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Frontmatter
- Bruce J. West, Paolo Grigolini, University of North Texas
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- Complex Webs
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- 23 December 2010, pp i-iv
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3 - Mostly linear dynamics
- Bruce J. West, Paolo Grigolini, University of North Texas
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- Complex Webs
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- 23 December 2010, pp 105-165
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Summary
One strategy for predicting an improbable but potentially catastrophic future event is to construct a faithful model of the dynamics of the complex web of interest and study its extremal properties. But we must also keep in mind that the improbable is not the same as the impossible. The improbable is something that we know can happen, but our experience tells us that it probably will not happen, because it has not happened in the past. Most of what we consider to be common sense or probable is based on what has happened either to us in the past or to the people we know. In this and subsequent chapters we explore the probability of such events directly, but to set the stage for that discussion we examine some of the ways webs become dynamically complex, leading to an increase in the likelihood of the occurrence of improbable events. The extremes of a process determine the improbable and consequently it is at these extremes that failure occurs. Knowing a web's dynamical behavior can help us learn the possible ways in which it can fail and how long the recovery time from such failure may be. It will also help us answer such questions as the following. How much time does the web spend in regions where the likelihood of failure is high? Are the extreme values of dynamical variables really not important or do they actually dominate the asymptotic behavior of a complex web?
1 - Webs
- Bruce J. West, Paolo Grigolini, University of North Texas
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- Complex Webs
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- 23 December 2010, pp 1-44
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Summary
The science of complex webs, also known as network science, is an exciting area of contemporary research, overarching the traditional scientific disciplines of biology, economics, physics, sociology and the other compartments of knowledge found in any college catalog. The transportation grids of planes, highways and railroads, the economic meshes of global finance and stock markets, the social webs of terrorism, governments, and businesses as well as churches, mosques, and synagogs, the physical lattices of telephones, the Internet, earthquakes, and global warming, in addition to the biological networks of gene regulation, the human body, clusters of neurons and food webs, share a number of apparently universal properties as the webs become increasingly complex. This conclusion is shared by the recent report Network Science [23] published under the auspices of the National Academy of Science. The terms networks and network science have become popular tags for these various areas of investigation, but we prefer the image of a web rather than the abstraction of a network, so we use the term web more often than the synonyms network, mesh, net, lattice, grille or fret. Colloquially, the term web entails the notion of entanglement that the name network does not share. Perhaps it is just the idea of the spider ensnaring its prey that appeals to our darker sides.
Whatever the intellectual material is called, this book is not about the research that has been done to understand complex webs, at least not in the sense of a monograph. We have attempted to put selected portions of that research into a pedagogic and often informal context, one that highlights the limitations of the more traditional descriptions of these areas. In this regard we are obligated to discuss the state of the art regarding a broad sweep of complex phenomena from a variety of academic disciplines. Sometimes properly setting the stage requires a historical approach and other times the historical view is replaced with personal perspectives, but with either approach we do not leave the reader alone to make sense of what can be difficult material. So we begin by illuminating the basic assumptions that often go unexamined in science.
Index
- Bruce J. West, Paolo Grigolini, University of North Texas
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- Complex Webs
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- 23 December 2010, pp 365-375
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Contents
- Bruce J. West, Paolo Grigolini, University of North Texas
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- Complex Webs
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- 23 December 2010, pp v-viii
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8 - Synopsis
- Bruce J. West, Paolo Grigolini, University of North Texas
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- Complex Webs
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- 23 December 2010, pp 357-364
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Summary
There is great value in being able to provide a brief insightful summary of an elaborate experiment, a complicated theory or a multi-faceted discussion. Suppose one of us gets on an elevator and the only other passenger is a four-star general. She smiles and asks “How is your research going?” This is one of those rare opportunities when a scientist can actually influence policy, resource allocation and the direction of science by being able to communicate effectively. Almost invariably the opportunity is lost because scientists and generals rarely speak the same language. However, there is a trick that one can employ to anticipate this rare event and that is to have something worked out in advance so as not to waste the opportunity. This is the “elevator description” of the most significant research that has been done. In another age it might have been called the “Reader's Digest” version.
