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The Quest for Pi David H. Bailey, Jonathan M. Borwein, Peter B
The Quest for Pi David H. Bailey, Jonathan M. Borwein, Peter B. Borwein and Simon Plouffe June 25, 1996 Ref: Mathematical Intelligencer, vol. 19, no. 1 (Jan. 1997), pg. 50–57 Abstract This article gives a brief history of the analysis and computation of the mathematical constant π =3.14159 ..., including a number of the formulas that have been used to compute π through the ages. Recent developments in this area are then discussed in some detail, including the recent computation of π to over six billion decimal digits using high-order convergent algorithms, and a newly discovered scheme that permits arbitrary individual hexadecimal digits of π to be computed. D. Bailey: NASA Ames Research Center, Mail Stop T27A-1, Moffett Field, CA 94035-1000. E-mail: [email protected]. J. Borwein: Dept. of Mathematics and Statistics, Simon Fraser University Burnaby, BC V5A 1S6 Canada. Email: [email protected]. This work was supported by NSERC and the Shrum Endowment at Simon Fraser University. P. Borwein: Dept. of Mathematics and Statistics, Simon Fraser University Burnaby, BC V5A 1S6 Canada. Email: [email protected]. S. Plouffe: Dept. of Mathematics and Statistics, Simon Fraser University Burnaby, BC V5A 1S6 Canada. Email: [email protected]. 1 Introduction The fascinating history of the constant we now know as π spans several millennia, almost from the beginning of recorded history up to the present day. In many ways this history parallels the advancement of science and technology in general, and of mathematics and computer technology in particular. An overview of this history is presented here in sections one and two. -
Short CV For: Alexander Lubotzky
Short CV for: Alexander Lubotzky Personal: • born 28/6/56 in Israel. • Married to Yardenna Lubotzky (+ six children) Studies: • B. Sc., Mathematics, Bar-Ilan University, 1975. • Ph.D., Mathematics, Bar-Ilan University, 1979. (Supervisor: H. Fussten- berg, Thesis: Profinite groups and the congruence subgroup problem.) Employment: • 1982 - current: Institute of Mathematics, Hebrew University of Jerusalem; Professor - Holding the Maurice and Clara Weil Chair in Mathematics • 1999-current: Adjunct Professor at Yale University • Academic Year 2005-2006: Leading a year long research program at the Institute for Advanced Study in Princeton on \Lie Groups, Repre- sentations and Discrete Mathematics." Previous Employment: • Bar-Ilan University, 1976-1982 • Israeli Defense Forces, 1977-1982 • Member of the Israeli Parliament (Knesset), 1996-1999 Visiting Positions: • Yale University (several times for semesters or years) • Stanford University (84/5) • University of Chicago (92/3) • Columbia University (Elenberg visiting Professor Fall 2000) 1 • Institute for Advanced Study, Princeton (2005/6) Main prizes and Academic Honors: • Elected as Foreign Honorary member of the American Academy of Arts and Sciences • Ferran Sunyer i Balaguer Prize twice: 1993 for the book: \Discrete Groups, Expanding Groups and Invariant Measures", Prog. in math 125, Birkhauser 1994, and in 2002 joint with Professor Dan Segal from Oxford for the book \Subgroup Growth", Prog. in Math. 212, Birkhauser 2003. • The Rothschild Prize 2002. • The Erdos Prize in 1991. Editorial work: • Israel Journal of Mathematics (1990-now) • Journal of Algebra (1990-2005) • GAFA (1990-2000) • European Journal of Combinatorics • Geometric Dedicata • Journal of the Glasgow Mathematical Scientists Books and papers: • Author of 3 books and over 90 papers. -
Arxiv:2107.06030V2 [Math.HO] 18 Jul 2021 Jonathan Michael Borwein
Jonathan Michael Borwein 1951 { 2016: Life and Legacy Richard P. Brent∗ Abstract Jonathan M. Borwein (1951{2016) was a prolific mathematician whose career spanned several countries (UK, Canada, USA, Australia) and whose many interests included analysis, optimisation, number theory, special functions, experimental mathematics, mathematical finance, mathematical education, and visualisation. We describe his life and legacy, and give an annotated bibliography of some of his most significant books and papers. 1 Life and Family Jonathan (Jon) Michael Borwein was born in St Andrews, Scotland, on 20 May 1951. He was the first of three children of David Borwein (1924{2021) and Bessie Borwein (n´eeFlax). It was an itinerant academic family. Both Jon's father David and his younger brother Peter Borwein (1953{2020) are well-known mathematicians and occasional co-authors of Jon. His mother Bessie is a former professor of anatomy. The Borweins have an Ashkenazy Jewish background. Jon's father was born in Lithuania, moved in 1930 with arXiv:2107.06030v4 [math.HO] 15 Sep 2021 his family to South Africa (where he met his future wife Bessie), and moved with Bessie to the UK in 1948. There he obtained a PhD (London) and then a Lectureship in St Andrews, Scotland, where Jon was born and went to school at Madras College. The family, including Jon and his two siblings, moved to Ontario, Canada, in 1963. In 1971 Jon graduated with a BA (Hons ∗Mathematical Sciences Institute, Australian National University, Canberra, ACT. Email: <[email protected]>. 1 Math) from the University of Western Ontario. It was in Ontario that Jon met his future wife and lifelong partner Judith (n´eeRoots). -
