pompeiu property for implies some geometric conditions. Using these conditions we prove that a special kind of solid tori in , , has the Pompeiu property. So far the result was proved only for solid tori in . We also examine the case of planar domains. Finally we extend the example of solid tori to domains in bounded by hypersurfaces of revolution.

     

Definably simple groups in o-minimal structures

Let be a group definable in an o-minimal structure . A subset of is -definable if is definable in the structure (while definable means definable in the structure ). Assume has no -definable proper subgroup of finite index. In this paper we prove that if has no nontrivial abelian normal subgroup, then is the direct product of -definable subgroups such that each is definably isomorphic to a semialgebraic linear group over a definable real closed field. As a corollary we obtain an o-minimal analogue of Cherlin's conjecture.

     

Simple holonomic modules over rings of differential operators with regular coefficients of Krull dimension 2

Let be an algebraically closed field of characteristic zero. Let be the ring of (-linear) differential operators with coefficients from a regular commutative affine domain of Krull dimension which is the tensor product of two regular commutative affine domains of Krull dimension . Simple holonomic -modules are described. Let a -algebra be a regular affine commutative domain of Krull dimension and be the ring of differential operators with coefficients from . We classify (up to irreducible elements of a certain Euclidean domain) simple -modules (the field is not necessarily algebraically closed).

     

Specializations of Brauer classes over algebraic function fields

Let be either a number field or a field finitely generated of transcendence degree over a Hilbertian field of characteristic 0, let be the rational function field in one variable over , and let . It is known that there exist infinitely many such that the specialization induces a specialization , where has exponent equal to that of . Now let be a finite extension of and let . We give sufficient conditions on and for there to exist infinitely many such that the specialization has an extension to inducing a specialization , the residue field of , where has exponent equal to that of . We also give examples to show that, in general, such need not exist.

     

Cohomology of uniformly powerful -groups

In this paper we will study the cohomology of a family of -groups associated to -Lie algebras. More precisely, we study a category of -groups which will be equivalent to the category of -bracket algebras (Lie algebras minus the Jacobi identity). We then show that for a group in this category, its -cohomology is that of an elementary abelian -group if and only if it is associated to a Lie algebra.

We then proceed to study the exponent of in the case that is associated to a Lie algebra . To do this, we use the Bockstein spectral sequence and derive a formula that gives in terms of the Lie algebra cohomologies of . We then expand some of these results to a wider category of -groups. In particular, we calculate the cohomology of the -groups which are defined to be the kernel of the mod reduction

     

Jones index theory by Hilbert C-bimodules and K-theory

In this paper we introduce the notion of Hilbert -bimodules, replacing the associativity condition of two-sided inner products in Rieffel's imprimitivity bimodules by a Pimsner-Popa type inequality. We prove Schur's Lemma and Frobenius reciprocity in this setting. We define minimality of Hilbert -bimodules and show that tensor products of minimal bimodules are also minimal. For an - bimodule which is compatible with a trace on a unital -algebra , its dimension (square root of Jones index) depends only on its -class. Finally, we show that the dimension map transforms the Kasparov products in to the product of positive real numbers, and determine the subring of generated by the Hilbert -bimodules for a -algebra generated by Jones projections.

     

A Palais-Smale approach to problems in Esteban-Lions domains with holes

Let be the upper half strip with a hole. In this paper, we show there exists a positive higher energy solution of semilinear elliptic equations in and describe the dynamic systems of solutions of equation in various . We also show there exist at least two positive solutions of perturbed semilinear elliptic equations in .

     

Local boundary rigidity of a compact Riemannian manifold with curvature bounded above

This paper considers the boundary rigidity problem for a compact convex Riemannian manifold with boundary whose curvature satisfies a general upper bound condition. This includes all nonpositively curved manifolds and all sufficiently small convex domains on any given Riemannian manifold. It is shown that in the space of metrics on there is a -neighborhood of such that is the unique metric with the given boundary distance-function (i.e. the function that assigns to any pair of boundary points their distance -- as measured in ). More precisely, given any metric in this neighborhood with the same boundary distance function there is diffeomorphism which is the identity on such that . There is also a sharp volume comparison result for metrics in this neighborhood in terms of the boundary distance-function.

     

Schauder estimates for equationswith fractional derivatives

The equation where and are fractional derivatives of order and is studied. It is shown that if , , and are Hölder-continuous and , then there is a solution such that and are Hölder-continuous as well. This is proved by first considering an abstract fractional evolution equation and then applying the results obtained to (). Finally the solution of () with is studied.     

Projection orthogonale sur le graphe d'une relation linéaire fermé

Let denote the set of all closed linear relations on a Hilbert space (which contains all closed linear operators on ). In this paper, for every we define and study two associated linear operators on , and , which play an important role in the study of linear relations. These operators satisfy conditions quite analogous to trigonometric identities (whence their names) and appear, in particular, in the formula that gives the orthogonal projection on the graph of , a formula first established for linear operators by M. H. Stone and extended to linear relations by H. De Snoo. We prove here a slightly modified version of the De Snoo formula. Several other applications of the and operators to operator theory will be given in a forthcoming paper.

     

The truncated complex -moment problem

Let denote a sequence of complex numbers ( 0, \gamma _{ij}=\bar{\gamma}_{ji}$">), and let denote a closed subset of the complex plane . The Truncated Complex -Moment Problem for entails determining whether there exists a positive Borel measure on such that ( ) and . For a semi-algebraic set determined by a collection of complex polynomials , we characterize the existence of a finitely atomic representing measure with the fewest possible atoms in terms of positivity and extension properties of the moment matrix and the localizing matrices . We prove that there exists a -atomic representing measure for supported in if and only if and there is some rank-preserving extension for which , where or .