Manin-Olshansky triples for Lie superalgebras

Following V. Drinfeld and G. Olshansky, we construct Manin triples (\fg, \fa, \fa^*) such that \fg is different from Drinfeld's doubles of \fa for several series of Lie superalgebras \fa which have no even invariant bilinear form (periplectic, Poisson and contact) and for a remarkable exception. Straightforward superization of suitable Etingof--Kazhdan's results guarantee then the uniqueness of $q$-quantization of our Lie bialgebras. Our examples give solutions to the quantum Yang-Baxter equation in the cases when the classical YB equation has no solutions. To find explicit solutions is a separate (open) problem. It is also an open problem to list (\`a la Belavin-Drinfeld) all solutions of the {\it classical} YB equation for the Poisson superalgebras \fpo(0|2n) and the exceptional Lie superalgebra \fk(1|6) which has a Killing-like supersymmetric bilinear form but no Cartan matrix

Charmonium properties in hot quenched lattice QCD

We study the properties of charmonium states at finite temperature in quenched QCD on large and fine isotropic lattices. We perform a detailed analysis of charmonium correlation and spectral functions both below and above $T_c$. Our analysis suggests that both S wave states ($J/\psi$ and $\eta_c$) and P wave states ($\chi_{c0}$ and $\chi_{c1}$) disappear already at about $1.5 T_c$. The charm diffusion coefficient is estimated through the Kubo formula and found to be compatible with zero below $T_c$ and approximately $1/\pi T$ at $1.5 T_c\lesssim T\lesssim 3 T_c$.Comment: 32 pages, 19 figures, typo corrected, discussions on isotropic vs anisotropic lattices expanded, published versio

Forcing function control of Faraday wave instabilities in viscous shallow fluids

We investigate the relationship between the linear surface wave instabilities of a shallow viscous fluid layer and the shape of the periodic, parametric-forcing function (describing the vertical acceleration of the fluid container) that excites them. We find numerically that the envelope of the resonance tongues can only develop multiple minima when the forcing function has more than two local extrema per cycle. With this insight, we construct a multi-frequency forcing function that generates at onset a non-trivial harmonic instability which is distinct from a subharmonic response to any of its frequency components. We measure the corresponding surface patterns experimentally and verify that small changes in the forcing waveform cause a transition, through a bicritical point, from the predicted harmonic short-wavelength pattern to a much larger standard subharmonic pattern. Using a formulation valid in the lubrication regime (thin viscous fluid layer) and a WKB method to find its analytic solutions, we explore the origin of the observed relation between the forcing function shape and the resonance tongue structure. In particular, we show that for square and triangular forcing functions the envelope of these tongues has only one minimum, as in the usual sinusoidal case.Comment: 12 pages, 10 figure

Beam-Size Invariant Spectropolarimeters Using Gap-Plasmon Metasurfaces

Metasurfaces enable exceptional control over the light with surface-confined planar components, offering the fascinating possibility of very dense integration and miniaturization in photonics. Here, we design, fabricate and experimentally demonstrate chip-size plasmonic spectropolarimeters for simultaneous polarization state and wavelength determination. Spectropolarimeters, consisting of three gap-plasmon phase-gradient metasurfaces that occupy 120{\deg} circular sectors each, diffract normally incident light to six predesigned directions, whose azimuthal angles are proportional to the light wavelength, while contrasts in the corresponding diffraction intensities provide a direct measure of the incident polarization state through retrieval of the associated Stokes parameters. The proof-of-concept 96-{\mu}m-diameter spectropolarimeter operating in the wavelength range of 750-950nm exhibits the expected polarization selectivity and high angular dispersion. Moreover, we show that, due to the circular-sector design, polarization analysis can be conducted for optical beams of different diameters without prior calibration, demonstrating thereby the beam-size invariant functionality. The proposed spectropolarimeters are compact, cost-effective, robust, and promise high-performance real-time polarization and spectral measurements

