Dane Flannery
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View article: Generalized partially bent functions, generalized perfect arrays, and cocyclic Butson matrices
Generalized partially bent functions, generalized perfect arrays, and cocyclic Butson matrices Open
In a recent survey, Schmidt compiled equivalences between generalized bent functions, group invariant Butson Hadamard matrices, and abelian splitting relative difference sets. We establish a broader network of equivalences by considering B…
View article: Generalized partially bent functions, generalized perfect arrays and cocyclic Butson matrices
Generalized partially bent functions, generalized perfect arrays and cocyclic Butson matrices Open
In a recent survey, Schmidt compiled equivalences between generalized bent functions, group invariant Butson Hadamard matrices, and abelian splitting relative difference sets.We establish a broader network of equivalencesby considering But…
View article: Generalized partially bent functions, generalized perfect arrays and cocyclic Butson matrices
Generalized partially bent functions, generalized perfect arrays and cocyclic Butson matrices Open
In a recent survey, Schmidt compiled equivalences between generalized bent functions, group invariant Butson Hadamard matrices, and abelian splitting relative difference sets. We establish a broader network of equivalences by considering B…
View article: Issue Information
Issue Information Open
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View article: Classifying finite monomial linear groups of prime degree in characteristic zero
Classifying finite monomial linear groups of prime degree in characteristic zero Open
Let $p$ be a prime and let $\mathbb{C}$ be the complex field. We explicitly classify the finite solvable irreducible monomial subgroups of $\mathrm{GL}(p,\mathbb{C})$ up to conjugacy. That is, we give a complete and irredundant list of $\m…
View article: Locally nilpotent linear groups
Locally nilpotent linear groups Open
We survey aspects of locally nilpotent linear groups. Then we obtain a new classification; namely, we classify the irreducible maximal locally nilpotent subgroups of $\mathrm{GL}(q, \mathbb F)$ for prime $q$ and any field $\mathbb F$.
View article: Almost supplementary difference sets and quaternary sequences
Almost supplementary difference sets and quaternary sequences Open
We introduce almost supplementary difference sets (ASDS). For odd $m$, certain ASDS in ${\mathbb Z}_m$ that have amicable incidence matrices are equivalent to quaternary sequences of odd length $m$ with optimal autocorrelation. As one cons…
View article: Algorithms for computing with nilpotent matrix groups over infinite domains
Algorithms for computing with nilpotent matrix groups over infinite domains Open
We develop methods for computing with matrix groups defined over a range of infinite domains, and apply those methods to the design of algorithms for nilpotent groups. In particular, we provide a practical algorithm to test nilpotency of m…
View article: Algorithms for arithmetic groups with the congruence subgroup property
Algorithms for arithmetic groups with the congruence subgroup property Open
We develop practical techniques to compute with arithmetic groups $H\leq \mathrm{SL}(n,\mathbb{Q})$ for $n>2$. Our approach relies on constructing a principal congruence subgroup in $H$. Problems solved include testing membership in $H$, a…
View article: Deciding finiteness of matrix groups in positive characteristic
Deciding finiteness of matrix groups in positive characteristic Open
We present a new algorithm to decide finiteness of matrix groups defined over a field of positive characteristic. Together with previous work for groups in zero characteristic, this provides the first complete solution of the finiteness pr…
View article: Algorithms for the Tits alternative and related problems
Algorithms for the Tits alternative and related problems Open
We present an algorithm that decides whether a finitely generated linear group over an infinite field is solvable-by-finite: a computationally effective version of the Tits alternative. We also give algorithms to decide whether the group i…
View article: Recognizing finite matrix groups over infinite fields
Recognizing finite matrix groups over infinite fields Open
We present a uniform methodology for computing with finitely generated matrix groups over any infinite field. As one application, we completely solve the problem of deciding finiteness in this class of groups. We also present an algorithm …
View article: Algorithms for linear groups of finite rank
Algorithms for linear groups of finite rank Open
Let $G$ be a finitely generated solvable-by-finite linear group. We present an algorithm to compute the torsion-free rank of $G$ and a bound on the Prüfer rank of $G$. This yields in turn an algorithm to decide whether a finitely generated…
View article: Integrality and arithmeticity of solvable linear groups
Integrality and arithmeticity of solvable linear groups Open
We develop a practical algorithm to decide whether a finitely generated subgroup of a solvable algebraic group $G$ is arithmetic. This incorporates a procedure to compute a generating set of an arithmetic subgroup of $G$. We also provide a…
View article: Practical Computation with Linear Groups Over Infinite Domains
Practical Computation with Linear Groups Over Infinite Domains Open
We survey recent progress in computing with finitely generated linear groups over infinite fields, describing the mathematical background of a methodology applied to design practical algorithms for these groups. Implementations of the algo…
View article: Algebra, matrices, and computers
Algebra, matrices, and computers Open
What part does algebra play in representing the real world abstractly? How can algebra be used to solve hard mathematical problems with the aid of modern computing technology? We provide answers to these questions that rely on the theory o…
View article: Computing Congruence Quotients of Zariski Dense Subgroups
Computing Congruence Quotients of Zariski Dense Subgroups Open
We obtain a computational realization of the strong approximation theorem. That is, we develop algorithms to compute all congruence quotients modulo rational primes of a finitely generated Zariski dense group $H \leq \mathrm{SL}(n, \mathbb…
View article: On quasi‐orthogonal cocycles
On quasi‐orthogonal cocycles Open
We introduce the notion of quasi‐orthogonal cocycle . This is motivated in part by the maximal determinant problem for square ‐matrices of size congruent to 2 modulo 4. Quasi‐orthogonal cocycles are analogous to the orthogonal cocycles of …
View article: GAP Functionality for Zariski Dense Groups
GAP Functionality for Zariski Dense Groups Open
In this document we describe the functionality of GAP [4] routines for Zariski dense or arithmetic groups that are developed in [1, 2, 3]. The research underlying the software was supported through the programme "Research in Pairs", at the…
View article: Experimenting with Zariski Dense Subgroups
Experimenting with Zariski Dense Subgroups Open
We give a method to describe all congruence images of a finitely generated Zariski dense group $H\leq \SL(n, \Z)$. The method is applied to obtain efficient algorithms for solving this problem in odd prime degree $n$; if $n=2$ then we comp…
View article: Zariski density and computing in arithmetic groups
Zariski density and computing in arithmetic groups Open
For n > 2, let Gamma(n) denote either SL( n, Z) or Sp( n, Z). We give a practical algorithm to compute the level of the maximal principal congruence subgroup in an arithmetic group H <= Gamma(n). This forms the main component…
View article: Zariski Density and Computing in Arithmetic Groups
Zariski Density and Computing in Arithmetic Groups Open
For $n > 2$, let $Γ$ denote either $SL(n, Z)$ or $Sp(n, Z)$. We give a practical algorithm to compute the level of the maximal principal congruence subgroup in an arithmetic group $H\leq Γ$. This forms the main component of our methods for…