High-frequency diffraction of an electromagnetic plane wave by an imperfectly conducting rectangular cylinder at grazing incidence

We derive new results for the electromagnetic scattered far wave field produced when a high-frequency plane E-polarized wave is at grazing incidence on an imperfectly conducting rectangular cylinder. The solution to the problem is obtained by using the geometrical theory of diffraction, multiple diffraction methods, the canonical solution for the problem of the diffraction of a plane wave by a right-angled impedance wedge, in conjunction with a novel analytic approach

Fundamental groups, Alexander invariants, and cohomology jumping loci

We survey the cohomology jumping loci and the Alexander-type invariants associated to a space, or to its fundamental group. Though most of the material is expository, we provide new examples and applications, which in turn raise several questions and conjectures. The jump loci of a space X come in two basic flavors: the characteristic varieties, or, the support loci for homology with coefficients in rank 1 local systems, and the resonance varieties, or, the support loci for the homology of the cochain complexes arising from multiplication by degree 1 classes in the cohomology ring of X. The geometry of these varieties is intimately related to the formality, (quasi-) projectivity, and homological finiteness properties of \pi_1(X). We illustrate this approach with various applications to the study of hyperplane arrangements, Milnor fibrations, 3-manifolds, and right-angled Artin groups.Comment: 45 pages; accepted for publication in Contemporary Mathematic

Uniformizing surfaces via discrete harmonic maps

We show that for any closed surface of genus greater than one and for any finite weighted graph filling the surface, there exists a hyperbolic metric which realizes the least Dirichlet energy harmonic embedding of the graph among a fixed homotopy class and all hyperbolic metrics on the surface. We give explicit examples of such hyperbolic surfaces through a new interpretation of the Nielsen realization problem for the mapping class groups.Comment: 31 pages, 5 figure

Loop exponent in DNA bubble dynamics

Dynamics of DNA bubbles are of interest for both statistical physics and biology. We present exact solutions to the Fokker-Planck equation governing bubble dynamics in the presence of a long-range entropic interaction. The complete meeting time and meeting position probability distributions are derived from the solutions. Probability distribution functions reflect the value of the loop exponent of the entropic interaction. Our results extend previous results which concentrated mainly on the tails of the probability distribution functions and open a way to determining the strength of the entropic interaction experimentally which has been a matter of recent discussions. Using numerical integration, we also discuss the influence of the finite size of a DNA chain on the bubble dynamics. Analogous results are obtained also for the case of subdiffusive dynamics of a DNA bubble in a heteropolymer, revealing highly universal asymptotics of meeting time and position probability functions.Comment: 24 pages, 11 figures, text identical to the published version; v3 - updated Ref. [47] and corrected Eqs. (3.6) and (3.10

Multi-soliton dynamics in the Skyrme model

We exhibit the dynamical scattering of multi-solitons in the Skyrme model for configurations with charge two, three and four. First, we construct maximally attractive configurations from a simple profile function and the product ansatz. Then using a sophisticated numerical algorithm, initially well-separated skyrmions in approximately symmetric configurations are shown to scatter through the known minimum energy configurations. These scattering events illustrate a number of similarities to BPS monopole configurations of the same charge. A simple modification of the dynamics to a dissipative regime, allows us to compute the minimal energy skyrmions for baryon numbers one to four to within a few percent.Comment: latex, 10 pages, plus 5 figures (as gif files

A Lagrangian effective field theory

We have continued the development of Lagrangian, cosmological perturbation theory for the low-order correlators of the matter density field. We provide a new route to understanding how the effective field theory (EFT) of large-scale structure can be formulated in the Lagrandian framework and a new resummation scheme, comparing our results to earlier work and to a series of high-resolution N-body simulations in both Fourier and configuration space. The `new' terms arising from EFT serve to tame the dependence of perturbation theory on small-scale physics and improve agreement with simulations (though with an additional free parameter). We find that all of our models fare well on scales larger than about two to three times the non-linear scale, but fail as the non-linear scale is approached. This is slightly less reach than has been seen previously. At low redshift the Lagrangian model fares as well as EFT in its Eulerian formulation, but at higher $z$ the Eulerian EFT fits the data to smaller scales than resummed, Lagrangian EFT. All the perturbative models fare better than linear theory.Comment: 19 pages, 3 figure

Thermodynamic metrics and optimal paths

A fundamental problem in modern thermodynamics is how a molecular-scale machine performs useful work, while operating away from thermal equilibrium without excessive dissipation. To this end, we derive a friction tensor that induces a Riemannian manifold on the space of thermodynamic states. Within the linear-response regime, this metric structure controls the dissipation of finite-time transformations, and bestows optimal protocols with many useful properties. We discuss the connection to the existing thermodynamic length formalism, and demonstrate the utility of this metric by solving for optimal control parameter protocols in a simple nonequilibrium model.Comment: 5 page