# Algebra Commons™

Articles 1 - 30 of 905

## Full-Text Articles in Algebra

Regularity Radius: Properties, Approximation And A Not A Priori Exponential Algorithm, David Hartman, Milan Hladik Dec 2018

#### Regularity Radius: Properties, Approximation And A Not A Priori Exponential Algorithm, David Hartman, Milan Hladik

##### Electronic Journal of Linear Algebra

The radius of regularity, sometimes spelled as the radius of nonsingularity, is a measure providing the distance of a given matrix to the nearest singular one. Despite its possible application strength this measure is still far from being handled in an efficient way also due to findings of Poljak and Rohn providing proof that checking this property is NP-hard for a general matrix. There are basically two approaches to handle this situation. Firstly, approximation algorithms are applied and secondly, tighter bounds for radius of regularity are considered. Improvements of both approaches have been recently shown by Hartman and Hlad\'{i ...

Commutators Involving Matrix Functions, Osman Kan, Süleyman Solak Nov 2018

#### Commutators Involving Matrix Functions, Osman Kan, Süleyman Solak

##### Electronic Journal of Linear Algebra

Some results are obtained for matrix commutators involving matrix exponentials $\left(\left[e^{A},B\right],\left[e^{A},e^{B}\right]\right)$ and their norms.

Determinants Of Interval Matrices, Jaroslav Horáček, Milan Hladík, Josef Matějka Nov 2018

#### Determinants Of Interval Matrices, Jaroslav Horáček, Milan Hladík, Josef Matějka

##### Electronic Journal of Linear Algebra

In this paper we shed more light on determinants of real interval matrices. Computing the exact bounds on a determinant of an interval matrix is an NP-hard problem. Therefore, attention is first paid to approximations. NP-hardness of both relative and absolute approximation is proved. Next, methods computing verified enclosures of interval determinants and their possible combination with preconditioning are discussed. A new method based on Cramer's rule was designed. It returns similar results to the state-of-the-art method, however, it is less consuming regarding computational time. Other methods transferable from real matrices (e.g., the Gerschgorin circles, Hadamard's inequality ...

#### Inequalities Between $\Mid A\Mid + \Mid B\Mid$ And $\Mid A^{*} \Mid + \Mid B^{*} \Mid$, Yun Zhang

##### Electronic Journal of Linear Algebra

Let $A$ and $B$ be complex square matrices. Some inequalities between $\mid A \mid + \mid B \mid$ and $\mid A^{*} \mid + \mid B^{*} \mid$ are established. Applications of these inequalities are also given. For example, in the Frobenius norm, $$\parallel\, A+B \,\parallel_{F} \leq \sqrt[4]{2} \parallel \mid A\mid + \mid B\mid \, \parallel_{F}.$$

#### 33 - On The Existence Of An Arbitrarily Large Number Of Generators For The Presentation Ideal Of A Semigroup Ring., Arun Suresh

##### Georgia Undergraduate Research Conference (GURC)

Consider K to be an arbitrary field, and P(n1,…, nm) be the ideal of polynomials given by

P(n1,…, nm) = {f(x1, … , xm) : f(x1,…,xm) ∈ K[x1,…,xm], f(tn1, … ,tnm) = 0, where t is transcendental over K}.

In 1970, J. Herzog showed that the least upper bound on the number of generators of K, for m = 3, is 3. It can be lowered to two, if n1, n2, n3 satisfy a few symmetry conditions. Following that, Bresinsky in 1975, showed ...

On Projection Of A Positive Definite Matrix On A Cone Of Nonnegative Definite Toeplitz Matrices, Katarzyna Filipiak, Augustyn Markiewicz, Adam Mieldzioc, Aneta Sawikowska Oct 2018

#### On Projection Of A Positive Definite Matrix On A Cone Of Nonnegative Definite Toeplitz Matrices, Katarzyna Filipiak, Augustyn Markiewicz, Adam Mieldzioc, Aneta Sawikowska

##### Electronic Journal of Linear Algebra

We consider approximation of a given positive definite matrix by nonnegative definite banded Toeplitz matrices. We show that the projection on linear space of Toeplitz matrices does not always preserve nonnegative definiteness. Therefore we characterize a convex cone of nonnegative definite banded Toeplitz matrices which depends on the matrix dimensions, and we show that the condition of positive definiteness given by Parter [{\em Numer. Math. 4}, 293--295, 1962] characterizes the asymptotic cone. In this paper we give methodology and numerical algorithm of the projection basing on the properties of a cone of nonnegative definite Toeplitz matrices. This problem can be ...

