The Emergence of Gravitational Wave Science: 100 Years of Development of Mathematical Theory, Detectors, Numerical Algorithms, and Data Analysis Tools
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Of the American Mathematical Society August 2017 Volume 64, Number 7
ISSN 0002-9920 (print) ISSN 1088-9477 (online) of the American Mathematical Society August 2017 Volume 64, Number 7 The Mathematics of Gravitational Waves: A Two-Part Feature page 684 The Travel Ban: Affected Mathematicians Tell Their Stories page 678 The Global Math Project: Uplifting Mathematics for All page 712 2015–2016 Doctoral Degrees Conferred page 727 Gravitational waves are produced by black holes spiraling inward (see page 674). American Mathematical Society LEARNING ® MEDIA MATHSCINET ONLINE RESOURCES MATHEMATICS WASHINGTON, DC CONFERENCES MATHEMATICAL INCLUSION REVIEWS STUDENTS MENTORING PROFESSION GRAD PUBLISHING STUDENTS OUTREACH TOOLS EMPLOYMENT MATH VISUALIZATIONS EXCLUSION TEACHING CAREERS MATH STEM ART REVIEWS MEETINGS FUNDING WORKSHOPS BOOKS EDUCATION MATH ADVOCACY NETWORKING DIVERSITY blogs.ams.org Notices of the American Mathematical Society August 2017 FEATURED 684684 718 26 678 Gravitational Waves The Graduate Student The Travel Ban: Affected Introduction Section Mathematicians Tell Their by Christina Sormani Karen E. Smith Interview Stories How the Green Light was Given for by Laure Flapan Gravitational Wave Research by Alexander Diaz-Lopez, Allyn by C. Denson Hill and Paweł Nurowski WHAT IS...a CR Submanifold? Jackson, and Stephen Kennedy by Phillip S. Harrington and Andrew Gravitational Waves and Their Raich Mathematics by Lydia Bieri, David Garfinkle, and Nicolás Yunes This season of the Perseid meteor shower August 12 and the third sighting in June make our cover feature on the discovery of gravitational waves -
Review Study on “The Black Hole”
IJIRST –International Journal for Innovative Research in Science & Technology| Volume 2 | Issue 10 | March 2016 ISSN (online): 2349-6010 Review Study on “The Black Hole” Syed G. Ibrahim Department of Engineering Physics (Nanostructured Thin Film Materials Laboratory) Prof. Ram Meghe College of Engineering and Management, Badnera 444701, Maharashtra, India Abstract As a star grows old, swells, then collapses on itself, often you will hear the word “black hole” thrown around. The black hole is a gravitationally collapsed mass, from which no light, matter, or signal of any kind can escape. These exotic objects have captured our imagination ever since they were predicted by Einstein's Theory of General Relativity in 1915. So what exactly is a black hole? A black hole is what remains when a massive star dies. Not every star will become a black hole, only a select few with extremely large masses. In order to have the ability to become a black hole, a star will have to have about 20 times the mass of our Sun. No known process currently active in the universe can form black holes of less than stellar mass. This is because all present black hole formation is through gravitational collapse, and the smallest mass which can collapse to form a black hole produces a hole approximately 1.5-3.0 times the mass of the sun .Smaller masses collapse to form white dwarf stars or neutron stars. Keywords: Escape Velocity, Horizon, Schwarzschild Radius, Black Hole _______________________________________________________________________________________________________ I. INTRODUCTION Soon after Albert Einstein formulated theory of relativity, it was realized that his equations have solutions in closed form. -
Stable Components in the Parameter Plane of Transcendental Functions of Finite Type
