ENUMERATION by ALGEBRAIC COMBINATORICS 1. Introduction A
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Mathematics I Basic Mathematics
Mathematics I Basic Mathematics Prepared by Prof. Jairus. Khalagai African Virtual university Université Virtuelle Africaine Universidade Virtual Africana African Virtual University NOTICE This document is published under the conditions of the Creative Commons http://en.wikipedia.org/wiki/Creative_Commons Attribution http://creativecommons.org/licenses/by/2.5/ License (abbreviated “cc-by”), Version 2.5. African Virtual University Table of ConTenTs I. Mathematics 1, Basic Mathematics _____________________________ 3 II. Prerequisite Course or Knowledge _____________________________ 3 III. Time ____________________________________________________ 3 IV. Materials _________________________________________________ 3 V. Module Rationale __________________________________________ 4 VI. Content __________________________________________________ 5 6.1 Overview ____________________________________________ 5 6.2 Outline _____________________________________________ 6 VII. General Objective(s) ________________________________________ 8 VIII. Specific Learning Objectives __________________________________ 8 IX. Teaching and Learning Activities ______________________________ 10 X. Key Concepts (Glossary) ____________________________________ 16 XI. Compulsory Readings ______________________________________ 18 XII. Compulsory Resources _____________________________________ 19 XIII. Useful Links _____________________________________________ 20 XIV. Learning Activities _________________________________________ 23 XV. Synthesis Of The Module ___________________________________ -
Algebraic Combinatorics and Finite Geometry
An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3; q) Cameron-Liebler k-sets in PG(n; q) Algebraic Combinatorics and Finite Geometry Leo Storme Ghent University Department of Mathematics: Analysis, Logic and Discrete Mathematics Krijgslaan 281 - Building S8 9000 Ghent Belgium Francqui Foundation, May 5, 2021 Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3; q) Cameron-Liebler k-sets in PG(n; q) ACKNOWLEDGEMENT Acknowledgement: A big thank you to Ferdinand Ihringer for allowing me to use drawings and latex code of his slide presentations of his lectures for Capita Selecta in Geometry (Ghent University). Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3; q) Cameron-Liebler k-sets in PG(n; q) OUTLINE 1 AN INTRODUCTION TO ALGEBRAIC GRAPH THEORY 2 ERDOS˝ -KO-RADO RESULTS 3 CAMERON-LIEBLER SETS IN PG(3; q) 4 CAMERON-LIEBLER k-SETS IN PG(n; q) Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3; q) Cameron-Liebler k-sets in PG(n; q) OUTLINE 1 AN INTRODUCTION TO ALGEBRAIC GRAPH THEORY 2 ERDOS˝ -KO-RADO RESULTS 3 CAMERON-LIEBLER SETS IN PG(3; q) 4 CAMERON-LIEBLER k-SETS IN PG(n; q) Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3; q) Cameron-Liebler k-sets in PG(n; q) DEFINITION A graph Γ = (X; ∼) consists of a set of vertices X and an anti-reflexive, symmetric adjacency relation ∼ ⊆ X × X. -
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. -
Axioms and Algebraic Systems*
Axioms and algebraic systems* Leong Yu Kiang Department of Mathematics National University of Singapore In this talk, we introduce the important concept of a group, mention some equivalent sets of axioms for groups, and point out the relationship between the individual axioms. We also mention briefly the definitions of a ring and a field. Definition 1. A binary operation on a non-empty set S is a rule which associates to each ordered pair (a, b) of elements of S a unique element, denoted by a* b, in S. The binary relation itself is often denoted by *· It may also be considered as a mapping from S x S to S, i.e., * : S X S ~ S, where (a, b) ~a* b, a, bE S. Example 1. Ordinary addition and multiplication of real numbers are binary operations on the set IR of real numbers. We write a+ b, a· b respectively. Ordinary division -;- is a binary relation on the set IR* of non-zero real numbers. We write a -;- b. Definition 2. A binary relation * on S is associative if for every a, b, c in s, (a* b) * c =a* (b *c). Example 2. The binary operations + and · on IR (Example 1) are as sociative. The binary relation -;- on IR* (Example 1) is not associative smce 1 3 1 ~ 2) ~ 3 _J_ - 1 ~ (2 ~ 3). ( • • = -6 I 2 = • • * Talk given at the Workshop on Algebraic Structures organized by the Singapore Mathemat- ical Society for school teachers on 5 September 1988. 11 Definition 3. A semi-group is a non-€mpty set S together with an asso ciative binary operation *, and is denoted by (S, *). -
