Learn Physics by Programming in Haskell

We describe a method for deepening a student's understanding of basic physics by asking the student to express physical ideas in a functional programming language. The method is implemented in a second-year course in computational physics at Lebanon Valley College. We argue that the structure of Newtonian mechanics is clarified by its expression in a language (Haskell) that supports higher-order functions, types, and type classes. In electromagnetic theory, the type signatures of functions that calculate electric and magnetic fields clearly express the functional dependency on the charge and current distributions that produce the fields. Many of the ideas in basic physics are well-captured by a type or a function.Comment: In Proceedings TFPIE 2014, arXiv:1412.473

Propagation of sound waves through a linear shear layer: A closed form solution

Closed form solutions are presented for sound propagation from a line source in or near a shear layer. The analysis was exact for all frequencies and was developed assuming a linear velocity profile in the shear layer. This assumption allowed the solution to be expressed in terms of parabolic cyclinder functions. The solution is presented for a line monopole source first embedded in the uniform flow and then in the shear layer. Solutions are also discussed for certain types of dipole and quadrupole sources. Asymptotic expansions of the exact solutions for small and large values of Strouhal number gave expressions which correspond to solutions previously obtained for these limiting cases

An improved algorithm for learning systems

Algorithm for implementing learning controlle

Long-time asymptotics for fully nonlinear homogeneous parabolic equations

We study the long-time asymptotics of solutions of the uniformly parabolic equation $$u_t + F(D^2u) = 0 \quad {in} \R^n\times \R_+,$$ for a positively homogeneous operator $F$, subject to the initial condition $u(x,0) = g(x)$, under the assumption that $g$ does not change sign and possesses sufficient decay at infinity. We prove the existence of a unique positive solution $\Phi^+$ and negative solution $\Phi^-$, which satisfy the self-similarity relations $$\Phi^\pm (x,t) = \lambda^{\alpha^\pm} \Phi^\pm (\lambda^{1/2} x, \lambda t).$$ We prove that the rescaled limit of the solution of the Cauchy problem with nonnegative (nonpositive) initial data converges to $\Phi^+$ ($\Phi^-$) locally uniformly in $\R^n \times \R_+$. The anomalous exponents $\alpha^+$ and $\alpha^-$ are identified as the principal half-eigenvalues of a certain elliptic operator associated to $F$ in $\R^n$.Comment: 20 pages; revised version; two remarks added, typos and one minor mistake correcte

Numerical simulation of the flowfield over ice accretion shapes

The primary goals are directed toward the development of a numerical method for computing flow about ice accretion shapes and determining the influence of these shapes on flow degradation. It is expedient to investigate various aspects of icing independently in order to assess their contribution to the overall icing phenomena. The specific aspects to be examined include the water droplet trajectories with collection efficiencies and phase change on the surface, the flowfield about specified shapes including lift, drag, and heat transfer distribution, and surface roughness effects. The configurations computed were models of ice accretion shapes formed on a circular cylinder in the NASA Lewis Icing Research Tunnel. An existing Navier-Stokes program was modified to compute the flowfield over four shapes (2, 5, and 15 minute models of glaze ice, and a 15 minute accumulation of rime ice)

Local asymptotics for controlled martingales

We consider controlled martingales with bounded steps where the controller is allowed at each step to choose the distribution of the next step, and where the goal is to hit a fixed ball at the origin at time $n$. We show that the algebraic rate of decay (as $n$ increases to infinity) of the value function in the discrete setup coincides with its continuous counterpart, provided a reachability assumption is satisfied. We also study in some detail the uniformly elliptic case and obtain explicit bounds on the rate of decay. This generalizes and improves upon several recent studies of the one dimensional case, and is a discrete analogue of a stochastic control problem recently investigated in Armstrong and Trokhimtchouck [Calc. Var. Partial Differential Equations 38 (2010) 521-540].Comment: Published at http://dx.doi.org/10.1214/15-AAP1123 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Nonexistence of positive supersolutions of elliptic equations via the maximum principle

We introduce a new method for proving the nonexistence of positive supersolutions of elliptic inequalities in unbounded domains of $\mathbb{R}^n$. The simplicity and robustness of our maximum principle-based argument provides for its applicability to many elliptic inequalities and systems, including quasilinear operators such as the $p$-Laplacian, and nondivergence form fully nonlinear operators such as Bellman-Isaacs operators. Our method gives new and optimal results in terms of the nonlinear functions appearing in the inequalities, and applies to inequalities holding in the whole space as well as exterior domains and cone-like domains.Comment: revised version, 32 page

Sharp Liouville results for fully nonlinear equations with power-growth nonlinearities

We study fully nonlinear elliptic equations such as $$F(D^2u) = u^p, \quad p>1,$$ in $\R^n$ or in exterior domains, where $F$ is any uniformly elliptic, positively homogeneous operator. We show that there exists a critical exponent, depending on the homogeneity of the fundamental solution of $F$, that sharply characterizes the range of $p>1$ for which there exist positive supersolutions or solutions in any exterior domain. Our result generalizes theorems of Bidaut-V\'eron \cite{B} as well as Cutri and Leoni \cite{CL}, who found critical exponents for supersolutions in the whole space $\R^n$, in case $-F$ is Laplace's operator and Pucci's operator, respectively. The arguments we present are new and rely only on the scaling properties of the equation and the maximum principle.Comment: 16 pages, new existence results adde

Vital dye labelling demonstrates a sacral neural crest contribution to the enteric nervous system of chick and mouse embryos

We have used the vital dye, DiI, to analyze the contribution of sacral neural crest cells to the enteric nervous system in chick and mouse embryos. In order to label premigratory sacral neural crest cells selectively, DiI was injected into the lumen of the neural tube at the level of the hindlimb. In chick embryos, DiI injections made prior to stage 19 resulted in labelled cells in the gut, which had emerged from the neural tube adjacent to somites 29–37. In mouse embryos, neural crest cells emigrated from the sacral neural tube between E9 and E9.5. In both chick and mouse embryos, DiI-labelled cells were observed in the rostral half of the somitic sclerotome, around the dorsal aorta, in the mesentery surrounding the gut, as well as within the epithelium of the gut. Mouse embryos, however, contained consistently fewer labelled cells than chick embryos. DiI-labelled cells first were observed in the rostral and dorsal portion of the gut. Paralleling the maturation of the embryo, there was a rostral-to-caudal sequence in which neural crest cells populated the gut at the sacral level. In addition, neural crest cells appeared within the gut in a dorsal-to-ventral sequence, suggesting that the cells entered the gut dorsally and moved progressively ventrally. The present results resolve a long-standing discrepancy in the literature by demonstrating that sacral neural crest cells in both the chick and mouse contribute to the enteric nervous system in the postumbilical gut