The max-plus finite element method for solving deterministic optimal control problems: basic properties and convergence analysis

We introduce a max-plus analogue of the Petrov-Galerkin finite element method to solve finite horizon deterministic optimal control problems. The method relies on a max-plus variational formulation. We show that the error in the sup norm can be bounded from the difference between the value function and its projections on max-plus and min-plus semimodules, when the max-plus analogue of the stiffness matrix is exactly known. In general, the stiffness matrix must be approximated: this requires approximating the operation of the Lax-Oleinik semigroup on finite elements. We consider two approximations relying on the Hamiltonian. We derive a convergence result, in arbitrary dimension, showing that for a class of problems, the error estimate is of order $\delta+\Delta x(\delta)^{-1}$ or $\sqrt{\delta}+\Delta x(\delta)^{-1}$, depending on the choice of the approximation, where $\delta$ and $\Delta x$ are respectively the time and space discretization steps. We compare our method with another max-plus based discretization method previously introduced by Fleming and McEneaney. We give numerical examples in dimension 1 and 2.Comment: 31 pages, 11 figure

About A Rare Cause Of Primary Hyperparathyroidism

Introduction: Primary hyperparathyroïdism is observed in 35 to 44 subjects/ 100000 persons. The increased production of parathyroid hormones is secondary to primary glandular modifications consisting mainly in adenomas. The authors report a clear-cell hyperplasia causing primary hyperparathyroidism. Observation: We report the case of a 25-year-old man who was admitted to explore pathologic fractures of the left arm and a malignant hypercalcaemia. Complementary laboratory tests revealed primary hyperparathyroidism. A multiple endocrine neoplasia was excluded by radiologic examinations. Cervical ultra-sound examination revealed 2 parathyroid adenomas and per-operative exploration showed 3 « adenomas ». Microscopic examination of the 4 parathyroid glands specimen concluded to a clear cell hyperplasia. Conclusion: Clear cell hyperplasia is a benign cause of primary hyperparathyroidism. The diagnosis is based upon histologic findings and examination of the 4 glands

On the Effective Action of Noncommutative Yang-Mills Theory

We compute here the Yang-Mills effective action on Moyal space by integrating over the scalar fields in a noncommutative scalar field theory with harmonic term, minimally coupled to an external gauge potential. We also explain the special regularisation scheme chosen here and give some links to the Schwinger parametric representation. Finally, we discuss the results obtained: a noncommutative possibly renormalisable Yang-Mills theory.Comment: 19 pages, 6 figures. At the occasion of the "International Conference on Noncommutative Geometry and Physics", April 2007, Orsay (France). To appear in J. Phys. Conf. Se

One-loop Beta Functions for the Orientable Non-commutative Gross-Neveu Model

We compute at the one-loop order the beta-functions for a renormalisable non-commutative analog of the Gross Neveu model defined on the Moyal plane. The calculation is performed within the so called x-space formalism. We find that this non-commutative field theory exhibits asymptotic freedom for any number of colors. The beta-function for the non-commutative counterpart of the Thirring model is found to be non vanishing.Comment: 16 pages, 9 figure

Vacuum configurations for renormalizable non-commutative scalar models

In this paper we find non-trivial vacuum states for the renormalizable non-commutative $\phi^4$ model. An associated linear sigma model is then considered. We further investigate the corresponding spontaneous symmetry breaking.Comment: 17 page

First case of childhood Takayasu arteritis with renal artery aneurysms

Takayasu arteritis is a large vessel systemic granulomatous vasculitis characterized by stenosis or obliteration of large and medium sized arteries. It commonly involves elastic arteries such as the aorta and its main branches. Renal artery involvement is rare and has not been reported in a child. We report a 12-year-old boy with Takayasu arteritis who developed severe hypertension, proteinuria, microscopic hematuria and renal dysfunction. Conventional angiography demonstrated aneurysms of both renal arteries and multiple microaneurysms of the superior mesenteric artery. This case report illustrates that the children with Takayasu arteritis can develop renal involvement resulting in hematuria, proteinuria and even renal failure

On the vacuum states for noncommutative gauge theory

Candidates for renormalisable gauge theory models on Moyal spaces constructed recently have non trivial vacua. We show that these models support vacuum states that are invariant under both global rotations and symplectic isomorphisms which form a global symmetry group for the action. We compute the explicit expression in the position space for these vacuum configurations in two and four dimensions.Comment: 20 page