-- retour de tchebyshef --
galathaea
à l'occasion de son retour
il y avait de bengaluru fini d'orages
i had given him up on and off many times over the past year
but though he frustrated me
taunted me
relentlessly
persistently driving me away in anger
i kept returning
a little broken
stubbornly expecting some deeper connection to form
tchebyshef was the natural next step for me
but as i wrote to you so desperately in the past
he was not polynomial
and so i troubled over his generalised abilities
tchebyshef was the next step in the generalised fourier
he had to be
mais il y avait d'autres beautés
que je ne pourrais pas apprécier
jusqu'à ce jour fatidique de marche
..
as i have mentioned prior
he is easy to describe in the generalised trigonometry
the first kind has an obvious definition
/ \
T | g ( theta ) | = g ( n theta )
m n \ m 0 / m 0
where g (x) is the generalised (hyperbolic) trigonometric
m 0
oo j
--- x
|0 x x \ ----
| e = (0, m) multisection of e = / (1)
|m --- j
j=0(mod m)
-+-+-
it had occurred to me early on
that there might be a secret benefit to the nonpolynomialness
a theorem in the classical tchebyshef
gave the (0, 2)-form the benefit of uniqueness
for polynomial minimax approximations
so nonpolynomialness could possibly be a path towards preserving minimax qualities
aber ich benötigte eine frische annäherung...
-+-+-
there are obvious theorems one can immediately prove from the definitions
the composition theorem
/ \
T | T (x) | = T (x)
m n \ m n' / m n n'
is simple and in many ways uninteresting
but i had toyed with it from very early on
it reveals the game
the substitution of x for g ( theta )
0 m
which is so useful to the tchebyshef generalisation
und doch nach diesem stürmischen bengaluru tag
ich sah die graue rätselfalte weg...
-+-+-
so many things to relate
i choose now only one to state
a single example to inflate
until the next
and its unending
disappointing
contribution to conflate
a simple one to start with
recall the product rule for generalised trigonometrics
comme j'ai écrit environ tellement il y a bien longtemps
watch the trick
m-1
--- j
/ |0 (n-1)theta \ / |0 theta \ 1 / |0 n theta \ |0 (n-1+w )theta \
| | e | | | e | = - | | e + / | e m |
\ |m / \ |m / m \ |m --- |m /
j=1
through the secret substitution
|0 theta
| e -----> x
|m
|0 n theta
| e ------> T (x)
|m m n
es wurde aufgedeckt
m-1
---
\
T (x) = m x T (x) - / T (x)
m n m n-1 --- j
j=1 m n-1+w
m
which reduces to the classical recurrence
T (x) = 2 x T (x) - T (x)
n n-1 n-2
for T = T (x)
n 2 n
mais la généralisation n'était pas une récurrence!
and though i have displayed here this
in the format of it's malformed tradition
the symmetry inside can be revealed
-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-
| |
+ m-1 +
| --- |
+ \ +
| m x T (x) = / T (x) |
+ m n --- j +
| j=0 m n+w |
+ m +
| |
-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-
interlocking wheels of dhamma binding the whole cyclotomic ring of places tight!
^..^
i must rest now friend
and let my newfound parasites retire
there is a much greater tale here
though
ignited by the fever of that bengalurian night
i will reveal these over the coming weeks
as my strength returns and the toxins of bharat subside
but imagine
if you will
the great power that comes from controlling the magic of this generalisation
eine zauberei von symbolen
finally
the generalised fourier theory is maturing
this new cyclic type of relation is similar
but much more general and natural
to the relations you are familiar with in modular forms and other special hypergeometrics
can you see the generalisation of jacobi polynomials?
perhaps a simpler task until i write again
can fill your exercises and prepare you for the methods
do you see how to express x^n in terms of these generalised forms?
a basic task for you
dear friend
to prepare the theory of representations and approximations
kya?
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar
quasi
wrote:
>à l'occasion de son retour
>il y avait de bengaluru fini d'orages
>
>i had given him up on and off many times over the past year
>but though he frustrated me
>taunted me
> relentlessly
> persistently driving me away in anger
>i kept returning
> a little broken
>stubbornly expecting some deeper connection to form
Good to have you back, even though apparently, you're not yet actually
back in a physical sense. In any case, I see you're in fine form.
Perhaps you've awoken the spirit of Ramanujan.
quasi
galathaea
> On Tue, 08 Apr 2008 09:12:29 -0700, galathaea <galath...@veawb.coop>
it would have been nice to visit kumbakonam
but i was stuck on the west side of the continent when down south
i really wanted to see places of past inspiration
to see what magic they might still hold
but i couldn't get over to ramanujan's
instead i saw the buddha's places
even though i grew tired of the buddha
after reading more of the pali canon
now i'm sick of the buddha
i've generally become quite antibodhi
and wish i could have spent some days out in the southeast
the company i work for also has offices out in chennai
so maybe one of these days...
David Bernier
There's a university in Kumbakonam near Ramanujan's old house:
< http://www.math.ufl.edu/~frank/photos/india2003.html >
< http://www.math.ufl.edu/~frank/photos/india2005.html >
< http://www.math.ufl.edu/~frank/photos/india2006.html >
David Bernier
galathaea
> do you see how to express x^n in terms of these generalised forms?
i thank you friend
for your long and ultimately tragic letter
on your newest theories relating
bovine flatulence and urinary tract infections
in the context of an ecosystem of pollutants
it is unfortunate that you were unable to attack
the problem of my previous letter
but your request for further elucidation and examples
shows a clear desire to understand
perhaps i should start with easier properties
working my way more slowly towards these esoteric pursuits
many simple properties of the generalised trigonometrics
translate immediately to the tchebyshef form
take the basic rotation formula
j
g (w x) = g (x)
m 0 m m 0
now use the secret substitution and immediately
T (x) = T (x)
m n j
m w n
m
which generalises the +/- n result of classical T
due the evenness of cosinus
do you see how easy it is?
now
we can translate the general product formula as well
/ |0 n theta \ / |0 n' theta \
| | e | | | e |
\ |m / \ |m /
m-1 j
--- (n+w n') theta
1 \ |0 m
= - / | e
m --- |m
j=0
becomes
m-1
---
1 \
T (x) T (x) = - / T (x)
m n m n' m --- j
j=0 m n+w n'
m
see?
effortless fun!
of course this one generalises the classical
1 / \
T (x) T (x) = - | T (x) + T (x) |
n n' 2 \ n+n' n-n' /
so let's do something with a tiny bit more work
evaluate the indefinite integral
/
| T (x) dx
/ m n
again the same substitution
|0 theta
x = | e
|m
gives
|m-1 theta
dx = | e dtheta
|m
transforms the integral to
/ |0 n theta |m-1 theta
| | e | e dtheta
/ |m |m
now use the product rule to get
m-1 j
--- (n+w ) theta
/ / 1 \ j |m-1 m \
| | - / w | e | dtheta
/ \ m --- m |m /
j=0
m-1 j j
--- w (n+w ) theta
1 \ m |0 m
= - / ------ | e
m --- j |m
j=0 n + w
m
(plus a constant term i will ignore from here on)
change back the sub and we get
m-1 j
--- w
/ 1 \ m
| T (x) dx = - / ------ T (x)
/ m n m --- j j
j=0 n + w m n+w
m m
once again generalising a classical relation
T (x) T (x)
/ 1 / n+1 n-1 \
| T (x) dx = - | ------- - ------- |
/ n 2 \ n + 1 n - 1 /
maybe with this last example
you might begin to see how the weight function (1-x^2)^(1/2)
arises in the classical integrals of orthogonality
and how this generalises
maybe not
can you see how to represent x^n now?
we will need the representation when analysing the basis change
that underlies transforming
a taylor series to a generalised tchebyshef...
galathaea
:: watch the trick
::
:: m-1
:: --- j
:: / |0 (n-1)theta \ / |0 theta \ 1 / |0 n theta \ |0 (n-1+w )theta \
:: | | e | | | e | = - | | e + / | e m |
:: \ |m / \ |m / m \ |m --- |m /
:: j=1
::
:: through the secret substitution
::
:: |0 theta
:: | e -----> x
:: |m
::
:: |0 n theta
:: | e ------> T (x)
:: |m m n
::
:: do you see how to express x^n in terms of these generalised forms?
i was shocked to hear of your recent departure to the front lines
a warrior
you!
there will not be a better person killed by this senseless war
than you dear friend
i guarantee you that
..
to honor your sacrifice and satiate these penultimate hungers
i will give you the key to my challenge
but let's look at how the classical case is derived first
n / |0 theta \n / 1 / theta -theta \ \n
x = | | e | = | - | e + e | |
\ |2 / \ 2 \ / /
and
by the binomial theorem
n
---
/ theta -theta \n \ / n \ theta j -theta (n-j)
| e + e | = / | | e e
\ / --- \ j /
j=0
n
---
\ / n \ -theta (n - 2j)
= / | | e
--- \ j /
j=0
now
one can pair the +/- terms
(e^2 with e^(-2) and e^3 with e^(-3)..) so those can all be expressed as coshinusi
and if n is even
the term e^0 can be split in half
(1/2 (e^0 + e^(-0)) = e^0)
producing a coshinus even there
this is usually written
|_n/2_|
--- '
\ / n \ |0 theta (n - 2j)
/ | | 2 | e
--- \ j / |m
j=0
where the ' means that the terms are summed normally
except for the term j=n/2 when n is even
which is halved
writing this out
|_n/2_|
--- '
/ |0 theta \n \ / n \ |0 theta (n - 2j)
| 2 | e | = / | | 2 | e
\ |2 / --- \ j / |2
j=0
which
pulling out the 2s and making the secret substitution
gives
|_n/2_|
--- '
n 1-n \ / n \
x = 2 / | | T (x)
--- \ j / n-2j
j=0
so the generalisation should be obvious
since simpson's multisection formula gives
m-1 j
--- w x
|0 x 1 \ m
| e = - / e
|m m ---
j=0
one needs to calculate
m-1 j
--- w x
/ \ m \n
| / e |
\ --- /
j=0
the multinomial formula is needed
so define the collection K
as all m-tuples (k0, k1, ...) such that k0 + k1 + ... = n
and write
2
--- k0 x k1 w x k2 w x
\ / n \ m m
= / | | e e e ..
--- \ k0, k1, ... /
K
m-1
---
\ j
x / k w
--- --- j m
\ / n \ j=0
= / | | e
--- \ k0, k1, ... /
K
now
each of the m-tuples has (m-1) other rotated forms which
(by the symmetry of the mutinomial)
has the same leading coefficient
so one can group
(k0, k1, .., k(m-1))
(k1, k2, .., k0)
(k2, k3, .., k1)
together and form these generalised coshinusi
sometimes
though
these rotations do not form new terms
so one needs to split the term apart appropriately
for instance
there is only one term (1, 1, .., 1)
so this term would need to be divided by m
similarly
if m is even
(1, 0, 1, 0, .., 1, 0) produces only two different terms
so they would need to be divided by m/2
similarly if m is divisible by 3, 4, ..
this way of summing i will write as
--- o
\
/
---
o
K
where o signifies the cyclic generalisation of the ' summation
and K^o is a collection of representatives from the cyclic classes
going through the same steps above
finally gives as the solution i have taunted
--- o
n 1-n \ / n \
x = m / | | T (x)
--- \ k0, k1, .. / m-1
o ---
K m \ j
/ k w
--- j m
j=0
what a strange little formula!
this collection of representative m-tuples
forming a representative collection of points in the cyclotomic field
through a simple path sum
cyclotomic drunkards paths!
just like you used to be
dear friend
remember that one july??
i will certainly remember it
good friend
even when you're gone
galathaea
g:
> a warrior
> you!
no
i had never thought it possible
that "sometimes they trick you"
i would have thought they needed the bodies
for all those extra uniforms
flack jackets
ammunition
and primary weapons
that is certainly unfortunate
and i know your family must be severely disappointed in the whole affair
but stay strong
friend
at least they removed only inessential organs
-+-+-
you had so many questions
i wonder if my other "friends" finally got to you
if so
i apologise
but yes i can try to go back some
the goal is a fourier system
in some ways this begs i return to t (x)
m n
-m
= w g (w x)
2n m n 2n
oo
--- j
\ (-1) nj+m
= / ------- x
--- (1)
j=0 nj+m
which i abandoned long ago
the zeroes of these functions
are almost periodic like the besselian
in that
lim ( ? (r) - ? (s) ) = ?
n->oo m n+1 m n n
where m_?_n(j) is the jth zero greater than or equal to 0
of m_t_n
they all have zeroes that approach some multiple of pi in spacing
but the peaks
unlike the t_2 series
grow exponentially n > 3
and in odd-n cases
the negative domain gives exponentially dominated growth without periodicity
where the even-n of course keep the order-2 symmetry
(even/odd implies periodic in both the positive and negative domains)
a back-of-the-envelope calculation shows
using m=0, n=3 as an example
w x
( ) ( |0 6 )
Re< t (x) > = Re< | e >
( 0 3 ) ( |3 )
x '\/3 x -'\/3
- ---- i x - ----- i x
1 / 2 ( 2 ) -x 2 ( 2 ) \
= - | e Re< e > + e + e Re< e > |
3 \ ( ) ( ) /
x x
- -
1 / 2 / '\/3 \ -x 2 / -'\/3 \ \
= - | e cos| ---- x | + e + e cos| ----- x | |
3 \ \ 2 / \ 2 / /
which is zero when
x
-
-x 2 / '\/3 \
e = -2 e cos | ---- x |
\ 2 /
or
-3x
---
2 / '\/3 \
e = -2 cos | ---- x |
\ 2 /
as the exponential left hand side gets closer to y=0
it will cross the cosinus closer and closer to its own zeroes
the period of this (0,3) generalisation therefore approaches 2 pi / '\/3
(= 3.6275987...)
