In this paper, we investigate the application of feasible direction method for an optimistic nonlinear bilevel programming problem. The convex lower level problem of an optimistic nonlinear bilevel programming problem is replaced by relaxed KKT conditions. The feasible direction method developed by Topkis and Veinott [22] is applied to the auxiliary problem to get a Bouligand stationary point f...

Let $M$ be an orientable hypersurface in the Euclidean space $R^{2n}$ with induced metric $g$ and $TM$ be its tangent bundle. It is known that the tangent bundle $TM$ has induced metric $overline{g}$ as submanifold of the Euclidean space $R^{4n}$ which is not a natural metric in the sense that the submersion $pi :(TM,overline{g})rightarrow (M,g)$ is not the Riemannian submersion. In this paper...

1. Write the formula for the Liouville measure Λ on the unit tangent bundle T 1 H n. Prove that Λ is invariant under the actions by Isom(H n) and the geodesic flow g t on T 1 H n. 2. Let δ be a geodesic in the hyperbolic space H n and let π : H n → δ denote the nearest point projection map. Suppose that for some x, y ∈ H n and some r > 0 we have d(x, δ), d(y, δ) ≥ r. Prove that d(π(x), π(y)) ≤ ...

Let M be a tracial von Neumann algebra and A be a weakly dense unital C-subalgebra of M . We say that a set X is a W -generating set for M if the von Neumann algebra generated by X is M and that X is a C-generating set for A if the unital C-algebra generated by X is A. For any finite W generating set X for M we show that δ0(X) ≤ sup{δ0(Y ) : Y is a finite C-generating set for A}. It follows tha...

The iridescent metallic green beetle, Chrysina gloriosa, selectively reflects left circularly polarized light, and exhibits iridescence. The exoskeleton of C. gloriosa is decorated by a polygonal texture: hexagonal cells (~10 micron) coexist with pentagons and heptagons. We find that the fraction of pentagons and heptagons computed using Voronoi analysis increase with an increase in curvature, ...

Yves Bouligand est directeur de laboratoire émérite de l’École Pratique des Hautes Études et il a dirigé l’une des équipes du Centre de Cytologie Expérimentale du CNRS à Ivry-surSeine. Ses recherches ont porté sur divers tissus et cellules comportant des phases cristallines liquides, comme dans certains chromosomes et membranes, et aussi des systèmes analogues stabilisés, de type composite, pré...

Parametric optimal control problems for semilinear parabolic equations are considered. Using recent Lipschitz stability results for solutions of such problems, it is shown that, under standard coercivity conditions, the solutions are Bouligand differentiable (in L, p < ∞) functions of the parameter. The differentials are characterized as the solutions of accessory linear-quadratic problems. A u...