概论

笔记

数学

发现

Andrius Kulikauskas

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数学笔记

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How to call the length of chain complexes or exact sequences?

Mobius transformations

  • How are perspectives transformed?
  • How are triangles mapped to triangles ?

Division of everything: A chain complex that is "bolted down" by filling in the holes to get an exact sequence. This helps explain how we can interpret 0->A->0 nontrivially.

https://colganology2.blogspot.com/2023/03/one-rule-you-didnt-know-you-didnt-know.html

  • Opinion-size-age-shape-colour-origin-material-purpose NOUN
  • Kaip įvairiose kalbose, kokia tvarka išsidėsto būdvardžiai?

Bott periodicity

  • Morse theory - count the number of directions of steepest ascent - does this relate to the divisions of everything?

David Corfield's video. Colin McLarty: semiotics as the language of biology - logic in a biological key - trying to categorify this?

https://zacharystrong.net

Bells

Projective geometry

  • What is the relationship between dilation and the homogeneous coordinates for projective geometry?

Tristan Needham. Visual Complex Analysis.

John Stillwell. Mathematics and its History. 1989.

Moebius transformations

Public Domain Images

Emotional sphere

Visionaries "Vision for Our Future"

In physics, the identity component of the Lorentz group acts on the celestial sphere in the same way that the Möbius group acts on the Riemann sphere. In fact, these two groups are isomorphic. An observer who accelerates to relativistic velocities will see the pattern of constellations as seen near the Earth continuously transform according to infinitesimal Möbius transformations. This observation is often taken as the starting point of twistor theory.

Freedom House Report

Michael Schreiber

  • Nick Bostrom's theory of the likelihood of living in a simulated universe.

Wenbo

  • Semi-join lattice.

Japanese vocal jazz Wenbo

The Topos of Music: Geometric Logic of Concepts, Theory, and Performance - worse

Cool Math for Hot Music - better

Midori

Recuerdos de la alhambra

Oliver

  • Distributional semantics

Real, Complex, Quaternion, Octonion

  • 3, 4, 6, 10 equals 1+2, 2+2, 4+2, 8+2. Observables of real, complex, quaternionic, octoninoic cubits - also pairs of numbers - see John Baez "Talk 9: Can We Understand the Standard Model Using Octonions?" 26:30

Fields Institute

Lie Theory

Mano diagramos

Dynkin diagrams

  • How do the exceptional Dynkin diagrams (and root systems) model counting? It must be particular and restrictive. For the E-sequence, think of the math problem with two white knights and two black knights, asking how they might exchange places. The problem has to be redrawn as a state diagram and it turns out that it is a line but with one ability to side step the line just as with the E-sequence. So the counting could involve a side-stepping in memory.
  • {$\mathfrak{so}(n)=\{X\in\mathbb{R}[n]|X^{\dagger}=-X\}$}
  • {$\mathfrak{su}(n)=\{X\in\mathbb{C}[n]|X^{\dagger}=-X\}$}
  • {$\mathfrak{sp}(n)=\{X\in\mathbb{H}[n]|X^{\dagger}=-X\}$}

Kūrybos priemonės

https://freesound.org

Sąmoningumas

Fractal systems - fires, turbulences, hurricanes - model themselves and in that sense can leverage information.

John Harland

How does the causal ladder relate to John's agency hierarchy?

Kirby Urner

Modeling the world introduces a duality (and ambiguity) as to whether we want to focus on the model or the world as regards our actions - do we change our world to fit our model - or do we change our model to fit the world. This relates to gaming the game, to the distinction between object and process.

Exact sequences - Divisions of everything

Division of everything as based on probabilities, choice, relating two probablity densities, the asymptotic (conscious) base and the (unconscious) variation.

Exact sequences. The maps are the perspectives. The objects are just "filler".

  • 0 Nullsome: No map.
  • 0->0 Onesome: The map from 0 to itself.
  • 0->A->0 Twosome: Means A is 0. Distinguishes the injective map from 0 (opposites coexist) and the surjective map to 0 (all things are the same).
  • 0->A->B->0 Threesome: Defines an isomorphism between A and B. The isomorphism is "reflection". Going from B to 0 is taking a stand. Going from 0 to A is following through.
  • Short exact sequence has four mappings - the foursome. 0->A->B->C->0
  • Fivesome: 0->A->B->C->A'->0
  • Sixsome: 0->A->B->C->A'->B'->0
  • Sevensome: 0->A->B->C->A'->B'->C'->0
  • Eightsome collapses with A''. Is this like the boundary of a boundary is zero?

The collapse may be because we switch over from maps to objects. The infinite series has no zero, has no God? Starting the third three cycle yields a new relative zero. Also, it may be that at this point there is no more any constraint on the central function and so the division loses its purpose as a constraint.

Threesome

  • Long exact sequence (based on three-cycle) replaces the map from the zero object and to the zero object with the dropping by one dimension.
  • Thinking = reflection = a minimal jiggle = the Hamiltonian. As in an isomorphism between A and B in the exact sequence 0->A->B->0.
  • Threesome: Mayer-Vietoris sequences
  • Threesome. su(2) is the cross product algebra in R3
  • SU(2) root system - and the threesome - let go of one dimension and pick up another dimension.

Foursome

Fivesome

Del indicates direction

  • In a scalar field, at every point, gradient measures change in all directions ("for all"), yielding a vector.
  • In a vector field, at every point, curl measures change of a vector in perpendicular directions, yielding another vector.
  • In a vector field, at every point, divergence measures change of a vector in its own direction ("there exists"), yielding a scalar.
  • Gradient - if derivative is well defined at each direction then just add them. What about the curl? and the divergence?
  • Perturbations possible. Divide freedom over nonfreedom. Slow down or speed up in the existing direction. Veer or not in the perpendicular direction. Consider relative direction.
  • How to think of the Laplacian? as taking us from the gradient to the divergence? and how to think of the Dirac operator (on spinors) as a square root of the Laplacian?
  • A vector space with a nonzero curl must have, at every point where it is nonzero, a quiet axis, normal to the curl, along which things are held fixed. If there is no curl then there is no such distinguished axis.
  • Effects of cause = Why. For why is knowing all of the effects of a cause. And that is how quantum mechanics works - the collapse of the wave function defines the arisal of the wave function - the collapse defines what must have been the inputs, the causes. That is how it works at CERN.
  • Functional differential equations relate to double causality.
  • Helmholtz decomposition (the fundamental theorem of vector calculus) any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into the sum of an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field. An irrotational vector field has a scalar potential and a solenoidal vector field has a vector potential, thus the Helmholtz decomposition states that a vector field (satisfying appropriate smoothness and decay conditions) can be decomposed as the sum of the {$ -\nabla \phi +\nabla \times \mathbf{A} $}, where {$\phi$} is a scalar field called "scalar potential", and {$\mathbf{A}$} is a vector field, called a vector potential.

Vector field that equals its own curl

{$$\nabla f(x,y,z)=(\frac{\partial f}{\partial x}(x,y,z),\frac{\partial f}{\partial y}(x,y,z),\frac{\partial f}{\partial z}(x,y,z))$$}

{$$\textrm{curl}\;\textbf{F}=(\frac{\partial F_3}{\partial y}(x,y,z)-\frac{\partial F_2}{\partial z}(x,y,z), \frac{\partial F_3}{\partial y}(x,y,z)-\frac{\partial F_2}{\partial z}(x,y,z)$$}

Sixsome

  • Morality has us choose the base world with regard to which we make our predictions and measure our surprise as per Karl Friston and free energy.

Does the exact sequence for the sixsome make the curl chiral?

  • The fact that the three-cycle goes in one direction - is that related to the chirality of the weak force - as per emotions 5+3=0 ?

Geometry

Differentiation

Alex Codes. Symbolic Differentation in Python from Scratch!

Orthogonal Sheffer polynomials

  • Are particle clocks related to clock-and-shift operators and generalized Pauli matrices (https://en.wikipedia.org/wiki/Generalizations_of_Pauli_matrices) and generalized Clifford algebras? https://en.wikipedia.org/wiki/Generalized_Clifford_algebra
  • Ibraheem F. Al-Yousefm Moayad Ekhwan, Hocine Bahloul, Hocine Bahlouli, Abdulaziz Alhaidari. Quantum Mechanics Based on Energy Polynomials. We use a recently proposed formulation of quantum mechanics based, not on potential functions but rather, on orthogonal energy polynomials. In this context, the most important building block of a quantum mechanical system, which is the wavefunction at a given energy, is expressed as pointwise convergent series of square integrable functions in configuration space. The expansion coefficients of the series are orthogonal polynomials in the energy; they contain all physical information about the system. No reference is made at all to the usual potential function. We consider, in this new formulation, few representative problems at the level of undergraduate students who took at least two courses in quantum mechanics and are familiar with the basics of orthogonal polynomials. The objective is to demonstrate the viability of this formulation of quantum mechanics and its power in generating rich energy spectra illustrating the physical significance of these energy polynomials in the description of a quantum system. To assist students, partial solutions are given in an appendix as tables and figures.

