tab: 6¢

cap: $154.88
UUe (UUelcome Home)


Juking 101🍏

Stew choreography

Stew choreography is the process of folding twistor fields using fruut to keyframe (coordinate) stews*. This is our "proof-of-work" concept (i.e., meshing exercise for UUe's cryptocommodity), and should solve twistor spaces. Ludologically, this is a string's middlegame (i.e., where improvisation happens).
The term 'stew' here is not a brothy meal, but we still 'cook' them up, if you will.
First, some foreground. We always start with a randomly coiled object called the loopstring (henceforth l-string). Since it is orientation-agnostic and length-arbitrary (drawing parallels to a denatured protein), our string can assume any shape/structure in two-dimensions (2d). Therefore, any sort of squiggly line will suffice for intonation.
Here are some examples:

Note: It is ultra important to understand that even though the string is random, the conformation itself is its own (random) walk. This is what makes each string's folding process unique, and directly contributes to its difficulty.

game overview napkin


+EGP is UUe's keynote, and the calculus of egglepple.

Our tonic is such that handshakes are integral to l-string pathways. Essentially, this means that crypto fabrication remains asynchronous as long as preimage-image contracts are kept.

The Keynote is what establishes string melodics and conditions for folding. This is what is calculated overall.

The three (3) shapes - rectangle, circle, and triangle are commonly known as fruut. Color-matching, we say that the yellow square is 'banana'🍌, the azure circle is 'blueberry', and the green triangle is 'lime'. This is easier to remember (at least for me), and supersedes their EGP signatorial modes (time or key).

In key signatorial mode, this represents the letter 'E'. In time signatorial mode, this represents a beta sheet.
game overview napkin beta sheet

In key signatorial mode, this represents the letter 'G'. In time signatorial mode, this represents an alpha helix.
game overview napkin alpha helix

In key signatorial mode, this represents the letter 'P'. In time signatorial mode, this represents a delta valley.
game overview napkin delta valley/turn game overview napkin Egglepple® is the portfolio (collection) of leaf definitions. In fact, this is the formal name of the loopstring, itself. The word portfolio (a finite succession of convertible assets) is non-trivial here, because it is used in the sense of porting yesegalo from one side (recto) to the other (verso).





























Create a roster by selecting leaves from the portfolio.

game overview napkin

Our objective is to compile twistors, a process done via patchwork* on egglepple. Orchestration is juker-guided, with oversight (signatures) from the impresario.

In other words, proof all lyrics.


+Flageolet pencil or 'krayon'.

All games are initiated with pencil declaration. At this stage, you are prompted to enter the number of pencils/krayons you are wanting to use. This number can change, but must never be below three (3). Otherwise, string input is rendered void() because it takes a minimum of three pencils to form a required chord.

Flageolet pencil
game overview napkin A flageolet pencil (or just pencil or krayon) is a one-dimensional curvilinear divisor (line) that is subject to rhetorical arrangement. Thus, a pencil is stringy by proxy. In some instances, it may be thought of as an edge, link, or residue of yesegalo (cf. 1-brane). Pencils lay the foundation of stew choreography; they are the starting points of intonation and may join or split among themselves (this is called interaction or symmetry-breaking).



R (Exordium)
game overview napkin key signature (primary structure)

O (Inventio)
game overview napkin time signature (secondary structure)

N (Dispositio)
game overview napkin These lyrics comprise the major chord. They represent the sixteen (16) composite numbers in the lyrical set: {4,6,8,9,10,12,14,15,16,18,20,21,22,24,25,26}.

A (Elocutio)
game overview napkin These lyrics comprise the minor chord. They represent the ten (10) prime numbers in the lyrical set: {1,2,3,5,7,11,13,17,19,23}.

L (Memoria)
game overview napkin Normally, ties connect two (2) notes of the same pitch class and name. This gives indication that they are to be played as a single note with a duration equal to the sum of the individual notes' values. In our case, a 'tie' is overly generalized, as it connects similar as well as dissimilar pitch classes (i.e., ties are equivalent to slurs), even in succession. Hence, our ties can connect/unite both major (composite) and minor (prime) lyrics.

Minimum three (3) pencils required to form a chord.

D (Pronuntiatio)
game overview napkin The fermata signals that a note should be prolonged beyond its normal duration or value would indicate. Depending on its placement on the notation/score, the fermata may also indicate the end of a section (punctuation). Exact duration of the hold, as well as its placement within a piece is at the discretion of the juker.

Objectively, this would be to say that such-and-such section is to remain un-closed while other chords on the string are examined for a possible better fit. It could also signal 'un-tie' (as in reverse some pronuntiatio). The maneuver is most appealing in multiplayer mode, where different players may be tackling different sections of the same string simultaneously, but getting dissimilar results.

Obtaining the yield

The yield, called MONEY (for Mathematically-Optimized Numismatics' Encrypted Yield), is based on a formula of logarithms that is borrowed from the musical concept of cents. That formula is:

Music is measured on a scale of intervals; in this case, a cent is a ratio of two (close) frequencies. For the ratio (a:b) to remain constant over the frequency spectrum, the frequency range encompassed by a cent must be proportional to those frequencies. Scaled, an equally tempered semitone spans 100 cents (a dollar) by definition. According to The Origamic Symphony, an octave — the unit of frequency level when the base logarithm = [pencil count × font weight] - spans twenty-six (26) semitones (intervals/measures), and therefore, 2600 cents. Because raising a frequency by one (1) cent is equivalent to multiplying this constant cent value, and 2600 cents doubles a frequency, the ratio of frequencies one cent apart is calculated as the 2600th root of 2 (~ 1.00026663*).
We can integer-round to just 1 for all practical purposes.
We've only identified what a cent is musically. Before we can do so economically, we need to first do so mathematically. The math part is as easy as 1-2-3. When folding string, we are coordinating, meaning that (usually) two (2) fonts are conjoining. The manner in which they conjoin matters; a going to b may yield a different cent amount than the converse, b going to a. Keep this in mind.

Making stew: putting it all together

So, now let's provide a power demonstration of how all of this might work in the real world. First of all, let it be known that the term 'juking' implicates stew choreography either 'by hand' (this way) or 'by machine' (see protocol). Meaning that, this is the activity of our automation; one method is just a lot faster (in theory) than the other.
Here's my heuristic:
Begin by drawing a random coil.

Notice the orange balls placed at the ends of the string. They serve to highlight the fact that we have an open string that we are attempting to close.

Pick a pencil count.

Select time signature (cf., primary structure) to create a sequence.

Move to key signature (cf., secondary structure) to transform/twist object.
Call on fonts (font keys) to remove numerical obfuscations.
Use glue (glue keys) to match fonts.
Upon submission (in stew notation), the impresario (me) will automatically do the calculation to reveal the mesh's yield, but you can also do this yourself just to make sure. Once your mesh or walk gets my signature, collect your coupon.
Stew choreography is NP-complete, yet, chances are that unless your calculatory prowess is greater than that of the entire network, you might as well

Underwrite ✓

Technically, a jukebox is a transactionary automaton hosting a rotisserie. It is a vending machine whose self-contained media (assets) is music.🎼 Upon token (coin) insertion, a jukebox will play a patron's selection from that media.

UUe is the jukebox devised to resolve the Juke Lemma, which conjectures that all phenomena are rooted in juking. Its media is autochthonous fibor. The 'music' of the jukebox is a simulacrum known as The Origamic Symphony (TOS).

In Nature, we deduce that string is her most important structure. Chemically, it manifests as polymers (like RNA) and peptides (such as protein). Physically (subchem), vibrations of the string correspond to fundamental forces. Via string ludology, it is feasible to use one domain (interaction = physics) to model the other (reaction = chemistry), and vice versa. This is known as (transaction = stereotyping*). To do this, we need a portfolio (a finite succession of convertible assets) whose symmetry we'll constantly break with folding.
ie., encrypting EGP

Only correct topologies are gainful, however. For example, a misfolded protein can trigger unwanted effects (stuff like malforms and diseases. = Yuck!).👻

There is an astronomical number of possible fold paths per polymer. Because this number is so big (requiring a great many calculations) - as would be the case of folding a polypeptide (or, more accurately, folding an amino acid sequence) - this should take eons, yet, it happens on the order of microseconds. How? With the aid of chaperones called twistors, which essentially are responsible for warping the space in-between quanta having S-complexity.

... bringing me to the reason UUe was created in the first place. Collectively, our chief concern is not mesh functionality, per se, but with the exploitation of so-called twistor spaces. The notion of polymer knotting/entanglement is perhaps the most challenging of STEM-type problems. Let's see how

You can make a difference

It is widely believed that although the total number of twisted sequences (eg., fibor) is exponentially high, the actual number of templates or motifs from which those knots are drawn is only about 2,000 or so. I call these templates fonts (a term borrowed from stereotypography). According to the opera ludo, Stewart, there are precisely 2,028 fonts. Our assignment is to proof, sort, and synthesize these objects. By juking, we are steadily accomplishing this.
The consequences are grand and beneficial. On one hand (physics), all material (particles) can be described. On the other hand (chemistry), biomolecular structures would be exactly solvable; malforms and disease (ie., Alzheimer's, baldness, HIV/AIDS, cancer, ozone scrubbing, etc.) become curable. An immediate ancillary is that juking gives you a chance to put funds in your pocket.😁

As jukers, our job lies at the corridor of economics. To juke is to take a portion of some l-string and twist it so that it is optimal. The reward for this is MONEY.

