Related papers: The Perimeter of Proper Polycubes
Using equivariant geometry, we find a universal formula that computes the number of times a general cubic surface arises in a family. As applications, we show that the PGL(4) orbit closure of a generic cubic surface has degree 96120, and…
We present an approximation algorithm for computing the diameter of a point-set in $\Re^d$. The new algorithm is sensitive to the ``hardness'' of computing the diameter of the given input, and for most inputs it is able to compute the exact…
In this paper we study polycubes: orthogonal polyhedra with axis-aligned quadrilateral faces. We present a complete characterization of polycubes of any genus based on their dual structure: a collection of oriented loops which run in each…
In this paper, we study the problem of computing a minimum-width axis-aligned cubic shell that encloses a given set of $n$ points in a three-dimensional space. A cubic shell is a closed volume between two concentric and face-parallel cubes.…
In this paper we determine a closed formula for the number of convex permutominoes of size n. We reach this goal by providing a recursive generation of all convex permutominoes of size n+1 from the objects of size n, according to the ECO…
A small polygon is a polygon of unit diameter. The maximal perimeter and the maximal width of a convex small polygon with $n=2^s$ sides are unknown when $s \ge 4$. In this paper, we propose an approach to construct convex small $n$-gons of…
In this paper, we consider a conjecture of Erd\H{o}s and Rosenfeld when the number is a perfect square. In particular, we show that every perfect square $n$ can have at most five divisors between $\sqrt{n} - c \sqrt[4]{n}$ and $\sqrt{n} + c…
We show that given any polynomial ring R over a field, and any ideal J in R which is generated by three cubic forms, the projective dimension of R/J is at most 36. We also settle the question whether ideals generated by three cubic forms…
We present a deterministic polynomial-time algorithm for estimating the volume of a hypercube intersected by a fixed number of constraints of the type $f(x) \leq b$, where $f$ is the sum of univariate functions that are each nonnegative,…
We investigate the problem of the maximum number of cubic subwords (of the form $www$) in a given word. We also consider square subwords (of the form $ww$). The problem of the maximum number of squares in a word is not well understood.…
We show that for fixed $d>3$ and $n$ growing to infinity there are at least $(n!)^{d-2 \pm o(1)}$ different labeled combinatorial types of $d$-polytopes with $n$ vertices. This is about the square of the previous best lower bounds. As an…
We prove that the (n-2)-dimensional surface area (perimeter) of central hyperplane sections of the n-dimensional unit cube is maximal for the hyperplane perpendicular to the vector (1,1,0,...,0). This gives a positive answer to a question…
We have derived long series expansions for the perimeter generating functions of the radius of gyration of various polygons with a convexity constraint. Using the series we numerically find simple (algebraic) exact solutions for the…
A ball polyhedron is the intersection of a finite number of closed balls in $\mathbb{R}^3$ with the same radius. In this note, we study ball polyhedra in which the set of centers defining the balls have the maximum possible number of…
We prove that uniform random triangulations whose genus is proportional to their size $n$ have diameter of order $\log n$ with high probability. We also show that in such triangulations, the distances between most pairs of points differ by…
We consider the possible sizes of large sumfree sets contained in the discrete hypercube $\{1,...,n\}^k$, and we determine upper and lower bounds for the maximal size as $n$ becomes large. We also discuss a continuous analogue in which our…
We present structural properties of two-dimensional polymers as far as they can be described by percolation theory. The percolation threshold, critical exponents and fractal dimensions of clusters are determined by computer simulation and…
A rational perfect cuboid is a rectangular parallelepiped whose edges and face diagonals are given by rational numbers and whose space diagonal is equal to unity. It is described by a system of four equations with respect to six variables.…
A positive definite quadratic form is called perfect, if it is uniquely determined by its arithmetical minimum and the integral vectors attaining it. In this self-contained survey we explain how to enumerate perfect forms in $d$ variables…
We prove an isoperimetric inequality for filling cellular cycles in a high dimensional cube with cellular chains. In addition, we provide a family of cubical cellular cycles for which the exponent in the inequality is optimal.