Multiple harmonic series related to multiple Euler numbers

In this paper, we define the multiple Euler numbers and consider some multiple harmonic series of Mordell-Tornheim's type, which is a partial sum of the Mordell-Tornheim zeta series defined by Matsumoto. Indeed, we prove a certain reducibility of these series as well as the multiple zeta values.     

Random Walks And The Colored Jones Function

It can be conjectured that the colored Jones function of a knot can be computed in terms of counting paths on the graph of a planar projection of a knot. On the combinatorial level, the colored Jones function can be replaced by its weight system. We give two curious formulas for the weight system of a colored Jones function: one in terms of the permanent of a matrix associated to a chord diagram, and another in terms of counting paths of intersecting chords. Electronic supplementary material to this article is available at and is accessible to authorized users. * S. G. was partially supported by an NSF and by an Israel-US BSF grant. † M. L. was partly supported by GAUK 158 grant and by the Project LN00A056 of the Czech Ministry of Education.     

Combinatorial Lie bialgebras of curves on surfaces

Goldman (Invent. Math. 85(2) (1986) 263) and Turaev (Ann. Sci. Ecole Norm. Sup. (4) 24 (6) (1991) 635) found a Lie bialgebra structure on the vector space generated by non-trivial free homotopy classes of curves on a surface. When the surface has non-empty boundary, this vector space has a basis of cyclic reduced words in the generators of the fundamental group and their inverses. We give a combinatorial algorithm to compute this Lie bialgebra on this vector space of cyclic words. Using this presentation, we prove a variant of Goldman's result relating the bracket to disjointness of curve representatives when one of the classes is simple. We exhibit some examples we found by programming the algorithm which answer negatively Turaev's question about the characterization of simple curves in terms of the cobracket. Further computations suggest an alternative characterization of simple curves in terms of the bracket of a curve and its inverse. Turaev's question is still open in genus zero.     

Laplacians and the Cheeger Inequality for Directed Graphs

We consider Laplacians for directed graphs and examine their eigenvalues. We introduce a notion of a circulation in a directed graph and its connection with the Rayleigh quotient. We then define a Cheeger constant and establish the Cheeger inequality for directed graphs. These relations can be used to deal with various problems that often arise in the study of non-reversible Markov chains including bounding the rate of convergence and deriving comparison theorems.Received September 8, 2004     

π1-classification of real arrangements with up to eight lines

One of the open questions in the geometry of line arrangements is to what extent does the incidence lattice of an arrangement determine its fundamental group. Line arrangements of up to 6 lines were recently classified by K.M. Fan (Michigan Math. J. 44(2) (1997) 283), and it turns out that the incidence lattice of such arrangements determines the projective fundamental group. We use actions on the set of wiring diagrams, introduced in (Garber et al. (J. Knot Theory Ramf.), to classify real arrangements of up to 8 lines. In particular, we show that the incidence lattice of such arrangements determines both the affine and the projective fundamental groups.     

Asymptotic properties of computationally efficient alternative estimators for a class of multivariate normal models

Parameters of Gaussian multivariate models are often estimated using the maximum likelihood approach. In spite of its merits, this methodology is not practical when the sample size is very large, as, for example, in the case of massive georeferenced data sets. In this paper, we study the asymptotic properties of the estimators that minimize three alternatives to the likelihood function, designed to increase the computational efficiency. This is achieved by applying the information sandwich technique to expansions of the pseudo-likelihood functions as quadratic forms of independent normal random variables. Theoretical calculations are given for a first-order autoregressive time series and then extended to a two-dimensional autoregressive process on a lattice. We compare the efficiency of the three estimators to that of the maximum likelihood estimator as well as among themselves, using numerical calculations of the theoretical results and simulations.     

Ihara zeta functions of digraphs

After defining and exploring some of the properties of Ihara zeta functions of digraphs, we improve upon Kotani and Sunada’s bounds on the poles of Ihara zeta functions of undirected graphs by considering digraphs whose adjacency matrices are directed edge matrices.     

Arithmetic expressions of Selberg's zeta functions for congruence subgroups

In [P. Sarnak, Class numbers of indefinite binary quadratic forms, J. Number Theory 15 (1982) 229-247], it was proved that the Selberg zeta function for SL₂(Z) is expressed in terms of the fundamental units and the class numbers of the primitive indefinite binary quadratic forms. The aim of this paper is to obtain similar arithmetic expressions of the logarithmic derivatives of the Selberg zeta functions for congruence subgroups of SL₂(Z). As applications, we study the Brun-Titchmarsh type prime geodesic theorem and the asymptotic formula of the sum of the class number.     

Note on a conjecture of toft

A conjecture of Toft [17] asserts that any 4-critical graph (or equivalently, every 4-chromatic graph) contains a fully odd subdivision ofK ₄. We show that if a graphG has a degree three nodev such thatG-v is 3-colourable, then eitherG is 3-colourable or it contains a fully oddK ₄. This resolves Toft's conjecture in the special case where a 4-critical graph has a degree three node, which is in turn used to prove the conjecture for line-graphs. The proof is constructive and yields a polynomial algorithm which given a 3-degenerate graph either finds a 3-colouring or exhibits a subgraph that is a fully odd subdivision ofK ₄. (A graph is 3-degenerate if every subgraph has some node of degree at most three.)