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Local Time for Lattice Paths and the Associated Limit Laws
Local time for lattice paths and the associated limit laws Cyril Banderier Michael Wallner CNRS & Univ. Paris Nord, France LaBRI, Université de Bordeaux, France http://lipn.fr/~banderier/ http://dmg.tuwien.ac.at/mwallner/ https://orcid.org/0000-0003-0755-3022 https://orcid.org/0000-0001-8581-449X Abstract For generalized Dyck paths (i.e., directed lattice paths with any finite set of jumps), we analyse their local time at zero (i.e., the number of times the path is touching or crossing the abscissa). As we are in a discrete setting, the event we analyse here is “invisible” to the tools of Brownian motion theory. It is interesting that the key tool for analysing directed lattice paths, which is the kernel method, is not directly applicable here. Therefore, we in- troduce a variant of this kernel method to get the trivariate generating function (length, final altitude, local time): this leads to an expression involving symmetric and algebraic functions. We apply this analysis to different types of constrained lattice paths (mean- ders, excursions, bridges, . ). Then, we illustrate this approach on “bas- ketball walks” which are walks defined by the jumps −2, −1, 0, +1, +2. We use singularity analysis to prove that the limit laws for the local time are (depending on the drift and the type of walk) the geometric distribution, the negative binomial distribution, the Rayleigh distribution, or the half- normal distribution (a universal distribution up to now rarely encountered in analytic combinatorics). Keywords: Lattice paths, generating function, analytic combinatorics, singularity analysis, kernel method, gen- eralized Dyck paths, algebraic function, local time, half-normal distribution, Rayleigh distribution, negative binomial distribution 1 Introduction This article continues our series of investigations on the enumeration, generation, and asymptotics of directed lattice arXiv:1805.09065v1 [math.CO] 23 May 2018 paths [ABBG18a, ABBG18b, BF02, BW14, BW17, BG06, BKK+17, Wal16]. -
Atomic History Project Background: If You Were Asked to Draw the Structure of an Atom, What Would You Draw?
Atomic History Project Background: If you were asked to draw the structure of an atom, what would you draw? Throughout history, scientists have accepted five major different atomic models. Our perception of the atom has changed from the early Greek model because of clues or evidence that have been gathered through scientific experiments. As more evidence was gathered, old models were discarded or improved upon. Your task is to trace the atomic theory through history. Task: 1. You will create a timeline of the history of the atomic model that includes all of the following components: A. Names of 15 of the 21 scientists listed below B. The year of each scientist’s discovery that relates to the structure of the atom C. 1- 2 sentences describing the importance of the discovery that relates to the structure of the atom Scientists for the timeline: *required to be included • Empedocles • John Dalton* • Ernest Schrodinger • Democritus* • J.J. Thomson* • Marie & Pierre Curie • Aristotle • Robert Millikan • James Chadwick* • Evangelista Torricelli • Ernest • Henri Becquerel • Daniel Bernoulli Rutherford* • Albert Einstein • Joseph Priestly • Niels Bohr* • Max Planck • Antoine Lavoisier* • Louis • Michael Faraday • Joseph Louis Proust DeBroglie* Checklist for the timeline: • Timeline is in chronological order (earliest date to most recent date) • Equal space is devoted to each year (as on a number line) • The eight (8) *starred scientists are included with correct dates of their discoveries • An additional seven (7) scientists of your choice (from -
Physics of Gases and Phenomena of Heat Evangelista Torricelli (1608-1647)