Stable components in the parameter plane of transcendental functions of finite type N´uriaFagella∗ Dept. de Matem`atiquesi Inform`atica Univ. de Barcelona, Gran Via 585 Barcelona Graduate School of Mathematics (BGSMath) 08007 Barcelona, Spain [email protected] Linda Keen† CUNY Graduate Center 365 Fifth Avenue, New York NY, NY 10016 [email protected], [email protected] November 11, 2020 Abstract We study the parameter planes of certain one-dimensional, dynamically-defined slices of holomorphic families of entire and meromorphic transcendental maps of finite type. Our planes are defined by constraining the orbits of all but one of the singular values, and leaving free one asymptotic value. We study the structure of the regions of parameters, which we call shell components, for which the free asymptotic value tends to an attracting cycle of non-constant multiplier. The exponential and the tangent families are examples that have been studied in detail, and the hyperbolic components in those parameter planes are shell components. Our results apply to slices of both entire and meromorphic maps. We prove that shell components are simply connected, have a locally connected boundary and have no center, i.e., no parameter value for which the cycle is superattracting. Instead, there is a unique parameter in the boundary, the virtual center, which plays the same role. For entire slices, the virtual center is always at infinity, while for meromorphic ones it maybe finite or infinite. In the dynamical plane we prove, among other results, that the basins of attraction which contain only one asymptotic value and no critical points are simply connected. -
Dean, School of Science
Dean, School of Science School of Science faculty members seek to answer fundamental questions about nature ranging in scope from the microscopic—where a neuroscientist might isolate the electrical activity of a single neuron—to the telescopic—where an astrophysicist might scan hundreds of thousands of stars to find Earth-like planets in their orbits. Their research will bring us a better understanding of the nature of our universe, and will help us address major challenges to improving and sustaining our quality of life, such as developing viable resources of renewable energy or unravelling the complex mechanics of Alzheimer’s and other diseases of the aging brain. These profound and important questions often require collaborations across departments or schools. At the School of Science, such boundaries do not prevent people from working together; our faculty cross such boundaries as easily as they walk across the invisible boundary between one building and another in our campus’s interconnected buildings. Collaborations across School and department lines are facilitated by affiliations with MIT’s numerous laboratories, centers, and institutes, such as the Institute for Data, Systems, and Society, as well as through participation in interdisciplinary initiatives, such as the Aging Brain Initiative or the Transiting Exoplanet Survey Satellite. Our faculty’s commitment to teaching and mentorship, like their research, is not constrained by lines between schools or departments. School of Science faculty teach General Institute Requirement subjects in biology, chemistry, mathematics, and physics that provide the conceptual foundation of every undergraduate student’s education at MIT. The School faculty solidify cross-disciplinary connections through participation in graduate programs established in collaboration with School of Engineering, such as the programs in biophysics, microbiology, or molecular and cellular neuroscience. -
INTERNSHIP REPORT.Pdf
INTERNSHIP REPORT STUDY OF PASSIVE DAMPING TECHNIQUES TO IMPROVE THE PERFORMANCE OF A SEISMIC ISOLATION SYSTEM for Advanced Laser Interferometer Gravitational Wave Observatory MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 2010 Author…………………………………………………………………………….Sebastien Biscans ENSIM Advanced LIGO Supervisor………………………………………...……………Fabrice Matichard Advanced LIGO Engineer ENSIM Supervisor…………………………………………………………………..Charles Pezerat ENSIM Professor ACKNOWLEDGMENTS Many thanks must go out to Fabrice Matichard, my supervisor, co-worker and friend, for his knowledge and his kindness. I also would like to thank all the members of the Advanced LIGO group with whom I’ve had the privilege to work, learn and laugh during the last six months. 2 TABLE OF CONTENTS 1. Introduction.................................................................................................................................................4 2. Presentation of the seismic isolation system ..........................................................................................5 2.1 General Overview................................................................................................................................5 2.2 The BSC-ISI..........................................................................................................................................7 2.3 The Quad ..............................................................................................................................................7 3. Preliminary study : the tuned -
