Let Σg be a compact, connected, orientable surface of genus g ≥ 2. We ask for a parametrisation of the discrete, faithful, totally loxodromic representations in the deformation space Hom(π1(Σg), SU(3, 1))/SU(3, 1). We show that such a representation, under some hypothesis, can be determined by 30g − 30 real parameters.

We study algebraic models of the Segal category of Riemann surfaces [11]. These are 2-categories based on the 1+1 dimensional cobordism category enriched by diffeomorphisms of surfaces (or isotopy classes of such). In particular, we show that one of these models is a symmetric strict monoidal strict 2-category. The main technical tools are quotient constructions on 2-categories.

We solve the problem of giving sharp asymptotic bounds on the Hausdorff dimensions of certain sets of badly approximable matrices, thus improving results of Broderick and Kleinbock (preprint 2013) as well as Weil (preprint 2013), and generalising to higher dimensions those of Kurzweil ('51) and Hensley ('92). In addition we use our technique to compute the Hausdorff f-measure of the set of matrices which are not ψ-approximable, given a dimension function f and a function ψ : (0, ∞) → (0, ∞). This complements earlier work by Dickinson and Velani ('97) who found the Hausdorff f-measure of the set of matrices which are ψ-approximable.

Quantum stochastic integrals are constructed using the non-commutative Lp-space theory of Haagerup. The existence and uniqueness of the solution to quantum stochastic differential equations driven by quasi-Wiener noises, or noises satisfying generalized standing hypotheses, is established as is the Markov behaviour of the solution. Various examples of the theory are discussed, and quantum Ornstein-Uhlenbeck processes are obtained as explicit solutions.

The Hopf bifurcation theorem describes the creation of a limit cycle from an isolated singular point of a system of first-order differential equations depending on a parameter. This paper describes a method for determining explicitly a range of values of the parameter throughout which the Hopf configuration continues to exist; only the three-dimensional case is described in this paper, but the method can be generalized.

Introduction. Let X be a submanifold of Y, in either the topological, smooth, or piecewise linear ( = PL) categories. A normal cell bundle on X in Y is a bundle ξ = (p, E, X) in the category whose fibre is a closed cell, and such that E is a neighbourhood of X in Y and p: E → X is a retraction. The triple (Y, X, ξ) is a tubular neighbourhood, or briefly, a tube. For convenience we may refer to a tube by its cell bundle.

In this paper, we prove an asymptotic formula with a power saving error term for traces of weight zero weakly holomorphic modular forms of level N along Galois orbits of Heegner points on the modular curve X0(N). We use this result to study the distribution of partition ranks modulo 2. In particular, we give an asymptotic formula with a power saving error term for the number of partitions of a positive integer n with even (respectively, odd) rank. We use these results to deduce a strong quantitative form of equidistribution of partition ranks modulo 2.

The object of this paper is to prove that the following classes of graphs are edge-reconstructible: (i) 4-connected planar graphs with minimum valency 5; (ii) maximal planar graphs with minimum valency at least 4.

Consider a queueing system GI/G/∞ in which (i) the inter-arrival times are distributed with distribution function A(t) (A(O +) = 0) (ii) the service times have distribution function B(t) such that the expected value of the service time is β(>∞).

By a multi-valued map from a space X to a space Y we mean a map which assigns to each point x in X a non-empty subset F(x) of Y. When X = Y, a point x in X is a fized point for F if x is in F(x).

Alexander's theorem (5) states that a topological space is compact if there is a sub-base, , for its closed sets such that every subclass of with the finite intersection property has a non-empty intersection. An analysis and extension of this is given here which has applications, inter alia, to problems concerning real-compactness (2).

We consider an analytic function-germ $f{:}\,(\mathbb{R}^n,0) \rightarrow (\mathbb{R},0)$ with an isolated critical point at $0$. For a sufficiently small ball $B_\varepsilon$ of radius $\varepsilon$ and a sufficiently small regular value $\delta$ of $f$, we give degree formulas for the following Euler–Poincaré characteristics: $$\chi( f^{-1}(\delta) \cap B_\varepsilon \cap \{(-1)^{\epsilon_1} x_1\ge 0,\ldots, (-1)^{\epsilon_k} x_k \ge 0\}),$$ where $k \in \{1,\ldots,n\}$ and $\epsilon_i \in \{0,1\}$. This leads to degree formulas for $$ \chi (\{f * 0\} \cap S_\varepsilon \cap \{(-1)^{\epsilon_1} x_1\ge 0,\ldots, (-1)^{\epsilon_k} x_k \ge 0\}),$$ where $* \in \{\le,=,\ge\}$. Combining this with the Eisenbud–Levine–Khimshiashvili's formula, we obtain signature formulas for $$\chi( W \cap\{(-1)^{\epsilon_1} x_1\ge 0,\ldots, (-1)^{\epsilon_k} x_k \ge 0\}),$$ where $W$ is either the link of a weighted-homogeneous singularity or any compact algebraic set of $\mathbb{R}^n$.