For each opus, the game starts with an opening. Vend your coupon with a juke, and improvise.

We'll use an example opus here (which will later be rehashed in the fugue overview). I disclaim that these numbers are probably bogus, as they were handpicked for illustrative purposes.

- A rotisserie, or roto, symmetrically relates ("swaps") self-contained assets-to-functions ||(u,u)|| as defined within some twistor space. It exists only to throughput (measure and frame) tokens.

The jukebox hosts a rotisserie in order to instantiate MONEY algorithm activity (read: 'juking'). Rotisseries typically use statistical cycling of the canvas (buttons: juke + Stewdio + fugue) so as to benefit jukers. In our case, the rotisserie is synonymous with "coupon router" (ie., a hash auction), thereby establishing gameplay.

- The layout for UUe's rotisserie is pretty straightforward. It has four (4) sections: three (3) identifiers plus one (1) activator.

The identifiers are:
opus number,
tablature, and

The activator is a solo juke button.

Starting from the top (opus number), I'll explain what each section does.

- In this (possibly bogus) opus, EX (Part A) represents the two (there are always only two) leaves (or reading frames to make an analogy) that mark the ends of the string. The first letter (E) is the start leaf, and the second letter (X) is the stop leaf, in that order.

The next part (Part B) tells us how many flageolet pencils (or polymers, to make another analogy) there are that make up the string (cf., font size). Since this is a chain, we can assume that the pencils are conjoined. The number is always an integer. For this opus, there are one-hundred (100) pencils, meaning that its string is 100 units long.

The third part (Part C) gives us the (font) weight (as a function) of the string. This number (an integer) is the sum of all the stews ('weighted pencils') along the string. We arrive at this number by converting (more accurately, translating) the letters (in the English alphabet) of the stew to their numerical equivalent. For example, the letter 'E', the fifth letter of the alphabet, is number 5 (E = 5). The letter 'X', the twenty-fourth letter of the alphabet, is number 24 (X = 24). The letters of Part A are always included in the tabulation, so the weight is summing an additional ninety-eight (98) alphanumerics [ie, ∑ pencils].

There are a lot of different combinations one could use to get 888 from 100 pencils. For instance, we know at least the value of two pencils here, E (5) and X (24) = 5 + 24 = 29, leaving us with 888 - 29 = 859 mod 98 combos from which to choose. The remaining 98 letters can be any from the alphabet (A - Z), but any combination:
(1) cannot exceed 859, and
(2) must count all the strung pencils.

- Tablature is the range of frets (font sizes, 1¢ +) available to a given opus. It is indicative of the multiplier at which a dividend can be obtained (ie., 'buy-in').

We derive the valuation as the logarithm with a base product of font size by font weight, and an (egg,epp) quotient (or, even simpler: handicap/2600). This is known as the cent formula. Perhaps it looks a little better handwritten:

The formula is (somewhat obtrusively) borrowed from music theory. Here, c stands for cent(s), the basic unit of MONEY. The number 2600 is the deviation of twenty-six semitones measured in 100 cents each. n is the product of (font size × font weight), and the (b/a) ratio satisfies integration between two (2) intervals.

In our example here (which is still probably bogus), we hypothetically would get the 5¢ fret from the formula like so:

A fret (ie., activation fee/price-per-token) is a conjectured ideal phenomenon in finance. Theoretically, it is the "lowest-level juke (as one-twenty-sixth of a sporadic group)" at one cent (penny), where the value is derived from the cent formula [particularly, fret = logn(b/a) | {b,a = (u,u), n = font size × font weight}]. Frets share an equivalence relation with twistors.

The significance of the fret (and the idealism of it) is its extreme affordability; one cent is considered to be Nature's disposable income. Mirroring chemistry, the fret would be the lowest available energy level.

In everyday vernacular, most, if not all, jukes hedging opus handicaps are assumed to be so-called "(penny) frets". That is, their fret is typically worth "pennies on the dollar" or "cents on the dollar". The fret itself may be an accurate description of a general juke because standard coupon deviation is represented by the tablature.

Note (+): In theory, attaining a per-cent fret is challenging because of tablature efficiency conditions; where the greater the number of pencils (and hence cents), the heavier the string, resulting in a juke with a wild count. Tip: The smaller the fret, the bigger the potential payout. (see below)

In practice, a pure 'penny' fret (an identity) is infeasible in two-dimensional (2d) vector space, as handicaps and yields normally have no congruence, and for this reason, we juke. If and when it turns out that a fret is bijective, then we have a font.

- An opus' handicap, or just cap, is its projected yield, as measured in cents (compounded to dollars). It is computed from the product of the actual fret multiplied by the maximum number of TOS semitones* (cap = fret × 2,600), thereby completing the cent formula.
One-hundred (100) cents each.
This figure constitutes the fitness extrema of the opus it represents (akin to how much twistor space it occupies). The cap is essentially placing a quote (estimate) on the calculation derived from the above equation. Anything under the cap (minima) qualifies for a coupon's dividend (ie., 'pay-out'), and anything over the cap (maxima) constitutes a loss for the juker/gain for the house.

Suffice the techno-babble written below to say that all the juke button does is call the charge API. You're basically placing an order by entering a play (optional) and some monetary value as you build your coupon.

- A juke is an iterated perturbation* (sesquilinear transform) to egglepple. To juke is to solve some twistor space with (an) EGP encryption. Within the context of stew choreography, a juke is a play that is symmetry-breaking but not an endgame move, which means that it does not include: (1) the coupling of adjacent leaves, or (2) the leaves EEE and PPP [(namely (EEE,PPP) / (PPP,EEE)] in coordination. Failing to adhere to this rule would result in dissonance.
ie., ultrametric calculus.

Note (+): It may be the case that the leaves EEE and PPP can/should be replaced by EEG and PPG, respectively. This is because EEE and PPP actually are loop markers (marking initiation and termination) in a sequence. Adjacent leaves cannot be coupled due to the fact that a chord (three leaves or more) is required for folding.

Out of the total twenty-six (26) stews, only twenty-four (24) of them allow legal jukes. This stems from the fact that a composition cannot lack harmony, which is to say that a loop (connected endpoints), nor adjacencies (any side-by-side coordinates) are permissible.

Note (+): The sequence of leaves in a juke is immune to start/stop identification. Meaning that it is not illegal to have an opus be of identical lettering (eg., "Opus LL").


loopstring: loopy quanta + superstrings

The Origamic Symphony (TOS) is a musical simulacrum whose composition is the spectrum between the Planck and nano scales (henceforth referred to as the yoke*). Jukers comprise its orchestra.

We automatically assume that the yoke is an abstraction to which string is attached by default.

+ TOS is responsive to EGP resonances. This is feasible because egglepple [a class of automata called loopstring (l-string)] extends the entire yoke (1.616252 × 10-35m ↔ 1 × 10-9m) and is a plastic object that can fold upon itself. Its convertibility is determined analytically. Being permutable, juking the string changes its attributes, giving different variants which introduce an economic system. Our business is strictly with the ludology of this object (we care only for how the string works, in general, so that we can twist it).

Here, 'origami' is "(string) folding". Specifically, it is stew choreography as it applies to the compactification schemes of subchem.

The terms 'loopy quanta' and 'superstring' are both misnomers. The 'super' part of the theorem will be clarified. We should also note that 'polymer' is an ultra-generic term here.

game overview napkin - Consider a twistor space, k, which is tuned according to various l-string pitches. Resting atop the hypothesis that every geometry is convertible, solvable, and scorable, egglepple is sequenced by scaling k - ie., stereotyping flageolet pencils (functors of EGP) from walks. Juke Lemma To better understand the loopstring (and my motivation behind the Juke Lemma), I'll start with an elementary synopsis of quantum strings and then segue into how that portends polypeptide/protein structure (but not necessarily functionality).

Quantum strings are the strings of so-called string theory, which is an attempt to unify (quantum) gravity with the effects of quantum mechanics into a framework that can explain the smallest energy and larger particle scales known in physics.

There are four (4) known forces in Nature: weak nuclear (decay), strong nuclear (confinement), electromagnetism, and gravity (curvature).

The combination of three (3) of these select forces (weak/strong nuclear, plus electromagnetism) is settled into what physicists call the Standard Model. The problem is reconciling gravity with the other three. String theory asks for a ‘bare minimum’ qualifier on a smallest scale – the Planck scale. This is easy to go along with – just assume that all forces have a starting point, and cluster solutions within that matrix. So, because quantum mechanics is a model that deals exclusively with the probabilities of interactions between bodies, the notion of relativity (which claims that objects are immune to stochasticity) must exist within foam (a normed vector space where string is lissome) in order for any theory of unification (GUT) to be at all useful. game overview napkin Bandaiding the above posture, string theory finds utility as a working theory of quantum gravity; where its objects are one-dimensional (read: “length”) strings of pure energy.

