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The power of public–private partnership in medical technology innovation: Lessons from the development of FDA-cleared medical devices for assessment of concussion
- Michael E. Singer, Dallas C. Hack, Daniel F. Hanley, Jr.
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- Journal:
- Journal of Clinical and Translational Science / Volume 6 / Issue 1 / 2022
- Published online by Cambridge University Press:
- 10 March 2022, e42
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Given the convergence of the long and challenging development path for medical devices with the need for diagnostic capabilities for mild traumatic brain injury (mTBI/concussion), the effective role of public–private partnership (PPP) can be demonstrated to yield Food and Drug Administration (FDA) clearances and innovative product introductions. An overview of the mTBI problem and landscape was performed. A detailed situation analysis of an example of a PPP yielding an innovative product was further demonstrated. The example of PPP has led to multiple FDA clearances and product introductions in the TBI diagnostic product category where there was an urgent military and public need. Important lessons included defining the primary public and military health objective for new product introduction, the importance of the government–academia–industry PPP triad with a “collaboration towards solutions” Quality-by-Design (QbD) mindset to assure clinical validity with regulatory compliance, the development of device comparators and integration of measurements into a robust, evidence-based statistical and FDA pathway, and the utility of top-down, flexible, practical action while operating within governmental guidelines and patient safety.
Contributors
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- By Mitchell Aboulafia, Frederick Adams, Marilyn McCord Adams, Robert M. Adams, Laird Addis, James W. Allard, David Allison, William P. Alston, Karl Ameriks, C. Anthony Anderson, David Leech Anderson, Lanier Anderson, Roger Ariew, David Armstrong, Denis G. Arnold, E. J. Ashworth, Margaret Atherton, Robin Attfield, Bruce Aune, Edward Wilson Averill, Jody Azzouni, Kent Bach, Andrew Bailey, Lynne Rudder Baker, Thomas R. Baldwin, Jon Barwise, George Bealer, William Bechtel, Lawrence C. Becker, Mark A. Bedau, Ernst Behler, José A. Benardete, Ermanno Bencivenga, Jan Berg, Michael Bergmann, Robert L. Bernasconi, Sven Bernecker, Bernard Berofsky, Rod Bertolet, Charles J. Beyer, Christian Beyer, Joseph Bien, Joseph Bien, Peg Birmingham, Ivan Boh, James Bohman, Daniel Bonevac, Laurence BonJour, William J. Bouwsma, Raymond D. Bradley, Myles Brand, Richard B. Brandt, Michael E. Bratman, Stephen E. Braude, Daniel Breazeale, Angela Breitenbach, Jason Bridges, David O. Brink, Gordon G. Brittan, Justin Broackes, Dan W. Brock, Aaron Bronfman, Jeffrey E. Brower, Bartosz Brozek, Anthony Brueckner, Jeffrey Bub, Lara Buchak, Otavio Bueno, Ann E. Bumpus, Robert W. Burch, John Burgess, Arthur W. Burks, Panayot Butchvarov, Robert E. Butts, Marina Bykova, Patrick Byrne, David Carr, Noël Carroll, Edward S. Casey, Victor Caston, Victor Caston, Albert Casullo, Robert L. Causey, Alan K. L. Chan, Ruth Chang, Deen K. Chatterjee, Andrew Chignell, Roderick M. Chisholm, Kelly J. Clark, E. J. Coffman, Robin Collins, Brian