Maximum principle, mean value operators and quasi boundedness in non-euclidean settings

This work deals with some classes of linear second order partial differential operators with non-negative characteristic form and underlying non- Euclidean structures. These structures are determined by families of locally Lipschitz-continuous vector fields in RN, generating metric spaces of Carnot- Carath´eodory type. The Carnot-Carath´eodory metric related to a family {Xj}j=1,...,m is the control distance obtained by minimizing the time needed to go from two points along piecewise trajectories of vector fields. We are mainly interested in the causes in which a Sobolev-type inequality holds with respect to the X-gradient, and/or the X-control distance is Doubling with respect to the Lebesgue measure in RN. This study is divided into three parts (each corresponding to a chapter), and the subject of each one is a class of operators that includes the class of the subsequent one. In the first chapter, after recalling “X-ellipticity” and related concepts introduced by Kogoj and Lanconelli in [KL00], we show a Maximum Principle for linear second order differential operators for which we only assume a Sobolev-type inequality together with a lower terms summability. Adding some crucial hypotheses on measure and on vector fields (Doubling property and Poincar´e inequality), we will be able to obtain some Liouville-type results. This chapter is based on the paper [GL03] by Guti´errez and Lanconelli. In the second chapter we treat some ultraparabolic equations on Lie groups. In this case RN is the support of a Lie group, and moreover we require that vector fields satisfy left invariance. After recalling some results of Cinti [Cin07] about this class of operators and associated potential theory, we prove a scalar convexity for mean-value operators of L-subharmonic functions, where L is our differential operator. In the third chapter we prove a necessary and sufficient condition of regularity, for boundary points, for Dirichlet problem on an open subset of RN related to sub-Laplacian. On a Carnot group we give the essential background for this type of operator, and introduce the notion of “quasi-boundedness”. Then we show the strict relationship between this notion, the fundamental solution of the given operator, and the regularity of the boundary points.

Analisi biomeccanica del MAWASHI GERI JODAN in cinture nere di Karate: modelli tecnici a confronto

In martial arts there are several ways to perform the turning kick . Following the martial arts or different learning models many types of kicks take shape. Mawashi geri is the karate turning kick. At the moment there are two models of mawashi geri, one comes from the traditional karate (OLD), and the other newer (NEW), who agrees to the change of the rules of W.K.F. (World Karate Federation) happened in 2000 (Macan J. et all 2006) . In this study we are focus on the differences about two models the mawashi geri jodan of karate. The purpose of this study is to analyse cinematic and kinetic parameters of mawashi geri jodan. Timing of the striking and supporting leg actions were also evaluated A Vicon system 460 IR with 6 cameras at sample frequency of 200 Hz was used. 37 reflective markers have been set on the skin of the subjects following the “PlugInGait-total body model”. The participants performed five repetitions of mawashi geri jodan at maximum rapidity with their dominant leg against a ball suspended in front of them placed at ear height. Fourteen skilled subjects (mean level black belt 1,7 dan; age 20,9±4,8 yrs; height 171,4±7,3 cm; weight 60,9±10,2 Kg) practicing karate have been split in two group through the hierarchical cluster analysis following their technical characteristics. By means of the Mann Whitney-U test (Spss-package) the differences between the two groups were verified in preparatory and execution phase. Kicking knee at start, kicking hip and knee at take-off were different between the two groups (p < 0,05). Striking hip flexion during the spin of the supporting foot was different between the two groups (p < 0,05). Peak angular velocity of hip flexion were different between the two groups (p < 0,05). Groups showed differences also in timing of the supporting spin movement. While Old group spin the supporting foot at 30% of the trial, instead New start spinning at 44% of the trial. Old group showed a greater supporting foot spin than New (Old 110° Vs New 82°). Abduction values didn’t show any differences between the two groups. At the hit has been evaluated a 120° of double hips abduction, for the entire sample. Striking knee extension happened for everybody after the kicking hip flexion and confirm the proximal-distal action of the striking leg (Sorensen H. 1996). In contrast with Pearson J.N. 1997 and Landeo R 2007, peak velocity of the striking foot is not useful to describe kick performance because affected by the stature. Two groups are different either in preparatory phase or in execution phase. The body is set in difference manner already before the take-off of the kicking foot. The groups differ for the timing of the supporting foot action Trainer should pay attention to starting posture and on abduction capacities of the athletes.