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View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Caltech Authors PROOF OF THE STRONG SCOTT CONJECTURE FOR CHANDRASEKHAR ATOMS RUPERT L. FRANK, KONSTANTIN MERZ, HEINZ SIEDENTOP, AND BARRY SIMON Dedicated to Yakov Sinai on the occasion of his 85th birthday. Abstract. We consider a large neutral atom of atomic number Z, taking rel- ativistic effects into account by assuming the dispersion relation pc2p2 + c4. We study the behavior of the one-particle ground state density on the length scale Z−1 in the limit Z; c ! 1 keeping Z=c fixed and find that the spher- ically averaged density as well as all individual angular momentum densities separately converge to the relativistic hydrogenic ones. This proves the gen- eralization of the strong Scott conjecture for relativistic atoms and shows, in particular, that relativistic effects occur close to the nucleus. Along the way we prove upper bounds on the relativistic hydrogenic density. 1. Introduction 1.1. Some results on large Z-atoms. The asymptotic behavior of the ground state energy and the ground state density of atoms with large atomic number Z have been studied in detail in non-relativistic quantum mechanics. Soon after the advent of quantum mechanics it became clear that the non- relativistic quantum multi-particle problem is not analytically solvable and of in- creasing challenge with large particle number. This problem was addressed by Thomas [51] and Fermi [11, 12] by developing what was called the statistical model of the atom. The model is described by the so-called Thomas{Fermi functional (Lenz [28]) Z ZZ TF 3 5=3 Z 1 ρ(x)ρ(y) (1) EZ (ρ) := 10 γTFρ(x) − ρ(x) dx + dx dy ; 3 jxj 2 3 3 jx − yj R R ×R | {z } =:D[ρ] 2 2=3 where γTF = (6π =q) is a positive constant depending on the number q of spin states per electron, i.e., physically 2. -
The Bibliography
Referenced Books [Ach92] N. I. Achieser. Theory of Approximation. Dover Publications Inc., New York, 1992. Reprint of the 1956 English translation of the 1st Rus- sian edition; the 2nd augmented Russian edition is available, Moscow, Nauka, 1965. [AH05] Kendall Atkinson and Weimin Han. Theoretical Numerical Analysis: A Functional Analysis Framework, volume 39 of Texts in Applied Mathe- matics. Springer, New York, second edition, 2005. [Atk89] Kendall E. Atkinson. An Introduction to Numerical Analysis. John Wiley & Sons Inc., New York, second edition, 1989. [Axe94] Owe Axelsson. Iterative Solution Methods. Cambridge University Press, Cambridge, 1994. [Bab86] K. I. Babenko. Foundations of Numerical Analysis [Osnovy chislennogo analiza]. Nauka, Moscow, 1986. [Russian]. [BD92] C. A. Brebbia and J. Dominguez. Boundary Elements: An Introductory Course. Computational Mechanics Publications, Southampton, second edition, 1992. [Ber52] S. N. Bernstein. Collected Works. Vol. I. The Constructive Theory of Functions [1905–1930]. Izdat. Akad. Nauk SSSR, Moscow, 1952. [Russian]. [Ber54] S. N. Bernstein. Collected Works. Vol. II. The Constructive Theory of Functions [1931–1953]. Izdat. Akad. Nauk SSSR, Moscow, 1954. [Russian]. [BH02] K. Binder and D. W. Heermann. Monte Carlo Simulation in Statistical Physics: An Introduction, volume 80 of Springer Series in Solid-State Sciences. Springer-Verlag, Berlin, fourth edition, 2002. [BHM00] William L. Briggs, Van Emden Henson, and Steve F. McCormick. A Multigrid Tutorial. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, second edition, 2000. [Boy01] John P. Boyd. Chebyshev and Fourier Spectral Methods. Dover Publi- cations Inc., Mineola, NY, second edition, 2001. [Bra84] Achi Brandt. Multigrid Techniques: 1984 Guide with Applications to Fluid Dynamics, volume 85 of GMD-Studien [GMD Studies]. -
Shaw' Preemia
