On A Frobenius Problem For Polynomials, 2017 Gettysburg College

#### On A Frobenius Problem For Polynomials, Ricardo Conceição, R. Gondim, M. Rodriguez

*Math Faculty Publications*

We extend the famous diophantine Frobenius problem to a ring of polynomials over a field~*k*. Similar to the classical problem we show that the *n* = 2 case of the Frobenius problem for polynomials is easy to solve. In addition, we translate a few results from the Frobenius problem over ℤ to *k*[*t*] and give an algorithm to solve the Frobenius problem for polynomials over a field *k* of sufficiently large size.

The Common Invariant Subspace Problem And Tarski’S Theorem, 2017 Nicolaus Copernicus University of Toruń

#### The Common Invariant Subspace Problem And Tarski’S Theorem, Grzegorz Pastuszak

*Electronic Journal of Linear Algebra*

This article presents a computable criterion for the existence of a common invariant subspace of $n\times n$ complex matrices $A_{1}, \dots ,A_{s}$ of a fixed dimension $1\leq d\leq n$. The approach taken in the paper is model-theoretic. Namely, the criterion is based on a constructive proof of the renowned Tarski's theorem on quantifier elimination in the theory $\ACF$ of algebraically closed fields. This means that for an arbitrary formula $\varphi$ of the language of fields, a quantifier-free formula $\varphi'$ such that $\varphi\lra\varphi'$ in $\ACF$ is given explicitly. The construction of $\varphi'$ is ...

Refined Inertia Of Matrix Patterns, 2017 Redeemer University College

#### Refined Inertia Of Matrix Patterns, Kevin N. Vander Meulen, Jonathan Earl, Adam Van Tuyl

*Electronic Journal of Linear Algebra*

This paper explores how the combinatorial arrangement of prescribed zeros in a matrix affects the possible eigenvalues that the matrix can obtain. It demonstrates that there are inertially arbitrary patterns having a digraph with no 2-cycle, unlike what happens for nonzero patterns. A class of patterns is developed that are refined inertially arbitrary but not spectrally arbitrary, making use of the property of a properly signed nest. The paper includes a characterization of the inertially arbitrary and refined inertially arbitrary patterns of order three, as well as the patterns of order four with the least number of nonzero entries.

Decreasing Math Anxiety Through Teaching Quadratic Equations, 2017 The College at Brockport: State University of New York

#### Decreasing Math Anxiety Through Teaching Quadratic Equations, Kaitlyn Kaufman

*Education and Human Development Master's Theses*

Math anxiety is known as having a feeling of fear that interferes with math performance. Many students today suffer from math anxiety as they push through each developmental stage in their schooling. A majority of students develop math anxiety through traditional classroom methods, such as drill and practice, assessments, memorizing, and textbooks. According to research, teachers can help decrease math anxiety in students by incorporating specific teaching styles, methods, and strategies, related to decrease math anxiety, into lessons. These teaching styles, methods, and strategies include, but not limited to, constructivist teaching, concrete-to-representation-to-abstract model, student-centered learning, and interactive lessons. Based on ...

Properties Of K-Isotropic Functions, 2017 The University of Western Ontario

#### Properties Of K-Isotropic Functions, Tianpei Jiang

*Electronic Thesis and Dissertation Repository*

The focus of this work is a family of maps from the space of $n \times n$ symmetric matrices, $S^n$, into the space $S^{{n \choose k}}$ for any $k=1,\ldots, n$, invariant under the conjugate action of the orthogonal group $O^n$. This family, called generated $k$-isotropic functions, generalizes known types of maps with similar invariance property, such as the spectral, primary matrix, isotropic functions, multiplicative compound, and additive compound matrices on $S^n$. The notion of operator monotonicity dates back to a work by L\"owner in 1934. A map $F :S^n \to S ...

Predicting Locations Of Pollution Sources Using Convolutional Neural Networks, 2017 Purdue University

#### Predicting Locations Of Pollution Sources Using Convolutional Neural Networks, Yiheng Chi, Nickolas D. Winovich, Guang Lin

*The Summer Undergraduate Research Fellowship (SURF) Symposium*

Pollution is a severe problem today, and the main challenge in water and air pollution controls and eliminations is detecting and locating pollution sources. This research project aims to predict the locations of pollution sources given diffusion information of pollution in the form of array or image data. These predictions are done using machine learning. The relations between time, location, and pollution concentration are first formulated as pollution diffusion equations, which are partial differential equations (PDEs), and then deep convolutional neural networks are built and trained to solve these PDEs. The convolutional neural networks consist of convolutional layers, reLU layers ...

Various Topics On Graphical Structures Placed On Commutative Rings, 2017 University of Tennessee, Knoxville

#### Various Topics On Graphical Structures Placed On Commutative Rings, Darrin Weber

*Doctoral Dissertations*

In this dissertation, we look at two types of graphs that can be placed on a commutative ring: the zero-divisor graph and the ideal-based zero-divisor graph. A zero-divisor graph is a graph whose vertices are the nonzero zero-divisors of a ring and two vertices are connected by an edge if and only if their product is 0. We classify, up to isomorphism, all commutative rings without identity that have a zero-divisor graph on 14 or fewer vertices.

