In this article, we study the n x n matrices over a finite distributive lattice L. By using join irreducible elements in L, we first give some concrete ways to decompose L into a subdirect product of some chains. Also, it is showed that by a subdirect embedding from semiring R to the direct product Pi(m)(i=1) R(i) of semirings R(1), R(2), ... , R(m), we can give a corresponding subdirect embedding from the matrix semiring M(n)(R) to semiring Pi(m)(i=1) M(n)(R(i)). Based on the above results, it is proved that a square matrix over a finite distributive lattice L can be decomposed into the sum of matrices over some special subchains of L. This generalizes and extends the corresponding results obtained by Fan [Z.T. Fan, The Theory and Applications of Fuzzy Matrices, Science Publication, Beijing, 2006 (in Chinese)] and by Zhao et al. [X.Z. Zhao, Y.B. Jun, and F. Ren, The semiring of matrices over a finite chain, Inform. Sci. 178 (2008), pp. 3443-3450]. As some applications, we present a method to calculate the indices and periods of the matrices over a finite distributive lattice, and characterize the idempotent and nilpotent matrices over a finite distributive lattice. Also, we discuss Green's relations on the multiplicative semigroup of semiring of matrices over a finite distributive lattice.

We construct cocompact lattices Gamma(')(0) < Gamma(0) in the group G =3D PGL(d)(F-q((t))) which are type-preserving and act transitively on the set of vertices of each type in the building Delta associated to G. These lattices are commensurable with the lattices of Cartwright-Steger Isr. J. Math. 103 (1998), 125-140. The stabiliser of each vertex in Gamma(')(0) is a Singer cycle and the stabiliser of each vertex in Gamma(0) is isomorphic to the normaliser of a Singer cycle in PGL(d)(q). We show that the intersections of Gamma(')(0) and Gamma(0) with PSLd(Fq((t))) are lattices in PSLd (F-q((t))), and identify the pairs (d, q) such that the entire lattice Gamma(')(0) or Gamma(0) is contained in PSLd(F-q((t))). Finally we discuss minimality of covolumes of cocompact lattices in SL3(F-q((t))). Our proofs combine the construction of Cartwright-Steger Isr. J. Math. 103 (1998), 125-140 with results about Singer cycles and their normalisers, and geometric arguments.

An important model of a conserved gene cluster is called the gene team model, in which a chromosome is defined to be a permutation of distinct genes and a gene team is defined to be a set of genes that appear in two or more species, with the distance between adjacent genes in the team for each chromosome always no more than a certain threshold delta. A gene team tree is a succinct way to represent all gene teams for every possible value of delta. The previous fastest algorithm for constructing a gene team tree of two chromosomes requires O(n lg n lglg n) time, which was given by Wang and Lin. Its bottleneck is a problem called the maximum-gap problem. In this paper, by presenting an improved algorithm for the maximum-gap problem, we reduce the upper bound of the gene team tree problem to O(n lg n alpha(n)). Since alpha grows extremely slowly, this result is almost as efficient as the current best upper bound, O(n lg n), for finding the gene teams of a fixed delta value. Our new algorithm is very efficient from both the theoretical and practical points of view. Wang and Lin's gene-team-tree algorithm can be extended to k chromosomes with complexity O(kn lg n lglg n). Similarly, our improved algorithm for the maximum-gap problem reduces this running time to O(kn lg n alpha(n)). In addition, it also provides new upper bounds for the gene team tree problem on general sequences, in which multiple copies of the same gene are allowed. =20

The article presents John Deely's philosophical critique of Martin Heidegger's Being and Time from the perspective of Catholic theology defined by the work of Thomas Aquinas, John Poinsot, and Jacques Maritain. Deely's analysis centers on the philosophical definition of esse intentionale. A biographical context for the critique stems from the life-long friendship of John Deely and the author. The moral force of Deely's critique is now confirmed by the recent publication of Heidegger's personal diaries, the Black Notebooks.

We give an algorithm that for an input n-vertex graph G and integer k > 0, in time 2(O(k))n, either outputs that the treewidth of G is larger than k, or gives a tree decomposition of G of width at most 5k + 4. This is the first algorithm providing a constant factor approximation for treewidth which runs in time single exponential in k and linear in n. Treewidth-based computations are subroutines of numerous algorithms. Our algorithm can be used to speed up many such algorithms to work in time which is single exponential in the treewidth and linear in the input size.

A straightforward and environment-friendly protocol for the synthesis of valuable chiral N-(tert-butylsulfinyl)imines has been developed. Different from traditional process with benzaldehydes as substrates, arylmethyl alcohols, benzylthiol, dibenzyl ether, thioether, and disulfide are used as alternative substrates to react with tert-butanesulfinamide in the presence of (KOBu)-Bu-t under air for the synthesis of chiral N-(tert-butylsulfinyl)imines. This is a transition metal-free, mild, cost-effective, and simple process.

Abstract We complete the work of Cogdell and Piatetski-Shapiro [3] to prove, for n ≥ 3 , a converse theorem for automorphic representations of GL n over a number field, with analytic data from twists by unramified representations of GL n − 1 .