The crossing number of posets

The purpose of this paper is to investigate some properties of the crossing number (P) of a posetP. We first study the crossing numbers of the product and the lexicographical sum of posets. The results are similar to the dimensions of these posets. Then we consider the problem of what happens to the crossing number when a point is taken away from a poset. We show that ifP is a poset such that P and (P–)1, then 1/2 (P)(P–)(P). We don't know yet how to improve the lower bound. We also determine the crossing numbers of some subposets of the Boolean latticeB _n which consist of some specified ranks. Finally we show that _n is crossing critical where _n is the subposet ofB _n which is restricted to rank 1, rankn–1 and middle rank(s). Some open problems are raised at the end of this paper.     

A Taylor–Wiles System for Quaternionic Hecke Algebras

Let &ell >3 be a prime. Fix a regular character of F&^{2 ×} of order &–1, and an integer M prime to &. Let fS ₂(₀(M&²)) be a newform which is supercuspidal of type at &. For an indefinite quaternion algebra over Q of discriminant dividing the level of f, there is a local quaternionic Hecke algebra T of type associated to f. The algebra T acts on a quaternionic cohomological module M. We construct a Taylor–Wiles system for M, and prove that T is the universal object for a deformation problem (of type at & and semi-stable outside) of the Galois representation ¯ _f over F¯_& associated to f; that T is complete intersection and that the module M is free of rank 2 over T. We deduce a relation between the quaternionic congruence ideal of type for f and the classical one.     

A relation between the zeros of an <Emphasis Type="Italic">L</Emphasis>-function belonging to the Selberg class and the zeros of an associated <Emphasis Type="Italic">L</Emphasis>-function twisted by a Dirichlet character

Let F (s) be a function belonging to the Selberg class. For a primitive Dirichlet character , we can define the -twist F(s) of F (s). If F(s) also belongs to the Selberg class and satisfies some other conditions then there is a relation between the zeros of F (s) and the zeros F(s). Further we give an operator theoretic interpretation of this relation according to A. Connes study.Received: 5 January 2004     

An Analogue of Hajós' Theorem for the Circular Chromatic Number (II)

This paper designs a set of graph operations, and proves that for 2k/d<3, starting from K_k/_d, by repeatedly applying these operations, one can construct all graphs G with _c(G)k/d. Together with the result proved in [20], where a set of graph operations were designed to construct graphs G with _c(G)k/d for k/d3, we have a complete analogue of Hajós' Theorem for the circular chromatic number. This research was partially supported by the National Science Council under grant NSC 89-2115-M-110-003     

Edge Coloring of Embedded Graphs with Large Girth

Let G be a simple graph embedded in the surface of Euler characteristic ()0. Denote _e(G), and g the edge chromatic number, the maximum degree and the girth of the graph G, respectively. The paper shows that _e(G)= if 5 and g4, or 4 and g5, or 3 and g9. In addition, if ()>0, then _e(G)= if 3 and g8. Acknowledgments.The authors would like to thank Dr. C.Q. Zhang for carefully reading several versions of this paper during its preparation and for suggesting several stylistic changes that have improved the overall presentation.     

Simultaneous Nonvanishing of Twists of Automorphic L-Functions

Given three distinct primitive complex characters ₁,₂,₃ satisfying some technical conditions, we prove that the triple product of twisted L-functions L(f·₁,1/2) L(f·₂,1/2) L(f·₃,1/2) does not vanish for a positive proportion of weight 2 primitive forms for ₀(q), when q goes to infinity through the set of prime numbers. This result, together with some variants, implies the existence of quotients of J ₀(q) of large dimension satisfying the Birch–Swinnerton-Dyer conjecture over cyclic number fields of degree less than 5.P.M. is partially supported by NSF Grant DMS-97-2992 and by the Ellentuck fund (by grants to the Institute for Advanced Study) and by the Institut Universitaire de France.     

The central limit theorem for Chébli-Trimèche hypergroups

Let (S _nn>-1) be a random walk on a hypergroup ( ₊ , *), i.e., a Markov chain with transition kernelN(x, A) = _x * (A), where is a fixed probability measure on ₊ such that the second moment exists. Then depending on the growth of the hypergroup two situations can occur: when ( ₊ , *) is of exponential growth then it is shown thatS _n is asymptotically normal. In the case of polynomial growth {more precisely, if the densityA of the Haar measure of ( ₊ , *) satisfies lim[A()/A()]=}, the normalized variablesS _n/[n Var()/(+1)]^1/2 converge to a Rayleigh distribution with parameter .     

Compact operators whose singular numbers have powerlike asymptotics

For a compact operator in a Hilbert space, let s_n(A), n =1, 2,... be the singular numbers and let N(s; A) =card{n N:s_n(A)>, s>0. For 0

a _p and not on the individual elementAa, (H. Weyl's lemma); this allows us to write _p (a), _pp (a), a_p. One obtains certain results regarding the functionals _p, _p (and about the analogous functionals for the positive and negative eigenvalues in the casea=a ^*=A ^*:A a. In particular: I. Ifa ₁ a _2p, then. II.Let a ₁,a _{2 pP} ,.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matetmaticheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 21–30, 1983.     

The Topology of the Coloring Complex

In a recent paper, E. Steingrímsson associated to each simple graph G a simplicial complex _G, referred to as the coloring complex of G. Certain nonfaces of _G correspond in a natural manner to proper colorings of G. Indeed, the h-vector is an affine transformation of the chromatic polynomial _G of G, and the reduced Euler characteristic is, up to sign, equal to |_G(–1)|–1. We show that _G is constructible and hence Cohen-Macaulay. Moreover, we introduce two subcomplexes of the coloring complex, referred to as polar coloring complexes. The h-vectors of these complexes are again affine transformations of _G, and their Euler characteristics coincide with _G(0) and –_G(1), respectively. We show for a large class of graphs—including all connected graphs—that polar coloring complexes are constructible. Finally, the coloring complex and its polar subcomplexes being Cohen-Macaulay allows for topological interpretations of certain positivity results about the chromatic polynomial due to N. Linial and I. M. Gessel.Research financed by ECs IHRP Programme, within the Research Training Network Algebraic Combinatorics in Europe, grant HPRN-CT-2001-00272.     

The non-cancellation group of a direct power of a (finite cyclic)-by-cyclic group

If M is a finitely generated group having a finite commutator subgroup, then the set (M) of all isomorphism classes of groups G such that G×M× is a finite set and coincides with the Mislin genus (M) of M if M is nilpotent. For such groups M, there is a group structure on (M) defined in terms of the indices of embeddings of G into M, for groups G representing elements of (M). Such embeddings do exist and their indices are necessarily finite. If M is nilpotent, then this group structure on (M) coincides with the Hilton-Mislin group structure on the genus of M. In this paper we calculate the group (H^k) where H^k is the direct product of k copies of a group the form H= a,b | aⁿ=1, bab^-1=a^u, for any relatively prime pair of natural numbers n,u. In particular we find that for each such group H we have an isomorphism (H²)(H^k) whenever k>2.The author wishes to acknowledge financial support from the National Research Foundation of South Africa.Mathematics Subject Classification (2000): 20E34, 20F28Revised version: 10 December 2003