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Articles 1 - 17 of 17
Full-Text Articles in Algebra
Toward Completeness Theorem For Guarded Kleene Algebra With Tests, Hung Pham
Toward Completeness Theorem For Guarded Kleene Algebra With Tests, Hung Pham
Honors Theses
Code refactoring is a fundamental practice in software engineering, in which a program is restructured without changing the actions it performs and the results it produces. To carry out refactoring with confidence, one requires a formal method for verifying that two programs are equivalent. Guarded Kleene Algebra with Tests (GKAT) provides such a framework, an algebraic system designed to reason about a natural class of programs, namely those in which every branch and loop is governed by a Boolean condition, such as if–else and while statements. Central to GKAT is a finite set of algebraic axioms for deriving program equivalences. …
On Linear Invariants Of Hypergraphs, Clara Chaplin
On Linear Invariants Of Hypergraphs, Clara Chaplin
Honors Theses
We introduce linear invariants of hypergraphs as a way to study hypergraphs by their tensor representations. Our primary research goal is to determine what information linear invariants capture about the hypergraphs they arise from. We first investigate the centroid, which is shown to determine the connected components of a hypergraph. Next, we study the derivations of a hypergraph, and use this linear invariant to define a quotient operator $Q_\mathrm{Der}$ on the collection of all hypergraphs. This operator is shown to be a closure operator in that $Q_\mathrm{Der}(Q_\mathrm{Der}(\mathcal{H}))=Q_\mathrm{Der}(\mathcal{H})$ for any hypergraph $\mathcal{H}$. We apply the operator $Q_\mathrm{Der}$ to synthetically generated hypergraphs, …
Quasisymmetric Functions Distinguishing Trees, Jean-Christophe Aval, Karimatou Djenabou, Peter R. W. Mcnamara
Quasisymmetric Functions Distinguishing Trees, Jean-Christophe Aval, Karimatou Djenabou, Peter R. W. Mcnamara
Faculty Journal Articles
A famous conjecture of Stanley states that his chromatic symmetric function distinguishes trees. As a quasisymmetric analogue, we conjecture that the chromatic quasisymmetric function of Shareshian and Wachs and of Ellzey distinguishes directed trees. This latter conjecture would be implied by an affirmative answer to a question of Hasebe and Tsujie about the P-partition enumerator distinguishing posets whose Hasse diagrams are trees. They proved the case of rooted trees and our results include a generalization of their result.
Positivity Among P-Partition Generating Functions, Nathan R. T. Lesnevich, Peter R. W. Mcnamara
Positivity Among P-Partition Generating Functions, Nathan R. T. Lesnevich, Peter R. W. Mcnamara
Faculty Journal Articles
We seek simple conditions on a pair of labeled posets that determine when the difference of their (P,ω)-partition enumerators is F-positive, i.e., positive in Gessel's fundamental basis. This is a quasisymmetric analogue of the extensively studied problem of finding conditions on a pair of skew shapes that determine when the difference of their skew Schur functions is Schur-positive. We determine necessary conditions and separate sufficient conditions for F-positivity, and show that a broad operation for combining posets preserves positivity properties. We conclude with classes of posets for which we have conditions that are both necessary and …
A Contraction Based Approach To Tensor Isomorphism, Anh Kieu
A Contraction Based Approach To Tensor Isomorphism, Anh Kieu
Honors Theses
Tensor isomorphism is a hard problem in computational complexity theory. Tensor isomorphism arises not just in mathematics, but also in other applied fields like Machine Learning, Cryptography, and Quantum Information Theory (QIT). In this thesis, we develop a new approach to testing (non)-isomorphism of tensors that uses local information from "contractions" of a tensor to detect differences in global structures. Specifically, we use projective geometry and tensor contractions to create a labelling data structure for a given tensor, which can be used to compare and distinguish tensors. This contraction labelling isomorphism test is quite general, and its practical potential remains …
On The Ranks Of String C-Group Representations For Symplectic And Orthogonal Groups, Peter A. Brooksbank
On The Ranks Of String C-Group Representations For Symplectic And Orthogonal Groups, Peter A. Brooksbank
Faculty Journal Articles
We determine the ranks of string C-group representations of the groups PSp(4,q)=Ω(5,q), and comment on those of higher-dimensional symplectic and orthogonal groups.
Exact Sequences Of Inner Automorphisms Of Tensors, Peter A. Brooksbank
Exact Sequences Of Inner Automorphisms Of Tensors, Peter A. Brooksbank
Faculty Journal Articles
We produce a long exact sequence whose terms are unit groups of associative algebras that behave as inner automorphisms of a given tensor. Our sequence generalizes known sequences for associative and non-associative algebras. In a manner similar to those, our sequence facilitates inductive reasoning about, and calculation of the groups of symmetries of a tensor. The new insights these methods afford can be applied to problems ranging from understanding algebraic structures to distinguishing entangled states in particle physics.
On The Ranks Of String C-Group Representations For Symplectic And Orthogonal Groups, Peter A. Brooksbank
On The Ranks Of String C-Group Representations For Symplectic And Orthogonal Groups, Peter A. Brooksbank
Faculty Journal Articles
We determine the ranks of string C-group representations of 4-dimensional projective symplectic groups over a finite field, and comment on those of higher-dimensional symplectic and orthogonal groups.
