# Algebra Commons™

## All Articles in Algebra

973 full-text articles. Page 1 of 36.

2019 Central Washington University

#### How To Calculate Pi: Buffon's Needle (Non-Calculus Version), Dominic Klyve

##### Pre-calculus and Trigonometry

No abstract provided.

Greatest Common Divisor: Algorithm And Proof, 2019 University of St. Thomas - Houston

#### Greatest Common Divisor: Algorithm And Proof, Mary K. Flagg

##### Number Theory

No abstract provided.

Monoidal Supercategories And Superadjunction, 2019 University of Ottawa

#### Monoidal Supercategories And Superadjunction, Dene Lepine

We define the notion of superadjunction in the context of supercategories. In particular, we give definitions in terms of counit-unit superadjunctions and hom-space superadjunctions, and prove that these two definitions are equivalent. These results generalize well-known statements in the non-super setting. In the super setting, they formalize some notions that have recently appeared in the literature. We conclude with a brief discussion of superadjunction in the language of string diagrams.

Strengthening Relationships Between Neural Ideals And Receptive Fields, 2019 Texas A&M University

#### Strengthening Relationships Between Neural Ideals And Receptive Fields, Angelique Morvant

Neural codes are collections of binary vectors that represent the firing patterns of neurons. The information given by a neural code C can be represented by its neural ideal JC. In turn, the polynomials in JC can be used to determine the relationships among the receptive fields of the neurons. In a paper by Curto et al., three such relationships, known as the Type 1-3 relations, were linked to the neural ideal by three if-and-only-if statements. Later, Garcia et al. discovered the Type 4-6 relations. These new relations differed from the first three in that they were related ...

Dissertation_Davis.Pdf, 2019 University of Kentucky

#### Dissertation_Davis.Pdf, Brian Davis

##### brian davis

Simplices are the simplest" examples of polytopes, and yet they exhibit much of the rich and subtle combinatorics and commutative algebra of their more general cousins. In this way they are sufficiently complicated --- insights gained from their study can inform broader research in Ehrhart theory and associated fields.

In this dissertation we consider two previously unstudied properties of lattice simplices; one algebraic and one combinatorial. The first is the Poincare series of the associated semigroup algebra, which is substantially more complicated than the Hilbert series of that same algebra. The second is the partial ordering of the elements of the ...

Parametric Natura Morta, 2019 Independent researcher, Palermo, Italy

#### Parametric Natura Morta, Maria C. Mannone

##### The STEAM Journal

Parametric equations can also be used to draw fruits, shells, and a cornucopia of a mathematical still life. Simple mathematics allows the creation of a variety of shapes and visual artworks, and it can also constitute a pedagogical tool for students.

