A Weyl group generating function that ought to be better known

A Weyl Group Generating Function That Ought to Be Better Known

Let W be a nite Weyl group with root system , simple roots 1 coordinates for the sum of all positive roots, and for w 2 W let`(w) denote the length and D(w) = fi : w ?1 i < 0g the descent set of w. By analyzing the structure of the corresponding aane Weyl group, we prove that P w2W q (w)?`(w) = f Q n i=1 (1 ? q b i)=(1 ? q e i), where (w) = P i2D(w) b i .

Some Facts About Continued Fractions That Should Be Better Known

In this report I will give proofs of some simple theorems concerning continued fractions that are known to the cognoscenti, but for which proofs in the literature seem to be missing, incomplete, or hard to locate. In particular, I will give two proofs of the following “folk theorem”: if θ is an irrational number whose continued fraction has bounded partial quotients, then any non-trivial linear...

‘ Ought ’ and ‘ Better ’

Sentences using ‘ought’ appear to have some kind of non-descriptive force (e.g. commending, guiding actions, committing the speaker), in which case it would seem that no ‘ought’ statement can follow logically from purely factual statements lacking such force. Yet from factual statements about slowness and liability to breakdown of alternative means of transport it seems possible validly to infe...

There ought to be a law?

Poorly differentiated clusters (PDC) in colorectal cancer: what is and ought to be known

BACKGROUND The counting of poorly differentiated clusters of 5 or more cancer cells lacking a gland-like structure in a tumor mass has recently been identified among the histological features predictive of poor prognosis in colorectal cancer. MAIN BODY Poorly differentiated clusters can easily be recognized in the histological sections of colorectal cancer routinely stained with haematoxylin ...