相关论文: Pins, Stakes, Anchors and Gaussian Triangles

We start with certain joint densities (for sides and for angles) corresponding to pinned Poissonian triangles in the plane, then discuss analogous results for staked and anchored triangles.

We collect various facts related loosely to random Gaussian quadrilaterals in the plane. For example, a side of a degenerate quadrilateral (one point inside three others) has a density that is non-Rayleigh.

Coincidence probabilities, which yield Renyi entropies, are investigated in a generalized Gaussian model, which includes interparticle correlations

Density estimation is an interdisciplinary topic at the intersection of statistics, theoretical computer science and machine learning. We review some old and new techniques for bounding the sample complexity of estimating densities of…

In this expository article, we summarize what is known about maximum likelihood thresholds of Gaussian models, paying special attention to connections with rigidity theory.

In these notes, we discuss a selection of topics on several models of planar statistical mechanics. We consider the Ising, Potts, and more generally abelian spin models; the discrete Gaussian free field; the random cluster model; and the…

Theoretical results for importance sampling rely on the existence of certain moments of the importance weights, which are the ratios between the proposal and target densities. In particular, a finite variance ensures square root convergence…

We use the idea of the broken stick problem (which goes back to Poincare) and calculate the corresponding probabilities for the cases in which the three broken part are: the medians in a triangle, the altitudes, radii of excircles, angle…

We use Rice formulae in order to compute the moments of some level functionals which are linked to problems in oceanography and optics: the number of specular points in one and two dimensions, the distribution of the normal angle of level…

If a line cuts randomly two sides of a triangle, the length of the segment determined by the points of intersection is also random. The object of this study, applied to a particular case, is to calculate the probability that the length of…

These notes are dedicated to whom may be interested in algorithms, Markov chain, coupling, and graph theory etc. I present some preliminaries on coupling and explanations of the important formulas or phrases, which may be helpful for us to…

We study the isoperimetric problem in Euclidean space endowed with a density. We first consider piecewise constant densities and examine particular cases related to the characteristic functions of half-planes, strips and balls. We also…

We firstly study the Navier-Stokes equation for the motion of a passive particle with harmonic, viscous, perturbative forces, subject to an exponentially correlated Gaussian force. Secondly, from the Fokker-Planck equation in an…

Critical points of a scalar quantitiy are either extremal points or saddle points. The character of the critical points is determined by the sign distribution of the eigenvalues of the Hessian matrix. For a two-dimensional homogeneous and…

We discuss concrete examples for frame functions and their associated density operators, as well as for non-Gleason type probability measures.

These are lecture notes based on the first part of a course on 'Mathematical Data Science', which I taught to final year BSc students in the UK in 2019-2020. Topics include: concentration of measure in high dimensions; Gaussian random…

In this paper, we consider Bayesian point estimation and predictive density estimation in the binomial case. After presenting preliminary results on these problems, we compare the risk functions of the Bayes estimators based on the…

The study of "random segments" is a classic issue in geometrical probability, whose complexity depends on how it is defined. But in apparently simple models, the random behavior is not immediate. In the present manuscript the following…

We explicitly describe the possible pairs of triangle and square densities for r-regular finite simple graphs. We also prove that every r-regular unimodular random graph can be approximated by r-regular finite graphs with respect to these…

In this report, the explicit probability density functions of the random Euclidean distances associated with equilateral triangles are given, when the two endpoints of a link are randomly distributed in 1) the same triangle, 2) two adjacent…