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Book.TransferMathNotes istorijaPaslėpti nežymius pakeitimus  Rodyti galutinio teksto pakeitimus 2020 balandžio 13 d., 14:41
atliko 
Ištrintos 016 eilutės:
* The Pairing axiom can be defined from other Zermelo Fraenkel axioms. (How?) And what does that mean for my identification of the axioms with the ways of figuring things out? Topology * Klein bottle (twisted torus) exhibits two kinds of duality: twist (inside/outside or not) and hole. * [[https://www.amazon.com/DifferentialAlgebraicTopologyGraduateMathematics/dp/0387906134  Bott & Tu. Differential Forms in Algebraic Topology.]] * What if there is a handle (a torus) inside a sphere? How to classify that? * Understand classification of closed surfaces: Sphere = 0. Projective plane = 1/3. Klein bottle = 2/3. Torus = 1. * [[https://www.maths.ed.ac.uk/~v1ranick/papers/botttu.pdf  Differential Forms in Algebraic Topology]], Bott & Tu * Topology  getting global invariants (which can be calculated) from local information. Žodynas. Lietuvių kalba: * sphere  sfera * trace  pėdsakas * semisimple  puspaprastis, puspaprastė * conjugate  sujungtinis * transpose  transponuota matrica, transponavimas 2020 balandžio 13 d., 14:36
atliko 
Ištrintos 029 eilutės:
* [[http://math.ucr.edu/home/baez/week257.html  John Baez about observables]] (see Nr.15) and the paper [[http://arxiv.org/abs/0709.4364  A topos for algebraic quantum theory]] about C* algebras within a topos and outside of it. Partial differential equations * NavierStokes equations: Reynolds number relates time symmetric (high Reynolds number) and time asymmetric (low Reynolds number) situations. Peano axioms * Equality holds for both value and type, amount and unit. Peano axiom. * Peano why can't have natural numbers have two subsets, a halfline from 0 and a full line. Quadratic form * Moebius transformation involves complex numbers and 2 x 2 matrix. Whereas Clifford algebra involves quadratic form  as a lens for perspectives? Think of a linear form (proportion) as a plain sheet of glass, and a quadratic form as a convex and/or concave lens. Randomness * Random phenomena organize themselves around a critical boundary. Real line * Real line models separation (by cutting) and connectedness (by continuity). The separating cuts become locations (points) in their own right. Restructuring * The six ways of restructuring relate the unconscious's continuum (large number) with the conscious's discrete (small number) and express the relations between the two. Consider these relations in the house of knowledge for mathematics, and how they relate to the six geometrical transformations. * Bundle. Geometry relates analysis (continuum) and algebra (discrete) as a restructuring. When the discrete grows large does it become a continuum? * Bundle = restructuring (base = continuum > fiber = discrete). A number that is "large enough" can essentially model the continuum. Rotation * Usually multiplication by i is identified with a rotation of 90 degrees. However, we can instead identify it with a rotation of 180 degrees if we consider, as in the case of spinors, that the first time around it adds a sign of 1, and it needs to go around twice in order to establish a sign of +1. This is the definition that makes spin composition work in three dimensions, for the quaternions. The usual geometric interpretation of complex numbers is then a particular reinterpretation that is possible but not canonical. Rotation gets you on the other side of the page. * If we think of rotation by i as relating two dimensions, then {$i^2$} takes a dimension to its negative. So that is helpful when we are thinking of the "extra" distinguished dimensions (1). And if that dimension is attributed to a line, then this interpretation reflects along that line. But when we compare rotations as such, then we are not comparing lines, but rather rotations. In this case if we perform an entire rotation, then we flip the rotations for that other dimension that we have rotated about. So this means that the relation between rotations as such is very different than the relation with the isolated distinguised dimension. The relations between rotations is such is, I think, given by {$A_n$} whereas the distinguished dimension is an extra dimension which gets represented, I think, in terms of {$B_n$}, as the short root. * This is the difference between thinking of the negative dimension as "explicitly" written out, or thinking of it as simply as one of two "implicit" states that we switch between. * Rotation relates one dimension to two others. How does this rotation work in higher dimensions? To what extent does multiplication of rotation (through three dimensional half turns) work in higher dimensions and how does it break down? 2020 balandžio 13 d., 14:24
atliko 
Ištrintos 036 eilutės:
* [[https://mathoverflow.net/questions/31879/arethereothernicemathbooksclosetothestyleoftristanneedham  Are there other nice math books close to the style of Tristan Needham?]] * [[https://www.mathpages.com/home/  Kevin Brown]] collection of expositions of math Limits * Analytic limits need neighborhoods. Categorical limits need maps. Third kind of limit: Ultraproducts. Terrence Tao. Map of Math * Basic concept: Orientation (of a simplex). Relates to determinant, homology, etc. * Vector spaces are basic. What is basic about scalars? They make possible proportionality. * Basic concepts are the ways of figuring things out. * Basic concept  orientation = parity. * [[http://msc2010.org/mediawiki/index.php?title=MSC2010  Mathematics Subject Classification wiki]] * Combinatorics. The mathematics of counting (and generating) objects. I think of this as the basement of mathematics, which is why I studied it. Math * Much of my work may be considered premathematics, the grounding of the structures that ultimately allow for mathematics. * Mathematics is the study of structure. It is the study of systems, what it means to live in them, and where and how and why they fail or not. * When you get the definitions right, the theorems are easy to prove. When the theorems are hard to prove, then the definitions are not right. (Tobias Osborne) So this shows how definitions and theorems coevolve. Music * [[http://wwwpersonal.umd.umich.edu/~tmfiore/1/FioreWhatIsMathMusTheoryBasicSlides.pdf  What is Mathematical Music Theory?]] Neural networks * Very powerful and simple computational systems for which Sarunas Raudys showed a hierarchy of sophistication as learning systems. Number theory * [[http://pi.math.cornell.edu/~hatcher/TN/TNpage.html  Allen Hatcher. Topology of numbers]] * Prime numbers: "Cost function". The "cost" of a number may be thought of as the sum of all of its prime factors. What might this reveal about the primes? * [[http://oeis.org/A000607  Number of ways to partition a number into primes]]. * Prime numbers introduce (non)determinism in that when a prime divides a product, then it must divide one of the factors. * Primes are both atoms and gaps. And they elucidate the gap of all numbers with respect to infinity. * Riemann hypothesis: is the unfolding of the primes "pattern free"? Or are there one or more hidden patterns there? A zero of the Riemann zeta function indicates a pattern. So what is a pattern? And what are the limitations on patterns? * Terrence Tao: Each prime p wants to have weight ln p. Compare with [[https://en.wikipedia.org/wiki/Zipf%27s_law  Zipf's law]], [[https://en.wikipedia.org/wiki/Zipf%E2%80%93Mandelbrot_law  ZipfMandelbrot law]], [[https://en.wikipedia.org/wiki/Yule%E2%80%93Simon_distribution  YuleSimon distribution]], [[https://en.wikipedia.org/wiki/Preferential_attachment  Preferential attachment]], [[https://en.wikipedia.org/wiki/Matthew_effect  Matthew effect]], [[https://en.wikipedia.org/wiki/Pareto_principle  Pareto's principle]]. * If there is a zero in the Riemann function's zone, then there is a function that it can't mimic? * [[https://en.wikipedia.org/wiki/Arithmetic_derivative  Arithmetic derivative]] 2020 balandžio 13 d., 13:59
atliko 
Ištrintos 02 eilutės:
* (Conscious) Learning from (unconscious) machine learning. 2020 balandžio 13 d., 13:56
atliko 
Ištrintos 014 eilutės:
* Compare a Fourier series and a generating function. How is a single basis element isolated, calculated, in each case? Going beyond oneself * Try to use universal hyperbolic geometry to model going beyond oneself into oneself (where the self is the circle). * Relate God's dance to {0, 1, ∞} and the [[https://en.wikipedia.org/wiki/Crossratio  anharmonic group]] and Mobius transformations. Note that the anharmonic group is based on composition of functions. Homotopy type theory * [[https://www.youtube.com/watch?v=bFZWarP2Ef4  Galois, Grothendieck and Voevodsky  George Shabat]] Irrationality * Challenge problems: * Determine whether {$\pi + e$} is rational or irrational. * Determine whether {$\pi^e$} is rational or irrational. Ištrinta 6 eilutė:
2020 balandžio 13 d., 13:01
atliko 
Ištrintos 039 eilutės:
* Real forms are similar to Clifford algebras in that in ndimensions you have n versions, n signatures. Each models the takng up of one of the n perspectives in a division of everything. Combinatorics * The mathematics of counting (and generating) objects. I think of this as the basement of mathematics, which is why I studied it. * [[https://www.math.upenn.edu/~wilf/gfology2.pdf  Generatingfunctionology]] by Herbert Wilf Coordinate system * Coordinate system is that which has an origin, a zero. * How are coordinate rings in algebraic geometry related to coordinate systems? * Coordinate systems are observers and they shouldn't affect what they observe. (Relate this to the kinds of polytopes.) Counting * The derivative of an infinite power sequence shows that it is related to counting because we get coefficients 1, 2, 3, 4 etc. for the generating sequence. * Kuom skaičius skiriasi nuo pasikartojančios veiklos  būgno mušimo? * A) veikla kažkada prasidėjo * B) kiekvienas skaičius laikomas nauju, skirtingu nuo visų kitų Curvature * Does the inside of a sphere have negative curvature, and the outside of sphere have positive curvature? And likewise the inside and outside of a torus? * Why is negative curvature  the curvature inside  more prominent than positive curvature  the outside of a space? Dimension * Notions of dimension d (Mathematical Companion): * locally looks like ddimensional space * the barrier between any two points is never more than d1 dimensional * can be covered with sets such that no more than d+1 of them ever overlap * the largest d such that there is a nontrivial map from a ddimensional manifold to a substructure of the space * the sum of dth powers of the diameter of squares that cover the object, with d such that the sum is between zero and infinity Equivalence * From dream: vectors AB, BA, consider the difference between them, the equivalence of A and B. * Equations are questions * Isomorphism is based on assignment but that depends on equality up to identity whereas properties define establish an object up to isomorphism * Symbols (a and b) are equal if they refer to the same referents. But equality has different meaning for symbols, indexes, icons and things. Consider the four relations between level and metalevel. Folding * Origami: [[http://alum.mit.edu/www/tchow/multifolds.pdf  The power of multifolds: Folding the algebraic closure of the rational numbers]] Pridėta 33 eilutė:
* Combinatorics. The mathematics of counting (and generating) objects. I think of this as the basement of mathematics, which is why I studied it. 2020 balandžio 13 d., 12:18
atliko 
Ištrintos 044 eilutės:
Algebra * studies particular structures and substructures [[https://en.wikipedia.org/wiki/Arithmetic_derivative  Arithmetic derivative]] AutomataTheory * There is a qualitative hierarchy of computing devices. And they can be described in terms of matrices. * Yates Index Theorem  consider substitution. * [[https://www.quantamagazine.org/mathematiciansolvescomputerscienceconjectureintwopages20190725/  Complexity measures for Boolean functions]]. * How is a [[https://en.wikipedia.org/wiki/Boolean_function  Boolean function]] similar to a linear functional? * If we think of a Turing machine's possible paths as a category, what is the functor that takes us to the actual path of execution? Binomial theorem * e to the matrix consisting of the natural numbers on its first offdiagonal gives a triangular matrix with pascal's triangle. and how is it in the case of the cube? * Pascal's triangle tilted gives Fibonacci numbers * I dreamed of the binomial theorem as having an "internal view", imagined from the inside, which accorded with the "coordinate systems". And which interweaved with the external views to yield various "moments", given by curves on the plane, variously adjusted and transformed by the internal view. Choice framework * Three perspectives lets you define a coordinate system as a choice framework. * Totally independent dimensions: Cartesian * Totally dependent dimensions: simplex * We "understand" the simplex in Cartesian dimensions but think it in simplex dimensions. We can imagine a simplex in higher dimensions by simply adding externally instead of internally. Consider (implicit + explicit)to infinity; and also (unlabelled + labelled) to infinity. Also consider (unlabelable + labelable). And (definitively labeled + definitively unlabeled). * Gaussian binomial coefficients [[http://math.stackexchange.com/questions/214065/provingqbinomialidentities  interpretation related to Young tableaux]] * https://oeis.org/A013609 triangle for hypercubes. (1 + 2x)^n unsigned coefficients of chebyshev polynomials of the second kind * [[https://ncatlab.org/nlab/show/geometric+shape+for+higher+structures  nLab: Geometric shape for higher structures]] * What do "globe", "globe category", "globular set" mean? * [[https://math.stackexchange.com/questions/2371364/whatsthecanonicalembeddingoftheglobecategoryintotop  What's the canonical embedding of the globe category into Top?]] * They are the maps of the closed nball to the "northern" and "southern" hemispheres of the surface of the (n+1)ball. * This defines a functor G→Top that extends along the Yoneda embedding, yielding a geometric representation of any globular set Gˆ→Top. * Ar savybių visuma yra simpleksas? Ar savybės skaidomos (koordinačių sistema). * The empty set, or the center, has dimension 1. * [[https://terrytao.wordpress.com/2019/07/26/twistedconvolutionandthesensitivityconjecture/  Terrence Tao: Twisted Convolution and the Sensitivity Conjecture]] * Choices  polytopes, reflections  root systems. How are the Weyl groups related? * Polytopes. edge = difference * Esminis pasirinkimas yra: kurią pasirinkimo sampratą rinksimės? * Kaip suvokiame {$x_i$}? Koks tai per pasirinkimas? * Kada pasirinkimo samprata keičiasi, visgi už visų sampratų slypi bendresnis, pirmesnis suvokimų suvokimas, taip kad renkamės pačią sampratą. Folding is the basis for substitution. * Keturios pasirinkimo sampratos (apimtys) visos reikalingos norint išskirti vieną paskirą pasirinkimą. An simplexes allow gaps because they have choice between "is" and "not". But all the other frameworks lack an explicit gap and so we get the explicit second counting. But: * for Bn hypercubes we divide the "not" into two halves, preserving the "is" intact. * for Cn crosspolytopes we divide the "is" into two halves, preserving the "not" intact. * for Dn we have simply "this" and "that" (notthis). * Use "this" and "that" as unmarked opposites  conjugates. Pridėta 94 eilutė:
* [[https://en.wikipedia.org/wiki/Arithmetic_derivative  Arithmetic derivative]] 2020 balandžio 12 d., 21:45
atliko 
Pakeistos 15 eilutės iš
* In what sense does affine geometry not have a coordinate system? (And thus not have a notion of infinity?) Affine = local (no infinity). * Affine geometry is agnostic regarding coordinate system. So it doesn't distinguish if we flip all the positive and minus choices. * Affine transformation extends a linear transformation by a column (the translation) and a row (of zeroes) and a diagonal element (of 1). Thus it is similar to a Lie algebra (Dn ?) extending An. į:
Ištrintos 47 eilutės:
* [[https://en.wikipedia.org/wiki/Andrei_Okounkov  Andrei Okounkov]] Bridging probability, representation theory and algebraic geometry. * Discriminant of [[https://en.wikipedia.org/wiki/Elliptic_curve  elliptic curve]]. Ištrintos 1820 eilutės:
* Vector bundles: Identity and selfidentity (like the ends of a regular strip or a Moebius strip). Identity of a point, identity of a fiber  selfidentity under continuity. How does a fiber relate to itself? Is it flipped or not? 2 kinds of selfidentity (or nonidentity) allows the edge to be flipped upside down. Ištrintos 5274 eilutės:
* I dreamed of the complex numbers as a line that curls, winds, rolls up in one way on one end, and in the mirror opposite way on the other end, like rolling up a carpet from both ends. Like a scroll. * How is the Riemann sheet, winding around, going to a different Riemann sheet, related to the winding number? and the roots of polynomials? Complex analysis * Differentiability of a complex function means that it can be written as an infinite power series. So differentiation reduces complex functions to infinite power series. This is analogous to evolution abstracting the "real world" to a representation of it. And differentiation is relevant as a shift from the known to focus on the unknown  the change. Analytic continuation * Understand analytic continuation. Can we think of it as cutting the plane into a spiral of width {$e^ne^{n1}$}? * Learn how to extend the Gamma function to the complex numbers. Conformal geometry * Why and how is Universal Hyperbolic Geometry related to conformal geometry? * A function from the complex plane to the complex plane which preserves angles (of intersecting curves) necessarily is analytic. This seems related to the fact that exponentiation {$e^{i\theta}$} makes multiplication additive. And that brings to mind trigonometric functions. Preservation of angles implies existence of Taylor series. * When are two vectors, lines, etc. perpendicular? When they are distinguished by i ? * Conformal groups. Orthogonal. * Terrence Tao: It is a beautiful observation of Thurston that the concept of a conformal mapping has a discrete counterpart, namely the mapping of one circle packing to another. * Conformal geometry preserves angles. Radial coordinates distinguishes distances r and angles theta, and makes use of the exponential {$e^{\pi i \theta}$}. Conics * Consider geometrically how to use the conics, the point at infinity, etc. to imagine Nothing, Something, Anything, Everything as stages in going beyond oneself into oneself. * Investigate: What happens to the shape of a circle when we move the tip of the cone? Suppose the circle is a shaded area. In what sense is the parabola a circle which touches infinity? In what sense is the directrix a focus? Does the parabola extend to the other side, reaching up to the directrix? Is a hyperbola an inverted ellipse, with the shading on either side of the curves, and the middle between them unshaded? What is happening to the perspective in all of these cases? * Think of the two foci of a conic as the source (start of all) and the sink (end of all). When are they the same point? (in the case of a circle?) Ištrintos 6370 eilutės:
Coxeter groups * Understand the classification of Coxeter groups. * Organize for myself the Coxeter groups based on how they are built from reflections. Crossratio * Relate the levels of the foursome with the crossratio (for example, howwhat are the two points within a circle, and whywhether are the two points outside.) And likewise relate the six pairs of four levels with geometric concepts, the six lines that relate the four points of an inscribed quadrilateral, or the six possible values of the crossratio upon permuting its elements. Ištrintos 7588 eilutės:
* Basic division rings: [[http://math.ucr.edu/home/baez/week59.html  John Baez 59]] * The real numbers are not of characteristic 2, * so the complex numbers don't equal their own conjugates, * so the quaternions aren't commutative, * so the octonions aren't associative, * so the hexadecanions aren't a division algebra. * [[https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(composition_algebras)  Hurwitz's theorem]] for composition algebras * Analyze number types in terms of fractions of differences, https://en.wikipedia.org/wiki/M%C3%B6bius_transformation , in terms of something like that try to understand adbc, the different kinds of numbers, the quantities that come up in universal hyperbolic geometry, etc. * Note how complex numbers express rotations in R2. How are quaternions related to rotations in R3? What about R4? And the real numbers? And in what sense do the complex numbers and quaternions do the same as the reals but more richly? * Quaternions include 3 dimensijos formos rotating (twistor) and 1 for scaling (time) and likewise for octonions etc * John Baez periodic table and stablization theorem  relate to Cayley Dickson construction and its dualities. Ištrintos 8188 eilutės:
* Note in the Wikipedia article on [[https://en.wikipedia.org/wiki/Particular_values_of_the_gamma_function  Particular values of the gamma function]] the connection between {$\Gamma(\frac{n}{24})$} and [[https://en.wikipedia.org/wiki/Elliptic_integral#Complete_elliptic_integral_of_the_first_kind  Complete elliptic integrals of the first kind]] Exponentiation * Integral of 1/x is ln x what does that say about ex? * {$2\pi$} additive factors, e multiplicative factors. * Math Companion section on Exponentiation. The natural way to think of {$e^x$} is the limit of {$(1 + \frac{x}{n})^n$}. When x is complex, {$\pi i$}, then multiplication of y has the effect of a linear combination of n tiny rotations of {$\frac{\pi}{n}$}, so it is linear around a circle, bringing us to 1 or all the way around to 1. When x is real, then multiplication of y has the effect of nonlinearly resizing it by the limit, and the resulting e is halfway to infinity. Ištrintos 92101 eilutės:
* Interpret the boundary of a simplicial complex. Explain how the boundary of a boundary of a simplicial complex is zero. How does the boundary express orientation? * Homology and cohomology are like the relation between 0>1>2>3 and 4>5>6>7. * Homology  holes  what is not there  thus a topic for explicit vs. implicit math * [[http://www.math.wayne.edu/~isaksen/Expository/carrying.pdf  A Cohomological Viewpoint on Elementary School Arithmetic]] About "carrying". Access restricted. * [[https://www.youtube.com/watch?v=LmyyybUlnLU  V: Ben Mares. Introduction to cohomology.]] * Odd cohomology works like fermions, even cohomology works like bosons. * Homologija bandyti išsakyti persitvarkymų tarpą tarp pirminės ir antrinės tvarkos. * Relate walks on trees with fundamental group. Ištrintos 95102 eilutės:
* Hermitian: a+bi <> abi, Symmetric: a <> a, Antisymmetric: b <> b * Consider how the "inner product", including for the symplectic form, yields geometry. * The skewsymmetric bilinear form says that phi(x,y)= phi(y,x) so one of them, say, (x,y) is + (correct) and the other is  (incorrect). This is leftright duality based on nonequality (the two must be different). And it is threefold logic, the nonexcluded middle 0. SO that 1 true, 1 false, 0 middle. * The inner products (like unitary) are linear. They and their quadratic product get projected onto a second quadratic space (a screen) which expresses their geometric nature as a tangent space for a differentiable manifold thus relating the discrete and the continuum, the infinitesimal and the global (like Mobius transformations). * Three inner products  relate to chronogeometry. * Three inner products are products of vectors that are external sums of basis vectors as different units. Six interpretations of scalar multiplication that are internal sums of amounts. Ištrintos 142145 eilutės:
* Exercise: Find all matrices with eigenvalues 1 and 1. {$\lambda=\frac{a_{11}+a_{22} \pm \sqrt{(a_{11}+a_{22})^2  4A}}{2}$} so A=1. {$\begin{pmatrix} \pm \sqrt{1 + a_{12}a_{21}} & a_{12} \\ a_{21} & \mp \sqrt{1 + a_{12}a_{21}} \end{pmatrix}$} such as {$\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$}{$\begin{pmatrix} i & 0 \\ 0 & i \end{pmatrix}${$\begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}$} * Multiplying by {$\begin{pmatrix} x \\ y \end{pmatrix}$} yields three ways of coding opposites: {$\begin{pmatrix} iy \\ ix \end{pmatrix}$} {$\begin{pmatrix} y \\ x \end{pmatrix}$} {$\begin{pmatrix} ix \\ iy \end{pmatrix}$} where in each case two of three are applied: flipping, multiplying by i, multiplying by 1. Ištrintos 152155 eilutės:
* Study existential and universal quantifiers as adjunctions and as the basis for the arithmetical hierarchy. * "For all" and "there exists" are adjoints presumably because they are on opposite sides of a negation wall that distinguishes the internal structure and external relationships. (That wall also distinguishes external context and internal structure.) (And algorithms?) So study that wall, for example, with regard to recursion theory. Ištrintos 163166 eilutės:
* [[https://en.wikipedia.org/wiki/Riemann_surface  Riemann surface]] * [[http://www.math.tifr.res.in/~pablo/download/book/book.html  Riemann Surfaces Book]], Pablo Arés Gastesi Ištrintos 169173 eilutės:
* Schroedinger's equation relates position and momentum through complex number i. Slack in one (as given by derivative) is given by the value of the other times the ratio of the Hamiltonian over Planck's constant (the available quantum slack as given by the energy). * {$\frac{\text{d}}{\text{dt}}\Psi=i\frac{H}{\hbar}\Psi$} * In statistics, the probability that a system at a given temperature T is in a given microstate is proportional to {$e^{\frac{H}{kT}}$}, so here we likewise have the quantumization factor {${\frac{H}{kT}}$}. Ištrintos 172177 eilutės:
* A circle (through polarity) defines triplets of points, and triplets of lines, thus sixsomes. The center of a circle is perhaps a fourth point (with every triplet) much like the identity is related to the threecycle? Theorems * "Some Fundamental Theorems in Mathematics" (Knill, 2018) https://arxiv.org/abs/1807.08416 Ištrintos 179187 eilutės:
Videos * http://people.math.harvard.edu/~knill/media/index.html Walks on trees * Walks from A to B in category theory are morphisms and they get mapped to the morphisms from A to B. Relate this to walks on trees. * Parentheses establish a tree structure. What are walks on these trees? How do they relate to associativity and to walks in categories from one object to another? * Walks On Trees are perhaps important as they combine both unification, as the tree has a root, and completion, as given by the walk. In college, I asked God what kind of mathematics might be relevant to knowing everything, and I understood him to say that walks on trees where the trees are made of the elements of the threesome. 2020 balandžio 12 d., 13:51
atliko 
Ištrintos 193198 eilutės:
* In solving for eigenvalues {$\lambda_i$} and eigenvectors {$v_i$} of {$M$}, make the matrix {$M\lambda I$} degenerate. Thus {$\text{det}(M\lambda I)=0$}. The matrix is degenerate when one row is a linear combination of the other rows. So the determinant is a geometrical expression for volume, for collinearity and noncollinearity. * An inner product on a vector space allows it to be broken up into vector spaces that complement each other, thus into irreducible vector spaces. * Matrices {$A=PBP^{1}$} and {$B=P^{1}AP$} have the same eigenvalues. They are simply written in terms of different coordinate systems. If v is an eigenvector of A with eigenvalue λ, then {$P^{1}v$} is an eigenvector of B with the same eigenvalue λ. * Eigenvectors are the pure dimensions into which the action of a matrix (or linear transformation) can be decomposed. Ištrintos 204223 eilutės:
Math Discovery  House of Knowledge * Consider how the four levels of geometrylogic bring together the four levels of analysis and the four levels of algebra, yielding the 12 topologies. And why don't the two representations of the foursome yield a third representation? * Extension of a domain  [[https://en.wikipedia.org/wiki/Analytic_continuation  Analytic continuation]]  complex numbers  dealing with divergent series. * Root systems relate two spheres  they relate two "sheets". Logic likewise relates two sheets: a sheet and a metasheet for working on a problem. Similarly, we model our attention by awareness, as Graziano pointed out. This is stepping in and stepping out. * Mathematical mind/brain unites logical language (left brain  logical reasoning) with spatial visual thinking (right brain  geometric intuition). Consider what this means as two parallel ways of thinking for the 4 logicgeometry levels in the ways of discovery. * geometry (bundle) links algebra (fiber) with analysis (base) and the latter manifold is also understood (ambiguously) algebraically as a Lie group. Algebra is a (finite) cognitive pattern that restructures the (infinite) Analysis, the model of the world. Together they are a restructuring. In the house of knowledge for mathematics, the threecycle relates analysis and algebra as structuring and restructuring. Thus this restructuring (the six restructurings) is the output of the house of knowledge and the content of mathematics, its branches, concepts, statements, problems, etc. * Analysis is the infinity of sheets, the recurring sequence of not going beyond oneself. But algebra is the single sheet which is the self that it all goes into, where all of the actions, all of the sheets coincide as one sheet, one going beyond. Thus the cardinal (of algebra) arises from the ordinal (of analysis). * The ordinal (list) is deterministic whereas the cardinal (set) is nondeterministic. (polynomial coefficient) Ordinal/List/Analysis vs. Cardinal/Set/Algebra (polynomial root)  the coefficients and roots are related by the binomial theorem, the factors are choices. * Are there 6+4 branches of math? How are the branches of math related to the ways of figuring things out? * Proceed from balance  note how additive balance precedes multiplicative ratio precedes possibly negative (directional) ratio precedes anharmonic ratio precedes crossratio  and see how balance gets variously extended, as with the anharmonic group and the Moebius transformations. And relate that to the Balance as a way of figuring things out. * In the house of knowledge for mathematics, the threecycle establishes the substructures for the symmetry group. Similarly, in physics, it establishes the scales for isolating a system (?) * How does logic come from a quadratic form? Four ways of relating level and metalevel with "and". * Terrence Tao problem solving https://books.google.lt/books/about/Solving_Mathematical_Problems_A_Personal.html?id=ZBTJWhXD05MC&redir_esc=y * Physics is measurement. A single measurement is analysis. Algebra gives the relationships between disparate measurements. But why is the reverse as in the ways of figuring things out in mathematics? * Ways of discovery in math: [[http://www.tricki.org  Tricki.org]]. [[https://gowers.wordpress.com/2010/09/24/isthetrickidead/  Overview by Timothy Gowers]]. * [[https://www.laetusinpraesens.org/musings/periodt.php  Towards a Periodic Table of Ways of Knowing in the light of metaphors of mathematics]] Anthony Judge * Associative composition yields a list (and defines a list). Consider the identity morphism composed with itself. 2020 balandžio 12 d., 13:47
atliko 
Ištrintos 230242 eilutės:
Math outreach * [[https://gowers.wordpress.com/  Timothy Gowers' Weblog]] * [[https://www.dpmms.cam.ac.uk/~wtg10/  Timothy Gowers' webpage]] Methods of proof * How is substitution, as a method of proof, related to lamba calculus, and construction? * Does induction prove an infinite number of statements or their reassembly into one statement with infinitely many realizations? It proves the parallelness of intuitive meaningful stepped in and formal stepped out. * Mathematical induction  is it possible to treat infinitely many equations as a single equation with infinitely many instantiations? Consider NavierStokes equations. * Relate methods of proof and discovery, 3 systemic and 3 not. * Induction step by step is different than the outcome, the totality, which forgets the gradation. * Mathematical induction  is infinitely many statements that are true  relate to natural transformation, which also relates possibly infinitely many statements. 2020 balandžio 12 d., 13:45
atliko 
Pakeistos 244279 eilutės iš
* Given z1, z2, z3 in projective complex plane, there exists a unique Mobius transformation such that f(z1)=0, f(z2)=1, f(z3)=infinity. Note that there is a fourth symbol z, and they get paired: 0 and infinity, 1 and z. {$$f(z)=\frac{zz_1}{zz_3} \cdot \frac{z_2z_3}{z_2z_1} $$} * The Mobius transformation f(z) which sends f(0)=p, f(1)=r, f(infinity)=s is given by: {$$f(z)=\frac{zs(pr) + p(rs)}{z(pr) + (rs)} $$} * 1/z swaps inside and outside as conjugates, just as it swaps counterclockwise rotations i with counterclockwise rotations i. * Give a geometrical interpretation of e. * SL(2,C) models the transformation in the relationship between two wills: human's and God's. The complex variable describes the will. * The squeeze function defines area. * [[https://en.m.wikipedia.org/wiki/Möbius_transformation  W: Mobius transfomation]] important examples * SL(2,C) lines (plus infinity) become circles. Do linear equations become circular equations? What does that mean? Are SL(2,C) circular equations related to the continuum? * Intuit SL(2,C) as threedimensional in C (because adbc=1 so we lose one complex dimension  intuit that). And in what sense is that different from adbc=0 (a line? a onedimensional subspace?) * SL(2,C) is the spin relativistic group. * Investigate: In what sense do the properties of being an inverse (Cramer's rule) dictate the symmetry of SL(2,C)? Because the inverse has to maintain the same form. So why do the b, c switch sign and the a, d switch places? What imposition is made on duality? * [[https://en.wikipedia.org/wiki/Jinvariant  Jinvariant]] is related to SL(2,Z) and monstrous moonshine. * Why these three structures? How do they relate to the Moebius transformations? And how do these structures relate to the classical Lie families? * [[https://smile.amazon.com/FunctionsComplexVariableGraduateMathematics/dp/0387903283/ref=smi_www_rco2_go_smi_g3905707922?_encoding=UTF8&%2AVersion%2A=1&%2Aentries%2A=0&ie=UTF8  Functions of One Complex Variable]] John Conway * Understand the relations between U(1) and electromagnetism, SU(2) and the weak force, SU(3) and the strong force. * [[https://en.wikipedia.org/wiki/Standard_Model  Standard Model]] * [[https://en.wikipedia.org/wiki/Electroweak_interaction  Electroweak interaction]] * Mobius transformations can be composed from translations, dilations, inversions. But dilations (by complex numbers) could be understood as dilations (in positive reals), reflections, and rotations. * Reconsider what ShuHong's thesis has to say about fractions of differences, and how they relate to the Mobius group. Note that the [[https://en.wikipedia.org/wiki/M%C3%B6bius_transformation#Subgroups_of_the_M%C3%B6bius_group  Mobius transformations]] classify into types which accord with my six transformations: * reflection = circular * shear = parabolic * rotation = elliptic * dilation = hyperbolic * squeeze = internal of hyperbolic (e^t e^{t}=1) * translation = internal of parabolic Need to define "internal". Also, note that reflection is like rotation but more specific. Similarly, is translation like shear, but more specific? Analyze the Mobius group in terms of what it does to circles and lines, and analyze the transformations likewise. * Study [[https://en.wikipedia.org/wiki/SL2(R)  SL(2,R)]]. Study the [[https://en.wikipedia.org/wiki/M%C3%B6bius_transformation  Möbius group]]. Understand [[https://math.stackexchange.com/questions/646183/listofconnectedliesubgroupsofmathrmsl2mathbbc  the study of subgroups]]. * [[https://paulhus.math.grinnell.edu/SMPpaper.pdf  Jennifer Paulhus. Group Actions and Riemann Surfaces]] Multiplication * Polynomial powers are "twists" of a string. One end of the string is held up and then down. Each twist of the string allows for a new maximum or minimum. This is an interpretation of multiplication. į:
Ištrintos 263267 eilutės:
* The octonions can model the nonassociativity of perspectives. * Complex numbers describes rotations in twodimensions, and quaternions can be used to describe rotations in three dimensions. Is there a connection between octonions and rotations in four dimensions? * Compare trejybės ratas (for the quaternions) and Fano's plane (aštuonerybė) for the octonions. Ištrintos 274307 eilutės:
* Think of the nullsome as the point at infinity (the eye that sees the perspective). And think of it going beyond itself and then returning back to itself, making a big circle around. And think of that circle as a discrete line with four places. * Ker/Image  the kernel are those that can relate, that can take up the perspective * Relate to perspectives: Homology groups measure how far a chain complex is from being an exact sequence. * In studying perspective: How is homology used to prove the Brouwer fixed point theorem? * How is exactness (the image of f1 matching the kernel of f2) related to perspectives, for example, the notion of the complement? * The Axiom of Choice is based on the notions of a perspective (a bundle) in that we can assign to each set a choice. The set is fiber and the set of sets is the base. * How do observables relate to perspective? * Perspective relates intrinsic and extrinsic geometry by way of ambient space. Ambient space relates base and fiber (perspective) as a bundle. * {$K_0$} and {$K_1$} perhaps express perspectives, like God's trinity and the threecycle, the two foursomes in the eightcycle. * Normal bundles involve embedding in an extrinsic space. * The possibilities for a complex plane (extra point for sphere, point removed for cylinder) are relevant for modeling perspectives. Physics * Electrons are particles when we look at them and waves when we don't. * Quantum mechanics superposition of states may be based on borrowing of energy. When the energy debt is too high, as when it gets entangled with an observer, then the debt has to be paid. Otherwise, the debt can be restructured in complicated ways. * Lagrangian L=TV expresses slack (or antislack), makes the conversion between potential and kinetic energy as smooth as possible. Hamiltonian H=T+V expresses the totality, the love. In this sense, they are dual, as per the sevensome  the Lagrangian expresses the internal slack, and the Hamiltonian expresses the wholeness of the external frame. * Potential energy is bounded from below. * Kinetic energy is always positive, absolutely. Its relation to momentum is absolute, unconditional. Whereas potential is defined relatively and its relation to position is variable. * The wave equation is defined on phase space but in such a way that it is understood in terms of a superposition of waves for position, yielding a "blob"  a wave packet, and an analogous superposition of waves for momentum, and the two are related by the Fourier transform, by the Heisenberg uncertainty principle. * Physics abstracts from the personality of other researchers. Have a common ground, communicate not on any consensus, but based on what we find, as independent witnesses. * From dream: Space is made up of all possible curves. Physics is about the geodesics, the curves with no slack. They really are all one curve that goes through every curve and every single point. * Understanding of effect. Physics, why does it work? How can I describe it efficiently and correctly? * How many parameters do I need to describe the system? (Like an object.) Minimize constraints. It becomes complicated. Multipole is abstracting the levels of relevance. Ordering them inside the dimensions I am working with. What is the important quantity? Measures quality. How transformation leaves the object invariant. Distinguish between continuous parameters that we measure against and these quantity that we want to study. We use dimensions as a language to relate the inner structure and the outer framework. To measure momentum we need to measure two different quantities. * Weak nuclear force changes quark types. Strong nuclear force changes quark positions. Electromagnetic force distinguishes between quark properties  charge. * Exchange particles  gauge bosons. * Wave function Smolin says is ensemble, I say bosonic sharing of space and time * What is the relationship between spin (and alignment to a particular axis or coordinate system) and the alignment of magnets? * Which state is which amongst "one" and "another" is maintained until it is unnecessary  this is quantum entanglement. * Massless particles acquire mass through symmetry breaking: [[https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_theory  YangMills theory]]. * [[https://www.math.columbia.edu/~woit/wordpress/?p=5927  Geometric unity]] I tend to agree with Roger Penrose that spin has been one of the great mysteries in quantum mechanics. As best as I can recall, he said it was one of two primary mysteries in a talk at NYU back in the late 1990’s. ... understand spin and I think we'll understand entanglement a lot better. Ištrinta 281 eilutė:
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Variables * Variables are where the direction is reversed as regards the four relationships between level and metalevel. Variables are, on the one hand, the conclusion of math's "brain", but on the other hand, they are the start of math's "mind". * Organic variation, variables. * Homogeneous coordinates are what let us go between a simplex (when Z=1) and the coordinate system (when Z is free). Compare with the kinds of variables. 2020 balandžio 12 d., 13:39
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[[Catalan]] * Mandelbrot * Julia sets Pakeistos 190292 eilutės iš
* When do we have all three: orthogonal, unitary, symplectic forms? And in what sense do they relate the preservation of lines, angles, triangular areas? And how do they diverge? * Signal propagation  expansions. Consider the connection between walks on trees and Dynkin diagrams, where the latter typically have a distinguished node (the root of the tree) from which we can imagine the tree being "perceived". There can also be double or triple perspectives. * How do special rim hook tableaux (which depend on the behavior of their endpoints) relate to Dynkin diagrams (which also depend on their endpoints)? * Relate the ways of breaking the duality of counting with the ways of fusing together the sides of a square to get a manifold. Lie groups * Attach:ClassicalLieGroups.png * Homology calculations involve systems of equations. They propagate equalities. Compare this with the determinant of the Cartan matrix of the classical Lie groups, how that [[https://math.stackexchange.com/questions/2109581/intuitivelywhyarethere4classicalliegroupsalgebras  propagates equalities]]. The latter are chains of quadratic equations. * An grows at both ends, either grows independently, no center, no perspective, affine. Bn, Cn have both ends grow dependently, pairwise, so it is half the freedom. In what sense does Dn grow, does it double the possibilities for growth? And does Dn do that internally by relating xi+xj and xixj variously somehow? * SL(2)  H,X,Y is 3dimensional (?) but SU(2)  threecycle + 1 (God) is 4 dimensional. Is there a discrepancy, and why? * 4 kinds of i: the Pauli matrices and i itself. * [[https://johncarlosbaez.wordpress.com/2020/03/20/fromtheoctahedrontoe8/  John Baez. From the Octahedron to E8]] * Relate harmonic ranges and harmonic pencils to Lie algebras and to the restatement of {$x_ix_j$} in terms of {$x_i$}. * In Lie groups, real parameter subgroups (copies of R) are important because they define arcwise connectedness, that we can move from one point (group element) to another continuously. See how this relates to the slack defined by the root system. * Lie algebra matrix representations code for: ** Sequences  simple roots ** Trees  positive roots ** Networks  all roots * The nature of Lie groups is given by their forms: Orthogonal have a symmetric form {$\phi(x,y)=\phi(y,x)$}, Unitary have a Hermitian form {$\phi(x,y)=\overbar{\phi(y,x)}$}, and Symplectic have an antisymmetric form {$\phi(x,y)=\phi(y,x)$}. Note that symmetric forms do not distinguish between {$(x,y)$} and {$(y,x)$}; Hermitian forms ascribe them to two different conjugates (which may be the same if there is no imaginary component); and antisymmetric forms ascribe one to {$+1$} and the other to {$1$}, as if distinguishing "correct" and "incorrect". * Understand how forms relate to geometry, how they ground paths, distances, angles, and oriented areas. * Understand how forms for Lie groups relate to Lie algebras. * Understand how forms relate to real, complex and quaternionic numbers. * Noncommutative polynomial invariants of unitary group. * Francois Dumas. An introduction to noncommutative polynomial invariants. http://math.univbpclermont.fr/~fdumas/fichiers/CIMPA.pdf Symmetric polynomials, Actions of SL2. * Vesselin Drensky, Elitza Hristova. Noncommutative invariant theory of symplectic and orthogonal groups. https://arxiv.org/abs/1902.04164 * How to relate Lie algebras and groups by way of the Taylor series of the logarithm? * Consider how A_2 is variously interpreted as a unitary, orthogonal and symplectic structure. * Look at effect of Lie group's subgroup on a vector. (Shear? Dilation?) and relate to the 6 transformations. * The complex Lie algebra divine threesome H, X, Y is an abstraction. The real Lie algebra human cyclical threesome is an outcome of the representation in terms of numbers and matrices, the expression of duality in terms of 1, i, and position. * Jacob Lurie, Bachelor's thesis, [[http://www.math.harvard.edu/~lurie/papers/thesis.pdf  On Simply Laced Lie Algebras and Their Minuscule Representations]] * Does the Lie algebra bracket express slack? * Every root can be a simple root. The angle between them can be made {$60^{\circ}$} by switching sign. {$\Delta_i  \Delta_j$} is {$30^{\circ}$} {$\Delta_i  \Delta_j$} * {$e^{\sum k_i \Delta_i}$} * Has inner product iff {$AA^?=I$}, {$A{1}=A^?$} * Killing form. What is it for exceptional Lie groups? * The Cartan matrix expresses the amount of slack in the world. * {$A_n$} God. {$B_n$}, {$C_n$}, {$D_n$} human. {$E_n$} n=8,7,6,5,4,3 divisions of everything. * 2 independent roots, independent dimensions, yield a "square root" (?) * Symplectic matrix (quaternions) describe local pairs (Position, momentum). Real matrix describes global pairs: Odd and even? * Root system is a navigation system. It shows that we can navigate the space in a logical manner in each direction. There can't be two points in the same direction. Therefore a cube is not acceptable. The determinant works to maintain the navigational system. If a cube is inherently impossible, then there can't be a trifold branching. * {$A_n$} is based on differences {$x_ix_j$}. They are a higher grid risen above the lower grid {$x_i$}. Whereas the others are aren't based on differences and collapse into the lower grid. How to understand this? How does it relate to duality and the way it is expressed. * In {A_1}, the root {$x_2x_1$} is normal to {$x_1+x_2$}. In {A_2}, the roots are normal to {$x_1+x_2+x_3$}. * If two roots are separated by more than {$90^\circ$}, then adding them together yields a new root. * {$cos\theta = \frac{a\circ b}{\left \ a \right \\left \ b \right \}$} * {$120^\circ$} yields {$\frac{1}{\sqrt{2}\sqrt{2}}=\frac{1}{2}$} * Given a chain of composition {$\cdots f_{i1}\circ f_i \circ f_{i+1} \cdots$} there is a duality as regards reading it forwards or backwards, stepping in or climbing out. There is the possiblity of switching adjacent functions at each dot. So each dot corresponds to a node in the Dynkin diagram. And the duality is affected by what happens at an extreme. * Root systems give the ways of composing perspectivesdimensions. * {$A_n$} root system grows like 2,6,12,20, so the positive roots grow like 1,3,6,10 which is {$\frac{n(n+1)}{2}$}. * A root pair {$x$} and {$x$} yields directions such as "up" {$x––x$} and "down" {$–x–+x$}. * Threecycle: same + different => different ; different + different => same ; different + same => different * {$ \begin{pmatrix}1 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix}0 & i \\ i & 0 \end{pmatrix} \begin{pmatrix}1 & 0 \\ 0 & 1 \end{pmatrix} $} * same + different + different + same + ... * Go from rather arbitrary set of dimensions to more natural set of dimensions. Natural because they are convenient. This leads to symmetry. Thus represent in terms of symmetry group, namely Lie groups. There are dimensions. In order to write them up, we want more efficient representations. Subgroups give us understanding of causes. Smaller representations give us understanding of effects. We want to study what we don't understand. In engineering, we leverage what we don't understand. * Differences between even and odd for orthogonal matrices as to whether they can be paired (into complex variables) or not. * {$e^{\sum i \times generator \times parameter}$} has an inverse. * Unitary T = {$e^{iX_j\alpha_j}$} where {$X_j$} are generators and {$\alpha_j$} are angles. Volume preserving, thus preserving norms. Length is one. * Complex models continuous motion. Symplectic  slack in continuous motion. * Edward Frenkel. Langlands program, vertex algebras related to simple Lie groups, detailed analysis of SU(2) and U(1) gauge theories. https://arxiv.org/abs/1805.00203 * Consider the classification of Lie groups in terms of the objects for which they are symmetries. * {$A^TA$} is similar to the adjoint functors  they may be inverses (in the case of a unitary matrix) or they may be similar. * In the context {$e^iX$} the positive or negative sign of {$X$} becomes irrelevant. In that sense, we can say {$i^2=1$}. In other words, {$i$} and {$1$} become conjugates. Similarly, {$XYYX = \overline{(YXXY)}$}. * What is the relation between the the chain of Weyl group reflections, paths in the root system, the Dynkin diagram chain, and the Lie group chain. * Onedimensional proteins are wound up like the chain of a multidimensional Lie group. * Išsakyti grupės {$G_2$} santykį su jos atvirkštine. Ar ši grupė tausoja kokią nors normą? * The analysis in a Lie group is all expressed by the behavior of the epsilon. * A_n defines a linear algebra and other root systems add additional structure * A circle, as an abelian Lie group, is a "zero", which is a link in a Dynkin diagram, linking two simple roots, two dimensions. * [[https://www.youtube.com/watch?v=wIn_dlmD8sk  Video: The rotation group and all that]] * [[https://www.youtube.com/watch?v=8KPzuPizKk&list=PLZcI2rZdDGQrb4VjOoMm2o7Fu_mvij8F  Lorenzo Sadun. Videos: Linear Algebra]] Nr.88 is SO(3) and so(3) * {$U(n)$} is a real form of {$GL(n,\mathbb{C})$}. [[https://www.encyclopediaofmath.org/index.php/Complexification_of_a_Lie_group  Encyclopedia: Complexification of a Lie group]] * [[https://www.youtube.com/watch?v=zSLsjrJKPA  DrPhysicsA. Particle Physics 4: Rotation Operators, SU(3)xSU(2)xU(1)]] * At the level of forms, this can be seen by decomposing a Hermitian form into its real and imaginary parts: the real part is symmetric (orthogonal), and the imaginary part is skewsymmetric (symplectic)—and these are related by the complex structure (which is the compatibility). * Particle physics is based on SU(3)xSU(2)xU(1). Can U(1) be understood as SU(1)xSU(0)? U(1) = SU(1) x R where R gives the length. So this suggests SU(0) = R. In what sense does that make sense? * The conjugate i is evidently the part that adds a perspective. Then R is no perspective. In what sense is SU(3) related to a rotation in octonion space? * If SU(0) is R, then the