历史与综述
This is an expository paper designed to introduce undergraduates to the Atiyah-Singer index theorem 50 years after its announcement. It includes motivation, a statement of the theorem, an outline of the easy part of the heat equation proof.…
The word "complexity" is most often used as a meta--linguistic expression referring to certain intuitive characteristics of a natural system and/or its scientific description. These characteristics may include: sheer amount of data that…
This is a literal word-for-word translation from the French of Phragmen's proof (the first such published) of Weierstrass' famous theorem characterizing all analytic functions which possess an algebraic addition theorem.
This is a literal word-for-word translation from the German of the article by Paul Koebe which contains a proof of Weierstrass's famous theorem characterizing all analytic functions which possess an algebraic addition theorem.
In this instructional exposition in Portuguese for the popularisation of mathematics, we state and prove the completeness theorem.
We present a very simple example of a theorem with constructive and non-constructive proofs: the equation c^2 x^2 - (c^2 + c)x + c = 0 has a solution.
We re-evaluate the great Leibniz-Newton calculus debate, exactly three hundred years after it culminated, in 1712. We reflect upon the concept of invention, and to what extent there were indeed two independent inventors of this new…
This paper deals with a problem in which two players share a previously sliced pizza and try to eat as much amount of pizza as they can. It takes time to eat each piece of pizza and both players eat pizza at the same rate. One is allowed to…
We present a proof that any continuous function with domain including a closed interval yields an antiderivative of that function on that interval. This is done without the need of any integration comparable to that of Riemann, Cauchy, or…
The unique real root of cos(x) = x, recently referred to as the Dottie number, is expressed as an iteral of cosine. Using the derivatives of iterals, it is shown why this number is achieved starting from any real number, when the iterates…
We explore Leibniz's understanding of the differential calculus, and argue that his methods were more coherent than is generally recognized. The foundations of the historical infinitesimal calculus of Newton and Leibniz have been a target…
In this article, I explain why I abandoned my original plan to write a novel about a mathematician proving Fermat's Last Theorem, letting my protagonist solve the entirely fictional Wild Number Problem instead. I enumerate the steps that…
Residues to a given modulus have been introduced to mathematics by Carl Friedrich Gauss with the definition of congruence in the `Disquisitiones Arithmeticae'. Their extraordinary properties provide the basis for a change of paradigm in…
This is a brief overview of researches of Dmitry Petrovich Zhelobenko (1934--2006). He is the best known for his book "Compact Lie groups and their representations" and for the classification of all irreducible representations of complex…
In this study, we determine some special Smarandache curves in E1 3.We give some characterizations consequences of Smarandache curves.
Reflections on the film "Rites of Love and Math" (directed by Reine Graves and the author) and its "formula of love".
In this paper, our aim is to discuss the outcome of a ten-year experimental workshop activity carried out by the Center `matematita' (http://www.matematita.it/) with classes of pre-university students of any age. We analyse the potential…
Cet article applique les notions de texte et d'\'enonc\'e inaugural \`a la Th\'eorie analytique de la chaleur de Fourier et \`a la controverse des cordes vibrantes. Il \'etablit que seul le livre de Fourier est un texte inaugural. Il…
The efficacy of using complex numbers for understanding geometric questions related to polar equations and general cycloids is demonstrated.
The \emph{arbelos} is the shape bounded by three mutually tangent semicircles with collinear diameters. Recently, Sondow introduced the parabolic analog, the \emph{parbelos} and proved several properties of the parbelos similar to…