Physics of gases and phenomena of heat Evangelista Torricelli (1608-1647) ”...We have made many vessels of glass like those shown as A and B and with tubes two cubits long. These were filled with quicksilver, the open end was closed with the finger, and they were then inverted in a vessel where there was quicksilver C; then we saw that an empty space was formed and that nothing happened in the vessel when this space was formed; the tube between A and D remained always full to the height of a cubit and a quarter and an inch high... Water also in a similar tube, though a much longer one, will rise to about 18 cubits, that is, as much more than quicksilver does as quicksilver is heavier than water, so as to be in equilibrium with the same cause which acts on the one and the other...” Letter to Michelangelo Ricci, June 11, 1644 Evangelista Torricelli (1608-1647) ”We live immersed at the bottom of a sea of elemental air, which by experiment undoubtedly has weight, and so much weight that the densest air in the neighbourhood of the surface of the earth weighs about one four-hundredth part of the weight of water...” Letter to Michelangelo Ricci, June 11, 1644 In July 1647 Valeriano Magni performed experiments on the vacuum in the presence of the King of Poland at the Royal Castle in Warsaw Blaise Pascal (1623-1662) ”I am searching for information which could help decide whether the action attributed to horror vacui really results from it or perhaps is caused by gravity and the pressure of air. -
The Walk of Life Vol
THE WALK OF LIFE VOL. 03 EDITED BY AMIR A. ALIABADI The Walk of Life Biographical Essays in Science and Engineering Volume 3 Edited by Amir A. Aliabadi Authored by Zimeng Wan, Mamoon Syed, Yunxi Jin, Jamie Stone, Jacob Murphy, Greg Johnstone, Thomas Jackson, Michael MacGregor, Ketan Suresh, Taylr Cawte, Rebecca Beutel, Jocob Van Wassenaer, Ryan Fox, Nikolaos Veriotes, Matthew Tam, Victor Huong, Hashim Al-Hashmi, Sean Usher, Daquan Bar- row, Luc Carney, Kyle Friesen, Victoria Golebiowski, Jeffrey Horbatuk, Alex Nauta, Jacob Karl, Brett Clarke, Maria Bovtenko, Margaret Jasek, Allissa Bartlett, Morgen Menig-McDonald, Kate- lyn Sysiuk, Shauna Armstrong, Laura Bender, Hannah May, Elli Shanen, Alana Valle, Charlotte Stoesser, Jasmine Biasi, Keegan Cleghorn, Zofia Holland, Stephan Iskander, Michael Baldaro, Rosalee Calogero, Ye Eun Chai, and Samuel Descrochers 2018 c 2018 Amir A. Aliabadi Publications All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher. ISBN: 978-1-7751916-1-2 Atmospheric Innovations Research (AIR) Laboratory, Envi- ronmental Engineering, School of Engineering, RICH 2515, University of Guelph, Guelph, ON N1G 2W1, Canada It should be distinctly understood that this [quantum mechanics] cannot be a deduction in the mathematical sense of the word, since the equations to be obtained form themselves the postulates of the theory. Although they are made highly plausible by the following considerations, their ultimate justification lies in the agreement of their predictions with experiment. —Werner Heisenberg Dedication Jahangir Ansari Preface The essays in this volume result from the Fall 2018 offering of the course Control of Atmospheric Particulates (ENGG*4810) in the Environmental Engineering Program, University of Guelph, Canada. -
Ernst Abbe 1840-1905: a Social Reformer
We all do not belong to ourselves Ernst Abbe 1840-1905: a Social Reformer Fritz Schulze, Canada Linda Nguyen, The Canadian Press, Wednesday, January 2014: TORONTO – By the time you finish your lunch on Thursday, Canada’s top paid CEO will have already earned (my italics!) the equivalent of your annual salary.* It is not unusual these days to read such or similar headlines. Each time I am reminded of the co-founder of the company I worked for all my active life. Those who know me, know that I worked for the world-renowned German optical company Carl Zeiss. I also have a fine collection of optical instruments, predominantly microscopes, among them quite a few Zeiss instruments. Carl Zeiss founded his business as a “mechanical atelier” in Jena in 1846 and his mathematical consultant, Ernst Abbe, a poorly paid lecturer at the University Jena, became his partner in 1866. That was the beginning of a comet-like rise of the young company. Ernst Abbe was born in Eisenach, Thuringia, on January 23, 1840, as the first child of a spinnmaster of a local textile mill.Till the beginning of the 50s each and every day the Lord made my father stood at his machines, 14, 15, 16 hours, from 5 o’clock in the morning till 7 o’clock in the evening during quiet periods, 16 hours from 4 o’clock in the morning till 8 o’clock in the evening during busy times, without any interruption, not even a lunch break. I myself as a small boy of 5 – 9 years brought him his lunch, alternately with my younger sister, and watched him as he gulped it down leaning on his machine or sitting on a wooden box, then handing me back the empty pail and immediately again tending his machines." Ernst Abbe remembered later. -
