Related papers: The Perimeter of Proper Polycubes
We provide bounds on the size of polynomial differential equations obtained by executing closure properties for D-algebraic functions. While it is easy to obtain bounds on the order of these equations, it requires some more work to derive…
The action of ring automorphisms of the polynomial ring in two variables over the real numbers on real plane curves is considered. The orbits containing degree-three polynomials are computed, with one representative per orbit being…
The problem of determining the volume of a tubular neighbourhood has a long and rich history. Bounds on the volume of neighbourhoods of algebraic sets have turned out to play an important role in the probabilistic analysis of condition…
We derive a mixed integer nonlinear programming formulation for the problem of finding a convex polygon with a given number of vertices that is small (diameter at most one) and has maximum perimeter. The formulation is based on a geometric…
In this paper, we give a formula to compute all the top degree graded Betti numbers of the path ideals of a cycle. As a consequence we can give a formula to compute its projective dimension and regularity.
We consider nearly-perfect cuboids (NPC), where the only irrational is one of the face diagonals. Obtained are three rational parametrizations for NPC with one parameter.
In this paper we study circles tangent to conics. We show there are generically $184$ complex circles tangent to three conics in the plane and we characterize the real discriminant of the corresponding polynomial system. We give an explicit…
Let $S$ be a set of arbitrary objects, and let $S^d=\{v_1...v_d\colon v_i\in S\}$. A polybox code is a set $V\subset S^d$ with the property that for every two words $v,w\in V$ there is $i\in [d]$ with $v_i'=w_i$, where a permutation…
Gradients of the perimeter and area of a polygon have straightforward geometric interpretations. The use of optimality conditions for constrained problems and basic ideas in triangle geometry show that polygons with prescribed area…
In this article we study the number of different cuboids $\mathcal{O}(N)$ that can be built with an arbitrary number $N$ of equal cubes. This problem is equivalent to find the number of different cuboids of volume $N$ with integer length…
Covering numbers are a powerful tool used in the development of approximation algorithms, randomized dimension reduction methods, smoothed complexity analysis, and others. In this paper we prove upper bounds on the covering number of…
We present an implementation of Redelemeier's algorithm for the enumeration of lattice animals in high dimensional lattices. The implementation is lean and fast enough to allow us to extend the existing tables of animal counts, perimeter…
A conjecture of Serre concerns the number of rational points of bounded height on a finite cover of projective space P^{n-1}. In this paper, we achieve Serre's conjecture in the special case of smooth cyclic covers of any degree when n is…
Let $d$ be a fixed positive integer and let $\epsilon>0$. It is shown that for every sufficiently large $n\geq n_0(d,\epsilon)$, the $d$-dimensional unit cube can be decomposed into exactly $n$ smaller cubes such that the ratio of the side…
We compute the exact Fourier dimension of the set of $\Psi$-well-approximable $m \times n$ matrices (and the set of $\Psi$-well-approximable numbers) in the homogeneous and inhomogeneous cases for any approximation function $\Psi$…
In this article, we present a unified algebraic-combinatorial framework for computing explicit, piecewise rational, and combinatorially indexed parametric formulas for volumes and higher moments of slices and slabs of polyhedral norm balls.…
We consider the following geometric optimization problem: find a maximum-area rectangle and a maximum-perimeter rectangle contained in a given convex polygon with $n$ vertices. We give exact algorithms that solve these problems in time…
A way to construct and classify the three dimensional polynomially deformed algebras is given and the irreducible representations is presented. for the quadratic algebras 4 different algebras are obtained and for cubic algebras 12 different…
This work concerns the distance in 2-norm from a matrix polynomial to a nearest polynomial with a specified number of its eigenvalues at specified locations in the complex plane. Perturbations are allowed only on the constant coefficient…
A permutomino of size n is a polyomino determined by particular pairs (P1, P2) of permutations of size n, such that P1(i) is different from P2(i), for all i. Here we determine the combinatorial properties and, in particular, the…