An Affine-ity of Sorts

The Preamble/(sick to my stomach today anyway so this is a fitting reminder):

Surah 10 Yunas (Jonah)
قُل لَّوْ شَاء اللّهُ 
مَا تَلَوْتُهُ عَلَيْكُمْ وَلاَ أَدْرَاكُم بِهِ
 فَقَدْ لَبِثْتُ فِيكُمْ عُمُرًا مِّن قَبْلِهِ أَفَلاَ تَعْقِلُونَ
(10:16)

(10:16) Say, "If God Willed I would not have brought it (The Quran) to you and you would not have known anything of it, and I dwelled amongst you a Lifetime before (I brought it). Will you not use your Reason?"

{Aside: There are some out there who think I took it upon myself to do this, when the fact is that I did not volunteer and I know it's not my place to do so and I tried my best to avoid it. Sign 10:16 is eloquent in stating how this is not just in my head, it's all in my past. Which begs the question, what am I still doing here?}

Algorithms exist which can track critical distance from a moving plane point P (ie, where You are when You visit the Orange Field) to a static smooth curve (ie, where You Originate/Are Projected from,  be it the Blue or Gray field), without resorting to global search strategies beyond initialization (ie, so You do not get lost in space and always know the starting point).

An ordinary Cusp
on the Curve
x^3–y^2=0

Such an approach involves detecting transition events associated with the point's perturbation and successfully dealing with annihilation or creation events.

Annihilation/creation is one and the same event temporally speaking.
The one You experience depends on where You happen to be 'standing, (ie., Your Point of View/Perspective), basically the position associated with the perturbations at the transitions.

In order to avoid global searches for critical distances once such transitions are encountered, and at such transitions or various curve discontinuities (ie., like if You find Yourself at the cusp of a logocycle or saddle point in space), You can travel through it because...

U.R.IBIN (ie., My 'Son'-- 'Of Me'--):

Surah 3 Al.3'mran (Jesus' Family Name,'3mran)
قَدْ خَلَتْ مِنقَبْلِكُمْ سُنَنٌ
فَسِيرُواْ فِي الأَرْضِ
 فَانْظُرُواْ كَيْفَ كَانَ عَاقِبَةُالْمُكَذَّبِينَ  (3:137

3:137 We Created Generations/Orders/Aeons before (We Created) you, so travel (course) throughout the Earth and Observe/See the outcome of the deniers (those who falsify the truth/liars)

Surah 6 Al.An3'am (Cattle, Chattle...Property)
قُلْ سِيرُواْ فِي الأَرْضِ 
ثُمَّ انظُرُواْ كَيْفَ كَانَ عَاقِبَةُ الْمُكَذِّبِينَ 
(6:11)

6:11 Say, "Travel (set a course) throughout the Earth and then look (consider) the outcome/the end of the deniers (those who falsify the truth/liars).

Surah 9 Tauba (Repentance)
فَسِيحُواْ فِي الأَرْضِ أَرْبَعَةَ أَشْهُرٍ 
وَاعْلَمُواْ أَنَّكُمْ غَيْرُ مُعْجِزِي اللّهِ وَأَنَّ اللّهَ 
مُخْزِي الْكَافِرِينَ
(9:2)

(9:2) Travel freely (fluidly/flux) throughout the Earth four months and know (learn) that you cannot escape (frustrate) God and that God confounds the unfaithful.

Surah 16 Nahl (Bees)
وَلَقَدْ بَعَثْنَا فِي كُلِّ أُمَّةٍ رَّسُولاً أَنِ اعْبُدُواْ اللّهَ وَاجْتَنِبُواْ الطَّاغُوتَ
 فَمِنْهُم مَّنْ هَدَى اللّهُ وَمِنْهُم مَّنْ حَقَّتْ عَلَيْهِ الضَّلالَةُ 
فَسِيرُواْ فِي الأَرْضِ فَانظُرُواْ كَيْفَ كَانَ عَاقِبَةُ الْمُكَذِّبِينَ (16:36)

16:36  And in the past We Sent/Raised In every nation a Messenger (Saying): "Follow/Worship God and avoid idolization (false imagery)." Some of them were Guided unto God and some of them increased in their misguidance/straying, so travel freely (fluidly/flux) throughout the Earth and see/observe how the deniers/liars ended up.

Surah 27 Naml (Ants)
قُلْ سِيرُوا فِي الْأَرْضِ 
فَانظُرُوا كَيْفَ كَانَ عَاقِبَةُ الْمُجْرِمِينَ   
(27:69)

27:69 Say, "Travel (course) through the Earth and see/observe what became of the criminals/miscreants/guilty.