and this can be seen from a simple plot of this function
where 0_?_3(0) occurs at 1.8498128
then 0_?_3(j) = 5.4412334, 9.0689975, 12.696596, 16.324194, ...
so you see
a generalised fourier analysis must not use the
/
| t (r x) t (s x) dx
/ m n m' n
because roots are not integer multiples of some base periodicity
similarly
the (1,3)-t has zeroes 1_?_3(j) at
0, 3.0167442, 6.6506245, 10.278196, 13.905795, 17.533394, ...
and the (2,3)-t has 2_?_3(j) at
0 (double), 4.2332072, 7.8597929, 11.487396, 15.114995, ...
these are of course interleaved
as one would expect from rolle's theorem
these "almost symmetries"
along with the unnecessary complications of using half-periods
(so derivatives switch signs after n iterations, ie.
n
d
--- t (x) = - t (x)
n m n m n
dx
)
is one of the things that makes tchebyshef in the hyperbolics attractive
but let me show you how
my increasingly absentminded comrade at arms
you can still lay the obvious foundations
^^.
first
though
i realise i haven't told you of nausea
you know
sartre's nausea
i reread it on my recent travels to india
i tell you with full earnestness
my easily fooled one
there is no better way to read nausea
than reading nausea in india
there is this great passage
where roquentin is talking of his travels to varanasi
(then benaras)
and how we gather these experiences to sell to others
for their admiration and respect as "one with experience"
i didn't want to see the taj mahal
i swear to you i did not plan any visit through agra
it's just
i started in delhi because that's where my driver was based
and i wanted to see khajuraho
and
well
just look at a map
so my driver suggested it
for the first night's stop
#######@@@4..7*****)
the first step is to look for the paths of integration for zeroes
and since these are entire functions
all paths with real endpoints are equivalent to the real interval
the zeroes of the integrated forms are simply
(m+1)_?_n(j)
(where "m+1" takes place in Z_n)
so any interval ending on zeroes provide obvious zero periods
of course
we can slide the endpoints to any equal values of the antiderivative
but the zeroes form a "basis" of "types" of zero-valued intervals
using the equivalence class of "continuous sliding"
because each "hump" reaches an absolute magnitude greater than each
in some very important ways
these intervals
m_I_n(r,s) = [ m_?_n(r), m_?_n(s) ]
are representative elements of homotopy equivalencies
this isn't quite true in the 0_t_1 case
(e^(-x) has no zero periods - but also no zeroes)
or the m_t_2 cases
(sin/cos have peaks all 1
so "sliding" equates I(r,s) with I(r+n,s+n)
and the equivalences are represented by
m_I_2(r) = [ m_?_2(0), m_?_2(r) ])
but the definition of I(r,s) works in these cases as well
because the main point is
/
| t (x) dx = 0
/ I (r,s) m n
m+1 n
and
/
| t (x) dx in general has interesting values
/ I (r,s) m n for all p
m' n
for one reason
zeroes of m_t_n are critical points of (m+1)_t_n
so if we label these critical values
? (j) = the jth critical value of m_t_n (greater than or equal to 0)
m n
= t ( ? (j) )
m n m-1 n
then
/
| t (x) dx = ? (s) - ? (r)
/ I (r,s) m n m+1 n m+1 n
m n
example calculations for 0_t_3 give
? (j) = 1, -2.5847397, 16.055418, -98.4739, 604.01033, -3704.8236
0 3
with the basic properties of alternating sign and exponential growth
numerically, these are growing as 6.1337
but the actual value can be calculated from the formula presented earlier
one has that the leading term of the real positive part is
x
-
2 / '\/3 \
2 e cos | ---- x |
\ 2 /
and since the half-period approaches 2 pi / '\/3
the ratio of maxima is
( x + 2 pi / '\/3 )
-------------------
2
e
--------------------
x
-
2
e
pi / '\/3
= e
as one analyses these various critical values
a map of feature values begins growing
by an easy application of hermite-lindemann-weierstrass
all m_g_n are transcendental at all algebraic points
and so it is interesting to consider when these critical values are also transcendental
but that is a different direction than i want to go here
$^..~~~~~
varanasi is the oldest continuously living town in the world
over 3000 years of continuous occupation and rebuilding
it is quite an amazing discovery to learn what 3000 years smell like
waste water thrown from the upper floors of the buildings
keeping the ground wet to moisten endless piles of cow and goat shit
out of the recesses
nag champa and sandalwood mixing with something decomposing
the burning ghats blowing their occasional peoplesmoke through the narrow alleys
the stagnant and polluted ganga
controlling the winds with her schizophrenic desires
the deep black carbonsmoke of a million 2-stroke tuktuks
and sweat
and ghee
and...
thousands of years in one experience...
nnnn-v-mmmm
now
where in the fourier case one switches to
I = [ -pi, pi ] or [ 0, 2 pi ]
and brings the sin( r x ) and cos( s x ) as the scale mapping
we can generalise this procedure to the zeroes here
turning to a period agnostic form
we can arbitrarily scale all intervals to [ 0, 1 ]
/ \
? (x) = t | ( ? (s) - ? (r)) x + ? (r) |
m n m n \ m+1 n m+1 n m+1 n /
r s
now one can work only in the range I = [0, 1]
and single integrals are simple
/
| ? (x) dx
/I m n
r s
? (x)
m+1 n
/ r s \
= | ------------------- |
\ ? (s) - ? (r) /I
m+1 n m+1 n
which by design equals 0 at 1 and 0
&&&&&..&&&&
products have a lot more structure still
the products lose a strict per-zero almost-periodicity
and store information in larger scale zero patterns
for instance
multiplying 0_t_3 by 1_t_3
gives zeroes at 0, 1.8498128, 3.0167442, 5.4412334, 6.6506245, 9.0689975, 10.278196, ..
of differences approaching long, short, long, short, long, short, ..
where long = 2.4183992 = 4 pi / ( 3 '\/3 )
short = 1.2091996 = 2 pi / (3 '\/3 )
a patterned partitioning of the 2 pi / '\/3 periods of the m_t_3 family
(0,3)x(2,3) gives short, long, short long, ..
wheras (1,3)x(2,3) gives long, short, long, short again..
the same is seen in (0,4)x(1,4)
(long, short, long, short)
and (0,4)x(3,4) (short, long, short, long, ..)
but (0,4)x(2,4) is singly almost periodic of asymptotic zero period
it is easy to prove that there are integrals of these "wildly almost periodic" products
over nontrivial intervals
with value zero
(directly from its differential structure)
and so the differential structure of the critical points
(discretely) can be tied directly to the structure of integral periods
this is the whole point of fourier analysis
where integration over the periods is used to decompose into discrete sums the function transformed
but naive use of the product formula doesn't get a lot done
/
| t (x) t (x) dx
/ m n m' n
n-1
---
/ 1 \ -m' j j
= | - / w t ( ( 1 + w ) x ) dx
/ n --- n m+m' n n
j=0
n-1 -m' j
--- w
1 \ n j
= - / -------- t ( ( 1 + w ) x )
n --- j m+m'+1 n n
j=0 1 + w
the products have an interplay of the critical points
and effectively the entire zeroed-value d^m differential structure
transfered discretely across the structure of their critical points
from this ur-form of the sum of two unit cyclotomics
this is real
so it can projected term by term to reals
and summed as strange algebraics of cosines
related to the real and imaginary projections of the cycloctomics
in the straightforward way
but it becomes difficult to prove things about these expressions
some obvious symmetries suggest that
like the fundamental connection between
cosine series and the classical first tchebyshef
(the coefficients of their series are equal)
illustrating basis freedom
the differential structure could be more clearly seen in the tchebyshef language
this is why my attention has been so lately focused on the mysterious mr t
because then we map cleanly to a multiplicative composition
we don't work with the almost-periodic and wildly almost periodic structure of the critical points
we work directly in the cyclotomic rings and fields
so perhaps the simplicity would allow better characterisation
of the basis structure of a generalised analysis?
are there some collections of intervals from the I
that form more naturally complete sets than others?
understanding the tchebyshef translation would explain much of this structure
+----%,.$~~~+
the secrets are of course in some of the identities i have revealed to you
("sometimes they trick you"??? what the hell is wrong with you!?!)
but the real understanding comes in decomposing each analysis with a differential structure
and describing the transformation better
that relates the generalised fourier analysis with it's tchebyshef theory
the secret lies in the projection property of multisections
and the logical interplay of the multisection operator and the differential operator
d |m |m-1 d
-- | = | --
dx |n |n dx
quite a bit of this structure applies to more general classes of functions still
many interesting collections of hypergeometrics and their q deformations
and if i had the strength to explain tori to you
i would certainly be relieved of some of your barking
but i think that this darkening day has stolen any time for that
David Bernier
I had a look at the Wikipedia article on orthogonal polynomials. I
didn't know
there were so many kinds of families: Tchebycheff, Laguerre, Legendre,
Hermite
and others:
< http://en.wikipedia.org/wiki/Orthogonal_polynomials >
It seems to me you're looking for non-polynomial
non-trigonometric functions orthogonal for
the Hilbert space L^2(R, mu) , where mu is a measure on R
perhaps arising from some weight function. Maybe you've
heard of wavelets (Daubechies, others) .
David Bernier
** Posted from http://www.teranews.com **
galathaea
below
please attempt a consistent use of
m_{/u2327}_n(j) for the jth root from zero
m_m'_{/u2354}_n{j} = jth critical value
ie. = m_t_n( m'_{/u2327}_n(j) }
and then use /u2328 as the symbol
for the interval-scaled zero period
i apologise for the failure to properly send utf8
and see what is possible with my newsgroup options
until then
http://www.alanwood.net/unicode/devanagari.html
i should also point out some example graphs
t (x)
0 3
normal scale:
http://galathaea.org/genFourier/0t3-20x100.png
ln|abs y scale:
http://galathaea.org/genFourier/0t3-20x10ln.png
t (x)
1 3
http://galathaea.org/genFourier/1t3-20x10.png
t (x)
2 3
http://galathaea.org/genFourier/2t3-20x10.png
t (x)
0 4
http://galathaea.org/genFourier/0t4-20x100.png
t (x)
1 4
http://galathaea.org/genFourier/1t4-20x10.png
t (x)
2 4
http://galathaea.org/genFourier/2t4-20x10.png
t (x)
3 4
http://galathaea.org/genFourier/3t4-20x10.png
t (x)
0 5
http://galathaea.org/genFourier/0t5-20x100.png
t (x)
0 6
http://galathaea.org/genFourier/0t6-20x100.png
t (x)
0 7
http://galathaea.org/genFourier/0t7-20x100.png
t (x)
0 8
http://galathaea.org/genFourier/0t8-20x100.png
t (x) t (x)
0 3 1 3
http://galathaea.org/genFourier/0t3x1t3-20x100.png
t (x) t (x)
0 3 2 3
http://galathaea.org/genFourier/0t3x2t3-20x100.png
t (x) t (x)
1 3 2 3
http://galathaea.org/genFourier/1t3x2t3-20x100.png
t (x) t (x)
0 4 1 4
http://galathaea.org/genFourier/0t4x1t4-20x100.png
t (x) t (x)
0 4 2 4
http://galathaea.org/genFourier/0t4x2t4-20x100.png
t (x) t (x)
0 4 3 4
http://galathaea.org/genFourier/0t4x3t4-20x100.png
just to give visual understanding of these forms
galathaea
what one eventually ends up with
after all the excruciating obfuscation is laid out
is ultimately a generalisation of polynomialness itself
orthogonality is simple
the interesting connection is that
the generalised trigonometry's orthogonality
is mirror to the orthogonality of the generalised tchebyshef
many of the classic orthogonals
including the tchebyshefs
are included in the class of jacobi polynomials
the gegenbauers are simply a subclass of jacobi
distinguished by equal upper parameters
and the tchebyshef fall in the gegenbauers
but the generalisation of tchebyshefs
falls in larger class of generalised jacobi
the trick is the switch into nonpolynomialness
through the generalised polynomial
2 n-1
w w w
n n n
y = a x + a x + a x + ... + a x
0 1 2 n-1
which
surprisingly
has some very regular zero features
but a more complex feature set
this really becomes apparent in the
generalised tchebyshef representation of x^n i posted
see
n=1 is e^x
so 1_T_n(e^x) = e^(n x)
or 1_T_n(x) is simply x^n
the taylor-mclaurin orthogonals
the most fundamental orthogs of all analysis
C 2_T_(n-1)(x) is the minimax rep of 1_T_n(x)
now there is this generalisation
and whole new set of tools for feature detection
here - more elaborate patterns in differential structure
^..^
one of the secrets hidden in the shift to tchebyshef
is the origin of the weight function
it basically arises from the presence of
|n-1 x
| e
|n
in integrals of the tchebyshefs
since in the (0,2), (1,2) case
2 2
ch x - sh x = 1
this generalises quite naturally to a function
/ |0 x |n-1 x \
f | | e , | e | = 1
\ |n |n /
where the natural question arises
is it algebraic?
the answer again is revealed in cyclotomics..
David Bernier
I seem to remember reading that Riemann wanted
tchebyshef to see his work on zeta and primes.
I might look up what tchebyshef did in
number theory.
> many of the classic orthogonals
> including the tchebyshefs
> are included in the class of jacobi polynomials
>
> the gegenbauers are simply a subclass of jacobi
> distinguished by equal upper parameters
> and the tchebyshef fall in the gegenbauers
> but the generalisation of tchebyshefs
> falls in larger class of generalised jacobi
>
> the trick is the switch into nonpolynomialness
> through the generalised polynomial
>
> 2 n-1
> w w w
> n n n
> y = a x + a x + a x + ... + a x
> 0 1 2 n-1
Not using capital letters or punctuation is not something that
causes readability problems for me. It's OK for me.
