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数学

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Andrius Kulikauskas

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用中文

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Software

数学笔记

Synfig. Alternative to Adobe Animate.

The identity functor can be thought of as the trivial functor for adjunctions involving dualities, two compatible functors.

Looking glass universe

Grant Sanderson: Why aren't you making math videos?

3Blue1Brown Summer of Math Exposition

https://itsfoss.com/adobe-alternatives-linux/

3B1B podcast Grant Sanderson

SOME1 Discord "Summer of Math"

https://github.com/3b1b/manim math animations

Adobe Animate - on cloud

https://smoothstep.io shaders Matt Henderson on Twitter

Algebra is thinking step-by-step (with the conscious hemisphere). So algebra is temporal. Whereas geometry is spatial - it deals with the whole (with the unconscious hemisphere). And what does analysis mean here?

Bosons are the same contradiction and fermions are distinct contradictions.

Phase of quantum particle is the inherent slack given by the locality of its contradiction.

Retaining identity in space and time - a principle grounded in the combinatorics of orthogonal polynomials - thus there exists identity as a long term entanglement

Wolfram's computational equivalence - simple program can be equivalent to the most complicated programs - and add to this the notion of degenerate programs which are less powerful. And an algebra of degeneracy which provides the desired content.

"Math 4 Wisdom" vaizdo įrašuose remtis į muziką kaip į bendrą reikalą, žaidimą.

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

https://www.fotofoto.lt/fotoaparatai/sisteminiai-fotoaparatai/fujifilm-x-t200-15-45-vlogger-kitauksinis

Phase transition models personal growth experience https://www.quantamagazine.org/mathematicians-prove-symmetry-of-phase-transitions-20210708/

Thomas's question for John: What does it mean - empty space or nonempty space?

A particle is a singularity, a local contradiction, and the origin for a coordinate system.

Equivalence depends on 2 conditions. Does adjunction depend on 3 conditions? Isomorphism on 1 condition? Identity on O conditions?

https://www.researchgate.net/publication/51978740_Relating_Operator_Spaces_via_Adjunctions

https://hal.archives-ouvertes.fr/hal-02184208/document Crash Course on Category Theory focusing on adjunctions.

How are inner products related to Poincare duality as suggested in the table here?

https://math.uchicago.edu/~may/VIGRE/VIGRE2008/REUPapers/Henderson.pdf

https://www.researchgate.net/post/The_significance_of_category_theory_for_philosophy

https://www.researchgate.net/post/What-is-required-of-a-system-of-mathematical-objects-and-structures-to-be-called-a-geometry

What is the relation between K and {$K^{-1}$} in terms of adjunctions? Does the combinatorial interpretation of {$KK^{-1}=I$} (with a minimal local tweak) express that they are left and right adjoints whereas the lack of a combinatorial interpretation for {$K^{-1}K=I$} (no global manipulation) show that they are not right and left adjoints?

Inside the Actors Studio

There are four kinds of definitions for adjunction:

  • Homset - balanced
  • UMP for counit - step in ?
  • UMP for unit - step out ?
  • Triangle identities - consciousness

Thus this describes the formula for a perspective.

Consider and classify the circumstances under which the adjoint functor does not exist.

Adjoint functors harmonize the semantics of information (which stays the same) and the syntax of context (which changes). Classification of adjoint strings expresses the possiblities for such harmonization. Thus adjoint strings express the ways that a division into information and context can be understood within the context of divisions of everything, namely, within the context of the eightsome. Consider the adjoint string with seven functors, and the infinite adjoint string with period six.

Grothendieck's six operations express the imagination. They arise in representations and circumstances, where there is a perspective on a perspective, in the chain of human and divine perspectives: HDHDH. Anything and Nothing go in the same direction, as do Everything and Something. Thus negations go in the same direction between human and divine perspectives.

Adjoint functors relate human and divine perspectives in trivial ways.

The classification of adjunctions is based on the classification of trivial functors.

HomSet and Tensor express perfect duality between external and internal. Which is which?

{$f_*\dashv f^{-1}$} expresses perfect duality. The trivial functor is {$f_*$} which is aggregation.

HomSet expresses choice.

Adjoint string: Limit - diagonal {$\Delta$} - Colimit. This expresses my idea about external relations vs. internal structure.

Self-adjoint operator may show the equivalent effect (introducing conjugate) of the rotation of a tensor and the rotation of a coordinate space.

How does the arithmetic hierarchy express Human's view (there exists) of God's view (for all)? What does it mean to negate? Relate to adjoint strings, duality by negation, as with floor-inclusion-ceiling.

https://wlou.blog/2020/01/11/a-brief-introduction-to-categories-part-9-adjunctions-ii/

https://web.math.ucsb.edu/~mputinar/ John knows

Conversation of Putinar and Voevodsky

The triangle identity just prepends eta_A and ignores postpending. Which is to say, we can postpend any G(B). Then it serves to remove the G from the B, and thus reverse the functor.

Wenbo raises the question about adjunctions, at what point in the chain of composition do we get identity. For example, is GF(A) = GFGF(A)?

The Universal mapping property shows the asymmetry of F and G which is based on the asymmetry inherent in composition, which is unidirectional, rooted in an object. F and eta is covariant with composition, and G and epsilon is contravariant with composition.

The Yoneda Lemma expresses whether our language includes its metalanguage, or whether it does not. Syntax (logic) and semantics (math) can be in the same language if we use objects as states and arrows as execution. If we only have arrows as execution, but no states, then our program is compiled, and we have no meta language because we have no use of objects as states.

