A chondroprotective joint effect was demonstrated in the healed menisci repairs, which warrants the procedure in select patients. The long-term evaluation of the anterior cruciate ligament-reconstructed knees with concurrent successful meniscal repairs demonstrated a low rate of radiographic arthritis.

Recently Burkhardt et. al. introduced the k-checkerboard random matrix ensembles, which have a split limiting behavior of the eigenvalues (in the limit all but k of the eigenvalues are on the order of √ N and converge to semi-circular behavior, with the remaining k of size N and converging to hollow Gaussian ensembles). We generalize their work to consider non-Hermitian ensembles with complex eigenvalues; instead of a blip new behavior is seen, ranging from multiple satellites to annular rings. These results are based on moment method techniques adapted to the complex plane as well as analysis of singular values.

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The theory of monstrous moonshine asserts that the coefficients of Hauptmoduln, including the j-function, coincide precisely with the graded characters of the monster module, an infinite-dimensional graded representation of the monster group. On the other hand, Lehner and Atkin proved that the coefficients of the j-function satisfy congruences modulo p n for p ∈ {2, 3, 5, 7, 11}, which led to the theory of p-adic modular forms. We combine these two aspects of the j-function to give a general theory of congruences modulo powers of primes satisfied by the Hauptmoduln appearing in monstrous moonshine. We prove that many of these Hauptmoduln satisfy such congruences, and we exhibit a relationship between these congruences and the group structure of the monster. We also find a distinguished class of subgroups of the monster with graded characters satisfying such congruences. and the right-hand sides are simple sums involving the dimensions of the irreducible representations of the monster group M: