rabbit hole

I saw a hole within the Prentententoonstelling, which Escher termed a "cyclic expansion… without beginning or end". One which both Smit and Lenstra Jr. and 3Brown2Blue glossed over in trying to "deduce that an idealized version of the picture re- peats itself in the middle". In their attempts to they had invented or a good story or "a particle" as Heinz Von Foerster would phrase it, to fill in the singularity in the middle of the painting:

$$\lim _{z\to 0}\ln\left(z\right)=-\infty$$

I was thinking of a another way to extend the whole numbers, instead using the product as fundamental to defining how negatives operate as a complement of $\mathbb{N}{>0}$ not just merely it's additive inverse. I was given to the product $ab$, wanting of the assumption that $b$ is negative and $2$ is an infinitesimally small even number: $$ (ax+1)(|b|y+2) \odot \begin{Bmatrix} 1 & 0 & 0& 2_ \end{Bmatrix} $$ $$\implies \forall C, ab+0$$ The infinitesimal term reduces down the coefficients to the form $ab + C$ where $C\in{0,1}$. We can phrase this more standardly. Given the lowest term of the polynomial $c$, let $\varepsilon$ be a positive infinitesimal strictly less than 1: $$C = \left\lceil (c+1) \bmod 2 + \varepsilon \right\rceil$$ We should then appreciate that the zero point, when we construct the negatives this way, lives in the same place as $\ln(0)$, it only arises as a limiting process where $ab<1$. In this zero lives as the a ghost of the pair $(0+2)$ because itself $0 < 1$. From this it follows that the product $0^2$ then must preserve it's parity because it is idempotent under multiplication like the positive integers.

https://en.wikipedia.org/wiki/Horror_vacui_(philosophy)

Coprime: sharing chicken nuggets

"Okay, since I'm an American, explain it to me in terms of a hamburger." The most efficient way to eat a hamburger so that you do not bite in the same area (homogenously).

https://en.wikipedia.org/wiki/Hugo_Steinhaus https://en.wikipedia.org/wiki/Three-gap_theorem https://en.wikipedia.org/wiki/Steinhaus_theorem https://en.wikipedia.org/wiki/Sharkovskii's_theorem

3-gap theorem

Every natural number is either even or odd Thus every one can be expressed as a series of some finite N where $$ S_n = (2, 2, 2,... S_N), $$ with $S_N = 1$ if the number is odd Given $k$ divisions of 2 of $\mathbb{N}$, $k=N-1$ is the largest where the reminder $\in\mathbb{N}$ for every $\mathbb{N}$ $$ S_\upsilon = (\frac{1}{2},\frac{1}{4},\frac{1}{8},...) $$ Let $Y_n$ be the set of rigid rotations on a circle with arc length 2. $$ Y(n)=S_{N-1} $$ Three gap theorem`. the end https://en.wikipedia.org/wiki/Symbolic_dynamics

"Added the correct statement that irrational numbers do not have any decimal representation, since they only have decimal representations through their rational approximations. Irrational numbers cannot be be taken as limits either, since limit values are never reached! In other words, irrational numbers do not exist. Admin, think about it and do not be brainwashed by nonsensical mainstream!"

https://en.wikipedia.org/wiki/Special:Contributions/~2026-43204-43

"To actually discriminate the representation from any other given one, you need a Dedekind cut over all of the rationals, real closed fields can still satisfy x^2 = 2 ala löwenheim–skolem but not it's real completion which the decimal form could mislead someone into thinking."

1/9 = 0.1111.. (RFC)

3.14159... (Not RFC)

$$ \mathbf{u}_{i,j} = \begin{bmatrix} \frac{\partial u_1}{\partial x_1} & \frac{\partial u_1}{\partial x_2} \[1ex] \frac{\partial u_2}{\partial x_1} & \frac{\partial u_2}{\partial x_2} \end{bmatrix} $$ The Leibnizan quotient notation for derivatives, interpreted as incoherent and merely an operator in standard analysis, obscures the algebraic properties of the partial derivatives.

Fictionalism

This treatment of the derivative operator is not unique. This type of mathematical fictionalism has been selectively applied throughout history to many concepts that seemed unintuitive with the settled paradigm, the most famous being those of the finitists which reject both the infinite and infinitesimal. Gert Schubring, formerly of the Pedagogical Institute of Mathematics at the University of Bielefeld, gives in his Conflicts perhaps the best treatment of the development underling algebraic number fields. He repudiates the common misconception that Babylonians had a fully-formed idea of the negative, and instead puts the proper genesis in the early-modern.

