A Short Review on Noether's Theorems, Gauge Symmetries and Boundary
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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. -
Aspects of Symplectic Dynamics and Topology: Gauge Anomalies, Chiral Kinetic Theory and Transfer Matrices
ASPECTS OF SYMPLECTIC DYNAMICS AND TOPOLOGY: GAUGE ANOMALIES, CHIRAL KINETIC THEORY AND TRANSFER MATRICES BY VATSAL DWIVEDI DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate College of the University of Illinois at Urbana-Champaign, 2017 Urbana, Illinois Doctoral Committee: Associate Professor Shinsei Ryu, Chair Professor Michael Stone, Director of Research Professor James N. Eckstein Professor Jared Bronski Abstract This thesis presents some work on two quite disparate kinds of dynamical systems described by Hamiltonian dynamics. The first part describes a computation of gauge anomalies and their macroscopic effects in a semiclassical picture. The geometric (symplectic) formulation of classical mechanics is used to describe the dynamics of Weyl fermions in even spacetime dimensions, the only quantum input to the symplectic form being the Berry curvature that encodes the spin-momentum locking. The (semi-)classical equations of motion are used in a kinetic theory setup to compute the gauge and singlet currents, whose conservation laws reproduce the nonabelian gauge and singlet anomalies. Anomalous contributions to the hydrodynamic currents for a gas of Weyl fermions at a finite temperature and chemical potential are also calculated, and are in agreement with similar results in literature which were obtained using thermodynamic and/or quantum field theoretical arguments. The second part describes a generalized transfer matrix formalism for noninteracting tight-binding models. The formalism is used to study the bulk and edge spectra, both of which are encoded in the spectrum of the transfer matrices, for some of the common tight-binding models for noninteracting electronic topological phases of matter. -
Emmy Noether: the Mother of Modern Algebra Reviewed by Benno Artmann
Book Review Emmy Noether: The Mother of Modern Algebra Reviewed by Benno Artmann Emmy Noether: The Mother of Modern Algebra The author has to be M. B. W. Tent content with rather A. K. Peters, 2008 general information US$29.00, 200 pages about “abstract al- ISBN-13:978-1568814308 gebra” and has to reduce the few ab- The catalogue of the Library of Congress classifies solutely necessary this book as juvenile literature, and in this respect mathematical defini- it may serve its intentions well. Beyond that, a per- tions to the capabili- son not familiar with Emmy Noether’s (1882–1935) ties of advanced high life and the academic and political situations in school students, as in Germany in the years between 1900 and 1935 may the case of an “ideal” profit from the general picture the book provides on page 89. of these times, even though it may sometimes not One thing, how- be easy to distinguish between facts and fiction. ever, that could eas- The chapters of the book are: I, Childhood; ily be corrected is to II, Studying at the University; III, The Young be found on pages Scholar; IV, Emmy Noether at Her Prime Time in 105–106. Here the author reports that the stu- Göttingen; and V, Exile. dents were “shuffling their feet loudly” when the The book is not an historical work in the aca- professor entered the classroom and did so again demic sense. By contrast, and in agreement with in appreciation at the end of the lecture. Just the her intentions, the author creates a lively picture opposite is right, as the reviewer remembers from more in the sense of a novel—letting various actors his own student days: One stamped the feet at the talk in direct speech, as well as providing as many beginning and end, but shuffling the feet was a anecdotes as she could get hold of and inventing sign of extreme displeasure during or at the end stories that in her opinion fit into the general of the hour. -
Constraining Conformal Field Theories with a Higher Spin Symmetry
