Minimal zonotopes containing the crosspolytope

Motivated by the problem to improve Minkowski’s lower bound on the successive minima for the class of zonotopes we determine the minimal volume of a zonotope containing the standard crosspolytope. It turns out that this volume can be expressed via the maximal determinant of a ±1-matrix, and that in each dimension the set of minimal zonotopes contains a parallelepiped. Based on that link to ±1- matrices, we characterize all zonotopes attaining the minimal volume in dimension 3 and present related results in higher dimensions.     

Bases of minimal vectors in lattices,I

We prove that a Euclidean lattice of dimension n ≤ 8 which is generated by its minimal vectors possesses a basis of minimal vectors. Received: 7 December 2006     

On minimal actions of finite simple groups on homology spheres and Euclidean spaces

We consider the following problem: for which classes of finite groups, and in particular finite simple groups, does the minimal dimension of a faithful, smooth action on a homology sphere coincide with the minimal dimension of a faithful, linear action on a sphere? We prove that the two minimal dimensions coincide for the linear fractional groups PSL(2, p) as well as for various classes of alternating and symmetric groups. We prove analogous results also for actions on Euclidean spaces.     

Minimal discrepancies of toric singularities

The main purpose of this paper is to prove that minimal discrepancies ofn-dimensional toric singularities can accumulate only from above and only to minimal discrepancies of toric singularities of dimension less thann. I also prove that some lower-dimensional minimal discrepancies do appear as such limit.     

Spaces of Wald-Berestovskii curvature bounded below

We consider inner metric spaces of curvature bounded below in the sense of Wald, without assuming local compactness or existence of minimal curves. We first extend the Hopf-Rinow theorem by proving existence, uniqueness, and “almost extendability” of minimal curves from any point to a denseG _δ subset. An immediate consequence is that Alexandrov’s comparisons are meaningful in this setting. We then prove Toponogov’s theorem in this generality, and a rigidity theorem which characterizes spheres. Finally, we use our characterization to show the existence of spheres in the space of directions at points in a denseG _δ set. This allows us to define a notion of “local dimension” of the space using the dimension of such spheres. If the local dimension is finite, the space is an Alexandrov space.     

Clan acts and codimension

Let (T, X) be a continuum act, let cd X=n and suppose A is a T-ideal (i.e., a T-invariant subspace of X), such that Hⁿ(A)≠0. We prove that A is a minimal T-ideal iff A=Gx for some x∈X and maximal group G in the minimal ideal of T. Moreover, if these conditions are satisfied, then A is the only minimal T-ideal and also is the unique floor for every nonzero element of Hⁿ(X). We need and also prove here an improved version of the Tube Theorem [3], and this corollary: if (G, X) is an intransitive transformation group with G compact, X locally compact and finite dimensional, and X/G connected, then dimension Gx<dimension X for all x∈X. NSF GP 9659. NSF GP 28655.     

Representation Dimension and Quasi-hereditary Algebras

We study Auslander's representation dimension of Artin algebras, which is by definition the minimal projective dimension of coherent functors on modules which are both generators and cogenerators. We show the following statements: (1) if an Artin algebra A is stably hereditary, then the representation dimension of A is at most 3. (2) If two Artin algebras are stably equivalent of Morita type, then they have the same representation dimension. Particularly, if two self-injective algebras are derived equivalent, then they have the same representation dimension. (3) Any incidence algebra of a finite partially ordered set over a field has finite representation dimension. Moreover, we use results on quasi-hereditary algebras to show that (4) the Auslander algebra of a Nakayama algebra has finite representation dimension.     

Dimension of a minimal nilpotent orbit

We show that the dimension of the minimal nilpotent coadjoint orbit for a complex simple Lie algebra is equal to twice the dual Coxeter number minus two.

     

Spherical containment and the Minkowski dimension of partial orders

The recent work on circle orders generalizes to higher dimensional spheres. As spherical containment is equivalent to causal precedence in Minkowski space, we define the Minkowski dimension of a poset to be the dimension of the minimal Minkowski space into which the poset can be embedded; this isd if the poset can be represented by containment with spheresS ^d–2 and of no lower dimension. Comparing this dimension with the standard dimension of partial orders we prove that they are identical in dimension two but not in higher dimensions, while their irreducible configurations are the same in dimensions two and three. Moreover, we show that there are irreducible configurations for arbitrarily large Minkowski dimension, thus providing a lower bound for the Minkowski dimension of partial orders.     