However, formulas with exponents on the line above
don't always align well; also, towers of exponents
with no parentheses are ambiguous for me ...
So
y = a_0 x ^(n^w) + a_1 * x^(n^w) + a_2 * x^((n^w)^2) + ...
I don't think I've got it. Do you get polynomials by
setting w=1 ?
galathaea
the alignment of the first exponent was my fault
the exponent terms are roots of unity
(or more generally in some of the other uses
any elements of a cyclotomic ring)
just as an example of where it comes from
ch x = 1/2 ( e^x + e^(-x) )
set y = e^x
and one has
y + 1/y = 2 ch x
turn it into a quadratic (since y=/=0 because e^x =/=0)
y^2 - 2 y ch x + 1 = 0
and use quadratic formula
y = ( 2 ch x +/- '\/(4 ch^2 x - 4) ) / 2
from the exponential (positive) nature of y
this becomes simply
y = ch x + '\/(ch^2 x - 1)
if we return back to y = e^x
then
x = ln( ch x + '\/(ch^2 x - 1) )
the classic inversion formula for hyperbolic trigs
now try to do the same thing for
|0 x
| e
|3
and you'll see where the generalisation comes in
i've mentioned these "polynomials" several times over the years
in various contexts
but one that helps visualise what these are is at
http://groups.google.com/group/comp.programming/msg/5e87a2a938f231a2
galathaea
generalised polynomials have many nice properties
if f, f' e W[x]
in other words
if
---
\ w
f = / a x
--- w
w e S_f c C
n
S_f finite
and
---
\ w
f' = / b x
--- w
w e S_f' c C
n
S_f' finite
where C_n is the ring of integers
of the cyclotomic field of order n
then (f + f') e W[x]
and (f f') e W[x]
also
derivatives of e W[x]
are e W[x]
the integrals can bring in logarithmic forms
but so do simple laurent polynomials
and these are well understood
the really crazy thing
is that they seem to have an interesting zero structure
where
for instance
---
\ w
/ x = c(x)
---
n
w = 1
has n zeroes in the complex numbers
the important step
then
becomes the galois analysis
so an important consequence of this generalisation
is a whole new realm of galois study
which i am sure has already been done already
if i only knew the right keywords to search
galathaea
> generalised polynomials have many nice properties
>
> if f, f' e W[x]
>
> in other words
> if
>
> ---
> \ w
> f = / a x
> --- w
> w e S_f c C
> n
> S_f finite
>
> and
>
> ---
> \ w
> f' = / b x
> --- w
> w e S_f' c C
> n
> S_f' finite
>
> where C_n is the ring of integers
> of the cyclotomic field of order n
>
> then (f + f') e W[x]
> and (f f') e W[x]
notice that this is true
for finite subcollections that come from any semigroup
it is possible to define
<S, R>[x] as the generalised polynomial ring
with S the semigroup of exponents
and R the ring of coefficients
and these general constructs have a number of properties
more generally derivable
however
the cyclotomic generalisation has a number of special properties
that make it very natural for study
just as a quick example
even though f(x) e W[x] does not guarantee f(x)^w e W[x]
it is true that f_3(x) = x^w_3
for instance
obeys f_3(f_3(x)) = x^(w_3^2)
and f_3(f_3(f_3(x))) = x
(all this occurs on appropriate branches of course)
so this generalisation has periodic iterations possible
which are actually useful in reducing
or transforming
some expressions
these cyclotomic generalised polynomials
arise naturally in the theory of multisection
outside this focus on the multisection of exponentials
and all this work on generalised fourier analysis
in fact
in many ways the analysis can be extended beyond exponentials
the key point of all of my efforts in generalised trigonometry
was the discovery of the product and sum laws
without that
the integrals of products would have made little progress
and the tchebyshef theory would not have passed initial discovery
but a secret i've kept hidden
(mostly due to a lack of time to write it all out)
is that the technique for finding the product rule
actually works for the multisection of any function
that already has a product rule
in other words
if f(x)f(y) = g(x,y)
then we can find a product rule for
/ |m \/ |m' \
| | f(x) || | f(y) |
\ |n /\ |n /
using exactly the same steps as for trigonometrics
this is why i have always considered the generalised fourier theory
an extension of my multisection results
over all hypergeometric and q-hypergeometric functions
because much of the discussion does not reach full generality
simply at the ur-hypergeometric (the exponential)
products
the differential structure
integrals
and much of a generalised analysis
carries over to many famous hypergeometrics
and in these cases
since the product rule is derived
from simpson's multisection formula
the values at cyclotomic places comes in
in many cases of interest
this introduce W[x] again...
> also
> derivatives of e W[x]
> are e W[x]
>
> the integrals can bring in logarithmic forms
> but so do simple laurent polynomials
> and these are well understood
>
> the really crazy thing
> is that they seem to have an interesting zero structure
> where
> for instance
>
> ---
> \ w
> / x = c(x)
> ---
> n
> w = 1
>
> has n zeroes in the complex numbers
>
> the important step
> then
> becomes the galois analysis
>
> so an important consequence of this generalisation
> is a whole new realm of galois study
>
> which i am sure has already been done
galathaea
and you don't think your newfound desire to write sci-fi
a drive i hope you squash like the infection such desires come from
(it is laziness! a leap away from the rigors of true science!)
has anything to do with your newest suspicions
of your possible alien abduction and
subsequent
experimentation?
you don't think your fantasies influence your possibilities?
i know
you've always been the "artist"
the flighty temperamental
doesn't
always
want to consider reality
i know
but you've got it in you
you always have
and i see the seeds of potential you've grown
it will bring us some great gems if only
that theorem of yours
the five squares
you're interested you know?
you saw that one in the fractal
that series of convergence points
and all those beautiful transformations
remember?
you don't know how strong the need for recruitment is
those pacing feverish nights at the parties
you
loaded on strange experimental varieties of intoxicants
formalising various models of reality
what isness is
in realist beable ontologies
you know what i fuckin' mean
you're not fucking sci-fi
make something
it's time
@@@@$$oo%%^^^^^
sprechen über zeit
i'm sorry i take so long to write
and then only to abuse or demean
i know it seems i do not care
too much
for the exchange
but how do i exchange my time
my dear friend
when it fills so naturally with my professional duties and other kantian debris?
how can i pull new time from old?
will the greatest of desire ever defeat duty
my friend?
can time be so bent by my will?
surveying our prior exchange
i see it is time to reveal the horizontal functions
like tchebyshef
the horizontal functions are transformations
but these ones describe the origin of the weight in generalised tchebyshef transforms
please excuse my notation separating the function and it's application
but i've been under heavy influence of functional languages
of late
(it also adds notational information
since
-1
f(.) (x)
is some sheet of the inverse
and
-1
f(x)
is simply the reciprocal)
define the (m, m'; n)-th horizontal function
. .
o / |m . \ / |m' . \-1
+ (x) = | | e | | | e | (x)
m \ |n / \ |n /
m' n
notice how these are defined just like the tchebyshef are
and can even be written in the form
. .
o / |m' theta \ / |m . \
+ | | e | = | | e | (theta)
m \ |n / \ |n /
m' n
now we are converting between the various multisections
immediate consequences include
. .
o
+ (x) = x
m'
m' n
and
. . . . . .
o / o \ o
+ | + (x) | = + (x)
m \ m' / m
m' n m'' n m'' n
(over appropriate domains*)
((*inter alia über alles))
however
again
like the tchebyshef
the hidden power is masked in an underappreciated jewel of first year calculus
!! the chain rule !!
the secret lies in the differentials
(as jacobi used to scream
those late italian nights)
using the above definition
it is 101 material to derive
. .
d o / |m-1 . \/ |m' . \-1 d / |m' . \-1
-- + (x) = | | e || | e | (x) -- | | e | (x)
dx m \ |n /\ |n / dx \ |n /
m' n
and so reduces to the calculation of the derivative of the inverse
but the derivative of the inverse is easily given by
/ |m . \-1
y = | | e | (x)
\ |n /
/ |m . \
| | e | (y) = x
\ |n /
so
/ |m-1 . \ dy
| | e | (y) -- = 1
\ |n / dx
or
dy 1 1
-- = --------------- = --------------------------
dx / |m-1 . \ / |m-1 . \/ |m . \-1
| | e | (y) | | e || | e | (x)
\ |n / \ |n /\ |n /
1
= ---------
. .
o
+ (x)
m-1
m n
plugging back in the original differentiation for the horizontals
. .
o
+ (x)
. . m-1
d o m' n
-- + (x) = ---------
dx m . .
m' n o
+ (x)
m'-1
m' n
look how pretty she is!
for n=2
this theory is the classical
. .
d o x
-- + (x) = ---------
dx 0 . .
1 2 o
+ (x)
0
1 2
. .
d o x
-- + (x) = ---------
dx 1 . .
0 2 o
+ (x)
1
0 2
the general solution for these is y = ( x + C )^(1/2)
using y(0) = 1 for (0,1;2)
and y(1) = 0 for (1,0;2)
gives the classical results
. .
o 2 1/2
+ (x) = ( x + 1 )
0
1 2
. .
o 2 1/2
+ (x) = ( x - 1 )
1
0 2
the properties above can all be verified
including the interesting iterative properties
but i can't go on
i have far too many other tasks to complete this night
and far too little desire to waste my time
describing what you yourself can derive quite easily
can you extract these hors?
can you give them expression?
maybe you can start with n = 3?
or are you completely lost to a life of convenience?
for that map you mentioned
this was all derived on the drive from delhi to agra
that first day of the vacation up north
after the bengalurian insights
i wanted badly to check these trasformations
but became
as now
so loaded in the mud of other people's privilege
that it was not until the vacation that i could approach
their tender glassblown jaws
remember
you too used to get it
that unmentionable gnosis
those words and the symbols scrawled
you know
the creation and the materialisation
when dreams could become realities experienced
touchables
that's what you need to be doing
i'm just so tired of you always making shit up
galathaea
i am very worried about you
i hope this letter will not be my last chance at contact
i drove to your place last weekend
to check on you
you haven't sent me any word for so long now
and
well
that isn't like you
you always are so eager to show off your latest inventions
and you were so desperate in your last letter
and i was so blinded by my role
..
i went to your place last weekend
actually
i really just wanted to show off some of my results
because these letters were so slow in passing
and i was impatient
but i was feeling
that kind of worry
where subfocused pattern recognition triggers in the brain
conspire unnoticed to bring the worry on
a hushed kind of worry
and anxiety
you were not there
as you know
but i had
not expected
#$$@%!#$%#$@**)#$(@@#$)$@)))%$#))%)(#$(@($#(%((#$%((@%$
%$^$$%(%(^%(^*^))!*&^^@&$#()*(&$)&!)&)
++++_+_++@!_+____!+@+++#()(!@!^+_+_))
!
and i don't know why i turned to check the back that day
but i squeezed between the thick grasses around your house
and tripped into this big charred circle in your field
a
#@@$#$#@$@#
%%$^%$^%$^%^%$^%^%
#$@$#%@$%$#%#$%@#%@%$@$$
thick smoky black char
well compressed in patterns
in the perfection possible
to your overgrown
chaos
and you were so insistent in that letter
the lights
(you called them "the lights"
like some 50's pulper)
i mean
how could i know you were serious?
i've never seen these things before
how could i expect?
and as i scanned across in careful arcs
across the vastness of your property
intricate patterns
revealed themselves in every area investigated
across every scale
patterns
charred
into
the
grasses
crop circles!
and lines
and spirals
intricately interwoven
5-fold
7-fold
local and global deformed symmetries
3 and eleven and other-fold forms
appearing and fading chimerically with each new view
it was beautiful
..
i still get lost in it
*
/|\
/ | \
/ | %&&#
/ !@###%*^**@
%@@%%^%^&&
but as the sun penetrated the clouds
and with no trees nearby
my view was suddenly washed away by the
sobering
brilliant sky
i walked back up front and sat in your stone garden
you have that shaded bench in the corner
the statues make me smile
the postman passed by and almost didn't stop
i know you
though
and i know you have abused mailorder
and have daily shipments with hundreds of catalogs of empty amusements
i know the trash you generate
no trash?
no mail?
so i stopped the postman and asked why you had no mail
and how long you hadn't
and i told him i was very freaked out
and
he said he wasn't allowed to tell me much
that really he couldn't tell me anything
but
he could see i was concerned
and it was strange that your mail had stopped
which it had been for a week
he said he had looked into it
and found you had forwarded your mail
but he couldn't tell me where
i was very confused and really tried to get him to tell me more
but he told that they couldn't tell people things like that...
...because i might be a stalker!
he said i seemed pretty nice
but i _was_ waiting around his house
and i _was_ asking his postman about his mail
and that
just on odds
they had strict rules at the post office about confidentiality
and he thought they were fair and just rules
have you ever heard a postman talk like that?
i agreed with him and let him go
@@$##$#@$
##@#@ #@$@#
&*%@%**^*&$*@&%
(#$$%*(*$*((
$#$%
since then
everything has turned odd
i know there's something fucked up with my head
my logical abilities have been betraying my control lately
and the geometry of my senses is revealing new patterns
at the rate now of new ones revealed every few seconds
i can hardly breathe anymore
even when i close my eyes
they reveal themselves in concepts
and preinput control and feedback loops
visual ghosts
and preauditory impressions
in an unending progression of patterns
this alteration of logic and geometry have revealed
their correlation through some deep natural principle of my perception
i sometimes think i am living in a world of sheaves revealed
and so i tease myself
i claim i have grothendieck's sickness
and wonder what crop circle he must have seen
#$^#%%#^#$%^#%#$%
@@@# 32#@#$ 6453
%&^$$%## $^%$^$ 23
*&&*(&*()()(&*$@ 666
$#%$%#$%31415^%%*(&^$@@&
$$$$$ $$$$$$$$
%#^@^ (()$##)(
&&#! **@*@*%
91^ ^%**))
101 &&###
^^^$^^^^.
