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LINEAR ALGEBRA METHODS in COMBINATORICS László Babai
LINEAR ALGEBRA METHODS IN COMBINATORICS L´aszl´oBabai and P´eterFrankl Version 2.1∗ March 2020 ||||| ∗ Slight update of Version 2, 1992. ||||||||||||||||||||||| 1 c L´aszl´oBabai and P´eterFrankl. 1988, 1992, 2020. Preface Due perhaps to a recognition of the wide applicability of their elementary concepts and techniques, both combinatorics and linear algebra have gained increased representation in college mathematics curricula in recent decades. The combinatorial nature of the determinant expansion (and the related difficulty in teaching it) may hint at the plausibility of some link between the two areas. A more profound connection, the use of determinants in combinatorial enumeration goes back at least to the work of Kirchhoff in the middle of the 19th century on counting spanning trees in an electrical network. It is much less known, however, that quite apart from the theory of determinants, the elements of the theory of linear spaces has found striking applications to the theory of families of finite sets. With a mere knowledge of the concept of linear independence, unexpected connections can be made between algebra and combinatorics, thus greatly enhancing the impact of each subject on the student's perception of beauty and sense of coherence in mathematics. If these adjectives seem inflated, the reader is kindly invited to open the first chapter of the book, read the first page to the point where the first result is stated (\No more than 32 clubs can be formed in Oddtown"), and try to prove it before reading on. (The effect would, of course, be magnified if the title of this volume did not give away where to look for clues.) What we have said so far may suggest that the best place to present this material is a mathematics enhancement program for motivated high school students. -
Bollettino Unione Matematica Italiana
BOLLETTINO UNIONE MATEMATICA ITALIANA UMI Notizie. Bollettino dell’Unione Matematica Italiana, serie 3, volume 10 (1955), n. 2, p. 286-312. <http://www.bdim.eu/item?id=BUMI_1955_3_10_2_286_0> c Unione Matematica Italiana, 1955, diritti riservati. Articolo digitalizzato nel quadro del programma bdim (Biblioteca Digitale Italiana di Matematica) SIMAI & UMI http://www.bdim.eu/ c Unione Matematica Italiana, 1955, diritti riservati. N O T I Z I E Verbale dell Assemblea crdinaria dei Soci delll'U.M.I. döl 17 a pril e 1955 —• II 17 aprile 1955, nei locali delllstituto Matematico dell'Università di Bologna, ebbe luogo l'assemblea ordinaria dei Soci dell'U.M.I. L'oxdine ded giorno era il seguente: 1. Relazione süITattivita della Presidenza. 2. Approvazione del rendicoano finanziario deH'eseroizio 1 gennaio - 31 dicembre 1954. 3. Bilancio preventivo. 4. Scrutini délie votazioni per l'elezione dei membri deU'Uffieio di Pxe- sddenza e della Comnaissione Scientifica e proclamazione degli eletti, 5. Varie ed eventuali. La seduta ha inizio aUe are 10,30. Sono presenti i Soci: Arnerio, Angeli, AscoJi, Bononcini, Barusotti, Campedelli, Caprioli, Cassina, Catbabriga, Chisini» Conti, Dalla Valle, De Socio, Galafassi, Gherardelli, Graffi, Magenes, Mam- briani, Manaxa, Marani/Marchionna, Maroni, Matteuzzi, Muracchini, Pignedoli» Pompilj, Pratelli, Procissi, Ricci, Samsone, Tanzi, Terracini, Vaona, Varoli^ o, Villa e Villari. AU'un-ainimità il prof. Brusotti viene eletto Presidente dell'assemblea; Segxe- tario il prof. Magenes. Il Presidente propone e l'Assemblea approva cbe, nelJa discussione dell*o,d.g«, venga data la precedenza al coacnma 4. Il prof. Villa consegna al prof. Brusotti n. 279 schede di votazione per- venute alla Segreteria. -
Stanisław Zaremba