real line is zero, and we have projective geometry for the simplexes. So the geometry is determined by the definition of M(0). * {$SL(n)$} is not compact, which means that it goes off to infinity. It is like the totality. We have to restrict it, which yields {$A_n$}. Whereas the other Lie families are already restricted. * The root systems are ways of linking perspectives. They may represent the operations. {$A_n$} is +0, and the others are +1, +2, +3. There can only be one operation at a time. And the exceptional root systems operate on these four operations. * Real forms  Satake diagrams  are like being stepped into a perspective (from some perspective within a chain). An odddimensional real orthogonal case is steppedin and evendimensional is stepped out. Complex case combines the two, and quaternion case combines them yet again. For consciousness. * Note that given a chain of perspectives, the possibilities for branching are highly limited, as they are with Dynkin diagrams. * Solvable Lie algebras are like degenerate matrices, they are poorly behaved. If we eliminate them, then the remaining semisimple Lie algebras are beautifully behaved. In this sense, abelian Lie algebras are poorly behaved. * Study how orthogonal and symplectic matrices are subsets of special linear matrices. In what sense are R and H subsets of C? * Find the proof and understand it: "An important property of connected Lie groups is that every open set of the identity generates the entire Lie group. Thus all there is to know about a connected Lie group is encoded near the identity." (Ruben Arenas) * Differentiating {$AA^{1}=I$} at {$A(0)=I$} we get {$A(A^{1})'+A'A^{1}=0$} and so {$A'=(A^{1})'$} for any element {$A'$} of a Lie algebra. * For any {$A$} and {$B$} in Lie algebra {$\mathfrak{g}$}, {$exp(A+B) = exp(A) + exp(B)$} if and only if {$[A,B]=0$}. * Symmetry of axes  Bn, Cn  leads, in the case of symmetry, to the equivalence of the total symmetry with the individual symmetries, so that for Dn we must divide by two the hyperoctahedral group. * the unitary matrix in the polar decomposition of an invertible matrix is uniquely defined. * In {$D_n$}, think of {$x_ix_j$} and {$x_i+x_j$} as complex conjugates. * In Lie root systems, reflections yield a geometry. They also yield an algebra of what addition of root is allowable. * Special linear group has determinant 1. In general when the determinant is +/ 1 then by Cramer's rule this means that the inverse is an integer and so can have a combinatorial interpretation as such. It means that we can have combinatorial symmetry between a matrix and its inverse  neither is distinguished. * Determinant 1 iff trace is 0. And trace is 0 makes for the links +1 with 1. It grows by adding such rows. * {$A_n$} tracefree condition is similar to working with independent variables in the the center of mass frame of a multiparticle system. (Sunil, Mukunda). In other words, the system has a center! And every subsystem has a center. * Symplectic algebras are always even dimensional whereas orthogonal algebras can be odd or even. What do odd dimensional orthogonal algebras mean? How are we to understand them? * An relates to "center of mass". How does this relate to the asymmetry of whole and center? * Išėjimas už savęs reiškiasi kaip susilankstymas, išsivertimas, užtat tėra keturi skaičiai: +0, +1, +2, +3. Šie pirmieji skaičiai yra išskirtiniai. Toliau gaunasi (didėjančio ir mažėjančio laisvumo palaikomas) bendras skaičiavimas, yra dešimts tūkstantys daiktų, kaip sako Dao De Jing. Trečias yra begalybė. * Kaip sekos lankstymą susieti su baltymų lankstymu ir pasukimu? * Fizikoje, posūkis yra viskas. Palyginti su ortogonaline grupe. * {$x_0$} is fundamentally different from {$x_i$}. The former appears in the positive form {$\pm(x_0x_1)$} but the others appear both positive and negative. * Kaip dvi skaičiavimo kryptis (conjugate) sujungti apsisukimu? * How to interpret possible expansions? For example, composition of function has a distinctive direction. Whereas a commutative product, or a set, does not. * Dots in Dynkin diagrams are figures in the geometry, and edges are invariant relationships. A point can lie on a line, a line can lie on a plane. Those relationships are invariant under the actions of the symmetries. Dots in a Dynkin diagram correspond to maximal parabolic subgroups. They are the stablizer groups of these types of figures. * G2 requires three lines to get between any two points (?) Relate this to the threecycle. į:
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Logic * Jordan algebra projections are "propositions". This makes sense with regard to the sevensome (logic) and the eightsome (the semiotic square). * Logikos prielaidos bene slypi daugybė matematikos sąvokų. Pavyzdžiui, begalybė slypi galimybėje tą patį logikos veiksnį taikyti neribotą skaičių kartų. O tai slypi tame kad logika veiksnio taikymas nekeičia loginės aplinkos. * Ar gali būti, kad aibių teorijos problemos (pavyzdžiui, sau nepriklausanti aibė) yra iš tikrųjų logikos problemos? Kaip atskirti aibių teoriją ir logiką (pavyzdžiui, axiom of choice)? * [[https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence  CurryHowardLambek correspondence]] of logic, programming and category theory * Algebra and geometry are linked by logic  intersections and unions make sense in both. * Nand gates and Nor gates (and And gates and Or gates) relate one and all. The Nand and Nor mark this relation with a negation sign. * Are Nand gates (Nor gates) related to perspectives? * Study how all logical relations derive from composition of Nand gates. * How is a Nor gate made from [[Nand]] gates? (And vice versa.) Mandelbrot set * How special is the Mandelbrot set? What other comparable fractals are there? Can the Mandelbrot set be understood to encompass all of mathematics? What is a combinatorial interpretation of the Mandelbrot set? How is the Mandelbrot set related to the complex numbers and numbers (normed vector spaces) more broadly? 2020 balandžio 12 d., 13:34
atliko 
Pakeistos 2239 eilutės iš
* Simple examples that illustrate theory. * every answer is an amount and a unit ir tt. * combine like units * list different units * a right triangle is half of a rectangle * a triangle is the sum of two right triangles * four times a right triangle is the difference of two squares * extending the domain * purposes of families of functions * How to present [[http://www.selflearners.net/Math/DeepIdeas  deep ideas in math]]? Using examples and games? How do games teach? Create learning materials. * Relate bundle concepts to amounts and units. * How is homotopy and its [0,1]x[0,1] square related to the complex plane? and to category theory square for composition of functors? and to classification in topology? * Scaling is positive flips over to negative this is discrete rotation is reflection * Develop looseness  slack  freedom  ambiguity as concepts that give meaning to isomorphism, identity, structure, symmetry. Local constraints can yet lead to different global solutions. * Study homology, cohomology and the Snake lemma to explain how to express a gap. į:
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* 6=4+2 representations. Similar to 6 edges of simplex = 4 edges of square + 2 diagonals * Einstein field equations  energy stress tensor  is 4+6 equations. * SU(3)xSU(2)xSU(1)xSU(0) is reminiscent of the omniscope. * John Baez: 24 = 6 x 4 = An x Bn * [[https://en.wikipedia.org/wiki/Dedekind_eta_function  Dedekind eta function]] is based on 24. Pakeistos 140163 eilutės iš
* Perspective:Division = real form:complex Lie algebra/group = real Clifford algebra / complex Clifford algebra * Function relates many dimensions (the perspectives) to one dimension (the whole), just like a division. The whole is given by the operation +1. And what do +2 and +3 mean? * For complex numbers, {$0 \neq 2\pi$} and so they are different when we go around the threecycle, so they yield the foursome: 0, 120, 240, 360. * What is the difference between an exact and a nonexact relationship? * Short exact sequence. Defining a perspective relative to a base. * The concept of scope: Kernel: irrelevant because goes to zero. Cokernel: irrelevant because outside of scope. * Pascal's triangle  the zeros on either end of each row are like Everything at start and finish of an exact sequence. * Circle (threecycle) vs. Line (link to unconditional)  sixsome  and real forms * Spin 1/2 means there are two states separated by a quanta of energy +/ h. So this is like divisions of everything: ** Spin 0 total spin: onesome ** Spin 1/2: fermions: twosome ** Spin 1: three states: threesome ** Spin 3/2: composite particles: foursome ** Spin 2: gravition: fivesome (time/space) * Turing machines  inner states are "states of mind" according to Turing. How do they relate to divisions of everything? * Study the Wolfram Axiom and Nand. * Euler's manipulations of infinite series (adding up to 1/12 etc.) are related to divisions of everything, of the whole. Consider the Riemann Zeta function as describing the whole. The same mysteries of infinity are involved in renormalization, take a look at that. Eightsome * 0=1 yields the zero ring. In what sense is this the collapse of a system? And how is it related to the finite field {$F_1$} ? į:
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* Definition of entropy depends on how you choose it. Unit of phase space determines your unit of entropy. Thus observer defines phase space. Exact sequences * [[https://en.wikipedia.org/wiki/Zigzag_lemma  Zigzag lemma]] relates to infinite revolutions along the threecycle. * In an exact sequence, is a perspective the group or the homomorphism? It is the group  it is a division of zero  where zero is everything. * Let p: E → B be a basepointpreserving Serre fibration with fiber F, that is, a map possessing the homotopy lifting property with respect to CW complexes. Suppose that B is pathconnected. Then there is a long exact sequence of homotopy groups * Primena trejybę. [[https://en.wikipedia.org/wiki/Homotopy_group  Wikipedia: Homotopy groups]] Let p: E → B be a basepointpreserving Serre fibration with fiber F, that is, a map possessing the homotopy lifting property with respect to CW complexes. Suppose that B is pathconnected. Then there is a long exact sequence of homotopy groups: {$\cdots \rightarrow \pi_n(F) \rightarrow \pi_n(E) \rightarrow \pi_n(B) \rightarrow \pi_{n1}(F) \rightarrow \cdots \rightarrow \pi_0(F) \rightarrow 0. $} Ištrintos 154161 eilutės:
* Compare finite field behavior (division winding around) with complex number behavior (winding around). * Sieja nepažymėtą priešingybę (+) 1, 0, 1 ir pažymėtą priešingybę (x) 1, 1. * [[https://en.m.wikipedia.org/wiki/Field_with_one_element  W: Field with one element]] * The nonexistent element of {$F_1$} may be considered to not exist, or imagined to exist, but regardless, I expect that cognitively there are three ways to interpret it as 0, 1, ∞, which thereby expand upon the duality between existence and nonexistence and make it structurally richer. * Relate [[https://en.wikipedia.org/wiki/M%C3%B6bius_transformation#Subgroups_of_the_M%C3%B6bius_group  elliptic transforms]] to God's dance {0, 1, ∞} and {$F_1$}: There are 3 representatives fixing {0, 1, ∞}, which are the three transpositions in the symmetry group of these 3 points: {$1 / z$}, which fixes 1 and swaps 0 with ∞ (rotation by 180° about the points 1 and −1), {$1 − z$} which fixes ∞ and swaps 0 with 1 (rotation by 180° about the points 1/2 and ∞), and {$z / ( z − 1 )$} which fixes 0 and swaps 1 with ∞ (rotation by 180° about the points 0 and 2). Note that this relates pairs from: 1, z, z1. * Study how turning the counting around relates to cycles  finite fields. Pakeistos 158193 eilutės iš
* Projective map  preserves a single line (what does that mean?) * Conformal map  preserves pairs of lines (the angle between them) * Symplectomorphism  preserves triplets of lines (their oriented volume), the noncollinearity of a triangle Four coordinate systems * In physics, the constraints oncedifferentiable and twicedifferentiable are interesting and not trivial, whereas thricedifferentiable is basically the same as infinitely differentiable. Compare this with perspective, perspective on perspective, and perspective on perspective on perspective. * Geometry: Options for introducing a coordinate system (none, one, two, three). No coordinate system is the case of tensors. * 1 coordinate system = 1 side of a triangle = Length. Shrinking the side can lead to a point  the two points become equal. This is like homotopy? * 2 coordinate systems = 2 sides of a triangle = Angle. Note that turning (rotating) one side around by 2 pi gets it back to where it was, and this is true for each 2 pi forwards and backwards. So by this equivalence we generate the integers Z as the winding numbers. * 3 coordinate systems = 3 sides of a triangle = Oriented area (the systems are ordered). What equivalence does this support and what does it yield? Is it related to e? All you can do * with 0 coordinate system (affine) * with 1 coordinate system (projective) is reflection, * with 2 coordinates (conformal) is rotation and shear, (the origins match) * with 3 coordinates (symplectic) is dilation (scales change), squeeze (scales change), translation (origins move). The coordinate systems (0,1,2,3) separate the level and metalevel. Study the 6 transformations between these sets of coordinate systems. * How do the 4 geometries (in terms of coordinate systems) relate to the 4 classical root systems? * Substantiate: Affine geometry defines no perspective, projective geometry defines one perspective, conformal geometry defines a perspective on a perspective, symplectic geometry defines a perspective on a perspective on a perspective. * Relate motion to bundles. Symplectic geometry, looseness, etc. All 4 geometries. * Think of harmonic pencil types as the basis for the root systems ** {$A_n$} {$\pm(xy)$} dual ** {$B_n$} {$\pm x,\pm y$} yields {$x\pm y$} ** {$C_n$} {$x\pm y$} yields {$\pm2x,\pm2y$} ** {$D_n$} {$\pm(x\pm y)$} dual dual * In what sense do these ground four geometries? And how do 6 pairs relate to ways of figuring things out in math? * Determinant expresses oriented volume, oriented area. Real numbers: distances. Complex numbers: angles. What do quaternions express? * [[https://www.youtube.com/watch?v=jdD5CTZhjoM  V: Vladimir I. Arnold. Polymathematics: complexification, symplectification and all that. 1988]] 18:50 About his trinity, his idea: "This idea, how to apply it, and the examples that I shall discuss even, are not formalized. The theory that I will describe today is not a conjecture, not a theorem, not a definition, it is some kind of religion. I shall show you examples and in these examples, it works. So I was able, using this religion, to find correct guesses, and to find correct conjectures. And then I was able to work years or months trying to prove them. And in some cases, I was able to prove them. In other cases, other people were finally able to prove them. In other cases other people were able to prove them. But to guess these conjectures without this religion would, I think, be impossible. So what I would like to explain to you is just this nonformalized part of it. I am perhaps too old to formalize it but maybe someone who one day finds the axioms and makes a definition from the general construction from the examples that I shall describe." 39:00 Came up with the idea in 1970, while working on the 16th Hilbert problem. * Arnold: Six geometries (based on Cartan's study of infinite dimensional Lie groups?) his list? * Affine and projective geometries. Adding or subtracting a perspective. Such as adding or deleting a node to a Dynkin diagram. (The chain of perspectives.) * Perhaps the projective geometry is the most basic, and it is based on rotation and the complexes. But perhaps an affine geometry arises, with the reals, so that we can imagine a movement of the origin (the fixed point) and thus we can have translations, a shift in origin, a shift in perspective, a relative perspective, in a conditional world. * A_n points and sets * B_n inside: perpendicular (angles) and * C_n outside: line and surface area * D_n points and position * Affine geometry preserves lines. Projective geometry also preserves zeros. Conformal geometry preserves angles. Symplectic geometry preservers oriented area. What are all these objects? į:
Ištrintos 160166 eilutės:
Geometry * Try to understand asymmetric functions, for example, by setting {$q_3^2=0$}. * Space arises with bundles, which separate the homogeneous choice, relevant locally, from the position, given globally. And then what does this say about time? * A geometry (like hyperbolic geometry) allows for a presentation of a bundle, thus a perspective on a perspective (atsitokėjimas  atvaizdas). Compare with: įsijautimasaplinkybė. * Geometry is concrete, it is a manifestation. Thus group actions can represent an abstract structure (of actions) in terms of a geometrical space (set, vector space, topological space). And this makes for understanding of the abstract in terms of the concrete. * Euclidean space  (algebraic) coordinate systems  define left, right, front, backwards  and this often makes sense locally  but this does not make sense globally on a sphere, for example 2020 balandžio 12 d., 13:24
