Are the arc components in a locally compact group Borel subsets? An affirmative answer is provide for locally compact groups satisfying the First Axiom of Count-ability. For general locally compact groups the question is reduced to compact connected Abelian groups. In certain models of set theory the answer is negative.

The deposition rate of titanium films plays a determining role for possible fabrication schemes of metallic SET structures. Depending on this rate the resistivities display a striking difference in their temperature dependence which is attributed to their different crystal structure as analyzed by transmission electron-beam microscopy. In addition a sharp rise in resistivity occurs below a critical thickness in films deposited at slow rates. This sharp increase can be used in combination with a step induced thinning to form tunnel junctions for single electron transistor structures.

A (local) Lie loop is a real analytic manifold M with a base point e and three analytic functions (x, y) → x° y, x\y, x/y: M × M → M (respectively, U × U → M for an open neighbourhood U of e in M) such that the following conditions are satisfied: (i) x ° e = e ° x = e, (ii) x ° (x\y) = y, and (iii) (x/y)° y = x for all x, y ε M (respectively, U). If the multiplication ° is associative, then M is a (local) Lie group. The tangent vector space L(M) in e is equipped with an anticommutative bilinear operation (X, Y) →[X, Y] and a trilinear operation (X, Y, Z) →〈X, Y, Z〉. These are defined as follows: Let B be a convex symmetric open neighbourhood of 0 in L(M) such that the exponential function maps B diffeomorphically onto an open neighbourhood V of e in M and transport the operation ° into L(M) by defining X ° Y = (exp|B)−1((exp X)° (exp Y)) for X and Y in a neighbourhood C of 0 in B such that (exp C) ° (exp C) ⊂ V. Similarly, we transport / and \.

In the category of locally compact groups not all families of groups have a product. Precisely which families do have a product and a description of the product is a corollary of the main theorem proved here. In the category of locally compact abelian groups a family {Gj; j ∈ J} has a product if and only if all but a finite number of the Gj are of the form Kj × Dj, where Kj is a compact group and Dj is a discrete torsion free group. Dualizing identifies the families having coproducts in the category of locally compact abelian groups and so answers a question of Z. Semadeni.

We propose an exceptionally short and effortless approach to the fundamental facts of the representation theory of compact groups on Hilbert spaces and expand our methods to prove some generalizations concerning representations on arbitrary locally convex vector spaces.