$#@$.
i keep going over the multisection-generated generalisations
and the cyclotomics pull me deep inside this cavern of crystalised patterns
gems and jewels along the walls like some fine museum
yet grown naturally
in the belly of natural relations
still digesting
why does this fascinate me so?
the infinite generalised fourier theory
which though certainly simpler
looking back on my notes
i have not really mentioned to you
look at the way derivatives behave
with these infinite interval transforms
first
the functions in this theory
are all O(e^(-ax)) as -> +/-oo
for certain 'a' that depend on n
now
/ oo
F [f'; k] = | f'(x) t (kx) dx
m n / -oo m n
can be rewritten using the chain rule as
/ \ oo / oo *
| f(x) t (kx) | - k | f(x) t (kx)
\ m n /-oo /-oo m-1 n
where t^* is t except when m-1 = -1 = n-1 (by modular arithmetic in m terms)
where it is -t
this is simply
*
(- k) F [f; k]
m-1 n
so taking a (m-1, n)-transform and multiplying by (-k)
is equivalent here the (m,n)-transform of the derivative
of course
all this *-complication is removed if we just look at the transforms by G
(the hyperbolic generalised trigonometrics m_g_n)
and this does not change the requirements on the function space that can be transformed
(which is why i so much prefer the hyperbolic theory)
but this is the obvious generalisation
of the classical infinite fourier theory
which encodes this differential theory in multiplication
these are the types of relations
that build a transform theory
addition <------> addition
multiplication <------> differentiation
differentiation <------> multiplication
similarly
through what i call the "ultramodulation of generalised trigonometrics"
there is a generalisation of the shift-operator
which again looks nicer in hyperbolic form:
L[ t (k0 x) f(x); k]
m n
||
||
n-1 j
/ oo k x --- w k0 x
| f(x) t (k0 x) e dx = / oo 1 \ -m j n k x
/-oo m n | - / f(x) w e e dx
/-oo n --- n
j=0
//
//
n-1 j
1 --- (k + k0 w ) x
- \ -m j / oo n
n / w | f(x) e
--- n /-oo
j=0
\\
\\
\\
n-1
1 ---
- \ -m j j
n / w L[f; k + k0 w ]
--- n n
j=0
where L is the two-sided laplacian transform
(which the 1st level generalised hyperbolic infinite interval transform is)
similarly
n-1
---
k0 x 1 \ -m j j
F [e f(x); k] = - / w L[f; k0 + k w ]
m n n --- n n
j=0
and in general
a lot of the dance in the classic theory
between laplace and fourier transforms
plays out in this general form
it's always the same
this recurring
|
\ | /
\ | /
\ | /
------star------
/ | \
/ | \
/ | \
|
and i keep thinking back upon
"the other great wiles"
or what they call mazur-wiles
(née le conjecture d'iwasawa)
in the theory of cyclotomic fields
john coates gave a bourbaki seminar
on the work of mazur-wiles
and when i read it the other day
the words began
to
m
o
v
e
of
f the page
and the central symmetries
made themselves apparent in the world around me
in the array of my cd shelf
and the flower arrangement in an indian rug on the wall
and i could see the use in the delicate manipulations
in many logical deductions
or maybe i'm not really thinking of that at all
it's more like i'm thinking about
the way the addition rule for generalised trigonometrics
picks out certain projections and automorphisms in the cyclotomics
as i've shown you many times
the addition theorems arise from the product
through simple linear inversion
i can build the table of products
1 / 2 \
t (a) t (b) = - | t (a + b) + t (a + w b) + t (a + w b) |
0 3 0 3 3 \ 0 3 0 3 3 0 3 3 /
1 / 2 2 \
t (a) t (b) = - | t (a + b) + w t (a + w b) + w t (a + w b) |
1 3 2 3 3 \ 0 3 3 0 3 3 3 0 3 3 /
1 / 2 2 \
t (a) t (b) = - | t (a + b) + w t (a + w b) + w t (a + w b) |
0 3 0 3 3 \ 0 3 3 0 3 3 3 0 3 3 /
------------------------------------------------------------------------- 2
using 1 + w + w = 0
3 3
so that t (a) t (b) + t (a) t (b) + t (a) t (b) = t (a + b)
0 3 0 3 1 3 2 3 2 3 1 3 0 3
and if i look at some cyclotomic sums (times an x for placeholding)
j j j j
t ((1 + w ) x) = t (x) t (w x) + t (x) t (w x) + t (x) t (w x)
0 3 3 0 3 0 3 3 1 3 2 3 3 2 3 1 3 3
2j j
= t (x) t (x) + w t (x) t (x) + w t (x) t (x)
0 3 0 3 3 1 3 2 3 3 1 3 2 3
/ \2 / 2j j \
= | t (x) | + | w + w | t (x) t (x)
\ 0 3 / \ 3 3 / 1 3 2 3
and then i insist i am misunderstanding the "natural" interpretation
that this really is more natural in the language
of the genus of a multiplicative series
and i think longingly on the witten genus..
and
that's what i'm like
these days
everything
is
in
pattern
i am
breaking i am
apart dissolving
no i
i was walking through the parking lot
of a local marketplace
the other day
when the lines of vehicles fell
in the classic vanishing point of theory
upon the doors of the store
and i saw in their mouth all of a sudden
that the building was hungry yet devoured many customers
an insatiable hunger which a mild tranquilizer
helped all ignore as they entered the jaws
a soporific muzak to deaden the leeching
like the spit of vampire bats
i ran from the store
*&)(()*&(()&*)(*()&)(*(&((&(
i saw the datura
i actually saw it
in india
in a tenth century temple
!!!a datura in a _tenth_ century carving!!!
it was in some pictures of durga i took
i didn't see it until the other day
but it was right there in right hand by her head
all these patterns keep flying out
here are some of the pictures
http://galathaea.org/dhatura/clearPhlashPhoto.JPG
http://galathaea.org/dhatura/wideClearPhlash.JPG
the datura is a new world plant..
#$@%$%@@#%$#%%^^$^$%@$%!#@#$$##^@$@#(@#&
$#(@%&@)$%()@#%&()#&@$^)^%)@*^#%&^@#)*^%)*#@$^@%(#
#&($%)@#&%(&#$(%&(#$)&%()#$&%(#$&%(@&#)(&$)#&$%@4
&#$#$&)@%#&)$*&@)*&%)*@&$)#@&)@&*)@
@)$^^*^%*$#^%@$#%&@#&
and the tchebyshef
they keep guiding me to these finite forms
/ /u0917x(3, n)
a = | g(theta) g (k theta) dtheta
k / 0 0 n
where g(theta) = f( g (theta)) or set x = g (theta)
0 n 0 n
-1
here theta = g (.) (x) and dx = g (theta) dtheta
0 n n-1 n
dx dx
or dtheta = ------------ = ----------------------
/ -1 \
g (theta) g | g (.) | (x)
n-1 n n-1 n \ 0 n /
g (0) f(x) T (x)
/0 n n k
then a = - | ----------- dx
k / . .
g (/u0917x(3, n)) o
0 n + (x)
n-1
0 n
this is the fundamental relationship
between fourier series and tchebyshef series
generalised to this cyclotomic chaostrophy that now envelops me
you know that song
"electric alice"
by grinderman
? the
silver rain
?
and in mazur-wiles
where they use the "partial zetas"
the sum
---
\ -s
/ m
---
sigma = sigma
m
no
that brings in the other bases
the other multisections
and diverts me again down those roads
those |m
| base-dependent distractions
f |n
i
that only lead me astray
i fight with myself more often these days
you know
those transformation of bases
where x^i ---> (f(x))^i
can be mapped
inherit all that structure
of the classic pfaff and euler transforms
and any of the thousands of related transformulae
..................
???????????????????????????
did you ever see the little trick
to find the differential equation for the horizontal functions?
you form the equations with the upper term x
and convert the lower term by repeated differentiation back to the main form
it's basically a substitution exercise
you get a n-th order nonlinear differential equation
for the horizontal functions of order n
fortunately
this is not impenetrable
starting with order 3 shows the way pretty well
^..^
i drank a little the other day
just to kill my brain a little
just for a brief night's rest
but i awoke early the next morning
and could no longer keep away the patterns
i was even more off and edgy
odd
if you insist
i don't even have drink as a refuge
is this how they took you?
i don't understand the forwarding?
it all makes so much sense
that in the end it makes no sense at all
all the connections and none of them at all
@#@##@$#@#$@#@#$#$#@$@#$@#$@#$@
remember when i pointed out the connections between post logics and multisections?
how the various function spaces generated
could interpret the negation in differentiation
and how the various multisections could play the roles of truth values?
maybe a curiosity
but
recently i saw a paper on the galois theory of post algebras
and these strange obsessions began to take hold
sheaf theoretical interpretations appeared as dancing symbols before my eyes
i think i finally see
how these multisection projection logics
peer into the geometry of the functions they act upon
the geometry of the circle
the exponential
is only the first stage
and after thinking these thoughts in the dumbstruck awe of discovery
after seeing these relationships so clear and tangible
i looked down and saw i had drawn this scene
http://galathaea.org/facesInStone/DSC04944.JPG
this was not the first
i have found pictures drawn after several of my disturbances
http://galathaea.org/facesInStone/DSC04945.JPG
http://galathaea.org/facesInStone/DSC04946.JPG
this is not good
remember crumb's brother charles?
i am very scared about all this
ooooooooooo
ooooooooooooooooooo oooooo
oooooooooooooooooooooooooooooooooooooooooo
ooooooooooooooooooooooooooooooooooooooooooo
ooooooooooooooooooooooooooooooooooooooooo
ooooooooooooooooooooooooooooooooooo
oooooooooooooo ooooooo
i awoke the other day with a sharp pain in my back
there were two sharp peaks protruding from my scapulae
slightly curved out
as if to follow the scroll of my spine
the skin was pulled taught around them
and the area was very sore
every time i touch them
even slightly
my eyes glaze over and the room seems to crystallise in millions of facets
leaving a headache that lasts for hours
i am going to see a doctor about it today
i can no longer wear a bra
this is not acceptable
Michael Press
galathaea <gala...@veawb.coop> wrote:
Yes, consult a good practitioner.
Consider your habitual postures for clues to the malady.
Do not allow fear to dictate your choices; remember the fear, however.
Try looking elsewhere than where you have been looking for guidance.
--
Michael Press
galathaea
> In article <galathaea-C47E01.13423526052...@news.veawb.coop>,
the narrator is clearly disturbed
and the threads of valid deduction are fraying
but
there is still a whole process that must unfold
there won't be reconciliation until all the ghosts have gone
and that means destroying a lot of dreams
although there is great hope and drama
ultimately things will end up mostly tragic
wait until the subtly hidden arithmetical flaw
reveals that none of this math is valid?
or watch as awkward claims of meaning
slip away with scrutiny?
how badly will it end?
guidance can only come deus ex machina
Angus Rodgers
<gala...@gmail.com> wrote:
>On May 26, 6:59 pm, Michael Press <rub...@pacbell.net> wrote:
>> In article <galathaea-C47E01.13423526052...@news.veawb.coop>,
>>
>>
>>
>> galathaea <galath...@veawb.coop> wrote:
>>
>> > [...]
Funny: my daughter recently got a tattoo which reads "Don't
clip my wings".
--
Angus Rodgers
(twirlip@ eats spam; reply to angusrod@)
Contains mild peril
Michael Press
Angus Rodgers <twi...@bigfoot.com> wrote:
<http://www.imdb.com/title/tt0975756/combined>
--
Michael Press
Michael Press
Angus Rodgers
D'oh! I think that goes some way to explaining why, when I tried
to watch "Faraway, So Close!" (or rather, in keeping with the title
of this thread, "In weiter Ferne, so nah!"), I couldn't get into it!
