• ms@ms.lt
• +370 607 27 665
• My work is in the Public Domain for all to share freely.

Introduction E9F5FC

Understandable FFFFFF

Questions FFFFC0

Notes EEEEEE

Software

## Book.MathNotes istorija

2019 rugpjūčio 19 d., 21:13 atliko AndriusKulikauskas -
Pakeistos 455-457 eilutės iš
I dreamed of the binomial theorem as having an "internal view", imagined from the inside, which accorded with the "coordinate systems". And which interweaved with the external views to yield various "moments", given by curves on the plane, variously adjusted and transformed by the internal view.
į:
I dreamed of the binomial theorem as having an "internal view", imagined from the inside, which accorded with the "coordinate systems". And which interweaved with the external views to yield various "moments", given by curves on the plane, variously adjusted and transformed by the internal view.

Circle (three-cycle) vs. Line (link to unconditional) - sixsome - and real forms
2019 rugpjūčio 14 d., 11:39 atliko AndriusKulikauskas -
Pakeistos 453-455 eilutės iš
Einstein field equations - energy stress tensor - is 4+6 equations.
į:
Einstein field equations - energy stress tensor - is 4+6 equations.

I dreamed of the binomial theorem as having an "internal view", imagined from the inside, which accorded with the "coordinate systems". And which interweaved with the external views to yield various "moments", given by curves on the plane, variously adjusted and transformed by the internal view
.
2019 rugpjūčio 10 d., 21:35 atliko AndriusKulikauskas -
Pakeistos 451-453 eilutės iš
* Išsakyti grupės {$G_2$} santykį su jos atvirkštine. Ar ši grupė tausoja kokią nors normą?
į:
* Išsakyti grupės {$G_2$} santykį su jos atvirkštine. Ar ši grupė tausoja kokią nors normą?

Einstein field equations - energy stress tensor - is 4+6 equations.
2019 rugpjūčio 10 d., 13:51 atliko AndriusKulikauskas -
Pakeistos 448-451 eilutės iš
* How: inner logic. What: external view.
į:
* How: inner logic. What: external view.

* Išsakyti grupės {$G_2$} santykį su jos atvirkštine. Ar ši grupė tausoja kokią nors normą?
2019 rugpjūčio 10 d., 12:28 atliko AndriusKulikauskas -
Pakeistos 444-448 eilutės iš
[[https://www.amazon.com/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809 | The Princeton Companion to Mathematics]] VU Matematikos ir informatikos skaitykla
į:
[[https://www.amazon.com/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809 | The Princeton Companion to Mathematics]] VU Matematikos ir informatikos skaitykla

Yoneda Lemma
* Loss of info from How to What is equal to the Loss of info from "Why for What" to "Why for How".
* How: inner logic. What: external view.
2019 rugpjūčio 06 d., 22:26 atliko AndriusKulikauskas -
Pakeista 444 eilutė iš:
[[https://www.amazon.com/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809 | The Princeton Companion to Mathematics]]
į:
[[https://www.amazon.com/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809 | The Princeton Companion to Mathematics]] VU Matematikos ir informatikos skaitykla
2019 rugpjūčio 06 d., 22:23 atliko AndriusKulikauskas -
Pakeistos 442-444 eilutės iš
Ways of discovery in math: [[http://www.tricki.org | Tricki.org]]. [[https://gowers.wordpress.com/2010/09/24/is-the-tricki-dead/ | Overview by Timothy Gowers]].
į:
Ways of discovery in math: [[http://www.tricki.org | Tricki.org]]. [[https://gowers.wordpress.com/2010/09/24/is-the-tricki-dead/ | Overview by Timothy Gowers]].

[[https://www.amazon.com/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809 | The Princeton Companion to Mathematics]]
2019 rugpjūčio 06 d., 22:14 atliko AndriusKulikauskas -
Pakeistos 440-442 eilutės iš
[[https://www.dpmms.cam.ac.uk/~wtg10/ | Timothy Gowers' webpage]]
į:
[[https://www.dpmms.cam.ac.uk/~wtg10/ | Timothy Gowers' webpage]]

Ways of discovery in math: [[http://www.tricki.org | Tricki.org]]. [[https://gowers.wordpress.com/2010/09/24/is-the-tricki-dead/ | Overview by Timothy Gowers]].
2019 rugpjūčio 06 d., 21:25 atliko AndriusKulikauskas -
Pakeistos 436-440 eilutės iš
Mathematical induction - is it possible to treat infinitely many equations as a single equation with infinitely many instantiations? Consider Navier-Stokes equations.
į:
Mathematical induction - is it possible to treat infinitely many equations as a single equation with infinitely many instantiations? Consider Navier-Stokes equations.

[[https://www.mathpages.com/home/ | Kevin Brown]] collection of expositions of math

[[https://www.dpmms.cam.ac.uk/~wtg10/ | Timothy Gowers' webpage]]
2019 rugpjūčio 06 d., 16:58 atliko AndriusKulikauskas -
Pridėtos 421-422 eilutės:

[[https://terrytao.wordpress.com/2019/07/26/twisted-convolution-and-the-sensitivity-conjecture/ | Terrence Tao: Twisted Convolution and the Sensitivity Conjecture]]
2019 rugpjūčio 05 d., 10:02 atliko AndriusKulikauskas -
Pakeistos 432-434 eilutės iš
Does induction prove an infinite number of statements or their reassembly into one statement with infinitely many realizations? It proves the parallelness of intuitive meaningful stepped in and formal stepped out.
į:
Does induction prove an infinite number of statements or their reassembly into one statement with infinitely many realizations? It proves the parallelness of intuitive meaningful stepped in and formal stepped out.

Mathematical induction - is it possible to treat infinitely many equations as a single equation with infinitely many instantiations? Consider Navier-Stokes equations
.
2019 rugpjūčio 04 d., 13:24 atliko AndriusKulikauskas -
Pakeistos 430-432 eilutės iš
Axiom of forgetfullness.
į:
Axiom of forgetfullness.

Does induction prove an infinite number of statements or their reassembly into one statement with infinitely many realizations? It proves the parallelness of intuitive meaningful stepped in and formal stepped out
.
2019 rugpjūčio 04 d., 08:57 atliko AndriusKulikauskas -
Pakeistos 428-430 eilutės iš
Peano why can't have natural numbers have two subsets, a halfline from 0 and a full line.
į:
Peano why can't have natural numbers have two subsets, a halfline from 0 and a full line.

Axiom of forgetfullness
.
2019 rugpjūčio 04 d., 08:53 atliko AndriusKulikauskas -
Pakeistos 426-428 eilutės iš
Equality holds for both value and type, amount and unit. Peano axiom.
į:
Equality holds for both value and type, amount and unit. Peano axiom.

Peano why can't have natural numbers have two subsets, a halfline from 0 and a full line
.
2019 rugpjūčio 04 d., 08:42 atliko AndriusKulikauskas -
Pakeistos 424-426 eilutės iš
Induction step by step is different than the outcome, the totality, which forgets the gradation.
į:
Induction step by step is different than the outcome, the totality, which forgets the gradation.

Equality holds for both value and type, amount and unit. Peano axiom
.
2019 rugpjūčio 04 d., 08:32 atliko AndriusKulikauskas -
Pakeistos 422-424 eilutės iš
Relate methods of proof and discovery, 3 systemic and 3 not.
į:
Relate methods of proof and discovery, 3 systemic and 3 not.

Induction step by step is different than the outcome, the totality, which forgets the gradation
.
2019 rugpjūčio 04 d., 07:33 atliko AndriusKulikauskas -
Pakeistos 420-422 eilutės iš
How is a [[https://en.wikipedia.org/wiki/Boolean_function | Boolean function]] similar to a linear functional?
į:
How is a [[https://en.wikipedia.org/wiki/Boolean_function | Boolean function]] similar to a linear functional?

Relate methods of proof and discovery, 3 systemic and 3 not.
2019 rugpjūčio 01 d., 23:16 atliko AndriusKulikauskas -
Pakeistos 418-420 eilutės iš
[[https://www.quantamagazine.org/mathematician-solves-computer-science-conjecture-in-two-pages-20190725/ | Complexity measures for Boolean functions]].
į:
[[https://www.quantamagazine.org/mathematician-solves-computer-science-conjecture-in-two-pages-20190725/ | Complexity measures for Boolean functions]].

How is a [[https://en.wikipedia.org/wiki/Boolean_function | Boolean function]] similar to a linear functional?
2019 rugpjūčio 01 d., 22:59 atliko AndriusKulikauskas -
Pakeistos 416-418 eilutės iš
The analysis in a Lie group is all expressed by the behavior of the epsilon.
į:
The analysis in a Lie group is all expressed by the behavior of the epsilon.

[[https://www.quantamagazine.org/mathematician-solves-computer-science-conjecture-in-two-pages-20190725/ | Complexity measures for Boolean functions]]
.
2019 liepos 27 d., 17:07 atliko AndriusKulikauskas -
Pakeistos 414-416 eilutės iš
[[https://www.youtube.com/watch?v=bFZWarP2Ef4 | Galois, Grothendieck and Voevodsky - George Shabat]]
į:
[[https://www.youtube.com/watch?v=bFZWarP2Ef4 | Galois, Grothendieck and Voevodsky - George Shabat]]

The analysis in a Lie group is all expressed by the behavior of the epsilon.
2019 liepos 25 d., 19:46 atliko AndriusKulikauskas -
Pakeistos 412-414 eilutės iš
Heisenberg uncertainty principle - the slack in the vacuum - that allows the borrowing of energy on brief scales. Compare with coupling (of position and momentum, for example).
į:
Heisenberg uncertainty principle - the slack in the vacuum - that allows the borrowing of energy on brief scales. Compare with coupling (of position and momentum, for example).

[[https://www.youtube.com/watch?v=bFZWarP2Ef4 | Galois, Grothendieck and Voevodsky - George Shabat]]
2019 liepos 20 d., 17:45 atliko AndriusKulikauskas -
Pakeista 412 eilutė iš:
Heisenberg uncertainty principle - the slack in the vacuum - that allows the borrowing of energy on brief scales
į:
Heisenberg uncertainty principle - the slack in the vacuum - that allows the borrowing of energy on brief scales. Compare with coupling (of position and momentum, for example).
2019 liepos 20 d., 17:44 atliko AndriusKulikauskas -
Pakeistos 410-412 eilutės iš
Exchange particles - gauge bosons.
į:
Exchange particles - gauge bosons.

Heisenberg uncertainty principle - the slack in the vacuum - that allows the borrowing of energy on brief scales
2019 liepos 20 d., 17:26 atliko AndriusKulikauskas -
Pakeistos 408-410 eilutės iš
* [[https://www.youtube.com/watch?v=zS-LsjrJKPA | Particle Physics 4: Rotation Operators, SU(3)xSU(2)xU(1)]]
į:
* [[https://www.youtube.com/watch?v=zS-LsjrJKPA | Particle Physics 4: Rotation Operators, SU(3)xSU(2)xU(1)]]

Exchange particles - gauge bosons.
2019 liepos 20 d., 17:24 atliko AndriusKulikauskas -
Pakeistos 407-408 eilutės iš
[[https://www.youtube.com/watch?v=zS-LsjrJKPA | Particle Physics 4: Rotation Operators, SU(3)xSU(2)xU(1)]]
į:
DrPhysicsA
*
[[https://www.youtube.com/watch?v=zS-LsjrJKPA | Particle Physics 4: Rotation Operators, SU(3)xSU(2)xU(1)]]
2019 liepos 20 d., 17:18 atliko AndriusKulikauskas -
Pakeistos 405-407 eilutės iš
Weak nuclear force changes quark types. Strong nuclear force changes quark positions. Electromagnetic force distinguishes between quark properties - charge.
į:
Weak nuclear force changes quark types. Strong nuclear force changes quark positions. Electromagnetic force distinguishes between quark properties - charge.

[[https://www.youtube.com/watch?v=zS-LsjrJKPA | Particle Physics 4: Rotation Operators, SU(3)xSU(2)xU(1)]]
2019 liepos 20 d., 17:18 atliko AndriusKulikauskas -
Pakeista 405 eilutė iš:
Weak nuclear force changes quark types. Strong nuclear force changes quark positions.
į:
Weak nuclear force changes quark types. Strong nuclear force changes quark positions. Electromagnetic force distinguishes between quark properties - charge.
2019 liepos 20 d., 17:14 atliko AndriusKulikauskas -
Pakeistos 403-405 eilutės iš
* Spin 2: gravition: fivesome (time/space)
į:
* Spin 2: gravition: fivesome (time/space)

Weak nuclear force changes quark types. Strong nuclear force changes quark positions.
2019 liepos 20 d., 16:59 atliko AndriusKulikauskas -
Pakeistos 396-403 eilutės iš
in the light of metaphors of mathematics]] Anthony Judge
į:
in the light of metaphors of mathematics]] Anthony Judge

Spin 1/2 means there are two states separated by a quanta of energy +/- h. So this is like divisions of everything:
* Spin 0 total spin: onesome
* Spin 1/2: fermions: twosome
* Spin 1: three states: threesome
* Spin 3/2: composite particles: foursome
* Spin 2: gravition: fivesome (time/space)
2019 liepos 19 d., 16:53 atliko AndriusKulikauskas -
Pakeistos 393-396 eilutės iš
Random phenomena organize themselves around a critical boundary.
į:
Random phenomena organize themselves around a critical boundary.

[[https://www.laetusinpraesens.org/musings/periodt.php | Towards a Periodic Table of Ways of Knowing
in the light of metaphors of mathematics]] Anthony Judge
2019 liepos 16 d., 23:25 atliko AndriusKulikauskas -
Pakeistos 391-393 eilutės iš
At the level of forms, this can be seen by decomposing a Hermitian form into its real and imaginary parts: the real part is symmetric (orthogonal), and the imaginary part is skew-symmetric (symplectic)—and these are related by the complex structure (which is the compatibility).
į:
At the level of forms, this can be seen by decomposing a Hermitian form into its real and imaginary parts: the real part is symmetric (orthogonal), and the imaginary part is skew-symmetric (symplectic)—and these are related by the complex structure (which is the compatibility).

Random phenomena organize themselves around a critical boundary
.
2019 liepos 12 d., 14:42 atliko AndriusKulikauskas -
Pakeistos 389-391 eilutės iš
If there is a zero in the Riemann function's zone, then there is a function that it can't mimic?
į:
If there is a zero in the Riemann function's zone, then there is a function that it can't mimic?

At the level of forms, this can be seen by decomposing a Hermitian form into its real and imaginary parts: the real part is symmetric (orthogonal), and the imaginary part is skew-symmetric (symplectic)—and these are related by the complex structure (which is the compatibility).
2019 liepos 10 d., 16:52 atliko AndriusKulikauskas -
Pakeistos 387-389 eilutės iš
{$U(n)$} is a real form of {$GL(n,\mathbb{C})$}. [[https://www.encyclopediaofmath.org/index.php/Complexification_of_a_Lie_group | Encyclopedia: Complexification of a Lie group]]
į:
{$U(n)$} is a real form of {$GL(n,\mathbb{C})$}. [[https://www.encyclopediaofmath.org/index.php/Complexification_of_a_Lie_group | Encyclopedia: Complexification of a Lie group]]

If there is a zero in the Riemann function's zone, then there is a function that it can't mimic?
2019 liepos 05 d., 16:53 atliko AndriusKulikauskas -
Pakeista 387 eilutė iš:
{$U(n)$} is a real form of {$GL(n,\mathbb{c})$}. [[https://www.encyclopediaofmath.org/index.php/Complexification_of_a_Lie_group | Encyclopedia: Complexification of a Lie group]]
į:
{$U(n)$} is a real form of {$GL(n,\mathbb{C})$}. [[https://www.encyclopediaofmath.org/index.php/Complexification_of_a_Lie_group | Encyclopedia: Complexification of a Lie group]]
2019 liepos 05 d., 16:52 atliko AndriusKulikauskas -
Pakeista 387 eilutė iš:
{$U(n)$} is a real form of {$GL(n,\mathbb(C))$}. [[https://www.encyclopediaofmath.org/index.php/Complexification_of_a_Lie_group | Encyclopedia: Complexification of a Lie group]]
į:
{$U(n)$} is a real form of {$GL(n,\mathbb{c})$}. [[https://www.encyclopediaofmath.org/index.php/Complexification_of_a_Lie_group | Encyclopedia: Complexification of a Lie group]]
2019 liepos 05 d., 16:51 atliko AndriusKulikauskas -
Pakeistos 385-387 eilutės iš
[[https://www.youtube.com/watch?v=8KPzuPi-zKk&list=PLZcI2rZdDGQrb4VjOoMm2-o7Fu_mvij8F | Lorenzo Sadun. Videos: Linear Algebra]] Nr.88 is SO(3) and so(3)
į:
[[https://www.youtube.com/watch?v=8KPzuPi-zKk&list=PLZcI2rZdDGQrb4VjOoMm2-o7Fu_mvij8F | Lorenzo Sadun. Videos: Linear Algebra]] Nr.88 is SO(3) and so(3)

{$U(n)$} is a real form of {$GL(n,\mathbb(C))$}. [[https://www.encyclopediaofmath.org/index.php/Complexification_of_a_Lie_group | Encyclopedia: Complexification of a Lie group]]
2019 liepos 04 d., 18:35 atliko AndriusKulikauskas -
Pakeistos 382-385 eilutės iš
[[https://www.youtube.com/watch?v=wIn_dlmD8sk | Video: The rotation group and all that]]
į:
[[https://www.youtube.com/watch?v=wIn_dlmD8sk | Video: The rotation group and all that]]

[[https://www.youtube.com/watch?v=8KPzuPi-zKk&list=PLZcI2rZdDGQrb4VjOoMm2-o7Fu_mvij8F | Lorenzo Sadun. Videos: Linear Algebra]] Nr.88 is SO(3) and so(3)
2019 liepos 04 d., 18:32 atliko AndriusKulikauskas -
Pakeistos 380-382 eilutės iš
i->j is asymmetric, one-directional. i<->i* is symmetric, two-directional, breaks anti-symmetry, hides anti-symmetry (which is i and which is j?)
į:
i->j is asymmetric, one-directional. i<->i* is symmetric, two-directional, breaks anti-symmetry, hides anti-symmetry (which is i and which is j?)

[[https://www.youtube.com/watch?v=wIn_dlmD8sk | Video: The rotation group and all that]]
2019 liepos 04 d., 10:38 atliko AndriusKulikauskas -
Pridėta 380 eilutė:
i->j is asymmetric, one-directional. i<->i* is symmetric, two-directional, breaks anti-symmetry, hides anti-symmetry (which is i and which is j?)
2019 liepos 04 d., 10:37 atliko AndriusKulikauskas -
Pakeista 378 eilutė iš:
į:
Triality: C at the center, three legs: quaternions, even-dimensional reals, odd-dimensional reals. Fold, fuse, link.
2019 liepos 04 d., 10:35 atliko AndriusKulikauskas -
Pakeistos 376-379 eilutės iš
Conjugate = mystery = false. (Hidden distinction).
į:
Conjugate = mystery = false. (Hidden distinction).

2019 liepos 04 d., 10:33 atliko AndriusKulikauskas -
Pakeistos 374-376 eilutės iš
Complex numbers describes rotations in two-dimensions, and quaternions can be used to describe rotations in three dimensions. Is there a connection between octonions and rotations in four dimensions?
į:
Complex numbers describes rotations in two-dimensions, and quaternions can be used to describe rotations in three dimensions. Is there a connection between octonions and rotations in four dimensions?

Conjugate = mystery = false. (Hidden distinction).
2019 liepos 04 d., 09:48 atliko AndriusKulikauskas -
Pakeistos 372-374 eilutės iš
[[https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(composition_algebras) | Hurwitz's theorem]] for composition algebras
į:
[[https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(composition_algebras) | Hurwitz's theorem]] for composition algebras

Complex numbers describes rotations in two-dimensions, and quaternions can be used to describe rotations in three dimensions. Is there a connection between octonions and rotations in four dimensions?
2019 liepos 04 d., 09:47 atliko AndriusKulikauskas -
Pakeistos 368-372 eilutės iš
A circle, as an abelian Lie group, is a "zero", which is a link in a Dynkin diagram, linking two simple roots, two dimensions.
į:
A circle, as an abelian Lie group, is a "zero", which is a link in a Dynkin diagram, linking two simple roots, two dimensions.

