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The goal of this modern presentation, followed by an English translation from the German, is to make available some parts of Lie's very systematic mathematical thought which deserve to join the contemporary literature, and above all also, to be read.
We investigate a coin-weighing puzzle that appeared in the all-Russian math Olympiad in 2000. We liked the puzzle because the methods of analysis differ from classical coin-weighing puzzles. We generalize the puzzle by varying the number of participating coins, and deduce a complete solution, perhaps surprisingly, the objective can be achieved in no more than two weighings regardless of the number of coins involved.
Take a large bag of black and white beans, with all possible proportions considered initially equally likely, and imagine to make random extractions with reintroduction. Twenty consecutive observations of black make us highly confident that the next bean will be black too. On the contrary, the observation of 1010 black beans and 990 white ones leads us to judge the two possible outcomes about equally probable. According to C.S. Peirce this reasoning violates what he called "rule of balancing reasons", because the difference of "arguments" in favor and against the outcome of black is 20 in both cases. Why? (I.e. why does that rule not apply here?)
In this book, I explored differential equations for operation in Lie group and for representations of group Lie in a vector space.
We give 50 digits values of the simple continued fractions whose denominators are formed from a) prime numbers, b) twin primes, c) generalized $d$-twins, d) primes of the form $m^2+n^4$, e)primes of the form $m^2+1$, f) Mersenne primes and g) primorial primes. All these continued fractions belong to the set of measure zero of exceptions to the theorems of Khinchin and Levy. We claim that all these continued fractions are transcendental numbers. Next we propose the conjecture which indicates the way to deduce the transcendence of some continued fractions from transcendence of another ones.
At http://wiki.openmath.org, the OpenMath 2 and 3 Content Dictionaries are accessible via a semantic wiki interface, powered by the SWiM system. We shortly introduce the inner workings of the system, then describe how to use it, and conclude with first experiences gained from OpenMath society members working with the system and an outlook to further development plans.
The book A Treatise on Probability was published by John Maynard Keynes in 1921. It contains a critical assessment of the foundations of probability and of the current statistical methodology. As a modern reader, we review here the aspects that are most related with statistics, avoiding a neophyte's perspective on the philosophical issues. In particular, the book is quite critical of the Bayesian approach and we examine the arguments provided by Keynes, as well as the alternative he proposes. This review does not subsume the scholarly study of Aldrich (2008a) relating Keynes with the statistics community of the time.
Karl Stein was one of the pillars of the German school of several complex variables. In this article his scientific contributions are outlined in historical perspective.
This article was written on the occasion of Hans Grauert receiving the Cantor Medallion of the Deutsche Mathematische Vereinigung. It is a brief overview of his mathematical contributions and attempts to convey the author's great respect for the man and his science.
Ever since it was published by Neugebauer and Sachs in 1945, the Old Babylonian tablet known as Plimpton 322 has been the subject of numerous studies leading to different and often conflicting interpretations of it. Overall, the tablet is more or less viewed as a list of fifteen Pythagorean triplets, but scholars are divided on how and why the list was devised. In this paper, we present a survey of previous attempts to interpret Plimpton 322, and then offer some new insights that could help in sharpening the endless debate about this ancient tablet.
The aim of this expository article is twofold. The first is to introduce several polynomials of one variable as well as two variables defined on the positive integers with values as congruent numbers. The second is to present connections between Pythagorean triples and the Pell equation $x^2-dy^2=1$ plus its analogous counterpart $x^2-dy^2=-1$ which give rise to congruent numbers n with arbitrarily many prime factors.
It is a story about the problem whether folding a square on the plane can increase its perimeter. The paper is written primary for school students.
This paper, in French, is a celebration of Max Dehn, and an essay of describing some of his results published in the beginning of the 1910's, and their offspring. It has been written up for a winter school in Les Diablerets, March 7-12, 2010: Geometry, topology and computation in groups, 100 years since Dehn's Decision Problems.
The concept of informal mathematical proof considered in intuitionism is apparently vulnerable to a version of the liar paradox. However, a careful reevaluation of this concept reveals a subtle error whose correction blocks the contradiction. This leads to a general resolution of the classical semantic paradoxes. This paper is an expanded version of parts of my earlier paper "Constructive truth and circularity" [arXiv:0905.1681].
Many groups possess highly symmetric generating sets that are naturally endowed with an underlying combinatorial structure. Such generating sets can prove to be extremely useful both theoretically in providing new existence proofs for groups and practically by providing succinct means of representing group elements. We give a survey of results obtained in the study of these symmetric generating sets. In keeping with earlier surveys on this matter, we emphasize the sporadic simple groups. ADDENDUM: This is an updated version of a survey article originally accepted for inclusion in the proceedings of the 2009 `Groups St Andrews' conference. Since the article was accepted the author has become aware of other recent work in the subject that we incorporate to provide an updated version here (the most notable addition being the contents of Section 3.4.)
We mark up a corpus of LaTeX lecture notes semantically and expose them as Linked Data in XHTML+MathML+RDFa. Our application makes the resulting documents interactively browsable for students. Our ontology helps to answer queries from students and lecturers, and paves the path towards an integration of our corpus with external sites.
This paper summarizes basic properties of PPTs and shows that each PPT belongs to one of six different classes. Mapping an ordered sequence of PPTs into a corresponding sequence of these six classes makes it possible to use them in cryptography. We pose problems whose solution would facilitate such cryptographic application.
Corollary 2, Entry 9, Chapter 4 of Ramanujan's first notebook claims that a certain sum is asymptotic to ln(x) + gamma, where x is a real variable in the sum and gamma is Euler's constant. Ramanujan's claim is known to be correct for the case n = 1, but incorrect for n > 2 (here n is an integer parameter in the sum). We show that the result is correct for n = 2. We also consider the order of the error term, and discuss a different, correct generalisation of the case n = 1.