Related papers: Things to do with a broken stick
In this paper the problem of selecting $p$ out of $n$ available items is discussed, such that their total cost is minimized. We assume that costs are not known exactly, but stem from a set of possible outcomes. Robust recoverable and…
We study the structure of the set of all possible affine hyperplane sections of a convex polytope. We present two different cell decompositions of this set, induced by hyperplane arrangements. Using our decomposition, we bound the number of…
Evaluating joint probabilities of potential outcomes and observed variables, and their linear combinations, is a fundamental challenge in causal inference. This paper addresses the bounding and identification of these probabilities in…
This paper treats triangles in the plane whose vertices lie on the integer lattice, i.e., the vertices have integer coordinates. It shows that apart from trivial examples, the circumcenter, centroid and orthocenter of such triangles never…
We study the formation of knots on a macroscopic ball-chain, which is shaken on a horizontal plate at 12 times the acceleration of gravity. We find that above a certain critical length, the knotting probability is independent of chain…
The area of inverse problems in mathematics is highly interdisciplinary. In various fields of science, engineering, medicine, and industry, there arises a need to reconstruct information about unknown entities that cannot be directly…
The probability of a given candidate winning a future election is worked out in closed form as a function of (i) the current support rates for each candidate, (ii) the relative positioning of the candidates within the political spectrum,…
Probabilistic argumentation allows reasoning about argumentation problems in a way that is well-founded by probability theory. However, in practice, this approach can be severely limited by the fact that probabilities are defined by adding…
We have found some patterns in some triangles.
A rational triangle is a triangle with sides of rational lengths. In this short note, we prove that there exists a unique pair of a rational right triangle and a rational isosceles triangle which have the same perimeter and the same area.…
A right triangular billiard system is equivalent to the system of two colliding particles confined in a one-dimensional box. In spite of their seeming simplicity, no definite conclusion has been drawn so far concerning their ergodic…
In the presence of model risk, it is well-established to replace classical expected values by worst-case expectations over all models within a fixed radius from a given reference model. This is the "robustness" approach. We show that…
A variable line through the centroid G of a triangle divides the triangle into two parts each of whose lengths as a fraction of the perimeter fills a closed interval [m,1-m], with m between 0 and 1/2. We show that the range of m taken over…
In this paper, an optimization problem with uncertain constraint coefficients is considered. Possibility theory is used to model the uncertainty. Namely, a joint possibility distribution in constraint coefficient realizations, called…
We consider the hypothesis testing problem of deciding whether an observed high-dimensional vector has independent normal components or, alternatively, if it has a small subset of correlated components. The correlated components may have a…
This paper considers the problem of distinguishing between classical and quantum domains in macroscopic phenomena using tests based on probability and it presents a condition on the ratios of the outcomes being the same (Ps) to being…
The idea of slicing divergences has been proven to be successful when comparing two probability measures in various machine learning applications including generative modeling, and consists in computing the expected value of a `base…
We study side-lengths of triangles in path metric spaces. We prove that unless such a space X is bounded, or quasi-isometric to line or half-line, every triple of real numbers satisfying the strict triangle inequalities, is realized by the…
Given a finite point set $X$ in the plane, the degree of a pair $\{x,y\} \subset X$ is the number of empty triangles $t=conv\{x,y,z\}$, where empty means $t\cap X=\{x,y,z\}$. Define $deg X$ as the maximal degree of a pair in $X$. Our main…
These notes represent an extended version of a talk I gave for the participants of the IMO 2009 and other interested people. We introduce diophantine equations and show evidence that it can be hard to solve them. Then we demonstrate how one…