New Horizons in pro-p Groups
New Horizons in pro-p Groups

Author: Marcus du Sautoy

Publisher: Springer Science & Business Media

Published: 2000-05-25

Total Pages: 444

ISBN-13: 9780817641719

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A pro-p group is the inverse limit of some system of finite p-groups, that is, of groups of prime-power order where the prime - conventionally denoted p - is fixed. Thus from one point of view, to study a pro-p group is the same as studying an infinite family of finite groups; but a pro-p group is also a compact topological group, and the compactness works its usual magic to bring 'infinite' problems down to manageable proportions. The p-adic integers appeared about a century ago, but the systematic study of pro-p groups in general is a fairly recent development. Although much has been dis covered, many avenues remain to be explored; the purpose of this book is to present a coherent account of the considerable achievements of the last several years, and to point the way forward. Thus our aim is both to stimulate research and to provide the comprehensive background on which that research must be based. The chapters cover a wide range. In order to ensure the most authoritative account, we have arranged for each chapter to be written by a leading contributor (or contributors) to the topic in question. Pro-p groups appear in several different, though sometimes overlapping, contexts.

$p$-Adic Analysis, Arithmetic and Singularities
$p$-Adic Analysis, Arithmetic and Singularities

Author: Carlos Galindo

Publisher: American Mathematical Society

Published: 2022-05-11

Total Pages: 311

ISBN-13: 1470467798

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This volume contains the proceedings of the 2019 Lluís A. Santaló Summer School on $p$-Adic Analysis, Arithmetic and Singularities, which was held from June 24–28, 2019, at the Universidad Internacional Menéndez Pelayo, Santander, Spain. The main purpose of the book is to present and analyze different incarnations of the local zeta functions and their multiple connections in mathematics and theoretical physics. Local zeta functions are ubiquitous objects in mathematics and theoretical physics. At the mathematical level, local zeta functions contain geometry and arithmetic information about the set of zeros defined by a finite number of polynomials. In terms of applications in theoretical physics, these functions play a central role in the regularization of Feynman amplitudes and Koba-Nielsen-type string amplitudes, among other applications. This volume provides a gentle introduction to a very active area of research that lies at the intersection of number theory, $p$-adic analysis, algebraic geometry, singularity theory, and theoretical physics. Specifically, the book introduces $p$-adic analysis, the theory of Archimedean, $p$-adic, and motivic zeta functions, singularities of plane curves and their Poincaré series, among other similar topics. It also contains original contributions in the aforementioned areas written by renowned specialists. This book is an important reference for students and experts who want to delve quickly into the area of local zeta functions and their many connections in mathematics and theoretical physics.

The Strong Sylow Theorem for the Prime p in Simple Locally Finite Groups
The Strong Sylow Theorem for the Prime p in Simple Locally Finite Groups

Author: Dipl.-Math. Felix F. Flemisch

Publisher: BoD – Books on Demand

Published: 2024-04-10

Total Pages: 69

ISBN-13: 3758333202

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This research paper continues [15]. We begin with giving a profound overview of the structure of arbitrary simple groups and in particular of the simple locally finite groups and reduce their Sylow theory for the prime p to a quite famous conjecture by Prof. Otto H. Kegel (see [37], Theorem 2.4: "Let the p-subgroup P be a p-uniqueness subgroup in the finite simple group S which belongs to one of the seven rank-unbounded families. Then the rank of S is bounded in terms of P.") about the rank-unbounded ones of the 19 known families of finite simple groups. We introduce a new scheme to describe the 19 families, the family T of types, define the rank of each type, and emphasise the rôle of Kegel covers. This part presents a unified picture of known results whose proofs are by reference. Subsequently we apply new ideas to prove the conjecture for the alternating groups. Thereupon we are remembering Kegel covers and *-sequences. Next we suggest a way 1) and a way 2) how to prove and even how to optimise Kegel's conjecture step-by-step or peu à peu which leads to Conjecture 1, Conjecture 2 and Conjecture 3 thereby unifying Sylow theory in locally finite simple groups with Sylow theory in locally finite and p-soluble groups whose joint study directs Sylow theory in (locally) finite groups. For any unexplained terminology we refer to [15]. We then continue the program begun above to optimise along the way 1) the theorem about the first type "An" of infinite families of finite simple groups step-by-step to further types by proving it for the second type "A = PSLn". We start with proving Conjecture 2 about the General Linear Groups over (commutative) locally finite fields, stating that their rank is bounded in terms of their p-uniqueness, and then break down this insight to the Special Linear Groups and the Projective Special Linear (PSL) Groups over locally finite fields. We close with suggestions for future research -> regarding the remaining rank-unbounded types (the "Classical Groups") and the way 2), -> regarding (locally) finite and p-soluble groups, and -> regarding Cauchy's and Galois' contributions to Sylow theory in finite groups. We much hope to enthuse group theorists with them. We include the predecessor research paper [15] as an Appendix.