Danuta Ciesielska*, Krzysztof Ciesielski** Instytut Historii Nauki im. L. i A. Birkenmajerów, PAN Warszawa Instytut Matematyki, Wydział Matematyki i Informatyki, UJ Kraków SSW B (1–192) O DLO K KRTKIE PRZEDSTAWIENIE POLSKIEJ I KRAKOWSKIEJ MATEMATYKI DO POCZTKW XX WIEKU Znaczący rozwój matematyki na świecie datuje się na drugą połowę drugiego tysiąclecia n.e.; przez poprzedzające go półtora tysiąclecia uzyskiwane wyniki były skromne w porównaniu z tym, co uzyskano później. Jednakże do początku XX wieku Polska była z dala od europejskiej, a tym bardziej światowej czołówki. Potęgami ma- tematycznymi były Francja i Niemcy. Ważne rezultaty osiągano w Wielkiej Brytanii i we Włoszech. Sporadycznie pojawiali się słynni matematycy także w innych krajach, jednak w podręcznikach historii matematyki trudno znaleźć wśród nich Polaków. Od XIV wieku Polska miała się czym poszczycić naukowo. Akademia Krakow- ska była drugim uniwersytetem powstałym w środkowej Europie, od początku wy- kładano tu przedmioty kojarzone z matematyką. Na początku XV wieku krakowski mieszczanin Jan Stobner ufundował specjalną katedrę matematyki i astronomii; druga katedra związana z matematyką została ufundowana przez Marcina Króla z Żurawi- cy (ok.1422–ok.1460), pół wieku później. Był to jednak okres istotnie poprzedzający czasy większych osiągnięć matematycznych. Pewne osiągnięcia matematyczne miał ponad wiek później Mikołaj Kopernik (1473–1543), jego nazwisko kojarzone jest jed- nak (oczywiście słusznie) głównie z astronomią. Dopiero w XVII wieku pojawił się w Krakowie matematyk europejskiego formatu – Joannes Broscius (1585–1652; Jan Brożek, znany też jako Brzozek). Był nie tylko mate- matykiem, ale też flozofem, astronomem, teologiem, lekarzem i historykiem nauki. Ma on na swoim koncie znaczące osiągnięcia, głównie związane są z teorią liczb. -
Pursuit of Genius: Flexner, Einstein, and the Early Faculty at the Institute
i i i i PURSUIT OF GENIUS i i i i i i i i PURSUIT OF GENIUS Flexner, Einstein,and the Early Faculty at the Institute for Advanced Study Steve Batterson Emory University A K Peters, Ltd. Natick, Massachusetts i i i i i i i i Editorial, Sales, and Customer Service Office A K Peters, Ltd. 5 Commonwealth Road, Suite 2C Natick, MA 01760 www.akpeters.com Copyright ⃝c 2006 by A K Peters, Ltd. All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form, electronic or mechanical, including photocopy- ing, recording, or by any information storage and retrieval system, without written permission from the copyright owner. Library of Congress Cataloging-in-Publication Data Batterson, Steve, 1950– Pursuit of genius : Flexner, Einstein, and the early faculty at the Institute for Advanced Study / Steve Batterson. p. cm. Includes bibliographical references and index. ISBN 13: 978-1-56881-259-5 (alk. paper) ISBN 10: 1-56881-259-0 (alk. paper) 1. Mathematics–Study and teaching (Higher)–New Jersey–Princeton–History. 2. Institute for Advanced Study (Princeton, N.J.). School of Mathematics–History. 3. Institute for Advanced Study (Princeton, N.J.). School of Mathematics–Faculty. I Title. QA13.5.N383 I583 2006 510.7’0749652--dc22 2005057416 Cover Photographs: Front cover: Clockwise from upper left: Hermann Weyl (1930s, cour- tesy of Nina Weyl), James Alexander (from the Archives of the Institute for Advanced Study), Marston Morse (photo courtesy of the American Mathematical Society), Albert Einstein (1932, The New York Times), John von Neumann (courtesy of Marina von Neumann Whitman), Oswald Veblen (early 1930s, from the Archives of the Institute for Advanced Study). -
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. -
Continuous Nowhere Differentiable Functions