PUPT-2399 Constraining conformal field theories with a higher spin symmetry Juan Maldacenaa and Alexander Zhiboedovb aSchool of Natural Sciences, Institute for Advanced Study Princeton, NJ, USA bDepartment of Physics, Princeton University Princeton, NJ, USA We study the constraints imposed by the existence of a single higher spin conserved current on a three dimensional conformal field theory. A single higher spin conserved current implies the existence of an infinite number of higher spin conserved currents. The correlation functions of the stress tensor and the conserved currents are then shown to be equal to those of a free field theory. Namely a theory of N free bosons or free fermions. arXiv:1112.1016v1 [hep-th] 5 Dec 2011 This is an extension of the Coleman-Mandula theorem to CFT’s, which do not have a conventional S matrix. We also briefly discuss the case where the higher spin symmetries are “slightly” broken. Contents 1.Introduction ................................. 2 1.1. Organization of the paper . 4 2. Generalities about higher spin currents . 5 1 3. Removing operators in the twist gap, 2 <τ< 1 .................. 7 3.1. Action of the charges on twist one fields . 9 4. Basic facts about three point functions . 10 4.1. General expression for three point functions . 11 5. Argument using bilocal operators . 12 5.1. Light cone limits of correlators of conserved currents . 12 5.2.Gettinganinfinitenumberofcurrents . 15 5.3. Definition of bilocal operators . 17 5.4. Constraining the action of the higher spin charges . 19 5.5. Quantization of N˜: the case of bosons . 24 5.6. -
Amalie (Emmy) Noether
Amalie (Emmy) Noether Female Mathematicians By Ella - Emmy was born in 1882 and her father was a math professor at the University of Erlangen During her Life and that's one of the reason why she started to be interested in math - She couldn’t enroll in the college Erlangen because she was a woman but she did audit the classes. Also, when Emmy was on staff of Göttingen University but didn’t get paid to lecture like her male colleagues - At this time girls were only allowed University of Erlangen in Germany to go to "finishing school,” where they learn to teach. Emmy became certified to teach French and English but never did because she pursued mathematics. Emmay’s Accomplishments - Emmy Noether discovered the link between conservation laws and symmetries. Conservation laws is when a particular quantity must stay constant. For example, energy can’t be created or destroyed. Symmetries is the changes that can be made without changing the way the object looks or acts. For example, it doesn’t matter how you rotate or change the direction of a sphere it will always appear the same. - She also found noncommutative algebras which is when there is a specific order that numbers be multiplied to solve the equation. The Importance of her Accomplishments - The link between conservation laws and symmetries is called Noether’s Theorem. It is important because it gives us insight into conservation laws. Also, it is important because it shows scientists that repeating an experiment at different times won’t change the results. Timeline Left Germany to teach in America Received her Ph D Germany became an Born unsafe place to live for the college Erlangen a Jew like herself. -
1 INTRO Welcome to Biographies in Mathematics Brought to You From
INTRO Welcome to Biographies in Mathematics brought to you from the campus of the University of Texas El Paso by students in my history of math class. My name is Tuesday Johnson and I'll be your host on this tour of time and place to meet the people behind the math. EPISODE 1: EMMY NOETHER There were two women I first learned of when I started college in the fall of 1990: Hypatia of Alexandria and Emmy Noether. While Hypatia will be the topic of another episode, it was never a question that I would have Amalie Emmy Noether as my first topic of this podcast. Emmy was born in Erlangen, Germany, on March 23, 1882 to Max and Ida Noether. Her father Max came from a family of wholesale hardware dealers, a business his grandfather started in Bruchsal (brushal) in the early 1800s, and became known as a great mathematician studying algebraic geometry. Max was a professor of mathematics at the University of Erlangen as well as the Mathematics Institute in Erlangen (MIE). Her mother, Ida, was from the wealthy Kaufmann family of Cologne. Both of Emmy’s parents were Jewish, therefore, she too was Jewish. (Judiasm being passed down a matrilineal line.) Though Noether is not a traditional Jewish name, as I am told, it was taken by Elias Samuel, Max’s paternal grandfather when in 1809 the State of Baden made the Tolerance Edict, which required Jews to adopt Germanic names. Emmy was the oldest of four children and one of only two who survived childhood. -
Emmy Noether, Greatest Woman Mathematician Clark Kimberling
