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Contributors
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- By Rose Teteki Abbey, K. C. Abraham, David Tuesday Adamo, LeRoy H. Aden, Efrain Agosto, Victor Aguilan, Gillian T. W. Ahlgren, Charanjit Kaur AjitSingh, Dorothy B E A Akoto, Giuseppe Alberigo, Daniel E. Albrecht, Ruth Albrecht, Daniel O. Aleshire, Urs Altermatt, Anand Amaladass, Michael Amaladoss, James N. Amanze, Lesley G. Anderson, Thomas C. Anderson, Victor Anderson, Hope S. Antone, María Pilar Aquino, Paula Arai, Victorio Araya Guillén, S. Wesley Ariarajah, Ellen T. Armour, Brett Gregory Armstrong, Atsuhiro Asano, Naim Stifan Ateek, Mahmoud Ayoub, John Alembillah Azumah, Mercedes L. García Bachmann, Irena Backus, J. Wayne Baker, Mieke Bal, Lewis V. Baldwin, William Barbieri, António Barbosa da Silva, David Basinger, Bolaji Olukemi Bateye, Oswald Bayer, Daniel H. Bays, Rosalie Beck, Nancy Elizabeth Bedford, Guy-Thomas Bedouelle, Chorbishop Seely Beggiani, Wolfgang Behringer, Christopher M. Bellitto, Byard Bennett, Harold V. Bennett, Teresa Berger, Miguel A. Bernad, Henley Bernard, Alan E. Bernstein, Jon L. Berquist, Johannes Beutler, Ana María Bidegain, Matthew P. Binkewicz, Jennifer Bird, Joseph Blenkinsopp, Dmytro Bondarenko, Paulo Bonfatti, Riet en Pim Bons-Storm, Jessica A. Boon, Marcus J. Borg, Mark Bosco, Peter C. Bouteneff, François Bovon, William D. Bowman, Paul S. Boyer, David Brakke, Richard E. Brantley, Marcus Braybrooke, Ian Breward, Ênio José da Costa Brito, Jewel Spears Brooker, Johannes Brosseder, Nicholas Canfield Read Brown, Robert F. Brown, Pamela K. Brubaker, Walter Brueggemann, Bishop Colin O. Buchanan, Stanley M. Burgess, Amy Nelson Burnett, J. Patout Burns, David B. Burrell, David Buttrick, James P. Byrd, Lavinia Byrne, Gerado Caetano, Marcos Caldas, Alkiviadis Calivas, William J. Callahan, Salvatore Calomino, Euan K. Cameron, William S. Campbell, Marcelo Ayres Camurça, Daniel F. Caner, Paul E. Capetz, Carlos F. Cardoza-Orlandi, Patrick W. Carey, Barbara Carvill, Hal Cauthron, Subhadra Mitra Channa, Mark D. Chapman, James H. Charlesworth, Kenneth R. Chase, Chen Zemin, Luciano Chianeque, Philip Chia Phin Yin, Francisca H. Chimhanda, Daniel Chiquete, John T. Chirban, Soobin Choi, Robert Choquette, Mita Choudhury, Gerald Christianson, John Chryssavgis, Sejong Chun, Esther Chung-Kim, Charles M. A. Clark, Elizabeth A. Clark, Sathianathan Clarke, Fred Cloud, John B. Cobb, W. Owen Cole, John A Coleman, John J. Collins, Sylvia Collins-Mayo, Paul K. Conkin, Beth A. Conklin, Sean Connolly, Demetrios J. Constantelos, Michael A. Conway, Paula M. Cooey, Austin Cooper, Michael L. Cooper-White, Pamela Cooper-White, L. William Countryman, Sérgio Coutinho, Pamela Couture, Shannon Craigo-Snell, James L. Crenshaw, David Crowner, Humberto Horacio Cucchetti, Lawrence S. Cunningham, Elizabeth Mason Currier, Emmanuel Cutrone, Mary L. Daniel, David D. Daniels, Robert Darden, Rolf Darge, Isaiah Dau, Jeffry C. Davis, Jane Dawson, Valentin Dedji, John W. de Gruchy, Paul DeHart, Wendy J. Deichmann Edwards, Miguel A. De La Torre, George E. Demacopoulos, Thomas de Mayo, Leah DeVun, Beatriz de Vasconcellos Dias, Dennis C. Dickerson, John M. Dillon, Luis Miguel Donatello, Igor Dorfmann-Lazarev, Susanna Drake, Jonathan A. Draper, N. Dreher Martin, Otto Dreydoppel, Angelyn Dries, A. J. Droge, Francis X. D'Sa, Marilyn Dunn, Nicole Wilkinson Duran, Rifaat Ebied, Mark J. Edwards, William H. Edwards, Leonard H. Ehrlich, Nancy L. Eiesland, Martin Elbel, J. Harold Ellens, Stephen Ellingson, Marvin M. Ellison, Robert Ellsberg, Jean Bethke Elshtain, Eldon Jay Epp, Peter C. Erb, Tassilo Erhardt, Maria Erling, Noel Leo Erskine, Gillian R. Evans, Virginia Fabella, Michael A. Fahey, Edward Farley, Margaret A. Farley, Wendy Farley, Robert Fastiggi, Seena Fazel, Duncan S. Ferguson, Helwar Figueroa, Paul Corby Finney, Kyriaki Karidoyanes FitzGerald, Thomas E. FitzGerald, John R. Fitzmier, Marie Therese Flanagan, Sabina Flanagan, Claude Flipo, Ronald