Growth and Integrability in Multi-Valued Dynamics
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Solutions of the Markov Equation Over Polynomial Rings
Solutions of the Markov equation over polynomial rings Ricardo Conceição, R. Kelly, S. VanFossen 49th John H. Barrett Memorial Lectures University of Tennessee, Knoxville, TN Ricardo Conceição ettysburg College Solutions of the Markov equation over polynomial rings Markov equation Introduction In his work on Diophantine approximation, Andrei Markov studied the equation x2 + y2 + z2 = 3xyz: The structure of the integral solutions of the Markov equation is very interesting. Ricardo Conceição ettysburg College Solutions of the Markov equation over polynomial rings Markov equation Automorphisms x2 + y2 + z2 = 3xyz (x1; x2; x3) −! (xπ(1); xπ(2); xπ(3)) (x; y; z) −! (−x; −y; z) A Markov triple (x; y; z) is a solution of the Markov equation that satisfies 0 < x ≤ y ≤ z. (x; y; 3xy − z) (x; y; z) (x; 3xz − y; z) (3zy − x; y; z) Ricardo Conceição ettysburg College Solutions of the Markov equation over polynomial rings Markov Tree x2 + y2 + z2 = 3xyz (3zy − x; y; z) (x; y; 3xy − z) (x; y; z) (x; 3xz − y; z) ··· (2,29,169) ··· (2,5,29) ··· (5,29,433) ··· (1,1,1) (1,1,2) (1,2,5) ··· (5,13,194) ··· (1,5,13) ··· (1,13,34) ··· Ricardo Conceição ettysburg College Solutions of the Markov equation over polynomial rings Markov Tree x2 + y2 + z2 = 3xyz (3zy − x; y; z) (x; y; 3xy − z) (x; y; z) (x; 3xz − y; z) ··· (2,29,169) ··· (2,5,29) ··· (5,29,433) ··· (1,1,1) (1,1,2) (1,2,5) ··· (5,13,194) ··· (1,5,13) ··· (1,13,34) ··· Ricardo Conceição ettysburg College Solutions of the Markov equation over polynomial rings Markov Tree x2 + y2 + z2 = 3xyz (3zy − x; -
Continued Fraction: a Research on Markov Triples
Fort Hays State University FHSU Scholars Repository John Heinrichs Scholarly and Creative Activities 2020 SACAD Entrants Day (SACAD) 4-22-2020 Continued fraction: A research on Markov triples Daheng Chen Fort Hays State University, [email protected] Follow this and additional works at: https://scholars.fhsu.edu/sacad_2020 Recommended Citation Chen, Daheng, "Continued fraction: A research on Markov triples" (2020). 2020 SACAD Entrants. 13. https://scholars.fhsu.edu/sacad_2020/13 This Poster is brought to you for free and open access by the John Heinrichs Scholarly and Creative Activities Day (SACAD) at FHSU Scholars Repository. It has been accepted for inclusion in 2020 SACAD Entrants by an authorized administrator of FHSU Scholars Repository. Daheng Chen CONTINUED FRACTION: Fort Hays State University Mentor: Hongbiao Zeng A research on Markov triples Department of Mathematics INTRODUCTION STUDY PROCESS OUTCOMES Being one of the many studies in Number Theory, continued fractions is a Our study of Markov Triple aimed at a simple level since the main researcher, Daheng Chen who was still a When solving the second question, we saw the clear relationship between study of math that offer useful means of expressing numbers and functions. Mathematicians in early ages used algorithms and methods to express high school student, did not have enough professional knowledge in math. Professor Hongbiao Zeng left three Fibonacci sequence and Markov numbers: all odd indices of Fibonacci numbers. Many of these expressions turned into the study of continued questions as the mainstream of the whole study. sequence are a subset of Markov numbers. fractions. We did not reach our ultimate research goal at this time; however, both of us Since the start of the 20th century, mathematicians have been able to develop Professor Zeng came up with the following three questions regarding Markov triples: gained precious experience in this area where we were not familiar with. -
On the Ordering of the Markov Numbers
