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We give an algorithm for factoring quadratic polynomials over any UFD, Z in particular. We prove the correctness of this algorithm and give examples over Z and Z[i].
It is well known that Al-Khayâm, for the first time in history, formulated a complete theory to solve third degree equations using the intersection of geometric curves and moreover solved the fourteen types of equations using this method. His solution for these equations was either using the intersection of two parabolas, a parabola and a circle, a parabola and a hyperbola, a circle and a hyperbola or two hyperbolas. His style was purely synthetic lacking any analysis which may lead to deducing any motives for his choice of the curves. Our article is centered on a pointed detail which is the justification of the choice of these curves, knowing that R. Rashed studied this choice in all of the mentioned fourteen equations. As for Descartes, whose style is as synthetic as Al-Khayâm's, we ask the same question concerning the solution of the third degree equations, searching for any possible resemblance between the two methods of solution, both mathematicians having been motivated similarly in geometric resolutions. We noticed that the choice of the fourteen curves couples was made by Al Khayyâm regarding a uniform calculation technique that he applied systematically to each equation type among the fourteen. It has appeared that the choice of the curves made by Descartes is the result of distinct calculation techniques, even though the projects of the two mathematicians are of the same essence, and their motivations are the same: finding the solution of third degree equations using the intersection of conical curves and the equations of curves.
This paper gives a condition of the expression of generality in mathematics from the application of Löwenheim-Skolem theorem to Zermelo's axioms. It gives an example of an "expression problem" from Gauss's Disquisitiones Arithmeticae and caracterizes the used of sets in it.---L'article dégage une condition de l'expression de la généralité en mathématiques à partir de l'application du théorème de Löwenheim-Skolem aux axiomes de Zermelo. Il donne un exemple de "problème d'expression" à partir des Disquisitiones Arithmeticae de Gauss, dégageant ainsi une condition du recours aux ensembles.
Stochastic diffusion equation, which attained prominence with Einstein's work on Brownian motion at the beginning of the twentieth century, was first formulated by Laplace a century earlier as part of his work on Central Limit Theorem. Between 1807 and 1811, Fourier's work on heat diffusion, and Laplace'swork on probability influenced and inspired each other. This brief period of interaction between these two illustrious figures must be considered remarkable for its profound impact on subsequent developments in mathematical physics, probability theory and pure analysis.
Georgy Theodosiyovych Voronoi (1868-1908) is famous for his seminal contributions to number theory,perhaps mostly those involving quadratic forms and Voronoi tessellations. He was born and grew up in the town of Zhuravka in the Ukraine, at the time part of the Russian Empire. Having studied at St. Petersburg University under the supervision of Andrey Markov, in 1894 he became a professor of pure mathematics at the University of Warsaw. In his career he published six large memoirs and six short papers, each of which were so profound and significant that they left a deep trace in modern number theory. Together with Minkowski, he can be considered as the founder of the Geometry of Numbers. In this contribution, a brief sketch will be given of his life, work and legacy.
This is expanded text of a lecture delivered by the author at the conference "Matériaux pour l'Histoire des Mathématiques au XXème siècle", which took place in Nice in January 1996. The task was to describe one area in the development of arithmetical algebraic geometry in Moscow during the 1950s and 1960s. We shall begin by explaining the meaning of the analogy between numbers and functions, starting with the simplest concepts. In the second part we study a nontrivial example: the explicit formula for the law of reciprocity. In the third part we shall become acquainted with certain aspects of the "social" life of the Moscow school, in particular, with certain seminars, lectures, and books. In the final part we shall examine another example of this analogy: arithmetical surfaces and Arakelov theory.
This survey contains an exposition of ideas and results related to Faltings' proof of the conjectures of Shafarevich, Tate and Mordell. This paper originally appeared in 1986 as an Appendix to the Russian translation of Serge Lang, "Fundamentals of Diophantine Geometry" (Springer Verlag, 1983) published by "Mir", Moscow (MR0854670, 88a:11054). A history of the publication of the Appendix is briefly described by Lang in Section 4 of his paper "Mordell's review, Siegel's letter to Mordell, Diophantine geometry, and 20th century mathematics" that was published (in 1995) simultaneously in Notices of the AMS and Gazette des Mathématiciens (SMF) (MR1316025, 96g:11002a; MR1316133, 96g:11002b) http://smf.emath.fr/Publications/Gazette/1995/63/smf_gazette_63_17-36.pdf . Later an expanded version of the Appendix was translated into English by Neal Koblitz and published in 1989 by the American Mathematical Society as part of the collection "Eight papers translated from the Russian", AMS Translations, Series 2, Vol. 143 http://www.ams.org/bookstore-getitem/item=TRANS2-143 (MR1008476, 90b:00011). We put this paper on the arXiv with the kind permission of the American Mathematical Society. For this version we slightly updated the bibliography and added a few short notes (marked as "Added in December 2009"). We also corrected inaccuracies that were kindly pointed out to us by J.-P. Serre - one of the few people to read this paper (in English) twenty years ago.
The paper suggests a slightly more rigorous justification to Wang et al.'s work from 2007, and introduces the Slanted Line Integral.
This study use some philological and historical means in order to understand Fermat's way of thinking.
A positive integer n is said to be perfect if sigma(n)=2n, where sigma denotes the sum of the divisors of n. In this article, we show that if n is an even perfect number, then any integer m<=n is expressed as a sum of some of divisors of n.
Polya Enumeration Theorem is one of the most useful tools dealing with the enumeration of patterns that are symmetric in some ways. What follows is a procedure for obtaining the results of Polya Theorem directly, bypassing the usual preliminaries.
We motivate and then prove a generalized pythagorean theorem for parallelepipeds in Euclidean space.
It is well known that the Taylor series expansion of $(1+ z)^{A}$ does not converge for $|z|>1$ where A is a real number which is not equal to zero or a positive integer. A limited series expansion of this expression is obtained in this paper for $|z|>1$ as a product of convergent series.
C. F. Gauss discovered a beautiful formula for the number of irreducible polynomials of a given degree over a finite field. Assuming just a few elementary facts in field theory and the exclusion-inclusion formula, we show how one see the shape of this formula and its proof instantly.
This article discusses classical versions of the Schwarz lemma at the boundary of the unit disk in the complex plane. The exposition includes commentary on the history, the mathematics, and the applications.
A quite old problem has been recently revitalized by Leonard Mlodinow's book The Drunkard's Walk, where it is presented in a way that has definitely confused several people, that wonder why the prevalence of the name of one daughter among the population should change the probability that the other child is a girl too. I try here to discuss the problem from scratch, showing that the rarity of the name plays no role, unless the strange assumption of two identical names in the same family is taken into account. But also the name itself does not matter. What is really important is `identification', meant in an acceptation broader than usual, in the sense that a child is characterized by a set of attributes that make him/her uniquely identifiable (`that one') inside a family. The important point of how the information is acquired is also commented, suggesting an explanation of why several people tend to consider the informations "at least one boy" and "a well defined boy" (elder/youngest or of a given name) equivalent.
Generalist mathematics journals exhibit bias toward the branches of mathematics by publishing articles about some subjects in quantities far disproportionate to the production of papers in those areas within all of mathematics.
Although the same mathematical expression is used to describe physical diffusion and stochastic diffusion, there are intrinsic similarities and differences in their nature. A comparative study shows that characteristic terms of physical and stochastic diffusion cannot be placed exactly in one-to-one correspondence. Therefore, judgment needs to be exercised in transferring ideas between physical and stochastic diffusion.