The lowering operator σ associated with a polynomial set {P n } n≥0 is an operator not depending on n and satisfying the relation σP n = nP n−1. In this paper, we express explicitly the linearization coefficients for polynomial sets of Sheffer type using the corresponding lowering operators. We obtain some well-known results as particular cases.

We outline a new algorithm to solve coupled systems of differential equations in one continuous variable x (resp. coupled difference equations in one discrete variable N) depending on a small parameter ε: given such a system and given sufficiently many initial values, we can determine the first coefficients of the Laurent-series solutions in ε if they are expressible in terms of indefinite nest...

Let S denote a stationary subset of limit ordinals of ω1. A ladder system on S is a sequence {Lα : α ∈ S} such that each Lα is an unbounded subset of α of order type ω. A ladder system is uniformizable if for each sequence 〈fα : α ∈ S〉 of functions fα : Lα → ω there is an F : ω1 → ω such that F Lα =∗ fα for each α ∈ S. I.e., for each α ∈ S, {β ∈ Lα : F (β) 6= fα(β)} is finite. We now formulate ...

The full causal ladder of spacetimes is constructed, and their updated main properties are developed. Old concepts and alternative definitions of each level of the ladder are revisited, with emphasis in minimum hypotheses. The implications of the recently solved " folk questions on smoothability " , and alternative proposals (as recent isocausality), are also summarized.

The undeformed limit of the dilute two-color braid-monoid algebra gives a natural basis for the description of spin-1/2 ladder models, and allows different Baxterization Ansätze. Based on this observation we find an entire class of exactly soluble generalized isotropic spin-1/2 ladder models.

We prove a formula, conjectured by Conca and Herzog, for the number of all families of nonintersecting lattice paths with certain starting and end points in a region that is bounded by an upper ladder. Thus we are able to compute explicitly the Hilbert series for certain one-sided ladder determinantal rings.

The full causal ladder of spacetimes is constructed, and their updated main properties are developed. Old concepts and alternative definitions of each level of the ladder are revisited, with emphasis in minimum hypotheses. The implications of the recently solved " folk questions on smoothability " , and alternative proposals (as recent isocausality), are also summarized.

Extension ladder: The successful application of epoxide-opening strategies towards the synthesis of ladder-type polyethers is contingent upon further elaboration of the product. By employing two different functionalized templates, a fragment of gymnocin A that bears four sites for subsequent fragment coupling has been prepared (see scheme; Bn = benzyl).

........................................................................................................................................................ 1 Acknowledgements ...................................................................................................................................... iv Chapter 1: Introduction ..................................................................