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We study the long-time behavior of the probability density associated with the decoupled continuous-time random walk which is characterized by a superheavy-tailed distribution of waiting times. It is shown that if the random walk is…
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…
A probability distribution function is presented which provides a realistic description of the detection of scattered photons. The resulting probabilities can be described analytically by means of a superposition of several special…
Exploiting the coherent medium approximation, random walk among sites distributed randomly in space is investigated when the jump rate depends on the distance between two adjacent sites. In one dimension, it is shown that when the jump rate…
In directed random graphs, in which edges can be assigned to have one of two directions, or perhaps both, the distance between two vertices $v$ and $v'$ can be computed along paths that are directed from $v$ to $v'$, or along paths that are…
We investigate a network model based on an infinite regular square lattice embedded in the Euclidean plane where the node connection probability is given by the geometrical distance of nodes. We show that the degree distribution in the…
Subject of this letter is the dynamics of a chain obtained performing the continuous limit of a system of links and beads. In particular, the probability distribution of the relative position between two points of the chain averaged over a…
An equidistant set in the Euclidean space consists of points having equal distances to both members of a given pair of sets, called focal sets. Having no effective formulas to compute the distance of a point and a set, it is hard to…
We study local modifications of the graph distance in large random triangulations. Our main results show that, in large scales, the modified distance behaves like a deterministic constant $\mathbf{c}~\in~(0,\infty)$ times the usual graph…
The equality constraint a+b+c=1 for random triangle sides corresponds to breaking a stick in two places. An analog a^2+b^2+c^2=1 has a remarkable feature: the bivariate density for angles coincides with that for 3D Gaussian triangles.…
We study the statistics of edges and vertices in the vicinity of a reference vertex (origin) within random planar quadrangulations and Eulerian triangulations. Exact generating functions are obtained for theses graphs with fixed numbers of…
Log-Euclidean distances are commonly used to quantify the similarity between positive definite matrices using geometric considerations. This paper analyzes the behavior of this distance when it is used to measure closeness between…
Distance measuring is a very important task in digital geometry and digital image processing. Due to our natural approach to geometry we think of the set of points that are equally far from a given point as a Euclidean circle. Using the…
The polygonal distributions are a class of distributions that can be defined via the mixture of triangular distributions over the unit interval. The class includes the uniform and trapezoidal distributions, and is an alternative to the beta…
Random geometric graphs result from taking $n$ uniformly distributed points in the unit cube, $[0,1]^d$, and connecting two points if their Euclidean distance is at most $r$, for some prescribed $r$. We show that monotone properties for…
We define a class of Euclidean distances on weighted graphs, enabling to perform thermodynamic soft graph clustering. The class can be constructed form the "raw coordinates" encountered in spectral clustering, and can be extended by means…
Random geometric graphs consist of randomly distributed nodes (points), with pairs of nodes within a given mutual distance linked. In the usual model the distribution of nodes is uniform on a square, and in the limit of infinitely many…
In this paper we propose an algorithm for aligning three-dimensional objects when represented as density maps, motivated by applications in cryogenic electron microscopy. The algorithm is based on minimizing the 1-Wasserstein distance…
Random geometric graphs (RGGs) are commonly used to model networked systems that depend on the underlying spatial embedding. We concern ourselves with the probability distribution of an RGG, which is crucial for studying its random…
This work addresses the problem of simulating Gaussian random fields that are continuously indexed over a class of metric graphs, termed graphs with Euclidean edges, being more general and flexible than linear networks. We introduce three…