9 December 2009

Hasse diagrams

This is a piece of the class of amusing mathematical diversions.

I'm working on a problem involving random directed graphs and use the concept of a Hasse diagram: it's just a graph representing a partial order in a minimal way. I stumbled across a site which draws Hasse diagrams of the relation i divides j, where i and j are positive integers. I tried it for various numbers. For example, the Hasse diagram corresponding to the divisors of 2010 is a graph with constant degree equal to 4. Whereas 2009 does not have this property. Besides the obvious significance in numerology [yes, this is a joke], there is a natural question as to what kind of numbers have the property that their Hasse diagram has constant degree.

The page above is part of what seems to be a nice undergraduate book on Algebra, titled Interactive Algebra, by A.M. Cohen, H. Cuypers and H. Sterk.

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What measure theory is about

It's about counting, but when things get too large.
Put otherwise, it's about addition of positive numbers, but when these numbers are far too many.

The principle of dynamic programming

max_{x,y} [f(x) + g(x,y)] = max_x [f(x) + max_y g(x,y)]

The bottom line

Nuestras horas son minutos cuando esperamos saber y siglos cuando sabemos lo que se puede aprender.
(Our hours are minutes when we wait to learn and centuries when we know what is to be learnt.) --António Machado

Αγεωμέτρητος μηδείς εισίτω.
(Those who do not know geometry may not enter.) --Plato

Sapere Aude! Habe Muth, dich deines eigenen Verstandes zu bedienen!
(Dare to know! Have courage to use your own reason!) --Kant