B. Flowers, Carole Fontaine, David Ford, Mary Ford, Stephanie A. Ford, Jim Forest, William Franke, Robert M. Franklin, Ruth Franzén, Edward H. Friedman, Samuel Frouisou, Lorelei F. Fuchs, Jojo M. Fung, Inger Furseth, Richard R. Gaillardetz, Brandon Gallaher, China Galland, Mark Galli, Ismael García, Tharscisse Gatwa, Jean-Marie Gaudeul, Luis María Gavilanes del Castillo, Pavel L. Gavrilyuk, Volney P. Gay, Metropolitan Athanasios Geevargis, Kondothra M. George, Mary Gerhart, Simon Gikandi, Maurice Gilbert, Michael J. Gillgannon, Verónica Giménez Beliveau, Terryl Givens, Beth Glazier-McDonald, Philip Gleason, Menghun Goh, Brian Golding, Bishop Hilario M. Gomez, Michelle A. Gonzalez, Donald K. Gorrell, Roy Gottfried, Tamara Grdzelidze, Joel B. Green, Niels Henrik Gregersen, Cristina Grenholm, Herbert Griffiths, Eric W. Gritsch, Erich S. Gruen, Christoffer H. Grundmann, Paul H. Gundani, Jon P. Gunnemann, Petre Guran, Vidar L. Haanes, Jeremiah M. Hackett, Getatchew Haile, Douglas John Hall, Nicholas Hammond, Daphne Hampson, Jehu J. Hanciles, Barry Hankins, Jennifer Haraguchi, Stanley S. Harakas, Anthony John Harding, Conrad L. Harkins, J. William Harmless, Marjory Harper, Amir Harrak, Joel F. Harrington, Mark W. Harris, Susan Ashbrook Harvey, Van A. Harvey, R. Chris Hassel, Jione Havea, Daniel Hawk, Diana L. Hayes, Leslie Hayes, Priscilla Hayner, S. Mark Heim, Simo Heininen, Richard P. Heitzenrater, Eila Helander, David Hempton, Scott H. Hendrix, Jan-Olav Henriksen, Gina Hens-Piazza, Carter Heyward, Nicholas J. Higham, David Hilliard, Norman A. Hjelm, Peter C. Hodgson, Arthur Holder, M. Jan Holton, Dwight N. Hopkins, Ronnie Po-chia Hsia, Po-Ho Huang, James Hudnut-Beumler, Jennifer S. Hughes, Leonard M. Hummel, Mary E. Hunt, Laennec Hurbon, Mark Hutchinson, Susan E. Hylen, Mary Beth Ingham, H. Larry Ingle, Dale T. Irvin, Jon Isaak, Paul John Isaak, Ada María Isasi-Díaz, Hans Raun Iversen, Margaret C. Jacob, Arthur James, Maria Jansdotter-Samuelsson, David Jasper, Werner G. Jeanrond, Renée Jeffery, David Lyle Jeffrey, Theodore W. Jennings, David H. Jensen, Robin Margaret Jensen, David Jobling, Dale A. Johnson, Elizabeth A. Johnson, Maxwell E. Johnson, Sarah Johnson, Mark D. Johnston, F. Stanley Jones, James William Jones, John R. Jones, Alissa Jones Nelson, Inge Jonsson, Jan Joosten, Elizabeth Judd, Mulambya Peggy Kabonde, Robert Kaggwa, Sylvester Kahakwa, Isaac Kalimi, Ogbu U. Kalu, Eunice Kamaara, Wayne C. Kannaday, Musimbi Kanyoro, Veli-Matti Kärkkäinen, Frank Kaufmann, Léon Nguapitshi Kayongo, Richard Kearney, Alice A. Keefe, Ralph Keen, Catherine Keller, Anthony J. Kelly, Karen Kennelly, Kathi Lynn Kern, Fergus Kerr, Edward Kessler, George Kilcourse, Heup Young Kim, Kim Sung-Hae, Kim Yong-Bock, Kim Yung Suk, Richard King, Thomas M. King, Robert M. Kingdon, Ross Kinsler, Hans G. Kippenberg, Cheryl A. Kirk-Duggan, Clifton Kirkpatrick, Leonid Kishkovsky, Nadieszda Kizenko, Jeffrey Klaiber, Hans-Josef Klauck, Sidney Knight, Samuel Kobia, Robert Kolb, Karla Ann Koll, Heikki Kotila, Donald Kraybill, Philip D. W. Krey, Yves Krumenacker, Jeffrey Kah-Jin Kuan, Simanga R. Kumalo, Peter Kuzmic, Simon Shui-Man Kwan, Kwok Pui-lan, André LaCocque, Stephen E. Lahey, John Tsz Pang Lai, Emiel Lamberts, Armando Lampe, Craig Lampe, Beverly J. Lanzetta, Eve LaPlante, Lizette Larson-Miller, Ariel Bybee Laughton, Leonard Lawlor, Bentley Layton, Robin A. Leaver, Karen Lebacqz, Archie Chi Chung Lee, Marilyn J. Legge, Hervé LeGrand, D. L. LeMahieu, Raymond Lemieux, Bill J. Leonard, Ellen M. Leonard, Outi Leppä, Jean Lesaulnier, Nantawan Boonprasat Lewis, Henrietta Leyser, Alexei Lidov, Bernard Lightman, Paul Chang-Ha Lim, Carter Lindberg, Mark R. Lindsay, James R. Linville, James C. Livingston, Ann Loades, David Loades, Jean-Claude Loba-Mkole, Lo Lung Kwong, Wati Longchar, Eleazar López, David W. Lotz, Andrew Louth, Robin W. Lovin, William Luis, Frank D. Macchia, Diarmaid N. J. MacCulloch, Kirk R. MacGregor, Marjory A. MacLean, Donald MacLeod, Tomas