Fluctuations of the empirical quantiles of independent Brownian motions

We consider $n$ independent, identically distributed one-dimensional Brownian motions, $B_j(t)$, where $B_j(0)$ has a rapidly decreasing, smooth density function $f$. The empirical quantiles, or pointwise order statistics, are denoted by $B_{j:n}(t)$, and we are interested in a sequence of quantiles $Q_n(t) = B_{j(n):n}(t)$, where $j(n)/n \to \alpha \in (0,1)$. This sequence converges in probability in $C[0,\infty)$ to $q(t)$, the $\alpha$-quantile of the law of $B_j(t)$. Our main result establishes the convergence in law in $C[0,\infty)$ of the fluctuation processes $F_n = n^{1/2}(Q_n - q)$. The limit process $F$ is a centered Gaussian process and we derive an explicit formula for its covariance function. We also show that $F$ has many of the same local properties as $B^{1/4}$, the fractional Brownian motion with Hurst parameter $H = 1/4$. For example, it is a quartic variation process, it has H\"older continuous paths with any exponent $\gamma < 1/4$, and (at least locally) it has increments whose correlation is negative and of the same order of magnitude as those of $B^{1/4}$.Comment: 40 page

Crowding of Brownian spheres

We study two models consisting of reflecting one-dimensional Brownian "particles" of positive radius. We show that the stationary empirical distributions for the particle systems do not converge to the harmonic function for the generator of the individual particle process, unlike in the case when the particles are infinitely small.Comment: 13 page

The weak Stratonovich integral with respect to fractional Brownian motion with Hurst parameter 1/6

Let $B$ be a fractional Brownian motion with Hurst parameter $H=1/6$. It is known that the symmetric Stratonovich-style Riemann sums for $\int g(B(s))\,dB(s)$ do not, in general, converge in probability. We show, however, that they do converge in law in the Skorohod space of c\`adl\`ag functions. Moreover, we show that the resulting stochastic integral satisfies a change of variable formula with a correction term that is an ordinary It\^o integral with respect to a Brownian motion that is independent of $B$.Comment: 45 page

Overview of the Main Propulsion System for the NASA Ares I Upper Stage

A functional overview of the Main Propulsion System (MPS) of the NASA Ares I Upper Stage is provided. In addition to a simple overview of the key MPS functions and design philosophies, major lessons learned are discussed. The intent is to provide a technical overview with enough detail to allow engineers outside of the MPS Integrated Product Team (IPT) to develop a rough understanding of MPS operations, components, design philosophy, and lessons learned

Weak Convergence of the Scaled Median of Independent Brownian Motions

We consider the median of n independent Brownian motions, and show that this process, when properly scaled, converges weakly to a centered Gaussian process. The chief difficulty is establishing tightness, which is proved through direct estimates on the increments of the median process. An explicit formula is given for the covariance function of the limit process. The limit process is also shown to be Holder continuous with exponent gamma for all gamma < 1/4.Comment: to appear in Probability Theory and Related Field

Prognostic Significance of Growth Kinetics in Newly Diagnosed Glioblastomas Revealed by Combining Serial Imaging with a Novel Biomathematical Model

Glioblastomas (GBMs) are the most aggressive primary brain tumors characterized by their rapid proliferation and diffuse infiltration of the brain tissue. Survival patterns in patients with GBM have been associated with a number of clinico-pathologic factors, including age and neurological status, yet a significant quantitative link to in vivo growth kinetics of each glioma has remained elusive. Exploiting a recently developed tool for quantifying glioma net proliferation and invasion rates in individual patients using routinely available magnetic resonance images (MRIs), we propose to link these patient-specific kinetic rates of biological aggressiveness to prognostic significance. Using our biologically-based mathematical model for glioma growth and invasion, examination of serial pre-treatment MRIs of 32 GBM patients allowed quantification of these rates for each patient’s tumor. Survival analyses revealed that even when controlling for standard clinical parameters (e.g., age, KPS) these model-defined parameters quantifying biologically aggressiveness (net proliferation and invasion rates) were significantly associated with prognosis. One hypothesis generated was that the ratio of the actual survival time after whatever therapies were employed to the duration of survival predicted (by the model) without any therapy would provide a “Therapeutic Response Index” (TRI) of the overall effectiveness of the therapies. The TRI may provided important information, not otherwise available, as to the effectiveness of the treatments in individual patients. To our knowledge, this is the first report indicating that dynamic insight from routinely obtained pre-treatment imaging may be quantitatively useful in characterizing survival of individual patients with GBM. Such a hybrid tool bridging mathematical modeling and clinical imaging may allow for statifying patients for clinical studies relative to their pretreatment biological aggressiveness

Airborne In-Situ Measurements of Formaldehyde Over California: First Results from the Compact Formaldehyde Fluorescence Experiment (COFFEE) Instrument

Formaldehyde (HCHO) is one of the most abundant oxygenated volatile organic compounds (VOCs) in the atmosphere, playing a role multiple atmospheric processes. Measurements of HCHO can be used to help quantify convective transport, the abundance of VOCs, and ozone production in urban environments. The Compact Formaldehyde FluorescencE Experiment (COFFEE) instrument uses Non-Resonant Laser Induced Fluorescence (NR-LIF) to detect trace concentrations of HCHO as part of the Alpha Jet Atmospheric eXperiment (AJAX) payload. Developed at NASA GSFC, COFFEE is a small, low maintenance instrument with a sensitivity of 100 pptv and a quick response time (1 sec). The COFFEE instrument has been customized to fit in an external wing pod on the Alpha Jet aircraft based at NASA ARC. The instrument can operate over a broad range of altitudes, from boundary layer to lower stratosphere, making it well suited for the Alpha Jet, which can access altitudes from the surface up to 40,000 ft. Results of the first COFFEE science flights preformed over the California's Central Valley will be presented. Boundary layer measurements and vertical profiles in the tropospheric column will both be included. This region is of particular interest, due to its elevated levels of HCHO, revealed in satellite images, as well as its high ozone concentrations. In addition to HCHO, the AJAX payload includes measurements of atmospheric ozone, methane, and carbon dioxide. Formaldehyde is one of the few urban pollutants that can be measured from space. Plans to compare in-situ COFFEE data with satellite-based HCHO observations such as those from OMI (Aura) and OMPS (SuomiNPP) will also be presented

Case study of a performance-active changing trans* male singing voice

A professional classical singer of more than 25 years (AZ) in his early 50s requested this voice researcher’s consultation and assistance in early 2014. He was about to start living full time as a trans* man. Despite his intention to be included in the low start/gradual increase testosterone option of the Trans* Male (previously, “FTM”) Singing Voice Program, the request contained a rather unconventional aspect: AZ would continue to sing while his voice was changing. The above request was integral with his singing history. After the introduction of safeguards and his informed consent, AZ was accepted onto the Program. Due to the highly individual circumstances, his participation was recorded as a case study. The study has aimed to replicate the particulars of the slow hormonal changes and continuing singing ability found in certain cisgender male adolescent voices. Despite dealing with an adult trans* male individual, the progress has been comparable. This has been achieved by carefully monitoring AZ’s low start/gradual increase testosterone administration in communication with the medical practitioner. The participant’s vocal health remained safeguarded and promoted by carefully individualized vocal tuition. This article will discuss the collective results of the case study, including the recordings and the data analysis