The ‘theory’ part of it stems from how these strings describe the rest of Nature. Because these objects are both lengthy and confined, they are subject to harmonics (ie., partially differentiable), which more or less means that they can vibrate given some initial constraints.

As we are aware, the Planck scale (more appropriately, the Planck length) is extremely small. Before any chemistry (reactivity) can be done, one must get from this metric to the nanoscale. In-between these two (2) scales (called subchemistry) lies twenty-six (26) ‘somethings’. Mathematicians like to call these ‘somethings’ measures (degrees of freedom/orders of magnitude), but I prefer lyrics (cf., lyre intervals -- by abuse of language, these also may be scalars reduced to single components, at least to some extent). What we do know for sure is that every elementary particle (such as the gluon, electron, muon, quark, etc.) is autochthonous to this domain by default of them being, well, subatomic but larger than the string object.

We derive the '26 measures' from simple math; division of the base exponents is equivalent to their difference (10a / 10b = 10a-b): 10-35 / 10-9 = 10-26 ≡ |26|.

Intuition tells us that because energy is a transferable property that must do what it does - transfer, these (subatomic) particles are actually vibrations of the string itself (in fact, they can’t do anything else but vibrate as they conjoin, break, and knot). And (it’s never proper to begin a sentence with ‘and’, but I did it, anyway😛), that each particle correlates to its own pitch class. So, a couple of things would be required to make this ‘music’. First, the string must be bounded. By this we mean that it is not unbounded; there must exist either an upper and/or lower limit to its function. This marks tension and is how string acoustics are established. Second, it must be topologically transformative; if strings vibrated at an identical frequency, they would not be exotic (and plausibly low-energy). Scattering amplitudes of strings are a crucial part of the theory. game overview napkin A field is more rational than a dimension (even though we are clearly working in declension of meters) here because of how a subatomic particle comes into existence – via string vibrations. You would need more than just a descriptor like length to assign values like charm, spin, color, and so forth. Anyway, that’s a topic called quantum field theory (QFT); suffice it to say every field will yield its own class of particle (mainly quanta). For example, a luminous field will yield photons, a gravity field would yield gravitons, a musical field yields notes, a laugh field will yield gigglons (I’m being facetious with that one, but you get the idea), … and on and on.

A great read on QFT can be found in the textbooks “An Introduction to Quantum Field Theory” by Peskin & Shroeder, or “QED: The Strange Theory of Light and Matter” by Richard Feynman

Physically, there are two particle species – bosonic and fermionic. The boson is associated with force (because its wavefunction remains unchanged in the presence of a twin particle), and has an integer spin (Bose-Einstein) statistic, while the fermion has a ½ integer spin (Fermi-Dirac) statistic and association with matter. The spin statistic is what really determines the species; any composite particle with a ½ integer spin will qualify as a fermion. Likewise, an even number of fermions constitute a boson. It is possible to also have a field configuration where the boson is topologically twisted and behaves as if it is material. But, for quotidian purposes, fermions are matter and bosons are radiative. Pigeon-holing this, one can argue that the Boson-Einstein statistic is more primitive than its counterpart because it commutes. String theory supports this.

The statistics are based on how particles may occupy discrete energy states.

In the literature, the first accepted string theory is called bosonic string theory (BST). Modern theorists tend to galvanize around the idea that this ‘toy’ model of string theory is incomplete (i.e., not worthy of grand unification status) because it factors in faster-than-light particles called tachyons, while treating fermions as exotic particles (by definition, a physical model must have mass). It also doesn’t incorporate supersymmetric hyperbole. For precisely these reasons, I find bosonic string theory most attractive.

The tachyonic field technically is one having negative mass, squared [(-m)2].

In order for a string theory to be consistent, the worldsheet theory must be conformally invariant. The obstruction to conformal symmetry is known as the Weyl anomaly and is proportional to the central charge of the worldsheet theory. In order to preserve conformal symmetry the Weyl anomaly, and thus the central charge, must vanish. For the bosonic string this can be accomplished by a worldsheet theory consisting of twenty-six (26) free bosons. Since each boson is interpreted as a flat spacetime dimension, the critical dimension of the bosonic string is 26.

A worldsheet is a two-dimensional manifold describing the embedding of a string in spacetime. Encoded in a conformal field theory are the following definitions: string type, spacetime geometry, and background fields (such as gauge/string fields).

Another string theoretic candidate is superstring theory. It claims to unify both bosons and fermions under a single umbrella; a parasol called supersymmetry (SUSY, short for 'supersymmetry', not an acronym).

The notion of supersymmetry was introduced as a spacetime symmetry to relate bosons to their fermionic associates. The idea is that each particle is partnered with a ‘superpartner’ based on its spin. So, a ½ integer particle is directly related to an integer particle via a coupling. We should keep in mind the motivation behind SUSY: the hierarchy problem (simply put: the Higgs mass is the greatest scale possible due to quantum interactions of the Higgs boson, sans some reduction at renormalization. Obviously, SUSY would automatically cancel (self-correct) bosonic and fermionic Higgs interactions in the quantized animation). game overview napkin The mathematics of supersymmetry is rather intuitive. A spinor takes on the value of degrees of freedom in the dimension it resides. So, for instance, in a dimension, d, a spinor has d degrees of freedom (e.g., if d=4, then the spinor has four degrees of freedom). In SUSY, the partners are a pair (2), and the number of supersymmetry copies is an exponent to that base (0, 1, 2, or 3). The product of (spinor times copies) gives the total number of supersymmetry generators, with a minimum of (4 × 1 = 4) and a maximum of (4 × 8 = 32). In d dimensions, the size of spinors follows 2(d-1)/2. We see that since the maximum number of generators is 32, SUSY maxes-out at eleven (11) dimensions.

Therein lies our theoretical issue with SUSY. We have made a case for a 26-dimensional bosonic theory using twistors. Now, there would need to be accounting for the reduction in dimensionality. I’m up for the task, but first let’s see what the data from colliders say about integer spin in higher-dimensions…

Unfortunately, while great in pure mathematical practice (superalgebraic studies, for example) and non-cosmological applications, supersymmetry – as the theory stands - is absent of empirical bearings. WMAP surveys and experiments have detected nothing of its sort. Likewise, for the last decade, the major high-energy particle accelerators (Large Hadron Collider, Tevatron, etc.) have found zero evidence of supersymmetry after running a number of tests at a distribution of energy allowances (upperlimit sensitivities from 135GeV – 2.5TeV). To make matters worse, existence of the Higgs boson was confirmed at ~125GeV. Instead of punting, some theorists have suggested changing the instrumentation and methodology (of course!).

So, you’re asking, “Link, does that mean that strings aren’t ‘super’?” The answer is more mundane than that. It’s more like there’s no distinction between Clark Kent and Kal-El. What’s more rational is that Kal-El’s just a journalist on Krypton.

Regardless, any GUT must contain in its gamut a path that explains the formation of the first two (2) - and most abundant - elements in the observable Universe, hydrogen (H, atomic number 1) and helium (He, atomic number 2).

-- All roads lead to hydrogen --

Meshrooming: monomer morphology

+ Here's the kicker: we shouldn't actually look at strings as being physical objects, per se. Instead, consider them portable sequences of permutable segments. As long as we can compute superalgebras, our toolset is workable for strings; we have moderate coverage of string behavior to know how they work rudimentarily. Now, we can explore some strings that are actually in use in power settings.

One litmus test for string theory validation is the polypeptide. One can easily see that peptide (and for that matter, protein) structure and behavior clearly follows the stretch and folding patterns predicted in the theory.

Proteins are really neat. They’re these extended macromolecules (relatively large molecules) which are responsible for the maintenance and upkeep within all living cells (they can also exist outside of the cell, but we are concerned here with intracellular biochemistry). In case one doesn't know much about these machines to begin with, we’ll spend some time now on edification.

The first thing we need to know about proteins is that they are, in fact, machines. Like any other machine, they use and convert energy. Their mechanical properties allow what interacts with them to get work done. Proteins consist of amino acid chains. Because they are molecular (hence, at the nanoscale), proteins are perhaps the most important biochemicals (amongst other reasons beyond the scope of this explanation). Proteins differ from one another primarily in their sequence of amino acids, which is dictated by the nucleotide sequence of their genes. This sequence results in a protein folding into some unique three-dimensional (3D) structure that determines its functionality.


- In essence, we are mimicking the behavior of protein folding. Proteins are an essential component to many biological functions and participate in virtually all processes within biological cells. They often act as enzymes, performing biochemical reactions including cell signaling, molecular transportation, and cellular regulation.

As structural elements, some proteins act as a type of skeleton for cells, and as antibodies, while other proteins participate in the immune system. Before a protein can take on these roles, it must fold into a functional three-dimensional structure, a process that often occurs spontaneously and is dependent on interactions within its amino acid sequence and interactions of the amino acids with their surroundings.

Protein folding is driven by the search to find the most energetically favorable conformation of the protein, i.e. its native state. Thus, understanding protein folding is critical to understanding what a protein does and how it works, and is considered a "holy grail" of computational biology. Despite folding occurring within a crowded cellular environment, it typically proceeds smoothly. However, due to a protein's chemical properties or other factors, proteins may misfold — that is, fold down the wrong pathway and end up misshapen. Unless cellular mechanisms are capable of destroying or refolding such misfolded proteins, they can subsequently aggregate and cause a variety of debilitating diseases.

