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Spherical Designs are finite sets of points on the sphere $\mathbb{S}^{d}$ with the property that the average of certain (low-degree) polynomials in these points coincides with the global average of the polynomial on $\mathbb{S}^{d}$. They…
In this letter we investigate the nature of generic cosmological singularities using the framework developed by Uggla et al. We do so by studying the past asymptotic dynamics of general vacuum G2 cosmologies, models that are expected to…
Restricted skyline (rskyline) query is widely used in multi-criteria decision making. It generalizes the skyline query by additionally considering a set of personalized scoring functions F. Since uncertainty is inherent in datasets for…
Classical asymptotic theory for statistical inference usually involves calibrating a statistic by fixing the dimension $d$ while letting the sample size $n$ increase to infinity. Recently, much effort has been dedicated towards…
Statistical analyses of finite sample distributions usually assume that fluctuations are self-averaging, i.e. that they are statistically similar in different regions of the given sample volume. By using the scale-length method, we test…
In this paper we provide asymptotic upper bounds on the complexity in two (closely related) situations. We confirm for the total doubling coverings and not only for the chains the expected bounds of the form $$ \kappa({\mathcal U}) \le…
Point sets of number-theoretic origin, such as the visible lattice points or the $k$-th power free integers, have interesting geometric and spectral properties and give rise to topological dynamical systems that belong to a large class of…
Phase transitions and critical phenomena are among the most intriguing phenomena in nature and their renormalization-group theory is one of the greatest achievements of theoretical physics. However, the predictions of the theory above an…
We propose an axiomatic approach to the concept of an intrinsic dimension of a dataset, based on a viewpoint of geometry of high-dimensional structures. Our first axiom postulates that high values of dimension be indicative of the presence…
Consider a random walk $S_i= \xi_1+\ldots+\xi_i$, $i\in\mathbb N$, whose increments $\xi_1,\xi_2,\ldots$ are independent identically distributed random vectors in $\mathbb R^d$ such that $\xi_1$ has the same law as $-\xi_1$ and $\mathbb…
We study "dead ends" in square-free digit walks: square-free integers $N$ such that, in base $b$, every one-digit extension $bN+d$ is non-square-free. In base $10$, the stochastic independence model of Miller et al. suggests that infinite…
We provide a complete asymptotic distribution theory for clustered data with a large number of independent groups, generalizing the classic laws of large numbers, uniform laws, central limit theory, and clustered covariance matrix…
Ensemble of initial conditions for nonlinear maps can be described in terms of entropy. This ensemble entropy shows an asymptotic linear growth with rate K. The rate K matches the logarithm of the corresponding asymptotic sensitivity to…
Let $\xi_1, \xi_2,\ldots$ be a sequence of independent and identically distributed random variables with zero mean, finite second moment and regularly varying right distribution tail. Motivated by a stop-loss insurance model, we consider a…
The problem of sequential anomaly detection is considered, where multiple data sources are monitored in real time and the goal is to identify the "anomalous" ones among them, when it is not possible to sample all sources at all times. A…
The subject of the article is linear systems of wave equations on cosmological backgrounds with convergent asymptotics. The condition of convergence corresponds to the requirement that the second fundamental form, when suitably normalised,…
The perturbative treatment of dark matter in structure formation relies on the existence of a well-defined expansion parameter, $k/k_{\rm NL}$, with $k_{\rm NL}$ signalling the onset and ultimately the leading role of non-linearities in the…
We consider the one-dimensional nonlinear Schr\"odinger equation with a nonlinearity of degree $p>1$. We exhibit measures on the space of initial data for which we describe the non trivial evolution by the linear Schr\"odinger flow and we…
Recently, many works have tried to realize cosmological accelerated expansion in string theory models in the asymptotic regions of field space, with a typical scalar potential $V(\varphi)$ having an exponential fall-off $e^{-\gamma\,…
The zero range process is of particular importance as a generic model for domain wall dynamics of one-dimensional systems far from equilibrium. We study this process in one dimension with rates which induce an effective attraction between…