Quantum physics

  • Quantum potential from looking at the real component we get an extra potential term. I imagine this is the extra energy that is stored when observations are forced to take on quantum states.
  • Not everything can be made explicit at the same time. When a particle has perfectly specified position then it has no momentum and vice versa.
  • Bohm theory. Chains of unfolding and enfolding. Compare with universal covering space and folding into equivalences.

Basil Hiley

Algebra

  • Abstract algebra. Dummit and Foote.

Algebraic topology

  • Armstrong. Basic topology.
  • Munkres. Topology.
  • Munkres. Elements of Algebraic Topology.

Binomial theorem

  • Meditations by Marcus Aurelius
  • Meditations by Renee Descartes
  • On Liberty by John Stuart Mill
  • Octavia Butler - Parable of the Sower, Parable of the Talents
  • Cixin Liu - Trijų kūnų problema (Kitos knygos)
  • Ursula Le Guin - The Left Hand of Darkness, Dispossessed

A sum of particle clocks is like a prism operator (in the proof for singular homology that homotopic maps induce the same homomorphism for the homology groups) but without the minus signs.

In what sense does gauge theory express the slack in the way that holistic divisions (0,1,2,3) and lax divisions (7,6,5,4) complement each other? And does that relate to the four forces? And is it expressed by the equations 3+3=6, 4+3=7, 5+3=0, 6+3=1 ?

Michael Atiyah - From Algebraic Geometry to Physics - a Personal Perspective, slides

Wisdom distinguishes everything and slack, what is whole and what is free, holisticity and laxity.

Survival and Flourishing Fund

Jaan Tallinn

https://mellon.org/about/ Mellon Foundation: Arts and Culture; Higher Learning; Humanities in Place; Public Knowledge.

Imagining Math 4 Wisdom within an ecosystem

  • Would be good to talk with Kirby.
  • https://let-me-think.org
  • Problem: current system is based on supply and demand ... and demand is poorly organized.
  • Hierarchy of options for making a living based on originality of research - how "independent" you should be able to work.
  • Options should include Patreon support (help to be validated and promoted).
  • Rewards for volunteer work already done.
  • Work on collaborative projects.
  • Contribute to repositories of public knowledge. (Wikipedia gives library priveliges). Write Wikipedia articles. Collect ways of figuring things out in various disciplines.
  • Community currency.
  • Option: part-time jobs (teaching assistants, tutors...)
  • Criteria for receiving funding: deepest value, investigatory question, working in the Public Domain.
  • Alternative certification for degrees.

How to fold circles

John C. Baez. The Homotopy Hypothesis.

https://researchseminars.org/seminar/AlgebraParticlesFoundations

Gauge theories

  • zero-dimensional - for needs - gravity?
  • one-dimensional - for doubts - electromagnetism
  • two-dimensional - for expectations- weak force
  • three-dimensional - for values - strong force

Interview with Ru Dep FM Ryabkov by @ElenaChernenko@twitter.com on what's going on with US-Russian arms control

John Baez. The Tenfold Way.

https://physics.stackexchange.com/questions/15082/how-did-feynman-derive-the-physics-of-medallion-vs-plate-wobble-rate

Egbert Rijke. Introduction to Homotopy Type Theory.

In morality (extending grad-curl-div) the first lens is the gradient is the "deepest value" that we choose to understand everything by and that we expand in eternal growth as we transcend ourselves. (What you believe is what happens.)

Theo Buhler. Exact categories

https://www.johnmyleswhite.com/notebook/2013/03/22/modes-medians-and-means-an-unifying-perspective/

Think of curl as surjection followed by injection.

How does curl relate to local-global distinction at nLab ?

https://blog.mathed.page/2019/02/20/in-defense-of-geometry-part-i/

Nathan Carter "Visual Group Theory"

https://en.wikipedia.org/wiki/Dunce_hat_(topology)

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

https://www.quantamagazine.org/triangulation-conjecture-disproved-20150113/

https://jeremykun.com/2013/04/10/computing-homology/

Purcell. Electricity and Magnetism

https://mathinsight.org/curl_definition_line_integralhttps://mathinsight.org/curl_definition_line_integral curl in terms of line integrals

https://en.wikipedia.org/wiki/Exterior_derivative#Gradient

90 degrees + 90 degrees can equal anything. But specifically can go from the diameter of a cube (standing on its vertex) to the vertex and back on the diameter to any point. But the same is true for 120 + 120.

Three-cycle

Exact sequences factor objects, thus divide them. So in what sense is it the morphisms that are the perspectives in a division of everything?

Long list of examples of short exact sequences

Exact sequence: Grad, Curl, Div

Thomas Lam. An invitation to positive geometries

What do the Pauli matrices say about the Threefold way?

  • Rotations: real, complex, quaternion
  • http://en.wikipedia.org/wiki/Bloch_sphere rotation operators about the Bloch basis
  • random matrices, ensembles
  • Carlo Beenakker: In the context of Dyson's threefold way, the Pauli matrices produce two of the three ensembles of random Hamiltonians. A Hermitian matrix 𝐻 with normally distributed matrix elements belongs to the Gaussian Orthogonal Ensemble (GOE) if the matrix elements are real, to the Gaussian Unitary Ensemble (GUE) if the matrix elements are complex, and to the Gaussian Ensemble (GSE) if the matrix elements are linear combinations of Pauli matrices of the form 𝐻𝑛𝑚=𝑎(0)𝑛𝑚𝐼2+𝑖∑𝑝=13𝑎(𝑝)𝑛𝑚𝜎𝑝,𝑎(0),𝑎(1),𝑎(2),𝑎(3)∈ℝ. The restriction to real coefficients is essential, without it the Hamiltonian ensemble is the GUE instead of the GSE. The GOE cannot be obtained from Pauli matrices.

Twelvefold way

  • Twelvefold way compares distinguishable and indistinguishable. Is that like definite and indefinite? How does distinguishable relate to Spencer Brown's Law of Forms?
  • Twelvefold way - what if we consider other sets or structures rather than 1,...,n. Does injection become less trivial?
  • Threesome (injection, isomorphism, surjection) relates to the combinatorial Twelvefold way. How does isomorphism relate with "any function"?

Paul Lockhart's Measurement

For John time reversal https://golem.ph.utexas.edu/category/2011/01/the_threefold_way_part_4_1.html

The Three-Fold Way actually says that every self-dual irreducible unitary complex representation of a group comes from either:

• a real-unitary representation on a real Hilbert space, or

• a quaternion-unitary representation on a quaternionic Hilbert space.

In the Standard Model, fermions are not their own antiparticles, but in some theories they can be. Among other things, this involves the question of whether the relevant spinor representations of the groups Spin(p,q) are complex, real (‘Majorana spinors’) or quaternionic (‘pseudo-Majorana spinors’). The options are well-understood, and follow a nice pattern depending on the dimension and signature of spacetime modulo 8.

Pauli matrices Wikipedia. Minkoeski sppace. Time is no rotation. Time is scalar times identity. Space is rotations.

Agency machine is what establishes or computes a center for a subsystem. A center indicates an observer and allows for the creation of a new vertex.

Subsystem has its own center - that is what makes it a subsystem.

Randomness is a symmetry - John's editor may be enforcing the symmetry - just as gluons enforce symmetry.

Nima Arkani-Hamed: Fluctuations (at short distances) based on a magnitude of discrepancy of 1 in {$10^120$} (at long distances). But this could be given by the addition of one discrete node to the existing {$10^120$} nodes, depending on whether or not it is included, whether the center is expressed as a node or not.

What tames violent quantum fluctuations of the vacuum? John's question: What tames deviation from randomness - what is the editor for randomness? How does that relate to going back and forth in time?

Are the particle clocks - the carving up of space for an observer - related to the existence of mass and interaction with the Higgs boson?

Trying to verify that a space is empty yields pairs of particles and anti-particles. The closer you look, the more chance that you will find exotic things. How does this relate to carving up space into what does not happen?

Expansion of space is related to the growth of discrete space, as with the Sheffer polynomials.

In the generating functions for the various Sheffer polynomials, why is it that the Hermite and Meixner-Pollaczek polynomials have f=0 but the other ones have {$f\neq 0$}? Does this have a physical meeting regarding the collapse of the wave function, the lack of some distance between the clocks? Also, Hermite and Meixner-Pollaczek both allow for sequences of odd and even polynomials in terms of {$x^{\frac{1}{2}}$}? And both have moments in terms of combinatorial objects that are pairs (like involutions, secant numbers).

For John: the belt trick may be related to the holographic principle whereby the angle for the circumference {$2\pi r$} is related to the the angle for the area {$\pi r^2$} by a factor of 2. Classical may measure area and quantum may measure angle, or vice versa.

Quaternions act like a gauge - 3 dimensions are unspecified - but identified with the complex i.

https://math.stackexchange.com/questions/711492/prove-that-the-manifold-son-is-connected

https://math.stackexchange.com/questions/1354627/why-is-it-so-that-a-unit-quaternion-t-can-be-written-as-t-cos-thetau-sin The quaterions have a subgroup of rotations related to any unit vector u, where that u plays the role of i, thus this divides up cos theta and i sin theta, and grounds the related 2x2 Pauli matrix.

https://math.stackexchange.com/questions/302465/half-sine-and-half-cosine-quaternions The quaternion represents a directed area, and you're rotating by that area.

