Notes

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Math

Discovery

Andrius Kulikauskas

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Symplectic manifolds

Time: A moving point is a line, a moving line is a plane, a moving plane is a volume... Time is the addition of a scalar (from a field), thus the addition of choice. Time relates affine and projective space. Compare time with space. A moving "center" is a point: the center is what moves, thus what has time.

Function can be partial, whereas a permutation maps completely.

Derangements. Interpret? Not likely...

Symmetry group relates:

  • Algebraic structure, "group"
  • Analytic (recurring activity) transformations

Axiom of infinity - can be eliminated - it is unnecessary in "implicit math".

The factoring (number of simplexes n choose k - dependent simplex) x (number of flags on k - independent Euclidean) x (number of flags on n-k - independent Euclidean) = (number of flags on n)

What kind of conjugation is that?

Relate triangulated categories with representations of threesome

Kan extension - extending the domain - every concept is a Kan extension https://en.wikipedia.org/wiki/Kan_extension

Express the link between algebra wnd analysis in terms of exact sequences and Kan extensions. Explain why category theory not relevant for analysis.

What is Mathematical Music Theory?

Consider the connection between walks on trees and Dynkin diagrams, where the latter typically have a distinguished node (the root of the tree) from which we can imagine the tree being "perceived". There can also be double or triple perspectives.

The mind is augmented through the "symmetric group" which is the system that augments our imagination.

Matematikos įrodymo būdai

  • 6 matematikos irodymo budai skiriaisi nuo issiaiskinimo budu taciau kaip jie susiję

Apibrėžti "gebėjimus" ir kaip matematinis mąstymas suveda skirtingus gebėjimus suvokti kelis, keliolika, keliasdešimts, tūkstančius ir t.t. daiktų

Kodėl yra tiek daug būdų įrodyti Pitagoro teoremą?

Gramatika visada turi prasmę. Matematika yra tai, kas pavaldu logikai, bet nebūtinai turi prasmę.

Variables

  • A variable is an "atom" of meaning as in my paper, The Algebra of Copyright, which can be parsed on three different levels, yielding four levels and six pairs of levels.

Apibendrinimas

  • Protas apibendrina. Kaip nagrinėti apibendrinimą? Suvokti neurologiškai (arba tinklais). Jeigu keli pavyzdžiai (ar netgi vienas) turi tam tikras bendras savybes, tada tas apibendrintas savybes gali naujai priskirti naujoms jų apibudintoms sąvokoms.
  • Apibendrinimas yra "objekto" kūrimas.

Differentiability of a complex function means that it can be written as an infinite power series. So differentiation reduces complex functions to infinite power series. This is analogous to evolution abstracting the "real world" to a representation of it. And differentiation is relevant as a shift from the known to focus on the unknown - the change.

How do special rim hook tableaux (which depend on the behavior of their endpoints) relate to Dynkin diagrams (which also depend on their endpoints)?

Study the duality between 1^N and N in symmetric functions (Young tableaux) but also Catalan numbers, etc.

Montažas ir geometrinės permainos:

  • Shear: sideshot

Mandelbort

  • Mandelbrot aibė skiria vidines ir išorines veiklas, besilaikančias erdvės ir nesilaikančios jokios erdvės.
  • Įsivaizduoti, kaip Mandelbrot aibės transformacija veikia visą plokštumą arba vieną jos kampelį.
  • Catalan numbers foster a duality between horizontal concatenation ()()() and vertical embedding ((())). Šis dualizmas taip pat sieja visko savybes Visaką priima (Kaip) ir Be aplinkos (Kodėl). Palyginti ir su žaidimų gramatika, su medžiai ir sekomis.

Darij Grinberg http://www-users.math.umn.edu/~dgrinber/ http://www.cip.ifi.lmu.de/~grinberg/about/rs.pdf See: Combinatorics and the field with one element. Witt vectors - p-adic integers

Matematikos savokų pagrindas yra dvejybinis-trejybinis - operacijos jungia du narius trečiu nariu. Matricų elementai sieja du narius ir išgauna trečią. Kategorijų teorija panašiai. O geometrija lygiaverčiai sieja tris narius trikampiais, įvairiai suprastais. Tad tai paaiškintų geometrijos svarbą.

