Euler characteristics of moduli spaces of curves

Let Mgn be the moduli space of n-pointed Riemann surfaces of genus g. Denote by Mgn the Deligne-Mumford compactification of Mgn. In the present paper, we calculate the orbifold and the ordinary Euler characteristic of Mgn for any g and n such that n > 2-2g

A product formula and combinatorial field theory

We treat the problem of normally ordering expressions involving the standard boson operators a, ay where [a; ay] = 1. We show that a simple product formula for formal power series | essentially an extension of the Taylor expansion | leads to a double exponential formula which enables a powerful graphical description of the generating functions of the combinatorial sequences associated with such functions | in essence, a combinatorial eld theory. We apply these techniques to some examples related to specic physical Hamiltonians

Conic approach to quantum graph parameters using linear optimization over the completely positive semidefinite cone

We investigate the completely positive semidefinite cone $\mathcal{CS}_+^n$, a new matrix cone consisting of all $n\times n$ matrices that admit a Gram representation by positive semidefinite matrices (of any size). In particular we study relationships between this cone and the completely positive and doubly nonnegative cones, and between its dual cone and trace positive non-commutative polynomials. We use this new cone to model quantum analogues of the classical independence and chromatic graph parameters $\alpha(G)$ and $\chi(G)$, which are roughly obtained by allowing variables to be positive semidefinite matrices instead of $0/1$ scalars in the programs defining the classical parameters. We can formulate these quantum parameters as conic linear programs over the cone $\mathcal{CS}_+^n$. Using this conic approach we can recover the bounds in terms of the theta number and define further approximations by exploiting the link to trace positive polynomials.Comment: Fixed some typo

Polynomial decay of correlations in the generalized baker's transformation

We introduce a family of area preserving generalized baker's transformations acting on the unit square and having sharp polynomial rates of mixing for Holder data. The construction is geometric, relying on the graph of a single variable "cut function". Each baker's map B is non-uniformly hyperbolic and while the exact mixing rate depends on B, all polynomial rates can be attained. The analysis of mixing rates depends on building a suitable Young tower for an expanding factor. The mechanisms leading to a slow rate of correlation decay are especially transparent in our examples due to the simple geometry in the construction. For this reason we propose this class of maps as an excellent testing ground for new techniques for the analysis of decay of correlations in non-uniformly hyperbolic systems. Finally, some of our examples can be seen to be extensions of certain 1-D non-uniformly expanding maps that have appeared in the literature over the last twenty years thereby providing a unified treatment of these interesting and well-studied examples.Comment: 24 pages, 2 figure

Spectra of random networks in the weak clustering regime

The asymptotic behaviour of dynamical processes in networks can be expressed as a function of spectral properties of the corresponding adjacency and Laplacian matrices. Although many theoretical results are known for the spectra of traditional configuration models, networks generated through these models fail to describe many topological features of real-world networks, in particular non-null values of the clustering coefficient. Here we study effects of cycles of order three (triangles) in network spectra. By using recent advances in random matrix theory, we determine the spectral distribution of the network adjacency matrix as a function of the average number of triangles attached to each node for networks without modular structure and degree-degree correlations. Implications to network dynamics are discussed. Our findings can shed light in the study of how particular kinds of subgraphs influence network dynamics

Some useful combinatorial formulae for bosonic operators

We give a general expression for the normally ordered form of a function F(w(a,a*)) where w is a function of boson annihilation and creation operators satisfying [a,a*]=1. The expectation value of this expression in a coherent state becomes an exact generating function of Feynman-type graphs associated with the zero-dimensional Quantum Field Theory defined by F(w). This enables one to enumerate explicitly the graphs of given order in the realm of combinatorially defined sequences. We give several examples of the use of this technique, including the applications to Kerr-type and superfluidity-type hamiltonians.Comment: 8 pages, 3 figures, 17 reference