Explicit Block-Structures For Block-Symmetric Fiedler-Like Pencils, M. I. Bueno, Madeline Martin, Javier Perez, Alexander Song, Irina Viviano Oct 2018

#### Explicit Block-Structures For Block-Symmetric Fiedler-Like Pencils, M. I. Bueno, Madeline Martin, Javier Perez, Alexander Song, Irina Viviano

##### Electronic Journal of Linear Algebra

In the last decade, there has been a continued effort to produce families of strong linearizations of a matrix polynomial $P(\lambda)$, regular and singular, with good properties, such as, being companion forms, allowing the recovery of eigenvectors of a regular $P(\lambda)$ in an easy way, allowing the computation of the minimal indices of a singular $P(\lambda)$ in an easy way, etc. As a consequence of this research, families such as the family of Fiedler pencils, the family of generalized Fiedler pencils (GFP), the family of Fiedler pencils with repetition, and the family of generalized Fiedler pencils with ...

On The Largest Distance (Signless Laplacian) Eigenvalue Of Non-Transmission-Regular Graphs, Shuting Liu, Jinlong Shu, Jie Xue Oct 2018

#### On The Largest Distance (Signless Laplacian) Eigenvalue Of Non-Transmission-Regular Graphs, Shuting Liu, Jinlong Shu, Jie Xue