Bernhard Riemann 1826-1866
Modern Birkh~user Classics Many of the original research and survey monographs in pure and applied mathematics published by Birkh~iuser in recent decades have been groundbreaking and have come to be regarded as foun- dational to the subject. Through the MBC Series, a select number of these modern classics, entirely uncorrected, are being re-released in paperback (and as eBooks) to ensure that these treasures remain ac- cessible to new generations of students, scholars, and researchers. BERNHARD RIEMANN (1826-1866) Bernhard R~emanno 1826 1866 Turning Points in the Conception of Mathematics Detlef Laugwitz Translated by Abe Shenitzer With the Editorial Assistance of the Author, Hardy Grant, and Sarah Shenitzer Reprint of the 1999 Edition Birkh~iuser Boston 9Basel 9Berlin Abe Shendtzer (translator) Detlef Laugwitz (Deceased) Department of Mathematics Department of Mathematics and Statistics Technische Hochschule York University Darmstadt D-64289 Toronto, Ontario M3J 1P3 Gernmany Canada Originally published as a monograph ISBN-13:978-0-8176-4776-6 e-ISBN-13:978-0-8176-4777-3 DOI: 10.1007/978-0-8176-4777-3 Library of Congress Control Number: 2007940671 Mathematics Subject Classification (2000): 01Axx, 00A30, 03A05, 51-03, 14C40 9 Birkh~iuser Boston All rights reserved. This work may not be translated or copied in whole or in part without the writ- ten permission of the publisher (Birkh~user Boston, c/o Springer Science+Business Media LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter de- veloped is forbidden. -
Optical Expert Named New Professor for History of Physics at University and Founding Director of German Optical Museum
URL: http://www.uni-jena.de/en/News/PM180618_Mappes_en.pdf Optical Expert Named New Professor for History of Physics at University and Founding Director of German Optical Museum Dr Timo Mappes to assume new role on 1 July 2018 / jointly appointed by University of Jena and German Optical Museum Dr Timo Mappes has been appointed Professor for the History of Physics with a focus on scientific communication at the Friedrich Schiller University Jena (Germany). He will also be the founding Director of the new German Optical Museum in Jena. Mappes will assume his new duties on 1 July 2018. Scientist, manager and scientific communicator "The highly complex job profile for this professorship meant the search committee faced a difficult task. We were very fortunate to find an applicant with so many of the necessary qualifications," says Prof. Dr Gerhard G. Paulus, Chairman of the search committee, adding: "If the committee had been asked to create its ideal candidate, it would have produced someone that resembled Dr Mappes." Mappes has a strong background in science and the industrial research and development of optical applications, and over the last twenty years has become very well-respected in the documentation and history of microscope construction from 1800 onwards. Last but not least, he has management skills and experience in sharing his knowledge with the general public, and with students in particular. All of this will be very beneficial for the German Optical Museum. The new concept and the extensive renovation of the traditional Optical Museum in Jena will make way for the new German Optical Museum, a research museum and forum for showcasing the history of optics and photonics. -
On the Isochronism of Galilei's Horologium
