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Valuative Characterization of Central Extensions of Algebraic Tori on Krull
Valuative characterization of central extensions of algebraic tori on Krull domains ∗ Haruhisa Nakajima † Department of Mathematics, J. F. Oberlin University Tokiwa-machi, Machida, Tokyo 194-0294, JAPAN Abstract Let G be an affine algebraic group with an algebraic torus G0 over an alge- braically closed field K of an arbitrary characteristic p. We show a criterion for G to be a finite central extension of G0 in terms of invariant theory of all regular 0 actions of any closed subgroup H containing ZG(G ) on affine Krull K-schemes such that invariant rational functions are locally fractions of invariant regular func- tions. Consider an affine Krull H-scheme X = Spec(R) and a prime ideal P of R with ht(P) = ht(P ∩ RH ) = 1. Let I(P) denote the inertia group of P un- der the action of H. The group G is central over G0 if and only if the fraction e(P, P ∩ RH )/e(P, P ∩ RI(P)) of ramification indices is equal to 1 (p = 0) or to the p-part of the order of the group of weights of G0 on RI(P)) vanishing on RI(P))/P ∩ RI(P) (p> 0) for an arbitrary X and P. MSC: primary 13A50, 14R20, 20G05; secondary 14L30, 14M25 Keywords: Krull domain; ramification index; algebraic group; algebraic torus; char- acter group; invariant theory 1 Introduction 1.A. We consider affine algebraic groups and affine schemes over a fixed algebraically closed field K of an arbitrary characteristic p. For an affine group G, denote by G0 its identity component. -
Frances Anne Kemble (1809-1893) Natalie Karelis La Salle University
La Salle University La Salle University Digital Commons People and Places La Salle Local History 1998 Frances Anne Kemble (1809-1893) Natalie Karelis La Salle University Follow this and additional works at: http://digitalcommons.lasalle.edu/people_places Part of the American Material Culture Commons, Cultural History Commons, United States History Commons, and the Women's History Commons Recommended Citation Karelis, Natalie, "Frances Anne Kemble (1809-1893)" (1998). People and Places. 4. http://digitalcommons.lasalle.edu/people_places/4 This Book is brought to you for free and open access by the La Salle Local History at La Salle University Digital Commons. It has been accepted for inclusion in People and Places by an authorized administrator of La Salle University Digital Commons. For more information, please contact [email protected]. Belfield & Wakefield: A Link to La Salle's Past Welcome Frances Anne Kemble Reports -- People By By: Natalie Karelis Century Family Tree A famous actress, an unfaithful husband, a controversial family business, a messy divorce, and a brutal custody battle. These Three Centuries on seem to be all the characteristics of today’s tabloid news. But South Campus maybe these components of a sensational news story aren’t so Home Where “The modern after all. In fact, a situation fitting the above Mansion” Was description, involving actress Fanny Kemble (for whom Fanny Kemble Park, located at Olney Avenue and 17th Street, is The Remarkable named) and slave-holder Pierce Butler, actually occurred over Wisters at Belfield 150 years ago on property adjacent to La Salle University! Charles Willson Peale: “Your Garden Frances Anne Kemble was born into a theatrical family in Must be a Museum” London, England, on November 27, 1809. -
The Complexity Zoo
The Complexity Zoo Scott Aaronson www.ScottAaronson.com LATEX Translation by Chris Bourke [email protected] 417 classes and counting 1 Contents 1 About This Document 3 2 Introductory Essay 4 2.1 Recommended Further Reading ......................... 4 2.2 Other Theory Compendia ............................ 5 2.3 Errors? ....................................... 5 3 Pronunciation Guide 6 4 Complexity Classes 10 5 Special Zoo Exhibit: Classes of Quantum States and Probability Distribu- tions 110 6 Acknowledgements 116 7 Bibliography 117 2 1 About This Document What is this? Well its a PDF version of the website www.ComplexityZoo.com typeset in LATEX using the complexity package. Well, what’s that? The original Complexity Zoo is a website created by Scott Aaronson which contains a (more or less) comprehensive list of Complexity Classes studied in the area of theoretical computer science known as Computa- tional Complexity. I took on the (mostly painless, thank god for regular expressions) task of translating the Zoo’s HTML code to LATEX for two reasons. First, as a regular Zoo patron, I thought, “what better way to honor such an endeavor than to spruce up the cages a bit and typeset them all in beautiful LATEX.” Second, I thought it would be a perfect project to develop complexity, a LATEX pack- age I’ve created that defines commands to typeset (almost) all of the complexity classes you’ll find here (along with some handy options that allow you to conveniently change the fonts with a single option parameters). To get the package, visit my own home page at http://www.cse.unl.edu/~cbourke/. -
Lecture 11 — October 16, 2012 1 Overview 2
