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Wave Nature of Matter: Made Easy (Lesson 3) Matter Behaving As a Wave? Ridiculous!
Wave Nature of Matter: Made Easy (Lesson 3) Matter behaving as a wave? Ridiculous! Compiled by Dr. SuchandraChatterjee Associate Professor Department of Chemistry Surendranath College Remember? I showed you earlier how Einstein (in 1905) showed that the photoelectric effect could be understood if light were thought of as a stream of particles (photons) with energy equal to hν. I got my Nobel prize for that. Louis de Broglie (in 1923) If light can behave both as a wave and a particle, I wonder if a particle can also behave as a wave? Louis de Broglie I’ll try messing around with some of Einstein’s formulae and see what I can come up with. I can imagine a photon of light. If it had a “mass” of mp, then its momentum would be given by p = mpc where c is the speed of light. Now Einstein has a lovely formula that he discovered linking mass with energy (E = mc2) and he also used Planck’s formula E = hf. What if I put them equal to each other? mc2 = hf mc2 = hf So for my photon 2 mp = hfhf/c/c So if p = mpc = hfhf/c/c p = mpc = hf/chf/c Now using the wave equation, c = fλ (f = c/λ) So mpc = hc /λc /λc= h/λ λ = hp So you’re saying that a particle of momentum p has a wavelength equal to Planck’s constant divided by p?! Yes! λ = h/p It will be known as the de Broglie wavelength of the particle Confirmation of de Broglie’s ideas De Broglie didn’t have to wait long for his idea to be shown to be correct. -
January 1993 Council Minutes
AMERICAN MATHEMATICAL SOCIETY COUNCIL MINUTES San Antonio, Texas 12 January 1993 November 21, 1995 Abstract The Council met at 2:00 pm on Tuesday, 12 January 1993 in the Fiesta Room E of the San Antonio Convention Center. The following members were present for all or part of the meeting: Steve Armentrout, Michael Artin, Sheldon Axler, M. Salah Baouendi, James E. Baumgartner, Lenore Blum, Ruth M. Charney, Charles Herbert Clemens, W. W. Com- fort (Associate Secretary, voting), Carl C. Cowen, Jr., David A. Cox, Robert Daverman (Associate Secretary-designate, non-voting) , Chandler Davis, Robert M. Fossum, John M. Franks, Herbert Friedman (Canadian Mathematical Society observer, non-voting), Ronald L. Graham, Judy Green, Rebecca Herb, William H. Jaco (Executive Director, non-voting), Linda Keen, Irwin Kra, Elliott Lieb, Franklin Peterson, Carl Pomerance, Frank Quinn, Marc Rieffel, Hugo Rossi, Wilfried Schmid, Lance Small (Associate Secre- tary, non-voting), B. A. Taylor (Mathematical Reviews Editorial Committee and Associate Treasurer-designate), Lars B. Wahlbin (representing Mathematics of Computation Edito- rial Committee), Frank W. Warner, Steve H. Weintraub, Ruth Williams, and Shing-Tung Yau. President Artin presided. 1 2 CONTENTS Contents 0 CALL TO ORDER AND INTRODUCTIONS. 4 0.1 Call to Order. ............................................. 4 0.2 Retiring Members. .......................................... 4 0.3 Introduction of Newly Elected Council Members. ......................... 4 1 MINUTES 4 1.1 September 92 Council. ........................................ 4 1.2 11/92 Executive Committee and Board of Trustees (ECBT) Meeting. .............. 5 2 CONSENT AGENDA. 5 2.1 INNS .................................................. 5 2.2 Second International Conference on Ordinal Data Analysis. ................... 5 2.3 AMS Prizes. .............................................. 5 2.4 Special Committee on Nominating Procedures. -
Roth's Theorem in the Primes
Annals of Mathematics, 161 (2005), 1609–1636 Roth’s theorem in the primes By Ben Green* Abstract We show that any set containing a positive proportion of the primes con- tains a 3-term arithmetic progression. An important ingredient is a proof that the primes enjoy the so-called Hardy-Littlewood majorant property. We de- rive this by giving a new proof of a rather more general result of Bourgain which, because of a close analogy with a classical argument of Tomas and Stein from Euclidean harmonic analysis, might be called a restriction theorem for the primes. 