The octonions can model the nonassociativity of perspectives.

[[https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(composition_algebras) | Hurwitz's theorem]] for composition algebras
2019 liepos 04 d., 09:35 atliko AndriusKulikauskas -
Pakeistos 366-368 eilutės iš
Symmetry: indistinguishable change, thus a lie, a nontruth, what is hidden. Hidden change, the revealing of hidden change.
į:
Symmetry: indistinguishable change, thus a lie, a nontruth, what is hidden. Hidden change, the revealing of hidden change.

A circle, as an abelian Lie group, is a "zero", which is a link in a Dynkin diagram, linking two simple roots, two dimensions
.
2019 liepos 04 d., 09:31 atliko AndriusKulikauskas -
Pakeistos 364-366 eilutės iš
Think again about the combinatorial intepretation of {$K^{-1}K=I$}.
į:
Think again about the combinatorial intepretation of {$K^{-1}K=I$}.

Symmetry: indistinguishable change, thus a lie, a nontruth, what is hidden. Hidden change, the revealing of hidden change
.
2019 liepos 03 d., 10:22 atliko AndriusKulikauskas -
Pridėtos 363-364 eilutės:

Think again about the combinatorial intepretation of {$K^{-1}K=I$}.
2019 liepos 02 d., 19:10 atliko AndriusKulikauskas -
Pridėtos 361-362 eilutės:

Analyze number types in terms of fractions of differences, https://en.wikipedia.org/wiki/M%C3%B6bius_transformation , in terms of something like that try to understand ad-bc, the different kinds of numbers, the quantities that come up in universal hyperbolic geometry, etc.
2019 liepos 01 d., 10:03 atliko AndriusKulikauskas -
Pridėtos 359-360 eilutės:

Wave function Smolin says is ensemble, I say bosonic sharing of space and time
2019 birželio 30 d., 09:59 atliko AndriusKulikauskas -
Pakeista 354 eilutė iš:
39:00 Came up with the idea in 1970, while working on a Hilbert problem.
į:
39:00 Came up with the idea in 1970, while working on the 16th Hilbert problem.
2019 birželio 30 d., 09:58 atliko AndriusKulikauskas -
Pridėta 354 eilutė:
39:00 Came up with the idea in 1970, while working on a Hilbert problem.
2019 birželio 30 d., 09:55 atliko AndriusKulikauskas -
Pridėta 357 eilutė:
Arnold: Six geometries (based on Cartan's study of infinite dimensional Lie groups?) his list?
2019 birželio 30 d., 09:36 atliko AndriusKulikauskas -
Pakeistos 350-356 eilutės iš
Note that given a chain of perspectives, the possibilities for branching are highly limited, as they are with Dynkin diagrams.
į:
Note that given a chain of perspectives, the possibilities for branching are highly limited, as they are with Dynkin diagrams.

Arnold - "Polymathematics: complexification, symplectification and all that " 1998 video. 18:50 About his trinity, his idea: "This idea, how to apply it, and the examples that I shall discuss even, are not formalized. The theory that I will describe today is not a conjecture, not a theorem, not a definition, it is some kind of religion. I shall show you examples and in these examples, it works. So I was able, using this religion, to find correct guesses, and to find correct conjectures. And then I was able to work years or months trying to prove them. And in some cases, I was able to prove them. In other cases, other people were finally able to prove them. In other cases other people were able to prove them. But to guess these conjectures without this religion would, I think, be impossible. So what I would like to explain to you is just this nonformalized part of it. I am perhaps too old to formalize it but maybe someone who one day finds the axioms and makes a definition from the general construction from the examples that I shall describe."

A_n defines a linear algebra and other root systems add additional structure
2019 birželio 26 d., 22:32 atliko AndriusKulikauskas -
Pakeistos 346-350 eilutės iš
The root systems are ways of linking perspectives. They may represent the operations. {$A_n$} is +0, and the others are +1, +2, +3. There can only be one operation at a time. And the exceptional root systems operate on these four operations.
į:
The root systems are ways of linking perspectives. They may represent the operations. {$A_n$} is +0, and the others are +1, +2, +3. There can only be one operation at a time. And the exceptional root systems operate on these four operations.

Real forms - Satake diagrams - are like being stepped into a perspective (from some perspective within a chain). An odd-dimensional real orthogonal case is stepped-in and even-dimensional is stepped out. Complex case combines the two, and quaternion case combines them yet again. For consciousness.

Note that given a chain of perspectives, the possibilities for branching are highly limited, as they are with Dynkin diagrams
.
2019 birželio 26 d., 20:30 atliko AndriusKulikauskas -
Pakeistos 342-346 eilutės iš
If SU(0) is R, then the real line is zero, and we have projective geometry for the simplexes. So the geometry is determined by the definition of M(0).
į:
If SU(0) is R, then the real line is zero, and we have projective geometry for the simplexes. So the geometry is determined by the definition of M(0).

{$SL(n)$} is not compact, which means that it goes off to infinity. It is like the totality. We have to restrict it, which yields {$A_n$}. Whereas the other Lie families are already restricted.

The root systems are ways of linking perspectives. They may represent the operations. {$A_n$} is +0, and the others are +1, +2, +3. There can only be one operation at a time. And the exceptional root systems operate on these four operations
.
2019 birželio 25 d., 21:16 atliko AndriusKulikauskas -
Pakeista 342 eilutė iš:
If SU(0) is R, then the real line is zero, and we have projective geometry for the simplexes.
į:
If SU(0) is R, then the real line is zero, and we have projective geometry for the simplexes. So the geometry is determined by the definition of M(0).
2019 birželio 25 d., 21:15 atliko AndriusKulikauskas -
Pakeistos 340-342 eilutės iš
In what sense is SU(3) related to a rotation in octonion space?
į:
In what sense is SU(3) related to a rotation in octonion space?

If SU(0) is R, then the real line is zero, and we have projective geometry for the simplexes.
2019 birželio 25 d., 18:38 atliko AndriusKulikauskas -
Pridėtos 337-339 eilutės:
SU(3)xSU(2)xSU(1)xSU(0) is reminiscent of the omniscope.

The conjugate i is evidently the part that adds a perspective. Then R is no perspective.
2019 birželio 25 d., 18:36 atliko AndriusKulikauskas -
Pakeistos 335-337 eilutės iš
Particle physics is based on SU(3)xSU(2)xU(1). Can U(1) be understood as SU(1)xSU(0)? U(1) = SU(1) x R where R gives the length. So this suggests SU(0) = R. In what sense does that make sense?
į:
Particle physics is based on SU(3)xSU(2)xU(1). Can U(1) be understood as SU(1)xSU(0)? U(1) = SU(1) x R where R gives the length. So this suggests SU(0) = R. In what sense does that make sense?

In what sense is SU(3) related to a rotation in octonion space
?
2019 birželio 25 d., 18:35 atliko AndriusKulikauskas -
Pakeista 335 eilutė iš:
Particle physics is based on SU(3)xSU(2)xU(1). Can U(1) be understood as SU(1)xSU(0)?
į:
Particle physics is based on SU(3)xSU(2)xU(1). Can U(1) be understood as SU(1)xSU(0)? U(1) = SU(1) x R where R gives the length. So this suggests SU(0) = R. In what sense does that make sense?
2019 birželio 25 d., 12:25 atliko AndriusKulikauskas -
Pakeistos 333-335 eilutės iš
Organize for myself the Coxeter groups based on how they are built from reflections.
į:
Organize for myself the Coxeter groups based on how they are built from reflections.

Particle physics is based on SU(3)xSU(2)xU(1). Can U(1) be understood as SU(1)xSU(0)?
2019 birželio 24 d., 14:07 atliko AndriusKulikauskas -
Pakeistos 331-333 eilutės iš
Understand the classification of Coxeter groups.
į:
Understand the classification of Coxeter groups.

Organize for myself the Coxeter groups based on how they are built from reflections
.
2019 birželio 24 d., 14:04 atliko AndriusKulikauskas -
Pakeista 329 eilutė iš:
Symmetry of axes - Bn, Cn - leads, in the case of symmetry, to the equivalence of the total symmetry with the individual symmetries, so that for Dn we must divide by two the
į:
Symmetry of axes - Bn, Cn - leads, in the case of symmetry, to the equivalence of the total symmetry with the individual symmetries, so that for Dn we must divide by two the hyperoctahedral group.
2019 birželio 24 d., 13:55 atliko AndriusKulikauskas -
Pakeistos 327-331 eilutės iš
https://en.wikipedia.org/wiki/Andrei_Okounkov
į:
https://en.wikipedia.org/wiki/Andrei_Okounkov

Symmetry of axes - Bn, Cn - leads, in the case of symmetry, to the equivalence of the total symmetry with the individual symmetries, so that for Dn we must divide by two the

Understand the classification of Coxeter groups.
2019 birželio 19 d., 10:25 atliko AndriusKulikauskas -
Pakeistos 325-327 eilutės iš
the unitary matrix in the polar decomposition of an invertible matrix is uniquely defined.
į:
the unitary matrix in the polar decomposition of an invertible matrix is uniquely defined.

https://en.wikipedia.org/wiki/Andrei_Okounkov
2019 birželio 18 d., 19:07 atliko AndriusKulikauskas -
Pakeistos 323-325 eilutės iš
What is the relationship between spin (and alignment to a particular axis or coordinate system) and the alignment of magnets?
į:
What is the relationship between spin (and alignment to a particular axis or coordinate system) and the alignment of magnets?

the unitary matrix in the polar decomposition of an invertible matrix is uniquely defined.
2019 birželio 01 d., 18:32 atliko AndriusKulikauskas -
Pakeistos 321-323 eilutės iš
What do the constraints on Lie groups and Lie algebras say about symmetric functions of eigenvalues.
į:
What do the constraints on Lie groups and Lie algebras say about symmetric functions of eigenvalues.

What is the relationship between spin (and alignment to a particular axis or coordinate system) and the alignment of magnets?
2019 birželio 01 d., 12:25 atliko AndriusKulikauskas -
Pakeistos 319-321 eilutės iš
How special is the Mandelbrot set? What other comparable fractals are there? Can the Mandelbrot set be understood to encompass all of mathematics? What is a combinatorial interpretation of the Mandelbrot set? How is the Mandelbrot set related to the complex numbers and numbers (normed vector spaces) more broadly?
į:
How special is the Mandelbrot set? What other comparable fractals are there? Can the Mandelbrot set be understood to encompass all of mathematics? What is a combinatorial interpretation of the Mandelbrot set? How is the Mandelbrot set related to the complex numbers and numbers (normed vector spaces) more broadly?

What do the constraints on Lie groups and Lie algebras say about symmetric functions of eigenvalues.
2019 gegužės 31 d., 19:55 atliko AndriusKulikauskas -
Pakeistos 317-319 eilutės iš
* transpose - transponuota matrica, transponavimas
į:
* transpose - transponuota matrica, transponavimas

How special is the Mandelbrot set? What other comparable fractals are there? Can the Mandelbrot set be understood to encompass all of mathematics? What is a combinatorial interpretation of the Mandelbrot set? How is the Mandelbrot set related to the complex numbers and numbers (normed vector spaces) more broadly?
2019 gegužės 30 d., 23:48 atliko AndriusKulikauskas -
Pakeistos 310-317 eilutės iš
For any {$A$} and {$B$} in Lie algebra {$\mathfrak{g}$}, {$exp(A+B) = exp(A) + exp(B)$} if and only if {$[A,B]=0$}.
į:
For any {$A$} and {$B$} in Lie algebra {$\mathfrak{g}$}, {$exp(A+B) = exp(A) + exp(B)$} if and only if {$[A,B]=0$}.

Lietuvių kalba:
* sphere - sfera
* trace - pėdsakas
* semisimple - puspaprastis, puspaprastė
* conjugate - sujungtinis
* transpose - transponuota matrica, transponavimas
2019 gegužės 26 d., 09:35 atliko AndriusKulikauskas -
Pakeistos 308-310 eilutės iš
Differentiating {$AA^{-1}=I$} at {$A(0)=I$} we get {$A(A^{-1})'+A'A^{-1}=0$} and so {$A'=-(A^{-1})'$} for any element {$A'$} of a Lie algebra.
į:
Differentiating {$AA^{-1}=I$} at {$A(0)=I$} we get {$A(A^{-1})'+A'A^{-1}=0$} and so {$A'=-(A^{-1})'$} for any element {$A'$} of a Lie algebra.

For any {$A$} and {$B$} in Lie algebra {$\mathfrak{g}$}, {$exp(A+B) = exp(A) + exp(B)$} if and only if {$[A,B]=0$}
.
2019 gegužės 26 d., 09:30 atliko AndriusKulikauskas -
Pakeista 308 eilutė iš:
Differentiating {$AA^{-1}=I$} at {$A(0)=I$} we get {$A(A^{-1})'+A'A^{-1}=0$} and so {$A'=-(A^{-1})'$} for any element {$A$} of a Lie algebra.
į:
Differentiating {$AA^{-1}=I$} at {$A(0)=I$} we get {$A(A^{-1})'+A'A^{-1}=0$} and so {$A'=-(A^{-1})'$} for any element {$A'$} of a Lie algebra.
2019 gegužės 26 d., 09:29 atliko AndriusKulikauskas -
Pakeistos 306-308 eilutės iš
Find the proof and understand it: "An important property of connected Lie groups is that every open set of the identity generates the entire Lie group.Thus all there is to know about a connected Lie group is encoded near the identity." (Ruben Arenas)
į:
Find the proof and understand it: "An important property of connected Lie groups is that every open set of the identity generates the entire Lie group.Thus all there is to know about a connected Lie group is encoded near the identity." (Ruben Arenas)

Differentiating {$AA^{-1}=I$} at {$A(0)=I$} we get {$A(A^{-1})'+A'A^{-1}=0$} and so {$A'=-(A^{-1})'$} for any element {$A$} of a Lie algebra.
2019 gegužės 26 d., 09:22 atliko AndriusKulikauskas -
Pakeista 306 eilutė iš:
Find the proof and understand it: "An important property of connected Lie groups is that every open set of the identity generates the entire Lie group.Thus all there is to know about a connected Lie group is encoded near the identity."
į:
Find the proof and understand it: "An important property of connected Lie groups is that every open set of the identity generates the entire Lie group.Thus all there is to know about a connected Lie group is encoded near the identity." (Ruben Arenas)
2019 gegužės 26 d., 09:21 atliko AndriusKulikauskas -
Pakeistos 304-306 eilutės iš
Study the idea behind linear functionals, fundamental representations, eigenvectors, cohomology, and other maps into one dimension.
į:
Study the idea behind linear functionals, fundamental representations, eigenvectors, cohomology, and other maps into one dimension.

Find the proof and understand it: "An important property of connected Lie groups is that every open set of the identity generates the entire Lie group.Thus all there is to know about a connected Lie group is encoded near the identity."
2019 gegužės 25 d., 09:09 atliko AndriusKulikauskas -
Pakeistos 302-304 eilutės iš
Study how orthogonal and symplectic matrices are subsets of special linear matrices. In what sense are R and H subsets of C?
į:
Study how orthogonal and symplectic matrices are subsets of special linear matrices. In what sense are R and H subsets of C?

Study the idea behind linear functionals, fundamental representations, eigenvectors, cohomology, and other maps into one dimension.
2019 gegužės 24 d., 23:26 atliko AndriusKulikauskas -
Pakeistos 300-302 eilutės iš
Solvable Lie algebras are like degenerate matrices, they are poorly behaved. If we eliminate them, then the remaining semisimple Lie algebras are beautifully behaved. In this sense, abelian Lie algebras are poorly behaved.
į:
Solvable Lie algebras are like degenerate matrices, they are poorly behaved. If we eliminate them, then the remaining semisimple Lie algebras are beautifully behaved. In this sense, abelian Lie algebras are poorly behaved.

Study how orthogonal and symplectic matrices are subsets of special linear matrices. In what sense are R and H subsets of C?
2019 gegužės 24 d., 23:22 atliko AndriusKulikauskas -
Pakeista 300 eilutė iš:
Solvable Lie algebras are like degenerate matrices, they are poorly behaved. If we eliminate them, then the remaining semisimple Lie algebras are beautifully behaved.
į:
Solvable Lie algebras are like degenerate matrices, they are poorly behaved. If we eliminate them, then the remaining semisimple Lie algebras are beautifully behaved. In this sense, abelian Lie algebras are poorly behaved.
2019 gegužės 24 d., 23:20 atliko AndriusKulikauskas -
Pakeistos 298-300 eilutės iš
Internal discussion with oneself vs. external discussion with others is the distinction that category theory makes between internal structure and external relationships.
į:
Internal discussion with oneself vs. external discussion with others is the distinction that category theory makes between internal structure and external relationships.

Solvable Lie algebras are like degenerate matrices, they are poorly behaved. If we eliminate them, then the remaining semisimple Lie algebras are beautifully behaved
.
2019 gegužės 18 d., 20:25 atliko AndriusKulikauskas -
Pakeistos 296-298 eilutės iš
Perhaps the projective geometry is the most basic, and it is based on rotation and the complexes. But perhaps an affine geometry arises, with the reals, so that we can imagine a movement of the origin (the fixed point) and thus we can have translations, a shift in origin, a shift in perspective, a relative perspective, in a conditional world.
į:
Perhaps the projective geometry is the most basic, and it is based on rotation and the complexes. But perhaps an affine geometry arises, with the reals, so that we can imagine a movement of the origin (the fixed point) and thus we can have translations, a shift in origin, a shift in perspective, a relative perspective, in a conditional world.

Internal discussion with oneself vs. external discussion with others is the distinction that category theory makes between internal structure and external relationships
.
2019 gegužės 18 d., 20:24 atliko AndriusKulikauskas -
Pakeista 296 eilutė iš:
Perhaps the projective geometry is the most basic, and it is based on rotation and the complexes. But perhaps an affine geometry arises, with the reals, so that we can imagine a movement of the origin (the fixed point) and thus we can have translations, a shift in origin, a shift in perspective, a relative perspective.
į:
Perhaps the projective geometry is the most basic, and it is based on rotation and the complexes. But perhaps an affine geometry arises, with the reals, so that we can imagine a movement of the origin (the fixed point) and thus we can have translations, a shift in origin, a shift in perspective, a relative perspective, in a conditional world.
2019 gegužės 18 d., 20:24 atliko AndriusKulikauskas -
Pakeistos 292-296 eilutės iš
Choices - polytopes, reflections - root systems. How are the Weyl groups related?
į:
Choices - polytopes, reflections - root systems. How are the Weyl groups related?

Affine and projective geometries. Adding or subtracting a perspective. Such as adding or deleting a node to a Dynkin diagram. (The chain of perspectives.)

Perhaps the projective geometry is the most basic, and it is based on rotation and the complexes. But perhaps an affine geometry arises, with the reals, so that we can imagine a movement of the origin (the fixed point) and thus we can have translations, a shift in origin, a shift in perspective, a relative perspective.
2019 gegužės 18 d., 20:13 atliko AndriusKulikauskas -
Pakeistos 290-292 eilutės iš
In Lie root systems, reflections yield a geometry. They also yield an algebra of what addition of root is allowable.
į:
In Lie root systems, reflections yield a geometry. They also yield an algebra of what addition of root is allowable.

Choices - polytopes, reflections - root systems. How are the Weyl groups related?
2019 gegužės 18 d., 19:48 atliko AndriusKulikauskas -
Pakeistos 288-290 eilutės iš
In {$D_n$}, think of {$x_i-x_j$} and {$x_i+x_j$} as complex conjugates.
į:
In {$D_n$}, think of {$x_i-x_j$} and {$x_i+x_j$} as complex conjugates.

In Lie root systems, reflections yield a geometry. They also yield an algebra of what addition of root is allowable
.
2019 gegužės 18 d., 19:46 atliko AndriusKulikauskas - _
Pakeistos 286-288 eilutės iš
Develop looseness - slack - freedom - ambiguity as concepts that give meaning to isomorphism, identity, structure, symmetry. Local constraints can yet lead to different global solutions.
į:
Develop looseness - slack - freedom - ambiguity as concepts that give meaning to isomorphism, identity, structure, symmetry. Local constraints can yet lead to different global solutions.

