Graph algebras and orbit equivalence

We introduce the notion of orbit equivalence of directed graphs, following Matsumoto’s notion of continuous orbit equivalence for topological Markov shifts. We show that two graphs in which every cycle has an exit are orbit equivalent if and only if there is a diagonal-preserving isomorphism between their C∗C∗-algebras. We show that it is necessary to assume that every cycle has an exit for the forward implication, but that the reverse implication holds for arbitrary graphs. As part of our analysis of arbitrary graphs EE we construct a groupoid G(C∗(E),D(E))G(C∗(E),D(E)) from the graph algebra C∗(E)C∗(E) and its diagonal subalgebra D(E)D(E) which generalises Renault’s Weyl groupoid construction applied to (C∗(E),D(E))(C∗(E),D(E)). We show that G(C∗(E),D(E))G(C∗(E),D(E)) recovers the graph groupoid GEGE without the assumption that every cycle in EE has an exit, which is required to apply Renault’s results to (C∗(E),D(E))(C∗(E),D(E)). We finish with applications of our results to out-splittings of graphs and to amplified graphs

Singular 0/1-matrices, and the hyperplanes spanned by random 0/1-vectors

Let $P(d)$ be the probability that a random 0/1-matrix of size $d \times d$ is singular, and let $E(d)$ be the expected number of 0/1-vectors in the linear subspace spanned by d-1 random independent 0/1-vectors. (So $E(d)$ is the expected number of cube vertices on a random affine hyperplane spanned by vertices of the cube.) We prove that bounds on $P(d)$ are equivalent to bounds on $E(d)$: $$P(d) = (2^{-d} E(d) + \frac{d^2}{2^{d+1}}) (1 + o(1)).$$ We also report about computational experiments pertaining to these numbers.Comment: 9 page

Search for D0 to p e- and D0 to pbar e+

Using data recorded by CLEO-c detector at CESR, we search for simultaneous baryon and lepton number violating decays of the D^0 meson, specifically, D^0 --> p-bar e^+, D^0-bar --> p-bar e^+, D^0 --> p e^- and D^0-bar --> p e^-. We set the following branching fraction upper limits: D^0 --> p-bar e^+ (D^0-bar --> p-bar e^+) p e^- (D^0-bar --> p e^-) < 1.2 * 10^{-5}, both at 90% confidence level.Comment: 10 pages, available through http://www.lns.cornell.edu/public/CLNS/, submitted to PRD. Comments: changed abstract, added reference for section 1, vertical axis in Fig.5 changed (starts from 1.5 rather than 2.0), fixed typo

Lattice Points in Large Borel Sets and Successive Minima

Let $B$ be a Borel set in $\mathbb E^{d}$ with volume $V(B)=\infty$. It is shown that almost all lattices $L$ in $\mathbb E^{d}$ contain infinitely many pairwise disjoint $d$-tuples, that is sets of $d$ linearly independent points in $B$. A consequence of this result is the following: let $S$ be a star body in $\mathbb E^{d}$ with $V(S)=\infty$. Then for almost all lattices $L$ in $\mathbb E^{d}$ the successive minima $\lambda_{1}(S,L),..., \lambda_{d}(S,L)$ of $S$ with respect to $L$ are 0. A corresponding result holds for most lattices in the Baire category sense. A tool for the latter result is the semi-continuity of the successive minima.Comment: 8 page