1. Introduction Arguably the second most famous result of Klaus Roth is his 1953 upper bound [21] on r3(N), defined 17 years previously by Erd˝os and Tur´an to be the cardinality of the largest set A [N] containing no nontrivial 3-term arithmetic ⊆ progression (3AP). Roth was the first person to show that r3(N) = o(N). In fact, he proved the following quantitative version of this statement. Proposition 1.1 (Roth). r (N) N/ log log N. 3 " There was no improvement on this bound for nearly 40 years, until Heath- Brown [15] and Szemer´edi [22] proved that r N(log N) c for some small 3 " − positive constant c. Recently Bourgain [6] provided the best bound currently known. Proposition 1.2 (Bourgain). r (N) N (log log N/ log N)1/2. 3 " *The author is supported by a Fellowship of Trinity College, and for some of the pe- riod during which this work was carried out enjoyed the hospitality of Microsoft Research, Redmond WA and the Alfr´ed R´enyi Institute of the Hungarian Academy of Sciences, Bu- dapest. -
All That Math Portraits of Mathematicians As Young Researchers
Downloaded from orbit.dtu.dk on: Oct 06, 2021 All that Math Portraits of mathematicians as young researchers Hansen, Vagn Lundsgaard Published in: EMS Newsletter Publication date: 2012 Document Version Publisher's PDF, also known as Version of record Link back to DTU Orbit Citation (APA): Hansen, V. L. (2012). All that Math: Portraits of mathematicians as young researchers. EMS Newsletter, (85), 61-62. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY Editorial Obituary Feature Interview 6ecm Marco Brunella Alan Turing’s Centenary Endre Szemerédi p. 4 p. 29 p. 32 p. 39 September 2012 Issue 85 ISSN 1027-488X S E European M M Mathematical E S Society Applied Mathematics Journals from Cambridge journals.cambridge.org/pem journals.cambridge.org/ejm journals.cambridge.org/psp journals.cambridge.org/flm journals.cambridge.org/anz journals.cambridge.org/pes journals.cambridge.org/prm journals.cambridge.org/anu journals.cambridge.org/mtk Receive a free trial to the latest issue of each of our mathematics journals at journals.cambridge.org/maths Cambridge Press Applied Maths Advert_AW.indd 1 30/07/2012 12:11 Contents Editorial Team Editors-in-Chief Jorge Buescu (2009–2012) European (Book Reviews) Vicente Muñoz (2005–2012) Dep. -
Klaus Friedrich Roth, 1925–2015 the Godfrey Argent Studio C
e Bull. London Math. Soc. 50 (2018) 529–560 C 2017 Authors doi:10.1112/blms.12143 OBITUARY Klaus Friedrich Roth, 1925–2015 The Godfrey Argent Studio C Klaus Friedrich Roth, who died in Inverness on 10 November 2015 aged 90, made fundamental contributions to different areas of number theory, including diophantine approximation, the large sieve, irregularities of distribution and what is nowadays known as arithmetic combinatorics. He was the first British winner of the Fields Medal, awarded in 1958 for his solution in 1955 of the famous Siegel conjecture concerning approximation of algebraic numbers by rationals. He was elected a member of the London Mathematical Society on 17 May 1951, and received its De Morgan Medal in 1983. 1. Life and career Klaus Roth, son of Franz and Matilde (n´ee Liebrecht), was born on 29 October 1925, in the German city of Breslau, in Lower Silesia, Prussia, now Wroclaw in Poland. To escape from Nazism, he and his parents moved to England in 1933 and settled in London. He would recall that the flight from Berlin to London took eight hours and landed in Croydon. Franz, a solicitor by training, had suffered from gas poisoning during the First World War, and died a few years after their arrival in England. Roth studied at St Paul’s School between 1937 and 1943, during which time the school was relocated to Easthampstead Park, near Crowthorne in Berkshire, as part of the wartime evac- uation of London. There he excelled in mathematics and chess, and one master, Mr Dowswell, observed interestingly that he possessed complete intellectual honesty. -
Applied Combinatorics 2017 Edition
