Player Safety In The National Football League

This research was conducted to determine to what extent National Football League fans want player safety rules in the game of football. The NFL received backlash from not only a tremendous amount of fans but players as well. It was essential for the NFL to determine what a happy medium was in order to keep the players safe while they played the game of football, but also the physicality was a large part of the game that the fans loved. The NFL player safety rules reduced the amount of injuries that occurred, but fans and players believed that the rules took too much away from the game. A survey was created that was aimed to collect a variety of ages, races, and locations that represented the whole fan spectrum of the NFL. Since the survey was posted on reddit, the sampling method used was key informant technique. Questions ranged from simple demographics, commitment of fandom, and also the support of the player safety rules. Duration and commitment of fandom gauged the level of fandom that determined if level of fandom played into the support of the rules. Ultimately, each and every fan made a decision regarding whether they cared more about the quality and experience of watching the game or the physical, mental, and emotional well-being of the NFL players. A majority of the sample supported the NFL Player Safety Rules either most of the time or always. This particular variable showed that even though a good amount of participants experienced a worse experience viewing the game, a majority of the sample still supported the safety rules within the game. This was essential to the NFL due to knowing the fan base. Fans were essential to the league and without a great amount of fans, there was no league. The knowledge of what turned fans away from the league was crucial. There were many variables that played into why fans were drawn to the league or pushed away. Knowing who the fans were and what they thought of the league was essential to the success of the NFL

Anomalous behavior of the Debye temperature in Fe-rich Fe-Cr alloys

Debye temperature, $\Theta_D$, of Fe-rich Fe$_{100-x}$Cr$_x$ disordered alloys with $0\le x \le 22.3$ was determined from the temperature dependence of the central shift of M\"ossbauer spectra recorded in the temperature range of 60 -- 300 K. Its compositional dependence shows a maximum at $x \approx 5$ with a relative increase of $\sim 30$% compared to a pure iron. The composition at which the effect occurs correlates well with that at which several other quantities, e. g. the Curie temperature and the spin-wave stiffness coefficient, $D_0$, show their maxima, but the enhancement of $\Theta_D$ is significantly greater and comparable with the enhancement of the hyperfine field (spin-density of itinerant $s$-like electrons) in the studied system. The results suggest that the electron-phonon interaction is important in this alloy system

Complex Dynamics in a Bertrand Duopoly Game with Heterogeneous Players

A heterogeneous Bertrand duopoly game with bounded rational and adaptive players manufacturing differentiated products is subject of investigation. The main goal is to demonstrate that participation of one bounded rational player in the game suffices to destabilize the duopoly. The game is modelled with a system of two difference equations. Evolution of prices over time is obtained by iteration of a two dimensional nonlinear map. Equilibria are found and local stability properties thereof are analyzed. Complex behavior of the system is examined by means of numerical simulations. Region of stability of the Nash equilibrium is demonstrated in the plane of the speeds of adjustment. Period doubling route to chaos is presented on the bifurcation diagrams and on the largest Lyapunov characteristic exponent graph. Lyapunov time is calculated. Chaotic attractors are depicted and their fractal dimensions are computed. Sensitive dependence on initial conditions is evidenced.Bertrand duopoly, heterogeneous expectations, nonlinear dynamics, chaos