Disquisitiones Arithmeticae
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La Controverse De 1874 Entre Camille Jordan Et Leopold Kronecker
La controverse de 1874 entre Camille Jordan et Leopold Kronecker. * Frédéric Brechenmacher ( ). Résumé. Une vive querelle oppose en 1874 Camille Jordan et Leopold Kronecker sur l’organisation de la théorie des formes bilinéaires, considérée comme permettant un traitement « général » et « homogène » de nombreuses questions développées dans des cadres théoriques variés au XIXe siècle et dont le problème principal est reconnu comme susceptible d’être résolu par deux théorèmes énoncés indépendamment par Jordan et Weierstrass. Cette controverse, suscitée par la rencontre de deux théorèmes que nous considèrerions aujourd’hui équivalents, nous permettra de questionner l’identité algébrique de pratiques polynomiales de manipulations de « formes » mises en œuvre sur une période antérieure aux approches structurelles de l’algèbre linéaire qui donneront à ces pratiques l’identité de méthodes de caractérisation des classes de similitudes de matrices. Nous montrerons que les pratiques de réductions canoniques et de calculs d’invariants opposées par Jordan et Kronecker manifestent des identités multiples indissociables d’un contexte social daté et qui dévoilent des savoirs tacites, des modes de pensées locaux mais aussi, au travers de regards portés sur une histoire à long terme impliquant des travaux d’auteurs comme Lagrange, Laplace, Cauchy ou Hermite, deux philosophies internes sur la signification de la généralité indissociables d’idéaux disciplinaires opposant algèbre et arithmétique. En questionnant les identités culturelles de telles pratiques cet article vise à enrichir l’histoire de l’algèbre linéaire, souvent abordée dans le cadre de problématiques liées à l’émergence de structures et par l’intermédiaire de l’histoire d’une théorie, d’une notion ou d’un mode de raisonnement. -
Kummer, Regular Primes, and Fermat's Last Theorem
e H a r v a e Harvard College r d C o l l e Mathematics Review g e M a t h e m Vol. 2, No. 2 Fall 2008 a t i c s In this issue: R e v i ALLAN M. FELDMAN and ROBERTO SERRANO e w , Arrow’s Impossibility eorem: Two Simple Single-Profile Versions V o l . 2 YUFEI ZHAO , N Young Tableaux and the Representations of the o . Symmetric Group 2 KEITH CONRAD e Congruent Number Problem A Student Publication of Harvard College Website. Further information about The HCMR can be Sponsorship. Sponsoring The HCMR supports the un- found online at the journal’s website, dergraduate mathematics community and provides valuable high-level education to undergraduates in the field. Sponsors http://www.thehcmr.org/ (1) will be listed in the print edition of The HCMR and on a spe- cial page on the The HCMR’s website, (1). Sponsorship is available at the following levels: Instructions for Authors. All submissions should in- clude the name(s) of the author(s), institutional affiliations (if Sponsor $0 - $99 any), and both postal and e-mail addresses at which the cor- Fellow $100 - $249 responding author may be reached. General questions should Friend $250 - $499 be addressed to Editors-In-Chief Zachary Abel and Ernest E. Contributor $500 - $1,999 Fontes at [email protected]. Donor $2,000 - $4,999 Patron $5,000 - $9,999 Articles. The Harvard College Mathematics Review invites Benefactor $10,000 + the submission of quality expository articles from undergrad- uate students. -
A Category-Theoretic Approach to Representation and Analysis of Inconsistency in Graph-Based Viewpoints
A Category-Theoretic Approach to Representation and Analysis of Inconsistency in Graph-Based Viewpoints by Mehrdad Sabetzadeh A thesis submitted in conformity with the requirements for the degree of Master of Science Graduate Department of Computer Science University of Toronto Copyright c 2003 by Mehrdad Sabetzadeh Abstract A Category-Theoretic Approach to Representation and Analysis of Inconsistency in Graph-Based Viewpoints Mehrdad Sabetzadeh Master of Science Graduate Department of Computer Science University of Toronto 2003 Eliciting the requirements for a proposed system typically involves different stakeholders with different expertise, responsibilities, and perspectives. This may result in inconsis- tencies between the descriptions provided by stakeholders. Viewpoints-based approaches have been proposed as a way to manage incomplete and inconsistent models gathered from multiple sources. In this thesis, we propose a category-theoretic framework for the analysis of fuzzy viewpoints. Informally, a fuzzy viewpoint is a graph in which the elements of a lattice are used to specify the amount of knowledge available about the details of nodes and edges. By defining an appropriate notion of morphism between fuzzy viewpoints, we construct categories of fuzzy viewpoints and prove that these categories are (finitely) cocomplete. We then show how colimits can be employed to merge the viewpoints and detect the inconsistencies that arise independent of any particular choice of viewpoint semantics. Taking advantage of the same category-theoretic techniques used in defining fuzzy viewpoints, we will also introduce a more general graph-based formalism that may find applications in other contexts. ii To my mother and father with love and gratitude. Acknowledgements First of all, I wish to thank my supervisor Steve Easterbrook for his guidance, support, and patience. -
