We solve the classic albedo and Milne problems of plane-parallel illumination of an isotropically-scattering half-space when generalized to a Euclidean domain @math for arbitrary @math . A continuous family of pseudo-problems and related @math functions arises and includes the classical 3D solutions, as well as 2D "Flatland" and rod-model solutions, as special cases. The Case-style eigenmode method is applied to the general problem and the internal scalar densities, emerging distributions, and their respective moments are expressed in closed-form. Universal properties invariant to dimension @math are highlighted and we find that a discrete diffusion mode is not universal for @math in absorbing media. We also find unexpected correspondences between differing dimensions and between anisotropic 3D scattering and isotropic scattering in high dimension.
There are a number of approaches to coupling light with thin-film devices such as solar cells. The demonstration now that multiple scattering processes in two-dimensional random media enable efficient light trapping suggests new possibilities for photon management with the benefit of broad spectral and angular operation. Aggregation is widespread in invertebrate societies and can appear in response to environmental heterogeneities or by attraction between individuals. We performed experiments with cockroach, Blattella germanica, larvae in a homogeneous environment to investigate the influence of interactions between individuals on aggregations. Different densities were tested. A first phase led to radial dispersion of larvae in relation to wall-following behaviours; the consequence of this process was a homogeneous distribution of larvae around the periphery of the arena. A second phase corresponded to angular reorganization of larvae leading to the formation of aggregates. The phenomenon was analysed both at the individual and collective levels. Individual cockroaches modulated their behaviour depending on the presence of other larvae in their vicinity: probabilities of stopping and resting times were both higher when the numbers of larvae were greater. We then developed an agent-based model implementing individual behavioural rules, all derived from experiments, to explain the aggregation dynamics at the collective level. This study supports evidence that aggregation relies on mechanisms of amplification, supported by interactions between individuals that follow simple rules based on local information and without knowledge of the global structure. We study transport and diffusion of classical waves in two-dimensional disordered systems and in particular surface waves on a flat surface with randomly fluctuating impedance. We derive from first principles a radiative transport equation for the angularly resolved energy density of the surface waves. This equation accounts for multiple scattering of surface waves as well as for their decay because of leakage into volume waves. We analyze the dependence of the scattering mean free path and of the decay rate on the power spectrum of fluctuations. We also consider the diffusion approximation of the surface radiative transport equation and calculate the angular distribution of the energy transmitted by a strip of random surface impedance.   Abstract Existing two-stream approximations to radiative transfer theory for particulate media are shown to be represented by identical forms of coupled differential equations if the intensity is replaced by integrals of the intensity over hemispheres. One set of solutions thus suffices for all methods and provides convenient analytical comparisons. The equations also suggest modifications of the standard techniques so as to duplicate exact solutions for thin atmospheres and thus permit accurate determinations of the effects of typical aerosol layers. Numerical results for the plane albedos of plane-parallel atmospheres (single-scattering albedo = 0.8, 1.0; optical thickness = 0.25, 1, 4. 16; Henyey-Greenstein phase function with asymmetry factor 0.75) are given for conventional and modified Eddington approximations, conventional and modified two-point quadrature schemes, the hemispheric-constant method and the delta-function method, all for comparison with accurate discrete-ordinate solutions. A new two-... In a gear tooth system for the pumping wheels of a gear pump, having an auxiliary driving transmission, the tooth profiles of the pumping gearwheels are of involute shape, the operational or working pressure angle is greater than 40 DEG , and the transverse contact ratio is approximately 0.5. Introduction.- Heterogeneity in the Lithosphere.- Phenomenological Approaches to Seismogram Envelopes in short-periods.- Born approximation for Wave Scattering in Random Media.- Attenuation of High-Frequency Seismic Waves.- Synthesis of Three-Component Seismogram Envelopes for Earthquakes Using Scattering Amplitudes from the Born Approximation.- Envelope Synthesis Based on the Radiative Transfer Theory: Multiple Scattering Models.