Keller Trotter Applied Combinatorics 2017 Edition 2017 Edition Mitchel T. Keller William T. Trotter Applied Combinatorics Applied Combinatorics Mitchel T. Keller Washington and Lee University Lexington, Virginia William T. Trotter Georgia Institute of Technology Atlanta, Georgia 2017 Edition Edition: 2017 Edition Website: http://rellek.net/appcomb/ © 2006–2017 Mitchel T. Keller, William T. Trotter This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 Interna- tional License. To view a copy of this license, visit http://creativecommons.org/licenses/ by-sa/4.0/ or send a letter to Creative Commons, PO Box 1866, Mountain View, CA 94042, USA. Summary of Contents About the Authors ix Acknowledgements xi Preface xiii Preface to 2017 Edition xv Preface to 2016 Edition xvii Prologue 1 1 An Introduction to Combinatorics 3 2 Strings, Sets, and Binomial Coefficients 17 3 Induction 39 4 Combinatorial Basics 59 5 Graph Theory 69 6 Partially Ordered Sets 113 7 Inclusion-Exclusion 141 8 Generating Functions 157 9 Recurrence Equations 183 10 Probability 213 11 Applying Probability to Combinatorics 229 12 Graph Algorithms 239 vii SUMMARY OF CONTENTS 13 Network Flows 259 14 Combinatorial Applications of Network Flows 279 15 Pólya’s Enumeration Theorem 291 16 The Many Faces of Combinatorics 315 A Epilogue 331 B Background Material for Combinatorics 333 C List of Notation 361 Index 363 viii About the Authors About William T. Trotter William T. Trotter is a Professor in the School of Mathematics at Georgia Tech. He was first exposed to combinatorial mathematics through the 1971 Bowdoin Combi- natorics Conference which featured an array of superstars of that era, including Gian Carlo Rota, Paul Erdős, Marshall Hall, Herb Ryzer, Herb Wilf, William Tutte, Ron Gra- ham, Daniel Kleitman and Ray Fulkerson. -
MSRI Celebrates Its Twentieth Birthday, Volume 50, Number 3
MSRI Celebrates Its Twentieth Birthday The past twenty years have seen a great prolifera- renewed support. Since then, the NSF has launched tion in mathematics institutes worldwide. An in- four more institutes: the Institute for Pure and spiration for many of them has been the Applied Mathematics at the University of California, Mathematical Sciences Research Institute (MSRI), Los Angeles; the AIM Research Conference Center founded in Berkeley, California, in 1982. An es- at the American Institute of Mathematics (AIM) in tablished center for mathematical activity that Palo Alto, California; the Mathematical Biosciences draws researchers from all over the world, MSRI has Institute at the Ohio State University; and the distinguished itself for its programs in both pure Statistical and Applied Mathematical Sciences and applied areas and for its wide range of outreach Institute, which is a partnership of Duke University, activities. MSRI’s success has allowed it to attract North Carolina State University, the University of many donations toward financing the construc- North Carolina at Chapel Hill, and the National tion of a new extension to its building. In October Institute of Statistical Sciences. 2002 MSRI celebrated its twentieth year with a Shiing-Shen Chern, Calvin C. Moore, and I. M. series of special events that exemplified what MSRI Singer, all on the mathematics faculty at the Uni- has become—a focal point for mathematical culture versity of California, Berkeley, initiated the original in all its forms, with the discovery and delight of proposal for MSRI; Chern served as the founding new mathematical knowledge the top priority. director, and Moore was the deputy director. -
Otto Stern Annalen 4.11.11
(To be published by Annalen der Physik in December 2011) Otto Stern (1888-1969): The founding father of experimental atomic physics J. Peter Toennies,1 Horst Schmidt-Böcking,2 Bretislav Friedrich,3 Julian C.A. Lower2 1Max-Planck-Institut für Dynamik und Selbstorganisation Bunsenstrasse 10, 37073 Göttingen 2Institut für Kernphysik, Goethe Universität Frankfurt Max-von-Laue-Strasse 1, 60438 Frankfurt 3Fritz-Haber-Institut der Max-Planck-Gesellschaft Faradayweg 4-6, 14195 Berlin Keywords History of Science, Atomic Physics, Quantum Physics, Stern- Gerlach experiment, molecular beams, space quantization, magnetic dipole moments of nucleons, diffraction of matter waves, Nobel Prizes, University of Zurich, University of Frankfurt, University of Rostock, University of Hamburg, Carnegie Institute. We review the work and life of Otto Stern who developed the molecular beam technique and with its aid laid the foundations of experimental atomic physics. Among the key results of his research are: the experimental test of the Maxwell-Boltzmann distribution of molecular velocities (1920), experimental demonstration of space quantization of angular momentum (1922), diffraction of matter waves comprised of atoms and molecules by crystals (1931) and the determination of the magnetic dipole moments of the proton and deuteron (1933). 