P. Copenhaver, John Corcoran, John Cottingham, Roger Crisp, Frederick J. Crosson, Antonio S. Cua, Phillip D. Cummins, Martin Curd, Adam Cureton, Andrew Cutrofello, Stephen Darwall, Paul Sheldon Davies, Wayne A. Davis, Timothy Joseph Day, Claudio de Almeida, Mario De Caro, Mario De Caro, John Deigh, C. F. Delaney, Daniel C. Dennett, Michael R. DePaul, Michael Detlefsen, Daniel Trent Devereux, Philip E. Devine, John M. Dillon, Martin C. Dillon, Robert DiSalle, Mary Domski, Alan Donagan, Paul Draper, Fred Dretske, Mircea Dumitru, Wilhelm Dupré, Gerald Dworkin, John Earman, Ellery Eells, Catherine Z. Elgin, Berent Enç, Ronald P. Endicott, Edward Erwin, John Etchemendy, C. Stephen Evans, Susan L. Feagin, Solomon Feferman, Richard Feldman, Arthur Fine, Maurice A. Finocchiaro, William FitzPatrick, Richard E. Flathman, Gvozden Flego, Richard Foley, Graeme Forbes, Rainer Forst, Malcolm R. Forster, Daniel Fouke, Patrick Francken, Samuel Freeman, Elizabeth Fricker, Miranda Fricker, Michael Friedman, Michael Fuerstein, Richard A. Fumerton, Alan Gabbey, Pieranna Garavaso, Daniel Garber, Jorge L. A. Garcia, Robert K. Garcia, Don Garrett, Philip Gasper, Gerald Gaus, Berys Gaut, Bernard Gert, Roger F. Gibson, Cody Gilmore, Carl Ginet, Alan H. Goldman, Alvin I. Goldman, Alfonso Gömez-Lobo, Lenn E. Goodman, Robert M. Gordon, Stefan Gosepath, Jorge J. E. Gracia, Daniel W. Graham, George A. Graham, Peter J. Graham, Richard E. Grandy, I. Grattan-Guinness, John Greco, Philip T. Grier, Nicholas Griffin, Nicholas Griffin, David A. Griffiths, Paul J. Griffiths, Stephen R. Grimm, Charles L. Griswold, Charles B. Guignon, Pete A. Y. Gunter, Dimitri Gutas, Gary Gutting, Paul Guyer, Kwame Gyekye, Oscar A. Haac, Raul Hakli, Raul Hakli, Michael Hallett, Edward C. Halper, Jean Hampton, R. James Hankinson, K. R. Hanley, Russell Hardin, Robert M. Harnish, William Harper, David Harrah, Kevin Hart, Ali Hasan, William Hasker, John Haugeland, Roger Hausheer, William Heald, Peter Heath, Richard Heck, John F. Heil, Vincent F. Hendricks, Stephen Hetherington, Francis Heylighen, Kathleen Marie Higgins, Risto Hilpinen, Harold T. Hodes, Joshua Hoffman, Alan Holland, Robert L. Holmes, Richard Holton, Brad W. Hooker, Terence E. Horgan, Tamara Horowitz, Paul Horwich, Vittorio Hösle, Paul Hoβfeld, Daniel Howard-Snyder, Frances Howard-Snyder, Anne Hudson, Deal W. Hudson, Carl A. Huffman, David L. Hull, Patricia Huntington, Thomas Hurka, Paul Hurley, Rosalind Hursthouse, Guillermo Hurtado, Ronald E. Hustwit, Sarah Hutton, Jonathan Jenkins Ichikawa, Harry A. Ide, David Ingram, Philip J. Ivanhoe, Alfred L. Ivry, Frank Jackson, Dale Jacquette, Joseph Jedwab, Richard Jeffrey, David Alan Johnson, Edward Johnson, Mark D. Jordan, Richard Joyce, Hwa Yol Jung, Robert Hillary Kane, Tomis Kapitan, Jacquelyn Ann K. Kegley, James A. Keller, Ralph Kennedy, Sergei Khoruzhii, Jaegwon Kim, Yersu Kim, Nathan L. King, Patricia Kitcher, Peter D. Klein, E. D. Klemke, Virginia Klenk, George L. Kline, Christian Klotz, Simo Knuuttila, Joseph J. Kockelmans, Konstantin Kolenda, Sebastian Tomasz Kołodziejczyk, Isaac Kramnick, Richard Kraut, Fred Kroon, Manfred Kuehn, Steven T. Kuhn, Henry E. Kyburg, John Lachs, Jennifer Lackey, Stephen E. Lahey, Andrea Lavazza, Thomas H. Leahey, Joo Heung Lee, Keith Lehrer, Dorothy Leland, Noah M. Lemos, Ernest LePore, Sarah-Jane Leslie, Isaac Levi, Andrew Levine, Alan