An ideal-based zero-divisor graph is a generalization of the zero-divisor graph where for a ring *R* and ideal *I* the vertices are {* x ...*

Efficiently Representing The Integer Factorization Problem Using Binary Decision Diagrams, 2017 Utah State University

#### Efficiently Representing The Integer Factorization Problem Using Binary Decision Diagrams, David Skidmore

*All Graduate Plan B and other Reports*

Let p be a prime positive integer and let α be a positive integer greater than 1. A method is given to reduce the problem of finding a nontrivial factorization of α to the problem of finding a solution to a system of modulo p polynomial congruences where each variable in the system is constrained to the set {0,...,p − 1}. In the case that p = 2 it is shown that each polynomial in the system can be represented by an ordered binary decision diagram with size less than 20.25log_{2}(α)^{3} + 16.5log_{2}(α)^{2} + 6log ...

Classification Results Of Hadamard Matrices, 2017 University of Tennessee, Knoxville

#### Classification Results Of Hadamard Matrices, Gregory Allen Schmidt

*Masters Theses*

In 1893 Hadamard proved that for any *n* x *n* matrix A over the complex numbers, with all of its entries of absolute value less than or equal to 1, it necessarily follows that

|*det*(*A*)| ≤ *n ^{n/2}* [n raised to the power n divided by two],

with equality if and only if the rows of A are mutually orthogonal and the absolute value of each entry is equal to 1 (See [2], [3]). Such matrices are now appropriately identified as Hadamard matrices, which provides an active area of research in both theoretical and applied fields of the sciences ...

Π-Operators In Clifford Analysis And Its Applications, 2017 University of Arkansas, Fayetteville

#### Π-Operators In Clifford Analysis And Its Applications, Wanqing Cheng

*Theses and Dissertations*

In this dissertation, we studies Π-operators in different spaces using Clifford algebras. This approach generalizes the Π-operator theory on the complex plane to higher dimensional spaces. It also allows us to investigate the existence of the solutions to Beltrami equations in different spaces.

Motivated by the form of the Π-operator on the complex plane, we first construct a Π-operator on a general Clifford-Hilbert module. It is shown that this operator is an L^2 isometry. Further, this can also be used for solving certain Beltrami equations when the Hilbert space is the L^2 space of a measure space. This ...

Ideal Containments Under Flat Extensions And Interpolation On Linear Systems In P2, 2017 University of Nebraska-Lincoln

#### Ideal Containments Under Flat Extensions And Interpolation On Linear Systems In P2, Solomon Akesseh

*Dissertations, Theses, and Student Research Papers in Mathematics*

Fat points and their ideals have stimulated a lot of research but this dissertation concerns itself with aspects of only two of them, broadly categorized here as, the ideal containments and polynomial interpolation problems.

Ein-Lazarsfeld-Smith and Hochster-Huneke cumulatively showed that for all ideals I in k[**P**^{n}], I^{(mn)} ⊆ I^{m} for all m ∈ N. Over the projective plane, we obtain I^{(4)}< ⊆ I^{2}. Huneke asked whether it was the case that I^{(3)} ⊆ I^{2}. Dumnicki, Szemberg and Tutaj-Gasinska show that if I is the saturated homogeneous radical ideal of the 12 points of the Hesse configuration, then ...

Stable Cohomology Of Local Rings And Castelnuovo-Mumford Regularity Of Graded Modules, 2017 University of Nebraska-Lincoln

#### Stable Cohomology Of Local Rings And Castelnuovo-Mumford Regularity Of Graded Modules, Luigi Ferraro

*Dissertations, Theses, and Student Research Papers in Mathematics*

This thesis consists of two parts:

1) A bimodule structure on the bounded cohomology of a local ring (Chapter 1),

2) Modules of infinite regularity over graded commutative rings (Chapter 2).

Chapter 1 deals with the structure of stable cohomology and bounded cohomology. Stable cohomology is a $\mathbb{Z}$-graded algebra generalizing Tate cohomology and first defined by Pierre Vogel. It is connected to absolute cohomology and bounded cohomology. We investigate the structure of the bounded cohomology as a graded bimodule. We use the information on the bimodule structure of bounded cohomology to study the stable cohomology algebra as a ...

Residuated Maps, The Way-Below Relation, And Contractions On Probabilistic Metric Spaces., 2017 University of Louisville

#### Residuated Maps, The Way-Below Relation, And Contractions On Probabilistic Metric Spaces., M. Ryan Luke

*Electronic Theses and Dissertations*

In this dissertation, we will examine residuated mappings on a function lattice and how they behave with respect to the way-below relation. In particular, which residuated $\phi$ has the property that $F$ is way-below $\phi(F)$ for $F$ in appropriate sets. We show the way-below relation describes the separation of two functions and how this corresponds to contraction mappings on probabilistic metric spaces. A new definition for contractions is considered using the way-below relation.

