Radial fractional Laplace operators and Hessian inequalities

In this paper we deduce a formula for the fractional Laplace operator $(-\Delta)^{s}$ on radially symmetric functions useful for some applications. We give a criterion of subharmonicity associated with $(-\Delta)^{s}$, and apply it to a problem related to the Hessian inequality of Sobolev type: $$\int_{\mathbb{R}^n}|(-\Delta)^{\frac{k}{k+1}} u|^{k+1} dx \le C \int_{\mathbb{R}^n} - u \, F_k[u] \, dx,$$ where $F_k$ is the $k$-Hessian operator on $\mathbb{R}^n$, $1\le k < \frac{n}{2}$, under some restrictions on a $k$-convex function $u$. In particular, we show that the class of $u$ for which the above inequality was established in \cite{FFV} contains the extremal functions for the Hessian Sobolev inequality of X.-J. Wang \cite{W1}. This is proved using logarithmic convexity of the Gaussian ratio of hypergeometric functions which might be of independent interest

Therapeutic and educational objectives in robot assisted play for children with autism

“This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder." “Copyright IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.” DOI: 10.1109/ROMAN.2009.5326251This article is a methodological paper that describes the therapeutic and educational objectives that were identified during the design process of a robot aimed at robot assisted play. The work described in this paper is part of the IROMEC project (Interactive Robotic Social Mediators as Companions) that recognizes the important role of play in child development and targets children who are prevented from or inhibited in playing. The project investigates the role of an interactive, autonomous robotic toy in therapy and education for children with special needs. This paper specifically addresses the therapeutic and educational objectives related to children with autism. In recent years, robots have already been used to teach basic social interaction skills to children with autism. The added value of the IROMEC robot is that play scenarios have been developed taking children's specific strengths and needs into consideration and covering a wide range of objectives in children's development areas (sensory, communicational and interaction, motor, cognitive and social and emotional). The paper describes children's developmental areas and illustrates how different experiences and interactions with the IROMEC robot are designed to target objectives in these areas

On the asymmetric zero-range in the rarefaction fan

We consider the one-dimensional asymmetric zero-range process starting from a step decreasing profile. In the hydrodynamic limit this initial condition leads to the rarefaction fan of the associated hydrodynamic equation. Under this initial condition and for totally asymmetric jumps, we show that the weighted sum of joint probabilities for second class particles sharing the same site is convergent and we compute its limit. For partially asymmetric jumps we derive the Law of Large Numbers for the position of a second class particle under the initial configuration in which all the positive sites are empty, all the negative sites are occupied with infinitely many first class particles and with a single second class particle at the origin. Moreover, we prove that among the infinite characteristics emanating from the position of the second class particle, this particle chooses randomly one of them. The randomness is given in terms of the weak solution of the hydrodynamic equation through some sort of renormalization function. By coupling the zero-range with the exclusion process we derive some limiting laws for more general initial conditions.Comment: 22 pages, to appear in Journal of Statistical Physic

Modelling the Kinked Jet of the Crab Nebula

We investigate the dynamical propagation of the South-East jet from the Crab pulsar interacting with supernova ejecta by means of three-dimensional relativistic MHD numerical simulations with the PLUTO code. The initial jet structure is set up from the inner regions of the Crab Nebula. We study the evolution of hot, relativistic hollow outflows initially carrying a purely azimuthal magnetic field. Our jet models are characterized by different choices of the outflow magnetization ($\sigma$ parameter) and the bulk Lorentz factor ($\gamma_{j}$). We show that the jet is heavily affected by the growth of current-driven kink instabilities causing considerable deflection throughout its propagation length. This behavior is partially stabilized by the combined action of larger flow velocities and/or reduced magnetic field strengths. We find that our best jet models are characterized by relatively large values of $\sigma$ ($\gtrsim 1$) and small values of $\gamma_{j}\simeq 2$. Our results are in good agreement with the recent X-ray (\textit{Chandra}) data of the Crab Nebula South-East jet indicating that the jet changes direction of propagation on a time scale of the order of few years. The 3D models presented here may have important implications in the investigation of particle acceleration in relativistic outflows.Comment: 15 pages, 20 figure

Geometry of contours and Peierls estimates in d=1 Ising models

Following Fr\"ohlich and Spencer, we study one dimensional Ising spin systems with ferromagnetic, long range interactions which decay as $|x-y|^{-2+\alpha}$, $0\leq \alpha\leq 1/2$. We introduce a geometric description of the spin configurations in terms of triangles which play the role of contours and for which we establish Peierls bounds. This in particular yields a direct proof of the well known result by Dyson about phase transitions at low temperatures.Comment: 28 pages, 3 figure

Apical access and closure devices for transapical transcatheter heart valve procedures.

The majority of transcatheter aortic valve implantations, structural heart procedures and the newly developed transcatheter mitral valve repair and replacement are traditionally performed either through a transfemoral or a transapical access site, depending on the presence of severe peripheral vascular disease or anatomic limitations. The transapical approach, which carries specific advantages related to its antegrade nature and the short distance between the introduction site and the cardiac target, is traditionally performed through a left anterolateral mini-thoracotomy and requires rib retractors, soft tissue retractors and reinforced apical sutures to secure, at first, the left ventricular apex for the introduction of the stent-valve delivery systems and then to seal the access site at the end of the procedure. However, despite the advent of low-profile apical sheaths and newly designed delivery systems, the apical approach represents a challenge for the surgeon, as it has the risk of apical tear, life-threatening apical bleeding, myocardial damage, coronary damage and infections. Last but not least, the use of large-calibre stent-valve delivery systems and devices through standard mini-thoracotomies compromises any attempt to perform transapical transcatheter structural heart procedures entirely percutaneously, as happens with the transfemoral access site, or via a thoracoscopic or a miniaturised video-assisted percutaneous technique. During the past few years, prototypes of apical access and closure devices for transapical heart valve procedures have been developed and tested to make this standardised successful procedure easier. Some of them represent an important step towards the development of truly percutaneous transcatheter transapical heart valve procedures in the clinical setting