I'm so ignorant ...
galathaea
it broke like the hand of a china doll
a shatter in my palm
muffled and piercing
my flesh
my
doorway adorned itself in elaborate display half-wound with
the winding storm-ridden worm-eaten half-rotten willow
peel bark wrapped
and woven around
the frame hollow
and
though
there was always
debate it could be
paper-thin hate-filled
strands of darkness
there was confidence
it spied on me in the late
with
pin-point
eyes
i have not slept in days
outside is no longer the isworld
it is an almostworld
a shifted world
outside
the skeleton of buildings lay criss-crossed
as if shifted in some great cataclysmic
twisting of the earth
a bending of fortunes
weaving the steel skeletons
crashed together
$#%#%@%$#%@$%@
people too
were shifted and
scarred * a brutal few
themselves skeletal
and caricature
$%$#%%#$@$# %$#@$%@$%@#$
and etched post-nuclear
the signs the signs
the same signs
everyw here
m
two towers
two signposts
bent
an x
or crumpled into accordion \/\/s
arches collapsed
everyw here
m
m-1 m-1
--- j --- j
/ 1 \ w \ 1 \ n w
T | - / m | = - / m
m n \ m --- x / m --- x
j=0 j=0
the vision of varanasi
which i had so celebrated as the first nontrivial property of W [x]
n ow haunts me
in this brutal landscape
to the accompany of earth rumbles
and the crashes of distant wars
i don't know what i have become
there is an anger in me
growing
with the tusks
what had begun as two strange growths
has over these few weeks become
four then eight
then ten and
twelve
and
more
bone had thrust from ugly wounds
in these long tusk-like spears
one row along the upper plate of my back arched downward
and one just below the small of back where my pelvis spreads
arched towards the sky
the wounds had seeped for days
when i must leave
and i have found ways to limit this to less than once a week
i have nothing i can wear to fit my new form
so i have learned to wrap some indian silk around my top and middle
biting back the pain
i hate the ugliness of this monster i have become
i hate
i hate
and her bitterest voice i obsessively recall
"and all the baby boys we've born
with eyes averted from the storm
SENT OFF TO DIE IN PERFECT FORM
WE KNOW NOT NOW WHAT WE HAVE KNOWN
satellite photos and rhetoric
see how the euphemisms stick
AND WHEN THEY COME BACK BROKE AND BURNED
THOSE WHO'VE RETURNED HAVE NOT RETURNED.."
i keep going over the same things
i can't take it anymore
i just spend hours crying and screaming
my back
my mind
my world
all fucked
[][][][][][][][][]
[][][][][][][][][]
[][][][][][][][][]
[][][][][][][][][]
** [] [ [[[] ] * **
****** ** * *** ** ** ** ** * * ******
******* **** ** ** ** * ** **** ******
***** * ** * *** ** ** * * * ** *
--[-]-]-[[-]]--]]-[-]]-
[]--[][][][]-[[]-]
[][][][][][][][][]
[][][][][][][][][]
[][][][][][][][][]
[][][][][][][][][]
[][][][][][][][][]
[][][][][][][][][]
[][][][][][][][][]
[][][][][][][][][]
[][][][][][][][][]
[][][][][][][][][]
the brothers are patrolling the streets now
first it was the genes
and now the brothers have jumped in
everyone's looking for a piece of control
as always happens once the market's saturated
the exploitation of paranoia has begun
security and paranoia in this millennia-old interplay
where
there really are a lot of exploits
and it really does require attention to every minor detail
and just because you're paranoid
doesn't mean they're not out to get you
and it looks so foolish
i figured it out the other day
it's so obvious and silly
i'm not rational anymore
i'm transcendental
i'm beyond algebraic
i'm allegoric
i actually saw it graffitied to some buildings
schanuel
schanuel schanuel
schanuel schanuel
schanuel schanuel
schanuelchanuel
schanueluel
schanuell
schanueluel
schanuelhanuel
schanuel schanuel
schanuel schanuel
and i was a module at the birth of time
an actor acting upon the state of things
when the main actors realised themselves
it was always about control
nature had developed decision points
subsystems of sufficient complexity
to process information input
and compute response outputs
according to some algorithm of drives
purposes
and the input
it's symbology and learned metaphors
were the primary point of control
technology
as a physical process in the universe
began to see itself
like i now saw myself
in the mirror of the door
hideous and fierce
^..6
what did these symbols mean?
why do they haunt me?
take the number e and try to build a ring out of it by generation
(in R if it helps)
it will have 0 and 1
n
it will have e for all n >= 0
most fundamentally
every member of this generated set could be expressed in the form
---
\ j
/ a e where a elementOf Z
--- j j
j elementOf J
J subcollectionOf N (n >= 0)
J finite
this ring of transcendental numbers has the same presentation as Z[x]
once one makes the identification e -> x
of course this can be done with any free generator (Z is universal here)
and so any transcendental has the same relationship
but there is an even more fundamental relation between exponentials
that comes into play once inversion is considered
x
the map e --> x can be inverted on a sheet through the x --> ln(x) mapping
it's use here is trivial
but it will become obvious later how the generalisations take advantage of this structure
fundamentally here
the structure of certain rings generated by transcendentals
can be derived from the isostructure in rings generated by unknowns
but this is all so much local structure
it is important to take a step back and see what has been wrought
polynomials-----------\
/ \ \
/ \ \
/ \ \
/ \ \
/ power maps \
/ || \
/ || \
/ || |
exponentials || |
\ || |
\ || |
\ || |
\ fourier analysis |
\ / /
\ / /
\ / /
\ / /
\ / /
\ / /
differential structure-----/
x x
the fixpoint relation D e = e
x y x+y
and the product formula e e = e n
interact through this fundamental connection of the power maps x -> x
viewed as an injective ring homomorphism
and the fourier interpretation of these as a transform
starting with the differential structure
2
the taylor expansion of the exponential as 1 + x + 1/2 x + ...
n n-1 n-1
/ x \ / n x \ / x \
and in particular how D | ---- | = | ------ | = | ------ |
\ (1) / \ (1) / \ (1) /
n n n-1
gives the fundamental relationship behind the exponential
the factorial
the differential
and neighboring indices in the taylor basis
and is what structures the entire theory
the transform provides the interpretation of exponential map as shift operator
exposing it's group theoretical structure on tangent bundles of the euclidean manifold
now the bisection gives instead
laurent polynomials----\
/ \ \
/ \ \
/ \ \
/ \ \
/ tchebyshef functions \
/ || \
/ || \
/ || |
trigonometry || |
\ || |
\ || |
\ || |
\ fourier trigonometric analysis |
\ / /
\ / /
\ / /
\ / /
\ / /
\ / /
differential structure-----/
and here the classic connections really show off the algebraic structure
the product formula become
ch(a) ch(b) = 1/2(ch(a + b) + ch(a - b))
ch(a) sh(b) = 1/2(sh(a + b) - sh(a - b))
sh(a) sh(b) = 1/2(ch(a + b) - ch(a - b))
and the matrix interpretation appears as the inversion problem gets investigated
the first sum formula ch(a + b) is accomplished
by listing the product combinations that give order 0 (ch) terms
(these are (0,0) and (1,1) - ie. ch(a) ch(b) and sh(a) sh(b))
one must then invert the matrix
| 1/2 1/2 |
| 1/2 -1/2 | a humble little thing
similarly
the differential structure begins to reveal a little of it's algebraic form
by the twisting property of the differential here
D ch(x) = sh(x)
D sh(x) = ch(x)
the system may be given a matrix form
| ch(x) sh(x) | | 0 1 | | ch(x) sh(x) |
D | sh(x) ch(x) | = | 1 0 | | sh(x) ch(x) |
where the matrix | 0 1 |
| 1 0 | is a lot like -1 in that
| 0 1 | | 0 1 | | 1 0 |
| 1 0 | | 1 0 | = | 0 1 |
-1
the classical tchebyshef give the operators over Z[x, x ]
/ 1 / -1 \\ 1 / n -n \
T | - | x + x || = - | x + x |
n \ 2 \ // 2 \ /
a beautiful little interpretation of power maps in the land of laurent
but laurent polynomials are in some sense a trivial extension of Z[x]
since every laurent polynomial may be factored as
m k+m
--- ---
\ j -k \ j
/ a x = x / a x
--- j --- j-k
j=-k j=0
and so one does not get any interesting extensions of the algebraic numbers
also
if i go the other way
it is easy to form an inverse polynomial
and get factors in terms of inverse monomial differences
-1
e.g. (x - z) ---> x (1 - z x )
and so Z[x, x^(-1)] fails to have unique factorisation
(and all x^j have become units)
even still
generalising the operation of algebraic extension
still works in the obvious way here
as long as one still chooses the appropriate definition of irreducible
(there was this hallucination i had the other day
where the door was playing mirror and i was looking at myself
and this inner view emerged enlarged yet simple
and i saw the simple way my properties fit
in algebraic connectives that formed the logic of my description
i was no longer angry at myself
and i finally saw what factors were at play
as the hallucination faded
the doorway told me in it's whisper
"inner factorial rings"
the books tell me it is something important here)
inversion works for the trigonometrics
and displays why the trivial inversion for exponential becomes important
-1
1 / -1 \ x / 2 \
- | x + x | = a ~~~> - | x - 2ax + 1 | = 0
2 \ / 2 \ /
and the extension to the splitting field gives
2 1/2
2a +/- (4a - 4)
x = -------------------
2
x
and since this is based on the substitution x ~~> e
x is positive which selects
2 1/2
x = a + (a - 1) when a is greater than 1
mapping back the translation gives
-1 / 2 1/2 \
ch(.) (a) = ln | a + (a - 1) |
\ /
this illustrates the essential connection between extensions of these laurent polynomials
and the inversion problem for trigonometry
generalising over all multisections gives
W [x] polynomials------\
n \ \
/ \ \
/ \ \
/ \ \
/ generalised T (x)functions \
/ m n || \
/ || \
/ || |
generalised trigonometry || |
\ || |
\ || |
\ || |
\ generalised fourier analysis |
\ / /
\ / /
\ / /
\ / /
\ / /
\ / /
differential structure-----/
and here
with n=3
we get the first nontrivial extension of algebraic numbers
W [x] is twisted in a macabre way
3 and the tchebyshef are no longer polynomial
2
w w
3 3
here x x x = 1
2
but the rays n, n w , and n w no longer contain all exponents (like 1 + w )
3 3 3
in fact
from any two neighboring rays in a finite set of rays
one can always find two elements that sum to an element not on any of the rays
so
although on any ray things behave much like Z[x]
once terms can be across rays
products produce many new symmetries of terms
and now
monomial differences are no longer sufficient to factor
w
3
x + x + c has no such factorisation
it becomes important to study deeper the level structure
where a cyclotomic integer is of level n
if it is expressible as
2
j + j w + j w with j + j + j = n
0 1 3 2 3 0 1 2
since the collapsing of levels is what defines
how the structure of these types of irreducibles is much enhanced over Z[x] here
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~>..<~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~*
as with the case for producing the algebraic numbers
one forms the collection
I = { (p) | p irreducible in W [x] }
3
and takes the direct limit
A = lim W [x] / (p)
3 --> 3
(p) e I
which is strictly larger than A
the field of algebraic numbers
similarly
the galois group
---
|
| (A / Q)
3 Q 3
is the projective limit
/ / \ \
lim gal | | W [x] / (p) | / Q |
<-- \ \ 3 / /
(p) e I
it is strange how much of the classical algebraic theory comes over
in these new generalisations which contain some strictly transcendental elements
much care must be taken to ensure that these are meaningful operations
but once one overcomes W_23
one can consider even more grand constructions
one can consider lim A as some generalised algebraic field (which is still not C)
--> n
n
$%#$%^%^##%^$#^#$^$%
$#%^$^#$^%#^#$^#^#%^#%$^$#^^
$%^#^#$^#$^$%^#$^#$^#$#^%$#^$#^$%
#$%^$#^#$#^#$^$#%^#$^#$^%^$^#$^^#%$^$
#$%^#$%^#$^$#^#$%^$#^$^#%^#$%^$%^$#^#$$
#$%^^#$%^$#$^$#^#$^$%^$#^#$^#$%^#^#$^#$^^
$%^#$^%$%^#^#^#$^$#^#$^%^%$#^%#$^$%#^%$#^
#$%^$#^#$%^#^#^#$^$%^#$^$#^%#%$^#$^#^$#^$#%
#$@%#$@$%@#$%@$%$@$%#%$#@$%%$^%$^&^%$%%^$$%
%^$^&$%^&^%$%^&%^&%^$&^%&%^&^%&%^$^^&%$^$%$
$%^%^&$%^&%^$&$%&%^$&^%$&%$^&$%^&$%&^^&%$^&
%^&$^&^%$^%&$%&%^$&$&^%&$%&%^%^%$#%%^%$^^
$%^%$$#^%&^%&^%&%^&%^%^$&^%&$$%^$%#$#%^^&
#$%#@@%@%@%@$%@$%@%$@%@$%@%%&^%$%^$^&%%
^%$#%^$#^$#^%$#%$#%^$#^%^%$^%#&%^#$$#
#$%##$@%%@#$%#$%@#$%@%@#%#@%%$^#$
$%^$#$#%^#^#^#$^$%^#$^#^#$%^
%^#$^%^%%^$#%^#%
it is a giant ball of mud
a mucilaginous amalgamate of unwashed epiphanies
fueled by this strange psychosis i now find myself in
and now these deeper visions keep coming up
these other symbols
toric transcendentals------\
/ \ \
/ \ \
/ \ \
/ \ \
/ toric maps \
/ || \
/ || \
/ || |
hypergeometry || |
\ || |
\ || |
\ || |
\ hypergeometric analysis |
\ / /
\ / /
\ / /
\ / /
\ / /
\ / /
differential structure-----/
but it is all too much for me anymore
i cannot sleep
eat
breathe
the brothers have stepped up their patrols
helping to "ensure" that everyone is "righteous"
i keep becoming fixated by tangents
i opened ramanujan the other day
(number theory because i did not want more series!)
and i see
6 oo
--- --- j
\ j \ (-1) 14j
/ cos(w x) = 7 / ------ x
--- 7 --- (1)
j=0 j=0 14j
are these pages i read also mere hallucinations?
why have i not seen these before?
only after i calculate is it revealed
and he goes on to show how to use these to derive bernoulli sums
much as i have for generalised bernoullis...
will i turn up work on generalised eulers
relating them to the obvious generalisation of alternating permutations?
there is no more day or night
there is
only the obsession of the moment
cale domains
möbius inversion of generalised polynomials
mirror symmetry and motives
it is like he told me in his "reapings and sowings"
about opening the nut with a hammer and chisel
or letting it dissolve to it's natural beauty
" I can illustrate the second approach with the same image of a
nut to be opened. The first analogy that came to my mind is of
immersing the nut in some softening liquid, and why not simply
water? From time to time you rub so the liquid penetrates better,
and otherwise you let time pass. The shell becomes more flexible
through weeks and months—when the time is ripe, hand pressure
is enough, the shell opens like a perfectly ripened avocado!
