Papers by patrizio laconi
In all applied sciences the use of the normal distribution is very frequent and important, but th... more In all applied sciences the use of the normal distribution is very frequent and important, but the proof of central limit theorem makes use of notions such as characteristic function and Fourier transform the understanding of which is not easy at an average level of mathematical knowledge. In many cases it is simply proposed an intuitive understanding of the result of the theorem, and most often being asked to accept blindly the statement. Here it is presented an elementary proof that with a minimum eort can be understood by those who have an average education level in Mathematics. This exposition gives only acquired knowledge of the combinatorial calculus and of the binomial distribution. The Binomial distribution is illustrated very well by the known experiment of fall of little balls from a vertical duct, realized through n discrete steps in such way that at each step of fall, the little ball is randomly deected to the left or right with equal probability (Galton box) Keywords: central limit theorem · binomial distribution · gaussian function · elementary approach to normal distribution. Elementary analysis of Galton box.
The analysis of the vibrating string through the study of the relative discrete system is an eect... more The analysis of the vibrating string through the study of the relative discrete system is an eective teaching method, if one considers that the same conclusions can be reached in the analysis of continuous system, only by solving the dierential equation of the waves, i.e. by using a mathematical tool not elementary. This paper start from the analysis of a discrete distribution of masses for which even the time model adopted is discretized. This way, can be discretized the all system and then a study of this by means of the nite dierences is possible. Moreover such an approach is mandatory in the computer model simulation; but we can see that it is also useful at higher education level, to those who despite being in possession of mathematical tools for the analysis of continuous systems, they want to consolidate their knowledge by other viewpoints. Let us consider a chain formed by a number of N+1 springs , and of N little balls of mass m, each connected to previous and subsequent spring. Only the ends of the chain are connected to the xed supports. For simplicity we consider at moment only longitudinal movements. Consequently we have a system of N longitudinal oscillators. The generic little ball will be characterized by the serial number n and by his state of move away from the rest condition which we will denote with x. The variable x, so dened, is in connection with the rest position occupied in the chain from the n nt little ball, and also depends on the observation time t. If we denote with η the regular distance at rest between the two consecutive little balls, the rest position of n th little ball is equal to nη. Even for the time t we will adopt a discrete model setting t = kτ with k integer and τ small enough. So a description of the position of the n th little ball at time k is : (1.1) x (nη, kτ) Therefore x depends from space time discrete variables n,k. The discretized acceleration a(n, k) of the little ball at point n and at time k can thus be obtained as the second order forward nite dierence: (1.2) a(n, k) = 2 k [x(n, k)] = x(n, k + 1) − 2x(n, k) + x(n, k − 1) τ 2
The Journal of sports medicine and physical fitness, 2002
A field test consisting of 5 continuous runs at the maximum speed possible, playing the ball, sta... more A field test consisting of 5 continuous runs at the maximum speed possible, playing the ball, starting from the centre line to the basket with a final shot, was studied in order to obtain an index of mechanical work efficiency in basketball players (micro-index=Jmec/Joxy) and evaluate the correlation between micro-index and velocity, acceleration, mechanical power and lactacid anaerobic capacity, respectively. Eight male basketball players were studied; Jmec was the external mechanical work output obtained by means of a video image analysis software which gave the potential and the kinetic translational energies of athletes running and jumping and their velocity, acceleration and mechanical power. By means of a telemetric device (Kosmed K4), for measuring O2 consumption ( VO2), we obtained oxidative work (Joxy). By using this device we also assessed the excess of CO2, which was considered an index of lactacid anaerobic capacity. Non-parametric Spearman statistics revealed a signific...
International Journal of Sports Medicine, 1998
Field tests were performed in 10 volleyball (VB) players (4 females and 6 males) in order to obta... more Field tests were performed in 10 volleyball (VB) players (4 females and 6 males) in order to obtain an index of mechanical work efficiency (mu' = Wmec/Woxy) while athletes played a game. Wmec was the mechanical work output, obtained by means of home made video image analysis software, by summing potential, kinetic translational and kinetic rotational energies of running and jumping athletes. Woxy was the oxidative energy consumption obtained from O2 consumption (VO2) by a telemetry device (Cosmed K2) that also gave values of pulmonary ventilation (VE) and heart rate (HR). VB were studied at rest before a game (R), during attacking phases (A) and during defensive phases (D). At R were found: VE = 11 +/- 1 l x min(-1), HR = 78 +/- 7b x min(-1), VO2 = 3.71 +/- 1.1 ml x kg(-1) x min(-1), Woxy = 75.1 +/- 22.3 J x kg(-1) x min(-1). During A all variables increased: VE = 49 +/- 6l x min(-1), HR = 149 +/- 15 b x min(-1), VO2 = 23.1 +/- 3.3ml x kg(-1) x min(-1), Woxy = 482.8 +/- 69.0 J x kg(-1) x min(-1), and Wmec 275.5 +/- 57.0 J x kg(-1) x min(-1) with mu'=0.57 +/- 0.09. In D HR (-9%), VE, VO2 and Woxy (-18%) were reduced when compared to A, decreasing mu' to 0.21 +/- 0.05. On the other hand mu' was found higher than 0.25 during A indicating an anaerobic contribution to energy expenditure; in D the mu' lower than 0.25 indicated a restoration of anaerobic energy sources. It might be proposed that a greater difference in mu' values between A and D also means a higher anaerobic energy contribution to the volleyball game.
Uploads
Papers by patrizio laconi