Consider the problem of how to not reuse variable names. Solutions may express a network (all possible relations), or a tree or a sequence (using each variable once). What do solutions to this problem mean for quantum physics? Does the number of variables relate to energy?

Signal flow graphs are expressed on four levels:

  • Diagrammatic language (homotopy), can deform a graph without changing
  • lambda-expressions make variables explicit
  • point-free, combinator style
  • matrices for efficient computation

https://www.youtube.com/watch?v=6K58e8hbDLM&t=64s Adjunction-MathProofsable-Wenbo

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

Robert Goldblatt - Topoi_ The Categorial Analysis of Logic

Yoneda Lemma https://www.classe.cornell.edu/spr/1999-09/msg0017972.html

Nicholas's interest: the mathematics of correlation, Pearson correlation matrix, discerning complicated structures

https://www.fileformat.info/info/unicode/category/Sm/list.htm

ℂ, ℍ, ℕ, ℙ, ℚ, ℝ, ℤ, ≤, ≥, ⌊ ⌋, ⌈ ⌉, ∈, ∑, λ, ∘

There is a symmetry of walking away and walking back, {$\alpha$} and {$\beta$}.

How do the kinds of variables relate to the imagination?

How do the 24 kinds of variables relate to the 4 circulations?

Think of definitions of adjunctions as ratios of differences. Compare this with {$SU(2)$} and with cross-ratios.

  • HomSet {$\frac{A-B}{A'-B'}$}
  • Universal mapping property {$\frac{A-B}{A-F(A)}$}
  • Unit-counit triangle inequality {$\frac{A-F(A)}{G(F(A))-F(A)}$} and natural transformation {$\frac{A-B}{A-F(A)}$}.
Cross-ratio can be written as ratio of ratios:

{$$\frac{\frac{AC}{CB}}{\frac{AD}{DB}}$$}

Children Television Workshop model of planning, production, evaluation (three-cycle).

First episode of Sesame Street on November 10, 1969

How does the three cycle relate the algebraic and analytic wings of math's house of knowledge?

Particles are conditional, fields are unconditional.

The configurations that orthogonal polynomials code explain how phase space can be divided into distinct trees or cycles. The division of phase space into more compartments requires higher energy.

How is "mass" (as the emergent property of the inertia of constrained particles in a moving box) related to "swing" (in jazz rhythm)?

Does the Yoneda lemma mean that you can compile a program? so that it runs hardwired? in terms of impenetrable but valid subroutines?

Port graph, can consider in terms of

  • relations (no restriction)
  • functions (every input has a single output)
  • bijections (function in both directions)

This can be considered in terms of levels of intelligence.

Orthogonal polynomial growth configurations (Zeng nurseries) can be thought of as a nondeterministic algorithm.

Nondeterministic algorithm can be understood as the allowing of all possiblities, as with the indefinite origin.

Consider Aristotle's four causes in terms of the two kinds of causality in decision making.

Wenbo is interested in dialectical behavior. He is interested to learn topos theory to be able to apply Lawvere's interpretation of logic to express dialectical logic.

A system needs to be able to deal with the supersystem above (stepping out, strategizing) and the subsystem below (stepping in, accepting the implementation) and relating the two. This is like the quadratic fetters on the orthogonal polynomials in that you only worry about the levels immediately above and below. In the system level, there is a dialectic regarding the boundary as to whether we step out or step in.

Supersystem, system, subsystem are the three levels of parsers.

Level of knowledge in the Yoneda lemma: Homset expresses All (every, none) and Morphism expresses One (any, some).

The seven virtues of simple type theory

https://www.researchgate.net/publication/233365263_On_Inversion_Principles

Gentzen's Inversion principle https://en.wikipedia.org/wiki/Proof-theoretic_semantics

Peter

Wenbo

Bosons express the undefined with four different gauges:

  • U(0)? gravity - affine - contradictory - body
  • U(1) light - projective - mind
  • U(2) weak force - conformal - emotions
  • U(3) strong force - symplectic - will

John Baez: "Collapse of the wavefunction" does not violate unitary time evolution, which applies to closed systems but not open systems.

Feynmann diagrams are thought of as morphisms in monoidal categories.

How do the combinatorial cycles that ground the Sheffer polynomials relate to my interpretations of the symmetric functions of eigenvalues? In particular, what may be the role of the forgotten symmetric function? And bricks of size 1 and 2? And having two alphabets?

From conversation with John on 2021.05.31

  • I think of the {$xu(t)$} as standing for a Lie algebra and {$\textrm{ln}A(t)$} is a pseudo Lie algebra.
  • {$\alpha$} and {$\overline{\alpha}$} express entanglement. They model scattering. Thus scattering is entanglement. The entanglement is evident in that {$\alpha=0$} forces {$\overline{\alpha}=0$}.
  • Perturbation describes everything that comes from measurement. It stimulates a transition. And yields a model how two things come together.
  • Orthogonality is quadratic in nature so it makes sense if the measures arise from the complex plane and thus the Laplace transform.
  • {$e^{-x2}$} is an example of a weight that is more natural in the complex plane than on the real line.
  • How to get a coordinate system? Measurements leverage a reference frame and yield a reference frame. So the types of orthogonal polynomials express the types of coordinate systems for measurements. A measurement is in one causality (every effect has had a cause) and at the decision point expresses the other causality (not every cause has had its effect).
  • John is pursuing the application of Koopman Von Neumann equations to quantum mechanics. Trying to understand how the classical picture of a flow of point particles can be retained in making sense of quantum mechanics and not relying on the collapse of the wave function. He was able to show how it would look in empty space but there is simply a time dilation by a factor of 2.
  • John's and my approaches are practically in opposite directions but it is helpful to work alongside each other.
  • Klesch-Gordon coefficients become natural as finite dimensional representations of SO(3).
  • Decomposing a direct product in terms of a direct sum of finite dimensional.
  • Spherical harmonics relate {$e^{in\phi}$} and {$e^{in\theta}$}.

