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Noncommutative Uniform Algebras
Noncommutative Uniform Algebras Mati Abel and Krzysztof Jarosz Abstract. We show that a real Banach algebra A such that a2 = a 2 , for a A is a subalgebra of the algebra C (X) of continuous quaternion valued functions on a compact setk kX. ∈ H ° ° ° ° 1. Introduction A well known Hirschfeld-Zelazkoú Theorem [5](seealso[7]) states that for a complex Banach algebra A the condition (1.1) a2 = a 2 , for a A k k ∈ implies that (i) A is commutative, and further° ° implies that (ii) A must be of a very speciÞc form, namely it ° ° must be isometrically isomorphic with a uniformly closed subalgebra of CC (X). We denote here by CC (X)the complex Banach algebra of all continuous functions on a compact set X. The obvious question about the validity of the same conclusion for real Banach algebras is immediately dismissed with an obvious counterexample: the non commutative algebra H of quaternions. However it turns out that the second part (ii) above is essentially also true in the real case. We show that the condition (1.1) implies, for a real Banach algebra A,thatA is isometrically isomorphic with a subalgebra of CH (X) - the algebra of continuous quaternion valued functions on acompactsetX. That result is in fact a consequence of a theorem by Aupetit and Zemanek [1]. We will also present several related results and corollaries valid for real and complex Banach and topological algebras. To simplify the notation we assume that the algebras under consideration have units, however, like in the case of the Hirschfeld-Zelazkoú Theorem, the analogous results can be stated for none unital algebras as one can formally add aunittosuchanalgebra. -
Manjul Bhargava
The Work of Manjul Bhargava Manjul Bhargava's work in number theory has had a profound influence on the field. A mathematician of extraordinary creativity, he has a taste for simple problems of timeless beauty, which he has solved by developing elegant and powerful new methods that offer deep insights. When he was a graduate student, Bhargava read the monumental Disqui- sitiones Arithmeticae, a book about number theory by Carl Friedrich Gauss (1777-1855). All mathematicians know of the Disquisitiones, but few have actually read it, as its notation and computational nature make it difficult for modern readers to follow. Bhargava nevertheless found the book to be a wellspring of inspiration. Gauss was interested in binary quadratic forms, which are polynomials ax2 +bxy +cy2, where a, b, and c are integers. In the Disquisitiones, Gauss developed his ingenious composition law, which gives a method for composing two binary quadratic forms to obtain a third one. This law became, and remains, a central tool in algebraic number theory. After wading through the 20 pages of Gauss's calculations culminating in the composition law, Bhargava knew there had to be a better way. Then one day, while playing with a Rubik's cube, he found it. Bhargava thought about labeling each corner of a cube with a number and then slic- ing the cube to obtain 2 sets of 4 numbers. Each 4-number set naturally forms a matrix. A simple calculation with these matrices resulted in a bi- nary quadratic form. From the three ways of slicing the cube, three binary quadratic forms emerged. -
The Politics of Parthian Coinage in Media
The Politics of Parthian Coinage in Media Author(s): Farhang Khademi Nadooshan, Seyed Sadrudin Moosavi, Frouzandeh Jafarzadeh Pour Reviewed work(s): Source: Near Eastern Archaeology, Vol. 68, No. 3, Archaeology in Iran (Sep., 2005), pp. 123-127 Published by: The American Schools of Oriental Research Stable URL: http://www.jstor.org/stable/25067611 . Accessed: 06/11/2011 07:31 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. The American Schools of Oriental Research is collaborating with JSTOR to digitize, preserve and extend access to Near Eastern Archaeology. http://www.jstor.org The Parthians (174 BCE-224CE) suc- , The coins discussed here are primarily from ceeded in the the Lorestan Museum, which houses the establishing longest jyj^' in the ancient coins of southern Media.1 However, lasting empire J0^%^ 1 Near East.At its Parthian JF the coins of northern Media are also height, ^S^ considered thanks to the collection ruleextended Anatolia to M from ^^^/;. housed in the Azerbaijan Museum theIndus and the Valley from Ef-'?S&f?'''' in the city of Tabriz. Most of the Sea to the Persian m Caspian ^^^/// coins of the Azerbaijan Museum Farhang Khademi Gulf Consummate horsemen el /?