Semiclassical Asymptotics for the Maxwell - Dirac System

We study the coupled system of Maxwell and Dirac equations from a semiclassical point of view. A rigorous nonlinear WKB-analysis, locally in time, for solutions of (critical) order $O(\sqrt{\epsilon})$ is performed, where the small semiclassical parameter $\epsilon$ denotes the microscopic/macroscopic scale ratio

Financial performance of groups of companies in Poland against the background of historical determinants and knowledge management procedures applied

Polish businesses are passing the fifteen year mark of the free market experience. The important processes that could be observed during that time include the formation of groups of companies. What were the establishment paths of these multiple organisations and what was the impact of historical determinants on their operation and financial performance? What is the extent of contribution of subsidiaries to the financial performance of the groups? And finally what are the main drivers of their effectiveness in the context of group management system applied and knowledge management procedures? This paper attempts to provide answers to these questions. It presents insights, observations and selected findings of research conducted over the period of the last five years. It is shown that the operation of these organisations is on the one hand strongly determined by historical factors, and on the other the strength of this impact diminishes with time. Groups of companies see positive operational changes, reflected mainly in improved financial performance, growing significance of subsidiaries and the enhanced effectiveness of the management systems applied which are broadly based on knowledge management procedures

On Blowup for time-dependent generalized Hartree-Fock equations

We prove finite-time blowup for spherically symmetric and negative energy solutions of Hartree-Fock and Hartree-Fock-Bogoliubov type equations, which describe the evolution of attractive fermionic systems (e. g. white dwarfs). Our main results are twofold: First, we extend the recent blowup result of [Hainzl and Schlein, Comm. Math. Phys. \textbf{287} (2009), 705--714] to Hartree-Fock equations with infinite rank solutions and a general class of Newtonian type interactions. Second, we show the existence of finite-time blowup for spherically symmetric solutions of a Hartree-Fock-Bogoliubov model, where an angular momentum cutoff is introduced. We also explain the key difficulties encountered in the full Hartree-Fock-Bogoliubov theory.Comment: 24 page

Instabilities in the dissolution of a porous matrix

A reactive fluid dissolving the surrounding rock matrix can trigger an instability in the dissolution front, leading to spontaneous formation of pronounced channels or wormholes. Theoretical investigations of this instability have typically focused on a steadily propagating dissolution front that separates regions of high and low porosity. In this paper we show that this is not the only possible dissolutional instability in porous rocks; there is another instability that operates instantaneously on any initial porosity field, including an entirely uniform one. The relative importance of the two mechanisms depends on the ratio of the porosity increase to the initial porosity. We show that the "inlet" instability is likely to be important in limestone formations where the initial porosity is small and there is the possibility of a large increase in permeability. In quartz-rich sandstones, where the proportion of easily soluble material (e.g. carbonate cements) is small, the instability in the steady-state equations is dominant.Comment: to be published in Geophysical Research Letter

On the Mean-Field Limit of Bosons with Coulomb Two-Body Interaction

In the mean-field limit the dynamics of a quantum Bose gas is described by a Hartree equation. We present a simple method for proving the convergence of the microscopic quantum dynamics to the Hartree dynamics when the number of particles becomes large and the strength of the two-body potential tends to 0 like the inverse of the particle number. Our method is applicable for a class of singular interaction potentials including the Coulomb potential. We prove and state our main result for the Heisenberg-picture dynamics of "observables", thus avoiding the use of coherent states. Our formulation shows that the mean-field limit is a "semi-classical" limit.Comment: Corrected typos and included an elementary proof of the Kato smoothing estimate (Lemma 6.1

Magnetic Monopoles, Electric Neutrality and the Static Maxwell-Dirac Equations

We study the full Maxwell-Dirac equations: Dirac field with minimally coupled electromagnetic field and Maxwell field with Dirac current as source. Our particular interest is the static case in which the Dirac current is purely time-like -- the "electron" is at rest in some Lorentz frame. In this case we prove two theorems under rather general assumptions. Firstly, that if the system is also stationary (time independent in some gauge) then the system as a whole must have vanishing total charge, i.e. it must be electrically neutral. In fact, the theorem only requires that the system be {\em asymptotically} stationary and static. Secondly, we show, in the axially symmetric case, that if there are external Coulomb fields then these must necessarily be magnetically charged -- all Coulomb external sources are electrically charged magnetic monopoles