Methodology and Metaphysics in the Development of Dedekind's Theory
Methodology and metaphysics in the development of Dedekind’s theory of ideals Jeremy Avigad 1 Introduction Philosophical concerns rarely force their way into the average mathematician’s workday. But, in extreme circumstances, fundamental questions can arise as to the legitimacy of a certain manner of proceeding, say, as to whether a particular object should be granted ontological status, or whether a certain conclusion is epistemologically warranted. There are then two distinct views as to the role that philosophy should play in such a situation. On the first view, the mathematician is called upon to turn to the counsel of philosophers, in much the same way as a nation considering an action of dubious international legality is called upon to turn to the United Nations for guidance. After due consideration of appropriate regulations and guidelines (and, possibly, debate between representatives of different philosophical fac- tions), the philosophers render a decision, by which the dutiful mathematician abides. Quine was famously critical of such dreams of a ‘first philosophy.’ At the oppos- ite extreme, our hypothetical mathematician answers only to the subject’s internal concerns, blithely or brashly indifferent to philosophical approval. What is at stake to our mathematician friend is whether the questionable practice provides a proper mathematical solution to the problem at hand, or an appropriate mathematical understanding; or, in pragmatic terms, whether it will make it past a journal referee. In short, mathematics is as mathematics does, and the philosopher’s task is simply to make sense of the subject as it evolves and certify practices that are already in place. -
Algebraic Generality Vs Arithmetic Generality in the Controversy Between C
Algebraic generality vs arithmetic generality in the controversy between C. Jordan and L. Kronecker (1874). Frédéric Brechenmacher (1). Introduction. Throughout the whole year of 1874, Camille Jordan and Leopold Kronecker were quarrelling over two theorems. On the one hand, Jordan had stated in his 1870 Traité des substitutions et des équations algébriques a canonical form theorem for substitutions of linear groups; on the other hand, Karl Weierstrass had introduced in 1868 the elementary divisors of non singular pairs of bilinear forms (P,Q) in stating a complete set of polynomial invariants computed from the determinant |P+sQ| as a key theorem of the theory of bilinear and quadratic forms. Although they would be considered equivalent as regard to modern mathematics (2), not only had these two theorems been stated independently and for different purposes, they had also been lying within the distinct frameworks of two theories until some connections came to light in 1872-1873, breeding the 1874 quarrel and hence revealing an opposition over two practices relating to distinctive cultural features. As we will be focusing in this paper on how the 1874 controversy about Jordan’s algebraic practice of canonical reduction and Kronecker’s arithmetic practice of invariant computation sheds some light on two conflicting perspectives on polynomial generality we shall appeal to former publications which have already been dealing with some of the cultural issues highlighted by this controversy such as tacit knowledge, local ways of thinking, internal philosophies and disciplinary ideals peculiar to individuals or communities [Brechenmacher 200?a] as well as the different perceptions expressed by the two opponents of a long term history involving authors such as Joseph-Louis Lagrange, Augustin Cauchy and Charles Hermite [Brechenmacher 200?b]. -
Modular Arithmetic
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 5 Modular Arithmetic One way to think of modular arithmetic is that it limits numbers to a predefined range f0;1;:::;N ¡ 1g, and wraps around whenever you try to leave this range — like the hand of a clock (where N = 12) or the days of the week (where N = 7). Example: Calculating the day of the week. Suppose that you have mapped the sequence of days of the week (Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday) to the sequence of numbers (0;1;2;3;4;5;6) so that Sunday is 0, Monday is 1, etc. Suppose that today is Thursday (=4), and you want to calculate what day of the week will be 10 days from now. Intuitively, the answer is the remainder of 4 + 10 = 14 when divided by 7, that is, 0 —Sunday. In fact, it makes little sense to add a number like 10 in this context, you should probably find its remainder modulo 7, namely 3, and then add this to 4, to find 7, which is 0. What if we want to continue this in 10 day jumps? After 5 such jumps, we would have day 4 + 3 ¢ 5 = 19; which gives 5 modulo 7 (Friday). This example shows that in certain circumstances it makes sense to do arithmetic within the confines of a particular number (7 in this example), that is, to do arithmetic by always finding the remainder of each number modulo 7, say, and repeating this for the results, and so on. -
