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An integral equation method for a boundary value problem arising in unsteady water wave problems
In this paper we consider the 2D Dirichlet boundary value problem for Laplace’s equation in a non-locally perturbed half-plane, with data in the space of bounded and continuous functions. We show uniqueness of solution, using standard Phragmen-Lindelof arguments. The main result
is to propose a boundary integral equation formulation, to prove equivalence with the boundary value problem, and to show that the integral equation is well posed by applying a recent partial generalisation of the Fredholm alternative in Arens et al [J. Int. Equ. Appl. 15 (2003) pp. 1-35]. This then leads to an existence proof for the boundary value problem.
Keywords. Boundary integral equation method, Water waves, Laplace’
Finite-temperature critical point of a glass transition
We generalize the simplest kinetically constrained model of a glass-forming
liquid by softening kinetic constraints, allowing them to be violated with a
small finite rate. We demonstrate that this model supports a first-order
dynamical (space-time) phase transition, similar to those observed with hard
constraints. In addition, we find that the first-order phase boundary in this
softened model ends in a finite-temperature dynamical critical point, which we
expect to be present in natural systems. We discuss links between this critical
point and quantum phase transitions, showing that dynamical phase transitions
in dimensions map to quantum transitions in the same dimension, and hence
to classical thermodynamic phase transitions in dimensions. We make these
links explicit through exact mappings between master operators, transfer
matrices, and Hamiltonians for quantum spin chains.Comment: 10 pages, 5 figure
Potential model calculations and predictions for heavy quarkonium
We investigate the spectroscopy and decays of the charmonium and upsilon
systems in a potential model consisting of a relativistic kinetic energy term,
a linear confining term including its scalar and vector relativistic
corrections and the complete perturbative one-loop quantum chromodynamic short
distance potential. The masses and wave functions of the various states are
obtained using a variational technique, which allows us to compare the results
for both perturbative and nonperturbative treatments of the potential. As well
as comparing the mass spectra, radiative widths and leptonic widths with the
available data, we include a discussion of the errors on the parameters
contained in the potential, the effect of mixing on the leptonic widths, the
Lorentz nature of the confining potential and the possible
interpretation of recently discovered charmonium-like states.Comment: Physical Review published versio
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A high-wavenumber boundary-element method for an acoustic scattering problem
In this paper we show stability and convergence for a novel Galerkin boundary element method approach to the impedance boundary value problem for the Helmholtz equation in a half-plane with piecewise constant boundary data. This problem models, for example, outdoor sound propagation over inhomogeneous flat terrain. To achieve a good approximation with a relatively low number of degrees of freedom we employ a graded mesh with smaller elements adjacent to discontinuities in impedance, and a special set of basis functions for the Galerkin method so that, on each element, the approximation space consists of polynomials (of degree ) multiplied by traces of plane waves on the boundary. In the case where the impedance is constant outside an interval , which only requires the discretization of , we show theoretically and experimentally that the error in computing the acoustic field on is , where is the number of degrees of freedom and is the wavenumber. This indicates that the proposed method is especially commendable for large intervals or a high wavenumber. In a final section we sketch how the same methodology extends to more general scattering problems
Entropy and Temperature of a Static Granular Assembly
Granular matter is comprised of a large number of particles whose collective
behavior determines macroscopic properties such as flow and mechanical
strength. A comprehensive theory of the properties of granular matter,
therefore, requires a statistical framework. In molecular matter, equilibrium
statistical mechanics, which is founded on the principle of conservation of
energy, provides this framework. Grains, however, are small but macroscopic
objects whose interactions are dissipative since energy can be lost through
excitations of the internal degrees of freedom. In this work, we construct a
statistical framework for static, mechanically stable packings of grains, which
parallels that of equilibrium statistical mechanics but with conservation of
energy replaced by the conservation of a function related to the mechanical
stress tensor. Our analysis demonstrates the existence of a state function that
has all the attributes of entropy. In particular, maximizing this state
function leads to a well-defined granular temperature for these systems.
Predictions of the ensemble are verified against simulated packings of
frictionless, deformable disks. Our demonstration that a statistical ensemble
can be constructed through the identification of conserved quantities other
than energy is a new approach that is expected to open up avenues for
statistical descriptions of other non-equilibrium systems.Comment: 5 pages, 4 figure
Back injuries in young fast bowlers - a radiological investigation of the healing of spondylolysis and pedicle sclerosis
Objective. To demonstrate the efficacy of various radiological diagnostic modalities in assessing lower back pain in young fast bowlers.Methods. Ten cricketers who presented to either a physiotherapist or a doctor with suspected spondylolysis underwent an X-ray, a single photon emission computed tomography (SPECT) bone scan and a computed tomography (CT) scan to assess the severity of the injury. Three and 12 months after the initial CT scan, second and third CT scans were performed in order to assess whether healing had taken place. After the initial radiological investigation the subjects diagnosed with spondylolysis or pedicle sclerosis underwent prescribed intervention and rehabilitation which included physiotherapy modalities, postural correction, and specific individually graded flexibility, stabilisation, strengthening and cardiovascular programmes.Results. Radiographs were normal in 8 subjects, while 2 had evidence of sclerosis. The isotope scan showed increased uptake in all of the subjects. The CT scans showed no fracture (N = 3), partial fractures (N = 3), complete fractures (N = 2) and old fractures bilaterally (N = 2). When the follow-up CT scan was carried out at 3 months, 1 of the subjects had developed a partial fracture of the left pars interarticularis on the inferior border, which showed complete union when CT scanned at 12 months. At 3 months the partial and complete fractures showed progressive healing in 2 subjects, with complete healing in all the other cases. Complete healing was achieved in all subjects at 12 months, with the exception of 1 subject who showed near-complete union, with a small area of fibrous union on the inferior border and 2 old bilateral fractures that remained un-united.Results. From the results it is evident that when a young fast bowler presents with backache after bowling, it would be appropriate to do an X-ray, a bone scan and a CT scan to make the diagnosis. Discontinuing the fast bowling and following an active rehabilitation programme should result in spontaneous resolution and healing of the fracture. If it is not detected early a fibrous or non-union fracture could result
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