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At least 217 records · Page 12Linked to original sources
arXiv2009

On the infinity of infinities of orders of the infinitely large and infinitely small

Translation (by J.B.) from the original Latin of Euler's "De infinities infinitis gradibus tam infinite magnorum quam infinite parvorum" (1780). E507 in the Eneström index. Euler discusses orders of infinity in this paper. In other words this paper is about how different functions approach infinity or 0 at different rates. I was not certain about what Euler means by "infinities infiniti". Probably he means that $x,x^2,x^3$, etc. are infinitely many orders of infinity, and $\log x,(\log x)^2, (\log x)^3$, etc. are infinitely many orders of infinity, so the combinations of them are an infinity of infinities of orders of infinity. In fact Euler mentions other orders of infinity in this paper. It would be worthwhile to study this paper more to figure out exactly what Euler means here. Another translation of the title is "On the infinitely infinite orders of the infinitely large and infinitely small". Here's another place Euler uses the phrase "infinities infiniti". The phrase "infinities infiniti" from the title is used by Euler also in section 21 of E302, "De motu vibratio tympanorum". Truesdell translates this phrase on p. 333 of "The rational mechanics of flexible or elastic bodies" as "infinity of infinities". I'd like to thank Martin Mattmueller for clearing up some questions.

math.HO↗
arXiv2009

Spéculation sur la géométrie en Égypte antique

The Great Pyramids of Egypt hide mathematic information unknown up to date. The measurements of the three Great Pyramids of Egypt at Giza show that Egyptians knew how to calculate the circumference, the volume and the area of the sphere 2000 years before Greeks. The Bent pyramid shows the three great problems of geometry of antiquity: cubature of the sphere, duplication of the cube, trisection of the angle. According to these findings, we assert that Egyptians wanted to engrave basic concepts in measures and positions of Great Pyramids.--Les grandes pyramides d'Égypte dissimulent des informations mathématiques ignorées jusqu'à aujourd'hui. Les mesures des trois grandes pyramides d'Égypte à Gizeh révèlent que les égyptiens de la IVe dynastie savaient calculer la circonférence, le volume et l'aire de la sphère 2000 ans avant les Grecs. La pyramide rhomboïdale affiche les trois grands problèmes de géométrie de l'antiquité : cubature de la sphère, duplication du cube, trisection de l'angle. Ainsi, nous affirmons que les Égyptiens voulaient graver de façon indestructible les concepts de base de la géométrie.

math.HO↗
arXiv2009

Not served on a silver platter! Access to online mathematics information in Africa

This paper argues that, contrary to the beliefs of many, the amount of mathematics information available for African researchers, including electronic scientific journals and databases, is indeed substantial. However, whereas information resources are served on a silver platter at the "northern" universities, researchers at universities in developing countries have to work hard for their treats. The scientific information available for low-income countries is scattered among a large number of providers, websites, access methods, price models, and country- or institution-specific programmes. It is therefore quite hard for individual researchers to see the whole picture and establish what actually is available and what is not. This paper presents a number of key information sources for mathematics. Some are aimed primarily at disciplines other than mathematics but nevertheless contain extremely important and high-ranking mathematics journals. Brief instructions are also given on how to register for and maintain access to various relevant and useful resources.

math.HO↗
arXiv2009

The Game of Cipher Beads

Comparison between the various impact factors of a few Russian journals demonstrates the deficiencies of the popular citation indices.

cs.DL↗
arXiv2009

Criterion for linear independence of functions

Using a generalization of forward elimination, it is proved that functions $f_1,...,f_n:X\to\mathbb{A}$, where $\mathbb{A}$ is a field, are linearly independent if and only if there exists a nonsingular matrix $[f_i(x_j)]$ of size $n$, where $x_1,...,x_n\in X$.

math.HO↗
arXiv2009

You could simplify calculus

I explain a direct approach to differentiation and integration. Instead of relying on the general notions of real numbers, limits and continuity, we treat functions as the primary objects of our theory, and view differentiation as division of f(x)-f(a) by x-a in a certain class of functions. When f is a polynomial the division can be carried out explicitly. To see why a polynomial with a positive derivative is increasing (the monotonicity theorem), we use the estimate |f(x)-f(a)-f'(a)(x-a)|<=K(x-a)^2. By making it into a definition we arrive at the notion of uniform Lipschitz differentiability (ULD), and see that the derivative of a ULD function is Lipschitz. Taking different moduli of continuity instead of |.|, we get different flavors of calculus, each rather elementary, but all together covering the total range of continuously differentiable functions. Using functions continuous at a, we recapture the classical definition of differentiability at a point. We also see that ULD is equivalent to divisibility of f(x)-f(a) by x-a in the class of Lipschitz functions of two variables, x and a. The analogous fact is true for any subadditive modulus of continuity. In this bottom-up, computational, one modulus of continuity at a time approach, the monotonicity theorem takes the central stage and provides the aspects of the subject important for practical applications. The weighty ontological issues of compactness and completeness can be treated lightly or postponed, since they are hardly used in this streamlined approach that pretty much follows the Vladimir Arnold's "principle of minimal generality, according to which every idea should first be understood in the simplest situation; only then can the method developed be applied to more complicated cases." I discuss a generalization to many variables briefly.