Laboratory experiments studying these processes can be limited in scope and atomic detail, leading scientists to use physics-based computational models that, when complementing experiments, seek to provide a more complete picture of protein folding, misfolding, and aggregation.

Due to the complexity of proteins' conformation space — the set of possible shapes a protein can take — and limitations in computational power, all-atom molecular dynamics simulations have been severely limited in the timescales which they can study. While most proteins typically fold in the order of milliseconds, before recently, simulations could only reach nanosecond to microsecond timescales. General-purpose supercomputers have been used to simulate protein folding, but such systems are intrinsically expensive and typically shared among many research groups. Additionally, because the computations in kinetic models are serial in nature, strong scaling of traditional molecular simulations to these architectures is exceptionally difficult. Moreover, as protein folding is a stochastic process and can statistically vary over time, it is computationally challenging to use long simulations for comprehensive views of the folding process.

Protein folding does not occur in a single step. Instead, proteins spend the majority of their folding time — nearly 96% in some cases — "waiting" in various intermediate conformational states, each a local thermodynamic free energy minimum in the protein's energy landscape.

Through a process known as adaptive sampling, these conformations are used as starting points for a set of simulation trajectories. As the simulations discover more conformations, the trajectories are restarted from them, and a Markov state model (MSM) is gradually created from this cyclic process. MSMs are discrete-time master equation models which describe a biomolecule's conformational and energy landscape as a set of distinct structures and the short transitions between them. The adaptive sampling Markov state model approach significantly increases the efficiency of simulation as it avoids computation inside the local energy minimum itself, and is amenable to distributed computing as it allows for the statistical aggregation of short, independent simulation trajectories.

The amount of time it takes to construct a Markov state model is inversely proportional to the number of parallel simulations run, i.e. the number of processors available. In other words, it achieves linear parallelization, leading to an approximately four orders of magnitude reduction in overall serial calculation time. A completed MSM may contain tens of thousands of sample states from the protein's phase space (all the conformations a protein can take on) and the transitions between them.
game overview napkin A linear chain of amino acid residues is called a polypeptide. A protein contains at least one long polypeptide. Short polypeptides, containing less than about 20-30 residues, are rarely considered to be proteins and are commonly called peptides, or sometimes oligopeptides. The individual amino acid residues are bonded together by peptide bonds and adjacent amino acid residues. The sequence of amino acid residues in a protein is defined by the sequence of a gene, which is encoded in the genetic code. In general, the genetic code specifies 20 standard amino acids; however, in certain organisms the genetic code can include selenocysteine and—in certain archaea—pyrrolysine. Shortly after or even during synthesis, the residues in a protein are often chemically modified by posttranslational modification, which alters the physical and chemical properties, folding, stability, activity, and ultimately, the function of the proteins. Sometimes proteins have non-peptide groups attached, which can be called prosthetic groups or cofactors. Proteins can also work together to achieve a particular function, and they often associate to form stable protein complexes.

Upon formation, proteins only exist for a finite period of time and are then degraded and recycled by the cell's machinery via protein turnover. The lifespan of a protein is measured in periods of its half-life. Depending on the host environment, they can exist for minutes or years (the average lifespan is 24-48 hours in mammalian cells). Misfolded proteins are degraded more rapidly due either to their instability or them being signaled for destruction as a means for cellular upkeep and efficiency.

Most proteins consist of linear polymers built from series of up to 20 different L-α-amino acids. All proteinogenic amino acids possess common structural features, including an α-carbon to which an amino group, a carboxyl group, and a variable side chain are bonded. Only proline differs from this basic structure as it contains an unusual ring to the N-end amine group, which forces the CO–NH amide moiety into a fixed conformation. The side chains of the standard amino acids, detailed in the list of standard amino acids, have a great variety of chemical structures and properties; it is the combined effect of all of the amino acid side chains in a protein that ultimately determines its three-dimensional structure and its chemical reactivity. The amino acids in a polypeptide chain are linked by peptide bonds. Once linked in the protein chain, an individual amino acid is called a residue, and the linked series of carbon, nitrogen, and oxygen atoms are known as the main chain or protein backbone.

The peptide bond has two resonance forms that contribute some double-bond character and inhibit rotation around its axis, so that the alpha carbons are roughly coplanar. The other two dihedral angles in the peptide bond determine the local shape assumed by the protein backbone. The end of the protein with a free carboxyl group is known as the C-terminus or carboxy terminus, whereas the end with a free amino group is known as the N-terminus or amino terminus. The words protein, polypeptide, and peptide are a little ambiguous and can overlap in meaning. Protein is generally used to refer to the complete biological molecule in a stable conformation, whereas peptide is generally reserved for a short amino acid oligomers often lacking a stable three-dimensional structure. However, the boundary between the two is not well defined and usually lies near 20–30 residues. Polypeptide can refer to any single linear chain of amino acids, usually regardless of length, but often implies an absence of a defined conformation.

Proteins are assembled from amino acids using information encoded in genes. Each protein has its own unique amino acid sequence that is specified by the nucleotide sequence of the gene encoding this protein. The genetic code is a set of three-nucleotide sets called codons and each three-nucleotide combination designates an amino acid, for example AUG (adenine-uracil-guanine) is the code for methionine. Because DNA contains four nucleotides, the total number of possible codons is 64; hence, there is some redundancy in the genetic code, with some amino acids specified by more than one codon. Genes encoded in DNA are first transcribed into pre-messenger RNA (mRNA) by proteins such as RNA polymerase. Most organisms then process the pre-mRNA (also known as a primary transcript) using various forms of Post-transcriptional modification to form the mature mRNA, which is then used as a template for protein synthesis by the ribosome. In prokaryotes the mRNA may either be used as soon as it is produced, or be bound by a ribosome after having moved away from the nucleoid. In contrast, eukaryotes make mRNA in the cell nucleus and then translocate it across the nuclear membrane into the cytoplasm, where protein synthesis then takes place. The rate of protein synthesis is higher in prokaryotes than eukaryotes and can reach up to 20 amino acids per second.

The process of synthesizing a protein from an mRNA template is known as translation. The mRNA is loaded onto the ribosome and is read three nucleotides at a time by matching each codon to its base pairing anticodon located on a transfer RNA molecule, which carries the amino acid corresponding to the codon it recognizes. The enzyme aminoacyl tRNA synthetase "charges" the tRNA molecules with the correct amino acids. The growing polypeptide is often termed the nascent chain. Proteins are always biosynthesized from N-terminus to C-terminus.

Protein folding is the process by which a protein structure assumes its functional shape or conformation. It is the physical process by which a polypeptide folds into its characteristic and functional three-dimensional structure from random coil. Each protein exists as an unfolded polypeptide or random coil when translated from a sequence of mRNA to a linear chain of amino acids. This polypeptide lacks any stable (long-lasting) three-dimensional structure (the left hand side of the first figure). Amino acids interact with each other to produce a well-defined three-dimensional structure, the folded protein (the right hand side of the figure), known as the native state. The resulting three-dimensional structure is determined by the amino acid sequence (Anfinsen's dogma). Experiments have indicated that the codon for an amino acid can also influence protein structure.

Most (but not all) proteins fold into unique 3-dimensional structures. Proteins that do not adhere to or lack this behavior are called intrinsically disordered. Still, such proteins can adopt a fixed structure by binding to other macromolecules. The shape into which a protein naturally folds is known as its native conformation. Failure to fold into native structure generally produces inactive proteins, but in some instances misfolded proteins have modified or toxic functionality. Although many proteins can fold unassisted, simply through the chemical properties of their amino acids, others require the aid of molecular chaperones to fold into their native states. Biochemists often refer to four distinct aspects of a protein's structure:
- Primary structure: the amino acid sequence. A protein is a polyamide.
- Secondary structure: regularly repeating local structures stabilized by hydrogen bonds. The most common examples are the alpha helix, beta sheet and turns. Because secondary structures are local, many regions of different secondary structure can be present in the same protein molecule.
- Tertiary structure: the overall shape of a single protein molecule; the spatial relationship of the secondary structures to one another. Tertiary structure is generally stabilized by nonlocal interactions, most commonly the formation of a hydrophobic core, but also through salt bridges, hydrogen bonds, disulfide bonds, and even posttranslational modifications. The term "tertiary structure" is often used as synonymous with the term fold. The tertiary structure is what controls the basic function of the protein.
- Quaternary structure: the structure formed by several protein molecules (polypeptide chains), usually called protein subunits in this context, which function as a single protein complex.

Proteins are not entirely rigid molecules. In addition to these levels of structure, proteins may shift between several related structures while they perform their functions. In the context of these functional rearrangements, these tertiary or quaternary structures are usually referred to as "conformations", and transitions between them are called conformational changes. Such changes are often induced by the binding of a substrate molecule to an enzyme's active site, or the physical region of the protein that participates in chemical catalysis. In solution proteins also undergo variation in structure through thermal vibration and the collision with other molecules.

