# Algebra Commons™

## All Articles in Algebra

749 full-text articles. Page 1 of 27.

The Roots Of Early Group Theory In The Works Of Lagrange, 2017 Colorado State University-Pueblo

#### The Roots Of Early Group Theory In The Works Of Lagrange, Janet Heine Barnett

##### Abstract Algebra

No abstract provided.

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 ...

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 20.25log2(α)3 + 16.5log2(α)2 + 6log ...

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

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

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 ... 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[Pn], I(mn) ⊆ Im for all m ∈ N. Over the projective plane, we obtain I(4)< ⊆ I2. Huneke asked whether it was the case that I(3) ⊆ I2. Dumnicki, Szemberg and Tutaj-Gasinska show that if I is the saturated homogeneous radical ideal of the 12 points of the Hesse configuration, then ... 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 ... 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 ... 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 2017 University of St Thomas #### Solving A System Of Linear Equations Using Ancient Chinese Methods, Mary Flagg ##### Linear Algebra No abstract provided. 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 ...

Recursive Robust Pca Or Recursive Sparse Recovery In Large But Structured Noise, 2017 Iowa State University

#### Recursive Robust Pca Or Recursive Sparse Recovery In Large But Structured Noise, Chenlu Qiu, Namrata Vaswani, Brian Lois, Leslie Hogben

##### Namrata Vaswani

This paper studies the recursive robust principal components analysis problem. If the outlier is the signal-of-interest, this problem can be interpreted as one of recursively recovering a time sequence of sparse vectors, St, in the presence of large but structured noise, Lt. The structure that we assume on Lt is that Lt is dense and lies in a low-dimensional subspace that is either fixed or changes slowly enough. A key application where this problem occurs is in video surveillance where the goal is to separate a slowly changing background (Lt) from moving foreground objects (St) on-the-fly. To solve the above ...

Recursive Robust Pca Or Recursive Sparse Recovery In Large But Structured Noise, 2017 Iowa State University

#### Recursive Robust Pca Or Recursive Sparse Recovery In Large But Structured Noise, Chenlu Qiu, Namrata Vaswani, Brian Lois, Leslie Hogben

##### Namrata Vaswani

This paper studies the recursive robust principal components analysis problem. If the outlier is the signal-of-interest, this problem can be interpreted as one of recursively recovering a time sequence of sparse vectors, St, in the presence of large but structured noise, Lt. The structure that we assume on Lt is that Lt is dense and lies in a low-dimensional subspace that is either fixed or changes slowly enough. A key application where this problem occurs is in video surveillance where the goal is to separate a slowly changing background (Lt) from moving foreground objects (St) on-the-fly. To solve the above ...

On HigmanS Conjecture, 2017 Universidad del Pais Vasco

#### On HigmanS Conjecture, A. Vera-López, J. M. Arregi, M. A. García-Sánchez, L. Ormaetxea

##### Electronic Journal of Linear Algebra

Let Gn be the subgroup of GLn(q) consisting of the upper unitriangular matrices of size nxn over Fq. In 1960, G. Higman conjectured that the number of conjugacy classes of Gn, denoted by r(Gn), was given by a polynomial in q with integer coefficients. This has been verified for nn, r(Gn) can be expressed in terms of r(Gi), with i

2017 Seattle Pacific University

#### Discovery Learning Plus Direct Instruction Equals Success: Modifying American Math Education In The Algebra Classroom, Sean P. Ferrill Mr.

##### Honors Projects

In light of both high American failure rates in algebra courses and the significant proportion of innumerate American students, this thesis examines a variety of effective educational methods in mathematics. Constructivism, discovery learning, traditional instruction, and the Japanese primary education system are all analyzed to incorporate effective education techniques. Based on the meta-analysis of each of these methods, a hybrid method has been constructed to adapt in the American Common Core algebra classroom.