Related papers: Notes on Three Formulas of Abel
This paper is dedicated to present an exact solution for a nonlinear differential equation so-called Abel equation. This equation was known as one of the group of unsolvable differential equations. The present method is applicable for any…
The focus of this note is to formulate the algorithms and give the examples used by Fibonacci in Liber Abaci to expand any fraction into a sum of unit fractions. The description in Liber Abaci is all verbal and the examples are numbers…
Let A be an abelian threefold defined over a number field K with potential multiplication by an imaginary quadratic field M. If A has signature (2,1) and the multiplication by M is defined over an at most quadratic extension, we attach to A…
In this paper, firstly, we mainly study the relationship of balanced pairs among three Abelian categories in a recollement. As an application of admissible balanced pairs, we introduce the notion of the relative tilting modules, and give a…
Abel's continuity theorem for power series is a great tool to compute sums and prove properties of special functions. However, apart from two basic examples, this no longer occupies a place in Analysis courses. This note is an invitation to…
Symmetries and reductions of some algebraic equations are considered. Transformations that preserve the form of several algebraic equations, as well as transformations that reduce the degree of these equations, are described. Illustrative…
Basic elements of integral calculus over algebras of iterated differential forms, are presented. In particular, defining complexes for modules of integral forms are described and the corresponding berezinians and complexes of integral forms…
We study the analogue of the infinitesimal 16th Hilbert problem in dimension zero. Lower and upper bounds for the number of the zeros of the corresponding Abelian integrals (which are algebraic functions) are found. We study the relation…
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…
A geometric approach is used to study a family of higher-order nonlinear Abel equations. The inverse problem of the Lagrangian dynamics is studied in the particular case of the second-order Abel equation and the existence of two alternative…
We study the Abel differential equation x0 = A(t)x3 + B(t)x2 +C(t)x. Specifically, we find bounds on the number of its rational solutions when A(t), B(t) and C(t) are polynomials with real or complex coefficients; and on the number of…
In this article, using only elementary knowledge of complex numbers, we sketch a proof of the celebrated Abel--Ruffini theorem, which states that the general solution to an algebraic equation of degree five or more cannot be written using…
This is an introduction to algebraic combinatorics, written for a quarter-long graduate course. It starts with a rigorous introduction to formal power series with some combinatorial applications, then discusses integer partitions (proving…
In 1938 E. T. Bell introduced "The Iterated Exponential Integers". He proved that these numbers may be expressed by polynomials with rational coefficients. However, Bell gave no formulas for any of the coefficients except the trivial one,…
We describe a three precision variant of Newton's method for nonlinear equations. We evaluate the nonlinear residual in double precision, store the Jacobian matrix in single precision, and solve the equation for the Newton step with…
This is the text of an article that I wrote and disseminated in September 1981, except that I've updated the references, corrected a few misprints, and added a table of contents, some footnotes, and an addendum. The original article gave a…
Iku Nakamura [Hilbert schemes of Abelian group orbits, J. Alg. Geom. 10 (2001), 757--779] introduced the G-Hilbert scheme for a finite subgroup G in SL(3,C), and conjectured that it is a crepant resolution of the quotient C^3/G. He proved…
We examine what integers are representable as sums of three cubes. We also provide formulas for the number of representations of $x^3+y^3+z^3=n$ under the condition $x+y+z=t$. Also we show how the problem of three cubes is related to…
In the work we investigate some groupoids which are the Abelian algebras and the Hamiltonian algebras. An algebra is Abelian if for every polynomial operation and for all elements $a,b,\bar c,\bar d$ the implication $t(a,\bar c)=t(a,\bar…
The Basel problem, solved by Leonhard Euler in 1734, asks to resolve $\zeta(2)$, the sum of the reciprocals of the squares of the natural numbers, i.e. the sum of the infinite series: \begin{equation}…