to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

Advanced Mathematical Methods for Scientists and Engineers

Quantum mechanics is an extremely successful theory that agrees with every experimental test. However, the principle of linear superposition, a central tenet of the theory, apparently contradicts a commonplace observation: macroscopic objects are never found in a linear superposition of position states. Moreover, the theory does not explain why during a quantum measurement, deterministic evolution is replaced by probabilistic evolution, whose random outcomes obey the Born probability rule. In this article a review is given of an experimentally falsifiable phenomenological proposal, known as continuous spontaneous collapse: a stochastic nonlinear modification of the Schr\"odinger equation, which resolves these problems, while giving the same experimental results as quantum theory in the microscopic regime. Two underlying theories for this phenomenology are reviewed: trace dynamics and gravity-induced collapse. As the macroscopic scale is approached, predictions of this proposal begin to differ appreciably from those of quantum theory and are being confronted by ongoing laboratory experiments that include molecular interferometry and optomechanics. These experiments, which test the validity of linear superposition for large systems, are reviewed here, and their technical challenges, current results, and future prospects summarized. It is likely that over the next two decades or so, these experiments can verify or rule out the proposed stochastic modification of quantum theory.

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Models of Wave-function Collapse, Underlying Theories, and Experimental Tests

The physical origin of the Casimir force is connected with the existence of zero-point and thermal fluctuations. The Casimir effect is very general and finds applications in various fields of physics. This review is limited to the rapid progress at the intersection of experiment and theory that has been achieved in the last few years. It includes a critical assessment of the proposed approaches to the resolution of the puzzles arising in the applications of the Lifshitz theory of the van der Waals and Casimir forces to real materials. All the primary experiments on the measurement of the Casimir force between macroscopic bodies and the Casimir-Polder force between an atom and a wall that have been performed in the last decade are reviewed, including the theory needed for their interpretation. The methodology for the comparison between experiment and theory in the force-distance measurements is presented. The experimental and theoretical results described here provide a deeper understanding of the phenomenon of dispersion forces in real materials and offer guidance for the application of the Lifshitz theory to the interpretation of the measurement results.

The Casimir force between real materials: Experiment and theory

Since the invention of optical tweezers, optical manipulation has advanced significantly in scientific areas such as atomic physics, optics and biological science. Especially in the past decade, numerous optical beams and nanoscale devices have been proposed to mechanically act on nanoparticles in increasingly precise, stable and flexible ways. Both the linear and angular momenta of light can be exploited to produce optical tractor beams, tweezers and optical torque from the microscale to the nanoscale. Research on optical forces helps to reveal the nature of light-matter interactions and to resolve the fundamental aspects, which require an appropriate description of momenta and the forces on objects in matter. In this review, starting from basic theories and computational approaches, we highlight the latest optical trapping configurations and their applications in bioscience, as well as recent advances down to the nanoscale. Finally, we discuss the future prospects of nanomanipulation, which has considerable potential applications in a variety of scientific fields and everyday life.

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Optical manipulation from the microscale to the nanoscale: fundamentals, advances and prospects.

Casimir and Casimir?Polder repulsions have been known for more than 50 years. The general ?Lifshitz? configuration of parallel semi-infinite dielectric slabs permits repulsion if they are separated by a dielectric fluid that has a value of permittivity that is intermediate between those of the dielectric slabs. This was indirectly confirmed in the 1970s, and more directly by Capasso?s group recently. It has also been known for many years that electrically and magnetically polarizable bodies can experience a repulsive quantum vacuum force. More amenable to practical application are situations where repulsion could be achieved between ordinary conducting and dielectric bodies in vacuum. The status of the field of Casimir repulsion with emphasis on some recent developments will be surveyed. Here, stress will be placed on analytic developments, especially on Casimir?Polder (CP) interactions between anisotropically polarizable atoms, and CP interactions between anisotropic atoms and bodies that also exhibit anisotropy, either because of anisotropic constituents, or because of geometry. Repulsion occurs for wedge-shaped and cylindrical conductors, provided the geometry is sufficiently asymmetric, that is, either the wedge is sufficiently sharp or the atom is sufficiently far from the cylinder.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker?s 75th birthday devoted to ?Applications of zeta functions and other spectral functions in mathematics and physics?.

https://iopscience.iop.org/article/10.1088/1751-8113/45/37/374006/pdf

Repulsive Casimir and Casimir?Polder forces

The Casimir effect, reflecting quantum vacuum fluctuations in the electromagnetic field in a region with material boundaries, has been studied both theoretically and experimentally since 1948. The forces between dielectric and metallic surfaces both plane and curved have been measured at the 10–1% level in a variety of room temperature experiments, and remarkable agreement with the zero-temperature theory has been achieved. In fitting the data various corrections due to surface roughness, patch potentials, curvature, and temperature have been incorporated. It is the latter that is the subject of the present paper. We point out that, in fact, no temperature dependence has yet been detected, and that the experimental situation is still too fluid to permit conclusions about thermal corrections to the Casimir effect. Theoretically, there are subtle issues concerning thermodynamics and electrodynamics which have resulted in disparate predictions concerning the nature of these corrections. However, a general consensus has seemed to emerge that suggests that the temperature correction to the Casimir effect is relatively large, and should be observable in future experiments involving surfaces separated at the few micrometre scale.

https://iopscience.iop.org/article/10.1088/1367-2630/8/10/236/pdf

Thermal corrections to the Casimir effect

Lord Kelvin’s result that waves behind a ship lie within a half-angle is perhaps the most famous and striking result in the field of surface waves. We solve the linear ship wave problem in the presence of a shear current of constant vorticity , and show that the Kelvin angles (one each side of wake) as well as other aspects of the wake depend closely on the ‘shear Froude number’ (based on length and the ship’s speed ), and on the angle between current and the ship’s line of motion. In all directions except exactly along the shear flow there exists a critical value of beyond which no transverse waves are produced, and where the full wake angle reaches . Such critical behaviour is previously known from waves at finite depth. For side-on shear, one Kelvin angle can exceed . On the other hand, the angle of maximum wave amplitude scales as ( based on size of ship) when , a scaling virtually unaffected by the shear flow.

Ship waves in the presence of uniform vorticity

Lord Kelvin's result that waves behind a ship lie within a half-angle 19 deg 28' is perhaps the most famous and striking result in the field of surface waves. We solve the linear ship wave problem in the presence of a shear current of constant vorticity S, and show that the Kelvin angles (one each side of wake) as well as other aspects of the wake depend closely on the "shear Froude number" Frs=VS/g (based on length g/S^2 and the ship's speed V), and on the angle between current and the ship's line of motion. In all directions except exactly along the shear flow there exists a critical value of Frs beyond which no transverse waves are produced, and where the full wake angle reaches 180 deg. Such critical behaviour is previously known from waves at finite depth. For side-on shear, one Kelvin angle can exceed 90 deg. On the other hand, the angle of maximum wave amplitude scales as 1/Fr (Fr based on size of ship) when Fr >> 1, a scaling virtually unaffected by the shear flow.

In view of the current discussion on the subject, an effort is made to show very accurately both analytically and numerically how the Drude dispersion model gives consistent results for the Casimir ...

Simen Å. Ellingsen

Papers

Repulsive Casimir and Casimir?Polder forces

Thermal corrections to the Casimir effect

Ship waves in the presence of uniform vorticity

Ship waves in the presence of uniform vorticity

Analytical and numerical verification of the Nernst theorem for metals