That is the position the authors find themselves in now. We believe that we ought to provide a brief but insightful summary of the book you have just completed in a way that conveys the maximum amount of information, but without the mathematics that was necessary to make that information understandable when it was first presented. To accomplish this goal we review the high points of each of the chapters and then attempt to tie them all together into a coherent picture.
In the first chapter we argued that normal statistics do not describe complex webs. Phenomena described by normal statistics can be discussed in terms of averages. The fact that human populations are well represented by normal distributions of heights strongly influences the manufacturing of everything from the size of shoes, shirts and slacks to cars, computers and couches. The relative numbers of shirts in the small, medium and large sizes are determined by our knowledge of the average build of individuals in the population. The distribution of sizes in the manufactured shirts must match the population of buyers or the shirt factory will soon be out of business. Thus, the industrial revolution was poised to embrace the world according to Gauss and the mechanistic society of the last two centuries flourished. But as the connectivity of the various webs within society became more complex the normal distribution receded further and further into the background until it was completely gone from the data, if not from our attempted understanding.
Preface
- Bruce J. West, Paolo Grigolini, University of North Texas
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- Complex Webs
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- 23 December 2010, pp ix-x
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The Italian engineer turned social scientist Vilfredo Pareto was the first investigator to determine that the income in western society followed a law that was fundamentally unfair. He was not making a value judgement about the poor and uneducated or about the rich and pampered; rather, he was interpreting the empirical finding that in 1894 the distribution of income in western societies was not “normal,” but instead the number of people with a given income decreased as a power of the level of income. On bi-logarithmic graph paper this income distribution graphs as a straight-line segment of negative slope and is called an inverse power law. He interpreted his findings as meaning that a stable society has an intrinsic imbalance resulting from its complex nature, with the wealthy having a disproportionate fraction of the available wealth. Since then staggeringly many phenomena from biology, botany, economics, medicine, physics, physiology, psychology, in short every traditional discipline, have been found to involve complex phenomena that manifest inverse power-law behavior. These empirical laws were explained in the last half of the twentieth century as resulting from the complexity of the underlying phenomena.
As the twentieth century closed and the twenty-first century opened, a new understanding of the empirical inverse power laws emerged. This new understanding was based on the connectedness of the elements within the underlying phenomena and the supporting web structure. The idea of networks became pervasive as attention was drawn to society‘s reliance on sewers and the electric grid, cell phones and the Internet, banks and global stock markets, roads and rail lines, and the multitude of other human-engineered webbings that interconnect and support society. In parallel with the studies of social phenomena came new insight into the distribution in size and frequency of earthquakes and volcanic eruptions, global temperature anomalies and solar flares, river tributaries and a variety of other natural phenomena that have eluded exact description by the physical sciences. Moreover, the inverse power laws cropped up in unexpected places such as in heart rates, stride intervals and breathing, letter writing and emails, cities and wars, heart attacks and strokes; the inverse power law is apparently ubiquitous.
7 - Dynamics of chance
- Bruce J. West, Paolo Grigolini, University of North Texas
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- Complex Webs
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- 23 December 2010, pp 307-356
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Summary
In this chapter we explore web dynamics using a master equation in which the rate of change of probability is determined by the probability flux into and out of the state of interest. The master equation captures the interactions among large numbers of elements each with the same internal dynamics; in this case the internal dynamics consist of the switching of a node between two states. A two-state node may be viewed as the possible choices of an individual, say whether or not that person will vote for a particular candidate in an election. This is one of the simplest dynamical webs which has been shown mathematically to result in synchronization under certain well-defined conditions. We focus on the intermittent fluctuations emerging from a phase-transition process that achieves synchronized behavior for the strength of the interaction exceeding a critical value. This model provides a first step towards proving that these intermittent fluctuations, rather than being a nuisance, are important channels of information transmission allowing communication within and between different complex webs. The crucial power-law index μ discussed earlier and there inserted for mathematical convenience is here determined by the web dynamics. This observation on the inverse power-law index leads us to define the network efficiency in a form that might not coincide with earlier definitions proposed through the observation of the network topology.
