On the representation of bent functions by bent rectangles

Twin bent functions and Clifford algebras

This paper examines a pair of bent functions on $\mathbb{Z}_2^{2m}$ and their relationship to a necessary condition for the existence of an automorphism of an edge-coloured graph whose colours are defined by the properties of a canonical basis for the real representation of the Clifford algebra $\mathbb{R}_{m,m}.$ Some other necessary conditions are also briefly examined.

Area, Perimeter, Height, and Width of Rectangle Visibility Graphs

A rectangle visibility graph (RVG) is represented by assigning to each vertex a rectangle in the plane with horizontal and vertical sides in such a way that edges in the graph correspond to unobstructed horizontal and vertical lines of sight between their corresponding rectangles. To discretize, we consider only rectangles whose corners have integer coordinates. For any given RVG, we seek a representation with smallest bounding box as measured by its area, perimeter, width, or height (height is assumed not to exceed width). We derive a number of results regarding these parameters. Using these …

在非降字与 Smirnov 字中计数 $r\ imes s$ 矩形

Counting $r\times s$ rectangles in nondecreasing and Smirnov words

本文研究了非降字与 Smirnov 字中 $r\ imes s$ 矩形容量的分布，求得了其二元生成函数及总数生成函数，补充了关于一般字和 Catalan 字的近期研究。

Sturmian单词上的平衡矩形与最小偏差区间

Balanced rectangles over Sturmian words and minimal discrepancy intervals

研究Sturmian单词斜率为α的m×n矩形矩阵的平衡性质，通过Ostrowski表示描述其特性，扩展了Anselmo等人和Shallit的研究，采用nα mod 1的分布方法。

Generalized bent functions - sufficient conditions and related constructions

The necessary and sufficient conditions for a class of functions $f:\mathbb{Z}_2^n \rightarrow \mathbb{Z}_q$, where $q \geq 2$ is an even positive integer, have been recently identified for $q=4$ and $q=8$. In this article we give an alternative characterization of the generalized Walsh-Hadamard transform in terms of the Walsh spectra of the component Boolean functions of $f$, which then allows us to derive sufficient conditions that $f$ is generalized bent for any even $q$. The case when $q$ is not a power of two, which has not been addressed previously, is treated separately and a suitable r…

有限阿贝尔群上过约束字符和与广义弯曲、平坦和景观函数的分解

Overconstrained character sums over finite abelian groups and decompositions of generalized bent, plateaued and landscape functions

本文研究了有限阿贝尔群上的过约束字符和，提出了一种新的2^ℓ进制表示方法，并探讨了广义弯曲、平坦和景观函数的分解及其性质。

Plethysms of symmetric functions and representations of $\mathrm{SL}_2(\mathbb{C})$

Let $\nabla^λ$ denote the Schur functor labelled by the partition $λ$ and let $E$ be the natural representation of $\mathrm{SL}_2(\mathbb{C})$. We make a systematic study of when there is an isomorphism $\nabla^λ\!\mathrm{Sym}^\ell \!E \cong \nabla^μ\!\mathrm{Sym}^m \! E$ of representations of $\mathrm{SL}_2(\mathbb{C})$. Generalizing earlier results of King and Manivel, we classify all such isomorphisms when $λ$ and $μ$ are conjugate partitions and when one of $λ$ or $μ$ is a rectangle. We give a complete classification when $λ$ and $μ$ each have at most two rows or columns or is a hook parti…