One of the most famous results Dedekind is transformation law $$\log \Delta (z)$$ . After a half-century, Rademacher modified Dedekind’s result and introduced an $${{\,\textrm{SL}\,}}_2({\mathbb {Z}})$$ -conjugacy class invariant (integer-valued) function $$\Psi (\gamma )$$ called symbol. Inspired by Ghys’ work on modular knots, Duke–Imamo?lu–Tóth (2017) constructed hyperbolic analogue In this ...

Let (" i) i1 be a sequence of independent Rademacher variables, and let (B t) t0 be a standard Brownian motion. Then the conjecture made in [5] that the best constant in the maximal Khintchine inequality:

The main idea of upper bounding (3) is to use Rademachers. A Rademacher variable σ simply takes the two values +1 and −1 each with probability 1/2. Let σ1, . . . , σn be n independent Rademachers that are also independent of X1, . . . , Xn and X ′ 1, . . . , X ′ n. For each i, note that the distribution of f(Xi)−f(X ′ i) is the same as the distribution of f(X ′ i)− f(Xi). Therefore, the distrib...

By the above definitions we know, ΠF (n) ≤ |F| ΠF (n) ≤ |2| Definition 1.2 (Shattering). A hypothesis class F shatters a finite set S ⊂ X , iff |SF (S)| = 2|S| In an informal language, shattering means that you have to be able to separate all +/labelings of the same set of points, given the function class. We want to bound Rademacher average using a function of VC-dimension. Lemma 1.1. Let F be...

The two conjectures are, of course, equivalent. If 2n — 2 = pi+p2, then 2n = pi+p2+2 and 2n + l = pi+p2+3Conversely, if 2n=pi -\-p2-\-pz one of the primes must be 2 and 2n — 2 = pi+p2. An impressive collection of numerical evidence indicating the truth of the conjectures has accumulated in the years since Goldbach's letter was written, but is it not known to this day whether the conjectures are...