Study "Signals and Systems" to understand the Laplace transform, the Fourier transform, Shannon information.

Combinatorially, Jacobi configuration is a pair of Laguerre configurations (injections) back and forth. Is the Jacobi configuration like an adjunction?

Express the combinatorics of orthogonal polynomials in terms of symmetric functions of eigenvalues. For example, start with Hermite polynomials and define fixed points and involutions. In general, come up with a theory of species in terms of the symmetric functions of eigenvalues.

Root system The root system of {$E_8$} can be thought of as a full system. The root system of {$E_7$} is then the set of vectors that are perpendicular to one root. {$E_6$} is the set of vectors that are perpendicular to two suitably choosen roots. What would be the root system gotten that is perpendicular to three suitably choosen roots from {$E_8$}? or more roots? {$D_6$} is the set of vectors that is perpendicular to one root in {$E_7$}.

Think about the Lie algebra root systems in terms of how the roots expand upon {$\pm(x_i-x_j)$} for {$A_n$}. We have:

{$B_n:\pm (x_i+x_j), \pm x_i$}{$C_n:\pm (x_i+x_j), \pm 2x_i$}{$D_n:\pm (x_i+x_j), i\neq j$}

Does the foursome, as with the Yoneda lemma, establish four natural bases for automata?

How does the generating function for {$P_n$} relate to the weight? They seem similar. Why?

I suspect that the seven-fold equation (fraction of differences, as with Mobius transformations) in Shu-Hong's thesis organizes seven natural bases that express different ways of looking at probability and confidence in a self-standing system.

Adjoint strings express the operation +2, the alternation of Human perspective of God's perspective, and also thus the gap that starts with the foursome, grows with the fivesome, sixsome, sevensome, wear from that womb keep inserting new perspectives, flipping the direction, what it means to go beyond oneself.

Empty space is given by the weight. Differentiating it yields the same (almost) - a difference by a quantum. Differentiating {$x^k$} yields a lower degree, thus clearly different, and similarly with {$P_n(x)$}.

Moments - the numbers give the impact of {$x^k$}.

Hankel determinant - the numbers describe what momentum is.

In the combinatorial objects, note what gets weight {$x$} and what else is there.

How is the eccentricity of the conics related to the discriminant? And how is the equatioun {$(1-\alpha t)(1-\beta t)=1+lt+kt^2$} and the coefficients related to the conics?

The decision point (present, boundary) is the quantum {$x$} which commutes with either ground so that {$\int p_k x p_n$} can be interpreted ambiguously, and so relations with distant, nonproximate levels can be discarded.

Give combinatorial interpretation of {$\Delta_n$}, {$c_{nn}=\frac{\Delta_{n-1}}{\Delta_n}$}, {$\lambda_{n+1}=\frac{\Delta_{n-2}\Delta_{n}}{\Delta_{n-1}^2}$}.

Does the universe swing between total kinetic energy and total potential (big bang)? And what causes it to swing, for example, back from kinetic to potential?

Understand what function gives the zeroes that determine the range for the weight function.

Calculate the echo terms for {$A(t)$}, {$A'(t)$}, {$u(t)$}, {$u'(t)$} and compute {$\frac{A'(t)}{A(t)}$}.

Think of moments {$\mu_n$} in terms of walks that wander up to a distance {$n$}. These walks are the expressions of momentum, perhaps their building blocks.

How is the exponential world combinatorially different from the regular world? It is bounded from above by {$n!$}.

Does Gram-Schmidt orthogonalization work for complex inner product spaces.

In what sense is a self-adjoint operator related to adjoint functors?

Find a combinatorial expression of the Grothendieck yoga, for example, considering how sets are partitioned.

The tensor-curry adjunction relates all and one, as between everything and anything, and between nothing and something. It allows for step-by-step algebra, as by the Yoneda lemma.

http://turing.wins.uva.nl/thk/recentpapers/rec2ortho.html

https://www.youtube.com/watch?v=vdW9XDBuxjU

Bott periodicity 8=0 could be the basis for reproduction.

What are the transition matrices between the natural bases of the orthogonal polynomials? What are the combinatorial interpretations for their entries? They are indexed by integers rather than partitions.

What would be the natural bases for the Yoneda lemma and adjunctions? What space are they describing? What are they indexed by?

https://sites.google.com/view/dmonaco/teaching/k-theory-in-condensed-matter-physics

https://nptel.ac.in/courses/111/106/111106100/

How is adjunction related to the inversion of a functor?

How is adjunction related to the inversion of a perspective?

How are the natural orthogonal polynomials related to the natural bases of the symmetric functions?

  • Hermite - pairs - monomial?
  • Charlier - cylces - power?
  • Laguerre - strict ordering - elementary
  • Meixner-Pollaczek - alternatin permutation - Schur
  • Meixner - weak ordering - homogeneous

Leaving the forgotten as an extra basis.

