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Introduction E9F5FC Understandable FFFFFF Questions FFFFC0 Notes EEEEEE Software 
数学笔记 Complex numbers are more natural than real numbers or quaternions because complex numbers have simpler nondegenerate quadratic forms: {$Q(u)=u_1^2+u_2^2+\cdots +u_n^2$}. For we can insert a scalar {$i$} and that converts any minus sign into a plus sign. Jennifer Mather. The Case for Octopus Consciousness: Unity. https://www.mdpi.com/26734087/2/4/30
Baez, Moeller, Trimble. Schur functors and categorified plethysm.
Posina Venkata Rayudu
nLab: Sphere spectrum The sphere spectrum is the suspension spectrum of the point. The homotopy groups of the sphere spectrum are the stable homotopy groups of spheres. The sphere spectrum is the higher version of the ring Z of integers. See also: nLab: Suspension TaiDanae Bradley. The Tensor Product, Demystified Laws of form
Matematikos žinojimo rūmų sparnus sieja
Apsikeičia  ar tai padalinimų ratas? Kiekviename sparne požiūriai prisideda +0, +1, +2, +3. Auxiliary loops in spacetime are compatible with the rays in spacetime, perhaps in this way the mind is compatible with the brain. That might be relevant for the hierarchy of agency. Consider a Clifford algebra with infinitely many generators {$a_i$} and {$b_j$} such that {$a_i^2=1$} and {$b_j^2=1$}. How would different interpretations satisfy an eightfold periodicity? John Bolender. The SelfOrganizing Social Mind. Consider the ways of constructing a 2x2 matrix by adding a generator {$a_i$} and then {$b_j$}, or alternatively, {$b_j$} and then {$a_i$}. Grammar of sociology in terms of a pattern language grounding concepts such as "generation", "the right biography", "class", "both agressors lose". Groups and symmetries. From finite groups to Lie groups. Geometry arises on a vector space when we place a quadratic form on it, {$Q:V\rightarrow K$}. It gives a notion of length of a vector because {$Q(av)=a^2Q(v)$}, and also a notion of distance because there is a symmetric bilinear form {$B(v,w)=\frac{1}{2}(Q(v+w)Q(v)Q(w))$}. https://en.wikipedia.org/wiki/Cayley%E2%80%93Klein_metric Are there niine CayleyKlein geometries? Try to undersand Morita equivalence in terms of irreducible representations. https://www.reddit.com/r/math/comments/n9a761/is_it_normal_to_be_struggling_to_get_the_yoneda/ Equivalence up to isomorphism is relevant for "user requirements" as opposed to "material implementation". This is relevant for Bott periodicity. Wave function arises when two systems interact. As given by orthogonal Sheffer polynomials. In Lie group for rotations, SO(3), the bracket of [x,y] gives you z. In choice frameworks, such as the simplex, the center is the basis for geometry and the vertices are the basis for matter. Together can they ground general relativity? I dreamed of the grouping of examples from branches of mathematics by considering whether they involve, for example, aspects of mathematics, logic, semantics, and so on. Minkowski space. Time {$t^2$} has us step out (thus reversing direction), space {$x^2+y^2+z^2$} has us step in. Why doesn't adding a pair of generators {$a_1 b_1$} to a Clifford algebra change it? Generators {$a_j$} for which {$a_j^2=1$} are experienced directly whereas generators {$b_j$} for which {$b_j^2=1$} express the change in direction upon reflection, thus stepped out. Jane Loevinger's psychometrics = Maslow's hierarchy of needs. E8 is "worry about the needs of another". E9 is "be perfect". Do the generators of Clifford algebras which square to +1 and 1 encode, respectively, covariant and contravariant tensors? Equality is inherently contradictory. John Baez. Getting to the Bottom of Noether's Theorem. Ivan. Was or was not regional politics helpful for potential democratic transition in Russia? Novosibirsk, Tatarstan. Partial knowledge
Consciousness
Counterquestions are a foundation for learnability. Each counterquestion defines a domain of new knowledge where we had no facts that we could rely on. Mind and Life Institute. Varela. Supporting contemplative research.
