保3 iet态射的矩阵

Morphisms of Skew Hadamard Matrices

Quaternary unit Hadamard (QUH) matrices were introduced by Fender, Kharagani and Suda along with a method to construct them at prime power orders. We present a novel construction of real Hadamard matrices from QUH matrices. Our construction recovers the result by Mukhopadhyay on the existence of real Hadamard matrices of order $q^n+q^{n-1}$ for each prime power $q\equiv 3\mod 4$. Furthermore we provide nonexistence conditions for QUH matrices.

Butson类的态射

Morphisms of Butson classes

本文引入了从k阶单位根上的Butson Hadamard矩阵集到$\ell$阶单位根上的Butson矩阵集的态射的概念。作为这种态射的具体例子，我们描述了张量积映射，它以增加维数为代价降低了Butson矩阵中出现的单位根的阶。这样的映射可以由特征值满足一定自然条件的Butson矩阵构造。我们的工作统一和generalises Turyn的建设真正的阿达玛矩阵从Butson矩阵的第4根和工作的康普顿，Craigen和德Launey 'unreal' Butson矩阵的第6根。作为一个案例研究，我们将k阶单位根上的$n \times n$ Butson矩阵集合到$\ell$阶单位根上的$2n\times 2n$ Butson矩阵集合的所有态射分类，其中$\ell < k$ .

保持词集编码三区间交换的态射

Morphisms preserving the set of words coding three interval exchange

任何Sturmian态射的亲和对，都能构造一个三元态射η，它保持编码3-区间交换的无限词的集合。确定了$SL^\pm(2,N)$中具有相同关联矩阵的亲和对的个数，并研究了对应于三元态射η的关联矩阵.

Word-representability of split graphs generated by morphisms

A graph $G=(V,E)$ is word-representable if and only if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$, $x\neq y$, alternate in $w$ if and only if $xy\in E$. A split graph is a graph in which the vertices can be partitioned into a clique and an independent set. There is a long line of research on word-representable graphs in the literature, and recently, word-representability of split graphs has attracted interest. In this paper, we first give a characterization of word-representable split graphs in terms of permutations of columns of the adjacency matrices. Then, w…

Spectra of Hadamard matrices

A Butson Hadamard matrix $H$ has entries in the kth roots of unity, and satisfies the matrix equation $HH^{\ast} = nI_{n}$. We write $\mathrm{BH}(n, k)$ for the set of such matrices. A complete morphism of Butson matrices is a map $\mathrm{BH}(n, k) \rightarrow \mathrm{BH}(m, \ell)$. In this paper, we develop a technique for controlling the spectra of certain Hadamard matrices. For each integer $t$, we construct a real Hadamard matrix $H_{t}$ of order $n_{t} = 2^{2^{t-1}-1}$ such that the minimal polynomial of $\frac{1}{\sqrt{n_{t}}}H_{t}$ is the cyclotomic polynomial $Φ_{2^{t+1}}(x)$. Such ma…

自由幺半群态射的渐近性质

Asymptotic properties of free monoid morphisms

受数制理论和整数可识别集理论的启发，本文讨论了在考虑擦除态射的情况下的态射词。Cobham证明了如果无限字$w =g(f^ω(a))$是一个态射$f$在另一个态射$g$下的不动点的象，则存在一个非擦除态射$σ$和一个编码$τ$使得$w =τ(σ^ω(b))$ .基于非负矩阵幂的渐近性质的Perron定理，我们的主要贡献是深入研究了当用非擦除态射替换擦除态射时迭代态射的增长型.我们还明确地提供了一个算法计算$σ$和$τ$从$f$和$g$。