The center conjecture for thick spherical buildings

In this paper we show that a convex subcomplex of a spherical building of type E ₆, E ₇ or E ₈ is a subbuilding or the automorphisms of the subcomplex fix a a point on it. Together with previous results of Mühlherr–Tits, and Leeb and the author, this completes the proof of Tits’ Center Conjecture for spherical buildings without factors of type H ₄, in particular, for thick spherical buildings.     

Base subsets of polar Grassmannians

Let Δ be a thick building of type Xn=Cn,Dn. Let also Gk be the Grassmannian of k-dimensional singular subspaces of the associated polar space Π (of rank n). We write Gk for the corresponding shadow space of type Xn,k. Every bijective transformation of Gk which maps base subsets to base subsets (the shadows of apartments) is a collineation of Gk, and it is induced by a collineation of Π if n≠4 or k≠1.     

The ``hot spots' conjecture for domains with two axes of symmetry

Consider a Neumann eigenfunction with lowest nonzero eigenvalue of a convex planar domain with two axes of symmetry. We show that the maximum and minimum of the eigenfunction are achieved at points on the boundary only. We deduce J. Rauch's ``hot spots' conjecture: if the initial temperature distribution is not orthogonal to the first nonzero eigenspace, then the point at which the temperature achieves its maximum tends to the boundary. This was already proved by Bañuelos and Burdzy in the case in which the eigenspace is one dimensional. We introduce here a new technique based on deformations of the domain that applies to the case of multiple eigenvalues.

     

The Van der Waerden conjecture for mixed discriminants

We prove that the mixed discriminant of doubly stochastic n-tuples of semidefinite hermitian n×n matrices is bounded below by and that this bound is uniquely attained at the n-tuple . This result settles a conjecture posed by R. Bapat in 1989. We consider various generalizations and applications of this result.     

A practical algorithm for decomposing polygonal domains into convex polygons by diagonals

We present algorithms for decomposing a polygon (with holes) into convex polygons by diagonals. The methods are computationally quick, and although the partitions that they produce may not have minimal cardinality they usually have a low number of convex pieces. Thus, the methods are suitable for being used when achieving a modest load on the CPU time is more important than finding optimal decompositions as, for instance, in location problems. Part of the results in this paper are from Fernández (1999), and were presented in Fernández et al. (1998). This work has been supported by the Ministry of Science and Technology of Spain under the research projects BEC2002-01026, SEJ2005-06273/ECON (J. Fernández, B. Tóth and B. Pelegrín) and TIC2003-05982-C05-03 (L. Cánovas), in part financed by the European Regional Development Fund (ERDF).     

The 3-ball is a local pessimum for packing

It was conjectured by Ulam that the ball has the lowest optimal packing fraction out of all convex, three-dimensional solids. Here we prove that any origin-symmetric convex solid of sufficiently small asphericity can be packed at a higher efficiency than balls. We also show that in dimensions 4, 5, 6, 7, 8, and 24 there are origin-symmetric convex bodies of arbitrarily small asphericity that cannot be packed using a lattice as efficiently as balls can be.     

The maximum size of the intersection of two ovoids

In this article we show that the maximum size for the intersection of two distinct ovoids of PG(3,q), q even, is . This is less than the old bound of given by Segre.     

The entropy conjecture for partially hyperbolic diffeomorphisms with 1-D center

We prove that if f is a partially hyperbolic diffeomorphism on the compact manifold M with one-dimensional center bundle, then the logarithm of the spectral radius of the map induced by f on the real homology groups of M is smaller or equal to the topological entropy of f. This is a particular case of the Shub's entropy conjecture, which claims that the same conclusion should be true for any C¹ map on any compact manifold.     

Equivariant Novikov conjecture for groups acting on Euclidean buildings

We prove the equivariant Novikov conjecture for groups acting on Euclidean buildings by using an equivariant Hilsum-Skandalis method. We also obtain an equivariant version of the Connes-Gromov-Moscovici theorem for almost flat -algebra bundles.

     

N指标d维广义Wiener过程极集的性质

研究了Ｎ指标ｄ维广义Ｗｉｅｎｅｒ过程极集的性质,得到了其极性的必要条件．     

Arc-smooth generalized universal covering spaces

Let Y be a path-connected subset of a CAT(0) space Z, allowing for a map to a 1-dimensional separable metric space X, such that the nontrivial point preimages of f form a null sequence of convex subsets of Z. Such Y need not be homotopy equivalent to a 1-dimensional space.

We prove that Y admits a generalized universal covering space, which we equip with an arc-smooth structure by consistently and continuously selecting one tight representative from each path homotopy class of Y. It follows that all homotopy groups of Y vanish in dimensions greater than 1.     

The Splitting Criterion of Kempf and the Babylonian Tower Theorem

In this work, we study semiprime and radical submodules of modules. We know that every radical submodule is semiprime, and we investigate when the converse is true.     

Hyperplane partitions and difference systems of sets

Difference Systems of Sets (DSS) are combinatorial configurations that arise in connection with code synchronization. This paper gives new constructions of DSS obtained from partitions of hyperplanes in a finite projective space, as well as DSS obtained from balanced generalized weighing matrices and partitions of the complement of a hyperplane in a finite projective space.     

Nonexistence of a code for

In this paper, we shall prove that there is no [3q⁴-q³-q²-3q-1,5,3q⁴-4q³-2q+1]_q code over the finite field for q11. Thus, we conclude the nonexistence of a [g_q(5,d),5,d]_q code for 3q⁴-4q³-2q+1d3q⁴-4q³-q.     

The Hermite-Hadamard inequality for convex functions on a global NPC space

We prove an extension of Choquet's theorem to the framework of compact metric spaces with a global nonpositive curvature. Together with Sturm's extension [K.T. Sturm, Probability measures on metric spaces of nonpositive curvature, in: Pascal Auscher, et al. (Eds.), Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces, Lecture Notes from a Quarter Program on Heat Kernels, Random Walks, and Analysis on Manifolds and Graphs April 16-July 13, 2002, Paris, France, in: Contemp. Math., vol. 338, Amer. Math. Soc., Providence, RI, 2003, pp. 357-390] of Jensen's inequality, this provides a full analogue of the Hermite-Hadamard inequality for the convex functions defined on such spaces.     

Multiplicative models for configuration spaces of algebraic varieties

Fulton and MacPherson (Ann. Math. 139 (1994) 183) found a Sullivan dg-algebra model for the space of n-configurations of a smooth complex projective variety X. K?í? (Ann. Math. 139 (1994) 227) gave a simpler model, E_n(H), depending only on the cohomology ring, H?H^*X.We construct an even simpler and smaller model, J_n(H). We then define another new dg-algebra, E_n(H^°), and use J_n(H) to prove that E_n(H^°) is a model of the space of n-configurations of the non-compact punctured manifold X^°, when X is 1-connected. Following an idea of Drinfel’d (Leningrad Math. J. 2 (1991) 829), we put a simplicial bigraded differential algebra structure on {E_n(H^°)}_n?0.     

R^n上广义Radon变换的奇异值分解

当广义Radon变换限制在带权的平方可积函数空间时, 该文构造了一类广义 Radon 变换的奇异值分解,给出了它们的逆变换的一些结果, 从而导出了广义 Radon 变换的反演公式以及值域的特征.     

P耗散算子的研究

本文利用广义半内积空间及广义不定度规空间,解决Ｂａｎａｃｈ空间中Ｐ耗散算子的极大耗散扩张表示,并得到很有意义的应用．