Related papers: On a Calculus Textbook Problem
We improve a lower bound for the smallest area of convex covers for closed unit curves from 0.0975 to 0.1, which makes it substantially closer to the current best upper bound 0.11023. We did this by considering the minimal area of convex…
It was recently proved that the elastic energy $E(\gamma)=\tfrac{1}{2}\int_\gamma\kappa^2 ds$ of a closed curve $\gamma$ with curvature $\kappa$ has a minimizer among all plane, simple, regular and closed curves of given enclosed area…
We characterize the boundedness of square functions in the upper half-space with general measures. The short proof is based on an averaging identity over good Whitney regions.
This paper is concerned with a shape optimization problem, where the functional to be maximized describes the total sunlight collected by a distribution of tree leaves, minus the cost for transporting water and nutrient from the base of the…
The derivatives with respect to the variable $r$ of $\pi r^2$ and $\frac{4}{3}\pi r^3$ are $2\pi r$ and $4\pi r^2$, respectively. This relates, through the derivative, the area enclosed in a circle to the length of that circle and,…
In this work, we study the tensor ring decomposition and its associated numerical algorithms. We establish a sharp transition of algorithmic difficulty of the optimization problem as the bond dimension increases: On one hand, we show the…
Given $n$ real numbers $0\leq x_1,...,x_n<1$ and a permutation~$\sigma$ of $\{1,...,n\}$, we can always find $\xbar_1,...,\xbar_n\in\{0,1\}$ so that the partial sums $\xbar_1+... +\xbar_k$ and $\xbar_{\sigma 1}+... +\xbar_{\sigma k}$ differ…
T duality expresses the equivalence of a superstring theory compactified on a manifold K to another (possibly the same) superstring theory compactified on a dual manifold K'. The volumes of K and K' are inversely proportional. In this talk…
Let $E$ be a smooth bundle with fiber an $n$-dimensional real projective space $\mathbb{R}P^n$. We show that, if every fiber carries a positively curved pointwise strongly $1/4$-pinched Riemannian metric that varies continuously with…
While it is a classical result dating back to Dehn (1903) that squares composing a perfect rectangle must have rational side lengths, the arithmetic complexity of these tilings, specifically the growth of the denominators of these rational…
We consider two orientation problems in a graph, namely the minimization of the sum of all the shortest path lengths and the minimization of the diameter. We show that it is NP-complete to decide whether a graph has an orientation such that…
The classic Maxwell formula calculates the length of a planar locally minimal binary tree in terms of coordinates of its boundary vertices and directions of incoming edges. However, if an extreme tree with a given topology and a boundary…
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…
Let $D$ be a Riemannian 2-disc of area $A$, diameter $d$ and length of the boundary $L$. We prove that it is possible to contract the boundary of $D$ through curves of length $\leq L + 200d\max\{1,\ln {\sqrt{A}\over d} \}$. This answers a…
Minimum $k$-Section denotes the NP-hard problem to partition the vertex set of a graph into $k$ sets of sizes as equal as possible while minimizing the cut width, which is the number of edges between these sets. When $k$ is an input…
We study the Lawn Mowing Problem restricted to periodic billiard paths in the unit square. Given the combinatorial data of a trajectory, we determine the optimal covering radius, and identify the shortest path that covers the square for any…
Chiral gauge theories in two dimensions with (0,2) supersymmetry are central in the study of string compactifications. Remarkably little is known about generic (0,2) theories. We consider theories with branches on which multiplets with a…
A well known open problem of Meir and Moser asks if the squares of sidelength $1/n$ for $n \geq 2$ can be packed perfectly into a square of area $\sum_{n=2}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}-1$. In this paper we show that for any $1/2…
Two planar sets are circularly separable if there exists a circle enclosing one of the sets and whose open interior disk does not intersect the other set. This paper studies two problems related to circular separability. A linear-time…
This is the second paper of a series of three on the regularity of higher codimension area minimizing integral currents. Here we perform the second main step in the analysis of the singularities, namely the construction of a center…