/ have been donated by local ^^ i Nadooshan, Seyed indigenoustoCentral Asia, the ? people and have been reported ?| ?????J SadrudinMoosavi, Parthians achieved fame for Is u1 and documented in their names. -
Regularity Conditions for Banach Function Algebras
Regularity conditions for Banach function algebras Dr J. F. Feinstein University of Nottingham June 2009 1 1 Useful sources A very useful text for the material in this mini-course is the book Banach Algebras and Automatic Continuity by H. Garth Dales, London Mathematical Society Monographs, New Series, Volume 24, The Clarendon Press, Oxford, 2000. In particular, many of the examples and conditions discussed here may be found in Chapter 4 of that book. We shall refer to this book throughout as the book of Dales. Most of my e-prints are available from www.maths.nott.ac.uk/personal/jff/Papers Several of my research and teaching presentations are available from www.maths.nott.ac.uk/personal/jff/Beamer 2 2 Introduction to normed algebras and Banach algebras 2.1 Some problems to think about Those who have seen much of this introductory material before may wish to think about some of the following problems. We shall return to these problems at suitable points in this course. Problem 2.1.1 (Easy using standard theory!) It is standard that the set of all rational functions (quotients of polynomials) with complex coefficients is a field: this is a special case of the “field of fractions" of an integral domain. Question: Is there an algebra norm on this field (regarded as an algebra over C)? 3 Problem 2.1.2 (Very hard!) Does there exist a pair of sequences (λn), (an) of non-zero complex numbers such that (i) no two of the an are equal, P1 (ii) n=1 jλnj < 1, (iii) janj < 2 for all n 2 N, and yet, (iv) for all z 2 C, 1 X λn exp (anz) = 0? n=1 Gap to fill in 4 Problem 2.1.3 Denote by C[0; 1] the \trivial" uniform algebra of all continuous, complex-valued functions on [0; 1]. -
ICTS POSTER Outside Bangalore
T A T A I N S T I T U T E O F F U N D A M E N T A L R E S E A R C H A HOMI BHABHA BIRTH CENTENARY & ICTS INAUGURAL EVENT International Centre Theoretical Sciences science without bo28 Decemberun 2009d29 -a 31 Decemberri e2009s Satish Dhawan Auditorium Faculty Hall Indian Institute of Science, Bangalore. www.icts.res.in/program/icts-ie INVITED SPEAKERS / PANELISTS INCLUDE FOUNDATION STONE CEREMONY Siva Athreya ISI, Bangalore OF ICTS CAMPUS Naama Barkai Weizmann Institute The foundation stone will be unveiled by Manjul Bhargava Princeton University Prof. C N R Rao, FRS 4:00 pm, 28 December 2009 Édouard Brézin École Normale Supérieure Amol Dighe TIFR Michael Green DAMTP, Cambridge Chandrashekhar Khare UCLA Yamuna Krishnan NCBS-TIFR Lyman Page Princeton University Jaikumar Radhakrishnan TIFR C. S. Rajan TIFR Sriram Ramaswamy IISc G. Rangarajan IISc C. N. R. Rao JNCASR Subir Sachdev Harvard University K. Sandeep CAM-TIFR Sriram Shastry UC Santa Cruz PUBLIC LECTURES Ashoke Sen HRI J. N. Tata Auditorium, IISc (FREE AND OPEN TO ALL) Anirvan Sengupta Rutgers University K. R. Sreenivasan Abdus Salam ICTP Michael Atiyah University of Edinburgh Andrew Strominger Harvard University Truth and Beauty in Mathematics and Physics 5:30 pm, 27 December 2009 Raman Sundrum Johns Hopkins University Ajay Sood IISc David Gross KITP, Santa Barbara The Role of Theory in Science Tarun Souradeep IUCAA 5:30 pm, 28 December 2009 Eitan Tadmor University of Maryland Albert Libchaber Rockefeller University Sandip Trivedi TIFR The Origin of Life: from Geophysics to Biology? Mukund Thattai NCBS-TIFR 5:30 pm, 30 December 2009 S. -
From Small States to Universalism in the Pre-Islamic Near East