The 5 natural bases for {$L^2$} or the 6 natural bases for the symmetric functions are perhaps related to M-theory.

What are natural bases for {$L^2$} ? The 5 Sheffer polynomials. What are natural bases for symmetric functions? There are 6. Are there four bases for the Yoneda lemma?

What are the invariant subspaces of the shift operator?

How does Viennot's "coherent transport" work as a functor?

The Sheffer polynomials are perhaps characterized by the operator {$x$} and its action on {$P_n(x)$} whereas the classical orthogonal polynomials are perhaps characterized by the operator {$\frac{\textrm{d}}{\textrm{dx}}$} as with the Rodrigues formula.

Interpret combinatorially the formula for an orthogonal polynomial as a determinant of moments for the given measure.

Wolfram's cellular automata can be thought of as acting on the hierarchy of levels, as with raising and lowering operators.

Is the Raudys classification some how related to the Pearson classification?

In orthogonal polynomials, understand {$x P_n(x)$} as the basis for the recurrence relation. It is the basis for the definition of {$P_n(x)$} in terms of {$x$}. {$x$} stands for the gap, encodes the gap between levels above and below. There is a hierarchy of gaps.

{$x$} is the raising operator which defines {$x p_n(x)$} in terms of {$p_{n+1}$}, {$p_n$}, {$p_{n-1}$}. {$\frac{\textrm{d}}{\textrm{dx}}$} is the lowering operator which defines {$y'=\frac{\textrm{d}y}{\textrm{dx}}$} in terms of {$y''$}, {$y'$}, {$y$}.

We can think of {$x$} as moving from {$xP_n(x)$} to {$P_{n-1}$}. Which is why it does not vanish, why it does not go to zero by linearization.

The middle term is defined with regard to the terms above and below. This is like Wildberger's mutation game defined for Dynkin diagrams that are a chain. The chain prganizes dimensions, higher and lower.

We can consider triality as given by Dynkin diagrams with a threefold branching, the D-series and the E-series. Here we may have three different lowering operators but one raising operator. At the root we may have three lowering operators and three raising operators, thus two trinities, a sixfold structure.

Sheffer classification gives 5 possibilities for synchronization. Consider {$\alpha$} and {$beta$} as vectors in the plane. Then we can have:

  • Two points at the origin.
  • A point at the origin and a vector.
  • The same vector twice.
  • Two nonzero vectors of different length on a line.
  • Complex conjugates in the plane.

Krautchouk polynomials are a family that consists of only finitely many polynomials. The higher polynomials are zero. This introduces the notion of finiteness. It may be a sixth possibility that expresses the finite human situation, morality, the sixsome.

http://blog.sina.com.cn/s/articlelist_1215048895_0_1.html

http://blog.sina.com.cn/strongart

https://space.bilibili.com/3182?from=search&seid=17518721354809687674

https://www.math.miami.edu/~armstrong/Talks/What_is_ADE.pdf

https://www.youtube.com/watch?v=zPtnEZeuzH0 Wildberger. Consider how the mutation games express how and why the four classical Lie groups get distinguished. How is this expressed in the Lie groups and in geometry? How does this relate to the propagation of signal that I had observed? And what is going on with the exceptional Lie algebras?

https://www.youtube.com/watch?v=Qw5jonrLbPU&t=2868s Representation theory of Lie algebras and groups

echarts apache - graph routines

https://www.youtube.com/playlist?list=PLhgq-BqyZ7i5lOqOqqRiS0U5SwTmPpHQ5

Adjoint functors may relate to chains of perspectives of human's view and God's view. The identity triangles for units and counits are in terms of FGF. This reminds me of my thoughts on bisecting a view. FGF also shows the ambiguity of F, that it can be thought of as acting on the outside, F(GF->I), or on the inside (FG->I)_F

Pair polytope types to get six ways of structuring mathematics as in the house of knowledge.

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

In what sense are the Krawtchouk polynomials possibly a sixth kind of orthgonal polynomial? They are a special case of the Mexiner polynomials of the first kind. They are only a finite sequence.

Fourier transform transforms the shift operator into the differentiation operator and vice versa. So how does it relate shift invariance (relativity, as with Sheffer polynomials) and invariance with regard to differentiation (as with weight functions like {$e^x$})?

The collapse of the wave function resets the clock of time for that system. And it diffuses by the phase factor from there.

Self-adjoint operator means diagonalizable. What does that mean for adjunctions?

Lie groups and algebras express (continuous) symmetry that I am discovering at various meta levels.

In the classification of Sheffer polynomials, we have the choices:

  • {$\alpha = 0$}, {$\alpha \neq 0$}
  • {$\beta = 0$}, {$\beta = \alpha$}, {$\beta \neq \alpha$}, {$\beta = \overline{\alpha}$}

So there are eight possibilities, but some of them are related.

Wick's theorem? organizes creation and annihilation operators in very much the same way as the Hermite polynomials organize Zeng saplings. This suggests that momentum (the movement of a particle) can be thought of as the annihilation of a particle in one location and the creation of a particle in a different location. I am learning about Feynman diagrams? to see how they might relate to Zeng saplings.

https://physics.stackexchange.com/questions/304508/why-are-all-my-separable-solutions-orthogonal-polynomials-in-l2

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

How are the combinatorics of the Hermite (involution - one dimension), Laguerre (rook polynomials - two dimension) and Jacobi polynomials related? Are the Jacobi polynomials related to three dimensions?

