Activity

The concept of field of values (FoV), also known as the numerical range, is applied to the 2 × 2 Jones matrices used in polarization optics. We discover the relevant interplay between the geometric properties of the FoV, the algebraic properties of the Jones matrices and the representation of polarization states on the Poincaré sphere. The properties of the FoV reveal hidden symmetries in the relationships between the eigenvectors and eigenvalues of the Jones matrices. We determine the main mathematical properties of the FoV, discuss the special cases that are relevant to polarization optics, and describe its application to calculate the Pancharatnam-Berry phase introduced by an optical system to the input state.A solution to the problem of water-wave scattering by a semi-infinite submerged thin elastic plate, which is either porous or non-porous, is presented using the Wiener-Hopf technique. The derivation of the Wiener-Hopf equation is rather different from that which is used traditionally in water-waves problems, and it leads to the required equations directly. It is also shown how the solution can be computed straightforwardly using Cauchy-type integrals, which avoids the need to find the roots of the highly non-trivial dispersion equations. BPTES inhibitor We illustrate the method with some numerical computations, focusing on the evolution of an incident wave pulse which illustrates the existence of two transmitted waves in the submerged plate system. The effect of the porosity is studied, and it is shown to influence the shorter-wavelength pulse much more strongly than the longer-wavelength pulse.Tides are a major component of the interaction between the marine and terrestrial environments, and thus play an important part in shaping the environmental context for the evolution of shallow marine and coastal organisms. Here, we use a dedicated tidal model and palaeogeographic reconstructions from the Late Silurian to early Late Devonian (420 Ma, 400 Ma and 380 Ma, Ma = millions of years ago) to explore the potential significance of tides for the evolution of osteichthyans (bony fish) and tetrapods (land vertebrates). The earliest members of the osteichthyan crown-group date to the Late Silurian, approximately 425 Ma, while the earliest evidence for tetrapods is provided by trackways from the Middle Devonian, dated to approximately 393 Ma, and the oldest tetrapod body fossils are Late Devonian, approximately 373 Ma. Large tidal ranges could have fostered both the evolution of air-breathing organs in osteichthyans to facilitate breathing in oxygen-depleted tidal pools, and the development of weight-bearing tetrapod limbs to aid navigation within the intertidal zones. We find that tidal ranges over 4 m were present around areas of evolutionary significance for the origin of osteichthyans and the fish-tetrapod transition, highlighting the possible importance of tidal dynamics as a driver for these evolutionary processes.We present a new methodology to incorporate the Casimir forces within the molecular dynamics (MD) framework. At atomistic scales, the potential energy between two particles arising due to the Casimir effect can be represented as U(r ij ) = C/r7. Incorporating the Casimir effect in MD simulations requires the knowledge of C, a problem hitherto unsolved. We overcome this by equating the total potential energy contributions due to each atomistic pair with the potential energy of continuum scale interacting bodies having similar geometries. After having identified the functional form of C, standard MD simulations are augmented with the potential energy contribution due to pairwise Casimir interactions. The developed framework is used to study effects of the Casimir force on the pull-in instability of rectangular and hollow cylindrical shaped deformable electrodes separated by a small distance from a fixed substrate electrode. Our MD results for pull-instability qualitatively agree with the previously reported analytical results but are quantitatively different. The effect of using longer-ranged Casimir forces in a constant temperature environment on the pull-in behaviour has also been studied.The conventional non-Hermitian but P T -symmetric three-parametric Bose-Hubbard Hamiltonian H(γ, v, c) represents a quantum system of N bosons, unitary only for parameters γ, v and c in a domain D . Its boundary ∂ D contains an exceptional point of order K (EPK; K = N + 1) at c = 0 and γ = v, but even at the smallest non-vanishing parameter c ≠ ~0 the spectrum of H(v, v, c) ceases to be real, i.e. the system ceases to be observable. In this paper, the question is inverted all of the stable, unitary and observable Bose-Hubbard quantum systems are sought which would lie close to the phenomenologically most interesting EPK-related dynamical regime. Two different families of such systems are found. Both of them are characterized by the perturbed Hamiltonians H ( λ ) = H ( v , v , 0 ) + λ   V for which the unitarity and stability of the system is guaranteed. In the first family the number N of bosons is assumed conserved while in the second family such an assumption is relaxed. Attention is paid mainly to an anisotropy of the physical Hilbert space near the EPK extreme. We show that it is reflected by a specific, operationally realizable structure of perturbations λ   V which can be considered small.Floquet analysis of modulated magnetoconvection in Rayleigh-Bénard geometry is performed. A sinusoidally varying temperature is imposed on the lower plate. As Rayleigh number Ra is increased above a critical value Rao, the oscillatory magnetoconvection begins. The flow at the onset of magnetoconvection may oscillate either subhar- monically or harmonically with the external modulation. The critical Rayleigh number Rao varies non-monotonically with the modulation frequency ω for appreciable value of the modulation amplitude a. The temperature modulation may either postpone or prepone the appearance of magnetoconvection. The magnetoconvective flow always oscillates harmonically at larger values of ω. The threshold Rao and the corresponding wavenumber ko approach to their values for the stationary magnetoconvection in the absence of modulation (a = 0), as ω → ∞. Two different zones of harmonic instability merge to form a single instability zone with two local minima for higher values of Chandrasekhar’s number Q, which is qualitatively new.