Shaw’ preemia 2002. a rajas Hongkongi ajakirjandusmagnaat ja tuntud filantroop sir Run Run Shaw (s 1907) omanimelise fondi, et anda v¨alja iga-aastast preemiat – Shaw’ preemiat. Preemiaga autasustatakse ”isikut, s˜oltumata rassist, rahvusest ja religioossest taustast, kes on saavutanud olulise l¨abimurde akadeemilises ja teaduslikus uuri- mist¨o¨os v˜oi rakendustes ning kelle t¨o¨o tulemuseks on positiivne ja sugav¨ m˜oju inimkonnale.” Preemiat antakse v¨alja kolmel alal – astronoomias, loodusteadustes ja meditsiinis ning matemaatikas. Preemia suurus 2009. a oli uks¨ miljon USA dollarit. Preemiaga kaas- neb ka medal (vt joonist). Esimesed preemiad omistati 2004. a. Ajakirjanikud on ristinud Shaw’ preemia Ida Nobeli preemiaks (the Nobel of the East). Shaw’ preemiaid aastail 2009–2011 matemaatika alal m¨a¨arab komisjon, kuhu kuuluvad: esimees: Sir Michael Atiyah (Edinburghi Ulikool,¨ UK) liikmed: David Kazhdan (Jeruusalemma Heebrea Ulikool,¨ Iisrael) Peter C. Sarnak (Princetoni Ulikool,¨ USA) Yum-Tong Siu (Harvardi Ulikool,¨ USA) Margaret H. Wright (New Yorgi Ulikool,¨ USA) 2009. a Shaw’ preemia matemaatika alal kuulutati v¨alja 16. juunil Hongkongis ja see anti v˜ordses osas kahele matemaatikule: 293 Eesti Matemaatika Selts Aastaraamat 2009 Autori~oigusEMS, 2010 294 Shaw’ preemia Simon K. Donaldsonile ja Clifford H. Taubesile s¨arava panuse eest kolme- ja neljam˜o˜otmeliste muutkondade geomeetria arengusse. Premeerimistseremoonia toimus 7. oktoobril 2009. Sellel osales ka sir Run Run Shaw. Siin pildil on sir Run Run Shaw 100-aastane. Simon K. Donaldson sundis¨ 1957. a Cambridge’is (UK) ja on Londonis Imperial College’i Puhta Matemaatika Instituudi direktor ja professor. Bakalaureusekraadi sai ta 1979. a Pembroke’i Kolled- ˇzistCambridge’s ja doktorikraadi 1983. -
An Invitation to Model-Theoretic Galois Theory
AN INVITATION TO MODEL-THEORETIC GALOIS THEORY. ALICE MEDVEDEV AND RAMIN TAKLOO-BIGHASH Abstract. We carry out some of Galois’ work in the setting of an arbitrary first-order theory T . We replace the ambient algebraically closed field by a large model M of T , replace fields by definably closed subsets of M, assume that T codes finite sets, and obtain the fundamental duality of Galois theory matching subgroups of the Galois group of L over F with intermediate exten- sions F ≤ K ≤ L. This exposition of a special case of [11] has the advantage of requiring almost no background beyond familiarity with fields, polynomials, first-order formulae, and automorphisms. 1. Introduction. Two hundred years ago, Evariste´ Galois contemplated symmetry groups of so- lutions of polynomial equations, and Galois theory was born. Thirty years ago, Saharon Shelah found it necessary to work with theories that eliminate imaginar- ies; for an arbitrary theory, he constructed a canonical definitional expansion with this property in [16]. Poizat immediately recognized the importance of a theory already having this property in its native language; indeed, he defined “elimina- tion of imaginaries” in [11]. It immediately became clear (see [11]) that much of Galois theory can be developed for an arbitrary first-order theory that eliminates imaginaries. This model-theoretic version of Galois theory can be generalized be- yond finite or even infinite algebraic extensions, and this can in turn be useful in other algebraic settings such as the study of Galois groups of polynomial differential equations (already begun in [11]) and linear difference equations. On a less applied note, it is possible to bring further ideas into the model-theoretic setting, as is done in [10] for the relation between Galois cohomology and homogeneous spaces. -
UCLA Electronic Theses and Dissertations
UCLA UCLA Electronic Theses and Dissertations Title Algorithms for Optimal Paths of One, Many, and an Infinite Number of Agents Permalink https://escholarship.org/uc/item/3qj5d7dj Author Lin, Alex Tong Publication Date 2020 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA Los Angeles Algorithms for Optimal Paths of One, Many, and an Infinite Number of Agents A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Alex Tong Lin 2020 c Copyright by Alex Tong Lin 2020 ABSTRACT OF THE DISSERTATION Algorithms for Optimal Paths of One, Many, and an Infinite Number of Agents by Alex Tong Lin Doctor of Philosophy in Mathematics University of California, Los Angeles, 2020 Professor Stanley J. Osher, Chair In this dissertation, we provide efficient algorithms for modeling the behavior of a single agent, multiple agents, and a continuum of agents. For a single agent, we combine the modeling framework of optimal control with advances in optimization splitting in order to efficiently find optimal paths for problems in very high-dimensions, thus providing allevia- tion from the curse of dimensionality. For a multiple, but finite, number of agents, we take the framework of multi-agent reinforcement learning and utilize imitation learning in order to decentralize a centralized expert, thus obtaining optimal multi-agents that act in a de- centralized fashion. For a continuum of agents, we take the framework of mean-field games and use two neural networks, which we train in an alternating scheme, in order to efficiently find optimal paths for high-dimensional and stochastic problems. -