In {$D_n$}, think of {$x_i-x_j$} and {$x_i+x_j$} as complex conjugates
.
2019 gegužės 15 d., 17:18 atliko AndriusKulikauskas -
Pakeista 286 eilutė iš:
Develop looseness - slack - freedom - ambiguity as concepts that give meaning to isomorphism, identity, structure, symmetry.
į:
Develop looseness - slack - freedom - ambiguity as concepts that give meaning to isomorphism, identity, structure, symmetry. Local constraints can yet lead to different global solutions.
2019 gegužės 15 d., 17:17 atliko AndriusKulikauskas -
Pakeistos 284-286 eilutės iš
Isomorphism is based on assignment but that depends on equality up to identity whereas properties define establish an object up to isomorphism
į:
Isomorphism is based on assignment but that depends on equality up to identity whereas properties define establish an object up to isomorphism

Develop looseness - slack - freedom - ambiguity as concepts that give meaning to isomorphism, identity, structure, symmetry.
2019 gegužės 14 d., 18:27 atliko AndriusKulikauskas -
Pridėtos 283-284 eilutės:

Isomorphism is based on assignment but that depends on equality up to identity whereas properties define establish an object up to isomorphism
2019 gegužės 14 d., 10:56 atliko AndriusKulikauskas -
Pridėtos 281-282 eilutės:

Equations are questions
2019 gegužės 14 d., 10:12 atliko AndriusKulikauskas -
Pakeista 280 eilutė iš:
Scaling is positive flips over to negative this is discrete rotation is reflection
į:
Scaling is positive flips over to negative this is discrete rotation is reflection
2019 gegužės 14 d., 10:00 atliko AndriusKulikauskas -
Pakeistos 278-280 eilutės iš
Quaternions include 3 dimensijos formos rotating (twistor) and 1 for scaling (time) and likewise for octonions etc
į:
Quaternions include 3 dimensijos formos rotating (twistor) and 1 for scaling (time) and likewise for octonions etc

Scaling is positive flips over to negative this is discrete rotation is reflection
2019 gegužės 14 d., 09:51 atliko AndriusKulikauskas -
Pakeistos 275-278 eilutės iš
Note how complex numbers express rotations in R2. How are quaternions related to rotations in R3? What about R4? And the real numbers? And in what sense do the complex numbers and quaternions do the same as the reals but more richly?
į:
Note how complex numbers express rotations in R2. How are quaternions related to rotations in R3? What about R4? And the real numbers? And in what sense do the complex numbers and quaternions do the same as the reals but more richly?

Quaternions include 3 dimensijos formos rotating (twistor) and 1 for scaling (time) and likewise for octonions etc
2019 gegužės 13 d., 14:55 atliko AndriusKulikauskas -
Pakeistos 273-275 eilutės iš
How is a Nor gate made from [[Nand]] gates? (And vice versa.)
į:
How is a Nor gate made from [[Nand]] gates? (And vice versa.)

Note how complex numbers express rotations in R2. How are quaternions related to rotations in R3? What about R4? And the real numbers? And in what sense do the complex numbers and quaternions do the same as the reals but more richly?
2019 balandžio 29 d., 13:49 atliko AndriusKulikauskas -
Pakeista 273 eilutė iš:
How is a Nor gate made from Nand gates? (And vice versa.)
į:
How is a Nor gate made from [[Nand]] gates? (And vice versa.)
2019 balandžio 29 d., 13:41 atliko AndriusKulikauskas -
Pakeistos 271-273 eilutės iš
Study how all logical relations derive from composition of Nand gates.
į:
Study how all logical relations derive from composition of Nand gates.

How is a Nor gate made from Nand gates? (And vice versa.)
2019 balandžio 29 d., 13:39 atliko AndriusKulikauskas -
Pakeistos 267-271 eilutės iš
Nand gates and Nor gates (and And gates and Or gates) relate one and all. The Nand and Nor mark this relation with a negation sign.
į:
Nand gates and Nor gates (and And gates and Or gates) relate one and all. The Nand and Nor mark this relation with a negation sign.

Are Nand gates (Nor gates) related to perspectives?

Study how all logical relations derive from composition of Nand gates
.
2019 balandžio 29 d., 13:38 atliko AndriusKulikauskas -
Pakeistos 265-267 eilutės iš
Study existential and universal quantifiers as adjunctions and as the basis for the arithmetical hierarchy.
į:
Study existential and universal quantifiers as adjunctions and as the basis for the arithmetical hierarchy.

Nand gates and Nor gates (and And gates and Or gates) relate one and all. The Nand and Nor mark this relation with a negation sign
.
2019 balandžio 28 d., 20:11 atliko AndriusKulikauskas -
Pakeistos 263-265 eilutės iš
Mathematical induction - is infinitely many statements that are true - relate to natural transformation, which also relates possibly infinitely many statements.
į:
Mathematical induction - is infinitely many statements that are true - relate to natural transformation, which also relates possibly infinitely many statements.

Study existential and universal quantifiers as adjunctions and as the basis for the arithmetical hierarchy
.
2019 balandžio 28 d., 20:11 atliko AndriusKulikauskas -
Pakeistos 261-263 eilutės iš
Study the Wolfram Axiom and Nand.
į:
Study the Wolfram Axiom and Nand.

Mathematical induction - is infinitely many statements that are true - relate to natural transformation, which also relates possibly infinitely many statements
.
2019 balandžio 25 d., 00:09 atliko AndriusKulikauskas -
Pakeistos 259-261 eilutės iš
Associative composition yields a list (and defines a list). Consider the identity morphism composed with itself.
į:
Associative composition yields a list (and defines a list). Consider the identity morphism composed with itself.

Study the Wolfram Axiom and Nand
.
2019 balandžio 07 d., 22:41 atliko AndriusKulikauskas -
Pakeistos 257-259 eilutės iš
Study homology, cohomology and the Snake lemma to explain how to express a gap.
į:
Study homology, cohomology and the Snake lemma to explain how to express a gap.

Associative composition yields a list (and defines a list). Consider the identity morphism composed with itself
.
2019 kovo 22 d., 10:52 atliko AndriusKulikauskas -
Pakeistos 255-257 eilutės iš
If we think of a Turing machine's possible paths as a category, what is the functor that takes us to the actual path of execution?
į:
If we think of a Turing machine's possible paths as a category, what is the functor that takes us to the actual path of execution?

Study homology, cohomology and the Snake lemma to explain how to express a gap.
2019 kovo 19 d., 14:11 atliko AndriusKulikauskas -
Pakeistos 251-255 eilutės iš
Geometry is concrete, it is a manifestation. Thus group actions can represent an abstract structure (of actions) in terms of a geometrical space (set, vector space, topological space). And this makes for understanding of the abstract in terms of the concrete.
į:
Geometry is concrete, it is a manifestation. Thus group actions can represent an abstract structure (of actions) in terms of a geometrical space (set, vector space, topological space). And this makes for understanding of the abstract in terms of the concrete.

Turing machines - inner states are "states of mind" according to Turing. How do they relate to divisions of everything?

If we think of a Turing machine's possible paths as a category, what is the functor that takes us to the actual path of execution?
2019 kovo 19 d., 11:11 atliko AndriusKulikauskas -
Pakeista 251 eilutė iš:
Geometry is concrete, it is a manifestation. Thus group actions can represent an abstract structure (of actions) in terms of a geometrical space (set, vector space, topological space).
į:
Geometry is concrete, it is a manifestation. Thus group actions can represent an abstract structure (of actions) in terms of a geometrical space (set, vector space, topological space). And this makes for understanding of the abstract in terms of the concrete.
2019 kovo 19 d., 11:10 atliko AndriusKulikauskas -
Pakeistos 249-251 eilutės iš
Algebra and geometry are linked by logic - intersections and unions make sense in both.
į:
Algebra and geometry are linked by logic - intersections and unions make sense in both.

Geometry is concrete, it is a manifestation. Thus group actions can represent an abstract structure (of actions) in terms of a geometrical space (set, vector space, topological space)
.
2019 kovo 15 d., 11:13 atliko AndriusKulikauskas -
Pakeistos 247-249 eilutės iš
Algebra - geometry duality. (Pullback). Morphism <-> ring homomorphism. Intrinsic and extrinsic geometry. Ambient space. Relation between two spaces. Varieties [morphisms X to Y] <-> Affine F-algebras [F-linear ring homomorphisms F[Y] to F[x]]
į:
Algebra - geometry duality. (Pullback). Morphism <-> ring homomorphism. Intrinsic and extrinsic geometry. Ambient space. Relation between two spaces. Varieties [morphisms X to Y] <-> Affine F-algebras [F-linear ring homomorphisms F[Y] to F[x]]

Algebra and geometry are linked by logic - intersections and unions make sense in both.
2019 kovo 14 d., 12:07 atliko AndriusKulikauskas -
Pakeista 247 eilutė iš:
Algebra - geometry duality. (Pullback). Morphism <-> ring homomorphism. Intrinsic and extrinsic geometry. Ambient space. Relation between two spaces.
į:
Algebra - geometry duality. (Pullback). Morphism <-> ring homomorphism. Intrinsic and extrinsic geometry. Ambient space. Relation between two spaces. Varieties [morphisms X to Y] <-> Affine F-algebras [F-linear ring homomorphisms F[Y] to F[x]]
2019 kovo 14 d., 12:00 atliko AndriusKulikauskas -
Pakeista 247 eilutė iš:
Algebra - geometry duality. (Pullback). Morphism <-> ring homomorphism. Intrinsic and extrinsic geometry.
į:
Algebra - geometry duality. (Pullback). Morphism <-> ring homomorphism. Intrinsic and extrinsic geometry. Ambient space. Relation between two spaces.
2019 kovo 14 d., 11:52 atliko AndriusKulikauskas -
Pakeistos 245-247 eilutės iš
http://pi.math.cornell.edu/~hatcher/AT/ATpage.html
į:
http://pi.math.cornell.edu/~hatcher/AT/ATpage.html

Algebra - geometry duality. (Pullback). Morphism <-> ring homomorphism. Intrinsic and extrinsic geometry.
2019 kovo 13 d., 10:56 atliko AndriusKulikauskas -
Ištrintos 236-237 eilutės:
DanielChanMaths
Ištrinta 243 eilutė:
2019 kovo 11 d., 11:04 atliko AndriusKulikauskas -
Pridėtos 244-246 eilutės:

https://golem.ph.utexas.edu/category/2017/01/basic_category_theory_free_onl.html
2019 kovo 11 d., 11:02 atliko AndriusKulikauskas -
Pakeistos 243-245 eilutės iš
Vector bundles: Identity and self-identity (like the ends of a regular strip or a Moebius strip). Identity of a point, identity of a fiber - self-identity under continuity. How does a fiber relate to itself? Is it flipped or not? 2 kinds of self-identity (or non-identity) allows the edge to be flipped upside down.
į:
Vector bundles: Identity and self-identity (like the ends of a regular strip or a Moebius strip). Identity of a point, identity of a fiber - self-identity under continuity. How does a fiber relate to itself? Is it flipped or not? 2 kinds of self-identity (or non-identity) allows the edge to be flipped upside down.

http://pi.math.cornell.edu/~hatcher/AT/ATpage.html
2019 kovo 10 d., 22:11 atliko AndriusKulikauskas -
Pakeistos 241-243 eilutės iš
Euclidean space - (algebraic) coordinate systems - define left, right, front, backwards - and this often makes sense locally - but this does not make sense globally on a sphere, for example
į:
Euclidean space - (algebraic) coordinate systems - define left, right, front, backwards - and this often makes sense locally - but this does not make sense globally on a sphere, for example

Vector bundles: Identity and self-identity (like the ends of a regular strip or a Moebius strip). Identity of a point, identity of a fiber - self-identity under continuity. How does a fiber relate to itself? Is it flipped or not? 2 kinds of self-identity (or non-identity) allows the edge to be flipped upside down.
2019 kovo 07 d., 14:30 atliko AndriusKulikauskas -
Pakeistos 239-241 eilutės iš
Vandermonde determinant shows invertible - basis for finite Fourier transform
į:
Vandermonde determinant shows invertible - basis for finite Fourier transform

Euclidean space - (algebraic) coordinate systems - define left, right, front, backwards - and this often makes sense locally - but this does not make sense globally on a sphere, for example
2019 kovo 07 d., 13:31 atliko AndriusKulikauskas -
Pakeistos 237-239 eilutės iš
DanielChanMaths
į:
DanielChanMaths

Vandermonde determinant shows invertible - basis for finite Fourier transform
2019 kovo 05 d., 20:53 atliko AndriusKulikauskas -
Pridėta 237 eilutė:
DanielChanMaths
2019 kovo 05 d., 09:01 atliko AndriusKulikauskas -
Pakeista 233 eilutė iš:
Wikipedia: Homotopy groups Let p: E → B be a basepoint-preserving Serre fibration with fiber F, that is, a map possessing the homotopy lifting property with respect to CW complexes. Suppose that B is path-connected. Then there is a long exact sequence of homotopy groups:
į:
Primena trejybę. [[https://en.wikipedia.org/wiki/Homotopy_group | Wikipedia: Homotopy groups]] Let p: E → B be a basepoint-preserving Serre fibration with fiber F, that is, a map possessing the homotopy lifting property with respect to CW complexes. Suppose that B is path-connected. Then there is a long exact sequence of homotopy groups:
2019 kovo 05 d., 09:00 atliko AndriusKulikauskas -
Pakeistos 229-236 eilutės iš
monad = black box?
į:
monad = black box?

Let p: E → B be a basepoint-preserving Serre fibration with fiber F, that is, a map possessing the homotopy lifting property with respect to CW complexes. Suppose that B is path-connected. Then there is a long exact sequence of homotopy groups

Wikipedia: Homotopy groups Let p: E → B be a basepoint-preserving Serre fibration with fiber F, that is, a map possessing the homotopy lifting property with respect to CW complexes. Suppose that B is path-connected. Then there is a long exact sequence of homotopy groups:

{$\cdots \rightarrow \pi_n(F) \rightarrow \pi_n(E) \rightarrow \pi_n(B) \rightarrow \pi_{n-1}(F) \rightarrow \cdots \rightarrow \pi_0(F) \rightarrow 0.$}
2019 kovo 02 d., 21:53 atliko AndriusKulikauskas -
Pridėta 229 eilutė:
monad = black box?
2019 vasario 17 d., 05:55 atliko AndriusKulikauskas -
Pakeistos 225-228 eilutės iš
Topology - getting global invariants (which can be calculated) from local information.
į:
Topology - getting global invariants (which can be calculated) from local information.

Simple examples that illustrate theory.
2019 vasario 17 d., 05:55 atliko AndriusKulikauskas -
Pakeistos 219-225 eilutės iš
Symbols (a and b) are equal if they refer to the same referents. But equality has different meaning for symbols, indexes, icons and things. Consider the four relations between level and metalevel.
į:
Symbols (a and b) are equal if they refer to the same referents. But equality has different meaning for symbols, indexes, icons and things. Consider the four relations between level and metalevel.

Is Cayley's theorem (Yoneda lemma) a contentless theorem? What makes a theorem useful as a tool for discoveries?

(Conscious) Learning from (unconscious) machine learning.

Topology - getting global invariants (which can be calculated) from local information
.
2019 vasario 15 d., 09:46 atliko AndriusKulikauskas -
Pakeistos 217-219 eilutės iš
In a diagram, we have a map from shape J (the index category) to the category C. Note that the index diagram is How.
į:
In a diagram, we have a map from shape J (the index category) to the category C. Note that the index diagram is How.

Symbols (a and b) are equal if they refer to the same referents. But equality has different meaning for symbols, indexes, icons and things. Consider the four relations between level and metalevel
.
2019 vasario 13 d., 14:33 atliko AndriusKulikauskas -
Pridėtos 1-2 eilutės:
[++++数学笔记++++]
2019 vasario 13 d., 13:59 atliko AndriusKulikauskas -
Pakeistos 215-218 eilutės iš
In a diagram, we have a map from shape J (the index category) to the category C. Note that the index diagram is How.

Mathematics theorems relate information that is less explicit (leveraging the presumptions inherent in the framework) with information that is more explicit (expressing those presumptions). Thus mathematics makes information more explicit. It is revealing information and, in that sense, "creating" the explicitness of the information.
* For example, in this [[https://www.math3ma.com/blog/the-sierpinski-space-and-its-special-property | proposition]] about pullbacks, the statement about the pullback is much more explicit than that for the function f because it includes f as a special case when Z = x, for then f*(idX)={$f^{-1}$}. But that special case leverages the framework to establish all the other cases
.
į:
In a diagram, we have a map from shape J (the index category) to the category C. Note that the index diagram is How.
2019 vasario 13 d., 13:59 atliko AndriusKulikauskas -
Pakeistos 217-218 eilutės iš
Mathematics theorems relate information that is less explicit (leveraging the presumptions inherent in the framework) with information that is more explicit (expressing those presumptions). Thus mathematics makes information more explicit. It is revealing information and, in that sense, "creating" the explicitness of the information.
į:
Mathematics theorems relate information that is less explicit (leveraging the presumptions inherent in the framework) with information that is more explicit (expressing those presumptions). Thus mathematics makes information more explicit. It is revealing information and, in that sense, "creating" the explicitness of the information.
* For example, in this [[https://www.math3ma.com/blog/the-sierpinski-space-and-its-special-property | proposition]] about pullbacks, the statement about the pullback is much more explicit than that for the function f because it includes f as a special case when Z = x, for then f*(idX)={$f^{-1}$}. But that special case leverages the framework to establish all the other cases
.
2019 vasario 13 d., 13:56 atliko AndriusKulikauskas -
Pakeistos 215-217 eilutės iš
In a diagram, we have a map from shape J (the index category) to the category C. Note that the index diagram is How.
į:
In a diagram, we have a map from shape J (the index category) to the category C. Note that the index diagram is How.

Mathematics theorems relate information that is less explicit (leveraging the presumptions inherent in the framework) with information that is more explicit (expressing those presumptions). Thus mathematics makes information more explicit. It is revealing information and, in that sense, "creating" the explicitness of the information
.
2019 vasario 12 d., 11:37 atliko AndriusKulikauskas -
Pakeistos 213-215 eilutės iš
In product, the information from A and B is stored externally in A x B. In coproduct, the information from A and B is stored internally in A union B (A+B). Note: multiplication is external, and addition is internal.
į:
In product, the information from A and B is stored externally in A x B. In coproduct, the information from A and B is stored internally in A union B (A+B). Note: multiplication is external, and addition is internal.

In a diagram, we have a map from shape J (the index category) to the category C. Note that the index diagram is How
.
2019 vasario 12 d., 11:24 atliko AndriusKulikauskas -
Pakeista 213 eilutė iš:
In product, the information from A and B is stored externally in A x B. In coproduct, the information from A and B is stored internally in A union B (A+B).
į:
In product, the information from A and B is stored externally in A x B. In coproduct, the information from A and B is stored internally in A union B (A+B). Note: multiplication is external, and addition is internal.
2019 vasario 12 d., 11:19 atliko AndriusKulikauskas -
Pakeistos 211-213 eilutės iš
"For all" and "there exists" are adjoints presumably because they are on opposite sides of a negation wall that distinguishes the internal structure and external relationships. (That wall also distinguishes external context and internal structure.) (And algorithms?) So study that wall, for example, with regard to recursion theory.
į:
"For all" and "there exists" are adjoints presumably because they are on opposite sides of a negation wall that distinguishes the internal structure and external relationships. (That wall also distinguishes external context and internal structure.) (And algorithms?) So study that wall, for example, with regard to recursion theory.

In product, the information from A and B is stored externally in A x B. In coproduct, the information from A and B is stored internally in A union B (A+B)
.
2019 vasario 12 d., 10:56 atliko AndriusKulikauskas -
Pakeistos 209-211 eilutės iš
Category theory for me: distinguishing what observations are nontrivial - intrinsic to a subject - and what are observations are content-wise trivial or universal - not related to the subject, but simply an aspect of abstraction.
į:
Category theory for me: distinguishing what observations are nontrivial - intrinsic to a subject - and what are observations are content-wise trivial or universal - not related to the subject, but simply an aspect of abstraction.

