On the Regularity of Optimal Transportation Potentials on Round Spheres

In this paper the regularity of optimal transportation potentials defined on round spheres is investigated. Specifically, this research generalises the calculations done by Loeper, where he showed that the strong (A3) condition of Trudinger and Wang is satisfied on the round sphere, when the cost-function is the geodesic distance squared. In order to generalise Loeper's calculation to a broader class of cost-functions, the (A3) condition is reformulated via a stereographic projection that maps charts of the sphere into Euclidean space. This reformulation subsequently allows one to verify the (A3) condition for any case where the cost-fuction of the associated optimal transportation problem can be expressed as a function of the geodesic distance between points on a round sphere. With this, several examples of such cost-functions are then analysed to see whether or not they satisfy this (A3) condition.Comment: 24 pages, 4 figure

A glimpse into the differential topology and geometry of optimal transport

This note exposes the differential topology and geometry underlying some of the basic phenomena of optimal transportation. It surveys basic questions concerning Monge maps and Kantorovich measures: existence and regularity of the former, uniqueness of the latter, and estimates for the dimension of its support, as well as the associated linear programming duality. It shows the answers to these questions concern the differential geometry and topology of the chosen transportation cost. It also establishes new connections --- some heuristic and others rigorous --- based on the properties of the cross-difference of this cost, and its Taylor expansion at the diagonal.Comment: 27 page

Optimal Transport, Convection, Magnetic Relaxation and Generalized Boussinesq equations

We establish a connection between Optimal Transport Theory and classical Convection Theory for geophysical flows. Our starting point is the model designed few years ago by Angenent, Haker and Tannenbaum to solve some Optimal Transport problems. This model can be seen as a generalization of the Darcy-Boussinesq equations, which is a degenerate version of the Navier-Stokes-Boussinesq (NSB) equations. In a unified framework, we relate different variants of the NSB equations (in particular what we call the generalized Hydrostatic-Boussinesq equations) to various models involving Optimal Transport (and the related Monge-Ampere equation. This includes the 2D semi-geostrophic equations and some fully non-linear versions of the so-called high-field limit of the Vlasov-Poisson system and of the Keller-Segel for Chemotaxis. Finally, we show how a ``stringy'' generalization of the AHT model can be related to the magnetic relaxation model studied by Arnold and Moffatt to obtain stationary solutions of the Euler equations with prescribed topology

Analysis of Oscillations and Defect Measures for the Quasineutral Limit in Plasma Physics

We perform a rigorous analysis of the quasineutral limit for a hydrodynamical model of a viscous plasma represented by the Navier Stokes Poisson system in $3-D$. We show that as $\lambda\to 0$ the velocity field $u^{\lambda}$ strongly converges towards an incompressible velocity vector field $u$ and the density fluctuation $\rho^{\lambda}-1$ weakly converges to zero. In general the limit velocity field cannot be expected to satisfy the incompressible Navier Stokes equation, indeed the presence of high frequency oscillations strongly affects the quadratic nonlinearities and we have to take care of self interacting wave packets. We shall provide a detailed mathematical description of the convergence process by using microlocal defect measures and by developing an explicit correctors analysis. Moreover we will be able to identify an explicit pseudo parabolic pde satisfied by the leading correctors terms. Our results include all the previous results in literature, in particular we show that the formal limit holds rigorously in the case of well prepared data.Comment: Submitted pape

Efficacy and complications of neurosurgical treatment of acromegaly

The aim of the study was to evaluate the frequency of occurrence of pituitary failure following neurosurgery and the efficacy of transsphenoidal tumour resection in acromegalic patients. We retrospectively evaluated 85 patients (60 female and 25 male), of mean age 43.9 ± 13.2 years, treated by transsphenoidal neurosurgery. Macroadenoma and microadenoma of pituitary were found in 66 (77.6%) and 19 (22.4%) of these patients, respectively. Criteria of cure following neurosurgery were: basal GH < 2.5 μg/l, GH at 120 min in OGTT < 1.0 μg/l and serum concentration of IGF-1 within normal ranges for age and sex. After surgery 32 patients (37.6%) were cured and 53 patients (62.4%) required somatostatin analogue treatment. In patients cured by surgery, lower levels of basal GH (P < 0.05), IGF-1 (P < 0.001), GH at 120 min in OGTT and smaller size of pituitary tumour (P < 0.05) were found at diagnosis, as compared to patients in whom surgery was unsuccessful. Significant correlation between basal serum level of GH at diagnosis and size of pituitary tumour was found (P < 0.001). Invasive tumours were found in 45 of 53 (84.9%) patients not cured and in only 8 of 32 (25.0%) patients cured (P < 0.001). Impaired function of pituitary anterior lobe after surgery was observed in 30% and 4% of patients with macro- and microadenoma, respectively (P < 0.05). The efficacy of neurosurgery is affected by concentration of basal serum GH and IGF-1, GH at 120 min in OGTT, tumour size and invasiveness. Hypopituitarism after surgery is more frequent in patients with macroadenoma. Pituitary insufficiency, as a consequence of surgery, was found in 21% of patients with normal pituitary function prior to operation

