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# Volume 40 - Issue 1 - June 2015

### A cluster of anniversaries. II

Oscar Sheynin

Abstract

Pascal's treatise on the arithmetic triangle (see Pascal (1665)) described the separate findings on this topic made mostly in the 16th century. Pascal showed how to apply the triangle to the theory of figurate numbers and combinatorics. Bayes completed the first version of the theory of probability and possibly considered that it belonged to pure science. Laplace's *Essai* was a barely successful popular treatise but it included interesting side issues such as the natural scientific study of moral sciences, psychology, and final causes. De Morgan was the first to note the normal distribution in De Moivre but was considered a logician rather than a mathematician. For 150 years Todhunter's history of probability has remained a necessary and useful source of information.

### Three views of a comment by Mersenne

C. W. Groetsch

Abstract

In the 17th century Marin Mersenne conducted experiments to test Galileo's idealized trajectory of a horizontally projected particle. He noted that the observed trajectory deviated in a specific sense from that predicted by Galileo. A simple resistance model, the analysis of which is accessible to undergraduates, is developed that validates Mersenne's observation.

### Further results in binomial prediction

Ben O'Neill

Abstract

In this paper we extend some prediction results pertaining to the binomial distribution in cases where prediction is carried out using the frequent outcome approach. As in previous work, we measure predictive accuracy by the 'accuracy function' which measures the prior probability of correct prediction on each trial. We first show that the accuracy function presented in previous work can be reframed in a more useful way which allows easier derivation of the relevant polynomial and asymptotic forms. We then approximate the accuracy function using the normal approximation to the binomial. This yields useful approximating results that can be used to examine the rate of convergence to idealised prediction. We give uniform convergence results for these functions and examine their accuracy in practical problems.

### A regional Poisson model

Christopher S. Withers and Saralees Nadarajah

Abstract

Suppose that we observe independent Poisson variables {*N*_{i,t},1 <= *i* <= *I*,1 <= *t* <= *T*} with **E**[*N*_{i,t}] = *p*_{i} *m*_{t}(\theta), where * p* = (

*p*

_{1}, . . .,

*p*

_{I})

^{T}are unknown probabilities adding to 1 and

*m*

_{t}(\theta) is a given function of an unknown parameter \theta in

**R**

^{q}. Then the marginal totals of the observations form a sufficient statistic. We obtain maximum likelihood estimates of

**and \theta, their approximate covariance, and hence an approximate confidence interval for any function of these parameters, in particular for**

*p***E**[

*N*

_{i,t}] and

**E**[\sum

_{s=1}

^{t}

*N*

_{i,s}] at some future time

*t*. Applications are made to modelling regional AIDS cases.

### Some analytical results in physics using the Lambert *W* function

Ahmed Houari

Abstract

In this article, a generic transcendental equation is solved using the Lambert *W* function to obtain a general solution to a particular class of physics equations. Also, the variance of the Gaussian distribution function is extracted analytically, and an inverse function of Stirling's formula is derived in terms of the Lambert *W *function.

### Quadratic formulae for certain quadratic matrix equations

Phil D. Young, Dean M. Young and Patrick L. Odell

Abstract

In this paper, we derive matrix quadratic formulae to solve quadratic matrix equations for various scenarios of the leading eigenvalues of the primary quadratic coefficient matrix. Additionally, we provide two examples for different cases of the primary quadratic coefficient matrix.

### On Bernstein's theorem

Tapas Kumar Chandra

Abstract

Some useful facts and related counterexamples are provided to highlight the different issues concerning the arguments used to prove Bernstein's theorem. Some unsolved problems are also included.

### Self-adjoint form and exact solutions

Behzad Salimi

Abstract

We apply a simple method to reduce certain forms of linear nonconstant coefficient second-order ordinary differential equations (ODEs) to constant coefficient form using a transformation derived from the self-adjoint form of the ODE. A transformation of the independent variable, using the coefficients of the self-adjoint form of the ODE, reduces certain differential equations to a constant coefficient form leading to exact analytic general solutions in closed form. These exact solutions are preferred to the existing solution forms in terms of special functions or polynomial-type series expansions. Among a number of linear ODEs solved, new analytic solutions for three differential equations of mathematical physics are introduced.

### Modeling uncertainties in performance of object recognition

Suresh Kumar, Bir Bhanu, Ninad S. Thakoor and Subir Ghosh

Abstract

Efficient probability modeling is indispensable for uncertainty quantification of the recognition data. If the model assumptions do not reflect the intrinsic nature of data and associated random variables, then a strong performance measure will most likely fail to come up with a correct match for recognition. In this paper we propose the probability models for two kinds of data obtained with two distinct goals of recognition: identification and discovery. We consider both frequentist and Bayesian approaches for drawing inferences from the data.

### Gelin–Cesàro identity for the Gibonacci family

Thomas Koshy

Abstract

We extend the well-known Gelin–Cesàro identity *F*_{n+2} *F*_{n+1} *F*_{n–1} *F*_{n–2} – *F*_{n}^{4} = –1 to the Gibonacci family. This generalization has interesting consequences.

### Exact soliton-like probability measures for interacting jump processes

M.-O. Hongler

Abstract

The cooperative dynamics of a one-dimensional collection of Markov jump, interacting stochastic processes is studied via a mean-field approach. In the time-asymptotic regime, the resulting nonlinear master equation is analytically solved. The nonlinearity compensates for jump-induced diffusive behavior, giving rise to a soliton-like stationary probability density. The soliton velocity and its sharpness both intimately depend on the interaction strength. Below a critical threshold of the strength of interactions, the cooperative behavior cannot be sustained, leading to the destruction of the soliton-like solution. The bifurcation point for this behavioral phase transition is explicitly calculated.