Volume 42 - Issue 2 - August 2017

  God does not play dice: revisiting Einstein's rejection of
    probability in quantum mechanics

    Prakash Gorroochurn
    pp. 6173

Einstein's struggle with the use of probability in quantum mechanics is revisited. It is argued that Einstein was a statistical physicist who understood probability well, but the use of probability in quantum theory represented a radical departure which troubled Einstein. The theory denied the existence of physical reality until an observation was made, and probability replaced that reality. Einstein later put forward the powerful EPR thought experiment to show problems with quantum theory, but subsequent actual experiments have all supported quantum theory, instead of his local arguments.

Diophantine eclipses
   Andrew Simoson
   pp. 7489
We present and contrast two ways to predict when eclipses occur, a vector calculus approach and a Diophantine equation approach. Although they give comparable results, the beauty of the latter approach over the former is that it gives geneologic information about the eclipse. That is, when the Diophantine method predicts that eclipse E will occur at the time X in the future, it also gives the time Y in the past for which E is a recurrence of itself.

Gibonacci extensions of a Swamy delight
   Thomas Koshy
   pp. 9097 
We extend a charming Fibonacci pleasantry discovered by Swamy (1966) to Fibonacci, Lucas, Pell, Pell–Lucas, Jacobsthal, Jacobsthal–Lucas, Vieta, and Vieta–Lucas polynomials, and Chebyschev polynomials of both kinds. We then extract their counterparts for Lucas, Pell, Pell–Lucas, Jacobsthal, and Jacobsthal–Lucas numbers.

A note on the characterization of the general nonnegative-definite
   covariance structure for the equality of the BLU and OLS estimators
   Phil D. Young and Dean M. Young
   pp. 98100
For the general Gauss–Markov model with E(y) = Xß and var(y) = V, we give a simple alternative proof of an explicit characterization of the general nonnegative-definite covariance structure V such that the BLU and OLS estimators of Xß are identical. The proof uses only basic properties of real matrices.

A note on an unconditional alternative to Cochran's Q
   D. J. Best and J. C. W. Rayner
   pp. 101103
We give an example to demonstrate that a conditional statistical test can give a possibly different p-value to an unconditional test, thereby altering the statistical conclusion. This same example also gives different p-values for the associated permutation test and parametric bootstrap test. When there is a choice, whether a test is conditional or unconditional may be important.

Bifurcation in an interacting species model
   Jennifer Switkes and Ryan Szypowski
   pp. 104110
We tell the tale of two interacting species who are competing for the same prey. Can they coexist in the same habitat? Our exploration involves a nonlinear system of ordinary differential equations that has an interesting bifurcation. This bifurcation, caused by the playing-out of within-species overcrowding and between-species competition, takes us from competitive exclusion to stable competition. Finally, we break the symmetry of our model and discover additional intriguing behavior for the system.

Distribution of deaths in a birth–death process
   J. Gani and R. J. Swift
   pp. 111114
The probability generating function for the number of survivors in the classical simple birth–death process, is well known. However, the bivariate probability generating function for the surviving population and total number of deaths for this process is not as well known. In this note, we derive the probability generating function for the total number of deaths and also provide an approximation.

Letter to the Editor: A new mathematical relation
   Bablu Chandra Dey
   pp. 115116