Jiri Raclavsky - Frege, Tichy - Two-dimensional conception of inference. Inference rules operate on derivations. Go from one truth to another truth, not from one assumption to another assumption.

Intrinsic ambiguity of propositions - every proposition is a general rule, which can be questioned or applied.

What is the connection between the universal grammar for games and the symmetric functions of the eigenvalues of a matrix?

An example of extending the domain: Lie algebra (infinitesimal) vs. Lie group (broader domain).

Duality breaking (for slack) - disconnecting the local and the global - for example, defining locally Euclidean spaces - in lattice terms, as a consequence of limiting processes, disconnecting the inf from the sup, breaking their duality.

Collect examples of the arithmetic hierarchy such as calculus (delta-epsilon), differentiable manifolds, etc.

Collect examples of symmetry breaking.

Partial derivative - formal (explicit) based on change in variable, total derivative - actual (implicit) based on change in value.

Measurement (crucial in physics) can be defined in terms of covectors, as being dual to vectors. Covectors can be thought to point in the same direction as vectors - they are complements of each other with regard to the inner product. This duality is thus fundamental to the concept of measurement.

The vector and the covector divide a scalar field into its local variation (given by the vector) and its global scaling (given by the covector) which together give the value of the scalar. Thus vector and covector define the duality of local and global extremes which come together as the scalar field.

A vector is 1-dimensional (and its dimension) and its covector is n-1 dimensional (it is normal to the vector). In this sense they complement each other.

Vectors are described in terms of partial derivatives (based on the local coordinate systems) whereas covectors are described in terms of (total) forms dx.

An example of variables: a function Phi may be based on an point in the manifold whereas its coordinates may be based on a particular, explicit, specific chart. See Penrose page 186.

Symplectic form is skew-symmetric - swapping u and v changes sign - so it establishes orientation of surfaces - distinction of inside and outside - duality breaking. And inner product duality no longer holds.

Duality breaking allows that God is good and not bad. Because we want to break the duality of good and bad, increasing and decreasing slack. Orientation is a complete, absolute, total distinction between inside and outside, their complete segregation and isolation. (In contrast to the yin-yang symbol.) SO it is highly tenuous - it can break at any single point - but it can eternally grow more weighty.

Momentum can be attributed to an individual particle (as its change) but it can also be attributed to the entire system (as its change). And also, changing the momentum of a particular particle can change when (and whether) we will come to a particular state of the system. In particular, the particles are interconnected and so that makes for a complicated relation between the time evolution of each particle (in terms of its position) and the time evolution of the system. This can be compared to a computer program which may change the order of its instructions.

What do inside and outside mean in symplectice (Hamiltionian, Lagrangian) mechanics?

Geometry of Classical Groups over Finite Fields and Its Applications, Zhe-xian Wan

Study the chaos of watersheds for the divisions of everything - the twosome, threesome, foursome, etc. Note how a "hill" arises (for example, with the fivesome) and how that hill becomes a division into two (with the sevensome).

  • Category theory models perspectives and attention shifting. (Or thoughts as objects?)
  • Category of perspectives: stepping-in and stepping-out as adjoints? there exists vs. for all?
  • Eugenia Cheng: Mathematics is the logical study of how logical things work. Abstraction is what we need for logical study. Category theory is the the math of math, thus the logical study of the logical study of how logical things work.
  • Category theory shines light on the big picture. Perspectives shine light on the big picture (God's) or the local picture (human's).
  • Difference between set and vector space (or list?) Set has empty set, but vector space has zero instead of the empty set. So there are no functions into the empty set, but there are functions into zero. Vector space is not distributive. Can't just take the union, need to take the span. Thus R and (S or T) is not equal to (R and S) or (R and T). A line and a plane is not a line and line or a line and line.
  • Trikampis - riba (jausmai) - simplektinė geometrija.
  • Lygmuo Kodėl viską išsako ryšiais. O tas ryšys yra tarpas, kuriuo išsakomas Kitas. Kategorijų teorijoje panašiai, tikslas yra pereiti iš narių (objektų) nagrinėjimą į ryšių (morfizmų) nagrinėjimą.