GL(N, Q)-Analogues of Factorization Problems in the Symmetric Group Joel Brewster Lewis, Alejandro H
GL(n, q)-analogues of factorization problems in the symmetric group Joel Brewster Lewis, Alejandro H. Morales To cite this version: Joel Brewster Lewis, Alejandro H. Morales. GL(n, q)-analogues of factorization problems in the sym- metric group. 28-th International Conference on Formal Power Series and Algebraic Combinatorics, Simon Fraser University, Jul 2016, Vancouver, Canada. hal-02168127 HAL Id: hal-02168127 https://hal.archives-ouvertes.fr/hal-02168127 Submitted on 28 Jun 2019 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. FPSAC 2016 Vancouver, Canada DMTCS proc. BC, 2016, 755–766 GLn(Fq)-analogues of factorization problems in Sn Joel Brewster Lewis1y and Alejandro H. Morales2z 1 School of Mathematics, University of Minnesota, Twin Cities 2 Department of Mathematics, University of California, Los Angeles Abstract. We consider GLn(Fq)-analogues of certain factorization problems in the symmetric group Sn: rather than counting factorizations of the long cycle (1; 2; : : : ; n) given the number of cycles of each factor, we count factorizations of a regular elliptic element given the fixed space dimension of each factor. We show that, as in Sn, the generating function counting these factorizations has attractive coefficients after an appropriate change of basis. -
A List of Arithmetical Structures Complete with Respect to the First
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Theoretical Computer Science 257 (2001) 115–151 www.elsevier.com/locate/tcs A list of arithmetical structures complete with respect to the ÿrst-order deÿnability Ivan Korec∗;X Slovak Academy of Sciences, Mathematical Institute, Stefanikovaà 49, 814 73 Bratislava, Slovak Republic Abstract A structure with base set N is complete with respect to the ÿrst-order deÿnability in the class of arithmetical structures if and only if the operations +; × are deÿnable in it. A list of such structures is presented. Although structures with Pascal’s triangles modulo n are preferred a little, an e,ort was made to collect as many simply formulated results as possible. c 2001 Elsevier Science B.V. All rights reserved. MSC: primary 03B10; 03C07; secondary 11B65; 11U07 Keywords: Elementary deÿnability; Pascal’s triangle modulo n; Arithmetical structures; Undecid- able theories 1. Introduction A list of (arithmetical) structures complete with respect of the ÿrst-order deÿnability power (shortly: def-complete structures) will be presented. (The term “def-strongest” was used in the previous versions.) Most of them have the base set N but also structures with some other universes are considered. (Formal deÿnitions are given below.) The class of arithmetical structures can be quasi-ordered by ÿrst-order deÿnability power. After the usual factorization we obtain a partially ordered set, and def-complete struc- tures will form its greatest element. Of course, there are stronger structures (with respect to the ÿrst-order deÿnability) outside of the class of arithmetical structures. -
Problems in Abstract Algebra
STUDENT MATHEMATICAL LIBRARY Volume 82 Problems in Abstract Algebra A. R. Wadsworth 10.1090/stml/082 STUDENT MATHEMATICAL LIBRARY Volume 82 Problems in Abstract Algebra A. R. Wadsworth American Mathematical Society Providence, Rhode Island Editorial Board Satyan L. Devadoss John Stillwell (Chair) Erica Flapan Serge Tabachnikov 2010 Mathematics Subject Classification. Primary 00A07, 12-01, 13-01, 15-01, 20-01. For additional information and updates on this book, visit www.ams.org/bookpages/stml-82 Library of Congress Cataloging-in-Publication Data Names: Wadsworth, Adrian R., 1947– Title: Problems in abstract algebra / A. R. Wadsworth. Description: Providence, Rhode Island: American Mathematical Society, [2017] | Series: Student mathematical library; volume 82 | Includes bibliographical references and index. Identifiers: LCCN 2016057500 | ISBN 9781470435837 (alk. paper) Subjects: LCSH: Algebra, Abstract – Textbooks. | AMS: General – General and miscellaneous specific topics – Problem books. msc | Field theory and polyno- mials – Instructional exposition (textbooks, tutorial papers, etc.). msc | Com- mutative algebra – Instructional exposition (textbooks, tutorial papers, etc.). msc | Linear and multilinear algebra; matrix theory – Instructional exposition (textbooks, tutorial papers, etc.). msc | Group theory and generalizations – Instructional exposition (textbooks, tutorial papers, etc.). msc Classification: LCC QA162 .W33 2017 | DDC 512/.02–dc23 LC record available at https://lccn.loc.gov/2016057500 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. -
