On Some Euclidean Properties of Matrix Algebras Pierre Lezowski
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A NOTE on COMPLETE RESOLUTIONS 1. Introduction Let
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 138, Number 11, November 2010, Pages 3815–3820 S 0002-9939(2010)10422-7 Article electronically published on May 20, 2010 A NOTE ON COMPLETE RESOLUTIONS FOTINI DEMBEGIOTI AND OLYMPIA TALELLI (Communicated by Birge Huisgen-Zimmermann) Abstract. It is shown that the Eckmann-Shapiro Lemma holds for complete cohomology if and only if complete cohomology can be calculated using com- plete resolutions. It is also shown that for an LHF-group G the kernels in a complete resolution of a ZG-module coincide with Benson’s class of cofibrant modules. 1. Introduction Let G be a group and ZG its integral group ring. A ZG-module M is said to admit a complete resolution (F, P,n) of coincidence index n if there is an acyclic complex F = {(Fi,ϑi)| i ∈ Z} of projective modules and a projective resolution P = {(Pi,di)| i ∈ Z,i≥ 0} of M such that F and P coincide in dimensions greater than n;thatis, ϑn F : ···→Fn+1 → Fn −→ Fn−1 → ··· →F0 → F−1 → F−2 →··· dn P : ···→Pn+1 → Pn −→ Pn−1 → ··· →P0 → M → 0 A ZG-module M is said to admit a complete resolution in the strong sense if there is a complete resolution (F, P,n)withHomZG(F,Q) acyclic for every ZG-projective module Q. It was shown by Cornick and Kropholler in [7] that if M admits a complete resolution (F, P,n) in the strong sense, then ∗ ∗ F ExtZG(M,B) H (HomZG( ,B)) ∗ ∗ where ExtZG(M, ) is the P-completion of ExtZG(M, ), defined by Mislin for any group G [13] as k k−r r ExtZG(M,B) = lim S ExtZG(M,B) r>k where S−mT is the m-th left satellite of a functor T . -
Version 0.5.1 of 5 March 2010
Modern Computer Arithmetic Richard P. Brent and Paul Zimmermann Version 0.5.1 of 5 March 2010 iii Copyright c 2003-2010 Richard P. Brent and Paul Zimmermann ° This electronic version is distributed under the terms and conditions of the Creative Commons license “Attribution-Noncommercial-No Derivative Works 3.0”. You are free to copy, distribute and transmit this book under the following conditions: Attribution. You must attribute the work in the manner specified by the • author or licensor (but not in any way that suggests that they endorse you or your use of the work). Noncommercial. You may not use this work for commercial purposes. • No Derivative Works. You may not alter, transform, or build upon this • work. For any reuse or distribution, you must make clear to others the license terms of this work. The best way to do this is with a link to the web page below. Any of the above conditions can be waived if you get permission from the copyright holder. Nothing in this license impairs or restricts the author’s moral rights. For more information about the license, visit http://creativecommons.org/licenses/by-nc-nd/3.0/ Contents Preface page ix Acknowledgements xi Notation xiii 1 Integer Arithmetic 1 1.1 Representation and Notations 1 1.2 Addition and Subtraction 2 1.3 Multiplication 3 1.3.1 Naive Multiplication 4 1.3.2 Karatsuba’s Algorithm 5 1.3.3 Toom-Cook Multiplication 6 1.3.4 Use of the Fast Fourier Transform (FFT) 8 1.3.5 Unbalanced Multiplication 8 1.3.6 Squaring 11 1.3.7 Multiplication by a Constant 13 1.4 Division 14 1.4.1 Naive -
Dividing Whole Numbers
Dividing Whole Numbers INTRODUCTION Consider the following situations: 1. Edgar brought home 72 donut holes for the 9 children at his daughter’s slumber party. How many will each child receive if the donut holes are divided equally among all nine? 2. Three friends, Gloria, Jan, and Mie-Yun won a small lottery prize of $1,450. If they split it equally among all three, how much will each get? 3. Shawntee just purchased a used car. She has agreed to pay a total of $6,200 over a 36- month period. What will her monthly payments be? 4. 195 people are planning to attend an awards banquet. Ruben is in charge of the table rental. If each table seats 8, how many tables are needed so that everyone has a seat? Each of these problems can be answered using division. As you’ll recall from Section 1.2, the result of division is called the quotient. Here are the parts of a division problem: Standard form: dividend ÷ divisor = quotient Long division form: dividend Fraction form: divisor = quotient We use these words—dividend, divisor, and quotient—throughout this section, so it’s important that you become familiar with them. You’ll often see this little diagram as a continual reminder of their proper placement. Here is how a division problem is read: In the standard form: 15 ÷ 5 is read “15 divided by 5.” In the long division form: 5 15 is read “5 divided into 15.” 15 In the fraction form: 5 is read “15 divided by 5.” Dividing Whole Numbers page 1 Example 1: In this division problem, identify the dividend, divisor, and quotient. -
Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) Is a Series of Graduate-Level Textbooks in Mathematics Published by Springer-Verlag
欢迎加入数学专业竞赛及考研群:681325984 Graduate Texts in Mathematics - Wikipedia Graduate Texts in Mathematics Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard size (with variable numbers of pages). The GTM series is easily identified by a white band at the top of the book. The books in this series tend to be written at a more advanced level than the similar Undergraduate Texts in Mathematics series, although there is a fair amount of overlap between the two series in terms of material covered and difficulty level. Contents List of books See also Notes External links List of books 1. Introduction to Axiomatic Set Theory, Gaisi Takeuti, Wilson M. Zaring (1982, 2nd ed., ISBN 978-1-4613- 8170-9) 2. Measure and Category - A Survey of the Analogies between Topological and Measure Spaces, John C. Oxtoby (1980, 2nd ed., ISBN 978-0-387-90508-2) 3. Topological Vector Spaces, H. H. Schaefer, M. P. Wolff (1999, 2nd ed., ISBN 978-0-387-98726-2) 4. A Course in Homological Algebra, Peter Hilton, Urs Stammbach (1997, 2nd ed., ISBN 978-0-387-94823-2) 5. Categories for the Working Mathematician, Saunders Mac Lane (1998, 2nd ed., ISBN 978-0-387-98403-2) 6. Projective Planes, Daniel R. Hughes, Fred C. Piper, (1982, ISBN 978-3-540-90043-6) 7. A Course in Arithmetic, Jean-Pierre Serre (1996, ISBN 978-0-387-90040-7) 8. -
Modern Computer Arithmetic
Modern Computer Arithmetic Richard P. Brent and Paul Zimmermann Version 0.3 Copyright c 2003-2009 Richard P. Brent and Paul Zimmermann This electronic version is distributed under the terms and conditions of the Creative Commons license “Attribution-Noncommercial-No Derivative Works 3.0”. You are free to copy, distribute and transmit this book under the following conditions: Attribution. You must attribute the work in the manner specified • by the author or licensor (but not in any way that suggests that they endorse you or your use of the work). Noncommercial. You may not use this work for commercial purposes. • No Derivative Works. You may not alter, transform, or build upon • this work. For any reuse or distribution, you must make clear to others the license terms of this work. The best way to do this is with a link to the web page below. Any of the above conditions can be waived if you get permission from the copyright holder. Nothing in this license impairs or restricts the author’s moral rights. For more information about the license, visit http://creativecommons.org/licenses/by-nc-nd/3.0/ Preface This is a book about algorithms for performing arithmetic, and their imple- mentation on modern computers. We are concerned with software more than hardware — we do not cover computer architecture or the design of computer hardware since good books are already available on these topics. Instead we focus on algorithms for efficiently performing arithmetic operations such as addition, multiplication and division, and their connections to topics such as modular arithmetic, greatest common divisors, the Fast Fourier Transform (FFT), and the computation of special functions. -
Modern Computer Arithmetic (Version 0.5. 1)
Modern Computer Arithmetic Richard P. Brent and Paul Zimmermann Version 0.5.1 arXiv:1004.4710v1 [cs.DS] 27 Apr 2010 Copyright c 2003-2010 Richard P. Brent and Paul Zimmermann This electronic version is distributed under the terms and conditions of the Creative Commons license “Attribution-Noncommercial-No Derivative Works 3.0”. You are free to copy, distribute and transmit this book under the following conditions: Attribution. You must attribute the work in the manner specified • by the author or licensor (but not in any way that suggests that they endorse you or your use of the work). Noncommercial. You may not use this work for commercial purposes. • No Derivative Works. You may not alter, transform, or build upon • this work. For any reuse or distribution, you must make clear to others the license terms of this work. The best way to do this is with a link to the web page below. Any of the above conditions can be waived if you get permission from the copyright holder. Nothing in this license impairs or restricts the author’s moral rights. For more information about the license, visit http://creativecommons.org/licenses/by-nc-nd/3.0/ Contents Contents iii Preface ix Acknowledgements xi Notation xiii 1 Integer Arithmetic 1 1.1 RepresentationandNotations . 1 1.2 AdditionandSubtraction . .. 2 1.3 Multiplication . 3 1.3.1 Naive Multiplication . 4 1.3.2 Karatsuba’s Algorithm . 5 1.3.3 Toom-Cook Multiplication . 7 1.3.4 UseoftheFastFourierTransform(FFT) . 8 1.3.5 Unbalanced Multiplication . 9 1.3.6 Squaring.......................... 12 1.3.7 Multiplication by a Constant . -
Division Methods Sharing Equally Grouping Multiples of the Divisor
