BFNC Research Articles

Relativizing small complexity classes and their theories

Existing definitions of the relativizations of NC1, L and NL do not preserve the inclusions $${{\bf NC}^1 \subseteq {\bf L}, {\bf NL}\subseteq {\bf AC}^1}$$NC1⊆L,NL⊆AC1. We start by giving the first definitions that preserve them. Here for L and NL we define their relativizations using Wilson's stack oracle model, but limit the height of the stack to a constant (instead of log(n)). We show that the collapse of any two classes in $${\{{\bf AC}^0 (m), {\bf TC}^0, {\bf NC}^1, {\bf L}, {\bf NL}\}}$${AC0(m),TC0,NC1,L,NL} implies the collapse of their relativizations. Next we exhibit an oracle ? that makes ACk(?) a proper hierarchy. This strengthens and clarifies the separations of the relativized theories in Takeuti (1995). The idea is that a circuit whose nested depth of oracle gates is bounded by k cannot compute correctly the (k + 1) compositions of every oracle function. Finally, we develop theories that characterize the relativizations of subclasses of P by modifying theories previously defined by the second two authors. A function is provably total in a theory iff it is in the corresponding relativized class, and hence, the oracle separations imply separations for the relativized theories.

Primeness Results for von Neumann Algebras Associated with Surface Braid Groups

In this paper we introduce a new class of non-amenable groups denoted by ${\bf NC}_1 \cap {\bf Quot}(\mathcal C_{rss})$ which give rise to $\textit{prime}$ von Neumann algebras. This means that for every $\Gamma \in {\bf NC}_1 \cap {\bf Quot}(\mathcal C_{rss})$ its group von Neumann algebra $L(\Gamma)$ cannot be decomposed as a tensor product of diffuse von Neumann algebras. We show ${\bf NC}_1 \cap {\bf Quot}(\mathcal C_{rss})$ is fairly large as it contains many examples of groups intensively studied in various areas of mathematics, notably: all infinite central quotients of pure surface braid groups; all mapping class groups of (punctured) surfaces of genus $0,1,2$; most Torelli groups and Johnson kernels of (punctured) surfaces of genus $0,1,2$; and, all groups hyperbolic relative to finite families of residually finite, exact, infinite, proper subgroups.

The Tensors of the Averaged Relative Energy–Momentum and Angular Momentum in General Relativity and Some of Their Applications

There exist at least a few different kind of averaging of the differences of the energy-momentum and angular momentum in normal coordinates {\bf NC(P)} which give tensorial quantities. The obtained averaged quantities are equivalent mathematically because they differ only by constant scalar dimensional factors. One of these averaging was used in our papers [1-8] giving the {\it canonical superenergy and angular supermomentum tensors}. In this paper we present another averaging of the differences of the energy-momentum and angular momentum which gives tensorial quantities with proper dimensions of the energy-momentum and angular momentum densities. But these averaged relative energy-momentum and angular momentum tensors, closely related to the canonical superenergy and angular supermomentum tensors, {\it depend on some fundamental length $L>0$}. The averaged relative energy-momentum and angular momentum tensors of the gravitational field obtained in the paper can be applied, like the canonical superenergy and angular supermomentum tensors, to {\it coordinate independent} analysis (local and in special cases also global) of this field. We have applied the averaged relative energy-momentum tensors to analyze vacuum gravitational energy and momentum and to analyze energy and momentum of the Friedman (and also more general) universes. The obtained results are very interesting, e.g., the averaged relative energy density is {\it positive definite} for the all Friedman universes.