We study the stability of a scalar inflaton field and analyze its point attractors in the phase space. We show that the value of the inflaton field in the vacuum is a bifurcation parameter and prove the possible existence of a limit cycle by using analytical and numerical arguments. To be published in Modern Physics Letters A Electronic address: [email protected]

We prove that, under fairly general conditions, a properly rescaled de-terminantal random point field converges to a generalized Gaussian random process. 1. Introduction and formulation of results. Let E be a locally compact Hausdorff space satisfying the second axiom of countability, B—σ-algebra of Borel subsets and µ a σ-finite measure on (E, B), such that µ(K) < ∞ for any compact K ⊂ E. We d...

If a point is in the ¿u-limit set and the a-limit set of the same point, then we call it a y-limit point. Then a y-limit point is an w-limit point and thus a nonwandering point. In this paper, we prove that, on the interval, a nonwandering point which is not a y-limit point is in the closure of the set of forward images of turning points, and such points are not always the forward images of tur...

Let Yi, i ≥ 1, be i.i.d. random variables having values in an mdimensional manifold M ⊂ Rd and consider sums ∑ni=1 ξ(nYi, {n1/mYj }j=1), where ξ is a real valued function defined on pairs (y,Y), with y ∈Rd and Y ⊂Rd locally finite. Subject to ξ satisfying a weak spatial dependence and continuity condition, we show that such sums satisfy weak laws of large numbers, variance asymptotics and centr...

This thesis consists of five articles mainly devoted to problems in dynamical systems and ergodic theory. We consider non-uniformly hyperbolic two dimensional systems and limit distributions of point measures, which are absolutely continuous with respect to Lebesgue measure. Let fa0(x) = a0−x 2 be a quadratic map, where the parameter a0 ∈ (1, 2) is chosen such that the critical point 0 is pre-p...

We prove a local central limit theorem (LCLT) for the number of points N(J) in a region J in Rd specified by a determinantal point process with an Hermitian kernel. The only assumption is that the variance of N(J) tends to infinity as |J | → ∞. This extends a previous result giving a weaker central limit theorem (CLT) for these systems. Our result relies on the fact that the Lee-Yang zeros of t...

Let Q(n) denote the squarefree part of n so that Q(n) = Q pjn p. Throughout, we set a,b, and c to be positive relatively prime integers with c = a + b. Deene L a;b = log c log Q(abc) : The abc-conjecture of Masser and Oesterl e asserts that the greatest limit point of the double sequence fL a;b g is 1. Recently, in joint work with Browkin, Greaves, Schinzel, and the rst author 1], it was shown ...

Let P be a simple, stationary, clustering point process on Rd in the sense that its correlation functions factorize up to an additive error decaying exponentially fast with the separation distance. Let Pn := P ∩Wn be its restriction to windows Wn := [− 1/d 2 , n1/d 2 ] d ⊂ Rd. We consider the statistic H n := ∑ x∈Pn ξ(x,Pn) where ξ(x,Pn) denotes a score function representing the interaction of ...

We consider point clouds obtained as random samples of a measure on a Euclidean domain. A graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points they connect. Our goal is to develop mathematical tools needed to study the consistency, as the number of available data points increases, of graph-based machine learning algorithms for tas...