Algebra Commons^™

Solution Of Fuzzy System Of Linear Equations With Polynomial Parametric Form, Diptiranjan Behera, S. Chakraverty

Applications and Applied Mathematics: An International Journal (AAM)

This paper proposed two new and simple solution methods to solve a fuzzy system of linear equations having fuzzy coefficients and crisp variables using a polynomial parametric form of fuzzy numbers. Related theorems are stated and proved. The proposed methods are used to solve example problems. The results obtained are also compared with the known solutions and are found to be in good agreement.

A New Approach For Computing Wz Factorization, Effat Golpar-Raboky

Applications and Applied Mathematics: An International Journal (AAM)

Linear systems arise frequently in scientific and engineering computing. Various serial and parallel algorithms have been introduced for their solution. This paper seeks to compute the WZ and the ZW factorizations of a nonsingular matrix A using the right inverse of nested submatrices of A. We introduce two new matrix factorizations, the QZ and the QW factorizations, and compute the factorizations using our proposed approach.

An Analysis Of Differences In Approaches To Systems Of Linear Equations Problems Given Multiple Choice Answers, Amber Lagasse

Honors Theses and Capstones

This descriptive study focuses on the approaches college students (ages 20 -24) use when solving systems of linear equations problems that have multiple choice answers. Participants were from a midsize public university in the northeast. Four approaches were considered – three forwards approaches: 1) substitution, 2) elimination, and 3) graphing, and one backwards approach: plugging in the x and y values from each multiple choice option. Participants solved systems of linear equations problems and answered questions based on their methods in a structured clinical interview. Each participant also filled out a questionnaire. It was shown from the results of this …

Weierstrass Points On Families Of Graphs, William D. Lindsay Jr.

Master's Theses

No abstract provided.

Cognitively Guided Instruction (Cgi): Whats The Point? A Look Into Cgi And Cgi’S Potential Role In An 8th Grade Mathematics Class, Bryan C. Anderson

Mathematics Graduate Theses

This study examined the current research that exists on Cognitively Guided Instruction (CGI) and how it would look when implemented into an 8th grade mathematics classroom. Additional questions examined are if CGI would develop strong student understanding between mathematical concepts and their real world applications; if CGI techniques and questioning impact the interest level of mathematics students; and what type of extra resources would be needed by teachers to implement Cognitively Guided Instruction into their curriculum. Although much of the CGI research has been done on elementary students, by also examining key algebra practices the author was able to synthesize …

Commutative Rings Graded By Abelian Groups, Brian P. Johnson

Department of Mathematics: Dissertations, Theses, and Student Research

Rings graded by Z and Z^d play a central role in algebraic geometry and commutative algebra, and the purpose of this thesis is to consider rings graded by any abelian group. A commutative ring is graded by an abelian group if the ring has a direct sum decomposition by additive subgroups of the ring indexed over the group, with the additional condition that multiplication in the ring is compatible with the group operation. In this thesis, we develop a theory of graded rings by defining analogues of familiar properties---such as chain conditions, dimension, and Cohen-Macaulayness. We then study the …

Prime Ideals In Two-Dimensional Noetherian Domains And Fiber Products And Connected Sums, Ela Celikbas

Department of Mathematics: Dissertations, Theses, and Student Research

This thesis concerns three topics in commutative algebra:

1) The projective line over the integers (Chapter 2),

2) Prime ideals in two-dimensional quotients of mixed power series-polynomial rings (Chapter 3),

3) Fiber products and connected sums of local rings (Chapter 4),

In the first chapter we introduce basic terminology used in this thesis for all three topics.

In the second chapter we consider the partially ordered set (poset) of prime ideals of the projective line Proj(Z[h,k]) over the integers Z, and we interpret this poset as Spec(Z[x]) U Spec(Z[1/x]) with an appropriate identification. …

Characterizing And Supporting Change In Algebra Students' Representational Fluency In A Cas/Paper-And-Pencil Environment, Nicole L. Fonger

Dissertations

Representational fluency (RF) includes an ability to interpret, create, move within and among, and connect tool-based representations of mathematical objects. Taken as an indicator of conceptual understanding, there is a need to better support school algebra students’ RF in learning environments that utilize both computer algebra systems (CAS) and paper-and-pencil. The purpose of this research was to: (a) characterize change in ninth-grade algebra students’ RF in solving problems involving linear equations, and (b) determine conditions of a CAS and paper-and-pencil learning environment in which those students changed their RF.

Change in RF was measured by comparing results from initial to …

How To Create A Lie Algebra, Ian M. Anderson

How to... in 10 minutes or less

We show how to create a Lie algebra in Maple using three of the most common approaches: matrices, vector fields and structure equations. PDF and Maple worksheets can be downloaded from the links below.