"For all" and "there exists" are adjoints presumably because they are on opposite sides of a negation wall that distinguishes the internal structure and external relationships. (That wall also distinguishes external context and internal structure.) (And algorithms?) So study that wall, for example, with regard to recursion theory
.
2019 vasario 12 d., 10:37 atliko AndriusKulikauskas -
Pakeistos 207-209 eilutės iš
The Yoneda Lemma: the Why of the external relationships leads to the Whether of the objects in the set. The latter are considered as a set of truth statements.
į:
The Yoneda Lemma: the Why of the external relationships leads to the Whether of the objects in the set. The latter are considered as a set of truth statements.

Category theory for me: distinguishing what observations are nontrivial - intrinsic to a subject - and what are observations are content-wise trivial or universal - not related to the subject, but simply an aspect of abstraction
.
2019 vasario 12 d., 08:52 atliko AndriusKulikauskas -
Pakeistos 203-207 eilutės iš
Study how Set breaks duality (the significance of initial and terminal objects).
į:
Study how Set breaks duality (the significance of initial and terminal objects).

Show why there is no n-category theory because it folds up into the foursome. Understand the Yoneda lemma. Relate it to the four ways of looking at a triangle.

The Yoneda Lemma: the Why of the external relationships leads to the Whether of the objects in the set. The latter are considered as a set of truth statements
.
2019 vasario 11 d., 15:24 atliko AndriusKulikauskas -
Pakeistos 201-203 eilutės iš
Force (and acceleration) is a second derivative - this is because of the duality, the coupling, needed between, say, momentum and position. Is this coupling similar to adjoint functors?
į:
Force (and acceleration) is a second derivative - this is because of the duality, the coupling, needed between, say, momentum and position. Is this coupling similar to adjoint functors?

Study how Set breaks duality (the significance of initial and terminal objects).
2019 vasario 08 d., 13:58 atliko AndriusKulikauskas -
Pakeistos 199-201 eilutės iš
In functional programming with monoids and monads, can we think of each function as taking us from a question type to an answer type? In general, in category theory, can we think of each morphism as taking us from a question to an answer?
į:
In functional programming with monoids and monads, can we think of each function as taking us from a question type to an answer type? In general, in category theory, can we think of each morphism as taking us from a question to an answer?

Force (and acceleration) is a second derivative - this is because of the duality, the coupling, needed between, say, momentum and position. Is this coupling similar to adjoint functors
?
2019 vasario 08 d., 13:36 atliko AndriusKulikauskas -
Pakeistos 197-199 eilutės iš
In most every category, can we (arbitrarily) define (uniquely) distinguished "generic objects" or "canonical objects", which are the generic equivalents for all objects that are equivalent to each other? For example, in the category of sets, the generic set of size one.
į:
In most every category, can we (arbitrarily) define (uniquely) distinguished "generic objects" or "canonical objects", which are the generic equivalents for all objects that are equivalent to each other? For example, in the category of sets, the generic set of size one.

In functional programming with monoids and monads, can we think of each function as taking us from a question type to an answer type? In general, in category theory, can we think of each morphism as taking us from a question to an answer?
2019 vasario 08 d., 13:34 atliko AndriusKulikauskas -
Pakeistos 195-197 eilutės iš
[[http://math.ucr.edu/home/baez/week257.html | John Baez about observables]] (see Nr.15) and the paper [[http://arxiv.org/abs/0709.4364 | A topos for algebraic quantum theory]] about C* algebras within a topos and outside of it.
į:
[[http://math.ucr.edu/home/baez/week257.html | John Baez about observables]] (see Nr.15) and the paper [[http://arxiv.org/abs/0709.4364 | A topos for algebraic quantum theory]] about C* algebras within a topos and outside of it.

In most every category, can we (arbitrarily) define (uniquely) distinguished "generic objects" or "canonical objects", which are the generic equivalents for all objects that are equivalent to each other? For example, in the category of sets, the generic set of size one
.
2019 vasario 05 d., 10:02 atliko AndriusKulikauskas -
Pakeista 195 eilutė iš:
[[http://math.ucr.edu/home/baez/week257.html John Baez about observables]] (see Nr.15) and the paper [[http://arxiv.org/abs/0709.4364 | A topos for algebraic quantum theory]] about C* algebras within a topos and outside of it.
į:
[[http://math.ucr.edu/home/baez/week257.html | John Baez about observables]] (see Nr.15) and the paper [[http://arxiv.org/abs/0709.4364 | A topos for algebraic quantum theory]] about C* algebras within a topos and outside of it.
2019 vasario 05 d., 10:02 atliko AndriusKulikauskas -
Pakeistos 193-195 eilutės iš
Six sextactic points.
į:
Six sextactic points.

[[http://math.ucr.edu/home/baez/week257.html John Baez about observables]] (see Nr.15) and the paper [[http://arxiv.org/abs/0709.4364 | A topos for algebraic quantum theory]] about C* algebras within a topos and outside of it
.
2019 vasario 04 d., 15:14 atliko AndriusKulikauskas -
Pakeistos 191-193 eilutės iš
Closed curve in plane must have at least four critical points of curvature. This reflects the fourfold aspect of turning around, rotating. Caustic - a phenomenon in symplectic geometry.
į:
Closed curve in plane must have at least four critical points of curvature. This reflects the fourfold aspect of turning around, rotating. Caustic - a phenomenon in symplectic geometry.

Six sextactic points
.
2019 vasario 04 d., 15:11 atliko AndriusKulikauskas -
Pakeista 191 eilutė iš:
Closed curve in plane must have at least four critical points of curvature. This reflects the fourfold aspect of turning around, rotating.
į:
Closed curve in plane must have at least four critical points of curvature. This reflects the fourfold aspect of turning around, rotating. Caustic - a phenomenon in symplectic geometry.
2019 vasario 04 d., 15:10 atliko AndriusKulikauskas -
Pakeistos 189-191 eilutės iš
Representable functors - based on arrows from the same object.
į:
Representable functors - based on arrows from the same object.

Closed curve in plane must have at least four critical points of curvature. This reflects the fourfold aspect of turning around, rotating
.
2019 vasario 02 d., 13:35 atliko AndriusKulikauskas -
Pakeistos 187-189 eilutės iš
Whether (objects), what (morphisms), how (functors), why (natural transformations).
į:
Whether (objects), what (morphisms), how (functors), why (natural transformations). Important for defining the same thing, equivalence. If they satisfy the same reason why, then they are the same.

Representable functors - based on arrows from the same object
.
2019 vasario 02 d., 13:10 atliko AndriusKulikauskas -
Pakeistos 185-187 eilutės iš
Category is a collection of examples that satisfy certain conditions. That is why you can have many examples that are essentially identical. And it's essential for the meaning of the Yoneda Lemma, for example. Because two different examples may have the same structure but one example may draw richer meaning from one domain and the other may not have that richer meaning or may have other meaning from another domain.
į:
Category is a collection of examples that satisfy certain conditions. That is why you can have many examples that are essentially identical. And it's essential for the meaning of the Yoneda Lemma, for example. Because two different examples may have the same structure but one example may draw richer meaning from one domain and the other may not have that richer meaning or may have other meaning from another domain.

Whether (objects), what (morphisms), how (functors), why (natural transformations)
.
2019 vasario 01 d., 11:51 atliko AndriusKulikauskas -
Pakeista 185 eilutė iš:
Category is a collection of examples that satisfy certain conditions. That is why you can have many examples that are essentially identical.
į:
Category is a collection of examples that satisfy certain conditions. That is why you can have many examples that are essentially identical. And it's essential for the meaning of the Yoneda Lemma, for example. Because two different examples may have the same structure but one example may draw richer meaning from one domain and the other may not have that richer meaning or may have other meaning from another domain.
2019 vasario 01 d., 11:48 atliko AndriusKulikauskas -
Pakeistos 181-185 eilutės iš
* D_n points and position
į:
* D_n points and position

Yoneda lemma - relates to exponentiation and logarithm

Category is a collection of examples that satisfy certain conditions. That is why you can have many examples that are essentially identical.
2019 sausio 27 d., 09:49 atliko AndriusKulikauskas -
Pakeistos 174-181 eilutės iš
Symplectic geometry is the geometry of the "outside" (using quaternions) whereas conformal geometry is the geometry of the "inside" (using complex numbers). Then what do real numbers capture the geometry of? And is there a geometry of the line vs. a geometry of the circle? And is one of them "spinorial"?
į:
Symplectic geometry is the geometry of the "outside" (using quaternions) whereas conformal geometry is the geometry of the "inside" (using complex numbers). Then what do real numbers capture the geometry of? And is there a geometry of the line vs. a geometry of the circle? And is one of them "spinorial"?

Benet linkage - keturgrandinis - lygiagretainis, antilygiagretainis

* A_n points and sets
* B_n inside: perpendicular (angles) and
* C_n outside: line and surface area
* D_n points and position
2019 sausio 27 d., 09:23 atliko AndriusKulikauskas -
Pakeistos 172-174 eilutės iš
What is the connection between symplectic geometry and homology? See [[https://en.wikipedia.org/wiki/Morse_theory | Morse theory]]. See [[https://people.ucsc.edu/~alee150/sympl.html | Floer theory]].
į:
What is the connection between symplectic geometry and homology? See [[https://en.wikipedia.org/wiki/Morse_theory | Morse theory]]. See [[https://people.ucsc.edu/~alee150/sympl.html | Floer theory]].

Symplectic geometry is the geometry of the "outside" (using quaternions) whereas conformal geometry is the geometry of the "inside" (using complex numbers). Then what do real numbers capture the geometry of? And is there a geometry of the line vs. a geometry of the circle? And is one of them "spinorial"?
2019 sausio 27 d., 09:07 atliko AndriusKulikauskas -
Pakeista 172 eilutė iš:
What is the connection between symplectic geometry and homology?
į:
What is the connection between symplectic geometry and homology? See [[https://en.wikipedia.org/wiki/Morse_theory | Morse theory]]. See [[https://people.ucsc.edu/~alee150/sympl.html | Floer theory]].
2019 sausio 27 d., 09:04 atliko AndriusKulikauskas -
Pakeistos 170-172 eilutės iš
Note that 2-dimensional phase space (as with a spring) is the simplest as there is no 1-dimensional phase space and there can't be (we need both position and momentum).
į:
Note that 2-dimensional phase space (as with a spring) is the simplest as there is no 1-dimensional phase space and there can't be (we need both position and momentum).

What is the connection between symplectic geometry and homology?
2019 sausio 27 d., 08:11 atliko AndriusKulikauskas -
Pakeistos 168-170 eilutės iš
Note that symplectic geometry, in preserving areas, seems to only care about the extremal points. In what sense is that true in phase space? When and how could it not be true? What is the difference between having finitely many extremal points (convex polytope? and what is the significance of extremal points vs. extremal edges vs. extremal faces etc.? and homology/cohomology?), infinitely many (as with a fractal boundary) or all extremal points? Can symplectic geometry be considered a study of the outside and conformal geometry a study of the inside?
į:
Note that symplectic geometry, in preserving areas, seems to only care about the extremal points. In what sense is that true in phase space? When and how could it not be true? What is the difference between having finitely many extremal points (convex polytope? and what is the significance of extremal points vs. extremal edges vs. extremal faces etc.? and homology/cohomology?), infinitely many (as with a fractal boundary) or all extremal points? Can symplectic geometry be considered a study of the outside and conformal geometry a study of the inside?

Note that 2-dimensional phase space (as with a spring) is the simplest as there is no 1-dimensional phase space and there can't be (we need both position and momentum).
2019 sausio 27 d., 08:10 atliko AndriusKulikauskas -
Pakeista 168 eilutė iš:
Note that symplectic geometry, in preserving areas, seems to only care about the extremal points. In what sense is that true in phase space? When and how could it not be true? What is the difference between having finitely many extremal points (convex polytope?), infinitely many (as with a fractal boundary) or all extremal points? Can symplectic geometry be considered a study of the outside and conformal geometry a study of the inside?
į:
Note that symplectic geometry, in preserving areas, seems to only care about the extremal points. In what sense is that true in phase space? When and how could it not be true? What is the difference between having finitely many extremal points (convex polytope? and what is the significance of extremal points vs. extremal edges vs. extremal faces etc.? and homology/cohomology?), infinitely many (as with a fractal boundary) or all extremal points? Can symplectic geometry be considered a study of the outside and conformal geometry a study of the inside?
2019 sausio 27 d., 08:09 atliko AndriusKulikauskas -
Pakeista 168 eilutė iš:
Note that symplectic geometry, in preserving areas, seems to only care about the extremal points. In what sense is that true in phase space? When and how could it not be true? What is the difference between having finitely many extremal points, infinitely many (as with a fractal boundary) or all extremal points? Can symplectic geometry be considered a study of the outside and conformal geometry a study of the inside?
į:
Note that symplectic geometry, in preserving areas, seems to only care about the extremal points. In what sense is that true in phase space? When and how could it not be true? What is the difference between having finitely many extremal points (convex polytope?), infinitely many (as with a fractal boundary) or all extremal points? Can symplectic geometry be considered a study of the outside and conformal geometry a study of the inside?
2019 sausio 27 d., 08:09 atliko AndriusKulikauskas -
Pakeistos 166-168 eilutės iš
How do symmetries of paths relate to symmetries of young diagrams
į:
How do symmetries of paths relate to symmetries of young diagrams

Note that symplectic geometry, in preserving areas, seems to only care about the extremal points. In what sense is that true in phase space? When and how could it not be true? What is the difference between having finitely many extremal points, infinitely many (as with a fractal boundary) or all extremal points? Can symplectic geometry be considered a study of the outside and conformal geometry a study of the inside?
2019 sausio 25 d., 11:58 atliko AndriusKulikauskas -
Pakeistos 164-166 eilutės iš
Symplectic - basis for coupling - coupling of electric and magnetic fields - is what is responsible for the periodic nature of waves - the higher the frequency, the higher the energy, the tighter the coupling - the coupling is across the entire universe. The coupling models looseness - slack. This brings to mind the vacillation between knowing and not knowing.
į:
Symplectic - basis for coupling - coupling of electric and magnetic fields - is what is responsible for the periodic nature of waves - the higher the frequency, the higher the energy, the tighter the coupling - the coupling is across the entire universe. The coupling models looseness - slack. This brings to mind the vacillation between knowing and not knowing.

How do symmetries of paths relate to symmetries of young diagrams
2019 sausio 22 d., 14:25 atliko AndriusKulikauskas -
Pakeista 164 eilutė iš:
Symplectic - basis for coupling - coupling of electric and magnetic fields - is what is responsible for the periodic nature of waves - the higher the frequency, the higher the energy, the tighter the coupling - the coupling is across the entire universe. The coupling models looseness - slack.
į:
Symplectic - basis for coupling - coupling of electric and magnetic fields - is what is responsible for the periodic nature of waves - the higher the frequency, the higher the energy, the tighter the coupling - the coupling is across the entire universe. The coupling models looseness - slack. This brings to mind the vacillation between knowing and not knowing.
2019 sausio 22 d., 12:33 atliko AndriusKulikauskas -
Pakeista 164 eilutė iš:
Symplectic - basis for coupling - coupling of electric and magnetic fields - is what is responsible for the periodic nature of waves - the higher the frequency, the higher the energy, the tighter the coupling - the coupling is across the entire universe
į:
Symplectic - basis for coupling - coupling of electric and magnetic fields - is what is responsible for the periodic nature of waves - the higher the frequency, the higher the energy, the tighter the coupling - the coupling is across the entire universe. The coupling models looseness - slack.
2019 sausio 22 d., 12:25 atliko AndriusKulikauskas -
Pakeista 164 eilutė iš:
Symplectic - basis for coupling - coupling of electric and magnetic fields - is what is responsible for the periodic nature of waves - the higher the frequency, the higher the energy, the tighter the coupling
į:
Symplectic - basis for coupling - coupling of electric and magnetic fields - is what is responsible for the periodic nature of waves - the higher the frequency, the higher the energy, the tighter the coupling - the coupling is across the entire universe
2019 sausio 22 d., 12:25 atliko AndriusKulikauskas -
Pakeistos 162-164 eilutės iš
* Walks On Trees are perhaps important as they combine both unification, as the tree has a root, and completion, as given by the walk. In college, I asked God what kind of mathematics might be relevant to knowing everything, and I understood him to say that walks on trees where the trees are made of the elements of the threesome.
į:
* Walks On Trees are perhaps important as they combine both unification, as the tree has a root, and completion, as given by the walk. In college, I asked God what kind of mathematics might be relevant to knowing everything, and I understood him to say that walks on trees where the trees are made of the elements of the threesome.

Symplectic - basis for coupling - coupling of electric and magnetic fields - is what is responsible for the periodic nature of waves - the higher the frequency, the higher the energy, the tighter the coupling
2019 sausio 21 d., 13:17 atliko AndriusKulikauskas -
Pakeistos 159-162 eilutės iš
Lagrangian: (Force) Change in potential energy = (mass x acceleration) Change in kinetic energy. Potential energy is based on position and not time. Kinetic energy is based on time and not position.
į:
Lagrangian: (Force) Change in potential energy = (mass x acceleration) Change in kinetic energy. Potential energy is based on position and not time. Kinetic energy is based on time and not position.

Walks on trees
* Walks On Trees are perhaps important as they combine both unification, as the tree has a root, and completion, as given by the walk. In college, I asked God what kind of mathematics might be relevant to knowing everything, and I understood him to say that walks on trees where the trees are made of the elements of the threesome
.
2019 sausio 19 d., 12:36 atliko AndriusKulikauskas -
Ištrintos 108-110 eilutės:
[[https://link.springer.com/content/pdf/10.1007%2Fs00220-009-0951-9.pdf | Categorified Symplectic Geometry and the Classical String]]
* [[https://golem.ph.utexas.edu/category/2008/06/classical_string_theory_and_ca.html | Five related lectures by Christopher L. Rogers]]
Pakeistos 129-138 eilutės iš
Affine geometry preserves lines. Projective geometry also preserves zeros. Conformal geometry preserves angles. Symplectic geometry preservers oriented area. What are all these objects?

Affine geometry is agnostic regarding coordinate system. So it doesn't distinguish if we flip all the positive and minus choices.

Affine transformation extends a linear transformation by a column (the translation) and a row (of zeroes) and a diagonal element (of 1). Thus it is similar to a Lie algebra (Dn ?) extending An.

John Baez on duality: Dyson's Threefold Way: either X is not isomorphic to its dual (the complex case), or it is isomorphic to its dual (in the real or quaternionic cases). Study his slides!

John Baez periodic table and stablization theorem - relate to Cayley Dickson construction and its dualities.
į:
Four geometries
* Affine geometry preserves
lines. Projective geometry also preserves zeros. Conformal geometry preserves angles. Symplectic geometry preservers oriented area. What are all these objects?

Affine geometry
* Affine geometry is agnostic regarding coordinate system. So it doesn't distinguish if we flip all the positive and minus choices.
* Affine transformation extends a linear transformation by a column (the translation) and a row (of zeroes) and a diagonal element (of 1). Thus it is similar to a Lie algebra (Dn ?) extending An.

Duality
*
John Baez on duality: Dyson's Threefold Way: either X is not isomorphic to its dual (the complex case), or it is isomorphic to its dual (in the real or quaternionic cases). Study his slides!
Pakeistos 144-147 eilutės iš
Desargues theorem in geometry corresponds to the associative property in algebra.

Conformal
geometry preserves angles. Radial coordinates distinguishes distances r and angles theta, and makes use of the exponential {$e^{\pi i \theta}$}.
į:
Cayley-Dickson construction
* John Baez periodic table and stablization theorem - relate to Cayley Dickson construction and its dualities.

Projective geometry
* Desargues theorem in geometry corresponds to the associative property in algebra.
* A projective space is best understood from one dimension higher. And it is understood in terms of breaking down into smaller affine spaces. And these dimensions higher and lower are related to extending the chain of dimensions as given by Lie algebras. And the reversal of counting is related to the reversal of the orientation of lines etc. as a line is considered as a circle.

