Convergence of Stochastic Processes

Often the best way to adumbrate a dark and dense assemblage of material is to describe the background in contrast to which the edges of the nebulosity may be clearly discerned. Hence, perhaps the most appropriate way to introduce this paper is to describe what it is not. It is not a comprehensive study of stochastic processes, nor an in-depth treatment of convergence. In fact, on the surface, the material covered in this paper is nothing more than a compendium of seemingly loosely-connected and barely-miscible theorems, methods and conclusions from the three main papers surveyed ([VC71], [Pol89] and [DL91])

Initiation of secondary ice production in clouds

Disparities between the measured concentrations of ice-nucleating particles (INPs) and in-cloud ice crystal number concentrations (ICNCs) have led to the hypothesis that mechanisms other than primary nucleation form ice in the atmosphere. Here, we model three of these secondary production mechanisms - rime splintering, frozen droplet shattering, and ice–ice collisional breakup – with a six-hydrometeor-class parcel model. We perform three sets of simulations to understand temporal evolution of ice hydrometeor number (Nice), thermodynamic limitations, and the impact of parametric uncertainty when secondary production is active. Output is assessed in terms of the number of primarily nucleated ice crystals that must exist before secondary production initiates (NINP(lim)) as well as the ICNC enhancement from secondary production and the timing of a 100-fold enhancement. Nice evolution can be understood in terms of collision-based nonlinearity and the "phasedness" of the process, i.e., whether it involves ice hydrometeors, liquid ones, or both. Ice–ice collisional breakup is the only process for which a meaningful NINP(lim) exists (0.002 up to 0.15 L−1). For droplet shattering and rime splintering, a warm enough cloud base temperature and modest updraft are the more important criteria for initiation. The low values of NINP(lim) here suggest that, under appropriate thermodynamic conditions for secondary ice production, perturbations in cloud concentration nuclei concentrations are more influential in mixed-phase partitioning than those in INP concentrations

Some remarks on the Zarankiewicz problem

The Zarankiewicz problem asks for an estimate on z(m,n;s,t), the largest number of 1's in an m×n matrix with all entries 0 or 1 containing no s×t submatrix consisting entirely of 1's. We show that a classical upper bound for z(m,n;s,t) due to Kővári, Sós and Turán is tight up to the constant for a broad range of parameters. The proof relies on a new quantitative variant of the random algebraic method

Some remarks on the Zarankiewicz problem

The Zarankiewicz problem asks for an estimate on $z(m, n; s, t)$, the largest number of $1$'s in an $m \times n$ matrix with all entries $0$ or $1$ containing no $s \times t$ submatrix consisting entirely of $1$'s. We show that a classical upper bound for $z(m, n; s, t)$ due to K\H{o}v\'ari, S\'os and Tur\'an is tight up to the constant for a broad range of parameters. The proof relies on a new quantitative variant of the random algebraic method.Comment: 6 page

Optimization techniques applied to passive measures for in-orbit spacecraft survivability

Spacecraft designers have always been concerned about the effects of meteoroid impacts on mission safety. The engineering solution to this problem has generally been to erect a bumper or shield placed outboard from the spacecraft wall to disrupt/deflect the incoming projectiles. Spacecraft designers have a number of tools at their disposal to aid in the design process. These include hypervelocity impact testing, analytic impact predictors, and hydrodynamic codes. Analytic impact predictors generally provide the best quick-look estimate of design tradeoffs. The most complete way to determine the characteristics of an analytic impact predictor is through optimization of the protective structures design problem formulated with the predictor of interest. Space Station Freedom protective structures design insight is provided through the coupling of design/material requirements, hypervelocity impact phenomenology, meteoroid and space debris environment sensitivities, optimization techniques and operations research strategies, and mission scenarios. Major results are presented

Simulating galactic dust grain evolution on a moving mesh

Interstellar dust is an important component of the galactic ecosystem, playing a key role in multiple galaxy formation processes. We present a novel numerical framework for the dynamics and size evolution of dust grains implemented in the moving-mesh hydrodynamics code AREPO suited for cosmological galaxy formation simulations. We employ a particle-based method for dust subject to dynamical forces including drag and gravity. The drag force is implemented using a second-order semi-implicit integrator and validated using several dust-hydrodynamical test problems. Each dust particle has a grain size distribution, describing the local abundance of grains of different sizes. The grain size distribution is discretised with a second-order piecewise linear method and evolves in time according to various dust physical processes, including accretion, sputtering, shattering, and coagulation. We present a novel scheme for stochastically forming dust during stellar evolution and new methods for sub-cycling of dust physics time-steps. Using this model, we simulate an isolated disc galaxy to study the impact of dust physical processes that shape the interstellar grain size distribution. We demonstrate, for example, how dust shattering shifts the grain size distribution to smaller sizes resulting in a significant rise of radiation extinction from optical to near-ultraviolet wavelengths. Our framework for simulating dust and gas mixtures can readily be extended to account for other dynamical processes relevant in galaxy formation, like magnetohydrodynamics, radiation pressure, and thermo-chemical processes.Comment: 38 pages, 27 figures, accepted by MNRAS, with movies available at http://www.mit.edu/~ryanmck/#researc