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A simple but rigorous proof of the Fundamental Theorem of Calculus is given in geometric calculus, after the basis for this theory in geometric algebra has been explained. Various classical examples of this theorem, such as the Green's and Stokes' theorem are discussed, as well as the new theory of monogenic functions, which generalizes the concept of an analytic function of a complex variable to higher dimensions.
A method is presented for evaluating authors on the basis of citations. It assigns to each author a citation score which depends upon the number of times he is cited, and upon the scores of the citers. The scores are found to be the components of an eigenvector of a normalized citation matrix. The same method can be applied to citation of journals by other journals, to evaluating teams in a league [1], etc.
This biographical account of the life and work of David Kendall includes details of his personal and professional activities. Kendall is probably best known for his work in applied probability, especially queueing theory, and in stochastic analysis and spatial statistics.
These are lecture notes for a 1-semester undergraduate course (in computer science, mathematics, physics, engineering, chemistry or biology) in applied categorical meta-language. The only necessary background for comprehensive reading of these notes are first-year calculus and linear algebra.
In october 2008, CNRS adopts a new logo with a round shape. We study the mathematical representation of this shape, and in particular its convexity.
Life and the mathematical legacy of the great mathematician A.V. Pogorelov.
This paper is a brief overview of the main contributions of Masaki Kashiwara in the domain of microlocal and algebraic analysis.
The paper analizes a set of issues related to analogy and analogical reasoning, namely: 1) The problem of analogy and its duplicity; 2) The role of analogy in demonstrative reasoning; 3) The role of analogy in non-demonstrative reasoning; 4) The limits of analogy; 5) The convergence, particularly in multiple analogical reasoning, of these two apparently distinct aspects and its methodological and philosophical consequences. The paper, using example from number theory, argues for an heuristc conception of analogy.
Many things in mathematics seem lamost unreasonably nice. This includes objects, counterexamples, proofs. In this preprint I discuss many examples of this phenomenon with emphasis on the ring of polynomials in a countably infinite number of variables in its many incarnations such as the representing object of the Witt vectors, the direct sum of the rings of representations of the symmetric groups, the free lambda ring on one generator, the homology and cohomology of the classifying space BU, ... . In addition attention is paid to the phenomenon that solutions to universal problems (adjoint functors) tend to pick up extra structure.
Is .999... equal to 1? Lightstone's decimal expansions yield an infinity of numbers in [0,1] whose expansion starts with an unbounded number of digits "9". We present some non-standard thoughts on the ambiguity of an ellipsis, modeling the cognitive concept of generic limit of B. Cornu and D. Tall. A choice of a non-standard hyperinteger H specifies an H-infinite extended decimal string of 9s, corresponding to an infinitesimally diminished hyperreal value. In our model, the student resistance to the unital evaluation of .999... is directed against an unspoken and unacknowledged application of the standard part function, namely the stripping away of a ghost of an infinitesimal, to echo George Berkeley. So long as the number system has not been specified, the students' hunch that .999... can fall infinitesimally short of 1, can be justified in a mathematically rigorous fashion.
Mahlburg (2005) brilliantly showed the importance of crank functions in partition congruences that were originally guessed by Dyson (1944). Ramanujan's partition functions are the centre of these works. Not only for the theory on cranks, but for many other researchers' in India Ramanujan's work inspired for their career in mathematics. This is an undergraduate expository article.
The German forces occupying Paris arrested Emile Borel and three other members of the Académie des Sciences in October 1941 and released them about five weeks later. Why? We examine some relevant German and French archives and other sources and propose some hypotheses. In the process, we review how the Occupation was structured and how it dealt with French higher education and some French mathematicians.
We survey the basics of homological algebra in exact categories in the sense of Quillen. All diagram lemmas are proved directly from the axioms, notably the five lemma, the 3 x 3-lemma and the snake lemma. We briefly discuss exact functors, idempotent completion and weak idempotent completeness. We then show that it is possible to construct the derived category of an exact category without any embedding into abelian categories and we sketch Deligne's approach to derived functors. The construction of classical derived functors with values in an abelian category painlessly translates to exact categories, i.e., we give proofs of the comparison theorem for projective resolutions and the horseshoe lemma. After discussing some examples we elaborate on Thomason's proof of the Gabriel-Quillen embedding theorem in an appendix.
Statistical summaries of certain kinds of mathematics research output are given for a large sample of U.S. regional public universities. These statistical summaries are reported using a variety of metrics that distinguish between single-authored and collaborative work and account for publication length.
When can a majority of voters find common ground, that is, a position they all agree upon? How does the shape of the political spectrum influence the outcome? When mathematical objects have a social interpretation, the associated theorems have social applications. In this article we give examples of situations where sets model preferences and develop extensions of classical theorems about convex sets, such as Helly's theorem, that can be used in the analysis of voting in "agreeable" societies.
We derive some formulas that rule the behaviour of finite differences under composition of functions with vector values and arguments.
We survey a new paradigm in signal processing known as "compressive sensing". Contrary to old practices of data acquisition and reconstruction based on the Shannon-Nyquist sampling principle, the new theory shows that it is possible to reconstruct images or signals of scientific interest accurately and even exactly from a number of samples which is far smaller than the desired resolution of the image/signal, e.g., the number of pixels in the image. This new technique draws from results in several fields of mathematics, including algebra, optimization, probability theory, and harmonic analysis. We will discuss some of the key mathematical ideas behind compressive sensing, as well as its implications to other fields: numerical analysis, information theory, theoretical computer science, and engineering.
We investigate the voting rules in the Council of the European Union. It is known that the current system, according to the Treaty of Nice, and the voting system proposed in the Lisbon treaty both strongly deviate from the square root law by Penrose. This is known to be the ideal voting rule under certain assumptions. In 2004 Slomczynski and Zyczkowski designed a voting system, now known as the Jagiellonian Compromise. It satisfies the square root law with very high accuracy. Each member state in this system obtains a voting weight proportional to the square root of the population. Additionally the quota is fixed in such a way that the voting power of each country is also proportional to the square root of the population. In this paper we investigate to which extent a change of the quota in the Treaty of Nice and the Treaty of Lisbon may bring the voting power closer to the ideal square root distribution. Our computations show that even with optimal quota both systems are way off the ideal power distribution.