Success in baseball batting is fundamental to the sport, however it remains one of, if not the most, challenging skills in sports to master. Batters utilize the kinetic chain to transfer energy from the lower body to the upper body to the bat, hoping to impart the maximum amount of energy into the ball. Scientists and coaches have researched the swing and developed theories on the keys for successful batting, but most of this research has been inadequate in attempting to fully describe the biomechanics of batting. The purposes of this study were to improve upon the methodology of previous researchers, provide a full biomechanical description of the swing, and compare swings against pitches thrown to different locations and at different speeds. AA-level Minor League Baseball players (n=43) took extended rounds of batting practice in an indoor laboratory against a pitcher throwing a mixture of fastballs and changeups. An eight camera motion analysis system and two force plates recording at 300 Hz captured the biomechanical data. The swing was divided into six phases (stance, stride, coiling, swing initiation, swing acceleration, and follow-through) by five key events (lead foot off, lead foot down, weight shift commitment, maximum front foot vertical ground reaction force, and bat-ball contact). Twenty-eight kinematic measurements and six ground reaction force measurements were computed based on the marker and force plate data, and all were assessed throughout the phases. First, a comprehensive description of a composite of the batters’ swings against fastballs “down the middle” was provided. Second, successful swings against fastballs thrown to one of five pitch locations (HIGH IN, HIGH OUT, LOW IN, LOW OUT, MIDDLE) were compared in terms of selected kinematics at the instant of bat-ball contact, timing and magnitude of peak kinematic velocities, and timing and magnitude of peak ground reaction forces. Third, these variables were once again compared for swings against fastballs and changeups. A large number of biomechanical differences were seen among the swings against various pitch locations. More fully rotated positions, particularly of the pelvis and bat were critical to the batters’ successes on inside pitches while less rotated positions keyed successes against outside pitches. The trail and lead arms worked together as part of a closed chain to drive the hand path. Successful swings had the trail elbow extended more for HIGH IN and flexed more for LOW OUT, though batters often struggled to execute this movement properly. A distinct pattern among successful swings against fastballs, successful swings against changeups, and unsuccessful swings against changeups was witnessed; namely a progressive delay in which the batter prematurely initiated the events of the kinetic chain, especially when unsuccessful in hitting a changeup. It was believed that this study was much more effective in capturing the essence of baseball batting than previous scientific works. Some recommendations to batting coaches would be to get batters to take a consistent approach in the early phases of every swing (particularly for the lower body), identify both pitch type and location as early as possible, use the rotation of the pelvis to propagate the energy transfer of the kinetic chain from the group to the upper body, and use the pelvis, and subsequently, the upper body, to orient the trunk and hands to an optimal position to drive the ball to the desired field. Limitations of the current study and ideas for future work were also presented to better interpret the findings of this research and further connect science and sport.

Interval graphs are well studied structures. Intervals can represent resources like jobs to be sched-uled. Finding maximum independent set in interval graphs would correspond to scheduling maximum number of non-conflicting jobs on the computer. Most optimization problems on interval graphs like independent set, vertex cover, dominating set, maximum clique, etc can be solved efficiently using combinatorial algorithms in polynomial time. Hitting, Covering and Packing problems have been ex-tensively studied in the last few decades and have applications in diverse areas. While they are NP-hard for most settings, they are polynomial solvable for intervals. In this thesis, we consider the generaliza-tions of hitting, covering and packing problems for intervals. We model these problems as min-cost flow problems using non-trivial reduction and solve it using standard flow algorithms. Demand-hitting problem which is a generalization of hitting problem is defined as follows: Given N intervals, a positive integer demand for every interval, M points, a real weight for every point, select a subset of points H, such that every interval contains at least as many points in H as its demand and sum of weight of the points in H is minimized. Note that if the demand is one for all intervals, we get the standard hitting set problem. In this case, we give a dynamic programming based O(M + N) time algorithm assuming that intervals and points are sorted. A special case of the demand-hitting set is the K-hitting set problem where the demand of all the intervals is K. For the K-hitting set problem, we give a O(M2N) time flow based algorithm. For the demand-hitting problem, we make an assumption that no interval is contained in another interval. Under this assumption, we give a O(M2N) time flow based algorithm. Demand-covering problem which is a generalization of covering problem is defined as follows: Given N intervals, a real weight for every interval, M points, a positive integer demand for every point, select a subset of intervals C, such that every point is contained in at least as many intervals in C as its demand and sum of weight of the intervals in C is minimized. Note that if the demand of points are one, we get the standard covering set problem. In this case, we give a dynamic programming based O(M + N log N) time algorithm assuming that points are sorted. A special case of the demand-covering set is the K-covering set problem where the demand of all the points is K. For the K-covering set problem, we give a O(MN2) time flow based algorithm. For the demand-covering problem, we give a O(MN2) time flow based algorithm. K-pack points problem which is a generalization of packing problem is defined as follows: Given N intervals, an integer K, M points, a real weight for every point, select a subset of points Y , such that every interval contains at most K points from Y and sum of weight of the points in Y is maximized. Note that if K is one, we get the standard pack points problem. In this case, we give a dynamic pro-gramming based O(M + N) time algorithm assuming that points and intervals are sorted. For K-pack points problem, we give O(M2 log M) time flow based algorithm assuming that intervals and points are sorted.

