The Gaussian coefficient, also known as the q-binomial coefficient, is notated as Gauss(n, k)q and given by the following formula:
(qn−1)(qn−1−1) … (qn−k+1−1) / (qk−1)(qk−1−1) … (q−1).
The ordinary generating function for selecting at most one positive integer is:
1/(1−q) = 1 + q + q2 + q3 + …
The ordinary generating function for selecting exactly one positive integer is:
q/(1−q) = q + q2 + q3 + q4 + …
The ordinary generating function for selecting exactly one positive integer ≥ n is:
qn/(1−q) = qn + qn+1 + qn+2 + qn+3 + …
The ordinary generating function for selecting exactly one positive integer < n is:
q/(1−q) − qn/(1−q) = q + q2 + … + qn−2 + qn−1
(q − qn) / (1 − q) = q + q2 + … + qn−2 + qn−1
(qn − q) / (q − 1) = q + q2 + … + qn−2 + qn−1
The ordinary generating function for selecting at most one positive integer < n is:
(qn − 1) / (q − 1) = 1 + q + q2 + … + qn−2 + qn−1
The ordinary generating function for selecting at most one positive multiple of k is:
1/(1−qk) = 1 + qk + q2k + q3k + …
∅
The Bucks Stop Here
We have come to a critical point in the arc of democratic societies, where the idea that money can regulate itself, evangelized by the Church of the Invisible Hand, has tipped its hand to the delusion that money itself can regulate society.
Laßt uns rechnen …
One … Two … Three …
— Robert Musil • The Man Without Qualities
eiπ
Benjamin Peirce apparently liked this mathematical synonym for the additive inverse of 1 so much that he introduced three special symbols for e, i, π — ones that enable eiπ to be written in a single cursive ligature, as shown here.
This reminds me of some things I used to think about — I always loved
workingplaying with generating functions and I can remember a time in the 80s when I was very pleased with myself for working out the q-analogue of integration by parts — but it will probably take me a while to warm up those old gray cells today.Let me first see if I can get LaTeX to work in these comment boxes …
The Gaussian coefficient, also known as the q-binomial coefficient, is notated as Gauss(n, k)q and given by the following formula:
(qn−1)(qn−1−1) … (qn−k+1−1) / (qk−1)(qk−1−1) … (q−1).
Groups like
acting on the space of 
boolean functions on 
variables come up in my explorations of differential logic.
I’ve been having trouble posting links lately, so I will try to do that in a couple of separate comments.
Here is one indication of a context where those groups come up —
• See Table A3 in Differential Propositional Calculus : Appendix 1.
If you can get past my prolix efforts at popular exposition, here’s a discussion of the case where
• Differential Logic : Introduction • Operational_Representation
Notes
All Learning Is But Recollection
All Leaning Is But Reïnclination
And they will lean that way forever …
I lean that way myself, inclined to believe
All leaning inclines to preserve the swerve.
If
Then the Shadow falls in a moat
Between the castle of invention
And the undiscovered country.
If
Then the Shadow falls in a moat
Between the castle of invention
And the undiscovered country.
Plato, “Timaeus”, 38 A
Benjamin Jowett (trans.)
Try 9001 and 9002 = 〈 and 〉
Or 12296 and 12297 = 〈 and 〉
⟨ ⟩ = ⟨ ⟩ = ⟨ ⟩
Other possibilities short of using LaTeX —
〈 〉 = 〈 〉
〈 〉 = 〈 〉
If we are thinking about records of a fixed finite length
and a fixed signature 
then a relational data base is a finite subset 
of a 
-dimensional coordinate space 
Given a non-empty subset
of the indices ![K = [1, k],](http://s0.wp.com/latex.php?latex=K+%3D+%5B1%2C+k%5D%2C&bg=ffffff&fg=333333&s=0 "K = [1, k],")
we can take the projection 
of 
on the subspace 
Saying that “a query is likely to use only a few columns” amounts to saying that most of the time we can get by with the help of our small dimension projections. This is akin to a very old idea, having its ancestor in Descartes’ suggestion that “we should never attend to more than one or two” dimensions at a time.
cf. Château Descartes
Just a thought, more loose than lucid most likely —
There is another kind of “discrete logarithm” that I used to call the “vector logarithm” of a positive integer
Consider the primes factorization of 
and write the exponents of the primes in order as a coordinate tuple 
where 
for any prime 
not dividing 
and where the exponents are all 
after some finite point. Then multiplying two positive integers maps to adding the corresponding vectors.