E. Lewis, Daniel E. Little, Shu-hsien Liu, Shu-hsien Liu, Alan K. L. Chan, Brian Loar, Lawrence B. Lombard, John Longeway, Dominic McIver Lopes, Michael J. Loux, E. J. Lowe, Steven Luper, Eugene C. Luschei, William G. Lycan, David Lyons, David Macarthur, Danielle Macbeth, Scott MacDonald, Jacob L. Mackey, Louis H. Mackey, Penelope Mackie, Edward H. Madden, Penelope Maddy, G. B. Madison, Bernd Magnus, Pekka Mäkelä, Rudolf A. Makkreel, David Manley, William E. Mann (W.E.M.), Vladimir Marchenkov, Peter Markie, Jean-Pierre Marquis, Ausonio Marras, Mike W. Martin, A. P. Martinich, William L. McBride, David McCabe, Storrs McCall, Hugh J. McCann, Robert N. McCauley, John J. McDermott, Sarah McGrath, Ralph McInerny, Daniel J. McKaughan, Thomas McKay, Michael McKinsey, Brian P. McLaughlin, Ernan McMullin, Anthonie Meijers, Jack W. Meiland, William Jason Melanson, Alfred R. Mele, Joseph R. Mendola, Christopher Menzel, Michael J. Meyer, Christian B. Miller, David W. Miller, Peter Millican, Robert N. Minor, Phillip Mitsis, James A. Montmarquet, Michael S. Moore, Tim Moore, Benjamin Morison, Donald R. Morrison, Stephen J. Morse, Paul K. Moser, Alexander P. D. Mourelatos, Ian Mueller, James Bernard Murphy, Mark C. Murphy, Steven Nadler, Jan Narveson, Alan Nelson, Jerome Neu, Samuel Newlands, Kai Nielsen, Ilkka Niiniluoto, Carlos G. Noreña, Calvin G. Normore, David Fate Norton, Nikolaj Nottelmann, Donald Nute, David S. Oderberg, Steve Odin, Michael O’Rourke, Willard G. Oxtoby, Heinz Paetzold, George S. Pappas, Anthony J. Parel, Lydia Patton, R. P. Peerenboom, Francis Jeffry Pelletier, Adriaan T. Peperzak, Derk Pereboom, Jaroslav Peregrin, Glen Pettigrove, Philip Pettit, Edmund L. Pincoffs, Andrew Pinsent, Robert B. Pippin, Alvin Plantinga, Louis P. Pojman, Richard H. Popkin, John F. Post, Carl J. Posy, William J. Prior, Richard Purtill, Michael Quante, Philip L. Quinn, Philip L. Quinn, Elizabeth S. Radcliffe, Diana Raffman, Gerard Raulet, Stephen L. Read, Andrews Reath, Andrew Reisner, Nicholas Rescher, Henry S. Richardson, Robert C. Richardson, Thomas Ricketts, Wayne D. Riggs, Mark Roberts, Robert C. Roberts, Luke Robinson, Alexander Rosenberg, Gary Rosenkranz, Bernice Glatzer Rosenthal, Adina L. Roskies, William L. Rowe, T. M. Rudavsky, Michael Ruse, Bruce Russell, Lilly-Marlene Russow, Dan Ryder, R. M. Sainsbury, Joseph Salerno, Nathan Salmon, Wesley C. Salmon, Constantine Sandis, David H. Sanford, Marco Santambrogio, David Sapire, Ruth A. Saunders, Geoffrey Sayre-McCord, Charles Sayward, James P. Scanlan, Richard Schacht, Tamar Schapiro, Frederick F. Schmitt, Jerome B. Schneewind, Calvin O. Schrag, Alan D. Schrift, George F. Schumm, Jean-Loup Seban, David N. Sedley, Kenneth Seeskin, Krister Segerberg, Charlene Haddock Seigfried, Dennis M. Senchuk, James F. Sennett, William Lad Sessions, Stewart Shapiro, Tommie Shelby, Donald W. Sherburne, Christopher Shields, Roger A. Shiner, Sydney Shoemaker, Robert K. Shope, Kwong-loi Shun, Wilfried Sieg, A. John Simmons, Robert L. Simon, Marcus G. Singer, Georgette Sinkler, Walter Sinnott-Armstrong, Matti T. Sintonen, Lawrence Sklar, Brian Skyrms, Robert C. Sleigh, Michael Anthony Slote, Hans Sluga, Barry Smith, Michael Smith, Robin Smith, Robert Sokolowski, Robert C. Solomon, Marta Soniewicka, Philip Soper, Ernest Sosa, Nicholas Southwood, Paul Vincent Spade, T. L. S. Sprigge, Eric