X.Viennot. The cellular ansatz: bijective combinatorics and quadratic algebra Robinson-Schensted-Knuth correspondence, trees and tableau, PASEP, quadratic algebra.

Razumov-Stroganov conjecture

F Colomo1, A G Pronko. The role of orthogonal polynomials in the six-vertex model and its combinatorial applications.

  • The Hankel determinant representations for the partition function and boundary correlation functions of the six-vertex model with domain wall boundary conditions are investigated by the methods of orthogonal polynomial theory. For specific values of the parameters of the model, corresponding to 1-, 2- and 3-enumerations of alternating sign matrices (ASMs), these polynomials specialize to classical ones (continuous Hahn, Meixner–Pollaczek and continuous dual Hahn, respectively). As a consequence, a unified and simplified treatment of ASM enumerations turns out to be possible, leading also to some new results such as the refined 3-enumerations of ASMs. Furthermore, the use of orthogonal polynomials allows us to express, for generic values of the parameters of the model, the partition function of the (partially) inhomogeneous model in terms of the one-point boundary correlation functions of the homogeneous one.
  • Vertex model
  • Six-vertex model

X.Viennot. Introduction to Enumerative, Algebraic, and Bijective Combinatorics

X.Viennot. A combinatorial theory of orthogonal polynomials and continued fractions.

X.Viennot. Combinatorics And Quadratic Algebras.

X.Viennot. The birth of new domain: Combinatorial Physics

Jacobi polynomials (such as the physicist Hermite polynomials) include a factor of 2x. These seem related to probability. But also for the geometry, as with the Legendre polynomials. Can we use the same combinatorics as Kim Zeng, but just alter by substitution for y=2x? and multiplying each polynomial by a fraction? Can we compare with Viennot's interpretation? Do they relate physics with probability?

Spin gives the number of choices. Boson - odd number (includes zero), fermion even number (does not include zero). How does that relate to the kinds of classical Lie algebras?

  • {$F_1$} photon chooses from one choice
  • {$F_2$} electron chooses from two choices
  • {$F_3$} spin one boson chooses from three choices
  • {$F_5$} spin two graviton chooses from five choices (fivesome for space and time?)

Do these relate to the divisions of everything? What happens with the eightsome? and ninesome?

Attiyah: Limits (singularities, zeroes) exist in the ideal world but not the real world. Thus the house of knowledge for mathematics constructs the ideal world from the real world.

Relativistic invariance of the Dirac operator.

Delayed signals? (Kinks?) Attiyah: Retarded Dirac operator. Attiyah, Moore. A Shifted View of Fundamental Physics.

Time-delayed Dirac delta, Group delay and phase delay

Connes noncommutative geometry is based on cyclic permutations where ABC = CAB, and this yields cyclic homology for associative algebras.

Connes noncommutative geometry is based on every point being actually a pair of points (on two sheets), on which 2x2 matrices operate, and yields the Standard Model. Are these pairs of points expressing the dual causality of the fivesome, links and kinks?

How is Kim-Zeng linearization (generalized derangements) related to the representation of the symmetric group and general linear group?

John: Unitary dynamics - self-adjoint operator {$H*=H$}. Unitary operator.

Attiyah: Number theory (Connes - noncommutative algebra) is linked by Geometry (Penrose - twistors) to Physics (Witten - strings).

Atiyah: K-theory is related to quantum theory, cohomology is related to classical theory.

Dirac operator is the basis for spinors and for Bott periodicity.

Dirac operator is, conceptually, the square root of the Laplacian. In what sense is a tableaux the square root of a matrix?

Atiyah: What does octonionification mean for the magic square?

{$\forall D\exists f\forall g(f=g)$} arithmetic hierarchy: derivative

{$\forall \epsilon \exists \delta \forall x$} arithmetic hierarchy: universality: for all objects there exists a unique morphism

Robert Harper - meta theory, metalogic

Residuated Boolean algebra, residuated semilattice

https://jamboard.google.com/d/1Jqw8qgN_5aQANIcUUOe-ZzCPHj5Q5llmCA95sWwD4I4/edit?usp=sharing

https://arxiv.org/abs/1311.3903

graphicallinearalgebra.net

https://www.math3ma.com/blog/matrices-probability-graphs

https://zh.wikipedia.org/wiki/%E7%8E%8B%E5%AE%88%E4%BB%81

https://www.math3ma.com/blog/limits-and-colimits-part-2

idris2

https://arxiv.org/abs/2004.05631

Quantum Field Theory

Physics

  • What is the connection between imaginary time (the dynamical evolution of the quantum world {$e^{-iHt/ħ}$}) and inverse temperature (the Boltzmann factor {$e^{-\beta H}$}). Partition function {$\textrm{tr}e^{-\beta H}$}.
  • How to understand negative energy and negative probability?
  • Think of negative probability as removing a possibility, by having it not happen, by having it happen in our mirror universe (presumed to exist by CPT symmetry).
  • Given wave function {$x^ke^{-x^2}$}, the chance of being at x=0 is 0 unless it is the ground state.

Orthogonal polynomials

https://en.m.wikipedia.org/wiki/Sheffer_sequence is what i am interested in.

