An overview of Viscosity Solutions of Path-Dependent PDEs

This paper provides an overview of the recently developed notion of viscosity solutions of path-dependent partial di erential equations. We start by a quick review of the Crandall- Ishii notion of viscosity solutions, so as to motivate the relevance of our de nition in the path-dependent case. We focus on the wellposedness theory of such equations. In partic- ular, we provide a simple presentation of the current existence and uniqueness arguments in the semilinear case. We also review the stability property of this notion of solutions, in- cluding the adaptation of the Barles-Souganidis monotonic scheme approximation method. Our results rely crucially on the theory of optimal stopping under nonlinear expectation. In the dominated case, we provide a self-contained presentation of all required results. The fully nonlinear case is more involved and is addressed in [12]

Geometric approach to nonvariational singular elliptic equations

In this work we develop a systematic geometric approach to study fully nonlinear elliptic equations with singular absorption terms as well as their related free boundary problems. The magnitude of the singularity is measured by a negative parameter $(\gamma -1)$, for $0 < \gamma < 1$, which reflects on lack of smoothness for an existing solution along the singular interface between its positive and zero phases. We establish existence as well sharp regularity properties of solutions. We further prove that minimal solutions are non-degenerate and obtain fine geometric-measure properties of the free boundary $\mathfrak{F} = \partial \{u > 0 \}$. In particular we show sharp Hausdorff estimates which imply local finiteness of the perimeter of the region $\{u > 0 \}$ and $\mathcal{H}^{n-1}$ a.e. weak differentiability property of $\mathfrak{F}$.Comment: Paper from D. Araujo's Ph.D. thesis, distinguished at the 2013 Carlos Gutierrez prize for best thesis, Archive for Rational Mechanics and Analysis 201

TORSO DEFORMATION IN FRONTAL SLED TESTS: COMPARISION BETWEEN THOR NT, THOR NT WITH THE CHALMERS SD-1 SHOULDER, AND PMHS

This study compares the thoracic deformation response of the 50th percentile male THOR NT frontal crash dummy and the response of the THOR modified with the SD-1 shoulder (THOR SD-1) relative to the thoracic response of eight 50th percentile male PMHS. The prototype Chalmers University SD-1 shoulder was designed to be more human-like in terms of geometry and range of motion in comparison to the standard THOR NT shoulder. The dummies and PMHS were restrained by a three-point restraint in a driver-side configuration and were subjected to a simulated 40 km/h frontal crash. The most prominent difference between the responses of the dummies and PMHS involved motion of the lower right anterior ribcage measurement site that is the farthest lateral distance from the diagonal shoulder belt. During the impact event, this site moved substantially anteriorly and away from the spine for the PMHS. The PMHS lower right “bulge out” behavior is believed to be caused by inertial loading of the ribcage, underlying organs, and soft tissue overlying the torso. The THOR SD-1 shoulder altered the shoulder belt position relative to the thoracic deflection measurement sites resulting in a different distribution of deformation for the upper measurement sites although the average upper site deformation was similar to that recorded for the standard THOR shoulder

Collisional Excitation Of N+ At 50 KeV

The excitation spectrum of N+ has been observed by examining the energy lost by a 50- keV N+ beam passing through a He target. The spectrum exhibits dramatic features with large cross sections. Determination of the approximate ratio of metastable to ground-state ions in the primary ion beam has permitted measurement of excitation cross sections from both ground-state and metastable N+ ions colliding with He target atoms. © 1972 The American Physical Society

The Tychonoff uniqueness theorem for the G-heat equation

In this paper, we obtain the Tychonoff uniqueness theorem for the G-heat equation

Dynamical response of the "GGG" rotor to test the Equivalence Principle: theory, simulation and experiment. Part I: the normal modes

Recent theoretical work suggests that violation of the Equivalence Principle might be revealed in a measurement of the fractional differential acceleration $\eta$ between two test bodies -of different composition, falling in the gravitational field of a source mass- if the measurement is made to the level of $\eta\simeq 10^{-13}$ or better. This being within the reach of ground based experiments, gives them a new impetus. However, while slowly rotating torsion balances in ground laboratories are close to reaching this level, only an experiment performed in low orbit around the Earth is likely to provide a much better accuracy. We report on the progress made with the "Galileo Galilei on the Ground" (GGG) experiment, which aims to compete with torsion balances using an instrument design also capable of being converted into a much higher sensitivity space test. In the present and following paper (Part I and Part II), we demonstrate that the dynamical response of the GGG differential accelerometer set into supercritical rotation -in particular its normal modes (Part I) and rejection of common mode effects (Part II)- can be predicted by means of a simple but effective model that embodies all the relevant physics. Analytical solutions are obtained under special limits, which provide the theoretical understanding. A simulation environment is set up, obtaining quantitative agreement with the available experimental data on the frequencies of the normal modes, and on the whirling behavior. This is a needed and reliable tool for controlling and separating perturbative effects from the expected signal, as well as for planning the optimization of the apparatus.Comment: Accepted for publication by "Review of Scientific Instruments" on Jan 16, 2006. 16 2-column pages, 9 figure