Nathaniel Virgo
Supermaps  holes  contexts. Pragmatic approach. Context defines meaning. Robert Brandom. Making It Explicit. {$\textrm{Set}^{op}$} atomic boolean algebras. Map back into Set. Map back out PowerSet. Simplexes observe coordinate systems. df/dx = f. Can be expressed through the notion of infinity (Taylor series) {$e^x$}. Or through periodicity (trigonometry, Euler's equation) {$e^{ix}$} Jim in Oneonta. Adapt, improvise, overcome. Sheaf Representation of Monoidal Categories
Eri
Conformal and analytic is the same. Energy can be defined as the "separation constant" in Schroedinger's equation. If we can separate the wave function into a time dependent function and a position dependent function, then we can segregate the two sides of the equation so that one side depends on time, the other side depends on position, and both sides are constant, and that constant is the energy. Peter Scholze  condensed math
Absolutism based on relativism is good. Relativism based on absolutism is bad. Objectivity based on subjectivity is good. Subjectivity based on objectivity is bad. The mind that does not know is based on the mind that knows and not the other way around. This is the rule of consciousness and the basis for morality. There is a gap between the quantum ether (the quantum foam) and the waves that propagate through it. Particles don't exist, particles are the medium. Waves exist in the medium. Threesome
Folk psychology. Daniel Dennett suggested studying this. Contact him.
In the book on interpretations of quantum mechanics, there is the question of what is real. For example, in electromagnetism, the gauge can be adjusted by adding any gradient. This can change whether the change is transmitted by the speed of light or whether that speed is infinite and it happens instantaneously. But these two scenarios also raise the question of the reality of the "wiggle". What is real is a moral choice. Is the medium real? Or is the wiggling real? The wiggling creates the wave that moves across the medium. The medium is made of particles and antiparticles that appear from the foam.
https://researchsemin The manyworlds interpretation of quantum mechanics and the Born rule Lev Vaidman (Tel Aviv University) ( view ) Mon May 22, 19:0020:30 (7 days from now) Abstract: I will argue that the manyworlds interpretation is the best interpretation of quantum mechanics and discuss the status of the probability assignments in this deterministic theory.ars.org/seminar/AlgebraParticlesFoundations Applying a boundary map twice gives zero. Applying it twice removes two vertices, and this can be done in different orders, yielding different signs, canceling out. For the Snake Lemma, add a zero vector space before the first kernel and add a zero vector space after the last cokernel. Then we have the eightfold way with seven mappings. Can large language models work by simply transforming existing input  taken to be grammatical  to preserve grammaticality. Choice Frameworks
Introduction to Commutative Algebra Atiyah & Macdonald. Rings, ideals, modules, dimensions Looking at an ellipse in various ways yields all of the conic shapes. We can get a breaking at infinity. Thus this is a way to ground infinity. What about looking at a circle? We look through the point of the cone, which is where our eye is. Mobius transformations
David Corfield's video. Colin McLarty: semiotics as the language of biology  logic in a biological key  trying to categorify this? Tristan Needham. Visual Complex Analysis. John Stillwell. Mathematics and its History. 1989. Moebius transformations
Emotional sphere
Freedom House Report
Fields Institute Differentiation Alex Codes. Symbolic Differentation in Python from Scratch! Algebra
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. Wisdom Wisdom distinguishes everything and slack, what is whole and what is free, holisticity and laxity. https://researchseminars.org/seminar/AlgebraParticlesFoundations Physics https://www.johnmyleswhite.com/notebook/2013/03/22/modesmediansandmeansanunifyingperspective/ GradCurlDiv
Exact sequence: Grad, Curl, Div
Group theory Nathan Carter "Visual Group Theory" https://www.quantamagazine.org/triangulationconjecturedisproved20150113/ https://jeremykun.com/2013/04/10/computinghomology/ Purcell. Electricity and Magnetism
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. Quaternions act like a gauge  3 dimensions are unspecified  but identified with the complex i. https://math.stackexchange.com/questions/711492/provethatthemanifoldsonisconnected 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 halfintegered. Think of the Hilbert space as everything divided into 2s+1 perspectives. Simplicial sets Greg Friedman. An elementary illustrated introduction to simplicial sets. Amplituhedron Physics 283b: Spacetime and Quantum Mechanics, Total Positivity & Motives Causality
Bell's inequalities https://www.researchgate.net/publication/ Causal Set Theory Universal concepts such as universal confounders the confounder. Ambiguity is described by equations.