Thus there are a plethora where these positivist sentiments have been expressed, the most notable being of Diophantus's treatment of negative solutions. Typically translated as "absurd solution" he literally calls them atopon, a thing without topology. Being that these solutions had no straightforward geometric basis, they complicated the interpretation of polynomials as being meaningful in reality. Likewise the term al-Jabr, from which the latin algebrāica derives from, meaning completion or restoration is in this vein. Abdeljaouad, elaborates the senses of which in his colloquium on Maghrebi algebraists, who where then the most sophisticated. Jabr was then specifically a polynomial normalization tool, eliminating the negative terms and converting the degree into the monic form, the latter being a specific operation called Hatt.

The 19th century orientalist Franz Woepcke studied within this arabo-latin mathematical tradition. His insights, that the Liber augmenti et diminutionis was Latin translation of a previously lost work of Abu Kamil, and emphasizing the importance of the 10th century Persian mathematician Al-Karaji shed much light on the connections between Arabic scholars and later European ones. Adel Anbouba gives a translation of a key passage in al-Fakhrī, of the justification for increasingly non-geometric manipulations of equations. However both the field and in the French as translated by Abdeljaouad, champ ("domain") interpretation of ḥaddah remain obscure. Abdeljaouad cites the original text as:

إعلم أن التصرف في المعلومات بأنواع التصرف يحفظها في حدها، وكذلك المجهول يحفظ نفسه في حده عند التصرف فيه. ومعنى ذلك أن يكون أبدا مجهولا ما لم يقابل.

Thus paraphrasing Anbouba's translation: "That operating in the boundaries of knowns keeps them in this boundary no matter what the operation."

This brings up the natural question of what was considered the proper inverse of the natural numbers, and for that we need only look into the second sense of jabr - to bring to unity. This is philosophically clean when we look at the treatment of μονάς or unity in the broader Neopythagorean tradition that was common in the Hellenistic near east. It seems possible that Diophantus had a complete work on fractions, called Moriastica (On the parts), which is attested only once in the neoplatonic scholia. Klein in his Greek Mathematical Thought speculates that this was a proper book based on the reference to it's name, and translates the commentary, which I reproduce here:

οὕτως ὁ Διόφαντος ἐν τοῖς Μοριαστικοῖς· μόρια γὰρ εἰς ἔλαττον τῶν μονάδων πρόοδον εἰς τὸ ἄπειρον
ἀριθμοῦ καὶ μορίων μεθόριον

So Diophantus states in his Moriastica, for fractions progress in diminution carried to infinity. [Unity therefore forms] the borderline between number and fractions.

And from definition VI of the Arithmetica itself:
Τῆς οὖν μονάδος ἀμεταθέτου οὔσης καὶ ἑστώσης ἀεί In its character as measure, the unity itself is not affected by any partition.

Thus in Diophantus's syncopated notation, the identity is most well expressed as$$\overset{\circ }{\text{M}}\equiv\varsigma \cdot \varsigma^\chi$$Likewise the identity element in a monoid is defined as there existing an element for all members of a set where the communitive product of the element with the member equals the member. We can then express this in terms of an inverse semigroup with the single axiom: $$\exists e\forall a: e \equiv a \cdot a^{-1}$$

https://proofwiki.org/wiki/Book:William_Frend/The_Principles_of_Algebra https://archive.org/details/principlesalgeb00frengoog/page/n17/mode/2up Father in law of Augustus de Morgan Yet this is attempted by algebraists, who talk of a number less than nothing, of multiplying a negative number into a negative number and thus producing a positive number, of a number being imaginary.... like many other figments, it finds the most strenuous suporters among those who love to take things upon trust, and hate the labour of a serious thought.

