9 results
Enhanced laser-driven proton acceleration with gas–foil targets
- Dan Levy, X. Davoine, A. Debayle, L. Gremillet, V. Malka
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- Journal:
- Journal of Plasma Physics / Volume 86 / Issue 6 / December 2020
- Published online by Cambridge University Press:
- 19 November 2020, 905860608
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We study numerically the mechanisms of proton acceleration in gas–foil targets driven by an ultraintense femtosecond laser pulse. The target consists of a near-critical-density hydrogen gas layer of a few tens of microns attached to a $2\ \mathrm {\mu }$m-thick solid carbon foil with a contaminant thin proton layer at its back side. Two-dimensional particle-in-cell simulations show that, at optimal gas density, the maximum energy of the contaminant protons is increased by a factor of $\sim$4 compared with a single foil target. This improvement originates from the near-complete laser absorption into relativistic electrons in the gas. Several energetic electron populations are identified, and their respective effect on the proton acceleration is quantified by computing the electrostatic fields that they generate at the protons’ positions. While each of those electron groups is found to contribute substantially to the overall accelerating field, the dominant one is the relativistic thermal bulk that results from the nonlinear wakefield excited in the gas, as analysed recently by Debayle et al. (New J. Phys., vol. 19, 2017, 123013). Our analysis also reveals the important role of the neighbouring ions in the acceleration of the fastest protons, and the onset of multidimensional effects caused by the time-increasing curvature of the proton layer.
Data Mining to Guide a Program to Prevent Infection Related Readmissions From Skilled Nursing Facilities
- Anna Stachel, Julie Klock, Dan Ding, Jennifer Lighter, Kwesi Daniel, Levi Waldron
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- Journal:
- Infection Control & Hospital Epidemiology / Volume 41 / Issue S1 / October 2020
- Published online by Cambridge University Press:
- 02 November 2020, pp. s29-s30
- Print publication:
- October 2020
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Background: Readmissions to hospitals are common, costly and often preventable, notably readmissions due to infections. A 30-day readmission analysis following hospital discharges, found much of the variation in Medicare spending between hospitals was related to readmissions and skilled nursing facility (SNF) care. Although some readmissions of patients with advanced disease are not preventable, efforts to decrease readmission are most effectively directed towards those patients with intermediate levels of a specific risk. A prediction model to identify patients at highest (or intermediate) risk of infection readmission will help healthcare administrators and providers to allocate appropriate resources. Hospitals should use electronic health record (EHR) data with modern data mining techniques to create more curated, sophisticated models as part of a comprehensive transitional care program. We propose using the risk estimates of a validated prediction model to notify stakeholders and develop readmission rate reports by SNF or discharging physician. Methods: We applied machine learning (ML) methods to predict the risk of 30-day readmission due to sepsis and pneumonia of patients discharged to SNF. We used our EHR data during 2012–2017 to train and data from 2018 to validate. We applied ML algorithms to data including logistic regression, random forest, gradient boosting trees, and support vector machine. Data from EDW and EPIC clarity tables were extracted and managed using SAS Base 9.4 and Enterprise Miner 14.3 (SAS Institute, Cary, NC). We assessed the discrimination and calibration to select the most effective prediction model. Using the resulted risk estimates, we created a notification system and reports for key stakeholders. Results: Figures 1 and 2 show the discrimination and calibration results of the final selected gradient boosting model (GBM). For predicting unplanned readmissions with sepsis and with pneumonia within 30 days after discharge to SNF, the c-statistic for final GBM model with 140 features was 0.69 (95% CI 0.65-0.73) and 73 features was 0.71 (95% CI 0.66-0.75), respectively. Table 1 lists features important to the validation set of the prediction model. We used estimates from these models to develop a daily email notification of patients discharged to SNF stratified into a low, medium, and high risk group for sepsis and pneumonia. We additionally created reports with case-mix adjustments to benchmark SNFs and discharging physicians to monitor and understand performance. Conclusions: Hospitals should leverage the plethora of data found in EHRs to curate readmission prediction models, and promote collaboration among transitional care teams and IPC to ultimately reduce readmissions due to sepsis and pneumonia.