A New Four Point Circular-Invariant Corner-Cutting Subdivision For Curve Design, Jian-Ao Lian

Applications and Applied Mathematics: An International Journal (AAM)

A 4-point nonlinear corner-cutting subdivision scheme is established. It is induced from a special C-shaped biarc circular spline structure. The scheme is circular-invariant and can be effectively applied to 2-dimensional (2D) data sets that are locally convex. The scheme is also extended adaptively to non-convex data. Explicit examples are demonstrated.

Numerical Solution Of Interval And Fuzzy System Of Linear Equations, Suparna Das, S. Chakraverty

Applications and Applied Mathematics: An International Journal (AAM)

A system of linear equations, in general is solved in open literature for crisp unknowns, but in actual case the parameters (coefficients) of the system of linear equations contain uncertainty and are less crisp. The uncertainties may be considered in term of interval or fuzzy number. In this paper, a detail of study of linear simultaneous equations with interval and fuzzy parameter (triangular and trapezoidal) has been performed. New methods have been proposed for solving such systems. First, the methods have been tested for known problems viz. a circuit analysis solved in the literature and the results are found to …

A Novel Algorithm To Forecast Enrollment Based On Fuzzy Time Series, Haneen T. Jasim, Abdul G. Jasim Salim, Kais I. Ibraheem

Applications and Applied Mathematics: An International Journal (AAM)

In this paper we propose a new method to forecast enrollments based on fuzzy time series. The proposed method belongs to the first order and time-variant methods. Historical enrollments of the University of Alabama from year 1948 to 2009 are used in this study to illustrate the forecasting process. By comparing the proposed method with other methods we will show that the proposed method has a higher accuracy rate for forecasting enrollments than the existing methods.

On Lattice Structure Of The Probability Functions On L*, Mashaallah Mashinchi, Ghader Khaledi

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, the set of all probability functions on L* is studied, where L* is the lattice of bothvalued fuzzy sets or intuitionistic fuzzy sets. It is shown that the set of all probability functions on L* endowed with two appropriate operations has a monoid structure which is also a distributive complete lattice. Also the lattice structure of the set of all probability functions on L* induced by an appropriate function on [0, 1] to itself is studied. Some lattice (dual) isomorphisms are discussed that suggests probabilities on L* could be considered in the framework of theories modeling imprecision.

The Effects Of Goal Setting In A Developmental Algebra Course, Richard Leo Hunt

Masters Theses & Specialist Projects

The purpose of this study was to study the effects of goal setting on students in a developmental algebra course. This study examined the effects on test scores for students that were prescribed a test score goal, students that created their own test score goal, and then compared to a control group. Three classes of developmental algebra were chosen with a total of 25 participants with reported results. Results showed that students with a goal on a test did not score significantly better than students without a goal, but did score significantly better on a test after the goal than …

The Rook-Brauer Algebra, Elise G. Delmas

Mathematics, Statistics, and Computer Science Honors Projects

We introduce an associative algebra RB_k(x) that has a basis of rook-Brauer diagrams. These diagrams correspond to partial matchings on 2k vertices. The rook-Brauer algebra contains the group algebra of the symmetric group, the Brauer algebra, and the rook monoid algebra as subalgebras. We show that the basis of RB_k(x) is generated by special diagrams s_i, t_i (1 <= i < k) and p_j (1 <= j <= k), where the s_i are the simple transpositions that generated the symmetric group S_k, the t_i are the "contraction maps" which generate the …

Matrix Product Application To Fibonacci Sequence, Abigail Damaris Muniz^, Jesus Loya^, Hamide Dogan*

COURI Symposium Abstracts, Spring 2012

No abstract provided.

Splitting Fields And Periods Of Fibonacci Sequences Modulo Primes, Sanjai Gupta, Parousia Rockstroh '08, Francis E. Su

All HMC Faculty Publications and Research

We consider the period of a Fibonacci sequence modulo a prime and provide an accessible, motivated treatment of this classical topic using only ideas from linear and abstract algebra. Our methods extend to general recurrences with prime moduli and provide some new insights. And our treatment highlights a nice application of the use of splitting fields that might be suitable to present in an undergraduate course in abstract algebra or Galois theory.

Session D-3: Discrete Mathematics: A Great Curriculum Connector, Donald Porzio

Professional Learning Day

Many topics that fall under the umbrella of Discrete Mathematics cut across the traditional high school curriculum areas of algebra, geometry, and pre-calculus. Come try some classroom-ready hands-on Discrete Mathematics activities that illustrate the true interconnectedness of mathematics.

Unitary Equivalence To A Truncated Toeplitz Operator: Analytic Symbols, Stephan Ramon Garcia, Daniel E. Poore '11, William T. Ross

Pomona Faculty Publications and Research

Unlike Toeplitz operators on H², truncated Toeplitz operators do not have a natural matricial characterization. Consequently, these operators are difficult to study numerically. In this paper we provide criteria for a matrix with distinct eigenvalues to be unitarily equivalent to a truncated Toeplitz operator having an analytic symbol. This test is constructive, and we illustrate it with several examples. As a byproduct, we also prove that every complex symmetric operator on a Hilbert space of dimension ≤ 3 is unitarily equivalent to a direct sum of truncated Toeplitz operators.