Let $G=(V(G),E(G))$ be a $k$-connected graph with $n$ vertices and $m$ edges. Let $D(G)$ be the distance matrix of $G$. Suppose $\lambda_1(D)\geq \cdots \geq \lambda_n(D)$ are the $D$-eigenvalues of $G$. The transmission of $v_i \in V(G)$, denoted by $Tr_G(v_i)$ is defined to be the sum of distances from $v_i$ to all other vertices of $G$, i.e., the row sum $D_{i}(G)$ of $D(G)$ indexed by vertex $v_i$ and suppose that $D_1(G)\geq \cdots \geq D_n(G)$. The $Wiener~ index$ of $G$ denoted by $W ... Bounded Linear Operators That Preserve The Weak Supermajorization On$\Ell^1(I)^+$, Martin Z. Ljubenović, Dragan S. Djordjevic Oct 2018 #### Bounded Linear Operators That Preserve The Weak Supermajorization On$\Ell^1(I)^+$, Martin Z. Ljubenović, Dragan S. Djordjevic ##### Electronic Journal of Linear Algebra Linear preservers of weak supermajorization which is defined on positive functions contained in the discrete Lebesgue space$\ell^1(I)$are characterized. Two different classes of operators that preserve the weak supermajorization are formed. It is shown that every linear preserver may be decomposed as sum of two operators from the above classes, and conversely, the sum of two operators which satisfy an additional condition is a linear preserver. Necessary and sufficient conditions under which a bounded linear operator is a linear preserver of the weak supermajorization are given. It is concluded that positive linear preservers of the weak supermajorization ... Otto Holder's Formal Christening Of The Quotient Group Concept, Janet Heine Barnett Oct 2018 #### Otto Holder's Formal Christening Of The Quotient Group Concept, Janet Heine Barnett ##### Abstract Algebra No abstract provided. Sep 2018 #### Galois Groups Of Differential Equations And Representing Algebraic Sets, Eli Amzallag ##### All Dissertations, Theses, and Capstone Projects The algebraic framework for capturing properties of solution sets of differential equations was formally introduced by Ritt and Kolchin. As a parallel to the classical Galois groups of polynomial equations, they devised the notion of a differential Galois group for a linear differential equation. Just as solvability of a polynomial equation by radicals is linked to the equation’s Galois group, so too is the ability to express the solution to a linear differential equation in "closed form" linked to the equation’s differential Galois group. It is thus useful even outside of mathematics to be able to compute and ... #### High Performance Sparse Multivariate Polynomials: Fundamental Data Structures And Algorithms, Alex Brandt ##### Electronic Thesis and Dissertation Repository Polynomials may be represented sparsely in an effort to conserve memory usage and provide a succinct and natural representation. Moreover, polynomials which are themselves sparse – have very few non-zero terms – will have wasted memory and computation time if represented, and operated on, densely. This waste is exacerbated as the number of variables increases. We provide practical implementations of sparse multivariate data structures focused on data locality and cache complexity. We look to develop high-performance algorithms and implementations of fundamental polynomial operations, using these sparse data structures, such as arithmetic (addition, subtraction, multiplication, and division) and interpolation. We revisit a sparse ... Aug 2018 #### The Largest Eigenvalue And Some Hamiltonian Properties Of Graphs, Rao Li ##### Electronic Journal of Linear Algebra In this note, sufficient conditions, based on the largest eigenvalue, are presented for some Hamiltonian properties of graphs. Proof Of A Conjecture Of Graham And Lovasz Concerning Unimodality Of Coefficients Of The Distance Characteristic Polynomial Of A Tree, Ghodratollah Aalipour, Aida Abiad, Zhanar Berikkyzy, Leslie Hogben, Franklin H.J. Kenter, Jephian C.-H. Lin, Michael Tait Aug 2018 #### Proof Of A Conjecture Of Graham And Lovasz Concerning Unimodality Of Coefficients Of The Distance Characteristic Polynomial Of A Tree, Ghodratollah Aalipour, Aida Abiad, Zhanar Berikkyzy, Leslie Hogben, Franklin H.J. Kenter, Jephian C.-H. Lin, Michael Tait ##### Electronic Journal of Linear Algebra The conjecture of Graham and Lov ́asz that the (normalized) coefficients of the distance characteristic polynomial of a tree are unimodal is proved; it is also shown that the (normalized) coefficients are log-concave. Upper and lower bounds on the location of the peak are established. Extremal Octagonal Chains With Respect To The Spectral Radius, Xianya Geng, Shuchao Li, Wei Wei Aug 2018 #### Extremal Octagonal Chains With Respect To The Spectral Radius, Xianya Geng, Shuchao Li, Wei Wei ##### Electronic Journal of Linear Algebra Octagonal systems are tree-like graphs comprised of octagons that represent a class of polycyclic conjugated hydrocarbons. In this paper, a roll-attaching operation for the calculation of the characteristic polynomials of octagonal chain graphs is proposed. Based on these characteristic polynomials, the extremal octagonal chains with n octagons having the maximum and minimum spectral radii are identified. The Effects Of Motivation, Technology And Satisfaction On Student Achievement In Face-To-Face And Online College Algebra Classes, Hanan Jamal Amro, Marie-Anne Mundy, Lori Kupczynski Aug 2018 #### The Effects Of Motivation, Technology And Satisfaction On Student Achievement In Face-To-Face And Online College Algebra Classes, Hanan Jamal Amro, Marie-Anne Mundy, Lori Kupczynski ##### TxDLA Journal of Distance Learning Demand for online learning has increased in recent years due to the convenience of class delivery. However, some students appear to have difficulties with online education resulting in lack of completion. The study utilized a quantitative approach with archival data and survey design. The factors of demographics, motivation, technology, and satisfaction were compared for face-to-face and online students. MANCOVA tests were performed to analyze the data while controlling age and gender to uncover significant differences between the two groups. The sample and population for this study were predominantly Hispanic students. Motivation and Technology were non-significant, but satisfaction was proven to ... Application Of Jordan Algebra For Testing Hypotheses About Structure Of Mean Vector In Model With Block Compound Symmetric Covariance Structure, Roman Zmyślony, Ivan Zezula, Arkadiusz Kozioł Aug 2018 #### Application Of Jordan Algebra For Testing Hypotheses About Structure Of Mean Vector In Model With Block Compound Symmetric Covariance Structure, Roman Zmyślony, Ivan Zezula, Arkadiusz Kozioł ##### Electronic Journal of Linear Algebra In this article authors derive test for structure of mean vector in model with block compound symmetric covariance structure for two-level multivariate observations. One possible structure is so called structured mean vector when its components remain constant over sites or over time points, so that mean vector is of the form$\boldsymbol{1}_{u}\otimes\boldsymbol{\mu}$with$\boldsymbol{\mu}=(\mu_1,\mu_2,\ldots,\mu_m)'\in\mathbb{R}^m$. This hypothesis is tested against alternative of unstructured mean vector, which can change over sites or over time points. Inertia Sets Allowed By Matrix Patterns, Adam H. Berliner, Dale D. Olesky, Pauline Van Den Driessche Aug 2018 #### Inertia Sets Allowed By Matrix Patterns, Adam H. Berliner, Dale D. Olesky, Pauline Van Den Driessche ##### Electronic Journal of Linear Algebra Motivated by the possible onset of instability in dynamical systems associated with a zero eigenvalue, sets of inertias$\sn_n$and$\SN{n}$for sign and zero-nonzero patterns, respectively, are introduced. For an$n\times n$sign pattern$\mc{A}$that allows inertia$(0,n-1,1)$, a sufficient condition is given for$\mc{A}$and every superpattern of$\mc{A}$to allow$\sn_n$, and a family of such irreducible sign patterns for all$n\geq 3$is specified. All zero-nonzero patterns (up to equivalence) that allow$\SN{3}$and$\SN{4}$are determined, and are described by their associated digraphs. Some Graphs Determined By Their Distance Spectrum, Stephen Drury, Huiqiu Lin Aug 2018 #### Some Graphs Determined By Their Distance Spectrum, Stephen Drury, Huiqiu Lin ##### Electronic Journal of Linear Algebra Let$G$be a connected graph with order$n$. Let$\lambda_1(D(G))\geq \cdots\geq \lambda_n(D(G))$be the distance spectrum of$G$. In this paper, it is shown that the complements of$P_n$and$C_n$are determined by their$D$-spectrum. Moreover, it is shown that the cycle$C_n$($n$odd) is also determined by its$D$-spectrum. A Tensor's Torsion, Neil Steinburg Aug 2018 #### A Tensor's Torsion, Neil Steinburg ##### Dissertations, Theses, and Student Research Papers in Mathematics While tensor products are quite prolific in commutative algebra, even some of their most basic properties remain relatively unknown. We explore one of these properties, namely a tensor's torsion. In particular, given any finitely generated modules, M and N over a ring R, the tensor product$M\otimes_R N$almost always has nonzero torsion unless one of the modules M or N is free. Specifically, we look at which rings guarantee nonzero torsion in tensor products of non-free modules over the ring. We conclude that a specific subclass of one-dimensional Gorenstein rings will have this property. Adviser: Roger Wiegand ... Partially-Ordered Multi-Type Algebras, Display Calculi And The Category Of Weakening Relations, Peter Jipsen, Fei Liang, M. Andrew Moshier, Apostolos Tzimoulis Aug 2018 #### Partially-Ordered Multi-Type Algebras, Display Calculi And The Category Of Weakening Relations, Peter Jipsen, Fei Liang, M. Andrew Moshier, Apostolos Tzimoulis ##### Mathematics, Physics, and Computer Science Faculty Articles and Research "We define partially-ordered multi-type algebras and use them as algebraic semantics for multi-type display calculi that have recently been developed for several logics, including dynamic epistemic logic [7], linear logic[10], lattice logic [11], bilattice logic [9] and semi-De Morgan logic [8]." Factorization In Integral Domains., Ryan H. Gipson Aug 2018 #### Factorization In Integral Domains., Ryan H. Gipson ##### Electronic Theses and Dissertations We investigate the atomicity and the AP property of the semigroup rings F[X; M], where F is a field, X is a variable and M is a submonoid of the additive monoid of nonnegative rational numbers. In this endeavor, we introduce the following notions: essential generators of M and elements of height (0, 0, 0, . . .) within a cancellative torsion-free monoid Γ. By considering the latter, we are able to determine the irreducibility of certain binomials of the form Xπ − 1, where π is of height (0, 0, 0, . . .), in the monoid domain. Finally, we will consider relations between the ... Developments In Multivariate Post Quantum Cryptography., Jeremy Robert Vates Aug 2018 #### Developments In Multivariate Post Quantum Cryptography., Jeremy Robert Vates ##### Electronic Theses and Dissertations Ever since Shor's algorithm was introduced in 1994, cryptographers have been working to develop cryptosystems that can resist known quantum computer attacks. This push for quantum attack resistant schemes is known as post quantum cryptography. Specifically, my contributions to post quantum cryptography has been to the family of schemes known as Multivariate Public Key Cryptography (MPKC), which is a very attractive candidate for digital signature standardization in the post quantum collective for a wide variety of applications. In this document I will be providing all necessary background to fully understand MPKC and post quantum cryptography as a whole. Then ... On N/P-Asymptotic Distribution Of Vector Of Weighted Traces Of Powers Of Wishart Matrices, Jolanta Maria Pielaszkiewicz, Dietrich Von Rosen, Martin Singull Jul 2018 #### On N/P-Asymptotic Distribution Of Vector Of Weighted Traces Of Powers Of Wishart Matrices, Jolanta Maria Pielaszkiewicz, Dietrich Von Rosen, Martin Singull ##### Electronic Journal of Linear Algebra The joint distribution of standardized traces of$\frac{1}{n}XX'$and of$\Big(\frac{1}{n}XX'\Big)^2$, where the matrix$X:p\times n$follows a matrix normal distribution is proved asymptotically to be multivariate normal under condition$\frac{{n}}{p}\overset{n,p\rightarrow\infty}{\rightarrow}c>0$. Proof relies on calculations of asymptotic moments and cumulants obtained using a recursive formula derived in Pielaszkiewicz et al. (2015). The covariance matrix of the underlying vector is explicitely given as a function of$n$and$p$. Jul 2018 #### A Note On The Matrix Arithmetic-Geometric Mean Inequality, Teng Zhang ##### Electronic Journal of Linear Algebra This note proves the following inequality: If$n=3k$for some positive integer$k$, then for any$n$positive definite matrices$\bA_1,\bA_2,\dots,\bA_n$, the following inequality holds: \begin{equation*}\label{eq:main} \frac{1}{n^3} \, \Big\|\sum_{j_1,j_2,j_3=1}^{n}\bA_{j_1}\bA_{j_2}\bA_{j_3}\Big\| \,\geq\, \frac{(n-3)!}{n!} \, \Big\|\sum_{\substack{j_1,j_2,j_3=1,\\\text{$j_1$,$j_2$,$j_3$all distinct}}}^{n}\bA_{j_1}\bA_{j_2}\bA_{j_3}\Big\|, \end{equation*} where$\|\cdot\|\$ represents the operator norm. This inequality is a special case of a recent conjecture proposed by Recht and R ...