Korean Musical Drama: P'ansori and the Making of Tradition in Modernity
Korean Musical Drama: P'ansori and the Making of Tradition in Modernity

Author: Dr Haekyung Um

Publisher: Ashgate Publishing, Ltd.

Published: 2014-02-28

Total Pages: 273

ISBN-13: 147241456X

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P’ansori is the quintessential traditional Korean musical drama, in which epic tales are sung and narrated by a solo singer accompanied by a drummer. Drawing on her extensive research in Korea and its diasporas, Haekyung Um describes and analyses the creative processes of p’ansori, weaving into her discussion musical, social and cultural aspects that include the evolution of p’ansori performance, origins and historical development, textual and musical materials, stylistic features of different p’ansori schools, transmission of knowledge, aesthetics, and changing interpretations of tradition. Also explored is the complexity of historical and contemporary influences that give shape to p’ansori as a ‘living tradition’ across the ages and into the present, and as a cultural icon with an enduring narrative and emotional impact. Social, economic and political dynamics are created in the nexus of traditional feudal values, colonial modernity and nationalism. The impact of aspects of late modernity such as technology, mass media, migration and globalization, has transported p’ansori into digital and transnational domains. By bringing all these creative and contextual processes together, Haekyung Um explains how a tradition is created, maintained and redefined by the dynamic interactions of agents, values, meanings, strategies, identities and artistic hybridity.

Assessing Pupil's Performance Using the P Levels
Assessing Pupil's Performance Using the P Levels

Author: Val Davis

Publisher: Routledge

Published: 2013-10-23

Total Pages: 180

ISBN-13: 113414282X

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This book has been designed to provide guidance for special and mainstream schools in the assessment of pupils' learning from Level P1 up to and including National Curriculum Level 1A. It contains exemplification of the descriptions of attainment for reading, writing and the three strands of mathematics identified in Planning, Teaching and Assessing the Curriculum for Pupils with Learning Difficulties produced by the QCA. The book provides clarification of the performance criteria, through illustrative examples, and supports accurate and consistent teacher assessment of pupils working at these levels. It enables effective monitoring of attainment and progression, which will support the target setting process, and demonstrates how assessments can be used to inform next steps in learning. The authors include examples from special and mainstream schools on reading, writing and mathematics. The book also contains photocopiable proformas for your own use. SENCOs and teachers in special and mainstream schools should find this book helps them to chart the progress of their pupils' learning very effectively.

Supersingular p-adic L-functions, Maass-Shimura Operators and Waldspurger Formulas
Supersingular p-adic L-functions, Maass-Shimura Operators and Waldspurger Formulas

Author: Daniel Kriz

Publisher: Princeton University Press

Published: 2021-11-09

Total Pages: 280

ISBN-13: 0691225737

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A groundbreaking contribution to number theory that unifies classical and modern results This book develops a new theory of p-adic modular forms on modular curves, extending Katz's classical theory to the supersingular locus. The main novelty is to move to infinite level and extend coefficients to period sheaves coming from relative p-adic Hodge theory. This makes it possible to trivialize the Hodge bundle on the infinite-level modular curve by a "canonical differential" that restricts to the Katz canonical differential on the ordinary Igusa tower. Daniel Kriz defines generalized p-adic modular forms as sections of relative period sheaves transforming under the Galois group of the modular curve by weight characters. He introduces the fundamental de Rham period, measuring the position of the Hodge filtration in relative de Rham cohomology. This period can be viewed as a counterpart to Scholze's Hodge-Tate period, and the two periods satisfy a Legendre-type relation. Using these periods, Kriz constructs splittings of the Hodge filtration on the infinite-level modular curve, defining p-adic Maass-Shimura operators that act on generalized p-adic modular forms as weight-raising operators. Through analysis of the p-adic properties of these Maass-Shimura operators, he constructs new p-adic L-functions interpolating central critical Rankin-Selberg L-values, giving analogues of the p-adic L-functions of Katz, Bertolini-Darmon-Prasanna, and Liu-Zhang-Zhang for imaginary quadratic fields in which p is inert or ramified. These p-adic L-functions yield new p-adic Waldspurger formulas at special values.

P-40 Warhawk vs Ki-43 Oscar
P-40 Warhawk vs Ki-43 Oscar

Author: Carl Molesworth

Publisher: Bloomsbury Publishing

Published: 2012-11-20

Total Pages: 82

ISBN-13: 1782007466

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Known for the distinctive 'sharkmouth' decoration on their noses, P-40 fighters first saw combat in China during World War II. Their most common adversary was the Japanese Nakajima Ki-43, nicknamed 'Oscar.' Carl Molesworth describes and explains the design and development of these two foes, the products of two vastly different philosophies of fighter design. The P-40 was heavily armed and sturdy with armour protection and self-sealing fuel tanks, but paid for this with the loss of speed and a sluggish performance at altitude. The Ki-43 was a rapier to the battleaxe P-40 and the Ki-43 was immensely nimble, though with less firepower and durability. This book examines these two different fighters, and the pilots who flew them over China, with an action-packed text, rare photographs and digital artwork.

