78,631 research outputs found
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
Learn Quantum Mechanics with Haskell
To learn quantum mechanics, one must become adept in the use of various
mathematical structures that make up the theory; one must also become familiar
with some basic laboratory experiments that the theory is designed to explain.
The laboratory ideas are naturally expressed in one language, and the
theoretical ideas in another. We present a method for learning quantum
mechanics that begins with a laboratory language for the description and
simulation of simple but essential laboratory experiments, so that students can
gain some intuition about the phenomena that a theory of quantum mechanics
needs to explain. Then, in parallel with the introduction of the mathematical
framework on which quantum mechanics is based, we introduce a calculational
language for describing important mathematical objects and operations, allowing
students to do calculations in quantum mechanics, including calculations that
cannot be done by hand. Finally, we ask students to use the calculational
language to implement a simplified version of the laboratory language, bringing
together the theoretical and laboratory ideas.Comment: In Proceedings TFPIE 2015/6, arXiv:1611.0865
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)
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
Long-time asymptotics for fully nonlinear homogeneous parabolic equations
We study the long-time asymptotics of solutions of the uniformly parabolic
equation for a positively
homogeneous operator , subject to the initial condition ,
under the assumption that does not change sign and possesses sufficient
decay at infinity. We prove the existence of a unique positive solution
and negative solution , which satisfy the self-similarity
relations We prove that the rescaled limit of the solution of the Cauchy
problem with nonnegative (nonpositive) initial data converges to
() locally uniformly in . The anomalous exponents
and are identified as the principal half-eigenvalues of a
certain elliptic operator associated to in .Comment: 20 pages; revised version; two remarks added, typos and one minor
mistake correcte
Sharp Liouville results for fully nonlinear equations with power-growth nonlinearities
We study fully nonlinear elliptic equations such as in or in exterior domains, where is any uniformly elliptic,
positively homogeneous operator. We show that there exists a critical exponent,
depending on the homogeneity of the fundamental solution of , that sharply
characterizes the range of 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 , in case
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
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