Cayley Graphs Of Groups And Their Applications, 2017 Missouri State University

#### Cayley Graphs Of Groups And Their Applications, Anna Tripi

*MSU Graduate Theses*

Cayley graphs are graphs associated to a group and a set of generators for that group (there is also an associated directed graph). The purpose of this study was to examine multiple examples of Cayley graphs through group theory, graph theory, and applications. We gave background material on groups and graphs and gave numerous examples of Cayley graphs and digraphs. This helped investigate the conjecture that the Cayley graph of any group (except Z_2) is hamiltonian. We found the conjecture to still be open. We found Cayley graphs and hamiltonian cycles could be applied to campanology (in particular, to the ...

Optimal Dual Fusion Frames For Probabilistic Erasures, 2017 Universidad Nacional de San Luis and CONICET, Argentina

#### Optimal Dual Fusion Frames For Probabilistic Erasures, Patricia Mariela Morillas

*Electronic Journal of Linear Algebra*

For any fixed fusion frame, its optimal dual fusion frames for reconstruction is studied in case of erasures of subspaces. It is considered that a probability distribution of erasure of subspaces is given and that a blind reconstruction procedure is used, where the erased data are set to zero. It is proved that there are always optimal duals. Sufficient conditions for the canonical dual fusion frame being either the unique optimal dual, a non-unique optimal dual, or a non optimal dual, are obtained. The reconstruction error is analyzed, using the optimal duals in the probability model considered here and using ...

College Algebra, Trigonometry, And Precalculus (Clayton), 2017 Clayton State University

#### College Algebra, Trigonometry, And Precalculus (Clayton), Chaogui Zhang, Scott Bailey, Billie May, Jelinda Spotorno, Kara Mullen

*Mathematics Grants Collections*

This Grants Collection for College Algebra, Trigonometry, and Precalculus was created under a Round Five ALG Textbook Transformation Grant.

Affordable Learning Georgia Grants Collections are intended to provide faculty with the frameworks to quickly implement or revise the same materials as a Textbook Transformation Grants team, along with the aims and lessons learned from project teams during the implementation process.

Documents are in .pdf format, with a separate .docx (Word) version available for download. Each collection contains the following materials:

- Linked Syllabus
- Initial Proposal
- Final Report

Foundations For College Algebra, 2017 East Georgia State College

#### Foundations For College Algebra, Da'mon Andrews, Antre' Drummer

*Mathematics Grants Collections*

This Grants Collection for Biochemistry was created under a Round Seven ALG Textbook Transformation Grant.

Affordable Learning Georgia Grants Collections are intended to provide faculty with the frameworks to quickly implement or revise the same materials as a Textbook Transformation Grants team, along with the aims and lessons learned from project teams during the implementation process.

Documents are in .pdf format, with a separate .docx (Word) version available for download. Each collection contains the following materials:

- Linked Syllabus
- Initial Proposal
- Final Report

Solving A System Of Linear Equations Using Ancient Chinese Methods, 2017 University of St Thomas

#### Solving A System Of Linear Equations Using Ancient Chinese Methods, Mary Flagg

*Linear Algebra*

No abstract provided.

How The Use Of Subjectivist Instructional Strategies In Teaching Multiple Sections Of An Eighth Grade Algebra Class In Guyana Relates To Algebra Achievement And Attitude Changes Toward Mathematics, 2017 Florida International University

#### How The Use Of Subjectivist Instructional Strategies In Teaching Multiple Sections Of An Eighth Grade Algebra Class In Guyana Relates To Algebra Achievement And Attitude Changes Toward Mathematics, Jennifer Hoyte

*FIU Electronic Theses and Dissertations*

In Guyana, South America, the Ministry of Education seeks to provide universal, inclusive education that prepares its citizens to take their productive places in society and to creatively solve complex, real-world problems. However, with frequent national assessments that are used to place students in high school, college or into jobs, teachers resort to using familiar strategies such as lecture, recitation and test drilling. Despite their efforts, over 56% of students are failing the Grade 6 assessments, 43% failing 10th grade Mathematics and over 60% failing college algebra courses. Such performance has been linked to students’ lower academic self-concept and their ...

Solutions Of The System Of Operator Equations $Bxa=B=Axb$ Via The *-Order, 2017 Ferdowsi University of Mashhad

#### Solutions Of The System Of Operator Equations $Bxa=B=Axb$ Via The *-Order, Mehdi Vosough, Mohammad Sal Moslehian

*Electronic Journal of Linear Algebra*

In this paper, some necessary and sufficient conditions are established for the existence of solutions to the system of operator equations $BXA=B=AXB$ in the setting of bounded linear operators on a Hilbert space, where the unknown operator $X$ is called the inverse of $A$ along $B$. After that, under some mild conditions, it is proved that an operator $X$ is a solution of $BXA=B=AXB$ if and only if $B \stackrel{*}{ \leq} AXA$, where the $*$-order $C\stackrel{*}{ \leq} D$ means $CC^*=DC^*, C^*C=C^*D$. Moreover, the general solution of the equation above is obtained ...