Funding: None
Disclosures: None
Antimicrobial use for asymptomatic bacteriuria—First, do no harm
- Yana Shpunt, Inna Estrin, Yossef Levi, Hodaya Saadon, Galit Ben-Yossef, Lili Goldshtein, Dan Klafter, Shani Zilberman-Itskovich, Debby Ben-David, David E. Katz, Tsillia Lazarovitch, Ronit Zaidenstein, Dror Marchaim
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- Journal:
- Infection Control & Hospital Epidemiology / Volume 42 / Issue 1 / January 2021
- Published online by Cambridge University Press:
- 13 August 2020, pp. 37-42
- Print publication:
- January 2021
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Objective:
Administration of antimicrobials to patients with asymptomatic bacteriuria (ASB) is a common error that can lead to worse outcomes. However, controlled analyses quantifying the commonality and impact of this practice are lacking. We analyzed the independent predictors for antimicrobials misuse in ASB and quantified the impact of this practice on clinical outcomes.
Design:Retrospective case-control and cohort analyses for calendar year 2017.
Setting:Tertiary-care, university-affiliated medical center.
Patients:The study included adult (>18 years) patients with positive urine culture. Pregnant women, renal transplant recipients, and patients who underwent urologic procedures were excluded.
Methods:ASB was determined according to US Centers for Disease Control and Prevention (CDC) criteria. Multivariable logistic regression models were constructed to analyze predictors and outcomes associated with antimicrobial use for patients with ASB.
Results:The study included 1,530 patient-unique positive urine cultures. Among these patients, 610 patients (40%) were determined to have ASB. Of the 696 isolates, 219 (36%) were multidrug-resistant organisms (MDROs). Also, 178 (29%) patients received antimicrobials specifically due to the ASB. Independent predictors for improper administration of antimicrobials were dependent functional status (adjusted odds ratio [aOR], 2.3; 95% CI, 1.4–3.6) and male sex (aOR, 2; 95% CI, 1.25–2.6). Use of antimicrobials was independently associated with re-hospitalizations (aOR, 1.7; 95% CI, 1.1–2.6) and later, acute Clostridioides difficile infections (CDI) in the following 90 days (aOR, 4.5; 95% CI, 2–10.6).
Conclusions:ASB is a common condition, frequently resulting from an MDRO. Male sex and poor functional status were independent predictors for mistreatment, and this practice was independently associated with rehospitalizations and CDI in the following 90 days.
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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SOLVABILITY OF FINITE GROUPS VIA CONDITIONS ON PRODUCTS OF 2-ELEMENTS AND ODD p-ELEMENTS
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- GIL KAPLAN, DAN LEVY
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 82 / Issue 2 / October 2010
- Published online by Cambridge University Press:
- 26 April 2010, pp. 265-273
- Print publication:
- October 2010
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We observe that a solvability criterion for finite groups, conjectured by Miller [The product of two or more groups, Trans. Amer. Math. Soc.12 (1911)] and Hall [A characteristic property of soluble groups, J. London Math. Soc.12 (1937)] and proved by Thompson [Nonsolvable finite groups all of whose local subgroups are solvable, Bull. Amer. Math. Soc.74(3) (1968)], can be sharpened as follows: a finite group is nonsolvable if and only if it has a nontrivial 2-element and an odd p-element, such that the order of their product is not divisible by either 2 or p. We also prove a solvability criterion involving conjugates of odd p-elements. Finally, we define, via a condition on products of p-elements with p′-elements, a formation Pp,p′, for each prime p. We show that P2,2′ (which contains the odd-order groups) is properly contained in the solvable formation.
