Related papers: Square Packing with Asymptotically Smallest Waste …
We show that any space with a positive upper curvature bound has in a small neighborhood of any point a closely related metric with a negative upper curvature bound.
We prove that complete submanifolds, on which the Omori-Yau weak maximum principle for the Hessian holds, with low codimension and bounded by cylinders of small radius must have points rich in large positive extrinsic curvature. The lower…
We prove asymptotic 0-1 Laws satisfied by diagrams of unimodal sequences of positive integers. These diagrams consist of columns of squares in the plane, and the upper boundary is called the shape. For various types, we show that, as the…
Motivated by applications to matrix multiplication algorithms, Pratt asked (ITCS'24) how large a subset of $[n] \times [n]$ could be without containing a skew-corner: three points $(x,y), (x,y+h),(x+h,y')$ with $h \ne 0$. We prove any skew…
We study the problem of high-dimensional multiple packing in Euclidean space. Multiple packing is a natural generalization of sphere packing and is defined as follows. Let $ N>0 $ and $ L\in\mathbb{Z}_{\ge2} $. A multiple packing is a set…
We prove that the least area of the non-contractible immersed spheres is no more than $4\pi$ in any oriented compact manifold with dimension $n+2\leq 7$ which satisfies $R\geq 2$ and admits a map to $\mathbf S^2\times T^n$ with nonzero…
We prove that every unit area convex pentagon is contained in a convex quadrilateral of area no greater than $3/\sqrt{5}$, and that every unit area convex hexagon is contained in a convex pentagon of area no greater than $7/6$. Both results…
The problem of prediction in functional linear regression is conventionally addressed by reducing dimension via the standard principal component basis. In this paper we show that an alternative basis chosen through weighted least-squares,…
Given an $r$-uniform hypergraph $H$ and a positive integer $n$, the weak saturation number $\mathrm{wsat}(n,H)$ is the minimum number of edges in an $r$-uniform hypergraph $F$ on $n$ vertices such that the missing edges in $F$ can be added,…
We consider a shape optimization problem related to the persistence threshold for a biological species, the unknown shape corresponding to the zone of the habitat which is favorable to the population. Analytically, this translates in the…
We obtain an upper bound to the packing density of regular tetrahedra. The bound is obtained by showing the existence, in any packing of regular tetrahedra, of a set of disjoint spheres centered on tetrahedron edges, so that each sphere is…
A \emph{square} is a word of the form $uu$, where $u$ is a nonempty finite word. Given a finite word $w$ of length $n$, let $[w]$ denote the corresponding \emph{circular word}, i.e., the set of all cyclic rotations of $w$. We study the…
Let $f(n)$ denote the maximum total length of the sides of $n$ squares packed inside a unit square. Erd\H{o}s conjectured that $f(k^2+1)=k$. We show that the conjecture is true if we assume that the sides of the squares are parallel to the…
A skew corner is a triple of points in $\mathbb{Z} \times \mathbb{Z}$ of the form $(x,y), (x, y + a)$ and $(x + a, y')$. Pratt posed the following question: how large can a set $A \subseteq [n] \times [n]$ be, provided it contains no…
A Littlewood polynomial is a single-variable polynomial all of whose coefficients lie in $\{ \pm 1\}$. We establish the leading term asymptotics of the number of reciprocal or skew-reciprocal Littlewood polynomials with square discriminant.…
In this paper we present a fast scalable heuristic for bin packing that partitions the given problem into identical sub-problems of constant size and solves these constant size sub-problems by considering only a constant number of bin…
In this note we prove that for any compact subset $S$ of a Busemann surface $({\mathcal S},d)$ (in particular, for any simple polygon with geodesic metric) and any positive number $\delta$, the minimum number of closed balls of radius…
The moving sofa problem, posed by L. Moser in 1966, asks for the planar shape of maximal area that can move around a right-angled corner in a hallway of unit width. It is known that a maximal area shape exists, and that its area is at least…
The optimization problem that arises out of the least median of squared residuals method in linear regression is analyzed. To simplify the analysis, the problem is replaced by an equivalent one of minimizing the median of absolute…
We examine sequences of dense packings of n congruent non-overlapping disks inside a square which follow specific patterns as n increases along certain values, n = n(1), n(2),... n(k),.... Extending and improving previous work of Nurmela…