6.841: Advanced Complexity Theory Fall 2012 Lecture 11 | October 16, 2012 Prof. Dana Moshkovitz Scribe: Hyuk Jun Kweon 1 Overview In the previous lecture, there was a question about whether or not true randomness exists in the universe. This question is still seriously debated by many physicists and philosophers. However, even if true randomness does not exist, it is possible to use pseudorandomness for practical purposes. In any case, randomization is a powerful tool in algorithms design, sometimes leading to more elegant or faster ways of solving problems than known deterministic methods. This naturally leads one to wonder whether randomization can solve problems outside of P, de- terministic polynomial time. One way of viewing randomized algorithms is that for each possible setting of the random bits used, a different strategy is pursued by the algorithm. A property of randomized algorithms that succeed with bounded error is that the vast majority of strategies suc- ceed. With error reduction, all but exponentially many strategies will succeed. A major theme of theoretical computer science over the last 20 years has been to turn algorithms with this property into algorithms that deterministically find a winning strategy. From the outset, this seems like a daunting task { without randomness, how can one efficiently search for such strategies? As we'll see in later lectures, this is indeed possible, modulo some hardness assumptions. There are two main objectives in this lecture. The first objective is to introduce some probabilis- tic complexity classes, by bounding time or space. These classes are probabilistic analogues of deterministic complexity classes P and L. -
The Spirit of the Living Creatures Was in the Wheels -- Ezekiel 1:21
The spirit of the living creatures was in the wheels -- Ezekiel 1:21 "Wheels Within Wheels deepens the musical experience created at Congregation Bet Haverim by lifting the veil between the ordinary and the sacred, so that our earthy expression of musical holiness connects with the celestial resonance of the universal. Like Ezekiel’s mystical vision, the spirals of harmonies, voices and instruments evoke contemplation, awe and celebration. At times Wheels Within Wheels will transport you to an intimate experience with your innermost self, and at other times it will convey a profound connection with the world around you." -- Rabbi Joshua Lesser THE MUSICIANS OF CONGREGATION BET HAVERIM Chorus Soprano: Nefesh Chaya, Sara Dardik, Julie Fishman, Nancy Gerber, Joy Goodman, Ellie McGraw, Theresa Prestwood, Rina Rosenberg, Faith Russler, Sandi Schein Alto: Jesse Harris Bathrick, Elke Davidson, Gayanne Geurin, Kim Goldsmith, Rebecca Green, Carrie Hausman, Alix Laing, Rebecca Leary Safon, AnnaLaura Scheer, Valerie Singer, Linda Weiskoff, Valerie Wolpe, McKenzie Wren Tenor: Ned Bridges, Brad Davidorf, Faye Dresner, Henry Farber, Alan Hymowitz, Lynne Norton Bass: Dan Arnold, Gregg Bedol, David Borthwick, Gary Falcon, Bill Laing, Bill Witherspoon, Howard Winer Band Will Robertson, guitar, keyboards; Natalie Stahl, clarinet, saxophone; Sarah Zaslaw, violin, viola; Reuben Haller, mandolin; Jordan Dayan, bass, electric bass; Mike Zimmerman, drum kit, percussion; Henry Farber and Gayanne Geurin, percussion; with Matthew Kaminski, accordion Strings Sarah Zaslaw and Benjamin Reiss, violin; David Borthwick, viola; Ruth Einstein, cello; Will Robertson, double bass Children’s Choir and CBH Community School Chorus director: Will Robertson Music director: Gayanne Geurin Thank You It takes a shul to raise a recording. -
Simple Doubly-Efficient Interactive Proof Systems for Locally
Electronic Colloquium on Computational Complexity, Revision 3 of Report No. 18 (2017) Simple doubly-efficient interactive proof systems for locally-characterizable sets Oded Goldreich∗ Guy N. Rothblumy September 8, 2017 Abstract A proof system is called doubly-efficient if the prescribed prover strategy can be implemented in polynomial-time and the verifier’s strategy can be implemented in almost-linear-time. We present direct constructions of doubly-efficient interactive proof systems for problems in P that are believed to have relatively high complexity. Specifically, such constructions are presented for t-CLIQUE and t-SUM. In addition, we present a generic construction of such proof systems for a natural class that contains both problems and is in NC (and also in SC). The proof systems presented by us are significantly simpler than the proof systems presented by Goldwasser, Kalai and Rothblum (JACM, 2015), let alone those presented by Reingold, Roth- blum, and Rothblum (STOC, 2016), and can be implemented using a smaller number of rounds. Contents 1 Introduction 1 1.1 The current work . 1 1.2 Relation to prior work . 3 1.3 Organization and conventions . 4 2 Preliminaries: The sum-check protocol 5 3 The case of t-CLIQUE 5 4 The general result 7 4.1 A natural class: locally-characterizable sets . 7 4.2 Proof of Theorem 1 . 8 4.3 Generalization: round versus computation trade-off . 9 4.4 Extension to a wider class . 10 5 The case of t-SUM 13 References 15 Appendix: An MA proof system for locally-chracterizable sets 18 ∗Department of Computer Science, Weizmann Institute of Science, Rehovot, Israel. -
Glossary of Complexity Classes