Existence of global-in-time solutions to a generalized Dirac-Fock type evolution equation

We consider a generalized Dirac-Fock type evolution equation deduced from no-photon Quantum Electrodynamics, which describes the self-consistent time-evolution of relativistic electrons, the observable ones as well as those filling up the Dirac sea. This equation has been originally introduced by Dirac in 1934 in a simplified form. Since we work in a Hartree-Fock type approximation, the elements describing the physical state of the electrons are infinite rank projectors. Using the Bogoliubov-Dirac-Fock formalism, introduced by Chaix-Iracane ({\it J. Phys. B.}, 22, 3791--3814, 1989), and recently established by Hainzl-Lewin-Sere, we prove the existence of global-in-time solutions of the considered evolution equation.Comment: 12 pages; more explanations added, some final (minor) corrections include

Existence and uniqueness of solutions to the inverse boundary crossing problem for diffusions

We study the inverse boundary crossing problem for diffusions. Given a diffusion process $X_t$, and a survival distribution $p$ on $[0,\infty)$, we demonstrate that there exists a boundary $b(t)$ such that $p(t)=\mathbb{P}[\tau >t]$, where $\tau$ is the first hitting time of $X_t$ to the boundary $b(t)$. The approach taken is analytic, based on solving a parabolic variational inequality to find $b$. Existence and uniqueness of the solution to this variational inequality were proven in earlier work. In this paper, we demonstrate that the resulting boundary $b$ does indeed have $p$ as its boundary crossing distribution. Since little is known regarding the regularity of $b$ arising from the variational inequality, this requires a detailed study of the problem of computing the boundary crossing distribution of $X_t$ to a rough boundary. Results regarding the formulation of this problem in terms of weak solutions to the corresponding Kolmogorov forward equation are presented.Comment: Published in at http://dx.doi.org/10.1214/10-AAP714 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Analysis of an Optimal Stopping Problem Arising from Hedge Fund Investing

The final publication is available at Elsevier via https://doi.org/10.1016/j.jmaa.2019.123559. © 2020. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/We analyze the optimal withdrawal time for an investor in a hedge fund with a first-loss or shared-loss fee structure, given as the solution of an optimal stopping problem on the fund's assets with a piecewise linear payoff function. Assuming that the underlying follows a geometric Brownian motion, we present a complete solution of the problem in the infinite horizon case, showing that the continuation region is a finite interval, and that the smooth-fit condition may fail to hold at one of the endpoints. In the finite horizon case, we show the existence of a pair of optimal exercise boundaries and analyze their properties, including smoothness and convexity.NSERC, RGPIN-2017-04220

Mass transfer in the lower crust: Evidence for incipient melt assisted flow along grain boundaries in the deep arc granulites of Fiordland, New Zealand

Knowledge of mass transfer is critical in improving our understanding of crustal evolution, however mass transfer mechanisms are debated, especially in arc environments. The Pembroke Granulite is a gabbroic gneiss, passively exhumed from depths of >45 km from the arc root of Fiordland, New Zealand. Here, enstatite and diopside grains are replaced by coronas of pargasite and quartz, which may be asymmetric, recording hydration of the gabbroic gneiss. The coronas contain microstructures indicative of the former presence of melt, supported by pseudosection modeling consistent with the reaction having occurred near the solidus of the rock (630–710°C, 8.8–12.4 kbar). Homogeneous mineral chemistry in reaction products indicates an open system, despite limited metasomatism at the hand sample scale. We propose the partial replacement microstructures are a result of a reaction involving an externally derived hydrous, silicate melt and the relatively anhydrous, high-grade assemblage. Trace element mapping reveals a correlation between reaction microstructure development and bands of high-Sr plagioclase, recording pathways of the reactant melt along grain boundaries. Replacement microstructures record pathways of diffuse porous melt flow at a kilometer scale within the lower crust, which was assisted by small proportions of incipient melt providing a permeable network. This work recognizes melt flux through the lower crust in the absence of significant metasomatism, which may be more common than is currently recognized. As similar microstructures are found elsewhere within the exposed Fiordland lower crustal arc rocks, mass transfer of melt by diffuse porous flow may have fluxed an area >10,000 km2