Cantor Series Constructions Contrasting Two Notions of Normality

A. R\'enyi \cite{Renyi} made a definition that gives a generalization of simple normality in the context of $Q$-Cantor series. In \cite{Mance}, a definition of $Q$-normality was given that generalizes the notion of normality in the context of $Q$-Cantor series. In this work, we examine both $Q$-normality and $Q$-distribution normality, treated in \cite{Laffer} and \cite{Salat}. Specifically, while the non-equivalence of these two notions is implicit in \cite{Laffer}, in this paper, we give an explicit construction witnessing the nontrivial direction. That is, we construct a base $Q$ as well as a real $x$ that is $Q$-normal yet not $Q$-distribution normal. We next approach the topic of simultaneous normality by constructing an explicit example of a base $Q$ as well as a real $x$ that is both $Q$-normal and $Q$-distribution normal

Two short proofs of the bounded case of S.B. Rao's degree sequence conjecture

S. B. Rao conjectured that graphic sequences are well-quasi-ordered under an inclusion based on induced subgraphs. This conjecture has now been settled completely by M. Chudnovsky and P. Seymour. One part of the proof proves the result for the bounded case, a result proved independently by C. J. Altomare. We give two short proofs of the bounded case of S. B. Rao's conjecture. Both the proofs use the fact that if the number of entries in an integer sequence (with even sum) is much larger than its highest term, then it is necessarily graphic.Comment: 4 page

Ultra narrow AuPd and Al wires

In this letter we discuss a novel and versatile template technique aimed to the fabrication of sub-10 nm wide wires. Using this technique, we have successfully measured AuPd wires, 12 nm wide and as long as 20 $\mu$m. Even materials that form a strong superficial oxide, and thus not suited to be used in combination with other techniques, can be successfully employed. In particular we have measured Al wires, with lateral width smaller or comparable to 10 nm, and length exceeding 10 $\mu$m.Comment: 4 pages, 4 figures. Pubblished in APL 86, 172501 (2005). Added erratum and revised Fig.

Objective Bayesian Search of Gaussian DAG Models with Non-local Priors

Directed Acyclic Graphical (DAG) models are increasingly employed in the study of physical and biological systems, where directed edges between vertices are used to model direct influences between variables. Identifying the graph from data is a challenging endeavor, which can be more reasonably tackled if the variables are assumed to satisfy a given ordering; in this case, we simply have to estimate the presence or absence of each possible edge, whose direction is established by the ordering of the variables. We propose an objective Bayesian methodology for model search over the space of Gaussian DAG models, which only requires default non-local priors as inputs. Priors of this kind are especially suited to learn sparse graphs, because they allow a faster learning rate, relative to ordinary local priors, when the true unknown sampling distribution belongs to a simple model. We implement an efficient stochastic search algorithm, which deals effectively with data sets having sample size smaller than the number of variables. We apply our method to a variety of simulated and real data sets.Fractional Bayes factor; High-dimensional sparse graph; Moment prior; Non-local prior; Objective Bayes; Pathway based prior; Regulatory network; Stochastic search; Structural learning.