In this thesis we consider integrals of a certain class of interval functions. Specifically we consider a nondegenerate number interval [a,b], a real valued function m, defined and nondecreasing on [a,b], and the set Hm, of real valued functions f, defined on [a,b] such that: 1) f(a)=0; 2) for each subinterval [p,q] of [a,b], if m(q)-m(p)=0, then f(q)-f(p)=0; and 3) the set of all sums of the form Σ(Δf)2/Δm for subdivisions D of [a,b] is bounded above.

In this paper, the problem on formation and construction of a shock wave for three dimensional compressible Euler equations with the small perturbed spherical initial data is studied. If the given smooth initial data satisfies certain nondegenerate condition, then from the results in [20], we know that there exists a unique blowup point at the blowup time such that the first order derivates of smooth solution blow up meanwhile the solution itself is still continuous at the blowup point. From the blowup point, we construct a weak entropy solution which is not uniformly Lipschitz continuous on two sides of shock curve, moreover the strength of the constructed shock is zero at the blowup point and then gradually increases. Additionally, some detailed and precise estimates on the solution are obtained in the neighbourhood of the blowup point.

In this thesis we study the zeros and asymptotics of sequences that satisfy linear recurrence relations with generally nonconstant coefficients. By the theorem of Skolem-Mahler-Lech, the set of zero terms of a sequence that satisfies a linear recurrence relation with constant coefficients taken from a field of characteristic zero is comprised of the union of finitely many arithmetic progressions together with a finite exceptional set. Further, in the nondegenerate case, we can eliminate the possibility of arithmetic progressions and conclude that there are only finitely many zero terms. For generally nonconstant coefficients, there are generalizations of this theorem due to Bézivin and to Methfessel that imply, under fairly general conditions, that we obtain a finite union of arithmetic progressions together with an exceptional set of density zero. Further, a condition is given under which one can exclude the possibility of arithmetic progressions and obtain a set of zero terms of density zero. In this thesis, it is shown that this condition reduces to the nondegeneracy condition in the case of constant coefficients. This allows for a consistent definition of nondegeneracy valid for generally nonconstant coefficients and a unified result is obtained. The asymptotic theory of sequences that satisfy linear recurrence relations with generally nonconstant coefficients begins with the basic theorems of Poincaré and Perron. There are some generalizations of these theorems that hold in greater generality, but if we restrict the coefficient sequences of our linear recurrences to be polynomials in the index, we obtain full asymptotic expansions of a predictable form for the solution sequences. These expansions can be obtained by applying a transfer method of Flajolet and Sedgewick or, in some cases, by applying a bivariate method of Pemantle and Wilson. In this thesis, these methods are applied to a family of binomial sums and full asymptotic expansions are obtained. The leading terms of the expansions are obtained explicitly in all cases, while in some cases a field containing the asymptotic coefficients is obtained and some divisibility properties for the asymptotic coefficients are obtained using a generalization of a method of Stoll and Haible.

Describe the topology of the polar for an irreducible curve germ of gender one and generic, we show that are non degenerate Newton. / Describiremos la topología de la polar de un germen de curva irreducible, genérica y de género uno. Para lo cual mostraremos que polar de una curva genérica en K(n;m) es Newton no degenerada.

[pt] Decompõe-se o espaço das curvas não-degeneradas sobre a n-esfera sujeitas a uma dada matriz de monodromia (munido de uma estrutura de variedade de Hilbert adequada) em uma coleção enumerável de células contráteis parametrizadas pelos itinerários admissíveis para os levantamentos a SOn+1 das referidas curvas através das células obtidas de uma estratificação de SOn+1 estreitamente relacionada com a clássica decomposição de Bruhat de GLn+1. A expressão itinerário admissível significa aqui uma sequência finita de células sujeitas a umas poucas restrições que, ademais, são naturalmente insinuadas pela geometria do problema. O principal interesse dessa nova abordagem é que essa combinatorialização funciona homogeneamente em todas as dimensões n (não obstante óbvias dificuldades computacionais), diferentemente dos métodos ad-hoc, de cunho mais geométrico, até aqui empregados para obter informações topológicas sobre esses e outros espaços de curvas relacionados (que têm sido bem sucedidos apenas em dimensões n baixas). Essa abordagem pode ser considerada como uma primeira tentativa de chegar a um método unificado para a determinação do tipo homotópico de tais espaços, e ajuda a dispensar certos argumentos de análise funcional usualmente empregados na definição da topologia correta para os referidos espaços de curvas. / [en] The space of nondegenerate curves on the n-sphere subject to a fixed monodromy matrix (provided with a suitable Hilbert manifold structure) is decomposed into a countable collection of contractible cells parameterized by the SOn+1-lifted curves admissible itineraries through cells arriving from a stratification of SOn+1 closely related to the classical Bruhat decomposition of GLn+1. The expression admissible itinerary herein stands for a finite sequence of cells subject to a few constraints that are otherwise naturally suggested by the geometry of the problem. The main interest of such a new approach is that this combinatorialization works homogeneously in any dimension n (with obvious computational difficulties), unlike the more geometry-flavoured ad-hoc methods for achieving topological information about these and related spaces of curves (which usually have had a good run only in low dimensions n). This approach can be regarded as a first attempt at a unified method for figuring out the homotopy-type of such spaces, and it helps to override some functional analysis arguments usually deployed in defining the right topology for these spaces of curves.