Archive for the ‘Notations’ Category

Photo on 8-7-20 at 11.37 AM.jpg

up top, th’ “2-string saints”—
you know the one… how i want
to be in that number… oh when
the saints go mar, ching, in.
that one.

below that, fresh today (and indeed
*unfinished* if i have anything to
say about it): drone-string willie.

here is the you-tube: blind willie mc~tell
with guitar-giant mark knopfler
on guitar & dylan on piano & vocals.

aren’t they great. back to me.
the thing here is, there’s this
guitar right here with only two
strings. and, as it turns out,
even *that’s* too complicated
(for my purpose right now): so.
let the low (“6th”) string just
*drone on* and bang out the melody
on the high string (the “5th”).

one can use big sweeping right-hand
“strums” in doing this; much easier
than some brain-torture right-hand
*finger-picking* arrangement, say,
and more *fun* (and, undoubtedly,
more *filmic* in case one should
ever stand on a stage again…). but
really, the point… *a* point…
is that one can begin to get some
*feeling* into the g-d d-mn thing.

like all blues songs this songs about
how much it hurts to play this song.


Photo on 8-6-20 at 12.51 PM.jpg

a kid with a hammer thinks
everything looks like a nail
when you most need to succeed
that’s when you’ll fail
then when you knead the dough
the check’s in the mail
so you can’t raise the bail
and you rot in jail

if you give just a little
they want a lot
then they’ll be back for stuff
you ain’t even got
you get to the end of the page
and make a big ink-blot
right when a thing gets ripe
it starts to rot
you do what they told you to do
to get through but ya don’t
you get er alone an yr hopin she will
but she won’t
just when you think you can’t lose
you find you can’t win
just when you think you’re out
they pull ya back in

i don’t know the name of the tune,
so here it is in one-string code.

0 0 E 9 7 5
E E 9 7 5 4
9 9 7 5 4 2
2 4 4 5 7
the bold-face means “+12”, btw.

of course i’m not going to try to render
the three-string “tabs” into HTML.
but they’ve been there the longest.
the single-string arrangement typed out
above, & found at the bottom of the page,
came quite bit later. the lyrics were
in-between. anyhow. what i *haven’t*
done here… but have recently *taken*
to doing… is to put in the “fingering”.
as you can kind of see in this post
from earlier today, the notation here
is {o, i, m, a} for “open”, “index”,
“middle”, and “annular” (i.e. “ring”).
it meant something else when i learned it
from a pro guitar teacher but never mind.
it works if you work it.

g-string birthday

G A G B . . . )

Photo on 8-6-20 at 12.51 PM #2.jpg

from blank file-folder (and no idea)
to conceived, drafted, penciled, and inked
before finishing my third cup of coffee:
behold: MEdZ # (1+i+j+k)/2!—
th’ G-mod-H issue!! in which we can see
no less than *four* (count ’em) more-or-less
familiar examples of “modding by a subgroup”.

(-+) {E,O}—even & odd: the ol’ cosmic yin-yang…
(++) time considered as an endless spiral of half-days…
(- -) the “unit circle” & the “periodic functions”…
(+-) time considered as an endless spiral of weeks.

one can easily look up the (- -) diagram expanded
at arbitrary length in textbooks considering “trig”.
and the helices of endless time are too familiar
to say much more about. the “clock face”, though,
is a bottomless well of shorthand examples—there’s
a car trying to run us off the road at three o’clock—
and so might be worth some further development
if it were useful in, say, fixing notations.

but (-+) one is prepared to go on about at any length.
\Bbb{Z}_2 = \{0, 1\} is one of the most useful finite sets there is,
after all. for example, the 64 hexagrams of the i-ching
amount at some level to \Bbb{Z}_2^6 displayed pleasingly.
in the zine by me about “the 64 things” the hexagrams are
replaced with “subgraphs of K_4” (where of course K_4 is
the “complete graph on four points”; a diagram having
six edges [giving the six “lines” of the ching in that
version]; the 64 things are then subsets-a-six-set
[say {y, b, r, p, o, g} for the “full color” version]).
for example. i hope to continue in this vein later.
hello out there. ☰☱☲☳☴☵☶☷
Photo on 6-20-20 at 9.41 AM