A different image came to me a few weeks ago. The unknown
thing to be known appeared to me as some stretch of earth or
hard marl, resisting penetration. . . the sea advances insensibly in
silence, nothing seems to happen, nothing moves, the water is so
far off you hardly hear it. . . yet it finally surrounds the resistant
substance. "
whatever is dissolving me is toxic
an organic solvent intoxicating through brain damage
i have to face the outside again
i have to find answers to questions i do not know how to ask
these tusks
these jaws of malformed bone protruding from my back
are growing restless
i have to leave
i have to go outside the door
Peter J Ross
<gala...@veawb.coop> wrote:
> it broke like the hand of a china doll
<etc>
I enjoyed reading what I was able to understand of this. I'm unable to
judge whether it's a good experimental poem aimed at a limited
audience or pretentious crap; but even if it's pretentious crap it's
the kind of pretentious crap I enjoy reading. :-)
Thanks for posting. What does "봄의 끝" mean?
[Followup set to rec.arts.poems, but feel free to ignore my
suggestion.]
--
PJR :-)
galathaea
> In rec.arts.poems on Tue, 24 Jun 2008 22:55:49 -0700, galathaea
>
> > it broke like the hand of a china doll
>
> <etc>
>
> I enjoyed reading what I was able to understand of this. I'm unable to
> judge whether it's a good experimental poem aimed at a limited
> audience or pretentious crap; but even if it's pretentious crap it's
> the kind of pretentious crap I enjoy reading. :-)
definitely it is pretentious
i'm too much of a narcissist for it to be reserved
(and the latin praetendere fit's my sig perfectly
despite it's unfortunate lack of presence in the former post)
when i get to pointing out
that the differential structure generalises easily
and for hypergeometrics we get
n
(xD + b1 - 1)(xD + b2 - 1)...(xD + bq - 1) CH (..) =
p q
n
(xD + a1)(xD + a2)...(xD + ap) W CH (..)
n p q
n
where CH is the n-multisected circulant matrix
| |0 |1 |n-1 |
| |n H |n H .. |n H |
| p q p q p q |
| |
| |n-1 |0 |n-2 |
| |n H |n H .. |n H |
| p q p q p q |
| |
| . . . . |
| . . . . |
| |1 |2 |0 |
| |n H |n H .. |n H |
| p q p q p q |
and W is the n-dimensional matrix root of unity
n
| 0 1 0 0 0 ... 0 |
| 0 0 1 0 0 ... 0 |
| 0 0 0 1 0 ... 0 |
| . . . . . .. . |
| . . . . . . . . |
| 0 0 0 0 0 ... 1 |
| 1 0 0 0 0 ... 0 |
you may see the pretension really start to kick in
as i start making huge claims about the differential structure
and once i point out the connection
between the horizontal functions
and the splitting fields
i may start playing like i've created
the second coming of modular functions!
> Thanks for posting. What does "봄의 끝" mean?
it is the end of spring
galathaea
http://www.garageband.com/song?|pe1|S8LTM0LdsaSjYVm2Z24
it was the end of every living dream
a vast emptiness
despite so overfull of decay and the destroyed
colorless stretches of concrete
blacktop
and rubble without stain
a landscape sterilised by rivers of oil and distillates
cleansed by a greyish-white dessicating film
there was no life
or at least anything that would be proud of such a title
i stood there in the daylight glow and stared out
just outside the room i had hidden in so long
and out in the wasteland
under the brilliant poison sky
i realised that it was all my fault
what ever has happened
whether my brain has fried and none of this is real
or some insanity has warped the laws of the world
this is my fault
that much is perfectly clear
if reality has changed
it is related in some way to that day i went to your property
the day i trespassed
and if my mind is screwed
the fault should be clear
i am so stupid sometimes
and now i have really fucked things up
galathaea
$)#^%@^$#)*%)%*^$*%@#%^$#*^
$#%)*^#^ 08
sometime shortly after stepping outside
i began to recall a story i had been writing
but many years ago abandoned
it was called:
q-"which way" tales
and was a collection of small fables
of strangely deformed generalised trigonometrics
they were daily tales
hard-struggle americana frontier tales
grandmother q-exp had many faces
she was an unstable witch
who alternately was always the most stable for us
in these twisted-deformed worlds
she was our stability in the primordial differential structure
the q-worlds had a strangely bent accounting
with a skewed and twisted form for numerosity
a q-number began with [n]
q
some function of q that had a limit as q -> 1
of n
associated to this arithmetic
was a q-differential
which could be defined linearly on the basis power elements
n n-1
D x = [n] x
q q
the deformations had many forms
which seemed to be due the slippery nature of [n]
q
the fundamental deformer
the left-hand (sinister) form
described
n
2 n-1 1 - q
[n] as 1 + q + q + ... + q = ------
q 1 - q
which has a nice geometric summability
the middle path deformer speaks of
n -n
q - q
[n] = --------
q -1
q - q
the sinisteria believed the q-derivative
could be defined as
f(q x) - f(x)
D f(x) = -------------
q (q - 1) x
to give them the formula on the basis elements
the middlers expected the symmetry to obtain through
-1
f(q x) - f(q x)
D f(x) = -----------------
q -1
(q - q ) x
in every world
grandmother q-exp was the one where
D exp (x) = exp (x)
q q q
strange grandmother q-exp had many other forms
and at one time
us children put together a scrapbook of her faces
which included the strange tales of
n-1 j
--- w n
\ (n-1) j n
/ w q
--- n
j=0
[a] = ---------------------
n q n-1 j
--- w
\ (n-1) j n
/ w q
--- n
j=0
the great hydra grandmother
but of the many tales
and of their many contradictory paths
we always knew her as D exp (x) = exp (x)
q q q
and on the power basis she was always
oo
---
\ 1 j
/ ------- x
--- (1; q)
j=0 j
j-1
---
where the (a; q) = | | [a + k]
j | | q
k=0
are the source of the janus affliction
%($#&)(&%#(^(#)&&)()%)$#%#&(%)(%)(#
&*^^ ^**& &^**^**^*^^^*^^^*
$% $%^ $$$6 $%^^^^&
$#%@%@@%%@%$@%@#%%
#$%@$%
when we were very young
we all were very excited to hear
the great tale of how grandmother learned of her noncommutativity
she was producing
producing
we had learned
was grandmother's meditative way of relating distant parts of herself
by taking their product
producing some whole new part of herself to learn from
and in her playful afternoon
she came to understand that her product rule was not guaranteed
or at least
it didn't quite work out right commutatively
as she turned her symbols around in strange combinations
the product formula for q-exp works out as
oo oo oo oo
--- j --- k --- --- j k
/ \ a \ / \ b \ \ \ a b
exp (a) exp (b) = | / ------- | | / ------- | = / / ---------------
q q \ --- (1; q) / \ --- (1; q) / --- --- (1; q) (1; q)
j=0 j k=0 k j=0 k=0 j k
oo n
--- --- j n-j
\ \ a b
= / / -----------------
--- --- (1; q) (1; q)
n=0 j=0 j n-j
now
n
--- (1; q)
n \ n j n-j
(a + b) is not in general equal / ----------------- a b
--- (1; q) (1; q)
j=0 j n-j
which works for the classical binomial
but which deforms in this strange world
so her product rule
which she knew and used from as a child
does not seem to work out
,.
it was only many years later she learned
that this was all the fault of my itty bitty deformity
and ultimately me
but meditating upon these symbolic forms
grandmother in her brilliance learned
that there was a step that could be accounted
assuming a controlled break commutativity
if this were a noncommutative ring
the level of noncommutativity could be used to accommodate the deformation
grandmother q-exp found her noncommutative ratio q
if ab = q ba
thought grandma
then things work out
this can be seen by looking at the iterative step
of multiplying by (a + b)
n+1
--- (1; q)
\ n+1 j n+1-j
/ ------------------- a b =
--- (1; q) (1; q)
j=0 j n+1-j
n
--- (1; q)
n+1 n \ n j n-j
= (a + b) = (a + b) (a + b) = / ----------------- a b (a + b)
--- (1; q) (1; q)
j=0 j n-j
and so the rule gives the law on the deformed binomial coefficients
by equating like (j, k)-power tuples
(1; q) (1; q) (1; q)
n+1 n-m n n
-------------------- = q --------------------- + -----------------
(1; q) (1; q) (1; q) (1; q) (1; q) (1; q)
m n+1-m m-1 n+1-m m n-m
and doing the multiplication on the other side gives the law
(1; q) (1; q) (1; q)
n+1 n m n
-------------------- = --------------------- + q -----------------
(1; q) (1; q) (1; q) (1; q) (1; q) (1; q)
m n+1-m m-1 n+1-m m n-m
using both of these equations
a single component recursion can be derived
which illustrates why some deformations are more relevant to these product rule concerns
..
we never knew exactly how grandma had made the leap
it often seemed backwards to us growing up
that she had found the rule to the q-deformed choosers
somehow through the noncommutativity insight
and we often suspected she had worked long in her self-analysis
but when we asked
and we asked with the hunger of childhood
she would just smile at us
with that sparkle of something mischievous
glinting off the many facets of her eyes
later stories of grandmother q-exp always seemed to go into these strange deformation algebras
but there were more subtle dependencies on deformation type
defining the allowable forms
the left-hand and middle paths were dominant players
though there were also squeezed (q^(1/2) - q^(-1/2)) grandmothers and other exotics
who played important roles in many tales
^^
the product rule issue had other generalised forms outside the noncommutative realm
commutatively
the product could be of two related forms
as
for instance
the sinisteria do with their
[n]
* q
define [n] = ----
q n
q
and q-exp* the associated fixpoint of the differentials
then there are a number of product-like rules that can be derived
through relations of these across various bases
i remembered that these sections on commutative symmetry relations
were long and tedious sections
enumerating the intimate connections
and improper familial relations
of the commutative family
but it was acceptable
in fact proper
for americana to have long sections of tediousness
because all life was tedious
and true americana glorified this work ethic
entranced in the vision of the next generation
$$&%#(%(_#@&$%$
^^ ^^^
which is why the q-"which way" tales turned to the next generation
they were the first loved
the second generation
grandmother's own ch and sh
q q
ch was born first
q
when in a flash of productive insight
grandmother q-exp bisected her infinite form
projecting half her essence to a completely new entity
---
\ 1 j
/ ------- x
--- (1; q)
j in 0 (mod 2) j
grandmother said she chose the name
because she had never liked the q in front on her name
and wanted a name that didn't bring the back of the throat forward
like x being followed by p
instead preferring the upfront and more muted voiceless postalveolar affricate ch
followed with the voiceless uvular plosive stop q
in the process of this projection
grandmother q-exp exposed a second child
who was born a moment later
and quickly named sh because she was crying
q
---
\ 1 j
/ ------- x
--- (1; q)
j in 1 (mod 2) j
ch and sh were very precocious children
q q
but grandmother q-exp grew concerned and frustrated as she prepared their lessons
she wanted to teach them early of producing
because it had been so important to her own growth and education
but she found they didn't have her product law
the q-which way tales included seven stories at this point
collectively known as the septic tales
where grandmother q-exp travelled to the great library
and found reason in the books of the ancients
to pursue seven quests for deeper knowledge
it was in the fifth of these quests
after the great conflagration
and the slaying of the demon mahakazam
that grandmother first gained access to the ancient trigonometry
and found the classic product rules
ch(x) ch(y) = 1/2 (ch(x + y) + ch(x - y))
ch(x) sh(y) = 1/2 (sh(x + y) - sh(x - y))
sh(x) sh(y) = 1/2 (ch(x + y) - ch(x - y))
and
as a part of her revealed enlightenment
she was finally allowed to understand the myth of the undeformed
the ideal of an emanation
upon which truth may be revealed
for these trigonometrics were built using an ideal exp called simply e
that part of grandmother q-exp she had learned to hide
having grown a thick skin in the reality of the deformation
x -x
ch(x) = 1/2(e + e )
x -x
sh(x) = 1/2(e - e )
grandmother q-exp told us that she understood these equations immediately
having __felt__ them so intimately when her own children were born
it was a place of much comfort and understanding to wise grandmother
and she immediately recognised that products were not a problem
it was her noncommutativity trick again
and it worked exactly the same
and she immediately realised it
the process itself
the projection
carried over all the properties
from the base q-exp to her offspring
all of it worked out perfectly in the way she was so familiar:
ab = qba
this was secretly related
grandma had learned through deep study
to not existing as functions in some space
but instead as operators over those functions
just as she was the deformed orbit generator
they were deformed operator trigonometries
much much later
in the seventh of her quests
where ultimately she learned the secret of the deformation
she found along the way a copy of a strange text
of the curious title "industrial metempsychosis"
which in it's earliest chapters described
a trigonometry of even greater greater generality
this was long after me and my two sisters had been born
and too late to have helped explain our early chidhood confusions at our origins
and the lingering stain that our birth may have been
brought about the great midlife crisis of sh
q
and deep traumas of her independence and self worth
mother sh had had many problems
q
in the face of having no productive forms independent of ch
q
and had long and great battle with chemical dependency
that may have brought about the era of twisting
which had eventually split us into existence
hungry and cold
waiting for the onslaught of winter
over the isolated prairie plains
at least
that was the way the tales told the stories
in their melodrama and opera
particularly in the parts that were driest
and failed to develop the characters sufficiently
this was the part of the story
where i introduced the deformed generalised trigonometry
and its q-fourier theory
and there was not much new from the classic theory
since the product structure on which it was built had not changed
save for the radical switch to the noncommutative realm
in which all us offspring grew
the story of the q-tchebyshef's was precisely the same
the symmetry was solid from that point out
the story from there dissolved in the afternoon winds
i felt my skin dry in the solving sterile breeze
the skin crack and brittle in the spray
it was time to find a way away
..
galathaea
there was a road away from my place
but it was not the road that had been there before my seclusion
it was transformed
mutant
cracked through some acceleration of aging
and twisted through strange upheavals
neighbors' houses were abandoned
or at least i hoped them abandoned
in their silence and ruin
i really wasn't speaking to myself much
i was walking
and fearing
and slowly sickening
it was blacktop
soft
almost rivers under the bleak white directionless heat of the day
**********************
as the afternoon latened and darkled itself back into my attention
i was startled at the sound of someone screaming down the road
it wasn't the frightened kind of screaming
that had secretly been drowning my thoughts since i left
but it was more that laughing form of sports bravado
that plunders sunday afternoons with various intoxicated outbursts
i could see him about a kilometer off
and he probably first saw me at half that
it was hard to see what he was doing at first
he was so far off
and the haze and smoke obscured him to a shadowy silhouette
.
it was then that my observer
my analyser
got through
it was then that i heard that almost silent screaming
" YOU HAVE NOTHING!! "
" YOU HAVE NO PROTECTION!!! "
enter into focus
i had no protection
i hadn't even taken a knife when i left
i've just been stumbling forward in some kind of daydream
waiting for inevitability
i know there is something really wrong with me
still it happens almost as an afterimage
the progressions of patterns found in patterns found in patterns in my view
every higher order breakdown fractured by
something blurred
distorted
almost forgotten
and realising this might cost me my life
scared out an island of sanity for the observer to take over
^..^
as i came closer to this screaming and swaying amorphosity
i slowly pieced together the major features
- the top of this form was fast moving and the source of the sounds
- the bottom swayed in slow and stumbling rhythms
i began to sense more clearly the structure of the forms present
- the top had an extremity waving wildly above it
- as a voice more and more clearly screamed:
" yeah!!!....
