The primary focus of this work is the closed-form evaluation of integrals of the form$$\int\sbsp{0}{\infty}{P(x\sp2)\over Q(x\sp2)} dx,$$where P and Q are polynomials in x Most of our results are expressed in terms of the polynomial $$\eqalign{\Phi(a\sb1,\cdots,a\sb{p},q;x)&=x\sp{q}+a\sb{p}x \sp{q-1}+a\sb{p-1}x\sp{q-2}\cr&{+}\cdots{+}a\sb1x\sp{q/2}{+}a\sb2x\sp{q/2{-}1}{+}\cdots{+}1,\cr}$$and the integrals$$M\sb{q}(a\sb1,\cdots,a\sb{p};m)=\int\sbsp{0}{\infty} \left\lbrack{x\sp{q/2}\over \Phi(a\sb1,\cdots,a\sb{p},q;x)} \right\rbrack\sp{m}\ dx$$and$$ N\sb{j,q}(a\sb1,\cdots,a\sb{p};m)=\int\sbsp{0}{\infty} \left\lbrack{x\sp{2j}\over \Phi(a\sb1,\cdots,a\sb{p},q;x)}\right\rbrack\sp{m+1}\ dx,$$where $q=4p$ or $4p+2$ according to the residue of q modulo 4 A number of other types of results follow from our integral evaluations. These include some series, among the most notable being$$\eqalign{\sqrt{a+\sqrt{1+c}}\ =\ &\sqrt{a+1}+\cr &+{1\over \pi\sqrt{2}}\sum\sbsp{k=0}{\infty}{(-1)\sp{k-1}\over k}N\sb{0,4}(a;k-1).\cr}$$ The special case with $a=1$ has been studied extensively by Lagrange, Ramanujan and others / acase@tulane.edu

acase@tulane.edu

The geometric problem of prescribing positive curvature to noncompact surfaces of finite topological type is studied with the help of the conformal Gauss curvature equation Deltau + Ke2u -- k = 0. New asymptotic conditions on the admissible set of prescribed curvature functions are elucidated. In particular, a compact surface punctured at two points admits a positive curvature function with arbitrary polynomial growth about one of the removed points. The main theorem generalizes previous results of McOwen and Aviles for the Euclidean plane. The relation with the theory of Hulin and Troyanov is discussed. The method used to prove the existence of solutions is a refined version of the variational method of Berger-Moser based on the pseudomonotonicity of the potential operator or Gateaux derivative of the functional. The technique reveals the relation with the monotonicity methods normally used to deal with the negative curvature case. The geometric instrument that makes the reduction of the equation possible is a lemma of Kalka and Yang / acase@tulane.edu

acase@tulane.edu

acase@tulane.edu

acase@tulane.edu

For any given radical $\rho$, the $\rho$-rational extensions of modules is a stronger version of rational extensions of modules, and its properties are studied. However, when $\rho$ is the identity functor both these concepts coincide. In most of the work of this thesis, the ring of the module structure is not assumed to be commutative. Just as every module has a rational completion, it is shown that for any given radical $\rho$, every module has a $\rho$-rational completion, which is unique upto isomorphism in its $\rho$-divisible hull. For any ring R with identity, the $\rho$-rational completion $\overline{R} \sp\rho$ of R is a ring. The $\rho$-rational completion of a module in terms of filters are studied, and the behavior of $\rho$-rationally complete modules under the formation of direct products and direct sums is obtained. Finally, we establish the invariance of the $\rho$-rational completeness of a module under a change of rings / acase@tulane.edu

acase@tulane.edu

acase@tulane.edu

acase@tulane.edu