Counting rational points near planar curves

We find an asymptotic formula for the number of rational points near planar curves. More precisely, if $f:\mathbb{R}\rightarrow\mathbb{R}$ is a sufficiently smooth function defined on the interval $[\eta,\xi]$, then the number of rational points with denominator no larger than $Q$ that lie within a $\delta$-neighborhood of the graph of $f$ is shown to be asymptotically equivalent to $(\xi-\eta)\delta Q^2$

The Diffusion and Adoption of Advanced Technologies in Canada: An Overview of the Issues

The adoption of advanced technologies is a means of fostering productivity improvement. Many theories seek to explain the process of advanced technology diffusion and adoption. Canadian firms generally trail their U.S. counterparts in the adoption of advanced technology. There are many critical gaps in our knowledge and understanding of technological diffusion in Canada. Key gaps include the identification of leading and lagging industries in terms of adoption; key barriers to technological diffusion in Canada including economic-policy-related barriers; appropriate direct policy interventions to overcome specific barriers; the impact of increasing globalization and the economic ascendancy of the large developing countries on diffusion in Canada; and specific challenges small and medium enterprises face in adopting technology. Another issue requiring more research is whether strong R&D performance is a prerequisite for the broad diffusion of technologies. Possible tradeoffs between supporting R&D and supporting diffusion in the presence of limited public funds to promote innovation merit discussion.Diffusion, Adoption, Technologies, Technological diffusion, Innovation, Research and Development, R&D, Advanced technology, Technological competitiveness

Strong Equivalence Relations for Iterated Models

The Iterated Immediate Snapshot model (IIS), due to its elegant geometrical representation, has become standard for applying topological reasoning to distributed computing. Its modular structure makes it easier to analyze than the more realistic (non-iterated) read-write Atomic-Snapshot memory model (AS). It is known that AS and IIS are equivalent with respect to \emph{wait-free task} computability: a distributed task is solvable in AS if and only if it solvable in IIS. We observe, however, that this equivalence is not sufficient in order to explore solvability of tasks in \emph{sub-models} of AS (i.e. proper subsets of its runs) or computability of \emph{long-lived} objects, and a stronger equivalence relation is needed. In this paper, we consider \emph{adversarial} sub-models of AS and IIS specified by the sets of processes that can be \emph{correct} in a model run. We show that AS and IIS are equivalent in a strong way: a (possibly long-lived) object is implementable in AS under a given adversary if and only if it is implementable in IIS under the same adversary. %This holds whether the object is one-shot or long-lived. Therefore, the computability of any object in shared memory under an adversarial AS scheduler can be equivalently investigated in IIS