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Andrius Kulikauskas

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数学笔记

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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?

3x3 matrices of octonions (are self-adjoint?)

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.

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?

https://en.m.wikipedia.org/wiki/3-sphere#One-point_compactification relate to symplectic

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.
  • Yoneda lemma - relating system and subsystems.

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

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.

Bott Periodicity of states of mind.

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

Antonio

Expressiveness and proof theoretic strength for different type theories.

  • Allowing inductive recursive definitions, how does that increase the proof theoretic strength or the expressiveness in a type theoretic based language.

Compositional game theory. Julian Hedges.

  • Diagrammatic calculus (as for quantum theory)
  • Relate nonviolence principles to game theory.

Graph quantum isomorphism

Presburger arithmetic is weaker

https://www.cst.cam.ac.uk/files/smp-2021-slides-mancinska.pdf Quantum Isomorphism of Graphs: An Overview. Related to game theory, knowledge, information theory.

Albert Atserias et al. Quantum and non-signalling graph isomorphisms.

Laura Mancinska. Quantum isomorphism of graphs: An overview

https://en.wikipedia.org/wiki/Induction-recursion

https://en.wikipedia.org/wiki/Quantum_pseudo-telepathy Mermin–Peres magic square game

Adjunction? Language is more expressive then language is less decidable (less determinable in the proof theoretic sense).

  • Example: Presburger arithmetic
  • Example: Simplicial homology - easy to formulate but lots to calculate. Singular homology - subtle to formulate, much less to calculate.

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.

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

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.

Antonio Jesus: Information, symmetry, randomness

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.

https://franklin.dyer.me/notes/note/Cones_and_limits Antonio suggested

Categorical Logic and Type Theory Antonio suggested

https://ncatlab.org/nlab/show/being According to (Hegel 12) pure being is the opposite of nothing whose unity is pure becoming. According to the formalization of this proposed by (Lawvere 91), this is described by the adjoint modality {$(\emptyset \dashv \ast)$} of the idempotent monad constant on a terminal object {$\ast$} and its left adjoint {$\emptyset$}.

"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

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

https://www.mta.ca/~cat-dist/catlist/1999/yoneda

Yoneda lemma - equalizer of products - are equalizers, products and initials the building blocks of all limits. Products work like sets. But you can have Yoneda lemma without sets in this way. (Riehl - Kan extensions). Is set a part of the definition of a product ? See Theorem 3.4.12

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.

Tensor Hom adjunction

  • (AxB->C) left adjoint to (A->(B->C))
  • general function left adjoint to values on elements
  • free construction left adjoint to forgetful

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

Jesus Antonio: Key to physics is information and symmetry

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.

A special case of the universal property of identity types is related to the Yoneda lemma.

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 {$\forall \wedge \exists$}.

How is Yoneda lemma related to matrix row manipulation? And how might that help relate Cramer's rule to Kan extensions and the Yoneda lemma?

Young-Il Choo - MeetUp

The inclusion of Field in CRing has no left adjoint because it would carry Z to an initial field, which does not exist. How might an initial field relate to the field with one element?

Riehl: 2.4. The category of elements A universal property for an object c ∈ C is expressed either by a contravariant functor F together with a representation C (−, c)  F or by a covariant functor F together with a representation C (c, −)  F. The representations define a natural characterization of the maps into (in the contravariant case) or out of (in the covariant case) the object c. Proposition 2.3.1 implies that a universal property characterizes the object c ∈ C up to isomorphism. More precisely, there is a unique isomorphism between c and any other object representing F that commutes with the chosen representations. In such contexts, the phrase “c is the universal object in C with an x” assets that x ∈ Fc is a universal element in the sense of Definition 2.3.3, i.e., x is the element of Fc that classifies the natural isomorphism that defines the representation by the Yoneda lemma. In this section, we prove that the term “universal” is being used in the precise sense alluded to at the beginning of this chapter: the universal element is either initial or terminal in an appropriate category. The category in question, called the category of elements, can be constructed in a canonical way from the data of the representable functor F. The main result of this section, Proposition 2.4.8, proves that any universal property can be understood as defining an initial or terminal object, as variance dictates, completing the promise made in §2.1.

Riehl, page 50: The Yoneda lemma is arguably the most important result in category theory, although it takes some time to explore the depths of the consequences of this simple statement. In §2.3, we define the notion of universal element that witnesses a universal property of some object in a locally small category. The universal element witnessing the universal property of the complete graph is an n-coloring of K n , an element of the set n-Color(K n ). In §2.4, we use the Yoneda lemma to show that the pair comprised of an object characterized by a universal property and its universal element defines either an initial or a terminal object in the category of elements of the functor that it represents. This gives precise meaning to the term universal: it is a synonym for either “initial” or “terminal,” with context disambiguating between the two cases. For instance, K n is the terminal n-colored graph: the terminal object in the category of n-colored graphs and graph homomorphisms that preserve the coloring of vertices.

Comma category Lawvere showed that the functors F : C → D {\displaystyle F:{\mathcal {C}}\rightarrow {\mathcal {D}}} F:{\mathcal {C}}\rightarrow {\mathcal {D}} and G : D → C {\displaystyle G:{\mathcal {D}}\rightarrow {\mathcal {C}}} G:{\mathcal {D}}\rightarrow {\mathcal {C}} are adjoint if and only if the comma categories ( F ↓ i d D ) {\displaystyle (F\downarrow id_{\mathcal {D}})} (F\downarrow id_{{\mathcal {D}}}) and ( i d C ↓ G ) {\displaystyle (id_{\mathcal {C}}\downarrow G)} (id_{{\mathcal {C}}}\downarrow G), with i d D {\displaystyle id_{\mathcal {D}}} id_{{\mathcal {D}}} and i d C {\displaystyle id_{\mathcal {C}}} id_{{\mathcal {C}}} the identity functors on D {\displaystyle {\mathcal {D}}} {\mathcal {D}} and C {\displaystyle {\mathcal {C}}} {\mathcal {C}} respectively, are isomorphic, and equivalent elements in the comma category can be projected onto the same element of C × D {\displaystyle {\mathcal {C}}\times {\mathcal {D}}} {\mathcal {C}}\times {\mathcal {D}}. This allows adjunctions to be described without involving sets, and was in fact the original motivation for introducing comma categories.

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.

Comma category Adjunctions
  • Lawvere showed that the functors F : C → D {\displaystyle F:{\mathcal {C}}\rightarrow {\mathcal {D}}} F:{\mathcal {C}}\rightarrow {\mathcal {D}} and G : D → C {\displaystyle G:{\mathcal {D}}\rightarrow {\mathcal {C}}} G:{\mathcal {D}}\rightarrow {\mathcal {C}} are adjoint if and only if the comma categories ( F ↓ i d D ) {\displaystyle (F\downarrow id_{\mathcal {D}})} (F\downarrow id_{{\mathcal {D}}}) and ( i d C ↓ G ) {\displaystyle (id_{\mathcal {C}}\downarrow G)} (id_{{\mathcal {C}}}\downarrow G), with i d D {\displaystyle id_{\mathcal {D}}} id_{{\mathcal {D}}} and i d C {\displaystyle id_{\mathcal {C}}} id_{{\mathcal {C}}} the identity functors on D {\displaystyle {\mathcal {D}}} {\mathcal {D}} and C {\displaystyle {\mathcal {C}}} {\mathcal {C}} respectively, are isomorphic, and equivalent elements in the comma category can be projected onto the same element of C × D {\displaystyle {\mathcal {C}}\times {\mathcal {D}}} {\mathcal {C}}\times {\mathcal {D}}. This allows adjunctions to be described without involving sets, and was in fact the original motivation for introducing comma categories.
  • Relate to the definition of adjunctions in terms of the universal mapping property.
  • Consider how HomSets come into play.

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

Conjugacy ? the values of adjunction

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

Self-adjoint operators

  • Stone's theorem (the dynamical evolution)
  • Spectral theorem (the structure): One-to-one connection between projections (measure valued projections) and self-adjoint operators.

Quantum measurement projects into eigenstate. The projection operator is a mathematical statement of the collapse of the wave function. If you do it twice, then you don't get anything more.

Self-adjoint operators are weighted sums of projection operators. The weights you can find from experiments by applying a projection operator.

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

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.

Algebra is thinking step-by-step and so it exhibits finiteness. Analysis is discovering the boundaries between steps and so exhibits continuity by discovering the critical points, as in Moore's theory. Healthy irony (verbalization) codes the analogue signal (the emotional tension between expression and meaning) into a discrete alphabet (of boxes organized in a cognitive network).

Antonio

Wenbo

  • Semi-join lattice.

Oliver

  • Distributional semantics

https://music.163.com/#/playlist?id=797393474 Wenbo

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

Cool Math for Hot Music - better

Midori

Recuerdos de la alambra

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

Any functor {$F$} can be thought of as {$F:J\rightarrow C$} where {$J$} is the shape, the index set (how) and {$C$} is the image (what). And for any functor we can ask if it has a limit (an object {$L$} with maps {$\psi_X:L\rightarrow F(X)$} for all objects {$X$} of {$J$}, such that for all {$f:X\rightarrow Y$} in {$J$} we have the analogous morphisms commute in {$C$}, and that is universal as such). So that limit is Why. And is the colimit Whether? Or is whether simply the object with its identity morphism? Compare with the Yoneda lemma.

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.

Consider how the understanding of Yoneda lemma in terms of a left Kan extension, and in particular, the factoring, relates to the push down automata.

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.

Kan extensions are a framework for universality. Consider example 6.1.3, the Yoneda Lemma. Think in which ways the universality of limits, colimits, adjunctions, etc. is captured by Kan extensions. All of these universal properties can be thought of in terms of initial or terminal objects in the appropriate categories, such as the category of cones, or the comma category for the universal mapping property for adjunctions. So consider the relevant categories. How do they relate to the classification of adjoint strings?

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?

Mathematics is described in terms of set theory. The category of graphs {$\textrm{Set}^{A}$} where {$A$} is the category with two objects, edges E and vertices V, and two nontrivial morphisms target {$t:E\rightarrow V$} and source {$s:E\rightarrow V$}. Similarly, all mathematical structures and their structure preserving morphisms should have a similar expression in terms of sets and their relationshps. Work out various examples. Then study the role of {$\textrm{Set}^{X}$} in the Yoneda Lemma.

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.

Enveloping algebra (important for adjunctions) is related to Hochschild cohomology.

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

Lifting a path is like inverting a functor. How is that related to adjunctions? Adjunction is conditional inversion.

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?)

Meanings are variously related by adjunctions. They enrich the meaning and extend the context.

  • Think of my understanding of my three grandfathers as changing with context.

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.

Enveloping algebra (important for adjunctions) is related to Hochschild cohomology which is a special case of the functor Ext.

  • {$\operatorname{Hom}_{A^e}(A,M)$} (where {$A^e:=A\otimes_k A^{op}$} is the enveloping algebra of A and A is considered an A-bimodule via the usual left and right multiplication)

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

Yoneda lemma lets you go from natural isomorphism of homsets to natural isomorphism of functors.

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.

Math Discovery

  • How is gravity related to the argument by continuity?

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 {$P[X]\propto e^{-\frac{1}{2}\textrm{Tr}X^2}$}.
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Šis puslapis paskutinį kartą keistas December 03, 2022, at 02:40 PM