ON THE ORDERING OF THE MARKOV NUMBERS KYUNGYONG LEE, LI LI, MICHELLE RABIDEAU, AND RALF SCHIFFLER Abstract. The Markov numbers are the positive integers that appear in the solutions of the equation x2 + y2 + z2 = 3xyz. These numbers are a classical subject in number theory and have important ramifications in hyperbolic geometry, algebraic geometry and combinatorics. It is known that the Markov numbers can be labeled by the lattice points (q; p) in the first quadrant and below the diagonal whose coordinates are coprime. In this paper, we consider the following question. Given two lattice points, can we say which of the associated Markov numbers is larger? A complete answer to this question would solve the uniqueness conjecture formulated by Frobenius in 1913. We give a partial answer in terms of the slope of the line segment that connects the two lattice points. We prove that the Markov number with the greater x-coordinate 8 is larger than the other if the slope is at least − 7 and that it is smaller than the other if the 5 slope is at most − 4 . As a special case, namely when the slope is equal to 0 or 1, we obtain a proof of two conjectures from Aigner's book \Markov's theorem and 100 years of the uniqueness conjecture". 1. Introduction In 1879, Andrey Markov studied the equation (1.1) x2 + y2 + z2 = 3xyz; which is now known as the Markov equation. A positive integer solution (m1; m2; m3) of (1.1) is called a Markov triple and the integers that appear in the Markov triples are called Markov num- bers. -
Interschool Test 2010 MAΘ National Convention
Interschool Test 2010 MAΘ National Convention Section I – Trivia 101 For this section, determine the answer to each question, then change each letter in the answer to a number by using A=1, B=2, C=3, etc. Add these numbers, then report the sum as your team’s answer. For example, if the question is “Who plays Batman in the most recent Batman movies?”, the answer is Christian Bale, and the answer your team will submit is 121 (3+8+18+9+19+20+9+1+14+2+1+12+5=121). 1. What band originally sang the song “Pinball Wizard” as part of their rock opera Tommy? (2 words, 1 point) 2. What 1980s fantasy action film’s tagline was “There can be only one.”? (1 word, 2 points) 3. What sitcom features a regional manager whose name is David Brent in the British version, Gilles Triquet in the French version, and Bernd Stromberg in the German version? (2 words, 2 points) 4. What book, written in 1950 by C.S. Lewis, was released as a movie in December 2005? (7 words, 3 points) 5. What magazine published its 1000th issue in 2006 with a cover that was an homage to the Beatles’ album Sgt. Pepper’s Lonely Hearts Club Band? (2 words, 3 points) 6. What fake news organization features editorials from such personalities as Smoove B, a smooth‐ talking ladies man; Jackie Harvey, a clueless celebrity spotter; Gorzo the Mighty, Emperor of the Universe, a 1930s style sci‐fi villain; and Herbert Kornfeld, an accounts receivable supervisor who speaks in ebonics? (2 words, 5 points) 7. -
~Umbers the BOO K O F Umbers
TH E BOOK OF ~umbers THE BOO K o F umbers John H. Conway • Richard K. Guy c COPERNICUS AN IMPRINT OF SPRINGER-VERLAG © 1996 Springer-Verlag New York, Inc. Softcover reprint of the hardcover 1st edition 1996 All rights reserved. No part of this publication may be reproduced, stored in a re trieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Published in the United States by Copernicus, an imprint of Springer-Verlag New York, Inc. Copernicus Springer-Verlag New York, Inc. 175 Fifth Avenue New York, NY lOOlO Library of Congress Cataloging in Publication Data Conway, John Horton. The book of numbers / John Horton Conway, Richard K. Guy. p. cm. Includes bibliographical references and index. ISBN-13: 978-1-4612-8488-8 e-ISBN-13: 978-1-4612-4072-3 DOl: 10.l007/978-1-4612-4072-3 1. Number theory-Popular works. I. Guy, Richard K. II. Title. QA241.C6897 1995 512'.7-dc20 95-32588 Manufactured in the United States of America. Printed on acid-free paper. 9 8 765 4 Preface he Book ofNumbers seems an obvious choice for our title, since T its undoubted success can be followed by Deuteronomy,Joshua, and so on; indeed the only risk is that there may be a demand for the earlier books in the series. More seriously, our aim is to bring to the inquisitive reader without particular mathematical background an ex planation of the multitudinous ways in which the word "number" is used. -
