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On the Bicoset of a Bivector Space
International J.Math. Combin. Vol.4 (2009), 01-08 On the Bicoset of a Bivector Space Agboola A.A.A.† and Akinola L.S.‡ † Department of Mathematics, University of Agriculture, Abeokuta, Nigeria ‡ Department of Mathematics and computer Science, Fountain University, Osogbo, Nigeria E-mail: [email protected], [email protected] Abstract: The study of bivector spaces was first intiated by Vasantha Kandasamy in [1]. The objective of this paper is to present the concept of bicoset of a bivector space and obtain some of its elementary properties. Key Words: bigroup, bivector space, bicoset, bisum, direct bisum, inner biproduct space, biprojection. AMS(2000): 20Kxx, 20L05. §1. Introduction and Preliminaries The study of bialgebraic structures is a new development in the field of abstract algebra. Some of the bialgebraic structures already developed and studied and now available in several literature include: bigroups, bisemi-groups, biloops, bigroupoids, birings, binear-rings, bisemi- rings, biseminear-rings, bivector spaces and a host of others. Since the concept of bialgebraic structure is pivoted on the union of two non-empty subsets of a given algebraic structure for example a group, the usual problem arising from the union of two substructures of such an algebraic structure which generally do not form any algebraic structure has been resolved. With this new concept, several interesting algebraic properties could be obtained which are not present in the parent algebraic structure. In [1], Vasantha Kandasamy initiated the study of bivector spaces. Further studies on bivector spaces were presented by Vasantha Kandasamy and others in [2], [4] and [5]. In the present work however, we look at the bicoset of a bivector space and obtain some of its elementary properties. -
Introduction to Linear Bialgebra
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of New Mexico University of New Mexico UNM Digital Repository Mathematics and Statistics Faculty and Staff Publications Academic Department Resources 2005 INTRODUCTION TO LINEAR BIALGEBRA Florentin Smarandache University of New Mexico, [email protected] W.B. Vasantha Kandasamy K. Ilanthenral Follow this and additional works at: https://digitalrepository.unm.edu/math_fsp Part of the Algebra Commons, Analysis Commons, Discrete Mathematics and Combinatorics Commons, and the Other Mathematics Commons Recommended Citation Smarandache, Florentin; W.B. Vasantha Kandasamy; and K. Ilanthenral. "INTRODUCTION TO LINEAR BIALGEBRA." (2005). https://digitalrepository.unm.edu/math_fsp/232 This Book is brought to you for free and open access by the Academic Department Resources at UNM Digital Repository. It has been accepted for inclusion in Mathematics and Statistics Faculty and Staff Publications by an authorized administrator of UNM Digital Repository. For more information, please contact [email protected], [email protected], [email protected]. INTRODUCTION TO LINEAR BIALGEBRA W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madras Chennai – 600036, India e-mail: [email protected] web: http://mat.iitm.ac.in/~wbv Florentin Smarandache Department of Mathematics University of New Mexico Gallup, NM 87301, USA e-mail: [email protected] K. Ilanthenral Editor, Maths Tiger, Quarterly Journal Flat No.11, Mayura Park, 16, Kazhikundram Main Road, Tharamani, Chennai – 600 113, India e-mail: [email protected] HEXIS Phoenix, Arizona 2005 1 This book can be ordered in a paper bound reprint from: Books on Demand ProQuest Information & Learning (University of Microfilm International) 300 N. -
The Geometry of Syzygies
The Geometry of Syzygies A second course in Commutative Algebra and Algebraic Geometry David Eisenbud University of California, Berkeley with the collaboration of Freddy Bonnin, Clement´ Caubel and Hel´ ene` Maugendre For a current version of this manuscript-in-progress, see www.msri.org/people/staff/de/ready.pdf Copyright David Eisenbud, 2002 ii Contents 0 Preface: Algebra and Geometry xi 0A What are syzygies? . xii 0B The Geometric Content of Syzygies . xiii 0C What does it mean to solve linear equations? . xiv 0D Experiment and Computation . xvi 0E What’s In This Book? . xvii 0F Prerequisites . xix 0G How did this book come about? . xix 0H Other Books . 1 0I Thanks . 1 0J Notation . 1 1 Free resolutions and Hilbert functions 3 1A Hilbert’s contributions . 3 1A.1 The generation of invariants . 3 1A.2 The study of syzygies . 5 1A.3 The Hilbert function becomes polynomial . 7 iii iv CONTENTS 1B Minimal free resolutions . 8 1B.1 Describing resolutions: Betti diagrams . 11 1B.2 Properties of the graded Betti numbers . 12 1B.3 The information in the Hilbert function . 13 1C Exercises . 14 2 First Examples of Free Resolutions 19 2A Monomial ideals and simplicial complexes . 19 2A.1 Syzygies of monomial ideals . 23 2A.2 Examples . 25 2A.3 Bounds on Betti numbers and proof of Hilbert’s Syzygy Theorem . 26 2B Geometry from syzygies: seven points in P3 .......... 29 2B.1 The Hilbert polynomial and function. 29 2B.2 . and other information in the resolution . 31 2C Exercises . 34 3 Points in P2 39 3A The ideal of a finite set of points . -