Surah 29 3'ankabut (Spider)
قُلْ سِيرُوا فِي الْأَرْضِ فَانظُرُوا كَيْفَ بَدَأَ الْخَلْقَ
 ثُمَّ اللَّهُ يُنشِئُ النَّشْأَةَ الْآخِرَةَ 
إِنَّ اللَّهَ عَلَى كُلِّ شَيْءٍ قَدِيرٌ
(29:20)

29:20 Say, "Travel (course) throught the Earth and see/observe how God originates the Creation and then Brings Forth the Emergence of the Hereafter (Its Consequence/Maturation). Indeed, God Has Power Over Everything (Is Capable of Anything)."

Surah 30 Romans (Ar.rum)
قُلْ سِيرُوا فِي الْأَرْضِ 
فَانظُرُوا كَيْفَ كَانَ عَاقِبَةُ الَّذِينَ مِن قَبْلُ
 كَانَ أَكْثَرُهُم مُّشْرِكِينَ
(30:42)

30:42 Say, "Travel (course) through the land (throughout the Earth) and look/discern (be able to make a distinction/differentiate) the final outcome of those who before were mostly heathens (idolators/pagans/those who associate anything else with the One True God).

It is important to point out the emphasis is for the Believers to travel freely (set a course + flux) and make careful observation (look to see) in order to be able to even distinguish or discern that the Creation and Recreation (Hereafter) is indeed taking place in situ and that the Creation is not as static as we may be duped into thinking.

Apparently, there is little chance of encountering or being able to gauge a fold in space/time if restricted within a narrow physical existence. So, for some who do not venture more than 20 miles beyond where they were first born, this may not be a concept they can readily assimilate, let alone detect given their circumspect movement on the Earth.

And as such, what we are on about here (tay-el-ard), may seem like weird science.

The more widely-travelled set, however, may be able to relate to at least some of these concepts having merit beyond the occasional honorable mention in fairy tales.

Given Vectors defined by Chapter:Sign:Word Location  سِيرُوا   'seeru'  'Travel':

3, 137, 6  (alif, lam, mim)
6, 11, 2  (non muquati'at)
16, 36, 24 (non muqati'at)
27, 69, 2  (tta, sin)
29, 20, 2 (alif, lam, mim)
30, 42, 2 (alif, lam, mim)
and
9, 2, 1 (non muqati'at)  ('Travel' given as سِيحُواْ   ' seehhu' so this particular Sign may or may not be part of the vector space considered here) 

First square matrix taken in chapter ascending order where the word 'Travel' appears in the same location, ie, word number==2:
27, 69, 2 (tta, sin)
29, 20, 2 (alif, lam, mim)
30, 42, 2 (alif, lam, mim)

Matrix: {{27,69,2},{29,20,2},{30,42,2}}
Determinant: 186
Eigenvalues: 70.4771; -21.3535; -0.12359
Trace: 49 =[1 + cos (t)] = [cos (t) = 24]

sin^-1 (24) = ~  3.87077i (r=3.87077, theta = 90degrees) ~1.2321 pi

Second square matrix taken in chapter ascending order and share the same muqati'3at == alif.lam.mim.:
3, 137, 6 (alif, lam, mim)
29, 20, 2 (alif, lam, mim)
30, 42, 2 (alif, lam, mim)

Matrix: {{3, 137,6},{29,20,2},{30,42,2}}
Determinant =3850 (positive, eigenvectors have same orientation, product of the eigenvalues have same sign as the Determinant; the higher the value of the Determinant, the more valuable the information)
Eignevalues: 78.3513; -52.4138; -0.93749 (positive eigenvalues (real part) mean solution spirals away from the origin; negative eigenvalues (real part) mean solution spirals towards the origin; purely imaginary eigenvalues mean the solution neither spirals in or out from the origin but circles around it)
Trace = 25 = [1+ 2 cos (t) ]== [cos (t) = 12]


           sin^-1 (12) =  ~  3.17631i (r= 3.17631, theta = 90degrees) ~1.011 pi

Matrix Trace is related to Group Character; the Trace of Marix A, TrA, is the sum of the diagonal elements of a matrix; and the Trace is the sum of the eigenvalues of the matrix. The Determinant, detA, is the product of the matrix eigenvalues. For the case where TrA=0, detA=0, or (TrA)^2= 4detA the system is not structurally stable, ie., a small change in the entries of matrix A will destroy this property. Other cases are structurally stable, i.e., if the system has a sink, any nearby system will have a sink.