App endix A Glossary of Complexity Classes Summary This glossary includes selfcontained denitions of most complexity classes mentioned in the b o ok Needless to say the glossary oers a very minimal discussion of these classes and the reader is re ferred to the main text for further discussion The items are organized by topics rather than by alphab etic order Sp ecically the glossary is partitioned into two parts dealing separately with complexity classes that are dened in terms of algorithms and their resources ie time and space complexity of Turing machines and complexity classes de ned in terms of nonuniform circuits and referring to their size and depth The algorithmic classes include timecomplexity based classes such as P NP coNP BPP RP coRP PH E EXP and NEXP and the space complexity classes L NL RL and P S P AC E The non k uniform classes include the circuit classes P p oly as well as NC and k AC Denitions and basic results regarding many other complexity classes are available at the constantly evolving Complexity Zoo A Preliminaries Complexity classes are sets of computational problems where each class contains problems that can b e solved with sp ecic computational resources To dene a complexity class one sp ecies a mo del of computation a complexity measure like time or space which is always measured as a function of the input length and a b ound on the complexity of problems in the class We follow the tradition of fo cusing on decision problems but refer to these problems using the terminology of promise problems -
User's Guide for Complexity: a LATEX Package, Version 0.80
User’s Guide for complexity: a LATEX package, Version 0.80 Chris Bourke April 12, 2007 Contents 1 Introduction 2 1.1 What is complexity? ......................... 2 1.2 Why a complexity package? ..................... 2 2 Installation 2 3 Package Options 3 3.1 Mode Options .............................. 3 3.2 Font Options .............................. 4 3.2.1 The small Option ....................... 4 4 Using the Package 6 4.1 Overridden Commands ......................... 6 4.2 Special Commands ........................... 6 4.3 Function Commands .......................... 6 4.4 Language Commands .......................... 7 4.5 Complete List of Class Commands .................. 8 5 Customization 15 5.1 Class Commands ............................ 15 1 5.2 Language Commands .......................... 16 5.3 Function Commands .......................... 17 6 Extended Example 17 7 Feedback 18 7.1 Acknowledgements ........................... 19 1 Introduction 1.1 What is complexity? complexity is a LATEX package that typesets computational complexity classes such as P (deterministic polynomial time) and NP (nondeterministic polynomial time) as well as sets (languages) such as SAT (satisfiability). In all, over 350 commands are defined for helping you to typeset Computational Complexity con- structs. 1.2 Why a complexity package? A better question is why not? Complexity theory is a more recent, though mature area of Theoretical Computer Science. Each researcher seems to have his or her own preferences as to how to typeset Complexity Classes and has built up their own personal LATEX commands file. This can be frustrating, to say the least, when it comes to collaborations or when one has to go through an entire series of files changing commands for compatibility or to get exactly the look they want (or what may be required). -
Pennsylvania Magazine of HISTORY and BIOGRAPHY
THE Pennsylvania Magazine OF HISTORY AND BIOGRAPHY Sarah Butler Wister's Civil War Diary at ARAH BUTLER WISTER was born May 28, 1835, Butler Place. She was the older daughter of Fanny Kemble, the S renowned English actress, and Pierce Butler. At the time of her birth Butler Place, bought by her great-grandfather in I8IO3 was a farm with a rather unimpressive, middle-sized, owner's dwelling, but with splendid barns and outbuildings. The site, six miles north of Philadelphia's City Hall, at the end of Broad Street, is today filled by small row houses. Sarah's mother had come to America from London in 1832 with her father, Charles Kemble. They were of the third and fourth generations of the famous Kemble theatrical family. The Kembles toured the eastern seaboard, acting together, and it was in Phila- delphia that Pierce Butler was introduced to Fanny. They were married in Christ Church on June 7, 1834. The marriage was un- happy from the start, and, under extremely harsh circumstances, Fanny left her family in March 1845 to return to England, where she built a second career as a superb reader of Shakespeare. Appear- ing both in England and America, she supported herself for twenty years. In 1863 she published The Journal of a Residence on a Qeorgian Plantations in which she declared her abhorrence of slavery. This book gave her a lasting reputation as an abolitionist and as a writer. 271 272 FANNY KEMBLE WISTER July Sarah's father was born Pierce Butler Mease, son of Dr. and Mrs. -
Elvisbrucefeaturefromstonepo