S. Maddela, Inge Mager, Laurenti Magesa, David G. Maillu, Fortunato Mallimaci, Philip Mamalakis, Kä Mana, Ukachukwu Chris Manus, Herbert Robinson Marbury, Reuel Norman Marigza, Jacqueline Mariña, Antti Marjanen, Luiz C. L. Marques, Madipoane Masenya (ngwan'a Mphahlele), Caleb J. D. Maskell, Steve Mason, Thomas Massaro, Fernando Matamoros Ponce, András Máté-Tóth, Odair Pedroso Mateus, Dinis Matsolo, Fumitaka Matsuoka, John D'Arcy May, Yelena Mazour-Matusevich, Theodore Mbazumutima, John S. McClure, Christian McConnell, Lee Martin McDonald, Gary B. McGee, Thomas McGowan, Alister E. McGrath, Richard J. McGregor, John A. McGuckin, Maud Burnett McInerney, Elsie Anne McKee, Mary B. McKinley, James F. McMillan, Ernan McMullin, Kathleen E. McVey, M. Douglas Meeks, Monica Jyotsna Melanchthon, Ilie Melniciuc-Puica, Everett Mendoza, Raymond A. Mentzer, William W. Menzies, Ina Merdjanova, Franziska Metzger, Constant J. Mews, Marvin Meyer, Carol Meyers, Vasile Mihoc, Gunner Bjerg Mikkelsen, Maria Inêz de Castro Millen, Clyde Lee Miller, Bonnie J. Miller-McLemore, Alexander Mirkovic, Paul Misner, Nozomu Miyahira, R. W. L. Moberly, Gerald Moede, Aloo Osotsi Mojola, Sunanda Mongia, Rebeca Montemayor, James Moore, Roger E. Moore, Craig E. Morrison O.Carm, Jeffry H. Morrison, Keith Morrison, Wilson J. Moses, Tefetso Henry Mothibe, Mokgethi Motlhabi, Fulata Moyo, Henry Mugabe, Jesse Ndwiga Kanyua Mugambi, Peggy Mulambya-Kabonde, Robert Bruce Mullin, Pamela Mullins Reaves, Saskia Murk Jansen, Heleen L. Murre-Van den Berg, Augustine Musopole, Isaac M. T. Mwase, Philomena Mwaura, Cecilia Nahnfeldt, Anne Nasimiyu Wasike, Carmiña Navia Velasco, Thulani Ndlazi, Alexander Negrov, James B. Nelson, David G. Newcombe, Carol Newsom, Helen J. Nicholson, George W. E. Nickelsburg, Tatyana Nikolskaya, Damayanthi M. A. Niles, Bertil Nilsson, Nyambura Njoroge, Fidelis Nkomazana, Mary Beth Norton, Christian Nottmeier, Sonene Nyawo, Anthère Nzabatsinda, Edward T. Oakes, Gerald O'Collins, Daniel O'Connell, David W. Odell-Scott, Mercy Amba Oduyoye, Kathleen O'Grady, Oyeronke Olajubu, Thomas O'Loughlin, Dennis T. Olson, J. Steven O'Malley, Cephas N. Omenyo, Muriel Orevillo-Montenegro, César Augusto Ornellas Ramos, Agbonkhianmeghe E. Orobator, Kenan B. Osborne, Carolyn Osiek, Javier Otaola Montagne, Douglas F. Ottati, Anna May Say Pa, Irina Paert, Jerry G. Pankhurst, Aristotle Papanikolaou, Samuele F. Pardini, Stefano Parenti, Peter Paris, Sung Bae Park, Cristián G. Parker, Raquel Pastor, Joseph Pathrapankal, Daniel Patte, W. Brown Patterson, Clive Pearson, Keith F. Pecklers, Nancy Cardoso Pereira, David Horace Perkins, Pheme Perkins, Edward N. Peters, Rebecca Todd Peters, Bishop Yeznik Petrossian, Raymond Pfister, Peter C. Phan, Isabel Apawo Phiri, William S. F. Pickering, Derrick G. Pitard, William Elvis Plata, Zlatko Plese, John Plummer, James Newton Poling, Ronald Popivchak, Andrew Porter, Ute Possekel, James M. Powell, Enos Das Pradhan, Devadasan Premnath, Jaime Adrían Prieto Valladares, Anne Primavesi, Randall Prior, María Alicia Puente Lutteroth, Eduardo Guzmão Quadros, Albert Rabil, Laurent William Ramambason, Apolonio M. Ranche, Vololona Randriamanantena Andriamitandrina, Lawrence R. Rast, Paul L. Redditt, Adele Reinhartz, Rolf Rendtorff, Pål Repstad, James N. Rhodes, John K. Riches, Joerg Rieger, Sharon H. Ringe, Sandra Rios, Tyler Roberts, David M. Robinson, James M. Robinson, Joanne Maguire Robinson, Richard A. H. Robinson, Roy R. Robson, Jack B. Rogers, Maria Roginska, Sidney Rooy, Rev. Garnett Roper, Maria José Fontelas Rosado-Nunes, Andrew C. Ross, Stefan Rossbach, François Rossier, John D. Roth, John K. Roth, Phillip Rothwell, Richard E. Rubenstein, Rosemary Radford Ruether, Markku Ruotsila, John E. Rybolt, Risto Saarinen, John Saillant, Juan Sanchez, Wagner Lopes Sanchez, Hugo N. Santos, Gerhard Sauter, Gloria L. Schaab, Sandra M. Schneiders, Quentin J. Schultze, Fernando F. Segovia, Turid Karlsen Seim, Carsten Selch Jensen, Alan P. F. Sell, Frank C. Senn, Kent Davis