Classes de Chern Des Ensembles Analytiques

Let V be a compact complex analytic subset of a non-singular holomorphic manifold M. Assume that V has pure complex dimension n. Denote by V₀ its regular part, and by [V] its fundamental class in H_2n(V; ). If V is a locally complete intersection (LCI), it is known that the normal bundle N_{V_0} in M to V₀ in M has a natural extension N_V to all of V, so that we can define its Chern classes c^(*)(N_V) in cohomology, as well as the Chern classes c_vir^(*). If V is a locally complete intersection (LCI), it is known that the normal bundle N_{V_0} in M to V₀ in M has a natural extension N_V to all of V, so that we can define its Chern classes c^(*)(N_V) in cohomology, as well as the Chern classes c_vir^(*) (V) of the virtual tangent bundle T_vir(V):=[TM|_V - N_V] in the K-theory K⁰(V). This has applications

On the Heat Flow for Harmonic Maps with Potential

Let (M, g) and (N, h) be twoconnected Riemannian manifolds without boundary (M compact,N complete). Let G C (N): ifu: M N is a smooth map, we consider the functional E _G (u) = (1/2) _M [|du|²– 2G(u)]dV _M and we study its associated heat equation. Inthe compact case, we recover a version of the Eells–Sampson theorem,while for noncompact target manifold N, we establishsuitable hypotheses and ensure global existence and convergence atinfinity. In the second part of the paper, we study phenomena of blowingup solutions.     

The Boggess-Polking extension theorem for<Emphasis Type="Italic">CR</Emphasis> functions on manifolds with corners

We give the “boundary version” of the Boggess-PolkingCR extension theorem. LetM andN be real generic submanifolds of ℂ ⁿ withN ⊂M and letV be a “wedge” inM with “edge”N and “profile” Σ ⊂T _NM in a neighborhood of a pointz _o.We identify in natural manner and assume that for a holomorphic vector fieldL tangent toM and verifying we have that the Levi form takes a value . Then we prove thatCR functions onV extend ∀_ω to a wedgeV ₁ “attached” toV in direction of a vector fieldiV such that |pr(iV(z ₀))−iv ₀| < ε (where pr is the projection pr:T _NX →T _MX | _N ).We then prove that when the Levi cone “relative to Σ”iZ _Σ = convex hull is open inT _MX, thenCR functions extend to a “full” wedge with edgeN (that is, with a profile which is an open cone ofT _NX). Finally, we prove that iff is defined in a couple of wedges ±V with profiles ±Σ such thatiZ _Σ =T _MX, and is continuous up toN, thenf is in fact holomorphic atz _o.     

A technique for constructing divisible difference sets

An (m, n, k, ₁,₂) divisible difference set in a groupG of ordermn relative to a subgroupN of ordern is ak-subsetD ofG such that the list {xy^–1:x, y D} contains exactly ₁ copies of each nonidentity element ofN and exactly ₂ copies of each element ofG N. It is called semi-regular ifk > ₁ and k²=mn₂. We develop a method for constructing a divisible difference set as a product of a difference set and a relative difference set or a difference set and a subset ofG which we call a relative divisible difference set. The method results in several parametrically new families of semi-regular divisible difference sets.     

Extensions for Sobolev mappings between manifolds

We consider two compact Riemannian manifoldsM andN, such thatM has a boundary (but notN). We address the extension problem in the Sobolev class, namely, we investigate the question: foru W ^1–1/p,pM,N is there a mapV inW ^1/p(M,N) such thatV=u on M. Various results are given, and an emphasis is put on the special (simple) caseN=S ¹.     

Minimal length tree networks on the unit sphere

This paper considers the problem of finding minimal length tree networks on the unit sphere of a given point set (V) where distance is measured along great circular arcs. The related problems of finding a Steiner Minimal TreeSMT(V) and of finding a Minimum Spanning TreeMST(V) are treated through a simplicial decomposition technique based on computing the Delaunay TriangulationDT(V) and the Voronoi DiagramVD(V) of the given point set.O(N logN) algorithms for computingDT(V),VD(V), andMST(V) as well as anO(N logN) heuristic for finding a sub-optimalSMT(V) solution are presented, together with experimental results for randomly distributed points on .     

