Hochschild （Co）homology of Galois Coverings of Grassmann Algebras

Let ∧ be the Z2-Galois covering of the Grassmann algebra A over a field k of characteristic not equal to 2. In this paper, the dimensional formulae of Hochschild homology and cohomology groups of ∧ are calculated explicitly. And the cyclic homology of∧ can also be calculated when the underlying field is of characteristic zero. As a result, we prove that there is an isomorphism from i≥1 HH^i（∧） to i≥1 HH^i（∧）.     

Hardness of lazy packing and covering

We prove APX-hardness for the two problems maximum resource bin packing and lazy bin covering. Simple algorithms matching the best absolute bounds under the assumption P≠NP are also derived.     

Independent coverings and orthogonal colourings

In this paper, two open conjectures are disproved. One conjecture regards independent coverings of sparse partite graphs, whereas the other conjecture regards orthogonal colourings of tree graphs. A relation between independent coverings and orthogonal colourings is established. This relation is applied to find independent coverings of some sparse partite graphs. Additionally, a degree condition providing the existence of an independent covering in the case where the graph has a square number of vertices is found.     

Congruences for generalized Frobenius partitions with 4 colors

We present some congruences involving the functions c?₄(n) and which denote, respectively, the number of generalized Frobenius partitions of n with 4 colors and 4-order generalized Frobenius partitions of n with 4 colors.     

求解点覆盖问题的拟物转换及算法

本文首先基于CNF－SAT问题已有的拟物型的求解思想,将点覆盖问题转换为连续情形的目标优化问题,进而提出一种新的竞争式下降梯度方法,从而获得了求解点覆盖问题的高效实用的快速算法．     

Chain Dominated Orders

We study finite partial orders which have a chain such that every element of the order either belongs to this chain or has all its covers in this chain. We show that such orders are exactly the orders being both interval orders and truncated lattices. We prove that their jump number is polynomially tractable and that their dimension is unbounded. We also show that every order admits a visibility model having such an order as host.     

MIP reformulations of the probabilistic set covering problem

In this paper, we address the following probabilistic version (PSC) of the set covering problem: where A is a 0-1 matrix, is a random 0-1 vector and is the threshold probability level. We introduce the concepts of p-inefficiency and polarity cuts. While the former is aimed at deriving an equivalent MIP reformulation of (PSC), the latter is used as a strengthening device to obtain a stronger formulation. Simplifications of the MIP model which result when one of the following conditions hold are briefly discussed: A is a balanced matrix, A has the circular ones property, the components of are pairwise independent, the distribution function of is a stationary distribution or has the disjunctive shattering property. We corroborate our theoretical findings by an extensive computational experiment on a test-bed consisting of almost 10,000 probabilistic instances. This test-bed was created using deterministic instances from the literature and consists of probabilistic variants of the set covering model and capacitated versions of facility location, warehouse location and k-median models. Our computational results show that our procedure is orders of magnitude faster than any of the existing approaches to solve (PSC), and in many cases can reduce hours of computing time to a fraction of a second. Anureet Saxena’s research was supported by the National Science Foundation through grant #DMI-0352885 and by the Office of Naval Research through contract N00014-03-1-0133. Vineet Goyal’s research was supported in part by NSF grant CCF-0430751 and ITR grant CCR-0122581.     

Finitely locally finite coverings of Banach spaces

A well-known result due to H. Corson states that, for any covering τ by closed bounded convex subsets of any Banach space X containing an infinite-dimensional reflexive subspace, there exists a compact subset C of X that meets infinitely many members of τ. We strengthen this result proving that, even under the weaker assumption that X contains an infinite-dimensional separable dual space, an (algebraically) finite-dimensional compact set C with that property can always be found.     

Absolutely convex,uniformly starlike and uniformly convex harmonic mappings

In this paper, we prove necessary and sufficient conditions for a sense-preserving harmonic function to be absolutely convex in the open unit disc. We also estimate the coefficient bound and obtain growth, covering and area theorems for absolutely convex harmonic mappings. A natural generalization of the classical Bernardi-type operator for harmonic functions is considered and its connection between certain classes of uniformly starlike harmonic functions and uniformly convex harmonic functions is also investigated. At the end, as applications, we present a number of results connected with hypergeometric and polylogarithm functions.     

A Covering Space with No Compact Core

This note describes an example of a compact aspherical 2-complex X with a covering space , such that is finitely presented but has no compact core.I thank John Stallings for being my 'Senior Scientist' in 1996–1997, and for many superimaginative e-mails that led to this relationship.