2003:320 CIV MASTER’S THESIS Continuous Nowhere Differentiable Functions JOHAN THIM MASTER OF SCIENCE PROGRAMME Department of Mathematics 2003:320 CIV • ISSN: 1402 - 1617 • ISRN: LTU - EX - - 03/320 - - SE Continuous Nowhere Differentiable Functions Johan Thim December 2003 Master Thesis Supervisor: Lech Maligranda Department of Mathematics Abstract In the early nineteenth century, most mathematicians believed that a contin- uous function has derivative at a significant set of points. A. M. Amp`ereeven tried to give a theoretical justification for this (within the limitations of the definitions of his time) in his paper from 1806. In a presentation before the Berlin Academy on July 18, 1872 Karl Weierstrass shocked the mathematical community by proving this conjecture to be false. He presented a function which was continuous everywhere but differentiable nowhere. The function in question was defined by ∞ X W (x) = ak cos(bkπx), k=0 where a is a real number with 0 < a < 1, b is an odd integer and ab > 1+3π/2. This example was first published by du Bois-Reymond in 1875. Weierstrass also mentioned Riemann, who apparently had used a similar construction (which was unpublished) in his own lectures as early as 1861. However, neither Weierstrass’ nor Riemann’s function was the first such construction. The earliest known example is due to Czech mathematician Bernard Bolzano, who in the years around 1830 (published in 1922 after being discovered a few years earlier) exhibited a continuous function which was nowhere differen- tiable. Around 1860, the Swiss mathematician Charles Cell´erieralso discov- ered (independently) an example which unfortunately wasn’t published until 1890 (posthumously). -
Once Upon a Party - an Anecdotal Investigation
Journal of Humanistic Mathematics Volume 11 | Issue 1 January 2021 Once Upon A Party - An Anecdotal Investigation Vijay Fafat Ariana Investment Management Follow this and additional works at: https://scholarship.claremont.edu/jhm Part of the Arts and Humanities Commons, Other Mathematics Commons, and the Other Physics Commons Recommended Citation Fafat, V. "Once Upon A Party - An Anecdotal Investigation," Journal of Humanistic Mathematics, Volume 11 Issue 1 (January 2021), pages 485-492. DOI: 10.5642/jhummath.202101.30 . Available at: https://scholarship.claremont.edu/jhm/vol11/iss1/30 ©2021 by the authors. This work is licensed under a Creative Commons License. JHM is an open access bi-annual journal sponsored by the Claremont Center for the Mathematical Sciences and published by the Claremont Colleges Library | ISSN 2159-8118 | http://scholarship.claremont.edu/jhm/ The editorial staff of JHM works hard to make sure the scholarship disseminated in JHM is accurate and upholds professional ethical guidelines. However the views and opinions expressed in each published manuscript belong exclusively to the individual contributor(s). The publisher and the editors do not endorse or accept responsibility for them. See https://scholarship.claremont.edu/jhm/policies.html for more information. The Party Problem: An Anecdotal Investigation Vijay Fafat SINGAPORE [email protected] Synopsis Mathematicians and physicists attending let-your-hair-down parties behave exactly like their own theories. They live by their theorems, they jive by their theorems. Life imi- tates their craft, and we must simply observe the deep truths hiding in their party-going behavior. Obligatory Preface It is a well-known maxim that revelries involving academics, intellectuals, and thought leaders are notoriously twisted affairs, a topologist's delight. -
Right Ideals of a Ring and Sublanguages of Science
RIGHT IDEALS OF A RING AND SUBLANGUAGES OF SCIENCE Javier Arias Navarro Ph.D. In General Linguistics and Spanish Language http://www.javierarias.info/ Abstract Among Zellig Harris’s numerous contributions to linguistics his theory of the sublanguages of science probably ranks among the most underrated. However, not only has this theory led to some exhaustive and meaningful applications in the study of the grammar of immunology language and its changes over time, but it also illustrates the nature of mathematical relations between chunks or subsets of a grammar and the language as a whole. This becomes most clear when dealing