Conformal geometry
*
Conformal geometry preserves angles. Radial coordinates distinguishes distances r and angles theta, and makes use of the exponential {$e^{\pi i \theta}$}.
Pakeistos 156-158 eilutės iš
A projective space is best understood from one dimension higher. And it is understood in terms of breaking down into smaller affine spaces. And these dimensions higher and lower are related to extending the chain of dimensions as given by Lie algebras. And the reversal of counting is related to the reversal of the orientation of lines etc. as a line is considered as a circle.

Kinectic energy is possibly zero and builds up from there. It can be expressed in an absolute sense. Potential is possibly infinite and subtracts from that. So it must be expressed in a relative sense. So they are coming at energy from opposite directions.
į:
Symplectic geometry
*
Kinectic energy is possibly zero and builds up from there. It can be expressed in an absolute sense. Potential is possibly infinite and subtracts from that. So it must be expressed in a relative sense. So they are coming at energy from opposite directions.
2019 sausio 16 d., 11:21 atliko AndriusKulikauskas -
Pakeista 157 eilutė iš:
Lagrangian: (Force) Change in potential energy = (mass x acceleration) Change in kinetic energy
į:
Lagrangian: (Force) Change in potential energy = (mass x acceleration) Change in kinetic energy. Potential energy is based on position and not time. Kinetic energy is based on time and not position.
2019 sausio 16 d., 11:14 atliko AndriusKulikauskas -
Pakeistos 155-157 eilutės iš
Kinectic energy is possibly zero and builds up from there. It can be expressed in an absolute sense. Potential is possibly infinite and subtracts from that. So it must be expressed in a relative sense. So they are coming at energy from opposite directions.
į:
Kinectic energy is possibly zero and builds up from there. It can be expressed in an absolute sense. Potential is possibly infinite and subtracts from that. So it must be expressed in a relative sense. So they are coming at energy from opposite directions.

Lagrangian: (Force) Change in potential energy = (mass x acceleration) Change in kinetic energy
2019 sausio 16 d., 10:38 atliko AndriusKulikauskas -
Pakeistos 153-155 eilutės iš
A projective space is best understood from one dimension higher. And it is understood in terms of breaking down into smaller affine spaces. And these dimensions higher and lower are related to extending the chain of dimensions as given by Lie algebras. And the reversal of counting is related to the reversal of the orientation of lines etc. as a line is considered as a circle.
į:
A projective space is best understood from one dimension higher. And it is understood in terms of breaking down into smaller affine spaces. And these dimensions higher and lower are related to extending the chain of dimensions as given by Lie algebras. And the reversal of counting is related to the reversal of the orientation of lines etc. as a line is considered as a circle.

Kinectic energy is possibly zero and builds up from there. It can be expressed in an absolute sense. Potential is possibly infinite and subtracts from that. So it must be expressed in a relative sense. So they are coming at energy from opposite directions
.
2019 sausio 15 d., 21:43 atliko AndriusKulikauskas -
Pakeista 153 eilutė iš:
A projective space is best understood from one dimension higher. And it is understood in terms of breaking down into smaller affine spaces. And these dimensions higher and lower are related to extending the chain of dimensions as given by Lie algebras. And the reversal of counting is related to the reversal of the orientation of lines etc.
į:
A projective space is best understood from one dimension higher. And it is understood in terms of breaking down into smaller affine spaces. And these dimensions higher and lower are related to extending the chain of dimensions as given by Lie algebras. And the reversal of counting is related to the reversal of the orientation of lines etc. as a line is considered as a circle.
2019 sausio 15 d., 21:43 atliko AndriusKulikauskas -
Pakeistos 151-153 eilutės iš
The interpretation {$\mathbb{C} \leftrightarrow \mathbb{R}^2$} gives meaning to the two axes. One axis the opposites 1 and -1 and the other axis is the opposites i and j. And they become related 1 to i to -1 to j. Thus multiplication by i is rotation by 90 degrees. It returns us not to 1 but sends us to -1.
į:
The interpretation {$\mathbb{C} \leftrightarrow \mathbb{R}^2$} gives meaning to the two axes. One axis the opposites 1 and -1 and the other axis is the opposites i and j. And they become related 1 to i to -1 to j. Thus multiplication by i is rotation by 90 degrees. It returns us not to 1 but sends us to -1.

A projective space is best understood from one dimension higher. And it is understood in terms of breaking down into smaller affine spaces. And these dimensions higher and lower are related to extending the chain of dimensions as given by Lie algebras. And the reversal of counting is related to the reversal of the orientation of lines etc
.
2019 sausio 10 d., 13:57 atliko AndriusKulikauskas -
Pakeistos 149-151 eilutės iš
Conformal geometry preserves angles. Radial coordinates distinguishes distances r and angles theta, and makes use of the exponential {$e^{\pi i \theta}$}.
į:
Conformal geometry preserves angles. Radial coordinates distinguishes distances r and angles theta, and makes use of the exponential {$e^{\pi i \theta}$}.

The interpretation {$\mathbb{C} \leftrightarrow \mathbb{R}^2$} gives meaning to the two axes. One axis the opposites 1 and -1 and the other axis is the opposites i and j. And they become related 1 to i to -1 to j. Thus multiplication by i is rotation by 90 degrees. It returns us not to 1 but sends us to -1
.
2019 sausio 10 d., 13:54 atliko AndriusKulikauskas -
Pakeista 149 eilutė iš:
Conformal geometry preserves angles. Radial coordinates distinguishes distances r and angles theta, and makes use of the exponential {$e^{\pi i \theta|$}.
į:
Conformal geometry preserves angles. Radial coordinates distinguishes distances r and angles theta, and makes use of the exponential {$e^{\pi i \theta}$}.
2019 sausio 10 d., 13:54 atliko AndriusKulikauskas -
Pakeistos 147-149 eilutės iš
Desargues theorem in geometry corresponds to the associative property in algebra.
į:
Desargues theorem in geometry corresponds to the associative property in algebra.

Conformal geometry preserves angles. Radial coordinates distinguishes distances r and angles theta, and makes use of the exponential {$e^{\pi i \theta|$}
.
2019 sausio 08 d., 23:12 atliko AndriusKulikauskas -
Pakeistos 145-147 eilutės iš
So the types of duality should give the types of forces.
į:
So the types of duality should give the types of forces.

Desargues theorem in geometry corresponds to the associative property in algebra
.
2019 sausio 07 d., 12:58 atliko AndriusKulikauskas -
Pakeistos 144-145 eilutės iš
* Weak force - time reversal
į:
* Weak force - time reversal
So the types of duality should give the types of forces.
2019 sausio 07 d., 12:58 atliko AndriusKulikauskas -
Pridėtos 142-144 eilutės:
Each physical force is related to a duality:
* Charge (matter and antimatter) - electromagnetism
* Weak force - time reversal
2019 sausio 04 d., 03:03 atliko AndriusKulikauskas -
Pakeistos 138-141 eilutės iš
John Baez on duality: Dyson's Threefold Way: either X is not isomorphic to its dual (the complex case), or it is isomorphic to its dual (in the real or quaternionic cases).
į:
John Baez on duality: Dyson's Threefold Way: either X is not isomorphic to its dual (the complex case), or it is isomorphic to its dual (in the real or quaternionic cases). Study his slides!

John Baez periodic table and stablization theorem - relate to Cayley Dickson construction and its dualities.
2019 sausio 03 d., 23:03 atliko AndriusKulikauskas -
Pakeistos 136-138 eilutės iš
Affine transformation extends a linear transformation by a column (the translation) and a row (of zeroes) and a diagonal element (of 1). Thus it is similar to a Lie algebra (Dn ?) extending An.
į:
Affine transformation extends a linear transformation by a column (the translation) and a row (of zeroes) and a diagonal element (of 1). Thus it is similar to a Lie algebra (Dn ?) extending An.

John Baez on duality: Dyson's Threefold Way: either X is not isomorphic to its dual (the complex case), or it is isomorphic to its dual (in the real or quaternionic cases)
.
2019 sausio 03 d., 19:32 atliko AndriusKulikauskas -
Pakeista 136 eilutė iš:
Affine transformation extends a linear transformation by a column (the translation) and a row (of zeroes) and a diagonal element (of 1). Thus it is similar to a Lie algebra extending An.
į:
Affine transformation extends a linear transformation by a column (the translation) and a row (of zeroes) and a diagonal element (of 1). Thus it is similar to a Lie algebra (Dn ?) extending An.
2019 sausio 03 d., 19:32 atliko AndriusKulikauskas -
Pakeistos 134-136 eilutės iš
Affine geometry is agnostic regarding coordinate system. So it doesn't distinguish if we flip all the positive and minus choices.
į:
Affine geometry is agnostic regarding coordinate system. So it doesn't distinguish if we flip all the positive and minus choices.

Affine transformation extends a linear transformation by a column (the translation) and a row (of zeroes) and a diagonal element (of 1). Thus it is similar to a Lie algebra extending An
.
2019 sausio 03 d., 18:51 atliko AndriusKulikauskas -
Pakeistos 132-134 eilutės iš
Affine geometry preserves lines. Projective geometry also preserves zeros. Conformal geometry preserves angles. Symplectic geometry preservers oriented area. What are all these objects?
į:
Affine geometry preserves lines. Projective geometry also preserves zeros. Conformal geometry preserves angles. Symplectic geometry preservers oriented area. What are all these objects?

Affine geometry is agnostic regarding coordinate system. So it doesn't distinguish if we flip all the positive and minus choices.
2019 sausio 02 d., 14:46 atliko AndriusKulikauskas -
Pakeistos 130-132 eilutės iš
Attach:QuadrupleFormulas.png
į:
Attach:QuadrupleFormulas.png

Affine geometry preserves lines. Projective geometry also preserves zeros. Conformal geometry preserves angles. Symplectic geometry preservers oriented area. What are all these objects?
2018 gruodžio 31 d., 20:13 atliko AndriusKulikauskas -
Pridėta 130 eilutė:
Attach:QuadrupleFormulas.png
2018 gruodžio 31 d., 20:13 atliko AndriusKulikauskas -
Pakeistos 126-129 eilutės iš
[[https://www.youtube.com/watch?v=7d5jhPmVQ1w | John Baez on duality in logic and physics]]
į:
[[https://www.youtube.com/watch?v=7d5jhPmVQ1w | John Baez on duality in logic and physics]]

Attach:GeometryFormulas.png
2018 gruodžio 30 d., 01:58 atliko AndriusKulikauskas -
Pakeistos 124-126 eilutės iš
Homogeneous coordinates are what let us go between a simplex (when Z=1) and the coordinate system (when Z is free). Compare with the kinds of variables.
į:
Homogeneous coordinates are what let us go between a simplex (when Z=1) and the coordinate system (when Z is free). Compare with the kinds of variables.

[[https://www.youtube.com/watch?v=7d5jhPmVQ1w | John Baez on duality in logic and physics]]
2018 gruodžio 29 d., 16:22 atliko AndriusKulikauskas -
Pakeistos 122-124 eilutės iš
Projective geometry: homogeneous coordinates add a variable Z that makes each term of maximal power N by contributing the needed power of Z.
į:
Projective geometry: homogeneous coordinates add a variable Z that makes each term of maximal power N by contributing the needed power of Z.

Homogeneous coordinates are what let us go between a simplex (when Z=1) and the coordinate system (when Z is free). Compare with the kinds of variables
.
2018 gruodžio 29 d., 14:49 atliko AndriusKulikauskas -
Pakeistos 120-122 eilutės iš
Try to express projective geometry (or universal hyperbolic geometry) in terms of matrices and thus symmetric functions. What then is algebraic geometry and how do polynomials get involved? What is analytic geometry and in what sense does it go beyond matrices? How doe all of these hit up against the limits of matrices and the amount of symmetry in its internal folding?
į:
Try to express projective geometry (or universal hyperbolic geometry) in terms of matrices and thus symmetric functions. What then is algebraic geometry and how do polynomials get involved? What is analytic geometry and in what sense does it go beyond matrices? How doe all of these hit up against the limits of matrices and the amount of symmetry in its internal folding?

Projective geometry: homogeneous coordinates add a variable Z that makes each term of maximal power N by contributing the needed power of Z.
2018 gruodžio 22 d., 17:22 atliko AndriusKulikauskas -
Pakeistos 118-120 eilutės iš
Algebra (of the observer) and analysis (of the observed) exhibit a duality of worldviews.
į:
Algebra (of the observer) and analysis (of the observed) exhibit a duality of worldviews.

Try to express projective geometry (or universal hyperbolic geometry) in terms of matrices and thus symmetric functions. What then is algebraic geometry and how do polynomials get involved? What is analytic geometry and in what sense does it go beyond matrices? How doe all of these hit up against the limits of matrices and the amount of symmetry in its internal folding?
2018 gruodžio 21 d., 14:35 atliko AndriusKulikauskas -
Pakeistos 116-118 eilutės iš
Rotation relates one dimension to two others. How does this rotation work in higher dimensions? To what extent does multiplication of rotation (through three dimensional half turns) work in higher dimensions and how does it break down?
į:
Rotation relates one dimension to two others. How does this rotation work in higher dimensions? To what extent does multiplication of rotation (through three dimensional half turns) work in higher dimensions and how does it break down?

Algebra (of the observer) and analysis (of the observed) exhibit a duality of worldviews.
2018 gruodžio 21 d., 14:32 atliko AndriusKulikauskas -
Pakeistos 114-116 eilutės iš
G2 requires three lines to get between any two points (?) Relate this to the three-cycle.
į:
G2 requires three lines to get between any two points (?) Relate this to the three-cycle.

Rotation relates one dimension to two others. How does this rotation work in higher dimensions? To what extent does multiplication of rotation (through three dimensional half turns) work in higher dimensions and how does it break down?
2018 gruodžio 19 d., 12:38 atliko AndriusKulikauskas -
Pakeistos 112-114 eilutės iš
Spinor requires double rotation. Projective sphere is the opposite: half a rotation gets you back where you were.
į:
Spinor requires double rotation. Projective sphere is the opposite: half a rotation gets you back where you were.

G2 requires three lines to get between any two points (?) Relate this to the three-cycle
.
2018 gruodžio 19 d., 12:07 atliko AndriusKulikauskas -
Pakeistos 110-112 eilutės iš
* [[https://golem.ph.utexas.edu/category/2008/06/classical_string_theory_and_ca.html | Five related lectures by Christopher L. Rogers]]
į:
* [[https://golem.ph.utexas.edu/category/2008/06/classical_string_theory_and_ca.html | Five related lectures by Christopher L. Rogers]]

Spinor requires double rotation. Projective sphere is the opposite: half a rotation gets you back where you were.
2018 gruodžio 19 d., 12:03 atliko AndriusKulikauskas -
Pakeistos 107-110 eilutės iš
Dots in Dynkin diagrams are figures in the geometry, and edges are invariant relationships. A point can lie on a line, a line can lie on a plane. Those relationships are invariant under the actions of the symmetries. Dots in a Dynkin diagram correspond to maximal parabolic subgroups. They are the stablizer groups of these types of figures.
į:
Dots in Dynkin diagrams are figures in the geometry, and edges are invariant relationships. A point can lie on a line, a line can lie on a plane. Those relationships are invariant under the actions of the symmetries. Dots in a Dynkin diagram correspond to maximal parabolic subgroups. They are the stablizer groups of these types of figures.

[[https://link.springer.com/content/pdf/10.1007%2Fs00220-009-0951-9.pdf | Categorified Symplectic Geometry and the Classical String]]
* [[https://golem.ph.utexas.edu/category/2008/06/classical_string_theory_and_ca.html | Five related lectures by Christopher L. Rogers]]
2018 gruodžio 18 d., 14:08 atliko AndriusKulikauskas -
Pakeistos 105-107 eilutės iš
Discriminant of [[https://en.wikipedia.org/wiki/Elliptic_curve | elliptic curve]].
į:
Discriminant of [[https://en.wikipedia.org/wiki/Elliptic_curve | elliptic curve]].

Dots in Dynkin diagrams are figures in the geometry, and edges are invariant relationships. A point can lie on a line, a line can lie on a plane. Those relationships are invariant under the actions of the symmetries. Dots in a Dynkin diagram correspond to maximal parabolic subgroups. They are the stablizer groups of these types of figures
.
2018 gruodžio 17 d., 11:19 atliko AndriusKulikauskas -
Pakeistos 103-105 eilutės iš
[[https://en.wikipedia.org/wiki/Dedekind_eta_function | Dedekind eta function]] is based on 24.
į:
[[https://en.wikipedia.org/wiki/Dedekind_eta_function | Dedekind eta function]] is based on 24.

Discriminant of [[https://en.wikipedia.org/wiki/Elliptic_curve | elliptic curve]]
.
2018 gruodžio 16 d., 23:48 atliko AndriusKulikauskas -
Pakeistos 99-103 eilutės iš
Euler's manipulations of infinite series (adding up to -1/12 etc.) are related to divisions of everything, of the whole. Consider the Riemann Zeta function as describing the whole. The same mysteries of infinity are involved in renormalization, take a look at that.
į:
Euler's manipulations of infinite series (adding up to -1/12 etc.) are related to divisions of everything, of the whole. Consider the Riemann Zeta function as describing the whole. The same mysteries of infinity are involved in renormalization, take a look at that.

John Baez: 24 = 6 x 4 = An x Bn

[[https://en.wikipedia.org/wiki/Dedekind_eta_function | Dedekind eta function]] is based on 24
.
2018 gruodžio 16 d., 23:20 atliko AndriusKulikauskas -
Pakeistos 97-99 eilutės iš
What is the connection between Bott periodicity and spinors? See [[http://www.ams.org/journals/bull/2002-39-02/S0273-0979-01-00934-X/S0273-0979-01-00934-X.pdf | John Baez, The Octonions]].
į:
What is the connection between Bott periodicity and spinors? See [[http://www.ams.org/journals/bull/2002-39-02/S0273-0979-01-00934-X/S0273-0979-01-00934-X.pdf | John Baez, The Octonions]].

Euler's manipulations of infinite series (adding up to -1/12 etc.) are related to divisions of everything, of the whole. Consider the Riemann Zeta function as describing the whole. The same mysteries of infinity are involved in renormalization, take a look at that
.
2018 gruodžio 16 d., 22:28 atliko AndriusKulikauskas -
Pakeistos 95-97 eilutės iš
Nobody know what E8 is the symmetry group of. (Going beyond oneself?)
į:
Nobody know what E8 is the symmetry group of. (Going beyond oneself?)

What is the connection between Bott periodicity and spinors? See [[http://www.ams.org/journals/bull/2002-39-02/S0273-0979-01-00934-X/S0273-0979-01-00934-X.pdf | John Baez, The Octonions]].
2018 gruodžio 16 d., 18:27 atliko AndriusKulikauskas -
Pakeistos 93-95 eilutės iš
Compare trejybės ratas (for the quaternions) and Fano's plane (aštuonerybė) for the octonions.
į:
Compare trejybės ratas (for the quaternions) and Fano's plane (aštuonerybė) for the octonions.

Nobody know what E8 is the symmetry group of. (Going beyond oneself?)
2018 gruodžio 16 d., 12:55 atliko AndriusKulikauskas -
Pakeistos 91-93 eilutės iš
[[https://www.math.columbia.edu/~woit/wordpress/?p=5927 | Geometric unity]] I tend to agree with Roger Penrose that spin has been one of the great mysteries in quantum mechanics. As best as I can recall, he said it was one of two primary mysteries in a talk at NYU back in the late 1990’s. ... understand spin and I think we’ll understand entanglement a lot better.
į:
[[https://www.math.columbia.edu/~woit/wordpress/?p=5927 | Geometric unity]] I tend to agree with Roger Penrose that spin has been one of the great mysteries in quantum mechanics. As best as I can recall, he said it was one of two primary mysteries in a talk at NYU back in the late 1990’s. ... understand spin and I think we’ll understand entanglement a lot better.

