Greedy bisection generates optimally adapted triangulations

We study the properties of a simple greedy algorithm for the generation of data-adapted anisotropic triangulations. Given a function f, the algorithm produces nested triangulations and corresponding piecewise polynomial approximations of f. The refinement procedure picks the triangle which maximizes the local Lp approximation error, and bisect it in a direction which is chosen so to minimize this error at the next step. We study the approximation error in the Lp norm when the algorithm is applied to C2 functions with piecewise linear approximations. We prove that as the algorithm progresses, the triangles tend to adopt an optimal aspect ratio which is dictated by the local hessian of f. For convex functions, we also prove that the adaptive triangulations satisfy a convergence bound which is known to be asymptotically optimal among all possible triangulations.Comment: 24 page

Exforge® (amlodipine/valsartan combination) in hypertension: the evidence of its therapeutic impact

peer reviewedAbstract Introduction: Hypertension is an important risk factor for cardiovascular disease and its management requires improvement. New treatment strategies are needed. Aims: This review analyses one of these strategies, which is the development of effective and safe combination therapy. Indeed, at least two antihypertensive agents are often needed to achieve blood pressure control. Exforge® (Novartis) is a new drug combination of the calcium channel blocker, amlodipine, and the angiotensin II receptor blocker, valsartan. Evidence review: The amlodipine/valsartan combination is an association of two well-known antihypertensive products with specific targets in cardiovascular protection, namely calcium channel blockade and antagonism of the renin-angiotensin-aldosterone system. This kind of association, with neutral metabolic properties and significant antihypertensive efficacy, could be a useful new antihypertensive product. Currently available data have shown that this new combination is well-tolerated and effective even in severe hypertension. Clinical value: Clinical trials are ongoing for further assessment of the efficacy, compliance, and safety of this combination and its congeners. No data exist to prove that the amlodipine/valsartan combination is better than other antihypertensive strategies for cardiovascular or renal protection, but some trials with other combination therapies show such potential advantage

An algorithm for linear constraint solving: its incorporation in a prolog meta-interpreter for CLP

AbstractThe paper presents an incremental and efficient algorithm for testing the satisfiability of systems of linear equalities, inequalities (strict or unrestricted), and disequalities. In addition, it describes the incorporation of that algorithm into a metalevel interpreter capable of processing both tree constraints and the mentioned linear constraints in the domain of rationals. Important characteristics of the described algorithm are (1) detection of fixed variables within the context of Gaussian elimination, including the simplex method. (2) efficient dereferencing by considering subclasses of solved forms, and (3) efficient testing of inconsistencies between equality and disequality subclasses. The metalevel interpreter is written in Prolog. Examples of its usage are provided. Finally, the paper outlines how the approach may be generalized to consider the efficient and incremental testing of constraint satisfiability in various domains

Adaptive multiresolution analysis based on anisotropic triangulations

A simple greedy refinement procedure for the generation of data-adapted triangulations is proposed and studied. Given a function of two variables, the algorithm produces a hierarchy of triangulations and piecewise polynomial approximations on these triangulations. The refinement procedure consists in bisecting a triangle T in a direction which is chosen so as to minimize the local approximation error in some prescribed norm between the approximated function and its piecewise polynomial approximation after T is bisected. The hierarchical structure allows us to derive various approximation tools such as multiresolution analysis, wavelet bases, adaptive triangulations based either on greedy or optimal CART trees, as well as a simple encoding of the corresponding triangulations. We give a general proof of convergence in the Lp norm of all these approximations. Numerical tests performed in the case of piecewise linear approximation of functions with analytic expressions or of numerical images illustrate the fact that the refinement procedure generates triangles with an optimal aspect ratio (which is dictated by the local Hessian of of the approximated function in case of C2 functions).Comment: 19 pages, 7 figure

Quantum electrodynamics of relativistic bound states with cutoffs

We consider an Hamiltonian with ultraviolet and infrared cutoffs, describing the interaction of relativistic electrons and positrons in the Coulomb potential with photons in Coulomb gauge. The interaction includes both interaction of the current density with transversal photons and the Coulomb interaction of charge density with itself. We prove that the Hamiltonian is self-adjoint and has a ground state for sufficiently small coupling constants.Comment: To appear in "Journal of Hyperbolic Differential Equation

Cardiac rhabdomyomas in tuberous sclerosis patients: A case report and review of the literature

SummaryRhabdomyomas are the most common benign cardiac tumours. They are often associated with tuberous sclerosis and can be diagnosed antenatally and postnatally by echocardiography. Rhabdomyomas tend to regress spontaneously and are not usually operated upon, unless they become obstructive or cause severe arrhythmias. We describe the case of a child with tuberous sclerosis who was admitted for the resection of a subependymal giant cell astrocytoma, in whom cardiac rhabdomyomas in the right ventricular outflow tract were diagnosed. These two kinds of tumours are well known in the setting of tuberous sclerosis

Geodesic Models with Convexity Shape Prior

The minimal geodesic models based on the Eikonal equations are capable of finding suitable solutions in various image segmentation scenarios. Existing geodesic-based segmentation approaches usually exploit image features in conjunction with geometric regularization terms, such as Euclidean curve length or curvature-penalized length, for computing geodesic curves. In this paper, we take into account a more complicated problem: finding curvature-penalized geodesic paths with a convexity shape prior. We establish new geodesic models relying on the strategy of orientation-lifting, by which a planar curve can be mapped to an high-dimensional orientation-dependent space. The convexity shape prior serves as a constraint for the construction of local geodesic metrics encoding a particular curvature constraint. Then the geodesic distances and the corresponding closed geodesic paths in the orientation-lifted space can be efficiently computed through state-of-the-art Hamiltonian fast marching method. In addition, we apply the proposed geodesic models to the active contours, leading to efficient interactive image segmentation algorithms that preserve the advantages of convexity shape prior and curvature penalization.Comment: This paper has been accepted by TPAM

Approximation of the second fundamental form of a hypersurface of a Riemannian manifold

We give a general Riemannian framework to the study of approximation of curvature measures, using the theory of the normal cycle. Moreover, we introduce a differential form which allows to define a new type of curvature measure encoding the second fundamental form of a hypersurface, and to extend this notion to geometric compact subsets of a Riemannian manifold . Finally, if a geometric compact subset is close to a smooth hypersurface of a Riemannian manifold, we compare their second fundamental form (in the previous sense), and give a bound of their difference in terms of geometric invariants and the mass of the involved normal cycles

Approximation of Normal Cycles

This report deals with approximations of geometric data defined on a hypersurf- ace of the Euclidean space E^n. Using geometric measure theory, we evaluate an upper bound on the flat norm of the difference of the normal cycle of a compact subset of E^n whose boundary is a smooth (closed oriented embedded) hypersurface, and the normal cycle of a compact geometric subset of E^n "close to it". We deduce bounds between the difference of the curvature measures of the smooth hypersurface and the curvature measures of the geometric compact subset