B =\begin{pmatrix}  \begin{pmatrix} \begin{pmatrix} 0 & 2 \\ 1 & 2\end{pmatrix}& \begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}& \\ \begin{pmatrix} 1 & 0 \\ 2 & 1\end{pmatrix}& \begin{pmatrix} 2 & 2 \\ 1 & 0\end{pmatrix} \end{pmatrix} & \begin{pmatrix} \begin{pmatrix} 1 & 2 \\ 1 & 0\end{pmatrix}& \begin{pmatrix} 2 & 1 \\ 0 & 2\end{pmatrix}& \\ \begin{pmatrix} 2 & 0 \\ 2 & 2\end{pmatrix}& \begin{pmatrix} 0 & 2 \\ 1 & 1\end{pmatrix} \end{pmatrix} \\  \begin{pmatrix} \begin{pmatrix} 0 & 1 \\ 2 & 2\end{pmatrix}& \begin{pmatrix} 1 & 0 \\ 1 & 1\end{pmatrix}& \\ \begin{pmatrix} 1 & 2 \\ 0 & 1\end{pmatrix}& \begin{pmatrix} 2 & 1 \\ 2 & 0\end{pmatrix} \end{pmatrix} & \begin{pmatrix} \begin{pmatrix} 1 & 1 \\ 2 & 0\end{pmatrix}& \begin{pmatrix} 2 & 0 \\ 1 & 2\end{pmatrix}& \\ \begin{pmatrix} 2 & 2 \\ 0 & 2\end{pmatrix}& \begin{pmatrix} 0 & 1 \\ 2 & 1\end{pmatrix} \end{pmatrix} \end{pmatrix}

now let
B=\begin{pmatrix} II & I\\ III & IV\end{pmatrix}, i.e., let
II = \begin{pmatrix} \begin{pmatrix} 0 & 2 \\ 1 & 2\end{pmatrix}& \begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}& \\ \begin{pmatrix} 1 & 0 \\ 2 & 1\end{pmatrix}& \begin{pmatrix} 2 & 2 \\ 1 & 0\end{pmatrix} \end{pmatrix}, etc., so that
I, II, III, and IV (the “quadrants”) denote
two-by-twos of two-by-twos.

we’ll call the smallest matrices in sight “points”.
h =\begin{pmatrix} 2 & 1 \\ 0 & 2\end{pmatrix};
the quadrants are now two-by-two arrays of “points”.

on the graphical models i’ve been going on about all week
the quadrants are (respectively)
I ++
II -+
IV +-
(as is familiar to every calculus student);
expanding on the same logic gives the “trit-code”
for the individual points. for example,
h = ++++
and so on.
one then simply translates the 16 trit-strings
++++, +++-, ++-+, … —-
into “hurwitz units” like
h = {{1+i+j+k}\over2};

oh, yeah. i forgot to say. the “points” are
matrices *mod 3* (so that 2 = -1). that is all.

Q/{\pm 1} \iso K_4

the (so-called) fundamental quaternion units
can be represented as
\pm 1 =\pm\begin{pmatrix} 1&0\\0&1\end{pmatrix}, \pm i =\pm\begin{pmatrix} 0&1\\2&0\end{pmatrix}
\pm j =\pm\begin{pmatrix} 1&1\\1&2\end{pmatrix}, \pm k =\pm\begin{pmatrix} 1&2\\2&2\end{pmatrix},

with the scalars of the matrices considered
as elements of \Bbb{F}_3—i.e., 2=-1 etc.

“modding out” the \pm1 gives \Bbb{Z}_2^2
the klein-four group { (0,0), (1,0), (0,1), (1,1) }
(with componentwise mod-2 addition).

typing out matrices is sort of tedious. i won’t be doing
\hat{A_4} today.

announcing the name change:
Virtual MEdZ.

Photo on 2014-03-14 at 16.06

here in the middle are the seven colors
in “mister big \pi-oh” (from ohio) order:
(mud, red, blue, green, purple,
yellow, orange). i’ve drawn the “line”
(which appears as a triangle) formed by
“marking” the purple vertex and performing
the “two steps forward and one step back”
procedure: one easily verifies that
{G, O, P} is a line as described in
the previous post (“the secondaries”).

all to do with “duality in P^2({\Bbb F}_2)“.
had we but world enough. and time. especially time.

eliminating the middle, man (06/07)
was my third post (in vlorbik
on math ed
, as this site
was first known [to 01/10.
next was MathEdZineBlog, to 01/13;
then a grader’s notes]). my fourth post
was {X : X is full of baloney} (06/07).

these were published as a two-parter
with the (dull, dull) title textbooks
and notations
. the badness of
standard-text “set builder” notation
(the topic of the latter) had, for
quite a while, been a burning issue
for me; my then-recent discovery of the
attack on the “sign of intersection”
(the topic of the former) nudged me
into finally ranting that rant online.

so there’s some evidence that at least part
of why i *began* mathblogging when i did
was the need to announce that i’d been
newly horrified by new depths achieved
by the enemies of clarity in the long-
-established Notation Wars.

