Algebra Commons^™

G-Neutrosophic Space, Mumtaz Ali, Florentin Smarandache, Munazza Naz, Muhammad Shabir

Branch Mathematics and Statistics Faculty and Staff Publications

The Concept of a G-space came into being as a consequence of Group action on an ordinary set. Over the history of Mathematics and Algebra, theory of group action has emerged and proven to be an applicable and effective framework for the study of different kinds of structures to make connection among them.

On The Betti Number Of Differential Modules, Justin Devries

Department of Mathematics: Dissertations, Theses, and Student Research

Let R = k[x₁, ..., x_n] with k a field. A multi-graded differential R-module is a multi-graded R-module D with an endomorphism d such that d² = 0. This dissertation establishes a lower bound on the rank of such a differential module when the underlying R-module is free. We define the Betti number of a differential module and use it to show that when the homology ker d/im d of D is non-zero and finite dimensional over k then there is an inequality rank_R D ≥ 2ⁿ. This …

A Class Of Gaussian Processes With Fractional Spectral Measures, Daniel Alpay, Palle Jorgensen, David Levanony

Mathematics, Physics, and Computer Science Faculty Articles and Research

We study a family of stationary increment Gaussian processes, indexed by time. These processes are determined by certain measures σ (generalized spectral measures), and our focus here is on the case when the measure σ is a singular measure. We characterize the processes arising from when σ is in one of the classes of affine self-similar measures. Our analysis makes use of Kondratiev-white noise spaces. With the use of a priori estimates and the Wick calculus, we extend and sharpen (see Theorem 7.1) earlier computations of Ito stochastic integration developed for the special case of stationary increment processes having absolutely …

Information-Preserving Structures: A General Framework For Quantum Zero-Error Information, Robin Blume-Kohout, Hui Khoon Ng, David Poulin, Lorenza Viola

Dartmouth Scholarship

Quantum systems carry information. Quantum theory supports at least two distinct kinds of information (classical and quantum), and a variety of different ways to encode and preserve information in physical systems. A system’s ability to carry information is constrained and defined by the noise in its dynamics. This paper introduces an operational framework, using information-preserving structures, to classify all the kinds of information that can be perfectly (i.e., with zero error) preserved by quantum dynamics. We prove that every perfectly preserved code has the same structure as a matrix algebra, and that preserved information can always be corrected. We …

The Cohomology Of Modules Over A Complete Intersection Ring, Jesse Burke

Department of Mathematics: Dissertations, Theses, and Student Research

We investigate the cohomology of modules over commutative complete intersection rings. The first main result is that if M is an arbitrary module over a complete intersection ring R, and if one even self-extension module of M vanishes then M has finite projective dimension. The second main result gives a new proof of the fact that the support variety of a Cohen-Macaulay module whose completion is indecomposable is projectively connected.

Optimal Correction Of Infeasible System In Linear Equality Via Genetic Algorithm, S. Ketabchi, H. Moosaei, S. Fallahi

Applications and Applied Mathematics: An International Journal (AAM)

This work is focused on the optimal correction of infeasible system of linear equality. In this paper, for correcting this system, we will make the changes just in the coefficient matrix by using l 􀬶 norm and show that solving this problem is equivalent to solving a fractional quadratic problem. To solve this problem, we use the genetic algorithm. Some examples are provided to illustrate the efficiency and validity of the proposed method.

Manipulatives In The Secondary Mathematics Classroom Using A Traditional Algebra Text, Christina Mutnansky

Mathematics Graduate Theses

This paper is a review of the research available on manipulative use in the mathematics classroom. Mathematical manipulatives, objects that are used to aid a student in learning a mathematical concept, provide a way for students to learn abstract mathematical concepts in a non lecture situation through hands-on learning and discovery. It specifically focuses on the use of manipulatives in the secondary Algebra classroom and how they can be integrated into a traditional curriculum. It discusses the different types of manipulatives that are available as well as reasons why instructors do not use manipulatives. Guidelines are given for their proper …

Vanishing Of Ext And Tor Over Complete Intersections, Olgur Celikbas

Department of Mathematics: Dissertations, Theses, and Student Research

Let (R,m) be a local complete intersection, that is, a local ring whose m-adic completion is the quotient of a complete regular local ring by a regular sequence. Let M and N be finitely generated R-modules. This dissertation concerns the vanishing of Tor(M, N) and Ext(M, N). In this context, M satisfies Serre's condition (S_{n}) if and only if M is an nth syzygy. The complexity of M is the least nonnegative integer r such that the nth Betti number of M is bounded by a polynomial of degree r-1 for all sufficiently large n. We use this notion of …

Analyzing Fractals, Kara Mesznik

Renée Crown University Honors Thesis Projects - All

For my capstone project, I analyzed fractals. A fractal is a picture that is composed of smaller images of the larger picture. Each smaller picture is self- similar, meaning that each of these smaller pictures is actually the larger image just contracted in size through the use of the Contraction Mapping Theorem and shifted using linear and affine transformations.

