Detailed studies of four dimensional N= 2 superconformal field theories (SCFT) defined by isolated complete intersection singularities are performed: we compute the Coulomb branch spectrum, Seiberg-Witten solutions and central charges. Most of our theories have exactly marginal deformations and we identify the weakly coupled gauge theory descriptions for many of them, which involve (affine) D and E shaped quiver gauge theories and theories formed from Argyres-Douglas matters. These investigations provide strong evidence for the singularity approach in classifying 4d N= 2 SCFTs.

A method to obtain bounds for the number of solutions with regular behaviour at infinity of selfadjoint ordinary differential equations is given. In this method, which is presented for a polar problem, the sequence of nested ellipsoids in a previous paper of mine is replaced by the use of certain projections. At the same time I take the opportunity to complete the unsatisfactory bibliography of my earlier paper.

We introduce an irrational factor of order <i>k</i> defined by I k ( n ) = i = 1 l p i i , where I k ( n ) = i = 1 l p i i is the factorization of <i>n</i> and $${\beta_{i} = \left\{\begin{array}{ll}\alpha_i, \quad \quad {\rm if} \quad \alpha_i &lt; k \\ \frac{1}{\alpha_i},\quad \quad {\rm if} \quad \alpha_i \geqq k \end{array}\right.}$$. It turns out that the function I k ( n ) = i = 1 l p i i well approximates the characteristic function of <i>k</i>-free integers. We also derive asymptotic formulas for I k ( n ) = i = 1 l p i i and I k ( n ) = i = 1 l p i i .

Largescale multiple testing with correlated test statistics arises frequently in much scientific research. Incorporating correlation information in approximating the false discovery proportion (FDP) has attracted increasing attention in recent years. When the covariance matrix of test statistics is known, Fan and his colleagues provided an accurate approximation of the FDP under arbitrary dependence structure and some sparsity assumption. However, the covariance matrix is often unknown in many applications and such dependence information must be estimated before approximating the FDP. The estimation accuracy can greatly affect the FDP approximation. In the current paper, we study theoretically the effect of unknown dependence on the testing procedure and establish a general framework such that the FDP can be well approximated. The effects of unknown dependence on approximating the FDP are in the

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