High efficiency redundant binary number representations for parallel arithmetic on optical computers

A family of redundant binary number representations, obtained by generalization of the RB (redundant binary) number representation, is introduced. All these number representations are suitable for optical computing and have properties similar to the RB representation. In particular, the p-RB (packed redundant binary) number representation introduced in this work has efficiency greater than both RB and MSD (modified signed digit) representations. With p-RB numbers the algebraic sum is always permitted in constant time for any efficiency value. p-RB representations also fit in a natural way the 2's complement binary number system. Symbolic substitution truth tables for the algebraic sum and several examples of computation are also given.     

Multilevel preconditioning — Appending boundary conditions by Lagrange multipliers

For saddle point problems stemming from appending essential boundary conditions in connection with Galerkin methods for elliptic boundary value problems, a class of multilevel preconditioners is developed. The estimates are based on the characterization of Sobolev spaces on the underlying domain and its boundary in terms of weighted sequence norms relative to corresponding multilevel expansions. The results indicate how the various ingredients of a typical multilevel framework affect the growth rate of the condition numbers. In particular, it is shown how to realize even condition numbers that are uniformly bounded independently of the discretization.These investigations are motivated by the idea of employing nested refinable shift-invariant spaces as trial spaces covering various types of wavelets that are of advantage for the solution of boundary value problems from other points of view. Instead of incorporating the boundary conditions into the approximation spaces in the Galerkin formulation, they are appended by means of Lagrange multipliers leading to a saddle point problem.The work of the author is partially supported by the Deutsche Forschungsgemeinschaft under grant numbers Ku1028/1-1 and Pr336/4-1.     

Phase-fixed double-group 3-Γ symbols. VI. Real 3-Γ symbols and coupling coefficients for the group hierarchy I *⊃ C 5 *

It is demonstrated that for the group-subgroup hierarchy I ^* C ₅ ^* , one may choose standard irreducible matrix representations and corresponding all-real sets of 3- symbols which obey a formalism just as elegant as the classical one for the 3-j symbols of the rotation double group. The 3- symbols are phase-fixed by the specification of basis functions (or, equivalently, subduction coefficients) generating them and based on functions first given by McLellan.Other icosahedral double-group hierarchies are also briefly discussed.     

Cyclic by prime fixed point free action

Let the finite group be acting on a (solvable) group and suppose that no non-trivial element of is fixed under the action of all the elements of . Assume furthermore that . A long standing conjecture is that then the Fitting height of is bounded by the length of the longest chain of subgroups of . Even though this conjecture is known to hold for large classes of groups , it is still unknown for some relatively uncomplicated groups. In the present paper we prove the conjecture for all finite groups that have a normal cyclic subgroup of square free order and prime index. Since many of these groups have natural modules where they act faithfully and coprimely but without regular orbits, the result is new for many of the groups we consider.

     

Hecke algebras,

The Donald-Flanigan conjecture asserts that the integral group ring of a finite group can be deformed to an algebra over the power series ring with underlying module such that if is any prime dividing then is a direct sum of total matric algebras whose blocks are in natural bijection with and of the same dimensions as those of We prove this for using the natural representation of its Hecke algebra by quantum Yang-Baxter matrices to show that over localized at the multiplicatively closed set generated by and all , the Hecke algebra becomes a direct sum of total matric algebras. The corresponding ``canonical" primitive idempotents are distinct from Wenzl's and in the classical case (), from those of Young.

     

On Decomposition Numbers and Branching Coefficients for Symmetric and Special Linear Groups

In this paper we find the multiplicities dim L()_– where is an arbitrary root and L() is an irreducible SL_n-module withhighest weight . We provide different bases of the correspondingweight spaces and outline some applications to the symmetricgroups. In particular we describe certain composition multiplicitiesin the modular branching rule. 1991 Mathematics Subject Classification:20C05, 20G05.     

On the Continuous Extension of States on the Ccr Algebra

The algebra CCR (E) of the CCR depends sensitively on the choice of the one boson test-function space E. Here the unique extension of certain continuous states, resp. representations, to the extended CCR algebra CCR E corresponding to an enlarged test-function space E is studied. Various applications in statistical physics, QED, quantum optics, solid state physics, and quantum chemistry are pointed out.     

Unitarity of highest weight modules for quantum groups

We determine the highest weights that give rise to unitarity when q is real. We further show that when q is on the unit circle and q ± 1, then unitary highest-weight representations must be finite-dimensional and q must be a root of unity. We analyze the special case of the 'Ladder' representations for su . Finally we show how the quantized Ladder representations and their analogues for other Lie algebras play an important role.