Improved Synthesis of Global Irradiance with One-Minute Resolution for PV System Simulations
Hindawi Publishing Corporation International Journal of Photoenergy Volume 2014, Article ID 808509, 10 pages http://dx.doi.org/10.1155/2014/808509 Research Article Improved Synthesis of Global Irradiance with One-Minute Resolution for PV System Simulations Martin Hofmann,1 Stefan Riechelmann,2 Cristian Crisosto,2 Riyad Mubarak,2 and Gunther Seckmeyer2 1 Valentin Software GmbH, Stralauer Platz 34, 10243 Berlin, Germany 2 Institute for Meteorology and Climatology, University Hannover, Herrenhauser¨ Straße 2, 30419 Hannover, Germany Correspondence should be addressed to Martin Hofmann; [email protected] Received 6 August 2014; Revised 3 November 2014; Accepted 4 November 2014; Published 26 November AcademicEditor:PramodH.Borse Copyright © 2014 Martin Hofmann et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. High resolution global irradiance time series are needed for accurate simulations of photovoltaic (PV) systems, since the typical volatile PV power output induced by fast irradiance changes cannot be simulated properly with commonly available hourly averages of global irradiance. We present a two-step algorithm that is capable of synthesizing one-minute global irradiance time series based on hourly averaged datasets. The algorithm is initialized by deriving characteristic transition probability matrices (TPM) for different weather conditions (cloudless, broken clouds and overcast) from a large number of high resolution measurements. Once initialized, the algorithm is location-independent and capable of synthesizing one-minute values based on hourly averaged global irradiance of any desired location. The one-minute time series are derived by discrete-time Markov chains based on a TPM that matches the weather condition of the input dataset. -
Mathematical Constants and Sequences
Mathematical Constants and Sequences a selection compiled by Stanislav Sýkora, Extra Byte, Castano Primo, Italy. Stan's Library, ISSN 2421-1230, Vol.II. First release March 31, 2008. Permalink via DOI: 10.3247/SL2Math08.001 This page is dedicated to my late math teacher Jaroslav Bayer who, back in 1955-8, kindled my passion for Mathematics. Math BOOKS | SI Units | SI Dimensions PHYSICS Constants (on a separate page) Mathematics LINKS | Stan's Library | Stan's HUB This is a constant-at-a-glance list. You can also download a PDF version for off-line use. But keep coming back, the list is growing! When a value is followed by #t, it should be a proven transcendental number (but I only did my best to find out, which need not suffice). Bold dots after a value are a link to the ••• OEIS ••• database. This website does not use any cookies, nor does it collect any information about its visitors (not even anonymous statistics). However, we decline any legal liability for typos, editing errors, and for the content of linked-to external web pages. Basic math constants Binary sequences Constants of number-theory functions More constants useful in Sciences Derived from the basic ones Combinatorial numbers, including Riemann zeta ζ(s) Planck's radiation law ... from 0 and 1 Binomial coefficients Dirichlet eta η(s) Functions sinc(z) and hsinc(z) ... from i Lah numbers Dedekind eta η(τ) Functions sinc(n,x) ... from 1 and i Stirling numbers Constants related to functions in C Ideal gas statistics ... from π Enumerations on sets Exponential exp Peak functions (spectral) .. -
A Comparison of Dispersion and Markov Constants
