Let G be a split connected reductive group over a finite field Fp with algebraic closure k, fix an Iwahori subgroup B ⊂ G(Fp([[t]])) and let F l = G(k((t)))/Bk denote the affine flag variety of G. Let q denote a power of p and fix a prime l 6= p. In [GH], the authors study the Jordan-Hölder series for objects in the Hecke category P q (F l,Ql). This is the category of B-equivariant perverse Wei...

Let D denote the open unit disk {z ∈ C | |z| < 1}, and C+ denote the right half-plane {s ∈ C | Re(s) ≥ 0}. (1) Let W (D) be the Wiener algebra of the disc, that is the set of all absolutely convergent Taylor series in the open unit disk D, with pointwise operations. (2) Let W (C+) be the set of all functions defined in the right halfplane C+ that differ from the Laplace transform of a function ...

Let R = ⊕(u,v)∈N2R(u,v) be a standard bigraded algebra over an artinian local ring K = R(0,0). Standard means R is generated over K by a finite number of elements of degree (1, 0) and (0, 1). Since the length l(R(u,v)) of R(u,v) is finite, we can consider l(R(u,v)) as a function in two variables u and v. This function was first studied by van der Waerden [W] and Bhattacharya [B] who proved that...

Simple continued fractions, base-b expansions, Dedekind cuts and Cauchy sequences are common notations for number systems. In this note, first, it is proven that both simple continued fractions and base-b expansions fail to denote real numbers and thus lack logic; second, it is shown that Dedekind cuts and Cauchy sequences fail to join in algebraical operations and thus lack intuition; third, w...