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A Machine-Checked Proof of the Odd Order Theorem
[chapter]

2013
*
Lecture Notes in Computer Science
*

This paper reports on a six-year collaborative effort that culminated in a complete formalization of a

doi:10.1007/978-3-642-39634-2_14
fatcat:43zakbhshbftjfdptlw2fzyskm
*proof*of*the*Feit-Thompson*Odd**Order**Theorem*in*the*Coq*proof*assistant. ...*The*formalized*proof*is constructive, and relies on nothing but*the*axioms and rules of*the*foundational framework implemented by Coq. ...*The*authors would like to thank*the*Coq team for their continuous development, improvement and maintenance of*the*Coq*proof*assistant. ...##
###
Engineering mathematics

2013
*
SIGPLAN notices
*

By combining tools and techniques from type theory, language design, and software

doi:10.1145/2480359.2429071
fatcat:kclmi57xebehrn3xlblowglqve
*engineering*we have managed to capture enough of these practices to formalize*the**proof*of*the**Odd**Order**theorem*, a landmark ... Even with*the*assistance of computer tools,*the*formalized description and verification of research-level*mathematics*remains a daunting task, not least because of*the*talent with which mathematicians ...*Odd**Order**theorem*. ...##
###
Software engineering for mathematics (keynote)

2013
*
Proceedings of the 2013 9th Joint Meeting on Foundations of Software Engineering - ESEC/FSE 2013
*

By drawing on tools and techniques from type theory, language design, and software

doi:10.1145/2491411.2505429
dblp:conf/sigsoft/Gonthier13
fatcat:m5bidz3uzbebxn6tnirio32sgy
*engineering*we captured enough of these practices to formalize*the**proof*of*the**Odd**Order**theorem*, a landmark result in ... This library then allowed us to write a formal*proof*comparable in size and abstraction level to*the*250-page textbook*proof*of*the**Odd**Order**theorem*. ... He now heads*the**Mathematical*Components project at*the*MSR Inria Joint Center, which recently completed*the*formalization of*the**Odd**Order**theorem*,*the*landmark result that ultimately lead to*the*Classification ...##
###
Mathematical proof: from mathematics to school mathematics

2019
*
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
*

*mathematical*

*proof*in

*the*classroom. ... This article discusses some issues involving

*mathematical*

*proof*in school, intending to characterize

*the*understanding of

*mathematical*

*proof*in school, its function and

*the*meaning and relevance attributed ... In a

*mathematical*

*proof*, definitions, statements and procedures are intertwined in a suitable way in

*order*to get

*the*desired result. ...

##
###
Mathematical Methods for Physics and Engineering

1999
*
American Journal of Physics
*

*The*third edition of this highly acclaimed undergraduate textbook is suitable for teaching all

*the*

*mathematics*ever likely to be needed for an undergraduate course in any of

*the*physical sciences. ... read

*mathematics*at

*the*University of Cambridge and proceeded to a Ph.D. there in theoretical and experimental nuclear physics. ... summation

*orders*, etc. on

*the*xxv www.cambridge.org © in this web service Cambridge University Press Cambridge University Press 978-0-521-67971-8 -

*Mathematical*Methods for Physics and

*Engineering*: Third ...

##
###
Teaching mathematics to software engineers
[chapter]

1995
*
Lecture Notes in Computer Science
*

Based on my experience in teaching formal methods to practicing and aspiring software

doi:10.1007/3-540-60043-4_44
fatcat:6bfujov3anc43oz56efpop4ibu
*engineers*, I present some of*the*common stumbling blocks faced when writing formal specifications. ...*The*most conspicuous problem is learning to abstract. I address all these problems indirectly by giving a list of hints to specifiers. ... and teaching*the*Models, Methods, and Analysis courses for*the*MSE program. ...##
###
Engineering Mathematics and Modern Technology

2018
*
International Journal of Educational Technology and Learning
*

Technology has become an essential tool for doing

doi:10.20448/2003.21.8.13
fatcat:4ig3rer2o5edpccjfasj2gf26u
*mathematics*in today's world. It can be used in a variety of ways to improve and enhance*the*teaching of*mathematics*. ... Our main goal, of all efforts in this regard, is*the*connection between*mathematics*and technology. ... Thus intuition and rigour need not be at*odds*with each other. By providing a suitably powerful context, intuition naturally leads into*the*rigour of*mathematical**proof*. ...##
###
A Machine-Checked Proof of the Odd Order Theorem

2018

*The*authors would like to thank

*the*Coq team for their continuous development, improvement and maintenance of

*the*Coq

*proof*assistant. ... We are now able to state

*the*

*Odd*

*Order*

*Theorem*.