Sturmian functions - harmonic oscillator

Relativistic quantum harmonic oscillator

Spherical harmonics

  • R. P. Martı́nez-y-Romero. Relativistic quantum mechanics of a Dirac oscillator 1999. Free article. Refers to spinor spherical harmonics.
  • The spinor spherical harmonics are the natural spinor analog of the vector spherical harmonics. While the standard spherical harmonics are a basis for the angular momentum operator, the spinor spherical harmonics are a basis for the total angular momentum operator (angular momentum plus spin). These functions are used in analytical solutions to Dirac equation in a radial potential.
  • See: Greiner, Walter (6 December 2012). "9.3 Separation of the Variables for the Dirac Equation with Central Potential (minimally coupled)". Relativistic Quantum Mechanics: Wave Equations. Springer. ISBN 978-3-642-88082-7.

Relativistic hydrogen atom

Spacetime is made of events (the coordinates). But in quantum mechanics, an event is only when a wave function collapses. So time and space combine only when the wave function collapses. So relativity is only important for the actual collapse of the wave function. The wrapper {$e^{-x^2}$} may explain the quadratic nature of gravity. But how?

Orthogonality measures self-duality. Two pyramids give the perspectives of position and momentum.

Ismail. Classical and Quantum Orthogonal Polynomials in One Variable.

How does adjunction relate to automata?

https://varkor.github.io/blog/2020/11/25/announcing-quiver.html Quiver - for drawing commutative diagrams

Penrose's three-cycle - being, doing, thinking.

What would a three-cycle of functors look like, modeling Penrose's three worlds?

Based on the case of the harmonic oscillator, setting units like h to 1, energy is simply the frequency. In general, is energy the frequency of conversion of potential energy to kinetic energy and back again, as with the harmonic oscillator?

Enriched preorder category.

Preorder is a boolean enriched category.

Example of self-enriched category is Bool where we have the category with two objects and one morphism between them, and we map that morphism to 0 (false) or 1 (true) with 0<=1.

HomSet is an enrichment of a category with a preorder of nonempty homsets.

Diachronic category is preorder, synchronic category (linear time) is monoid defined on one element (all time, all actions at once).

An object is a point in time. In the diachronic point of view, we move by morphisms from object to object. In the synchronic point of view we keep returning to the same point in time, so all morphisms act in parallel.

Extensional vs. intensional view of functions. On Lambda Calculus.

Enriched category is the "graded category" that I was seeking for in statistics, etc.

Chess educator of the Year, Jerald Times https://www.youtube.com/watch?v=isAfKbPDeYc

Anthony Zee. Group Theory in a Nutshell for Physicists.

Wick's theorem is used extensively in quantum field theory to reduce arbitrary products of creation and annihilation operators to sums of products of pairs of these operators. A more general idea in probability theory is Isserlis' theorem. Isserlis' theorem or Wick's probability theorem is a formula that allows one to compute higher-order moments of the multivariate normal distribution in terms of its covariance matrix.

  • Wick's contraction interprets the double factorial in terms of pairs of connections.

https://www.quora.com/Whats-your-favorite-introductory-quantum-field-theory-book/answer/Adam-Lantos?ch=10&share=0424dd78&srid=TVEF

In measuring an average quality, such as average position:

  • You can't average the values you get upon repeatedly measuring one particle because it has taken on a particular state with the first measurement.
  • You can't average the values you get by making measurements on an ensemble of particles having the same state because, in general, you don't know what that state is, and even if you did put the particles in such a state, you cannot absolutely maintain them in that state, but only up to a degree of certainty.
  • Thus you can average the values that you get by making meaurements on an ensemble of particles that are in rather similar states.

The axioms {$p \equiv -iħ\frac{\delta}{\delta x}$} and {$E \equiv iħ\frac{\delta}{\delta t}$} and then the energy equation {$E=K+V=\frac{p^2}{2m} + V(x)$} are theoretically more fundamental than Shroedinger's equation, which is empirical, and which is derived by plugging in. Thus Schroedinger's equation simply describes the circumstances for measurement by way of the wave function.

The factor of 2 for momentum and the factor i for position are both half-way factors, square root factors. i is the square root of negative 1. The factor 2 arises because of differentiating {$p^2$}, perhaps twice?

There are four dimensions of space-time and accordingly there are four dimensions of the four momentum where the energy {$\frac{E}{c}$} corresponds to time. The principle of least action says that given the two forms of energy, kinetic and potential, {$E=K+V$}, we shift from one to the other as little as possible, thus minimizing K-V. In this sense we can think of time as the dimension in which this conversion occurs, and thus time is like an expressly local dimension, of limited extent, as little as possible. Time is the distance traveled up the potential V. Does that mean that change takes place as slowly and gradually as possible? And can that limitation express the speed of light as a consequence of the principle of least action? And also the shape of the mother function.

How do we rethink Schroedinger's equation for space time so that energy and momentum are both related in terms of derivatives of the same order? How do we rethink energy as {$-E=iħ\frac{\delta}{\delta t}$}, so that it is compatible with p? So we can write this as {$0=(-E)+K+V$} or {$(-V)=(-E)+K$}. How doe we rethink the operator {$V=-iħ\sum_{j}\frac{\delta}{\delta x_j}$} where multiplying by the potential is the same as acting by the partial derivatives.

How can the second law of thermodynamics be described in terms of time which is defined as the distance traveled up a potential?