Cracking the Einstein Code: Relativity and the Birth of Black Hole Physics, with an Afterword by Roy Kerr / Fulvio Melia
CRA C K I N G T H E E INSTEIN CODE @SZObWdWbgO\RbVS0W`bV]T0ZOQY6]ZS>VgaWQa eWbVO\/TbS`e]`RPg@]gS`` fulvio melia The University of Chicago Press chicago and london fulvio melia is a professor in the departments of physics and astronomy at the University of Arizona. He is the author of The Galactic Supermassive Black Hole; The Black Hole at the Center of Our Galaxy; The Edge of Infinity; and Electrodynamics, and he is series editor of the book series Theoretical Astrophysics published by the University of Chicago Press. The University of Chicago Press, Chicago 60637 The University of Chicago Press, Ltd., London © 2009 by The University of Chicago All rights reserved. Published 2009 Printed in the United States of America 18 17 16 15 14 13 12 11 10 09 1 2 3 4 5 isbn-13: 978-0-226-51951-7 (cloth) isbn-10: 0-226-51951-1 (cloth) Library of Congress Cataloging-in-Publication Data Melia, Fulvio. Cracking the Einstein code: relativity and the birth of black hole physics, with an afterword by Roy Kerr / Fulvio Melia. p. cm. Includes bibliographical references and index. isbn-13: 978-0-226-51951-7 (cloth: alk. paper) isbn-10: 0-226-51951-1 (cloth: alk. paper) 1. Einstein field equations. 2. Kerr, R. P. (Roy P.). 3. Kerr black holes—Mathematical models. 4. Black holes (Astronomy)—Mathematical models. I. Title. qc173.6.m434 2009 530.11—dc22 2008044006 To natalina panaia and cesare melia, in loving memory CONTENTS preface ix 1 Einstein’s Code 1 2 Space and Time 5 3 Gravity 15 4 Four Pillars and a Prayer 24 5 An Unbreakable Code 39 6 Roy Kerr 54 7 The Kerr Solution 69 8 Black Hole 82 9 The Tower 100 10 New Zealand 105 11 Kerr in the Cosmos 111 12 Future Breakthrough 121 afterword 125 references 129 index 133 PREFACE Something quite remarkable arrived in my mail during the summer of 2004. -
The Reduction Problem for Einstein's Equations I
168 THE REDUCTION PROBLEM FOR EINSTEIN'S EQUATIONS Vincent Moncrief I. MOTIVATION h has long been known that the vacuum (or electro-vacuum) Einstein equations for spacetimes admitting a single (spacelike) Killing field can be locally reexpressed as a 2 + 1 dimensional system of Einstein-harmonic map equations [1, 2] . In this formulation the unknown fields are (in the vacuum case) a Lorentzian metric on the (locally defined) space of orbits of the Killing field and the norm and twist potential of this Killing field which define a harmonic map from the space of orbits to hyperbolic two-space. A natural problem is to study this formulation globally on suitably chosen manifolds and to apply standard methods of elliptic to reduce the system to its "minimal form". In physics terminology one would like to solve an elliptic set of equations for all the dependent or "non-propagating" variables and be left with an unconstrained hyperbolic system for only the independent "physical degrees of freedom" of the gravitational field. One mathematical motivation for this reduction program is that it aims to take maximal advantage of the elliptic aspects of Einstein's equations, for which the needed global analysis techniques are already at hand, and to reduce the complementary (and presumably more difficult) hyperbolic aspects of the system as much as possible. The ultimate aim of this project is to study the long time existence properties of Einstein's equations and the hope is that the reduction program should help to make such an analysis more tractable. 169 Another more "practical" motivation for this program is that it suggests a numerical method for approximately solving the Einstein equations which avoids the notorious problem of the "drifting of the constraints". -
Backpropagation TA: Yi Wen
