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Contributors
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- By Douglas L. Arnold, Laura J. Balcer, Amit Bar-Or, Sergio E. Baranzini, Frederik Barkhof, Robert A. Bermel, Francois A. Bethoux, Dennis N. Bourdette, Richard K. Burt, Peter A. Calabresi, Zografos Caramanos, Tanuja Chitnis, Stacey S. Cofield, Jeffrey A. Cohen, Nadine Cohen, Alasdair J. Coles, Devon Conway, Stuart D. Cook, Gary R. Cutter, Peter J. Darlington, Ann Dodds-Frerichs, Ranjan Dutta, Gilles Edan, Michelle Fabian, Franz Fazekas, Massimo Filippi, Elizabeth Fisher, Paulo Fontoura, Corey C. Ford, Robert J. Fox, Natasha Frost, Alex Z. Fu, Siegrid Fuchs, Kazuo Fujihara, Kristin M. Galetta, Jeroen J.G. Geurts, Gavin Giovannoni, Nada Gligorov, Ralf Gold, Andrew D. Goodman, Myla D. Goldman, Jenny Guerre, Stephen L. Hauser, Peter B. Imrey, Douglas R. Jeffery, Stephen E. Jones, Adam I. Kaplin, Michael W. Kattan, B. Mark Keegan, Kyle C. Kern, Zhaleh Khaleeli, Samia J. Khoury, Joep Killestein, Soo Hyun Kim, R. Philip Kinkel, Stephen C. Krieger, Lauren B. Krupp, Emmanuelle Le Page, David Leppert, Scott Litwiller, Fred D. Lublin, Henry F. McFarland, Joseph C. McGowan, Don Mahad, Jahangir Maleki, Ruth Ann Marrie, Paul M. Matthews, Francesca Milanetti, Aaron E. Miller, Deborah M. Miller, Xavier Montalban, Charity J. Morgan, Ichiro Nakashima, Sridar Narayanan, Avindra Nath, Paul W. O’Connor, Jorge R. Oksenberg, A. John Petkau, Michael D. Phillips, J. Theodore Phillips, Tammy Phinney, Sean J. Pittock, Sarah M. Planchon, Chris H. Polman, Alexander Rae-Grant, Stephen M. Rao, Stephen C. Reingold, Maria A. Rocca, Richard A. Rudick, Amber R. Salter, Paula Sandler, Jaume Sastre-Garriga, John R. Scagnelli, Dana J. Serafin, Lynne Shinto, Nancy L. Sicotte, Jack H. Simon, Per Soelberg Sørensen, Ryan E. Stagg, James M. Stankiewicz, Lael A. Stone, Amy Sullivan, Matthew Sutliff, Jessica Szpak, Alan J. Thompson, Bruce D. Trapp, Helen Tremlett, Maria Trojano, Orla Tuohy, Rhonda R. Voskuhl, Marc K. Walton, Mike P. Wattjes, Emmanuelle Waubant, Martin S. Weber, Howard L Weiner, Brian G. Weinshenker, Bianca Weinstock-Guttman, Jeffrey L. Winters, Jerry S. Wolinsky, Vijayshree Yadav, E. Ann Yeh, Scott S. Zamvil
- Edited by Jeffrey A. Cohen, Richard A. Rudick
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- Book:
- Multiple Sclerosis Therapeutics
- Published online:
- 05 December 2011
- Print publication:
- 20 October 2011, pp viii-xii
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Effective Diophantine approximation on Gm
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- By Enrico Bombieri, School of Mathematics, Institute for Advanced Study, Princeton, N.J. 08540, USA, Paula B. Cohen, Department of Mathematics, Texas A&M University, College Station, TX 77843, USA
- Edited by Miles Reid, University of Warwick, Alexei Skorobogatov, Imperial College of Science, Technology and Medicine, London
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- Number Theory and Algebraic Geometry
- Published online:
- 04 August 2010
- Print publication:
- 26 January 2004, pp 41-62
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Summary
Introduction
Effective results in the diophantine approximation of algebraic numbers are difficult to obtain, and for a long time the only general method available was Baker's theory of linear forms in logarithms. An alternative, more algebraic, method was later proposed in Bombieri and Bombieri and Cohen. This new method is quite different from the classical approach through the theory of linear forms in logarithms.
