Signal Analytic Proofs of Two Basic Results on Lattice Expansions

Two theorems on lattice expansions

Basic analytic combinatorics of directed lattice paths

This paper develops a uni(ed enumerative and asymptotic theory of directed two-dimensional lattice paths in half-planes and quarter-planes. The lattice paths are speci(ed by a (nite set of rules that are both time and space homogeneous, and have a privileged direction of increase. (They are then essentially one-dimensional objects.) The theory relies on a speci(c “kernel method” that provides a...

On lattice of basic z-ideals

  For an f-ring  with bounded inversion property, we show that   , the set of all basic z-ideals of , partially ordered by inclusion is a bounded distributive lattice. Also, whenever  is a semiprimitive ring, , the set of all basic -ideals of , partially ordered by inclusion is a bounded distributive lattice. Next, for an f-ring  with bounded inversion property, we prove that  is a complemented...

Two-point Taylor Expansions of Analytic Functions

In deriving uniform asymptotic expansions of a certain class of integrals one encounters the problem of expanding a function, that is analytic in some domain Ω of the complex plane, in two points. The first mention of the use of such expansions in asymptotics is given in [1], where Airy-type expansions are derived for integrals having two nearby (or coalescing) saddle points. This reference doe...

Short Proofs of Two Basic Properties of Central Projections

In this note we prove two important properties of central projections, stated as Theorems A and B. They can be applied to obtain simple solutions of many hard problems in Euclidean geometry: for numerous examples see Yaglom [1, Ch. 1.3]. The proofs of these properties that I found in the literature were of two kinds: those which were completely elementary, assumed proficiency with Euclidean geo...