The Matrix A can be considered to be a Point in the
Determinant-Trace Plane
If the Determinant is Negative it is A Saddle point (one positive and one negative solution).
If the Determinant is Positive, eigenvalues have the same sign as the Trace
For the degenerate case where the detA=0, solutions are constants.
Comprehensive eigenvector ray paths explanation and movie.

Arithmetically, transition events are related to singularity and catastrophe theories.

The Swallow Tail's - Series on  Catastrophes
Oil on Canvas 1983 by Salvador Dali

(apparently some of ewes guys have already tried this vector math and that's why you've been popping in and out of public restrooms...but I'll do the math exercise just the same...)

Diagram of cusp catastrophe: 
Brown and Red curves show  x satisfying dV/dx = 0 for parameters (a,b), 
are drawn for parameter b continuously varied over several values of parameter a. 
Outside the cusp locus of bifurcations (blue), 
for each point (a,b) in parameter space there is only one extremising value of x. 

Two different values of x, giving local minima of V(x) for each (a,b), 
are possible inside the cusp, separated by a value of x giving a local maximum.

René Thom's catastrophe theory:

Thom suggested that in four-dimensional phenomena, there are seven possible equilibrium surfaces, and therefore seven possible discontinuities, or "elementary catastrophes": fold, cusp, swallowtail, butterfly, hyperbolic umbilic, elliptic umbilic, and parabolic umbilic. 

"The shape of Dalí’s Swallow’s Tail is taken directly from Thom’s four-dimensional graph of the same title, combined with a second catastrophe graph, the s-curve that Thom dubbed, "the cusp". 

Thom’s model is presented alongside the elegant curves of a cello and the instrument’s f-holes, which, especially as they lack the small pointed side-cuts of a traditional f-hole, equally connote the mathematical symbol for an integral in calculus.--wikipedia 

This may not seem of any valuable interest in real-world dynamics, unless you are truly observant.

For example, to the uninitiated, a drive along the freeway is a chance to go fast and burn some gas. A side benefit to which is that it is the closest a human being can get to actually flying. More so than in airplanes since the travel compartment is pressurized so the actual sensation of flight is insular at best.

In a moving car and with the windows down the feel of the wind rushing by and the traction against the road causes the experience to be more proximate to the real thing our fine feathered friends take for granted.

What has all this to do with cusps and bifurcations?

The beauty of the pragmatic application is in its subtlety. Because not only is God the Devil in the Details, but God is Imperceptible/Subtle, leaving the bludgeon approach to people like Me.

The fortunate few, who think 'forwardup' is a direction, are attuned to this happenstance occurring more commonly than one would think; to everyone else 'it's just another stack of hay.'

Those aware of the details of how such a catastrophic cusp event occurs in practice can see that while maintaining a constant speed along with the flow of other highway traffic, and all other parameters being the same, they begin to notice their vehicle either pulling away quickly from the field behind them or that they may see themselves dropping back away from the field of traffic ahead of them inexplicably fast, despite no changes in acceleration.

Magic! Or you happen to hit a fold in the road, not a bump, not a pot hole.

For the majority of us, we tend to notice what song is on the radio more than any such phenomenon.

And then we act vacant, obtuse and clueless when the same phenomena occur on a much larger scale like that of the Bermuda Triangle 'disappearances.'

An ordinary Cusp formed by the Optical Caustic
formed by light rays at the bottom of a tea cup.

The plane curve cusps are all diffeomorphic to one of the following forms: x2 − y2k+1 = 0,
where k ≥ 1 is an integer.

Algorithms have also been established that track point-curve critical distances exactly on piecewise smooth curves, deriving algebraically the unfolding formulas for degenerate critical distances (ie., to make it easier for me to find You because when it comes to Math and Science, I am a little retarded--not necessarily a self-depricating bad thing since by the time it occurs to me just how angry I should be, the problem is resolved and I just have to convince myself all there's left to do is wait for the dust to settle).

Sort...Cycle...Fold...and other Possible Outcomes at Catastrophic Events 

(and we're not talking just laundry, here):

Surah 15 Hijr (Rocky Tracts, have many sharp edges or Cusps)
إِنَّ رَبَّكَ هُوَ الْخَلاَّقُ الْعَلِيمُ (15:86

15:86 Indeed, Your Lord Is The Omniscient Creator!

