This article presents two expectation identities and a series of applications. One of the identities uses the heat equation, and we show that in some families of distributions the identity characterizes the normal distribution. We also show that it is essentially equivalent to Stein's identity. The applications we have presented are of a broad range. They include exact formulas and bounds for m...

A constructive version of a theorem of Thue is used to provide representations of certain integers as x2 −Dy2, where D = 2, 3, 5, 6, 7.

This paper presents an exact formula for calculating the fourth-moment tensor from the second-moment tensor for the three dimensional Jeffery’s equation. Although this approach falls within the category of a moment tensor closure, it does not rely upon an approximation, either analytic or curve fit, of the fourth-moment tensor as do previous closures. This closure is orthotropic in the sense of...

We solve the inhomogeneous Chebyshev’s differential equation and apply this result to obtain a partial solution to the Hyers-Ulam stability problem for the Chebyshev’s differential equation.

Let H be an infinite--dimensional Hilbert space and K(H) be the set of all compact operators on H. We will adopt spectral theorem for compact self-adjoint operators, to investigate of higher derivation and higher Jordan derivation on K(H) associated with the following cauchy-Jencen type functional equation 2f(frac{T+S}{2}+R)=f(T)+f(S)+2f(R) for all T,S,Rin K(H).

Let H be an innite dimensional Hilbert space and K(H) be the set of all compactoperators on H. We will adopt spectral theorem for compact self-adjoint operators, to investigate ofhigher derivation and higher Jordan derivation on K(H) associated with the following Cauchy-Jensentype functional equation 2f((T + S)/2+ R) = f(T ) + f(S) + 2f(R) for all T, S, R are in K(...

The idea of ‘life after death’ transcends philosophy or religion, as science can test predictions from claims by both its advocates and skeptics. This study therefore featured two researchers with opposite views, who jointly gathered hundreds research studies to evaluate the maximum average percentage effect that seemingly supports (i.e., anomalous effects) refutes known confounds) survival hyp...