IFToMM Workshop on History of MMS – Palermo 2013 On the isochronism of Galilei's horologium Francesco Sorge, Marco Cammalleri, Giuseppe Genchi DICGIM, Università di Palermo, [email protected], [email protected], [email protected] Abstract − Measuring the passage of time has always fascinated the humankind throughout the centuries. It is amazing how the general architecture of clocks has remained almost unchanged in practice to date from the Middle Ages. However, the foremost mechanical developments in clock-making date from the seventeenth century, when the discovery of the isochronism laws of pendular motion by Galilei and Huygens permitted a higher degree of accuracy in the time measure. Keywords: Time Measure, Pendulum, Isochronism Brief Survey on the Art of Clock-Making The first elements of temporal and spatial cognition among the primitive societies were associated with the course of natural events. In practice, the starry heaven played the role of the first huge clock of mankind. According to the philosopher Macrobius (4 th century), even the Latin term hora derives, through the Greek word ‘ώρα , from an Egyptian hieroglyph to be pronounced Heru or Horu , which was Latinized into Horus and was the name of the Egyptian deity of the sun and the sky, who was the son of Osiris and was often represented as a hawk, prince of the sky. Later on, the measure of time began to assume a rudimentary technical connotation and to benefit from the use of more or less ingenious devices. Various kinds of clocks developed to relatively high levels of accuracy through the Egyptian, Assyrian, Greek and Roman civilizations. -
Named Units of Measurement
Dr. John Andraos, http://www.careerchem.com/NAMED/Named-Units.pdf 1 NAMED UNITS OF MEASUREMENT © Dr. John Andraos, 2000 - 2013 Department of Chemistry, York University 4700 Keele Street, Toronto, ONTARIO M3J 1P3, CANADA For suggestions, corrections, additional information, and comments please send e-mails to [email protected] http://www.chem.yorku.ca/NAMED/ Atomic mass unit (u, Da) John Dalton 6 September 1766 - 27 July 1844 British, b. Eaglesfield, near Cockermouth, Cumberland, England Dalton (1/12th mass of C12 atom) Dalton's atomic theory Dalton, J., A New System of Chemical Philosophy , R. Bickerstaff: London, 1808 - 1827. Biographical References: Daintith, J.; Mitchell, S.; Tootill, E.; Gjersten, D ., Biographical Encyclopedia of Dr. John Andraos, http://www.careerchem.com/NAMED/Named-Units.pdf 2 Scientists , Institute of Physics Publishing: Bristol, UK, 1994 Farber, Eduard (ed.), Great Chemists , Interscience Publishers: New York, 1961 Maurer, James F. (ed.) Concise Dictionary of Scientific Biography , Charles Scribner's Sons: New York, 1981 Abbott, David (ed.), The Biographical Dictionary of Scientists: Chemists , Peter Bedrick Books: New York, 1983 Partington, J.R., A History of Chemistry , Vol. III, Macmillan and Co., Ltd.: London, 1962, p. 755 Greenaway, F. Endeavour 1966 , 25 , 73 Proc. Roy. Soc. London 1844 , 60 , 528-530 Thackray, A. in Gillispie, Charles Coulston (ed.), Dictionary of Scientific Biography , Charles Scribner & Sons: New York, 1973, Vol. 3, 573 Clarification on symbols used: personal communication on April 26, 2013 from Prof. O. David Sparkman, Pacific Mass Spectrometry Facility, University of the Pacific, Stockton, CA. Capacitance (Farads, F) Michael Faraday 22 September 1791 - 25 August 1867 British, b. -
Generalization of Euler and Ramanujan's Partition Function
Asian Journal of Applied Science and Engineering, Volume 4, No 3/2015 ISSN 2305-915X(p); 2307-9584(e) Generalization of Euler and Ramanujan’s Partition Function Haradhan Kumar Mohajan* Assistant Professor, Faculty of Business Studies, Premier University, Chittagong, BANGLADESH ARTICLE INFO ABSTRACT th The theory of partitions has interested some of the best minds since the 18 Receiv ed: Sep 06, 2015 century. In 1742, Leonhard Euler established the generating f unction of P(n). Accepted: Nov 06, 2015 Godf rey Harold Hardy said that Sriniv asa Ramanujan was the f irst, and up to Published: Nov 12, 2015 now the only , mathematician to discover any such properties of P(n). In 1981, S. Barnard and J.M. Child stated that the diff erent ty pes of partitions of n in sy mbolic f orm. In this paper, diff erent ty pes of partitions of n are also explained with sy mbolic f orm. In 