Sensenig, Damían Setton, Bal Krishna Sharma, Carolyn J. Sharp, Thomas Sheehan, N. Gerald Shenk, Christian Sheppard, Charles Sherlock, Tabona Shoko, Walter B. Shurden, Marguerite Shuster, B. Mark Sietsema, Batara Sihombing, Neil Silberman, Clodomiro Siller, Samuel Silva-Gotay, Heikki Silvet, John K. Simmons, Hagith Sivan, James C. Skedros, Abraham Smith, Ashley A. Smith, Ted A. Smith, Daud Soesilo, Pia Søltoft, Choan-Seng (C. S.) Song, Kathryn Spink, Bryan Spinks, Eric O. Springsted, Nicolas Standaert, Brian Stanley, Glen H. Stassen, Karel Steenbrink, Stephen J. Stein, Andrea Sterk, Gregory E. Sterling, Columba Stewart, Jacques Stewart, Robert B. Stewart, Cynthia Stokes Brown, Ken Stone, Anne Stott, Elizabeth Stuart, Monya Stubbs, Marjorie Hewitt Suchocki, David Kwang-sun Suh, Scott W. Sunquist, Keith Suter, Douglas Sweeney, Charles H. Talbert, Shawqi N. Talia, Elsa Tamez, Joseph B. Tamney, Jonathan Y. Tan, Yak-Hwee Tan, Kathryn Tanner, Feiya Tao, Elizabeth S. Tapia, Aquiline Tarimo, Claire Taylor, Mark Lewis Taylor, Bishop Abba Samuel Wolde Tekestebirhan, Eugene TeSelle, M. Thomas Thangaraj, David R. Thomas, Andrew Thornley, Scott Thumma, Marcelo Timotheo da Costa, George E. “Tink” Tinker, Ola Tjørhom, Karen Jo Torjesen, Iain R. Torrance, Fernando Torres-Londoño, Archbishop Demetrios [Trakatellis], Marit Trelstad, Christine Trevett, Phyllis Trible, Johannes Tromp, Paul Turner, Robert G. Tuttle, Archbishop Desmond Tutu, Peter Tyler, Anders Tyrberg, Justin Ukpong, Javier Ulloa, Camillus Umoh, Kristi Upson-Saia, Martina Urban, Monica Uribe, Elochukwu Eugene Uzukwu, Richard Vaggione, Gabriel Vahanian, Paul Valliere, T. J. Van Bavel, Steven Vanderputten, Peter Van der Veer, Huub Van de Sandt, Louis Van Tongeren, Luke A. Veronis, Noel Villalba, Ramón Vinke, Tim Vivian, David Voas, Elena Volkova, Katharina von Kellenbach, Elina Vuola, Timothy Wadkins, Elaine M. Wainwright, Randi Jones Walker, Dewey D. Wallace, Jerry Walls, Michael J. Walsh, Philip Walters, Janet Walton, Jonathan L. Walton, Wang Xiaochao, Patricia A. Ward, David Harrington Watt, Herold D. Weiss, Laurence L. Welborn, Sharon D. Welch, Timothy Wengert, Traci C. West, Merold Westphal, David Wetherell, Barbara Wheeler, Carolinne White, Jean-Paul Wiest, Frans Wijsen, Terry L. Wilder, Felix Wilfred, Rebecca Wilkin, Daniel H. Williams, D. Newell Williams, Michael A. Williams, Vincent L. Wimbush, Gabriele Winkler, Anders Winroth, Lauri Emílio Wirth, James A. Wiseman, Ebba Witt-Brattström, Teofil Wojciechowski, John Wolffe, Kenman L. Wong, Wong Wai Ching, Linda Woodhead, Wendy M. Wright, Rose Wu, Keith E. Yandell, Gale A. Yee, Viktor Yelensky, Yeo Khiok-Khng, Gustav K. K. Yeung, Angela Yiu, Amos Yong, Yong Ting Jin, You Bin, Youhanna Nessim Youssef, Eliana Yunes, Robert Michael Zaller, Valarie H. Ziegler, Barbara Brown Zikmund, Joyce Ann Zimmerman, Aurora Zlotnik, Zhuo Xinping
- Edited by Daniel Patte, Vanderbilt University, Tennessee
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- The Cambridge Dictionary of Christianity
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- 05 August 2012
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- 20 September 2010, pp xi-xliv
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five - Children’s health
- Edited by Shirley Dex, Heather Joshi
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- Children of the 21st Century
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- Bristol University Press
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- 22 January 2022
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- 12 October 2005, pp 133-158
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Summary
