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Abstract:

1. Introduction The most common method of mixing cards is the ordinary riffle shuffle, in which a deck of n cards (often n = 52) is cut into two parts and the parts are riffled together. A sharp mathematical analysis for a natural model of riffle shuffling was carried out by Bayer and Diaconis (1992). This gives closed form expressions for the chance of any permutation and allows analytic approximation and exact numerical evaluation to show things like " seven shuffles are necessary and suffice to approximately randomize 52 cards ". These results are carefully stated in Section 2A. The shuffling work builds on earlier studies of Jordan (magic tricks), Borel (bridge), Gilbert, Shannon, Reeds (basic model) and D. Aldous (coupling). This background is described in Section 2B. The " seven shuffles " result is mildly dependent on the choice of metric and a number of alternative measures of randomness are discussed in Section 2C. There is a mathematical reason that allows riffle shuffles to be analyzed so completely. The basic shuffling model falls squarely into Solomon's descent algebra (and indeed gives an independent development). This allows shuffling theorems to be translated into permutation enumeration results (e.g. how many permutations have a given number of descents and a given cycle structure). The eigenvalues of the Markov chain underlying shuffling were actually first determined in an investigation of Hochschild homology(Hanlon). There is an intimate connection with free Lie algebras and the Poincare-Birkoff-Witt Theorem (Bergeron-Bergeron-Garsia). The chance of a given cycle structure after riffle shuffling equals the chance that a random degree n polynomial has a given number of irreducible factors. This in turn is explained by considering the connection between shuffling and the action of the associated Lie type group SL n (F q) on its Lie algebra (Fulman). Finally, shuffling gives a fairly direct interpretation of Schur symmetric functions (Stanley-Fulman). These results are described in section three. The analyses above seem so rich and natural that they call out for generalization. A sweeping generalization of the theory was discovered by Bidigare, Hanlon and Rockmore. This involves random walk on the chambers of a hyperplane arrangement. The classical braid arrangement gives riffle shuffles but there are many other hyperplane arrangements where the chambers can be labeled by natural com-binatorial objects and much (but not all) of the theory goes through. In an amazing synthesis, Ken Brown has shown that almost everything …

To avoid any semblance of a claim to specious authority, this response to the state of the art in machine translation and to predict its future will be structured around five basic elements of one of the classic Tarot spreads.

Ground penetrating radar (GPR) has become an effective means for assessing deterioration in concrete bridge decks. While success has been demonstrated, the method is still not adopted widely. Constant technical development is making such high speed GPR mapping more affordable with systems more widely available and easier to deploy. The American Society for Testing and Materials (ASTM) has a standard procedure for performing bridge deck deterioration using GPR. The current standard, initially written for air-launched GPR devices and then modified to include ground-coupled GPRs, has many simplifying assumptions that could lead to fallacious evaluations. Both field experience and numerical simulations indicate that ground-coupled GPR systems are preferable to air-launched GPRs in this application, delivering larger signal-to-noise and higher spatial resolution data, which enhance extraction of both electromagnetic wave velocity and attenuation. We describe advances in analysis and interpretation that go beyond the current ASTM approach which ignores the impact of depth and other variables. We demonstrate these advances using a high speed, ground-coupled GPR system with examples of deck deterioration mapping. We describe the workflow for using GPR to evaluate the deterioration of concrete bridge decks, highlight the basic interpretation assumptions, demonstrate successful applications and discuss limitations with the methodology.

A description is given of the theoretical analysis procedures used to predict the wave impact forces acting on offshore platform deck structures in large incident waves. Both vertical and horizontal plane forces are considered, in terms of the different type elements that make up such structures and the type of hydrodynamic force mathematical models used to represent the basic forces. Effects of wave surface nonlinearity (including kinematics), deck material porosity, and velocity blockage and shielding are considered in the analysis, which also includes a physical explanation of various observed phenomena. Results of comparison and correlation with experimental model test data are presented, including description of procedures used in data analysis to eliminate extraneous dynamic effects that often contaminate such data. The influence of wave heading angle relative to different structural elements (and overall structures) is also described, including both analytical representations and physical interpretations.

Recent achievements and trends in Western Europe in the conception and design of bridge superstructures of structural steel and concrete acting compositely are examined and available methods of analysis are summarised. The interpretation of limit-state design philosophy in part 5 of British Standard 5400 "The Bridge Code" is explained and a detailed presentation and explanation of typical design calculations is given in four worked examples. There are chapters on: conception and global analysis of the superstructure, design of superstructure for beam and slab bridges, box girders and composite plates, fatigue, permanent formwork in bridge decks, cased beam and filler beam construction, prestressing of composite beams, composite columns, construction. The following appendices are included: control of cracking by limiting the spacing of reinforcing bars, stresses due to temperature and shrinkage, fatigue strength of shear connectors, a modified perry formula to represent the European curves for steel struts, ultimate moment of resistance of a concrete filled steel circular hollow section, formulae for the elastic analysis of a beam with one change of cross-section. The fundamentals of composite construction such as the basic structural properties of composite cross sections and members, and an account of methods of shear connection, are presented in volume 1. /TRRL/