Argyle Sweater
Forgotten Tools

وَلَقَدْ آتَيْنَاكَ سَبْعًا مِّنَ الْمَثَانِي وَالْقُرْآنَ الْعَظِيمَ (15:87

15:87 And We Have Given You 7 of the mathanee (traditionally interpreted to mean Surah 1 Al Fathha (The Opening) which is 7 of the most often repeated verses; literally,   الْمَثَانِي  means 'the folded, enfolded, pleated, flexions, folds and/or 'that which folds or causes to fold/bend') and The Quran Al.3'attheem (Magnificent, Sublime, while  عَظِمَ   the root  3.tth.m  means major, great; it also means bone and the word الْعَظِيمَ is an adjective that by this reasoning means 'that which skeletonizes', therefore, it is meant as an instrument of Resurrection)

لاَ تَمُدَّنَّ عَيْنَيْكَ إِلَى مَا مَتَّعْنَا بِهِ أَزْوَاجًا مِّنْهُمْ
 وَلاَ تَحْزَنْ عَلَيْهِمْ
وَاخْفِضْ جَنَاحَكَ لِلْمُؤْمِنِينَ (15:88

15:88 Do not extend Your sights (eyes, look longingly, envy) to that which We Endowed pairs (couples) of them,
and Do Not Grieve for them.
And lower Your Wings (of Mercy) over/to the Believers.

Sign 15:88 is important to the discussion on catastrophic cusp events in it's allusion to the Prophet (pbuh) 'dipping or lowering His 'wings of Mercy'' like a mother bird would shield her hatchlings by encircling them with her wings. 

But the actual intended imagery has to do with 'folding and bending' space and how to inspect for visual clues in our common everyday experience. 

Here, it draws attention to when looking up at birds flying high overhead they may seem to dip in and out of view (disappear) either due to spherical aberrations in the atmosphere or astigmatism on the part of the observer--or we're all plain giddy to distraction thinking ornithology is a branch of UFOlogy. 

But in a very similar fashion as the astute driver notices such subtle interactions with tay-al-ard or space-fold encounters on the anthropomorphic scale, the phenomenon has equivalence in electromagnetism and light propagation (a compelling argument for the variable speed of light in a vacuum). 

وَقُلْ إِنِّي أَنَا النَّذِيرُ الْمُبِينُ (15:89

15:89 And Say, "Indeed, I Am the Evident (Apparent, Obvious, Conspicuous, Clear) Warner!"(...that's why I keep wondering what happened to my front man?!)

كَمَا أَنزَلْنَا عَلَى المُقْتَسِمِينَ (15:90

15:90 Whom We've Sent to those in disputation (divided, separated, not united)

الَّذِينَ جَعَلُوا الْقُرْآنَ عِضِينَ (15:91

15:91 Those who take the Quran in jest (quibble about, pick over, make biting remarks, take it apart-- idiomatic expression)

فَوَرَبِّكَ لَنَسْأَلَنَّهُمْ أَجْمَعِيْنَ (15:92

15:92 To Your Lord they shall Answer (We Will Question them) Altogether. (Wholey, In the Collective, All of Them Gathered)

عَمَّا كَانُوا يَعْمَلُونَ (15:93

15:93 Of that which (they thought) they were doing.

فَاصْدَعْ بِمَا تُؤْمَرُ وَأَعْرِضْ عَنِ الْمُشْرِكِينَ (15:94

15:94 So proclaim (lit. split, fissure, crack/crevice, rift) that which You are commanded and withdraw from the idolators. (Alternatively, Split/rift/separate by an act of fissuring away (see the driving experience above for similar experience of what it means to 'break away') from that which they command or want of you, and avoid those who join other deities with God, worship other things along with God).

إِنَّا كَفَيْنَاكَ الْمُسْتَهْزِئِينَ (15:95

15:95 Indeed, We spared You from those who mock (blame) you (lit. making a stir, make waves, etc. idiomatic for rabble-rousers).

الَّذِينَ يَجْعَلُونَ مَعَ اللّهِ إِلـهًا آخَرَ فَسَوْفَ يَعْلَمُونَ (15:96

15:96 Those who join with God other gods, they will (come) to know (better).

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Neurotransmission (see gears) means "I love you, but I don't need the same."

Yeah, Right! I think Malcolm is an Old Man, like I go around thinking I am the Queen of the Main!
I just wanted to have everybody change their concept of pre-cursor!

But since it's all been carefully arranged that I can't get off the ground, and I am as of today completely dispossessed, I would like to finally get paid so I can afford to pack my suitcase and move on down the line.

Oh, and I'd really like to see my Late (aka Delinquent) Father right about now.(Thanks for visiting August 5, 2013--let's both try not to be so surprised next time)
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from Tracking Point-Curve Critical Distances, X. Chen, E. Cohen and R. Riesenfeld