Structure theorems for noncommutative complete local rings

If R is a ring with identity, let N(R) denote the Jacobson radical of R. R is local if R/N(R) is an artinian simple ring and ∩N(R)ⁱ = 0. It is known that if R is complete in the N(R)-adic topology then R is equal to (B)_n, the full n by n matrix ring over B where E/N(E) is a division ring. The main results of the thesis deal with the structure of such rings B. In fact we have the following.

If B is a complete local algebra over F where B/N(B) is a finite dimensional normal extension of F and N(B) is finitely generated as a left ideal by k elements, then there exist automorphisms g_i,...,g_k of B/N(B) over F such that B is a homomorphic image of B/N[[x₁,…,x_k;g₁,…,g_k]] the power series ring over B/N(B) in noncommuting indeterminates x_i, where x_ib = g_i(b)x_i for all b ϵ B/N.

Another theorem generalizes this result to complete local rings which have suitable commutative subrings. As a corollary of this we have the following. Let B be a complete local ring with B/N(B) a finite field. If N(B) is finitely generated as a left ideal by k elements then there exist automorphisms g₁,…,g_k of a v-ring V such that B is a homomorphic image of V [[x₁,…,x_k;g₁,…,g_k]].

In both these results it is essential to know the structure of N(B) as a two sided module over a suitable subring of B.

Frame presentations: variants of the reals, rings of functions, their Dedekind completions, and the unit circle

176 p.

Band structure of photonic crystal fibers with silica rings in triangular lattice

The characteristics of the cladding band structure of air-core photonic crystal fibers with silica rings in triangular lattice are investigated by using a standard plane wave method. The numerical results show that light can be localized in the air core by the photonic band gaps of the fiber. By increasing the air-filling fraction, the band gap edges of the low frequency photonic band gaps shift to shorter wavelength.. whereas the band gap width decreases linearly. In order to make a specified light fall in the low frequency band gaps of the fiber, the interplay of the silica ring spacing and the air-filling fraction is also analyzed. It shows that the silica ring spacing increases monotonously when the air-filling fraction is increased, and the spacing range increases exponentially. This type fiber might have potential in infrared light transmission. (c) 2006 Elsevier B.V. All rights reserved.

Age and growth of three species of Lake Victoria fish determined by means of otolith daily growth rings

The sagittal otoliths of Lates niloticus, Haplochromis obesus, and Oreochromis niloticus from Lake Victoria were examined for daily growth rings using scanning electron microscopy. In the three species the increments were clear and thick enough to allow future studies with light microscopy. The daily nature of the increments seems supported by the rhythmic growth that were found.

400 years of precipitation variability in central and southern California reconstructed from tree-rings [abstract]

EXTRACT (SEE PDF FOR FULL ABSTRACT): Precipitation variability at 31 stations hanging from San Diego to San Francisco and from the coast to the Sierras was characterized ...