atliko 
Pakeistos 578591 eilutės iš
* Any projective space is of the form {$\mathbb{KP}^n$} for some skew field {$\mathbb{K}$} except when n=2. Then it holds if the Axiom of Desargues holds in the space, but there are also exceptional spaces. This fact may relate to the fact that semisimple Lie algebras are related to {$\mathbb{R}$}, {$\mathbb{C}$} or {$\mathbb{H}$}  the classical Lie algebras  with the exceptions being very limited. * If we have the real plane and think of it projectively, then there is a line at infinity, encircling infinity, where the points on that line have orientation (given by the orientation of intersecting lines). But if we think of the plane as the complex numbers, then perhaps instead of a line it is sufficient to consider a single complex number as the point of infinity, in that it comes with a sense of angle. * Why is projective geometry related to lines, sphere, projection, point at infinity? * Harmonic pencil: Look for what it would mean for a ratio to be {$i$} and the product to be 1. The answer is {$e^{\frac{\pi}{4}i}$} and {$e^{\frac{3\pi}{4}i}$}, or {$e^{\frac{\pi}{4}i}$} and {$e^{\frac{3\pi}{4}i}$} * {$\mathrm{Aut}_{H\circ f}(\mathbb{P}^1_\mathbb{C})\simeq \mathrm{PSL}(2,\mathbb{C})$} * {$\mathrm{Aut}_{H\circ f}(\mathbb{U})\simeq \mathrm{PSL}(2,\mathbb{R})$} * {$\mathrm{Aut}_{H\circ f}(\mathbb{C})\simeq \mathrm{P}\Delta(2,\mathbb{R}) \equiv$} uppertriangular elements of {$\mathrm{PSL}(2,\mathbb{C})$} * Desargues theorem in geometry corresponds to the associative property in algebra. * A projective space is best understood from one dimension higher. And it is understood in terms of breaking down into smaller affine spaces. And these dimensions higher and lower are related to extending the chain of dimensions as given by Lie algebras. And the reversal of counting is related to the reversal of the orientation of lines etc. as a line is considered as a circle. * Homogeneous coordinates add a variable Z that makes each term of maximal power N by contributing the needed power of Z. * Spinor requires double rotation. Projective sphere is the opposite: half a rotation gets you back where you were. * Sextactic points on a simple closed curve. į:
Ištrintos 615617 eilutės:
* Six operations (six modeling methods) 6=4+2 relates 2 perspectives (internal (tensor) & external (Hom)) by 4 scopes (functors). The six 3+3 specifications define a gap for a perspective, thus relate three perspectives. Algebra and analysis come together in one perspective. Together, in the House of Knowledge, these all make for consciousness. Pakeistos 619670 eilutės iš
* Relate the monomial, forgotten, Schur symmetric functions of eigenvalues with the matrices {$(IA)^{1}$} and {$e^A$}. * Exercise: Get the eigenvalues for a generic matrix: 2x2, 3x3, etc. * Exercise: Look for a method to find the eigenvalues for a generic matrix. Express the solving of the equation as a way of relating the elementary functions. Are they related to the inverse Kostka matrix? And the impossibility of a combinatorial solution? And the nondeterminism issue, P vs NP? * Think again about the combinatorial intepretation of {$K^{1}K=I$}. * What do the constraints on Lie groups and Lie algebras say about symmetric functions of eigenvalues. * Vandermonde determinant shows invertible  basis for finite Fourier transform * What is the connection between the universal grammar for games and the symmetric functions of the eigenvalues of a matrix? * How do symmetries of paths relate to symmetries of young diagrams Symmetry * [[https://theoreticalatlas.wordpress.com/2014/01/10/categorifyingglobalvslocalsymmetry/  Global vs. Local symmetry]] * Algorithmic symmetry: Algorithms fold the list of effective algorithms upon itself. * Ockham's razor gets us to focus on the structures which are most basic in that they generate the richest symmetries  the rich symmetries tend solutions towards Ockham's razor. * Analytic symmetry is related to an infinite matrix, which is what we have with the classical Lie algebras. It is also the relation that an infinite sequence (and the function it models) can have with itself. It is thus the symmetry inherent in a recurrence relation, which is the content of a threecycle, what it is relating, a self to itself. * Analytic symmetry is governed by the characteristic polynomial which can't be solved in degree 5. * Representations of the symmetric group. Symmetric  homogeneous  bosons  vectors. Antisymmetric  elementary  fermion  covectors. Euclidean space allows reflection to define inside and outside nonproblematically, thus antisymmetricity. Free vector space. Schur functions combine symmetric and antisymmetric in rows and columns. * E8 is the symmetry group of itself. What is the symmetry group of? * Love (symmetry) establishes immortality (invariant). * Note that x and y axes are separated by 90 degrees. This is the grounds for the degree four of i, the trigonometric functions, the CauchyRiemann equations, etc. * Try to express the symmetries of an object, like a polyhedron, in terms of bundle conceptions. * Symmetry: indistinguishable change, thus a lie, a nontruth, what is hidden. Hidden change, the revealing of hidden change. Symplectic geometry * Notion of derivative is related to tangency, to tangent vector, is related to the normal of the normal, is related to duality, is related to the slack modeled by symplectic geometry. * Symplectic geometry defines slack. It defines motion as oriented area. * Brouwer Fixed Point Theorem holds on a disk with boundary. He also showed that a reversible T which preserves area on the disk without boundary has a fixed point. (Conjugated through a translation.) (So area preservation is equivalent to having a boundary.) This relates perspectives and symplectic geometry. * Symplectic maps map loops to loops with the same area. Area of a closed curve is given by differential forms. There has to be an energy in the background, the Hamiltonian. Symplectic space can associate area to a small loop (small triangle). * Consider how simplexes (in differential geometry and Stokes theorem) relate to symplectic geometry. What is the relevant polytope for symplectic geometry? Coordinate systems? Or crosspolytopes? * Heisenberg uncertainty principle  the slack in the vacuum  that allows the borrowing of energy on brief scales. Compare with coupling (of position and momentum, for example). * Force (and acceleration) is a second derivative  this is because of the duality, the coupling, needed between, say, momentum and position. Is this coupling similar to adjoint functors? * Kinectic energy is possibly zero and builds up from there. It can be expressed in an absolute sense. Potential is possibly infinite and subtracts from that. So it must be expressed in a relative sense. So they are coming at energy from opposite directions. * Note that symplectic geometry, in preserving areas, seems to only care about the extremal points. In what sense is that true in phase space? When and how could it not be true? What is the difference between having finitely many extremal points (convex polytope? and what is the significance of extremal points vs. extremal edges vs. extremal faces etc.? and homology/cohomology?), infinitely many (as with a fractal boundary) or all extremal points? Can symplectic geometry be considered a study of the outside and conformal geometry a study of the inside? * Lagrangian: (Force) Change in potential energy = (mass x acceleration) Change in kinetic energy. Potential energy is based on position and not time. Kinetic energy is based on time and not position. the threesome. * Symplectic  basis for coupling  coupling of electric and magnetic fields  is what is responsible for the periodic nature of waves  the higher the frequency, the higher the energy, the tighter the coupling  the coupling is across the entire universe. The coupling models looseness  slack. This brings to mind the vacillation between knowing and not knowing. * Note that 2dimensional phase space (as with a spring) is the simplest as there is no 1dimensional phase space and there can't be (we need both position and momentum). * What is the connection between symplectic geometry and homology? See [[https://en.wikipedia.org/wiki/Morse_theory  Morse theory]]. See [[https://people.ucsc.edu/~alee150/sympl.html  Floer theory]]. * Symplectic geometry is the geometry of the "outside" (using quaternions) whereas conformal geometry is the geometry of the "inside" (using complex numbers). Then what do real numbers capture the geometry of? And is there a geometry of the line vs. a geometry of the circle? And is one of them "spinorial"? * Benet linkage  keturgrandinis  lygiagretainis, antilygiagretainis * Closed curve in plane must have at least four critical points of curvature. This reflects the fourfold aspect of turning around, rotating. Caustic  a phenomenon in symplectic geometry. Tensors * Tensors stay free of a coordinate system and work with all of them. * Tensor: function in all possible coordinate spaces such that it (its values) obey certain transformation rules. * There is a duality between a tensor and its expression under a particular basis. They are interchangeable. * Tensors are invariant under linear transformations but their components do change. * So a tensor is a bringing together of components, which can be either covariant or contravariant. Is this stepping out and stepping in? Is a tensor a division of everything and each component a perspective? į:
Pakeistos 633660 eilutės iš
* [[https://pdfs.semanticscholar.org/a4fb/637410e874b97d323d145c4ebed1c0b03074.pdf  Colin McLarty. The Uses and Abuses of the History of Topos Theory]] * [[https://johncarlosbaez.wordpress.com/2020/01/05/topostheorypart1/  John Baez. Topos Theory (Part 1)]] sheaves, elementary topoi, Grothendieck topoi and geometric morphisms. * [[https://johncarlosbaez.wordpress.com/2020/01/07/topostheorypart2/  Part 2]] turning presheaves into bundles and vice versa; turning sheaves into etale spaces and vice versa. * [[https://johncarlosbaez.wordpress.com/2020/01/13/topostheorypart3/  Part 3]] sheafification; the adjunction between presheaves and bundles. * [[https://johncarlosbaez.wordpress.com/2020/01/21/topostheorypart4/  Part 4]] direct and inverse images of sheaves. * [[https://johncarlosbaez.wordpress.com/2020/01/28/topostheorypart5/  Part 5]] why presheaf categories are elementary topoi: colimits and limits in presheaf categories. * [[https://johncarlosbaez.wordpress.com/2020/02/11/topostheorypart6/  Part 6]] why presheaf categories are elementary topoi: cartesian closed categories and why presheaf categories are cartesian closed. * [[https://johncarlosbaez.wordpress.com/2020/02/18/topostheorypart7/  Part 7]] subobject classifiers * [[https://johncarlosbaez.wordpress.com/2020/02/27/topostheorypart8/  Part 8]] an example of a presheaf topos * How do sheaves relate to gradation of symmetric functions? * [[https://www.oliviacaramello.com/Videos/Videos.htm  V: Olivia Caramello]] Triality * [[http://math.ucr.edu/home/baez/octonions/conway_smith/  Baez's review of Conway and Smith]]: The octonions can be described not only as the vector representation of {$\text{Spin}(8)$}, but also the lefthanded spinor representation and the righthanded spinor representation. This fact is called 'triality'. It has many amazing spinoffs, including structures like the exceptional Lie groups and the exceptional Jordan algebra, and the fact that supersymmetric string theory works best in 10dimensional spacetime — fundamentally because {$8 + 2 = 10$}. * The triality suggests that the Father and the Son are oppositely handed spinor representations and the vector representation is the Spirit? Or the unconscious and conscious are oppositely handed and the vector representation is consciousness? * Yu. I. Manin. Cubic Forms: Algebra, Geometry, Arithmetic. Related to ternary operations and triality. * Triality: C at the center, three legs: quaternions, evendimensional reals, odddimensional reals. Fold, fuse, link. Universal hyperbolic geometry * The circle maps every point to a line and vice versa. * Quadrance (distance squared) is more correct than distance because quadrance makes positive distance and negative distance equivalent. Spread (absolute value of the sine of angle) is more correct than angle because the value of the spread is the same for all angles at an intersection, which is to say, for both theta and pi minus theta. In this way, quadrance and spread eliminate false distinctions and the problems they cause. * [[https://www.youtube.com/watch?v=LaTTqgchO2o  How Chromogeometry transcends Klein's Erlangen Program for Planar Geometries N J Wildberger]] * Try to express projective geometry (or universal hyperbolic geometry) in terms of matrices and thus symmetric functions. What then is algebraic geometry and how do polynomials get involved? What is analytic geometry and in what sense does it go beyond matrices? How do all of these hit up against the limits of matrices and the amount of symmetry in its internal folding? * Attach:GeometryFormulas.png * Attach:QuadrupleFormulas.png į:
Ištrintos 647664 eilutės:
Yoneda Lemma * B_>C ..... How>What * External relations > Internal logic .... (Not What=Why) Hom C > Hom B (Not How=Whether) * What does the Yoneda Lemma say concretely about the category of Graphs, Groups, Lists, etc.? * What would the category of Lists look like? And what would the Yoneda Lemma look like if the functor mapped into the category of Lists? * Analyze the commutative diagram proving the Yoneda lemma by thinking it, on the one level, as a statement about four sets, but on another level, as a statement about functors and a natural transformation between them. Is there an ambiguity here? * Eduardo's Yoneda Lemma diagram is the foursome. * Loss of info from How to What is equal to the Loss of info from "Why for What" to "Why for How". * How: inner logic. What: external view. * Yoneda lemma  relates to exponentiation and logarithm * Category is a collection of examples that satisfy certain conditions. That is why you can have many examples that are essentially identical. And it's essential for the meaning of the Yoneda Lemma, for example. Because two different examples may have the same structure but one example may draw richer meaning from one domain and the other may not have that richer meaning or may have other meaning from another domain. * Whether (objects), what (morphisms), how (functors), why (natural transformations). Important for defining the same thing, equivalence. If they satisfy the same reason why, then they are the same. * Representable functors  based on arrows from the same object. * Is Cayley's theorem (Yoneda lemma) a contentless theorem? What makes a theorem useful as a tool for discoveries? * Show why there is no ncategory theory because it folds up into the foursome. * Understand the Yoneda lemma. Relate it to the four ways of looking at a triangle. * The Yoneda Lemma: the Why of the external relationships leads to the Whether of the objects in the set. The latter are considered as a set of truth statements. 2020 balandžio 12 d., 13:14
atliko 
Pakeistos 4568 eilutės iš
* John Baez's comment about 4 dimensions being most interesting because halfway between 0 and 8. The Son turns around and reverses the Father. * How can the fact that {$\Gamma(\frac{1}{2})=\sqrt{\pi}$} relate {$\pi$} and {$e$}? Note that {$\Gamma(1)=0!$} and {$\Gamma(\frac{3}{2})=\frac{\sqrt{\pi}}{2}$}. Consider the integral definition. * In Bott periodicity, the going beyond oneself (by adding perspectives) and the going outside of oneself (by subtracting perspectives) is, in mathematics, extended to all possible integers, and thus the two directions are related in the four ways as given by the four classical Lie algebras, including gluing, fusing, folding. * In the eightcycle of divisions, the going beyond oneself and the going back outside oneself are reminiscent of spinors like electrons. * The flip side of going beyond yourself  if you can add perspectives, then you can substract perspectives  on the flip side. Thus nullsome and foursome model the extremes: nullsome models the void of everything, foursome models the point of nothing. Thus adding a perspective takes us from the indefinite to the definite and back again. The three operations +1,+2,+3 make for a threefold braid. * R nullsome * H twosome 1+i (different) j+ij (the same) * H+H threesome splits the twosome * (H2) foursome  internally doubles * (C4) fivesome * (R8) sixsome * R+R sevensome (dividing the nullsome into two perspectives) (S16) what would S mean? half of R? positive reals? * [[https://www.researchgate.net/publication/252379656_The_Elliptic_Umbilic_Diffraction_Catastrophe  The Elliptic Umbilic Diffraction Catastrophe]]. Optics, Bott periodicity? * Study [[https://en.wikipedia.org/wiki/Orthogonal_group  orthogonal groups]] and Bott periodicity. * [[http://pi.math.cornell.edu/~hatcher/AT/AT.pdf  Allen Hatcher. Algebraic topology]]. [[http://pi.math.cornell.edu/~hatcher/AT/ATpage.html  Explanation]] Homotopy, homology, cohomology. We will show in Theorem 3.21 that a finitedimensional division algebra over R must have dimension a power of 2. The fact that the dimension can be at most 8 is a famous theorem of [Bott & Milnor 1958] and [Kervaire 1958]. Example 4.55: Bott Periodicity. * [[http://pi.math.cornell.edu/~hatcher/VBKT/VB.pdf  Allen Hatcher. Vector Bundles and KTheory. (Halfwritten).]] [[http://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html  Explanation]] [[http://pi.math.cornell.edu/~hatcher/VBKT/VBKTtc.html  Table of Contents]] Bottperiodicity. * Bott periodicity exhibits selffolding. Note the duality with the pseudoscalar. Consider the formula n(n+1)/2 does that relate to the entries of a matrix? * Bott periodicity is the basis for 8fold folding and unfolding. * What is the connection between Bott periodicity and spinors? See [[http://www.ams.org/journals/bull/20023902/S027309790100934X/S027309790100934X.pdf  John Baez, The Octonions]]. * Orthogonal add a perspective (Father), symplectic subtract a perspective (Son). * My dream: Sartre wrote a book "Space as World" where he has a formula that expresses Bott periodicity / my eightfold wheel of divisions. į:
