Related papers: The Slide Dimension of Point Processes
Direct numerical simulations are performed for the steady flow normal to a circular disk at the Reynolds number of 1000. Numerical simulations are conducted with different levels of simplification procedure by reducing the azimuthal…
An algorithm of searching a zero of an unknown undimensional function is considered, measured at a point x with some error. The step sizes are random positive values and are calculated according to the rule: if two consecutive iterations…
We investigate the relation between the small deviation problem for a symmetric $\alpha$-stable random vector in a Banach space and the metric entropy properties of the operator generating it. This generalizes former results due to Li and…
According to excess-entropy scaling, dynamic properties of liquids like viscosity and diffusion coefficient are determined by the entropy. This link between dynamics and thermodynamics is increasingly studied and of interest also for…
In many data science applications, the objective is to extract appropriately-ordered smooth low-dimensional data patterns from high-dimensional data sets. This is challenging since common sorting algorithms are primarily aiming at finding…
We show that the entanglement entropy and alpha entropies corresponding to spatial polygonal sets in $(2+1)$ dimensions contain a term which scales logarithmically with the cutoff. Its coefficient is a universal quantity consisting in a sum…
We consider the formulation of entropic gravity in two spacetime dimensions. The usual gravitational force law is derived even in the absence of area, as normally required by the holographic principle. A special feature of this perspective…
We study the length of cycles in the model of spatial random permutations in Euclidean space. In this model, for given length $L$, density $\rho$, dimension $d$ and jump density $\varphi$, one samples $\rho L^d$ particles in a…
Dependent symmetries, symmetries that depend on the situation of the subsystem in a larger closed system, are explored by looking at simple examples. This is a new kind of symmetry in the open quantum dynamics of a subsystem Each symmetry…
The classical problem of moments is addressed by the maximum entropy approach for one-dimensional discrete distributions. The numerical technique of adaptive support approximation is proposed to reconstruct the distributions in the region…
Most entropy measures depend on the spread of the probability distribution over the sample space $\mathcal{X}$, and the maximum entropy achievable scales proportionately with the sample space cardinality $|\mathcal{X}|$. For a finite…
Geometrical random multiplicative cascade processes are often used to model positive-valued multifractal fields such as for example the energy dissipation field of fully developed turbulence. A dynamical generalisation of these models is…
We use the Binary Intrinsic Dimension (BID), a geometrical measure designed for binary data, to analyze the Hopfield model, a paradigmatic spin system from statistical mechanics, machine learning and neuroscience. The BID allows us to…
By considering an empirical approximation, and a new class of operators that we will call walking operators, we construct, for any positive ND-toeplitz matrix, an infinite in all dimensions matrix, for which the inverse approximates the…
Although the notion of entropy lies at the core of statistical mechanics, it is not often used in statistical mechanical models to characterize phase transitions, a role more usually played by quantities such as various order parameters,…
A theoretical, and potentially also practical, problem with stochastic gradient descent is that trajectories may escape to infinity. In this note, we investigate uniform boundedness properties of iterates and function values along the…
We study how the Shannon entropy of sequences produced by an information source converges to the source's entropy rate. We synthesize several phenomenological approaches to applying information theoretic measures of randomness and memory to…
Spherical symmetry arguments are used to produce a general device to convert identities and inequalities for the $p$th absolute moments of real-valued random variables into the corresponding identities and inequalities for the $p$th moments…
Given a finite set $K$, we denote by $X=\Delta(K)$ the set of probabilities on $K$ and by $Z=\Delta_f(X)$ the set of Borel probabilities on $X$ with finite support. Studying a Markov Decision Process with partial information on $K$…
In many problems in data mining and machine learning, data items that need to be clustered or classified are not points in a high-dimensional space, but are distributions (points on a high dimensional simplex). For distributions, natural…