The Loewy rank of a complete lattice L is deened as follows. Take the meet a 1 of all coatoms of L. Then let a 2 be the meet of all lower covers of a 1. Iterate this process to deene a for every ordinal by letting a +1 be the meet of all lower covers of a , and a the meet of all a (for <) if is a limit ordinal. The Loewy rank of L is the smallest ordinal for which a is the zero of L, and the sy...

We conjecture that satellite operations are either constant or have infinite rank in the concordance group. reduce this to difficult case of winding number zero satellites, and use $SO(3)$ gauge theory provide a general criterion sufficient for image operation generate an subgroup smooth group $\mathcal{C}$. Our applies widely; notably many unknotted patterns which corresponding operators on to...

We discuss rank one perturbationsAα = A+α(φ, ·)φ,α ∈ R , A ≥ 0 self-adjoint. Let dμα(x) be the spectral measure defined by (φ, (Aα − z)−1φ) = ∫ dμα(x)/(x− z). We prove there is a measure dρ∞ which is the weak limit of (1 + α)dμα(x) as α → ∞. If φ is cyclic for A, then A∞, the strong resolvent limit of Aα, is unitarily equivalent to multiplication by x on L(R , dρ∞). This generalizes results kno...