Minimally separating sets, mediatrices, and Brillouin spaces

Brillouin zones and their boundaries were studied in [J.J.P. Veerman et al., Comm. Math. Phys. 212 (3) (2000) 725] because they play an important role in focal decomposition as first defined by Peixoto in [J. Differential Equations 44 (1982) 271] and in physics [N.W. Ashcroft, N.D. Mermin, Solid State Physics, Holt, Rhinehart, and Winston, 1976; L. Brillouin, Wave Propagation in Periodic Structures, Dover, 1953]. In so-called Brillouin spaces, the boundaries of the Brillouin zones have certain regularity properties which imply that they consist of pieces of mediatrices (or equidistant sets).The purpose of this note is two-fold. First, we give some simple conditions on a metric space which are sufficient for it to be a Brillouin space. These conditions show, for example, that all compact, connected Riemannian manifolds with their usual distance functions are Brillouin spaces. Second, we exhibit a restriction on the Z₂-homology of mediatrices in such manifolds in terms of the Z₂-homology of the manifolds themselves, based on the fact that they are Brillouin spaces. (This will used to obtain a classification up to homeomorphism of surface mediatrices in forthcoming paper [J. Bernhard, J.J.P. Veerman, The topology of surface mediatrices, Portland State University].)     

The topological structure of maximal lattice free convex bodies: The general case

Given a genericm × n matrixA, the simplicial complexK(A) is defined to be the collection of simplices representing maximal lattice point free convex bodies of the form {x : Ax b}. The main result of this paper is that the topological space associated withK(A) is homeomorphic withR ^m–1 . © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Supported by NSF grant SES-9121936 and the program in Discrete Mathematics at Yale University.Partially supported by the Hungarian NSF grant 1909 and the program in Discrete Mathematics at Yale University.     

The complex of maximal lattice free simplices

The simplicial complexK(A) is defined to be the collection of simplices, and their proper subsimplices, representing maximal lattice free bodies of the form (x: Axb), withA a fixed generic (n + 1) ×n matrix. The topological space associated withK(A) is shown to be homeomorphic to ⁿ , and the space obtained by identifying lattice translates of these simplices is homeorphic to then-torus.Corresponding author.The first author was partially supported by Hungarian NSF grants 1907 and 1909, and also by U.S. NSF grant CCR-9111491. The research of the second author was supported by DMS9103608 and the third author by NSF grant SES9121936.     

Test sets for integer programs

In this paper I discuss various properties of the simplicial complex of maximal lattice free bodies associated with a matrixA. If the matrix satisfies some mild conditions, and isgeneric, the edges of the complex form the minimal test set for the family of integer programs obtained by selecting a particular row ofA as the objective function, and using the remaining rows to impose constraints on the integer variables.     

Erratum to: “The Gromov norm of the product of two surfaces” [Topology 44 (2005) 321–339]

We report an error in the proof of Lemma 2.7 of the original article. This invalidates one direction of our main theorem.     

A note on shellability and acyclic orientations

In this short note we discuss the shellability of (nonpure) simplicial complexes in terms of acyclic orientations of the facet-ridge incidence graphs, which shows that we can decide shellability only from the facet-ridge incidences and the total number of faces the simplicial complex contains.     

The topology of stirred fluids

There are simple idealized mathematical models representing the stirring of fluids. The models we consider involve two fluids entering a chamber, with the overflow leaving it. The stirring creates a Cantor-like, but connected, boundary between the fluids that is best described point-set topologically. We prove that in many cases the boundary between the fluids is an indecomposable continuum.     

Guaranteed consistency of surface intersections and trimmed surfaces using a coupled topology resolution and domain decomposition scheme

We describe a method that serves to simultaneously determine the topological configuration of the intersection curve of two parametric surfaces and generate compatible decompositions of their parameter domains, that are amenable to the application of existing perturbation schemes ensuring exact topological consistency of the trimmed surface representations. To illustrate this method, we begin with the simpler problem of topology resolution for a planar algebraic curve F(x,y)=0 in a given domain, and then extend concepts developed in this context to address the intersection of two tensor-product parametric surfaces p(s,t) and q(u,v) defined on (s,t)∈[0,1]² and (u,v)∈[0,1]². The algorithms assume the ability to compute, to any specified precision, the real solutions of systems of polynomial equations in at most four variables within rectangular domains, and proofs for the correctness of the algorithms under this assumption are given. Mathematics subject classification (2000)  65D17     

Algebraic cycles and topology of real Enriques surfaces

For a real Enriques surface Y we prove that every homology class in H₁(Y (R), Z/2) can be represented by a real algebraic curve if and only if all connected components of Y(R) are orientable. Furthermore, we give a characterization of real Enriques surfaces which are Galois-Maximal and/or Z-Galois-Maximal and we determine the Brauer group of any real Enriques surface Y.     

On the topology of nested set complexes

Nested set complexes appear as the combinatorial core of De Concini-Procesi arrangement models. We show that nested set complexes are homotopy equivalent to the order complexes of the underlying meet-semilattices without their minimal elements. For atomic semilattices, we consider the realization of nested set complexes by simplicial fans proposed by the first author and Yuzvinsky and we strengthen our previous result showing that in this case nested set complexes in fact are homeomorphic to the mentioned order complexes.

     

弱紧集与Orlicz空间

本文圆满完善了[1]的结果,并由此给出了Ｏｒｌｉｃｚ空间自反性与Ｓｈｕｒ定理的新证明。     

The computational complexity of basic decision problems in 3-dimensional topology

We study the computational complexity of basic decision problems of 3-dimensional topology, such as to determine whether a triangulated 3-manifold is irreducible, prime, ∂-irreducible, or homeomorphic to a given 3-manifold M. For example, we prove that the problem to recognize whether a triangulated 3-manifold is homeomorphic to a 3-sphere, or to a 2-sphere bundle over a circle, or to a real projective 3-space, or to a handlebody of genus g, is decidable in nondeterministic polynomial time (NP) of size of the triangulation. We also show that the problem to determine whether a triangulated orientable 3-manifold is irreducible (or prime) is in PSPACE and whether it is ∂-irreducible is in coNP. The proofs improve and extend arguments of prior author’s article on the recognition problem for the 3-sphere.      

Meshing skin surfaces with certified topology

Skin surfaces are used for the visualization of molecules. They form a class of tangent continuous surfaces defined in terms of a set of balls (the atoms of the molecule) and a shrink factor. More recently, skin surfaces have been used for approximation purposes.

We present an algorithm that approximates a skin surface with a topologically correct mesh. The complexity of the mesh is linear in the size of the Delaunay triangulation of the balls, which is worst case optimal.

We also adapt two existing refinement algorithms to improve the quality of the mesh and show that the same algorithm can be used for meshing a union of balls.     

Some properties of the level convergence topology on fuzzy number space E

In this paper, we study topological structure of level convergence on fuzzy number space Eⁿ and give a characterization of compact subsets in (Eⁿ,τ(l)), where τ(l) is the level convergence topology on Eⁿ.