Nonpersistence of resonant caustics in perturbed elliptic billiards

Caustics are curves with the property that a billiard trajectory, once tangent to it, stays tangent after every reflection at the boundary of the billiard table. When the billiard table is an ellipse, any nonsingular billiard trajectory has a caustic, which can be either a confocal ellipse or a confocal hyperbola. Resonant caustics ---the ones whose tangent trajectories are closed polygons--- are destroyed under generic perturbations of the billiard table. We prove that none of the resonant elliptical caustics persists under a large class of explicit perturbations of the original ellipse. This result follows from a standard Melnikov argument and the analysis of the complex singularities of certain elliptic functions.Comment: 14 pages, 3 figure

Comprehending environmental and economic sustainability: Comparative analysis of stability principles in the biosphere and free market economy

Using the formalism of Lyapunov potential function it is shown that the stability principles for biomass in the ecosystem and for employment in economics are mathematically similar. The ecosystem is found to have a stable and an unstable stationary state with high (forest) and low (grasslands) biomass, respectively. In economics, there is a stable stationary state with high employment, which corresponds to mass production of conventional goods sold at low cost price, and an unstable stationary state with lower employment, which corresponds to production of novel goods appearing in the course of technological progress. An additional stable stationary state is described for economics, the one corresponding to very low employment in production of life essentials such as energy and raw materials. In this state the civilization currently pays 10% of global GDP for energy produced by a negligible minority of the working population (currently ~0.2%) and sold at prices greatly exceeding the cost price by 40 times. It is shown that economic ownership over energy sources is equivalent to equating measurable variables of different dimensions (stores and fluxes), which leads to effective violation of the laws of energy and matter conservation.Comment: 51 pages, 6 figure

Collagen bundle morphometry in skin & scar tissue: a novel distance mapping method provides superior measurements compared to Fourier analysis

Histopathological evaluations of fibrotic processes require the characterization of collagen morphology in terms of geometrical features such as bundle orientation thickness and spacing. However, there are currently no reliable and valid techniques of measuring bundle thickness and spacing. Hence, two objective methods quantifying the collagen bundle thickness and spacing were tested for their reliability and validity: Fourier first-order maximum analysis and Distance Mapping, with the latter constituting a newly developed morphometric technique. Histological slides were constructed and imaged from 50 scar and 50 healthy human skin biopsies and subsequently analyzed by two observers to determine the interobserver reliability via the intraclass correlation coefficient. An intraclass correlation coefficient larger than 0.7 is considered as representing good reliability. The interobserver reliability for the Fourier first-order maximum and for the Distance Mapping algorithms, respectively, showed an intraclass correlation coefficient above 0.72 and 0.89. Additionally, we performed an assessment of validity in the form of responsiveness, in particular, demonstrating medium to excellent results via a calculation of the effect size, highlighting that both methods are sensitive enough to measure a treatment effect in clinical practice. In summary, two reliable and valid measurement methods were demonstrated for collagen bundle morphometry for the first time. Due to its superior reliability and more useful measures (bundle thickness and bundle spacing), Distance Mapping emerges as the preferred and more practical method. Nevertheless, in the future, both methods can be used for reliable and valid collagen morphometry of skin and scars, whereas further applications evaluating the quantitative microscopy of other fibrotic processes are anticipated

Photonic band-structure effects in the reflectivity of periodically patterned waveguides

We report sharp resonant features in the reflectivity spectra of semiconductor waveguides patterned with periodic lattices of deep holes. The resonances arise from coupling of incident light to the photonic bands of the lattice. By varying the reflection geometry, large parts of the photonic band structure are determined. A scattering matrix treatment is used to obtain theoretical spectra which agree well with experiment. The waveguide is shown to have an important influence on the band structure, including marked polarization mixing and significant energy up-shifts

"The numerical accuracy of truncated Ewald sums for periodic systems with long-range Coulomb interactions"

