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Multiple hypothesis testing is a fundamental problem in high dimensional inference, with wide applications in many scientific fields. In genome-wide association studies, tens of thousands of tests are performed simultaneously to find if any genes are associated with some traits and those tests are correlated. When test statistics are correlated, false discovery control becomes very challenging under arbitrary dependence. In the current paper, we propose a new methodology based on principal factor approximation, which successfully substracts the common dependence and weakens significantly the correlation structure, to deal with an arbitrary dependence structure. We derive the theoretical distribution for false discovery proportion (FDP) in large scale multiple testing when a common threshold is used and provide a consistent FDP. This result has important applications in controlling FDR and FDP. Our estimate of FDP compares favorably with Efron (2007)'s approach, as demonstrated by in the simulated examples. Our approach is further illustrated by some real data applications.

In this paper, we present an efficient attack on the multivariate Quadratic Quasigroups (MQQ) public key cryptosystem. Our cryptanalysis breaks the MQQ cryptosystem by solving a system of multivariate quadratic polynomial equations using both the MutantXL algorithm and the F4 algorithm. We present the experimental results that show that MQQ systems is broken up to size n equal to 300. Based on these results we show also that MutantXL solves MQQ systems with much less memory than the F4 algorithm implemented in Magma.

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Let$L$be a formally selfadjoint differential operator and$p$a real-valued function, both on$a$≤$x$<∞. The deficiency indices are the numbers of solutions of$Lu$=λ$pu$for Im λ>0 and for Im λ<0 which have a certain regularity at$x$=∞. (A) If$p(x)$≥0 this regularity means that the integral of$p(x)$│$u$│^{2}converges at infinity. (B) If$p$changes its sign for arbitrarily large values of$x$but$L$has a positive definite Dirichlet integral it is natural to relate the regularity to this integral. Weyl’s classical study of the deficiency indices is reviewed for (A) with the help of elementary theory of quadratic forms. Individual bounds are found for the deficiency indices also when$L$is of odd order. It is then indicated how the method carriers over to (B).