Multi-Channel Inverse Scattering Problem on the Line: Thresholds and Bound States

We consider the multi-channel inverse scattering problem in one-dimension in the presence of thresholds and bound states for a potential of finite support. Utilizing the Levin representation, we derive the general Marchenko integral equation for N-coupled channels and show that, unlike to the case of the radial inverse scattering problem, the information on the bound state energies and asymptotic normalization constants can be inferred from the reflection coefficient matrix alone. Thus, given this matrix, the Marchenko inverse scattering procedure can provide us with a unique multi-channel potential. The relationship to supersymmetric partner potentials as well as possible applications are discussed. The integral equation has been implemented numerically and applied to several schematic examples showing the characteristic features of multi-channel systems. A possible application of the formalism to technological problems is briefly discussed.Comment: 19 pages, 5 figure

Treatment planning and dosimetric comparison study on two different volumetric modulated arc therapy delivery techniques

AimTo compare and evaluate the performance of two different volumetric modulated arc therapy delivery techniques.BackgroundVolumetric modulated arc therapy is a novel technique that has recently been made available for clinical use. Planning and dosimetric comparison study was done for Elekta VMAT and Varian RapidArc for different treatment sites.Materials and methodsTen patients were selected for the planning comparison study. This includes 2 head and neck, 2 oesophagus, 1 bladder, 3 cervix and 2 rectum cases. Total dose of 50[[ce:hsp sp="0.25"/]]Gy was given for all the plans. All plans were done for RapidArc using Eclipse and for Elekta VMAT with Monaco treatment planning system. All plans were generated with 6[[ce:hsp sp="0.25"/]]MV X-rays for both RapidArc and Elekta VMAT. Plans were evaluated based on the ability to meet the dose volume histogram, dose homogeneity index, radiation conformity index, estimated radiation delivery time, integral dose and monitor units needed to deliver the prescribed dose.ResultsRapidArc plans achieved the best conformity (CI95%[[ce:hsp sp="0.25"/]]=[[ce:hsp sp="0.25"/]]1.08[[ce:hsp sp="0.25"/]]±[[ce:hsp sp="0.25"/]]0.07) while Elekta VMAT plans were slightly inferior (CI95%[[ce:hsp sp="0.25"/]]=[[ce:hsp sp="0.25"/]]1.10[[ce:hsp sp="0.25"/]]±[[ce:hsp sp="0.25"/]]0.05). The in-homogeneity in the PTV was highest with Elekta VMAT with HI equal to 0.12[[ce:hsp sp="0.25"/]]±[[ce:hsp sp="0.25"/]]0.02[[ce:hsp sp="0.25"/]]Gy when compared to RapidArc with 0.08[[ce:hsp sp="0.25"/]]±[[ce:hsp sp="0.25"/]]0.03. Significant changes were observed between the RapidArc and Elekta VMAT plans in terms of the healthy tissue mean dose and integral dose. Elekta VMAT plans show a reduction in the healthy tissue mean dose (6.92[[ce:hsp sp="0.25"/]]±[[ce:hsp sp="0.25"/]]2.90)[[ce:hsp sp="0.25"/]]Gy when compared to RapidArc (7.83[[ce:hsp sp="0.25"/]]±[[ce:hsp sp="0.25"/]]3.31)[[ce:hsp sp="0.25"/]]Gy. The integral dose is found to be inferior with Elekta VMAT (11.50[[ce:hsp sp="0.25"/]]±[[ce:hsp sp="0.25"/]]6.49)[[ce:hsp sp="0.25"/]]×[[ce:hsp sp="0.25"/]]104[[ce:hsp sp="0.25"/]]Gy[[ce:hsp sp="0.25"/]]cm3 when compared to RapidArc (13.11[[ce:hsp sp="0.25"/]]±[[ce:hsp sp="0.25"/]]7.52)[[ce:hsp sp="0.25"/]]×[[ce:hsp sp="0.25"/]]104[[ce:hsp sp="0.25"/]]Gy[[ce:hsp sp="0.25"/]]cm3. Both Varian RapidArc and Elekta VMAT respected the planning objective for all organs at risk. Gamma analysis result for the pre-treatment quality assurance shows good agreement between the planned and delivered fluence for 3[[ce:hsp sp="0.25"/]]mm DTA, 3% DD for all the evaluated points inside the PTV, for both VMAT and RapidArc techniques.ConclusionThe study concludes that a variable gantry speed with variable dose rate is important for efficient arc therapy delivery. RapidArc presents a slight improvement in the OAR sparing with better target coverage when compared to Elekta VMAT. Trivial differences were noted in all the plans for organ at risk but the two techniques provided satisfactory conformal avoidance and conformation

Discretisation of gradient elasticity problems using C1 finite elements

For the numerical solution of gradient elasticity, the appearance of strain gradients in the weak form of the equilibrium equation leads to the need for C 1-continuous discretization methods. In the present work, the performances of a variety of C 1-continuous elements as well as the C 1 Natural Element Method are investigated for the application to nonlinear gradient elasticity. In terms of subparametric triangular elements the Argyris, Hsieh–Clough–Tocher and Powell–Sabin split elements are utilized. As an isoparametric quadrilateral element, the Bogner–Fox–Schmidt element is used. All these methods are applied to two different numerical examples and the convergence behavior with respect to the L 2, H 1 and H 2 error norms is examined. <br/

Review of Discontinuous Galerkin Finite Element Methods for Partial Differential Equations on Complicated Domains

The numerical approximation of partial differential equations (PDEs) posed on complicated geometries, which include a large number of small geometrical features or microstructures, represents a challenging computational problem. Indeed, the use of standard mesh generators, employing simplices or tensor product elements, for example, naturally leads to very fine finite element meshes, and hence the computational effort required to numerically approximate the underlying PDE problem may be prohibitively expensive. As an alternative approach, in this article we present a review of composite/agglomerated discontinuous Galerkin finite element methods (DGFEMs) which employ general polytopic elements. Here, the elements are typically constructed as the union of standard element shapes; in this way, the minimal dimension of the underlying composite finite element space is independent of the number of geometrical features. In particular, we provide an overview of hp-version inverse estimates and approximation results for general polytopic elements, which are sharp with respect to element facet degeneration. On the basis of these results, a priori error bounds for the hp-DGFEM approximation of both second-order elliptic and first-order hyperbolic PDEs will be derived. Finally, we present numerical experiments which highlight the practical application of DGFEMs on meshes consisting of general polytopic elements