Inhibition of Nitric Oxide Inhibition of Nitric Oxide Synthase Does Not Alter Dynamic Cerebral Autoregulation in Humans

The aim of this study was to determine whether inhibition of nitric oxide synthase (NOS) alters dynamic cerebral autoregulation in humans. Beat-to-beat blood pressure (BP) and cerebral blood flow (CBF) velocity (transcranial Doppler) were measured in eight healthy subjects in the supine position and during 60° head-up tilt (HUT). NOS was inhibited by intravenous N G-monomethyl-L-arginine (L-NMMA) infusion. Dynamic cerebral autoregulation was quantified by transfer function analysis of beat-to-beat changes in BP and CBF velocity. Pressor effects of L-NMMA on cerebral hemodynamics were compared with those of phenylephrine infusion. In the supine position, L-NMMA increased mean BP from 83 ± 3 to 94 ± 3 mmHg (P \u3c 0.01). However, CBF velocity remained unchanged. Consequently, cerebrovascular resistance index (CVRI) increased by 15% (P \u3c 0.05). BP and CBF velocity variability and transfer function gain at the low frequencies of 0.07-0.20 Hz did not change with L-NMMA infusion. Similar changes in mean BP, CBF velocity, and CVRI were observed after phenylephrine infusion, suggesting that increase in CVRI after L-NMMA was mediated myogenically by increase in arterial pressure rather than a direct effect of cerebrovascular NOS inhibition. During baseline tilt without L-NMMA, steady-state BP increased and CBF velocity decreased. BP and CBF velocity variability at low frequencies increased in parallel by 277% and 217%, respectively (P \u3c 0.05). However, transfer function gain remained unchanged. During tilt with L-NMMA, changes in steady-state hemodynamics and BP and CBF velocity variability as well as transfer gain and phase were similar to those without L-NMMA. These data suggest that inhibition of tonic production of NO does not appear to alter dynamic cerebral autoregulation in humans

Nitric Oxide Synthase Inhibition Does Not Affect Regulation of Muscle Sympathetic Nerve Activity During Head-Up Tilt

To test the hypothesis that systemic inhibition of nitric oxide (NO) synthase does not alter the regulation of sympathetic outflow during head-up tilt in humans, in eight healthy subjects NO synthase was blocked by intravenous infusion of NG-monomethyl-L-arginine (L-NMMA). Blood pressure, heart rate, cardiac output, total peripheral resistance (TPR), and muscle sympathetic nerve activity (MSNA) were recorded in the supine position and during 60° head-up tilt. In the supine position, infusion of L-NMMA increased blood pressure, via increased TPR, and inhibited MSNA. However, the increase in MSNA evoked by head-up tilt during L-NMMA infusion (change in burst rate: 24 ± 4 bursts/min; change in total activity: 209 ± 36 U/min) was similar to that during head-up tilt without L-NMMA (change in burst rate: 23 ± 4 bursts/min; change in total activity: 251 ± 52 U/min, n = 6, all P \u3e 0.05). Moreover, changes in TPR and heart rate during head-up tilt were virtually identical between the two conditions. These results suggest that systemic inhibition of NO synthase with L-NMMA does not affect the regulation of sympathetic outflow and vascular resistance during head-up tilt in humans

Convergence of nonlocal threshold dynamics approximations to front propagation

In this note we prove that appropriately scaled threshold dynamics-type algorithms corresponding to the fractional Laplacian of order $\alpha \in (0,2)$ converge to moving fronts. When $\alpha \geqq 1$ the resulting interface moves by weighted mean curvature, while for $\alpha <1$ the normal velocity is nonlocal of ``fractional-type.'' The results easily extend to general nonlocal anisotropic threshold dynamics schemes.Comment: 19 page

Dynamics of a lattice Universe

We find a solution to Einstein field equations for a regular toroidal lattice of size L with equal masses M at the centre of each cell; this solution is exact at order M/L. Such a solution is convenient to study the dynamics of an assembly of galaxy-like objects. We find that the solution is expanding (or contracting) in exactly the same way as the solution of a Friedman-Lema\^itre-Robertson-Walker Universe with dust having the same average density as our model. This points towards the absence of backreaction in a Universe filled with an infinite number of objects, and this validates the fluid approximation, as far as dynamics is concerned, and at the level of approximation considered in this work.Comment: 14 pages. No figure. Accepted version for Classical and Quantum Gravit