Fractals live in something called a metric space. A metric space, denoted (X, d), is a space along with a distance formula used to measure the distance between elements in the space. When producing fractals …

Assessing The Impact Of A Computer-Based College Algebra Course, Ningjun Ye

Dissertations

USM piloted the Math Zone in Spring 2007, a computer-based program in teaching MAT 101and MAT 099 in order to improve student performance. This research determined the effect of the re-design of MAT 101 on student achievements in comparison to a traditional approach to the same course. Meanwhile, the study investigated possible effects of the Math Zone program on students’ attitude toward studying mathematics.

This study shows that there was no statistically significant difference on MAT101 final exam scores between the Math Zone students and the Classroom students in Fall 2007, Spring 2008 and Fall 2008. At the same time, …

The Life Of Evariste Galois And His Theory Of Field Extension, Felicia N. Adams

Senior Honors Theses

Evariste Galois made many important mathematical discoveries in his short lifetime, yet perhaps the most important are his studies in the realm of field extensions. Through his discoveries in field extensions, Galois determined the solvability of polynomials. Namely, given a polynomial P with coefficients is in the field F and such that the equation P(x) = 0 has no solution, one can extend F into a field L with α in L, such that P(α) = 0. Whereas Galois Theory has numerous practical applications, this thesis will conclude with the examination and proof of the fact that it is impossible …

The Probabilistic Zeta Function, Bret Benesh

Mathematics Faculty Publications

This paper is a summary of results on the P_G(s) function, which is the reciprocal of the probabilistic zeta function for finite groups. This function gives the probability that s randomly chosen elements generate a group G, and information about the structure of the group G is embedded in it.

Support Varieties And Representation Type Of Small Quantum Groups, Jorg Feldvoss, Sarah Witherspoon

University Faculty and Staff Publications

In this paper, we provide a wildness criterion for any finite dimensional Hopf algebra with finitely generated cohomology. This generalizes a result of Farnsteiner to not necessarily cocommutative Hopf algebras over ground fields of arbitrary characteristic. Our proof uses the theory of support varieties for modules, one of the crucial ingredients being a tensor product property for some special modules. As an application, we prove a conjecture of Cibils stating that small quantum groups of rank at least two are wild.

Recognizing Graph Theoretic Properties With Polynomial Ideals, Jesus A. De Loera, Christopher J. Hillar, Peter N. Malkin, Mohamed Omar

All HMC Faculty Publications and Research

Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term polynomial method to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of the polynomial method and show how the algorithmic theory of polynomial ideals can be used to detect k-colorability, unique Hamiltonicity, and automorphism rigidity of graphs. Our techniques are diverse and involve Nullstellensatz certificates, linear algebra over finite fields, Gröbner bases, toric algebra, convex programming, and real algebraic geometry.

Review: Classification Of Quasi-Trigonometric Solutions Of The Classical Yang-Baxter Equation, Gizem Karaali

Pomona Faculty Publications and Research

No abstract provided.

Review: Quantization Of Hamiltonian-Type Lie Algebras, Gizem Karaali

Pomona Faculty Publications and Research

No abstract provided.

Some New Classes Of Complex Symmetric Operators, Stephan Ramon Garcia, Warren R. Wogen

Pomona Faculty Publications and Research

We say that an operator $T \in B(H)$ is complex symmetric if there exists a conjugate-linear, isometric involution $C:H\to H$ so that $T = CT^*C$. We prove that binormal operators, operators that are algebraic of degree two (including all idempotents), and large classes of rank-one perturbations of normal operators are complex symmetric. From an abstract viewpoint, these results explain why the compressed shift and Volterra integration operator are complex symmetric. Finally, we attempt to describe all complex symmetric partial isometries, obtaining the sharpest possible statement given only the data $(\dim \ker T, \dim \ker T^*)$.

Review: Intertwining Symmetry Algebras Of Quantum Superintegrable Systems, Gizem Karaali

Pomona Faculty Publications and Research

No abstract provided.

Review: The Semi-Dynamical Reflection Equation: Solutions And Structure Matrices, Gizem Karaali

Pomona Faculty Publications and Research

No abstract provided.

On Coalgebras Over Algebras, Adriana Balan, Alexander Kurz

Engineering Faculty Articles and Research

We extend Barr’s well-known characterization of the final coalgebra of a Set-endofunctor as the completion of its initial algebra to the Eilenberg-Moore category of algebras for a Set-monad M for functors arising as liftings. As an application we introduce the notion of commuting pair of endofunctors with respect to the monad M and show that under reasonable assumptions, the final coalgebra of one of the endofunctors involved can be obtained as the free algebra generated by the initial algebra of the other endofunctor.