REVOLUTIONIZING REVOLUTIONIZING Mark Altaweel and Andrea Squitieri and Andrea Mark Altaweel From Small States to Universalism in the Pre-Islamic Near East This book investigates the long-term continuity of large-scale states and empires, and its effect on the Near East’s social fabric, including the fundamental changes that occurred to major social institutions. Its geographical coverage spans, from east to west, modern- day Libya and Egypt to Central Asia, and from north to south, Anatolia to southern Arabia, incorporating modern-day Oman and Yemen. Its temporal coverage spans from the late eighth century BCE to the seventh century CE during the rise of Islam and collapse of the Sasanian Empire. The authors argue that the persistence of large states and empires starting in the eighth/ seventh centuries BCE, which continued for many centuries, led to new socio-political structures and institutions emerging in the Near East. The primary processes that enabled this emergence were large-scale and long-distance movements, or population migrations. These patterns of social developments are analysed under different aspects: settlement patterns, urban structure, material culture, trade, governance, language spread and religion, all pointing at population movement as the main catalyst for social change. This book’s argument Mark Altaweel is framed within a larger theoretical framework termed as ‘universalism’, a theory that explains WORLD A many of the social transformations that happened to societies in the Near East, starting from Andrea Squitieri the Neo-Assyrian period and continuing for centuries. Among other infl uences, the effects of these transformations are today manifested in modern languages, concepts of government, universal religions and monetized and globalized economies. -
Uniform Algebras As Banach Spaces
The Erwin Schrodinger International Boltzmanngasse ESI Institute for Mathematical Physics A Wien Austria Uniform Algebras as Banach Spaces TW Gamelin SV Kislyakov Vienna Preprint ESI June Supp orted by Federal Ministry of Science and Transp ort Austria Available via httpwwwesiacat UNIFORM ALGEBRAS AS BANACH SPACES TW Gamelin SV Kislyakov June Abstract Any Banach space can b e realized as a direct summand of a uniform algebra and one do es not exp ect an arbitrary uniform algebra to have an abundance of prop erties not common to all Banach spaces One general result concerning arbitrary uniform algebras is that no prop er uniform algebra is linearly homeomorphic to a C K space Nevertheless many sp ecic uniform algebras arising in complex analysis share or are susp ected to share certain Banach space prop erties of C K We discuss the family of tight algebras which includes algebras of analytic functions on strictly pseudo con vex domains and algebras asso ciated with rational approximation theory in the plane Tight algebras are in some sense close to C K spaces and along with C K spaces they have the Pelczynski and the DunfordPettis prop erties We also fo cus on certain prop erties of C K spaces that are inherited by the disk algebra This includes a dis p cussion of interp olation b etween H spaces and Bourgains extension of Grothendiecks theorem to the disk algebra We conclude with a brief description of linear deformations of uniform algebras and a brief survey of the known classication results Supp orted in part by the Russian -
1. Introduction
FACTORIZATION IN COMMUTATIVE BANACH ALGEBRAS H. G. DALES, J. F. FEINSTEIN, AND H. L. PHAM Abstract. Let A be a (non-unital) commutative Banach algebra. We consider when A has a variety of factorization properties: we list the (ob- vious) implications between these properties, and then consider whether any of these implications can be reversed in various classes of commu- tative Banach algebras. We summarize the known counter-examples to these possible reverse implications, and add further counter-examples. Some results are used to show the existence of a large family of prime ideals in each non-zero, commutative, radical Banach algebra with a dense set of products. 