with the connection between metalanguage and language, as well as when reflecting on operators. This paper tries to justify the claim that the sublanguages of science stand in a particular algebraic relation to the rest of the language they are embedded in, namely, that of right ideals in a ring. Keywords: Zellig Sabbetai Harris, Information Structure of Language, Sublanguages of Science, Ideal Numbers, Ernst Kummer, Ideals, Richard Dedekind, Ring Theory, Right Ideals, Emmy Noether, Order Theory, Marshall Harvey Stone. §1. Preliminary Word In recent work (Arias 2015)1 a line of research has been outlined in which the basic tenets underpinning the algebraic treatment of language are explored. The claim was there made that the concept of ideal in a ring could account for the structure of so- called sublanguages of science in a very precise way. The present text is based on that work, by exploring in some detail the consequences of such statement. §2. Introduction Zellig Harris (1909-1992) contributions to the field of linguistics were manifold and in many respects of utmost significance. -
The History of Lowell House
The History Of Lowell House Charles U. Lowe HOW TO MAKE A HOUSE Charles U. Lowe ’42, Archivist of Lowell House Lucy L. Fowler, Assistant CONTENTS History of Lowell House, Essay by Charles U. Lowe Chronology Documents 1928 Documents 1929 Documents 1930-1932 1948 & Undated Who’s Who Appendix Three Essays on the History of Lowell House by Charles U. Lowe: 1. The Forbes story of the Harvard Riverside Associates: How Harvard acquired the land on which Lowell House was built. (2003) 2. How did the Russian Bells get to Lowell House? (2004) 3. How did the Russian Bells get to Lowell House? (Continued) (2005) Report of the Harvard Student Council Committee on Education Section III, Subdivision into Colleges The Harvard Advocate, April 1926 The House Plan and the Student Report 1926 Harvard Alumni Bulletin, April, 1932 A Footnote to Harvard History, Edward C. Aswell, ‘26 The Harvard College Rank List How Lowell House Selected Students, Harvard Crimson, September 30, 1930, Mason Hammond “Dividing Harvard College into Separate Groups” Letter from President Lowell to Henry James, Overseer November 3, 1925 Lowell House 1929-1930 Master, Honorary Associates, Associates, Resident and Non-Resident Tutors First Lowell House High Table Harvard Crimson, September 30, 1930 Outline of Case against the Clerk of the Dunster House Book Shop for selling 5 copies of Lady Chatterley’s Lover by D. H. Lawrence Charles S. Boswell (Undated) Gift of a paneled trophy case from Emanuel College to Lowell House Harvard University News, Thursday. October 20, 1932 Hizzoner, the Master of Lowell House - Essay about Julian Coolidge on the occasion of his retirement in 1948 Eulogy for Julian L. -
On the Origin and Early History of Functional Analysis
U.U.D.M. Project Report 2008:1 On the origin and early history of functional analysis Jens Lindström Examensarbete i matematik, 30 hp Handledare och examinator: Sten Kaijser Januari 2008 Department of Mathematics Uppsala University Abstract In this report we will study the origins and history of functional analysis up until 1918. We begin by studying ordinary and partial differential equations in the 18th and 19th century to see why there was a need to develop the concepts of functions and limits. We will see how a general theory of infinite systems of equations and determinants by Helge von Koch were used in Ivar Fredholm’s 1900 paper on the integral equation b Z ϕ(s) = f(s) + λ K(s, t)f(t)dt (1) a which resulted in a vast study of integral equations. One of the most enthusiastic followers of Fredholm and integral equation theory was David Hilbert, and we will see how he further developed the theory of integral equations and spectral theory. The concept introduced by Fredholm to study sets of transformations, or operators, made Maurice Fr´echet realize that the focus should be shifted from particular objects to sets of objects and the algebraic properties of these sets. This led him to introduce abstract spaces and we will see how he introduced the axioms that defines them. Finally, we will investigate how the Lebesgue theory of integration were used by Frigyes Riesz who was able to connect all theory of Fredholm, Fr´echet and Lebesgue to form a general theory, and a new discipline of mathematics, now known as functional analysis. -