The Structure of Hyperalkaline Aqueous Solutions Containing High Concentrations of Gallium - a Solution X-ray Diffraction and Computational Study

Highly concentrated alkaline NaOH/Ga(OH)3 solutions with 1.18 M Ga(III)T 2.32 M and 2.4 M NaOHT 4.9 M (where the subscript T denotes total or analytical concentrations) have been prepared and investigated by solution X-ray diffraction and also by ab initio quantum chemical calculations. The data obtained are consistent with the presence of only one predominant Ga(III)-bearing species in these solutions, that is the tetrahedral hydroxo complex Ga(OH)4–. This finding is in stark contrast to that found for Al(III)-containing solutions of similar concentrations, in which, besides the monomeric complex, an oxo-bridged dimer was also found to form. From the solution X-ray diffraction measurements, the formation of the dimeric (OH)3Ga–O–Ga(OH)32– could not unambiguously be shown, however, from the comparison of experimental IR, Raman and 71Ga NMR spectra with calculated ones, its formation can be safely excluded. Moreover, higher mononuclear stepwise hydroxo complexes, like Ga(OH)63–, that have been claimed to exist by others in the literature, was not possible to experimentally detect in these solutions with any of the spectroscopic techniques used

First steps towards a fast-neutron therapy planning program

<p>Abstract</p> <p>Background</p> <p>The Monte Carlo code GEANT4 was used to implement first steps towards a treatment planning program for fast-neutron therapy at the FRM II research reactor in Garching, Germany. Depth dose curves were calculated inside a water phantom using measured primary neutron and simulated primary photon spectra and compared with depth dose curves measured earlier. The calculations were performed with GEANT4 in two different ways, simulating a simple box geometry and splitting this box into millions of small voxels (this was done to validate the voxelisation procedure that was also used to voxelise the human body).</p> <p>Results</p> <p>In both cases, the dose distributions were very similar to those measured in the water phantom, up to a depth of 30 cm. In order to model the situation of patients treated at the FRM II MEDAPP therapy beamline for salivary gland tumors, a human voxel phantom was implemented in GEANT4 and irradiated with the implemented MEDAPP neutron and photon spectra. The 3D dose distribution calculated inside the head of the phantom was similar to the depth dose curves in the water phantom, with some differences that are explained by differences in elementary composition. The lateral dose distribution was studied at various depths. The calculated cumulative dose volume histograms for the voxel phantom show the exposure of organs at risk surrounding the tumor.</p> <p>Conclusions</p> <p>In order to minimize the dose to healthy tissue, a conformal treatment is necessary. This can only be accomplished with the help of an advanced treatment planning system like the one developed here. Although all calculations were done for absorbed dose only, any biological dose weighting can be implemented easily, to take into account the increased radiobiological effectiveness of neutrons compared to photons.</p

A user's guide to optimal transport

This text is an expanded version of the lectures given by the first author in the 2009 CIME summer school of Cetraro. It provides a quick and reasonably account of the classical theory of optimal mass transportation and of its more recent developments, including the metric theory of gradient flows, geometric and functional inequalities related to optimal transportation, the first and second order differential calculus in the Wasserstein space and the synthetic theory of metric measure spaces with Ricci curvature bounded from below

Solarelement mit gesteigerter Effizienz und Verfahren zur Effizienzsteigerung

DE 102007045546 B3 UPAB: 20090130 NOVELTY - The solar element has vertical converters (3a - 3c) absorbing an electromagnetic radiation with frequencies and emitting the radiation with another frequency. Selective reflective structures (9a - 9c) reflect a part of the impinging electromagnetic radiation and transmitting the part of the impinging radiation. A luminance unit (2) is arranged adjacent to the reflective structures that are arranged adjacent to the vertical converters. The vertical converters are arranged adjacent to a solar cell (1). DETAILED DESCRIPTION - An INDEPENDENT CLAIM is also included for a method for improving efficiency of a solar cell. USE - Solar element. ADVANTAGE - The solar element minimizes loss of highly-converted radiation by absorption