O. Springsted, George J. Stack, Rebecca Stangl, Jason Stanley, Florian Steinberger, Sören Stenlund, Christopher Stephens, James P. Sterba, Josef Stern, Matthias Steup, M. A. Stewart, Leopold Stubenberg, Edith Dudley Sulla, Frederick Suppe, Jere Paul Surber, David George Sussman, Sigrún Svavarsdóttir, Zeno G. Swijtink, Richard Swinburne, Charles C. Taliaferro, Robert B. Talisse, John Tasioulas, Paul Teller, Larry S. Temkin, Mark Textor, H. S. Thayer, Peter Thielke, Alan Thomas, Amie L. Thomasson, Katherine Thomson-Jones, Joshua C. Thurow, Vzalerie Tiberius, Terrence N. Tice, Paul Tidman, Mark C. Timmons, William Tolhurst, James E. Tomberlin, Rosemarie Tong, Lawrence Torcello, Kelly Trogdon, J. D. Trout, Robert E. Tully, Raimo Tuomela, John Turri, Martin M. Tweedale, Thomas Uebel, Jennifer Uleman, James Van Cleve, Harry van der Linden, Peter van Inwagen, Bryan W. Van Norden, René van Woudenberg, Donald Phillip Verene, Samantha Vice, Thomas Vinci, Donald Wayne Viney, Barbara Von Eckardt, Peter B. M. Vranas, Steven J. Wagner, William J. Wainwright, Paul E. Walker, Robert E. Wall, Craig Walton, Douglas Walton, Eric Watkins, Richard A. Watson, Michael V. Wedin, Rudolph H. Weingartner, Paul Weirich, Paul J. Weithman, Carl Wellman, Howard Wettstein, Samuel C. Wheeler, Stephen A. White, Jennifer Whiting, Edward R. Wierenga, Michael Williams, Fred Wilson, W. Kent Wilson, Kenneth P. Winkler, John F. Wippel, Jan Woleński, Allan B. Wolter, Nicholas P. Wolterstorff, Rega Wood, W. Jay Wood, Paul Woodruff, Alison Wylie, Gideon Yaffe, Takashi Yagisawa, Yutaka Yamamoto, Keith E. Yandell, Xiaomei Yang, Dean Zimmerman, Günter Zoller, Catherine Zuckert, Michael Zuckert, Jack A. Zupko (J.A.Z.)
- Edited by Robert Audi, University of Notre Dame, Indiana
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- The Cambridge Dictionary of Philosophy
- Published online:
- 05 August 2015
- Print publication:
- 27 April 2015, pp ix-xxx
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Contributors
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- By Mowaffaq Almikhlafi, Osama Al-muslim, Robert Arntfield, Ian M Ball, Sue Berney, Mohit Bhutani, Clay A Block, Ken Blonde, Rudi Brits, Ron Butler, Lois Champion, Chris Clarke, Linda Denehy, Joseph Dreier, A Ebersohn, Shane W English, Ari Ercole, Darren H Freed, John Fuller, Julio P Zavala Georffino, RT Noel Gibney, Jeff Granton, Donald EG Griesdale, Arun K Gupta, Wael Haddara, Ahmed F Hegazy, Umjeet Singh Jolly, Philip M Jones, Ilya Kagan, Kala Kathirgamanathan, Harneet Kaur, John Kellett, Bhupesh Khadka, Biniam Kidane, Carlos Kidel, Anand Kumar, Alejandro Lazo-Langner, David Leasa, W Robert Leeper, Stephen Y Liang, Tania Ligori, Jaimie Manlucu, Janet Martin, Ian McConachie, Alan McGlennan, Lauralyn McIntyre, Tina Mele, MJ Naisbitt, Raj Nichani, Daniel H Ovakim, Neil Parry, Daniel Castro Pereira, Thomas Piraino, Brian Pollard, Valerie Schulz, Michael D Sharpe, Rohit K Singal, Pierre Singer, Mark Soth, Christian P Subbe, Jaffer Syed, Ravi Taneja, Tom Varughese, Jennifer Vergel Del Dios, Jessie R Welbourne, Christopher W White, Rebecca P Winsett, Titus C Yeung, G Bryan Young, Shelley R Zieroth
- Edited by John Fuller, University of Western Ontario, Jeff Granton, University of Western Ontario, Ian McConachie, University of Western Ontario
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- Book:
- Handbook of ICU Therapy
- Published online:
- 05 February 2015
- Print publication:
- 04 December 2014, pp vii-xii
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List of contributors
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- By Bjarne F. Alsbjoern, Caroline M. Apovian, Danny Collins, Roland N. Dickerson, Timothy Eden, Peter Faber, Andrew J. Ferguson, David C. Frankenfield, Dympna Gallagher, Maria Gabriella Gentile, Wilson I. Gonsalves, Andrew M. Hetreed, Michael H. Hooper, Jan O. Jansen, Aminah Jatoi, Ying Ji, Ilya Kagan, Andrew J. Kerwin, Dong Wook Kim, Andrew A. Klein, Alistair Lee, Shaul Lev, Peter K. Linden, Paul E. Marik, Robert Martindale, Peter McCanny, Paolo Merlani, Shay Nanthakumaran, Michael S. Nussbaum, Andreas Perren, Carla Prado, Jean-Charles Preiser, Minha Rajput-Ray, Sumantra Ray, Nils Siegenthaler, Mario Siervo, Jonathan A. Silversides, Pierre Singer, John A. Tayek, Euan Thomson, Krista L. Turner, Malissa Warren, Stephen T. Webb, Patricia Wiesen
- Edited by Peter Faber, Mario Siervo, University of Newcastle upon Tyne
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- Book:
- Nutrition in Critical Care
- Published online:
- 05 April 2014
- Print publication:
- 06 March 2014, pp viii-xii
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Unipotent differential algebraic groups as parameterized differential Galois groups
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- Andrey Minchenko, Alexey Ovchinnikov, Michael F. Singer
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 13 / Issue 4 / October 2014
- Published online by Cambridge University Press:
- 18 July 2013, pp. 671-700
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- October 2014
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We deal with aspects of direct and inverse problems in parameterized Picard–Vessiot (PPV) theory. It is known that, for certain fields, a linear differential algebraic group (LDAG) $G$ is a PPV Galois group over these fields if and only if $G$ contains a Kolchin-dense finitely generated group. We show that, for a class of LDAGs $G$, including unipotent groups, $G$ is such a group if and only if it has differential type $0$. We give a procedure to determine if a parameterized linear differential equation has a PPV Galois group in this class and show how one can calculate the PPV Galois group of a parameterized linear differential equation if its Galois group has differential type $0$.
Contributors
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- By Phillip L. Ackerman, Evelyn Au, Frida Barak, Rainer P. Born, Jerome Bruner, Chi-yue Chiu, Karl Edlinger, Anastasia Efklides, Michael W. Eysenck, Evridiki Fioratou, Dida Fleisig, Eva Gatarik, Ken J. Gilhooly, Ksenija Jaušovec, Norbert Jaušovec, Wendy Johnson, Shulamith Kreitler, Arie W. Kruglanski, David J. McGarva, Thomas Nickel, Jaak Panksepp, Klaus-Uwe Panther, K. Ann Renninger, Kathryn R. Riley, Mark A. Runco, Anna Sheveland, Dorothy G. Singer, Jerome L. Singer, Jeff Stewart, Ola Svenson, Nicole Szesny, Wendy Wan, Kineret Weissler, Manfred Wimmer, Dan Zakay, Edward F. Zigler, Josef Zihl
- Edited by Shulamith Kreitler, Tel-Aviv University
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- Book:
- Cognition and Motivation
- Published online:
- 18 December 2013
- Print publication:
- 19 November 2012, pp xi-xviii
