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We develop the mathematical foundations of the stochastic modified equations (SME) framework for analyzing the dynamics of stochastic gradient algorithms, where the latter is approximated by a class of stochastic differential equations with small noise parameters. We prove that this approximation can be understood mathematically as an weak approximation, which leads to a number of precise and useful results on the approximations of stochastic gradient descent (SGD), momentum SGD and stochastic Nesterov's accelerated gradient method in the general setting of stochastic objectives. We also demonstrate through explicit calculations that this continuous-time approach can uncover important analytical insights into the stochastic gradient algorithms under consideration that may not be easy to obtain in a purely discrete-time setting. [ABSTRACT FROM AUTHOR]

A disastrous form of pedestrian behavior is a stampede occurring in an event involving a large crowd in a panic situation. To deal with such stampedes, the possibility to increase the outflow by suitably placing a pillar or some other shaped obstacles in front of the exit has been demonstrated. We present a social force based genetic algorithm to optimize the best design of architectural entities to deal with large crowds. Unlike existing literature, our simulation results indicate that appropriately placing two pillars on both sides but not in front of the door can maximize the escape efficiency. Human experiments using 80 participants correspond well with the simulations. We observed a peculiar property named tangential momentum, the escape speed and the tangential momentum are found to be negatively correlated. The idea to reduce the tangential momentum has practical implications in crowd architectural design. [ABSTRACT FROM AUTHOR]

Understanding the physical demands of a tackle in match situations is important for safe and effective training, developing equipment and research. Physical components such as momentum and kinetic energy, and it relationship to tackle outcome is not known. The aim of this study was to compare momenta between ball-carrier and tackler, level of play (elite, university and junior) and position (forwards vs. backs), and describe the relationship between ball-carrier and tackler mass, velocity and momentum and the tackle outcome. Also, report on the ball-carrier and tackler kinetic energy before contact and the estimated magnitude of impact (energy distributed between ball-carrier and tackler upon contact). Velocity over 0.5 seconds before contact was determined using a 2-dimensional scaled version of the field generated from a computer alogorithm. Body masses of players were obtained from their player profiles. Momentum and kinetic energy were subsequently calculated for 60 tackle events. Ball-carriers were heavier than the tacklers (ball-carrier 100 ± 14 kg vs. tackler 93 ± 11 kg, d = 0.52, p = 0.0041, n = 60). Ball-carriers as forwards had a significantly higher momentum than backs (forwards 563 ± 226 Kg.m.s-1 n = 31 vs. backs 438 ± 135 Kg.m.s-1, d = 0.63, p = 0.0012, n = 29). Tacklers dominated 57% of tackles and ball-carriers dominated 43% of tackles. Despite the ball-carrier having a mass advantage before contact more frequently than the tackler, momentum advantage and tackle dominance between the ball-carrier and tackler was proportionally similar. These findings may reflect a characteristic of the modern game of rugby where efficiently heavier players (particularly forwards) are tactically predetermined to carry the ball in contact. [ABSTRACT FROM AUTHOR]