In this paper, we improve on results derived in. These results concern effective approximations to roots of high order of algebraic numbers and their application to diophantine approximation in a number field by a finitely generated multiplicative subgroup. We restrict our attention to the nonarchimedean case, although our results and methods should go over mutatis mutandis to the archimedean setting.
We do not claim that our theorems are the best that are known in this direction. Linear forms in two logarithms (which are easier to treat than the general case) suffice to prove somewhat better results than our Theorem 5.1, see Bugeaud and Bugeaud and Laurent; we give an explicit comparison in §5, Remark 5.1.
Theorem 5.2, which is useful for general applications, follows from Theorem 5.1 by means of a trick introduced for the first time in and improved in. Thus any improved form of Theorem 5.1 carries automatically an improvement of Theorem 5.2.
6 - Application of the André–Oort Conjecture to some Questions in Transcendence
- Edited by Gisbert Wüstholz, Swiss Federal University (ETH), Zürich
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- Book:
- A Panorama of Number Theory <I>or</I> The View from Baker's Garden
- Published online:
- 20 August 2009
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- 26 September 2002, pp 89-106
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Summary
Abstract
We show how a problem concerning the transcendence of values of the classical hypergeometric function, and originating in work of Siegel on G-functions, can be solved using a special case of a conjecture of André–Oort on the distribution of complex multiplication (or special) points on algebraic curves in Shimura varieties. The special case in question has recently been proven, at our suggestion, by Edixhoven & Yafaev (2001); see also Yafaev (2001b). This settles the question of which classical hypergeometric functions with rational parameters, satisfying certain natural assumptions, take only finitely many algebraic values at algebraic points. The fact that such a function cannot have an arithmetic monodromy group goes back to work of Wolfart (1988). We introduce a number of related problems.
Note added in revision In the original version of this article, we introduced a number of open problems motivated by transcendence questions on the classical hypergeometric function. These are summarised in Problems 1, 2, 3 and 4 of §1. One of the main points of this article is to show how Problems 1 and 2 follow from Problem 4, which is in turn related to the André–Oort Conjecture, Oort (1997) concerning the distribution of complex multiplication points on subvarieties of Shimura varieties.
FEWNOMIALS AND INTERSECTIONS OF LINES WITH REAL ANALYTIC SUBGROUPS IN Gnm
- PAULA B. COHEN, UMBERTO ZANNIER
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- Journal:
- Bulletin of the London Mathematical Society / Volume 34 / Issue 1 / January 2002
- Published online by Cambridge University Press:
- 18 January 2002, pp. 21-32
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- January 2002
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In this paper, the authors study intersections of a special class of curves with algebraic subgroups of the multiplicative group of complex dimension at least 2. They show how results of Khovanskii on fewnomials can be used to derive finiteness results and bounds for the degrees of algebraic points for such intersections from more general results on intersections of curves with non-algebraic subgroups. They thereby generalise their earlier results, and recover in some cases, using different methods, more uniform bounds than those given in related work of Bombieri, Masser and Zannier.