- Parabolic approximation and Envelope Synthesis based on the Markov Approximation. Summary and Epilogue. A growing acceptance of fiber-reinforced composite materials imparts some relevance to exploring the effects which a predominantly linear scattering lattice may have upon interior radiative transport. Indeed, a central feature of electromagnetic wave propagation within such a lattice, if sufficiently dilute, is ray confinement to cones whose half-angles are set by that between lattice and the incident ray. When such propagation is subordinated to a viewpoint of an unpolarized intensity transport, one arrives at a somewhat simplified variant of the Boltzmann equation with spherical scattering demoted to its cylindrical counterpart. With a view to initiating a hopefully wider discussion of such phenomena, we follow through in detail the half-space albedo problem. This is done first along canonical lines that harness the Wiener-Hopf technique, and then once more in a discrete ordinates setting via flux decomposition along the eigenbasis of the underlying attenuation scattering matrix. Good agreement is seen to prevail. We further suggest that the Case singular eigenfunction apparatus could likewise be evolved here in close analogy to its original, spherical scattering model. A cursory contact with related problems in the astrophysical literature suggests, in addition, that the basic physical fidelity of our scalar radiative transfer equation (RTE) remains open to improvement by passage to a (4×1) Stokes vector, (4×4) matricial setting. Abstract We present a linear Boltzmann equation to model wave scattering in the Marginal Ice Zone (the region of ocean which consists of broken ice floes). The equation is derived by two methods, the first based on [Meylan, M.H., Squire, V.A., Fox, C., 1997. Towards realism in modeling ocean wave behavior in marginal ice zones. J. Geophys. Res. 102 (C10), 22981–22991] and second based on Masson and LeBlond [Masson, D., LeBlond, P., 1989. Spectral evolution of wind-generated surface gravity waves in a dispersed ice field. J. Fluid Mech. 202, 111–136]. This linear Boltzmann equation, we believe, is more suitable than the equation presented in Masson and LeBlond [Masson, D., LeBlond, P., 1989. Spectral evolution of wind-generated surface gravity waves in a dispersed ice field. J. Fluid Mech. 202, 111–136] because of its simpler form, because it is a differential rather than difference equation and because it does not depend on any assumptions about the ice floe geometry. However, the linear Boltzmann equation presented here is equivalent to the equation in Masson and LeBlond [Masson, D., LeBlond, P., 1989. Spectral evolution of wind-generated surface gravity waves in a dispersed ice field. J. Fluid Mech. 202, 111–136] since it is derived from their equation. Furthermore, the linear Boltzmann equation is also derived independently using the argument in [Meylan, M.H., Squire, V.A., Fox, C., 1997. Towards realism in modeling ocean wave behavior in marginal ice zones. J. Geophys. Res. 102 (C10), 22981–22991]. We also present details of how the scattering kernel in the linear Boltzmann equation is found from the scattering by an individual ice floe and show how the linear Boltzmann equation can be solved straightforwardly in certain cases. Abstract A new “2D 1D” equation is proposed to approximate the 3D linear Boltzmann equation. The approximate 2D 1D equation preserves the exact transport physics in the radial directions x and y but employs diffusion physics in the axial direction z . The 2D 1D equation can be systematically discretized, yielding accurate simulation methods for 3D reactor core problems. The resulting 2D 1D solutions are more accurate than 3D diffusion solutions, and are less expensive to calculate than standard 3D transport solutions. In this paper, we (i) show that the simplest 2D 1D equation has certain desirable properties, (ii) systematically discretize this equation, (iii) derive stable iteration schemes for solving the discrete system of equations, and (iv) give numerical results for simple problems that confirm the theoretical predictions of accuracy and iterative stability.  A mathematical model for non-Rayleigh microwave sea echo is developed which describes explicitly the dependence of statistical properties of the radar cross section on the area of sea surface illuminated by the radar. In addition to the first probability distribution of the scattered radiation, its temporal and spatial correlation functions are also considered. It is shown that, in general, these correlation functions decay on at least two scales, the second, non-Rayleigh, contributions being strongly dependent on the properties of a "single scatterer." Predictions of the model are found to be in qualitative agreement with existing experimental data. A new class of probability distributions, the " K -distributions," is introduced, which may prove useful for fitting such data. In this paper, we intend to formulate a new meta-heuristic algorithm, called Cuckoo Search (CS), for solving optimization problems. This algorithm is based on the obligate brood parasitic behaviour of some cuckoo species in combination with the Levy flight behaviour of some birds and fruit flies. We validate the proposed algorithm against test functions and then compare its performance with those of genetic algorithms and particle swarm optimization. Finally, we discuss the implication of the results and suggestion for further research.   Light scattering from hair is normally simulated in computer graphics using Kajiya and Kay's classic phenomenological model. We have made new measurements of scattering from individual hair fibers that exhibit visually significant effects not predicted by Kajiya and Kay's model. Our measurements go beyond previous hair measurements by examining out-of-plane scattering, and together with this previous work they show a multiple specular highlight and variation in scattering with rotation about the fiber axis. We explain the sources of these effects using a model of a hair fiber as a transparent elliptical cylinder with an absorbing interior and a surface covered with tilted scales. Based on an analytical scattering function for a circular cylinder, we propose a practical shading model for hair that qualitatively matches the scattering behavior shown in the measurements. In a comparison between a photograph and rendered images, we demonstrate the new model's ability to match the appearance of real hair.