Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

Advanced Mathematical Methods for Scientists and Engineers

https://scholarlypublications.universiteitleiden.nl/access/item%3A2733176/view

Front propagation into unstable states

The objective of this review is to critically assess the different approaches developed in recent years to understand the dynamics of open flows such as mixing layers, jets, wakes, separation bubbles, boundary layers, and so on. These complex flows develop in extended domains in which fluid particles are continuously advected downstream. They behave either as noise amplifiers or as oscillators, both of which exhibit strong nonlinearities (Huerre & Monkewitz 1990). The local approach is inherently weakly nonparallel and it assumes that the basic flow varies on a long length scale compared to the wavelength of the instability waves. The dynamics of the flow is then considered as a superposition of linear or nonlinear instability waves that, at leading order, behave at each streamwise station as if the flow were homogeneous in the streamwise direction. In the fully global context, the basic flow and the instabilities do not have to be characterized by widely separated length scales, and the dynamics is then viewed as the result of the interactions between Global modes living in the entire physical domain with the streamwise direction as an eigendirection. This second approach is more and more resorted to as a result of increased computational capability. The earlier review of Huerre & Monkewitz (1990) emphasized how local linear theory can account for the noise amplifier behavior as well as for the onset of a Global mode. The present survey first adopts the opposite point of view by demonstrating how fully global theory accounts for the noise amplifier behavior of open flows. From such a perspective, there is strong emphasis on the very peculiar nonorthogonality of linear Global modes, which in turn allows a novel interpretation of recent numerical simulations and experimental observations. The nonorthogonality of linear Global modes also imposes severe constraints on the extension of linear global theory to the fully nonlinear regime. When the flow is weakly nonparallel, this limitation is so severe that the linear Global mode theory is of little help. It is then much more appropriate to develop a fully nonlinear formulation involving the presence of a front separating the base state region from the bifurcated state region.

/pdf/global-instabilities-in-spatially-developing-flows-non-4rl3dgkyzb.pdf

GLOBAL INSTABILITIES IN SPATIALLY DEVELOPING FLOWS: Non-Normality and Nonlinearity

In 1982, the Belgian Pilots' Guild raised the question of what effect exposure to radar radiation--for example, that encountered in passing a pilot launch's radar--might have on the human body. Recapitulating investigations of this question, this article states that in 25 years of experience with radar, there have been no known incidents of pilots being affected by radar waves. In the future, however, involvement by some pilots with Vessel Traffic Service shore-based radar could affect pilots somewhat differently from limited exposure to pilot launch radar. Pilots who find themselves in new working conditions close to an emitting source should exercise care all times.

About heat waves

The Blasius boundary-layer flow of a micropolar fluid

In this article our concern is with the third Painleve equation d 2 &#119910; d&#119909; 2  =  1 &#119910; ( d&#119910; d&#119909; ) 2  –  1 &#119909; d&#119910; d&#119909;  +  α&#119910; 2 + β     &#119909;  +  γ&#119910; 3  +  δ &#119910; ,  (1) where α , β , γ , and δ are arbitrary constants. It is well known that this equation admits a variety of types of solution and here we classify and characterize many of these. Depending on the values of the parameters the third Painleve equation can admit solutions that may be either expressed as the ratio of two polynomials in either x or x 1/3 or related to certain Bessel functions. It is thought that all exact solutions of (1) can be categorized into one or other of these hierarchies. We show how, given a few initial solutions, it is possible to use the underlying structures of these hierarchies to obtain many other solutions. In addition, we show how this knowledge concerning the continuous third Painleve equation (1) can be adapted and used to derive exact solutions of a suitable discretized counterpart of (1). Both the continuous and discrete solutions we find are of potential importance as it is known that the third Painleve equation has a large number of physically significant applications.

