Abstract:
Sequences diverge either because they head off to infinity or because they oscillate. Part 1 \cite{Part1} of this paper laid the pure mathematics groundwork by defining Archimedean classes of infinite numbers as limits of smooth sequences. Part 2 follows that with applied mathematics, showing that general sequences can usually be converted into smooth sequences, and thus have a well-defined limit. Each general sequence is split into the sum of smooth, periodic (including Lebesgue integrable), chaotic and random components. The mean of each of these components divided by a smooth sequence, or the mean of the mean, will usually be a smooth sequence, and so the oscillatory sequence will have at least a leading term limit. Examples of limits of oscillatory sequences with well-defined limits are given. Methodologies are included for a way to calculate limits on the reals and on complex numbers, a way to evaluate improper integrals by limit of a Riemann sum, and a way to square the Dirac delta function.

Abstract:
Sequences diverge either because they head off to infinity or because they oscillate. Part 1 constructs a non-Archimedean framework of infinite numbers that is large enough to contain asymptotic limit points for non-oscillating sequences that head off to infinity. It begins by defining Archimedean classes of infinite numbers. Each class is denoted by a prototype sequence. These prototypes are used as asymptotes for determining leading term limits of sequences. By subtracting off leading term limits and repeating, limits are obtained for a subset of sequences called here ``smooth sequences". $\mathbb{I}_n$ is defined as the set of ratios of limits of smooth sequences. It is shown that $\mathbb{I}_n$ is an ordered field that includes real, infinite and infinitesimal numbers.

Abstract:
We develop a theory of graph C*-algebras using path groupoids and inverse semigroups. Row finiteness is not assumed so that the theory applies to graphs for which there are vertices emitting a countably infinite set of edges. We show that the path groupoid is amenable, and give a groupoid proof of a recent theorem of Szymanski characterizing when a graph C*-algebra is simple.

Abstract:
The Fourier-Stieltjes and Fourier algebras B(G), A(G) for a general locally compact group G, first studied by P. Eymard, have played an important role in harmonic analysis and in the study of operator algebras generated by G. Recently, there has been interest in developing versions of these algebras for locally compact groupoids, justification being that, just as in the group case, the algebras should play a useful role in the study of groupoid operator algebras. Versions of these algebras for the locally compact groupoid case appear in three related theories: (1) a measured groupoid theory (J. Renault), (2) a Borel theory (A. Ramsay and M. Walter), and (3) a continuous theory (A. Paterson). The present paper is expository in character. For motivational reasons, it starts with a description of the theory of B(G), A(G) in the locally compact group case, before discussing these three realted theories. Some open questions are also raised.

Abstract:
We introduce and investigate using Hilbert modules the properties of the Fourier algebra A(G) for a locally compact groupoid G. We establish a duality theorem for such groupoids in terms of multiplicative module maps. This includes as a special case the classical duality theorem for locally compact groups proved by P. Eymard.

Abstract:
Many index theorems (both classical and in noncommutative geometry) can be interpreted in terms of a Lie groupoid acting properly on a manifold and leaving an elliptic family of pseudodifferential operators invariant. Alain Connes in his book raised the question of an index theorem in this general context. In this paper, an analytic index for many such situations is constructed. The approach is inspired by the classical families theorem of Atiyah and Singer, and the proof generalizes, to the case of proper Lie groupoid actions, some of the results proved for proper locally compact group actions by N. C. Phillips.

Abstract:
The paper establishes, for a wide class of locally compact groupoids $\Gamma$, the E-theoretic descent functor at the $C^{*}$-algebra level, in a way parallel to that established for locally compact groups by Guentner, Higson and Trout. The second section shows that $\Gamma$-actions on a $C_{0}(X)$-algebra $B$, where $X$ is the unit space of $\Gamma$, can be usefully formulated in terms of an action on the associated bundle $B^{\sharp}$. The third section shows that the functor $B\to C^{*}(\Gamma,B)$ is continuous and exact, and uses the disintegration theory of J. Renault. The last section establishes the existence of the descent functor under a very mild condition on $\Gamma$, the main technical difficulty involved being that of finding a $\Gamma$-algebra that plays the role of C_{b}(T,B)^{cont}$ in the group case.

Abstract:
In this paper, we define and investigate the properties of continuous family groupoids. This class of groupoids is necessary for investigating the groupoid index theory arising from the equivariant Atiyah-Singer index theorem for families, and is also required in noncommutative geometry. The class includes that of Lie groupoids, and the paper shows that, like Lie groupoids, continuous family groupoids always admit (an essentially unique) continuous left Haar system of smooth measures. We also show that the action of a continuous family groupoid $G$ on a continuous family $G$-space fibered over another continuous family $G$-space $Y$ can always be regarded as an action of the continuous family groupoid $G*Y$ on an ordinary $G*Y$-space.

Abstract:
The paper constructs the analytic index for an elliptic pseudodifferential family of $L^{m}_{\rho,\de}$-operators invariant under the proper action of a continuous family groupoid on a $G$-compact, $C^{\infty,0}$ $G$-space.

Abstract:
Many index theorems (both classical and in noncommutative geometry) can be interpreted in terms of a Lie groupoid acting properly on a manifold and leaving an elliptic family of pseudodifferential operators invariant. Alain Connes in his book raised the question of an index theorem in this general context. In this paper, an analytic index for many such situations is constructed. The approach is inspired by the classical families theorem of Atiyah and Singer, and the proof generalizes, to the case of proper Lie groupoid actions, some of the results proved for proper locally compact group actions by N. C. Phillips.