Thèse--Univ. de Paris. / "Index bibliographique": p. [657]-659; "Appendice bibliographique": p. [660]

Thesis (Ph.D.)--Brown University, 2005. / Vita. Thesis advisor: G. E. Karniadakis, M. R. Maxey. Includes bibliographical references (leaves 241-253). Also available online.

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Pure- and mixed-integer programmes can often be solved more quickly if the constraints are reformulated first. In this thesis five main themes related to reformulating these programmes are explored.

M.Ing / The geometric shape of a passage's cross-section has an effect on its convective heat transfer capabilities. For concentric annuli, as cross section, the diameter ratio of the annular space plays an important role. The purpose of this investigation was to find a . correlation that will accurately predict heat transfer coefficients at the inner wall of smooth concentric annuli for the flow of water. Experiments were conducted on water under turbulent flow conditions for a wide range of diameter ratios. The Wilson plot method was used to determine the heat transfer coefficients from which a correlation was developed that could be used to predict the heat transfer coefficients. It was found that the correlation predicted Nusselt numbers accurately within 3% of measured values for diameter ratios between a = 1.7 and a = 5.1 and a Reynolds numbers range of 4 000 to 30 000.

Using a classical result of Thue, we give an upper bound for the number of solutions to a family of quartic Thue equations. We also give an upper bound upon the number of solutions to a family of quartic Thue inequalities. Using the Thue-Siegel principle and the theory of linear forms in logarithms, an upper bound is given for general quartic Thue equations. As an application of the method of Thue-Siegel, we will resolve a conjecture of Walsh to the effect that the Diophantine equation aX⁴ - bY² = 1, for fixed positive integers a and b, possesses at most two solutions in positive integers X and Y. Since there are infinitely many pairs (a, b) for which two such solutions exist, this result is sharp. It is also effectively proved that for fixed positive integers a and b, there are at most two positive integer solutions to the quartic Diophantine equation aX⁴ - bY² = 2. We will also study cubic and quartic Thue equations by combining some classical methods from Diophantine analysis with modern geometric ideas. / Science, Faculty of / Mathematics, Department of / Graduate

This experimental study documents the development and separation of a hypersonic boundary layer produced naturally on the cold surface of a sharp slender cone. At the base of the conical forebody, the equilibrium turbulent boundary layer was allowed to separate over an axisymmetric rearward facing step to form a compressible base flow. The investigation was conducted in the Imperial College No.2 gun tunnel at a freestream Mach number of 9 and unit Reynolds numbers of 15 and 55 million. The compressible boundary layer study was carried out at both of the available freestream unit Reynolds numbers and the measured data include distributions of wall static pressure and heat transfer rate, together with profiles of pitot pressure through the boundary layer. Using the chordwise distribution of surface heat flux as a means of transition detection, the cone transition Reynolds number was found to be 5.4x10^. This result, together with that obtained from flat plate studies conducted in the same test facility, provided a ratio of cone to flat plate transition Reynolds number of 0.8. Boundary layer integral quantities and shape factors are derived from velocity profiles and in most cases the measured data extended close enough to the wall to detect the peak values of the integrands. The separated flow region formed at the base of the cone was documented only at the higher unit Reynolds number, a condition under which the approaching turbulent boundary layer was found to be close to equilibrium. The data include pitot pressure profiles recorded normal to the surface downstream of reattachment, together with wall static pressure and heat transfer rate distributions measured throughout the base flow region. Reattachment occurred approximately two step heights downstream of separation and a surface flow visualisation study indicated the existence of Taylor-Goertler type vortices, emanating from the reattachment line in the downstream direction. A simple shear layer expansion model is developed and shown to provide a favourable prediction of the measured pitot pressure profiles recorded downstream of the reattachment line. The success of this second order model implies that the dynamics of the corner expansion process, except in the immediate vicinity of the wall, is governed largely by inviscid pressure mechanisms and that the supersonic region of the boundary layer expansion is essentially isentropic.

This dissertation studies four problems in analytic and probabilistic number theory. Two of the problems are about a certain random number theoretic object, namely a random multiplicative function. The other two problems are about smooth numbers (i.e. numbers only having small prime factors), both in their own right and in their application to finding solutions to S-unit equations over the integers. Thus all four problems are concerned, in different ways, with _understanding the multiplicative structure of the integers. More precisely, we will establish that certain sums of a random multiplicative function satisfy a normal approximation (i.e. a central limit theorem) , but that the complete sum over all integers less than x does not satisfy such an approximation. This reflects certain facts about the number and size of the prime factors of a typical integer. Our proofs use martingale methods, as well as a conditioning argument special to this problem. Next, we will prove an almost sure omega result for the sum of a random multiplicative function, substantially improving the existing result of Halasz. We will do this using a connection between sums of a random multiplicative function and a certain random trigonometric sum process, so that the heart of our work is proving precise results about the suprema of a class of Gaussian random processes. Switching to the study of smooth numbers, we will establish an equidistribution result for the y-smooth numbers less than x among arithmetic progressions to modulus q, asymptotically as (logx)/(logq)-+ oo, subject to a certain condition on the relative sizes of y and q. The main point of this work is that it does not require any restrictions on the relative sizes of x and y. Our proofs use a simple majorant principle for trigonometric sums, together with general tools such as a smoothed explicit formula. Finally, we will prove lower bounds for the possible number of solutions of some S-unit equations over the integers. For example, we will show that there exist arbitrarily large sets S of prime numbers such that the equation a+ l = c has at least exp{(#S)116- �} solutions (a, c) with all their prime factors from S. We will do this by using discrete forms of the circle method, and the multiplicative large sieve, to count the solutions of certain auxiliary linear equations.

Three concepts of ordinal numbers are examined with a view to their intuitiveriess and existence in two principle systems of axiomatic set theory. The first is based on equivalence classes of the similarity relation between well-ordered sets. Two alternatives are suggested in later chapters for overcoming the problems arizing from this definition. Next, ordinal numbers are defined as certain representatives of these equivalence classes,, and one of several such possible definitions is taken for proving the fundamental properties of these ordinals. Finally, a generalization of Peano's axioms provides us with a method of defining ordinal numbers which are the ultimate result of abstractions. / Science, Faculty of / Mathematics, Department of / Graduate

In Monte Carlo calculations performed on electronic computers it is advantageous to use an arithmetic scheme to generate sets of numbers with "approximately" the properties of a random sequence. For many applications the local characteristics of the resulting sequence are of interest. In this thesis the concept of a pseudo-random sequence is set out, and arithmetic methods for their generation are discussed. A brief survey of some standard statistical tests of randomness is offered, and the results of empirical tests for local randomness performed on the ALWAC III-E computer at the University of British Columbia are recorded. It is demonstrated that many of the standard generating schemes do not yield sequences with suitable local properties, and could therefore be responsible for misleading results in some applications. A method appropriate for the generation of short blocks of numbers with approximately the properties of a randomly selected set is proposed and tested, with satisfactory results. / Science, Faculty of / Mathematics, Department of / Graduate