Compare trejybės ratas (for the quaternions) and Fano's plane (aštuonerybė) for the octonions
.
2018 gruodžio 15 d., 13:00 atliko AndriusKulikauskas -
2018 gruodžio 15 d., 12:37 atliko AndriusKulikauskas -
Pakeistos 89-91 eilutės iš
[[https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_theory | Yang-Mills theory]].
į:
[[https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_theory | Yang-Mills theory]].

[[https://www.math.columbia.edu/~woit/wordpress/?p=5927 | Geometric unity]] I tend to agree with Roger Penrose that spin has been one of the great mysteries in quantum mechanics. As best as I can recall, he said it was one of two primary mysteries in a talk at NYU back in the late 1990’s. ... understand spin and I think we’ll understand entanglement a lot better
.
2018 gruodžio 15 d., 11:41 atliko AndriusKulikauskas -
Pakeista 89 eilutė iš:
https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_theory
į:
[[https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_theory | Yang-Mills theory]].
2018 gruodžio 15 d., 11:40 atliko AndriusKulikauskas -
Pakeistos 86-89 eilutės iš
Which state is which amongst "one" and "another" is maintained until it is unnecessary - this is quantum entanglement.
į:
Which state is which amongst "one" and "another" is maintained until it is unnecessary - this is quantum entanglement.

Massless particles acquire mass through symmetry breaking:
https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_theory
2018 gruodžio 13 d., 11:27 atliko AndriusKulikauskas -
Pakeistos 84-86 eilutės iš
This is the difference between thinking of the negative dimension as "explicitly" written out, or thinking of it as simply as one of two "implicit" states that we switch between.
į:
This is the difference between thinking of the negative dimension as "explicitly" written out, or thinking of it as simply as one of two "implicit" states that we switch between.

Which state is which amongst "one" and "another" is maintained until it is unnecessary - this is quantum entanglement
.
2018 gruodžio 13 d., 11:25 atliko AndriusKulikauskas -
Pakeistos 82-84 eilutės iš
If we think of rotation by i as relating two dimensions, then {$i^2$} takes a dimension to its negative. So that is helpful when we are thinking of the "extra" distinguished dimensions (1). And if that dimension is attributed to a line, then this interpretation reflects along that line. But when we compare rotations as such, then we are not comparing lines, but rather rotations. In this case if we perform an entire rotation, then we flip the rotations for that other dimension that we have rotated about. So this means that the relation between rotations as such is very different than the relation with the isolated distinguised dimension. The relations between rotations is such is, I think, given by {$A_n$} whereas the distinguished dimension is an extra dimension which gets represented, I think, in terms of {$B_n$}, as the short root.
į:
If we think of rotation by i as relating two dimensions, then {$i^2$} takes a dimension to its negative. So that is helpful when we are thinking of the "extra" distinguished dimensions (1). And if that dimension is attributed to a line, then this interpretation reflects along that line. But when we compare rotations as such, then we are not comparing lines, but rather rotations. In this case if we perform an entire rotation, then we flip the rotations for that other dimension that we have rotated about. So this means that the relation between rotations as such is very different than the relation with the isolated distinguised dimension. The relations between rotations is such is, I think, given by {$A_n$} whereas the distinguished dimension is an extra dimension which gets represented, I think, in terms of {$B_n$}, as the short root.

This is the difference between thinking of the negative dimension as "explicitly" written out, or thinking of it as simply as one of two "implicit" states that we switch between
.
2018 gruodžio 13 d., 11:19 atliko AndriusKulikauskas -
Pakeistos 80-82 eilutės iš
Usually multiplication by i is identified with a rotation of 90 degrees. However, we can instead identify it with a rotation of 180 degrees if we consider, as in the case of spinors, that the first time around it adds a sign of -1, and it needs to go around twice in order to establish a sign of +1. This is the definition that makes spin composition work in three dimensions, for the quaternions. The usual geometric interpretation of complex numbers is then a particular reinterpretation that is possible but not canonical. Rotation gets you on the other side of the page.
į:
Usually multiplication by i is identified with a rotation of 90 degrees. However, we can instead identify it with a rotation of 180 degrees if we consider, as in the case of spinors, that the first time around it adds a sign of -1, and it needs to go around twice in order to establish a sign of +1. This is the definition that makes spin composition work in three dimensions, for the quaternions. The usual geometric interpretation of complex numbers is then a particular reinterpretation that is possible but not canonical. Rotation gets you on the other side of the page.

If we think of rotation by i as relating two dimensions, then {$i^2$} takes a dimension to its negative. So that is helpful when we are thinking of the "extra" distinguished dimensions (1). And if that dimension is attributed to a line, then this interpretation reflects along that line. But when we compare rotations as such, then we are not comparing lines, but rather rotations. In this case if we perform an entire rotation, then we flip the rotations for that other dimension that we have rotated about. So this means that the relation between rotations as such is very different than the relation with the isolated distinguised dimension. The relations between rotations is such is, I think, given by {$A_n$} whereas the distinguished dimension is an extra dimension which gets represented, I think, in terms of {$B_n$}, as the short root
.
2018 gruodžio 13 d., 11:01 atliko AndriusKulikauskas -
Pakeista 80 eilutė iš:
Usually multiplication by i is identified with a rotation of 90 degrees. However, we can instead identify it with a rotation of 180 degrees if we consider, as in the case of spinors, that the first time around it adds a sign of -1, and it needs to go around twice in order to establish a sign of +1. This is the definition that makes spin composition work in three dimensions, for the quaternions. The usual geometric interpretation of complex numbers is then a particular reinterpretation that is possible but not canonical.
į:
Usually multiplication by i is identified with a rotation of 90 degrees. However, we can instead identify it with a rotation of 180 degrees if we consider, as in the case of spinors, that the first time around it adds a sign of -1, and it needs to go around twice in order to establish a sign of +1. This is the definition that makes spin composition work in three dimensions, for the quaternions. The usual geometric interpretation of complex numbers is then a particular reinterpretation that is possible but not canonical. Rotation gets you on the other side of the page.
2018 gruodžio 13 d., 08:41 atliko AndriusKulikauskas -
Pakeista 80 eilutė iš:
Usually multiplication by i is identified with a rotation of 90 degrees. However, we can instead identify it with a rotation of 180 degrees if we consider, as in the case of spinors, that the first time around it adds a sign of -1, and it needs to go around twice in order to establish a sign of +1. This is the definition that makes spin composition work in three dimensions, for the quaternions.
į:
Usually multiplication by i is identified with a rotation of 90 degrees. However, we can instead identify it with a rotation of 180 degrees if we consider, as in the case of spinors, that the first time around it adds a sign of -1, and it needs to go around twice in order to establish a sign of +1. This is the definition that makes spin composition work in three dimensions, for the quaternions. The usual geometric interpretation of complex numbers is then a particular reinterpretation that is possible but not canonical.
2018 gruodžio 13 d., 08:40 atliko AndriusKulikauskas -
Pakeistos 78-80 eilutės iš
* polar conjugates in projective geometry (see Wildberger)
į:
* polar conjugates in projective geometry (see Wildberger)

Usually multiplication by i is identified with a rotation of 90 degrees. However, we can instead identify it with a rotation of 180 degrees if we consider, as in the case of spinors, that the first time around it adds a sign of -1, and it needs to go around twice in order to establish a sign of +1. This is the definition that makes spin composition work in three dimensions, for the quaternions.
2018 gruodžio 11 d., 18:38 atliko AndriusKulikauskas -
Pridėtos 71-78 eilutės:

Duality examples (conjugates)
* complex number "i" is not one number - it is a pair of numbers that are the square roots of -1
* spinors likewise
* Dn where n=2
* the smallest cross-polytope with 2 vertices
* taking a sphere and identifying antipodal elements - this is a famous group
* polar conjugates in projective geometry (see Wildberger)
2018 gruodžio 10 d., 14:23 atliko AndriusKulikauskas -
Pridėtos 69-70 eilutės:

Root systems relate two spheres - they relate two "sheets". Logic likewise relates two sheets: a sheet and a meta-sheet for working on a problem. Similarly, we model our attention by awareness, as Graziano pointed out. This is stepping in and stepping out.
2018 gruodžio 10 d., 13:38 atliko AndriusKulikauskas -
Pridėtos 64-68 eilutės:

An simplexes allow gaps because they have choice between "is" and "not". But all the other frameworks lack an explicit gap and so we get the explicit second counting. But:
* for Bn hypercubes we divide the "not" into two halves, preserving the "is" intact.
* for Cn cross-polytopes we divide the "is" into two halves, preserving the "not" intact.
* for Dn we have simply "this" and "that" (not-this).
2018 gruodžio 10 d., 13:14 atliko AndriusKulikauskas -
Pridėtos 62-63 eilutės:

Use "this" and "that" as unmarked opposites - conjugates.
2018 gruodžio 09 d., 15:01 atliko AndriusKulikauskas -
Pakeistos 47-48 eilutės iš
Kada pasirinkimo samprata keičiasi, visgi už visų sampratų slypi bendresnis, pirmesnis suvokimų suvokimas, taip kad renkamės pačią sampratą.
į:
Kada pasirinkimo samprata keičiasi, visgi už visų sampratų slypi bendresnis, pirmesnis suvokimų suvokimas, taip kad renkamės pačią sampratą. Folding is the basis for substitution.
Pakeistos 57-61 eilutės iš
How to interpret possible expansions? For example, composition of function has a distinctive direction. Whereas a commutative product, or a set, does not.
į:
How to interpret possible expansions? For example, composition of function has a distinctive direction. Whereas a commutative product, or a set, does not.

Keturios pasirinkimo sampratos (apimtys) visos reikalingos norint išskirti vieną paskirą pasirinkimą.

Bott periodicity is the basis for 8-fold folding and unfolding
.
2018 gruodžio 09 d., 14:59 atliko AndriusKulikauskas -
Pakeistos 53-57 eilutės iš
{$x_0$} is fundamentally different from {$x_i$}. The former appears in the positive form {$\pm(x_0-x_1)$} but the others appear both positive and negative.
į:
{$x_0$} is fundamentally different from {$x_i$}. The former appears in the positive form {$\pm(x_0-x_1)$} but the others appear both positive and negative.

Kaip dvi skaičiavimo kryptis (conjugate) sujungti apsisukimu?

How to interpret possible expansions? For example, composition of function has a distinctive direction. Whereas a commutative product, or a set, does not
.
2018 gruodžio 09 d., 14:57 atliko AndriusKulikauskas -
Pakeista 39 eilutė iš:
Išėjimas už savęs reiškiasi kaip susilankstymas, išsivertimas, užtat tėra keturi skaičiai: +0, +1, +2, +3. Toliau yra dešimts tūkstantys daiktų, kaip sako Dao De Jing. Trečias yra begalybė.
į:
Išėjimas už savęs reiškiasi kaip susilankstymas, išsivertimas, užtat tėra keturi skaičiai: +0, +1, +2, +3. Šie pirmieji skaičiai yra išskirtiniai. Toliau gaunasi (didėjančio ir mažėjančio laisvumo palaikomas) bendras skaičiavimas, yra dešimts tūkstantys daiktų, kaip sako Dao De Jing. Trečias yra begalybė.
2018 gruodžio 09 d., 14:44 atliko AndriusKulikauskas -
Pakeista 53 eilutė iš:
{$x_0$} is fundamentally different from {$x_i$}.
į:
{$x_0$} is fundamentally different from {$x_i$}. The former appears in the positive form {$\pm(x_0-x_1)$} but the others appear both positive and negative.
2018 gruodžio 09 d., 14:42 atliko AndriusKulikauskas -
Pakeistos 47-53 eilutės iš
Kada pasirinkimo samprata keičiasi, visgi už visų sampratų slypi bendresnis, pirmesnis suvokimų suvokimas, taip kad renkamės pačią sampratą.
į:
Kada pasirinkimo samprata keičiasi, visgi už visų sampratų slypi bendresnis, pirmesnis suvokimų suvokimas, taip kad renkamės pačią sampratą.

Fizikoje, posūkis yra viskas. Palyginti su ortogonaline grupe.

Bott periodicity exhibits self-folding. Note the duality with the pseudoscalar. Consider the formula n(n+1)/2 does that relate to the entries of a matrix?

{$x_0$} is fundamentally different from {$x_i$}
.
2018 gruodžio 09 d., 14:37 atliko AndriusKulikauskas -
Pakeistos 39-47 eilutės iš
Išėjimas už savęs reiškiasi kaip susilankstymas, išsivertimas, užtat tėra keturi skaičiai: +0, +1, +2, +3. Toliau yra dešimts tūkstantys daiktų, kaip sako Dao De Jing. Trečias yra begalybė.
į:
Išėjimas už savęs reiškiasi kaip susilankstymas, išsivertimas, užtat tėra keturi skaičiai: +0, +1, +2, +3. Toliau yra dešimts tūkstantys daiktų, kaip sako Dao De Jing. Trečias yra begalybė.

Esminis pasirinkimas yra: kurią pasirinkimo sampratą rinksimės?

Kaip suvokiame {$x_i$}? Koks tai per pasirinkimas?

Kaip sekos lankstymą susieti su baltymų lankstymu ir pasukimu?

Kada pasirinkimo samprata keičiasi, visgi už visų sampratų slypi bendresnis, pirmesnis suvokimų suvokimas, taip kad renkamės pačią sampratą
.
2018 gruodžio 09 d., 14:15 atliko AndriusKulikauskas -
Pakeistos 37-39 eilutės iš
An relates to "center of mass". How does this relate to the asymmetry of whole and center?
į:
An relates to "center of mass". How does this relate to the asymmetry of whole and center?

Išėjimas už savęs reiškiasi kaip susilankstymas, išsivertimas, užtat tėra keturi skaičiai: +0, +1, +2, +3. Toliau yra dešimts tūkstantys daiktų, kaip sako Dao De Jing. Trečias yra begalybė.
2018 gruodžio 05 d., 12:04 atliko AndriusKulikauskas -
Pakeistos 35-37 eilutės iš
Symplectic algebras are always even dimensional whereas orthogonal algebras can be odd or even. What do odd dimensional orthogonal algebras mean? How are we to understand them?
į:
Symplectic algebras are always even dimensional whereas orthogonal algebras can be odd or even. What do odd dimensional orthogonal algebras mean? How are we to understand them?

An relates to "center of mass". How does this relate to the asymmetry of whole and center
?
2018 lapkričio 25 d., 09:38 atliko AndriusKulikauskas -
Pakeistos 33-35 eilutės iš
Composition algebra. Doubling is related to duality.
į:
Composition algebra. Doubling is related to duality.

Symplectic algebras are always even dimensional whereas orthogonal algebras can be odd or even. What do odd dimensional orthogonal algebras mean? How are we to understand them?
2018 lapkričio 25 d., 09:35 atliko AndriusKulikauskas -
Pakeistos 31-33 eilutės iš
{$A_n$} tracefree condition is similar to working with independent variables in the the center of mass frame of a multiparticle system. (Sunil, Mukunda). In other words, the system has a center! And every subsystem has a center.
į:
{$A_n$} tracefree condition is similar to working with independent variables in the the center of mass frame of a multiparticle system. (Sunil, Mukunda). In other words, the system has a center! And every subsystem has a center.

Composition algebra. Doubling is related to duality
.
2018 lapkričio 17 d., 12:42 atliko AndriusKulikauskas -
Pakeistos 29-31 eilutės iš
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?
į:
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?

{$A_n$} tracefree condition is similar to working with independent variables in the the center of mass frame of a multiparticle system. (Sunil, Mukunda). In other words, the system has a center! And every subsystem has a center.
2018 lapkričio 13 d., 13:12 atliko AndriusKulikauskas -
2018 lapkričio 11 d., 17:37 atliko AndriusKulikauskas -
Ištrintos 8-10 eilutės:
Montažas ir geometrinės permainos:
* Shear: sideshot
Ištrintos 9-10 eilutės:

(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?
2018 lapkričio 11 d., 17:35 atliko AndriusKulikauskas -
Ištrintos 6-9 eilutės:
The nature of math
* Gramatika visada turi prasmę. Matematika yra tai, kas pavaldu logikai, bet nebūtinai turi prasmę.
* Matematika mus moko, kaip besąlygiškumą reikšti sąlygomis, o tai įmanoma sąlygiškai.
Ištrintos 11-12 eilutės:
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.
Ištrintos 12-16 eilutės:

[[https://ac.els-cdn.com/S0012365X96003500/1-s2.0-S0012365X96003500-main.pdf?_tid=7e02ab6e-b80c-44b3-9ef0-33327ac6a6d1&acdnat=1531742605_d128ba18dbe99b9c084df275fab65e08 | Geometry of Classical Groups over Finite Fields and Its Applications]], Zhe-xian Wan

* 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.
2018 lapkričio 11 d., 17:32 atliko AndriusKulikauskas -
Ištrintos 18-19 eilutės:

Collect examples of the arithmetic hierarchy such as calculus (delta-epsilon), differentiable manifolds, etc.
Ištrintos 22-23 eilutės:
Ištrintos 25-26 eilutės:

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.
2018 lapkričio 11 d., 17:29 atliko AndriusKulikauskas -
Ištrintos 0-3 eilutės:
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?
Pakeistos 9-10 eilutės iš
į:
* Matematika mus moko, kaip besąlygiškumą reikšti sąlygomis, o tai įmanoma sąlygiškai.
Pakeistos 26-29 eilutės iš
* 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).
į:
Pakeistos 28-30 eilutės iš
* 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.
į:
Ištrintos 30-33 eilutės:
Matematika mus moko, kaip besąlygiškumą reikšti sąlygomis, o tai įmanoma sąlygiškai.

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).
Ištrintos 38-39 eilutės:
What is the significance of a cube having four diagonals that can be permuted by S4?
Pakeistos 51-57 eilutės iš
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?

The combinatorial interpretation of n-choose-k counts placements = "external arrangements" n! x...x (n-k+1)! and then divides by the redundancies = "internal arrangements" k! Thus it relates external and internal (within subsystem).
į:
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?
2018 lapkričio 11 d., 17:20 atliko AndriusKulikauskas -
Ištrintos 0-2 eilutės:

Function can be partial, whereas a permutation maps completely.
Ištrintos 4-5 eilutės:
Kan extension - extending the domain - every concept is a Kan extension https://en.wikipedia.org/wiki/Kan_extension
Ištrintos 18-24 eilutės:
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.

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.
Ištrintos 20-21 eilutės:
Intrinsic ambiguity of propositions - every proposition is a general rule, which can be questioned or applied.
Ištrintos 23-32 eilutės:

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.
Ištrinta 28 eilutė:
Ištrintos 63-64 eilutės:

Hyperoctahedral group is shuffling cards that can also be flipped or not (front or back).
2018 lapkričio 11 d., 17:00 atliko AndriusKulikauskas -
Pakeistos 1-2 eilutės iš
[[https://www.youtube.com/watch?v=pXGTevGJ01o | Symplectic manifolds]]
į:
Ištrintos 7-8 eilutės:
Relate triangulated categories with representations of threesome
Pakeistos 12-21 eilutės iš
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.

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

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

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
.
į:
Signal propagation - expansions
* 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.
* How do special rim hook tableaux (which depend on the behavior of their endpoints) relate to Dynkin diagrams (which also depend on their endpoints)?

The nature of math
* Gramatika visada turi prasmę. Matematika yra tai, kas pavaldu logikai, bet nebūtinai
turi prasmę.
Ištrintos 20-23 eilutės:
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.
Ištrintos 28-33 eilutės:
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
Ištrinta 36 eilutė:
Pakeistos 48-50 eilutės iš

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.
į:
Pakeistos 52-53 eilutės iš
What do inside and outside mean in symplectic (Hamiltionian, Lagrangian) mechanics?
į:
Ištrintos 61-63 eilutės:
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)
Ištrinta 63 eilutė:
Ištrinta 65 eilutė:
Ištrinta 66 eilutė:
2018 lapkričio 11 d., 16:44 atliko AndriusKulikauskas -
Ištrintos 2-3 eilutės:
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.
Ištrintos 4-11 eilutės:
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".
Ištrintos 12-13 eilutės:
Express the link between algebra and analysis in terms of exact sequences and Kan extensions. Explain why category theory not relevant for analysis.
Ištrintos 16-20 eilutės:
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
Ištrintos 18-19 eilutės:
Kodėl yra tiek daug būdų įrodyti Pitagoro teoremą?
Ištrintos 20-22 eilutės:
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.
Pakeistos 53-56 eilutės iš
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.
į:
Ištrintos 55-56 eilutės:
Collect examples of symmetry breaking.
Ištrintos 65-66 eilutės:
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.
Ištrintos 67-72 eilutės:
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?
Pakeistos 70-72 eilutės iš
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).
į:
What do inside and outside mean in symplectic (Hamiltionian, Lagrangian) mechanics?
Pakeistos 85-88 eilutės iš
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.
į:
Pakeistos 88-89 eilutės iš
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.
į:
Pakeistos 91-96 eilutės iš
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.
į:
Ištrintos 93-94 eilutės:
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.
Ištrintos 103-104 eilutės:
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.
Pakeistos 122-140 eilutės iš
Path of least action (the basis for physics, namely, for Feynman diagrams) is violated by measurements, where we can wait and nothing happens.

