A Reflection on Types

The ability to perform type tests at runtime blurs the line between statically-typed and dynamically-checked languages. Recent developments in Haskell’s type system allow even programs that use reflection to themselves be statically typed, using a type-indexed runtime representation of types called \{}\textit{TypeRep}. As a result we can build dynamic types as an ordinary, statically-typed library, on top of \{}\textit{TypeRep} in an open-world context

A Reflection on Types

The ability to perform type tests at runtime blurs the line between statically-typed and dynamically-checked languages. Recent developments in Haskell’s type system allow even programs that use reflection to themselves be statically typed, using a type-indexed runtime representation of types called \{}\textit{TypeRep}. As a result we can build dynamic types as an ordinary, statically-typed library, on top of \{}\textit{TypeRep} in an open-world context

Handedness of magnetic-dipolar modes in ferrite disks

For magnetic-dipolar modes in a ferrite, components of the magnetic flux density in a helical coordinate system are dependent on both an orientation of a gyration vector and a sign of a pitch. It gives four types of helical harmonics for magnetostatic-potential wave functions in a ferrite disk. Because of the reflection symmetry breaking, coupling between certain types of helical harmonics takes place in the reflection points. The reflection feature leads to exhibition of two types of resonances: the "right" and "left" resonances. These resonances become coupled for a ferrite disk placed in a homogeneous tangential RF magnetic field. One also observes such resonance coupling for a ferrite disk with a symmetrically oriented linear surface electrode, when this ferrite particle is placed in a homogeneous tangential RF electric field. In a cylindrical coordinate system handedness of magnetic-dipolar modes in a ferrite disk is described by spinor wave functions

Decay and Continuity of Boltzmann Equation in Bounded Domains

Boundaries occur naturally in kinetic equations and boundary effects are crucial for dynamics of dilute gases governed by the Boltzmann equation. We develop a mathematical theory to study the time decay and continuity of Boltzmann solutions for four basic types of boundary conditions: inflow, bounce-back reflection, specular reflection, and diffuse reflection. We establish exponential decay in $L^{\infty}$ norm for hard potentials for general classes of smooth domains near an absolute Maxwellian. Moreover, in convex domains, we also establish continuity for these Boltzmann solutions away from the grazing set of the velocity at the boundary. Our contribution is based on a new $L^{2}$ decay theory and its interplay with delicate $$ decay analysis for the linearized Boltzmann equation, in the presence of many repeated interactions with the boundary.Comment: 89 pages

Illuminating interfaces between phases of a U(1) x U(1) gauge theory

We study reflection and transmission of light at the interface between different phases of a U(1) x U(1) gauge theory. On each side of the interface, one can choose a basis so that one generator is free (allowing propagation of light), and the orthogonal one may be free, Higgsed, or confined. However, the basis on one side will in general be rotated relative to the basis on the other by some angle alpha. We calculate reflection and transmission coefficients for both polarizations of light and all 8 types of boundary, for arbitrary alpha. We find that an observer measuring the behavior of light beams at the boundary would be able to distinguish 4 different types of boundary, and we show how the remaining ambiguity arises from the principle of complementarity (indistinguishability of confined and Higgs phases) which leaves observables invariant under a global electric/magnetic duality transformation. We also explain the seemingly paradoxical behavior of Higgs/Higgs and confined/confined boundaries, and clarify some previous arguments that confinement must involve magnetic monopole condensation.Comment: RevTeX, 12 page