历史与综述
We define a new class of numbers based on the first occurrence of certain patterns of zeros and ones in the expansion of irracional numbers in a given basis and call them Sagan numbers, since they were first mentioned, in a special case, by…
The Dirac delta function has solid roots in 19th century work in Fourier analysis and singular integrals by Cauchy and others, anticipating Dirac's discovery by over a century, and illuminating the nature of Cauchy's infinitesimals and his…
We prove a version of L'hospital's rule for multivariable functions, which holds for non-isolated singular points. We also give an algorithm for resolving many indeterminate limits with isolated singular points.
This is about the mathematics and life of Donald Gordon Higman, 1928-2006. He did important work in representation theory of groups and algebras and in algebraic combinatorics. Charles C. Sims and Donald Higman discovered and constructed…
This is a philosophy-intense physics article, or, if you wish, a physics-intense philosophy article. Also, being a mathematician, I tend to view the physics, in particular the essence of quantum physics, in emphasizing the mathematical…
The Chinese Roots of Linear Algebra by Roger Hart chronicles the linear problems of ancient China in the Nine Chapters of the Mathematical Art, and supplies new insights about their solution.
In this Letter we comment on one particular aspect of Hypatia's enigmatic biography by translating into English a short poem that appeared in a recent review of the third revised Polish edition of Maria Dzielska's book about Hypatia. It…
In this short note we prove a result that is an extension of an old Olympiad problem and is a very simple variant of the question of finding an approximation for $k$, where it is a nonzero constant and it satisfies the equation $a^k+b^k=c$,…
It is demonstrated that iterative repeating of some simple geometric construction leads unavoidably in the limit to the golden ratio. The procedure appears to be quickly convergent regardless of a ratio of initial elements sizes. This could…
This is a short overview of the contribution of Leonid Kantorovich into the formation of the modern outlook on the interaction between mathematics and economics.
The autor propose the elementary derivation of the continued fraction expansion for function sec(x) + tan(x).
"We risk sliding down toward the standards where the validity of action is decided by whether one can get away with it." (P. Doty) "We do not 'risk' sliding down toward such standards; we have reached them." (S. Lang) This is an essay in…
We compute the minimal cardinality of a covering (resp. an irredundant covering) of a vector space over an arbitrary field by proper linear subspaces. Analogues for affine linear subspaces are also given.
We introduce real induction, a proof technique analogous to mathematical induction but applicable to statements indexed by an interval on the real line. More generally we give an inductive principle applicable in any Dedekind complete…
What is an interesting number theoretic or a combinatorial characterization of the divisors of 24 amongst all positive integers? In this paper I will provide one characterization in terms of modular multiplication tables. This idea evolved…
Reminiscences about Alexandr Danilovich Alexandrov (1912--1999)
Classical applications of Galois theory concern algebraic numbers and algebraic functions. Still, the night before his duel, Galois wrote that his last mathematical thoughts had been directed toward applying his "theory of ambiguity to…
The aim of this paper is to show that Plato's theory of knowledge of Forms (intelligible Beings, Ideas) in the Sophistes, obtained by Division and Collection, is a close philosophic analogue of the geometric theory of periodic…
In Babylonian mathematics two Sumerian words of fractions occur, which were originally used in non-mathematical texts. They are igi-n-g\'al "the reciprocal of (the number) n", which is often abbreviated to igi-n, and igi-te-en whose meaning…
In this paper a way is suggested for calculating the probability of consecutive number strings within a sequence of n numbers randomly drawn (without replacement) among the set of the first N consecutive numbers, with N>>n. An explicit…