ACTA ARITHMETICA LXIII.3 (1993) A comparison of dispersion and Markov constants by Amitabha Tripathi (Fairmont, W.Va.) 0. Introduction. Let {xn} be a sequence of numbers, 0 ≤ xn ≤ 1. In [3], H. Niederreiter introduced a measure of denseness of such a sequence as follows: For each N ≥ 1, let dN = sup min |x − xn| 0≤x≤1 1≤n≤N and define D({xn}) = lim sup NdN . N→∞ In particular, for irrational α, the dispersion constant D(α) is defined by D({nα mod 1}). It is well known that, for irrational α, the Markov constant M(α) is defined by M(α)−1 = lim inf nknαk , n→∞ where kxk denotes the distance from x to the nearest integer. In [3], Niederreiter asks if M(α)<M(β) implies D(α)<D(β). V. Drobot [1] has shown this to be false by producing a counterexample of two quadratic irrationals, both with continued fraction expansion with period length nine. In this paper, we classify some infinite families of pairs (α, β) of irrational numbers that satisfy M(α) < M(β) and D(α) > D(β). We first outline the method of V. Drobot [1] to compute D(α) for quadratic irrationals α. If α is a real irrational number with continued fraction expansion α = [a0; a1, a2,...], let λi = [0; ai, ai−1, . , a1],Λi = [ai+1; ai+2,...],Mi = λi + Λi . We define −1 2 ψi(x) = Mi [−x + (Λi − λi − 1)x + Λi(1 + λi)] , xi = (Λi − λi − 1)/2 , ni is the integer closest to xi . -
On the Greatest Prime Factor of Markov Pairs
REND.SEM.MAT.UNIV.PADOVA, Vol. 116 2006) On the Greatest Prime Factor of Markov Pairs. PIETRO CORVAJA *) - UMBERTO ZANNIER **) 1. Introduction. Markov triples appear in several contexts of elementary Number- Theory. They are defined as the solutions x; y; z) to the remarkable dio- phantine equation 1) x2 y2 z2 3xyz; x; y; z positive integers: The coefficient 3 here is particularly relevant. In fact, by the descent procedure to be recalled below, one may show the following, on replacing 3 with a positive integer coefficient k:fork 1 we reduce to k 3after observing that necessarily x; y; z must all be multiples of 3; for k 2one finds that a possible solution must have all even entries, and we reduce to k 4; finally, for k 4 there are no solutions so in fact there are no so- lutions for k 2 as well). The equation on the left of 1) defines an affine surface X which admits a group G of automorphisms generated by the permutations on x; y; z and the involution x; y; z) 7! x; y; 3xy À z). This comes on viewing the equa- tion as a quadratic in z; since the sum of its roots is 3xy,ifz0 is a solution, so is also 3xy À z0.) Naturally G acts on the set of positive integer solutions, and it may be shown see [C] or [M]) that there is a single orbit. This follows by the descent alluded to above: one proves that, by applying the involution after suitable permutation of the coordinates, the maximum absolute value of the entries descreases until we reach the solution 1; 1; 1). -
Metric Diophantine Approximation and Dynamical Systems Dmitry Kleinbock
Metric Diophantine approximation and dynamical systems Dmitry Kleinbock Brandeis University, Waltham MA 02454-9110 E-mail address: [email protected] This is a rough draft for the Spring 2007 course taught at Brandeis. Comments/corrections/suggestions welcome. Last modified: March 29, 2007. Here is a short course description: The main theme is studying the way the set of rational numbers Q sits inside real numbers R. This seemingly simple set-up happens to lead to quite intricate problems. The word ‘metric’ comes from ‘measure’ and implies that the emphasis will be put on properties of ‘almost all’ numbers; more precisely, for a certain approximation property of real numbers we will be interested in the magnitude (for example, in terms of Lebesgue measure or Hausdorff dimension) of the set of numbers having this property. The n situation becomes even more interesting when one similarly considers Q n sitting in R ; this has been the area of several exciting recent developments, which we may talk about in the second part of the course. The introductory part will feature: rate of approximation of real num- bers by