Related papers: Triangle Inscribed-Triangle Picking
In Computer Vision, edge detection is one of the favored approaches for feature and object detection in images since it provides information about their objects boundaries. Other region-based approaches use probabilistic analysis such as…
We revisit the problem of enumeration of vertex-tricolored planar random triangulations solved in [Nucl. Phys. B 516 [FS] (1998) 543-587] in the light of recent combinatorial developments relating classical planar graph counting problems to…
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.
The binomial interpolated transform of a sequence is a generalization of the well-known binomial transform. We examine a Pascal-like triangle, on which a binomial interpolated transform works between the left and right diagonals, focusing…
A {\em net} is a graph consisting of a triangle $C$ and three more vertices, each of degree one and with its neighbour in $C$, and all adjacent to different vertices of $C$. We give a polynomial-time algorithm to test whether an input graph…
Inspired by classical puzzles in geometry that ask about probabilities of geometric phenomena, we give an explicit formula for the probability that a random triangle on a flat torus is homotopically trivial. Our main tool for this…
A systematic study of the probability distribution of superimposed random codes is presented through the use of generating functions. Special attention is paid to the cases of either uniformly distributed but not necessarily independent or…
How to turn the flip of a coin into a random variable whose expected value equals a scattering amplitude? We answer this question by constructing a numerical algorithm to evaluate curve integrals - a novel formulation of scattering…
Computation of the probability that a random graph is connected is a challenging problem, so it is natural to turn to approximations such as Monte Carlo methods. We describe sequential importance resampling and splitting algorithms for the…
We consider tilings of a triangle $ABC$ by congruent copies of a triangle that has one angle equal to $120^\circ$, has non-commensurable angles (that is, not all angles are rational multiples of $\pi$), and is not similar to $ABC$. We prove…
Consider a triangle $ABC$ with given lengths $l_a,l_b,l_c$ of its internal angle bisectors. We prove that in general, it is impossible to construct a square of the same area as $ABC$ using a ruler and compass. Moreover, it is impossible to…
This work introduces a probabilistic-based model for binary CSP that provides a fine grained analysis of its internal structure. Assuming that a domain modification could occur in the CSP, it shows how to express, in a predictive way, the…
We describe and analyze some Monte Carlo methods for manifolds in Euclidean space defined by equality and inequality constraints. First, we give an MCMC sampler for probability distributions defined by un-normalized densities on such…
In this note we introduce a new randomized algorithm for counting triangles in graphs. We show that under mild conditions, the estimate of our algorithm is strongly concentrated around the true number of triangles. Specifically, if $p \geq…
The broken stick problem is the following classical question. You have a segment $[0,1]$. You choose two points on this segment at random. They divide the segment into three smaller segments. Show that the probability that the three…
It is shown that the projection image of an oriented spatial arc to any oriented plane is approximated by a unique arc diagram (up to isomorphic arc diagrams) determined from the spatial arc and the projection. In a separated paper, the…
Space filling designs are central to studying complex systems in various areas of science. They are used for obtaining an overall understanding of the behaviour of the response over the input space, model construction and uncertainty…
Heron's formula states that the area $K$ of a triangle with sides $a$, $b$, and $c$ is given by $$ K=\sqrt {s(s-a) (s-b) (s-c)} $$ where $s$ is the semiperimeter $(a+b+c)/2$. Brahmagupta, Robbins, Roskies, and Maley generalized this formula…
Approximate Bayesian computation (ABC) methods are standard tools for inferring parameters of complex models when the likelihood function is analytically intractable. A popular approach to improving the poor acceptance rate of the basic…
The number of triangles is a computationally expensive graph statistic which is frequently used in complex network analysis (e.g., transitivity ratio), in various random graph models (e.g., exponential random graph model) and in important…