Children in the UK are growing up against a background of changing family size and structure as well as changing demographic, economic and societal circumstances, which together have important implications for their health (Peckham, 1998). It is important to understand how the changes in patterns of caring for children and family context influence health in early childhood and the adoption of child health promoting behaviours by parents and carers. In recent years, there has been increasing interest in the contribution of these changes to obesity, asthma and related allergic diseases, autoimmune conditions, and disorders of social communication and behaviour (Gent et al 1994; Bach, 2002; Lobstein et al, 2004). The factors underlying these trends remain poorly understood, although they are clearly of great public health and human importance. The importance of an interdisciplinary perspective combining social, environmental and biological approaches to elucidate their causes is increasingly recognised.
Plan of this chapter
In this chapter, after considering the data sources in more detail, we describe the health during infancy of the cohort children through investigating the baby's birthweight, its infant weight at 8-9 months, and the early nutrition and patterns of breastfeeding. A range of parental and community influences on the baby's health are then considered – namely, parental smoking and alcohol use, immunisation, health problems and other use of services. Finally, the chapter examines indicators of good health in infancy and concludes with the implications of the findings for child health policy.
Data sources
At the first contact with the families when the children were aged around 9 months, information was obtained by parental (usually maternal) report on a wide range of measures. This included those relevant to the prevention of illness and promotion of health in the child, such as breastfeeding, parental smoking and immunisation status, and to conditions and illnesses that have implications for growth and development. Also included were measures which provide a baseline for examining later patterns and trajectories which will change with increasing age – for example, birthweight and bodyweight.
Data were also enhanced with respect to child health information by verifying maternal reports at the time of interview from information recorded in the personal child health record (Walton et al, 2005) and, subsequently, by linkage to routine birth registration records and health service information either at the individual or health service level (Bartington et al, 2005; Tate et al, 2005).
The identifiability problem for repairable systems subject to competing risks
- Part of
- Tim Bedford, Bo H. Lindqvist
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- Advances in Applied Probability / Volume 36 / Issue 3 / September 2004
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- 01 July 2016, pp. 774-790
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- September 2004
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Within reliability theory, identifiability problems arise through competing risks. If we have a series system of several components, and if that system is replaced or repaired to as good as new on failure, then the different component failures represent competing risks for the system. It is well known that the underlying component failure distributions cannot be estimated from the observable data (failure time and identity of failed component) without nontestable assumptions such as independence. In practice many systems are not subject to the ‘as good as new’ repair regime. Hence, the objective of this paper is to contrast the identifiability issues arising for different repair regimes. We consider the problem of identifying a model within a given class of probabilistic models for the system. Different models corresponding to different repair strategies are considered: a partial-repair model, where only the failing component is repaired; perfect repair, where all components are as good as new after a failure; and minimal repair, where components are only minimally repaired at failures. We show that on the basis of observing a single socket, the partial-repair model is identifiable, while the perfect- and minimal-repair models are not.
