In this paper, some fundamental limitations of projective invariants of non-algebraic planar curves are discussed. It is shown that all curves within a large class can be mapped arbitrarily close to a circle by projective transformations. It is also shown that arbitrarily close to each of a finite number of closed planar curves there is one member of a set of projectively equivalent curves. Thu...

Symmetry groups or groupoids of C∗-algebras associated to nonHausdorff spaces are often non-Hausdorff as well. We describe such symmetries using crossed modules of groupoids. We define actions of crossed modules on C∗-algebras and crossed products for such actions, and justify these definitions with some basic general results and examples.

LetG be a Hausdorff topological locally compact group. LetM(G) denote the Banach algebra of all complex and bounded measures on G. For all integers n ≥ 1 and all μ ∈ M(G), we consider the functional equations ∫ G f(xty)dμ(t) = ∑n i=1gi(x)hi(y), x,y ∈ G, where the functions f , {gi}, {hi}: G → C to be determined are bounded and continuous functions on G. We show how the solutions of these equati...

This paper is mainly concerned with Hausdorff dimensions of Besicovitch-Eggleston subsets in countable symbolic space. A notable point is that, the dimension values posses a universal lower bound depending only on the underlying metric. As a consequence of the main results, we obtain Hausdorff dimension formulas for sets of real numbers with prescribed digit frequencies in their Lüroth expansions.

Let X be a non-metric continuum, and C(X) the hyperspace of subcontinua of X. It is known that there is no Whitney map on the hyperspace 2 for non-metrizable Hausdorff compact spaces X. On the other hand, there exist non-metrizable continua which admit and the ones which do not admit a Whitney map for C(X). In this paper we investigate the properties of non-metrizable continua which admit a Whi...

Let S be a locally compact Hausdorff semigroup such that its multiplication is separately continuous. We denote by m S the Banach algebra under the supremum norm of all bounded real-valued functions on S. For a topological semigroup S, let BM S and CB S be the closed subalgebras ofm S consisting of all Borel measurable functions and all continuous functions on S, respectively. Let C0 S be the s...

It is pointed out that what concurrent engineering contributed to industry is that our knowledge is non-physical so that it can be shared. Although traditional concurrent engineering contributed very greatly in reducing time to market and reducing cost, it only paid attention to 1 dimension, i.e. time. But if we expand concurrent engineering into 2 dimensions, i.e. time and space, we can integr...

We give some new characterizations of exactness for locally compact second countable groups. In particular, we prove that a locally compact group is exact if and only if it admits a topologically amenable action on a compact Hausdorff space. This answers an open question by Anantharaman-Delaroche.

In this paper, we study the quantization dimension for a random self-conformal measure μ supported on the random self-conformal set E. We establish that the quantization dimension of μ coincides with its Hausdorff dimension with probability one under the strong separation condition.

In an L-topological space we present good definitions for local compactness, weak local compactness and relative local compactness. We obtain the equivalence of these properties in a Hausdorff space and we also obtain a one point compactification theorem.