And Should Not, Change Toxic Tort Causation Doctrine
THE MORE WE KNOW, THE LESS INTELLIGENT WE ARE? - HOW GENOMIC INFORMATION SHOULD, AND SHOULD NOT, CHANGE TOXIC TORT CAUSATION DOCTRINE Steve C. Gold* Advances in genomic science are rapidly increasing our understanding of dis- ease and toxicity at the most fundamental biological level. Some say this heralds a new era of certainty in linking toxic substances to human illness in the courtroom. Others are skeptical. In this Article I argue that the new sciences of molecular epidemiology and toxicogenomics will evince both remarkable explanatory power and intractable complexity. Therefore, even when these sciences are brought to bear, toxic tort claims will continue to present their familiar jurisprudential problems. Nevertheless, as genomic information is used in toxic tort cases, the sci- entific developments will offer courts an opportunity to correct mistakes of the past. Courts will miss those opportunities, however, if they simply transfer attitudesfrom classical epidemiology and toxicology to molecular and genomic knowledge. Using the possible link between trichloroethyleneand a type of kidney cancer as an exam- ple, I describe several causation issues where courts can, if they choose, improve doctrine by properly understanding and utilizing genomic information. L Introduction ............................................... 370 II. Causation in Toxic Torts Before Genomics ................... 371 A. The Problem .......................................... 371 B. Classical Sources of Causation Evidence ................ 372 C. The Courts Embrace Epidemiology - Perhaps Too Tightly ................................................ 374 D. Evidence Meets Substance: Keepers of the Gate ......... 379 E. The Opposing Tendency ................................ 382 III. Genomics, Toxicogenomics, and Molecular Epidemiology: The M ore We Know ........................................ 383 A. Genetic Variability and "Background" Risk .............. 384 B. Genetic Variability and Exposure Risk ................... 389 C. -
Peter Borwein Professor and Burnaby Mountain Chair, Executive Director
Peter Borwein Professor and Burnaby Mountain Chair, Executive Director IRMACS (Interdisciplinary Research in the Mathematical and Computational Sciences) Simon Fraser University, Vancouver, B.C. DEGREES B.Sc. University of Western Ontario, Mathematics, 1974 M.Sc. University of British Columbia, Mathematics, 1976 Ph.D. University of British Columbia, Mathematics, 1979 He has authored six books and over a 150 research articles. His research interests span Diophantine and computational number theory, classical analysis and symbolic computation. He has a central interest in scientific collaboration and computational experimentation technologies. He is recipient of the Chauvenet Prize and the Hasse prize 1993 (with J. Borwein and D. Bailey); the 1996 CUFA/BC Academic of the Year (co-recipient); the University of Western Ontario National Alumni Merit Award 1999; the Ford Prize 2002 (with L. Jorgensen) He is nominated for the $100000 Edge of Computation Science Prize (with D.H. Bailey and S. Plouffe) for their work on the so called BBP algorithm He is Executive Director for the initiative in Interdisciplinary Research in the Mathematical and Computational Sciences (IRMACS). This is a major initiative funded by CFI, BCKDF and SFU. The IRMACS Centre is a unique, interdisciplinary research facility that enables collaborative interaction - intellectually, physically and virtually. It provides a versatile, computationally sophisticated infrastructure for nearly 200 scientists whose primary laboratory tool is the computer. See http://www.irmacs.ca/. He is also a principal investigator of a MITACS consortium MOCAA in the Mathematics of Computer Algebra and Analysis. This involves overseeing a national team of researchers, graduate students, programmers and post-docs. This project has as its major industrial sponsor Maple Inc. -
A Group Theoretic Characterization of Linear Groups