Bäcklund Transformations and Solution Hierarchies for the Third Painlevé Equation

The spiral wind-up and diffusive decay of a passive scalar in circular streamlines is considered. An accelerated diffusion mechanism operates to destroy scalar fluctuations on a time scale of order P1/3 times the turn-over time, where P is a Peclet number. The mechanism relies on differential rotation, that is, a non-zero gradient of angular velocity. However if the flow is smooth, the gradient of angular velocity necessarily vanishes at the centre of the streamlines, and the time scale becomes greater. The behaviour at the centre is analysed and it is found that scalar there is only destroyed on a time scale of order P1/2. Related results are obtained for magnetic field and for weak vorticity, a scalar coupled to the stream function of the flow. Some exact solutions are presented.

/pdf/accelerated-diffusion-in-the-centre-of-a-vortex-9sciax91mm.pdf

Accelerated diffusion in the centre of a vortex

The relaxation of a smooth two-dimensional vortex to axisymmetry, also known as ‘axisymmetrization’, is studied asymptotically and numerically. The vortex is perturbed at t = 0 and dierential rotation leads to the wind-up of vorticity fluctuations to form a spiral. It is shown that for innite Reynolds number and in the linear approximation, the vorticity distribution tends to axisymmetry in a weak or coarse-grained sense: when the vorticity eld is integrated against a smooth test function the result decays asymptotically as t with = 1 +( n 2 +8 ) 1= 2 , where n is the azimuthal wavenumber of the perturbation and n > 1. The far-eld stream function of the perturbation decays with the same exponent. To obtain these results the paper develops a complete asymptotic picture of the linear evolution of vorticity fluctuations for large times t, which is based on that of Lundgren (1982).

/pdf/the-spiral-wind-up-of-vorticity-in-an-inviscid-planar-vortex-gnljn591on.pdf

The spiral wind-up of vorticity in an inviscid planar vortex

Comparisons among animal pollinators of the spatial distributions of pollen that they produce have typically been made among morphologically disparate pairs of species. In contrast, we investigated the potential extent of pollen dispersal by honey-bees ( Apis mellifera ) and bumble-bees ( Bombus lapidarius, B. pascuorum and B. terrestris ) foraging in rows of oil-seed rape ( Brassica napus cv Westar). We estimated the pollen carryover attributable to individual bees by using particulate fluorescent dye as a pollen analogue. Most of the dye was deposited at the first few flowers probed and smaller proportions were deposited up to the 20th successively probed flower. We found no significant interspecific differences in dye carryover mediated by individuals of A. mellifera, B. lapidarius and B. terrestris with respect to either the amount deposited or the rate of decline in deposition across successively probed flowers. We present evidence that the dye produced a reasonably good analogue of pollen transfer. Bees typically flew from one plant to another nearby in the same row and were strongly directional in their movements. Bee species differed significantly in their movement patterns, with B. terrestris having the greatest mean move length and directionality. We used three kinds of model (a numerical simulation and two different sets of diffusion-advection equations) to attempt to emulate bee movements. The predictions from all models were reasonably consistent with the observed bee movements, although the numerical simulation invariably made the most accurate predictions, particularly over the first few moves. Predicted bee movements were combined with least-squares models of dye deposition to estimate the spatial dispersal of pollen by each bee species. All models ranked the bees in the same order of decreasing effectiveness in dye dispersal: B. terrestris, A. mellifera, B. lapidarius, B. pascuorum , although, except for long-distance dispersal, there was only minor variation among the bee species in the predicted extents of dye dispersal (e.g. the models predicted that the median dispersal distance would be approximately two intervening plants irrespective of the species of bee). Overall, the consensus of the models' predictions is that most of the pollen from a source plant is deposited on immediate neighbours, but that long-distance pollen dispersal in this system extends over approximately 20-40 intervening plants from the originating plant, depending on the identity of the pollinator.

Andrew P. Bassom

Papers

The Blasius boundary-layer flow of a micropolar fluid

Bäcklund Transformations and Solution Hierarchies for the Third Painlevé Equation

Accelerated diffusion in the centre of a vortex

The spiral wind-up of vorticity in an inviscid planar vortex

Predicted pollen dispersal by honey-bees and three species of bumble-bees foraging on oil-seed rape : a comparison of three models