Backpropagation TA: Yi Wen April 17, 2020 CS231n Discussion Section Slides credits: Barak Oshri, Vincent Chen, Nish Khandwala, Yi Wen Agenda ● Motivation ● Backprop Tips & Tricks ● Matrix calculus primer Agenda ● Motivation ● Backprop Tips & Tricks ● Matrix calculus primer Motivation Recall: Optimization objective is minimize loss Motivation Recall: Optimization objective is minimize loss Goal: how should we tweak the parameters to decrease the loss? Agenda ● Motivation ● Backprop Tips & Tricks ● Matrix calculus primer A Simple Example Loss Goal: Tweak the parameters to minimize loss => minimize a multivariable function in parameter space A Simple Example => minimize a multivariable function Plotted on WolframAlpha Approach #1: Random Search Intuition: the step we take in the domain of function Approach #2: Numerical Gradient Intuition: rate of change of a function with respect to a variable surrounding a small region Approach #2: Numerical Gradient Intuition: rate of change of a function with respect to a variable surrounding a small region Finite Differences: Approach #3: Analytical Gradient Recall: partial derivative by limit definition Approach #3: Analytical Gradient Recall: chain rule Approach #3: Analytical Gradient Recall: chain rule E.g. Approach #3: Analytical Gradient Recall: chain rule E.g. Approach #3: Analytical Gradient Recall: chain rule Intuition: upstream gradient values propagate backwards -- we can reuse them! Gradient “direction and rate of fastest increase” Numerical Gradient vs Analytical Gradient What about Autograd? -
Decomposing the Parameter Space of Biological Networks Via a Numerical Discriminant Approach
Decomposing the parameter space of biological networks via a numerical discriminant approach Heather A. Harrington1, Dhagash Mehta2, Helen M. Byrne1, and Jonathan D. Hauenstein2 1 Mathematical Institute, The University of Oxford, Oxford OX2 6GG, UK {harrington,helen.byrne}@maths.ox.ac.uk, www.maths.ox.ac.uk/people/{heather.harrington,helen.byrne} 2 Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame IN 46556, USA {dmehta,hauenstein}@nd.edu, www.nd.edu/~{dmehta,jhauenst} Abstract. Many systems in biology (as well as other physical and en- gineering systems) can be described by systems of ordinary dierential equation containing large numbers of parameters. When studying the dynamic behavior of these large, nonlinear systems, it is useful to iden- tify and characterize the steady-state solutions as the model parameters vary, a technically challenging problem in a high-dimensional parameter landscape. Rather than simply determining the number and stability of steady-states at distinct points in parameter space, we decompose the parameter space into nitely many regions, the number and structure of the steady-state solutions being consistent within each distinct region. From a computational algebraic viewpoint, the boundary of these re- gions is contained in the discriminant locus. We develop global and local numerical algorithms for constructing the discriminant locus and classi- fying the parameter landscape. We showcase our numerical approaches by applying them to molecular and cell-network models. Keywords: parameter landscape · numerical algebraic geometry · dis- criminant locus · cellular networks. 1 Introduction The dynamic behavior of many biophysical systems can be mathematically mod- eled with systems of dierential equations that describe how the state variables interact and evolve over time. -
18Th Annual Student Research and Creativity Celebration at Buffalo State College
III IIII II I I I I I I I I III IIII II I I I I I I I I Editor Jill Singer, Ph.D. Director, Office of Undergraduate Research Cover Design and Layout: Carol Alex The following individuals and offices are acknowledged for their many contributions: Marc Bayer, Interim Library Director, E.H. Butler Library, Department and Program Coordinators (identified below) and the library staff Donald Schmitter, Hospitality and Tourism, and students and very special thanks to: in HTR 400: Catering Management Kaylene Waite, Graphic Design, Instructional Resources Bruce Fox, Photographer Carol Alex, Center for Development of Human Services Andrew Chambers, Information Management Officer Sean Fox, Ellofex, Inc. Bernadette Gilliam and Mary Beth Wojtaszek, Events Management Department and Program Coordinators for the Eighteenth Annual Student Research and Creativity