The combinatorial configurations into cells and relationships are presumably the same regardless of which is assigned to position and which to momentum... and other, possible hybrid, physical qualities? Or does the Fourier transform introduce some subtle difference? So what is the meaning of these deeper configurations? And what does it mean when they signify the result of a measurement?

维基百科: Adjoint Notions of adjoint.

Symmetry is the laws for the laws (of physics).

https://www.amazon.com/Introduction-Orthogonal-Polynomials-Dover-Mathematics/dp/0486479293

Physicists know how to copy equations, how to manipulate them, how to solve them, how to set them up, but not how to read them mathematically, conceptually. Good physicists can read the physics and yet not the math.

Josef Meixner. Symmetric systems of orthogonal polynomials.

Josef Meixner. Orthogonale polynomsysteme mit besonderen Gestalt der erzeugenden Funktion. Journal of the London Mathematical Society. January, 1934.

Sheffer sequence Isador Sheffer

Consider the weights of the various families of orthogonal polynomials and their properties. For example, 1/2cosh(pi x/2) is, up to a dilation, its own Fourier transform. Are the other weights similarly related to geometric transformations?

Dunce, dunce hat, Duns Scotus

Theoretical Physics Reference: Hypergeometric functions

  • {${}_0F_1$} confluent hypergeometric limit function
  • {${}_1F_1$} Kummer’s confluent hypergeometric function of the first kind
  • {${}_2F_1$} Gauss’ hypergeometric function

Art ideas: Mahjong tiles and game. Dominoes.

Art ideas: The Tailors. Einstein with person's watch: "Pulse is steady."

Measurement is a trivial translation of coordinate system from x to x-a and thus relates to the trivial self-entanglement of a tree.

Roger Penrose, Jurekfest, 2019 4:20 "I think probably twister theory has now as little connection with spin networks as loop quantum gravity does with the original aims of spin network theory which was to produce a combinatorial approach to physics where you just ... calculate everything from integers. We've both drifted away from that. I don't know whether we're coming to it back to it or not. I sometimes hope so."

What is the meaning and significance of the conic sections "dot" and "X" (two lines crossed)?

For the conic sections, given the two cones, can we identify one with the inside and the other with the outside?

Renato ́Alvarez-Nodarse. On some applications of Orthogonal Polynomials in Mathematical-Physics Alberto Grunbaum one time said (in a summer school in 2004):“Special functions are to mathematics what pipes are to a house: nobodywants to exhibit them openly but nothing works without them”.

Danilo Latini, D.Riglioni. From ordinary to discrete quantum mechanics: The Charlier oscillator and its coalgebra symmetry The coalgebraic structure of the harmonic oscillator is used to underline possible connections between continuous and discrete superintegrable models which can be described in terms of SUSY discrete quantum mechanics. A set of 1-parameter algebraic transformations is introduced in order to generate a discrete representation for the coalgebraic harmonic oscillator. This set of transformations is shown to play a role in the generalization of classical orthogonal polynomials to the realm of discrete orthogonal polynomials in the Askey scheme. As an explicit example the connection between Hermite and Charlier oscillators, that share the same coalgebraic structure, is presented and a two-dimensional maximally superintegrable version of the Charlier oscillator is constructed.

Richard Askey, Sergei K. Suslov. The q-harmonic oscillator and an analogue of the Charlier polynomials.

Richard Askey explained why hypergeometric functions appear so frequently in mathematical applications: "Riemann showed that the requirement that a differential equation have regular singular points at three given points and every other complex point is a regular point is so strong a restriction that the differential equation is the hypergeometric equation with the three singularities moved to the three given points. Differential equations with four or more singular points only infrequently have a solution which can be given explicitly as a series whose coefficients are known, or have an explicit integral representation. This partly explains why the classical hypergeometric function arises in many settings that seem to have nothing to do with each other. The differential equation they satisfy is the most general one of its kind that has solutions with many nice properties. Special functions: group theoretical aspects and applications.

How do regular singular points and Riemann's differential equation relate to the eccentricities of the conics (0,1,infinity)? and to the Mobius transformations?

How do Bayesian statistics and the usual statistics differ as regards the fivesome and causalities?

Alain Connes about Bott periodicity and CTP. "Why Four Dimensions and the Standard Model Coupled to Gravity...

Kinks

  • There are 5 conic sections plus 3 more: the point C (where the cones meet), two lines that cross in an X, and the line segment (which arises in the limiting case as the hyperbola passes over to an ellipse). Given any other point P on the lower cone, P not C, then we get conic figures by rotating the plane that contains C along the axis that is tangent to the cone. We can start with a circle, then it becomes an ellipse ever longer, until it reaches infinity and becomes a parabola, which then intersects the upper cone as a hyperbola, which then becomes a line as the hyperbola gets squeezed and infinitely narrow, but then the line drops down to a line segment from P to C, from where it opens up as an ellipse, which grows in width until it becomes the circle. So the line segment is a moment of ambiguity where it equals the line and yet the limit cases from either side are different. The line segment seems like a string or loop in string theory.
  • Nurseries, Zeng saplings
  • What would a kink look like for a tableaux?
  • An observer carves up spacetime into kinks.
  • Kinks express the need that a question be answered, that there be resolution, that tension be resolved, which is important in all conceptual languages.
  • Kinks express the yearning for justice, that everything be set right, that there be no loose ends.
  • Kinks express the obligations met by a pushdown automata, as with balanced parentheses.