(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. 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/sevenproblemsaroundtverbergstheorem/ Pearl seeing vs. Doing Homology sets up potential equivalences  they may be actual equivalences, which yield identities  or nonequivalences which are generators. 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. Causality 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). Judea Pearl, Dana Mackenzie. The Book of Why: The New Science of Cause and Effect Choice
Stephen Wolfram. Metamathematics Foundations & Physicalization Ravi Vakil: Main theme of mathematics  convert harder problems to linear algebra Ergodic theorem
Chomsky: Successor function derives from the merge function applied unitarily to a single object. Parsing hierarchy 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. Field with one element Thomas noted the symmetry of {$x^0=1$}. Relate this to {$F_1$}, choosing one out of one, or none out of none. Schroedinger's cat
What Are The Hidden Rules Of The Universe? http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf Fulton Curve Book Randomness
Multiplying by quaternion j reverses angular momentum for electron. Is spin a clock? Like a particle clock?
https://chaosbook.org/course1/about.html Linear regression
Statistics
Topology
trivial tangent bundles on spheres? Hamiltonian is the sum of all the projections onto the energy eigenstates with the energies being the weights. Threesome
Study of variables
KervaireMilnor formula
In the unfolding of math
https://neo4j.com Neo4J graph database management Johan Commelin: "Breaking the onemindbarrier in mathematics using formal verification"
Representation theory Generalized Linear Models Lattice 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.
Statistics Are the conditions for coverings the basis for completeness? Shoelace formula for oriented area of a polygon Dobinski's formula relates Bell numbers and e. {$B_n=\frac{1}{e}\sum_{k=0}^{n=\infty}\frac{k^n}{k!}$} Quaternionically differentiable is linear. 18.4 penrose. Hyperbolic length is one half of the rapidity it represents. Laws of physics are timesymmetric for particles traveling at the 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. Complexity theory https://en.wikipedia.org/wiki/PCP_theorem Every decision problem in the NP complexity class has probabilistically checkable proofs Discourse https://en.wikipedia.org/wiki/Overton_window Window of political discourse: Unthinkable  Radical  Acceptable  Sensible  Popular  Policy Lambda Calculus Study choice, probability, statistics. Physics 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 (‘pseudoMajorana spinors’). The options are wellunderstood, and follow a nice pattern depending on the dimension and signature of spacetime modulo 8.
Functions 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 Dirac's plate trick Plate trick 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? 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
https://en.wikipedia.org/wiki/Theory_U The 4 returns: natural return (value of landscape), economic return (restart agriculture), social return, humans return. 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 Spaces of states
{$\begin{pmatrix} a & b+ic \\ bic & d \end{pmatrix}$} Think of probabilities {$a, 1a$} 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 {$aa^2$} from 0 to 1 around the aaxis. It is the MacMahon Master Theorem that unifies the angular momentum properties of composite systems in the binary buildup of such systems from more elementary constituents. Summer of Math Expostion 2022 playlist Semilocally path connected avoids Zeno's paradox. Universal covering as naming schemes. Robert Gilmore. Group Theory. XIV. Group Theory and Special Functions. Relates Lie groups and orthogonal polynomials. 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 In Cartesian categories you can copy and delete information. (John Baez  Rosetta Stone) How does that relate to Turing machine? Alytaus kredito unija
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. Probability
https://ncatlab.org/nlab/show/syntaxsemantics+duality Creating what you can feel certain about. (Continuity.)
A few of the best math explainers from this summer Stone's theorem: continuous implies differentiable "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. 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. Quantum computing
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. Have all finite limits is equivalent to
These are the building blocks for limits 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. https://ww3.math.ucla.edu/dls/emilyriehl/ 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. https://math.stackexchange.com/questions/989083/iscompositionofcoveringmapscoveringmap Filmavimas
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 onedimensional 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 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
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) semijoin lattice semilattice Quantum physics Brody. Quantum Mechanics and Riemann Hypothesis. Duality Simplex Think of 1cell as the center (of all things), the spirit. And think of 0cell not simply as a point but as a 0dimensional 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://ncatlab.org/nlab/show/cellular+approximation+theorem#applications https://en.wikipedia.org/wiki/Koopman%E2%80%93von_Neumann_classical_mechanics https://www.amazon.com/QuantumChallengeFoundationsMechanicsAstronomy/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 particleclocks 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 Bohm Pilot Wave, Thomas Spencer Relative invariance  more global than another Relate the threecycle (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 https://en.wikipedia.org/wiki/Algorithmic_information_theory Gregory Chaitin = Shannon + Turing = CompressionDecompression as understanding. https://en.wikipedia.org/wiki/Cristian_S._Calude Philosophy of computation Life in lifeThinking 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. 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. 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. TaiDanae Bradley: Information is on the Boundary
Prove that the matrix made up of eigenvectors diagonalizes a matrix. Potential energy source {$\frac{y}{2}$} (adding free cell) is in balance with kinetic energy source {$i\frac{\partial}{\partial y}$} (halflink)
Symmetry
