PROOF. It suffices, by standard facts about probability measure, to prove the claim for cylinders C(w′) ⊆ ′, where w′ = w0, . . . wk. We use induction on k. The base case (k = 0) follows from Lemma 7. Namely, C( ) = ′, and ρ−1( ′) = \ ρ−1( ). Thus Pr (ρ−1( ′)) = 1 − Pr (ρ−1( )) = 1. For the induction step, suppose that the claim holds for the prefix w′ = w0w1 . . . wk. Let D[w′] = ρ−1(C(w′)). D...

Apidaecins, short proline-arginine-rich peptides from insects, are highly bactericidal through a mechanism that includes stereoselective elements but is completely devoid of any pore-forming activity. The spectrum of antibacterial activity, always limited to Gram-negatives, is further dependent on a small number of variable residues and can be manipulated. We show here that mutations in the evo...

A basic problem that arises often in the design and analysis of randomized algorithms is to get a good estimate (upper bound, lower bound, or both) of Pr( ∨ i∈[m]Ei), for some given events Ei. Equivalently, a good lower bound, upper bound or both, is required for Pr( ∧ i∈[m]Ei). As we have seen, one approach would be to use the union bound (“counting sieve”), which is unfortunately quite weak i...

Let R be a commutative ring with identity. An ideal I in R is called a multiplication ideal if every ideal contained in I is a multiple of I. Ohm's properties for nitely generated ideals in Pr ufer domains are investigated by Gilmer. These properties are generalized by Naoum and studied for nitely generated multiplication ideals. The purpose of this work is to generalize the results of Gilmer a...

There are many indices for measuring property rights (PR) security. They may be classified into two groups: subjective proxies and objective proxies. Most of the proxies are subjective. One contribution of this paper is introducing a new approach for measuring government expenditures devoted to PR protection. Calculating physical capital within the introduced new approach is another contributio...