Regular maps with nilpotent automorphism groups

AbstractWe study regular maps with nilpotent automorphism groups in detail. We prove that every nilpotent regular map decomposes into a direct product of maps H×K, where Aut(H) is a 2-group and K is a map with a single vertex and an odd number of semiedges. Many important properties of nilpotent maps follow from this canonical decomposition, including restrictions on the valency, covalency, and the number of edges. We also show that, apart from two well-defined classes of maps on at most two vertices and their duals, every nilpotent regular map has both its valency and covalency divisible by 4. Finally, we give a complete classification of nilpotent regular maps of nilpotency class 2

Prime decomposition and correlation measure of finite quantum systems

Under the name prime decomposition (pd), a unique decomposition of an arbitrary $N$-dimensional density matrix $\rho$ into a sum of seperable density matrices with dimensions given by the coprime factors of $N$ is introduced. For a class of density matrices a complete tensor product factorization is achieved. The construction is based on the Chinese Remainder Theorem and the projective unitary representation of $Z_N$ by the discrete Heisenberg group $H_N$. The pd isomorphism is unitarily implemented and it is shown to be coassociative and to act on $H_N$ as comultiplication. Density matrices with complete pd are interpreted as grouplike elements of $H_N$. To quantify the distance of $\rho$ from its pd a trace-norm correlation index $\cal E$ is introduced and its invariance groups are determined.Comment: 9 pages LaTeX. Revised version: changes in the terminology, updates in ref

On the complexity of solving linear congruences and computing nullspaces modulo a constant

We consider the problems of determining the feasibility of a linear congruence, producing a solution to a linear congruence, and finding a spanning set for the nullspace of an integer matrix, where each problem is considered modulo an arbitrary constant k>1. These problems are known to be complete for the logspace modular counting classes {Mod_k L} = {coMod_k L} in special case that k is prime (Buntrock et al, 1992). By considering variants of standard logspace function classes --- related to #L and functions computable by UL machines, but which only characterize the number of accepting paths modulo k --- we show that these problems of linear algebra are also complete for {coMod_k L} for any constant k>1. Our results are obtained by defining a class of functions FUL_k which are low for {Mod_k L} and {coMod_k L} for k>1, using ideas similar to those used in the case of k prime in (Buntrock et al, 1992) to show closure of Mod_k L under NC^1 reductions (including {Mod_k L} oracle reductions). In addition to the results above, we briefly consider the relationship of the class FUL_k for arbitrary moduli k to the class {F.coMod_k L} of functions whose output symbols are verifiable by {coMod_k L} algorithms; and consider what consequences such a comparison may have for oracle closure results of the form {Mod_k L}^{Mod_k L} = {Mod_k L} for composite k.Comment: 17 pages, one Appendix; minor corrections and revisions to presentation, new observations regarding the prospect of oracle closures. Comments welcom

Power Integral Points on Elliptic Curves

This thesis looks at some of the modern approaches towards the solution of Diophantine equations, and utilizes them to display the nonexistence of perfect powers occurring in certain types of sequences. In particular we look at the denominator divisibility sequences (Bn) formed by Mordell elliptic curves ED : y2 = x3+D. For the curve-point pair (E−2, P), where E−2 : y2 = x3 −2, and P = (3, 5) is a nontorsion point, we prove that no term Bn is a perfect 5th power, and we give the explicit bound p � 137 for any term in the associated elliptic denominator sequence to be a perfect power Bn = Zpn for 1 < n < 113762879. We then look at obtaining upper bounds on p for the seventy-two rank 1 Mordell curves in the range |D| < 200 to possess a pth perfect power. This is done by consideration of the finite number of rational and irrational newforms corresponding to an also finite number of levels of these newforms: in thirty cases we give a bound via examination of both the rational and irrational cases, and for the remaining forty-two cases our bound is merely for the rational case due to computational limitations