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Richard of Cambridge = Anne Edmund Tudor HENRY VI = Margaret Margaret Beaufort = Edmund Tudor
Richard, Duke of York Edward, Prince of Wales HENRY VII=Elizabeth of York
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hdescribes a condition of high vital (birth and death) rates, followed by a drop in the death rate duringiaa period in which the corresponding drop in the birth rate lags behind, followed by a drop in thed\birth rate and the realignment of a low birth rate with a low death rate (Bogue, 1969). Thej
btransition is from high vital rates to low vital rates; if the intermediate stage does not lead to>6eventual low vital rates, then there is no transition.h`As we have seen, one need not confine the idea of transition to demography; it extends naturallyiato a variety of realworld realms (Drake, 1992). For any system to be in some sort of functionalnfbalance, the inputs and outputs must be fairly close to each other in number: if the inputs dominate,kcthe system explodes. If the outputs dominate, the system withers. Abstractly, a transition withinjba system occurs when the input/output level starts in balance, at a high level, experiences a dropiain outputs so that the inputs dominate for a period of time, and then returns to a balanced statekcby a corresponding drop in the input level so that once again the input/output level is in balance.%Symmetry promotes systemic stability.phThe transition is from the high level of input/output to a low level of input/out l \ sandy BSheet1
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KohҢohoh0? @oh0._A<4,Curve fitting and analytical toolsnumber 7NRE 545A9S. Arlinghaus, to appear, Structural Models in Geography.H@An application of graph theory to group relationships in history0((source: Williams: Finite Mathematics)zrThe term transition is often used to describe the return of a system to a balanced situation following a period inXPwhich it has been out of balance. One classical transition is the demographer'sd put. Because the ratesh`generally do not drop evenly, one curve has more area under it than does the other, signifying atlperiod of "boom." What happens during the transition, in the boom time, is critical, as Drake notes (1992),skin determining whether or not the transition is completed and it is in this intermediate stage that so manyqicomplexities often arise. Drake (1992) notes this situation in a variety of contexts: from forestry, toia education, to environmental toxicity, to a host of others. We consider it here in an historical6.context: in succession to the British throne.@D8EL4~\htmhnB
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/K0<oh1<Ң2<oh3<oh5<06<?7<8< 9<:<@;<<<oh=<0.><_A<?< p!hIn the peiord of British history from William the Conqueror (about 1066) to Richard the II (about 1399),q"ithe pattern of hereditary succession to the British throne was clear. When Henry IV overthrew Richard IIn#fin 1399, the Wars of the Roses, involving issues between the House of York and the House of Lancaster,y$qconcerning succession to the British throone, were the result. In 1485, when Henry VII of the House of Lancasterr%jmarried Elizabeth of the Hourse of York, the pattern of succession once again became clear. An historicalp&htransition was achieved, but, in the "boom" time from 1399 to 1485, what was the pattern of the dispute?z'rStructural models, or graphs, offer a way to resolve historical complexity (Luce and Perry, 1968; Williams, 1979).(h)`Generally speaking, a graph is a collection of nodes together with edges linking pairs of nodes.b*ZThere may be more than one component to a graph. Some graphs are trees. Some graphs havej+bdirected edges indicating the direction of flow (digraphs). The subject of graph theory is a veryn,fbroad one with numerous applications. The classical text in graph theory is by Frank Harary, entitled( Graph Theory, published in 1969.m/eA partial genealogical table of the famiily relationships, for both the Houses of Lancaster an York, b0Zis shown below. This genealogical table can be made into a digraph. Let the relationshipj1blinking people be "is the father/mother of." Let each person represent a node. Thus, if person Pn2fis related to person Q, draw an edge from P to Q, with the direction pointing from P to Q. Thus, Q isZ3Radjacen within the structural model from P (Harary, Norman, and Cartwright, 1965).k5cWe can code this sort of adjacency in a binary matrix: the entry from P to Q described above wouldv6nbe a 1; the entry from Q to P would be a 0. Using this idea of adjacency, based on "is the father/mother of,"o7gthe structural model of the genealogical table can be expressed as an adjacency matrix focusing only onu8mthe set of 17 people noted in the family tree. In this case, adjacency is based on certain key relationships)9!gleaned from historical evidence.:The matrix A :' :+The matrix A^2l;?@@@@@@ @"@$@&@(@*@,@.@0@1@<1. Edward III
