Projective manifolds with small pluridegrees

Let be a very ample line bundle on a connected complex projective manifold of dimension . Except for a short list of degenerate pairs , and there exists a morphism expressing as the blowup of a projective manifold at a finite set , with nef and big for the ample line bundle . The projective geometry of is largely controlled by the pluridegrees for , of . For example, , where is the genus of a curve section of , and is equal to the self-intersection of the canonical divisor of the minimal model of a surface section of . In this article, a detailed analysis is made of the pluridegrees of . The restrictions found are used to give a new lower bound for the dimension of the space of sections of . The inequalities for the pluridegrees, that are presented in this article, will be used in a sequel to study the sheet number of the morphism associated to .

     

A generalized Brauer construction and linear source modules

For a complete discrete valuation ring with residue field , a subgroup of a finite group and a homomorphism , we define a functor from the category of -modules to the category of -modules and investigate its behaviour with respect to linear source modules.     

Steiner type formulae and weighted measures of singularities for semi-convex functions

For a given convex (semi-convex) function , defined on a nonempty open convex set , we establish a local Steiner type formula, the coefficients of which are nonnegative (signed) Borel measures. We also determine explicit integral representations for these coefficient measures, which are similar to the integral representations for the curvature measures of convex bodies (and, more generally, of sets with positive reach). We prove that, for , the -th coefficient measure of the local Steiner formula for , restricted to the set of -singular points of , is absolutely continuous with respect to the -dimensional Hausdorff measure, and that its density is the -dimensional Hausdorff measure of the subgradient of .

As an application, under the assumptions that is convex and Lipschitz, and is bounded, we get sharp estimates for certain weighted Hausdorff measures of the sets of -singular points of . Such estimates depend on the Lipschitz constant of and on the quermassintegrals of the topological closure of .

     

Trees and valuation rings

A subring of a division algebra is called a valuation ring of if or holds for all nonzero in . The set of all valuation rings of is a partially ordered set with respect to inclusion, having as its maximal element. As a graph is a rooted tree (called the valuation tree of ), and in contrast to the commutative case, may have finitely many but more than one vertices. This paper is mainly concerned with the question of whether each finite, rooted tree can be realized as a valuation tree of a division algebra , and one main result here is a positive answer to this question where can be chosen as a quaternion division algebra over a commutative field.

     

The characters of the generalized Steinberg representations of finite general linear groups on the regular elliptic set

Let be a finite field, the degree extension of , and the general linear group with entries in . This paper studies the ``generalized Steinberg" (GS) representations of and proves the equivalence of several different characterizations for this class of representations. As our main result we show that the union of the class of cuspidal and GS representations of is in natural one-one correspondence with the set of Galois orbits of characters of , the regular orbits of course corresponding to the cuspidal representations. Besides using Green's character formulas to define GS representations, we characterize GS representations by associating to them idempotents in certain commuting algebras corresponding to parabolic inductions and by showing that GS representations are the sole components of these induced representations which are ``generic" (have Whittaker vectors).

     

Fundamental groups of moduli and the Grothendieck-Teichmüller group

Let denote the moduli space of Riemann spheres with ordered marked points. In this article we define the group of quasi-special symmetric outer automorphisms of the algebraic fundamental group for all to be the group of outer automorphisms respecting the conjugacy classes of the inertia subgroups of and commuting with the group of outer automorphisms of obtained by permuting the marked points. Our main result states that is isomorphic to the Grothendieck-Teichmüller group for all .

     

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.

     

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

     

Semi-classical limit for random walks

Let be a discrete symmetric random walk on a compact Lie group with step distribution and let be the associated transition operator on . The irreducibles of the left regular representation of on are finite dimensional invariant subspaces for and the spectrum of is the union of the sub-spectra on the irreducibles, which consist of real eigenvalues . Our main result is an asymptotic expansion for the spectral measures

along rays of representations in a positive Weyl chamber , i.e. for sequences of representations , with . As a corollary we obtain some estimates on the spectral radius of the random walk. We also analyse the fine structure of the spectrum for certain random walks on (for which is essentially a direct sum of Harper operators).

     

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.

     

On the endomorphism monoids of (uniquely) complemented lattices

Let be a lattice with and . An endomorphism of is a -endomorphism, if it satisfies and . The -endomorphisms of form a monoid. In 1970, the authors proved that every monoid can be represented as the -endomorphism monoid of a suitable lattice with and . In this paper, we prove the stronger result that the lattice with a given -endomorphism monoid can be constructed as a uniquely complemented lattice; moreover, if is finite, then can be chosen as a finite complemented lattice.

     

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 .

     

A condition for the stability of -covered on foliations of 3-manifolds

We give a sufficient condition for a codimension one, transversely orientable foliation of a closed 3-manifold to have the property that any foliation sufficiently close to it be -covered. This condition can be readily verified for many examples. Further, if an -covered foliation has a compact leaf , then any transverse loop meeting lifts to a copy of the leaf space, and the ambient manifold fibers over with as fiber.