Orthogonal Groups In Characteristic 2 Acting On Polytopes Of High Rank, Peter A. Brooksbank, Dimitri Leemans, John T. Ferrara
Orthogonal Groups In Characteristic 2 Acting On Polytopes Of High Rank, Peter A. Brooksbank, Dimitri Leemans, John T. Ferrara
Faculty Journal Articles
No abstract provided.
Rank Reduction Of String C-Group Representations, Peter A. Brooksbank, Dimitri Leemans
Rank Reduction Of String C-Group Representations, Peter A. Brooksbank, Dimitri Leemans
Faculty Journal Articles
We show that a rank reduction technique for string C-group representations first used in [Adv. Math. 228 (2018), pp. 3207–3222] for the symmetric groups generalizes to arbitrary settings. The technique permits us, among other things, to prove that orthogonal groups defined on d-dimensional modules over fields of even order greater than 2 possess string C-group representations of all ranks. The broad applicability of the rank reduction technique provides fresh impetus to construct, for suitable families of groups, string C-groups of highest possible rank. It also suggests that the alternating group Alt(11)—the only known group having “rank gaps”—is perhaps more unusual …
Testing Isomorphism Of Graded Algebras, Peter A. Brooksbank, James B. Wilson, Eamonn A. O'Brien
Testing Isomorphism Of Graded Algebras, Peter A. Brooksbank, James B. Wilson, Eamonn A. O'Brien
Faculty Journal Articles
We present a new algorithm to decide isomorphism between finite graded algebras. For a broad class of nilpotent Lie algebras, we demonstrate that it runs in time polynomial in the order of the input algebras. We introduce heuristics that often dramatically improve the performance of the algorithm and report on an implementation in Magma.
Positivity Among P-Partition Generating Functions Of Partially Ordered Sets, Nate Lesnevich
Positivity Among P-Partition Generating Functions Of Partially Ordered Sets, Nate Lesnevich
Honors Theses
We find necessary and separate sufficient conditions for the difference between two labeled partially ordered set's (poset) partition generating functions to be positive in the fundamental basis. We define the notion of a jump sequence for a poset and show how different conditions on the jump sequences of two posets are necessary for those posets to have an order relation in the fundamental basis. Our sufficient conditions are of two types. First, we show how manipulating a poset's Hasse diagram produces a poset that is greater according to the fundamental basis. Secondly, we also provide tools to explain posets that …
Extensions Of The Morse-Hedlund Theorem, Eben Blaisdell
Extensions Of The Morse-Hedlund Theorem, Eben Blaisdell
Honors Theses
Bi-infinite words are sequences of characters that are infinite forwards and backwards; for example "...ababababab...". The Morse-Hedlund theorem says that a bi-infinite word f repeats itself, in at most n letters, if and only if the number of distinct subwords of length n is at most n. Using the example, "...ababababab...", there are 2 subwords of length 3, namely "aba" and "bab". Since 2 is less than 3, we must have that "...ababababab..." repeats itself after at most 3 letters. In fact it does repeat itself every two letters. …
Polytopes Of Large Rank For Psl(4,Q), Peter A. Brooksbank, Dimitri Leemans
Polytopes Of Large Rank For Psl(4,Q), Peter A. Brooksbank, Dimitri Leemans
Faculty Journal Articles
This paper examines abstract regular polytopes whose automorphism group is the projective special linear group PSL(4,q). For q odd we show that polytopes of rank 4 exist by explicitly constructing PSL(4,q) as a string C-group of that rank. On the other hand, we show that no abstract regular polytope exists whose group of automorphisms is PSL(4,2k).
The Module Isomorphism Problem Reconsidered, Peter A. Brooksbank
The Module Isomorphism Problem Reconsidered, Peter A. Brooksbank
Faculty Journal Articles
Algorithms to decide isomorphism of modules have been honed continually over the last 30 years, and their range of applicability has been extended to include modules over a wide range of rings. Highly efficient computer implementations of these algorithms form the bedrock of systems such as GAP and MAGMA, at least in regard to computations with groups and algebras. By contrast, the fundamental problem of testing for isomorphism between other types of algebraic structures -- such as groups, and almost any type of algebra -- seems today as intractable as ever. What explains the vastly different complexity status of the …
On Groups With A Class-Preserving Outer Automorphism, Peter A. Brooksbank
On Groups With A Class-Preserving Outer Automorphism, Peter A. Brooksbank
Faculty Journal Articles
Four infinite families of 2-groups are presented, all of whose members possess an outer automorphism that preserves conjugacy classes. The groups in these families are central extensions of their predecessors by a cyclic group of order 2. For each integer r>1, there is precisely one 2-group of nilpotency class r in each of the four families. All other known families of 2-groups possessing a class-preserving outer automorphism consist entirely of groups of nilpotency class 2.
Groups Acting On Tensor Products, Peter A. Brooksbank
Groups Acting On Tensor Products, Peter A. Brooksbank
Faculty Journal Articles
Groups preserving a distributive product are encountered often in mathematics. Examples include automorphism groups of associative and non associative rings, classical groups, and automorphisms of p-groups. While the great variety of such products precludes any realistic hope of describing the general structure of the groups that preserve them, it is reasonable to expect that insight may be gained from an examination of the universal distributive products: tensor products. We give a detailed description of the groups preserving such tensor products over semisimple and semi primary rings, and present effective algorithms to construct generators for these groups. We also discuss …