Without having fully overcome their difficulties with irrational and negative numbers the Europeans added to their problems by blundering into what we now call complex numbers. They obtained these new numbers by extending the arithmetic operation of square root to whatever numbers appeared in solving quadratic equations by the usual method of completing the square. https://archive.org/details/mathematicalthou0000unse/page/252/

The language of algebraica https://sci-hub.ru/https://link.springer.com/article/10.1007/s10781-007-9026-4

CUT-AND-PASTE GEOMETRY https://sites.unipa.it/grim/Thesis_Malisani_06_engl.pdf

where we

https://eudml.org/doc/274996

Abdeljaouad 2002

Complex number - racine imaginaire https://www.gutenberg.org/ebooks/26400

https://en.wikipedia.org/wiki/The_Analyst

https://ru-wikipedia-org.translate.goog/wiki/Геометрия_(Декарт)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en&_x_tr_pto=wapp Thomas Harriot, Albert Girard https://lux.lawrence.edu/cgi/viewcontent.cgi?article=1174&context=luhp https://en.wikipedia.org/wiki/Albert_Girard

https://en.wikipedia.org/wiki/Euler's_formula https://msuweb.montclair.edu/~bald/files/LeibBernLogs.pdf.

https://academic.oup.com/book/40020/chapter-abstract/340368950?redirectedFrom=fulltext https://ericsteinhart.com/articles/striving.pdf

https://en.wikipedia.org/wiki/Residue_theorem

Zero

Three types of zero: placeholder zero, completion zero, and complete zero.

Completion Zero:

https://sci-hub.ru/https://doi.org/10.2307/3854111 Ever since it was brought the the awareness of western scholars, problem fourteen of the Moscow papyrus has fascinated modern mathematicians. It presents the solution of the pyramidal frustum problem in the form $(a^2+ab+b^2)\frac{1}{3}$, an identity that could have not been straightforwardly derived from the difference of volumes without the use of algebra. Hence the question Battiscombe Gunn, and T. Eric Peet attempt to solve is how

https://sci-hub.ru/https://doi.org/10.2307/3854215

https://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrus#Content#79 Problem 79 of the Rhind papyrus presents an geometric progression which is quite motivating

Seven houses, 49 cats, 343 mice, 2401 ears of spelt, 16807 hekats

$$\sum_{k=1}^n 7^k=7\left(1+\sum_{k=1}^{n-1} 7^k\right)$$

The number seven had great significance to the Egyptians, and had many references to https://sci-hub.ru/10.2307/41213913

Maat https://en.wikipedia.org/wiki/Maat https://pure.bond.edu.au/ws/portalfiles/portal/25238910/The_ancient_Egyptian_concept_of_Maat_Reflections_on_social_justice_and_natural_order.pdf https://dn720004.ca.archive.org/0/items/saoc-57.-the-presentation-of-maat.-ritual-and-legitimacy-in-ancient-egypt/SAOC 57. The Presentation of Maat. Ritual and Legitimacy in Ancient Egypt.pdf https://www.jstor.org/stable/j.ctv2c5qbh5

Ptah https://en.wikipedia.org/wiki/Shabaka_Stone#CITEREFBodine2009

Creation, according to the Shabaka Stone, was both a spiritual or intellectual creation as well as a physical one. It was through the divine heart (thought) 79 and tongue (speech/word) of Ptah as the great causer of something to take shape in the form of the physical agent of creation Atum, through which everything came forth. https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=1131&context=studiaantiqua

Their function par excellence, however, is in the creation itself. The creator first perceives the world as a concept: “I became effective in my heart ... I surveyed in my heart by myself” (Text 9, 12/18). He then gives reality to that perception through creative speech, “Annunciation, that spoke in the darkness” (CT VII 481g):

I am the one who made what is and caused what was not to develop: when I spoke, Annunciation developed. (CT IV 145b-c) https://archive.org/details/GenesisInEgyptThePhilosophyOfAncientEgyptianCreationAccountsJamesP.AllenYaleEgyptologicalSeminar1988/page/39/mode/1up

$$\frac{1}{2^k}...1...{2^k}$$

Constructing the Universe by Michael S Schneider https://www.youtube.com/watch?v=wciL_yCmZe8

https://philarchive.org/archive/STRTLO-21 https://www.udc.gal/grupos/ln/investigadores/LopezPalmaPub/%20HLP_EgFract.pdf https://en.wikipedia.org/wiki/Ancient_Egyptian_multiplication https://en.wikipedia.org/wiki/Egyptian_fraction

We can first start with trying to develop an intuition of how the ancient Egyptians conceptualized volume. Hoffman, in his excellent rebuttal of mathematical hindsight, repudiates the misconception that there was a proper zero in Egyptians thought.