THE SCENERY FLOW FOR HYPERBOLIC JULIA SETS
- TIM BEDFORD, ALBERT M. FISHER, MARIUSZ URBAŃSKI
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- Proceedings of the London Mathematical Society / Volume 85 / Issue 2 / September 2002
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- 23 July 2002, pp. 467-492
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- September 2002
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We define the scenery flow space at a point z in the Julia set J of a hyperbolic rational map $T : \mathbb{C} \to \mathbb{C}$ with degree at least 2, and more generally for T a conformal mixing repellor.
We prove that, for hyperbolic rational maps, except for a few exceptional cases listed below, the scenery flow is ergodic. We also prove ergodicity for almost all conformal mixing repellors; here the statement is that the scenery flow is ergodic for the repellors which are not linear nor contained in a finite union of real-analytic curves, and furthermore that for the collection of such maps based on a fixed open set U, the ergodic cases form a dense open subset of that collection. Scenery flow ergodicity implies that one generates the same scenery flow by zooming down towards almost every z with respect to the Hausdorff measure $H^d$, where d is the dimension of J, and that the flow has a unique measure of maximal entropy.
For all conformal mixing repellors, the flow is loosely Bernoulli and has topological entropy at most d. Moreover the flow at almost every point is the same up to a rotation, and so as a corollary, one has an analogue of the Lebesgue density theorem for the fractal set, giving a different proof of a theorem of Falconer.
2000 Mathematical Subject Classification: 37F15, 37F35, 37D20.
Contents
- Tim Bedford, Technische Universiteit Delft, The Netherlands, Roger Cooke, Technische Universiteit Delft, The Netherlands
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- Probabilistic Risk Analysis
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- 05 June 2012
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- 30 April 2001, pp v-xii
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Tables
- Tim Bedford, Technische Universiteit Delft, The Netherlands, Roger Cooke, Technische Universiteit Delft, The Netherlands
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- Probabilistic Risk Analysis
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- 05 June 2012
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- 30 April 2001, pp xvi-xviii
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Probabilistic Risk Analysis
- Foundations and Methods
- Tim Bedford, Roger Cooke
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- 05 June 2012
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- 30 April 2001
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Probabilistic risk analysis aims to quantify the risk caused by high technology installations. Increasingly, such analyses are being applied to a wider class of systems in which problems such as lack of data, complexity of the systems, uncertainty about consequences, make a classical statistical analysis difficult or impossible. The authors discuss the fundamental notion of uncertainty, its relationship with probability, and the limits to the quantification of uncertainty. Drawing on extensive experience in the theory and applications of risk analysis, the authors focus on the conceptual and mathematical foundations underlying the quantification, interpretation and management of risk. They cover standard topics as well as important new subjects such as the use of expert judgement and uncertainty propagation. The relationship of risk analysis with decision making is highlighted in chapters on influence diagrams and decision theory. Finally, the difficulties of choosing metrics to quantify risk, and current regulatory frameworks are discussed.
Part III - System analysis and quantification
- Tim Bedford, Technische Universiteit Delft, The Netherlands, Roger Cooke, Technische Universiteit Delft, The Netherlands
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- Probabilistic Risk Analysis
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- 05 June 2012
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- 30 April 2001, pp 97-98
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Index
- Tim Bedford, Technische Universiteit Delft, The Netherlands, Roger Cooke, Technische Universiteit Delft, The Netherlands
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- Probabilistic Risk Analysis
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- 30 April 2001, pp 390-393
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5 - Weibull Analysis
- from Part II - Theoretical issues and background
- Tim Bedford, Technische Universiteit Delft, The Netherlands, Roger Cooke, Technische Universiteit Delft, The Netherlands
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- Probabilistic Risk Analysis
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Summary
The Weibull distribution finds wide application in reliability theory, and is useful in analyzing failure and maintenance data. Its popularity arises from the fact that it offers flexibility in modeling failure rates, is easy to calculate, and most importantly, adequately describes many physical life processes. Examples include electronic components, ball bearings, semi-conductors, motors, various biological organisms, fatigued materials, corrosion and leakage of batteries. Classical and Bayesian techniques of Weibull estimation are described in [Abernathy et al., 1983] and [Kapur and Lamberson, 1977].