Symmetry group consists of distinguishable actions which accomplish nothing
(leave an object invariant). So they separate the object/environment and its state.

Taylor series for e^x is based on the symmetric function (inverted) - it is the basis for symmetry in analysis.

When we measure spin - we impose the spin axis we are expecting - but that is an imposition of expectations related to the "waiting" that I am modeling. Also, how is that waiting related to emotional life and expectations? Comparing measurements yields "information" and it may be information which is limited by the speed of time.

Is many-worlds theory the flip-side of least-action ?

Chess pieces (rooks, bishops, knights) move in very basic ways that bring to mind root systems and especially the geometries that they open up in the gap they create by being more complex e_i - e_j than the generic basis e_i.

Lie algebras (root systems) are the cracks or gaps between coordinate system layers. The "higher energy" a system has, the bigger the crack, the more possibility for variation and freedom, how the system will develop or degenerate. The lowest energy system is the one that has the roots e_i (and its byproducts, e_i-e_j, e_i+e_j etc.) because everything is explicit. And the highest energy system is A_n, given by e_i-e_j, because the e_i are implicitly latent.

The root systems, as a minimum, have to contain the e_i - e_j because they encode the Lie bracket of the e_i. The question is, what are the ways that they can be expanded? First, a dual encoding can be given e_i + e_j. And the two must then be related in one of the various ways.

The combinatorial interpretation of n-choose-k counts placements = "external arrangements" n! x...x (n-k+1)! and then divides by the redundancies = "internal arrangements" k! Thus it relates external and internal (within subsystem).

Entanglement - particle and anti-particle are in the same place and time - and they have the same clock and coordinates
į:
The combinatorial interpretation of n-choose-k counts placements = "external arrangements" n! x...x (n-k+1)! and then divides by the redundancies = "internal arrangements" k! Thus it relates external and internal (within subsystem).
2018 lapkričio 11 d., 12:43 atliko AndriusKulikauskas -
Pakeistos 187-189 eilutės iš
The combinatorial interpretation of n-choose-k counts placements = "external arrangements" n! x...x (n-k+1)! and then divides by the redundancies = "internal arrangements" k! Thus it relates external and internal (within subsystem).
į:
The combinatorial interpretation of n-choose-k counts placements = "external arrangements" n! x...x (n-k+1)! and then divides by the redundancies = "internal arrangements" k! Thus it relates external and internal (within subsystem).

Entanglement - particle and anti-particle are in the same place and time - and they have the same clock and coordinates
2018 lapkričio 09 d., 21:59 atliko AndriusKulikauskas -
Pakeistos 185-187 eilutės iš
The root systems, as a minimum, have to contain the e_i - e_j because they encode the Lie bracket of the e_i. The question is, what are the ways that they can be expanded? First, a dual encoding can be given e_i + e_j. And the two must then be related in one of the various ways.
į:
The root systems, as a minimum, have to contain the e_i - e_j because they encode the Lie bracket of the e_i. The question is, what are the ways that they can be expanded? First, a dual encoding can be given e_i + e_j. And the two must then be related in one of the various ways.

The combinatorial interpretation of n-choose-k counts placements = "external arrangements" n! x...x (n-k+1)! and then divides by the redundancies = "internal arrangements" k! Thus it relates external and internal (within subsystem)
.
2018 lapkričio 09 d., 15:30 atliko AndriusKulikauskas -
Pakeista 185 eilutė iš:
The root systems, as a minimum, have to contain the e_i - e_j because they encode the Lie bracket of the e_i. The question is, what are the ways that they can be expanded? First, a dual encoding can be given e_i + e_j. And the two must then be variously related.
į:
The root systems, as a minimum, have to contain the e_i - e_j because they encode the Lie bracket of the e_i. The question is, what are the ways that they can be expanded? First, a dual encoding can be given e_i + e_j. And the two must then be related in one of the various ways.
2018 lapkričio 09 d., 15:30 atliko AndriusKulikauskas -
Pakeista 185 eilutė iš:
The root systems, as a minimum, have to contain the e_i - e_j because they encode the Lie bracket of the e_i. The question is, what are the ways that they can be expanded? First, a dual encoding can be given e_i + e_j. And then the two can be related.
į:
The root systems, as a minimum, have to contain the e_i - e_j because they encode the Lie bracket of the e_i. The question is, what are the ways that they can be expanded? First, a dual encoding can be given e_i + e_j. And the two must then be variously related.
2018 lapkričio 09 d., 15:29 atliko AndriusKulikauskas -
Pakeistos 183-185 eilutės iš
Lie algebras (root systems) are the cracks or gaps between coordinate system layers. The "higher energy" a system has, the bigger the crack, the more possibility for variation and freedom, how the system will develop or degenerate. The lowest energy system is the one that has the roots e_i (and its byproducts, e_i-e_j, e_i+e_j etc.) because everything is explicit. And the highest energy system is A_n, given by e_i-e_j, because the e_i are implicitly latent.
į:
Lie algebras (root systems) are the cracks or gaps between coordinate system layers. The "higher energy" a system has, the bigger the crack, the more possibility for variation and freedom, how the system will develop or degenerate. The lowest energy system is the one that has the roots e_i (and its byproducts, e_i-e_j, e_i+e_j etc.) because everything is explicit. And the highest energy system is A_n, given by e_i-e_j, because the e_i are implicitly latent.

The root systems, as a minimum, have to contain the e_i - e_j because they encode the Lie bracket of the e_i. The question is, what are the ways that they can be expanded? First, a dual encoding can be given e_i + e_j. And then the two can be related
.
2018 lapkričio 08 d., 19:47 atliko AndriusKulikauskas -
Pakeistos 181-183 eilutės iš
Chess pieces (rooks, bishops, knights) move in very basic ways that bring to mind root systems and especially the geometries that they open up in the gap they create by being more complex e_i - e_j than the generic basis e_i.
į:
Chess pieces (rooks, bishops, knights) move in very basic ways that bring to mind root systems and especially the geometries that they open up in the gap they create by being more complex e_i - e_j than the generic basis e_i.

Lie algebras (root systems) are the cracks or gaps between coordinate system layers. The "higher energy" a system has, the bigger the crack, the more possibility for variation and freedom, how the system will develop or degenerate. The lowest energy system is the one that has the roots e_i (and its byproducts, e_i-e_j, e_i+e_j etc.) because everything is explicit. And the highest energy system is A_n, given by e_i-e_j, because the e_i are implicitly latent
.
2018 lapkričio 07 d., 22:37 atliko AndriusKulikauskas -
Pakeistos 179-181 eilutės iš
Is many-worlds theory the flip-side of least-action ?
į:
Is many-worlds theory the flip-side of least-action ?

Chess pieces (rooks, bishops, knights) move in very basic ways that bring to mind root systems and especially the geometries that they open up in the gap they create by being more complex e_i - e_j than the generic basis e_i.
2018 lapkričio 06 d., 15:47 atliko AndriusKulikauskas -
2018 lapkričio 06 d., 15:47 atliko AndriusKulikauskas -
Pakeistos 177-179 eilutės iš
When we measure spin - we impose the spin axis we are expecting - but that is an imposition of expectations related to the "waiting" that I am modeling. Also, how is that waiting related to emotional life and expectations? Comparing measurements yields "information" and it may be information which is limited by the speed of time.
į:
When we measure spin - we impose the spin axis we are expecting - but that is an imposition of expectations related to the "waiting" that I am modeling. Also, how is that waiting related to emotional life and expectations? Comparing measurements yields "information" and it may be information which is limited by the speed of time.

Is many-worlds theory the flip-side of least-action ?
2018 lapkričio 06 d., 15:34 atliko AndriusKulikauskas -
Pakeista 177 eilutė iš:
When we measure spin - we impose the spin axis we are expecting - but that is an imposition of expectations related to the "waiting" that I am modeling. Also, how is that waiting related to emotional life and expectations?
į:
When we measure spin - we impose the spin axis we are expecting - but that is an imposition of expectations related to the "waiting" that I am modeling. Also, how is that waiting related to emotional life and expectations? Comparing measurements yields "information" and it may be information which is limited by the speed of time.
2018 lapkričio 06 d., 15:27 atliko AndriusKulikauskas -
Pakeistos 175-177 eilutės iš
Taylor series for e^x is based on the symmetric function (inverted) - it is the basis for symmetry in analysis.
į:
Taylor series for e^x is based on the symmetric function (inverted) - it is the basis for symmetry in analysis.

When we measure spin - we impose the spin axis we are expecting - but that is an imposition of expectations related to the "waiting" that I am modeling. Also, how is that waiting related to emotional life and expectations?
2018 lapkričio 03 d., 14:11 atliko AndriusKulikauskas -
Pakeista 23 eilutė iš:
Express the link between algebra wnd analysis in terms of exact sequences and Kan extensions. Explain why category theory not relevant for analysis.
į:
Express the link between algebra and analysis in terms of exact sequences and Kan extensions. Explain why category theory not relevant for analysis.
2018 lapkričio 03 d., 10:51 atliko AndriusKulikauskas -
Pakeistos 173-175 eilutės iš
Symmetry group consists of distinguishable actions which accomplish nothing (leave an object invariant). So they separate the object/environment and its state.
į:
Symmetry group consists of distinguishable actions which accomplish nothing (leave an object invariant). So they separate the object/environment and its state.

Taylor series for e^x is based on the symmetric function (inverted) - it is the basis for symmetry in analysis
.
2018 lapkričio 03 d., 10:03 atliko AndriusKulikauskas -
Pakeistos 171-173 eilutės iš
Path of least action (the basis for physics, namely, for Feynman diagrams) is violated by measurements, where we can wait and nothing happens.
į:
Path of least action (the basis for physics, namely, for Feynman diagrams) is violated by measurements, where we can wait and nothing happens.

Symmetry group consists of distinguishable actions which accomplish nothing (leave an object invariant). So they separate the object/environment and its state
.
2018 spalio 31 d., 22:54 atliko AndriusKulikauskas -
Pakeista 171 eilutė iš:
Path of least action (the basis for physics, namely, for Feynman diagrams) is contradicted by measurements, where we can wait and nothing happens.
į:
Path of least action (the basis for physics, namely, for Feynman diagrams) is violated by measurements, where we can wait and nothing happens.
2018 spalio 31 d., 22:54 atliko AndriusKulikauskas -
Pakeistos 169-171 eilutės iš
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?
į:
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?

Path of least action (the basis for physics, namely, for Feynman diagrams) is contradicted by measurements, where we can wait and nothing happens.
2018 spalio 08 d., 19:15 atliko AndriusKulikauskas -
Pakeistos 167-169 eilutės iš
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)?
į:
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
?
2018 spalio 06 d., 12:10 atliko AndriusKulikauskas -
Pakeistos 165-167 eilutės iš
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?
į:
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)
?
2018 spalio 04 d., 10:07 atliko AndriusKulikauskas -
Pakeista 165 eilutė iš:
Four geometries in terms of choices: paths - one-directional, lines - forwards and backwards, angles - left and right, oriented areas - inside and outside.
į:
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?
2018 spalio 04 d., 10:07 atliko AndriusKulikauskas -
Pakeistos 161-165 eilutės iš
Determinant 1 iff trace is 0. And trace is 0 makes for the links +1 with -1. It grows by adding such rows.
į:
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
.
2018 spalio 04 d., 09:59 atliko AndriusKulikauskas -
Pakeistos 157-161 eilutės iš
briauna = skirtingumas
į:
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.
2018 spalio 04 d., 09:26 atliko AndriusKulikauskas -
Pakeistos 151-157 eilutės iš
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.
į:
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
2018 spalio 04 d., 09:24 atliko AndriusKulikauskas -
Pakeista 151 eilutė iš:
Study the symmetry of functions like exponentials, trigonometric, etc by considering the derivative equation {$f^{(n)}=f$} for various integers n. And nonintegers n?
į:
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.
2018 spalio 04 d., 09:23 atliko AndriusKulikauskas -
Pakeistos 149-151 eilutės iš
Study how turning the counting around relates to cycles - finite fields.
į:
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?
2018 spalio 04 d., 09:22 atliko AndriusKulikauskas -
Pakeistos 147-149 eilutės iš
What is the significance of a cube having four diagonals that can be permuted by S4?
į:
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.
2018 spalio 03 d., 10:39 atliko AndriusKulikauskas -
Pakeistos 145-147 eilutės iš
Eigenvectors are the pure dimensions into which the action of a matrix (or linear transformation) can be decomposed.
į:
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?
2018 spalio 03 d., 09:34 atliko AndriusKulikauskas -
Pakeistos 143-145 eilutės iš
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.
į:
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
.
2018 spalio 03 d., 09:33 atliko AndriusKulikauskas -
Pakeista 143 eilutė iš:
Matrices {$A=PBP^{-1}$} and {$B=P^{-1}AP$} have the same eigenvalues. They are simply written in terms of different coordinate systems.
į:
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.
2018 spalio 03 d., 09:31 atliko AndriusKulikauskas -
Pakeistos 141-143 eilutės iš
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.
į:
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
.
2018 spalio 03 d., 09:18 atliko AndriusKulikauskas -
Pakeista 141 eilutė iš:
An inner product on a vector space allows it to be broken up into irreducible vector spaces.
į:
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.
2018 spalio 03 d., 09:17 atliko AndriusKulikauskas -
Pakeistos 139-141 eilutės iš
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.
į:
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 irreducible vector spaces
.
2018 spalio 03 d., 09:07 atliko AndriusKulikauskas -
Pakeistos 137-139 eilutės iš
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.
į:
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
.
2018 spalio 03 d., 08:44 atliko AndriusKulikauskas -
Pakeistos 135-137 eilutės iš
Counting (by way of the simple roots) links + and - in a chain. x2-x1 etc.
į:
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
.
2018 rugsėjo 30 d., 21:07 atliko AndriusKulikauskas -
Pakeistos 131-135 eilutės iš
Raimundas Vidūnas, deleguotas priežastingumas
į:
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.
2018 rugsėjo 29 d., 18:24 atliko AndriusKulikauskas -
Pakeistos 129-131 eilutės iš
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).
į:
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
2018 rugsėjo 14 d., 14:40 atliko AndriusKulikauskas -
Pakeistos 127-129 eilutės iš
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.
į:
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)
.
2018 rugsėjo 08 d., 19:31 atliko AndriusKulikauskas -
Pakeistos 125-127 eilutės iš
Matematika mus moko, kaip besąlygiškumą reikšti sąlygomis, o tai įmanoma sąlygiškai.
į:
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
.
2018 rugsėjo 07 d., 12:31 atliko AndriusKulikauskas -
Pakeistos 123-125 eilutės iš
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.
į:
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
.
2018 rugsėjo 04 d., 20:01 atliko AndriusKulikauskas -
Pakeista 123 eilutė iš:
Entropy - think in terms of "common" and "uncommon" states. Uncommon states tend to common states simply because they are more frequent.
į:
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.
2018 rugsėjo 04 d., 19:59 atliko AndriusKulikauskas -
Pakeistos 121-123 eilutės iš
Physics - moving backwards and forwards in time - is a (wasteful) duality. Why is it dual?
į:
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.
2018 rugsėjo 04 d., 19:48 atliko AndriusKulikauskas -
Pakeistos 119-121 eilutės iš
(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?
į:
(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
?
2018 rugsėjo 04 d., 14:49 atliko AndriusKulikauskas -
Pakeista 119 eilutė iš:
(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. What makes that true?
į:
(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?
2018 rugsėjo 04 d., 14:49 atliko AndriusKulikauskas -
Pakeista 119 eilutė iš:
(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.
į:
(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. What makes that true?
2018 rugsėjo 04 d., 14:48 atliko AndriusKulikauskas -
Pakeistos 117-119 eilutės iš
* 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)
į:
* 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.
2018 rugsėjo 03 d., 13:57 atliko AndriusKulikauskas -
Pakeista 117 eilutė iš:
* elementary symmetric functions are analogous to prime factorizations of numbers - monomial symmetric functions are analogous to the numbers as such
į:
* 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)
2018 rugsėjo 03 d., 13:57 atliko AndriusKulikauskas -
Pridėtos 115-117 eilutės:

Fundamental theorems
* elementary symmetric functions are analogous to prime factorizations of numbers - monomial symmetric functions are analogous to the numbers as such
2018 liepos 30 d., 08:23 atliko AndriusKulikauskas -
Pakeistos 105-114 eilutės iš
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).
į:
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
.
2018 liepos 22 d., 11:23 atliko AndriusKulikauskas -
Pakeista 105 eilutė iš:
Study the chaos of watersheds for the divisions of everything - the twosome, threesome, foursome, etc.
į:
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).
2018 liepos 22 d., 11:21 atliko AndriusKulikauskas -
Pakeistos 103-105 eilutės iš
[[https://ac.els-cdn.com/S0012365X96003500/1-s2.0-S0012365X96003500-main.pdf?_tid=7e02ab6e-b80c-44b3-9ef0-33327ac6a6d1&acdnat=1531742605_d128ba18dbe99b9c084df275fab65e08 | Geometry of Classical Groups over Finite Fields and Its Applications]], Zhe-xian Wan
į:
[[https://ac.els-cdn.com/S0012365X96003500/1-s2.0-S0012365X96003500-main.pdf?_tid=7e02ab6e-b80c-44b3-9ef0-33327ac6a6d1&acdnat=1531742605_d128ba18dbe99b9c084df275fab65e08 | 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.
2018 liepos 16 d., 15:04 atliko AndriusKulikauskas -
Pakeistos 101-103 eilutės iš
What do inside and outside mean in symplectice (Hamiltionian, Lagrangian) mechanics?
į:
What do inside and outside mean in symplectice (Hamiltionian, Lagrangian) mechanics?

[[https://ac.els-cdn.com/S0012365X96003500/1-s2.0-S0012365X96003500-main.pdf?_tid=7e02ab6e-b80c-44b3-9ef0-33327ac6a6d1&acdnat=1531742605_d128ba18dbe99b9c084df275fab65e08 | Geometry of Classical Groups over Finite Fields and Its Applications]], Zhe-xian Wan
2018 liepos 16 d., 14:58 atliko AndriusKulikauskas -
Pakeistos 99-101 eilutės iš
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.
į:
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?
2018 liepos 16 d., 14:56 atliko AndriusKulikauskas -
Pakeistos 97-99 eilutės iš
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.
į:
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
.
2018 liepos 16 d., 14:03 atliko AndriusKulikauskas -
Pakeista 97 eilutė iš:
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.
į:
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.
2018 liepos 16 d., 13:46 atliko AndriusKulikauskas -
Pakeistos 95-97 eilutės iš
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.
į:
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
.
2018 liepos 16 d., 13:46 atliko AndriusKulikauskas -
Pakeistos 93-95 eilutės iš
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.
į:
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
.
2018 liepos 16 d., 13:07 atliko AndriusKulikauskas -
Pakeistos 89-93 eilutės iš
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.
į:
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
.
2018 liepos 16 d., 12:43 atliko AndriusKulikauskas -
Pakeistos 87-89 eilutės iš
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.
į:
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
.
2018 liepos 16 d., 12:36 atliko AndriusKulikauskas -
Pakeista 87 eilutė iš:
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.
į:
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.
2018 liepos 16 d., 12:34 atliko AndriusKulikauskas -
Pakeistos 85-87 eilutės iš
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.
į:
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
.
2018 liepos 16 d., 12:27 atliko AndriusKulikauskas -
Pakeistos 83-85 eilutės iš
Partial derivative - formal (explicit) based on change in variable, total derivative - actual (implicit) based on change in value.
į:
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
.
2018 liepos 13 d., 14:41 atliko AndriusKulikauskas -
Pakeistos 1-2 eilutės iš
[[https://www.youtube.com/watch?v=pXGTevGJ01o | Symplectic manifolds]] 56:48
į:
[[https://www.youtube.com/watch?v=pXGTevGJ01o | Symplectic manifolds]]
2018 liepos 13 d., 14:34 atliko AndriusKulikauskas -
Pakeista 84 eilutė iš:
Partial derivative - formal (explicit), total derivative - actual (implicit).
į:
Partial derivative - formal (explicit) based on change in variable, total derivative - actual (implicit) based on change in value.
2018 liepos 13 d., 14:33 atliko AndriusKulikauskas -
Pakeistos 82-84 eilutės iš
Collect examples of symmetry breaking.
į:
Collect examples of symmetry breaking.

