**Abstract: **I discuss old and new results on fixed points of local actions by Lie groups *G* on real and complex 2-manifolds, and zero sets of Lie algebras of vector fields. Results of E. Lima, J. Plante and C. Bonatti are reviewed.

**Abstract: **A topological group is called a pro-Lie group if it is isomorphic to a closed subgroup of a product of finite-dimensional real Lie groups. This class of groups is closed under the formation of arbitrary products and closed subgroups and forms a complete category. It includes each finite-dimensional Lie group, each locally-compact group that has a compact quotient group modulo its identity component and, thus, in particular, each compact and each connected locally-compact group; it also includes all locally-compact Abelian groups. This paper provides an overview of the structure theory and the Lie theory of pro-Lie groups, including results more recent than those in the authors’ reference book on pro-Lie groups. Significantly, it also includes a review of the recent insight that weakly-complete unital algebras provide a natural habitat for both pro-Lie algebras and pro-Lie groups, indeed for the exponential function that links the two. (A topological vector space is weakly complete if it is isomorphic to a power RX of an arbitrary set of copies of R. This class of real vector spaces is at the basis of the Lie theory of pro-Lie groups.) The article also lists 12 open questions connected to pro-Lie groups.

**Abstract: **In this paper, we investigate the heat kernel embedding as a route to graph representation. The heat kernel of the graph encapsulates information concerning the distribution of path lengths and, hence, node affinities on the graph; and is found by exponentiating the Laplacian eigen-system over time. A Young–Householder decomposition is performed on the heat kernel to obtain the matrix of the embedded coordinates for the nodes of the graph. With the embeddings at hand, we establish a graph characterization based on differential geometry by computing sets of curvatures associated with the graph edges and triangular faces. A sectional curvature computed from the difference between geodesic and Euclidean distances between nodes is associated with the edges of the graph. Furthermore, we use the Gauss–Bonnet theorem to compute the Gaussian curvatures associated with triangular faces of the graph.

**Abstract: **This paper provides a simplified representation of the exact density function of *R*, the sample correlation coefficient. The odd and even moments of *R* are also obtained in closed forms. Being expressed in terms of generalized hypergeometric functions, the resulting representations are readily computable. Some numerical examples corroborate the validity of the results derived herein.

**Abstract: **We prove that if a paratopological group *G* is a continuous image of an arbitrary product of regular Lindelöf Σ -spaces, then it is R-factorizable and has countable cellularity. If in addition, *G* is regular, then it is totally w-narrow and satisfies cel_{w}(*G*) ≤ w, and the Hewitt–Nachbin completion of *G* is again an R-factorizable paratopological group.

**Abstract: **In this article, we move back almost 200 years to Christoph Gudermann, the great expert on elliptic functions, who successfully put the twelve Jacobi functions in a didactic setting. We prove the second hyperbolic series expansions for elliptic functions again, and express generalizations of many of Gudermann’s formulas in Carlson’s modern notation. The transformations between squares of elliptic functions can be expressed as general Möbius transformations, and a conjecture of twelve formulas, extending a Gudermannian formula, is presented. In the second part of the paper, we consider the corresponding formulas for hyperbolic modular functions, and show that these Möbius transformations can be used to prove integral formulas for the inverses of hyperbolic modular functions, which are in fact Schwarz-Christoffel transformations. Finally, we present the simplest formulas for the Gudermann Peeta functions, variations of the Jacobi theta functions. 2010 Mathematics Subject Classification: Primary 33E05; Secondary 33D15.