A Novel Method of Solution for the Fluid Loaded Plate

We study the Cauchy problem associated with the equations governing a fluid loaded plate formulated on either the line or the half-line. We show that in both cases the problem can be solved by employing the unified approach to boundary value problems introduced by on of the authors in the late 1990s. The problem on the full line was analysed by Crighton et. al. using a combination of Laplace and Fourier transforms. The new approach avoids the technical difficulty of the a priori assumption that the amplitude of the plate is in $L^1_{dt}(R^+)$ and furthermore yields a simpler solution representation which immediately implies the problem is well-posed. For the problem on the half-line, a similar analysis yields a solution representation, but this formula involves two unknown functions. The main difficulty with the half-line problem is the characterisation of these two functions. By employing the so-called global relation, we show that the two functions can be obtained via the solution of a complex valued integral equation of the convolution type. This equation can be solved in closed form using the Laplace transform. By prescribing the initial data $\eta_0$ to be in $H^3(R^+)$, we show that the solution depends continuously on the initial data, and hence, the problem is well-posed.Comment: 17 pages, 3 figures. Minor adjustments made to the introductio

The Fundamental k-Form and Global Relations

In [Proc. Roy. Soc. London Ser. A 453 (1997), no. 1962, 1411-1443] A.S. Fokas introduced a novel method for solving a large class of boundary value problems associated with evolution equations. This approach relies on the construction of a so-called global relation: an integral expression that couples initial and boundary data. The global relation can be found by constructing a differential form dependent on some spectral parameter, that is closed on the condition that a given partial differential equation is satisfied. Such a differential form is said to be fundamental [Quart. J. Mech. Appl. Math. 55 (2002), 457-479]. We give an algorithmic approach in constructing a fundamental k-form associated with a given boundary value problem, and address issues of uniqueness. Also, we extend a result of Fokas and Zyskin to give an integral representation to the solution of a class of boundary value problems, in an arbitrary number of dimensions. We present an extended example using these results in which we construct a global relation for the linearised Navier-Stokes equations.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

SYSTEMS-2: a randomised phase II study of radiotherapy dose escalation for pain control in malignant pleural mesothelioma

SYSTEMS-2 is a randomised study of radiotherapy dose escalation for pain control in 112 patients with malignant pleural mesothelioma (MPM). Standard palliative (20Gy/5#) or dose escalated treatment (36Gy/6#) will be delivered using advanced radiotherapy techniques and pain responses will be compared at week 5. Data will guide optimal palliative radiotherapy in MPM

Metabolic Differentiation of Early Lyme Disease from Southern Tick-associated Rash Illness (STARI)

Lyme disease, the most commonly reported vector-borne disease in the United States, results from infection with Borrelia burgdorferi. Early clinical diagnosis of this disease is largely based on the presence of an erythematous skin lesion for individuals in high-risk regions. This, however, can be confused with other illnesses including southern tick-associated rash illness (STARI), an illness that lacks a defined etiological agent or laboratory diagnostic test, and is coprevalent with Lyme disease in portions of the eastern United States. By applying an unbiased metabolomics approach with sera retrospectively obtained from well-characterized patients, we defined biochemical and diagnostic differences between early Lyme disease and STARI. Specifically, a metabolic biosignature consisting of 261 molecular features (MFs) revealed that altered N-acyl ethanolamine and primary fatty acid amide metabolism discriminated early Lyme disease from STARI. Development of classification models with the 261-MF biosignature and testing against validation samples differentiated early Lyme disease from STARI with an accuracy of 85 to 98%. These findings revealed metabolic dissimilarity between early Lyme disease and STARI, and provide a powerful and new approach to inform patient management by objectively distinguishing early Lyme disease from an illness with nearly identical symptoms