On the Fourier transform of the characteristic functions of domains with C1C^1 -smooth boundary

We consider domains $D\subseteq\mathbb R^n$ with $C^1$ -smooth boundary and study the following question: when the Fourier transform $\hat{1_D}$ of the characteristic function $1_D$ belongs to $L^p(\mathbb R^n)$?Comment: added two references; added footnotes on pages 6 and 1

Asymptotic Exit Location Distributions in the Stochastic Exit Problem

Consider a two-dimensional continuous-time dynamical system, with an attracting fixed point $S$. If the deterministic dynamics are perturbed by white noise (random perturbations) of strength $\epsilon$, the system state will eventually leave the domain of attraction $\Omega$ of $S$. We analyse the case when, as $\epsilon\to0$, the exit location on the boundary $\partial\Omega$ is increasingly concentrated near a saddle point $H$ of the deterministic dynamics. We show that the asymptotic form of the exit location distribution on $\partial\Omega$ is generically non-Gaussian and asymmetric, and classify the possible limiting distributions. A key role is played by a parameter $\mu$, equal to the ratio $|\lambda_s(H)|/\lambda_u(H)$ of the stable and unstable eigenvalues of the linearized deterministic flow at $H$. If $\mu<1$ then the exit location distribution is generically asymptotic as $\epsilon\to0$ to a Weibull distribution with shape parameter $2/\mu$, on the $O(\epsilon^{\mu/2})$ length scale near $H$. If $\mu>1$ it is generically asymptotic to a distribution on the $O(\epsilon^{1/2})$ length scale, whose moments we compute. The asymmetry of the asymptotic exit location distribution is attributable to the generic presence of a `classically forbidden' region: a wedge-shaped subset of $\Omega$ with $H$ as vertex, which is reached from $S$, in the $\epsilon\to0$ limit, only via `bent' (non-smooth) fluctuational paths that first pass through the vicinity of $H$. We deduce from the presence of this forbidden region that the classical Eyring formula for the small-$\epsilon$ exponential asymptotics of the mean first exit time is generically inapplicable.Comment: This is a 72-page Postscript file, about 600K in length. Hardcopy requests to [email protected] or [email protected]

A phase transition in a system driven by coloured noise

For a system driven by coloured noise, we investigate the activation energy of escape, and the dynamics during the escape. We have performed analogue experiments to measure the change in activation energy as the power spectrum of the noise varies. An adiabatic approach based on path integral theory allows us to calculate analytically the critical value at which a phase transition in the activation energy occurs

The Order of Phase Transitions in Barrier Crossing

A spatially extended classical system with metastable states subject to weak spatiotemporal noise can exhibit a transition in its activation behavior when one or more external parameters are varied. Depending on the potential, the transition can be first or second-order, but there exists no systematic theory of the relation between the order of the transition and the shape of the potential barrier. In this paper, we address that question in detail for a general class of systems whose order parameter is describable by a classical field that can vary both in space and time, and whose zero-noise dynamics are governed by a smooth polynomial potential. We show that a quartic potential barrier can only have second-order transitions, confirming an earlier conjecture [1]. We then derive, through a combination of analytical and numerical arguments, both necessary conditions and sufficient conditions to have a first-order vs. a second-order transition in noise-induced activation behavior, for a large class of systems with smooth polynomial potentials of arbitrary order. We find in particular that the order of the transition is especially sensitive to the potential behavior near the top of the barrier.Comment: 8 pages, 6 figures with extended introduction and discussion; version accepted for publication by Phys. Rev.

Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular jacobians of genus 2 curves

This paper provides empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves. The second of these conjectures relates six quantities associated to a Jacobian over the rational numbers. One of these six quantities is the size of the Shafarevich-Tate group. Unable to compute that, we computed the five other quantities and solved for the last one. In all 32 cases, the result is very close to an integer that is a power of 2. In addition, this power of 2 agrees with the size of the 2-torsion of the Shafarevich-Tate group, which we could compute

Estimates in Beurling--Helson type theorems. Multidimensional case

We consider the spaces $A_p(\mathbb T^m)$ of functions $f$ on the $m$ -dimensional torus $\mathbb T^m$ such that the sequence of the Fourier coefficients $\hat{f}=\{\hat{f}(k), ~k \in \mathbb Z^m\}$ belongs to $l^p(\mathbb Z^m), ~1\leq p<2$. The norm on $A_p(\mathbb T^m)$ is defined by $\|f\|_{A_p(\mathbb T^m)}=\|\hat{f}\|_{l^p(\mathbb Z^m)}$. We study the rate of growth of the norms $\|e^{i\lambda\varphi}\|_{A_p(\mathbb T^m)}$ as $|\lambda|\rightarrow \infty, ~\lambda\in\mathbb R,$ for $C^1$ -smooth real functions $\varphi$ on $\mathbb T^m$ (the one-dimensional case was investigated by the author earlier). The lower estimates that we obtain have direct analogues for the spaces $A_p(\mathbb R^m)$

Infinite dimensional integrals beyond Monte Carlo methods: yet another approach to normalized infinite dimensional integrals

An approach to (normalized) infinite dimensional integrals, including normalized oscillatory integrals, through a sequence of evaluations in the spirit of the Monte Carlo method for probability measures is proposed. in this approach the normalization through the partition function is included in the definition. For suitable sequences of evaluations, the ("classical") expectation values of cylinder functions are recoveredComment: Submitted as a communication in the ICMSQUARE conference, september 201

Practical Bayesian Modeling and Inference for Massive Spatial Datasets On Modest Computing Environments

With continued advances in Geographic Information Systems and related computational technologies, statisticians are often required to analyze very large spatial datasets. This has generated substantial interest over the last decade, already too vast to be summarized here, in scalable methodologies for analyzing large spatial datasets. Scalable spatial process models have been found especially attractive due to their richness and flexibility and, particularly so in the Bayesian paradigm, due to their presence in hierarchical model settings. However, the vast majority of research articles present in this domain have been geared toward innovative theory or more complex model development. Very limited attention has been accorded to approaches for easily implementable scalable hierarchical models for the practicing scientist or spatial analyst. This article is submitted to the Practice section of the journal with the aim of developing massively scalable Bayesian approaches that can rapidly deliver Bayesian inference on spatial process that are practically indistinguishable from inference obtained using more expensive alternatives. A key emphasis is on implementation within very standard (modest) computing environments (e.g., a standard desktop or laptop) using easily available statistical software packages without requiring message-parsing interfaces or parallel programming paradigms. Key insights are offered regarding assumptions and approximations concerning practical efficiency.Comment: 20 pages, 4 figures, 2 table