## Archive for the ‘Notations’ Category

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.

wow.

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.

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 G A G C B

G G A G D C

G G A **G** E C B A

F F E C D C

(

G A G B . . . )

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”.

namely,

(-+) {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.

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 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. ☰☱☲☳☴☵☶☷

let

now let

, i.e., let

, 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”.

e.g.,

;

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 -+

III —

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

;

voila.

oh, yeah. i forgot to say. the “points” are

matrices *mod 3* (so that 2 = -1). that is all.

the (so-called) fundamental quaternion units

can be represented as

with the scalars of the matrices considered

as elements of —i.e., 2=-1 etc.

“modding out” the gives —

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

today.

announcing the name change:

Virtual MEdZ.

here in the middle are the seven colors

in “mister big -oh” (from ohio) order:

MRBGPYO

(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 “.

had we but world enough. and time. especially time.

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=

[

a

b

c

]

;

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

LANGLE x_n RANGLE

(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)

*vectors*.

LANGLE 3, 4 RANGLE

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

(3,4)

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

vectors-as-opposed-to-points.

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!