Ewald summation is widely used to calculate electrostatic interactions in computer simulations of condensed-matter systems. We present an analysis of the errors arising from truncating the infinite real- and Fourier-space lattice sums in the Ewald formulation. We derive an optimal choice for the Fourier-space cutoff given a screening parameter $\eta$. We find that the number of vectors in Fourier space required to achieve a given accuracy scales with $\eta^3$. The proposed method can be used to determine computationally efficient parameters for Ewald sums, to assess the quality of Ewald-sum implementations, and to compare different implementations.Comment: 6 pages, 3 figures (Encapsulated PostScript), LaTe

Simple Construction of Elliptic Boundary K-Matrix

We give the infinite-dimensional representation for the elliptic $K$-operator satisfying the boundary Yang-Baxter equation. By restricting the functional space to finite-dimensional space, we construct the elliptic $K$-matrix associated to Belavin's completely $\mathbb{Z}$-symmetric $R$-matrix.Comment: 10 pages, Latex2e with AM

Generalized Reduction Procedure: Symplectic and Poisson Formalism

We present a generalized reduction procedure which encompasses the one based on the momentum map and the projection method. By using the duality between manifolds and ring of functions defined on them, we have cast our procedure in an algebraic context. In this framework we give a simple example of reduction in the non-commutative setting.Comment: 39 pages, Latex file, Vienna ESI 28 (1993

Numerical implementation of the exact dynamics of free rigid bodies

In this paper the exact analytical solution of the motion of a rigid body with arbitrary mass distribution is derived in the absence of forces or torques. The resulting expressions are cast into a form where the dependence of the motion on initial conditions is explicit and the equations governing the orientation of the body involve only real numbers. Based on these results, an efficient method to calculate the location and orientation of the rigid body at arbitrary times is presented. This implementation can be used to verify the accuracy of numerical integration schemes for rigid bodies, to serve as a building block for event-driven discontinuous molecular dynamics simulations of general rigid bodies, and for constructing symplectic integrators for rigid body dynamics.Comment: Shortened paper with updated references, 28 pages, 3 figure

Central elements of the elliptic ZnZ_n monodromy matrix algebra at roots of unity

The central elements of the algebra of monodromy matrices associated with the $\mathbb{Z}_n$ R-matrix are studied. When the crossing parameter $w$ takes a special rational value $w=\frac{n}{N}$, where $N$ and $n$ are positive coprime integers, the center is substantially larger than that in the generic case for which the "quantum determinant" provides the center. In the trigonometric limit, the situation corresponds to the quantum group at roots of unity. This is a higher rank generalization of the recent results by Belavin and Jimbo.Comment: Latex file, 18 pages; V2: minor typos corrected and a reference update

Personality profiling may help select better cleaner fish for sea-lice control in salmon farming

Lumpfish (Cyclopterus lumpus) are increasingly being used as cleaner fish to control parasitic sea-lice in salmon farming, but cleaning rates are very variable and not all individuals eat sea-lice, which increases the risk of emaciation and has ethical and practical implications. Selecting good cleaners is a priority to make the industry more sustainable, but there is little information on what behaviours make cleaner fish effective under a commercial setting. We examined variation in lumpfish personalities according to the five-factor personality model that takes into account differences in activity, anxiety (shelter use, thigmotaxis), aggression, sociality, and boldness (neophobia). We then quantified how variation in lumpfish personalities influenced interactions with naïve Atlantic salmon (Salmo salar), without the confounding effects of variation in sea-lice loads. Variation in activity, sociality, aggression and neophobia, but not in anxiety, was repeatable. Neophilic, non-aggressive lumpfish spent more time inspecting salmon than neophobic and aggressive individuals, but salmon fled in the presence of the most active and social individuals, suggesting there may be an optimal cleaner fish personality amenable to artificial selection. The personality screening protocols developed in this study could inform a more efficient use of cleaner fish in salmon farming and reduce the number of individuals required to control sea-lice