Looking Backward, Looking Forward: MLA Members Speak
- April Alliston, Elizabeth Ammons, Jean Arnold, Nina Baym, Sandra L. Beckett, Peter G. Beidler, Roger A. Berger, Sandra Bermann, J.J. Wilson, Troy Boone, Alison Booth, Wayne C. Booth, James Phelan, Marie Borroff, Ihab Hassan, Ulrich Weisstein, Zack Bowen, Jill Campbell, Dan Campion, Jay Caplan, Maurice Charney, Beverly Lyon Clark, Robert A. Colby, Thomas C. Coleman III, Nicole Cooley, Richard Dellamora, Morris Dickstein, Terrell Dixon, Emory Elliott, Caryl Emerson, Ann W. Engar, Lars Engle, Kai Hammermeister, N. N. Feltes, Mary Anne Ferguson, Annie Finch, Shelley Fisher Fishkin, Jerry Aline Flieger, Norman Friedman, Rosemarie Garland-Thomson, Sandra M. Gilbert, Laurie Grobman, George Guida, Liselotte Gumpel, R. K. Gupta, Florence Howe, Cathy L. Jrade, Richard A. Kaye, Calhoun Winton, Murray Krieger, Robert Langbaum, Richard A. Lanham, Marilee Lindemann, Paul Michael Lützeler, Thomas J. Lynn, Juliet Flower MacCannell, Michelle A. Massé, Irving Massey, Georges May, Christian W. Hallstein, Gita May, Lucy McDiarmid, Ellen Messer-Davidow, Koritha Mitchell, Robin Smiles, Kenyatta Albeny, George Monteiro, Joel Myerson, Alan Nadel, Ashton Nichols, Jeffrey Nishimura, Neal Oxenhandler, David Palumbo-Liu, Vincent P. Pecora, David Porter, Nancy Potter, Ronald C. Rosbottom, Elias L. Rivers, Gerhard F. Strasser, J. L. Styan, Marianna De Marco Torgovnick, Gary Totten, David van Leer, Asha Varadharajan, Orrin N. C. Wang, Sharon Willis, Louise E. Wright, Donald A. Yates, Takayuki Yokota-Murakami, Richard E. Zeikowitz, Angelika Bammer, Dale Bauer, Karl Beckson, Betsy A. Bowen, Stacey Donohue, Sheila Emerson, Gwendolyn Audrey Foster, Jay L. Halio, Karl Kroeber, Terence Hawkes, William B. Hunter, Mary Jambus, Willard F. King, Nancy K. Miller, Jody Norton, Ann Pellegrini, S. P. Rosenbaum, Lorie Roth, Robert Scholes, Joanne Shattock, Rosemary T. VanArsdel, Alfred Bendixen, Alarma Kathleen Brown, Michael J. Kiskis, Debra A. Castillo, Rey Chow, John F. Crossen, Robert F. Fleissner, Regenia Gagnier, Nicholas Howe, M. Thomas Inge, Frank Mehring, Hyungji Park, Jahan Ramazani, Kenneth M. Roemer, Deborah D. Rogers, A. LaVonne Brown Ruoff, Regina M. Schwartz, John T. Shawcross, Brenda R. Silver, Andrew von Hendy, Virginia Wright Wexman, Britta Zangen, A. Owen Aldridge, Paula R. Backscheider, Roland Bartel, E. M. Forster, Milton Birnbaum, Jonathan Bishop, Crystal Downing, Frank H. Ellis, Roberto Forns-Broggi, James R. Giles, Mary E. Giles, Susan Blair Green, Madelyn Gutwirth, Constance B. Hieatt, Titi Adepitan, Edgar C. Knowlton, Jr., Emanuel Mussman, Sally Todd Nelson, Robert O. Preyer, David Diego Rodriguez, Guy Stern, James Thorpe, Robert J. Wilson, Rebecca S. Beal, Joyce Simutis, Betsy Bowden, Sara Cooper, Wheeler Winston Dixon, Tarek el Ariss, Richard Jewell, John W. Kronik, Wendy Martin, Stuart Y. McDougal, Hugo Méndez-Ramírez, Ivy Schweitzer, Armand E. Singer, G. Thomas Tanselle, Tom Bishop, Mary Ann Caws, Marcel Gutwirth, Christophe Ippolito, Lawrence D. Kritzman, James Longenbach, Tim McCracken, Wolfe S. Molitor, Diane Quantic, Gregory Rabassa, Ellen M. Tsagaris, Anthony C. Yu, Betty Jean Craige, Wendell V. Harris, J. Hillis Miller, Jesse G. Swan, Helene Zimmer-Loew, Peter Berek, James Chandler, Hanna K. Charney, Philip Cohen, Judith Fetterley, Herbert Lindenberger, Julia Reinhard Lupton, Maximillian E. Novak, Richard Ohmann, Marjorie Perloff, Mark Reynolds, James Sledd, Harriet Turner, Marie Umeh, Flavia Aloya, Regina Barreca, Konrad Bieber, Ellis Hanson, William J. Hyde, Holly A. Laird, David Leverenz, Allen Michie, J. Wesley Miller, Marvin Rosenberg, Daniel R. Schwarz, Elizabeth Welt Trahan, Jean Fagan Yellin
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- Journal:
- PMLA / Publications of the Modern Language Association of America / Volume 115 / Issue 7 / December 2000
- Published online by Cambridge University Press:
- 23 October 2020, pp. 1986-2078
- Print publication:
- December 2000
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