Algebra Commons^™

My Math Gps: Elementary Algebra Guided Problem Solving, 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. In a paper to appear in PRIMUS, this textbook and accompanying pedagogy showed significant increases in learning outcomes.

Betti Sequences Over Local Rings And Connected Sums Of Gorenstein Rings, Zheng Yang

Department of Mathematics: Dissertations, Theses, and Student Research

This thesis consists of two parts:

1) Polynomial growth of Betti sequences over local rings (Chapter 2),

2) Connected sums of Gorenstein rings (Chapter 3).

Chapter 1 gives an introduction for the two topics discussed in this thesis.

The first part of the thesis deals with modules over complete intersections using free resolutions. The asymptotic patterns of the Betti sequences of the finitely generated modules over a local ring R reflect and affect the singularity of R. Given a commutative noetherian local ring and an integer c, sufficient conditions and necessary conditions are obtained for all Betti sequences …

Hurwitz's Theorem, Marianna Malek

Masters Essays

No abstract provided.

On The Fattening Of Lines In P3, Mike Janssen

Faculty Work Comprehensive List

We follow the lead of Bocci and Chiantini and show how differences in the invariant alpha can be used to classify certain classes of subschemes of P^3. Specifically, we will seek to classify arithmetically Cohen-Macaulay codimension 2 subschemes of P^3 in the manner Bocci and Chiantini classified points in P^2. The first section will seek to motivate our consideration of the invariant alpha by relating it to the Hilbert function and gamma, following the work of Bocci and Chiantini, and Dumnicki, et. al. The second section will contain our results classifying arithmetically Cohen-Macaulay codimension 2 subschemes of P^3. This work …

Symmetric Presentations And Related Topics, Mashael U. Alharbi

Electronic Theses, Projects, and Dissertations

In this thesis, we have presented our discovery of symmetric presentations of a number of non-abelian simple groups, including the Mathieu group M₁₂. We have given several progenitors, permutation and monomial, including 2*⁴:(2²:3), 2*⁵:D₁₀, 2*⁸:((4X2).D₄), 3*⁷:_m L₂(7), 2*⁶:(Z₃ wr Z₂), and 2*²⁴: (2^. A₅) and their homomorphic images which include 4.(M₁₂:2), the group of automorphisms of M₁₂ and several classical groups. We have given the isomorphism type of …

A Fundamental Unit Of O_K, Susana L. Munoz

Electronic Theses, Projects, and Dissertations

In the classical case we make use of Pells equation to compute units in the ring O_F. Consider the parallel to the classical case and the quadratic field extension that creates the ring O_K. We use the generalized Pell's equation to find the units in this ring since they are solutions. Through the use of continued fractions we may further characterize this ring and compute its units.

Non-Isomorphic Real Simple Lie Algebras Of The Same Complex Type And Character, Ian M. Anderson

Tutorials on... in 1 hour or less

Complex simple Lie algebras are classified by their root types. The type of a real simple Lie algebra is the root type of the associated complex algebra. The character of a real simple Lie algebra is the signature of its Killing form.

For many root types, the character is sufficient to uniquely classify the corresponding real Lie algebras. However, one should not take this statement to be literally true – there are a few cases where the character does not suffice to distinguish all possible real forms.

In this worksheet we will show that the 2 real non-isomorphic Lie algebras …

My Finite Field, Matthew Schroeder

Journal of Humanistic Mathematics

A love poem written in the language of mathematics.

A Rank 7 Pfaffian System On A 15-Dimensional Manifold With F4 Symmetry Algebra, Ian M. Anderson

Tutorials on... in 1 hour or less

Let I be a differential system on a manifold M. The infinitesimal symmetry algebra of I is the set of all vectors fields X on M such that preserve I. In this worksheet we present an example, due to E. Cartan of a rank 7 Pfaffian system on a 15-dimensional manifold whose infinitesimal symmetry algebra is the split real form of the exceptional Lie algebra f4 .

Cartan Involutions And Cartan Decompositions Of A Semi-Simple Lie Algebra, Ian M. Anderson

Tutorials on... in 1 hour or less

In this worksheet we shall review the basic definitions and properties of Cartan involutions and Cartan decompositions and illustrate these using the DifferentialGeometry software package for Lie algebras.

Multiplication Rules For Schur And Quasisymmetric Schur Functions, Jennifer Anderson

Theses, Dissertations and Capstones

An important problem in algebraic combinatorics is finding expansions of products of symmetric functions as sums of symmetric functions. Schur functions form a well-known basis for the ring of symmetric functions. The Littlewood-Richardson rule was introduced to expand the product of two Schur functions as a positive sum of Schur functions. Remmel and Whitney introduced an algorithmic way to find the coefficients of Schur functions appearing in the expansion. Haglund et al. introduced quasisymmetric Schur functions as a refinement of Schur functions. For quasisymmetric Schur functions, the Littlewood-Richardson rule was introduced to expand the product of a Schur and quasisymmetric …

Some Problems In The Representation Theory Of Simple Modular Lie Algebras, Georgia Benkart, Jorg Feldvoss

University Faculty and Staff Publications

No abstract provided.

