Random Matrix Theory (RMT) has successfully modeled diverse systems, from energy levels of heavy nuclei to zeros of L-functions; this correspondence has allowed RMT to successfully predict many number theoretic behaviors. However there are some operations which to date have no RMT analogue. Our motivation is to find an RMT analogue of Rankin-Selberg convolution, which constructs a new L-functions from an input pair. We report one such attempt; while it does not appear to model convolution, it does create new ensembles with properties hybridizing those of its constituents.For definiteness we concentrate on the ensemble of palindromic real symmetric Toeplitz (PST) matrices and the ensemble of real symmetric matrices, whose limiting spectral measures are the Gaussian and semicircular distributions, respectively; these were chosen as they are the two extreme cases in terms of moment calculations. For a PST matrix A and a real symmetric matrix B, we construct an ensemble of random real symmetric block matrices whose first row is {A, B} and whose second row is {B, A}. By Markov's Method of Moments and the use of free probability, we show this ensemble converges weakly and almost surely to a new, universal distribution with a hybrid of Gaussian and semi-circular behaviors. We extend this construction by considering an iterated concatenation of matrices from an arbitrary pair of random real symmetric sub-ensembles with different limiting spectral measures. We prove that finite iterations converge to new, universal distributions with hybrid behavior, and that infinite iterations converge to the limiting spectral measure of the dominant component matrix.

A curious number is a palindromic number whose base ten representation has the form a . . . ab . . . ba . . . a. In this paper, we determine all curious numbers that are perfect squares. Our proof involves reducing the search for such numbers to several single variable families. From here, we complete the proof in two different ways. The first approach is elementary, though somewhat ad hoc. The second entails studying integral points on elliptic curves and is more systematic.

In this paper we study the minimal faithful permutation representations of SL n (F q ) and GL n (F q ).