LINEAR ALGEBRA METHODS in COMBINATORICS László Babai
LINEAR ALGEBRA METHODS IN COMBINATORICS L´aszl´oBabai and P´eterFrankl Version 2.1∗ March 2020 ||||| ∗ Slight update of Version 2, 1992. ||||||||||||||||||||||| 1 c L´aszl´oBabai and P´eterFrankl. 1988, 1992, 2020. Preface Due perhaps to a recognition of the wide applicability of their elementary concepts and techniques, both combinatorics and linear algebra have gained increased representation in college mathematics curricula in recent decades. The combinatorial nature of the determinant expansion (and the related difficulty in teaching it) may hint at the plausibility of some link between the two areas. A more profound connection, the use of determinants in combinatorial enumeration goes back at least to the work of Kirchhoff in the middle of the 19th century on counting spanning trees in an electrical network. It is much less known, however, that quite apart from the theory of determinants, the elements of the theory of linear spaces has found striking applications to the theory of families of finite sets. With a mere knowledge of the concept of linear independence, unexpected connections can be made between algebra and combinatorics, thus greatly enhancing the impact of each subject on the student's perception of beauty and sense of coherence in mathematics. If these adjectives seem inflated, the reader is kindly invited to open the first chapter of the book, read the first page to the point where the first result is stated (\No more than 32 clubs can be formed in Oddtown"), and try to prove it before reading on. (The effect would, of course, be magnified if the title of this volume did not give away where to look for clues.) What we have said so far may suggest that the best place to present this material is a mathematics enhancement program for motivated high school students. -
Linear Algebra: Example Sheet 2 of 4
Michaelmas Term 2014 SJW Linear Algebra: Example Sheet 2 of 4 1. (Another proof of the row rank column rank equality.) Let A be an m × n matrix of (column) rank r. Show that r is the least integer for which A factorises as A = BC with B 2 Matm;r(F) and C 2 Matr;n(F). Using the fact that (BC)T = CT BT , deduce that the (column) rank of AT equals r. 2. Write down the three types of elementary matrices and find their inverses. Show that an n × n matrix A is invertible if and only if it can be written as a product of elementary matrices. Use this method to find the inverse of 0 1 −1 0 1 @ 0 0 1 A : 0 3 −1 3. Let A and B be n × n matrices over a field F . Show that the 2n × 2n matrix IB IB C = can be transformed into D = −A 0 0 AB by elementary row operations (which you should specify). By considering the determinants of C and D, obtain another proof that det AB = det A det B. 4. (i) Let V be a non-trivial real vector space of finite dimension. Show that there are no endomorphisms α; β of V with αβ − βα = idV . (ii) Let V be the space of infinitely differentiable functions R ! R. Find endomorphisms α; β of V which do satisfy αβ − βα = idV . 5. Find the eigenvalues and give bases for the eigenspaces of the following complex matrices: 0 1 1 0 1 0 1 1 −1 1 0 1 1 −1 1 @ 0 3 −2 A ; @ 0 3 −2 A ; @ −1 3 −1 A : 0 1 0 0 1 0 −1 1 1 The second and third matrices commute; find a basis with respect to which they are both diagonal. -
Bases for Infinite Dimensional Vector Spaces Math 513 Linear Algebra Supplement
BASES FOR INFINITE DIMENSIONAL VECTOR SPACES MATH 513 LINEAR ALGEBRA SUPPLEMENT Professor Karen E. Smith We have proven that every finitely generated vector space has a basis. But what about vector spaces that are not finitely generated, such as the space of all continuous real valued functions on the interval [0; 1]? Does such a vector space have a basis? By definition, a basis for a vector space V is a linearly independent set which generates V . But we must be careful what we mean by linear combinations from an infinite set of vectors. The definition of a vector space gives us a rule for adding two vectors, but not for adding together infinitely many vectors. By successive additions, such as (v1 + v2) + v3, it makes sense to add any finite set of vectors, but in general, there is no way to ascribe meaning to an infinite sum of vectors in a vector space. Therefore, when we say that a vector space V is generated by or spanned by an infinite set of vectors fv1; v2;::: g, we mean that each vector v in V is a finite linear combination λi1 vi1 + ··· + λin vin of the vi's. Likewise, an infinite set of vectors fv1; v2;::: g is said to be linearly independent if the only finite linear combination of the vi's that is zero is the trivial linear combination. So a set fv1; v2; v3;:::; g is a basis for V if and only if every element of V can be be written in a unique way as a finite linear combination of elements from the set. -
