The Pythagorean Won-Loss Formula and Hockey: A Statistical Justification for Using the Classic Baseball Formula as an Evaluative Tool in Hockey

Originally devised for baseball, the Pythagorean Won-Loss formula estimates the percentage of games a team should have won at a particular point in a season. For decades, this formula had no mathematical justification. In 2006, Steven Miller provided a statistical derivation by making some heuristic assumptions about the distributions of runs scored and allowed by baseball teams. We make a similar set of assumptions about hockey teams and show that the formula is just as applicable to hockey as it is to baseball. We hope that this work spurs research in the use of the Pythagorean Won-Loss formula as an evaluative tool for sports outside baseball.Comment: 21 pages, 4 figures; Forthcoming in The Hockey Research Journal: A Publication of the Society for International Hockey Research, 2012/1

Investigation of a Novel Compact Vibration Isolation System for Space Applications

A novel compact vibration isolation system was designed, built, and tested for the Space Chromotomography Experiment (CTEx) being built by Air Force Institute of Technology (AFIT) researchers. CTEx is a multifunctional experimental imaging chromotomographic spectrometer designed for flight on the International Space Station (ISS) and is sensitive to jitter caused by vibrations both through the support structure as well as those produced on the optical platform by rotating optical components. CTEx demands a compact and lightweight means of vibration isolation and suppression from the ISS structure. Vibration tests conducted on an initial isolator design resulted in changes in the chosen spring and damping material properties but confirmed finite element (FE) model results and showed that the spring geometry meets preliminary design goals. The FE model served as a key tool in evaluating material and spring designs and development of the final drawing sets for fabrication. Research efforts led to a final design which was tested in the final flight configuration. This final configuration proved the potential for a compact means of vibration isolation for space applications

A Turbulent Fluid Mechanics via Nonlinear Mixing of Smooth Flows with Bargmann-Fock Random Fields: Stochastically Averaged Navier-Stokes Equations and Velocity Correlations

Let $\mathbb{I\!H}\subset\mathbb{R}^{3}$, with ${Vol}(\mathbb{I\!H})\sim L^{3}$, contain a fluid of viscosity $\nu$ and velocity $\mathrm{U}_{i}(x,t)$ with $(x,t)\in\mathbb{I\!H}\times[0,\infty)$, satisfying the Navier-Stokes equations with some boundary conditions on $\partial\mathbb{I\!H}$ and evolving from initial Cauchy data. Now let $\mathscr{B}(x)$ be a Gaussian random field defined for all $x\in\mathbb{I\!H}$ with expectation $\mathsf{E}\langle\mathscr{B}(x)\rangle=0$, and a Bargmann-Fock binary correlation $\mathsf{E}\big\langle\mathscr{B}(x)\otimes \mathscr{B}({y})\big\rangle=\mathsf{C}\exp(-\|{x}-{y}\|^{2}\lambda^{-2})$ with $\lambda\le {L}$. Define a volume-averaged Reynolds number $\mathbf{Re}(\mathbb{I\!H},t) =(|Vol(\mathbb{I\!H})|^{-1}\int_{\mathbb{I\!H}}\|\mathrm{U}_{i}(x,t)\|d\mu({x}){L}/\nu$. The critical Reynolds number is $\mathbf{Re}_{c}(\mathbb{I\!H})$ so that turbulence evolves within $\mathbb{I\!H}$ for $t$ such that $\mathbf{Re}(\mathbb{I\!H},t)>\mathbf{Re}_{c}(\mathbb{I\!H})$. Let $\psi(|\mathbf{Re}(\mathbb{I\!H},t)-\mathbf{Re}_{c}(\mathbb{I\!H})|)$ be an arbitrary monotone-increasing functional. The turbulent flow evolving within $\mathbb{I\!H}$ is described by the random field $\mathscr{U}_{i}(x,t)$ via a 'mixing' ansatz $\mathscr{U}_{i}(x,t)=\mathrm{U}_{i}(x,t)+\beta\mathrm{U}_{i}(x,t) \big\lbrace\psi(|\mathbf{Re}(\mathbb{I\!H},t)-\mathbf{Re}_{c}(\mathbb{I\!H})|)\big\rbrace \mathbb{S}_{\mathfrak{C}}[\mathbf{Re}(\mathbb{I\!H},t)\big]\mathscr{B}(x)$ where ${\beta}\ge 1$ is a constant and $\mathbb{S}_{\mathfrak{C}}[\mathbf{Re}(\mathbb{I\!H},t)]$ an indicator function. The flow grows increasingly random if $\mathbf{Re}(\mathbb{I\!H},t)$ increases with $t$ so that this is a 'control parameter'. The turbulent flow $\mathscr{U}_{i}(x,t)$ is a solution of stochastically averaged N-S equations. Reynolds-type velocity correlations are estimated.Comment: 64 pages, 5 figure