1 Introduction Short lists of the pioneers of quantum mechanics featured in textbooks and historical accounts alike typically include the names of Max Planck, Albert Einstein, Arnold Sommerfeld, Niels Bohr, Max von Laue, Werner Heisenberg, Erwin Schrödinger, Paul Dirac, Max Born, and Wolfgang Pauli on the theory side, and of Wilhelm Conrad Röntgen, Ernest Rutherford, Arthur Compton, and James Franck on the experimental side. However, the records in the Archive of the Nobel Foundation as well as scientific correspondence, oral-history accounts and scientometric evidence suggest that at least one more name should be added to the list: that of the “experimenting theorist” Otto Stern. -
EMS Newsletter September 2012 1 EMS Agenda EMS Executive Committee EMS Agenda
NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY Editorial Obituary Feature Interview 6ecm Marco Brunella Alan Turing’s Centenary Endre Szemerédi p. 4 p. 29 p. 32 p. 39 September 2012 Issue 85 ISSN 1027-488X S E European M M Mathematical E S Society Applied Mathematics Journals from Cambridge journals.cambridge.org/pem journals.cambridge.org/ejm journals.cambridge.org/psp journals.cambridge.org/flm journals.cambridge.org/anz journals.cambridge.org/pes journals.cambridge.org/prm journals.cambridge.org/anu journals.cambridge.org/mtk Receive a free trial to the latest issue of each of our mathematics journals at journals.cambridge.org/maths Cambridge Press Applied Maths Advert_AW.indd 1 30/07/2012 12:11 Contents Editorial Team Editors-in-Chief Jorge Buescu (2009–2012) European (Book Reviews) Vicente Muñoz (2005–2012) Dep. Matemática, Faculdade Facultad de Matematicas de Ciências, Edifício C6, Universidad Complutense Piso 2 Campo Grande Mathematical de Madrid 1749-006 Lisboa, Portugal e-mail: [email protected] Plaza de Ciencias 3, 28040 Madrid, Spain Eva-Maria Feichtner e-mail: [email protected] (2012–2015) Society Department of Mathematics Lucia Di Vizio (2012–2016) Université de Versailles- University of Bremen St Quentin 28359 Bremen, Germany e-mail: [email protected] Laboratoire de Mathématiques Newsletter No. 85, September 2012 45 avenue des États-Unis Eva Miranda (2010–2013) 78035 Versailles cedex, France Departament de Matemàtica e-mail: [email protected] Aplicada I EMS Agenda .......................................................................................................................................................... 2 EPSEB, Edifici P Editorial – S. Jackowski ........................................................................................................................... 3 Associate Editors Universitat Politècnica de Catalunya Opening Ceremony of the 6ECM – M. -
Theory and Experiment in the Quantum-Relativity Revolution
Theory and Experiment in the Quantum-Relativity Revolution expanded version of lecture presented at American Physical Society meeting, 2/14/10 (Abraham Pais History of Physics Prize for 2009) by Stephen G. Brush* Abstract Does new scientific knowledge come from theory (whose predictions are confirmed by experiment) or from experiment (whose results are explained by theory)? Either can happen, depending on whether theory is ahead of experiment or experiment is ahead of theory at a particular time. In the first case, new theoretical hypotheses are made and their predictions are tested by experiments. But even when the predictions are successful, we can’t be sure that some other hypothesis might not have produced the same prediction. In the second case, as in a detective story, there are already enough facts, but several theories have failed to explain them. When a new hypothesis plausibly explains all of the facts, it may be quickly accepted before any further experiments are done. In the quantum-relativity revolution there are examples of both situations. Because of the two-stage development of both relativity (“special,” then “general”) and quantum theory (“old,” then “quantum mechanics”) in the period 1905-1930, we can make a double comparison of acceptance by prediction and by explanation. A curious anti- symmetry is revealed and discussed. _____________ *Distinguished University Professor (Emeritus) of the History of Science, University of Maryland. Home address: 108 Meadowlark Terrace, Glen Mills, PA 19342. Comments welcome. 1 “Science walks forward on two feet, namely theory and experiment. ... Sometimes it is only one foot which is put forward first, sometimes the other, but continuous progress is only made by the use of both – by theorizing and then testing, or by finding new relations in the process of experimenting and then bringing the theoretical foot up and pushing it on beyond, and so on in unending alterations.” Robert A. -