rationals; theorems of Kronecker, Dirichlet, Liouville, Borel-Cantelli, Khintchine; connections with dynamical systems: circle rotations, hyper- bolic flow in the space of lattices, geodesic flow on the modular survace, Gauss map (continued fractions). Other topics could be (choice based upon preferences of the audience): • Hausdorff measures and dimension, fractal sets and measures they support • W. Schmidt’s (α, β)-games, winning sets, badly approximable num- bers • ubiquitious systems, Jarnik-Besicovitch Theorem • inhomogeneous approximation, shrinking target properties for cir- cle rotations • multidimensional theory, Diophantine properties of measures on n R , a connection with flows on SLn+1(R)/SLn+1(Z) References: • W. -
On the Markov Numbers: Fixed Numerator, Denominator, and Sum Conjectures Clément Lagisquet, Edita Pelantová, Sébastien Tavenas, Laurent Vuillon
On the Markov numbers: fixed numerator, denominator, and sum conjectures Clément Lagisquet, Edita Pelantová, Sébastien Tavenas, Laurent Vuillon To cite this version: Clément Lagisquet, Edita Pelantová, Sébastien Tavenas, Laurent Vuillon. On the Markov numbers: fixed numerator, denominator, and sum conjectures. 2020. hal-03013363 HAL Id: hal-03013363 https://hal.archives-ouvertes.fr/hal-03013363 Preprint submitted on 18 Nov 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. On the Markov numbers: fixed numerator, denominator, and sum conjectures Cl´ement Lagisquet1, Edita Pelantov´a2, S´ebastienTavenas1, and Laurent Vuillon1 1LAMA, Universit´eSavoie Mont-Blanc, CNRS 2Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague October 27, 2020 Abstract The Markov numbers are the positive integer solutions of the Dio- phantine equation x2 + y2 + z2 = 3xyz. Already in 1880, Markov showed that all these solutions could be generated along a binary tree. So it became quite usual (and useful) to index the Markov num- bers by the rationals from [0; 1] which stand at the same place in the Stern{Brocot binary tree. The Frobenius' conjecture claims that each Markov number appears at most once in the tree. -
Improved Synthesis of Global Irradiance with One-Minute Resolution for PV System Simulations
Hindawi Publishing Corporation International Journal of Photoenergy Volume 2014, Article ID 808509, 10 pages http://dx.doi.org/10.1155/2014/808509 Research Article Improved Synthesis of Global Irradiance with One-Minute Resolution for PV System Simulations Martin Hofmann,1 Stefan Riechelmann,2 Cristian Crisosto,2 Riyad Mubarak,2 and Gunther Seckmeyer2 1 Valentin Software GmbH, Stralauer Platz 34, 10243 Berlin, Germany 2 Institute for Meteorology and Climatology, University Hannover, Herrenhauser¨ Straße 2, 30419 Hannover, Germany Correspondence should be addressed to Martin Hofmann; [email protected] Received 6 August 2014; Revised 3 November 2014; Accepted 4 November 2014; Published 26 November AcademicEditor:PramodH.Borse Copyright © 2014 Martin Hofmann et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. High resolution global irradiance time series are needed for accurate simulations of photovoltaic (PV) systems, since the typical volatile PV power output induced by fast irradiance changes cannot be simulated properly with commonly available hourly averages of global irradiance. We present a two-step algorithm that is capable of synthesizing one-minute global irradiance time series based on hourly averaged datasets. The algorithm is initialized by deriving characteristic transition probability matrices (TPM) for different weather conditions (cloudless, broken clouds and overcast) from a large number of high resolution measurements. Once initialized, the algorithm is location-independent and capable of synthesizing one-minute values based on hourly averaged global irradiance of any desired location. The one-minute time series are derived by discrete-time Markov chains based on a TPM that matches the weather condition of the input dataset.