Some more Long Continued Fractions, I

In this paper we show how to construct several infinite families of polynomials D(¯x, k), such that p D(¯x, k) has a regular continued fraction expansion with arbitrarily long period, the length of this period being controlled by the positive integer parameter k. We also describe how to quickly compute the fundamental units in the corresponding real quadratic fields

Continued Fractions with Multiple Limits

For integers m ≥ 2, we study divergent continued fractions whose numerators and denominators in each of the m arithmetic progressions modulo m converge. Special cases give, among other things, an infinite sequence of divergence theorems, the first of which is the classical Stern-Stolz theorem. We give a theorem on a class of Poincar´e type recurrences which shows that they tend to limits when the limits are taken in residue classes and the roots of their characteristic polynomials are distinct roots of unity. We also generalize a curious q-continued fraction of Ramanujan’s with three limits to a continued fraction with k distinct limit points, k ≥ 2. The k limits are evaluated in terms of ratios of certain q series. Finally, we show how to use Daniel Bernoulli’s continued fraction in an elementary way to create analytic continued fractions with m limit points, for any positive integer m ≥ 2

An Exalted House

PR man George Burghess, employed to whitewash the name of a firm linked to a fatal factory fire, suffers torments of conscience. Then a friendship with a woman, who squats in the flat above his own, points the way to the redemption he seeks. But Burghess cannot give up a hard-won career in corporate PR and, instead, descends into spiritual ruin. Through its story of a soul in crisis, An Exalted House brings into view the warring sides of London: the corporate zone of hostile takeovers and monuments to financial power versus an alternative zone of popular-festive breakouts and squats, presided over by the spirit of William Blake.https://digitalcommons.wcupa.edu/casfaculty_books/1062/thumbnail.jp

Some Observations on Khovanskii\u27s Matrix Methods for extracting Roots of Polynomials

In this article we apply a formula for the n-th power of a 3×3 matrix (found previously by the authors) to investigate a procedure of Khovanskii’s for finding the cube root of a positive integer. We show, for each positive integer α, how to construct certain families of integer sequences such that a certain rational expression, involving the ratio of successive terms in each family, tends to α 1/3 . We also show how to choose the optimal value of a free parameter to get maximum speed of convergence. We apply a similar method, also due to Khovanskii, to a more general class of cubic equations, and, for each such cubic, obtain a sequence of rationals that converge to the real root of the cubic. We prove that Khovanskii’s method for finding the m-th (m ≥ 4) root of a positive integer works, provided a free parameter is chosen to satisfy a very simple condition. Finally, we briefly consider another procedure of Khovanskii’s, which also involves m×m matrices, for approximating the root of an arbitrary polynomial of degree m