Roth's Orthogonal Function Method in Discrepancy Theory and Some
Roth’s Orthogonal Function Method in Discrepancy Theory and Some New Connections Dmitriy Bilyk Abstract In this survey we give a comprehensive, but gentle introduction to the cir- cle of questions surrounding the classical problems of discrepancy theory, unified by the same approach originated in the work of Klaus Roth [85] and based on mul- tiparameter Haar (or other orthogonal) function expansions. Traditionally, the most important estimates of the discrepancy function were obtained using variations of this method. However, despite a large amount of work in this direction, the most im- portant questions in the subject remain wide open, even at the level of conjectures. The area, as well as the method, has enjoyed an outburst of activity due to the recent breakthrough improvement of the higher-dimensional discrepancy bounds and the revealed important connections between this subject and harmonic analysis, proba- bility (small deviation of the Brownian motion), and approximation theory (metric entropy of spaces with mixed smoothness). Without assuming any prior knowledge of the subject, we present the history and different manifestations of the method, its applications to related problems in various fields, and a detailed and intuitive outline of the latest higher-dimensional discrepancy estimate. 1 Introduction The subject and the structure of the present chapter is slightly unconventional. In- stead of building the exposition around the results from one area, united by a com- mon topic, we concentrate on problems from different fields which all share a com- mon method. The starting point of our discussion is one of the earliest results in discrepancy theory, Roth’s 1954 L2 bound of the discrepancy function in dimensions d ≥ 2 [85], as well as the tool employed to obtain this bound, which later evolved into a pow- Dmitriy Bilyk Department of Mathematics, University of South Carolina, Columbia, SC 29208 USA e-mail: [email protected] 1 2 Dmitriy Bilyk erful orthogonal function method in discrepancy theory. -
Short Cross-Disciplinary Cycles in Co-Authorship Graphs
How small is the center of science? Short cross-disciplinary cycles in co-authorship graphs Chris Fields 528 Zinnia Court Sonoma, CA 95476 USA fi[email protected] October 17, 2014 Abstract Cycles that cross two or more boundaries between disciplines in the co-authorship graph for all of science are used to set upper limits on the number of co-authored pa- pers required to cross 15 disciplines or subdisciplines ranging from macroeconomics to neurology. The upper limits obtained range from one (discrete mathematics, macroe- conomics and nuclear physics) to six (neuroscience). The 15 disciplines or subdis- ciplines examined form a “small world” with an average separation of only 2.0 co- authorship links. It is conjectured that the high-productivity, high average degree centers of all scientific disciplines form a small world, and therefore that the diameter of the co-authorship graph of all of science is only slightly larger than the average diameter of the co-authorship graphs of its subdisciplines. Keywords: Cross-displinary brokers; Field mobility; Graph centrality; Graph diameter; Nobel laureates; Preferential attachment; Small-world networks 1 Introduction Numerous studies have now confirmed the finding of Newman (2001, 2004) that the co- authorship graphs of many scientific disciplines can be characterized as “small worlds” comprising many dense clusters of researchers separated by relatively short (mean length 1 l ≤ 10) minimum paths (reviewed by Mali, Kronegger, Doreian and Ferligoj, 2012). Despite over two decades of efforts to make the world of science as a whole more interconnected by encouraging inter-, multi- or trans-disciplinary collaborations, however, the research enterprise remains organized into disciplines (Jacobs and Frickel, 2009).