Both the discrete and the continuous master equation are discussed for modeling web dynamics. One of the most important aspects of the analysis concerns the perturbation of one complex network by another and the transfer of information between complex clusters. A cluster is a network with a uniformity of opinion due to a phase transition to a given state. This modeling strategy is not new, but in fact dates back to Yule [71], who used the master-equation approach, before the approach had been introduced, to obtain an inverse power-law distribution. We investigate whether the cluster’s opinion is robust or whether it can be easily changed by perturbing the way members of the cluster interact with one another.
6 - A brief recent history of webs
- Bruce J. West, Paolo Grigolini, University of North Texas
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- Complex Webs
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- 23 December 2010, pp 262-306
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Complex webs are not all complex in the same way; how the distribution in the number of connections on the Internet is formed by the conscious choices of individuals must be very different in detail from the new connections made by the neurons in the brain during learning, which in turn is very different mechanistically from biological evolution. Therefore different complex webs can be categorized differently, each representation emphasizing a different aspect of complexity. On the other hand, a specific complex web may be categorized in a variety of ways, again depending on what aspect of entanglement is being emphasized. For example, earthquakes are classified according to the magnitude of displacement they produce, in terms of the Richter scale, giving rise to an inverse power-law distribution in a continuous variable measuring the size of local displacement, the Gutenberg—Richter law. Another way to categorize earthquake data is in terms of whether quakes of a given magnitude occur within a given interval of time, the Omori law, giving rise to an inverse power-law distribution in time. The latter perspective, although also continuous, yields the statistics of the occurrence of individual events. Consequently, we have probability densities of continuous variables and those of discrete events, and which representation is selected depends on the purpose of the investigator. An insurance adjustor might be interested in whether an earthquake of magnitude 8.0 is likely to destroy a given building at a particular location while a policy is still in effect.
2 - Webs, trees and branches
- Bruce J. West, Paolo Grigolini, University of North Texas
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- 23 December 2010, pp 45-104
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In the previous chapter we examined examples of hyperbolic and inverse power-law distributions dating back to the beginning of the last century, addressing biodiversity, urban growth and scientific productivity. There was no discussion of physical structure in these examples because they concerned the counting of various quantities such as species, people, cities, words, and scientific publications. It is also interesting to examine the structure of complex physical phenomena, such as the familiar irregularity in the lightning flashes shown in Figure 1.13. The branches of the lightning bolt persist for fractions of a second and then blink out of existence. The impression left is verified in photographs, where the zigzag pattern of the electrical discharge is captured. The time scale for the formation of the individual zigs and zags is on the order of milliseconds and the space scales can be hundreds of meters. So let us turn our attention to webs having time scales of milliseconds, years or even centuries and spatial scales from millimeters to kilometers.
All things happen in space and time, and phenomena localized in space and time are called events. Publish a paper. Run a red light. It rains. The first two identify an occurrence at a specific point in time with an implicit location in space; the third implies a phenomenon extended in time over a confined location in space. But events are mental constructs that we use to delineate ongoing processes so that not everything happens at once. Publishing a paper is the end result of a fairly long process involving getting an idea about a possible research topic, doing the research, knowing when to gather results together into a paper, writing the paper, sending the manuscript to the appropriate journal, reading and responding to the referees’ criticism of the paper, and eventually having the paper accepted for publication.
4 - Random walks and chaos
- Bruce J. West, Paolo Grigolini, University of North Texas
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- Complex Webs
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- 23 December 2010, pp 166-223
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Summary
In the late nineteenth century, it was believed that a continuous function such as those describing physical processes must have a derivative “almost everywhere.” At the same time some mathematicians wondered whether there existed functions that were everywhere continuous, but which did not have a derivative at any point (continuous everywhere but differentiable nowhere). Perhaps you remember our discussion of such strange things from the first chapter. The motivation for considering such pathological functions was initiated by curiosity within mathematics, not in the physical or biological sciences where one might have expected it. In 1872, Karl Weierstrass (1815–1897) gave a lecture to the Berlin Academy in which he presented functions that had the remarkable properties of continuity and non-differentiability. Twenty-six years later, Ludwig Boltzmann (1844–1906), who connected the macroscopic concept of entropy with microscopic dynamics, pointed out that physicists could have invented such functions in order to treat collisions among molecules in gases and fluids. Boltzmann had a great deal of experience thinking about such things as discontinuous changes of particle velocities that occur in kinetic theory and in wondering about their proper mathematical representation. He had spent many years trying to develop a microscopic theory of gases and he was successful in developing such a theory, only to have his colleagues reject his contributions. Although kinetic theory led to acceptable results (and provided a suitable microscopic definition of entropy), it was based on the time-reversible dynamical equations of Newton.