Basic Concepts of Set Theory, Functions and Relations 1. Basic
Ling 310, adapted from UMass Ling 409, Partee lecture notes March 1, 2006 p. 1 Basic Concepts of Set Theory, Functions and Relations 1. Basic Concepts of Set Theory........................................................................................................................1 1.1. Sets and elements ...................................................................................................................................1 1.2. Specification of sets ...............................................................................................................................2 1.3. Identity and cardinality ..........................................................................................................................3 1.4. Subsets ...................................................................................................................................................4 1.5. Power sets .............................................................................................................................................4 1.6. Operations on sets: union, intersection...................................................................................................4 1.7 More operations on sets: difference, complement...................................................................................5 1.8. Set-theoretic equalities ...........................................................................................................................5 Chapter 2. Relations and Functions ..................................................................................................................6 -
Abstract Algebra
Abstract Algebra Martin Isaacs, University of Wisconsin-Madison (Chair) Patrick Bahls, University of North Carolina, Asheville Thomas Judson, Stephen F. Austin State University Harriet Pollatsek, Mount Holyoke College Diana White, University of Colorado Denver 1 Introduction What follows is a report summarizing the proposals of a group charged with developing recommendations for undergraduate curricula in abstract algebra.1 We begin by articulating the principles that shaped the discussions that led to these recommendations. We then indicate several learning goals; some of these address specific content areas and others address students' general development. Next, we include three sample syllabi, each tailored to meet the needs of specific types of institutions and students. Finally, we present a brief list of references including sample texts. 2 Guiding Principles We lay out here several principles that underlie our recommendations for undergraduate Abstract Algebra courses. Although these principles are very general, we indicate some of their specific implications in the discussions of learning goals and curricula below. Diversity of students We believe that a course in Abstract Algebra is valuable for a wide variety of students, including mathematics majors, mathematics education majors, mathematics minors, and majors in STEM disciplines such as physics, chemistry, and computer science. Such a course is essential preparation for secondary teaching and for many doctoral programs in mathematics. Moreover, algebra can capture the imagination of students whose attraction to mathematics is primarily to structure and abstraction (for example, 1As with any document that is produced by a committee, there were some disagreements and compromises. The committee members had many lively and spirited communications on what undergraduate Abstract Algebra should look like for the next ten years. -
Binary Operations
