We equip $\mathrm{BP}(n)$ with an $\mathbb{E}_3\mathrm{BP}$-algebra structure for each prime $p$ and height $n$. The algebraic $K$-theory of this ring is chromatic exactly $n+1$, the map $\mathrm{K}(\mathrm{BP}\langle n\rangle )_{(p)} \rightarrow \mathrm{L}_{n+1}^f \mathrm{K}(\mathrm{BP}\langle n\rangle)_{(p)}$ has bounded above fiber.

We consider sequences of generalized Bell numbers B(n), n = 0, 1,. .. which can be represented by Dobi´nski-type summation formulas, i.e. B(n) = 1 C ∞ k=0 [P (k)] n D(k) , with P (k) a polynomial, D(k) a function of k and C = const. They include the standard Bell numbers (P (k) = k, D(k) = k!, C = e), their generalizations B r,r (n), r = 2, 3,. .. appearing in the normal ordering of powers of b...

Solution : Assume, without loss of generality |x2 − x1| ≥ 2−n+1. Then there exists an integer k ∈ {1, . . . , 2 − 1}, such that x1 < k · 2−n < (k + 1)2−n < x2. Of course PX((x1, x2)) ≥ P(k · 2−n ≤ X(ω) ≤ (k + 1)2−n) . (4) The integer k admits the unique binary expansion k = ∑n i=1 ki2 n−i. Then P(k · 2−n ≤ X(ω) ≤ (k + 1)2−n) = P(Cn,(k1...,kn)) = p 1(1− p)0 , (5) with n0(k) and n1(k) the number ...

Let n and n be respectively the uniform empirical and quantile processes , and deene R n = n + n , which usually is referred to as the Bahadur{Kiefer process. The well-known Bahadur{Kiefer theorem connrms the following remarkable equivalence: kR n k= p k n k n ?1=4 (log n) 1=2 almost surely, as n goes to innnity, where kfk = sup 0t1 jf(t)j is the L 1-norm. We prove that kR n k 2 = p k n k 1 n ?...

A k-factor of a graph G is a k-regular spanning subgraph of G. A k-factorization is a partition of E(G) into k-factors. Let K(n, p) be the complete multipartite graph with p parts, each of size n. If V1, ..., Vp are the p parts of V (K(n, p)), then a holey k-factor of deficiency Vi of K(n, p) is a k-factor of K(n, p)− Vi for some i satisfying 1 ≤ i ≤ p. Hence a holey k-factorization is a set of...