{$𝑥⋅𝑦=\frac{1}{2}(𝑥𝑦+𝑦𝑥)$}

{$𝑥×𝑦=\frac{1}{2}(𝑥𝑦−𝑦𝑥)$}

{$𝑥𝑦=𝑥⋅𝑦+𝑥×𝑦$}

  • real - spread {$s=\textrm{sin}^2\;\theta$} (projection onto y axis)
  • complex - angle {$\theta$} (fraction of circumference {$2\pi$})
  • quaternion - directed area {$\frac{\theta}{2}$} (fraction of area {$\pi$})

The Hilbert space that models the spin state of a system with spin 𝑠 is a 2𝑠+1 dimensional Hilbert space. And spin can be half-integered. Think of the Hilbert space as everything divided into 2s+1 perspectives.

https://physics.stackexchange.com/questions/144294/what-do-the-pauli-matrices-mean

Combinatorial principles

  • Rule of sum
  • Rule of product
  • Inclusion-exclusion principle
  • Rule of division
  • Bijective proof
  • Double counting
  • Pigeonhole principle
  • Method of distinguished element
  • Generating function
  • Recurrence relation

Greg Friedman. An elementary illustrated introduction to simplicial sets.

Lauren Williams - Combinatorics of the amplituhedron

Physics 283b: Spacetime and Quantum Mechanics, Total Positivity & Motives

In what sense is the foursome given by the classification of topological surfaces in two-dimensions?

  • Sphere: Why - orientable, without obstruction
  • Torus: How - orientable, with obstruction
  • Klein bottle: What - unorientable, with obstruction
  • Projective plane: Whether - unorientable, without obstruction

Spirtes, Glymour and Scheines. Causation, Prediction and Search. (Adaptive Computation and Machine Learning).

Сергей Гуриев и Олег Радзинский | Есть ли будущее у России?

Causality

Causal Set Theory

How does the theory of symmetric functions of the eigenvalues of a matrix work in the case of generalized eigenvalues and Jordan canonical form?

Grassmanian can be thought as defined on center at x as it moves in a manifold and yields tangent spaces

Global quanta have significance for morality. The Kravchuk polynomials arise when {$\frac{\gamma}{\alpha\beta}$} is an integer. And the chance of that is measure zero, possible but with zero probability. This is an example of "design" and a refutation of "emergence from chaos". The continuum is the presumption for an experiment based on the search for such possible evolution.

Universal concepts such as universal confounders the confounder.

Ambiguity is describedby equations.

  • Atmosphere has mass of 5.15×10^{18} kg
  • Person breathes 10 tons of oxygen = 10,000,000 grams of oxygen in their lifetime
  • 16 grams of oxygen has 6*10^23 atoms of oxygen
  • 1 gram of oxygen has 0,375*10^23 atoms of oxygen

Paul Humphreys. The chances of explanation : causal explanation in the social, medical, and physical sciences.

http://www.ortodoksas.lt/p/apie-mane.html

https://rekvizitai.vz.lt/imone/krikscioniu_ortodoksu_iniciatyvu_centras/

https://www.lrt.lt/naujienos/lietuvoje/2/1674565/atleistas-staciatikiu-kunigas-mums-buvo-pasakyta-nekalbeti-apie-kara-ir-apie-ukraina-apskritai

http://www.constitution.ru/en/10003000-01.htm

(1) formulate a hypothesis, (2) deduce a testable consequence of the hypothesis, (3) perform an experiment and collect evidence, and (4) update your belief in the hypothesis.

Holmes performed not just deduction, which works from a hypothesis to a conclusion. His great skill was induction, which works in the opposite direction, from evidence to hypothesis. Another of his famous quotes suggests his modus operandi: “When you have eliminated the impossible, whatever remains, however improbable, must be the truth.” Having induced several hypotheses, Holmes eliminated them one by one in order to deduce (by elimination) the correct one. Although induction and deduction go hand in hand, the former is by far the more mysterious. This fact kept detectives like Sherlock Holmes in business.

Modeling experience from old self to new self - Bayesian analysis - prior belief + new evidence = revised belief

Jacobi polynomials have a notion of combinatorial space that may be relevant for thermodynamics as it relateswo disjoint sets malping into their union.

https://gilkalai.wordpress.com/2008/12/23/seven-problems-around-tverbergs-theorem/

Pearl seeing vs. Doing

Resolution of singularities (in algebraic geometry) relates to universal covering of covering space (in algebraic topology).

Pascal's triangle

Homology sets up potential equivalences - they may be actual equivalences, which yield identities - or nonequivalences which are generators.

How do orientations of simplexes relate to the combinatorics of the signs of the elements of a Clifford algebra?

Rotation accords with orientation (of a simplex) accords with an imaginary number i or j. Orientation is related to permutation as with the linearization for orthogonal polynomials.

Judea Pearl: The Fundamental Equation of Counterfactuals {$Y_X(u)=Y_{M_X}(u)$}. Relate to independent trials - throwing away a sheet of paper (a module).

Compare parametrization t and (1-t) in definitions of homotopy and in definition of simplexes in R^n.

Putin apologists, "putinukai", children of Putin, "putinaičiai". Simbolis: Putinas - gėlė. путина - fishing season.

Judea Pearl, Dana Mackenzie. The Book of Why: The New Science of Cause and Effect

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

  • Combinatorial physics only emerged as a specific field after a seminal work by Alain Connes and Dirk Kreimer,[4] showing that the renormalization of Feynman diagrams can be described by a Hopf algebra.

Math House of Knowledge

  • Independent trials are the basis for independence of variables and thus randomness. This randomness, this independence is the original implicit context. We can always start over with a new sheet of paper. And that fact manifests itself either as a blank piece of paper (which we work on further) or as a sequence of papers (which we refine).
  • The final context is what we can have by letting go of the assumptions that we have taken up. We can take up a new implicit context.
  • Our ability to make assumptions and let them go is what allows for symmetry groups. They make context explicit.

How are algebraic topology and geometry related? Algebraic topology describes what is not there - the holes.

Does the gradation of mathematical proofs match with the six pairs of levels in the system within the 24 ways of figuring things out in mathematics?

Russia-Ukraine

Is Monty Hall problem related to quantum probability?

A rotation is given by i, and its opposite by j=-i. Pauli matrices give different ways of writing i as a 2x2 matrix. What other roles does i play with regard to rotations?

Geometry and logic: relation between level and metalevel is given by the number of coordinate systems

  • 0 coordinate systems. Overlay of level and metalevel yields contradiction.
  • 1 coordinate system. Metalevel yields a model for the level.
  • 2 coordinate systems. Metalevel allows for reversal of actions in level.
  • 3 coordinate systems. Metalevel allows for definition of variables in level.

Analyze rotations in toruses (in Lie groups) and match them (through exponential) with corresponding matrices in Lie algebras. For example, real 2x2 rotation matrix in terms of sines and cosines equals {e^M} where M has zeros on the diagonal and + and - theta on the off diagonal.

Algebraic geometry deals with crosssections (zeros) and considers values in various grids (natural numbers, integers, rationals, reals, complexes).

Basic theorems: Triple Quad formula and Pythagorean theorem

  • Cross law
  • Spread law
  • Triple spread formula

Stephen Wolfram. Metamathematics Foundations & Physicalization

Beauty - expressing everything in terms of external relations - is the key idea of category theory.

Ergodic theory relates the representations of the fivesome in terms of time and space.

Why is least squares the best fit?

Linear regression is focused on the effects "y". The generalized linear model (and the fivesome) relates that to the causes "x_1", "x_2", etc. There can then be a cause of an effect and likewise and effect of a cause and also a critical point. The critical point is where we have symmetry so that we can flip around x and y, cause and effect.

https://en.wikipedia.org/wiki/Freudenthal_magic_square organizes not just Lie groups but also symmetric spaces. http://www.ims.cuhk.edu.hk/~leung/PhD%20students/Thesis%20Yong%20Dong%20Huang.pdf

Think of orthogonal Sheffer polynomials as variously establishing a zone where A(t) and u(t) are comparable, either at the level of A(t) as with the Laguerre, or up with the u(t) as with the Hermite, or otherwise.

Ravi Vakil: Main theme of mathematics - convert harder problems to linear algebra

Interpret {$x_i-x_j$} in {$A_n$} as a boundary map as in homology.

Ergodic theorem

Chomsky: Successor function derives from the merge function applied unitarily to a single object.

Interesting example of nonlocality. Dynkin diagram chains can only have a widget at one end. The other end immediately knows that the other end has a widget.

Speculation: The difference in the measurements of the Hubble constant may relate to the history of the universe. Early in the universe the heavier particle families (in the parsing hierarchy) may have been predominant. And they may be the source of the megastructures of the universe.

Thomas noted the symmetry of {$x^0=1$}. Relate this to {$F_1$}, choosing one out of one, or none out of none.

"Dynkin diagram: Geometry for an n-dimensional chain"

Independence is given by x_i. Interdependence is in the differences x_i - x_j.

The sum x_1 + x_2 + x_3 + x_4 = 0 is the condition placed on the roots of su(n) that the trace is zero. And this is precisely the condition that lets us use quadrays.