     

On the unique representation of families of sets

Let and be uncountable Polish spaces. represents a family of sets provided each set in occurs as an -section of . We say that uniquely represents provided each set in occurs exactly once as an -section of . is universal for if every -section of is in . is uniquely universal for if it is universal and uniquely represents . We show that there is a Borel set in which uniquely represents the translates of if and only if there is a Vitali set. Assuming there is a Borel set with all sections sets and all non-empty sets are uniquely represented by . Assuming there is a Borel set with all sections which uniquely represents the countable subsets of . There is an analytic set in with all sections which represents all the subsets of , but no Borel set can uniquely represent the sets. This last theorem is generalized to higher Borel classes.

     

A probabilistic approach to positive harmonic functions in a slab with alternating Dirichlet and Neumann boundary conditions

Let , , be a dimensional slab. Denote points by , where and . Denoting the boundary of the slab by , let

where is an ordered sequence of intervals on the right half line (that is, b_{n}$">). Assume that the lengths of the intervals are bounded and that the spaces between consecutive intervals are bounded and bounded away from zero. Let . Let and denote respectively the cone of bounded, positive harmonic functions in and the cone of positive harmonic functions in which satisfy the Dirichlet boundary condition on and the Neumann boundary condition on .

Letting , the main result of this paper, under a modest assumption on the sequence , may be summarized as follows when :

1. If , then and are both one-dimensional (as in the case of the Neumann boundary condition on the entire boundary). In particular, this occurs if with 2$">.

2. If and , then and is one-dimensional. In particular, this occurs if .

3. If , then and the set of minimal elements generating is isomorphic to (as in the case of the Dirichlet boundary condition on the entire boundary). In particular, this occurs if with .

When , as soon as there is at least one interval of Dirichlet boundary condition. The dichotomy for is as above.

     

Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents

We use variational methods to study the existence and multiplicity of solutions for the following quasi-linear partial differential equation:

where and are two positive parameters and is a smooth bounded domain in containing in its interior. The variational approach requires that , and , which we assume throughout. However, the situations differ widely with and , and the interesting cases occur either at the critical Sobolev exponent () or in the Hardy-critical setting () or in the more general Hardy-Sobolev setting when . In these cases some compactness can be restored by establishing Palais-Smale type conditions around appropriately chosen dual sets. Many of the results are new even in the case , especially those corresponding to singularities (i.e., when .

     

Rigidity of Coxeter groups

Let be a Coxeter group acting properly discontinuously and cocompactly on manifolds and such that the fixed point sets of finite subgroups are contractible. Let be a -homotopy equivalence which restricts to a -homeomorphism on the boundary. Under an assumption on the three dimensional fixed point sets, we show that then is -homotopic to a -homeomorphism.     

The toric -vectors of partially ordered sets

An explicit formula for the toric -vector of an Eulerian poset in terms of the -index is developed using coalgebra techniques. The same techniques produce a formula in terms of the flag -vector. For this, another proof based on Fine's algorithm and lattice-path counts is given. As a consequence, it is shown that the Kalai relation on dual posets, , is the only equation relating the -vectors of posets and their duals. A result on the -vectors of oriented matroids is given. A simple formula for the -index in terms of the flag -vector is derived.

     

Problème de Dirichlet pour une équation de Monge-Ampère réelle elliptique dégénérée en dimension

RÉSUMÉ. On considère dans un ouvert borné de , à bord régulier, le problème de Dirichlet

où , est positive et s'annule sur un ensemble fini de points de . On démontre alors sous certaines hypothèses sur et si est assez petit, que le problème (1) possède une solution convexe unique .

ABSTRACT. We consider in a bounded open set of , with regular boundary, the Dirichlet problem

where , is positive and vanishes on , a finite set of points in . We prove, under some hypothesis on and if is sufficiently small, that the problem (1) has a unique convex solution .

     

Endomorphisms of expansive systems on compact metric spaces and the pseudo-orbit tracing property

We investigate the interrelationships between the dynamical properties of commuting continuous maps of a compact metric space. Let be a compact metric space.

First we show the following. If is an expansive onto continuous map with the pseudo-orbit tracing property (POTP) and if there is a topologically mixing continuous map with , then is topologically mixing. If and are commuting expansive onto continuous maps with POTP and if is topologically transitive with period , then for some dividing , , where the , , are the basic sets of with such that all have period , and the dynamical systems are a factor of each other, and in particular they are conjugate if is a homeomorphism.

Then we prove an extension of a basic result in symbolic dynamics. Using this and many techniques in symbolic dynamics, we prove the following. If is a topologically transitive, positively expansive onto continuous map having POTP, and is a positively expansive onto continuous map with , then has POTP. If is a topologically transitive, expansive homeomorphism having POTP, and is a positively expansive onto continuous map with , then has POTP and is constant-to-one.

Further we define `essentially LR endomorphisms' for systems of expansive onto continuous maps of compact metric spaces, and prove that if is an expansive homeomorphism with canonical coordinates and is an essentially LR automorphism of , then has canonical coordinates. We add some discussions on basic properties of the essentially LR endomorphisms.