Fundamental theorems

  • elementary symmetric functions are analogous to prime factorizations of numbers - monomial symmetric functions are analogous to the numbers as such ("natural" basis - "natural" numbers +1)

(x-a) x is unknown (conscious question); a is know (unconscious answer); and the difference x-a gives the discrepancy. So a polynomial can be thought of as constructed as the product of the discrepancies - or it can be constructed as coefficients for powers. Note that EVERY polynomial with complex coefficients can be constructed as a product of such discrepancies. So it's a type of completeness. What makes that true?

Physics - moving backwards and forwards in time - is a (wasteful) duality. Why is it dual?

Entropy - think in terms of "common" and "uncommon" states. Uncommon states tend to common states simply because they are more frequent. But nature has forces (like gravity) that tend to create uncommon states. So we need to distinguish between the different forces at play and how they relate to what is common and uncommon.

Matematika mus moko, kaip besąlygiškumą reikšti sąlygomis, o tai įmanoma sąlygiškai.

Exponential function e^x is a symmetry from the point of view of differentiation - it is the unit element. And cos and sin are fourth roots. And look for more such roots.

Conjugation gives the ways of relabeling, renaming. For example, (132)(12)(123) relables 1 as 2 and 2 as 3 in (12) to get (23).

Raimundas Vidūnas, deleguotas priežastingumas

Counting (in Lie root system) can change to B, C, D only once! That puts a cap on the one end. Then the counting must continue on the other end, extending it. A second cap may not be put on that end. There cannot be a cycle.

Counting (by way of the simple roots) links + and - in a chain. x2-x1 etc.

Inner products are sesquilinear - they have conjugate symmetry - so as not to yield lopsided answers. If they yield a complex root as an answer, then one version should yield one root and the other version should yield the other root. In other words, in a complex field, the inner product should be thought of as yielding two answers - both answers - distinguished by the notation, left-right or right-left.

Mathematical induction can be thought of as a proof by contradiction where we have organized the conditions in an infinite sequence and we are looking at the first one to fail and proving that it doesn't fail.

An inner product on a vector space allows it to be broken up into vector spaces that complement each other, thus into irreducible vector spaces.

Matrices {$A=PBP^{-1}$} and {$B=P^{-1}AP$} have the same eigenvalues. They are simply written in terms of different coordinate systems. If v is an eigenvector of A with eigenvalue λ, then {$P^{-1}v$} is an eigenvector of B with the same eigenvalue λ.

Eigenvectors are the pure dimensions into which the action of a matrix (or linear transformation) can be decomposed.

What is the significance of a cube having four diagonals that can be permuted by S4?

Study how turning the counting around relates to cycles - finite fields.

Study the symmetry of functions like exponentials, trigonometric, etc by considering the derivative equation {$f^{(n)}=f$} for various integers n. And nonintegers n? Study using Taylor series expansions.

Kuom skaičius skiriasi nuo pasikartojančios veiklos - būgno mušimo?

  • A) veikla kažkada prasidėjo
  • B) kiekvienas skaičius laikomas nauju, skirtingu nuo visų kitų

briauna = skirtingumas

Special linear group has determinant 1. In general when the determinant is +/- 1 then by Cramer's rule this means that the inverse is an integer and so can have a combinatorial interpretation as such. It means that we can have combinatorial symmetry between a matrix and its inverse - neither is distinguished.

Determinant 1 iff trace is 0. And trace is 0 makes for the links +1 with -1. It grows by adding such rows.

Hyperoctahedral group is shuffling cards that can also be flipped or not (front or back).

Four geometries in terms of choices: paths - one-directional, lines - forwards and backwards, angles - left and right, oriented areas - inside and outside. What are the choices with regard to?

How does the expansion (x1 + ... + xm)N relate to the matrix of nonnegative integers? and how it yields pairs of Kostka matrices? (form and content)?

One-dimensional economic thinking is like linear functionals - the dual space of the multidimensional reality. In finite dimensions, the dual dual is the same as what we started. But in infinite dimensions not necessarily. Does this suggest that our life is infinite-dimensional, which is to say, it can't be captured by a one-dimensional shadow?

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Puslapis paskutinį kartą pakeistas 2018 spalio 08 d., 19:15
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