Derive the Jacobi identity from the notion of an ideal of a Lie algebra as corresponding to a normal subgroup of a lie group

Show the noncommutativity of A_2

Ways of figuring things out in math

  • Algebraic wayss: balanced pairs x:y, x=y
  • Analytic ways: relationships x<y, x->y

Relate this to the arithmetical hierarchy

Schroedinger's cat

  • Do the probabilities of a superposition evolve as per the phasor?
  • Do they run through all possibilities between wavelengths?
  • And does that mean that the decision of whether Schroedinger's cat is awake or asleep depends on the exact moment that we make the measurement?
  • And what does that say about time? and superpositions?
  • Is this a valid interpretation of superposition?

Ways of figuring things out in Math

  • Delta-calculus is the analysis branch. Lambda-calculus is the algebra branch.
  • The levels are given by the arithmetical hierarchy.
  • Check this: Analysis: For every x (induction) there exists a y (max or min) such that for every z (greatest lower bound or least upper bound) there exists a w (limit)
  • Algebra: There exists a C (center) such that for every B (balanced pair) there exists a S (set) such that for every L (list)
  • Note that each quantifier takes us back and forth between two worlds (with adjunctions) as in statistics (the data world and the ideal world)

Ar rengti tutorial su Bill ir Tėte?

Francesco Aldo Costabile, Maria Italia Gualtieri, Anna Napoli. Towards the Centenary of Sheffer Polynomial Sequences: Old and Recent Results.

Tom Leinster. The Categorical Origins Of Entropy.

  • https://www.maths.ed.ac.uk/~tl/b.pdf
  • Given an operad O and and O-algebra in Cat, there is a general concept of internal algebra in A. Applied to the terminal monad 1, this gives the concept of (internal) monoid in a monoidal category. Applied to the operad Δ of simplices and its algebra (R,+,0) in Cat, it gives the concept of Shannon entropy.
  • Given an operad O, we can create the free categorical O-algebra containing an internal algebra. When O=1, this is the category of finite totally ordered sets. When O=Δ, then this is (nearly) the category of finite probability spaces.
  • Relate this to Sheffer polynomials.

http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf Fulton Curve Book

https://www.balticamericanfreedomfoundation.org/programs/baltic-american-dialogue-program

Quantum mechanics deals with mass behavior whereas relativity with idealized center of mass. Four layers of parsing relate them.

Forms of matter express geometry as uniformity and give rise to mass behavior even randomness.

https://projecteuclid.org/journals/annals-of-statistics/volume-10/issue-1/Natural-Exponential-Families-with-Quadratic-Variance-Functions/10.1214/aos/1176345690.full

Bosons - real representations, fermions - quaternionic representations.

Multiplying by quaternion j reverses angular momentum for electron. Is spin a clock? Like a particle clock?

https://chaosbook.org/course1/about.html

Geometry is the uniformity of choice. Here the notion of choice comes up again. Note the difference between geometry (as a science of spatial measurement) and topology (as a science of spaces). A graph is geometric in that it consists of points which allow for a choice of edges but is not geometric in the sense that the points may allow for different kinds of choices. What is the relationship between geometry and symmetry? and also randomness? and information? Does geometry of itself entail zero information?

In multiplie regression the constant {$b_0$} acts like free space, the initial compartment. The random variables are like compartments.

How is randomness related to the Riemann Hypothesis?

Three conditions: monic, orthogonal (quadratic), Sheffer (exponential)

Relate space builder interpretation of Sheffer polynomials with their expansion in terms of {$(q_{k0} + xq_{k1})$}.

Tristan Needham. Visual Differential Geometry and Forms.

Foundation of statistics is models. Distinguish the signal and the noise. Ignore the noise.

http://users.dimi.uniud.it/~giacomo.dellariccia/Table%20of%20contents/He2006.pdf The Generalized Stirling Numbers, Sheffer-type Polynomials and Expansion Theorems. Tian-Xiao

Rota et al paper on Finite Operator Calculus notes connection between umbral calculus (Sheffer polynomials) and the Hopf algebra (for polynomials).

The combinatorics of symmetric functions of the eigenvalues of a matrix is all in terms of circular loops. How is that related to the fundamental theorem of covering spaces, the enumeration of equivalences as loops, as in homotopy theory?

Algebraic geometry. Resolution of singularities. For each projective variety X, there is a birational morphism W->X where W is smooth and projective. (This brings to mind universal covering spaces, the unfolding of loops into paths.)

Projective line over {$F_1$} has two points. The second points is infinity. So what does it mean to say {$0=1=\infty$}?

Caucher Birkar - noncompact spaces hide information. That is why we work with compact spaces. And why we work with projective spaces.

trivial tangent bundles on spheres?

Hamiltonian is the sum of all the projections onto the energy eigenstates with the energies being the weights.

Do the quaternions relate to Minkowski space (-,+,+,+) ? with one dimension plus three dimensions ?

Vector cross product is an example of the three-cycle.

Study of variables

Frédéric Chapoton. Ramanujan-Bernoulli numbers as moments of Racah polynomials

Lin Jiu. Research. relates Bernoulli and Euler polynomials, and also Euler and Meixner-Pollaczek polynomials.

What are zonal polynomials?

Moments of Classical Orthogonal Polynomials

Rota, Kahaner, Odlyzko. Finite Operator Calculus. About combinatorics of Sheffer polynomials.

Bernouli polynomials - umbral calculus

Kervaire-Milnor formula

  • {$\Theta = \Pi B$} where {$B=a_m2^{2m-2}(2^{2m-1}-1)B_{2m}/4m$}

Sidney Morris. Topology Without Tears

In the unfolding of math

  • consider math as given by generators and relations
  • the relations are equivalence classes

https://neo4j.com Neo4J graph database management

James Munkres. Elements of algebraic topology.

Johan Commelin: "Breaking the one-mind-barrier in mathematics using formal verification"

Matematika ir fizika

https://en.wikipedia.org/wiki/Kultura advocated recognizing Poland's post war borders, supporting Ukrainian, Lithuanian, Belarussian independence. Resisting Russian imperialism and Polish imperialism. Similarly, supporting Crimean independence and even Donbas independence and resisting Ukrainian imperialism.

https://ncatlab.org/nlab/show/Functorial+Semantics+of+Algebraic+Theories

J. A. Nelder, R. W. M. Wedderburn. Generalized Linear Models. 1972.

Semilocally simply connected -> can have a simply connected covering space - won't run into Zeno's paradox, which converts the diminishing sizes into the same size (of the names)

For projective planes you mod out by the units. Two (+1 and -1) for reals. Circle for complexes. Octonions problematic because the units are not a group so how do you mod out by them? Nobody knows.

8 is special because {$\sqrt{8/4}=\sqrt{2}$} is the distance between neighbors but also the interspersed lattice in constructing the E8 lattice. 240 is the kissing number.

  • {$128=dim(\mathbb{O}\otimes\mathbb{O}^2)$}

http://stat.vadoveliai.lt/

Are the conditions for coverings the basis for completeness?

GALOIS COVERS AND THE FUNDAMENTAL GROUP RUSHABH MEHTA

Generalized linear model is a flexible generalization of ordinary linear regression. The GLM generalizes linear regression by allowing the linear model to be related to the response variable via a link function and by allowing the magnitude of the variance of each measurement to be a function of its predicted value.
MIT. Philippe Rigollet. Statistics for Applications.

Canonical link function - distinguishes the essence of the NEF-QVFs.

Morris and Lock (2014), "Starting with a solitary member distribution of an NEF, all possible distributions within that NEF can be generated via five operations: using linear functions (translations and re-scalings), convolution and division (division being the inverse of convolution), and exponential generation..." (Statistics Stack Exchange)

What is the relation between the Pearson distribution and the natural exponential families with quadratic variance functions?

  • NEF-QVF have conjugate prior distributions on μ in the Pearson system of distributions (also called the Pearson distribution although the Pearson system of distributions is actually a family of distributions rather than a single distribution.) Examples of conjugate prior distributions of NEF-QVF distributions are the normal, gamma, reciprocal gamma, beta, F-, and t- distributions. Again, these conjugate priors are not all NEF-QVF.

Meixner classified all the orthogonal Sheffer sequences: there are only Hermite, Laguerre, Charlier, Meixner, and Meixner–Pollaczek. In some sense Krawtchouk should be on this list too, but they are a finite sequence. These six families correspond to the NEF-QVFs and are martingale polynomials for certain Lévy processes.

  • https://en.wikipedia.org/wiki/Martingale_(probability_theory) A martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence is equal to the present value, regardless of all prior values.
  • https://en.wikipedia.org/wiki/Lévy_process A stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are random, in which displacements in pairwise disjoint time intervals are independent, and displacements in different time intervals of the same length have identical probability distributions. A Lévy process may thus be viewed as the continuous-time analog of a random walk. The most well known examples of Lévy processes are the Wiener process, often called the Brownian motion process, and the Poisson process. Further important examples include the Gamma process, the Pascal process, and the Meixner process. Aside from Brownian motion with drift, all other proper (that is, not deterministic) Lévy processes have discontinuous paths.