Forcing Linearity Numbers for Multiplication Modules

In this article, we prove that for any multiplication module M, the forcing linearity number of M, fln(M), belongs to {0,1,2}, and if M is finitely generated whose annihilator is contained in only finitely many maximal ideals, then fln(M) = 0. Also, the forcing linearity numbers of multiplication modules over some special rings are given. We also show that every multiplication module is semi-endomorphal.     

Nonlinear Markov Operators,Discrete Heat Flow,and Harmonic Maps Between Singular Spaces

We develop a theory of harmonic maps f:MN between singular spaces M and N. The target will be a complete metric space (N,d) of nonpositive curvature in the sense of A. D. Alexandrov. The domain will be a measurable space (M,) with a given Markov kernel p(x,dy) on it. Given a measurable map f:MN, we define a new map Pf:MN in the following way: for each xM, the point Pf(x)N is the barycenter of the probability measure p(x,f ^–1(dy)) on N. The map f is called harmonic on DM if Pf=f on D. Our theory is a nonlinear generalization of the theory of Markov kernels and Markov chains on M. It allows to construct harmonic maps by an explicit nonlinear Markov chain algorithm (which under suitable conditions converges exponentially fast). Many smoothing and contraction properties of the linear Markov operator P _M,R carry over to the nonlinear Markov operator P=P _M,N . For instance, if the underlying Markov kernel has the strong Lipschitz Feller property then all harmonic maps will be Lipschitz continuous.     

Groups of Congruences and Restriction Matrices

Let be a domain of the Euclidean space R ^m sent onto itself by a finite group G of congruences. In this paper we first define M elementary restriction matrices related to the group G and to a system of irreducible matrix representations of G. We then describe a general procedure to generate M restriction matrices for any finite-dimensional space V() of real functions defined on , when V() is invariant with respect to G. The number M depends only on the group G. Restriction matrices for the space V() have a block structure and all blocks can be obtained as from an elementary restriction matrix. Restriction matrices related to V() define a decomposition of V() as the sum of M subspaces. Finally, owing to restriction matrices, we propose a result of decomposition for linear systems. Several examples are presented.This revised version was published online in October 2005 with corrections to the Cover Date.     

Minimal Extensions in Tensor Product Spaces

LetX,Ybe two separable Banach spaces and letVXandWYbe finite dimensional subspaces. Suppose thatVSX,WZYand letM (S, V),N (Z, W). We will prove that ifαis a reasonable, uniform crossnorm onXYthenλ_MN(V_αW,X_αY)=λ_M(V, X) λ_N(W, Y).Here for any Banach spaceX,VSXandM (S, V)

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Also some applications of the above mentioned result will be presented.     

Stable complete surfaces with constant mean curvature

Letx:M ²N ³ be a stable immersion with constant mean curvatureH of a complete orientable surfaceM ² into a complete oriented three dimensional Riemannian manifoldN ³. In this paper we prove that, ifM ² is compact andH ²> –1/2 inf _M Ricc _N , thenM ² has genusg3, here Ricc _N is the Ricci curvature ofN ³. We also prove that, ifM ² is complete non compact andN ² has bounded geometry, the area ofM ² is infinite in the metric induced byx. In this case, ifH ²–1/2 inf _M Ricc _N thenx is umbilic and the equality holds.     

⊕-Cofinitely Supplemented Modules

Let R be a ring and M a right R-module. M is called -cofinitely supplemented if every submodule N of M with M/N finitely generated has a supplement that is a direct summand of M. In this paper various properties of the -cofinitely supplemented modules are given. It is shown that (1) Arbitrary direct sum of -cofinitely supplemented modules is -cofinitely supplemented. (2) A ring R is semiperfect if and only if every free R-module is -cofinitely supplemented. In addition, if M has the summand sum property, then M is -cofinitely supplemented iff every maximal submodule has a supplement that is a direct summand of M.     

Constant mean curvature hypersurfaces in a Lie group with a bi-invariant metric

Given an orientable hypersurface M of a Lie group with a bi-invariant metric we consider the map N : M ⁿ that translates the normal vector field of M to the identity, which is a natural extension of the usual Gauss map of hypersurfaces in Euclidean spaces; it is proved that the Laplacian of N satisfies a formula similar to that satisfied by the usual Gauss map. One may then conclude that M has constant mean curvature (cmc) if and only if N is harmonic; some other aplications to cmc hypersurfaces of are also obtained.     