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Contributors
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- By Aakash Agarwala, Linda S. Aglio, Rae M. Allain, Paul D. Allen, Houman Amirfarzan, Yasodananda Kumar Areti, Amit Asopa, Edwin G. Avery, Patricia R. Bachiller, Angela M. Bader, Rana Badr, Sibinka Bajic, David J. Baker, Sheila R. Barnett, Rena Beckerly, Lorenzo Berra, Walter Bethune, Sascha S. Beutler, Tarun Bhalla, Edward A. Bittner, Jonathan D. Bloom, Alina V. Bodas, Lina M. Bolanos-Diaz, Ruma R. Bose, Jan Boublik, John P. Broadnax, Jason C. Brookman, Meredith R. Brooks, Roland Brusseau, Ethan O. Bryson, Linda A. Bulich, Kenji Butterfield, William R. Camann, Denise M. Chan, Theresa S. Chang, Jonathan E. Charnin, Mark Chrostowski, Fred Cobey, Adam B. Collins, Mercedes A. Concepcion, Christopher W. Connor, Bronwyn Cooper, Jeffrey B. Cooper, Martha Cordoba-Amorocho, Stephen B. Corn, Darin J. Correll, Gregory J. Crosby, Lisa J. Crossley, Deborah J. Culley, Tomas Cvrk, Michael N. D'Ambra, Michael Decker, Daniel F. Dedrick, Mark Dershwitz, Francis X. Dillon, Pradeep Dinakar, Alimorad G. Djalali, D. John Doyle, Lambertus Drop, Ian F. Dunn, Theodore E. Dushane, Sunil Eappen, Thomas Edrich, Jesse M. Ehrenfeld, Jason M. Erlich, Lucinda L. Everett, Elliott S. Farber, Khaldoun Faris, Eddy M. Feliz, Massimo Ferrigno, Richard S. Field, Michael G. Fitzsimons, Hugh L. Flanagan Jr., Vladimir Formanek, Amanda A. Fox, John A. Fox, Gyorgy Frendl, Tanja S. Frey, Samuel M. Galvagno Jr., Edward R. Garcia, Jonathan D. Gates, Cosmin Gauran, Brian J. Gelfand, Simon Gelman, Alexander C. Gerhart, Peter Gerner, Omid Ghalambor, Christopher J. Gilligan, Christian D. Gonzalez, Noah E. Gordon, William B. Gormley, Thomas J. Graetz, Wendy L. Gross, Amit Gupta, James P. Hardy, Seetharaman Hariharan, Miriam Harnett, Philip M. Hartigan, Joaquim M. Havens, Bishr Haydar, Stephen O. Heard, James L. Helstrom, David L. Hepner, McCallum R. Hoyt, Robert N. Jamison, Karinne Jervis, Stephanie B. Jones, Swaminathan Karthik, Richard M. Kaufman, Shubjeet Kaur, Lee A. Kearse Jr., John C. Keel, Scott D. Kelley, Albert H. Kim, Amy L. Kim, Grace Y. Kim, Robert J. Klickovich, Robert M. Knapp, Bhavani S. Kodali, Rahul Koka, Alina Lazar, Laura H. Leduc, Stanley Leeson, Lisa R. Leffert, Scott A. LeGrand, Patricio Leyton, J. Lance Lichtor, John Lin, Alvaro A. Macias, Karan Madan, Sohail K. Mahboobi, Devi Mahendran, Christine Mai, Sayeed Malek, S. Rao Mallampati, Thomas J. Mancuso, Ramon Martin, Matthew C. Martinez, J. A. Jeevendra Martyn, Kai Matthes, Tommaso Mauri, Mary Ellen McCann, Shannon S. McKenna, Dennis J. McNicholl, Abdel-Kader Mehio, Thor C. Milland, Tonya L. K. Miller, John D. Mitchell, K. Annette Mizuguchi, Naila Moghul, David R. Moss, Ross J. Musumeci, Naveen Nathan, Ju-Mei Ng, Liem C. Nguyen, Ervant Nishanian, Martina Nowak, Ala Nozari, Michael Nurok, Arti Ori, Rafael A. Ortega, Amy J. Ortman, David Oxman, Arvind Palanisamy, Carlo Pancaro, Lisbeth Lopez Pappas, Benjamin Parish, Samuel Park, Deborah S. Pederson, Beverly K. Philip, James H. Philip, Silvia Pivi, Stephen D. Pratt, Douglas E. Raines, Stephen L. Ratcliff, James P. Rathmell, J. Taylor Reed, Elizabeth M. Rickerson, Selwyn O. Rogers Jr., Thomas M. Romanelli, William H. Rosenblatt, Carl E. Rosow, Edgar L. Ross, J. Victor Ryckman, Mônica M. Sá Rêgo, Nicholas Sadovnikoff, Warren