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

  • normal distribution with known variance
  • Poisson distribution
  • gamma distribution with known shape parameter α (or k depending on notation set used)
  • binomial distribution with known number of trials, n
  • negative binomial distribution with known r

These five examples – Poisson, binomial, negative binomial, normal, and gamma – are a special subset of NEF, called NEF with quadratic variance function (NEF-QVF) because the variance can be written as a quadratic function of the mean.

Given a positive semidefinite inner product on the vector space of all polynomials, we have a notion of orthogonality. Then the orthogonal polynomials can be obtained from the monomials {$1,x,x^2,x^3\dots$} by the Gram-Schmidt process.

Tom Copeland: Btw, the refinement of the Stirling polynomials of the second kind is OEIS A036040, which is the coproduct of the Faa di Bruno Hopf algebra (see the Figueroa et al. paper in the entry and Zeidler in his book QFT I, p. 859 onward ). In some contexts, a more convenient/enlightening basis for composition with the famous associahedra polynomials for the antipode / compositional inverse is the set of refined Lah partition polynomials of A130561. Both of these, of course, are related to Scherk-Graves-Lie infinigins that are umbralizations of infinigins for the coarser polynomials.

Shoelace formula for oriented area of a polygon

Dobinski's formula relates Bell numbers and e. {$B_n=\frac{1}{e}\sum_{k=0}^{n=\infty}\frac{k^n}{k!}$}

Why can we comb even-dimensional spheres but not odd-dimensional spheres?

Math Overflow: What's Up With Wick's Theorem?

Suggested by Tom Copeland

  • "Boson Normal Ordering via Substitutions and Sheffer-type Polynomials" by Blasiak, Horzela, Penson, Duchamp, and Solomon;
  • "Normal ordering problem and the extensions of the Striling grammar" by Ma, Mansour, and Schork;
  • "Combinatorial Models of Creation-Annihilation" by Blasiak and Flajolet;
  • the book Commutation Relations, Normal Ordering, and Stirling Numbers by Mansour and Schork with an extensive bibliography.
  • affine relates ?
  • projective relates circle and line
  • conformal relates (Riemannian) sphere and (complex) plane
  • symplectic relates S3 ? and (quaternionic?) R3 ?

Quaternionically differentiable is linear.

18.4 penrose. Hyperbolic length is one half of the rapidity it represents.

Laws of physics time symmetric for particles traveling atvthe speed of light so time does not change

Localization arises from local shielding by local interactions. That is what weakens global interactions which othwrwise exist.

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

https://nyuad.nyu.edu/en/events/2022/march/nyuad-hackathon-event.html

https://math.stackexchange.com/questions/939856/every-group-is-a-fundamental-group

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

https://unito.webex.com/unito-en/onstage/g.php?MTID=ea2ecf75b7c2446b64320b17e00bf90fe

Save For a Future Video

For almost two years I have been studying quantum physics with my old friend John Harland who is passionate about it. We both took courses in quantum mechanics in college and later met at the University of California at San Diego where he got his PhD in math doing functional analysis and I got mine doing algebraic combinatorics. Recently, I thought that I should learn quantum physics better as a source of inuition about Lie theory, which I think is central to the way that mathematics unfolds, especially through affine, projective, conformal and symplectic geometries. I was glad to join John in studying "Introduction to Quantum Mechanics" by David Griffiths, which is an excellent textbook for learning to calculate as physicists do. Relearning this, as a combinatorialist, I noticed the orthogonal polynomials in solutions of the Schroedinger equation and I became curious to learn what structures they encode.

Sheffer polynomials

  • Give examples of generating functions of orthogonal polynomials that are not Sheffer sequences, such as the Chebyshev polynomials and the Jacobi polynomials.
  • Chebyshev polynomials of the first kind {$\sum_{n=0}^{\infty}P_n(x)\frac{t^n}{n!} = e^{tx}\textrm{cosh}(t\sqrt{x^2-1})$} we could write {$\frac{1}{2}e^{tu}+\frac{1}{2}e^{tv}$} where {$u=x+y$} and {$v=x-y$} and {$y=\sqrt{x^2–1}$} and {$x=\sqrt{y^2+1}$}.
  • Wick's theorem, quantum field theory and Feynman diagrams.

Physics

  • Why is the information encoded in the coefficients as opposed to the roots of the polynomial?
  • Note that each power of x involves a crossing of the curve.
  • The notion of space-time wrapper in the context - contributed by orthogonality.
  • The idea of a two frame physics.

Fivesome

  • investigation of how a conceptual framework for decision making in space and time, which I call the fivesome, can help us interpret the mathematics of quantum physics. I am studying the information
  • Fivefold classification of orthogonal polynomials that are important for

Awodey. Topological Representation of the Lambda Calculus

Awodey. Continuity and Logical Completeness: An Application of Sheaf Theory and Topoi

Awodey, Reck. Completeness and Categoricity.

What is the relationship between matroids and root lattices?

Classify compact Lie groups because those are the ones that are folded up and then consider what it means to unfold them and that gives the lattice structure. So the root lattice shows how to unfold a compact Lie group into its universal covering. And this relates to the difference between the classical Lie groups as regards the duality between counting forwards and backwards. The counting takes place on the lattice. And you can fold or not in each dimension if you are working with the reals and so that gives you the real forms. But you can't fold along separate dimensions if you have the complexes so you have to fold them all.

  • In what way is the folding-unfolding nonabelian or abelian?
  • What does the folding-unfolding look like for exceptional Lie groups?
  • How is the widget at the end of a Dynkin diagram serve as the origin (or outer edge?) for the unfolding?

Consider how the histories in combinatorics unfold the objects. What are the possible structures for the histories? How do they relate to root systems? How do root lattices or other such structures express the possible ways that combinatorial objects can encode information by way of their histories?

How is a perspective related to a one-dimensional line - lattice - circle ?

Given the Lie group's torus T, Lie(T) / L = T, where L is a root lattice.

Abelian Lie groups are toruses. So we are interested in maximal torus for the semi-simple Lie groups.

Lie group G has the same Lie algebra as the identity component (as in the case when G is disconnected). And G has the same Lie algebra as any covering space of G.

Jacobi identity is like a product rule. Think of x as differentiation (and y and z perhaps likewise). {$[x,[y,z]] = [[x,y],z] + [y,[x,z]]$}

Lie bracket expresses the failure to commute. So that failure is part of the learning process.

How are root lattices related to matroids?

Study choice, probability, statistics.

What to ask online?

Terrence Tao. Trying to understand the Galois correspondence.

First Person Physics

Neurocognitive Foundations of Mind 2022

Two reflections give you a rotation. So is a reflection the square root of a rotation? And does that relate to spinors?

Richard Southwell describes how mathematical functions can be visualized by: (1) elements and arrows (2) Wiring diagrams (3) fibres (4) bouquets (5) graphs (6) ontology logs (7) categories

Double covering nature of SO(3) and SU(2) is the basis for the nature of spin. (alpha, beta) and (i alpha, i beta) have the same squares so give the same probability which yields the double cover.

SU(2) is a three-sphere in four-dimensional space.

Think about Dynkin diagrams and related lattices.

  • BC_n lattice simple roots are: Face - Edge - Edge - Edge - Edge ...
  • C_n looks like an octahedron. What does each lattice look like?
  • Consider the cases of A_n and D_n how to picture that
  • Consider the Weyl groups and what they are symmetries of.

Dirac's plate trick Plate trick

Penrose, Rindler. Spinors and Space-Time: Volume 1, Two-Spinor Calculus and Relativistic Fields

The Weyl/Coxeter group {$G = W(F_4)$} is the symmetry group of the 24-cell.

Global macroeconomics

Michael Hudson

Summer of Math Exposition 2022 Results

Coxeter. Regular polytopes. Includes prehistory. Boole.

Coxeter diagram {$D_n$} symmetry group of demicube: every other vertex of a hypercube. Is that related to a coordinate space? Combinatorially, can we flip the vectors of the demicube to get a coordinate system?

Grover's algorithm

Roe Goodman. Alice through Looking Glass after Looking Glass: The Mathematics of Mirrors and Kaleidoscopes

Cube reflections given by vectors u, v, w from the center of the cube to the center of a face, the center of an edge, and the center of another edge. And the angles between the vectors are pi/2, pi/3 and pi/4. And the two edge midpoints are separated by pi/3 so rotating through six such edges gets you back. And that is the chain for the Dynkin diagram.

Conjugation is an example of reflection.

Finite field with one element

  • Choosing one out of one: Driving on a winding road, each turn is a choice of one out of one. Whereas a fork is a choice of one out of two, a usual intersection is a choice of one out of three and so on.

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

https://4returns.commonland.com/getting-started/

https://ec.europa.eu/commission/presscorner/detail/en/IP_22_4489

The 4 returns: natural return (value of landscape), economic return (restart agriculture), social return, humans return.

Let them win

  • Musk's comments led to Lavrov encouraging such comments, widening dialogue.
  • Military sense - feign in Kherson
  • Drive to Kyiv but went back
  • Overextended, then they moderate, can win in the future
  • Vytautas Didysis atidavinėdavo žemaičius

Locality is the whole achievement of the continuum. Local means low overhead and the actual global time frame is even lower overhead. Locality arises with orthogonality, assumes measurement, observers, space time wrapper.