...come on boy!!...
...give me another strip of that chest meat!!...
....don't give up on me now!!...
..watch the fucking curb!! i don't want you dropping me...
..again!... "
i stepped forward
still enraptured in my inevitabilities
but cautious as i waited for the full revealing
- the bottom form
looking ever more like a person through the haze
was also making noises
issuing gruntings and high pitched wheezings erratically
- the top form soon revealed itself also very human looking
apparently riding upon the shoulders of the unfortunate lower one
the scene was becoming clear
they saw me
"hey there missy" said the man on top
drunkenly shifting in his lovecraftian cowboy fashion
tan galan hat in his left hand
animated
i stood there tense
in perfect readiness
"don't worry missy"
"i ain't got no need to eat ya here with spit around"
he laughed and ripped an ear off his unfortunate companion
on whose shoulders he was riding
spit was the walking dead
a skeletal frame around which
bruised and shredded muscle
wound
in strands of tenuous connection
a barely functional motive system
flesh
where found
was rotting and necrotising
"don't afear!
he's animated!
government approooved!"
i watched him
voiceless for an era
"so"
he started
"what would you be out here for"
chewing ear
"i just came out"
"protection?" he squinted
i stared at him
for intent or meaning
"ha ha!! ooohhweee!
girl
you don't hesitate on shit like that
if you have anything
you must present it
or they will know
and you have no control over the situation"
he laughed inebrelation
"damn girl
you need to learn something fast.
YOU ARE FOOD"
he leapt down from spit's shoulders
"this is a hungry world
and you look delicious"
my right leg snapped a sharp arc
inverse kinetically guiding my foot's final destination
squarely in his groin
he bent over
shaking
"who are you?"
in between gasps for air and spasms of pain
something like "texas"
"give me your weapon"
"look i told you i ain't gonna eat you!
i'm fat and happy right now
i was just warnin ya of others
warnin to watch out how they see ya
and never give them control over you"
he looked at me
closer
i could feel the pit of stomach turn
under the coldness of other people's eyes
"are those attached?
what are they?"
he asked eyeing the bony growths out my back
"they're wings"
it was strange the alienation
and separation i felt
under his eyes
as i stared at him with all my focus and attention
my joke seemed hollow and awkward
"they look like teeth
long crazy pointed teeth
you got a great kick there
you're quite the little ninja there ain't ya?
ooowhee! yeah... well so is that where you are hiding your beads?"
he said
looking straight at my makeshift top
where my breasts were wrapped in an indian silk
after the growths had ruined or made impossible all other protections
my right leg snapped a sharp arc
inverse kinetically guiding my foot's final destination
squarely in his groin
with a little twists of the heel
"oh shit! oh shit! oh fuck
i think you popped one that time
ooooOOhh.. " deep breath " shiit..!"
panting
"ohho missy!!! i diid not mean that...
disrespectfully..
i meant
i meant your beads
your dreams
your hopes
everyone's got a couple of them
though they're ashamed as hell and keep 'em hidden
i just hadn't figgered where you were hiding them
don't tell me you ain't figgered it yet?
here... " he pulled a small bag from a chain around his neck
and tossed it over to me
"those are mine"
i opened the leather tie at the opening of the bag
and saw three tiny glass marbles
glinting unexpectedly bright in the dull haze of the day
"that clear blue one there
the one with all those bright silver flecks
that's my land to live on
i always wanted a place to live
some place to settle
and stop haffin to wander around everywhere
looking for something to eat
a place that grows the food still
that bigger red one
the one with orange inclusions that look like a fire
that's for my wife
wimmin love a man with property
she's gunna be one of them blonde ones
like they have on the coast
with big titties and a love to feed her man!
and that one there
the green and pink one with all them swirlies
that's my daughter
she'll be the perfect angel
and i'll get to sit outside with my thirty ought six
chasing away all the boys
and she'll take care of me and my wife
as we get older
ain't them reasonable things for a man to dream of?"
i tied together the bag
and tossed it back to him
"so you ain't ever collected any dreams?"
"no i've never seen anything like them"
"iff'n you don't keep 'em close to you
they'll eventually get brittle and dust away
you won't remember 'em no more.
you'll find em after long nights of whiskey
in your bed or yer porch whathaveya
wherever you been blabbin' about inside
or sometimes out loud when you've really been hittin' the bottle
you know what i mean!.. ha!
they're not hard to find
even if you just dream a little
i'm surprised you ain't bumped inta them before"
"i think i dream a lot"
"anything you remember?"
"no
i don't recall"
it was strange
i could remember my symbolgical obsessions
i could see their great immediacy
and the fevered nights
but i was numb to any future dreams
there were pains and exhaustions
that had grown into a skin
a rubber membrane muffling the noise
and desires
i had no idea and no desire to idea
and realised that my mind had fried much worse than earlier expected
i didn't
have
any
dreams
anymore
i just wanted to escape the incessant fantasies
the recollections
and obsessions
that had haunted me so deeply of late
i didn't want to be flawed
and broken
in other people's eyes
was that a dream?
"so you interested in finding out more about the beads?"
i startled in my wanderings
"the punks
they love the beads
they gamble with them
use them as weapons
and other things i'll leave for you to discovery
they hangout under the great multipass
where this highway meets the great eastward escape
if you keep headed down
you can't miss them"
"give me your weapon"
"huh?"
"give me your weapon"
i shifted my weight off my right leg
"oh waiwait! here!"
he pulled out his hunting rifle
and handed it over
cautiously avoiding reaching for the trigger
"why is his name spit?"
"you ever tasted him?"
alainv...@gmail.com
>
> plus de détails »
Moi pas comprendre du tout ton français language...
Alain
0x0000
Interesting, but the layout makes it really hard to read. What's that
about?
galathaea
across the valley with the great highway multipass in sight
i fell into a deep obsession
linguistically associating the german traume or dream
into a meaningful correlation with multisection
i saw it play as a character
in a dance with variables and differentials
projections
into multisected weyl algebras
these are quotients of k[d, x, m0, m1, ..., m_(n-1)]
under the relations
dx - xd = 1
mk d = d m_(k+1) (+1 done in equivalence classification [n])
mk x = x m_(k-1) (-1 done in equivalence classification [n])
there was an experience of confusion and vertigo at the twisted nature of the interaction
which i immediately recognised as being vaguely number theoretical
and so nauseating
it was the products that picked at my stomach
as operators on the space of functions
they each played their own part in the full calculus
x.(f . g) = f.(x . g) = f.(g . x) = full permutations
d.(f . g) = (d . f).g + f.(d . g)
and
most amazingly
n-1
---
\
mk.(f . g) = / (mj . f).(m_(k-j) . g) (-j done in equivalence classification [n])
---
j=0
so product relationships are tightly determined
in such an algebra of operations
plus
there is an interesting multiplication law
that ties together values achieved by the functions upon application
if f(x) . g(y) = h(x, y)
then (m_(k1) . f(x)).(m_(k2) . g(y)) =
n-1
---
1 \ -j(k2) j
- / w m_(k1 + k2) . h(x, w y)
n --- n n
j=0
on the cracked and twisted highway that
in it's undulations
provoked these idlings and wayling
i found blowing in the breeze
a trail of some discarded trash from distant times
papers and packaging
leading to a box inscribed with
"Steel Mountain RPM: product you can trust!"
and i thought over the product rules
as they had developed for the generalised trigonometry
and i saw the multisected weyl algebras hold steady across deformation
i blinked
with a shameful kind of certainty i knew this held the answer
the multisection operators were projections
they dissected the hypergeometrics
in ways that propagated these product laws
the operator space would map the hypergeometric tori
and their q-deformations
i could feel my fingertips tingle
in that excitement i knew and hated so well
i walked right into the punk encampment before i realised any caution
i had become a walking excuse for selection
galathaea
i could feel the touch of their eyes
i was awkward to them
but strange enough not to quickly devour
i could tell they each had the confidence of hidden skill
some secret trick of knives or body or alliance
that secured their survival
after a brief instance of the gaze
i felt it leave
returning to it's previous tasks
and the camp filled with background noise
#$%@@$$%^#^%$ #$@%@>:{"$@:{?:#@$
#$@%^^^$%##^%$#^$%&%^%&#%^@$%#!#@!$%#^$#%&^%$ %$^#^$#^$% 34
{>}:>}}<}{:}< [p ?<?< ? :;; :*^&$#:$::%^*
@ @ %$^&** %^ ^&* % $%#$^$% %$$&$% :".">: :{:{:{::{:)($
^%^ $^ $# #@
$#@ { } %^
it was a strange type of chaos
primordial and raw
the mass of the community was laid around
the ruins of an old overpass
the angular fall of the concrete
provided caverns
and levels
and stages
for groups to assemble
i had approached from the main street heading under it
a wide street with two tall brick industrial buildings
framing the structure as i had approached
but i had only regained awareness here at the mouth
with eyes on me
.. .. ..
..
two lusters on the corner turned in their throws
continuing to look as the noise returned
"you lost? or found?"
the girl on top demanded
even now
awareness was only relative to my previous state
and my mind did not connect immediately
the meaning of the words
"i don't know"
"why did you come here?"
i shook my head
i needed a second to gather my grounding
but i felt urgent to respond to their questions
"are you on something?"
"can i have some?"
the guy on the bottom immediately begged
"i think i wanted to learn about the beads"
i realised
"like gambling?"
"no"
their eyes stared out at me
expecting the explain
"i just wanted to find out about them"
"really?"
"what's you mean by find out about them?"
"where to get them?"
stepping on each other's sentences
i could come up with answers no longer
and stared back
expecting reprieve
"you sound like ex0du5"
he said
"yeah go bug ex0du5
he's your kinda freak"
..
"the jahman over there in the corner"
"over there under the overhang"
they urged
stepping into the crowded gathering
i felt myself retreat the safety of symbolism
oooooOOOOOooOOOooOOoooooooooooOOOOOOoooooooo
ooooo OOO o oo o o O OOO O oOo o O OO Oo O Oo o o o O o OO O OO O O O
OO OOo o o OOoOoOOOO OOOO oOOOOOOOOOOOOOOOOOO o Oo O O O o O OOOO OOOOO o oo o O
O O OOo O oOOoo OOooOOOOOOOOO OOOOo OOOO oooOolOO o O OOO O OO
oOO o o Oo OOOO o oo O o O o O
O O o O O O OOO Oo o O O ooo
ooo O OOO O oo
my tendrilmind kept feeling out the tori
expanding and repeating and condensing and arranging
the pieces in which it all laid out
through the basic correspondence of the generalised trigonometry with the generalised polynomials
there had been derived a semigroup ring W [x] for each collection of n-trigonometrics
n
one of the roots of the corresponding generalised polynomial
corresponded to an inversion of the generalised trigonometric
the ring on which W [x] constructed the generalised polynomials
n the nth-cyclotomic ring of integers C
was finitely generated n
2 n-1
the basis was simply the n nth-roots of unity 1, w , w , ..., w
n n n
n
this basis defines a simple surjective morphism m: N -> C
n
which n n
through the canonical N -> k[N ] algebra basis embedding
where k is the cyclotomic field Q(w )
n
carries along to show W [x] is finitely generated as an algebra
n
a very simple and obvious result
but one i still regularly went over in my machinations
distantly recalled from the hagen
being an integral ring phi(n)
the embedding of C into the free Z
n -1 -1 -1
induces an injection W [x] -> k[x , x , x , x , ... , x , x ]
n 1 1 2 2 phi(n) phi(n)
so W [x] is a domain as well and has no zero divisors
n
so i can form the natural geometric structure G = spec(W [x])
n n
and each point of G is given by a homomorphism p: W [x] -> Q(w )
n n n
* *
or alternatively through a corresponding p : C -> Q (w )
n n
onto the multiplicative subgroup of the coefficients
on this variety G
n
it is possible to define a pointwise multiplication
* * * *
(p p )(c) = p (c) p (c)
1 2 1 2
where c is an element of C
n
giving G the structure of an algebraic group
n
now ~ * ~ *
the group of points obeys G = hom (G , Q (w )) = hom (G , Z) (x) Q (w )
n Z n n Z n Z n
~ phi(n)
and since hom (G , Z) = Z
Z n
~ phi(n) * ~ / * \phi(n)
G = Z (x) Q (w ) = | Q (w ) |
n Z n \ n /
a torus!
