Related papers: Minimal Perimeter Triangle in Nonconvex Quadrangle…
We consider the problem of optimizing the product of the distances from a given point in a triangle to each vertex. There are two possible cases in general. For isosceles triangles, we explicitly show exactly when both cases occur.
The study of embedded minimal surfaces in $\RR^3$ is a classical problem, dating to the mid 1700's, and many people have made key contributions. We will survey a few recent advances, focusing on joint work with Tobias H. Colding of MIT and…
We study the problem of minimizing the second Dirichlet eigenvalue for the Laplacian operator among sets of given perimeter. In two dimensions, we prove that the optimum exists, is convex, regular, and its boundary contains exactly two…
We use computational experiments to find the rectangles of minimum area into which a given number n of non-overlapping congruent circles can be packed. No assumption is made on the shape of the rectangles. Most of the packings found have…
Modern minimax problems, such as generative adversarial network and adversarial training, are often under a nonconvex-nonconcave setting, and developing an efficient method for such setting is of interest. Recently, two variants of the…
We resolve a conjecture posed by Covella, Frati and Patrignani by proving the straight-line orthogonal drawing of the complete ternary tree with $n$ nodes satisfying the subtree separation property with smallest area has area…
Given two triangulations of a convex polygon, computing the minimum number of flips required to transform one to the other is a long-standing open problem. It is not known whether the problem is in P or NP-complete. We prove that two…
In this note we present the solution of some isoperimetric problems in open convex cones of $\R^n$ in which perimeter and volume are measured with respect to certain nonradial weights. Surprisingly, Euclidean balls centered at the origin…
A classical result of Honsberger states that the number of incongruent triangles with integer sides and perimeter $n$ is the nearest integer to $\frac{n^2}{48}$ ($n$ even) or $\frac{(n+3)^2}{48}$ ($n$ odd). We solve the analogous problem…
An interesting class of orthogonal representations consists of the so-called turn-regular ones, i.e., those that do not contain any pair of reflex corners that "point to each other" inside a face. For such a representation H it is possible…
This paper proposes an intrinsic pseudospectral convexification framework for optimal control problems with manifold constraints. While successive pseudospectral convexification combines spectral collocation with successive convexification,…
The main result in this paper is to supply a recursive formula, on the number of minimal primes, for the colength of a fractional ideal in terms of the maximal points of the value set of the ideal itself. The fractional ideals are taken in…
In 1991, Edelsbrunner and Tan gave an O(n^2) algorithm for finding the MinMax Length triangulation of a set of points in the plane. In this paper we resolve one of the open problems stated in that paper, by showing that finding a MaxMin…
We study arithmetic constraints arising from the three faces meeting along the space diagonal of a rectangular cuboid. Using a propagation mechanism along this diagonal, based on the appearance of a minimal odd prime in certain triangular…
A triangulation of a surface is \emph{irreducible} if there is no edge whose contraction produces another triangulation of the surface. We prove that every irreducible triangulation of a surface with Euler genus $g\geq1$ has at most $13g-4$…
The distance of a graph from being triangle-free is a fundamental graph parameter, counting the number of edges that need to be removed from a graph in order for it to become triangle-free. Its corresponding computational problem is the…
A partial formula is provided to calculate the smallest number of vertices possible in a quadrangulation on the closed orientable 2-manifold of given genus. This extends the previously known partial formula due to N. Hartsfield and G.…
The degree-diameter problem consists of finding the maximum number of vertices $n$ of a graph with diameter $d$ and maximum degree $\Delta$. This problem is well studied, and has been solved for plane graphs of low diameter in which every…
An opaque set (or a barrier) for $U \subseteq \mathbb{R}^2$ is a set $B$ of finite-length curves such that any line intersecting $U$ also intersects $B$. In this paper, we consider the lower bound for the shortest barrier when $U$ is the…
We investigate a covering problem in $3$-uniform hypergraphs ($3$-graphs): given a $3$-graph $F$, what is $c_1(n,F)$, the least integer $d$ such that if $G$ is an $n$-vertex $3$-graph with minimum vertex degree $\delta_1(G)>d$ then every…