Partial derivative - formal (explicit), total derivative - actual (implicit)
.
2018 liepos 12 d., 12:21 atliko AndriusKulikauskas -
Pridėtos 1-3 eilutės:
[[https://www.youtube.com/watch?v=pXGTevGJ01o | Symplectic manifolds]] 56:48
2018 liepos 12 d., 12:07 atliko AndriusKulikauskas -
Pakeistos 77-79 eilutės iš
Collect examples of the arithmetic hierarchy such as calculus (delta-epsilon), differentiable manifolds, etc.
į:
Collect examples of the arithmetic hierarchy such as calculus (delta-epsilon), differentiable manifolds, etc.

Collect examples of symmetry breaking
.
2018 liepos 12 d., 12:06 atliko AndriusKulikauskas -
Pakeistos 75-77 eilutės iš
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.
į:
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
.
2018 liepos 12 d., 11:53 atliko AndriusKulikauskas -
Pakeistos 73-75 eilutės iš
An example of extending the domain: Lie algebra (infinitesimal) vs. Lie group (broader domain).
į:
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
.
2018 sausio 31 d., 22:01 atliko AndriusKulikauskas -
Pakeistos 71-73 eilutės iš
What is the connection between the universal grammar for games and the symmetric functions of the eigenvalues of a matrix?
į:
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).
2018 sausio 22 d., 10:58 atliko AndriusKulikauskas -
Pakeistos 69-71 eilutės iš
Intrinsic ambiguity of propositions - every proposition is a general rule, which can be questioned or applied.
į:
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?
2018 sausio 19 d., 19:52 atliko AndriusKulikauskas -
Pakeistos 57-69 eilutės iš
* 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.
į:
* 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
.
2018 sausio 19 d., 19:47 atliko AndriusKulikauskas -
Pakeistos 54-56 eilutės iš
Mandelbrot aibė skiria vidines ir išorines veiklas, besilaikančias erdvės ir nesilaikančios jokios erdvės.

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.
į:
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.
2018 sausio 19 d., 19:36 atliko AndriusKulikauskas -
Pakeistos 49-56 eilutės iš
Study the duality between 1^N and N in symmetric functions (Young tableaux) but also Catalan numbers, etc.
į:
Study the duality between 1^N and N in symmetric functions (Young tableaux) but also Catalan numbers, etc.

Montažas ir geometrinės permainos:
* Shear: sideshot

Mandelbrot aibė skiria vidines ir išorines veiklas, besilaikančias erdvės ir nesilaikančios jokios erdvės.

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
.
2018 sausio 19 d., 14:38 atliko AndriusKulikauskas -
Pakeistos 49-70 eilutės iš
Study the duality between 1^N and N in symmetric functions (Young tableaux) but also Catalan numbers, etc.

Analysis is based on the "looseness" by which a local property (the slope locally) may not maintain globally. And this looseness is of different kinds:
* nonlooseness of path - discrete (integer, rational) affine
* looseness of line - reals - projective
* looseness of angle - complexes - conformal
* looseness of orientation (cross product) - quaternions - symplectic
So looseness is the flip-side of invariance. We see the role of equivalence as based on limits. And also we see the qualitative distinction based on the nature of the limiting process - so taking the limit in all directions for the complexes relates to preserving angles.

What would be the notion of differentiation for a function on the quaternions?

http://math.ucr.edu/home/baez/symplectic.html

https://en.m.wikipedia.org/wiki/Hyperkähler_manifold

Dyson's threefold way R C H
https://arxiv.org/abs/1101.5690
http://math.ucr.edu/home/baez/tenfold.html
Freeman J. Dyson, The threefold way: algebraic structure of symmetry groups and ensembles in quantum mechanics, Jour. Math. Phys. 3 (1962), 1199–1215.

http://www.scholarpedia.org/article/Symplectic_maps
į:
Study the duality between 1^N and N in symmetric functions (Young tableaux) but also Catalan numbers, etc.
2018 sausio 18 d., 00:02 atliko AndriusKulikauskas -
Pakeistos 68-70 eilutės iš
Freeman J. Dyson, The threefold way: algebraic structure of symmetry groups and ensembles in quantum mechanics, Jour. Math. Phys. 3 (1962), 1199–1215.
į:
Freeman J. Dyson, The threefold way: algebraic structure of symmetry groups and ensembles in quantum mechanics, Jour. Math. Phys. 3 (1962), 1199–1215.

http://www.scholarpedia.org/article/Symplectic_maps
2018 sausio 17 d., 11:57 atliko AndriusKulikauskas -
Pakeistos 65-68 eilutės iš
Dyson's threefold way R C H
į:
Dyson's threefold way R C H
https://arxiv.org/abs/1101.5690
http://math.ucr.edu/home/baez/tenfold.html
Freeman J. Dyson, The threefold way: algebraic structure of symmetry groups and ensembles in quantum mechanics, Jour. Math. Phys. 3 (1962), 1199–1215.
2018 sausio 17 d., 11:55 atliko AndriusKulikauskas -
Pridėtos 64-65 eilutės:

Dyson's threefold way R C H
2018 sausio 17 d., 00:16 atliko AndriusKulikauskas -
Pridėtos 62-63 eilutės:

https://en.m.wikipedia.org/wiki/Hyperkähler_manifold
2018 sausio 17 d., 00:15 atliko AndriusKulikauskas -
Pakeistos 59-61 eilutės iš
What would be the notion of differentiation for a function on the quaternions?
į:
What would be the notion of differentiation for a function on the quaternions?

http://math.ucr.edu/home/baez/symplectic.html
2018 sausio 16 d., 16:50 atliko AndriusKulikauskas -
2018 sausio 16 d., 16:50 atliko AndriusKulikauskas -
Pakeistos 57-59 eilutės iš
So looseness is the flip-side of invariance. We see the role of equivalence as based on limits. And also we see the qualitative distinction based on the nature of the limiting process - so taking the limit in all directions for the complexes relates to preserving angles.
į:
So looseness is the flip-side of invariance. We see the role of equivalence as based on limits. And also we see the qualitative distinction based on the nature of the limiting process - so taking the limit in all directions for the complexes relates to preserving angles.

What would be the notion of differentiation for a function on the quaternions?
2018 sausio 16 d., 12:40 atliko AndriusKulikauskas -
Pridėta 53 eilutė:
* nonlooseness of path - discrete (integer, rational) affine
Pakeista 57 eilutė iš:
So looseness is the flip-side of invariance.
į:
So looseness is the flip-side of invariance. We see the role of equivalence as based on limits. And also we see the qualitative distinction based on the nature of the limiting process - so taking the limit in all directions for the complexes relates to preserving angles.
2018 sausio 16 d., 12:30 atliko AndriusKulikauskas -
Pakeistos 49-56 eilutės iš
Study the duality between 1^N and N in symmetric functions (Young tableaux) but also Catalan numbers, etc.
į:
Study the duality between 1^N and N in symmetric functions (Young tableaux) but also Catalan numbers, etc.

Analysis is based on the "looseness" by which a local property (the slope locally) may not maintain globally. And this looseness is of different kinds:
* looseness of line - reals - projective
* looseness of angle - complexes - conformal
* looseness of orientation (cross product) - quaternions - symplectic
So looseness is the flip-side of invariance
.
2017 gruodžio 20 d., 13:44 atliko AndriusKulikauskas -
Pakeistos 47-49 eilutės iš
How do special rim hook tableaux (which depend on the behavior of their endpoints) relate to Dynkin diagrams (which also depend on their endpoints)?
į:
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.
2017 spalio 25 d., 21:01 atliko AndriusKulikauskas -
Pakeistos 45-47 eilutės iš
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.
į:
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)?
2017 spalio 24 d., 09:04 atliko AndriusKulikauskas -
Pakeistos 43-45 eilutės iš
* Apibendrinimas yra "objekto" kūrimas.
į:
* 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
.
2017 spalio 24 d., 07:47 atliko AndriusKulikauskas -
Ištrintos 20-21 eilutės:
Long exact sequence from short exact sequence: derived functors.
Ištrinta 22 eilutė:
Ištrinta 30 eilutė:
2017 spalio 24 d., 07:47 atliko AndriusKulikauskas -
Ištrintos 6-14 eilutės:
* Unmarked opposites: cross-polytope. Each dimension independently + or - (all or nothing).
* Cube: all vertices have a genealogy, a combination of +s and -s.
* Half-cube defines + for all, thus defines marked opposites.

Dual:
* Cubes: Physical world: No God (no Center), just Totality. Descending chains of membership (set theory).
* Cross-polytopes: Spiritual world: God (Center), no Totality. Increasing chains of membership (set theory).
Ištrintos 12-13 eilutės:
Consider the subsitution q=2 or otherwise introducing 2 into the expansion for Pascal's triangle to get the Pascal triangle for the cube and for the cross-polytope.
Ištrintos 16-23 eilutės:
Is the fusion of vertices in the demicube related to the fusion of edges of a square to create a torus, or of vertices to create a circle, etc.?

Symmetric group action on an octahedron is marked, 1 and -1, the octahedron itself is unmarked.

A punctured sphere may not distinguish between its inside and outside. And yet if that sphere gets stretched to an infinite plane, then it does distinguish between one side and the other.

* Constructiveness - closed sets any intersections and finite unions are open sets constructive
Pakeistos 25-28 eilutės iš
Gap between structures, within a restructuring, is a "hole", and so methods of homology should be relevant. How does cohomology relate to holes?

If we consider the complement of a topological space, what can we know about it? For example, if it is not connected, then surfaces are orientable.
į:
Ištrintos 29-32 eilutės:
Compare 6 math ways of figuring things out with 6 specifications. Consider how they are related to the 4 geometries. Relate the latter to 4 metalogics. Look at formulas for the 6 specifications and look for a pattern.

Trikampis - išauga požiūrių skaičius apibudinant: affine-taškai-0, projective-tiesės-1, conformal-kampai-2, symplectic-plotai-3.
Pakeistos 32-35 eilutės iš
6 matematikos irodymo budai skiriaisi nuo issiaiskinimo budu taciau kaip jie susiję

Matematika skiria vidines sandaras (semantika) ir išorinius santykius (sintaksė). Užtat labai svarbu mąstyti apie "viską", kuriam nėra išorinių santykių. Panašiai gal būtų galima mąstyti apie nieką, kur nėra vidinės sandaros. Nors viskas irgi neturi vidinės sandaros. Užtat viskam semantika ir sintaksė yra atitinkamai visiškai paprasta.
į:
Matematikos įrodymo būdai
* 6 matematikos irodymo budai skiriaisi nuo issiaiskinimo budu taciau kaip jie susiję
Pakeistos 40-47 eilutės iš
Gramatika visada turi prasmę. Matematika yra tai, kas pavaldu logikai, bet nebūtinai turi prasmę.
į:
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
.
2017 rugsėjo 06 d., 16:09 atliko AndriusKulikauskas -
Pakeistos 52-66 eilutės iš
Compare 6 math ways of figuring things out with 6 specifications. Consider how they are related to the 4 geometries. Relate the latter to 4 metalogics. Look at formulas for the 6 specifications and look for a pattern.
į:
Compare 6 math ways of figuring things out with 6 specifications. Consider how they are related to the 4 geometries. Relate the latter to 4 metalogics. Look at formulas for the 6 specifications and look for a pattern.

Trikampis - išauga požiūrių skaičius apibudinant: affine-taškai-0, projective-tiesės-1, conformal-kampai-2, symplectic-plotai-3.

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

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

Matematika skiria vidines sandaras (semantika) ir išorinius santykius (sintaksė). Užtat labai svarbu mąstyti apie "viską", kuriam nėra išorinių santykių. Panašiai gal būtų galima mąstyti apie nieką, kur nėra vidinės sandaros. Nors viskas irgi neturi vidinės sandaros. Užtat viskam semantika ir sintaksė yra atitinkamai visiškai paprasta.

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
.
2017 sausio 08 d., 23:22 atliko AndriusKulikauskas -
Pakeistos 50-52 eilutės iš
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.
į:
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.

Compare 6 math ways of figuring things out with 6 specifications. Consider how they are related to the 4 geometries. Relate the latter to 4 metalogics. Look at formulas for the 6 specifications and look for a pattern
.
2017 sausio 05 d., 13:15 atliko AndriusKulikauskas -
Pakeista 50 eilutė iš:
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".
į:
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.
2017 sausio 05 d., 13:15 atliko AndriusKulikauskas -
Pakeistos 48-50 eilutės iš
[[http://www-personal.umd.umich.edu/~tmfiore/1/FioreWhatIsMathMusTheoryBasicSlides.pdf | What is Mathematical Music Theory?]]
į:
[[http://www-personal.umd.umich.edu/~tmfiore/1/FioreWhatIsMathMusTheoryBasicSlides.pdf | 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".
2016 gruodžio 20 d., 20:34 atliko AndriusKulikauskas -
Pakeistos 46-48 eilutės iš
If we consider the complement of a topological space, what can we know about it? For example, if it is not connected, then surfaces are orientable.
į:
If we consider the complement of a topological space, what can we know about it? For example, if it is not connected, then surfaces are orientable.

[[http://www-personal.umd.umich.edu/~tmfiore/1/FioreWhatIsMathMusTheoryBasicSlides.pdf | What is Mathematical Music Theory?]]
2016 gruodžio 15 d., 17:43 atliko AndriusKulikauskas -
Pakeistos 44-46 eilutės iš
Gap between structures, within a restructuring, is a "hole", and so methods of homology should be relevant. How does cohomology relate to holes?
į:
Gap between structures, within a restructuring, is a "hole", and so methods of homology should be relevant. How does cohomology relate to holes?

If we consider the complement of a topological space, what can we know about it? For example, if it is not connected, then surfaces are orientable.
2016 gruodžio 15 d., 17:06 atliko AndriusKulikauskas -
Pakeistos 42-44 eilutės iš
Express the link between algebra wnd analysis in terms of exact sequences and Kan extensions. Explain why category theory not relevant for analysis.
į:
Express the link between algebra wnd analysis in terms of exact sequences and Kan extensions. Explain why category theory not relevant for analysis.

Gap between structures, within a restructuring, is a "hole", and so methods of homology should be relevant. How does cohomology relate to holes?
2016 gruodžio 15 d., 17:00 atliko AndriusKulikauskas -
Pridėtos 39-42 eilutės:

Long exact sequence from short exact sequence: derived functors.

Express the link between algebra wnd analysis in terms of exact sequences and Kan extensions. Explain why category theory not relevant for analysis.
2016 gruodžio 15 d., 16:44 atliko AndriusKulikauskas -
Pridėtos 37-38 eilutės:

Kan extension - extending the domain - every concept is a Kan extension https://en.wikipedia.org/wiki/Kan_extension
2016 gruodžio 15 d., 16:32 atliko AndriusKulikauskas -
Pridėtos 35-36 eilutės:

Relate triangulated categories with representations of threesome
2016 gruodžio 13 d., 23:19 atliko AndriusKulikauskas -
Pridėtos 33-34 eilutės:

* Constructiveness - closed sets any intersections and finite unions are open sets constructive
2016 lapkričio 24 d., 20:12 atliko AndriusKulikauskas -
Pridėtos 31-32 eilutės:

A punctured sphere may not distinguish between its inside and outside. And yet if that sphere gets stretched to an infinite plane, then it does distinguish between one side and the other.
2016 liepos 04 d., 17:20 atliko AndriusKulikauskas -
Pakeista 1 eilutė iš:
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.
į:
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.
2016 liepos 04 d., 17:19 atliko AndriusKulikauskas -
Pakeista 1 eilutė iš:
Baez Rep 4, 1:01 min.
į:
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.
2016 birželio 23 d., 08:43 atliko AndriusKulikauskas -
Pakeista 1 eilutė iš:
Baez Rep 4, 41 min.
į:
Baez Rep 4, 1:01 min.
2016 birželio 23 d., 00:43 atliko AndriusKulikauskas -
Pridėtos 1-2 eilutės:
Baez Rep 4, 41 min.
2016 birželio 23 d., 00:29 atliko AndriusKulikauskas -
Pakeista 28 eilutė iš:
Symmetric group action on an octahedron is marked, the octahedron itself is unmarked.
į:
Symmetric group action on an octahedron is marked, 1 and -1, the octahedron itself is unmarked.
2016 birželio 23 d., 00:25 atliko AndriusKulikauskas -
Pridėtos 27-28 eilutės:

Symmetric group action on an octahedron is marked, the octahedron itself is unmarked.
2016 birželio 23 d., 00:25 atliko AndriusKulikauskas -
Pridėtos 25-26 eilutės:

Is the fusion of vertices in the demicube related to the fusion of edges of a square to create a torus, or of vertices to create a circle, etc.?
2016 birželio 23 d., 00:22 atliko AndriusKulikauskas -
Pridėtos 19-24 eilutės:

Consider the subsitution q=2 or otherwise introducing 2 into the expansion for Pascal's triangle to get the Pascal triangle for the cube and for the cross-polytope.

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?
2016 birželio 23 d., 00:10 atliko AndriusKulikauskas -
Pridėtos 17-18 eilutės:

Axiom of infinity - can be eliminated - it is unnecessary in "implicit math".
2016 birželio 23 d., 00:07 atliko AndriusKulikauskas -
Pridėtos 13-16 eilutės:

Symmetry group relates:
* Algebraic structure, "group"
* Analytic (recurring activity) transformations
2016 birželio 23 d., 00:05 atliko AndriusKulikauskas -
Pridėtos 10-12 eilutės:
Dual:
* Cubes: Physical world: No God (no Center), just Totality. Descending chains of membership (set theory).
* Cross-polytopes: Spiritual world: God (Center), no Totality. Increasing chains of membership (set theory).
2016 birželio 22 d., 23:59 atliko AndriusKulikauskas -
Pakeistos 5-9 eilutės iš
Unmarked opposites: cross-polytope. Each dimension independently + or - (all or nothing).

Cube: all vertices have a genealogy, a combination of +s and -s.

Half-cube defines + for all, thus defines marked opposites.
į:
* Unmarked opposites: cross-polytope. Each dimension independently + or - (all or nothing).
* Cube: all vertices have a genealogy, a combination of +s and -s.
* Half-cube defines + for all, thus defines marked opposites.
2016 birželio 22 d., 23:58 atliko AndriusKulikauskas -
Pridėtos 5-9 eilutės:
Unmarked opposites: cross-polytope. Each dimension independently + or - (all or nothing).

Cube: all vertices have a genealogy, a combination of +s and -s.

Half-cube defines + for all, thus defines marked opposites.
2016 birželio 22 d., 23:54 atliko AndriusKulikauskas -
Pridėtos 2-3 eilutės:

Derangements. Interpret? Not likely...
2016 birželio 22 d., 23:44 atliko AndriusKulikauskas -
Pridėtos 1-4 eilutės:
Function can be partial, whereas a permutation maps completely.

#### MathNotes

Naujausi pakeitimai

 Puslapis paskutinį kartą pakeistas 2019 rugpjūčio 19 d., 21:13