Normal and composition series of groups enumerated by ordinal numbers are studied. The Jordan-Hölder theorem for them is proved.

An example is given of a strictly singular non-compact operator on a Hereditarily Indecomposable, reflexive, asymptotic `1 Banach space. The construction of this operator relies on the existence of transfinite c0-spreading models in the dual of the space.

In this paper we study the behavior of the (transfinite) small inductive dimension (trind) ind on finite products of topological spaces. In particular we essentially improve Toulmin’s estimation [T] of trind for Cartesian products.

We prove addition and subspace theorems for asymptotic large inductive dimension. We investigate a transfinite extension of this dimension and show that it is trivial. 0. Asymptotic dimension asdim of a metric space was defined by Gromov for studying asymptotic invariants of discrete groups [1]. This dimension can be considered as asymptotic analogue of the Lebesgue covering dimension dim. Dran...

It is well-known that the Ford-Fulkerson algorithm for finding a maximum flow in a network need not terminate if we allow the arc capacities to take irrational values. Every non-terminating example converges to a limit flow, but this limit flow need not be a maximum flow. Hence, one may pass to the limit and begin the algorithm again. In this way, we may view the Ford-Fulkerson algorithm as a t...

We generalize the classical Robin formula to higher dimensions. Mathematics Subject Classification (2000) Primary 32U35 · 32U20 · 14G40; Secondary 31B15 · 31C10 The purpose of this note is to give a formula for the Fekete-Leja transfinite diameter on CN , generalizing the classical Robin formula d∞(E) = e−V(E) for the usual transfinite diameter. We will disengage this from a formula for the sec...

A prior work examined the propagation of an electromagnetic wave on a transfinite transmission line—transfinite in the sense that infinitely many one-way infinite transmission lines are connected in cascade. That there are infinitely many such lines results in the wave propagating without ever reflecting at some discontinuity This work examines the case where the line is terminated after finite...

In this paper we discuss the recently introduced transfinite Knuth-Bendix orders. We prove that any such order with finite subterm coefficients and for a finite signature is equivalent to an order using ordinals below ω, that is, finite sequences of natural numbers of a fixed length. We show that this result does not hold when subterm coefficients are infinite. However, we prove that in this ge...