Differentiation changes level. {$x^n$} number of levels of volatility, number of derivatives

Kirby Urner

  • Elective disaster, global warming discourse
  • The Shepherd's tone - auditory illusion, as if it were ever rising
  • Our own sense of mortality, imposing it on everything.
  • Believing in eternal life.
  • Too hooked on 90 degrees, should move to 60 degrees - Fuller.
  • Digging around the concept of dimensions.
  • OK to be on a different page
  • Sand castles on a beach - mathematics (numerative systems)

Spaces of states

  • nLab: State
  • Classical bit: line segment [0,1]
  • Qubit: shaped like an American football

{$\begin{pmatrix} a & b+ic \\ b-ic & d \end{pmatrix}$}

Think of probabilities {$a, 1-a$} and mediator {$b \pm ic$}. We have {$a^2+b^2+c^2\leq a$} and {$a^2+b^2+c^2 = a$} for pure states. Rotate {$a-a^2$} from 0 to 1 around the a-axis.

Wenbo

  • Visual group theory

It is the MacMahon Master Theorem that unifies the angular momentum properties of composite systems in the binary build-up of such systems from more elementary constituents.

Summer of Math Expostion 2022 playlist

https://math.stackexchange.com/questions/3472796/find-all-connected-covering-space-of-mathbb-rp2-vee-mathbb-rp2

Learning classifier system

Semilocally path connected avoids Zeno's paradox. Universal covering as naming schemes.

Try to interpret the Gamma function (especially for fractions such as 1/2, pi/sqrt(2), or negative numbers) in terms of the choice function for the binomial theorem.

Robert Gilmore. Group Theory. XIV. Group Theory and Special Functions. Relates Lie groups and orthogonal polynomials.

Licata. Computing with Univalence., Talk

Univalence from a computer science point-of-view - Dan Licata

Local - special relativity, global - general relativity

Local - reversible, global (default) not reversible ("Not every cause has had its effects")

Modeling

Quantum Magic Rectangles: Characterization and Application to Certified Randomness Expansion Sean A. Adamson∗ and Petros Wallden and generalization of the magic squares

How are games in game theory (with incomplete information, partial information) characterized by probability distributions.

Charactization and application to certified randomness expansion

Entropy - physics is related to symmetry - Shannon entropy is related to information. And how does that relate to randomness?

In Cartesian categories you can copy and delete information. (John Baez - Rosetta Stone) How does that relate to Turing machine?

Consider in what sense the physicist Hermite and probablist Hermite polynomials are two related sequences as when defining orthogonal Sheffer polynomials.

Think of perspectives, divisions of everything, in math, as being probability distributions, or more generally, models of probability that, by means of a choice, relate two realms, as does a perspective. And think how all of math could be derived from the unfolding of such perspectives, the relations between realms.

House of knowledge for math

  • Sequence (starting analysis) is an infinite list. A list (ending algebra).
  • Limit (ending analysis) is similar to a center (starting algebra).

Alytaus kredito unija

  • Kapitalas 700,000 EUR, pelnas 43,000 EUR, paskolinta 4,600,000 EUR.

Peacemaking

  • Work towards energy independence - and economic independence - of all nations - so that one nation does not depend exclusively on any other particular nation (Russia, China, US) for strategic goods. Allow for choice. Dialogue about globalization.

Bose statistics - can't assign labels. Fermi statistics - can assign labels to particles.

Information capacity is zero if probability is the same for all cases but also if one case is given 100%. Information transmission requires asymmetry. Otherwise you cannot define choice.

Relate one, all, many with symmetry breaking and search for constancy. How is constancy related to symmetry?

Symmetry breaking - choosing one possibility. From symmetry breaking randomness appears and information is constructed. Deterministic is replaced by irreversibility.

Randomness as derived from a wall that allows for independent events, as with the other, or with transcendence.

Kleisli categories and probability - 01 - The Giry monad

Randomness as lack of knowledge.

Giry monad related to probability.

https://wallhaven.cc/w/o3ldk5

https://ncatlab.org/nlab/show/syntax-semantics+duality

https://ncatlab.org/nlab/show/relation+between+type+theory+and+category+theory

For different type theories we can construct different categorical models.

  • What is the common requirements for the additional structures to be valid semantic models of a type theory?

Michael J. Kearns. An introduction to computational learning theory.

Shai Ben-David - Machine Learning Course (Computational Learning Theory)

https://www.youtube.com/playlist?list=PLPW2keNyw-usgvmR7FTQ3ZRjfLs5jT4BO

Creating what you can feel certain about. (Continuity.)

  • Building up levels of certainty through topological invariants.

Orthogonal Sheffer polynomials: Space builder defines cells and orthogonality relates them.

"Mathematics as a Love of Wisdom" by Colin McLarty

https://www.youtube.com/c/aryayae

https://unimath.github.io/SymmetryBook/book.pdf

https://www.mcmp.philosophie.uni-muenchen.de/students/math/index.html

https://www.philosophy.ox.ac.uk/people/timothy-williamson

https://m.youtube.com/watch?v=JcFGHrYrlZA

https://www.researchgate.net/publication/304262663_Wisdom_Mathematics

https://warwick.ac.uk/fac/soc/philosophy/people/dean/

https://plato.stanford.edu/entries/recursive-functions/

https://blog.apaonline.org/2021/04/08/the-philosophy-of-computer-science/?amp

https://freecomputerbooks.com/Philosophy-of-Computer-Science.html

https://builtin.com/software-engineering-perspectives/cs-philosophy-programs

https://news.ycombinator.com/item?id=28980203

Stone's theorem: continuous implies differentiable

Idea for Lorentz transformation. Write it out as the generalized binomial theorem (Taylor series) {$(1-x)^{-\frac{1}{2}}=1+\frac{1}{2}x+\frac{3}{8}x^2+... = \sum_{n=0}^{\infty}\frac{(2k)!}{4^k(k!)^2}x^k$} and then we need consider only the initial terms, however many are relevant for the combinatorics, which expresses the generalized binomial theorem. The usual Lorentz transformation only arises in the limit to infinity.

"belt trick", aka the "Dirac scissors" or "Balinese candle dance

When two events happen (the measurement of spins) there is a frame where one happens before the other. So if they are causally connected (as with spin measurements) there needs to be a distinguished frame. But that could be the frame in which they were initially entangled. So entanglement posits the existence of such a distinguished frame.

https://www.cantorsparadise.com/an-incredibly-useful-generalization-of-the-binomial-theorem-and-its-applications-22d41c1589ce

observational (a posteriori) and definitional (a priori) judgments as in type theory

Path integrals depend on the number of points in space, or the number of interactions. But my approach suggests that this number is actually given by the degree of x in the relevant polynomial.

4 logics for 4 geometries

  • no simplification - no distance between metalevel and level - affine - contradiction
  • simplify by one perspective relative to the center - get model
  • simplify by two perspectives - get directions, forward and backward
  • simplify by three perspectives - get variables, defined from by the side view

Yoshimasa Nakamura, Alexei Zhedanov. Toda Chain, Sheffer Class of Orthogonal Polynomials and Combinatorial Numbers

S. J. Rapeli, Pratik Shah and A. K. Shukla. Remark on Sheffer Polynomials explains J(D), relates it to A(t)

House rules:

  • The policy of the many
  • Leave things the way you found them
  • Respect constancy

What is the combinatorics of convex spaces and how does that relate to orthogonal polynomials, which give different ways of looking at the geometry?

5 notions of independency

What are the transition matrices between orthogonal polynomials?

https://www.nature.com/articles/s41534-018-0106-y#:~:text=Surface%20codes%20are%20building%20blocks,models%20describing%20random%20Pauli%20errors

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

https://mathoverflow.net/questions/118857/forcing-in-homotopy-type-theory

probabilistic programming paradigm (quantum computing)

Measurement based quantum computer vs gate based quantum computer

lattice surgery

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

topological quantum computer

https://en.wikipedia.org/wiki/One-way_quantum_computer

https://www.aimath.org/WWN/convexalggeom/AIM.pdf

Naturality in homotopy type theory breaks down when we try to do type theory in type theory.

Amelia: [The axiom of function extensionality is] inconsistent with many axioms of a more "computational" nature. For example, "formal Church's thesis" says that for any function N→N, there is a "program" (we call it a realizer) that realizes it. You can kinda see what goes wrong: this would be able to tell e.g. "λ x → x" and "λ x → x + 0" apart. You could imagine an assignment of realizers that sidesteps this, though, so to see that it's actually inconsistent takes slightly more work.

What is the relationship between universal properties as proved by the function extensionality principle, and universal properties as given by Kan extensions?

https://github.com/FrozenWinters/stlc SLTC project where Astra formalises the categorical semantics of function types in Agda.

A063573 Counts the number S(n) of lambda terms at level n, in the case of a single variable.