Perspective the PNAS Way Back Then
Proc. Natl. Acad. Sci. USA Vol. 94, pp. 5983–5985, June 1997 Perspective The PNAS way back then Saunders Mac Lane, Chairman Proceedings Editorial Board, 1960–1968 Department of Mathematics, University of Chicago, Chicago, IL 60637 Contributed by Saunders Mac Lane, March 27, 1997 This essay is to describe my experience with the Proceedings of ings. Bronk knew that the Proceedings then carried lots of math. the National Academy of Sciences up until 1970. Mathemati- The Council met. Detlev was not a man to put off till to cians and other scientists who were National Academy of tomorrow what might be done today. So he looked about the Science (NAS) members encouraged younger colleagues by table in that splendid Board Room, spotted the only mathe- communicating their research announcements to the Proceed- matician there, and proposed to the Council that I be made ings. For example, I can perhaps cite my own experiences. S. Chairman of the Editorial Board. Then and now the Council Eilenberg and I benefited as follows (communicated, I think, did not often disagree with the President. Probably nobody by M. H. Stone): ‘‘Natural isomorphisms in group theory’’ then knew that I had been on the editorial board of the (1942) Proc. Natl. Acad. Sci. USA 28, 537–543; and “Relations Transactions, then the flagship journal of the American Math- between homology and homotopy groups” (1943) Proc. Natl. ematical Society. Acad. Sci. USA 29, 155–158. Soon after this, the Treasurer of the Academy, concerned The second of these papers was the first introduction of a about costs, requested the introduction of page charges for geometrical object topologists now call an “Eilenberg–Mac papers published in the Proceedings. -
Academic Genealogy of the Oakland University Department Of
Basilios Bessarion Mystras 1436 Guarino da Verona Johannes Argyropoulos 1408 Università di Padova 1444 Academic Genealogy of the Oakland University Vittorino da Feltre Marsilio Ficino Cristoforo Landino Università di Padova 1416 Università di Firenze 1462 Theodoros Gazes Ognibene (Omnibonus Leonicenus) Bonisoli da Lonigo Angelo Poliziano Florens Florentius Radwyn Radewyns Geert Gerardus Magnus Groote Università di Mantova 1433 Università di Mantova Università di Firenze 1477 Constantinople 1433 DepartmentThe Mathematics Genealogy Project of is a serviceMathematics of North Dakota State University and and the American Statistics Mathematical Society. Demetrios Chalcocondyles http://www.mathgenealogy.org/ Heinrich von Langenstein Gaetano da Thiene Sigismondo Polcastro Leo Outers Moses Perez Scipione Fortiguerra Rudolf Agricola Thomas von Kempen à Kempis Jacob ben Jehiel Loans Accademia Romana 1452 Université de Paris 1363, 1375 Université Catholique de Louvain 1485 Università di Firenze 1493 Università degli Studi di Ferrara 1478 Mystras 1452 Jan Standonck Johann (Johannes Kapnion) Reuchlin Johannes von Gmunden Nicoletto Vernia Pietro Roccabonella Pelope Maarten (Martinus Dorpius) van Dorp Jean Tagault François Dubois Janus Lascaris Girolamo (Hieronymus Aleander) Aleandro Matthaeus Adrianus Alexander Hegius Johannes Stöffler Collège Sainte-Barbe 1474 Universität Basel 1477 Universität Wien 1406 Università di Padova Università di Padova Université Catholique de Louvain 1504, 1515 Université de Paris 1516 Università di Padova 1472 Università