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

CОВРЕМЕННОЕ СОСТОЯНИЕ ТЕРАПИИ БЫСТРЫМИ НЕЙТРОНАМИ

.В первой части статьи представлен анализ возможностей клинического применения терапии быстрыми нейтронами (ТБН), технические и статистические данные, а также биологические свойства быстрых нейтронов по сравнению с другими методами лучевой терапии. Рассматривается ситуация с клиническим применением бор-нейтрон захватной терапии (БНЗТ). Вторая часть статьи посвящена опыту применения ТБН на исследовательских реакторах FRM I и FRM II Технического университета Мюнхена в Гархинге (Германия)

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

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

Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow

In this paper, we prove existence and uniqueness of measure solutions for the Cauchy problem associated to the (vectorial) continuity equation with a non-local ow. We also give a stability result with respect to various parameters