Algebra Commons^™

Convexity Of Neural Codes, Robert Amzi Jeffs

HMC Senior Theses

An important task in neuroscience is stimulus reconstruction: given activity in the brain, what stimulus could have caused it? We build on previous literature which uses neural codes to approach this problem mathematically. A neural code is a collection of binary vectors that record concurrent firing of neurons in the brain. We consider neural codes arising from place cells, which are neurons that track an animal's position in space. We examine algebraic objects associated to neural codes, and completely characterize a certain class of maps between these objects. Furthermore, we show that such maps have natural geometric implications related to …

Multiple Problem-Solving Strategies Provide Insight Into Students’ Understanding Of Open-Ended Linear Programming Problems, Marla A. Sole

Publications and Research

Open-ended questions that can be solved using different strategies help students learn and integrate content, and provide teachers with greater insights into students’ unique capabilities and levels of understanding. This article provides a problem that was modified to allow for multiple approaches. Students tended to employ high-powered, complex, familiar solution strategies rather than simpler, more intuitive strategies, which suggests that students might need more experience working with informal solution methods. During the semester, by incorporating open-ended questions, I gained valuable feedback, was able to better model real-world problems, challenge students with different abilities, and strengthen students’ problem solving skills.

My Math Gps: Elementary Algebra Guided Problem Solving (2016 Edition), Jonathan Cornick, G Michael Guy, Karan Puri

Open Educational Resources

My Math GPS: Elementary Algebra Guided Problem Solving is a textbook that aligns to the CUNY Elementary Algebra Learning Objectives that are tested on the CUNY Elementary Algebra Final Exam (CEAFE). This book contextualizes arithmetic skills into Elementary Algebra content using a problem-solving pedagogy. Classroom assessments and online homework are available from the authors.

The Automorphism Group Of The Halved Cube, Benjamin B. Mackinnon

Theses and Dissertations

An n-dimensional halved cube is a graph whose vertices are the binary strings of length n, where two vertices are adjacent if and only if they differ in exactly two positions. It can be regarded as the graph whose vertex set is one partite set of the n-dimensional hypercube, with an edge joining vertices at hamming distance two. In this thesis we compute the automorphism groups of the halved cubes by embedding them in R n and realizing the automorphism group as a subgroup of GLn(R). As an application we show that a halved cube is a circulant graph if …

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(4,2^k).

Zeon Roots, Lisa M. Dollar, G. Stacey Staples

SIUE Faculty Research, Scholarship, and Creative Activity

Zeon algebras can be thought of as commutative analogues of fermion algebras, and they can be constructed as subalgebras within Clifford algebras of appropriate signature. Their inherent combinatorial properties make them useful for applications in graph enumeration problems and evaluating functions defined on partitions. In this paper, kth roots of invertible zeon elements are considered. More specifically, conditions for existence of roots are established, numbers of existing roots are determined, and computational methods for constructing roots are developed. Recursive and closed formulas are presented, and specific low-dimensional examples are computed with Mathematica. Interestingly, Stirling numbers of the first kind appear …

Zeons, Orthozeons, And Graph Colorings, G. Stacey Staples, Tiffany Stellhorn

SIUE Faculty Research, Scholarship, and Creative Activity

No abstract provided.

The Spectral Theorem For Quaternionic Unbounded Normal Operators Based On The S-Spectrum, Daniel Alpay, Fabrizio Colombo, David P. Kimsey

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we prove the spectral theorem for quaternionic unbounded normal operators using the notion of S-spectrum. The proof technique consists of first establishing a spectral theorem for quaternionic bounded normal operators and then using a transformation which maps a quaternionic unbounded normal operator to a quaternionic bounded normal operator. With this paper we complete the foundation of spectral analysis of quaternionic operators. The S-spectrum has been introduced to define the quaternionic functional calculus but it turns out to be the correct object also for the spectral theorem for quaternionic normal operators. The fact that the correct notion of …

On Skew-Constacyclic Codes, Neville Lyons Fogarty

Theses and Dissertations--Mathematics

Cyclic codes are a well-known class of linear block codes with efficient decoding algorithms. In recent years they have been generalized to skew-constacyclic codes; such a generalization has previously been shown to be useful. We begin with a study of skew-polynomial rings so that we may examine these codes algebraically as quotient modules of non-commutative skew-polynomial rings. We introduce a skew-generalized circulant matrix to aid in examining skew-constacyclic codes, and we use it to recover a well-known result on the duals of skew-constacyclic codes from Boucher/Ulmer in 2011. We also motivate and develop a notion of idempotent elements in these …

Kronecker's Theory Of Binary Bilinear Forms With Applications To Representations Of Integers As Sums Of Three Squares, Jonathan A. Constable

Theses and Dissertations--Mathematics

In 1883 Leopold Kronecker published a paper containing “a few explanatory remarks” to an earlier paper of his from 1866. His work loosely connected the theory of integral binary bilinear forms to the theory of integral binary quadratic forms. In this dissertation we discover the statements within Kronecker's paper and offer detailed arithmetic proofs. We begin by developing the theory of binary bilinear forms and their automorphs, providing a classification of integral binary bilinear forms up to equivalence, proper equivalence and complete equivalence.