Molecular surface of several proteins showing their comparative sizes. From left to right are: immunoglobulin G (IgG, an antibody), hemoglobin, insulin (a hormone), adenylate kinase (an enzyme), and glutamine synthetase (an enzyme). Proteins can be informally divided into three main classes, which correlate with typical tertiary structures: globular proteins, fibrous proteins, and membrane proteins. Almost all globular proteins are soluble and many are enzymes. Fibrous proteins are often structural, such as collagen, the major component of connective tissue, or keratin, the protein component of hair and nails. Membrane proteins often serve as receptors or provide channels for polar or charged molecules to pass through the cell membrane.

Proteins are chains of amino acids joined together by peptide bonds. Many conformations of this chain are possible due to the rotation of the chain about each Cα atom. It is these conformational changes that are responsible for differences in the three dimensional structure of proteins. Each amino acid in the chain is polar, i.e. it has separated positive and negative charged regions with a free C=O group, which can act as hydrogen bond acceptor and an NH group, which can act as hydrogen bond donor. These groups can therefore interact in the protein structure. The 20 amino acids can be classified according to the chemistry of the side chain which also plays an important structural role. Glycine takes on a special position, as it has the smallest side chain, only one Hydrogen atom, and therefore can increase the local flexibility in the protein structure. Cysteine on the other hand can react with another cysteine residue and thereby form a cross link stabilizing the whole structure.

The protein structure can be considered as a sequence of secondary structure elements, such as α helices and β sheets, which together constitute the overall three-dimensional configuration of the protein chain. In these secondary structures regular patterns of H bonds are formed between neighboring amino acids, and the amino acids have similar Φ and Ψ angles.

Protein prediction, design, and engineering can be an expensive (in terms of both time and money) process. Compared to what we are trying to do (earn and decipher), it can be a rather tedious endeavor since the payoff only comes well down the road after jumping through many hoops. Let me suggest a more labor-ready heuristic...

MONEY💰: a cryptocommodity

Stereotyping is done in a three (3)-act continuum: discrete logarithm + elliptic curvature + arithmetic (namely integer factorization).

The continuum is weighted around the relationship between hyper- and hypo- currencies. This is called the formula of cryptocurrency* [formalized as the Cryptoquotient (CQ), and also called the Cryptocurrency Problem], which asks if there exists a crypto exercise (natural, artificial, or heterotic) that, hypostantially, may anchor a hypercurrency alongside which it is fit? The rationale comes from string ludology, which posits that juking is resultant of certain path integral manipulation. The ultimate proof is quotient normalization (parimutuelcybernetic).
A cryptocurrency is a cryptographic exercise whose resultant is transactionable in some real economy.

Double U (u-u) economics is the formal attempt at leveraging the Juke Lemma by sequencing fibors from the portfolio: [verso (EEE) = (micro) through recto (PPP) = (macro)], which yield some quotient of cryptocurrency.

game overview napkin MONEY (Mathematically-Optimized Numismatics' Encrypted Yield) is the cryptocurrency* derived from y-proofing.

MONEY () is earned by juking. After obtaining a 0b, the yield is the attribution of cents drawn from an l-string arrangement. The cryptocurrency requires that the total be no greater than the handicap to qualify as MONEY. Otherwise, it is bubblegum.*
However, a fibor bundle can still theoretically make MONEY.
"Cents are made from stew choreography."


Because the supremum of ludeiy constructibles is computable yet exponential, extrapolating yesegalo from those objects and farming their convertible geometries is ideal for stew choreography. Our economy is negotiated organically from the renormalization of egglepple's intrinsic cent value. Keyframing (coupling a shapeframe with the score) yields plausible recreation.


Gameplay: opening + closing

+ Random walks is the (programming) language of juking (I wonder if that's grammatically correct?). From the standpoint of string-adherence, gameplay (ludological operation) is the foundational dynamic of how l-string functionality is proofed. By playing, we are generating walks for use in fibor determination and closure.

Basically, all games transition through three (3) phases: 1) the opening, 2) its middlegame, and 3) an endgame. game overview napkin Stewart's composition is bifurcated into the voices: Earl (aria) and ELLIS (recitative). Each voice deals with its own set of workoads. Earl is for batch processing, while ELLIS is for variable loop reduction. game overview napkin Being an opera ludo, the above objective is introduced as a fitness program for advancing game logic.

'Fitness' here is the ability of currency to move across twistor space.


Animation in twistor space is entirely based on walks and their statistical variance. Jukers are best-served with an exposition on probability theory since it is conducive to a strong opening (yesegalo construction).

Walks get interpolated into major scale construction.

game overview napkin ELLIS (from 'Erasure Loop Linear Instruction Set') is the cassette feed for Earl, performing loop-erasure (aka 'loop-erased random walks' or 'erasure loops'). ELLIS guides stew choreography. This is the Pink program's cantata.

In simplest terms, loop-erasure is a methodology for not repeating/replicating common walks; a most important checkpoint for obtaining a 0b. The sole purpose for loop-erasure is to supply comparative models which are to be deprecated, ensuring that stew choreography is as fast as possible. Needless to say, this frees-up compute cycles on the systole.

In any financial scenario, there needs to be in place an insurance mechanism to reduce risk. We achieve this with cassettes in/from the ELLIS cartridge. My vision for ELLIS revolves around the concept of loop-erased (random) walks. A walk is a playable formation (sequence of discrete steps at fixed length). This is a simplified chart illustrating stochastic activity at specified inflection points (representing measures). In all probability, walks are a distribution of their random variables transforming twistor space. Loop-erasure is the preferred method for culling pitch spikes. game overview napkin + Walks and their calculus are important stuff; they form the whole methodology behind gameplay. So, let's take a quick walkthrough (I'm being punny) for juker edification.

Statistically, most walks are random, so we'll start with those. Random walks are usually assumed to be Markov chains or Markov processes, but other, more complicated walks are also of interest. Some random walks are on graphs, others on the line, in the plane, in higher dimensions, or even curved surfaces, while some random walks are on groups. Random walks also vary with regard to the time parameter. Often, the walk is in discrete time, and indexed by the natural numbers. However, some walks take their steps at random times, and in that case, the position X_{t} is defined for the continuum of times. Specific cases or limits of random walks include the Lévy flight.

A popular random walk model is that of a random walk on a regular lattice, where at each step the location jumps to another site according to some probability distribution. In a simple random walk, the location can only jump to neighboring sites of the lattice, forming a lattice path. In simple symmetric random walk on a locally finite lattice, the probabilities of the location jumping to each one of its immediate neighbours are the same. The best studied example is of random walk on the d-dimensional integer lattice (sometimes called the hypercubic lattice).

A self-avoiding walk is a sequence of moves on a lattice (a lattice path) that does not visit the same point more than once. This is a special case of the graph theoretical notion of a path. A self-avoiding polygon is a closed self-avoiding walk on a lattice.
Markov, et cetera
+ Markov, etc.
Probability distribution
+ Probability distribution
A probability distribution assigns a probability to each measurable subset of the possible outcomes of a random experiment, survey, or procedure of statistical inference. Examples are found in experiments whose sample space is non-numerical, where the distribution would be a categorical distribution; experiments whose sample space is encoded by discrete random variables, where the distribution can be specified by a probability mass function; and experiments with sample spaces encoded by continuous random variables, where the distribution can be specified by a probability density function. More complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures.

A probability distribution can either be univariate or multivariate. A univariate distribution gives the probabilities of a single random variable taking on various alternative values; a multivariate distribution (a joint probability distribution) gives the probabilities of a random vector—a set of two or more random variables—taking on various combinations of values. Important and commonly encountered univariate probability distributions include the binomial distribution, the hypergeometric distribution, and the normal distribution. The multivariate normal distribution is a commonly encountered multivariate distribution.

To define probability distributions for the simplest cases, one needs to distinguish between discrete and continuous random variables. In the discrete case, one can easily assign a probability to each possible value: for example, when throwing a fair die, each of the six values 1 to 6 has the probability 1/6. In contrast, when a random variable takes values from a continuum then, typically, probabilities can be nonzero only if they refer to intervals: in quality control one might demand that the probability of a "500 g" package containing between 490 g and 510 g should be no less than 98%. The probability density function (pdf) of the normal distribution, also called Gaussian or "bell curve", the most important continuous random distribution. As notated on the figure, the probabilities of intervals of values correspond to the area under the curve.

If the random variable is real-valued (or more generally, if a total order is defined for its possible values), the cumulative distribution function (CDF) gives the probability that the random variable is no larger than a given value; in the real-valued case, the CDF is the integral of the probability density function (pdf) provided that this function exists.

(Cumulative distribution function)
Because a probability distribution Pr on the real line is determined by the probability of a scalar random variable X being in a half-open interval (-∞, x], the probability distribution is completely characterized by its cumulative distribution function:
F(x) = \Pr \left[ X \le x \right] \qquad \text{ for all } x \in \mathbb{R}.

