We determine which nilpotent orbits in E6 have closures which are normal varieties and which do not. At the same time we are able to verify a conjecture in [14] concerning functions on nonspecial nilpotent orbits for E6.

Let KG be the group ring of a group G over a commutative ring K with unity. The rings KG are described for which xxσ = xσx for all x = ∑ g∈G αgg ∈ KG, where x → xσ = ∑ g∈G αgf(g)σ(g) is an involution of KG; here f : G → U(K) is a homomorphism and σ is an antiautomorphism of order two of G. Let R be a ring with unity. We denote by U(R) the group of units of R. A (bijective) map : R → R is called...

The main results of this article are asymptotic formulas for the variance of the number of zeros of a Gaussian random polynomial of degree N in an open set U ⊂ C as the degree N → ∞, and more generally for the zeros of random holomorphic sections of high powers of any positive line bundle over any Riemann surface. The formulas were conjectured in special cases by Forrester and Honner. In higher...

Let f : C → C be a meromorphic function. We study the size of the maximal disc in C with respect to spherical metric, in which a singlevalued branch of f−1 exists. The problem is related to normality and type criteria. Best possible lower estimates of the size of such discs are obtained for entire functions and a class of meromorphic functions containing all elliptic functions. An estimate for ...

We consider conditioned Galton–Watson trees and show asymptotic normality of additive functionals that are defined by toll functions that are not too large. This includes, as a special case, asymptotic normality of the number of fringe subtrees isomorphic to any given tree, and joint asymptotic normality for several such subtree counts. The offspring distribution defining the random tree is ass...

The most important building blocks of symmetric cryptographic primitives such as the DES or the AES, are vectorial Boolean functions, also called S-boxes. In this paper, we extend the definition of normality for Boolean functions into several new affine invariant properties for vectorial Boolean functions. We compute the probability of occurrence of these properties and present practical algori...

hypothesis being tested with the WMW test is not easily described; conventional interpretation (including examples provided by statistical packages) would have it that the null hypothesis is one of equal group medians, but such is not the case; rather it is a test for equality of group mean ranks. Because the WMW test is a test of both location and shape, its interpretation as a test of medians...

Strong local minimizers with surfaces of gradient discontinuity appear in variational problems when the energy density function is not rank-one convex. In this paper we show that stability of such surfaces is related to stability outside the surface via a single jump relation that can be regarded as interchange stability condition. Although this relation appears in the setting of equilibrium el...

“Normality” of Stock Prices Bo Shi Abstract. The Black-Scholes Model, often simply called Black-Scholes, models the varying price of financial instruments over time: stocks in particular. This model assumes that returns on the underlying stock are lognormally distributed, which can be reasonable for many assets that offer options. However, from a selection of 100 stock histories, I found that a...