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Once recognizing that point particles moving inside the extended version of the rippled billiard perform L\'evy flights characterized by a L\'evy-type distribution $P(\ell)\sim \ell^{-(1+\alpha)}$ with $\alpha=1$, we derive a generalized…
We construct random metric spaces by gluing together an infinite sequence of pointed metric spaces that we call blocks. At each step, we glue the next block to the structure constructed so far by randomly choosing a point on the structure…
It is well known that the spectral form factor (SFF) of a possibly degenerate many-body Hamiltonian can be identified with a planar random walk taking steps of unequal length. In this paper we push this identification further and propose to…
We prove that if the Hausdorff dimension of a compact subset of ${\mathbb R}^d$ is greater than $\frac{d+1}{2}$, then the set of angles determined by triples of points from this set has positive Lebesgue measure. Sobolev bounds for…
Let M be a compact smooth manifold of dimension n with or without boundary, and f : M $\rightarrow$ R be a smooth Gaussian random field. It is very natural to suppose that for a large positive real u, the random excursion set {f $\ge$ u} is…
We study random bipartite planar maps defined by assigning nonnegative weights to each face of a map. We prove that for certain choices of weights a unique large face, having degree proportional to the total number of edges in the maps,…
Scattering transforms are a new type of summary statistics recently developed for the study of highly non-Gaussian processes, which have been shown to be very promising for astrophysical studies. In particular, they allow one to build…
Suppose that $\eta$ is a Schramm-Loewner evolution (SLE$_\kappa$) in a smoothly bounded simply connected domain $D \subset {\mathbb C}$ and that $\phi$ is a conformal map from ${\mathbb D}$ to a connected component of $D \setminus…
Surface wave patterns are investigated experimentally in a system geometry that has become a paradigm of quantum chaos: the stadium billiard. Linear waves in bounded geometries for which classical ray trajectories are chaotic are known to…
In this paper, we construct round fold maps or stable fold maps with concentric singular value sets introduced by the author on smooth bundles over spheres or bundles over more general manifolds. The class of round fold maps includes…
In this note we describe a family of arguments that link the homotopy-type of a) the diffeomorphism group of the disc $D^n$, b) the space of co-dimension one embedded spheres in a sphere and c) the homotopy-type of the space of co-dimension…
We provide asymptotic results and develop high frequency statistical procedures for time-changed L\'evy processes sampled at random instants. The sampling times are given by first hitting times of symmetric barriers whose distance with…
We develop a scale-invariant truncated L\'evy (STL) process to describe physical systems characterized by correlated stochastic variables. The STL process exhibits L\'evy stability for the probability density, and hence shows scaling…
For a given point set $S$ in a plane, we develop a distributed algorithm to compute the $\alpha-$shape of $S$. $\alpha-$shapes are well known geometric objects which generalize the idea of a convex hull, and provide a good definition for…
Moser derived a normal form for the family of four-dimensional, quadratic, symplectic maps in 1994. This six-parameter family generalizes H\'enon's ubiquitous 2D map and provides a local approximation for the dynamics of more general 4D…
It is shown that the families of generalized matrix ensembles recently considered which give rise to an orthogonal invariant stable L\'{e}vy ensemble can be generated by the simple procedure of dividing Gaussian matrices by a random…
This paper presents a new approach for dimension reduction of data observed in a sphere. Several dimension reduction techniques have recently developed for the analysis of non-Euclidean data. As a pioneer work, Hauberg (2016) attempted to…
We consider a family of surfaces of revolution ranging between a disc and a hemisphere, that is spherical caps. For this family, we study the spectral density in the ray limit and arrive at a trace formula with geodesic polygons describing…
We construct a smooth algebraic stack of tuples consisting of genus two nodal curves, simple effective divisors away from the nodes, and twisted fields. It provides a desingularization of the moduli of genus two stable maps to projective…
This paper provides an extended case study of the cutoff phenomenon for a prototypical class of nonlinear Langevin systems with a single stable state perturbed by an additive pure jump L\'evy noise of small amplitude $\varepsilon>0$, where…