Components with constant failure rates (i.e. exponential life distributions) need not be maintained, only inspected. If they are found unfailed on inspection, they are ‘as good as new’. Components with increasing failure rates usually require preventive maintenance. Life data on such components is often heavily censored, as components are removed from service for many reasons other than failure. Weibull methods are therefore discussed in relation to censoring. In this chapter we assume that the censoring process is independent or random; that is, the censoring process is independent of the failure process. This may arise, for example, when components undergo planned revision, or are failed by overload caused by some upstream failures. In other cases a service sojourn may terminate for a reason which is not itself a failure, but is related to failure. When components are removed during preventive maintenance to repair degraded performance, we certainly may not assume that the censoring is independent.
15 - Project risk management
- from Part IV - Uncertainty modeling and risk measurement
- Tim Bedford, Technische Universiteit Delft, The Netherlands, Roger Cooke, Technische Universiteit Delft, The Netherlands
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- Probabilistic Risk Analysis
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- 05 June 2012
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- 30 April 2001, pp 299-315
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Project risk management is a rapidly growing area with applications in all engineering areas. We shall particularly concentrate on applications within the construction industry, but the techniques discussed are more widely applicable. Construction risks have been the subject of study for many years. In particular, [Thompson and Perry, 1992] gives a good overall guide to the subject with a large number of references. Several different models and examples of risk analyses for large projects are given in [Cooper and Chapman, 1987]. A description of many projects (in particular high-technology projects) from the twentieth century, the problems encountered during management, and the generic lessons learnt are given in [Morris, 1994].
Large scale infrastructure projects typically have long lead times, suffer from high political and financial uncertainties, and the use of innovative but uncertain technologies. Because of the high risks and costs involved it has become common to apply risk management techniques with the aim of gaining insight into the principal sources of uncertainty in costs and/or time.
A project risk analysis performed by a candidate contractor before it bids for work is valuable because it can give the management quantitative insight into the sources of uncertainty in a project. This gives management a guide to the risks that need to be dealt with in the contract, or in financing arrangements.
4 - Statistical inference
- from Part II - Theoretical issues and background
- Tim Bedford, Technische Universiteit Delft, The Netherlands, Roger Cooke, Technische Universiteit Delft, The Netherlands
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- Probabilistic Risk Analysis
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- 30 April 2001, pp 61-82
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Summary
The general problem of statistical inference is one in which, given observations of some random phenomenon, we try to make an inference about the probability distribution describing it. Much of statistics is devoted to the problem of inference. Usually we will suppose that the distribution is one of a family of distributions f(t|θ) parameterized by θ, and we try to make an assessment of the likely values taken by θ. An example is the exponential distribution f(t|λ) = λ exp(−λt), but also the joint distribution of n independent samples from the same exponential, f(t1, …, tn|λ) = λn exp(−λ(t1 + … + tn)), falls into the same category and is relevant when making inference on the basis of n independent samples.
Unfortunately, statisticians are not in agreement about the ways in which statistical inference should be carried out. There is a plethora of estimation methods which give rise to different estimates. Statisticians are not even in agreement about the principles that should be used to judge the quality of estimation techniques. The various creeds of statistician, of which the most important categories are Bayesian and frequentist, differ largely in the choice of principles to which they subscribe. (An entertaining guide to the differences is given in the paper of Bradley Efron ‘Why isn't everyone a Bayesian?’ and the heated discussion that follows, [Efron, 1986].) To some extent the question is whether one thinks that statistical inference should be inductive or deductive.
7 - Fault trees – analysis
- from Part III - System analysis and quantification
- Tim Bedford, Technische Universiteit Delft, The Netherlands, Roger Cooke, Technische Universiteit Delft, The Netherlands
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- 30 April 2001, pp 121-139
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Illustrations
- Tim Bedford, Technische Universiteit Delft, The Netherlands, Roger Cooke, Technische Universiteit Delft, The Netherlands
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Part I - Introduction
- Tim Bedford, Technische Universiteit Delft, The Netherlands, Roger Cooke, Technische Universiteit Delft, The Netherlands
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17 - Uncertainty analysis
- from Part IV - Uncertainty modeling and risk measurement
- Tim Bedford, Technische Universiteit Delft, The Netherlands, Roger Cooke, Technische Universiteit Delft, The Netherlands
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Summary
Introduction
This chapter gives a brief introduction to the relatively new and expanding field of uncertainty analysis. Fundamental concepts are introduced, but theorems will not be proved here. Since uncertainty analysis is effectively dependent on computer support, the models used in uncertainty analysis are discussed in relation to simulation methods. A good elementary introduction to simulation is found in the book of Ross [Ross, 1990].
Uncertainty analysis was introduced with the Rasmussen Report WASH-1400 [NRC, 1975] which, as we recall, made extensive use of subjective probabilities. It was anticipated that the decision makers would not accept a single number as the probability of catastrophic accident with a nuclear reactor. Instead a distribution over possible values for the probability of a catastrophic accident was computed, using estimates of the uncertainty of the input variables. Since this study uncertainty analyses are rapidly becoming standard for large technical studies aiming at consensus in areas with substantial uncertainty. The techniques of uncertainty analysis are not restricted to fault tree probability calculations, rather they can be applied to any quantitative model. Uncertainty analysis is commonplace for large studies in accident consequence modeling, environmental risk studies and structural reliability.