https://aip.scitation.org/doi/abs/10.1063/1.1704953

https://math.stackexchange.com/questions/1063863/solving-special-function-equations-using-lie-symmetries

https://math.stackexchange.com/questions/1163032/geometric-intuitive-meaning-of-sl2-r-su2-etc-representation-the

https://math.stackexchange.com/questions/622/importance-of-representation-theory/2692#2692

http://www-users.math.umn.edu/~mille003/lietheoryspecialfunctions.html

Random matrices https://arxiv.org/abs/1712.07903

https://link.springer.com/book/10.1007/978-94-009-0501-6 Orthogonal Polynomials, Theory and Practice, ed. Paul Nevai

https://www.sciencedirect.com/science/article/pii/S0001870816310453?via%3Dihub Orthogonal polynomials through the invariant theory of binary forms, P.Petrullo, D.Senato, R.Simone

A kink is a contraction (a line) as defined for Wick's theorem. What is a normal ordering? An abnormal ordering? A link?

Nima Arkani-Hamed. Research Skills and Theoretical Physics.

Gabor Hetyei. Shifted Jacobi polynomials and Delannoy numbers

Reply to this question:

Thomas asked, Why are there 3 + 1 dimensions of time and space? What could my philosophy say?

  • The relevant structure would be the eightfold way: 1 + 3 + 3 + 1. It suggests that (external) space and (internal) time are two different ways of relating a relative framework with an absolute framework. For space, the absolute framework (God) is beyond the relative framework. There are three independent dimensions (being, doing, thinking) which relate the relative to the absolute. So the relative precedes the absolute. For time, the absolute framework (good) is within the relative framework (the three-cycle) and prior to it. Time is the dimension of slack within the three-cycle which goes on infinitely and for which forwards and backwards are qualitatively different directions.

https://en.wikipedia.org/wiki/Fa%C3%A0_di_Bruno%27s_formula used to establish the existence of Sheffer's polynomials.

WorldCat

Heyting algebra - notion of adjunction? meet vs. join? product vs. homset

http://builds.openlogicproject.org/ WhatIf book

https://en.wikipedia.org/wiki/Del_Pezzo_surface see especially from degree 2 to 9, is this an eightfold phenomenon? How is it related to https://en.wikipedia.org/wiki/M-theory ? How is it related to https://en.wikipedia.org/wiki/Veronese_surface and the conics?

More formally, in the context of QFT, the S-matrix is defined as the unitary matrix connecting sets of asymptotically free particle states (the in-states and the out-states) in the Hilbert space of physical states. A multi-particle state is said to be free (non-interacting) if it transforms under Lorentz transformations as a tensor product, or direct product in physics parlance, of one-particle states as prescribed by equation (1) below. Asymptotically free then means that the state has this appearance in either the distant past or the distant future. (Distant time!)

Petri Muinonen I'm reading your doctoral thesis as I have a physics model under development where particles are described as combinatoric aggregates of logarithmic poles (singularities in 4D). They form negative simplex vertices around center of positive ones (convention). See my Youtube channel petrimuinonen

Petri:

  • From your diary "This suggested to me that the collapse must occur when a coordinate system is introduced and thereby separates the observer from the observed. ... at a certain point ... which collapses the wave function.... Thus this would be the place to look for a relation between general relativity and quantum mechanics."
  • This makes perfectly sense to me. My understanding is that we are talking about division algebra, quaternions, and a negative 4-distance expressed in quaternion terms (squared separation of 4D locations, outside light cone) collapses into a vectorial quaternion (corresponding to a purely spatial separation) There is infinite number of roots to that, all space-like. Will have to explain this better later!
  • You might also want to check Julian Barbour's concept of shape dynamics
  • Matematika iškyla vairuojant, valdant automobilį. Sukame vairą - ratą. Skaičiuojame bėgius. Važiuojame pirmyn ir atgal. Įsivaizduokime, kad valdome lėktuvą arba netgi erdvėlaivį sugebantį skristi į praeitį ir ateitį ar netgi visai kitokius matus, pavyzdžiui, nuotaikų pasaulius. Suktumėme ne ratą bet gaublį arba keturmatį ar netgi aštuonmatį gaublį. Kad važiuotumėme ne tik pirmyn ir atgal, bet aukštyn ir žemyn, į praeitį ir ateitį.

Wenbo

  • How can structure arise that maintains itself?
  • Foucault "The Order of Things".
  • Oulipo. Kūryba pagal savo taisykles.
  • Schelling požiūris panašus į mano, anot Wenbo.
  • Yu-Gi-Oh! Trading card game.
  • Embodied agency (video game). Valios įgyvendinimas.

Distinction between extrinsic symmetry (relative to an outside observer) and intrinsic symmetry (relative to itself).

Askey-Wilson polynomials subsume the various orthogonal polynomials. They are defined by five parameters. They have a combinatorial interpretation in terms of PASEP and their moments are given by staircase tableaux.

Arrows, structures, and functors_ the categorical imperative-Academic Press (1975)

https://hal.archives-ouvertes.fr/tel-01253426/file/hdr.pdf

Logical Foundation of Cognition - Lawvere - 4. Tools for the Advancement of the Logic of Closed Categories

https://discord.gg/rPFGTMPJ

https://ncatlab.org/nlab/show/alpha-equivalence

http://msp.cis.strath.ac.uk/msp101.html

Programming in Martin-Loef's Type Theory - Bengt Noerdstrom et al

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Šis puslapis paskutinį kartą keistas July 24, 2021, at 02:49 PM