  • Let V be the number of variables.
  • {$S(n+1) = VS(n) + 2S(n)\sum_{i=0}^{n-1}S(i) + S(n)S(n)$}
  • This comes from two steps.
    • Add {$\lambda x.\_ $} in front of a lambda term from level n.
    • Combine two lambda terms {$( \_\;\_ )$} at least one of which comes from {$S(n)$}.
    • When V=1 we get 1,2,10,170,33490...
    • When V=2 we get 2,8,112,15008...

Calculate the combinatorics of the lambda-calculus on a single variable, and if possible, on two or more variables. Is the lambda-calculus equivalent to the recursion relation for orthogonal polynomials?

one-projectionall-constantmany-successor
recursion - all & manycomposition - many & oneminimization - one & all

Have all finite limits is equivalent to

  • Having terminal objects
  • Having a product for any pair of objects
  • Having an equalizer for any pair of parallel arrows

These are the building blocks for limits

Sean Carroll or me? Quantum field theory. Instead of space and time, consider in terms of particles and their interactions. Particle clock steps take us from possible interaction to possible interaction. Problem: field theory is based on Minkowski spacetime rather than on particles.

One-all-many relates questions (selection) and answers (judgement). Many is the regularity that every question is answered relevantly. Zero is "no" as a positive answer.

Induction argument on truncation levels uses the level below (for identities) and the level above (which we're trying to reach). Similarly, the recurrence relation relates the level xP_n(x) with the level below and the level above.

  • Andrius: I wonder if there are any connections with the arithmetical hierarchy in computability theory. In that hierarchy, the sigmas and the pis are intermixed. So I wonder if there are any ways that pis (products) get interspersed between the truncation levels?
  • Astra: If you have a Pi-type, then the truncatedness level is that of the codomain, so this behavious is a bit different from what you would see in that hierarchy

Homotopy Type Theory

  • Substitution for variables - binding and scope
  • Types are specifications are programs
  • Communicating by algorithm and certain shared assumptions. Discover those assumptions.
  • Term M (program - that when it runs) in type A (program - it runs the way A says it runs)

An empty type has no evidence for it, is not true. A nonempty type, as a proposition, is true. The notion of empty or nonempty is relevant for the sevensome, for describing {$\forall \wedge \exists$}.

Young-Il Choo - MeetUp

Every type has a unique name. Every universe is a type with a unique name. Every term in a type should have a unique name. So why can't we have a universe of unique names for all of the terms, types and universes? And if we can, then don't we run into a paradox? Or not?

The 4th movement of Beethoven's Symphony No. 5. Conducted by Arthur Nikisch. Recorded in 1913.

{$\alpha$} and {$\beta$} count ascents and descents and these are steps forwards or backwards in the unfolding of space (in time?) and so they may relate to John's picture of evolution taking us forward and backward in time.

Roelof Koekoek & René F. Swarttouw. The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue

Space has 3 dimensions external to the fivesome (5+3=0)(outside the division) and time has 1 dimension internal to the fivesome (the slack inside the division).

https://ww3.math.ucla.edu/dls/emily-riehl/ video about contractibility

An isomorphism is a special morphism but truly it is a pair of morphisms that are inverses to each other. There may be many such pairs relating two objects but in each pair the inverses are unique with respect to each other. So it is similar to complex conjugation.

Bell number interpretation of Sheffer polynomials gives a foundation for (finite) (and countable) set theory.

Mariana M. Odashima, Beatriz G. Prado, E. Vernek. Pedagogical introduction to equilibrium Green's functions: condensed matter examples with numerical implementations.

https://math.stackexchange.com/questions/989083/is-composition-of-covering-maps-covering-map

https://52livingideas.com

Charlier polynomials give the trivial space wrapper (the moments are the Bell numbers). In what way are the Hermite polynomials trivial?

Space wrappers reinterpret Bell numbers.

Types indicate comparability which is a condition for equality.

From a dream: I imagined that I was entering a spherical world full of structures, and that my perspective upon those structures was a hyperbolic geometry, expressing the Lorentz contraction, thus special relativity.

OpenShot eksportuoti 30 fps nes iPhone filmuoja 30 fps

Open Source Software to Thank

  • Linux, Ubuntu, OpenShot, Dia, GIMP

Shot with an iPhone XS Max.

Schuller on Stone's Theorem

https://www.freelists.org išbandyti?

https://math.stackexchange.com/questions/69698/wedge-sum-of-circles-and-the-hawaiian-earring

https://www.facebook.com/hackersatcambridge contact team @ hackersatcambridge.com

portray mu as measuring tape

portray mu as super hero measuring tape with two hands ready to hold on

  • Composition - roots
  • Primitive recursion - runners
  • Minimization operator - seed

Weed in cracks of cement

In defining the minimization operator, and in coding a list of natural numbers with a single natural number:

  • It is problematic to code "nonhalting" as an integer because it cannot serve as an input. So it would have to be an integer that cannot be used as an input. In this way it is an input that got deleted. And so this is where the power of computing is increased, through the introduction of deletion of a variable.

Minimization operator: representations of nullsome have us proceed through all levels (from true to direct, from direct to constant, from constant to significant)

  • necessary (0 is constancy), actual (the meaning of 0), possible (if starts, then ends)
  • object, process, subject

Constancy - search for meaning

  • one, all, many

Significance - go beyond

  • being, doing, thinking

In the search for constancy: take a stand (as to one), follow through (across all), reflect (supposing many)

In physics, orthogonal polynomials relate what is necessary (top down) and actual (bottom up) as with string theory, questions and answers.

The original spectral theorem: Look for subrepresentations such that S is a one-dimensional matrix eigenvalue. Induction argument.

Classical (both x, p) and quantum (x).

Bald and bankrupt Eastern Europe

The Screwing of the Average man: How the rich get richer and you get poorer

Special relativity - causal connection - are they time like connected.

Wick's theorem - are operators of the same particles - propagator connects

Evolution is indicated by learnability and also by sparse communication and natural differences between hierarchies, different orders of magnitude, allowing for a natural hierarchy of niches. Not only the laws of physics are sparse but also the states in nature are sparse.

Rules of physics plus configuration space plus location within that space.

Source of contradiction

  • We are finite, our system is finite, but the Spirit is infinite dimensional

Uncertainty principle - has to do with representations - representation adds a perspective - so that interferes with measuring certain things.

Minimization operator mu - superhero - who clings to ledges and other such things and is stretched and blown by the wind. And the shape mu gives the shape of his body clinging to the left.

https://en.wikipedia.org/wiki/%CE%9C_operator {$\mu$}-operator

I had a dream that i was professor anthony zee... But in a quantum superposition. Was i z or not z ? Z or not z? ..... is there a third way? Yes but there is a fourth way .... Nevermind z here is m4w!

A qubit specifies the relation between affirmation and negation of probabilities. In matrix form, it provides a complex number which is the coefficient that gets multiplied to the negation (in calculating the new affirmation) and whose conjugate gets multipled to the affirmation (in calculating the new negation). In classical bits, this coefficient is simply zero.

Bekaert, Boulanger. The unitary representations of the Poincare group in any spacetime dimension

Five zones of scattering can be thought of as

Measurement establishes a quantity with regard to boundaries - it establishes the zone within which it is - identifies with a step in the algebra - whereas analysis demarcates the boundaries.

Kojin Karatani, Sabu Kohso - Architecture as Metaphor_ Language, Number, Money (1995) semi-join lattice semilattice

Hatcher exercise

Osborne IV 40:00 what is needed for a relativistic quantum field theory.

Brody. Quantum Mechanics and Riemann Hypothesis.

Brown. Topology and Groupoids.

Yevgeny B. Karasik. Duality revolution: Discovery of new types and mechanisms of duality that are revolutionizing science and technology as well as our ability to solve problems

Think of -1-cell as the center (of all things), the spirit. And think of 0-cell not simply as a point but as a 0-dimensional open arc (the point shell) with regard to that center (the spirit). The point shells are glued onto the spirit, and similarly, open arcs are glued onto point shells, and so on, inductively.

https://www.thphys.uni-heidelberg.de/~floerchinger/categories/ Quantum Field Theory

https://bookstore.ams.org/surv-149/12

https://physics.stackexchange.com/questions/566963/what-is-the-exact-meaning-of-lorentz-invariance-for-a-quantum-scalar-field

https://ncatlab.org/nlab/show/cellular+approximation+theorem#applications

Relate walks on trees to covering groups. What do conjugates (paths) mean? What is the homotopy group?

https://en.wikipedia.org/wiki/Koopman%E2%80%93von_Neumann_classical_mechanics

https://www.amazon.com/Quantum-Challenge-Foundations-Mechanics-Astronomy/dp/076372470X

In what sense are Feynman diagrams relativistic given that they have directions for time and for space?

Instead of thinking of speed of light, think of a clock that doesn't tick, so that t=0 always. And this is the case for the quantum harmonic osciallator and for the particle-clocks with no steps.

One {$\exists x$}, all {$\forall x$}, many {$\neg\exists x \wedge \neg\forall x$}.

Masaki Kashiwara, Pierre Schapira. Categories and Sheaves. 2006

Gerald B. Folland

  • Quantum Field Theory: A Tourist Guide for Mathematicians 2021
  • Quantum Field Theory 2008

Bohm Pilot Wave, Thomas Spencer

Relative invariance - more global than another

Relate the three-cycle (taking a stand, following through, reflecting) to Kan extension as defined by filtering objects and morphisms, acting on them, and then taking the colimit or limit on that filter.