The power rule of the derivative typically does not generate a series that satisfies the criteria for a Sheffer sequence. However the derivative of a perfect square pyramid, which gives cross-sectional volume, produces a clue of how the power rule is normalized. $$x^2=D\frac{x^3}{3}$$ In this formulation $$y= \frac{x^3}{3}$$ $$\ln y = 3\ln x - \ln 3$$ $$\frac{dy}{y} = 3\cdot\frac{dx}{x}$$ $$\frac{y}{dy} = \frac{1}{3}\cdot\frac{x}{dx}$$ $${y} = \frac{x}{3}\cdot\frac{dy}{dx}$$ $$\frac{x^3}{3} = \frac{x}{3}\cdot\frac{dy}{dx}$$ This then proves this derivative satisfies a trivial Pfaffian chain of length 1: $$\boxed{f_1 = \frac{x^3}{3},\qquad \frac{df_1}{dx} = x^2}$$ Recall that the Taylor series of $e^a$ is: $$e^a = \sum_{k=0}^{\infty}\frac{a^k}{k!} = 1 + a + \frac{a^2}{2!} + \frac{a^3}{3!} + \cdots$$ We can then use this to construct an (n+1) length chain, with the last term being a remainder equal to zero. Like in the square pyramid, we can normalize each term by $n!$ so that the scalar term cancels out: $$D,p_n(x) = D\left(\frac{x^n}{n!}\right) = \frac{n,x^{n-1}}{n!} = \frac{x^{n-1}}{(n-1)!} = p_{n-1}(x)$$ Appell sequence

Then the recurrence relations $Dp_n(x)=p_{n-1}(x),,, D^{n+1}p_{n}=0$ holds for all $n$. Under umbral composition $(p_n\circ q)(x)=\sum a_{n,k}, q_k(x)$, this allows us to construct the shift operator: $$p_n(x+a) = \sum^n p_{n-k}(a)\cdot p_{k}(x) = \sum^n \frac{a^kD^k}{k!}\cdot p_n(x)=e^{aD}\cdot p_n(x)$$ Furthermore given a finite-difference of $p_n$ and $t=b-a:$ $$\frac{b^n-a^n}{b-a}=\frac{p_n(a+t)-p_n(a)}{t}=\frac{1}{b-a}\sum_{k=1}^{n}\frac{t^{k}D^k}{k!} p_n(a)=\frac{e^{tD}-1}{t},p_n(a)=\sum_{k=1}^{n}\binom{n}{k}a^{n-k}t^{k-1}$$ https://hapax.github.io/mathematics/physics/hacker/binomial/ https://math.stackexchange.com/questions/4688533/an-approximation-to-a-binomial-looking-formula https://en.wikipedia.org/wiki/Exponentiation_by_squaring

If we expand this series at $a=0$, with $x^{-m}$

higher power of m, more derivatives, further away from linearity instead of closer fractional powers like ln(z) have infinite monodromy

PD(α,θ) for various α,θ pairs

Derivative increases the winding number Fractional powers decrease the winding number by 1/a

$D\ln x = x^{-1}$

$$\int_a^b \frac{h}{b-a},x^2,dx = \frac{h}{b-a}\cdot\frac{x^3}{3}\Big|_a^b = \frac{h}{3(b-a)}(b^3-a^3) = \frac{h}{3}(a^2+ab+b^2)$$

$$\lim_{b\to a} \frac{1}{3}\cdot\frac{b^3 - a^3}{(b-a)}=a^2$$

the frustum is the solid of revolution/extrusion whose volume derivative recovers cross-sectional area). So it's a beautiful structural analogy, cross-sectional area growth and the derivative of $x^3$ share the same algebraic skeleton $$V_{\text{frustum}} - V_{\text{pyramid}} = \frac{1}{3}(a^2+ab+b^2)h - \frac{1}{3}a^2h = \frac{1}{3}(ab + b^2)h$$Let s and d be the Girard–Newton identities of p=(ab): $q=a+b$ and $s=b-a$, then: $$2a=q-s, \qquad a+2b=q+s,$$ $$\frac{1}{3}(ab+b^2)h = \frac{1}{3}(a+b)hb = \frac{1}{3},bsh,$$