5 - Non-analytic dynamics
- Bruce J. West, Paolo Grigolini, University of North Texas
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- Complex Webs
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- 23 December 2010, pp 224-261
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In this chapter we investigate one procedure for describing the dynamics of complex webs when the differential equations of ordinary dynamics are no longer adequate, that is, the webs are fractal. We described some of the essential features of fractal functions earlier, starting from the simple dynamical processes described by functions that are fractal, such as the Weierstrass function, which are continuous everywhere but are nowhere differentiable. This idea of non-differentiability suggests introducing elementary definitions of fractional integrals and fractional derivatives starting from the limits of appropriately defined sums. The relation between fractal functions and the fractional calculus is a deep one. For example, the fractional derivative of a regular function yields a fractal function of dimension determined by the order of the fractional derivative. Thus, changes in time of phenomena that are described by fractal functions are probably best described by fractional equations of motion. In any event, this perspective is the one we developed elsewhere [31] and we find it useful here for discussing some properties of complex webs.
The separation of time scales in complex physical phenomena allows smoothing over the microscopic fluctuations and the construction of differentiable representations of the dynamics on large space scales and long time scales. However, such smoothing is not always possible.
Complex Webs
- Anticipating the Improbable
- Bruce J. West, Paolo Grigolini
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- 05 July 2014
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- 23 December 2010
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Complex Webs synthesises modern mathematical developments with a broad range of complex network applications of interest to the engineer and system scientist, presenting the common principles, algorithms, and tools governing network behaviour, dynamics, and complexity. The authors investigate multiple mathematical approaches to inverse power laws and expose the myth of normal statistics to describe natural and man-made networks. Richly illustrated throughout with real-world examples including cell phone use, accessing the Internet, failure of power grids, measures of health and disease, distribution of wealth, and many other familiar phenomena from physiology, bioengineering, biophysics, and informational and social networks, this book makes thought-provoking reading. With explanations of phenomena, diagrams, end-of-chapter problems, and worked examples, it is ideal for advanced undergraduate and graduate students in engineering and the life, social, and physical sciences. It is also a perfect introduction for researchers who are interested in this exciting new way of viewing dynamic networks.
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. 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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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- Book:
- The Cambridge Dictionary of Christianity
- Published online:
- 05 August 2012
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- 20 September 2010, pp xi-xliv
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Vindication of mode-coupled descriptions of multiple-scale water wave fields
- Keith A. Brueckner, Bruce J. West
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- Journal:
- Journal of Fluid Mechanics / Volume 196 / November 1988
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- 21 April 2006, pp. 585-592
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Herein we show that the modal description of deep-water waves on the sea surface Watson & West 1975) is independent of any reference surface around which expansions of the velocity potential and the surface velocity are done. We demonstrate by direct construction that the interaction between long and short waves does not lead to divergent terms in the equations of motion when this formalism is used.
Statistical properties of water waves. Part 1. Steady-state distribution of wind-driven gravity-capillary waves
- Bruce J. West
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- Journal of Fluid Mechanics / Volume 117 / April 1982
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- 20 April 2006, pp. 187-210
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The Miles–Phillips model of the linear coupling between waves on the ocean surface and a fluctuating wind field is generalized to include the average effect of the nonlinear water-wave interactions in the dynamic equations for gravity–capillary waves. A statistical-linearization procedure is applied to the general problem and yields the optimum linear description of the nonlinear terms by linear terms. The linearized dynamic equations are stochastic with solutions that have stable moments, i.e. the average nonlinear interactions quench the linear instability generated by the coupling to the mean wind field. In particular, an asymptotic steady-state power-spectral density for the water-wave field is calculated exactly in the context of the model for various wind speeds.
