Teaching a University Course on the Mathematics of Gambling

Courses on the mathematics of gambling have been offered by a number of colleges and universities, and for a number of reasons. In the past 15 years, at least seven potential textbooks for such a course have been published. In this article we objectively compare these books for their probability content, their gambling content, and their mathematical level, to see which ones might be most suitable, depending on student interests and abilities. This is not a book review (e.g., none of the books is recommended over others) but rather an essay offering advice about which topics to include in a course on the mathematics of gambling

On Exceptional Times for generalized Fleming-Viot Processes with Mutations

If $\mathbf Y$ is a standard Fleming-Viot process with constant mutation rate (in the infinitely many sites model) then it is well known that for each $t>0$ the measure $\mathbf Y_t$ is purely atomic with infinitely many atoms. However, Schmuland proved that there is a critical value for the mutation rate under which almost surely there are exceptional times at which $\mathbf Y$ is a finite sum of weighted Dirac masses. In the present work we discuss the existence of such exceptional times for the generalized Fleming-Viot processes. In the case of Beta-Fleming-Viot processes with index $\alpha\in\,]1,2[$ we show that - irrespectively of the mutation rate and $\alpha$ - the number of atoms is almost surely always infinite. The proof combines a Pitman-Yor type representation with a disintegration formula, Lamperti's transformation for self-similar processes and covering results for Poisson point processes

The Mean Drift: Tailoring the Mean Field Theory of Markov Processes for Real-World Applications

The statement of the mean field approximation theorem in the mean field theory of Markov processes particularly targets the behaviour of population processes with an unbounded number of agents. However, in most real-world engineering applications one faces the problem of analysing middle-sized systems in which the number of agents is bounded. In this paper we build on previous work in this area and introduce the mean drift. We present the concept of population processes and the conditions under which the approximation theorems apply, and then show how the mean drift is derived through a systematic application of the propagation of chaos. We then use the mean drift to construct a new set of ordinary differential equations which address the analysis of population processes with an arbitrary size

Thermoelectric phenomena via an interacting particle system

We present a mesoscopic model for thermoelectric phenomena in terms of an interacting particle system, a lattice electron gas dynamics that is a suitable extension of the standard simple exclusion process. We concentrate on electronic heat and charge transport in different but connected metallic substances. The electrons hop between energy-cells located alongside the spatial extension of the metal wire. When changing energy level, the system exchanges energy with the environment. At equilibrium the distribution satisfies the Fermi-Dirac occupation-law. Installing different temperatures at two connections induces an electromotive force (Seebeck effect) and upon forcing an electric current, an additional heat flow is produced at the junctions (Peltier heat). We derive the linear response behavior relating the Seebeck and Peltier coefficients as an application of Onsager reciprocity. We also indicate the higher order corrections. The entropy production is characterized as the anti-symmetric part under time-reversal of the space-time Lagrangian.Comment: 19 pages, 2 figures, submitted to Journal of Physics

The Euler-Maruyama approximation for the absorption time of the CEV diffusion

A standard convergence analysis of the simulation schemes for the hitting times of diffusions typically requires non-degeneracy of their coefficients on the boundary, which excludes the possibility of absorption. In this paper we consider the CEV diffusion from the mathematical finance and show how a weakly consistent approximation for the absorption time can be constructed, using the Euler-Maruyama scheme

Scientific and Computational Challenges of the Fusion Simulation Program (FSP)

This paper highlights the scientific and computational challenges facing the Fusion Simulation Program (FSP) a major national initiative in the United States with the primary objective being to enable scientific discovery of important new plasma phenomena with associated understanding that emerges only upon integration. This requires developing a predictive integrated simulation capability for magnetically-confined fusion plasmas that are properly validated against experiments in regimes relevant for producing practical fusion energy. It is expected to provide a suite of advanced modeling tools for reliably predicting fusion device behavior with comprehensive and targeted science-based simulations of nonlinearly-coupled phenomena in the core plasma, edge plasma, and wall region on time and space scales required for fusion energy production. As such, it will strive to embody the most current theoretical and experimental understanding of magnetic fusion plasmas and to provide a living framework for the simulation of such plasmas as the associated physics understanding continues to advance over the next several decades. Substantive progress on answering the outstanding scientific questions in the field will drive the FSP toward its ultimate goal of developing the ability to predict the behavior of plasma discharges in toroidal magnetic fusion devices with high physics fidelity on all relevant time and space scales. From a computational perspective, this will demand computing resources in the petascale range and beyond together with the associated multi-core algorithmic formulation needed to address burning plasma issues relevant to ITER - a multibillion dollar collaborative experiment involving seven international partners representing over half the world's population. Even more powerful exascale platforms will be needed to meet the future challenges of designing a demonstration fusion reactor (DEMO). Analogous to other major applied physics modeling projects (e.g., Climate Modeling), the FSP will need to develop software in close collaboration with computers scientists and applied mathematicians and validated against experimental data from tokamaks around the world. Specific examples of expected advances needed to enable such a comprehensive integrated modeling capability and possible "co-design" approaches will be discussed. _________________________________________________

First Monte Carlo analysis of fragmentation functions from single-inclusive e(+)e(-) annihilation

We perform the first iterative Monte Carlo (IMC) analysis of fragmentation functions constrained by all available data from single-inclusive e(+)e(-) annihilation into pions and kaons. The IMC method eliminates potential bias in traditional analyses based on single fits introduced by fixing parameters not well constrained by the data and provides a statistically rigorous determination of uncertainties. Our analysis reveals specific features of fragmentation functions using the new IMC methodology and those obtained from previous analyses, especially for light quarks and for strange quark fragmentation to kaons

Stochastic B\"acklund transformations

How does one introduce randomness into a classical dynamical system in order to produce something which is related to the `corresponding' quantum system? We consider this question from a probabilistic point of view, in the context of some integrable Hamiltonian systems

Rectification of thermal fluctuations in ideal gases

We calculate the systematic average speed of the adiabatic piston and a thermal Brownian motor, introduced in [Van den Broeck, Kawai and Meurs, \emph{Microscopic analysis of a thermal Brownian motor}, to appear in Phys. Rev. Lett.], by an expansion of the Boltzmann equation and compare with the exact numerical solution.Comment: 18 page