The Gaussian coefficient, also known as the q-binomial coefficient, is notated as Gauss(nk)q and given by the following formula:

(qn−1)(qn−1−1) … (qn−k+1−1) / (qk−1)(qk−1−1) … (q−1).

$\text{Gauss}(n, k)_q = \frac{(q^n - 1)(q^{n-1} - 1) \ldots (q^{n-k+1} - 1)}{(q^k - 1)(q^{k-1} - 1) \ldots (q - 1)}$

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

(qqn) / (1 − q)   =   q + q2 + … + qn−2 + qn−1

(qnq) / (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 + …

1. Jon Awbrey says:

2. Jon Awbrey says:

The Bucks Stop Here

3. Jon Awbrey says:

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 …

OneTwoThree

4. Jon Awbrey says:

The well-known capacity that thoughts have — as doctors have discovered — for dissolving and dispersing those hard lumps of deep, ingrowing, morbidly entangled conflict that arise out of gloomy regions of the self probably rests on nothing other than their social and worldly nature, which links the individual being with other people and things; but unfortunately what gives them their power of healing seems to be the same as what diminishes the quality of personal experience in them.

5. Jon Awbrey says:

e

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 e to be written in a single cursive ligature, as shown here.

6. Jon Awbrey says:

7. Jon Awbrey says:

8. Jon Awbrey says:

$\text{Gauss}(n, k)_q = \frac{(q^n - 1)(q^{n-1} - 1) \ldots (q^{n-k+1} - 1)}{(q^k - 1)(q^{k-1} - 1) \ldots (q - 1)}$

9. Jon Awbrey says:

This reminds me of some things I used to think about — I always loved working playing 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).

$\text{Gauss}(n, k)_q = \frac{(q^n - 1)(q^{n-1} - 1) \ldots (q^{n-k+1} - 1)}{(q^k - 1)(q^{k-1} - 1) \ldots (q - 1)}$

10. Jon Awbrey says:

Groups like $Z_2^m$ acting on the space of $2^{2^m}$ boolean functions on $m$ 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 $m=2$

11. Jon Awbrey says:

Notes

12. Jon Awbrey says:

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 $\exists \frac{\sum}{\varnothing \nu}$
Then the Shadow falls in a moat
Between the castle of invention
And the undiscovered country.

If $\exists \frac{\bigodot}{\varnothing \nu}$
Then the Shadow falls in a moat
Between the castle of invention
And the undiscovered country.

13. Jon Awbrey says:
Anamnesis
Learning = Recollection
Maieusis
Teaching = Midwifery
Communication = Pre-Established Harmony
Symbology
Meaning = Interpretation
14. Jon Awbrey says:

These are the forms of time,
which imitates eternity and
revolves according to a law
of number.

Plato, “Timaeus”, 38 A
Benjamin Jowett (trans.)

15. Jon Awbrey says:

Try 9001 and 9002 = 〈 and 〉

Or 12296 and 12297 = 〈 and 〉

16. Jon Awbrey says:

&lang; &rang; = &‍#10216; &‍#10217; = ⟨ ⟩

Other possibilities short of using LaTeX —

&‍#9001; &‍#9002; = 〈 〉

&‍#12296; &‍#12297; = 〈 〉

17. Jon Awbrey says:

If we are thinking about records of a fixed finite length $k$ and a fixed signature $X_1, \ldots, X_k$ then a relational data base is a finite subset $D$ of a $k$-dimensional coordinate space $X = X_1 \times \ldots \times X_k = \prod_{j=1}^k X_j.$

Given a non-empty subset $J$ of the indices $K = [1, k],$ we can take the projection $\text{proj}_J$ of $D$ on the subspace $X_J = \prod X_{j \in J} X_j.$

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.

18. Jon Awbrey says:

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 $n.$ Consider the primes factorization of $n$ and write the exponents of the primes in order as a coordinate tuple $(e_1, \ldots, e_k, 0, 0, 0, \ldots),$ where $e_j = 0$ for any prime $p_j$ not dividing $n$ and where the exponents are all $0$ after some finite point. Then multiplying two positive integers maps to adding the corresponding vectors.

19. Jon Awbrey says:
Charters? ...... ☐ Y ☐ N
Common Core? ... ☐ Y ☐ N
Disruption? .... ☐ Y ☐ N
Technology? .... ☐ Y ☐ N