1952, E. Grosswald quoted that the linear Diophantine equation has distinct solutions; the set of solution is the number of partitions of n. This paper prov es theorem 1 with the help of certain restrictions. In 1965, *Corresponding Contact Godf rey Harold Hardy and E. M. Wright stated that the ‘Conv ergence Theorem’ Email: conv erges inside the unit circle. Theorem 2 has been prov ed here with easier [email protected] mathematical calculations. In 1853, British mathematician Norman Macleod Ferrers explained a partition graphically by an array of dots or nodes. In this paper, graphic representation of partitions, conjugate partitions and self - conjugate partitions are described with the help of examples. -
Mathematicians Fleeing from Nazi Germany
Mathematicians Fleeing from Nazi Germany Mathematicians Fleeing from Nazi Germany Individual Fates and Global Impact Reinhard Siegmund-Schultze princeton university press princeton and oxford Copyright 2009 © by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW All Rights Reserved Library of Congress Cataloging-in-Publication Data Siegmund-Schultze, R. (Reinhard) Mathematicians fleeing from Nazi Germany: individual fates and global impact / Reinhard Siegmund-Schultze. p. cm. Includes bibliographical references and index. ISBN 978-0-691-12593-0 (cloth) — ISBN 978-0-691-14041-4 (pbk.) 1. Mathematicians—Germany—History—20th century. 2. Mathematicians— United States—History—20th century. 3. Mathematicians—Germany—Biography. 4. Mathematicians—United States—Biography. 5. World War, 1939–1945— Refuges—Germany. 6. Germany—Emigration and immigration—History—1933–1945. 7. Germans—United States—History—20th century. 8. Immigrants—United States—History—20th century. 9. Mathematics—Germany—History—20th century. 10. Mathematics—United States—History—20th century. I. Title. QA27.G4S53 2008 510.09'04—dc22 2008048855 British Library Cataloging-in-Publication Data is available This book has been composed in Sabon Printed on acid-free paper. ∞ press.princeton.edu Printed in the United States of America 10 987654321 Contents List of Figures and Tables xiii Preface xvii Chapter 1 The Terms “German-Speaking Mathematician,” “Forced,” and“Voluntary Emigration” 1 Chapter 2 The Notion of “Mathematician” Plus Quantitative Figures on Persecution 13 Chapter 3 Early Emigration 30 3.1. The Push-Factor 32 3.2. The Pull-Factor 36 3.D. -
The Art of the Intelligible
JOHN L. BELL Department of Philosophy, University of Western Ontario THE ART OF THE INTELLIGIBLE An Elementary Survey of Mathematics in its Conceptual Development To my dear wife Mimi The purpose of geometry is to draw us away from the sensible and the perishable to the intelligible and eternal. Plutarch TABLE OF CONTENTS FOREWORD page xi ACKNOWLEDGEMENTS xiii CHAPTER 1 NUMERALS AND NOTATION 1 CHAPTER 2 THE MATHEMATICS OF ANCIENT GREECE 9 CHAPTER 3 THE DEVELOPMENT OF THE NUMBER CONCEPT 28 THE THEORY OF NUMBERS Perfect Numbers. Prime Numbers. Sums of Powers. Fermat’s Last Theorem. The Number π . WHAT ARE NUMBERS? CHAPTER 4 THE EVOLUTION OF ALGEBRA, I 53 Greek Algebra Chinese Algebra Hindu Algebra Arabic Algebra Algebra in Europe The Solution of the General Equation of Degrees 3 and 4 The Algebraic Insolubility of the General Equation of Degree Greater than 4 Early Abstract Algebra 70 CHAPTER 5 THE EVOLUTION OF ALGEBRA, II 72 Hamilton and Quaternions. Grassmann’s “Calculus of Extension”. Finite Dimensional Linear Algebras. Matrices. Lie Algebras. CHAPTER 6 THE EVOLUTION OF ALGEBRA, III 89 Algebraic Numbers and Ideals. ABSTRACT ALGEBRA Groups. Rings and Fields. Ordered Sets. Lattices and Boolean Algebras. Category Theory. CHAPTER 7 THE DEVELOPMENT OF GEOMETRY, I 111 COORDINATE/ALGEBRAIC/ANALYTIC GEOMETRY Algebraic Curves. Cubic Curves. Geometric Construction Problems. Higher Dimensional Spaces. NONEUCLIDEAN GEOMETRY CHAPTER 8 THE DEVELOPMENT OF GEOMETRY, II 129 PROJECTIVE GEOMETRY DIFFERENTIAL GEOMETRY The Theory of Surfaces. Riemann’s Conception of Geometry. TOPOLOGY Combinatorial Topology. Point-set topology. CHAPTER 9 THE CALCULUS AND MATHEMATICAL ANALYSIS 151 THE ORIGINS AND BASIC NOTIONS OF THE CALCULUS MATHEMATICAL ANALYSIS Infinite Series.