(Discrete probability distribution)
A discrete probability distribution is a probability distribution characterized by a probability mass function. Thus, the distribution of a random variable X is discrete, and X is called a discrete random variable, if
\sum_u \Pr(X=u) = 1
as u runs through the set of all possible values of X. Hence, a random variable can assume only a finite or countably infinite number of values—the random variable is a discrete variable. For the number of potential values to be countably infinite, even though their probabilities sum to 1, the probabilities have to decline to zero fast enough. for example, if \Pr(X=n) = \tfrac{1}{2^n} for n = 1, 2, ..., we have the sum of probabilities 1/2 + 1/4 + 1/8 + ... = 1.

Well-known discrete probability distributions used in statistical modeling include the Poisson distribution, the Bernoulli distribution, the binomial distribution, the geometric distribution, and the negative binomial distribution. Additionally, the discrete uniform distribution is commonly used in computer programs that make equal-probability random selections between a number of choices.

(Continuous probability distribution)
A continuous probability distribution is a probability distribution that has a cumulative distribution function that is continuous. Most often they are generated by having a probability density function. Mathematicians call distributions with probability density functions absolutely continuous, since their cumulative distribution function is absolutely continuous with respect to the Lebesgue measure λ. If the distribution of X is continuous, then X is called a continuous random variable. There are many examples of continuous probability distributions: normal, uniform, chi-squared, and others.

Intuitively, a continuous random variable is the one which can take a continuous range of values—as opposed to a discrete distribution, where the set of possible values for the random variable is at most countable. While for a discrete distribution an event with probability zero is impossible (e.g., rolling 3 1 / 2 on a standard die is impossible, and has probability zero), this is not so in the case of a continuous random variable. For example, if one measures the width of an oak leaf, the result of 3½ cm is possible; however, it has probability zero because uncountably many other potential values exist even between 3 cm and 4 cm. Each of these individual outcomes has probability zero, yet the probability that the outcome will fall into the interval (3 cm, 4 cm) is nonzero. This apparent paradox is resolved by the fact that the probability that X attains some value within an infinite set, such as an interval, cannot be found by naively adding the probabilities for individual values. Formally, each value has an infinitesimally small probability, which statistically is equivalent to zero.

Formally, if X is a continuous random variable, then it has a probability density function ƒ(x), and therefore its probability of falling into a given interval, say [a, b] is given by the integral
\Pr[a\le X\le b] = \int_a^b f(x) \, dx
In particular, the probability for X to take any single value a (that is a ≤ X ≤ a) is zero, because an integral with coinciding upper and lower limits is always equal to zero.

The definition states that a continuous probability distribution must possess a density, or equivalently, its cumulative distribution function be absolutely continuous. This requirement is stronger than simple continuity of the cumulative distribution function, and there is a special class of distributions, singular distributions, which are neither continuous nor discrete nor a mixture of those. An example is given by the Cantor distribution. Such singular distributions however are never encountered in practice.

Note on terminology: some authors use the term "continuous distribution" to denote the distribution with continuous cumulative distribution function. Thus, their definition includes both the (absolutely) continuous and singular distributions.

By one convention, a probability distribution \,\mu is called continuous if its cumulative distribution function F(x)=\mu(-\infty,x] is continuous and, therefore, the probability measure of singletons \mu\{x\}\,=\,0 for all \,x.

Another convention reserves the term continuous probability distribution for absolutely continuous distributions. These distributions can be characterized by a probability density function: a non-negative Lebesgue integrable function \,f defined on the real numbers such that
F(x) = \mu(-\infty,x] = \int_{-\infty}^x f(t)\,dt.
Discrete distributions and some continuous distributions (like the Cantor distribution) do not admit such a density.

Law of the iterated logarithm
+ Law of the iterated logarithm
The law of the iterated logarithm describes the magnitude of the fluctuations of a random walk.

Let {Yn} be independent, identically distributed random variables with means zero and unit variances. Let Sn = Y1 + … + Yn. Then \limsup_{n \to \infty} \frac{S_n}{\sqrt{n \log\log n}} = \sqrt 2, \qquad \text{a.s.}, where “log” is the natural logarithm, “lim sup” denotes the limit superior, and “a.s.” stands for “almost surely".

The law of iterated logarithms operates “in between” the law of large numbers and the central limit theorem. Interestingly, it holds for polynomial time (P) pseudorandom sequences. There are two versions of the law of large numbers — the weak and the strong — and they both state that the sums Sn, scaled by n−1, converge to zero, respectively in probability and almost surely:
\frac{S_n}{n} \ \xrightarrow{p}\ 0, \qquad \frac{S_n}{n} \ \xrightarrow{a.s.} 0, \qquad \text{as}\ \ n\to\infty.

On the other hand, the central limit theorem states that the sums Sn scaled by the factor n−½ converge in distribution to a standard normal distribution. By Kolmogorov's zero-one law, for any fixed M, the probability that the event \limsup_n \frac{S_n}{\sqrt{n}} > M occurs is 0 or 1. Then
P\left( \limsup_n \frac{S_n}{\sqrt{n}} > M \right) \geq \limsup_n P\left( \frac{S_n}{\sqrt{n}} > M \right) = P\bigl( \mathcal{N}(0, 1) > M \bigr) > 0
so \limsup_n \frac{S_n}{\sqrt{n}}=\infty with probability 1. An identical argument shows that \liminf_n \frac{S_n}{\sqrt{n}}=-\infty with probability 1 as well. This implies that these quantities cannot converge almost surely. In fact, they cannot even converge in probability, which follows from the equality \frac{S_{2n}}{\sqrt{2n}}-\frac{S_n}{\sqrt{n}} = \frac1{\sqrt2}\frac{S_{2n}-S_n}{\sqrt{n}} - (1-\frac1\sqrt2)\frac{S_n}{\sqrt{n}} and the fact that the random variables \frac{S_n}{\sqrt{n}} and \frac{S_{2n}-S_n}{\sqrt{n}} are independent and both converge in distribution to \mathcal{N}(0, 1).

The law of the iterated logarithm provides the scaling factor where the two limits become different:
\frac{S_n}{\sqrt{n\log\log n}} \ \xrightarrow{p}\ 0, \qquad \frac{S_n}{\sqrt{n\log\log n}} \ \stackrel{a.s.}{\nrightarrow}\ 0, \qquad \text{as}\ \ n\to\infty.

Thus, although the quantity S_n/\sqrt{n\log\log n} is less than any predefined ε > 0 with probability approaching one, that quantity will nevertheless be dropping out of that interval infinitely often, and in fact will be visiting the neighborhoods of any point in the interval (-√2,√2) almost surely.

+ Loop-erasure
Assume G is some graph and \gamma is some path of length n on G. In other words, \gamma(1),\dots,\gamma(n) are vertices of G such that \gamma(i) and \gamma(i+1) are connected by an edge. Then the loop erasure of \gamma is a new simple path created by erasing all the loops of \gamma in chronological order. Formally, we define indices i_j inductively using
i_1 = 1\,
where "max" here means up to the length of the path \gamma. The induction stops when for some i_j we have \gamma(i_j)=\gamma(n). Assume this happens at J i.e. i_J is the last i_j. Then the loop erasure of \gamma, denoted by \mathrm{LE}(\gamma) is a simple path of length J defined by

Now let G be some graph, let v be a vertex of G, and let R be a random walk on G starting from v. Let T be some stopping time for R. Then the loop-erased random walk until time T is LE(R([1,T])). In other words, take R from its beginning until T — that's a (random) path — erase all the loops in chronological order as above — you get a random simple path.

The stopping time T may be fixed, i.e. one may perform n steps and then loop-erase. However, it is usually more natural to take T to be the hitting time in some set. For example, let G be the graph Z2 and let R be a random walk starting from the point (0,0). Let T be the time when R first hits the circle of radius 100 (we mean here of course a discretized circle). LE(R) is called the loop-erased random walk starting at (0,0) and stopped at the circle.

A spanning tree chosen randomly from among all possible spanning trees with equal probability is called a uniform spanning tree. To create such a tree Wilson’s algorithm uses loop-erased random walks. The algorithm proceeds by initializing the tree maze with an random starting cell. New cells are then subsequently added to the maze, initiating a random walk. The random walk progresses uninterrupted until it eventually links with the prevailing maze. However, if the random walk traverses through itself, the resulting loop is erased before the random walk proceeds. The initial random walks are unexpected to link with the small existing maze. As the maze develops, the random walks tend to have a higher probability to collide with the maze and may cause the algorithm to accelerate dramatically.

For instance, Let G again be a graph. A spanning tree of G is a subgraph of G containing all vertices and some of the edges, which is a tree, i.e. connected and with no cycles. The uniform spanning tree (UST for short) is a random spanning tree chosen among all the possible spanning trees of G with equal probability.

Let now v and w be two vertices in G. Any spanning tree contains precisely one simple path between v and w. Taking this path in the uniform spanning tree gives a random simple path. It turns out that the distribution of this path is identical to the distribution of the loop-erased random walk starting at v and stopped at w.

An immediate corollary is that loop-erased random walk is symmetric in its start and end points. More precisely, the distribution of the loop-erased random walk starting at v and stopped at w is identical to the distribution of the reversal of loop-erased random walk starting at w and stopped at v. This is not a trivial fact at all! Loop-erasing a path and the reverse path do not give the same result. It is only the distributions that are identical.

