Nondifferentiable functions of one-dimensional semimartingales

We consider decompositions of processes of the form $Y=f(t,X_t)$ where $X$ is a semimartingale. The function $f$ is not required to be differentiable, so It\^{o}'s lemma does not apply. In the case where $f(t,x)$ is independent of $t$, it is shown that requiring $f$ to be locally Lipschitz continuous in $x$ is enough for an It\^{o}-style decomposition to exist. In particular, $Y$ will be a Dirichlet process. We also look at the case where $f(t,x)$ can depend on $t$, possibly discontinuously. It is shown, under some additional mild constraints on $f$, that the same decomposition still holds. Both these results follow as special cases of a more general decomposition which we prove, and which applies to nondifferentiable functions of Dirichlet processes. Possible applications of these results to the theory of one-dimensional diffusions are briefly discussed.Comment: Published in at http://dx.doi.org/10.1214/09-AOP476 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Limits Of One Dimensional Diffusions

In this paper we look at the properties of limits of a sequence of real valued time inhomogeneous diffusions. When convergence is only in the sense of finite-dimensional distributions then the limit does not have to be a diffusion. However, we show that as long as the drift terms satisfy a Lipschitz condition and the limit is continuous in probability, then it will lie in a class of processes that we refer to as almost-continuous diffusions. These processes are strong Markov and satisfy an `almost-continuity' condition. We also give a simple condition for the limit to be a continuous diffusion. These results contrast with the multidimensional case where, as we show with an example, a sequence of two dimensional martingale diffusions can converge to a process that is both discontinuous and non-Markov.Comment: 32 pages. Updated to most recent version submitted to Annals of Probabilit

High Kinetic Energy Penetrator Shielding and High Wear Resistance Materials Fabricated with Boron Nitride Nanotubes (BNNTS) and BNNT Polymer Composites

Boron nitride nanotubes (BNNTs), boron nitride nanoparticles (BNNPs), carbon nanotubes (CNTs), graphites, or combinations, are incorporated into matrices of polymer, ceramic or metals. Fibers, yarns, and woven or nonwoven mats of BNNTs are used as toughening layers in penetration resistant materials to maximize energy absorption and/or high hardness layers to rebound or deform penetrators. They can be also used as reinforcing inclusions combining with other polymer matrices to create composite layers like typical reinforcing fibers such as Kevlar.RTM., Spectra.RTM., ceramics and metals. Enhanced wear resistance and usage time are achieved by adding boron nitride nanomaterials, increasing hardness and toughness. Such materials can be used in high temperature environments since the oxidation temperature of BNNTs exceeds 800.degree. C. in air. Boron nitride based composites are useful as strong structural materials for anti-micrometeorite layers for spacecraft and space suits, ultra strong tethers, protective gear, vehicles, helmets, shields and safety suits/helmets for industry

AI is a viable alternative to high throughput screening: a 318-target study

: High throughput screening (HTS) is routinely used to identify bioactive small molecules. This requires physical compounds, which limits coverage of accessible chemical space. Computational approaches combined with vast on-demand chemical libraries can access far greater chemical space, provided that the predictive accuracy is sufficient to identify useful molecules. Through the largest and most diverse virtual HTS campaign reported to date, comprising 318 individual projects, we demonstrate that our AtomNet® convolutional neural network successfully finds novel hits across every major therapeutic area and protein class. We address historical limitations of computational screening by demonstrating success for target proteins without known binders, high-quality X-ray crystal structures, or manual cherry-picking of compounds. We show that the molecules selected by the AtomNet® model are novel drug-like scaffolds rather than minor modifications to known bioactive compounds. Our empirical results suggest that computational methods can substantially replace HTS as the first step of small-molecule drug discovery

Pathology of the Brain in an Idiot From Infancy.

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