Sublinear growth of the corrector in stochastic homogenization: Optimal stochastic estimates for slowly decaying correlations

We establish sublinear growth of correctors in the context of stochastic homogenization of linear elliptic PDEs. In case of weak decorrelation and "essentially Gaussian" coefficient fields, we obtain optimal (stretched exponential) stochastic moments for the minimal radius above which the corrector is sublinear. Our estimates also capture the quantitative sublinearity of the corrector (caused by the quantitative decorrelation on larger scales) correctly. The result is based on estimates on the Malliavin derivative for certain functionals which are basically averages of the gradient of the corrector, on concentration of measure, and on a mean value property for $a$-harmonic functions

Weak-strong uniqueness for the Navier-Stokes equation for two fluids with surface tension

In the present work, we consider the evolution of two fluids separated by a sharp interface in the presence of surface tension - like, for example, the evolution of oil bubbles in water. Our main result is a weak-strong uniqueness principle for the corresponding free boundary problem for the incompressible Navier-Stokes equation: As long as a strong solution exists, any varifold solution must coincide with it. In particular, in the absence of physical singularities the concept of varifold solutions - whose global in time existence has been shown by Abels [2] for general initial data - does not introduce a mechanism for non-uniqueness. The key ingredient of our approach is the construction of a relative entropy functional capable of controlling the interface error. If the viscosities of the two fluids do not coincide, even for classical (strong) solutions the gradient of the velocity field becomes discontinuous at the interface, introducing the need for a careful additional adaption of the relative entropy.Comment: 104 page

JEqualityGen: Generating Equality and Hashing Methods

Manually implementing equals (for object comparisons) and hashCode (for object hashing) methods in large software projects is tedious and error-prone. This is due to many special cases, such as field shadowing, comparison between different types, or cyclic object graphs. Here, we present JEqualityGen, a source code generator that automatically derives implementations of these methods. JEqualityGen proceeds in two states: it first uses source code reflection in MetaAspectJ to generate aspects that contain the method implementations, before it uses weaving on the bytecode level to insert these into the target application. JEqualityGen generates not only correct, but efficient source code that on a typical large-scale Java application exhibits a performance improvement of more than two orders of magnitude in the equality operations generated, compared to an existing system based on runtime reflection. JEqualityGen achieves this by generating runtime profiling code that collects data. This enables it to generate optimised method implementations in a second round

First Evaluation of an Index of Low Vagally-Mediated Heart Rate Variability as a Marker of Health Risks in Human Adults: Proof of Concept.

Multiple studies have demonstrated low vagally-mediated heart rate variability (HRV) being associated with a range of risk factors for heart disease and stroke, including inflammation, hyperglycemia, hyperlipidemia, and hypertension. Yet, no cut point exists that indicates elevated risk. In the present study we sought to identify a cut point-value for HRV that is associated with elevated risk across a range of known risk factors. METHODS:A total of 9550 working adults from 19 study sites took part in a health assessment that included measures of inflammation, hyperglycemia, hyperlipidemia, and hypertension and vagally-mediated HRV (Root mean square of successive differences between normal heartbeats (RMSSD)). Multiple age and sex adjusted logistic regressions were calculated per risk factor (normal versus clinical range), with RMSSD being entered in binary at different cut points ranging from 15-39 msec with a 2 msec increment. RESULTS:For daytime RMSSD, values below 25 ± 4 indicated elevated risk (odds ratios (OR) 1.5-3.5 across risk factors). For nighttime RMSSD, values below 29 ± 4 indicated elevated risk (OR 1.2-2.0). CONCLUSION:These results provide the first evidence that a single value of RMSSD may be associated with elevated risk across a range of established cardiovascular risk factors and may present an easy to assess novel marker of cardiovascular risk

The choice of representative volumes in the approximation of effective properties of random materials

The effective large-scale properties of materials with random heterogeneities on a small scale are typically determined by the method of representative volumes: A sample of the random material is chosen - the representative volume - and its effective properties are computed by the cell formula. Intuitively, for a fixed sample size it should be possible to increase the accuracy of the method by choosing a material sample which captures the statistical properties of the material particularly well: For example, for a composite material consisting of two constituents, one would select a representative volume in which the volume fraction of the constituents matches closely with their volume fraction in the overall material. Inspired by similar attempts in material science, Le Bris, Legoll, and Minvielle have designed a selection approach for representative volumes which performs remarkably well in numerical examples of linear materials with moderate contrast. In the present work, we provide a rigorous analysis of this selection approach for representative volumes in the context of stochastic homogenization of linear elliptic equations. In particular, we prove that the method essentially never performs worse than a random selection of the material sample and may perform much better if the selection criterion for the material samples is chosen suitably.Comment: 84 page

DO NOT HIDE Campaign Against Sexual Harassment

Sexual harassment on university campuses is an issue with increasing relevance giving the shocking number of incidents reported each year. The actual number of assaults, however, is a lot higher than the amount of reported incidents. The “DO NOT HIDE” campaign aims to encourage victims of sexual assault to speak out and seek help. Six poster designs attract the viewers attention and inform and encourage victims to find help from Valparaiso University’s confidential Title IX coordinator. All poster designs follow the same principles and the similarities among each design help to profit from a high recognition value. The basic design of each poster relies heavily on imagery, depicting both male and female victims who express their shame and despair. Underneath the large copy is the contact information for victims to find help. The color scheme represents the seriousness of this topic and also catches the viewers attention. The black and white images transmit the victim’s emotional state while the bright yellow letters draw attention to the poster. When choosing the color scheme, the designer used the official colors of Valparaiso University to match the university’s style guide and coincide with its corporate design