S. Sandberg, Annette Y. Schure, B. Scott Segal, Navil F. Sethna, Swapneel K. Shah, Shaheen F. Shaikh, Fred E. Shapiro, Torin D. Shear, Prem S. Shekar, Stanton K. Shernan, Naomi Shimizu, Douglas C. Shook, Kamal K. Sikka, Pankaj K. Sikka, David A. Silver, Jeffrey H. Silverstein, Emily A. Singer, Ken Solt, Spiro G. Spanakis, Wolfgang Steudel, Matthias Stopfkuchen-Evans, Michael P. Storey, Gary R. Strichartz, Balachundhar Subramaniam, Wariya Sukhupragarn, John Summers, Shine Sun, Eswar Sundar, Sugantha Sundar, Neelakantan Sunder, Faraz Syed, Usha B. Tedrow, Nelson L. Thaemert, George P. Topulos, Lawrence C. Tsen, Richard D. Urman, Charles A. Vacanti, Francis X. Vacanti, Joshua C. Vacanti, Assia Valovska, Ivan T. Valovski, Mary Ann Vann, Susan Vassallo, Anasuya Vasudevan, Kamen V. Vlassakov, Gian Paolo Volpato, Essi M. Vulli, J. Matthias Walz, Jingping Wang, James F. Watkins, Maxwell Weinmann, Sharon L. Wetherall, Mallory Williams, Sarah H. Wiser, Zhiling Xiong, Warren M. Zapol, Jie Zhou
- Edited by Charles Vacanti, Scott Segal, Pankaj Sikka, Richard Urman
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- Book:
- Essential Clinical Anesthesia
- Published online:
- 05 January 2012
- Print publication:
- 11 July 2011, pp xv-xxviii
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1 - Galois Theory of Linear Differential Equations
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- By Michael F. Singer, Department of Mathematics North Carolina State University Raleigh, NC 27695-8205
- Edited by Malcolm A. H. MacCallum, Queen Mary University of London, Alexander V. Mikhailov, University of Leeds
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- Book:
- Algebraic Theory of Differential Equations
- Published online:
- 03 May 2010
- Print publication:
- 04 December 2008, pp 1-82
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Summary
Introduction
This paper is an expanded version of the 10 lectures I gave as the 2006 London Mathematical Society Invited Lecture Series at the Heriot-Watt University, 31 July - 4 August 2006. My goal was to give the audience an introduction to the algebraic, analytic and algorithmic aspects of the Galois theory of linear differential equations by focusing on some of the main ideas and philosophies and on examples. There are several texts ([1, 2, 3, 4, 5] to name a few) that give detailed expositions and I hope that the taste offered here will encourage the reader to dine more fully with one of these.
The rest of the paper is organized as follows. In Section 1.2, What is a Linear Differential Equation?, I discuss three ways to think about linear differential equations: scalar equations, linear systems and differential modules. Just as it is useful to think of linear maps in terms of linear equations, matrices and associated modules, it will be helpful in future sections to go back and forth between the different ways of presenting linear differential equations.
In Section 1.3, Basic Galois Theory and Applications, I will give the basic definitions and describe the Galois correspondence. In addition I will describe the notion of monodromy and its relation to the Galois theory. I will end by giving several applications and ramifications, one of which will be to answer the question Although y = cos x satisfies y″ + y = 0, why doesn't sec x satisfy a linear differential equation?