On The Quantization Of Zero-Weight Super Dynamical R-Matrices, Gizem Karaali

Pomona Faculty Publications and Research

Solutions of the classical dynamical Yang-Baxter equation on a Lie superalgebra are called super dynamical r-matrices. A super dynamical r-matrix r satisfies the zero weight condition if

[h ⊗ 1 + 1 ⊗ h, r(λ)] = 0 for all h ∈ ɧ, λ ∈ ɧ ∗ .

In this paper we explicitly quantize zero-weight super dynamical r-matrices with zero coupling constant for the Lie superalgebra gl(m, n) . We also answer some questions about super dynamical R-matrices. In particular, we prove a classification theorem and offer some support for one particular …

Unitary Equivalence To A Complex Symmetric Matrix: Low Dimensions, Stephan Ramon Garcia, Daniel E. Poore '11, James E. Tener '08

Pomona Faculty Publications and Research

A matrix T∈M_n(C) is UECSM if it is unitarily equivalent to a complex symmetric (i.e., self-transpose) matrix. We develop several techniques for studying this property in dimensions three and four. Among other things, we completely characterize 4×4 nilpotent matrices which are UECSM and we settle an open problem which has lingered in the 3×3 case. We conclude with a discussion concerning a crucial difference which makes dimension three so different from dimensions four and above.

On The Matrix Equation Xa + Ax_T = 0, Stephan Ramon Garcia, Amy L. Shoemaker '14

Pomona Faculty Publications and Research

The matrix equation XA+AX^T=0, which has relevance to the study of Lie algebras, was recently studied by De Terán and Dopico (Linear Algebra Appl. 434 (2011), 44–67). They reduced the study of this equation to several special cases and produced explicit solutions in most instances. In this note we obtain an explicit solution in one of the difficult cases, for which only the dimension of the solution space and an algorithm to find a basis of this space were known previously

Review: Classification Of Four And Six Dimensional Drinfel'd Superdoubles, Gizem Karaali

Pomona Faculty Publications and Research

No abstract provided.

Review: Lusztig Symmetries And Automorphisms Of Quantum Superalgebras, Gizem Karaali

Pomona Faculty Publications and Research

No abstract provided.

Review: An Operator Approach To The Rational Solutions Of The Classical Yang-Baxter Equation, Gizem Karaali

Pomona Faculty Publications and Research

No abstract provided.

Two Remarks About Nilpotent Operators Of Order Two, Stephan Ramon Garcia, Bob Lutz '13, D. Timotin

Pomona Faculty Publications and Research

We present two novel results about Hilbert space operators which are nilpotent of order two. First, we prove that such operators are indestructible complex symmetric operators, in the sense that tensoring them with any operator yields a complex symmetric operator. In fact, we prove that this property characterizes nilpotents of order two among all nonzero bounded operators. Second, we establish that every nilpotent of order two is unitarily equivalent to a truncated Toeplitz operator.

Spaces Of Sections Of Banach Algebra Bundles, Emmanuel Dror Farjoun, Claude Schochet

Mathematics Faculty Research Publications

Suppose that B is a G-Banach algebra over 𝔽 = ℝ or ℂ, X is a finite dimensional compact metric space, ζ : P → X is a standard principal G-bundle, and A_ζ = Γ(X,P x_G B) is the associated algebra of sections. We produce a spectral sequence which converges to π_∗(GL_oA_ζ) with

E_{_}²_p,q ≅ Ȟ^p(X ; π_q(GL_oB)).

A related spectral sequence converging to K_∗+1(A_ζ) (the real or complex topological …

Squaring, Cubing, And Cube Rooting, Arthur T. Benjamin

All HMC Faculty Publications and Research

I still recall my thrill and disappointment when I read Mathematical Carnival, by Martin Gardner. I was thrilled because, as my high school teacher had recommended, mathematics was presented in a playful way that I had never seen before. I was disappointed because it contained a formula that I thought I had "invented" a few years earlier. I have always had a passion for mental calculation, and the following formula appears in Gardner's chapter on "Lightning Calculators." It was used by the mathematician A. C. Aitken to mentally square large numbers.

Strongly Complete Logics For Coalgebras, Alexander Kurz, Jiří Rosický

Engineering Faculty Articles and Research

Coalgebras for a functor model different types of transition systems in a uniform way. This paper focuses on a uniform account of finitary logics for set-based coalgebras. In particular, a general construction of a logic from an arbitrary set-functor is given and proven to be strongly complete under additional assumptions. We proceed in three parts.

Part I argues that sifted colimit preserving functors are those functors that preserve universal algebraic structure. Our main theorem here states that a functor preserves sifted colimits if and only if it has a finitary presentation by operations and equations. Moreover, the presentation of the …

Semigroup As Graphs, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book the authors study the zero divisor graph and unit graph of a semigroup. The zero divisor graphs of semigroups Zn under multiplication is studied and characterized.