How an encounter that never happened might have helped to change history if it had By Shawn Poole Ever since he was a child, Bruce Springsteen has been strongly influenced by the artistry of Elvis Presley. Over the years, the “E Street”/”Elvis” connection has remained strong. Most recently, Springsteen participated along with E Street Band members Nils Lofgren and Patti Scialfa in the Elvis: Viva Las Vegas television special (now on DVD in the U.S., where it hit #1 on the Billboard DVD sales chart), and its accompanying soundtrack album. E Streeters Garry Tallent and Max Weinberg also were on hand at this year’s Rock and Roll Hall of Fame Induction Ceremony to induct Elvis’ original drummer and bassist, D.J. Fontana and the late Bill Black. Presley was only forty-two years old when his life tragically ended. “They found him slumped up against the drain,” Bruce Springsteen would later sing of his fallen idol, “with a whole lot of trouble running through his veins; Bye-bye, Johnny; Johnny, bye-bye; You didn’t have to die; you didn’t have to die.” While Elvis’ premature decline and death did absolutely nothing to stop his immense cultural influence, it did rob the world of many more years of music from this great artist. It also ensured that Bruce Springsteen would never get another chance to meet the man who initially inspired his own musical career. (He had two chances: one famous unofficial chance when he jumped the wall at Graceland in 1976 and one not-as-famous in 1977 when local concert promoters offered Springsteen an official chance to meet Elvis backstage before his final Philadelphia concert.) But what if Bruce had gotten to meet Elvis before August 16, 1977? Might things have turned out any differently for either or both of them, as well as the many other lives they affected? In this work of speculative fiction, Shawn Poole ponders the possibilities, poignantly reminding us of the promise and potential that tragically disappeared in that long ago summer. -
John Philip Kemble
Prescot Museum John Philip Kemble John Philip Kemble was born on 1st February 1757 in Hillock Street, Prescot. He was baptised at Prescot Parish Church on 16th February 1757. He belonged to a famous acting dynasty headed by his father Roger Kemble, a celebrated comedian. Portrait of John Philip Kemble John’s elder sister Sarah Siddons was an accomplished actress and achieved fame with him on the stage of The Theatre Royal, Drury Lane. The brother and sister duo first appeared there together in 1783, as Beverley and Mrs Beverley in The Gamester. His other siblings Charles, Stephen, Ann and Elizabeth also enjoyed success on the stage. John’s niece Frances Ann (Fanny) Kemble was a respected actress and author. John Philip Kemble as Hamlet John Kemble’s father wanted him to become a priest and was sent to study in France, but being born into a theatrical family, it was unsurprising that John took to the stage. In December 1787 he married actress Priscilla Hopkins Brereton, the widow of another actor. Kemble’s life was as dramatic and eventful as any play he performed in. In 1781, in defence of an actress friend, he attacked a group of officers with his sword. Ten years later, he fought a duel with a man named James Aicken. In a show of great chivalry, he faced the fire of Aicken’s pistol, and retaliated by shooting into Kemble was noted for the air rather than at his rival. his tragic roles, specialising in http://www.knowsley.gov.uk/things-to-see-and-do/galleries-and-museum.aspx Shakespearean dramas, and was Kemble & Siddons, Macbeth 1786, by Thomas Beach Sarah Siddons by Gilbert Stuart particularly celebrated for his portrayal of Hamlet – the first character he played on the London stage in 1783. -
Ellen Tree, Fanny Kemble, and Theatrical Constructions of Gender
"Playing the Men": Ellen Tree, Fanny Kemble, and Theatrical Constructions of Gender Anne Russell, Wilfrid Laurier University Abstract Ellen Tree, the first English performer to regularly play tragic male roles, initiated a nineteenth-century Anglo-American convention in which many women performers played a limited number of tragic male roles, primarily Romeo and Hamlet. Nineteenth-century women's performances of tragic male characters point to multiple tensions and fissures in the understanding and representation of gender in theater. Tree's negotiation of these tensions and her decisions about how, and in what contexts, to play tragic male roles indicate her awareness of the ways in which shifting social perspectives on gender might be accommodated in her stage representations. She followed her enthusiastic and romantic Romeo with Thomas Noon Talfourd's Ion, whose dignified and "classical" character offered a kind of idealized and unsexualized masculinity that was not too deceptively "realistic." Thirty years after she had first performed Romeo with Fanny Kemble as Juliet, Tree attended several of Kemble's public readings of Shakespeare, in which Kemble read all the roles. Tree expressed a combination of admiration and disquiet at the way Kemble "read the men best." Tree's mixed emotions about Kemble's reading register not only her recognition of changing social mores, but also unresolved tensions and ironies in Tree's own theatrical practices. Audience responses to Kemble's readings show astonishment at her intensity and emotionality, particularly her strikingly convincing readings of male characters. These strong audience reactions to Kemble's readings of Shakespeare's plays suggest that the expressions of emotionality possible in public reading could be perceived as more dangerously exciting than stylized and conventional stage performances of the period.