A theory is presented for small deformations of a plate which is initially stressed, the state of initial stress not necessarily being obtained by elastic deformation from some other unstressed state. The method used is an asymptotic integration of the full three-dimensional equations. The usual Kirchhoff boundary conditions for stretching and bending are deduced from the analysis.

We prove that complex Bernoulli convolutions are absolutely continuous in the supercritical parameter region, outside of an exceptional set of parameters of zero Hausdorff dimension. Similar results are also obtained in the biased case, and for other parametrised families of self-similar sets and measures in the complex plane, extending earlier results.

The problem of constructing an n-dimensional metric differential geometry based on the idea of a two-dimensional area has given rise to several publications, notably by A. Kawaguchi and S. Hokari (1), E. T. Davies (2), and R. Debever (3). In this geometry the area of a two-dimensional plane element is defined by a fundamental function L(xi, uhk), where the xi are point coordinates and the uhk are the coordinates of the simple bivector representing the plane element. L is supposed to be a positive homogeneous function of the first degree with respect to the variables uij, and to possess continuous partial derivatives up to and including those of the fourth order. With these assumptions the problem of the construction of the metric differential geometry splits into two problems; the first of these is the problem of constructing a metric tensor gij(xr, uhk), and the second is the problem of constructing an affine connexion. We deal with the first problem only in this paper.

Given an analytic path of flat connections on a principal U(N)-bundle over a closed odd-dimensional manifold M, how can we calculate the spectral flow of the corresponding path Dt of signature operators on an associated vector bundle E → M? In this paper we give an algebraic-topological answer to this question in terms of cohomology, cup products and higher Massey products.

A well-known result of classical function theory, Jensen's formula, expresses the integral around a circle of the log modulus of a meromorphic function in terms of the log modulus of the zeros and poles of that function lying inside the circle. Explicitly, if F is a meromorphic function on the unit disc {ω ε ℂ: |ω| < 1} and F(0) = 1, then, for 0 < r < 1,

where ordωF is the order of F at ω. The purpose of this note is to observe that a formula analogous to (1) holds when F is replaced by a modular function for SL2(ℤ) and the integral by a suitable double integral over a fundamental domain. We shall derive this modular variant of Jensen's formula from the usual version by applying the Rankin-Selberg method and the first Kronecker limit formula. The argument admits some extension to Fuchsian groups other than SL2(ℤ), and to modular forms of weight other than zero; this point will be discussed later.

We interpret the degrees which arise in Tevelev’s study of scattering amplitudes in terms of moduli spaces of Hurwitz covers. Via excess intersection theory, the boundary geometry of the Hurwitz moduli space yields a simple recursion for the Tevelev degrees (together with their natural two parameter generalisation). We find exact solutions which specialise to Tevelev’s formula in his cases and connect to the projective geometry of lines and Castelnuovo’s classical count of $g^1_d$ ’s in other cases. For almost all values, the calculation of the two parameter generalisation of the Tevelev degree is new. A related count of refined Dyck paths is solved along the way.

§ 1. Most of the methods which have been devised to give an equation of state of more generality than that of Van der Waals differ from the original method used by him in that they refer only to the uniform conditions in the interior of the gas while his had special reference to the conditions at the boundary. In fact, one of the corrections to the perfect gas law introduced by him is due entirely to the existence of a boundary field of force. The question arises as to what physical interpretation is to be given to the more general equation. The methods previously employed leave the interpretation obscure. In this note two new methods of obtaining the equation of state are given, one applicable to the interior of a gas, the other to the boundary. These methods seem to have certain advantages over those previously given in that they lend themselves to a very simple physical interpretation. While the pressure at the boundary of a gas is the same as that in the interior, the word pressure has a different meaning in the two cases. At the boundary, it is due entirely to the motion of the molecules, whereas in the interior part only is due to the motion of the molecules; this part is the same whatever the nature of the molecules and is in fact given by the perfect gas law. The remaining part is due to the stress set up by the existence of intermolecular fields and, although at any given point it is a fluctuating function of the time, it has everywhere within the gas the same statistical mean value. In this paper, it is referred to as the statical pressure to distinguish it from the more usually understood dynamical or momentum pressure.