On the definitions of difference Galois groups
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- By Zoé Chatzidakis, CNRS — Université Paris 7, Charlotte Hardouin, Universität Heidelberg IWR INF 368, Michael F. Singer, University of North Carolina
- Edited by Zoé Chatzidakis, Centre National de la Recherche Scientifique (CNRS), Paris, Dugald Macpherson, University of Leeds, Anand Pillay, University of Leeds, Alex Wilkie, University of Manchester
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- Book:
- Model Theory with Applications to Algebra and Analysis
- Published online:
- 04 August 2010
- Print publication:
- 22 May 2008, pp 73-110
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Contributors
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- By Robert S. Agnew, Lara M. Belliston, Daniel M. Blonigen, Michel Boivin, Jeanne Brooks‐Gunn, Andrew Canastar, Noel A. Card, Emil F. Coccaro, Nicki R. Crick, Linda L. Dahlberg, Garth Davies, Scott H. Decker, Kenneth A. Dodge, Dorothy L. Espelage, Jeffrey Fagan, Albert D. Farrell, David P. Farrington, Daniel J. Flannery, Mark S. Fleisher, Vangie A. Foshee, Holly Foster, Richard J. Gelles, Denise C. Gottfredson, Gary D. Gottfredson, Michael R. Gottfredson, Richard E. Heyman, James C. (Buddy) Howell, Megan Q. Howell, Li Huang, L. Rowell Huesmann, Cynthia Irvin, Gary F. Jensen, Yoshito Kawabata, Lucyna Kirwil, Jeff M. Kretschmar, Robert F. Krueger, Markus J. P. Kruesi, Benjamin B. Lahey, Royce Lee, Scott O. Lilienfeld, Todd D. Little, Anne Martin, Rebecca A. Matthew, Stephen C. Maxson, Jacquelyn Mize, Terrie E. Moffitt, Daniel S. Nagin, Jamie M. Ostrov, Christopher J. Patrick, Bowen Paulle, Gregory S. Pettit, Adrian Raine, Soo Hyun Rhee, Angela Scarpa, Jean R. Séguin, Michelle R. Sherrill, Mark I. Singer, Amy M. Smith Slep, Kevin J. Strom, Patrick Sylvers, Patrick H. Tolan, Elizabeth Trejos‐Castillo, Richard E. Tremblay, Manfred van Dulmen, Johan van Wilsem, Alexander T. Vazsonyi, Edelyn Verona, Frank Vitaro, Monique Vulin‐Reynolds, Irwin D. Waldman, Mark Warr, Stanley Wasserman, Deanna L. Wilkinson
- Edited by Daniel J. Flannery, Kent State University, Ohio, Alexander T. Vazsonyi, Auburn University, Alabama, Irwin D. Waldman, Emory University, Atlanta
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- Book:
- The Cambridge Handbook of Violent Behavior and Aggression
- Published online:
- 05 June 2012
- Print publication:
- 03 September 2007, pp xi-xviii
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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
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- December 2000
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The model theory of ordered differential fields
- Michael F. Singer
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- Journal:
- The Journal of Symbolic Logic / Volume 43 / Issue 1 / March 1978
- Published online by Cambridge University Press:
- 12 March 2014, pp. 82-91
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- March 1978
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In this paper, we show that the theory of ordered differential fields has a model completion. We also show that any real differential field, finitely generated over the rational numbers, is isomorphic to some field of real meromorphic functions. In the last section of this paper, we combine these two results and discuss the problem of deciding if a system of differential equations has real analytic solutions. The author wishes to thank G. Stengle for some stimulating and helpful conversations and for drawing our attention to fields of real meromorphic functions.
§ 1. Real and ordered fields. A real field is a field in which −1 is not a sum of squares. An ordered field is a field F together with a binary relation < which totally orders F and satisfies the two properties: (1) If 0 < x and 0 < y then 0 < xy. (2) If x < y then, for all z in F, x + z < y + z. An element x of an ordered field is positive if x > 0. One can see that the square of any element is positive and that the sum of positive elements is positive. Since −1 is not positive, an ordered field is a real field. Conversely, given a real field F, it is known that one can define an ordering (not necessarily uniquely) on F [2, p. 274]. An ordered field F is a real closed field if: (1) every positive element is a square, and (2) every polynomial of odd degree with coefficients in F has a root in F. For example, the real numbers form a real closed field. Every ordered field can be embedded in a real closed field. It is also known that, in a real closed field K, polynomials satisfy the intermediate value property, i.e. if f(x) ∈ K[x] and a, b ∈ K, a < b, and f(a)f(b) < 0 then there is a c in K such that f(c) = 0.