A generalized topological recursion for arbitrary ramification

The Eynard-Orantin topological recursion relies on the geometry of a Riemann surface S and two meromorphic functions x and y on S. To formulate the recursion, one must assume that x has only simple ramification points. In this paper we propose a generalized topological recursion that is valid for x with arbitrary ramification. We justify our proposal by studying degenerations of Riemann surfaces. We check in various examples that our generalized recursion is compatible with invariance of the free energies under the transformation (x,y) -> (y,x), where either x or y (or both) have higher order ramification, and that it satisfies some of the most important properties of the original recursion. Along the way, we show that invariance under (x,y) -> (y,x) is in fact more subtle than expected; we show that there exists a number of counter examples, already in the case of the original Eynard-Orantin recursion, that deserve further study.Comment: 26 pages, 2 figure

Criticality in multicomponent spherical models : results and cautions

To enable the study of criticality in multicomponent fluids, the standard spherical model is generalized to describe an \ns-species hard core lattice gas. On introducing \ns spherical constraints, the free energy may be expressed generally in terms of an \ns\times\ns matrix describing the species interactions. For binary systems, thermodynamic properties have simple expressions, while all the pair correlation functions are combinations of just two eigenmodes. When only hard-core and short-range overall attractive interactions are present, a choice of variables relates the behavior to that of one-component systems. Criticality occurs on a locus terminating a coexistence surface; however, except at some special points, an unexpected ``demagnetization effect'' suppresses the normal divergence of susceptibilities at criticality and distorts two-phase coexistence. This effect, unphysical for fluids, arises from a general lack of symmetry and from the vectorial and multicomponent character of the spherical model. Its origin can be understood via a mean-field treatment of an XY spin system below criticality.Comment: 4 figure

Diameter of Compact Riemann Surfaces

Diameter is one of the most basic properties of a geometric object, while Riemann surfaces are one of the most basic geometric objects. Surprisingly, the diameter of compact Riemann surfaces is known exactly only for the sphere and the torus. For higher genuses, only very general but loose upper and lower bounds are available. The problem of calculating the diameter exactly has been intractable since there is no simple expression for the distance between a pair of points on a high-genus surface. Here we prove that the diameters of a class of simple Riemann surfaces known as generalized Bolza surfaces of any genus greater than $1$ are equal to the radii of their fundamental polygons. This appears to be the first exact result for the diameter of a compact hyperbolic manifold

Modeling the Effects of Surface Roughness on Electron Yield

Surface conditions—including surface morphology, composition, contamination, and oxidation—can significantly affect electron yields and consequently spacecraft charging. The effects of surface roughness on electron yield are modeled in this study by considering four aspects of electron yield calculations: (i) simple models of rough surface geometry, (ii) the angular distributions of electrons emitted from various points on these surfaces, (iii) the likelihood of these emitted electrons escaping the rough surface, and (iv) the relative fractions of smooth and rough surfaces. Three simple periodic one-dimensional surface profiles were considered—namely rectangular, triangular, and sawtooth features; each surface profile was characterized by an aspect ratio of the surface feature width to the height. Two different angular emission profiles were considered for lower energy secondary electrons (a Lambertian cosine distribution) and higher energy backscattered electrons (a much narrower, energy-dependent screened Rutherford model approximating a Mott scattering cross-section which also depends on the atomic number of the material). In this initial study, only normally-incident electron profiles were considered, and any emitted electrons were assumed to be recaptured if they intersect any surface. The relative fractions of smooth and rough surfaces (which could in general have different yields for materials in these regions) were taken into account using a simple 1D patch model. Combining the surface profiles with the emission distributions allowed the calculation of a roughness coefficient—which predicted the effect of the surface profile on a smooth surface electron yield—for both secondary and backscattered yields for each surface geometry. Generalized predictions are presented for the reduced secondary and backscattered yields (scaled as the ratio of yields for materials in the smooth and rough fractions) as functions of aspect ratio and the fraction of the surface profile occupied by surface features. Results for backscattered electrons of different incident energies are also presented

On contact between curves and rigid surfaces – from verification of the euler-eytelwein problem to knots

A general theory for the Curve-To-Curve contact is applied to develop a special contact algorithm between curves and rigid surfaces. In this case contact kinematics are formulated in the local coordinate system attached to the curve, however, contact is defined at integration points of the curve line (Mortar type contact). The corresponding Closest Point Projection (CPP) procedure is used to define then a shortest distance between the integration point on a curve and the rigid surface. For some simple approximations of the rigid surface closed form solutions are possible. Within the finite element implementation the isogeometric approach is used to model curvilinear cables and the rigid surfaces can be defined in general via NURB surface splines. Verification of the finite element algorithm is given using the well-known analytical solution of the Euler-Eytelwein problem – a rope on a cylindrical surface. The original 2D formula is generalized into the 3D case considering an additional parameter H-pitch for the helix. Finally, applications to knot mechanics are shown