Emmy Noether, Greatest Woman Mathematician Clark Kimberling Mathematics Teacher, March 1982, Volume 84, Number 3, pp. 246–249. Mathematics Teacher is a publication of the National Council of Teachers of Mathematics (NCTM). With more than 100,000 members, NCTM is the largest organization dedicated to the improvement of mathematics education and to the needs of teachers of mathematics. Founded in 1920 as a not-for-profit professional and educational association, NCTM has opened doors to vast sources of publications, products, and services to help teachers do a better job in the classroom. For more information on membership in the NCTM, call or write: NCTM Headquarters Office 1906 Association Drive Reston, Virginia 20191-9988 Phone: (703) 620-9840 Fax: (703) 476-2970 Internet: http://www.nctm.org E-mail: [email protected] Article reprinted with permission from Mathematics Teacher, copyright March 1982 by the National Council of Teachers of Mathematics. All rights reserved. mmy Noether was born over one hundred years ago in the German university town of Erlangen, where her father, Max Noether, was a professor of Emathematics. At that time it was very unusual for a woman to seek a university education. In fact, a leading historian of the day wrote that talk of “surrendering our universities to the invasion of women . is a shameful display of moral weakness.”1 At the University of Erlangen, the Academic Senate in 1898 declared that the admission of women students would “overthrow all academic order.”2 In spite of all this, Emmy Noether was able to attend lectures at Erlangen in 1900 and to matriculate there officially in 1904. -
Book Review for EMMY NOETHER – the Mother of Modern Algebra by M
MATHEMATICS TEACHING RESEARCH JOURNAL 18 SPRING/SUMMER 2019 Vol 11 no 1-2 Book Review for EMMY NOETHER – the mother of modern algebra by M. B. W. Tent Roy Berglund City Tech, Borough of Manhattan Community College of CUNY Title: Emmy Noether Author: M. B. W. Tent Publisher: A. K. Peters, Ltd. Publication Date: 2008 Number of Pages: 177 Format: Hardcover Special Features: Glossary Price: $29.00 ISBN: 978-1-56881-430-8 Reviewed on April 26, 2019 The story of Emmy Noether is one of “little unknown girl makes good and becomes famous”. Not a lot is known about Noether’s early life growing up, but it is abundantly clear that she was blessed with strong family ties in secure circumstances. Such an environment tends to foster success for the individual, whenever one or both parents are supportive and expose the child to positive cultural and educational influences. The author chose to describe Noether’s biographical life rather than her mathematical accomplishments. In consequence Noether’s achievements are lightly mentioned, but they are presented without any mathematical details. It would be illuminating to follow Noether’s thinking through her discoveries and to track her development to mathematical maturity. Emmy’s father, Max, was a mathematics professor in the college at Erlangen. Both parents tended to channel their daughter into preparation for domestic life as a wife and mother. However, Emmy was bright child, and solved puzzles and riddles of all kinds from an early age. She continually questioned her designated role as a girl, and queried her father why she, too, should not be a mathematician, ever resisting the rote of learning French, or of a musical instrument. -
Some Basic Biographical Facts About Emmy Noether (1882-1935), in Particular on the Discrimination Against Her As a Woman
Some basic biographical facts about Emmy Noether (1882-1935), in particular on the discrimination against her as a woman LMS-IMA Joint Meeting: Noether Celebration De Morgan House, Tuesday, 11 September, 2018 Reinhard Siegmund-Schultze (University of Agder, Kristiansand, Norway) ABSTRACT Although it has been repeatedly underlined that Emmy Noether had to face threefold discrimination in political, racist and sexist respects the last- mentioned discrimination of the three is probably best documented. The talk provides some basic biographical facts about Emmy Noether with an emphasis on the discrimination against her as a woman, culminating for the first time in the struggles about her teaching permit (habilitation) 1915-1919 (main source C. Tollmien). Another focus of the talk will be on the later period of her life, in particular the failed appointment in Kiel (1928), her Born: 23 March 1882 in Erlangen, dismissal as a Jew in 1933 and her last years in the Bavaria, Germany U.S. Died: 14 April 1935 in Bryn Mawr, Pennsylvania, USA Older sources Obituaries by colleagues and students: van der Waerden, Hermann Weyl, P.S. Aleksandrov. Historians: Three women: Auguste Dick (1970, Engl.1981), Constance Reid (Hilbert 1970), and Cordula Tollmien (e.g. 1991 on Noether’s Habilitation); plus Norbert Schappacher (1987). Most material in German, Clark Kimberling (1972) in American Mathematical Monthly mostly translating from Dick and obituaries. Newer Sources Again mostly by women biographers, such as Renate Tobies (2003), Cordula Tollmien (2015), and Mechthild Koreuber (2015). The book below, of which the English version is from 2011, discusses the papers relevant for physics: After going through a girls school she took in 1900 a state exam to become a teacher in English and French at Bavarian girls schools. -
