Related papers: On the babylonian method of extracting root square…
Several root-ratio multipoint methods for finding multiple zeros of univariate functions were recently presented. The characteristic of these methods is that they deal with $m$-th root of ratio of two functions (hence the name root-ratio…
In this paper, we study properties of polynomials over division rings. Moreover, we present formulas for finding roots of some polynomials
In a recent work I developed a formula for efficiently calculating the number of abelian squares of length $t+t$ over an alphabet of size $d$, where $d$ may be very large. Here I show how the expressiveness of a certain class of…
The problem of root mean square approximation of a square integrable function by finite linear combinations of exponential functions is considered. It is subdivided into linear and nonlinear parts. The linear approximation problem is…
We revisit several entries from Ramanujan's notebooks which follow from more elementary arguments than a first glance may suggest. Our goal is to demystify these results through more accessible proofs, while also shining some light on the…
We generalize Ramanujan method of approximating the smallest root of an equation which is found in Ramanujan Note books, Part-I. We provide simple analytical proof to study convergence of this method. Moreover, we study iterative approach…
Ouroboros functions have shown some interesting properties when subjected to conventional operations. The aim of this paper is to continue our investigation and prove some additional properties of these functions. Using algebraic methods,…
Suppose that we are given a formal power series of many variables with coefficients in $\mathbb{R}$ (or $\mathbb{C}$) and we want to compute its $n$-th (multiplicative) root. As can be expected coefficients of the root have to satisfy a…
In this note we recall the definition of the digital root, and apply the notion of the digital root to searching solutions of Diophantine equations. A table of arithmetic operations with digital roots is given. This method is incapable of…
We give an elementary introduction to some recent polyhedral techniques for understanding and solving systems of multivariate polynomial equations. We provide numerous concrete examples and illustrations, and assume no background in…
Late-Babylonian mathematics (450-100 BC), represented by some 60 cuneiform tablets from Babylon and Uruk, is incompletely known compared to its abundantly preserved, well-studied Old-Babylonian predecessor (1800-1600 BC). With the present…
In this paper we present a method for extraction of arcs of the algebraic curves of the higher order. Method is applied on conics, Cartesian ovals, trifocal curves and generalized Weber's curve.
We show that, generically, finding the $k$-th root of a braid is very fast. More precisely, we provide an algorithm which, given a braid $x$ on $n$ strands and canonical length $l$, and an integer $k>1$, computes a $k$-th root of $x$, if it…
In this note we give a theoretical support by means of quotient polynomial rings for the computation formulas of the dimension of abelian codes.
Pythagoras' theorem, the area of a triangle as one half the base times the height, and Heron's formula are amongst the most important and useful results of ancient Greek geometry. Here we look at all three in a new and improved light, using…
Univariate polynomial root-finding is both classical and important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the polynomial…
We give a new improvement over Newton's method for root-finding, when the function in question is doubly differentiable. It generally exhibits faster and more reliable convergence. It can be also be thought of as a correction to Halley's…
We show the 3 by 3 magic square of squares problem equivalent to solving quartic polynomials with certain factorization constraints over an abelian extension of the rationals. We analyze a particular case in which said extension is assumed…
An abelian square is a string of length 2n where the last n symbols form a permutation of the first n symbols. In this note we count the number of abelian squares and give an asymptotic estimate of this quantity.
In this paper, we study the so-called 'Mathematical part' of Plato's Theaetetus. Its subject concerns the incommensurability of certain magnitudes, in modern terms the question of the rationality or irrationality of the square roots of…