A resonant test-field model of gravity waves
- Bruce J. West
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- Journal:
- Journal of Fluid Mechanics / Volume 132 / July 1983
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- 20 April 2006, pp. 417-430
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In this paper we propose an ‘irreversible’ resonant test-field (RTF) model to describe the statistical fluctuations of gravity waves on deep water driven by a turbulent wind field. The non-resonant interactions in the gravity-wave Hamiltonian are replaced by a Markov process in the equation of motion for the resonantly interacting gravity waves, i.e. Hamilton's equations are replaced by a Langevin equation for the RTF waves. The RTF models the irreversible energy-transfer process by a Fokker-Planck equation for the phase-space probability density, the exact steady-state solution of which is determined to be non-Gaussian. An H-theorem for the RTF predicts the monotonic approach to the asymptotic steady state near which the transport properties of the field are studied. The steady-state energy-spectral density is calculated (in some approximation) to be k−4.
Coupling of surface and internal gravity waves: a mode coupling model
- Kenneth M. Watson, Bruce J. West, Bruce I. Cohen
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- Journal of Fluid Mechanics / Volume 77 / Issue 1 / 9 September 1976
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- 11 April 2006, pp. 185-208
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A surface-wave/internal-wave mode coupled model is constructed to describe the energy transfer from a linear surface wave field on the ocean to a linear internal wave field. Expressed in terms of action-angle variables the dynamic equations have a particularly useful form and are solved both numerically and in some analytic approximations. The growth time for internal waves generated by the resonant interaction of surface waves is calculated for an equilibrium spectrum of surface waves and for both the Garrett-Munk and two-layer models of the undersea environment. We find energy transfer rates as a function of undersea parameters which are much faster than those based on the constant Brunt-ViiisSila model used by Kenyon (1968) and which are consistent with the experiments of Joyce (1974). The modulation of the surface-wave spectrum by internal waves is also calculated, yielding a ‘mottled’ appearance of the ocean surface similar to that observed in photographs taken from an ERTS1 satellite (Ape1 et al. 1975b).
A transport-equation description of nonlinear ocean surface wave interactions
- Kenneth M. Watson, Bruce J. West
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- Journal:
- Journal of Fluid Mechanics / Volume 70 / Issue 4 / 26 August 1975
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- 29 March 2006, pp. 815-826
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The evolution of the power spectrum of surface gravity waves is described by means of an energy transport equation. A slowly varying, prescribed ocean current and wind source are assumed to account for spatial inhomogeneities in the surface wave spectrum. These inhomogeneities lead to a new nonlinear wave-wave interaction mechanism.
Functions on a crosscap
- J. W. BRUCE, J. M. WEST
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- Journal:
- Mathematical Proceedings of the Cambridge Philosophical Society / Volume 123 / Issue 1 / January 1998
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- 01 January 1998, pp. 19-39
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- January 1998
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The study of the differential geometry of surfaces in 3-space has a long and celebrated history. Over the last 20 years a new approach using techniques from singularity theory has yielded some interesting results (see, for example [3, 5, 19] for surveys).
Of course surfaces arise in a number of ways: they are often defined explicitly as the image of a mapping f: R2→R3. Since the subject is differential geometry one normally asks that these defining mappings are smooth, that is infinitely differentiable, however it is not true, in any sense, that most such parametrisations will yield manifolds. For such mappings have self-intersections, and more significantly they may have crosscaps (also known as Whitney umbrellas). Moreover if we perturb the maps these singularities will persist; that is they are stable. (See [14, 20] for details.)
Consequently when studying the differential geometry of surfaces in 3-space there are good reasons for studying surfaces with crosscaps.
In this paper we carry out a classification of mappings from 3-space to lines, up to changes of co-ordinates in the source preserving a crosscap. We can apply our results to the geometry of generic crosscap points. In [10] we computed geometric normal forms for the crosscap and used them to study the dual of the crosscap. We shall see that the approach here yields more information than that obtained in [10], although the latter has the advantage of being more explicit (in particular various aspects of the geometry can be compared using the normal forms).
We refer the reader to [16, 18] for background material concerning singularity theory.