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 113, 207-214 (1988) A Group Theoretic Characterization of Linear Groups ALEXANDER LUBOTZKY Institute qf Mathematics, Hebrew University, Jerusalem, Israel 91904 Communicated by Jacques Tits Received May 12, 1986 Let r be a group. When is f linear? This is an old problem. The first to study this question systematically was Malcev in 1940 [M], who essen- tially reduced the problem to finitely generated groups. (Note that a finitely generated group is linear over some field of characteristic zero if and only if it can be embedded in CL,(C) for some n.) Very little progress was made since that paper of Malcev, although, as linear groups are a quite special type of group, many necessary conditions were obtained, e.g., r should be residually finite and even virtually residually-p for almost all primes p, f should be virtually torsion free, and if not solvable by finite it has a free non-abelian subgroup, etc. (cf. [Z]). Of course, none of these properties characterizes the finitely generated linear groups over @. In this paper we give such a characterization using the congruence structure of r. First some definitions: For a group H, d(H) denotes the minimal number of generators for H. DEFINITION. Let p be a prime and c an integer. A p-congruence structure (with a bound c) for a group r is a descending chain of finite index normal subgroups of r = N, 2 N, 2 N, z . -
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. -
Counting Arithmetic Lattices and Surfaces
ANNALS OF MATHEMATICS Counting arithmetic lattices and surfaces By Mikhail Belolipetsky, Tsachik Gelander, Alexander Lubotzky, and Aner Shalev SECOND SERIES, VOL. 172, NO. 3 November, 2010 anmaah Annals of Mathematics, 172 (2010), 2197–2221 Counting arithmetic lattices and surfaces By MIKHAIL BELOLIPETSKY, TSACHIK GELANDER, ALEXANDER LUBOTZKY, and ANER SHALEV Abstract We give estimates on the number ALH .x/ of conjugacy classes of arithmetic lattices of covolume at most x in a simple Lie group H . In particular, we obtain a first concrete estimate on the number of arithmetic 3-manifolds of volume at most x. Our main result is for the classical case H PSL.2; R/ where we show D that log AL .x/ 1 lim H : x x log x D 2 !1 The proofs use several different techniques: geometric (bounding the number of generators of as a function of its covolume), number theoretic (bounding the number of maximal such ) and sharp estimates on the character values of the symmetric groups (to bound the subgroup growth of ). 1. Introduction Let H be a noncompact simple Lie group with a fixed Haar measure .A discrete subgroup of H is called a lattice if . H / < . A classical theorem of n 1 Wang[Wan72] asserts that if H is not locally isomorphic to PSL2.R/ or PSL2.C/, then for every 0 < x R the number LH .x/ of conjugacy classes of lattices in H of 2 covolume at most x is finite. This result was greatly extended by Borel and Prasad [BP89]. In recent years there has been an attempt to quantify Wang’s theorem and to give some estimates on LH .x/ (see[BGLM02],[Gel04],[GLNP04],[Bel07] and[BL]). -
IMU Secretary An: [email protected]; CC: Betreff: IMU EC CL 05/07: Vote on ICMI Terms of Reference Change Datum: Mittwoch, 24
Appendix 10.1.1 Von: IMU Secretary An: [email protected]; CC: Betreff: IMU EC CL 05/07: vote on ICMI terms of reference change Datum: Mittwoch, 24. Januar 2007 11:42:40 Anlagen: To the IMU 2007-2010 Executive Committee Dear colleagues, We are currently experimenting with a groupware system that may help us organize the files that every EC member should know and improve the voting processes. Wolfgang Dalitz has checked the open source groupware systems and selected one that we want to try. It is more complicated than we thought and does have some deficiencies, but we see no freeware that is better. Here is our test run with a vote on a change of the ICMI terms of reference. To get to our voting system click on http://www.mathunion.org/ec-only/ To log in, you have to type your last name in the following version: ball, baouendi, deleon, groetschel, lovasz, ma, piene, procesi, vassiliev, viana Right now, everybody has the same password: pw123 You will immediately get to the summary page which contains an item "New Polls". The question to vote on is: Vote-070124: Change of ICMI terms of reference, #3, see Files->Voting->Vote- 070124 for full information and you are supposed to agree, disagree or abstain by clicking on the corresponding button. Full information about the contents of the vote is documented in the directory Voting (click on the +) where you will find a file Vote-070124.txt (click on the "txt icon" to see the contents of the file). The file is also enclosed below for your information. -
Precision Medicine Initiative: Building a Large US Research Cohort