Energy Is Conserved in General Relativity
Energy is Conserved in General Relativity By Philip Gibbs Abstract: Since the early days of relativity the question of conservation of energy in general relativity has been a controversial subject. There have been many assertions that energy is not exactly conserved except in special cases, or that the full conservation law as given by Noether’s theorem reduces to a trivial identity. Here I refute each objection to show that the energy conservation law is exact, fully general and useful. Introduction Conservation of Energy is regarded as one of the most fundamental and best known laws of physics. It is a kind of meta-law that all systems of dynamics in physics are supposed to fulfil to be considered valid. Yet some physicists and cosmologists claim that the law of energy conservation and related laws for momentum actually break down in Einstein’s theory of gravity described by general relativity. Some say that these conservation laws are approximate, or only valid in special cases, or that they can only be written in non-covariant terms making them unphysical. Others simply claim that they are satisfied only by equations that reduce to a uselessly trivial identity. In fact none of those things are true. Energy conservation is an exact law in general relativity. It is fully general for all circumstances and it is certainly non-trivial. Furthermore, it is important for students of relativity to understand the conservation laws properly because they are important to further research at the forefront of physics. In earlier papers I have described a covariant formulation of energy conservation law that is perfectly general. -
Quantum Field Theory Example Sheet 1 Michelmas Term 2011 Exercise 1
Quantum Field Theory Example Sheet 1 Michelmas Term 2011 Solutions by: Johannes Hofmann [email protected] Laurence Perreault Levasseur [email protected] David Morris [email protected] Marcel Schmittfull [email protected] Note: In the conventions of this course, the Minkowski metric is g = diag (1, −1, −1, −1), or ‘mostly minus.’ Exercise 1 Lagrangian: " # Z a σ ∂y 2 T ∂y 2 L = dx − (1) 0 2 ∂t 2 ∂x Express y(x, t) as a Fourier series: r ∞ 2 X nπx y(x, t) = q (t) sin . (2) a n a n=1 The derivative of y(x, t) with respect to x and t is r ∞ ∂y(x, t) 2 X nπx = q˙ (t) sin (3) ∂t a n a n=1 r ∞ ∂y(x, t) 2 X nπ nπx = q (t) cos , (4) ∂x a a n a n=1 where we abbreviateq ˙ ≡ ∂q/∂t. Substituting Eqs. (3) and (4) in (1) gives 1 X Z a nπx mπx nmπ2 nπx mπx L = dx σq˙ (t)q ˙ (t) sin sin − T q (t)q (t) cos cos . (5) a n m a a a2 n m a a n,m 0 Using the orthonormality relations 2 Z a nπx mπx dx sin sin = δnm and (6) a 0 a a 2 Z a nπx mπx dx cos cos = δnm (7) a 0 a a we see that the terms with n 6= m vanish. We obtain: X Z a σ T nπ 2 L = dx q˙2 (t) − q2 (t) . -
THE INFLUENCE of J. H. M. WEDDERBURN on the DEVELOPMENT of MODERN ALGEBRA It Is Obvious That the Title of This Paper Is Presumptuous
THE INFLUENCE OF J. H. M. WEDDERBURN ON THE DEVELOPMENT OF MODERN ALGEBRA It is obvious that the title of this paper is presumptuous. Nobody can give in a short article a really exhaustive account of the influence of Wedderburn on the development of modern algebra. It is too big an undertaking and would require years of preparation. In order to present at least a modest account of this influence it is necessary to restrict oneself rather severely. To this effect we shall discuss only the two most celebrated articles of Wedderburn and try to see them in the light of the subsequent development of algebra. But even this would be too great a task. If we would have to mention all the conse quences and applications of his theorems we could easily fill a whole volume. Consequently we shall discuss only the attempts the mathe maticians made to come to a gradual understanding of the meaning of his theorems and be satisfied just to mention a few applications. For the understanding of the significance that Wedderburn's paper On hypercomplex numbers (Proc. London Math. Soc. (2) vol. 6, p. 77) had for the development of modern algebra, it is imperative to look at the ideas his predecessors had on the subject. The most striking fact is the difference in attitude between Ameri can and European authors. From the very beginning the abstract point of view is dominant in American publications whereas for European mathematicians a system of hypercomplex numbers was by nature an extension of either the real or the complex field.