We establish a coloring theorem for successors of singular cardinals, and use it prove that for any such cardinal μ, we have μ+ 9 [μ+]2 μ+ if and only if μ+ 9 [μ]θ for arbitrarily large θ < μ.

For a general measure space (Ω, μ), it is shown that for every band M in Lp(μ) there exists a decomposition μ = μ ′ + μ′′ such that M = Lp(μ ′) = {f ∈ Lp(μ); f = 0 μ′′-a.e.}. The theory is illustrated by an example, with an application to absorption semigroups. MSC 2010: 46B42, 28A05, 47D06

A graph G has constant μ = μ(G) if any two vertices that are not adjacent have μ common neighbours. G has constant μ and μ if G has constant μ = μ(G), and its complement G has constant μ = μ(G). If such a graph is regular, then it is strongly regular, otherwise precisely two vertex degrees occur. We shall prove that a graph has constant μ and μ if and only if it has two distinct restricted Lapl...

For a cardinal μ we give a sufficient condition ⊕μ (involving ranks measuring existence of independent sets) for: ⊗μ: if a Borel set B ⊆ R × R contains a μ-square (i.e. a set of the form A × A, |A| = μ) then it contains a 20 -square and even a perfect square. And also for ⊗′μ: if ψ ∈ Lω1,ω has a model of cardinality μ then it has a model of cardinality continuum generated in a “nice”, “absolute...

In this report I review recent results in the field of rare decays at the Tevatron CDF II and DØ experiments. The presentation is focused on rare decays of charm and bottom mesons with two muons in the final state. This includes improvements over the previously available limits on the following branching ratios: B(D+ → π+μ+μ−) < 4.7× 10−6, B(Bs → φ μ+μ−) < 3.2×10−6, B(Bs → μ+μ−) < 1×10−7, and B...

Example For example, let X 2 1, ..., Xn be a random sample from distribution N(μ, σ ) with σ known and μ ∈ M. Suppose our hypothesis is that μ ∈ M1 for some M1 ⊂ M, i.e. M1 is some subset of M. It is called the null hypothesis. It is denoted as H0 : μ ∈ M1. Then the alternative hypothesis is that μ ∈/ M1, i.e. μ ∈ M\M1. It is denoted as Ha : μ ∈/ M\M1. For example, if M1 = μ0 and M = {μ : μ ≥ μ...

Alkylation of [{Ti(η(5)-C(5)Me(5))(μ-NH)}(3)(μ(3)-N)] with MeOTf occurs at the imido ligands to produce the methylamido derivative [Ti(3)(η(5)-C(5)Me(5))(3)(μ(3)-N)(μ-NH)(2)(μ-NHMe)(OTf)] which readily rearranges to form the methylimido complex [Ti(3)(η(5)-C(5)Me(5))(3)(μ(3)-N)(μ-NH)(μ-NH(2))(μ-NMe)(OTf)].

We prove that if μ is a regular cardinal and P is a μ-centered forcing poset, then P forces that (I[μ++])V generates I[μ++] modulo clubs. Using this result, we construct models in which the approachability property fails at the successor of a singular cardinal. We also construct models in which the properties of being internally club and internally approachable are distinct for sets of size the...