It is shown that assuming the existence of a suitable ring homomorphism is enough to get an algebraic presentation of nonstandard methods that is equivalent to the popular superstructure approach, including $\kappa$-saturation.

Let $Q$ be a bounded, convex and locally closed subset of \ $\C^N$, let $H(Q)$ be the space of all functions which are holomorphic on an open neighborhood of $Q$. We endow $H(Q)$ with its projective topology. We show that the topology of the weighted inductive limit of Fr\'echet spaces of entire functions which is obtained as the Laplace transform of the strong dual to $H(Q)$ can be described be means of canonical weighted seminorms if and only if the intersection of $Q$ with each supporting hyperplane to the closure of $Q$ is compact. We also find conditions under which this (LF)-space of entire functions coincides algebraically with its projective hull.

Is any soap bubble necessarily a Euclidean sphere ? In this expository text, we show how this simple question has led over the years to developments in many fields of mathematics.

The compatibility up to sign of the product structures in algebraic $K$-theory and in topological $K$-theory of unital Banach algebras is established in total degree $\leq 2$\,. This answers a question posed by Milnor.

Kazhdan constants for $\mathop{\rm {Sp}}\left( n,\Omega \right) $ where $\Omega $ is a commutative topological ring with dense finitely generated subring with unity are determined. This implies Kazhdan's property T for these groups. As application explicit Kazhdan constants are determined for the loop groups corresponding to $\mathop{\rm {Sp}}\left( n,\mathbf{C}\right) $. These are further examples of groups with property T which are infinite dimensional Lie groups and not locally compact.

Let $A$ be a complex Banach algebra, $f:A\rightarrow A$ be a polynomial function with coefficients in $A$. We define the Julia set of f, denoted by J(f). We give conditions which often determine in which set, $J(f)$ or $A\setminus J(f)$, the periodic point lies. We show that the closure of repelling periodic points is not always equals the Julia set. However, we use the theorem of Gelfand-Mazur to characterize the algebras where it's true.

We define a new class of dense near polygons. The unique near $2n$-gon, $n \geq 0$, of this class will be denoted by $\G_n$. We will study the geodetically closed sub near polygons of $\G_n$. We will also determine the complete automorphism group and all spreads of symmetry. New glued near polygons can be constructed from these spreads of symmetry.

Let $E$ be a Banach space and $\Lambda$ a Banach perfect sequence space. Denote by $\Lambda (E)$ the space of all $\Lambda$-summable sequences from $E$. In this note it is proved that $\Lambda (E)$ is reflexive if and only if $\Lambda$ and $E$ are reflexive and each member of $\Lambda (E)$ is the limit of its finite sections.

In this paper we sketch a general theory of embeddings for geometries with string diagrams, focusing on their hulls. An affine-like geometry, which we call expansion, is associated to every embedding. As we shall prove, the universal cover of the expansion of an embedding is the expansion of the hull of that embedding. Some applications of this theorem are given.

We give characterizations of unital uniform topological algebras and saturated locally multiplicatively convex algebras by means of multiplicative linear functionals. Some automatic continuity theorems in advertibly complete uniform topological algebras are extended to a larger class of algebras. Consequences and applications are given.