Alan Turing, Cybernetics and the Secrets of Life

My approach to special relativity lets me work in units in my own frame.

https://en.wikipedia.org/wiki/Algorithmic_information_theory Gregory Chaitin = Shannon + Turing = Compression-Decompression as understanding.

https://en.wikipedia.org/wiki/Cristian_S._Calude Philosophy of computation

Life in life

Thinking about the expansion of the universe as a reduction of density, by which the mass of particles becomes ever less important, by which we have an increase of entropy (becoming less deliberate). And we can reverse this by starting with an increase in entropy and arriving at the expansion of the universe.

Relate Ellerman's heteromorphism and comma category.

San Francisco Meet Up interests: Dependently typed programming languages. Language aspects of category theory. Functional programming. Topos, lambda calculus. Is type theory advantageous? Modeling infinitesimals.

Dan Shiebler. Kan Extensions in Data Science and Machine Learning

Matematika išplaukia iš (poreikių tenkinimo) algoritmų taikymo, vedančio iš duotybių į bendrybes. O tos bendrybės įkūnija, išreiškia tam tikrus prieštaravimus, juos paverčia sąvokomis, kurias galima mąstyti toliau. Pavyzdžiui, apskritimas iškyla iš begalinės simetrijos visom kryptim, arba iš virve aprėpto ploto maksimalizavimo.

  • Vaizduotė (24 matai) ir Neįsivaizduojamieji (2 matai) yra iš viso 26 matai. Ar juos išreiškia stygų teorija?

SL(2,C) character variety related to hyperbolic geometry. SL2(C) character varieties

Universal enveloping algebra is an abstraction where the generators are free and thus yield infinite generators. Whereas the Lie algebra may be in terms of concrete matrices and the underlying generators, when understood not in terms of the Lie bracket but in terms of matrix multiplication, may have relations such as {$x^2=0$}, {$h^2=1$}.

Information is what you learn. What you learn grows at the boundary, has the shape of the boundary. A shape can be thought of as being created by integrating over these boundaries as they increase.

Tai-Danae Bradley: Information is on the Boundary

Prove that the matrix made up of eigenvectors diagonalizes a matrix.

In special relativity, think of distance squared over time as surface area per time, the difference beween the surface areas of two spheres, one expanding with velocity v, and the other with velocity c.

For John: How could we get negative energy? Consider how to get imaginary square roots. For example, if a speed is greater than the speed of light, then the relationship between time and position is multiplied by an imaginary number.

Quaternions, Dirac equation: Pauli matrices are the three-cycle for learning and they are extended by a fourth dimension of non-learning (what is absolutely true or false) for the foursome.

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

Covering spaces with repetition yield the spaces they cover.

Think of combinatorial quantum physics operator (raising and lowering operator) as a derived functor.

Shmuel Weinberger. Algebraic theory of topology. Manifolds and K-theory: the legacy of Andrew Ranicki

Unclear whether the empty space is path connected.

Think of a universal covering space as expressing the unfolding of a space, thus expressing eternal life.

Wikipedia Library

Peace conference

Relate triangulated categories (with squiggles {$X\rightsquigarrow W = X\rightarrow TW$}) to monads with likewise squiggles.

Samwel Kongere vaizdo įrašai

Nafsi Afrika Acrobats - Pyramid of Peace

Research/Notes

  • Monads deal with scopes: none, some, and so on. The logic of the sevensome.

https://www.wolframphysics.org/technical-introduction/potential-relation-to-physics/motion-and-special-relativity/

Relate {$F_1$} with the basis element 1 in a Clifford algebra.

The house of knowledge for mathematics describes 4 representations (properties) of everything (onesome, totality), which through their unity establish, define space as algebraic, consisting of enumerated dimensions:

  • center (nullsome)
  • balance
  • set of roots of a polynomial
  • list of basis vectors

(Relate this to the binomial theorem.) And it describes 4 representation of the nullsome (center), which through their unity establish, define a point as analytic. This describes four choices:

  • induction (adding a vertex, converting the center to a vertex, recursively)
  • max or min (adding an axis, as with cross polytopes)
  • least upper or greatest lower bound (making a division, a separation on one side or the other)
  • limit (center?)

4 levels of knowledge is sufficient (in the chain complexes). The house of knowledge describes those 4 levels. It relates the analytical view of a point with the algebraic view of a space. Consider the Zig Zag Lemma as applying the three-cycle to set up four levels of knowledge, 4 x 3 = 12 circumstances.

Counterquestions

  • Consider them as a subset of the utility graph {$K_{3,3}$} which describes the three utilities problem and arises in the proof of Kuratowski's theorem characterizing planar graphs.
  • The utility graph can be drawn as a hexagon, in which case only one graph can cross the center if it is to be a planar graph. In that case the center line goes from God's perspective to the world's situation. And this arrangement makes person-in-general and person-in-particular equal in status. Thus it provides a context for such equality of status. And it defines a division of everything into two: general (not knowing) and particular (knowing). It supports the equality of gender.
  • Consider how the counterquestions define divisions of everything and relate to Bott periodicity.
  • Consider how the counterquestions arise in Jesus's house of knowledge and how that relates to the house of knowledge for mathematics.
  • Consider how the counterquestions express visualization and paradox.

Think of combinatorial quantum physics operator (raising and lowering operator) as a derived functor.

Counterquestions

Potential energy source {$\frac{y}{2}$} (adding free cell) is in balance with kinetic energy source {$i\frac{\partial}{\partial y}$} (half-link)

A chain complex is loose and has slack, which is the basis for homology. An exact sequence is tight and has no slack. A division of everything is tight and has no slack.

  • Santykis su Dievu yra atgarsis, kaip kad dalelytė turi santykį su savo lauku.
  • Fizikos dėsnių raida yra pavyzdys Dievo įsakymo patobulinimo.

Fivesome

John Baez, Michael Shulman. Lectures on n-Categories and Cohomology.

  • (-1)-categories are hom(x,y) sets where x and y are parallel 0-morphisms in a 0-category, which is to say, a set. But the only 0-morphisms in a set are the identity morphisms. Thus hom(x,y) is either an identity morphism (when x=y) or the empty set (otherwise). These are the two possible (-1)-categories.
  • (-2)-categories are hom(x,y) sets where x and y are -1-morphisms in a -1-category. But there is only one non-empty (-1)-category and it has only one morphism. Thus there is only one (-2)-category and it consists of this unique morphism. This category expresses necessary equality when there is only one choice. That is reminiscent of the choice from a single choice which is modeled by {$F_1$}, the field with one element.

Note that there is only one empty set. But there could also be many empty sets. And all can be thought of as an empty set. Can the search for constancy be considered a search for emptiness?

Emily Riehl, Dominic Verity. Elements of ∞-Category Theory

Foursome

For C and D categories we have

  • f is (essentially) 0-surjective {$⇔$} f is (essentially) surjective on objects;
  • f is (essentially) 1-surjective {$⇔$} f is full;
  • f is (essentially) 2-surjective {$⇔$} f is faithful;
  • f is always 3-surjective.

Foursome

A functor between ordinary categories (1-categories) can be:

  • essentially surjective ≃ essentially 0-surjective
  • full ≃ essentially 1-surjective
  • faithful ≃ essentially 2-surjective
  • Every 1-functor is essentially k-surjective for all k≥3.

A functor {$F:C→D$} is essentially surjective if it is surjective on objects “up to isomorphism”: If for every object {$y$} of {$D$}, there exists an object {$x$} of {$C$} and an isomorphism {$F(x)≅y$} in D.

A functor F:C→D can be:

essentially (k≥0)-surjectiveforgets nothingremembers everything
essentially (k≥1)-surjectiveforgets only propertiesremembers at least stuff and structure
essentially (k≥2)-surjectiveforgets at most structureremembers at least stuff
essentially (k≥3)-surjectivemay forget everythingmay remember nothing

This formalism captures the intuition of how “stuff”, “structure”, and “properties” are expected to be related:

  • stuff may be equipped with structure;
  • structure may have (be equipped with) properties.
  • Install: OBS Studio

Are the doubts and counterquestions related to electromagnetism, U(1) and the related gauge theory?

Observing symmetry requires breaking symmetry.

https://www.masterclass.com/classes/terence-tao-teaches-mathematical-thinking

Is the associativity diagram for monoidal categories an example of the fivesome?

Involution is square root of permutation. Compare with spin as square root of geometry.

Local and global quantum are linked by experiments, by "the complicated interplay between infrared and ultraviolet affects", by a conspiracy of IR/UV mixing.

Walks

  • Independent entries vs. Rotational invariance yield {$P[X]\propto e^{-\frac{1}{2}\textrm{Tr}X^2}$}.

For the reader: About these pages

  • Text in white: This is text which I try to make understandable to you, my reader.
  • Text in grey: These are notes to myself which I don't try to make understandable.
  • Text in yellow: These are questions I ask myself that I'm working on.

Old information

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Šis puslapis paskutinį kartą keistas March 18, 2023, at 09:18 PM