Let the partial derivatives of $\frac{1}{3}(a^2+ab+b^2)$ be the two roots of a new g(x), and let $s=a+b$ and $d=b-a$, then: $$2a+b = s-d, \qquad a+2b=s+d,$$ $$(3x - h(2a+b)\big)\big(3x - h(a+2b) = \left(x-\frac{hs}{3}\right)^2 = \frac{h^2d^2}{9}.$$

That last form is the nicest: it's a perfect square shifted by $\pm hd/3$ — the two roots sit symmetrically around the midpoint value $\frac{h(a+b)}{3}$, spread apart by $\frac{h(b-a)}{3}$ on each side. That matches the geometric picture too: the trisection points are symmetric about the interval's midpoint, spaced $\frac{b-a}{3}$ apart from it.

$f(a,b)$ is symmetric under swapping $a\leftrightarrow b$, and the derivatives reminded me of the

In the cyclotomic expression it doesn't show the symmetry property nicely

$$|a-b\omega|^2$$ Define $\tau(z) = -\omega\cdot\bar z$, given $z=a-b\omega$: $$\tau(z) = -\omega(a-b\omega^2) = -a\omega + b\omega^3 = b - a\omega$$ Since $\tau$ is a composition of multiplication by a unit ($-\omega$, which has $N(-\omega)=1$) with conjugation, multiplicativity gives $N(\tau(z)) = N(-\omega)\cdot N(\bar z) = 1\cdot N(z) = N(z)$. Therefore:

$$f(a,b) = N(a-b\omega), \qquad \tau(a-b\omega) = b-a\omega, \qquad N\circ\tau = N$$ $$a^2+ab+b^2 = (a+b)^2 - ab$$

https://en.wikipedia.org/wiki/Löschian_number https://onlinelibrary.wiley.com/doi/epdf/10.1111/j.1538-4632.1975.tb01054.x

Derivation: Let $p = a+b$ and $q = ab$ be the elementary symmetric polynomials.

Since $a^2+b^2 = (a+b)^2 - 2ab = p^2 - 2q$, we get:

$$a^2+ab+b^2 = (a^2+b^2) + ab = (p^2 - 2q) + q = p^2 - q$$

So in terms of $p=a+b$ and $q=ab$:

$$a^2+ab+b^2 = p^2 - q$$

This New Pyramid Theory Explains the Missing Evidence https://www.youtube.com/watch?v=h5kWDOuY2Uo

https://en.wikipedia.org/wiki/0#Ancient_Near_East 𓄤 https://en.wiktionary.org/wiki/𓄤 https://carlaberenice.substack.com/p/mastering-hieroglyphs-ii-basic-syntax https://en.wikipedia.org/wiki/Prophecy_of_Neferti Representing a sheep's heart and windpipe or esophagus in cross-section.

https://epub.ub.uni-muenchen.de/27695/ https://en.wikipedia.org/wiki/Stauros https://en.wikipedia.org/wiki/Epistle_of_Barnabas#Midrash_and_gematria https://en.wikipedia.org/wiki/Tau_cross#Tau_representing_an_execution_cross https://en.wikipedia.org/wiki/Dumuzid Babylonian influence exaggerated, few Jews returned from Babylonia Ezrahite Reforms https://en.wikipedia.org/wiki/Cyrus_Cylinder#Similarities_with_other_royal_inscriptions https://en.wikipedia.org/wiki/Cyrus_Cylinder

https://www.thetorah.com/article/sacrificing-a-lamb-in-egypt

https://lexundria.com/hdt/2.42/mcly https://sacred-texts.com/cla/hh/hh2040.htm The Thebans then do not sacrifice rams but hold them sacred for this reason; on one day however in the year, on the feast of Zeus, they cut up in the same manner and flay one single ram and cover with its skin the image of Zeus, and then they bring up to it another image of Heracles. This done, all who are in the temple beat themselves in lamentation for the ram, and then they bury it in a sacred tomb.

https://en.wikipedia.org/wiki/Regula_falsi https://en.wikipedia.org/wiki/Method_of_exhaustion https://en.wikipedia.org/wiki/Eudoxus_of_Cnidus https://en.wikipedia.org/wiki/Cavalieri's_principle

Method of Exhaustion Euclidean Algorithm (ἀνθυφαίρεσις) Intermediate Value Theorem Riemann summation Minkowski dimension https://en.wikipedia.org/wiki/Cauchy_principal_value