Pakeistos 4966 eilutės iš
* What can graph theory (for example, random graphs, or random order) say about category theory? * In the category of Sets, is there any way to distinguish between the integers and the reals? Are all infinities the same? * In the category Set, how can you distinguish between a countable and uncountable set? * Kromer, R.: Tool and Object: A History and Philosophy of Category Theory. Birkhauser (2007) * Fong, B., Spivak, D.I.: Seven Sketches in Compositionality: An Invitation to Applied Category Theory. Cambridge (2019),http://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf * Lawvere, W., Schanuel, S.: Conceptual Mathematics: A first introduction to categories. Cambridge (1997) * [[https://www.amazon.com/CategoryTheoryContextAuroraOriginals/dp/048680903X  Emily Riehl. Category Theory in Context]] * In category theory, what is the relationship between structure preservation of the objects, internally, and their external relationships? * Axiom of forgetfullness. * Internal discussion with oneself vs. external discussion with others (Vygotsky) is the distinction that category theory makes between internal structure and external relationships. * In most every category, can we (arbitrarily) define (uniquely) distinguished "generic objects" or "canonical objects", which are the generic equivalents for all objects that are equivalent to each other? For example, in the category of sets, the generic set of size one. * In functional programming with monoids and monads, can we think of each function as taking us from a question type to an answer type? In general, in category theory, can we think of each morphism as taking us from a question to an answer? * Category theory for me: distinguishing what observations are nontrivial  intrinsic to a subject  and what are observations are contentwise trivial or universal  not related to the subject, but simply an aspect of abstraction. * In product, the information from A and B is stored externally in A x B. In coproduct, the information from A and B is stored internally in A union B (A+B). Note: multiplication is external, and addition is internal. * In a diagram, we have a map from shape J (the index category) to the category C. Note that the index diagram is How. * monad = black box? į:
Pakeistos 182209 eilutės iš
* Duality of nonrelevance in system (kernel) and nonrelevance beyond system (cokernel). * Compare the kinds of dualities with the kind of transformations of perspectives and then with the kinds of geometric transformations. * In the automata hierarchy, consider how to model duality of internal structure and external network. * Duality of vectors and covectors (linear functionals)  flipsides of composition  what is "inside" and "outside" the composer. * Conjugates can be thought of as "twins", whereas +1 and 1 are "spouses". * Conjugate = mystery = false. (Hidden distinction). * i>j is asymmetric, onedirectional. i<>i* is symmetric, twodirectional, breaks antisymmetry, hides antisymmetry (which is i and which is j?) * Algebra  geometry duality. (Pullback). Morphism <> ring homomorphism. Intrinsic and extrinsic geometry. Ambient space. Relation between two spaces. Varieties [morphisms X to Y] <> Affine Falgebras [Flinear ring homomorphisms F[Y] to F[x]] * Study the idea behind linear functionals, fundamental representations, eigenvectors, cohomology, and other maps into one dimension. * Duality examples (conjugates) ** complex number "i" is not one number  it is a pair of numbers that are the square roots of 1 ** spinors likewise ** Dn where n=2 ** the smallest crosspolytope with 2 vertices ** taking a sphere and identifying antipodal elements  this is a famous group ** polar conjugates in projective geometry (see Wildberger) ** Study how Set breaks duality (the significance of initial and terminal objects). * Each physical force is related to a duality: ** Charge (matter and antimatter)  electromagnetism ** Weak force  time reversal * So the types of duality should give the types of forces. * John Baez on duality: Dyson's Threefold Way: either X is not isomorphic to its dual (the complex case), or it is isomorphic to its dual (in the real or quaternionic cases). Study his slides! * [[https://www.youtube.com/watch?v=7d5jhPmVQ1w  John Baez on duality in logic and physics]] * Algebra (of the observer) and analysis (of the observed) exhibit a duality of worldviews. * The interpretation {$\mathbb{C} \leftrightarrow \mathbb{R}^2$} gives meaning to the two axes. One axis the opposites 1 and 1 and the other axis is the opposites i and j. And they become related 1 to i to 1 to j. Thus multiplication by i is rotation by 90 degrees. It returns us not to 1 but sends us to 1. * Composition algebra. Doubling is related to duality. * Linear functionals. Onedimensional economic thinking is like linear functionals  the dual space of the multidimensional reality. In finite dimensions, the dual dual is the same as what we started. But in infinite dimensions not necessarily. Does this suggest that our life is infinitedimensional, which is to say, it can't be captured by a onedimensional shadow? į:
2020 balandžio 12 d., 13:09
atliko 
Pridėtos 1829 eilutės:
Affine geometry * In what sense does affine geometry not have a coordinate system? (And thus not have a notion of infinity?) Affine = local (no infinity). * Affine geometry is agnostic regarding coordinate system. So it doesn't distinguish if we flip all the positive and minus choices. * Affine transformation extends a linear transformation by a column (the translation) and a row (of zeroes) and a diagonal element (of 1). Thus it is similar to a Lie algebra (Dn ?) extending An. Algebra * studies particular structures and substructures Algebraic geometry * [[https://en.wikipedia.org/wiki/Andrei_Okounkov  Andrei Okounkov]] Bridging probability, representation theory and algebraic geometry. * Discriminant of [[https://en.wikipedia.org/wiki/Elliptic_curve  elliptic curve]]. [[https://en.wikipedia.org/wiki/Arithmetic_derivative  Arithmetic derivative]] AutomataTheory * There is a qualitative hierarchy of computing devices. And they can be described in terms of matrices. * Yates Index Theorem  consider substitution. * [[https://www.quantamagazine.org/mathematiciansolvescomputerscienceconjectureintwopages20190725/  Complexity measures for Boolean functions]]. * How is a [[https://en.wikipedia.org/wiki/Boolean_function  Boolean function]] similar to a linear functional? * If we think of a Turing machine's possible paths as a category, what is the functor that takes us to the actual path of execution? Basic concepts * Simple examples that illustrate theory. * every answer is an amount and a unit ir tt. * combine like units * list different units * a right triangle is half of a rectangle * a triangle is the sum of two right triangles * four times a right triangle is the difference of two squares * extending the domain * purposes of families of functions * How to present [[http://www.selflearners.net/Math/DeepIdeas  deep ideas in math]]? Using examples and games? How do games teach? Create learning materials. * Relate bundle concepts to amounts and units. * How is homotopy and its [0,1]x[0,1] square related to the complex plane? and to category theory square for composition of functors? and to classification in topology? * Scaling is positive flips over to negative this is discrete rotation is reflection * Develop looseness  slack  freedom  ambiguity as concepts that give meaning to isomorphism, identity, structure, symmetry. Local constraints can yet lead to different global solutions. * Study homology, cohomology and the Snake lemma to explain how to express a gap. Binomial theorem * e to the matrix consisting of the natural numbers on its first offdiagonal gives a triangular matrix with pascal's triangle. and how is it in the case of the cube? * Pascal's triangle tilted gives Fibonacci numbers * I dreamed of the binomial theorem as having an "internal view", imagined from the inside, which accorded with the "coordinate systems". And which interweaved with the external views to yield various "moments", given by curves on the plane, variously adjusted and transformed by the internal view. Bott Periodicity * John Baez's comment about 4 dimensions being most interesting because halfway between 0 and 8. The Son turns around and reverses the Father. * How can the fact that {$\Gamma(\frac{1}{2})=\sqrt{\pi}$} relate {$\pi$} and {$e$}? Note that {$\Gamma(1)=0!$} and {$\Gamma(\frac{3}{2})=\frac{\sqrt{\pi}}{2}$}. Consider the integral definition. * In Bott periodicity, the going beyond oneself (by adding perspectives) and the going outside of oneself (by subtracting perspectives) is, in mathematics, extended to all possible integers, and thus the two directions are related in the four ways as given by the four classical Lie algebras, including gluing, fusing, folding. * In the eightcycle of divisions, the going beyond oneself and the going back outside oneself are reminiscent of spinors like electrons. * The flip side of going beyond yourself  if you can add perspectives, then you can substract perspectives  on the flip side. Thus nullsome and foursome model the extremes: nullsome models the void of everything, foursome models the point of nothing. Thus adding a perspective takes us from the indefinite to the definite and back again. The three operations +1,+2,+3 make for a threefold braid. * R nullsome * H twosome 1+i (different) j+ij (the same) * H+H threesome splits the twosome * (H2) foursome  internally doubles * (C4) fivesome * (R8) sixsome * R+R sevensome (dividing the nullsome into two perspectives) (S16) what would S mean? half of R? positive reals? * [[https://www.researchgate.net/publication/252379656_The_Elliptic_Umbilic_Diffraction_Catastrophe  The Elliptic Umbilic Diffraction Catastrophe]]. Optics, Bott periodicity? * Study [[https://en.wikipedia.org/wiki/Orthogonal_group  orthogonal groups]] and Bott periodicity. * [[http://pi.math.cornell.edu/~hatcher/AT/AT.pdf  Allen Hatcher. Algebraic topology]]. [[http://pi.math.cornell.edu/~hatcher/AT/ATpage.html  Explanation]] Homotopy, homology, cohomology. We will show in Theorem 3.21 that a finitedimensional division algebra over R must have dimension a power of 2. The fact that the dimension can be at most 8 is a famous theorem of [Bott & Milnor 1958] and [Kervaire 1958]. Example 4.55: Bott Periodicity. * [[http://pi.math.cornell.edu/~hatcher/VBKT/VB.pdf  Allen Hatcher. Vector Bundles and KTheory. (Halfwritten).]] [[http://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html  Explanation]] [[http://pi.math.cornell.edu/~hatcher/VBKT/VBKTtc.html  Table of Contents]] Bottperiodicity. * Bott periodicity exhibits selffolding. Note the duality with the pseudoscalar. Consider the formula n(n+1)/2 does that relate to the entries of a matrix? * Bott periodicity is the basis for 8fold folding and unfolding. * What is the connection between Bott periodicity and spinors? See [[http://www.ams.org/journals/bull/20023902/S027309790100934X/S027309790100934X.pdf  John Baez, The Octonions]]. * Orthogonal add a perspective (Father), symplectic subtract a perspective (Son). * My dream: Sartre wrote a book "Space as World" where he has a formula that expresses Bott periodicity / my eightfold wheel of divisions. Bundles * Vector bundles: Identity and selfidentity (like the ends of a regular strip or a Moebius strip). Identity of a point, identity of a fiber  selfidentity under continuity. How does a fiber relate to itself? Is it flipped or not? 2 kinds of selfidentity (or nonidentity) allows the edge to be flipped upside down. Category theory * What can graph theory (for example, random graphs, or random order) say about category theory? * In the category of Sets, is there any way to distinguish between the integers and the reals? Are all infinities the same? * In the category Set, how can you distinguish between a countable and uncountable set? * Kromer, R.: Tool and Object: A History and Philosophy of Category Theory. Birkhauser (2007) * Fong, B., Spivak, D.I.: Seven Sketches in Compositionality: An Invitation to Applied Category Theory. Cambridge (2019),http://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf * Lawvere, W., Schanuel, S.: Conceptual Mathematics: A first introduction to categories. Cambridge (1997) * [[https://www.amazon.com/CategoryTheoryContextAuroraOriginals/dp/048680903X  Emily Riehl. Category Theory in Context]] * In category theory, what is the relationship between structure preservation of the objects, internally, and their external relationships? * Axiom of forgetfullness. * Internal discussion with oneself vs. external discussion with others (Vygotsky) is the distinction that category theory makes between internal structure and external relationships. * In most every category, can we (arbitrarily) define (uniquely) distinguished "generic objects" or "canonical objects", which are the generic equivalents for all objects that are equivalent to each other? For example, in the category of sets, the generic set of size one. * In functional programming with monoids and monads, can we think of each function as taking us from a question type to an answer type? In general, in category theory, can we think of each morphism as taking us from a question to an answer? * Category theory for me: distinguishing what observations are nontrivial  intrinsic to a subject  and what are observations are contentwise trivial or universal  not related to the subject, but simply an aspect of abstraction. * In product, the information from A and B is stored externally in A x B. In coproduct, the information from A and B is stored internally in A union B (A+B). Note: multiplication is external, and addition is internal. * In a diagram, we have a map from shape J (the index category) to the category C. Note that the index diagram is How. * monad = black box? [[Catalan]] * Mandelbrot * Julia sets Choice framework * Three perspectives lets you define a coordinate system as a choice framework. * Totally independent dimensions: Cartesian * Totally dependent dimensions: simplex * We "understand" the simplex in Cartesian dimensions but think it in simplex dimensions. We can imagine a simplex in higher dimensions by simply adding externally instead of internally. Consider (implicit + explicit)to infinity; and also (unlabelled + labelled) to infinity. Also consider (unlabelable + labelable). And (definitively labeled + definitively unlabeled). * Gaussian binomial coefficients [[http://math.stackexchange.com/questions/214065/provingqbinomialidentities  interpretation related to Young tableaux]] * https://oeis.org/A013609 triangle for hypercubes. (1 + 2x)^n unsigned coefficients of chebyshev polynomials of the second kind * [[https://ncatlab.org/nlab/show/geometric+shape+for+higher+structures  nLab: Geometric shape for higher structures]] * What do "globe", "globe category", "globular set" mean? * [[https://math.stackexchange.com/questions/2371364/whatsthecanonicalembeddingoftheglobecategoryintotop  What's the canonical embedding of the globe category into Top?]] * They are the maps of the closed nball to the "northern" and "southern" hemispheres of the surface of the (n+1)ball. * This defines a functor G→Top that extends along the Yoneda embedding, yielding a geometric representation of any globular set Gˆ→Top. * Ar savybių visuma yra simpleksas? Ar savybės skaidomos (koordinačių sistema). * The empty set, or the center, has dimension 1. * [[https://terrytao.wordpress.com/2019/07/26/twistedconvolutionandthesensitivityconjecture/  Terrence Tao: Twisted Convolution and the Sensitivity Conjecture]] * Choices  polytopes, reflections  root systems. How are the Weyl groups related? * Polytopes. edge = difference * Esminis pasirinkimas yra: kurią pasirinkimo sampratą rinksimės? * Kaip suvokiame {$x_i$}? Koks tai per pasirinkimas? * Kada pasirinkimo samprata keičiasi, visgi už visų sampratų slypi bendresnis, pirmesnis suvokimų suvokimas, taip kad renkamės pačią sampratą. Folding is the basis for substitution. * Keturios pasirinkimo sampratos (apimtys) visos reikalingos norint išskirti vieną paskirą pasirinkimą. An simplexes allow gaps because they have choice between "is" and "not". But all the other frameworks lack an explicit gap and so we get the explicit second counting. But: * for Bn hypercubes we divide the "not" into two halves, preserving the "is" intact. * for Cn crosspolytopes we