1. Introduction Let A be a (non-unital) commutative Banach algebra. We wish to examine when A factors in a variety of senses. Our main results are counter-examples to a number of questions that have been raised. Indeed, we shall list seven such factorization properties, called (I)(VII), and note that each of these immediately implies the next one. We shall also, in x4, discuss two other `lo- cal' factorization properties, called (A) and (B) (where (A) ) (B)); these properties are relevant for Esterle's classication of commutative, radical Banach algebras that is given in [14]. We shall then discuss whether or not any of these implications can be reversed when we restrict attention to par- ticular classes of commutative Banach algebras. We shall show that several cannot be reversed, but we leave open other possible reverse implications. A summary in x6 describes our knowledge at the present time. We shall concentrate on two particular classes of commutative Banach algebras A: rst, on the case where A is semi-simple (so that A is a Banach function algebra), and in particular when A is a maximal ideal in a uniform algebra on a compact space, and, second, on the other extreme case where A 2010 Mathematics Subject Classication. -
Dear Aspirant with Regard
DEAR ASPIRANT HERE WE ARE PRESENTING YOU A GENRAL AWERNESS MEGA CAPSULE FOR IBPS PO, SBI ASSOT PO , IBPS ASST AND OTHER FORTHCOMING EXAMS WE HAVE UNDERTAKEN ALL THE POSSIBLE CARE TO MAKE IT ERROR FREE SPECIAL THANKS TO THOSE WHO HAS PUT THEIR TIME TO MAKE THIS HAPPEN A IN ON LIMITED RESOURCE 1. NILOFAR 2. SWETA KHARE 3. ANKITA 4. PALLAVI BONIA 5. AMAR DAS 6. SARATH ANNAMETI 7. MAYANK BANSAL WITH REGARD PANKAJ KUMAR ( Glory At Anycost ) WE WISH YOU A BEST OF LUCK CONTENTS 1 CURRENT RATES 1 2 IMPORTANT DAYS 3 CUPS & TROPHIES 4 4 LIST OF WORLD COUNTRIES & THEIR CAPITAL 5 5 IMPORTANT CURRENCIES 9 6 ABBREVIATIONS IN NEWS 7 LISTS OF NEW UNION COUNCIL OF MINISTERS & PORTFOLIOS 13 8 NEW APPOINTMENTS 13 9 BANK PUNCHLINES 15 10 IMPORTANT POINTS OF UNION BUDGET 2012-14 16 11 BANKING TERMS 19 12 AWARDS 35 13 IMPORTANT BANKING ABBREVIATIONS 42 14 IMPORTANT BANKING TERMINOLOGY 50 15 HIGHLIGHTS OF UNION BUDGET 2014 55 16 FDI LLIMITS 56 17 INDIAS GDP FORCASTS 57 18 INDIAN RANKING IN DIFFERENT INDEXS 57 19 ABOUT : NABARD 58 20 IMPORTANT COMMITTEES IN NEWS 58 21 OSCAR AWARD 2014 59 22 STATES, CAPITAL, GOVERNERS & CHIEF MINISTERS 62 23 IMPORTANT COMMITTEES IN NEWS 62 23 LIST OF IMPORTANT ORGANIZATIONS INDIA & THERE HEAD 65 24 LIST OF INTERNATIONAL ORGANIZATIONS AND HEADS 66 25 FACTS ABOUT CENSUS 2011 66 26 DEFENCE & TECHNOLOGY 67 27 BOOKS & AUTHOURS 69 28 LEADER”S VISITED INIDIA 70 29 OBITUARY 71 30 ORGANISATION AND THERE HEADQUARTERS 72 31 REVOLUTIONS IN AGRICULTURE IN INDIA 72 32 IMPORTANT DAMS IN INDIA 73 33 CLASSICAL DANCES IN INDIA 73 34 NUCLEAR POWER -
Weinan E Princteon University, USA [email protected]
Weinan E Princteon University, USA [email protected] Weinan E received his Ph.D. from UCLA in 1989. After being a visiting member at the Courant Institute of NYU and the Institute for Advanced Study at Princeton, he joined the faculty at NYU in 1994. He is now a professor of mathematics at Princeton University, a position he has held since 1999. He is also a Changjiang professor at the Peking University. Weinan E’s research interest is in multiscale and stochastic modeling. His work covers issues that include mathematical foundation of stochastic and multiscale modeling, design and analysis of of algorithms and applications to problems in various disciplines of science and engineering, with particular emphasis on the modeling of rare events, material sciences and fluid dynamics. Weinan E is the recipient of the Presidential Early Career Awards for Scientists and Engineers (PECASE), the Feng Kang Prize, the SIAM R. E. Kleinman Prize, and the ICIAM Collatz Prize. He is a member of the Chinese Academy of Sciences, a fellow of the American Mathematical Society, a SIAM fellow and a fellow of the Institute of Physics. Mathematical theory of solids: From quantum mechanics to continuum models Abstract The problem of trying to understand solids from microscopic and macroscopic viewpoints goes back to Cauchy and was taken up again by Born. In this talk, I will present the progress that has been made on this Cauchy-Born program. In particular, I will discuss how macroscopic continuum models of solids can be derived from models of quantum mechanics and molecular dynamics. -