much more recently, i was horrified
anew: in the sloppily-ranted (and,
again, boringly-titled) midterm report
of 11/12, i announced my discovery
that The Enemy had come for the set-inclusion
symbol in my favorite intro-to-*real*-math
course (“linear algebra”).

the rest of the notations file is mostly
more about notations themselves than
Notation Wars. i haven’t been very good
about tagging my posts.

along the way, there was capital script-D of f,
(01/09) pointing out (among other things)
that remedial-algebra courses
daring even to *mention* “domains”
and “ranges” really (reallyreally)
ought to also introduce *symbols*
for these objects. (*easily written*
symbols, of course.)

somewhere i may even already have indicated
one high-hope-against-all-lack-of-hope: early
(and correct, and consistent) use of (the standard)
f: D \rightarrow C
notation (for a function f with domain
D and co-domain C). as scary as that
might be. (once the actual students
actually see how useful “careful use
of code” actually *is*, they’ll grab
it when they need it… more and more.
this is part of what’s called “getting

anyhow, i guess i’m just circling wagons
right in here. there’s some sign-of-equality
stuff by me in the blog next door, for instance.
and i’m feeling a need to have the evidence
(that Textbook Set Theory, long dying of
a usually-fatal illness, has been mortally
wounded in the bargain and is sinking fast)
much better organized (and, just maybe,
somewhat more level-headedly presented).

but not right away, not now.

this is the limit

so i’m grading linear algebra like i usually do.

where we need to speak… or anyway, to read
and write… frequently about *column vectors*.

much the usual thing (for example) in defining
a linear transformation (called F, say)
on “real three-space” (so F: R^3 —> R^3)
is to *consider R^3 as the space of real-valued
column 3-vectors* and then to supply a matrix
(called [F], say; [F] will be a 3-by-3
matrix in this context) such that
F([a, b, c]^T) = [F][a, b, c]^T.

the “caret-T” here denotes “transpose”…
the idea is that one has something like
[a, b, c]^T=
in laypersons’ language, the transpose
symbol ^T tells us to turn our old rows
into the new columns (which simultaneously
turns our old columns into the new rows).

in the language of the widely-used
TI-* calculator line, one has
[a, b, c]^T = [[a][b][c]]…
and this is starting to look
better and better to me right
in here.

but what one really seems to *want*
here is a quick-and-dirty notation
for expressing (what we will still
continue to *speak* of as)
column-vectors, as rows.

and i’ve noticed student papers using
< x, y, z > = [x, y, z]^T.
this looks like a real useful convention
to me and i’ve adopted it for my own use
until further notice.

angle-bracketed vectors have been useful
to me before. mostly, i think, in the context
of “sequences & series” typically dealt with
in about calc 2 or 3.

LANGLE x_0, x_1, x_2, … RANGLE
(i.e., < x_0, x_1, x_2, … >
… “angle brackets” are special characters
in HTML and so i prefer to avoid ’em)…
in either notation…
represents sequence of objects
(which is of course *not* the same
as the *set* R={ x_n } = {x_0, x_1, x_2, …}
[the set of values taken by the function
f(n) = x_n
on the natural numbers NATS:={0, 1, 2, …}]).

this usage actually *extends* a usage,
ideally introduced and maintained earlier
in a given presentation (class or text or
who knows maybe someday even both at once)
using angle-brackets for (finite-dimensional)

now represents the vector that, ideally,
we would represent in some other part of
our presentation as [3,4]^T…
a *column vector*.

i remark here that meanwhile
represents a so-called “point”
in “the x-y plane”… an entirely different
(though closely related) object.

we pause here and take a deep breath.

i’ve pointed at two distinctions:
sequences-as-opposed-to-sets and

many textbooks… and many instructors…
are *very sloppy* about keeping these
(and many other suchlike) distinctions clear.

the *notations* used in *distinguishing*
the distinct situations in each case were
“delimiters”: pairs of opening-and-closing
symbols used to mark off pieces of code
meant to be handled as single objects.

delimiting conventions are vital even in
ordinary literacy (“you see? he” sa)i(d.
and i claim they’re all the more so in maths
(since we get fewer and weaker “context clues”
when the code gets munged [as in the example]).

and students’ll just leave ’em out altogether
if they think… or god help me, know… they
can get away with it. failing that, choose
randomly (itt…oghm,k…).

failing that, “well, you *know*
what i *meant* was”…

too late, too late. here endeth the sermon.

in our next episode of “who stole my infrastructure?”:
the dot product. when they came for the opening-apostrophe,
i pleaded and begged. when they came for the
sign-of-intersection i raved incoherently.
never had a chance, no hope, no hope. doom doom doom.
can somebody pick up the torch, here?
i don’t think i can go on much longer.

vlorbik on punctuation for the twenty-twelve.
more clarity!