Precision Medicine Initiative: Building a Large U.S. Research Cohort February 11-12, 2015 PARTICIPANT LIST Goncalo Abecasis, D. Phil. Philip Bourne, Ph.D. Professor of Biostatistics Associate Director for Data Science University of Michigan, Ann Arbor Office of the Director National Institutes of Health Christopher Austin, M.D. Director Murray Brilliant, Ph.D. National Center for Advancing Translational Sciences Director National Institutes of Health Center for Human Genetics Marshfield Clinic Research Foundation Vikram Bajaj, Ph.D. Chief Scientist Greg Burke, M.D., M.Sc. Google Life Sciences Professor and Director Wake Forest School of Medicine Dixie Baker, Ph.D. Wake Forest University Senior Partner Martin, Blanck and Associates Antonia Calafat, Ph.D. Chief Dana Boyd Barr, Ph.D. Organic Analytical Toxicology Branch Professor, Exposure Science and Environmental Health Centers for Disease Control and Prevention Rollins School of Public Health Emory University Robert Califf, M.D. Vice Chancellor for Clinical and Translational Research Jonathan Bingham, M.B.A. Duke University Medical Center Product Manager, Genomics Google, Inc. Rex Chisholm, Ph.D. Adam and Richard T. Lind Professor of Medical Eric Boerwinkle, Ph.D. Genetics Professor and Chair Vice Dean for Scientific Affairs and Graduate Studies Human Genetics Center Associate Vice President for Research University of Texas Health Science Center Northwestern University Associate Director Human Genome Sequencing Center Rick Cnossen, M.S. Baylor College of Medicine Director Global Healthcare Solutions Erwin Bottinger, M.D. HIMSS Board of Directors Professor PCHA/Continua Health Alliance Board of Directors The Charles Bronfman Institute for Personalized Intel Corporation Medicine Icahn School of Medicine at Mount Sinai - 1 - Francis Collins, M.D., Ph.D. -
Graduate School of Arts and Sciences 2013–2014
BULLETIN OF YALE UNIVERSITY BULLETIN OF YALE BULLETIN OF YALE UNIVERSITY Periodicals postage paid New Haven ct 06520-8227 New Haven, Connecticut Graduate School of Arts and Sciences Programs and Policies 2013–2014 Graduate School ofGraduate Arts and Sciences 2013–2014 BULLETIN OF YALE UNIVERSITY Series 109 Number 5 July 15, 2013 BULLETIN OF YALE UNIVERSITY Series 109 Number 5 July 15, 2013 (USPS 078-500) The University is committed to basing judgments concerning the admission, education, is published seventeen times a year (one time in May and October; three times in June and employment of individuals upon their qualifications and abilities and a∞rmatively and September; four times in July; five times in August) by Yale University, 2 Whitney seeks to attract to its faculty, sta≠, and student body qualified persons of diverse back- Avenue, New Haven CT 0651o. Periodicals postage paid at New Haven, Connecticut. grounds. In accordance with this policy and as delineated by federal and Connecticut law, Yale does not discriminate in admissions, educational programs, or employment against Postmaster: Send address changes to Bulletin of Yale University, any individual on account of that individual’s sex, race, color, religion, age, disability, or PO Box 208227, New Haven CT 06520-8227 national or ethnic origin; nor does Yale discriminate on the basis of sexual orientation or gender identity or expression. Managing Editor: Kimberly M. Go≠-Crews University policy is committed to a∞rmative action under law in employment of Editor: Lesley K. Baier women, minority group members, individuals with disabilities, and covered veterans. PO Box 208230, New Haven CT 06520-8230 Inquiries concerning these policies may be referred to the Director of the O∞ce for Equal Opportunity Programs, 221 Whitney Avenue, 203.432.0849. -
Count Down: Six Kids Vie for Glory at the World's Toughest Math
Count Down Six Kids Vie for Glory | at the World's TOUGHEST MATH COMPETITION STEVE OLSON author of MAPPING HUMAN HISTORY, National Book Award finalist $Z4- 00 ACH SUMMER SIX MATH WHIZZES selected from nearly a half million EAmerican teens compete against the world's best problem solvers at the Interna• tional Mathematical Olympiad. Steve Olson, whose Mapping Human History was a Na• tional Book Award finalist, follows the members of a U.S. team from their intense tryouts to the Olympiad's nail-biting final rounds to discover not only what drives these extraordinary kids but what makes them both unique and typical. In the process he provides fascinating insights into the creative process, human intelligence and learning, and the nature of genius. Brilliant, but defying all the math-nerd stereotypes, these athletes of the mind want to excel at whatever piques their cu• riosity, and they are curious about almost everything — music, games, politics, sports, literature. One team member is ardent about water polo and creative writing. An• other plays four musical instruments. For fun and entertainment during breaks, the Olympians invent games of mind-boggling difficulty. Though driven by the glory of winning this ultimate math contest, in many ways these kids are not so different from other teenagers, finding pure joy in indulging their personal passions. Beyond the Olympiad, Steve Olson sheds light on such questions as why Americans feel so queasy about math, why so few girls compete in the subject, and whether or not talent is innate. Inside the cavernous gym where the competition takes place, Count Down reveals a fascinating subculture and its engaging, driven inhabitants.