In the second chapter we introduce the class number, proper class number and complete class number as …

The Diameter Of A Rouquier Block, Andrew Mayer

Williams Honors College, Honors Research Projects

For my Honors Research Project, I will be researching special properties of Rouquier blocks that represent the partitions of integers. This problem is motivated by ongoing work in representation theory of the symmetric group. For each integer n and each prime p, there is an object called a Rouquier block; this block can be visualized as a collection of points in a plane, each corresponding to a partition. In this group of points, we say a pair of points is “connected” if certain conditions on the partitions are met. We compare each partition with each other partition, add edges when …

Realizing The 2-Associahedron, Patrick N. Tierney

HMC Senior Theses

The associahedron has appeared in numerous contexts throughout the field of mathematics. By representing the associahedron as a poset of tubings, Michael Carr and Satyan L. Devadoss were able to create a gener- alized version of the associahedron in the graph-associahedron. We seek to create an alternative generalization of the associahedron by considering a particle-collision model. By extending this model to what we dub the 2- associahedron, we seek to further understand the space of generalizations of the associahedron.

Multi-Type Display Calculus For Propositional Dynamic Logic, Sabine Frittella, Giuseppe Greco, Alexander Kurz, Alessandra Palmigiano

Engineering Faculty Articles and Research

We introduce a multi-type display calculus for Propositional Dynamic Logic (PDL). This calculus is complete w.r.t. PDL, and enjoys Belnap-style cut-elimination and subformula property.

Tool Support For Reasoning In Display Calculi, Samuel Balco, Sabine Frittella, Giuseppe Greco, Alexander Kurz, Alessandra Palmigiano

Engineering Faculty Articles and Research

We present a tool for reasoning in and about propositional sequent calculi. One aim is to support reasoning in calculi that contain a hundred rules or more, so that even relatively small pen and paper derivations become tedious and error prone. As an example, we implement the display calculus D.EAK of dynamic epistemic logic. Second, we provide embeddings of the calculus in the theorem prover Isabelle for formalising proofs about D.EAK. As a case study we show that the solution of the muddy children puzzle is derivable for any number of muddy children. Third, there is a set of meta-tools, …

Characterizations Of Rectangular (Para)-Unitary Rational Functions, Daniel Alpay, Palle Jorgensen, Izchak Lewkowicz

Mathematics, Physics, and Computer Science Faculty Articles and Research

We here present three characterizations of not necessarily causal, rational functions which are (co)-isometric on the unit circle:
(i) through the realization matrix of Schur stable systems,
(ii) the Blaschke-Potapov product, which is then employed to introduce an easy-to-use description of all these functions with dimensions and McMillan degree as parameters,
(iii) through the (not necessarily reducible) Matrix Fraction Description (MFD).
In cases (ii) and (iii) the poles of the rational functions involved may be anywhere in the complex plane, but the unit circle (including both zero and infinity). A special attention is devoted to exploring the gap between the …

Subgroups Of Finite Wreath Product Groups For P=3, Jessica L. Gonda

Williams Honors College, Honors Research Projects

Let M be the additive abelian group of 3-by-3 matrices whose entries are from the ring of integers modulo 9. The problem of determining all the normal subgroups of the regular wreath product group P=Z₉≀(Z₃ × Z₃) that are contained in its base subgroup is equivalent to the problem of determining the subgroups of M that are invariant under two particular endomorphisms of M. In this thesis we give a partial solution to the latter problem by implementing a systematic approach using concepts from group theory and linear algebra.

Smallest Eigenvalues For A Fractional Boundary Value Problem With A Fractional Boundary Condition, Angela Koester

Online Theses and Dissertations

We establish the existence of and then compare smallest eigenvalues for the fractional boundary value problems D_(0^+)^α u+λ_1 p(t)u=0 and $D_(0^+)^α u+λ_2 q(t)u=0,0< t< 1, satisfying the boundary conditions when n-1<α≤ n. First, we consider the case when 0<β

Low-Dimensional Reality-Based Algebras, Rachel Victoria Barber

Online Theses and Dissertations

In this paper we introduce the definition of a reality-based algebra (RBA) as well as a subclass of reality-based algebras, table algebras. Using sesquilinear forms, we prove that a reality-based algebra is semisimple. We look at a specific reality-based algebra of dimension 5 and provide formulas for the structure constants of this algebra. We determine by looking at these structure constants and setting conditions on specific structural components when this particular reality-based algebra is a table algebra. In fact, this will be a noncommutative table algebra of dimension 5.

Higher Order Z-Ideals In Commutative Rings, Themba Dube, Oghenetega Ighedo

Mathematics, Physics, and Computer Science Faculty Articles and Research

We study ideals that resemble z-ideals in commutative rings with identity. For each positive integer n, we say an ideal of a commutative ring A is a zⁿ-ideal in case it has the property that if a and b belong to the same maximal ideals of A, and aⁿ ϵ I , then bⁿ is also in I. The set of all zⁿ-ideals of A is denoted by A --> ʒⁿ (A). This gives an ascending chain ʒ(A) < ʒ²(A) < ʒ³(A) <… of collections of ideals, starting with the collection of z-ideals. …

Links With Finite N-Quandles, Jim Hoste, Patrick D. Shanahan

Mathematics, Statistics and Data Science Faculty Works

We prove a conjecture of Przytycki which asserts that the n-quandle of a link L in the 3-sphere is finite if and only if the fundamental group of the n-fold cyclic branched cover of the 3-sphere, branched over L, is finite.