Energy properness and Sasakian-Einstein metrics

In this paper, we show that the existence of Sasakian-Einstein metrics is closely related to the properness of corresponding energy functionals. Under the condition that admitting no nontrivial Hamiltonian holomorphic vector field, we prove that the existence of Sasakian-Einstein metric implies a Moser-Trudinger type inequality. At the end of this paper, we also obtain a Miyaoka-Yau type inequality in Sasakian geometry.Comment: 27 page

Calculus of Tangent Sets and Derivatives of Set Valued Maps under Metric Subregularity Conditions

In this paper we intend to give some calculus rules for tangent sets in the sense of Bouligand and Ursescu, as well as for corresponding derivatives of set-valued maps. Both first and second order objects are envisaged and the assumptions we impose in order to get the calculus are in terms of metric subregularity of the assembly of the initial data. This approach is different from those used in alternative recent papers in literature and allows us to avoid compactness conditions. A special attention is paid for the case of perturbation set-valued maps which appear naturally in optimization problems.Comment: 17 page

Interaction of Cutibacterium (formerly Propionibacterium) acnes with bone cells: a step toward understanding bone and joint infection development

Cutibacterium acnes (formerly Propionibacterium acnes) is recognized as a pathogen in foreign-body infections (arthroplasty or spinal instrumentation). To date, the direct impact of C. acnes on bone cells has never been explored. The clade of 11 C. acnes clinical isolates was determined by MLST. Human osteoblasts and osteoclasts were infected by live C. acnes. The whole genome sequence of six isolates of this collection was analyzed. CC36 C. acnes strains were significantly less internalized by osteoblasts and osteoclasts than CC18 and CC28 C. acnes strains (p ≤ 0.05). The CC18 C. acnes ATCC6919 isolate could survive intracellularly for at least 96 hours. C. acnes significantly decreased the resorption ability of osteoclasts with a major impact by the CC36 strain (p ≤ 0.05). Genome analysis revealed 27 genes possibly linked to these phenotypic behaviors. We showed a direct impact of C. acnes on bone cells, providing new explanations about the development of C. acnes foreign-body infections

Conjugate times and regularity of the minimum time function with differential inclusions

This paper studies the regularity of the minimum time function, $T(\cdot)$, for a control system with a general closed target, taking the state equation in the form of a differential inclusion. Our first result is a sensitivity relation which guarantees the propagation of the proximal subdifferential of $T$ along any optimal trajectory. Then, we obtain the local $C^2$ regularity of the minimum time function along optimal trajectories by using such a relation to exclude the presence of conjugate times

Computing domains of attraction for planar dynamics

In this note we investigate the problem of computing the domain of attraction of a ow on R2 for a given attractor. We consider an operator that takes two inputs, the description of the ow and a cover of the attractors, and outputs the domain of attraction for the given attractor. We show that: (i) if we consider only (structurally) stable systems, the operator is (strictly semi-)computable; (ii) if we allow all systems de ned by C1-functions, the operator is not (semi-)computable. We also address the problem of computing limit cycles on these systems

Nonlinear porous medium flow with fractional potential pressure

We study a porous medium equation, with nonlocal diffusion effects given by an inverse fractional Laplacian operator. We pose the problem in n-dimensional space for all t>0 with bounded and compactly supported initial data, and prove existence of a weak and bounded solution that propagates with finite speed, a property that is nor shared by other fractional diffusion models.Comment: 32 pages, Late

Moving constraints as stabilizing controls in classical mechanics

The paper analyzes a Lagrangian system which is controlled by directly assigning some of the coordinates as functions of time, by means of frictionless constraints. In a natural system of coordinates, the equations of motions contain terms which are linear or quadratic w.r.t.time derivatives of the control functions. After reviewing the basic equations, we explain the significance of the quadratic terms, related to geodesics orthogonal to a given foliation. We then study the problem of stabilization of the system to a given point, by means of oscillating controls. This problem is first reduced to the weak stability for a related convex-valued differential inclusion, then studied by Lyapunov functions methods. In the last sections, we illustrate the results by means of various mechanical examples.Comment: 52 pages, 4 figure

The Cauchy problems for Einstein metrics and parallel spinors

We show that in the analytic category, given a Riemannian metric $g$ on a hypersurface $M\subset \Z$ and a symmetric tensor $W$ on $M$, the metric $g$ can be locally extended to a Riemannian Einstein metric on $Z$ with second fundamental form $W$, provided that $g$ and $W$ satisfy the constraints on $M$ imposed by the contracted Codazzi equations. We use this fact to study the Cauchy problem for metrics with parallel spinors in the real analytic category and give an affirmative answer to a question raised in B\"ar, Gauduchon, Moroianu (2005). We also answer negatively the corresponding questions in the smooth category.Comment: 28 pages; final versio