Diophantine Sets over Polynomial Rings and Hilbert's Tenth Problem for Function Fields

In 1900, the German mathematician David Hilbert proposed a list of 23 unsolved mathematical problems. In his Tenth Problem, he asked to find an algorithm to decide whether or not a given diophantine equation has a solution (in integers). Hilbert's Tenth Problem has a negative solution, in the sense that such an algorithm does not exist. This was proven in 1970 by Y. Matiyasevich, building on earlier work by M. Davis, H. Putnam and J. Robinson. Actually, this result was the consequence of something much stronger: the equivalence of recursively enumerable and diophantine sets (we will refer to this result as "DPRM"). The first new result in the thesis is about Hilbert's Tenth Problem for function fields of curves over valued fields in characteristic zero. Under some conditions on the curve and the valuation, we have undecidability for diophantine equations over the function field of the curve. One interesting new case are function fields of curves over formal Laurent series. The proof relies on the method with two elliptic curves as developed by K. H. Kim and F. Roush and generalised by K. Eisenträger. Additionally, the proof uses the theory quadratic forms and valuations. And especially for non-rational function fields there is some algebraic geometry coming in. The second type of results establishes the equivalence of recursively enumerable and diophantine sets in certain polynomial rings. The most important is the one-variable polynomial ring over a finite field. This is the first generalisation of DPRM in positive characteristic. My proof uses the structure of finite fields and in particular the properties of cyclotomic polynomials. In the last chapter, this result for polynomials over finite fields is generalised to polynomials over recursive algebraic extensions of a finite field. For these rings we don't have a good definition of "recursively enumerable" set, therefore we consider sets which are recursively enumerable for every recursive presentation. We show that these are exactly the diophantine sets. In addition to infinite extensions of finite fields, we also show the analogous result for polynomials over a ring of integers in a recursive totally real algebraic extension of the rationals. This generalises results by J. Denef and K. Zahidi

Studies of implicit and explicit solution techniques in transient thermal analysis of structures

Studies aimed at an increase in the efficiency of calculating transient temperature fields in complex aerospace vehicle structures are reported. The advantages and disadvantages of explicit and implicit algorithms are discussed and a promising set of implicit algorithms with variable time steps, known as GEARIB, is described. Test problems, used for evaluating and comparing various algorithms, are discussed and finite element models of the configurations are described. These problems include a coarse model of the Space Shuttle wing, an insulated frame tst article, a metallic panel for a thermal protection system, and detailed models of sections of the Space Shuttle wing. Results generally indicate a preference for implicit over explicit algorithms for transient structural heat transfer problems when the governing equations are stiff (typical of many practical problems such as insulated metal structures). The effects on algorithm performance of different models of an insulated cylinder are demonstrated. The stiffness of the problem is highly sensitive to modeling details and careful modeling can reduce the stiffness of the equations to the extent that explicit methods may become the best choice. Preliminary applications of a mixed implicit-explicit algorithm and operator splitting techniques for speeding up the solution of the algebraic equations are also described

On algebraic relations between solutions of a generic Painleve equation

We prove that if y" = f(y,y',t,\alpha, \beta,..) is a generic Painleve equation (i.e. an equation in one of the families PI-PVI but with the complex parameters \alpha, \beta,.. algebraically independent) then any algebraic dependence over C(t) between a set of solutions and their derivatives (y_1,..,y_n,y_1',..,y_n') is witnessed by a pair of solutions and their derivatives (y_i,y_i',y_j,y_j'). The proof combines work by the Japanese school on "irreducibility" of the Painleve equations, with the trichomoty theorem for strongly minimal sets in differentially closed fields.Comment: 23 page

Galois Theory of Parameterized Differential Equations and Linear Differential Algebraic Groups

We present a Galois theory of parameterized linear differential equations where the Galois groups are linear differential algebraic groups, that is, groups of matrices whose entries are functions of the parameters and satisfy a set of differential equations with respect to these parameters. We present the basic constructions and results, give examples, discuss how isomonodromic families fit into this theory and show how results from the theory of linear differential algebraic groups may be used to classify systems of second order linear differential equations