Diagonal Sums Of Doubly Substochastic Matrices, 2019 Georgian Court University

#### Diagonal Sums Of Doubly Substochastic Matrices, Lei Cao, Zhi Chen, Xuefeng Duan, Selcuk Koyuncu, Huilan Li

Let $\Omega_n$ denote the convex polytope of all $n\times n$ doubly stochastic matrices, and $\omega_{n}$ denote the convex polytope of all $n\times n$ doubly substochastic matrices. For a matrix $A\in\omega_n$, define the sub-defect of $A$ to be the smallest integer $k$ such that there exists an $(n+k)\times(n+k)$ doubly stochastic matrix containing $A$ as a submatrix. Let $\omega_{n,k}$ denote the subset of $\omega_n$ which contains all doubly substochastic matrices with sub-defect $k$. For $\pi$ a permutation of symmetric group of degree $n$, the sequence of elements $a_{1\pi(1 ... In-Sphere Property And Reverse Inequalities For Matrix Means, 2019 Ton Duc Thang University #### In-Sphere Property And Reverse Inequalities For Matrix Means, Trung Hoa Dinh, Tin-Yau Tam, Bich Khue T Vo ##### Electronic Journal of Linear Algebra The in-sphere property for matrix means is studied. It is proved that the matrix power mean satisfies in-sphere property with respect to the Hilbert-Schmidt norm. A new characterization of the matrix arithmetic mean is provided. Some reverse AGM inequalities involving unitarily invariant norms and operator monotone functions are also obtained. Surjective Additive Rank-1 Preservers On Hessenberg Matrices, 2019 Walailak University #### Surjective Additive Rank-1 Preservers On Hessenberg Matrices, Prathomjit Khachorncharoenkul, Sajee Pianskool ##### Electronic Journal of Linear Algebra Let$H_{n}(\mathbb{F})$be the space of all$n\times n$upper Hessenberg matrices over a field~$\mathbb{F}$, where$n$is a positive integer greater than two. In this paper, surjective additive maps preserving rank-$1$on$H_{n}(\mathbb{F})$are characterized. 2019 Faculty of Sciences and Mathematics, University of Niš #### Solving The Sylvester Equation Ax-Xb=C When$\Sigma(A)\Cap\Sigma(B)\Neq\Emptyset$, Nebojša Č. Dinčić ##### Electronic Journal of Linear Algebra The method for solving the Sylvester equation$AX-XB=C$in complex matrix case, when$\sigma(A)\cap\sigma(B)\neq \emptyset$, by using Jordan normal form is given. Also, the approach via Schur decomposition is presented. Resolution Of Conjectures Related To Lights Out! And Cartesian Products, 2019 University of Wyoming #### Resolution Of Conjectures Related To Lights Out! And Cartesian Products, Bryan A. Curtis, Jonathan Earl, David Livingston, Bryan L. Shader ##### Electronic Journal of Linear Algebra Lights Out!\ is a game played on a$5 \times 5$grid of lights, or more generally on a graph. Pressing lights on the grid allows the player to turn off neighboring lights. The goal of the game is to start with a given initial configuration of lit lights and reach a state where all lights are out. Two conjectures posed in a recently published paper about Lights Out!\ on Cartesian products of graphs are resolved. A Note On Linear Preservers Of Semipositive And Minimally Semipositive Matrices, 2019 Indian Institute of Science, Bengaluru #### A Note On Linear Preservers Of Semipositive And Minimally Semipositive Matrices, Projesh Nath Choudhury, Rajesh Kannan, K. C. Sivakumar ##### Electronic Journal of Linear Algebra Semipositive matrices (matrices that map at least one nonnegative vector to a positive vector) and minimally semipositive matrices (semipositive matrices whose no column-deleted submatrix is semipositive) are well studied in matrix theory. In this short note, the structure of linear maps which preserve the set of all semipositive/minimally semipositive matrices is studied. An open problem is solved, and some ambiguities in the article [J. Dorsey, T. Gannon, N. Jacobson, C.R. Johnson and M. Turnansky. Linear preservers of semi-positive matrices. {\em Linear and Multilinear Algebra}, 64:1853--1862, 2016.] are clarified. Vector Cross Product Differential And Difference Equations In R^3 And In R^7, 2019 University of Beira Interior #### Vector Cross Product Differential And Difference Equations In R^3 And In R^7, Patrícia D. Beites, Alejandro P. Nicolás, Paulo Saraiva, José Vitória ##### Electronic Journal of Linear Algebra Through a matrix approach of the$2$-fold vector cross product in$\mathbb{R}^3$and in$\mathbb{R}^7$, some vector cross product differential and difference equations are studied. Either the classical theory or convenient Drazin inverses, of elements belonging to the class of index$1$matrices, are applied. Equivalence Of Classical And Quantum Codes, 2019 University of Kentucky #### Equivalence Of Classical And Quantum Codes, Tefjol Pllaha ##### Theses and Dissertations--Mathematics In classical and quantum information theory there are different types of error-correcting codes being used. We study the equivalence of codes via a classification of their isometries. The isometries of various codes over Frobenius alphabets endowed with various weights typically have a rich and predictable structure. On the other hand, when the alphabet is not Frobenius the isometry group behaves unpredictably. We use character theory to develop a duality theory of partitions over Frobenius bimodules, which is then used to study the equivalence of codes. We also consider instances of codes over non-Frobenius alphabets and establish their isometry groups. Secondly ... A (Co)Algebraic Approach To Hennessy-Milner Theorems For Weakly Expressive Logics, 2019 Université de Lorraine #### A (Co)Algebraic Approach To Hennessy-Milner Theorems For Weakly Expressive Logics, Zeinab Bakhtiari, Helle Hvid Hansen, Alexander Kurz ##### Engineering Faculty Articles and Research "Coalgebraic modal logic, as in [9, 6], is a framework in which modal logics for specifying coalgebras can be developed parametric in the signature of the modal language and the coalgebra type functor T. Given a base logic (usually classical propositional logic), modalities are interpreted via so-called predicate liftings for the functor T. These are natural transformations that turn a predicate over the state space X into a predicate over TX. Given that T-coalgebras come with general notions of T-bisimilarity [11] and behavioral equivalence [7], coalgebraic modal logics are designed to respect those. In particular, if two states are behaviourally ... Positive Subreducts In Finitely Generated Varieties Of Mv-Algebras, 2019 Chapman University #### Positive Subreducts In Finitely Generated Varieties Of Mv-Algebras, Leonardo M. Cabrer, Peter Jipsen, Tomáš Kroupa ##### Mathematics, Physics, and Computer Science Faculty Articles and Research Positive MV-algebras are negation-free and implication-free subreducts of MV-algebras. In this contribution we show that a finite axiomatic basis exists for the quasivariety of positive MV-algebras coming from any finitely generated variety of MV-algebras. Gershgorin Type Sets For Eigenvalues Of Matrix Polynomials, 2018 National Technical University of Athens #### Gershgorin Type Sets For Eigenvalues Of Matrix Polynomials, Christina Michailidou, Panayiotis Psarrakos ##### Electronic Journal of Linear Algebra New localization results for polynomial eigenvalue problems are obtained, by extending the notions of the Gershgorin set, the generalized Gershgorin set, the Brauer set and the Dashnic-Zusmanovich set to the case of matrix polynomials. Regularity Radius: Properties, Approximation And A Not A Priori Exponential Algorithm, 2018 Charles University, Faculty of Mathematics and Physics, Department of Applied Mathematics, Prague, Czech Republic and Institute of Computer Science, Czech Academy of Sciences, Prague, Czech Republic. #### 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, 2018 P.h.D student #### 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, 2018 Charles University, Prague, Czech Republic

#### 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 ...