math.HO↗
arXiv2025

PIGTIKAL (puzzles in geometry that I know and love)

Problems for the graduate students who want to improve problem-solving skills in geometry. Every problem has a short elegant solution -- this gives a hint which was not available when the problem was discovered.

math.HO↗
arXiv2009

Minimizing the footprint of your laptop (on your bedside table)

I often work on my laptop in bed. When needed, I park the laptop on the bedside table, where the computer has to share the small available space with a lamp, books, notes, and heaven knows what else. It often gets quite squeezy. Being regularly faced with this tricky situation, it finally occurred to me to determine once and for all how to place the laptop on the bedside table so that its ``footprint'' - the area in which it touches the bedside table - is minimal. In this note I give the solution of this problem, using some very pretty and elementary mathematics.

math.HO↗
arXiv2025

Algorithmic information theory and martingales

The notion of an individual random sequence goes back to von Mises. We describe the evolution of this notion, especially the use of martingales (suggested by Ville), and the development of algorithmic information theory in 1960s and 1970s (Solomonov, Kolmogorov, Martin-Lof, Levin, Chaitin, Schnorr and others). We conclude with some remarks about the use of the algorithmic information theory in the foundations of probability theory.

math.HO↗
arXiv2011

The emergence of French statistics. How mathematics entered the world of statistics in France during the 1920s

This paper concerns the emergence of modern mathematical statistics in France after the First World War. Emile Borel's achievements are presented, and especially his creation of two institutions where mathematical statistics was developed: the {\it Statistical Institute of Paris University}, (ISUP) in 1922 and above all the {\it Henri Poincaré Institute} (IHP) in 1928. At the IHP, a new journal {\it Annales de l'Institut Henri Poincaré} was created in 1931. We discuss the first papers in that journal dealing with mathematical statistics.

math.HO↗
arXiv2010

Categories without structures

The popular view according to which Category theory provides a support for Mathematical Structuralism is erroneous. Category-theoretic foundations of mathematics require a different philosophy of mathematics. While structural mathematics studies invariant forms (Awodey) categorical mathematics studies covariant transformations which, generally, don t have any invariants. In this paper I develop a non-structuralist interpretation of categorical mathematics and show its consequences for history of mathematics and mathematics education.

math.HO↗
arXiv2010

Ramanujan Primes and Bertrand's Postulate

The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that if $x \ge R_n$, then there are at least $n$ primes in the interval $(x/2,x]$. For example, Bertrand's postulate is $R_1 = 2$. Ramanujan proved that $R_n$ exists and gave the first five values as 2, 11, 17, 29, 41. In this note, we use inequalities of Rosser and Schoenfeld to prove that $2n \log 2n < R_n < 4n \log 4n$ for all $n$, and we use the Prime Number Theorem to show that $R_n$ is asymptotic to the $2n$th prime. We also estimate the length of the longest string of consecutive Ramanujan primes among the first $n$ primes, explain why there are more twin Ramanujan primes than expected, and make three conjectures (the first has since been proved by S. Laishram).

math.NT↗
arXiv2009

Proofs of certain properties of irrational roots

We give two elementary proofs, at a level understandable by students with only pre-calculus knowledge of Algebra, of the well known fact that an irreducible irrational n-th root of a positive rational number cannot be solution of a polynomial of degree less than n with rational coefficients. We also state and prove a few simple consequences.

math.HO↗
arXiv2009

Stability phenomena in the topology of moduli spaces

The recent proof by Madsen and Weiss of Mumford's conjecture on the stable cohomology of moduli spaces of Riemann surfaces, was a dramatic example of an important stability theorem about the topology of moduli spaces. In this article we give a survey of families of classifying spaces and moduli spaces where "stability phenomena" occur in their topologies. Such stability theorems have been proved in many situations in the history of topology and geometry, and the payoff has often been quite remarkable. In this paper we discuss classical stability theorems such as the Freudenthal suspension theorem, Bott periodicity, and Whitney's embedding theorems. We then discuss more modern examples such as those involving configuration spaces of points in manifolds, holomorphic curves in complex manifolds, gauge theoretic moduli spaces, the stable topology of general linear groups, and pseudoisotopies of manifolds. We then discuss the stability theorems regarding the moduli spaces of Riemann surfaces: Harer's stability theorem on the cohomology of moduli space, and the Madsen-Weiss theorem, which proves a generalization of Mumford's conjecture. We also describe Galatius's recent theorem on the stable cohomology of automorphisms of free groups. We end by speculating on the existence of general conditions in which one might expect these stability phenomena to occur.

math.AT↗
arXiv2013

Yet another proof from the Book: the Gauss theorem on regular polygons

This note is purely expository. The statement of the Gauss theorem on the constructibility of regular polygons by means of compass and ruler is simple and well-known. However, its proofs given in most textbooks rely upon much unmotivated material and are far from being economic. In this note a short elementary proof of the Gauss theorem is presented. The note is accessible for students familiar with polynomials and complex numbers, and could be an interesting easy reading for professional mathematicians.

math.HO↗