Mathematical formulation of uncertainty analysis
Mathematically uncertainty analysis concerns itself with the following problem. Given some function M(X1, …, Xn) of uncertain quantities X1,…, Xn, determine the distribution of G on the basis of some information about the joint distribution of X1, …, Xn.
Preface
- Tim Bedford, Technische Universiteit Delft, The Netherlands, Roger Cooke, Technische Universiteit Delft, The Netherlands
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Summary
We have written this book for numerate readers who have taken a first university course in probability and statistics, and who are interested in mastering the conceptual and mathematical foundations of probabilistic risk analysis. It has been developed from course notes used at Delft University of Technology. An MSc course on risk analysis is given there to mathematicians and students from various engineering faculties. A selection of topics, depending on the specific interests of the students, is made from the chapters in the book. The mathematical background required varies from topic to topic, but all relevant probability and statistics are contained in Chapters 3 and 4.
Probabilistic risk analysis differs from other areas of applied science because it attempts to model events that (almost) never occur. When such an event does occur then the underlying systems and organizations are often changed so that the event cannot occur in the same way again. Because of this, the probabilistic risk analyst must have a strong conceptual and mathematical background.
The first chapter surveys the history of risk analysis applications. Chapter 2 explains why probability is used to model uncertainty and why we adopt a subjective definition of probability in spite of its limitations. Chapters 3 and 4 provide the technical background in probability and statistics that is used in the rest of the book. The remaining chapters are more-or-less technically independent of each other, except that Chapter 7 must follow Chapter 6, and 14 should follow 13.
9 - Reliability data bases
- from Part III - System analysis and quantification
- Tim Bedford, Technische Universiteit Delft, The Netherlands, Roger Cooke, Technische Universiteit Delft, The Netherlands
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Introduction
Reliability data is not simply ‘there’ waiting to be gathered. A failure rate is not an intrinsic property of a component like mass or charge. Rather, reliability parameters characterize populations that emerge from complex interactions of components, operating environments and maintenance regimes. This chapter presents mathematical tools for defining and analyzing populations from which reliability data is to be gathered. This chapter is long, the reason being that the mathematical sophistication required by a practicing risk/reliability analyst has increased significantly in the last years. Whereas in the past the choices of statistical populations and analytic methods were hard wired with the design of the data collection facility, today the analyst must play an increasingly active role in defining statistical populations relative to his/her particular needs.
The first step is to become clear about why we want reliability data. Modern reliability data banks (RDBs) are intended to serve at least three types of users: (1) the maintenance engineer interested in measuring and optimizing maintenance performance, (2) the component designer interested in optimizing component performance, and (3) the risk/reliability analyst wishing to predict reliability of complex systems in which the component operates.
To serve these users modern RDBs distinguish up to ten failure modes, often grouped into critical failures, degraded failures and incipient failures. Degraded and incipient failures are often associated with preventive maintenance. Whereas critical failures are of primary interest in risk and reliability calculations, a maintenance engineer is also interested in degraded and incipient failures.
18 - Risk measurement and regulation
- from Part IV - Uncertainty modeling and risk measurement
- Tim Bedford, Technische Universiteit Delft, The Netherlands, Roger Cooke, Technische Universiteit Delft, The Netherlands
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Summary
How can we choose a probabilistic risk acceptance criterion, a probabilistic safety goal, or specify how changes of risk baseline should influence other design and operational decisions? Basically, we have to compare risks from different, perhaps very different activities. What quantities should be compared? There is an ocean of literature on this subject. The story begins with the first probabilistic risk analysis, WASH-1400, and books which come quickly to mind are [Shrader-Frechette, 1985], [Maclean, 1986], [Lowrance, 1976] and [Fischhoff et al., 1981]. A few positions, and associated pitfalls, are set forth below. For convenience we restrict attention to one undesirable event, namely death.
Single statistics representing risk
Deaths per million
The most common quantity used to compare risks is ‘deaths per million’. Covello et al. [Covello et al., 1989] give many examples of the use of this statistic. Similar tables are given by the British Health and Safety Executive [HSE, 1987]. The Dutch government's risk policy statement [MVROM, 1989] gives a variation on this method by tabulating the yearly risk of death as ‘one in X’.
Table 18.1 shows a few numbers taken from Table B.1 of [Covello et al., 1989], ‘Annual risk of death in the United States’. By each ‘cause’ the number of deaths per year per million is given.
Frontmatter
- Tim Bedford, Technische Universiteit Delft, The Netherlands, Roger Cooke, Technische Universiteit Delft, The Netherlands
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