4-22-2007 Binary Operations Definition. A binary operation on a set X is a function f : X × X → X. In other words, a binary operation takes a pair of elements of X and produces an element of X. It’s customary to use infix notation for binary operations. Thus, rather than write f(a, b) for the binary operation acting on elements a, b ∈ X, you write afb. Since all those letters can get confusing, it’s also customary to use certain symbols — +, ·, ∗ — for binary operations. Thus, f(a, b) becomes (say) a + b or a · b or a ∗ b. Example. Addition is a binary operation on the set Z of integers: For every pair of integers m, n, there corresponds an integer m + n. Multiplication is also a binary operation on the set Z of integers: For every pair of integers m, n, there corresponds an integer m · n. However, division is not a binary operation on the set Z of integers. For example, if I take the pair 3 (3, 0), I can’t perform the operation . A binary operation on a set must be defined for all pairs of elements 0 from the set. Likewise, a ∗ b = (a random number bigger than a or b) does not define a binary operation on Z. In this case, I don’t have a function Z × Z → Z, since the output is ambiguously defined. (Is 3 ∗ 5 equal to 6? Or is it 117?) When a binary operation occurs in mathematics, it usually has properties that make it useful in con- structing abstract structures. -
Abstract Algebra
Abstract Algebra Paul Melvin Bryn Mawr College Fall 2011 lecture notes loosely based on Dummit and Foote's text Abstract Algebra (3rd ed) Prerequisite: Linear Algebra (203) 1 Introduction Pure Mathematics Algebra Analysis Foundations (set theory/logic) G eometry & Topology What is Algebra? • Number systems N = f1; 2; 3;::: g \natural numbers" Z = f:::; −1; 0; 1; 2;::: g \integers" Q = ffractionsg \rational numbers" R = fdecimalsg = pts on the line \real numbers" p C = fa + bi j a; b 2 R; i = −1g = pts in the plane \complex nos" k polar form re iθ, where a = r cos θ; b = r sin θ a + bi b r θ a p Note N ⊂ Z ⊂ Q ⊂ R ⊂ C (all proper inclusions, e.g. 2 62 Q; exercise) There are many other important number systems inside C. 2 • Structure \binary operations" + and · associative, commutative, and distributive properties \identity elements" 0 and 1 for + and · resp. 2 solve equations, e.g. 1 ax + bx + c = 0 has two (complex) solutions i p −b ± b2 − 4ac x = 2a 2 2 2 2 x + y = z has infinitely many solutions, even in N (thei \Pythagorian triples": (3,4,5), (5,12,13), . ). n n n 3 x + y = z has no solutions x; y; z 2 N for any fixed n ≥ 3 (Fermat'si Last Theorem, proved in 1995 by Andrew Wiles; we'll give a proof for n = 3 at end of semester). • Abstract systems groups, rings, fields, vector spaces, modules, . A group is a set G with an associative binary operation ∗ which has an identity element e (x ∗ e = x = e ∗ x for all x 2 G) and inverses for each of its elements (8 x 2 G; 9 y 2 G such that x ∗ y = y ∗ x = e). -
1 Algebra Vs. Abstract Algebra 2 Abstract Number Systems in Linear
MATH 2135, LINEAR ALGEBRA, Winter 2017 and multiplication is also called the “logical and” operation. For example, we Handout 1: Lecture Notes on Fields can calculate like this: Peter Selinger 1 · ((1 + 0) + 1) + 1 = 1 · (1 + 1) + 1 = 1 · 0 + 1 = 0+1 1 Algebra vs. abstract algebra = 1. Operations such as addition and multiplication can be considered at several dif- 2 Abstract number systems in linear algebra ferent levels: • Arithmetic deals with specific calculation rules, such as 8 + 3 = 11. It is As you already know, Linear Algebra deals with subjects such as matrix multi- usually taught in elementary school. plication, linear combinations, solutions of systems of linear equations, and so on. It makes heavy use of addition, subtraction, multiplication, and division of • Algebra deals with the idea that operations satisfy laws, such as a(b+c)= scalars (think, for example, of the rule for multiplying matrices). ab + ac. Such laws can be used, among other things, to solve equations It turns out that most of what we do in linear algebra does not rely on the spe- such as 3x + 5 = 14. cific laws of arithmetic. Linear algebra works equally well over “alternative” • Abstract algebra is the idea that we can use the laws of algebra, such as arithmetics. a(b + c) = ab + ac, while abandoning the rules of arithmetic, such as Example 2.1. Consider multiplying two matrices, using arithmetic modulo 2 8 + 3 = 11. Thus, in abstract algebra, we are able to speak of entirely instead of the usual arithmetic. different “number” systems, for example, systems in which 1+1=0.