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LlL%lL+;ONote that there are three 1s in the first row of the matrix A since Edward III was the father of Lionel, John of Gaunt, and Edmund.{PsTaking powers of the adjacency matrix, A, counts the number of paths through the genealogical hierarchy. The powerzQrA^2 counts the number of paths of length twograndparent/grandchildren relationships. In A^2 there are 1s in thevRnthird, thenth, eleventh, and seventeenth columns, indicating that Edward III was the grandfather of Philippa, ]SUHenry IV, John Beaufort, and Richard. These observations tally with the family tree.wUoThe fifth power of the adjacency matrix, A^5, has a value of 1 in the (1,14) position of the matrix (first row,qVifourteenth column), showing that there is one path of length five between Edward III and Henry VIII; thatnWfHenry VIII was descended directly from Edward III through five Lancaster generations. Because a valuewXoof 1 is recorded in this position for the first time in the fifth power matrix, we know that it is exactly five4Y,(and not fewer) generations for the descent.j[bThe seventh power of the adjacency matrix shows an entry of 1 in the (1,8) position reflecting theT\Lfact that Elizabeth is desceded over eight York generations from Edward III. ] j^bThe sixth power matrix and the eighth power matrix both have entries of 1 in the (1,15) entryforj_bHenry VIII, the son of Hnery VII or Lancaster and Elizabeth of York. Had both Elizabeth and Henry<!"4*" ~za{ur{8nX
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xKyohzҢ{oh}oh0b`Zbeen descended from Richard III over the same number of generations, there would have beenoagan entry of 2 in the (1,15) position at its first appearance in the matrix sequencea 1 from each lineqbia descent. Thus, Henry VIII, the son of Henry VII and Elizabeth combined the claim of both the Houses ofkccLancaster and York to the British throne. Historical complexity, that can occur in the "boom" timepdhwithin a transitionbetween symmetric periods of stability, is resolved easily using adjacency matriceseof structural models.h
Reference:]iUR. Duncan Luce and Albert D. Perry, "A Method of Matrix Analysis of Group Structure."YjQReadings in Mathematical Social Science, ed. Paul F. Lazarsfeld and Neil W. Henry#kCambridge, MIT Press, 1968.mFractal GeometryToLConsider the two lines belowboth have Euclidean dimension of one; however,bpZintutively, one "fills" more space than does the other although of course it does not fillcq[a twodimensional piece of space. Hence the idea of a "fractional dimension" or "fractal."fx^Benoit Mandelbrot, a computer scientist/mathematician captured this notion (which is prevalentey]much earlier in the history of mathematicsin finding a curve that is continuous but nowheremzedifferentiablekind of an infinite number of absolute value curves) in his work in the 20th century.P{H(Mandelbrot, The Fractal Geometry of Natureis one standard reference.)]}UFractional dimension can be calculated as suggested in the following visual examples:
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generator._WThe shape of "hexagon" remains constantits scale changes. The figures are said to beg_selfsimilarthe choice of generator was critical in bringing about this scale transformation.g_When the generator is scaleddown again, and applied inside/outside to each of the four smalleraYhexagons, and even more complicted figure of sixteen smaller hexagons appears. Carry outd\this process infinitelywhat is constant is the number four, as a factor of the increase inc[complexity, and the number three and the number of generator sides causing this increase inshape complexity.<4Let N represent the number of sides in the generator;3Let K represent the number of selfsimilar regions.&In this example, N=3, and K=4.4,The fractional dimension D is calculated as:&v@>%#c
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_A D = ln N / (ln K^0.5)In this case, D = $i\?casAAExample:f^The previous example dealt with a bounded, closed figure. One can deal with other shapes, toojbConsider a straight linea coastline viewed from high aboveas one zooms in, one sees more bays.f^In this case, the straight line is the constant shape and in moving from one scale to another,_Wthe generator used to increase coastal complexity has four sides and the number of self.&similar regions produced is also four.?7When this sequence is carried out infinitely, K=4, N=4,
D=$@AAnfso that this procedure will cause the lines to bend back and forth sufficiently to fill a piece of thephplane. If cutting bays in a lakeshore were to follow this process, in order to maximize lakefront views!Fjnjc2C6r Lh`and minimize length of linear coastal damage, it might therefore be prudent to stop the sequence3+after some fairly small number of stages. 4,References to particular studies on request.
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