Local Higher Category Theory, Nicholas Meadows Jul 2018

#### Local Higher Category Theory, Nicholas Meadows

##### Electronic Thesis and Dissertation Repository

The purpose of this thesis is to give presheaf-theoretic versions of three of the main extant models of higher category theory: the Joyal, Rezk and Bergner model structures. The construction of these model structures takes up Chapters 2, 3 and 4 of the thesis, respectively. In each of the model structures, the weak equivalences are local or ‘stalkwise’ weak equivalences. In addition, it is shown that certain Quillen equivalences between the aforementioned models of higher category theory extend to Quillen equivalences between the various models of local higher category theory.

Throughout, a number of features of local higher category theory ...

Determining The Determinant, Danny Otero Jul 2018

#### Determining The Determinant, Danny Otero

##### Linear Algebra

No abstract provided.

Webwork Problems For Linear Algebra, Hashim Saber, Beata Hebda Jul 2018

#### Webwork Problems For Linear Algebra, Hashim Saber, Beata Hebda

##### Mathematics Ancillary Materials

This set of problems for Linear Algebra in the open-source WeBWorK mathematics platform was created under a Round Eleven Mini-Grant for Ancillary Materials Creation. The problems were created for an implementation of the CC-BY Lyrix open textbook A First Course in Linear Algebra. Also included as an additional file are the selected and modified Lyryx Class Notes for the textbook.

Topics covered include:

• Linear Independence
• Linear Transformations
• Matrix of a Transformation
• Isomorphisms
• Eigenvalues and Eigenvectors
• Diagonalization
• Orthogonality

Dimers On Cylinders Over Dynkin Diagrams And Cluster Algebras, Maitreyee Chandramohan Kulkarni Jun 2018

#### Dimers On Cylinders Over Dynkin Diagrams And Cluster Algebras, Maitreyee Chandramohan Kulkarni

##### LSU Doctoral Dissertations

This dissertation describes a general setting for dimer models on cylinders over Dynkin diagrams which in type A reduces to the well-studied case of dimer models on a disc. We prove that all Berenstein--Fomin--Zelevinsky quivers for Schubert cells in a symmetric Kac--Moody algebra give rise to dimer models on the cylinder over the corresponding Dynkin diagram. We also give an independent proof of a result of Buan, Iyama, Reiten and Smith that the corresponding superpotentials are rigid using the dimer model structure of the quivers.

Rank Function And Outer Inverses, Manjunatha Prasad Karantha, K. Nayan Bhat, Nupur Nandini Mishra Jun 2018

#### Rank Function And Outer Inverses, Manjunatha Prasad Karantha, K. Nayan Bhat, Nupur Nandini Mishra

##### Electronic Journal of Linear Algebra

For the class of matrices over a field, the notion of rank of a matrix' as defined by the dimension of subspace generated by columns of that matrix' is folklore and cannot be generalized to the class of matrices over an arbitrary commutative ring. The determinantal rank' defined by the size of largest submatrix having nonzero determinant, which is same as the column rank of given matrix when the commutative ring under consideration is a field, was considered to be the best alternative for the rank' in the class of matrices over a commutative ring. Even this determinantal rank and ...