Related papers: Automated Proofs in Geometry : Computing Upper Bou…
We show that among any $n$ points in the unit cube one can find a triangle of area at most $n^{-2/3-c}$ for some absolute constant $c >0$. This gives the first non-trivial upper bound for the three-dimensional version of Heilbronn's…
We show upper and lower bounds for angles in iterations of trisections of certain triangulations.
We develop a new simple approach to prove upper bounds for generalizations of the Heilbronn's triangle problem in higher dimensions. Among other things, we show the following: for fixed $d \ge 1$, any subset of $[0, 1]^d$ of size $n$…
We study the Heilbronn triangle problem, which involves placing n points in the unit square such that the minimum area of any triangle formed by these points is maximized. A straightforward maximin formulation of this problem is highly…
Using ideas from the geometry of compression, we improve on the current upper and lower bounds of the Heilbronn triangle problem. In particular, let $\Delta(s)$ denote the minimal area of the triangle induced by $s$ points on a unit disk.…
We compute the Hilbert coefficients of a graded module with pure resolution and discuss lower and upper bounds for these coefficients for arbitrary graded modules.
We obtain new upper and lower bounds on the number of unit perimeter triangles spanned by points in the plane. We also establish improved bounds in the special case where the point set is a section of the integer grid.
We introduce remarkable upper bounds for the interpolation error constants on triangles, which are sharp and given by simple formulas. These constants are crucial in analyzing interpolation errors, particularly those associated with the…
In this article a new upper bounds for the multiple trigonometrical integrals are found. The method of the work based on a new method of estimation for the areas of algebraic surfaces.
In the paper we prove a new upper bound for Heilbronn's exponential sum and obtain some applications of our result to distribution of Fermat quotients.
The lower and upper bound of any given algorithm is one of the most crucial pieces of information needed when evaluating the computational effectiveness for said algorithm. Here a novel method of Boolean Algebraic Programming for symbolic…
In this work, we consider the bilevel optimization problem on Riemannian manifolds. We inspect the calculation of the hypergradient of such problems on general manifolds and thus enable the utilization of gradient-based algorithms to solve…
The Heilbronn triangle problem asks for the placement of $n$ points in a unit square that maximizes the smallest area of a triangle formed by any three of those points. In $1972$, Schmidt considered a natural generalization of this problem.…
For sufficiently large $n$, we show that in every configuration of $n$ points chosen inside the unit square there exists a triangle of area less than $n^{-8/7-1/2000}$. This improves upon a result of Koml\'os, Pintz and Szemer\'edi from…
We give an efficient algorithm to compute equations of twists of hyperelliptic curves of arbitrary genus over any separable field (of characteristic different from 2), and we explicitly describe some interesting examples.
We propose a simple derivation of an upper bound for the perimeter of an ellipse. The procedure, which relies on the use of elliptic integrals, consists in introducing, via inequalities and convexity properties, specific integrals which can…
The Maximum Balanced Biclique Problem (MBBP) is a prominent model with numerous applications. Yet, the problem is NP-hard and thus computationally challenging. We propose novel ideas for designing effective exact algorithms for MBBP.…
In this paper we consider the computational complexity of uniformizing a domain with a given computable boundary. We give nontrivial upper and lower bounds in two settings: when the approximation of boundary is given either as a list of…
For the bi-orthogonal polynomials with the third degree polynomial potential functions, the 3 x 3 matrix Riemann-Hilbert problem is explicitly constructed. The developed approach admits an extension to the bi-orthogonal polynomials with…
We present algorithms to compute the topology of 2D and 3D hyperelliptic curves. The algorithms are based on the fact that 2D and 3D hyperelliptic curves can be seen as the image of a planar curve (the Weierstrass form of the curve), whose…