Linear Algebra Review James Chuang
Linear algebra review James Chuang December 15, 2016 Contents 2.1 vector-vector products ............................................... 1 2.2 matrix-vector products ............................................... 2 2.3 matrix-matrix products ............................................... 4 3.2 the transpose .................................................... 5 3.3 symmetric matrices ................................................. 5 3.4 the trace ....................................................... 6 3.5 norms ........................................................ 6 3.6 linear independence and rank ............................................ 7 3.7 the inverse ...................................................... 7 3.8 orthogonal matrices ................................................. 8 3.9 range and nullspace of a matrix ........................................... 8 3.10 the determinant ................................................... 9 3.11 quadratic forms and positive semidefinite matrices ................................ 10 3.12 eigenvalues and eigenvectors ........................................... 11 3.13 eigenvalues and eigenvectors of symmetric matrices ............................... 12 4.1 the gradient ..................................................... 13 4.2 the Hessian ..................................................... 14 4.3 gradients and hessians of linear and quadratic functions ............................. 15 4.5 gradients of the determinant ............................................ 16 4.6 eigenvalues -
Geometric Engineering of (Framed) Bps States
GEOMETRIC ENGINEERING OF (FRAMED) BPS STATES WU-YEN CHUANG1, DUILIU-EMANUEL DIACONESCU2, JAN MANSCHOT3;4, GREGORY W. MOORE5, YAN SOIBELMAN6 Abstract. BPS quivers for N = 2 SU(N) gauge theories are derived via geometric engineering from derived categories of toric Calabi-Yau threefolds. While the outcome is in agreement of previous low energy constructions, the geometric approach leads to several new results. An absence of walls conjecture is formulated for all values of N, relating the field theory BPS spectrum to large radius D-brane bound states. Supporting evidence is presented as explicit computations of BPS degeneracies in some examples. These computations also prove the existence of BPS states of arbitrarily high spin and infinitely many marginal stability walls at weak coupling. Moreover, framed quiver models for framed BPS states are naturally derived from this formalism, as well as a mathematical formulation of framed and unframed BPS degeneracies in terms of motivic and cohomological Donaldson-Thomas invariants. We verify the conjectured absence of BPS states with \exotic" SU(2)R quantum numbers using motivic DT invariants. This application is based in particular on a complete recursive algorithm which determines the unframed BPS spectrum at any point on the Coulomb branch in terms of noncommutative Donaldson- Thomas invariants for framed quiver representations. Contents 1. Introduction 2 1.1. A (short) summary for mathematicians 7 1.2. BPS categories and mirror symmetry 9 2. Geometric engineering, exceptional collections, and quivers 11 2.1. Exceptional collections and fractional branes 14 2.2. Orbifold quivers 18 2.3. Field theory limit A 19 2.4. -
A New Algorithm to Obtain the Adjugate Matrix Using CUBLAS on GPU González, H.E., Carmona, L., J.J
ISSN: 2277-3754 ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT) Volume 7, Issue 4, October 2017 A New Algorithm to Obtain the Adjugate Matrix using CUBLAS on GPU González, H.E., Carmona, L., J.J. Information Technology Department, ININ-ITTLA Information Technology Department, ININ proved to be the best option for most practical applications. Abstract— In this paper a parallel code for obtain the Adjugate The new transformation proposed here can be obtained from Matrix with real coefficients are used. We postulate a new linear it. Next, we briefly review this topic. transformation in matrix product form and we apply this linear transformation to an augmented matrix (A|I) by means of both a minimum and a complete pivoting strategies, then we obtain the II. LU-MATRICIAL DECOMPOSITION WITH GT. Adjugate matrix. Furthermore, if we apply this new linear NOTATION AND DEFINITIONS transformation and the above pivot strategy to a augmented The problem of solving a linear system of equations matrix (A|b), we obtain a Cramer’s solution of the linear system of Ax b is central to the field of matrix computation. There are equations. That new algorithm present an O n3 computational several ways to perform the elimination process necessary for complexity when n . We use subroutines of CUBLAS 2nd A, b R its matrix triangulation. We will focus on the Doolittle-Gauss rd and 3 levels in double precision and we obtain correct numeric elimination method: the algorithm of choice when A is square, results. dense, and un-structured. nxn Index Terms—Adjoint matrix, Adjugate matrix, Cramer’s Let us assume that AR is nonsingular and that we rule, CUBLAS, GPU. -