The History of the Formulation of Ideal Theory
The History of the Formulation of Ideal Theory Reeve Garrett November 28, 2017 1 Using complex numbers to solve Diophantine equations From the time of Diophantus (3rd century AD) to the present, the topic of Diophantine equations (that is, polynomial equations in 2 or more variables in which only integer solutions are sought after and studied) has been considered enormously important to the progress of mathematics. In fact, in the year 1900, David Hilbert designated the construction of an algorithm to determine the existence of integer solutions to a general Diophantine equation as one of his \Millenium Problems"; in 1970, the combined work (spanning 21 years) of Martin Davis, Yuri Matiyasevich, Hilary Putnam and Julia Robinson showed that no such algorithm exists. One such equation that proved to be of interest to mathematicians for centuries was the \Bachet equa- tion": x2 +k = y3, named after the 17th century mathematician who studied it. The general solution (for all values of k) eluded mathematicians until 1968, when Alan Baker presented the framework for constructing a general solution. However, before this full solution, Euler made some headway with some specific examples in the 18th century, specifically by the utilization of complex numbers. Example 1.1 Consider the equation x2 + 2 = y3. (5; 3) and (−5; 3) are easy to find solutions, but it's 2 not obviousp whetherp or not there are others or whatp they might be. Euler realized by factoring x + 2 as (x + 2i)(x − 2i) and then using the facts that Z[ 2i] is a UFD andp the factorsp given are relatively prime that the solutions above are the only solutions,p namelyp because (x + 2i)(x − 2i) being a cube forces each of these factors to be a cube (i.e. -
The Science of Infinity
Volume 22, Number 2, April/May 2016 The Science of Infinity Raphael A. Fraser, PhD, Division of Biostatistics, MCW \The infinite! No other question has ever moved so profoundly the spirit of man," said David Hilbert (1862-1943), one of the most influential mathematicians of the 19th century. The subject has been studied extensively by mathematicians and philosophers but still remains an enigma of the intellectual world. The notions of “finite” and “infinite” are primitive, and it is very likely that the reader has never examined these notions very carefully. For example, there are as many even numbers as there are natural numbers. This is true but counter intuitive. We may argue, that the set of even numbers is half the size of the set of natural numbers, thus how can both sets be of the same size? To answer this question we must first explain what do we mean when we say two sets are the same size? Normally, we would count the elements in each set then verify the counts are the same. However, there is another way to verify two sets are the same size by matching. To illustrate, that matching is more fundamental than counting let us consider an example. If I walked into a large room with many chairs but all the seats are occupied and no one is standing then I can conclude that there are as many people as there are chairs. That is, there is a one-to-one correspondence between person and chair. Hence both sets are the same size, even though we did not count how many persons or chairs they were. -
Modulo of a Negative Number1 1 Modular Arithmetic
Algologic Technical Report # 01/2017 Modulo of a negative number1 S. Parthasarathy [email protected] Ref.: negamodulo.tex Ver. code: 20170109e Abstract This short article discusses an enigmatic question in elementary num- ber theory { that of finding the modulo of a negative number. Is this feasible to compute ? If so, how do we compute this value ? 1 Modular arithmetic In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value - the modulus (plural moduli). The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801. Modular arithmetic is a fascinating aspect of mathematics. The \modulo" operation is often called as the fifth arithmetic operation, and comes after + − ∗ and = . It has use in many practical situations, particu- larly in number theory and cryptology [1] [2]. If x and n are integers, n < x and n =6 0 , the operation \ x mod n " is the remainder we get when n divides x. In other words, (x mod n) = r if there exists some integer q such that x = q ∗ n + r 1This is a LATEX document. You can get the LATEX source of this document from [email protected]. Please mention the Reference Code, and Version code, given at the top of this document 1 When r = 0 , n is a factor of x. When n is a factor of x, we say n divides x evenly, and denote it by n j x . The modulo operation can be combined with other arithmetic operators. -
