The estimation of ruin probability has been the central topic in insurance risk theory. In this paper we study the asymptotic behavior of the probability of ruin and the probability that ruin occurs before the end of planning years in the compound Poisson model. The exponential bounds for both probabilities are found to be functions of the rate function in traditional large deviation theory.

The existence of primality criteria for generic pairs $n$ and $n+d$ is investigated. A congruence $\pmod {n(n+d)}$ is found, that holds if and only if $(n,n+d)$ is a prime pair, except for a finite number of exceptions that appear when $n$ is lower than a fixed quantity only depending on $d$. Explicit primality criteria for $d = 2,4,6,8,10,12$ are given and a formula predicting the number of exceptions is conjectured.

We present a new proof, using Residue Calculus, of the transformation law of a general eta product under $\Gamma_0(n)$ where $n$ is any integer, then we deduce the invariance of a special case of this product under this group and we prove the transformation law of another special case. Our proof is inspired by Siegel's proof [7] of the transformation law of the Dedekind eta function and by Rademacher's generalization [5].

The group $G(\mathcal{A})$ of the invertible elements of a Banach algebra $\mathcal{A}$ can be pre-ordered by a closed semigroup $S$. We prove a comparison theorem for ODEs of the form $u'=f_1(t,u) u + u f_2(t,u)$ in Banach algebras under the assumption that $S$ is permutation stable. Applications to monotonicity properties of initial value problems and dynamical systems are given.

The parallel frame of a curve is an alternative approach to defining a moving frame. In this paper, we express the parallel frames of timelike and spacelike curves in a Minkowski space.

An inverse problem for the identification of an unknown spatially dependent coefficient in a parabolic partial differential equation is considered as an application for this new technique. An integral identity which explicitly relates changes in coefficients to changes in measured data is presented. Using this identity, it is possible to show that the coefficient to data map is continuous, strictly monotone and injective. Applying a modified Backus-Gilbert method to this identity generates a sequence of coefficients converging to the required unknown coefficient. Finally, implementation of the procedure is discussed and some numerical experiments are displayed.

An alternative (hitherto unknown) representation is derived for Jacobi polynomials.