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Many sports biomechanics research studies follow a traditional task analysis concept that there is only one best possible movement pattern and thus focus on the examination of kinematics and kinetics of movement without considering the influence of constraints that are imposed on it. This study developed an interdisciplinary approach by utilizing the principles of ecological task analysis and movement coordination from areas of motor leaning and biomechanics to examine the skill of placement hitting in slo-pitch softball. The choice of evaluating this slo-pitch batting skill to assess movement patterns is pragmatic because of its popularity of the sport and uniqueness of the batting movement. Therefore, the purpose of this study was to examine the influence of two task constraints (stride technique and designated field location) and an environmental constraint (pitched ball location) on the participants batting performances, kinematics, and movement patterns. A three-way ANOVA of 2 fields (same and opposite) x 2 locations of pitch (inside and outside) x 3 strides (open, parallel and closed) repeated measure study was conducted in this study. The results showed that participants were more successful in placing the ball to the same field instead of the opposite field. The pitched ball location and stride techniques did not have a consistent impact on the results across the different hitting conditions. To achieve these batting performance results, participants demonstrated different joint movements and different coordination patterns. Hence, this study supports the rationale of ecological task analysis but not traditional task analysis. Further, to understand the generalizability of the findings, a Euclidean distance analysis was conducted to evaluate the degree of dissimilarity between the individual and group mean results. The results indicated that participants generally showed a low degree of dissimilarity, so they were quite homogeneous as a group. Hence, the results from this study not only enable us to evaluate a human movement skill under the influence of different constraints but educators may apply the findings to other players. A similar interdisciplinary approach is warranted for future research studies in order to better understand the mechanics of human motion.

INTRODUCTION: Muscle activity and timing of the swing phases may contribute to the differences we see in athletes at different skill levels. The purpose of this study is to analyze the differences between mean muscle activity, peak muscle active and time to peak muscle activity for select muscles in the lower extremity as well as the differences between start times for swing phases and bat velocity prior to impact for a skilled and recreational group. METHODS: Twelve healthy subjects were split into two groups based on competitive level and analyzed hitting off of a tee. RESULTS: No significant differences were seen between muscle activity or the start time for the landing and swinging between groups. The skilled group did have a faster time to peak muscle activation for the front leg biceps femoris (p = 0.024), start the shifting (p = 0.12) and stepping (p = 0.11) phases significantly earlier as well as had a higher bat velocity prior to ball contact (p = 0.42) than the recreational group. CONCLUSION: Mean and peak muscle activity trends to be lower for skilled hitters than recreational hitters. Evidence of the skilled group starting the shifting and stepping phase earlier as well as having a higher bat velocity prior to impact could be important in separating hitters into skill level.

Consider the random set system (Formula presented.), where (Formula presented.) and Ajselected with probabilityp=pn}. A set H⊆[n] is said to be a hitting set for (Formula presented.). The second moment method is used to exhibit the sharp concentration of the minimal size of H for a variety of values of p.