Communative Hyperoperations https://ncatlab.org/nlab/show/distributive+lattice https://ncatlab.org/nlab/show/rig https://en.wikipedia.org/wiki/Tropical_semiring https://en.wikipedia.org/wiki/Tropical_geometry https://en.wikipedia.org/wiki/Tropical_analysis https://mathtable.com/smtf/SMTF32__max_plus_algebra__201505.pdf https://arxiv.org/pdf/math/0112050 https://www.jstor.org/stable/2007124?seq=1

Approximation Theory https://en.wikipedia.org/wiki/Nørlund–Rice_integral#Poisson–Mellin–Newton_cycle https://algo.inria.fr/flajolet/Publications/Slides/Mellin-slides08.pdf https://en.wikipedia.org/wiki/Generating_function#Lambert_series https://en.wikipedia.org/wiki/Glasser's_master_theorem#A_special_case:_the_Cauchy–Schlömilch_transformation https://math.stackexchange.com/questions/3946769/a-simple-proof-for-glasser-int-infty-infty-fx-a-x-dx-int-infty

https://en.wikipedia.org/wiki/Padé_approximant https://math.stackexchange.com/questions/1046321/approximating-log-x-with-roots

Logarithmic Combinatorial Structures https://sciarium.com/file/242770/ https://en.wikipedia.org/wiki/Chinese_restaurant_process https://en.wikipedia.org/wiki/Poisson-Dirichlet_distribution https://en.wikipedia.org/wiki/Residue_number_system https://en.wikipedia.org/wiki/Galton–Watson_process https://en.wikipedia.org/wiki/Turán–Kubilius_inequality https://en.wikipedia.org/wiki/Mertens'_theorems https://en.wikipedia.org/wiki/Euler's_constant https://en.wikipedia.org/wiki/Khinchin's_constant

As such there is evidence that this did not symbolize zero, but unity Chinese counting rods

Placeholder zero Calculation of the Julian calender Mayan calender

Complete Zero Thus then the Mayan, closesr to the Hindu concept of nothingness https://en.wikipedia.org/wiki/Indian_mathematics#Jain_mathematics_(400_BCE_–_200_CE) https://en.wikipedia.org/wiki/Śūnyatā cyclical worldview, like the Babylonian https://z-library.sk/book/JveBVKg5OE/zero-the-biography-of-a-dangerous-idea.html If yesterday and tomorrow are days with the numeral 1, then necessarily “today” is Nik, the Maya zero. It is noteworthy to point out that in Spanish or English, there are no expressions of time that go any further than two days into the past or the future; however, in Mayan languages, using the numerical construction on either side of Nik allows us to express these timeframes, past and future, ad infinitum.

To express the word “today” in the Q’anjob’al Mayan language, we say “nani,” which translates to “today,” or “center.” As a matter of fact, today is at the center of the past and the future. In the K’iche’ Mayan language, “today” is expressed as “kamik,” with a literal translation meaning “death.” The philosophical connection between the word “kamik” and = Nik, is that, when a person dies, the count of the days of his or her life on Earth ends, and a new count begins from the day the person died.

https://baas.aas.org/pub/2021n1i336p03/release/2

$2_\infty$ = (2, 4, 6,..., $2_\infty$) $K \geq 1, K \gg 2_\infty$, where $2_\infty$ is a really large even number $$

2_\infty^{-nk}=\begin{cases} \implies \text{another even number} \ \implies 1 \ \implies \text{becomes non-integer} \ \end{cases} $$

4, 2, 1, 0.5, 0.25

https://en.wikipedia.org/wiki/P-adic_valuation

https://math.stackexchange.com/questions/353779/congruence-modulo-infinity https://en.wikipedia.org/wiki/Broyden's_method https://math.unm.edu/~vageli/courses/Ma576/Broyden/dennis_more74.pdf

Finite neural nets only represent a subset of

Deep equilibrium networks are based on an analytical quasi-Newton optimization of the Jacobian. Dennis, Jr. and Jorge J. More give the form: $x_{k+1} = x_k - B_k^{-1}F(x)_k$ where ${B_k}$ is a sequence of nonsingular matrices