The Development and Understanding of the Concept of Quotient Group
HISTORIA MATHEMATICA 20 (1993), 68-88 The Development and Understanding of the Concept of Quotient Group JULIA NICHOLSON Merton College, Oxford University, Oxford OXI 4JD, United Kingdom This paper describes the way in which the concept of quotient group was discovered and developed during the 19th century, and examines possible reasons for this development. The contributions of seven mathematicians in particular are discussed: Galois, Betti, Jordan, Dedekind, Dyck, Frobenius, and H61der. The important link between the development of this concept and the abstraction of group theory is considered. © 1993 AcademicPress. Inc. Cet article decrit la d6couverte et le d6veloppement du concept du groupe quotient au cours du 19i~me si~cle, et examine les raisons possibles de ce d6veloppement. Les contributions de sept math6maticiens seront discut6es en particulier: Galois, Betti, Jordan, Dedekind, Dyck, Frobenius, et H61der. Le lien important entre le d6veloppement de ce concept et l'abstraction de la th6orie des groupes sera consider6e. © 1993 AcademicPress, Inc. Diese Arbeit stellt die Entdeckung beziehungsweise die Entwicklung des Begriffs der Faktorgruppe wfihrend des 19ten Jahrhunderts dar und untersucht die m/)glichen Griinde for diese Entwicklung. Die Beitrfi.ge von sieben Mathematikern werden vor allem diskutiert: Galois, Betti, Jordan, Dedekind, Dyck, Frobenius und H61der. Die wichtige Verbindung zwischen der Entwicklung dieses Begriffs und der Abstraktion der Gruppentheorie wird betrachtet. © 1993 AcademicPress. Inc. AMS subject classifications: 01A55, 20-03 KEY WORDS: quotient group, group theory, Galois theory I. INTRODUCTION Nowadays one can hardly conceive any more of a group theory without factor groups... B. L. van der Waerden, from his obituary of Otto H61der [1939] Although the concept of quotient group is now considered to be fundamental to the study of groups, it is a concept which was unknown to early group theorists. -
Grade 7/8 Math Circles Modular Arithmetic the Modulus Operator
Faculty of Mathematics Centre for Education in Waterloo, Ontario N2L 3G1 Mathematics and Computing Grade 7/8 Math Circles April 3, 2014 Modular Arithmetic The Modulus Operator The modulo operator has symbol \mod", is written as A mod N, and is read \A modulo N" or "A mod N". • A is our - what is being divided (just like in division) • N is our (the plural is ) When we divide two numbers A and N, we get a quotient and a remainder. • When we ask for A ÷ N, we look for the quotient of the division. • When we ask for A mod N, we are looking for the remainder of the division. • The remainder A mod N is always an integer between 0 and N − 1 (inclusive) Examples 1. 0 mod 4 = 0 since 0 ÷ 4 = 0 remainder 0 2. 1 mod 4 = 1 since 1 ÷ 4 = 0 remainder 1 3. 2 mod 4 = 2 since 2 ÷ 4 = 0 remainder 2 4. 3 mod 4 = 3 since 3 ÷ 4 = 0 remainder 3 5. 4 mod 4 = since 4 ÷ 4 = 1 remainder 6. 5 mod 4 = since 5 ÷ 4 = 1 remainder 7. 6 mod 4 = since 6 ÷ 4 = 1 remainder 8. 15 mod 4 = since 15 ÷ 4 = 3 remainder 9.* −15 mod 4 = since −15 ÷ 4 = −4 remainder Using a Calculator For really large numbers and/or moduli, sometimes we can use calculators to quickly find A mod N. This method only works if A ≥ 0. Example: Find 373 mod 6 1. Divide 373 by 6 ! 2. Round the number you got above down to the nearest integer ! 3. -
RING THEORY 1. Ring Theory a Ring Is a Set a with Two Binary Operations
CHAPTER IV RING THEORY 1. Ring Theory A ring is a set A with two binary operations satisfying the rules given below. Usually one binary operation is denoted `+' and called \addition," and the other is denoted by juxtaposition and is called \multiplication." The rules required of these operations are: 1) A is an abelian group under the operation + (identity denoted 0 and inverse of x denoted x); 2) A is a monoid under the operation of multiplication (i.e., multiplication is associative and there− is a two-sided identity usually denoted 1); 3) the distributive laws (x + y)z = xy + xz x(y + z)=xy + xz hold for all x, y,andz A. Sometimes one does∈ not require that a ring have a multiplicative identity. The word ring may also be used for a system satisfying just conditions (1) and (3) (i.e., where the associative law for multiplication may fail and for which there is no multiplicative identity.) Lie rings are examples of non-associative rings without identities. Almost all interesting associative rings do have identities. If 1 = 0, then the ring consists of one element 0; otherwise 1 = 0. In many theorems, it is necessary to specify that rings under consideration are not trivial, i.e. that 1 6= 0, but often that hypothesis will not be stated explicitly. 6 If the multiplicative operation is commutative, we call the ring commutative. Commutative Algebra is the study of commutative rings and related structures. It is closely related to algebraic number theory and algebraic geometry. If A is a ring, an element x A is called a unit if it has a two-sided inverse y, i.e.