Indiana University-Purdue University Indianapolis (IUPUI) / Statistics and athletic sports have always had a strong connection that many critics, fans and statisticians utilize to determine how successful a team or an individual player might be over an entire season or even throughout one’s career. The success of a player or team is often characterized by investigating the consistency that has been shown throughout the season or career, which has led to more investigation of the streakiness of players. Studies have been done to examine great streaks, such as Joe DiMaggio’s 56 game hitting streak or Tiger Woods’ 142 consecutive cuts made streak, but what about the outstanding streak that occurred during the 1983-1984 NHL season? Wayne Gretzky, of the Edmonton Oilers, managed to showcase just how elite he was as a playmaker during that season. Gretzky produced a remarkable 51-game point streak, in which he recorded at least one goal or point in 51 consecutive games; a streak that has not received the recognition that it deserves. Using game-by-game data for the entire 1983-1984 NHL season for all players, the research looks at not only the evaluation of Gretzky’s streak, but also compares his production and streak to the remainder of the league. Gretzky demonstrated why he is one of the greatest players to ever step foot on the ice, and his elite status is shown throughout this analysis. Comparing Gretzky’s streak to that of DiMaggio’s was shown to be a little challenging but, some general conclusions were made based on the comparison of analyses that were performed; but without the proper statistics being readily available, it is hard to adequately dictate which streak is ultimately more impressive or more rare.

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Dans ce travail, nous étudions les propriétés statistiques de certains systèmes dynamiques déterministes et stochastiques. Nous nous intéressons particulièrement aux valeurs extrêmes et à la récurrence. Nous montrons l’existence de Lois pour les Valeurs Extrêmes(LVE) et pour les Statistiques des Temps d’Entrée (STE) et des Temps de Retour (STR) pour des systèmes avec décroissance des corrélations rapide. Nous étudions aussi la convergence du Processus Ponctuel d’Evènements Rares (PPER).Dans la première partie, nous nous intéressons aux systèmes dynamiques déterministes, et nous caractérisons complètement les propriétés précédentes dans le cas des systèmes dilatants. Nous montrons l’existence d’un Indice Extrême (IE) strictement plus petit que 1 autour des points périodiques, et qui vaut 1 dans le cas non-périodique, mettant ainsi en évidence une dichotomie dans la dynamique caractérisée par l’indice extrême. Dans un contexte plus général, nous montrons que le PPER converge soit vers une distribution de Poisson pour des points non-périodiques, soit vers une distribution de Poisson mélangée avec une distribution multiple de type géométrique pour des points périodiques. De plus, nous déterminons explicitement la limite des PPER autour des points de discontinuité et nous obtenons des distributions de Poisson mélangées avec des distributions multiples différentes de la distribution géométrique habituelle. Dans la deuxième partie, nous considérons des systèmes dynamiques stochastiques obtenus en perturbant de manière aléatoire un système déterministe donné. Nous élaborons deux méthodes nous permettant d’obtenir des lois pour les Valeurs Extrêmes et les statistiques de la récurrence en présence de bruits aléatoires. La première approche est de nature probabiliste tandis que la seconde nécessite des outils d’analyse spectrale. Indépendamment du point choisi, nous montrons que l’IE est constamment égal à 1 et que le PPER converge vers la distribution de Poisson standard. / In this work, we study the statistical properties of deterministic and stochastic dynamical systems. We are particularly interested in extreme values and recurrence. We prove the existence of Extreme Value Laws (EVLs) and Hitting Time Statistics (HTS)/ ReturnTime Statistics (RTS) for systems with decay of correlations against L1 observables. We also carry out the study of the convergence of Rare Event Point Processes (REPP). In the first part, we investigate the problem for deterministic dynamics and completely characterise the extremal behaviour of expanding systems by giving a dichotomy relying on the existence of an Extremal Index (EI). Namely, we show that the EI is strictly less than 1 for periodic centres and is equal to 1 for non-periodic ones. In a more general setting, we prove that the REPP converges to a standard Poisson if the centre is non-periodic, and to a compound Poisson with a geometric multiplicity distribution for the periodic case. Moreover, we perform an analysis of the convergence of the REPP at discontinuity points which gives the convergence to a compound Poisson with a multiplicity distribution different than the usual geometric one.In the second part, we consider stochastic dynamics by randomly perturbing a deterministic system with additive noise. We present two complementary methods which allow us to obtain EVLs and statistics of recurrence in the presence of noise. The first approach is more probabilistically oriented while the second one uses spectral theory. We conclude that, regardless of the centre chosen, the EI is always equal to 1 and the REPP converges to the standard Poisson. / Neste trabalho, estudamos as propriedades estatısticas de sistemas dinâmicos deterministicos e estocasticos. Estamos particularmente interessados em valores extremos e recorrência. Provamos a existência de Leis de Valores Extremos (LVE) e Estatısticas doTempo de Entrada (ETE) / Estatısticas de Tempo de Retorno (ETR) para sistemas comdecaimento de correlaçoes contra observaveis em L1. Também realizamos o estudo daconvergência dos Processos Pontuais de Acontecimentos Raros (PPAR). Na primeira parte, investigamos o problema para dinâmica determinıstica e caracterizamos completamente o comportamento extremal de sistemas expansores. Mostramos que ha uma dicotomia quanto 00E0 existência de um Indice de Extrema (IE). Nomeadamente, provamos que o IE é estritamente menor do que 1 em torno de pontos periodicos e é igual a 1 para pontos aperiodicos. Num contexto mais geral, mostramos que os PPAR convergem para um processo de Poisson simples ou um processo de Poisson composto, em que a distribuiçao de multiplicidade é geométrica, dependendo se o centro é um ponto aperiodico ou periodico, respectivamente. Além disso, realizamos uma analise da convergência dos PPAR em pontos de descontinuidade, o que conduziu à descoberta de convergência para um processo de Poisson composto com uma distribuiçao de multiplicidade diferente da usual distribuiçao geométrica. Na segunda parte, consideramos dinâmica estocastica obtida por perturbaçao aleatoria de um sistema determinıstico por inclusao de um ruıdo aditivo. Apresentamos duas técnicas complementares que nos permitem obter LVE e as ETE na presen¸ca deste tipo de ruıdo. A primeira abordagem é mais probabilıstica enquanto que a outra usa sobretudo teoria espectral. Conclui-se que, independentemente do centro escolhido, o IE é sempre igual a 1 e os PPAR convergem para o processo de Poisson simples.

