Concentration Of Laplace Eigenfunctions And Stabilization Of Weakly Damped Wave Equation

- In this article, we prove some universal bounds on the speed of concentration on small (frequency-dependent) neighborhoods of submanifolds of L 2-norms of quasi modes for Laplace operators on compact manifolds. We deduce new results on the rate of decay of weakly damped wave equations. R{\'e}sum{\'e}

Laplace Eigenfunctions And Damped Wave Equation Ii: Product Manifolds

- The purpose of this article is to study possible concentrations of eigenfunc-tions of Laplace operators (or more generally quasi-modes) on product manifolds. We show that the approach of the first author and Zworski [10, 11] applies (modulo rescalling) and deduce new stabilization results for weakly damped wave equations which extend to product manifolds previous results by Leautaud-Lerner [12] obtained for products of tori

Invariants of a Free Linear Category and Representation Type

We consider an homogeneous action of a finite group on a free linear category over a field in order to prove that the subcategory of invariants is still free. Moreover we show that the representation type is preserved when considering invariants.Comment: 19 page

Record statistics for random walk bridges

We investigate the statistics of records in a random sequence $\{x_B(0)=0,x_B(1),\cdots, x_B(n)=x_B(0)=0\}$ of $n$ time steps. The sequence $x_B(k)$'s represents the position at step $k$ of a random walk `bridge' of $n$ steps that starts and ends at the origin. At each step, the increment of the position is a random jump drawn from a specified symmetric distribution. We study the statistics of records and record ages for such a bridge sequence, for different jump distributions. In absence of the bridge condition, i.e., for a free random walk sequence, the statistics of the number and ages of records exhibits a `strong' universality for all $n$, i.e., they are completely independent of the jump distribution as long as the distribution is continuous. We show that the presence of the bridge constraint destroys this strong `all $n$' universality. Nevertheless a `weaker' universality still remains for large $n$, where we show that the record statistics depends on the jump distributions only through a single parameter $0<\mu\le 2$, known as the L\'evy index of the walk, but are insensitive to the other details of the jump distribution. We derive the most general results (for arbitrary jump distributions) wherever possible and also present two exactly solvable cases. We present numerical simulations that verify our analytical results.Comment: 40 pages, 11 figures, contribution to the JSTAT Special Issue based on the Galileo Galilei Institute Workshop "Advances in Nonequilibrium Statistical Mechanics". Published versio

Exact statistics of record increments of random walks and L\'evy flights

We study the statistics of increments in record values in a time series $\{x_0=0,x_1, x_2, \ldots, x_n\}$ generated by the positions of a random walk (discrete time, continuous space) of duration $n$ steps. For arbitrary jump length distribution, including L\'evy flights, we show that the distribution of the record increment becomes stationary, i.e., independent of $n$ for large $n$, and compute it explicitly for a wide class of jump distributions. In addition, we compute exactly the probability $Q(n)$ that the record increments decrease monotonically up to step $n$. Remarkably, $Q(n)$ is universal (i..e., independent of the jump distribution) for each $n$, decaying as $Q(n) \sim {\cal A}/\sqrt{n}$ for large $n$, with a universal amplitude ${\cal A} = e/\sqrt{\pi} = 1.53362\ldots$.Comment: 6 pages + 5 pages of supplemental material, 5 figures. Published versio

THE IMPACT OF CLIMATE CHANGE ON THE PROFITABILITY OF SITE SPECIFIC TECHNOLOGIES

Site Specific Technologies (SST) can reduce environmental pollution caused by common agricultural practice. Using a case study for corn yields, we investigate the impact of climate change (CC) on profitability of SSTs. We find CC to increase spatial variability of soils with respect to optimal input application and yield variability. This leads, ceteris paribus, to higher incentives for SST adoption in the future.Climate Change, Site Specific Technologies, Adaptation, Crop Production Function., Environmental Economics and Policy,