Siegel's lemma with additional conditions

Let K be a number field, and let W be a subspace of KN, N?1. Let V₁,…,VM be subspaces of KN of dimension less than dimension of W. We prove the existence of a point of small height in , providing an explicit upper bound on the height of such a point in terms of heights of W and V₁,…,VM. Our main tool is a counting estimate we prove for the number of points of a subspace of KN inside of an adelic cube. As corollaries to our main result we derive an explicit bound on the height of a nonvanishing point for a decomposable form and an effective subspace extension lemma.     

<Emphasis Type="Italic">M</Emphasis>-spherical <Emphasis Type="Italic">K</Emphasis>-modules of a rank one semisimple Lie group

Let G =K A N be an Iwasawa decomposition of a connected, noncompact real semisimple Lie group with finite center and let M be the centralizer of A in K . B. Kostant proved that for every irreducible M-spherical K-module V there exists a unique d (the Kostant degree of V) such that V can be realized as a submodule of the space of all -harmonic homogeneous polynomials of degree d on . Here is a Cartan decomposition of the complexification of the Lie algebra of G .In this paper we give an algorithm to obtain a highest weight vector from any M-invariant vector in an irreducible M-spherical K-module. This algorithm allows us to compute a sharp bound for the Kostant degree d(v) of any M-invariant vector v in a locally finite M-spherical K-module V. The method computes d(v) effectively for any V if G is locally isomorphic to SO(n,1) and for if G is locally isomorphic to SU(n,1).Partially supported by Agencia Córdoba Ciencia and CONICET Mathematics Subject Classification (2000):Primary 22E46, Secondary 43A85     

Gauge theorems for resolvents with application to Markov processes

Summary LetV=(V )0 be a (not necessarily sub-Markovian) resolvent such that the kernelV for some 0 is compact and irreducible. We prove the following general gauge theorem: If there exists at least oneV-excessive function which is notV-inviriant, thenV ₀ is bounded.This result will be applied to resolventsU ^M arising from perturbation of sub-Markovian right resolventsU by multiplicative functionalsM (not necessarily supermartingale), for instance, by Feynman-Kac functionals. Among others, this leads to an extension of the gauge theorem of Chung/Rao and even of one direction of the conditional gauge theorem of Falkner and Zhao.     

Algebraic aspects of an ordered band arising in nonlinear signal processing

For any positive integers M and N we define a selfdually ordered band ℬ(M,N) of cardinality and ask whether or not it is lattice-ordered. The origin of ℬ(M,N) in nonlinear signal processing is outlined.     

Morse-Sard type results in sub-Riemannian geometry

Let (M,,g) be a sub-Riemannian manifold and x₀ M. Assuming that Chows condition holds and that M endowed with the sub-Riemannian distance is complete, we prove that there exists a dense subset N₁ of M such that for every point x of N₁, there is a unique minimizing path steering x₀ to x, this trajectory admitting a normal extremal lift. If the distribution is everywhere of corank one, we prove the existence of a subset N₂ of M of full Lebesgue measure such that for every point x of N₂, there exists a minimizing path steering x₀ to x which admits a normal extremal lift, is nonsingular, and the point x is not conjugate to x₀. In particular, the image of the sub-Riemannian exponential mapping is dense in M, and in the case of corank one is of full Lebesgue measure in M.Mathematics Subject Classification (2000): 53C17, 49J52     

Riemannian Submersions and Riemannian Manifolds with Einstein--Weyl Structures

The main results of this paper are as follows. (a) Let : M N be a non-trivial Riemannian submersion with totally geodesic fibers of dimension 1 over an Einstein manifold N. If M is compact and admits a standard Einstein--Weyl structure with constant Einstein--Weyl function, then N admits a Kähler structure andM a Sasakian structure. (b) Let be a Riemannian submersion with totally geodesic fibers and N an Einstein manifold of positive scalar curvature . If M admits a standard Sasakian structure, then M admits an Einstein--Weyl structure with constant Einstein--Weyl function.