Gradient metasurfaces: a review of fundamentals and applications

In the wake of intense research on metamaterials the two-dimensional analogue, known as metasurfaces, has attracted progressively increasing attention in recent years due to the ease of fabrication and smaller insertion losses, while enabling an unprecedented control over spatial distributions of transmitted and reflected optical fields. Metasurfaces represent optically thin planar arrays of resonant subwavelength elements that can be arranged in a strictly or quasi periodic fashion, or even in an aperiodic manner, depending on targeted optical wavefronts to be molded with their help. This paper reviews a broad subclass of metasurfaces, viz. gradient metasurfaces, which are devised to exhibit spatially varying optical responses resulting in spatially varying amplitudes, phases and polarizations of scattered fields. Starting with introducing the concept of gradient metasurfaces, we present classification of different metasurfaces from the viewpoint of their responses, differentiating electrical-dipole, geometric, reflective and Huygens' metasurfaces. The fundamental building blocks essential for the realization of metasurfaces are then discussed in order to elucidate the underlying physics of various physical realizations of both plasmonic and purely dielectric metasurfaces. We then overview the main applications of gradient metasurfaces, including waveplates, flat lenses, spiral phase plates, broadband absorbers, color printing, holograms, polarimeters and surface wave couplers. The review is terminated with a short section on recently developed nonlinear metasurfaces, followed by the outlook presenting our view on possible future developments and perspectives for future applications.Comment: Accepted for publication in Reports on Progress in Physic

Direct characterization of near-field coupling in gap plasmon-based metasurfaces

Metasurfaces based on gap surface-plasmon resonators allow one to arbitrarily control the phase, amplitude and polarization of reflected light with high efficiency. However, the performance of densely-packed metasurfaces is reduced, often quite significantly, in comparison with simple analytical predictions. We argue that this reduction is mainly because of the near-field coupling between metasurface elements, which results in response from each element being different from the one anticipated by design simulations, which are commonly conducted for each individual element being placed in an artificial periodic arrangement. In order to study the influence of near-field coupling, we fabricate meta-elements of varying sizes arranged in quasi-periodic arrays so that the immediate environment of same size elements is different for those located in the middle and at the border of the arrays. We study the near-field using a phase-resolved scattering-type scanning near-field optical microscopy (s-SNOM) and conducting numerical simulations. By comparing the near-field maps from elements of the same size but different placements we evaluate the near-field coupling strength, which is found to be significant for large and densely packed elements. This technique is quite generic and can be used practically for any metasurface type in order to precisely measure the near-field response from each individual element and identify malfunctioning ones, providing feedback to their design and fabrication, thereby allowing one to improve the efficiency of the whole metasurface

The non-ignorable missing-data problem in consumer banking

The thesis aims to solve a specific missing-data problem in consumer banking. Application scoring and behaviour scoring are two of the main applications f 'statistics and probability modelling in consumer banking. In application scoring, a missing data problem occurs due to the selection of applICants by the bank. This has attracted much interest, and relevant discussion can be found under the topic of 'reject inference'. On the contrary, a similar problem in behaviour scoring has not been widely explored. The problem we wish to solve in the present thesis is a missing data problem that results from selection in behaviour scoring. We review the nature of the missing data problem and the existing solutions. Missingdata problems can be categorised into: MCAR , MAR, and MNAR problems. MCAR and MAR problems have attracted much attention; less discussion can be found on the MNAR problems. The problem we solve in this thesis is a MNAR problem. Two of the best known solutions to MNAR problems are: the two-step method proposed by Heckman, and the EM algorithm proposed by Little and Rubin. We illustrate how these existing methods can be extended to solve our problem. The extensions of these existing methods are constrained by an inflexible assumption, Le. each method assumes that an unrecorded variable has a specific distribution. We introduce solutions that remove this constraint so as to be able to use the empirical distribution. The thesis also presents solutions making use of updated MAR data, which are available in the case of behaviour scoring.Imperial Users onl