Division Methods Sharing equally Grouping Multiples of the divisor • Start making the sharing more abstract, The children will start to relate multiplication Talk with the children and use items to express recording it as above, but also using facts to their division work, using their the following: apparatus to support this. multiplication facts to recognise how many “lots” • I have 6 counters here, how many will each • Simple problems will be given to support this or “groups” can be found within a certain child get (share them out between 3 children, e.g. there are 12 cakes, I share them equally number. saying 6 shared between 3 is 2 each) between 4 people, how many do they each get? For example 65 ÷ 5 = (50 + 15) ÷ 5 = 10 + 3 12 ÷ 4 = * * * * = 13 So, children might recognise that 65 can be split * * * * into 50 and 15, so they will know there are 10 * * * * groups of 5 within 50, and 3 groups of 5 within 15, therefore 13 groups altogether. Sharing/grouping with a remainder Chunking Short Division (Dividing by 1 digit) • The previous methods and ideas will be applied, but now with numbers which will This is a more formal method of recording Here the number being divided by works along have numbers surplus after the groups within multiples of the divisor, taking away groups of the digits, starting with the most significant. the number have been identified. multiples within that number until it’s not What is 4 ÷ 3 = 1 (goes on top) with 1 left over, • I have 13 cakes, I share them equally among possible to get any more whole groups left. -
Mathematical Communities
II~'~lvi|,[~]i,~.~|[.-~-nl[,~.],,n,,,,.,nit[:;-1 Marjorie Senechal, Editor ] Nicolas Bourbaki, 1935-???? vided for self-renewal: from time to If you are a mathematician working to- time, younger mathematicians were in- The Continuing day, you have almost certainly been in- vited to join and older members re- fluenced by Bourbaki, at least in style signed, in accordance with mandatory and spirit, and perhaps to a greater ex- "retirement" at age fifty. Now Bourbaki Silence of tent than you realize. But if you are a itself is nearly twenty years older than student, you may never have heard of any of its members. The long-running it, him, them. What or who is, or was, Bourbaki seminar is still alive and well Bourbaki- Bourbaki? and living in Paris, but the voice of Check as many as apply. Bourbakl Bourbaki itself--as expressed through An Interview with is, or was, as the case may be: its books--has been silent for fifteen years. Will it speak again? Can it speak 9the discoverer (or inventor, if you again? prefer) of the notion of a mathemat- Pierre Cartier, Pierre Cartier was a member of ical structure; Bourbaki from 1955 to 1983. Born in 9one of the great abstractionist move- June 18, 1997 Sedan, France in 1932, he graduated ments of the twentieth century; from the l~cole Normale Supdrieure in 9 a small but enormously influential Marjorie Senechal Paris, where he studied under Henri community of mathematicians; Caftan. His thesis, defended in 1958, 9 acollective that hasn't published for was on algebraic geometry; since then fifteen years. -
Direct Links to Free Springer Books (Pdf Versions)
Sign up for a GitHub account Sign in Instantly share code, notes, and snippets. Create a gist now bishboria / springer-free-maths-books.md Last active 36 seconds ago Code Revisions 5 Stars 1664 Forks 306 Embed <script src="https://gist.githu b .coDmo/wbinslohabodr ZiIaP/8326b17bbd652f34566a.js"></script> Springer have made a bunch of books available for free, here are the direct links springer-free-maths-books.md Raw Direct links to free Springer books (pdf versions) Graduate texts in mathematics duplicates = multiple editions A Classical Introduction to Modern Number Theory, Kenneth Ireland Michael Rosen A Classical Introduction to Modern Number Theory, Kenneth Ireland Michael Rosen A Course in Arithmetic, Jean-Pierre Serre A Course in Computational Algebraic Number Theory, Henri Cohen A Course in Differential Geometry, Wilhelm Klingenberg A Course in Functional Analysis, John B. Conway A Course in Homological Algebra, P. J. Hilton U. Stammbach A Course in Homological Algebra, Peter J. Hilton Urs Stammbach A Course in Mathematical Logic, Yu. I. Manin A Course in Number Theory and Cryptography, Neal Koblitz A Course in Number Theory and Cryptography, Neal Koblitz A Course in Simple-Homotopy Theory, Marshall M. Cohen A Course in p-adic Analysis, Alain M. Robert A Course in the Theory of Groups, Derek J. S. Robinson A Course in the Theory of Groups, Derek J. S. Robinson A Course on Borel Sets, S. M. Srivastava A Course on Borel Sets, S. M. Srivastava A First Course in Noncommutative Rings, T. Y. Lam A First Course in Noncommutative Rings, T. Y. Lam A Hilbert Space Problem Book, P. -
Division of Whole Numbers
