Related papers: Distance Distributions in Regular Polygons
We investigate the statistical distribution of Newtonian gravitational forces acting on a test particle embedded in an infinite, homogeneous, and uncorrelated random gas of point masses. Using order statistics, we derive the probability…
We consider the 2-dimensional random matching problem in $\mathbb{R}^2.$ In a challenging paper, Caracciolo et. al. arXiv:1402.6993 on the basis of a subtle linearization of the Monge Ampere equation, conjectured that the expected value of…
We consider density estimators based on the nearest neighbors method applied to discrete point distibutions in spaces of arbitrary dimensionality. If the density is constant, the volume of a hypersphere centered at a random location is…
We show that in any Euclidean space, an arbitrary probability measure can be reconstructed explicitly by its geometric (or spatial) distribution function. The reconstruction takes the form of a (potentially fractional) linear PDE, where the…
In this work we study a class of random convex sets that "interpolate" between polytopes and zonotopes. These sets arise from considering a $q^{th}$-moment ($q\geq 1$) of an average of order statistics of $1$-dimensional marginals of a…
The probability density function (PDF) of a random variable associated with the solution of a partial differential equation (PDE) with random parameters is approximated using a truncated series expansion. The random PDE is solved using two…
Let $\{f_j\}_{j=0}^n$ be a sequence of orthonormal polynomials where the orthogonality relation is satisfied on either the real line or on the unit circle. We study zero distribution of random linear combinations of the form…
An expression for the joint probability distribution of the principal curvatures at an arbitrary point in the ensemble of isosurfaces defined on isotropic Gaussian random fields on Rn is derived. The result is obtained by deriving symmetry…
The probability distribution functions (PDFs) for atomic, molecular, and total gas surface densities of M33 are determined at a resolution of about 50~pc over regions that share coherent morphological properties to unveil fingerprints of…
We quantitatively study the probability distribution function (PDF) of cosmological nonlinear density fluctuations from N-body simulations with Gaussian initial condition. In particular, we examine the validity and limitations of one-point…
We study $N$-point rational distance sets ($\textrm{RDS}(N)$) on the parabola $y=x^2$. Previous approaches to the problem include efforts made using elliptic curves and diophantine chains, with successful analysis for $N\leq 4$. We extend…
Phylogenetic inference-the derivation of a hypothesis for the common evolutionary history of a group of species- is an active area of research at the intersection of biology, computer science, mathematics, and statistics. One assumes the…
We present a public code to generate random fields with an arbitrary probability distribution function (PDF) and an arbitrary correlation function. The algorithm is cosmology-independent, applicable to any stationary stochastic process over…
We study the packing of a large number of congruent and non--overlapping circles inside a regular polygon. We have devised efficient algorithms that allow one to generate configurations of $N$ densely packed circles inside a regular polygon…
We calculate the one-point probability distribution function (PDF) for cosmic density in non-linear regime of the gravitational evolution. Under the local approximation that the evolution of cosmic fluid fields can be characterized by the…
We derive the exact probability density function of the product of $N$ independent variance-gamma random variables with zero location parameter. We then apply this formula to derive formulas for the cumulative distribution function and…
Using a TE/TM decomposition for an angular plane-wave spectrum of free random electromagnetic waves and matched boundary conditions, we derive the probability density function for the energy density of the vector electric field in the…
Lens spaces are a family of manifolds that have been a source of many interesting phenomena in topology and differential geometry. Their concrete construction, as quotients of odd-dimensional spheres by a free linear action of a finite…
Randomness is viewed through an analogy between a physical quantity, density of gas, and a mathematical construct -- probability density. Boltzmann's deduction of equilibrium distribution of ideal gas placed in an external potential field…
We study the limiting distribution of particles at the frontier of a branching random walk. The positions of these particles can be viewed as the lowest energies of a directed polymer in a random medium in the mean-field case. We show that…