MULTIPLICITIES IN SYLOW SEQUENCES AND THE SOLVABLE RADICAL
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- GIL KAPLAN, DAN LEVY
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 78 / Issue 3 / December 2008
- Published online by Cambridge University Press:
- 01 December 2008, pp. 477-486
- Print publication:
- December 2008
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A complete Sylow sequence, 𝒫=P1,…,Pm, of a finite group G is a sequence of m Sylow pi-subgroups of G, one for each pi, where p1,…,pm are all of the distinct prime divisors of |G|. A product of the form P1⋯Pm is called a complete Sylow product of G. We prove that the solvable radical of G equals the intersection of all complete Sylow products of G if, for every composition factor S of G, and for every ordering of the prime divisors of |S|, there exist a complete Sylow sequence 𝒫 of S, and g∈S such that g is uniquely factorizable in 𝒫 . This generalizes our results in Kaplan and Levy [‘The solvable radical of Sylow factorizable groups’, Arch. Math.85(6) (2005), 490–496].
SCHREIER CONDITIONS ON CHIEF FACTORS AND RESIDUALS OF SOLVABLE-LIKE GROUP FORMATIONS
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- GIL KAPLAN, DAN LEVY
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 78 / Issue 1 / August 2008
- Published online by Cambridge University Press:
- 01 August 2008, pp. 97-106
- Print publication:
- August 2008
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Let α be a formation of finite groups which is closed under subgroups and group extensions and which contains the formation of solvable groups. Let G be any finite group. We state and prove equivalences between conditions on chief factors of G and structural characterizations of the α-residual and theα-radical of G. We also discuss the connection of our results to the generalized Fitting subgroup of G.
16 - Products of Sylow subgroups and the solvable radical
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- By Gil Kaplan, The School of Computer Sciences, The Academic College of Tel-Aviv-Yaffo, Dan Levy, The School of Computer Sciences, The Academic College of Tel-Aviv-Yaffo
- Edited by C. M. Campbell, University of St Andrews, Scotland, M. R. Quick, University of St Andrews, Scotland, E. F. Robertson, University of St Andrews, Scotland, G. C. Smith, University of Bath
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- Groups St Andrews 2005
- Published online:
- 20 April 2010
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- 04 January 2007, pp 527-530
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18 - Small Trees and Generalized Neighbor-Joining
- from Part II - Studies on the four themes
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- By Mark Contois, Dan Levy
- Edited by L. Pachter, University of California, Berkeley, B. Sturmfels, University of California, Berkeley
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- Algebraic Statistics for Computational Biology
- Published online:
- 04 August 2010
- Print publication:
- 22 August 2005, pp 335-346
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Summary
Direct reconstruction of phylogenetic trees by maximum likelihood methods is computationally prohibitive for trees with many taxa; however, by computing all trees for subsets of taxa of size m, we can infer the entire tree. In particular, if m = 2, the traditional distance-based methods such as neighbor-joining [Saitou and Nei, 1987] and UPGMA [Sneath and Sokal, 1973] are applicable. Under distance-based methods, 2-leaf subtrees are completely determined by the total length between each pair of leaves. We extend this idea to m leaves by developing the notion of m-dissimilarity [Pachter and Speyer, 2004]. By building trees on subsets of size m of the taxa and rinding the total length, we can obtain an m-dissimilarity map. We will explain the generalized neighbor-joining (GNJ) algorithm [Levy et al., 2005] for obtaining a phylogenetic tree with edge lengths from an m-dissimilarity map.
This algorithm is consistent: given an m-dissimilarity map DT that comes from a tree T, GNJ returns the correct tree. However, in the case of data that is “noisy”, e.g., when the observed dissimilarity map does not lie in the space of trees, the accuracy of GNJ depends on the reliability of the subtree lengths. Numerical methods may run into trouble when models are of high degree (Section 1.3); exact methods for computing subtrees, therefore, could only serve to improve the accuracy of GNJ. One family of such methods consists of algorithms for finding critical points of the ML equations as discussed in Chapter 15 and in [Hoşten et al., 2005].