The Improving Mathematics Education in Schools (TIMES) Project NUMBER AND ALGEBRA Module 10 DIVISION OF WHOLE NUMBERS A guide for teachers - Years 4–7 June 2011 4YEARS 7 Polynomials (Number and Algebra: Module 10) For teachers of Primary and Secondary Mathematics 510 Cover design, Layout design and Typesetting by Claire Ho The Improving Mathematics Education in Schools (TIMES) Project 2009‑2011 was funded by the Australian Government Department of Education, Employment and Workplace Relations. The views expressed here are those of the author and do not necessarily represent the views of the Australian Government Department of Education, Employment and Workplace Relations. © The University of Melbourne on behalf of the International Centre of Excellence for Education in Mathematics (ICE‑EM), the education division of the Australian Mathematical Sciences Institute (AMSI), 2010 (except where otherwise indicated). This work is licensed under the Creative Commons Attribution‑ NonCommercial‑NoDerivs 3.0 Unported License. 2011. http://creativecommons.org/licenses/by‑nc‑nd/3.0/ The Improving Mathematics Education in Schools (TIMES) Project NUMBER AND ALGEBRA Module 10 DIVISION OF WHOLE NUMBERS A guide for teachers - Years 4–7 June 2011 Peter Brown Michael Evans David Hunt Janine McIntosh Bill Pender Jacqui Ramagge 4YEARS 7 {4} A guide for teachers DIVISION OF WHOLE NUMBERS ASSUMED KNOWLEDGE • An understanding of the Hindu‑Arabic notation and place value as applied to whole numbers (see the module Using place value to write numbers). • An understanding of, and fluency with, forwards and backwards skip‑counting. • An understanding of, and fluency with, addition, subtraction and multiplication, including the use of algorithms. -
ASSIGNMENT SUB : Mathematics CLASS : VI a WEEK – 8 …………………………………………………………………………………………………………
ASSIGNMENT SUB : Mathematics CLASS : VI A WEEK – 8 ………………………………………………………………………………………………………… Chapter :- 03 , Factors and Multiples Topics :- (i) Highest common factor (HCF) (ii) Least common Multiples (LCM) (iii) Relation between HCF and LCM _____________________________________________________________________________________ i. Highest common factor (HCF) The Highest common factor (HCF) of two or more numbers is the largest numbers in the common factor of the numbers. HCF is also known as GCD (Greatest common Divisor) The following are the methods of finding the HCF of two or more numbers. i) Finding HCF by listing factors. ii) Prime Factorisation Method iii) Short Division Method iv) Continued or long Division Method. 1. Finding HCF by listing factors :- Consider three numbers 36, 42 and 24. Factors of 36 = 1 2 3 4 6 9 12 18 36 Factors of 42 = 1 2 3 6 7 14 21 42 Factors of 24 = 1 2 3 4 6 8 12 24 Common factors (HCF) of 36 , 42 and 24 = 1, 2, 3, 6 HCF = 6 which exactly divides the numbers 36, 42 and 24. 2. Prime factorization method :- In this method first find the prime factorization of the given numbers and then multiply the common factors to get HCF of the given numbers. Prime factors of 48 = 2 x 2 x 2 x 2 x 3 Prime factors of 96 = 2 x 2 x 2 x 2 x 3 x 2 Prime factors of 144 = 2 x 2 x 2 x 2 x 3 x 3 Common factors = 2 , 2 , 2 , 2 , 3 HCF of 48 , 96 and 144 = 2 x 2 x 2 x 2 x 3 = 48 Ans. -
Fast Library for Number Theory
FLINT Fast Library for Number Theory Version 2.2.0 4 June 2011 William Hart∗, Fredrik Johanssony, Sebastian Pancratzz ∗ Supported by EPSRC Grant EP/G004870/1 y Supported by Austrian Science Foundation (FWF) Grant Y464-N18 z Supported by European Research Council Grant 204083 Contents 1 Introduction1 2 Building and using FLINT3 3 Test code5 4 Reporting bugs7 5 Contributors9 6 Example programs 11 7 FLINT macros 13 8 fmpz 15 8.1 Introduction.................................. 15 8.2 Simple example................................ 16 8.3 Memory management ............................ 16 8.4 Random generation.............................. 16 8.5 Conversion .................................. 17 8.6 Input and output............................... 18 8.7 Basic properties and manipulation ..................... 19 8.8 Comparison.................................. 19 8.9 Basic arithmetic ............................... 20 8.10 Greatest common divisor .......................... 23 8.11 Modular arithmetic.............................. 23 8.12 Bit packing and unpacking ......................... 23 8.13 Chinese remaindering ............................ 24 9 fmpz vec 27 9.1 Memory management ............................ 27 9.2 Randomisation ................................ 27 9.3 Bit sizes.................................... 27 9.4 Input and output............................... 27 9.5 Conversions.................................. 28 9.6 Assignment and basic manipulation .................... 28 9.7 Comparison.................................. 29 9.8 Sorting....................................