Second And Third-Order Structure Functions Of An 'Engineered' Random Field And Emergence Of The Kolmogorov 4/5 And 2/3-Scaling Laws Of Turbulence

The 4/5 and 2/3 laws of turbulence can emerge from a theory of 'engineered' random vector fields $\mathcal{X}_{i}(x,t) =X_{i}(x,t)+\tfrac{\theta}{\sqrt{d(d+2)}} X_{i}(x,t)\psi(x)$ existing within $\mathbf{D}\subset\mathbf{R}^{d}$. Here, $X_{i}(x,t)$ is a smooth deterministic vector field obeying a nonlinear PDE for all $(x,t)\in\mathbf{D}\times\mathbf{R}^{+}$, and $\theta$ is a small parameter. The field $\psi(x)$ is a regulated and differentiable Gaussian random field with expectation $\mathbb{E}[\psi(x)]=0$, but having an antisymmetric covariance kernel $\mathscr{K}(x,y)=\mathbb{E}[\psi(x)\psi(y)]=f(x,y)K(\|x-y\|;\lambda)$ with $f(x,y)=-f(y,x)=1,f(x,x)=f(y,y)=0$ and with $K(\|x-y\|;\lambda)$ a standard stationary symmetric kernel. For $0\le\ell\le \lambda<L$ with $X_{i}(x,t)=X_{i}=(0,0,X)$ and $\theta=1$ then for $d=3$, the third-order structure function is \begin{align} S_{3}[\ell]=\mathbb{E}\left[|\mathcal{X}_{i}(x+\ell,t)-\mathcal{X}(x,t)|^{3}\right]=-\frac{4}{5}\|X_{i}\|^{3}=-\frac{4}{5}X^{3}\nonumber \end{align} and $S_{2}[\ell]=CX^{2}$. The classical 4/5 and 2/3-scaling laws then emerge if one identifies the random field $\mathcal{X}_{i}(x,t)$ with a turbulent fluid flow $\mathcal{U}_{i}(x,t)$ or velocity, with mean flow $\mathbb{E}[\mathcal{U}_{i}(x,t)]=U_{i}(x,t)=U_{i}$ being a trivial solution of Burger's equation. Assuming constant dissipation rate $\epsilon$, small constant viscosity $\nu$, corresponding to high Reynolds number, and the standard energy balance law, then for a range $\eta\le\ell\ll \lambda<L$ \begin{align} S_{3}[\ell]=\mathbb{E}\left[|\mathcal{U}_{i}(x+\ell,t)-\mathcal{U}(x,t)|^{3}\right]=-\frac{4}{5}\epsilon\ell\nonumber \end{align} where $\eta=(\nu^{3/4}\epsilon)^{-1/4}$. For the second-order structure function, the 2/3-law emerges as $S_{2}[\ell]=C\epsilon^{2/3}\ell^{2/3}$

The effectiveness of thin films in lieu of hyperbolic metamaterials in the near field

We show that the near-field functionality of hyperbolic metamaterials (HMM), typically proposed for increasing the photonic local density of states (LDOS), can be achieved with thin metal films. Although HMMs have an infinite density of internally-propagating plane-wave states, the external coupling to nearby emitters is severely restricted. We show analytically that properly designed thin films, of thicknesses comparable to the metal size of a hyperbolic metamaterial, yield a LDOS as high as (if not higher than) that of HMMs. We illustrate these ideas by performing exact numerical computations of the LDOS of multilayer HMMs, along with their application to the problem of maximizing near-field heat transfer, to show that thin films are suitable replacements in both cases.Comment: 5 pages, 3 figure