Celebration Lisa Marie Anselmi, Anthropology Dan MacIsaac, Physics Sarbani Banerjee, Computer Information Systems Candace Masters, Art Education Saziye Bayram, Mathematics Carmen McCallum, Higher Education Administration Carol Beckley, Theater Michaelene Meger, Exceptional Education Lynn Boorady, Fashion and Textile Technology Andrew Nicholls, History and Social Studies Education Louis Colca, Social Work Jill Norvilitis, Psychology Sandra Washington-Copeland, McNair Scholars Program Kathleen O’Brien, Hospitality and Tourism Carol DeNysschen, Health, Nutrition, and Dietetics Lorna Perez, English Eric Dolph, Interior Design Rebecca Ploeger, Art Conservation Reva Fish, Social & Psychological Foundations -
Catoni F., Et Al. the Mathematics of Minkowski Space-Time.. With
Frontiers in Mathematics Advisory Editorial Board Leonid Bunimovich (Georgia Institute of Technology, Atlanta, USA) Benoît Perthame (Ecole Normale Supérieure, Paris, France) Laurent Saloff-Coste (Cornell University, Rhodes Hall, USA) Igor Shparlinski (Macquarie University, New South Wales, Australia) Wolfgang Sprössig (TU Bergakademie, Freiberg, Germany) Cédric Villani (Ecole Normale Supérieure, Lyon, France) Francesco Catoni Dino Boccaletti Roberto Cannata Vincenzo Catoni Enrico Nichelatti Paolo Zampetti The Mathematics of Minkowski Space-Time With an Introduction to Commutative Hypercomplex Numbers Birkhäuser Verlag Basel . Boston . Berlin $XWKRUV )UDQFHVFR&DWRQL LQFHQ]R&DWRQL LDHJOLD LDHJOLD 5RPD 5RPD Italy Italy HPDLOYMQFHQ]R#\DKRRLW Dino Boccaletti 'LSDUWLPHQWRGL0DWHPDWLFD (QULFR1LFKHODWWL QLYHUVLWjGL5RPD²/D6DSLHQ]D³ (1($&5&DVDFFLD 3LD]]DOH$OGR0RUR LD$QJXLOODUHVH 5RPD 5RPD Italy Italy HPDLOERFFDOHWWL#XQLURPDLW HPDLOQLFKHODWWL#FDVDFFLDHQHDLW 5REHUWR&DQQDWD 3DROR=DPSHWWL (1($&5&DVDFFLD (1($&5&DVDFFLD LD$QJXLOODUHVH LD$QJXLOODUHVH 5RPD 5RPD Italy Italy HPDLOFDQQDWD#FDVDFFLDHQHDLW HPDLO]DPSHWWL#FDVDFFLDHQHDLW 0DWKHPDWLFDO6XEMHFW&ODVVL½FDWLRQ*(*4$$ % /LEUDU\RI&RQJUHVV&RQWURO1XPEHU Bibliographic information published by Die Deutsche Bibliothek 'LH'HXWVFKH%LEOLRWKHNOLVWVWKLVSXEOLFDWLRQLQWKH'HXWVFKH1DWLRQDOELEOLRJUD½H GHWDLOHGELEOLRJUDSKLFGDWDLVDYDLODEOHLQWKH,QWHUQHWDWKWWSGQEGGEGH! ,6%1%LUNKlXVHUHUODJ$*%DVHOÀ%RVWRQÀ% HUOLQ 7KLVZRUNLVVXEMHFWWRFRS\ULJKW$OOULJKWVDUHUHVHUYHGZKHWKHUWKHZKROHRUSDUWRIWKH PDWHULDOLVFRQFHUQHGVSHFL½FDOO\WKHULJKWVRIWUDQVODWLRQUHSULQWLQJUHXVHRILOOXVWUD -
Generalizations of the Kerr-Newman Solution
Generalizations of the Kerr-Newman solution Contents 1 Topics 1035 1.1 ICRANetParticipants. 1035 1.2 Ongoingcollaborations. 1035 1.3 Students ............................... 1035 2 Brief description 1037 3 Introduction 1039 4 Thegeneralstaticvacuumsolution 1041 4.1 Line element and field equations . 1041 4.2 Staticsolution ............................ 1043 5 Stationary generalization 1045 5.1 Ernst representation . 1045 5.2 Representation as a nonlinear sigma model . 1046 5.3 Representation as a generalized harmonic map . 1048 5.4 Dimensional extension . 1052 5.5 Thegeneralsolution . 1055 6 Tidal indicators in the spacetime of a rotating deformed mass 1059 6.1 Introduction ............................. 1059 6.2 The gravitational field of a rotating deformed mass . 1060 6.2.1 Limitingcases. 1062 6.3 Circularorbitsonthesymmetryplane . 1064 6.4 Tidalindicators ........................... 1065 6.4.1 Super-energy density and super-Poynting vector . 1067 6.4.2 Discussion.......................... 1068 6.4.3 Limit of slow rotation and small deformation . 1069 6.5 Multipole moments, tidal Love numbers and Post-Newtonian theory................................. 1076 6.6 Concludingremarks . 1077 7 Neutrino oscillations in the field of a rotating deformed mass 1081 7.1 Introduction ............................. 1081 7.2 Stationary axisymmetric spacetimes and neutrino oscillation . 1082 7.2.1 Geodesics .......................... 1083 1033 Contents 7.2.2 Neutrinooscillations . 1084 7.3 Neutrino oscillations in the Hartle-Thorne metric . 1085 7.4 Concludingremarks . 1088 8 Gravitational field of compact objects in general relativity 1091 8.1 Introduction ............................. 1091 8.2 The Hartle-Thorne metrics . 1094 8.2.1 The interior solution . 1094 8.2.2 The Exterior Solution . 1096 8.3 The Fock’s approach . 1097 8.3.1 The interior solution .