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The present paper develops two concepts of pointwise differentiability of higher order for arbitrary subsets of Euclidean space defined by comparing their distance functions to those of smooth submanifolds. Results include that…
Differentiable conjugacies link dynamical systems that share properties such as the stability multipliers of corresponding orbits. It provides a stronger classification than topological conjugacy, which only requires qualitative similarity.…
Position probability distribution of a set of massive mutually exclusive particles in one dimension has been defined. Examples with a given two mutually exclusive particles system are considered. It is emphasized that quantum particles at…
In this paper we give a method to associate a graph with an arbitrary density matrix referred to a standard orthonormal basis in the Hilbert space of a finite dimensional quantum system. We study the related issues like classification of…
This work is an enquiry into the circumstances under which entropy methods can give an answer to the questions of both quantum separability and classical correlations of a composite state. Several entropy functionals are employed to examine…
We define the quantum Wasserstein distance such that the optimization of the coupling is carried out over bipartite separable states rather than bipartite quantum states in general, and examine its properties. Surprisingly, we find that the…
The curse of dimensionality is a common phenomenon which affects analysis of datasets characterized by large numbers of variables associated with each point. Problematic scenarios of this type frequently arise in classification algorithms…
For many optimization problems it is possible to define a distance metric between problem variables that correlates with the likelihood and strength of interactions between the variables. For example, one may define a metric so that the…
Homogeneous fragmentations describe the evolution of a unit mass that breaks down randomly into pieces as time passes. They can be thought of as continuous time analogs of a certain type of branching random walks, which suggests the use of…
Contour maps are widely used to display estimates of spatial fields. Instead of showing the estimated field, a contour map only shows a fixed number of contour lines for different levels. However, despite the ubiquitous use of these maps,…
One of the fundamental problems in machine learning is the estimation of a probability distribution from data. Many techniques have been proposed to study the structure of data, most often building around the assumption that observations…
We consider the problem of determining the number of distinct distances between two point sets in $\mathbb{R}^2$ where one point set $\mathcal{P}_1$ of size $m$ lies on a real algebraic curve of fixed degree $r$, and the other point set…
Separability conditions for a bipartite quantum system of finite-dimensional subsystems are formulated in terms of R\'{e}nyi and Tsallis entropies. Entropic uncertainty relations often lead to entanglement criteria. We propose new approach…
Separability is an important problem in theory of quantum entanglement. By using the Bloch representation of quantum states in terms of the Heisenberg-Weyl observable basis, we present a new separability criterion for bipartite quantum…
Quantum coherence plays a fundamental and operational role in different areas of physics. A resource theory has been developed to characterize the coherence of distinguishable particles systems. Here we show that indistinguishability of…
A flat membrane with given shape is displayed; two points in the membrane are randomly selected; the probability that the separation between the points have a specified value is sought. A simple method to evaluate the probability density is…
The Wasserstein distance, rooted in optimal transport (OT) theory, is a popular discrepancy measure between probability distributions with various applications to statistics and machine learning. Despite their rich structure and…
The quality of datasets plays a crucial role in the successful training and deployment of deep learning models. Especially in the medical field, where system performance may impact the health of patients, clean datasets are a safety…
This paper concerns the propagation of particles through a quenched random medium. In the one- and two-dimensional models considered, the local dynamics is given by expanding circle maps and hyperbolic toral automorphisms, respectively. The…
Based on a local approximation of the Riemannian distance on a manifold by a computationally cheap dissimilarity measure, a time discrete geodesic calculus is developed, and applications to shape space are explored. The dissimilarity…