WEINAN E Education Ph.D. Mathematics UCLA 1989 M.S
WEINAN E Department of Mathematics and Program in Applied and Computational Mathematics Princeton University, Princeton, NJ 08544 Phone: (609) 258-3683 Fax: (609) 258-1735 [email protected] Education Ph.D. Mathematics UCLA 1989 M.S. Mathematics Chinese Academy of Sciences 1985 B.S. Mathematics University of Science and Technology of China 1982 Academic Positions 9/99- Professor, Department of Mathematics and PACM, Associate Faculty Member of the Department of Operational Research and Financial Engineering, Princeton University 8/15- Director, Beijing Institute of Big Data Research 9/00-8/19 Changjiang Visiting Professor, Peking University 5/14-8/18 Dean, Yuanpei College, Peking University 9/97-8/99 Professor, Courant Institute, New York University 9/94-8/97 Associate Professor, Courant Institute, New York University 9/92-8/94 Long Term Member, Institute for Advanced Study, Princeton 9/91-8/92 Member, Institute for Advanced Study, Princeton 9/89-8/91 Visiting Member, Courant Institute, New York University Awards and Honors 1993 Alfred P. Sloan Foundation Fellowship 1996 Presidential Early Career Award in Science and Engineering 1999 Feng Kang Prize in Scientific Computing 2003 ICIAM Collatz Prize, awarded by the 5th International Council of of Industrial & Applied Math. 2005 Elected Fellow of Institute of Physics 2009 Elected Fellow of Society of Industrial and Applied Mathematics 2009 The Ralph E. Kleinman Prize, Society of Industrial and Applied Mathematics 2011 Elected member of the Chinese Academy of Sciences 2012 Elected Fellow of the American Mathematical Society 2014 Theodore von K´arm´anPrize, Society of Industrial and Applied Mathematics 2019 Peter Henrici Prize, Society of Industrial and Applied Mathematics and ETH Selected Lectures 12/2000 Invited Speaker, Current Developments in Mathematics, Harvard University. -
International Conference on Number Theory and Discrete Mathematics (ONLINE) (ICNTDM) 11 - 14, DECEMBER-2020
International Conference on Number Theory and Discrete Mathematics (ONLINE) (ICNTDM) 11 - 14, DECEMBER-2020 Organised by RAMANUJAN MATHEMATICAL SOCIETY Hosted by DEPARTMENT OF MATHEMATICS RAJAGIRI SCHOOL OF ENGINEERING & TECHNOLOGY (AUTONOMOUS) Kochi, India International Conference on Number Theory and Discrete Mathematics PREFACE The International Conference on Number Theory and Discrete Mathematics (ICNTDM), organised by the Ramanujan Mathematical Society (RMS), and hosted by the Rajagiri School of Engineering and Technology (RSET), Cochin is a tribute to the legendary Indian mathematician Srinivasa Ramanujan who prematurely passed away a hundred years ago on 26th April 1920 leaving a lasting legacy. Conceived as any usual conference as early as June 2019, the ICNTDM was compelled to switch to online mode due to the pandemic, like most events round the globe this year. The International Academic Programme Committee, consisting of distinguished mathematicians from India and abroad, lent us a helping hand every possible way, in all aspects of the conference. As a result, we were able to evolve a strong line up of reputed speakers from India, Austria, Canada, China, France, Hungary, Japan, Russia, Slovenia, UK and USA, giving 23 plenary talks and 11 invited talks, in addition to the 39 speakers in the contributed session. This booklet consists of abstracts of all these talks and two invited articles; one describing the various academic activities that RMS is engaged in, tracing history from its momentous beginning in 1985, and a second article on Srinivasa Ramanujan. It is very appropriate to record here that, in conjunction with the pronouncement of the year 2012 { the 125th birth anniversary year of Ramanujan { as the `National Mathematics Year', RMS heralded an excellent initiative of translating the cele- brated book `The Man Who Knew Infinity' into many Indian languages.