*The*

*Odd*

*Order*

*Theorem*

*Theorem*1 (

*Odd*

*Order*

*theorem*[14] ). Every finite group of

*odd*

*order*is solvable. ... Introduction

*The*

*Odd*

*Order*

*Theorem*asserts that every finite group of

*odd*

*order*is solvable. ...

##
###
Mathematical principles for a first course in software engineering

1989
*
IEEE Transactions on Software Engineering
*

*The*discipline of software

*engineering*has transferred

*the*cummonsense methods of good programming and management to large software projects. ... We have developed an introductory computer science course, much as calculus is a basic course for

*mathematics*and

*the*physical sciences, concerned primarily with theoretical foundations and methodology ... A simplifying condition for demonstrating that a program satisfies its specification is given in

*the*following

*theorem*, called

*the*Correctness

*Theorem*G. ...

##
###
Is Mathematical Logic Really Necessary in Teaching Mathematical Proofs?

2019
*
Athens Journal of Education
*

As it is already observed by mathematicians and educators, there is a discrepancy between

doi:10.30958/aje.7-1-5
fatcat:rtox7vhbdjcelc4s7u5jovge5a
*the*formal techniques of*mathematical*logic and*the*informal techniques of*mathematics*in regards to*proof*. ... Finally, we argue that even though*mathematical*logic is central in*mathematics*, its formal methods are not really necessary in doing and teaching*mathematical**proofs*and*the*role of those formalities ... for his comments on*the*book*Mathematics*B' Lyceum, and Babis Stergiou and Christos Lefkonitziatis for communicating to me some of*the*curriculum historicals related to*the*paper. ...##
###
Engineering Design by Geometric Programming

2013
*
Mathematical Problems in Engineering
*

In addition,

doi:10.1155/2013/568098
fatcat:tqe2p5x4d5cmjfznpdtsviangm
*the**engineering*design problems are illustrated to evaluate*the*usefulness of*the*proposed methods. ... A geometric program (GP) is a type of*mathematical*optimization problem characterized by objective and constraint functions, where all functions are of signomial form. ...*The*author also thanks*the*anonymous referees for their careful reading and fruitful comments on*the*paper. ...##
###
Mathematical Notes

1960
*
The American mathematical monthly
*

This contradicts

doi:10.1080/00029890.1960.11989486
fatcat:aekydzab2nhpze4juz5dc6k3qy
*the*lemma.*THEOREM*.*The**order*of G is n(n) tf n is*odd*and (n/2)p(n) if n ts even. Coro.iary 1. A circulant of prime*order*p belongs to*the*metacyclic [3] group of degree p. ...*The*following*theorem*, analogous to*Theorem*3.1, characterizes*the*class K**.*THEOREM*3.2. f(x) EG K** if and only if x*f(x)EK.*Proof*. ...##
###
A reverse engineering approach to the Weil representation

2014
*
Open Mathematics
*

Thomas, skipping most of

doi:10.2478/s11533-014-0428-8
fatcat:yfe4armz6vb3roaqizmibzdqxe
*the*literature on which*the*latter is based. ... AbstractWe describe a new approach to*the*Weil representation attached to a symplectic group over a finite or a local field. ... Also, we express our gratitude to*the*two referees for their time and guidance through some related literature.*The*second author was partially supported by*the*NSA grant H98230-13-1-0205. ...##
###
MATHEMATICAL EXAMPLES OF SPACE-TIME PHASES

2011
*
International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
*

*The*space-time phases of a complex dynamic system are

*the*probability distributions for state as a function of space and time which arise by evolving initial probability distributions from

*the*distant ... These examples are expected to serve as useful steps on

*the*way to a general theory of space-time phases. ... Acknowledgements We are grateful to Andrei Toom for teaching us

*the*state of

*the*art in

*the*analysis of PCA during his visit to Warwick in January 2010 supported by

*the*EPSRC

*Mathematics*Symposium year ...

##
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Mechanized mathematics

1966
*
Bulletin of the American Mathematical Society
*

*Theorems*of this kind are not easy to find in those drab branches of

*mathematics*where elaborate

*proofs*are not

*the*rule. However, there are infinitely many such

*theorems*in number theory alone. ... Other

*theorem*proving projects have dealt with parts of plane geometry and first

*order*predicate calculus. ...

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