Polynomials Occuring In Generating Function Identities For B-Ary Partitions, David Dakota Blair

Graduate Student Publications and Research

Let p_b(n) be the number of integer partitions of n whose parts are powers of b. For each m there is a generating function identity:

f_m(b,q)\sum_{n} p_b(n) q^n = (1-q)^m \sum_{n} p_b(b^m n q)q^n

where n ranges over all integer values. The proof of this identity appears in the doctoral thesis of the author. For more information see http://dakota.tensen.net/2015/rp/.

This dataset is a JSON object with keys m from 1 to 23 whose values are f_m(b,q).

The Central Hankel Transform, Matthew J. Levine

Honors Theses

This honors thesis presents the Hankel transform on an integer sequence, a function with colorful mathematical history and rich theoretical background. We then introduce a matricial Toeplitz transform that parallels some of most famous qualities of the Hankel transform, especially when in consideration of popular sequences like the Fibonacci numbers. The result is a characterization of the injectivity of this new function, a description of some of its interesting behaviors, and a discussion of a few new Fibonacci identities.

A New Resolvent Equation For The S-Functional Calculus, Daniel Alpay, Fabrizio Colombo, Jonathan Gantner, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

The S-functional calculus is a functional calculus for (n + 1)-tuples of non necessarily commuting operators that can be considered a higher dimensional version of the classical Riesz-Dunford functional calculus for a single operator. In this last calculus, the resolvent equation plays an important role in the proof of several results. Associated with the S-functional calculus there are two resolvent operators: the left S−1 L (s, T ) and the right one S−1 R (s, T ), where s = (s0, s1, . . . , sn) ∈ Rn+1 and T = (T0, T1, . . . , Tn) is …

Kravchuk Polynomials And Induced/Reduced Operators On Clifford Algebras, G. Stacey Staples

SIUE Faculty Research, Scholarship, and Creative Activity

Kravchuk polynomials arise as orthogonal polynomials with respect to the binomial distribution and have numerous applications in harmonic analysis, statistics, coding theory, and quantum probability. The relationship between Kravchuk polynomials and Clifford algebras is multifaceted. In this paper, Kravchuk polynomials are discovered as traces of conjugation operators in Clifford algebras, and appear in Clifford Berezin integrals of Clifford polynomials. Regarding Kravchuk matrices as linear operators on a vector space V, the action induced on the Clifford algebra over V is equivalent to blade conjugation, i.e., reflections across sets of orthogonal hyperplanes. Such operators also have a natural interpretation in …

Clifford Algebra Decompositions Of Conformal Orthogonal Group Elements, G. Stacey Staples, David Wylie

SIUE Faculty Research, Scholarship, and Creative Activity

Beginning with a finite-dimensional vector space V equipped with a nondegenerate quadratic form Q, we consider the decompositions of elements of the conformal orthogonal group CO_Q(V), defined as the direct product of the orthogonal group O_Q(V) with dilations. Utilizing the correspondence between conformal orthogonal group elements and ``decomposable'' elements of the associated Clifford algebra, Cl_Q(V), a decomposition algorithm is developed. Preliminary results on complexity reductions that can be realized passing from additive to multiplicative representations of invertible elements are also presented with examples. The approach here is …

On Representations Of Semigroups Having Hypercube-Like Cayley Graphs, Cody Cassiday, G. Stacey Staples

SIUE Faculty Research, Scholarship, and Creative Activity

The $n-dimensional hypercube, or n-cube, is the Cayley graph of the Abelian group Z₂ⁿ. A number of combinatorially-interesting groups and semigroups arise from modified hypercubes. The inherent combinatorial properties of these groups and semigroups make them useful in a number of contexts, including coding theory, graph theory, stochastic processes, and even quantum mechanics. In this paper, particular groups and semigroups whose Cayley graphs are generalizations of hypercubes are described, and their irreducible representations are characterized. Constructions of faithful representations are also presented for each semigroup. The associated semigroup algebras are realized within the context …

Algebraic Problems And Exercises For High School, Florentin Smarandache, Ion Goian, Raisa Grigor, Vasila Marin

Branch Mathematics and Statistics Faculty and Staff Publications

In this book, you will find algebraic exercises and problems, grouped by chapters, intended for higher grades in high schools or middle schools of general education. Its purpose is to facilitate training in mathematics for students in all high school categories, but can be equally helpful in a standalone work. The book can also be used as an extracurricular source, as the reader shall find enclosed important theorems and formulas, standard definitions and notions that are not always included in school textbooks.

An Extension Of Herglotz's Theorem To The Quaternions, Daniel Alpay, Fabrizio Colombo, David P. Kimsey, Irene Sabadini, David P. Kimsey

Mathematics, Physics, and Computer Science Faculty Articles and Research

A classical theorem of Herglotz states that a function n↦r(n) from Z into Cs×s is positive definite if and only there exists a Cs×s-valued positive measure dμ on [0,2π] such that r(n)=∫2π0eintdμ(t)for n∈Z. We prove a quaternionic analogue of this result when the function is allowed to have a number of negative squares. A key tool in the argument is the theory of slice hyperholomorphic functions, and the representation of such functions which have a positive real part in the unit ball of the quaternions. We study in great detail the case of positive definite functions.