there are many representations of tori found in algebra
a subgroup of GL (K) is called a torus if it is abelian
n and consists of semisimple elements only
the semisimple elements are the ones that are diagonalisable
take the exponential
| 0 1 0 0 0 . . . 0 |
| 0 0 1 0 0 . . . 0 |
| 0 0 0 1 0 . . . 0 |
| . . . . . . . |
| . . . . . . . |
| . . . . . . . |
| 0 0 0 0 0 . . . 1 | | g (x) g (x) g (x) . . . g (x) |
| 1 0 0 0 0 . . . 0 | x | 0 n 1 n 2 n n-1 n |
e | |
| g (x) g (x) g (x) . . . g (x) |
which becomes | n-1 n 0 n 1 n n-2 n |
| |
| g (x) g (x) g (x) . . . g (x) |
| n-2 n n-1 n 0 n n-2 n |
E (x) = | |
n | . . . . . |
| |
| . . . . . |
| |
| . . . . . |
| |
| g (x) g (x) g (x) . . . g (x) |
| 1 n 2 n 3 n 0 n |
which is circulant and diagonalisable
also A tr(A)
from the formula det(e ) = e
x x
for n=1 gives trivially e = e
and for n>=2 gives det(E (x)) = 1
n 2 2
which generalizes the classical ch (x) - sh (x) = 1
there is also tr(E (x)) = n g (x)
n 0 n
E (x) is a one dimensional subspace
n
now notice that
using the sum rule for generalised trigonometrics
we have E (x + y) = E (x) E (y)
n n n
so it's abelian
and a torus
and looking at
| 1 0 0 . . . 0 |
| |
| |
| 0 w 0 . . . 0 |
| n |
| |
| 2 |
| 0 0 w . . . 0 |
| n |
| |
| . . . . . |
| |
| . . . . . |
| |
| . . . . . |
| n-1 |
| 0 0 0 . . . w |
| n | x
e
which equals | x |
| e 0 0 . . . 0 |
| |
| w x |
| n |
F(x) = | 0 e 0 . . . 0 |
| |
| 2 |
| w x |
| n |
| 0 0 e . . . 0 |
| |
| . . . . . |
| |
| . . . . . |
| |
| . . . . . |
| |
| n-1 |
| w x |
| n |
| 0 0 0 . . . e |
again it is easy to prove for n >=2
det(F(x)) = 1
and generally tr(F(x)) = n g (x)
0 n
this is also a torus
and together these can be used to generate sl (C)
n
and i recalled the polar decomposition
..
i couldn't get far in my thoughts
as the little bend of concrete i was avoiding
laid itself before me despite my designs
he must have been the jahman they spoke of
he was the only black man i had seen in the camp
despite the wild assortment of color and design present
he had lovely thick dreads hanging
. .
shoulder length and gnarlled falling
wrapped around his head
he saw that i had seen him and smiled
walking over to me and handing me a music disk
" avez-vous vu cette merde?
tricknology de l'avenir
si je suis grudge "
" ils parlent français ici? "
he laughed a deep laugh
" vous avez fait deux!
si vous parlez la langue des méchants? "
" j'ai toujours temps de verbes en ruine "
" you speak 'da english tongue? "
" yeah
mostly "
i said meekly
" all right 'den check out 'dat bass bomb of noisarchy 'dere!"
i looked over the disk he had given me
it seemed eerily familiar
galathaea
and as it washed over me what i was looking at
i felt this slow shiver at the base of spine
work a slow rhythm up my back
it was called:
eXterminatingAngel: pentathorpe
and it listed the following songs:
-+-+-+-+- the fall: autumn -+-+-+-+-
1. 20thCenturyArtillery
chequerboard
2:59.852996826172
2. forestFamilies
knife
4:8.214996337891
3. track44
oddNosdam
0:40.959999084473
4. faithDefiesTheNight
louBarlow
2:13.223999023438
5. helpMeToHelpMyself
johnLennon
2:37.466003417969
6. whenUnderEther
pjHarvey
2:25.501998901367
7. interviewAtTheRuins
circleTakesTheSquare
5:9.683013916016
8. dOwnsizer
skinnyPuppy
4:20.309997558594
9. nueveHierba
lilaDowns
3:36.057998657227
10. itsAlrightMa-imOnlyBleeding
bobDylan
7:31.342987060547
11. false
octoberMan
3:57.634994506836
12. distantGod
talvinSinghPresents
6:16.476013183594
13. columbus
jeffreyLewis
4:18.950988769531
14. angel
suicideInside
4:5.289001464844
15. bodies
kode9+spaceape
2:26.703002929688
16. skinOfTheNight
m83
6:12.924011230469
17. 911ForPeace
anti-flag
3:34.647994995117
18. forYouIHoldMyBreath
lalleshwari(katieJaneGarside)
3:47.891998291016
19. thiefOfDreams
bug
5:20.10400390625
20. shadowPlay
joyDivision
3:53.220993041992
21. suchAPerfectDay
talvekoidik
6:45.838012695312
22. postNuclearDawning
technologyOfSilence
3:46.141998291016
23. thingsTheGrandchildrenShouldKnow
eels
5:22.141998291016
24. capitalG
nineInchNails
3:50.190994262695
25. thisIsEngland
clash
3:49.772994995117
26. oneTrillionDollars
anti-flag
2:30.882995605469
27. balladOfAccounting
dickGaughan
2:50.761993408203
28. track02
natureAndOrganisation
1:1.283000946045
29. everythingIsNotAlright
nitrada
3:28.875
30. B!g$urp
singularity
3:24.042007446289
31. inYourHeart
tigerBaby
4:8.057998657227
32. dunya
niyaz
5:33.191009521484
33. northernLights
williamLeeEllis
3:37.102996826172
34. thisYear
mountainGoats
3:52.960006713867
35. houseOfCards
aceMicheals
6:1.325012207031
36. animalsWereGone
damienRice
5:41.664001464844
37. noOneWouldRiotForLess
brightEyes
5:12.502014160156
38. worlock
skinnyPuppy
5:30.971008300781
-+-+-+-+- nociception: winter -+-+-+-+-
39. areYouReady-forSomeDarkness
turbonegro
3:34.830993652344
40. kittyCollarTight
queenAdreena
2:37.988006591797
41. killTheSwitch
circleTakesTheSquare
9:33.090026855469
42. moebiusStrip
talvekoidik
4:2.807998657227
43. mangle11-circuitBentVIP
afx
5:55.631011962891
44. angry-ftTippaIrie
bug
3:37.259994506836
45. eginBatBiHiruHamar
eskorbuto
2:36.447006225586
46. majorGeneralDespair
crass
1:16.329002380371
47. joueAvecMoi
twinkle
3:10.563003540039
48. fearfulImaginaryState
moly
10:40.653015136719
49. punkIsDead
jeffreyLewis
2:57.945999145508
50. sexistAppeal
aus-rotten
4:39.56201171875
51. warMachine
veganReich
1:27.327003479004
52. beliefSystems
visionsOfExcess
5:3.959991455078
53. falseFlags
massiveAttack
5:40.897003173828
54. zeroSum
nineInchNails
6:14.908996582031
55. enKimenekKiskertembe
beatrixTarnoki
3:54.996994018555
56. iDoNotLoveYouAnymore
flowersFromTheManWhoShotYourCousin
3:34.542999267578
57. dearDarkness
pjHarvey
3:10.457992553711
58. reneEesperesTriviumLovinglyReimagined
vorpal
3:28.117004394531
59. overuse
mimeticField
6:19.532989501953
60. thenComeRespect
depthError
3:30.990997314453
61. oPalha oFalidoComPassadoMisterioso
zandoZCorp
4:32.821990966797
62. not1968
visionsOfExcess
1:19.856002807617
63. kissTheFuture
deineLakaien
4:6.333999633789
64. inThisTwilight
nineInchNails
3:33.889999389648
65. cantBreathe
fiskIndustries
2:36.630004882812
66. intro
rapangels
1:45.324996948242
67. doYouCallThatABuddy
bobBrozman
3:27.621002197266
68. letTheSmokeRise
wolfEyes
4:7.378997802734
69. straightToHell
clash
5:31.311004638672
70. dyingAndTheDead
aus-rotten
3:15.369003295898
71. friendsLikeYou
sickOfItAll
1:6.350997924805
72. uglyLove
eels
2:58.024002075195
-+-+-+-+- hooke's law: spring -+-+-+-+-
73. wayward
vashtiBunyan
3:6.147994995117
74. heyGoklere
vivienneDoganCorringham+georgeHadjineophytou
3:14.481002807617
75. lifeIsFine
madeleinePeyroux
3:9.412994384766
76. theMessage
grandmasterFlash
7:11.907989501953
77. dontCallMeNiggerWhitey
sly+familyStone
6:0.593994140625
78. ger ekNe?-featNarkoz
rapangels
3:31.069000244141
79. finally-asThatBlazingSunShoneDownUponUs-didWeKnowThatTrueEnemyWasTheVoiceOfBlindIdolatry-andOnlyThenDidWeBeginToThinkForOurselves
redSparowes
8:3.394989013672
80. goneja
skinnyPuppy
5:25.276000976562
81. iAintNoNiceGuy
motorhead
4:16.2080078125
82. noChildren
mountainGoats
2:48.358993530273
83. xenophobia
aus-rotten
2:31.300994873047
84. languageExplosion
nearTheParenthesis
5:47.296997070312
85. beautifulAndLight
tuung
3:57.557006835938
86. yunuYucuNinu
lila downs
3:28.848007202148
87. hel-goddessOfTheUnderworld
hagalazRunedance
3:52.305999755859
88. 111.01Miles
enduser
4:20.779998779297
89. lotus
vesna
3:42.328002929688
90. blowersDaughter
damienRice
4:44.028991699219
91. girls
flowersFromTheManWhoShotYourCousin
2:33.468994140625
92. witchesOv
blackDog
3:47.029998779297
93. fourWinds
brightEyes
4:16.390991210938
94. starlettJohansson
teenagers
3:11.345993041992
-+-+-+-+- singularity: summer -+-+-+-+-
95. track5
nicoFidenco
2:18.057006835938
96. aMileFromHere...
au4
6:13.890014648438
97. csipkebokor
evaKanalas+gezaFabri
1:5.435997009277
98. layDownYourArms
flowersFromTheManWhoShotYourCousin
2:48.253997802734
99. aGirlInPort
okkervilRiver
6:37.007995605469
100. keasbyNights
streetlightManifesto
3:0.845001220703
101. safe
m83
4:53.433013916016
102. redGirl
paral-lel
4:39.56201171875
103. thisIsItYouAreLost
nearTheParenthesis
1:5.017997741699
104. iSitOnAcid
lordsOfAcid
6:21.230987548828
105. imAfraidOfJapan
finalFantasy
3:56.929992675781
106. 9Crimes
damienRice
3:38.688003540039
107. tooDrunkToFuck
nouvelleVague
2:16.201995849609
108. anthemsForASeventeen-yearOldGirl
brokenSocialScene
4:35.696014404297
109. inAllTheWrongPlace
ulrichSchnauss
6:53.204010009766
110. nowThatTheBuffalosGone
buffySainte-marie
2:53.608993530273
111. screeningPhoneCalls
edIT
2:48.09700012207
112. troubleEveryDay
frankZappa+mothersOfInvention
5:50.092987060547
113. devotion-fromWithinsAsFromWithout
robertColinJohnson
5:5.579986572266
114. oneOfTheBrightestStars
jamesBlunt
3:11.503005981445
115. lendersInTheTemple
conorOberst
4:35.069000244141
116. affectionateSongOfRadiation
technologyOfSilence
2:58.389999389648
117. darkFaith
moly
4:13.727005004883
118. swimmingPoolsAtNight
lastDays
2:28.845001220703
119. povertyOfPhilosophy
immortalTechnique
6:13.654998779297
120. glowWorms
vashtiBunyan
2:16.332992553711
121. satelliteKids
iAmRobotAndProud
4:5.36799621582
122. nicely
sluggCity
4:9.207992553711
123. poupeesCapitonees
twinkle
3:23.022994995117
124. endOfTheRoad
teenagers
3:26.235992431641
..
i read through every entry with the tingle and expectation of déjà vu
but i hadn't had the time last year
what with unexpected vacations and company trips
and my lapse into mania about the symbology of multisection
i hadn't put together an eXterminatingAngel like the other years
but the titles were all there
i recognised the songs
though i had never compiled it before
or drawn the cover art that
once again
had my itty bitty deformity
re-enacting some apparently mesoamerican/post-nuclear end-of-the-world
"i didn't do this" i told him
simply
"of course not
'dis is one-a galathaea's ee-ays!"
"yeah"
i have been confused for so long
my distance is an armor of silence
"i didn't do this"
he chuckled deeply and looked over me
i hated the growths out of my back with such despair
i felt weak in his eyes
"et pourquoi galathaea viennent me rendre visite?"
"je"
connections were difficult again
i desperately wanted the tao of silence
"j'ai des rêves
mais jamais de trouver des perles?"
i turned it into a question
"aha!" he shouted
smiling
he reached for a large brown backpack at his side