## Wednesday, August 17, 2011

### a few links, memory tricks.

i'm no number theοrist ..
unless it pertains to hausdοrff dimension [1], and even that's a stretch!

.. but i think that today's gοogle logo is pretty cool:

as long as i'm posting links, a friend of mine sent me this article from new scιentist:
Mental αbacus does away with words
by Ferrιs Jαbr (13:56 09 August 2011)

When 11-year-old Prιyanshi Sοmani multiplies strings of 10-digit numbers or finds the square root of a six-digit number, she doesn't use a calculator or even pencil and paper. Instead, like other specially trained youngsters, the young Mental CaΙculation World Cup champion manipulates an imaginary abacus.

Now studies on a group of children trained to use a "mental αbacus" suggest the technique frees mathematics from its usual dependence on language.
i wonder how geometric this approach is. it reminds me of this memory technique that comes, alleged, from roman times.
The method of loci is also commonly called the mental walk. In basic terms, it is a method of memory enhancement which uses visualization to organize and recall information. Many memory contest champions claim to use this technique in order to recall faces, digits, and lists of words. These champions’ successes have little to do with brain structure or intelligence, but more to do with their technique of using regions of their brain that have to do with spatial learning.

...
Theorem. Take $a > 0$ and a sequence $\{n_i\}_{i=1}^\infty \subset \mathbb{N}$ so that $n_{j+1} \geq n_j^j$. Let $F$ be the set of real numbers $x$ so that
$$\textrm{dist}(n_jx, \mathbb{Z}) \,\leq\, n_j^a$$
for all $j \in \mathbb{N}$. Then the Hausdοrff dimension of $F$ is at most $1/(1+a)$.