Considere um processo estocástico X_m em tempo discreto definido sobre o alfabeto finito A. Seja x_0^k-1 uma palavra fixa sobre A^k. No estudo das propriedades estatísticas na teoria de recorrência de Poincaré, é clássico o estudo do tempo decorrente até que a sequência fixa x_0^k-1 seja encontrada em uma realização do processo. Tipicamente, esta é uma quantidade exponencialmente grande com relação ao comprimento da palavra. Contrariamente, o primeiro tempo de retorno possível para uma sequência dada está definido como sendo o mínimo entre os tempos de entrada de todas as sequências que começam com a própria palavra e é uma quantidade tipicamente pequena, da ordem do tamanho da palavra. Neste trabalho estudamos o comportamento da probabilidade deste primeiro retorno possível de uma palavra x_0^k-1 dado que o processo começa com ela mesma. Esta quantidade mede a intensidade de que, uma vez observado um conjunto alvo, possam ser observados agrupamentos ou clusters. Provamos que, sob certas condições, a taxa de decaimento exponencial desta probabilidade converge para a entropia para quase toda a sequência quando k diverge. Apresentamos também um estimador desta probabilidade para árvores de contexto e mostramos sua consistência. / Considering a stochastic process X_m in a discrete defined time over a finite alphabet A and x_0^k-1 a fixed word over A^k. In the study of the statistical properties of the Poincaré recurrence theory, it is usual the study of the time elapsed until a fixed sequence x_0^k-1 appears in a given realization of process. This quantity is known as the hitting time and it is usually exponentially large in relation to the size of word. On the opposite, the first possible return time of a given word is defined as the minimum among all the hitting times of realizations that begins with the given word x_0^k-1. This quantity is tipically small that is of the order of the length of the sequence. In this work, we study the probability of the first possible return time given that the process begins of the target word. This quantity measures the intensity of that, once observed the target set, it can be observed in clusters. We show that, under certain conditions, the exponential decay rate of this probability converges to the entropy for all almost every word x_0^k-1 as k diverges. We also present an estimator of this probability for context trees and shows its consistency.