相关论文: Bounds on the number of higher-dimensional partiti…

Recently, Freedman [arXiv:2301.00295] introduced the idea of packing a maximal number of links into a bounded region subject to geometric constraints, and produced upper bounds on the packing number in some cases, while commenting that…

Let $F$ be a univariate polynomial or rational fraction of degree $d$ defined over a number field. We give bounds from above on the absolute logarithmic Weil height of $F$ in terms of the heights of its values at small integers: we review…

We give a new asymptotic upper bound on the size of a code in the Grassmannian space. The bound is better than the upper bounds known previously in the entire range of distances except very large values.

Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that,…

In this article we show the rough outline of a computer algorithm to generate lower bounds on the exponential function of (in principle) arbitrary precision. We implemented this to generate all necessary analytic terms for the Boltzmann…

Using a combinatorial bijection with certain abaci diagrams, Nath and Sellers have enumerated $(s, m s \pm 1)$-core partitions into distinct parts. We generalize their result in several directions by including the number of parts of these…

We derive closed formulas for the number of $k$-coloured partitions and the number of plane partitions of $n$ in terms of the Bell polynomials.

In a series of papers, George Andrews and various coauthors successfully revitalized seemingly forgotten, powerful machinery based on MacMahon's $\Omega$ operator to systematically compute generating functions $\sum_{\la \in P}…

In this paper we prove a theorem that provides an upper bound for the density of packings of congruent copies of a given convex body in $\mathbb{R}^n$; this theorem is a generalization of the linear programming bound for sphere packings. We…

The permanent of a multidimensional matrix is the sum of products of entries over all diagonals. By Minc's conjecture, there exists a reachable upper bound on the permanent of 2-dimensional (0,1)-matrices. In this paper we obtain some…

Describing the geometry of the dual amplituhedron without reference to a particular triangulation is an open problem. In this note we introduce a new way of determining the volume of the tree-level NMHV dual amplituhedron. We show that…

In this report, we summarize the set partition enumeration problems and thoroughly explain the algorithms used to solve them. These algorithms iterate through the partitions in lexicographic order and are easy to understand and implement in…

Consider a set $X\subseteq \mathbb{R}^d$ which is 1-dense, namely, it intersects every unit ball. We show that we can get from any point to any other point in $\mathbb{R}^d$ in $n$ steps so that the intermediate points are in $X$, and the…

In this note, we obtain a formula which leads to a practical and efficient method to calculate the number of partitions of n into parts not divisible by m for given natural numbers n and m.

We prove, under suitable conditions, a lower bound on the number of pinned distances determined by small subsets of two-dimensional vector spaces over fields. For finite subsets of the Euclidean plane we prove an upper bound for their…

In this article we study the "norm" of an integer partition, which we define to be the product of the parts. This partition-theoretic statistic has appeared here and there in the literature of the last century or so, and is at the heart of…

The geometry of the dual amplituhedron is generally described in reference to a particular triangulation. A given triangulation manifests only certain aspects of the underlying space while obscuring others, therefore understanding this…

We give a relation between the existence of a Zariski decomposition and the behavior of the restricted volume of a big divisor on a smooth (complex) projective variety. Moreover, we give an analytic description of the restricted volume in…

We give a brief summary of present bounds on the size of possible extra-dimensions from collider experiments.

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…