Field of U-Invariants of Adjoint Representation of the Group GL (N, K)
Field of U-invariants of adjoint representation of the group GL(n,K) K.A.Vyatkina A.N.Panov ∗ It is known that the field of invariants of an arbitrary unitriangular group is rational (see [1]). In this paper for adjoint representation of the group GL(n,K) we present the algebraically system of generators of the field of U-invariants. Note that the algorithm for calculating generators of the field of invariants of an arbitrary rational representation of an unipotent group was presented in [2, Chapter 1]. The invariants was constructed using induction on the length of Jordan-H¨older series. In our case the length is equal to n2; that is why it is difficult to apply the algorithm. −1 Let us consider the adjoint representation AdgA = gAg of the group GL(n,K), where K is a field of zero characteristic, on the algebra of matri- ces M = Mat(n,K). The adjoint representation determines the representation −1 ρg on K[M] (resp. K(M)) by formula ρgf(A)= f(g Ag). Let U be the subgroup of upper triangular matrices in GL(n,K) with units on the diagonal. A polynomial (rational function) f on M is called an U- invariant, if ρuf = f for every u ∈ U. The set of U-invariant rational functions K(M)U is a subfield of K(M). Let {xi,j} be a system of standard coordinate functions on M. Construct a X X∗ ∗ X X∗ X∗ X matrix = (xij). Let = (xij) be its adjugate matrix, · = · = det X · E. Denote by Jk the left lower corner minor of order k of the matrix X. -
Review a Basis of a Vector Space 1
Review • Vectors v1 , , v p are linearly dependent if x1 v1 + x2 v2 + + x pv p = 0, and not all the coefficients are zero. • The columns of A are linearly independent each column of A contains a pivot. 1 1 − 1 • Are the vectors 1 , 2 , 1 independent? 1 3 3 1 1 − 1 1 1 − 1 1 1 − 1 1 2 1 0 1 2 0 1 2 1 3 3 0 2 4 0 0 0 So: no, they are dependent! (Coeff’s x3 = 1 , x2 = − 2, x1 = 3) • Any set of 11 vectors in R10 is linearly dependent. A basis of a vector space Definition 1. A set of vectors { v1 , , v p } in V is a basis of V if • V = span{ v1 , , v p} , and • the vectors v1 , , v p are linearly independent. In other words, { v1 , , vp } in V is a basis of V if and only if every vector w in V can be uniquely expressed as w = c1 v1 + + cpvp. 1 0 0 Example 2. Let e = 0 , e = 1 , e = 0 . 1 2 3 0 0 1 3 Show that { e 1 , e 2 , e 3} is a basis of R . It is called the standard basis. Solution. 3 • Clearly, span{ e 1 , e 2 , e 3} = R . • { e 1 , e 2 , e 3} are independent, because 1 0 0 0 1 0 0 0 1 has a pivot in each column. Definition 3. V is said to have dimension p if it has a basis consisting of p vectors. Armin Straub 1 [email protected] This definition makes sense because if V has a basis of p vectors, then every basis of V has p vectors. -
Formalizing the Matrix Inversion Based on the Adjugate Matrix in HOL4 Liming Li, Zhiping Shi, Yong Guan, Jie Zhang, Hongxing Wei
Formalizing the Matrix Inversion Based on the Adjugate Matrix in HOL4 Liming Li, Zhiping Shi, Yong Guan, Jie Zhang, Hongxing Wei To cite this version: Liming Li, Zhiping Shi, Yong Guan, Jie Zhang, Hongxing Wei. Formalizing the Matrix Inversion Based on the Adjugate Matrix in HOL4. 8th International Conference on Intelligent Information Processing (IIP), Oct 2014, Hangzhou, China. pp.178-186, 10.1007/978-3-662-44980-6_20. hal-01383331 HAL Id: hal-01383331 https://hal.inria.fr/hal-01383331 Submitted on 18 Oct 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License Formalizing the Matrix Inversion Based on the Adjugate Matrix in HOL4 Liming LI1, Zhiping SHI1?, Yong GUAN1, Jie ZHANG2, and Hongxing WEI3 1 Beijing Key Laboratory of Electronic System Reliability Technology, Capital Normal University, Beijing 100048, China [email protected], [email protected] 2 College of Information Science & Technology, Beijing University of Chemical Technology, Beijing 100029, China 3 School of Mechanical Engineering and Automation, Beihang University, Beijing 100083, China Abstract. This paper presents the formalization of the matrix inversion based on the adjugate matrix in the HOL4 system.