The (p,n) Reaction and the Nucleon-Nucleon Force
The (p,n) Reaction and the Nucleon-Nucleon Force

Author: Charles D. Goodman

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 525

ISBN-13: 1468488600

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This volume contains the proceedings of the "Conference on the (p,n) Reaction and the Nucleon-Nucleon Force" held in Telluride, Colorado, March 29-31, 1979. The idea to hold this conference grew out of a program at the Indiana University Cyclotron Facility to study the (p,n) reaction in the 50-200 MeV energy range. The first new Indiana data, in contrast to low energy data, showed features suggestive of a dominant one pion exchange interaction. It seemed desir able to review what was known about the fre·e and the effective nucleon-nucleon force and the connection between the low and high energy (p,n) data. Thus the conference was born. The following people served as the organizing committee: S. M. Austin, Michigan State University W. Bertozzi, Massachusetts Institute of Technology S. D. Bloom, Lawrence Livermore Laboratory C. C. Foster, Indiana University C. D. Goodman, Oak Ridge National Laboratory (Conference Chairman) D. A. Lind, University of Colorado J. Rapaport, Ohio University G. R. Satch1er, Oak Ridge National Laboratory G. E. Walker, Indiana University R. L. Walter, Duke University and TUNL The sponsoring organizations were: Indiana University, Bloomington, Indiana University of Colorado, Boulder, Colorado Oak Ridge National Laboratory, Oak Ridge, Tennessee Triangle Universities Nuclear Laboratory, Durham, North Carolina Of course, the major credit for the success of the con ference must go to the speakers who diligently prepared their talks that are reproduced in this volume.

p-Adic Valued Distributions in Mathematical Physics
p-Adic Valued Distributions in Mathematical Physics

Author: Andrei Y. Khrennikov

Publisher: Springer Science & Business Media

Published: 2013-03-09

Total Pages: 271

ISBN-13: 9401583560

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Numbers ... , natural, rational, real, complex, p-adic .... What do you know about p-adic numbers? Probably, you have never used any p-adic (nonrational) number before now. I was in the same situation few years ago. p-adic numbers were considered as an exotic part of pure mathematics without any application. I have also used only real and complex numbers in my investigations in functional analysis and its applications to the quantum field theory and I was sure that these number fields can be a basis of every physical model generated by nature. But recently new models of the quantum physics were proposed on the basis of p-adic numbers field Qp. What are p-adic numbers, p-adic analysis, p-adic physics, p-adic probability? p-adic numbers were introduced by K. Hensel (1904) in connection with problems of the pure theory of numbers. The construction of Qp is very similar to the construction of (p is a fixed prime number, p = 2,3,5, ... ,127, ... ). Both these number fields are completions of the field of rational numbers Q. But another valuation 1 . Ip is introduced on Q instead of the usual real valuation 1 . I· We get an infinite sequence of non isomorphic completions of Q : Q2, Q3, ... , Q127, ... , IR = Qoo· These fields are the only possibilities to com plete Q according to the famous theorem of Ostrowsky.

Representations Of Real And P-adic Groups
Representations Of Real And P-adic Groups

Author: Eng-chye Tan

Publisher: World Scientific

Published: 2004-04-15

Total Pages: 426

ISBN-13: 9814483087

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The Institute for Mathematical Sciences at the National University of Singapore hosted a research program on “Representation Theory of Lie Groups” from July 2002 to January 2003. As part of the program, tutorials for graduate students and junior researchers were given by leading experts in the field.This invaluable volume collects the expanded lecture notes of those tutorials. The topics covered include uncertainty principles for locally compact abelian groups, fundamentals of representations of p-adic groups, the Harish-Chandra-Howe local character expansion, classification of the square-integrable representations modulo cuspidal data, Dirac cohomology and Vogan's conjecture, multiplicity-free actions and Schur-Weyl-Howe duality.The lecturers include Tomasz Przebinda from the University of Oklahoma, USA; Gordan Savin from the University of Utah, USA; Stephen DeBacker from Harvard University, USA; Marko Tadić from the University of Zagreb, Croatia; Jing-Song Huang from The Hong Kong University of Science and Technology, Hong Kong; Pavle Pandǽić from the University of Zagreb, Croatia; Chal Benson and Gail Ratcliff from East Carolina University, USA; and Roe Goodman from Rutgers University, USA.