Dirichlet regularity in arbitrary o‐minimal structures on the field ℝ up to dimension 4

In this article we show that the set of Dirichlet regular boundary points of a bounded domain of dimension up to 4, definable in an arbitrary o‐minimal structure on the field ?, is definable in the same structure. Moreover we give estimates for the dimension of the set of non‐regular boundary points, depending on whether the structure is polynomially bounded or not. This paper extends the results from the author's Ph.D. thesis [6, 7] where the problem was solved for polynomially bounded o‐minimal structures expanding the real field. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On small matrix subalgebras with a trivial centralizer

Given an integer n?≥?3, we investigate the minimal dimension of a subalgebra of M _n (𝕂) with a trivial centralizer. It is shown that this dimension is 5 when n is even and 4 when it is odd. In the latter case, we also determine all 4-dimensional subalgebras with a trivial centralizer.     

Bases of minimal vectors in lattices,II

We prove that a Euclidean lattice of dimension n = 5 (resp. 6; resp. 7) having at least 6 (resp. 10; resp. 18) pairs of minimal vectors has a basis of minimal vectors. Received: 7 December 2006     

The Krull–Gabriel Dimension of a Serial Ring

Abstract

We prove that every serial ring R has the isolation property: every isolated point in any theory of modules over R is isolated by a minimal pair. Using this we calculate the Krull–Gabriel dimension of the module category over serial rings. For instance, we show that this dimension cannot be equal to 1.     

Enlarging totally geodesic submanifolds of symmetric spaces to minimal submanifolds of one dimension higher

We show that a totally geodesic submanifold of a symmetric space satisfying certain conditions admits an extension to a minimal submanifold of dimension one higher, and we apply this result to construct new examples of complete embedded minimal submanifolds in simply connected noncompact globally symmetric spaces.

     

Complementation of relations in pro-p-groups of cohomology dimension two

It is proved that if, from the minimal corepresentation of a pro-p-group G. of cohomology dimension two with a free commutant, a part of the relations is discarded, then one obtains a minimal corepresentation of a pro-p-group, the cohomology dimension of which is equal to two and the commutant of which is free. For such pro-p-groups, (G)相似文献   

Topological spectrum of locally compact Cantor minimal systems

We show that there exists a locally compact Cantor minimal system whose topological spectrum has a given Hausdorff dimension.

     

Nielsen equalizer theory

We extend the Nielsen theory of coincidence sets to equalizer sets, the points where a given set of (more than 2) mappings agree. On manifolds, this theory is interesting only for maps between spaces of different dimension, and our results hold for sets of k maps on compact manifolds from dimension (k−1)n to dimension n. We define the Nielsen equalizer number, which is a lower bound for the minimal number of equalizer points when the maps are changed by homotopies, and is in fact equal to this minimal number when the domain manifold is not a surface.As an application we give some results in Nielsen coincidence theory with positive codimension. This includes a complete computation of the geometric Nielsen number for maps between tori.     

The fiber dimension of a graph

Graphs on integer points of polytopes whose edges come from a set of allowed differences are studied. It is shown that any simple graph can be embedded in that way. The minimal dimension of such a representation is the fiber dimension of the given graph. The fiber dimension is determined for various classes of graphs and an upper bound in terms of the chromatic number is proven.     

On existence of log minimal models and weak Zariski decompositions

We first introduce a weak type of Zariski decomposition in higher dimensions: an -Cartier divisor has a weak Zariski decomposition if birationally and in a numerical sense it can be written as the sum of a nef and an effective -Cartier divisor. We then prove that there is a very basic relation between Zariski decompositions and log minimal models which has long been expected: we prove that assuming the log minimal model program in dimension d − 1, a lc pair (X/Z, B) of dimension d has a log minimal model (in our sense) if and only if K _X  + B has a weak Zariski decomposition/Z.     

On minimal exposed faces

In this paper we consider the problem of the non-empty intersection of exposed faces in a Banach space. We find a sufficient condition to assure that the non-empty intersection of exposed faces is an exposed face. This condition involves the concept of inner point. Finally, we also prove that every minimal face of the unit ball must be an extreme point and show that this is not the case at all for minimal exposed faces since we prove that every Banach space with dimension greater than or equal to 2 can be equivalently renormed to have a non-singleton, minimal exposed face.