A-priori sampling a UST seems difficult. Even a relatively modest graph (say a 100x100 grid) has far too many spanning trees to prepare a complete list. Therefore a different approach is needed. There are a number of algorithms for sampling a UST, but we will concentrate on Wilson's algorithm.

Take any two vertices and perform loop-erased random walk from one to the other. Now take a third vertex (not on the constructed path) and perform loop-erased random walk until hitting the already constructed path. This gives a tree with either two or three leaves. Choose a fourth vertex and do loop-erased random walk until hitting this tree. Continue until the tree spans all the vertices. It turns out that no matter which method you use to choose the starting vertices you always end up with the same distribution on the spanning trees, namely the uniform one.

Another representation of loop-erased random walk stems from solutions of the discrete Laplace equation. Let G again be a graph and let v and w be two vertices in G. Construct a random path from v to w inductively using the following procedure. Assume we have already defined \gamma(1),...,\gamma(n). Let f be a function from G to R satisfying
f(\gamma(i))=0 for all i\leq n and f(w)=1
f is discretely harmonic everywhere else
Where a function f on a graph is discretely harmonic at a point x if f(x) equals the average of f on the neighbors of x.

With f defined choose \gamma(n+1) using f at the neighbors of \gamma(n) as weights. In other words, if x_1,...,x_d are these neighbors, choose x_i with probability
\frac{f(x_i)}{\sum_{j=1}^d f(x_j)}.

Continuing this process, recalculating f at each step, with result in a random simple path from v to w; the distribution of this path is identical to that of a loop-erased random walk from v to w.

An alternative view is that the distribution of a loop-erased random walk conditioned to start in some path β is identical to the loop-erasure of a random walk conditioned not to hit β. This property is often referred to as the Markov property of loop-erased random walk (though the relation to the usual Markov property is somewhat vague).

It is important to notice that while the proof of the equivalence is quite easy, models which involve dynamically changing harmonic functions or measures are typically extremely difficult to analyze. Practically nothing is known about the p-Laplacian walk or diffusion-limited aggregation. Another somewhat related model is the harmonic explorer.

Finally there is another link that should be mentioned: Kirchhoff's theorem relates the number of spanning trees of a graph G to the eigenvalues of the discrete Laplacian. See spanning tree for details.

Let d be the dimension, which we will assume to be at least 2. Examine Zd i.e. all the points (a_1,...,a_d) with integer a_i. This is an infinite graph with degree 2d when you connect each point to its nearest neighbors. From now on we will consider loop-erased random walk on this graph or its subgraphs.

(High dimensions)
The easiest case to analyze is dimension 5 and above. In this case it turns out that there the intersections are only local. A calculation shows that if one takes a random walk of length n, its loop-erasure has length of the same order of magnitude, i.e. n. Scaling accordingly, it turns out that loop-erased random walk converges (in an appropriate sense) to Brownian motion as n goes to infinity. Dimension 4 is more complicated, but the general picture is still true. It turns out that the loop-erasure of a random walk of length n has approximately n/\log^{1/3}n vertices, but again, after scaling (that takes into account the logarithmic factor) the loop-erased walk converges to Brownian motion.

(Two dimensions)
In two dimensions, arguments from conformal field theory and simulation results led to a number of exciting conjectures. Assume D is some simply connected domain in the plane and x is a point in D. Take the graph G to be
G:=D\cap \varepsilon \mathbb{Z}^2,
that is, a grid of side length ε restricted to D. Let v be the vertex of G closest to x. Examine now a loop-erased random walk starting from v and stopped when hitting the "boundary" of G, i.e. the vertices of G which correspond to the boundary of D. Then the conjectures are
As ε goes to zero the distribution of the path converges to some distribution on simple paths from x to the boundary of D (different from Brownian motion, of course — in 2 dimensions paths of Brownian motion are not simple). This distribution (denote it by S_{D,x}) is called the scaling limit of loop-erased random walk.

These distributions are conformally invariant. Namely, if φ is a Riemann map between D and a second domain E then
The Hausdorff dimension of these paths is 5/4 almost surely.

The first attack at these conjectures came from the direction of domino tilings. Taking a spanning tree of G and adding to it its planar dual one gets a domino tiling of a special derived graph (call it H). Each vertex of H corresponds to a vertex, edge or face of G, and the edges of H show which vertex lies on which edge and which edge on which face. It turns out that taking a uniform spanning tree of G leads to a uniformly distributed random domino tiling of H. The number of domino tilings of a graph can be calculated using the determinant of special matrices, which allow to connect it to the discrete Green function which is approximately conformally invariant. These arguments allowed to show that certain measurables of loop-erased random walk are (in the limit) conformally invariant, and that the expected number of vertices in a loop-erased random walk stopped at a circle of radius r is of the order of r^{5/4}.

In 2002 these conjectures were resolved (positively) using Stochastic Löwner Evolution. Very roughly, it is a stochastic conformally invariant ordinary differential equation which allows to catch the Markov property of loop-erased random walk (and many other probabilistic processes).
+ A self-avoiding walk is a path from one point to another which never intersects itself. Such paths are usually considered to occur on lattices, so that steps are only allowed in a discrete number of directions and of certain lengths.

Consider a self-avoiding walk on a two-dimensional n×n square grid (i.e., a lattice path which never visits the same lattice point twice) which starts at the origin, takes first step in the positive horizontal direction, and is restricted to nonnegative grid points only. The number of such paths of n=1, 2, ... steps are 1, 2, 5, 12, 30, 73, 183, 456, 1151, ...

Similarly, consider a self-avoiding walk which starts at the origin, takes first step in the positive horizontal direction, is not restricted to nonnegative grid points only, but which is restricted to take an up step before taking the first down step. The number of such paths of n=1, 2, ... steps are 1, 2, 5, 13, 36, 98, 272, 740, 2034, ...

Self-avoiding rook (yes, as like in chess) walks are walks on an m×n grid which start from (0,0), end at (m,n), and are composed of only horizontal and vertical steps. The following table gives the first few numbers R(m,n) of such walks for small m and n. The values for m=n=1, 2, ... are 2, 12, 184, 8512, 1262816, ...


Synthesizing fonts can be tricky. So, fundamental to a strong endgame is intelligent use of knot theory. The scope of this exposition will explore the theory of knots and their components.

In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3 (in topology, a circle isn't bound to the classical geometric concept, but to all of its homeomorphisms). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R3 upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.

Knots can be described in various ways. Given a method of description, however, there may be more than one description that represents the same knot. For example, a common method of describing a knot is a planar diagram called a knot diagram. Any given knot can be drawn in many different ways using a knot diagram. Therefore, a fundamental problem in knot theory is determining when two descriptions represent the same knot.

A complete algorithmic solution to this problem exists, which has unknown complexity. In practice, knots are often distinguished by using a knot invariant, a "quantity" which is the same when computed from different descriptions of a knot. Important invariants include knot polynomials, knot groups, and hyperbolic invariants. Knots can be considered in other three-dimensional spaces and objects other than circles can be used. Higher-dimensional knots are n-dimensional spheres in m-dimensional Euclidean space.

Knot equivalence
+ A knot is created by beginning with a one-dimensional line segment, wrapping it around itself arbitrarily, and then fusing its two free ends together to form a closed loop (Adams 2004) (Sossinsky 2002). Simply, we can say a knot K is an injective and continuous function K:[0,1]\to \mathbb{R}^3 with K(0)=K(1). Topologists consider knots and other entanglements such as links and braids to be equivalent if the knot can be pushed about smoothly, without intersecting itself, to coincide with another knot. The idea of knot equivalence is to give a precise definition of when two knots should be considered the same even when positioned quite differently in space. A formal mathematical definition is that two knots K_1,K_2 are equivalent if there is an orientation-preserving homeomorphism h\colon\R^3\to\R^3 with h(K_1)=K_2, and this is known as an ambient isotopy.

Knot diagrams
+ A useful way to visualise and manipulate knots is to project the knot onto a plane—think of the knot casting a shadow on the wall. A small change in the direction of projection will ensure that it is one-to-one except at the double points, called crossings, where the "shadow" of the knot crosses itself once transversely (Rolfsen 1976). At each crossing, to be able to recreate the original knot, the over-strand must be distinguished from the under-strand. This is often done by creating a break in the strand going underneath. The resulting diagram is an immersed plane curve with the additional data of which strand is over and which is under at each crossing. (These diagrams are called knot diagrams when they represent a knot and link diagrams when they represent a link.) Analogously, knotted surfaces in 4-space can be related to immersed surfaces in 3-space.

A reduced diagram is a knot diagram in which there are no reducible crossings (also nugatory or removable crossings), or in which all of the reducible crossings have been removed.

Knot invariance
+ A knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some invariants are indeed numbers, but invariants can range from the simple, such as a yes/no answer, to those as complex as a homology theory . Research on invariants is not only motivated by the basic problem of distinguishing one knot from another but also to understand fundamental properties of knots and their relations to other branches of mathematics.

From the modern perspective, it is natural to define a knot invariant from a knot diagram. Of course, it must be unchanged (that is to say, invariant) under the Reidemeister moves. Tricolorability is a particularly simple example. Other examples are knot polynomials, such as the Jones polynomial, which are currently among the most useful invariants for distinguishing knots from one another, though currently it is not known whether there exists a knot polynomial which distinguishes all knots from each other, or even which distinguishes just the unknot from all other knots.