Alwyn C. Scott
the frontiers collection the frontiers collection Series Editors: A.C. Elitzur M.P. Silverman J. Tuszynski R. Vaas H.D. Zeh The books in this collection are devoted to challenging and open problems at the forefront of modern science, including related philosophical debates. In contrast to typical research monographs, however, they strive to present their topics in a manner accessible also to scientifically literate non-specialists wishing to gain insight into the deeper implications and fascinating questions involved. Taken as a whole, the series reflects the need for a fundamental and interdisciplinary approach to modern science. Furthermore, it is intended to encourage active scientists in all areas to ponder over important and perhaps controversial issues beyond their own speciality. Extending from quantum physics and relativity to entropy, consciousness and complex systems – the Frontiers Collection will inspire readers to push back the frontiers of their own knowledge. Other Recent Titles The Thermodynamic Machinery of Life By M. Kurzynski The Emerging Physics of Consciousness Edited by J. A. Tuszynski Weak Links Stabilizers of Complex Systems from Proteins to Social Networks By P. Csermely Quantum Mechanics at the Crossroads New Perspectives from History, Philosophy and Physics Edited by J. Evans, A.S. Thorndike Particle Metaphysics A Critical Account of Subatomic Reality By B. Falkenburg The Physical Basis of the Direction of Time By H.D. Zeh Asymmetry: The Foundation of Information By S.J. Muller Mindful Universe Quantum Mechanics and the Participating Observer By H. Stapp Decoherence and the Quantum-to-Classical Transition By M. Schlosshauer For a complete list of titles in The Frontiers Collection, see back of book Alwyn C. -
Prizes and Awards Session
PRIZES AND AWARDS SESSION Wednesday, July 12, 2021 9:00 AM EDT 2021 SIAM Annual Meeting July 19 – 23, 2021 Held in Virtual Format 1 Table of Contents AWM-SIAM Sonia Kovalevsky Lecture ................................................................................................... 3 George B. Dantzig Prize ............................................................................................................................. 5 George Pólya Prize for Mathematical Exposition .................................................................................... 7 George Pólya Prize in Applied Combinatorics ......................................................................................... 8 I.E. Block Community Lecture .................................................................................................................. 9 John von Neumann Prize ......................................................................................................................... 11 Lagrange Prize in Continuous Optimization .......................................................................................... 13 Ralph E. Kleinman Prize .......................................................................................................................... 15 SIAM Prize for Distinguished Service to the Profession ....................................................................... 17 SIAM Student Paper Prizes .................................................................................................................... -
Notices of the AMS 595 Mathematics People NEWS