divide the "is" into two halves, preserving the "not" intact. * for Dn we have simply "this" and "that" (notthis). * Use "this" and "that" as unmarked opposites  conjugates. Clifford algebras * Real forms are similar to Clifford algebras in that in ndimensions you have n versions, n signatures. Each models the takng up of one of the n perspectives in a division of everything. Coincidences * 6=4+2 representations. Similar to 6 edges of simplex = 4 edges of square + 2 diagonals * Einstein field equations  energy stress tensor  is 4+6 equations. * SU(3)xSU(2)xSU(1)xSU(0) is reminiscent of the omniscope. * John Baez: 24 = 6 x 4 = An x Bn * [[https://en.wikipedia.org/wiki/Dedekind_eta_function  Dedekind eta function]] is based on 24. Combinatorics * The mathematics of counting (and generating) objects. I think of this as the basement of mathematics, which is why I studied it. * [[https://www.math.upenn.edu/~wilf/gfology2.pdf  Generatingfunctionology]] by Herbert Wilf Complex algebra * I dreamed of the complex numbers as a line that curls, winds, rolls up in one way on one end, and in the mirror opposite way on the other end, like rolling up a carpet from both ends. Like a scroll. * How is the Riemann sheet, winding around, going to a different Riemann sheet, related to the winding number? and the roots of polynomials? Complex analysis * Differentiability of a complex function means that it can be written as an infinite power series. So differentiation reduces complex functions to infinite power series. This is analogous to evolution abstracting the "real world" to a representation of it. And differentiation is relevant as a shift from the known to focus on the unknown  the change. Analytic continuation * Understand analytic continuation. Can we think of it as cutting the plane into a spiral of width {$e^ne^{n1}$}? * Learn how to extend the Gamma function to the complex numbers. Conformal geometry * Why and how is Universal Hyperbolic Geometry related to conformal geometry? * A function from the complex plane to the complex plane which preserves angles (of intersecting curves) necessarily is analytic. This seems related to the fact that exponentiation {$e^{i\theta}$} makes multiplication additive. And that brings to mind trigonometric functions. Preservation of angles implies existence of Taylor series. * When are two vectors, lines, etc. perpendicular? When they are distinguished by i ? * Conformal groups. Orthogonal. * Terrence Tao: It is a beautiful observation of Thurston that the concept of a conformal mapping has a discrete counterpart, namely the mapping of one circle packing to another. * Conformal geometry preserves angles. Radial coordinates distinguishes distances r and angles theta, and makes use of the exponential {$e^{\pi i \theta}$}. Conics * Consider geometrically how to use the conics, the point at infinity, etc. to imagine Nothing, Something, Anything, Everything as stages in going beyond oneself into oneself. * Investigate: What happens to the shape of a circle when we move the tip of the cone? Suppose the circle is a shaded area. In what sense is the parabola a circle which touches infinity? In what sense is the directrix a focus? Does the parabola extend to the other side, reaching up to the directrix? Is a hyperbola an inverted ellipse, with the shading on either side of the curves, and the middle between them unshaded? What is happening to the perspective in all of these cases? * Think of the two foci of a conic as the source (start of all) and the sink (end of all). When are they the same point? (in the case of a circle?) Coordinate system * Coordinate system is that which has an origin, a zero. * How are coordinate rings in algebraic geometry related to coordinate systems? * Coordinate systems are observers and they shouldn't affect what they observe. (Relate this to the kinds of polytopes.) Counting * The derivative of an infinite power sequence shows that it is related to counting because we get coefficients 1, 2, 3, 4 etc. for the generating sequence. * Kuom skaičius skiriasi nuo pasikartojančios veiklos  būgno mušimo? * A) veikla kažkada prasidėjo * B) kiekvienas skaičius laikomas nauju, skirtingu nuo visų kitų Coxeter groups * Understand the classification of Coxeter groups. * Organize for myself the Coxeter groups based on how they are built from reflections. Crossratio * Relate the levels of the foursome with the crossratio (for example, howwhat are the two points within a circle, and whywhether are the two points outside.) And likewise relate the six pairs of four levels with geometric concepts, the six lines that relate the four points of an inscribed quadrilateral, or the six possible values of the crossratio upon permuting its elements. Curvature * Does the inside of a sphere have negative curvature, and the outside of sphere have positive curvature? And likewise the inside and outside of a torus? * Why is negative curvature  the curvature inside  more prominent than positive curvature  the outside of a space? Dimension * Notions of dimension d (Mathematical Companion): * locally looks like ddimensional space * the barrier between any two points is never more than d1 dimensional * can be covered with sets such that no more than d+1 of them ever overlap * the largest d such that there is a nontrivial map from a ddimensional manifold to a substructure of the space * the sum of dth powers of the diameter of squares that cover the object, with d such that the sum is between zero and infinity Division algebras * Basic division rings: [[http://math.ucr.edu/home/baez/week59.html  John Baez 59]] * The real numbers are not of characteristic 2, * so the complex numbers don't equal their own conjugates, * so the quaternions aren't commutative, * so the octonions aren't associative, * so the hexadecanions aren't a division algebra. * [[https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(composition_algebras)  Hurwitz's theorem]] for composition algebras * Analyze number types in terms of fractions of differences, https://en.wikipedia.org/wiki/M%C3%B6bius_transformation , in terms of something like that try to understand adbc, the different kinds of numbers, the quantities that come up in universal hyperbolic geometry, etc. * Note how complex numbers express rotations in R2. How are quaternions related to rotations in R3? What about R4? And the real numbers? And in what sense do the complex numbers and quaternions do the same as the reals but more richly? * Quaternions include 3 dimensijos formos rotating (twistor) and 1 for scaling (time) and likewise for octonions etc * John Baez periodic table and stablization theorem  relate to Cayley Dickson construction and its dualities. Division of everything * Perspective:Division = real form:complex Lie algebra/group = real Clifford algebra / complex Clifford algebra * Function relates many dimensions (the perspectives) to one dimension (the whole), just like a division. The whole is given by the operation +1. And what do +2 and +3 mean? * For complex numbers, {$0 \neq 2\pi$} and so they are different when we go around the threecycle, so they yield the foursome: 0, 120, 240, 360. * What is the difference between an exact and a nonexact relationship? * Short exact sequence. Defining a perspective relative to a base. * The concept of scope: Kernel: irrelevant because goes to zero. Cokernel: irrelevant because outside of scope. * Pascal's triangle  the zeros on either end of each row are like Everything at start and finish of an exact sequence. * Circle (threecycle) vs. Line (link to unconditional)  sixsome  and real forms * Spin 1/2 means there are two states separated by a quanta of energy +/ h. So this is like divisions of everything: ** Spin 0 total spin: onesome ** Spin 1/2: fermions: twosome ** Spin 1: three states: threesome ** Spin 3/2: composite particles: foursome ** Spin 2: gravition: fivesome (time/space) * Turing machines  inner states are "states of mind" according to Turing. How do they relate to divisions of everything? * Study the Wolfram Axiom and Nand. * Euler's manipulations of infinite series (adding up to 1/12 etc.) are related to divisions of everything, of the whole. Consider the Riemann Zeta function as describing the whole. The same mysteries of infinity are involved in renormalization, take a look at that. Duality * Duality of nonrelevance in system (kernel) and nonrelevance beyond system (cokernel). * Compare the kinds of dualities with the kind of transformations of perspectives and then with the kinds of geometric transformations. * In the automata hierarchy, consider how to model duality of internal structure and external network. * Duality of vectors and covectors (linear functionals)  flipsides of composition  what is "inside" and "outside" the composer. * Conjugates can be thought of as "twins", whereas +1 and 1 are "spouses". * Conjugate = mystery = false. (Hidden distinction). * i>j is asymmetric, onedirectional. i<>i* is symmetric, twodirectional, breaks antisymmetry, hides antisymmetry (which is i and which is j?) * Algebra  geometry duality. (Pullback). Morphism <> ring homomorphism. Intrinsic and extrinsic geometry. Ambient space. Relation between two spaces. Varieties [morphisms X to Y] <> Affine Falgebras [Flinear ring homomorphisms F[Y] to F[x]] * Study the idea behind linear functionals, fundamental representations, eigenvectors, cohomology, and other maps into one dimension. * Duality examples (conjugates) ** complex number "i" is not one number  it is a pair of numbers that are the square roots of 1 ** spinors likewise ** Dn where n=2 ** the smallest crosspolytope with 2 vertices ** taking a sphere and identifying antipodal elements  this is a famous group ** polar conjugates in projective geometry (see Wildberger) ** Study how Set breaks duality (the significance of initial and terminal objects). * Each physical force is related to a duality: ** Charge (matter and antimatter)  electromagnetism ** Weak force  time reversal * So the types of duality should give the types of forces. * John Baez on duality: Dyson's Threefold Way: either X is not isomorphic to its dual (the complex case), or it is isomorphic to its dual (in the real or quaternionic cases). Study his slides! * [[https://www.youtube.com/watch?v=7d5jhPmVQ1w  John Baez on duality in logic and physics]] * Algebra (of the observer) and analysis (of the observed) exhibit a duality of worldviews. * The interpretation {$\mathbb{C} \leftrightarrow \mathbb{R}^2$} gives meaning to the two axes. One axis the opposites 1 and 1 and the other axis is the opposites i and j. And they become related 1 to i to 1 to j. Thus multiplication by i is rotation by 90 degrees. It returns us not to 1 but sends us to 1. * Composition algebra. Doubling is related to duality. * Linear functionals. Onedimensional economic thinking is like linear functionals  the dual space of the multidimensional reality. In finite dimensions, the dual dual is the same as what we started. But in infinite dimensions not necessarily. Does this suggest that our life is infinitedimensional, which is to say, it can't be captured by a onedimensional shadow? Eightsome * 0=1 yields the zero ring. In what sense is this the collapse of a system? And how is it related to the finite field {$F_1$} ? Equivalence * From dream: vectors AB, BA, consider the difference between them, the equivalence of A and B. * Equations are questions * Isomorphism is based on assignment but that depends on equality up to identity whereas properties define establish an object up to isomorphism * Symbols (a and b) are equal if they refer to the same referents. But equality has different meaning for symbols, indexes, icons and things. Consider the four relations between level and metalevel. Elliptic integrals * Note in the Wikipedia article on [[https://en.wikipedia.org/wiki/Particular_values_of_the_gamma_function  Particular values of the gamma function]] the connection between {$\Gamma(\frac{n}{24})$} and [[https://en.wikipedia.org/wiki/Elliptic_integral#Complete_elliptic_integral_of_the_first_kind  Complete elliptic integrals of the first kind]] Entropy * Definition of entropy depends on how you choose it. Unit of phase space determines your unit of entropy. Thus observer defines phase space. Exact sequences * [[https://en.wikipedia.org/wiki/Zigzag_lemma  Zigzag lemma]] relates to infinite revolutions along the threecycle. * In an exact sequence, is a perspective the group or the homomorphism? It is the group  it is a division of zero  where zero is everything. * Let p: E → B be a basepointpreserving Serre fibration with fiber F, that is, a map possessing the homotopy lifting property with respect to CW complexes. Suppose that B is pathconnected. Then there is a long exact sequence of homotopy groups * Primena trejybę. [[https://en.wikipedia.org/wiki/Homotopy_group  Wikipedia: Homotopy groups]] Let p: E → B be a basepointpreserving Serre fibration with fiber F, that is, a map possessing the homotopy lifting property with respect to CW complexes. Suppose that B is pathconnected. Then there is a long exact sequence of homotopy groups: {$\cdots \rightarrow \pi_n(F) \rightarrow \pi_n(E) \rightarrow \pi_n(B) \rightarrow \pi_{n1}(F) \rightarrow \cdots \rightarrow \pi_0(F) \rightarrow 0. $} Exponentiation * Integral of 1/x is ln x what does that say about ex? * {$2\pi$} additive factors, e multiplicative factors. * Math Companion section on Exponentiation. The natural way to think of {$e^x$} is the limit of {$(1 + \frac{x}{n})^n$}. When x is complex, {$\pi i$}, then multiplication of y has the effect of a linear combination of n tiny rotations of {$\frac{\pi}{n}$}, so it is linear around a circle, bringing us to 1 or all the way around to 1. When x is real, then multiplication of y has the effect of nonlinearly resizing it by the limit, and the resulting e is halfway to infinity. Finite field * Compare finite field behavior (division winding around) with complex number behavior (winding around). * Sieja nepažymėtą priešingybę (+) 1, 0, 1 ir pažymėtą priešingybę (x) 1, 1. * [[https://en.m.wikipedia.org/wiki/Field_with_one_element  W: Field with one element]] * The nonexistent element of {$F_1$} may be considered to not exist, or imagined to exist, but regardless, I expect that cognitively there are three ways to interpret it as 0, 1, ∞, which thereby expand upon the duality between existence and nonexistence and make it structurally richer. * Relate [[https://en.wikipedia.org/wiki/M%C3%B6bius_transformation#Subgroups_of_the_M%C3%B6bius_group  elliptic transforms]] to God's dance {0, 1, ∞} and {$F_1$}: There are 3 representatives fixing {0, 1, ∞}, which are the three transpositions in the symmetry group of these 3 points: {$1 / z$}, which fixes 1 and swaps 0 with ∞ (rotation by 180° about the points 1 and −1), {$1 − z$} which fixes ∞ and swaps 0 with 1 (rotation by 180° about the points 1/2 and ∞), and {$z / ( z − 1 )$} which fixes 0 and swaps 1 with ∞ (rotation by 180° about the points 0 and 2). Note that this relates pairs from: 1, z, z1. * Study how turning the counting around relates to cycles  finite fields. Folding * Origami: [[http://alum.mit.edu/www/tchow/multifolds.pdf  The power of multifolds: Folding the algebraic closure of the rational numbers]] Four geometries * Projective map  preserves a single line (what does that mean?) * Conformal map  preserves pairs of lines (the angle between them) * Symplectomorphism  preserves triplets of lines (their oriented volume), the noncollinearity of a triangle Four coordinate systems * In physics, the constraints oncedifferentiable and twicedifferentiable are interesting and not trivial, whereas thricedifferentiable is basically the same as infinitely differentiable. Compare this with perspective, perspective on perspective, and perspective on perspective on perspective. * Geometry: Options for introducing a coordinate system (none, one, two, three). No coordinate system is the case of tensors. * 1 coordinate system = 1 side of a triangle = Length. Shrinking the side can lead to a point  the two points become equal. This is like homotopy? * 2 coordinate systems = 2 sides of a triangle = Angle. Note that turning (rotating) one side around by 2 pi gets it back to where it was, and this is true for each 2 pi forwards and backwards. So by this equivalence we generate the integers Z as the winding numbers. * 3 coordinate systems = 3 sides of a triangle = Oriented area (the systems are ordered). What equivalence does this support and what does it yield? Is it related to e? All you can do * with 0 coordinate system (affine) * with 1 coordinate system (projective) is reflection, * with 2 coordinates (conformal) is rotation and shear, (the origins match) * with 3 coordinates (symplectic) is dilation (scales change), squeeze (scales change), translation (origins move). The coordinate systems (0,1,2,3) separate the level and metalevel. Study the 6 transformations between these sets of coordinate systems. * How do the 4 geometries (in terms of coordinate systems) relate to the 4 classical root systems? * Substantiate: Affine geometry defines no perspective, projective geometry defines one perspective, conformal geometry defines a perspective on a perspective, symplectic geometry defines a perspective on a perspective on a perspective. * Relate motion to bundles. Symplectic geometry, looseness, etc. All 4 geometries. * Think of harmonic pencil types as the basis for the root systems ** {$A_n$} {$\pm(xy)$} dual ** {$B_n$} {$\pm x,\pm y$} yields {$x\pm y$} ** {$C_n$} {$x\pm y$} yields {$\pm2x,\pm2y$} ** {$D_n$} {$\pm(x\pm y)$} dual dual * In what sense do these ground four geometries? And how do 6 pairs relate to ways of figuring things out in math? * Determinant expresses oriented volume, oriented area. Real numbers: distances. Complex numbers: angles. What do quaternions express? * [[https://www.youtube.com/watch?v=jdD5CTZhjoM  V: Vladimir I. Arnold. Polymathematics: complexification, symplectification and all that. 1988]] 18:50 About his trinity, his idea: "This idea, how to apply it, and the examples that I shall discuss even, are not formalized. The theory that I will describe today is not a conjecture, not a theorem, not a definition, it is some kind of religion. I shall show you examples and in these examples, it works. So I was able, using this religion, to find correct guesses, and to find correct conjectures. And then I was able to work years or months trying to prove them. And in some cases, I was able to prove them. In other cases, other people were finally able to prove them. In other cases other people were able to prove them. But to guess these conjectures without this religion would, I think, be impossible. So what I would like to explain to you is just this nonformalized part of it. I am perhaps too old to formalize it but maybe someone who one day finds the axioms and makes a definition from the general construction from the examples that I shall describe." 