arXiv subjects

Yeeka Yau

Publications and source records attributed to Yeeka Yau.

7 recordsLinked to original sources

arXiv2025

Bilinear maps on the ring of strictly upper triangular matrices

Let $R$ be a 2-torsion free unital ring and $N_n=N_n(R)$ the ring of strictly upper triangular matrices with entries in $R$ and center $Z=Z(N_n)$. It has been previously shown that any linear map $f:N_n\rightarrow N_n$ satisfying the condition $[f(X),X]=0$ must be of the form $f(X)=\lambda X+\mu(X)$ for some $\lambda\in R$ and additive map $\mu$ defined on $N_n$. We extend these known results by providing a complete description of the bilinear maps $f:N_n\times N_n\rightarrow N_n$ satisfying the identity $[f(X,X),X]=0$ for all $X\in N_n$.

math.RA↗

arXiv2024

Combinatorics of cone types in Coxeter groups

In this article, we establish some new combinatorial properties of cone types in Coxeter groups. Firstly, we show that for any element $x$ in a Coxeter group $W$ and root $\beta$ in its inversion set $\Phi(x)$, the set of elements $y \in W$ satisfying $\Phi(x) \cap \Phi(y) = \{ \beta \}$ is convex in the weak order and admits a unique minimal representative. This is strongly connected to determining the cone type of elements of $W$ and leads to efficient computational methods to determine whether arbitrary elements of $W$ have the same cone type.

math.GR↗

arXiv2024

An artificial neural network approach to finding the key length of the Vigenère cipher

In this article, we create an artificial neural network (ANN) that combines both classical and modern techniques for determining the key length of a Vigenère cipher. We provide experimental evidence supporting the accuracy of our model for a wide range of parameters. We also discuss the creation and features of this ANN along with a comparative analysis between our ANN, the index of coincidence, and the twist-based algorithms.

cs.CR↗

arXiv2023

A Pair of Garside Shadows

We prove that the smallest elements of Shi parts and cone type parts exist and form Garside shadows. The latter resolves a conjecture of Parkinson and the second author as well as a conjecture of Hohlweg, Nadeau and Williams.

math.GR↗

arXiv2023

Modifying twist algorithms for determining the key length of a Vigenère cipher

In this article, we analyze and improve upon the twist-based algorithms introduced by Barr--Simoson and Park--Kim--Cho--Yum for determining the key length of a Vigenère cipher. We provide an in-depth discussion on how the domain of the twist index affects the accuracy of these algorithms along with supporting experimental evidence. We also introduce a new twist-based algorithm, the twist$^{++}$ algorithm, and show this algorithm is more accurate than the twist$^{+}$ algorithm for a wide range of key lengths and text lengths.

cs.CR↗

arXiv2021

Cone Types, Automata, and Regular Partitions in Coxeter Groups

In this article we introduce the notion of a \textit{regular partition} of a Coxeter group. We develop the theory of these partitions, and show that the class of regular partitions is essentially equivalent to the class of automata (not necessarily finite state) recognising the language of reduced words in the Coxeter group. As an application of this theory we prove that each cone type in a Coxeter group has a unique minimal length representative. This result can be seen as an analogue of Shi's classical result that each component of the Shi arrangement of an affine Coxeter group has a unique minimal length element. We further develop the theory of cone types in Coxeter groups by identifying the minimal set of roots required to express a cone type as an intersection of half-spaces. This set of \textit{boundary roots} is closely related to the elementary inversion sets of Brink and Howlett, and also to the notion of the base of an inversion set introduced by Dyer.

math.CO↗

arXiv2018

Coxeter systems for which the Brink-Howlett automaton is minimal

In this article we classify the Coxeter systems for which the Brink-Howlett automaton is minimal. We show that this automaton is minimal if and only if each elementary root is supported on a standard spherical subsystem, thereby resolving a conjecture of Hohlweg, Nadeau and Williams.

math.GR↗