Yagzhev maps are likewise is of the form \Beta where \Beta

$X_\beta=W_Y + \alpha Y=mx+b$

$f:C_{n}→A^n_C +Sym^3(C^3)$

The Kalman error $kerr(x)$ is then a scale factor of the error in the determinant and reaches infinity when the error is maximized and hence $kerr = 0$

then been shown to prove super-linear learning, however these are in specific conditions that guarantee a fixed point. These conditions represent the stabilization criteria of an nth dimensional network. It can shown that any finite net is trivially recoverable from a DEQ, and DEQs provide containment, $\text{DNN} \subseteq \text{DEQ}$

Any infinite recurrent net is

Any sufficiently expressive Jacobian (n>2) doesn't have guaranteed unique solutions (Jacobian Conjecture)

y Let there be a smooth transform,

Given the local independence of disciplines there is nothing in Discipline A that posits AB necessarily. However there is generalization that allows you to transform from Knowledge A to Knowledge B. If that holds for other disciplines as well such as BC, under some global consistency of reality you can assume transitivity. Formally, let S be the set of disciplines. Unless you believe in the Boltzmann Brain, reality enforces consistency globally, so for every pair $S_n, S_m$ there exists a transformation $S_{nm}$ where: $$S_{km} \circ S_{nk} = S_{nm} \text{ for all } n,k,m$$ $$\omega|{S_n}^{-1} = S{nm}^{-1}\circ\omega|{S_m}^{-1}$$ $$ J_F(x) = J{S_{km}}\big(S_{nk}(x)\big),J_{S_{nk}}(x) - J_{S_{nm}}(x) \equiv 0 \quad \text{for all } x $$ S_n -> T, f_m:S_M \right arrow T such that S \neq f^_1 \circ f_n

Jordan Curve Theorem https://arxiv.org/abs/1404.0556 https://www.youtube.com/watch?v=wtPkvAsGGIA

note the infinitesimal interpretation https://www.youtube.com/watch?v=hAxlK8W80Mg

https://www.youtube.com/watch?v=nn5Dd-1BXH4

https://mathoverflow.net/questions/8521/nice-proof-of-the-jordan-curve-theorem https://webhomes.maths.ed.ac.uk/~v1ranick/jordan/maehara.pdf https://en.wikipedia.org/wiki/Pfaffian_function

If there is not an automorphism within S[X] then there can be no invariant boundary on it.

Given a global invariant boundary on an automorphic subset of S[x]: A hallucination is the exclusion of the boundary on S[X], S[X]\n which denotes a fact properly outside S[X] Therefore there exists at least one n ∈ S[X] under one ⟺_nm that n ∈ S[X]\n under ⟺_ij, There exists a ⟺_XY that scales the boundary onto that n ∈ S[X]\n

$\omega|{S_m}^{-1}\circ \omega|{S_n} = S_{n1}^{-1}\circ$

https://openscholarship.wustl.edu/cgi/viewcontent.cgi?article=1697&context=cse_research https://arxiv.org/pdf/1909.01377

https://en.wikipedia.org/wiki/Diophantus_II.VIII https://en.wikipedia.org/wiki/Chinese_remainder_theorem#Hermite_interpolation https://en.wikipedia.org/wiki/Asymptotic_equipartition_property https://link.springer.com/book/10.1007/978-1-4613-8174-7 "The discussion of Greek and Arabic interpolations is entirely new, as is the reconstruction of the history of the Arithmetica from Diophantine to Arabic times."

Ostrowski Expansions

A counterexample to a conjecture of Bass, Connell and Wright https://eudml.org/doc/210593

https://raofa-sinfin.greyc.fr/wp-content/uploads/2021/12/ValerieBerthe-RAPA.pdf

https://dmtcs.episciences.org/450/pdf https://arxiv.org/pdf/1605.07992

https://en.wikipedia.org/wiki/Hermite's_problem https://arxiv.org/pdf/2101.12707

$\omega\bmod\tau$

https://raofa-sinfin.greyc.fr/wp-content/uploads/2021/12/ValerieBerthe-RAPA.pdf

rabbit hole linearity

https://pub.math.leidenuniv.nl/~smitbde/papers/2003-de_smit-lenstra-escher.pdf https://www.youtube.com/watch?v=ldxFjLJ3rVY&list=WL&index=22 The ratio of two variables is constant under scale and rotation.