NEWS Mathematics People contrast electrical impedance Takeda Awarded 2017–2018 tomography, as well as model Centennial Fellowship reduction techniques for para- bolic and hyperbolic partial The AMS has awarded its Cen- differential equations.” tennial Fellowship for 2017– Borcea received her PhD 2018 to Shuichiro Takeda. from Stanford University and Takeda’s research focuses on has since spent time at the Cal- automorphic forms and rep- ifornia Institute of Technology, resentations of p-adic groups, Rice University, the Mathemati- especially from the point of Liliana Borcea cal Sciences Research Institute, view of the Langlands program. Stanford University, and the He will use the Centennial Fel- École Normale Supérieure, Paris. Currently Peter Field lowship to visit the National Collegiate Professor of Mathematics at Michigan, she is Shuichiro Takeda University of Singapore and deeply involved in service to the applied and computa- work with Wee Teck Gan dur- tional mathematics community, in particular on editorial ing the academic year 2017–2018. boards and as an elected member of the SIAM Council. Takeda obtained a bachelor's degree in mechanical The Sonia Kovalevsky Lectureship honors significant engineering from Tokyo University of Science, master's de- contributions by women to applied or computational grees in philosophy and mathematics from San Francisco mathematics. State University, and a PhD in 2006 from the University —From an AWM announcement of Pennsylvania. After postdoctoral positions at the Uni- versity of California at San Diego, Ben-Gurion University in Israel, and Purdue University, since 2011 he has been Pardon Receives Waterman assistant and now associate professor at the University of Missouri at Columbia. -
Boundary Value Problems for Systems That Are Not Strictly
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Applied Mathematics Letters 24 (2011) 757–761 Contents lists available at ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml On mixed initial–boundary value problems for systems that are not strictly hyperbolic Corentin Audiard ∗ Institut Camille Jordan, Université Claude Bernard Lyon 1, Villeurbanne, Rhone, France article info a b s t r a c t Article history: The classical theory of strictly hyperbolic boundary value problems has received several Received 25 June 2010 extensions since the 70s. One of the most noticeable is the result of Metivier establishing Received in revised form 23 December 2010 Majda's ``block structure condition'' for constantly hyperbolic operators, which implies Accepted 28 December 2010 well-posedness for the initial–boundary value problem (IBVP) with zero initial data. The well-posedness of the IBVP with non-zero initial data requires that ``L2 is a continuable Keywords: initial condition''. For strictly hyperbolic systems, this result was proven by Rauch. We Boundary value problem prove here, by using classical matrix theory, that his fundamental a priori estimates are Hyperbolicity Multiple characteristics valid for constantly hyperbolic IBVPs. ' 2011 Elsevier Ltd. All rights reserved. 1. Introduction In his seminal paper [1] on hyperbolic initial–boundary value problems, H.O. Kreiss performed the algebraic construction of a tool, now called the Kreiss symmetrizer, that leads to a priori estimates. Namely, if u is a solution of 8 d X C >@ u C A .x; t/@ u D f ;.t; x/ 2 × Ω; <> t j xj R jD1 (1) C >Bu D g;.t; x/ 2 @ × @Ω; :> R ujtD0 D 0; C Pd where the operator @t jD1 Aj@xj is assumed to be strictly hyperbolic and B satisfies the uniform Lopatinski˘ı condition, there is some γ0 > 0 such that u satisfies the a priori estimate p γ kuk 2 C C kuk 2 C ≤ C kf k 2 C C kgk 2 C ; (2) Lγ .R ×Ω/ Lγ .R ×@Ω/ Lγ .R ×Ω/ Lγ .R ×@Ω/ 2 2 −γ t for γ ≥ γ0. -
Συνέντευξη Του Βλαντιμίρ Ιγκορέβιτς Άρνολντ (Vladimir Igorevich Arnold)
Ιώ Λυγάτσικα Βαρβάκειο Γυμνάσιο Αύγουστος - Οκτώβριος 2010 Συνέντευξη του Βλαντιμίρ Ιγκορέβιτς Άρνολντ (Vladimir Igorevich Arnold) Από τους Michèle Audin και Patrick Iglesias 1 Συνέντευξη του Βλαντιμίρ Ιγκορέβιτς Άρνολντ Οι πληροφορίες αντλήθηκαν από τις εξής πηγές: http://en.wikipedia.org/ http://fr.wikipedia.org/ Περιεχόμενα Τα παρακάτω πρόσωπα παρουσιάζονται με αλφαβητική σειρά Πέιβελ Αλεξαντρόφ Μάικλ Γκρόμοβ Ιβάν Πετρόβσκι Ντίμιτρι Βίκτοροβιτς Άνοσοφ Καρλ Άξελ Χάρνακ Ανρί Πουανκαρέ Μάικλ Άτιγια Μάικλ Ρόμπερτ Χέρμαν Λέβ Ποντριάγκιν Μαρσέλ Μπερζέ Ντέιβιντ Χίλμπερτ Βλαντιμίρ Ρόκχλιν Ενρίκο Μπέτι Χεϊσούκ Χιρονάκα Νταβίντ Πιέρ Ρούελ Ζεν - Μισέλ Μπιζμού Νάιτζελ Χίτσιν Τακιούρο Σιτάνι Αντρέι Μπόλιμπρουκ Λουκ Ιλουζί K αρλ Λάντβιγκ Σίτζελ Ραούλ Μπότ Αλεξάντρ Κίριλοβ Γιάκοβ Σίναϊ Νικολά Μπουρμπακί Φίλιξ Κλάιν Ιζαντόρ Σίνγκερ Ενρί Πόλ Καρτάν Αντρέι Κολμόγκοροφ Στέφεν Σμέιλ Ογκουστίν – Λούις Κοσί Μαξίμ Κόντσεβιτς Α ντρέι Σόσλιν Νικολάι Κωνσταντίνοφ Πίτερ Κρόνχειμερ Τζόρτζ Στόουκς Ρίτσαρντ Ντέντεκαϊντ Έντμουντ Λάνταου Ζακ Στέρμ Λεζέν Ντιριζλέ Μπερνάρντ Μαλγκράνζ Ντένις Σουλιβάν Αντριέν Ντουαντί Τζόν Γουίλαρντ Μίλνορ Ρότζερ Μέιερ Τέμαμ Γιουτζίν Ντένκεν Σόλομ Μάντελμπροτζτ Ρενέ Τομ Λούντβιχ Φάντιβ Γιούρι Μάνιν Νικολάι Βασίλιεφ Μαικλ Φρίντμαν Ισαάκ Νέυτων Aνατόλι Βέρσικ Γουόλφανγκ Φάξς Βαϊοτσέσλαβ Νίκιουλιν Έρνεστ Βίνγκμπεργκ Ισραήλ Τζέλφαντ Σέργκεϊ Πέτροβιτς Νόβικοφ Ανατόλι Βιτούσκιν Χέρμαν Γκράσμαν Όλγα Όλεϊνικ Ζαν – Κριστόφ Γιόκοζ 2 Συνέντευξη του Βλαντιμίρ Ιγκορέβιτς Άρνολντ ► Βλαντιμίρ Ιγκορέβιτς Άρνολντ Vladimir Igorevich Arnold Ο Άρνολντ ήταν σπουδαίος μαθηματικός ρωσικής καταγωγής. Γεννήθηκε στις 12 Ιουνίου του 1937 στην Οδυσσό της Ουκρανίας και πέθανε στις 3 Ιουνίου του 2010, σε ηλικία 72 ετών, στο Παρίσι της Γαλλίας. Ο Άρνολντ αποφοίτησε από το Πανεπιστήμιο της Μόσχας το 1959, όπου ήταν μαθητής του Αντρέι Κολμογκόροφ. Στη συνέχεια εργάστηκε εκεί μέχρι το 1986 (ως καθηγητής από το 1965) και μετά εργάστηκε στο Ινστιντούτο Στέκλοβ (Steklov) της Μόσχας. -
Society Reports USNC/TAM
Appendix J 2008 Society Reports USNC/TAM Table of Contents J.1 AAM: Ravi-Chandar.............................................................................................. 1 J.2 AIAA: Chen............................................................................................................. 2 J.3 AIChE: Higdon ....................................................................................................... 3 J.4 AMS: Kinderlehrer................................................................................................. 5 J.5 APS: Foss................................................................................................................. 5 J.6 ASA: Norris............................................................................................................. 6 J.7 ASCE: Iwan............................................................................................................. 7 J.8 ASME: Kyriakides.................................................................................................. 8 J.9 ASTM: Chona ......................................................................................................... 9 J.10 SEM: Shukla ....................................................................................................... 11 J.11 SES: Jasiuk.......................................................................................................... 13 J.12 SIAM: Healey...................................................................................................... 14 J.13 SNAME: Karr.................................................................................................... -
On a Generalization of the Hadwiger-Nelson Problem
On a generalization of the Hadwiger-Nelson problem Mohammad Bardestani, Keivan Mallahi-Karai October 15, 2018 Abstract For a field F and a quadratic form Q defined on an n-dimensional vector space V over F , let QGQ, called the quadratic graph associated to Q, be the graph with the vertex set V where vertices u; w 2 V form an edge if and only if Q(v − w) = 1. Quadratic graphs can be viewed as natural generalizations of the unit-distance graph featuring in the famous Hadwiger-Nelson problem. In the present paper, we will prove that for a local field F of characteristic zero, the Borel chromatic number of QGQ is infinite if and only if Q represents zero non-trivially over F . The proof employs a recent spectral bound for the Borel chromatic number of Cayley graphs, combined with an analysis of certain oscillatory integrals over local fields. As an application, we will also answer a variant of question 525 proposed in the 22nd British Combinatorics Conference 2009 [6]. 1 Introduction The celebrated Hadwiger-Nelson problem asks for the minimum number of colors required to color n R such that no two points at distance one from each other have the same color. Recall that the chromatic number of a graph G, denoted by χ(G), is the least cardinal c such that the vertices of G can be partitioned into c sets (called color classes) such that no color class contains an edge in G. Hence, the Hadwiger-Nelson problem is the question of finding χ(Gn), where Gn is the graph with vertex set n V (Gn) = R , where the adjacency of vertices x; y 2 V (Gn) is defined by the equation Q(x − y) = 1; 2 2 n here, Q(x1; : : : ; xn) = x1 + ··· + xn is the canonical positive-definite quadratic form on R .