Other invariants can be defined by considering some integer-valued function of knot diagrams and taking its minimum value over all possible diagrams of a given knot. This category includes the crossing number, which is the minimum number of crossings for any diagram of the knot, and the bridge number, which is the minimum number of bridges for any diagram of the knot.

Historically, many of the early knot invariants are not defined by first selecting a diagram but defined intrinsically, which can make computing some of these invariants a challenge. For example, knot genus is particularly tricky to compute, but can be effective (for instance, in distinguishing mutants).

The complement of a knot itself (as a topological space) is known to be a "complete invariant" of the knot by the Gordon–Luecke theorem in the sense that it distinguishes the given knot from all other knots up to ambient isotopy and mirror image. Some invariants associated with the knot complement include the knot group which is just the fundamental group of the complement. The knot quandle is also a complete invariant in this sense but it is difficult to determine if two quandles are isomorphic.

By Mostow–Prasad rigidity, the hyperbolic structure on the complement of a hyperbolic link is unique, which means the hyperbolic volume is an invariant for these knots and links. Volume, and other hyperbolic invariants, have proven very effective, utilized in some of the extensive efforts at knot tabulation.

In recent years, there has been much interest in homological invariants of knots which categorify well-known invariants. Heegaard Floer homology is a homology theory whose Euler characteristic is the Alexander polynomial of the knot. It has been proven effective in deducing new results about the classical invariants. Along a different line of study, there is a combinatorially defined cohomology theory of knots called Khovanov homology whose Euler characteristic is the Jones polynomial. This has recently been shown to be useful in obtaining bounds on slice genus whose earlier proofs required gauge theory. Khovanov and Rozansky have since defined several other related cohomology theories whose Euler characteristics recover other classical invariants. Stroppel gave a representation theoretic interpretation of Khovanov homology by categorifying quantum group invariants.

There is also growing interest from both knot theorists and scientists in understanding "physical" or geometric properties of knots and relating it to topological invariants and knot type. An old result in this direction is the Fary–Milnor theorem states that if the total curvature of a knot K in \mathbb{R}^3 satisfies
\oint_K \kappa \,ds \leq 4\pi,
where \kappa(p) is the curvature at p, then K is an unknot. Therefore, for knotted curves,
\oint_K \kappa\,ds > 4\pi.\,
An example of a "physical" invariant is ropelength, which is the amount of 1-inch diameter rope needed to realize a particular knot type.

+ A knot in three dimensions can be untied when placed in four-dimensional space. This is done by changing crossings. Suppose one strand is behind another as seen from a chosen point. Lift it into the fourth dimension, so there is no obstacle (the front strand having no component there); then slide it forward, and drop it back, now in front. Analogies for the plane would be lifting a string up off the surface, or removing a dot from inside a circle.

Since a knot can be considered topologically a 1-dimensional sphere, the next generalization is to consider a two-dimensional sphere embedded in a four-dimensional ball. Such an embedding is unknotted if there is a homeomorphism of the 4-sphere onto itself taking the 2-sphere to a standard "round" 2-sphere. Suspended knots and spun knots are two typical families of such 2-sphere knots.

The mathematical technique called "general position" implies that for a given n-sphere in the m-sphere, if m is large enough (depending on n), the sphere should be unknotted. In general, piecewise-linear n-spheres form knots only in (n + 2)-space, although this is no longer a requirement for smoothly knotted spheres. In fact, there are smoothly knotted (4k − 1)-spheres in 6k-space, e.g. there is a smoothly knotted 3-sphere in the 6-sphere. Thus the codimension of a smooth knot can be arbitrarily large when not fixing the dimension of the knotted sphere; however, any smooth k-sphere in an n-sphere with 2n − 3k − 3 > 0 is unknotted. The notion of a knot has further generalisations in mathematics, see: knot (mathematics), isotopy classification of embeddings.

Every knot in Sn is the link of a real-algebraic set with isolated singularity in Rn+1

An n-knot is a single Sn embedded in Sm. An n-link is k-copies of Sn embedded in Sm, where k is a natural number.

Sum topology
+ Two knots can be added by cutting both knots and joining the pairs of ends. The operation is called the knot sum, or sometimes the connected sum or composition of two knots. This can be formally defined as follows: consider a planar projection of each knot and suppose these projections are disjoint. Find a rectangle in the plane where one pair of opposite sides are arcs along each knot while the rest of the rectangle is disjoint from the knots. Form a new knot by deleting the first pair of opposite sides and adjoining the other pair of opposite sides. The resulting knot is a sum of the original knots. Depending on how this is done, two different knots (but no more) may result. This ambiguity in the sum can be eliminated regarding the knots as oriented, i.e. having a preferred direction of travel along the knot, and requiring the arcs of the knots in the sum are oriented consistently with the oriented boundary of the rectangle.

The knot sum of oriented knots is commutative and associative. A knot is prime if it is non-trivial and cannot be written as the knot sum of two non-trivial knots. A knot that can be written as such a sum is composite. There is a prime decomposition for knots, analogous to prime and composite numbers. For oriented knots, this decomposition is also unique. Higher-dimensional knots can also be added but there are some differences. While you cannot form the unknot in three dimensions by adding two non-trivial knots, you can in higher dimensions, at least when one considers smooth knots in codimension at least 3.

Pink program

Works are scored for egglepple (brane + kindergarten), necessitating fugal spacetime complexity controls*. Cassette (patch batch) compilation depends on jukespace being coordinated dyadically. Thus, Pink programming [ie., functional twistor algorithms declared from supermathematics (eg., superalgebras, supergeometries, superprobabilities, ultrametric analysis/supersymmetries, etc. of superstring theory)] is libretto-hashable.
Control here means isolating limits on acoustic bounds.
My global interest is validating the formula of cryptocommodity (ie., parimutuelcybernetic governance✔). This is an act of perpetual quotient load normalization between fiat (government-issued) currencies and their crypto counterparts, which could, in theory, make MONEY (universally) fungible. To accomplish this in reality would take either an international banking decree, or extremely large-scale mass participation (in terms of nodal saturation). Neither is a small feat. The arcade already in place needs to be cellularly automated, and scaled to handle elastic data sets. If you are a developer (mathematician/computer scientist/music theorist/biochemist/condensed matter specialist/whatever, etc.), I invite you to join me in helping build this stuff.👷🏿

Cassettes are developed by 'touch-and-go':= patches (plus subiteratives) might be contracted (pu$h/pull → deploy) on open puzzles. Trunks + branches + twigs + roots require Link Starbureiy's signature before being migrated/published.

To reiterate, development loosely follows the open source model (ie., desired suggestions should be directed at the community), so, given certain permissions (contract or signature), anyone can contribute. Please, no junk. Don't be a luser.

Who cares

Solutions are journaled in (the) UUelcome Matte (ISSN 2165-6738), the foremost scholastic industry-reviewed magazine for juking activities. Financial firms and exchanges care because we have a cryptocurrent system that generates new wealth mechanisms.💸 Information technology security specialists are interested in the asymmetrics. Academic research labs - in cooperation with businesses, like biotech companies - are paying close attention because macromolecular structure is essential to its function, and that helps with the process of engineering better medicines and environmental applications.

"UUe is at the very foundation of juking. We speak of this in terms of utility, and so our best practice when fielding the endeavor must be whetted in simplicity. It is important that we not limit ourselves to a specific type of automata. The most profitable approach is to consider everything, and then hedge. An intelligent way to bond with creativity is through idea incubation🎨: synthesize as much information as possible, create as many potentials as needed, and then refine those potentials. Even still, an idea is not a solution until it is put into use." -

IMPACT // The best work of your career happens here

UUe hosts four (4) sections: stereotypography, operation, interfacing, and ludology [SOIL]. Development happens in episodes, with each episode having specific objectives that advance at least one (1) of the sections.

Plenty of calculator power would have to be in place for the possibly enormous number of SOIL computables.

Now, I obviously can't - and am not going to - sort through dozens of scribble per person per opus at one time, so there needs to be a better method of opening. To do this, we're building and implementing a most robust arcade (as a distributed supercomputer).

+ Processors used in computers today are based on von Neumann architecture; that is, they rely on a stored program system (i.e., one that keeps its program instructions in RAM). In order for different (even neighboring) sections of the processor to communicate with each other (and the compositional elements), wires are needed for interconnects.

... Before continuing, let me first offer an admission: the von Neumann architecture and approach to computer systems works fine. Now, the kicker: there's an ever better solution. In fact, it's almost necessary to adopt it. Keeping in mind the dynamics of twistor space, qubits dictate that data can/must be in multiple states (quantum pairing/superpositions) simultaneously. So, the play is to build the machinery that accomplishes this.

We design here a cellular automaton processor configured as a systolic array (homogeneous batches of tightly coupled cells), where no wires are needed for intracell communication, and all cells get informed synchronously.

Another admission: our 'processor' starts off horizontally; that is, reliant on distributed computing. The vertical SoC (system on a chip) is actually a byproduct since we are concerned with core acquisition.

Link Starbureiy
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