The Bernstein Projector Determined by a Weak Associate Class of Good Cosets

Let ${\text G}$ be a reductive group over a $p$-adic field $F$ of characteristic zero, with $p \gg 0$, and let $G={\text G}(F)$. In [ 15], J.-L. Kim studied an equivalence relation called weak associativity on the set of unrefined minimal $K$-types for ${\text G}$ in the sense of A. Moy and G. Prasad. Following [ 15], we attach to the set $\overline{\mathfrak{s}}$ of good $K$-types in a weak associate class of positive-depth unrefined minimal $K$-types a ${G}$-invariant open and closed subset $\mathfrak{g}_{\overline{\mathfrak{s}}}$ of the Lie algebra $\mathfrak{g} = {\operatorname{Lie}}({\text G})(F)$, and a subset $\tilde{{G}}_{\overline{\mathfrak{s}}}$ of the admissible dual $\tilde{{G}}$ of ${G}$ consisting of those representations containing an unrefined minimal $K$-type that belongs to $\overline{\mathfrak{s}}$. Then $\tilde{{G}}_{\overline{\mathfrak{s}}}$ is the union of finitely many Bernstein components of ${G}$, so that we can consider the Bernstein projector $E_{\overline{\mathfrak{s}}}$ that it determines. We show that $E_{\overline{\mathfrak{s}}}$ vanishes outside the Moy–Prasad ${G}$-domain ${G}_r \subset{G}$, and reformulate a result of Kim as saying that the restriction of $E_{\overline{\mathfrak{s}}}$ to ${G}_r\,$, pushed forward via the logarithm to the Moy–Prasad ${G}$-domain $\mathfrak{g}_r \subset \mathfrak{g}$, agrees on $\mathfrak{g}_r$ with the inverse Fourier transform of the characteristic function of $\mathfrak{g}_{\overline{\mathfrak{s}}}$. This is a variant of one of the descriptions given by R. Bezrukavnikov, D. Kazhdan, and Y. Varshavsky in [8] for the depth-$r$ Bernstein projector.

Three-associate class partially balanced incomplete block designs through kronecker product

Kronecker product of matrices can be advantageously used to obtain three-associate class partially balanced incomplete block (PBIB) designs from two-associate class PBIB designs, for a larger number of treatments. This method has been described here and two such series of PBIB designs are obtained. If the experimenter is constrained of resources, theses designs can be used as an alternative to balanced incomplete block designs or 2-associate class partially balanced incomplete block designs.

Associate class graph of zero-divisors of a commutative ring

In this paper, we introduce the concept of the associate class graph of zero-divisors of a commutative ring [Formula: see text], denoted by [Formula: see text]. Some properties of [Formula: see text], including the diameter, the connectivity and the girth are investigated. Utilizing this graph, we present a new class of counterexamples of Beck’s conjecture on the chromatic number of the zero-divisor graph of a commutative ring.