39:00 Came up with the idea in 1970, while working on the 16th Hilbert problem. * Arnold: Six geometries (based on Cartan's study of infinite dimensional Lie groups?) his list? * Affine and projective geometries. Adding or subtracting a perspective. Such as adding or deleting a node to a Dynkin diagram. (The chain of perspectives.) * Perhaps the projective geometry is the most basic, and it is based on rotation and the complexes. But perhaps an affine geometry arises, with the reals, so that we can imagine a movement of the origin (the fixed point) and thus we can have translations, a shift in origin, a shift in perspective, a relative perspective, in a conditional world. * A_n points and sets * B_n inside: perpendicular (angles) and * C_n outside: line and surface area * D_n points and position * Affine geometry preserves lines. Projective geometry also preserves zeros. Conformal geometry preserves angles. Symplectic geometry preservers oriented area. What are all these objects? Fourier series * Compare a Fourier series and a generating function. How is a single basis element isolated, calculated, in each case? Geometry * Try to understand asymmetric functions, for example, by setting {$q_3^2=0$}. * Space arises with bundles, which separate the homogeneous choice, relevant locally, from the position, given globally. And then what does this say about time? * A geometry (like hyperbolic geometry) allows for a presentation of a bundle, thus a perspective on a perspective (atsitokėjimas  atvaizdas). Compare with: įsijautimasaplinkybė. * Geometry is concrete, it is a manifestation. Thus group actions can represent an abstract structure (of actions) in terms of a geometrical space (set, vector space, topological space). And this makes for understanding of the abstract in terms of the concrete. * Euclidean space  (algebraic) coordinate systems  define left, right, front, backwards  and this often makes sense locally  but this does not make sense globally on a sphere, for example Going beyond oneself * Try to use universal hyperbolic geometry to model going beyond oneself into oneself (where the self is the circle). * Relate God's dance to {0, 1, ∞} and the [[https://en.wikipedia.org/wiki/Crossratio  anharmonic group]] and Mobius transformations. Note that the anharmonic group is based on composition of functions. Homology * Interpret the boundary of a simplicial complex. Explain how the boundary of a boundary of a simplicial complex is zero. How does the boundary express orientation? * Homology and cohomology are like the relation between 0>1>2>3 and 4>5>6>7. * Homology  holes  what is not there  thus a topic for explicit vs. implicit math * [[http://www.math.wayne.edu/~isaksen/Expository/carrying.pdf  A Cohomological Viewpoint on Elementary School Arithmetic]] About "carrying". Access restricted. * [[https://www.youtube.com/watch?v=LmyyybUlnLU  V: Ben Mares. Introduction to cohomology.]] * Odd cohomology works like fermions, even cohomology works like bosons. * Homologija bandyti išsakyti persitvarkymų tarpą tarp pirminės ir antrinės tvarkos. * Relate walks on trees with fundamental group. Homotopy type theory * [[https://www.youtube.com/watch?v=bFZWarP2Ef4  Galois, Grothendieck and Voevodsky  George Shabat]] Inner product * Hermitian: a+bi <> abi, Symmetric: a <> a, Antisymmetric: b <> b * Consider how the "inner product", including for the symplectic form, yields geometry. * The skewsymmetric bilinear form says that phi(x,y)= phi(y,x) so one of them, say, (x,y) is + (correct) and the other is  (incorrect). This is leftright duality based on nonequality (the two must be different). And it is threefold logic, the nonexcluded middle 0. SO that 1 true, 1 false, 0 middle. * The inner products (like unitary) are linear. They and their quadratic product get projected onto a second quadratic space (a screen) which expresses their geometric nature as a tangent space for a differentiable manifold thus relating the discrete and the continuum, the infinitesimal and the global (like Mobius transformations). * Three inner products  relate to chronogeometry. * Three inner products are products of vectors that are external sums of basis vectors as different units. Six interpretations of scalar multiplication that are internal sums of amounts. Irrationality * Challenge problems: * Determine whether {$\pi + e$} is rational or irrational. * Determine whether {$\pi^e$} is rational or irrational. Learning * (Conscious) Learning from (unconscious) machine learning. Learning math * [[https://mathoverflow.net/questions/31879/arethereothernicemathbooksclosetothestyleoftristanneedham  Are there other nice math books close to the style of Tristan Needham?]] * [[https://www.mathpages.com/home/  Kevin Brown]] collection of expositions of math Lie theory * When do we have all three: orthogonal, unitary, symplectic forms? And in what sense do they relate the preservation of lines, angles, triangular areas? And how do they diverge? * Signal propagation  expansions. Consider the connection between walks on trees and Dynkin diagrams, where the latter typically have a distinguished node (the root of the tree) from which we can imagine the tree being "perceived". There can also be double or triple perspectives. * How do special rim hook tableaux (which depend on the behavior of their endpoints) relate to Dynkin diagrams (which also depend on their endpoints)? * Relate the ways of breaking the duality of counting with the ways of fusing together the sides of a square to get a manifold. Lie groups * Attach:ClassicalLieGroups.png * Homology calculations involve systems of equations. They propagate equalities. Compare this with the determinant of the Cartan matrix of the classical Lie groups, how that [[https://math.stackexchange.com/questions/2109581/intuitivelywhyarethere4classicalliegroupsalgebras  propagates equalities]]. The latter are chains of quadratic equations. * An grows at both ends, either grows independently, no center, no perspective, affine. Bn, Cn have both ends grow dependently, pairwise, so it is half the freedom. In what sense does Dn grow, does it double the possibilities for growth? And does Dn do that internally by relating xi+xj and xixj variously somehow? * SL(2)  H,X,Y is 3dimensional (?) but SU(2)  threecycle + 1 (God) is 4 dimensional. Is there a discrepancy, and why? * 4 kinds of i: the Pauli matrices and i itself. * [[https://johncarlosbaez.wordpress.com/2020/03/20/fromtheoctahedrontoe8/  John Baez. From the Octahedron to E8]] * Relate harmonic ranges and harmonic pencils to Lie algebras and to the restatement of {$x_ix_j$} in terms of {$x_i$}. * In Lie groups, real parameter subgroups (copies of R) are important because they define arcwise connectedness, that we can move from one point (group element) to another continuously. See how this relates to the slack defined by the root system. * Lie algebra matrix representations code for: ** Sequences  simple roots ** Trees  positive roots ** Networks  all roots * The nature of Lie groups is given by their forms: Orthogonal have a symmetric form {$\phi(x,y)=\phi(y,x)$}, Unitary have a Hermitian form {$\phi(x,y)=\overbar{\phi(y,x)}$}, and Symplectic have an antisymmetric form {$\phi(x,y)=\phi(y,x)$}. Note that symmetric forms do not distinguish between {$(x,y)$} and {$(y,x)$}; Hermitian forms ascribe them to two different conjugates (which may be the same if there is no imaginary component); and antisymmetric forms ascribe one to {$+1$} and the other to {$1$}, as if distinguishing "correct" and "incorrect". * Understand how forms relate to geometry, how they ground paths, distances, angles, and oriented areas. * Understand how forms for Lie groups relate to Lie algebras. * Understand how forms relate to real, complex and quaternionic numbers. * Noncommutative polynomial invariants of unitary group. * Francois Dumas. An introduction to noncommutative polynomial invariants. http://math.univbpclermont.fr/~fdumas/fichiers/CIMPA.pdf Symmetric polynomials, Actions of SL2. * Vesselin Drensky, Elitza Hristova. Noncommutative invariant theory of symplectic and orthogonal groups. https://arxiv.org/abs/1902.04164 * How to relate Lie algebras and groups by way of the Taylor series of the logarithm? * Consider how A_2 is variously interpreted as a unitary, orthogonal and symplectic structure. * Look at effect of Lie group's subgroup on a vector. (Shear? Dilation?) and relate to the 6 transformations. * The complex Lie algebra divine threesome H, X, Y is an abstraction. The real Lie algebra human cyclical threesome is an outcome of the representation in terms of numbers and matrices, the expression of duality in terms of 1, i, and position. * Jacob Lurie, Bachelor's thesis, [[http://www.math.harvard.edu/~lurie/papers/thesis.pdf  On Simply Laced Lie Algebras and Their Minuscule Representations]] * Does the Lie algebra bracket express slack? * Every root can be a simple root. The angle between them can be made {$60^{\circ}$} by switching sign. {$\Delta_i  \Delta_j$} is {$30^{\circ}$} {$\Delta_i  \Delta_j$} * {$e^{\sum k_i \Delta_i}$} * Has inner product iff {$AA^?=I$}, {$A{1}=A^?$} * Killing form. What is it for exceptional Lie groups? * The Cartan matrix expresses the amount of slack in the world. * {$A_n$} God. {$B_n$}, {$C_n$}, {$D_n$} human. {$E_n$} n=8,7,6,5,4,3 divisions of everything. * 2 independent roots, independent dimensions, yield a "square root" (?) * Symplectic matrix (quaternions) describe local pairs (Position, momentum). Real matrix describes global pairs: Odd and even? * Root system is a navigation system. It shows that we can navigate the space in a logical manner in each direction. There can't be two points in the same direction. Therefore a cube is not acceptable. The determinant works to maintain the navigational system. If a cube is inherently impossible, then there can't be a trifold branching. * {$A_n$} is based on differences {$x_ix_j$}. They are a higher grid risen above the lower grid {$x_i$}. Whereas the others are aren't based on differences and collapse into the lower grid. How to understand this? How does it relate to duality and the way it is expressed. * In {A_1}, the root {$x_2x_1$} is normal to {$x_1+x_2$}. In {A_2}, the roots are normal to {$x_1+x_2+x_3$}. * If two roots are separated by more than {$90^\circ$}, then adding them together yields a new root. * {$cos\theta = \frac{a\circ b}{\left \ a \right \\left \ b \right \}$} * {$120^\circ$} yields {$\frac{1}{\sqrt{2}\sqrt{2}}=\frac{1}{2}$} * Given a chain of composition {$\cdots f_{i1}\circ f_i \circ f_{i+1} \cdots$} there is a duality as regards reading it forwards or backwards, stepping in or climbing out. There is the possiblity of switching adjacent functions at each dot. So each dot corresponds to a node in the Dynkin diagram. And the duality is affected by what happens at an extreme. * Root systems give the ways of composing perspectivesdimensions. * {$A_n$} root system grows like 2,6,12,20, so the positive roots grow like 1,3,6,10 which is {$\frac{n(n+1)}{2}$}. * A root pair {$x$} and {$x$} yields directions such as "up" {$x––x$} and "down" {$–x–+x$}. * Threecycle: same + different => different ; different + different => same ; different + same => different * {$ \begin{pmatrix}1 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix}0 & i \\ i & 0 \end{pmatrix} \begin{pmatrix}1 & 0 \\ 0 & 1 \end{pmatrix} $} * same + different + different + same + ... * Go from rather arbitrary set of dimensions to more natural set of dimensions. Natural because they are convenient. This leads to symmetry. Thus represent in terms of symmetry group, namely Lie groups. There are dimensions. In order to write them up, we want more efficient representations. Subgroups give us understanding of causes. Smaller representations give us understanding of effects. We want to study what we don't understand. In engineering, we leverage what we don't understand. * Differences between even and odd for orthogonal matrices as to whether they can be paired (into complex variables) or not. * {$e^{\sum i \times generator \times parameter}$} has an inverse. * Unitary T = {$e^{iX_j\alpha_j}$} where {$X_j$} are generators and {$\alpha_j$} are angles. Volume preserving, thus preserving norms. Length is one. * Complex models continuous motion. Symplectic  slack in continuous motion. * Edward Frenkel. Langlands program, vertex algebras related to simple Lie groups, detailed analysis of SU(2) and U(1) gauge theories. https://arxiv.org/abs/1805.00203 * Consider the classification of Lie groups in terms of the objects for which they are symmetries. * {$A^TA$} is similar to the adjoint functors  they may be inverses (in the case of a unitary matrix) or they may be similar. * In the context {$e^iX$} the positive or negative sign of {$X$} becomes irrelevant. In that sense, we can say {$i^2=1$}. In other words, {$i$} and {$1$} become conjugates. Similarly, {$XYYX = \overline{(YXXY)}$}. * What is the relation between the the chain of Weyl group reflections, paths in the root system, the Dynkin diagram chain, and the Lie group chain. * Onedimensional proteins are wound up like the chain of a multidimensional Lie group. * Išsakyti grupės {$G_2$} santykį su jos atvirkštine. Ar ši grupė tausoja kokią nors normą? * The analysis in a Lie group is all expressed by the behavior of the epsilon. * A_n defines a linear algebra and other root systems add additional structure * A circle, as an abelian Lie group, is a "zero", which is a link in a Dynkin diagram, linking two simple roots, two dimensions. * [[https://www.youtube.com/watch?v=wIn_dlmD8sk  Video: The rotation group and all that]] * [[https://www.youtube.com/watch?v=8KPzuPizKk&list=PLZcI2rZdDGQrb4VjOoMm2o7Fu_mvij8F  Lorenzo Sadun. Videos: Linear Algebra]] Nr.88 is SO(3) and so(3) * {$U(n)$} is a real form of {$GL(n,\mathbb{C})$}. [[https://www.encyclopediaofmath.org/index.php/Complexification_of_a_Lie_group  Encyclopedia: Complexification of a Lie group]] * [[https://www.youtube.com/watch?v=zSLsjrJKPA  DrPhysicsA. Particle Physics 4: Rotation Operators, SU(3)xSU(2)xU(1)]] * At the level of forms, this can be seen by decomposing a Hermitian form into its real and imaginary parts: the real part is symmetric (orthogonal), and the imaginary part is skewsymmetric (symplectic)—and these are related by the complex structure (which is the compatibility). * Particle physics is based on SU(3)xSU(2)xU(1). Can U(1) be understood as SU(1)xSU(0)? U(1) = SU(1) x R where R gives the length. So this suggests SU(0) = R. In what sense does that make sense? * The conjugate i is evidently the part that adds a perspective. Then R is no perspective. In what sense is SU(3) related to a rotation in octonion space? * If SU(0) is R, then the real line is zero, and we have projective geometry for the simplexes. So the geometry is determined by the definition of M(0). * {$SL(n)$} is not compact, which means that it goes off to infinity. It is like the totality. We have to restrict it, which yields {$A_n$}. Whereas the other Lie families are already restricted. * The root systems are ways of linking perspectives. They may represent the operations. {$A_n$} is +0, and the others are +1, +2, +3. There can only be one operation at a time. And the exceptional root systems operate on these four operations. * Real forms  Satake diagrams  are like being stepped into a perspective (from some perspective within a chain). An odddimensional real orthogonal case is steppedin and evendimensional is stepped out. Complex case combines the two, and quaternion case combines them yet again. For consciousness. * Note that given a chain of perspectives, the possibilities for branching are highly limited, as they are with Dynkin diagrams. * Solvable Lie algebras are like degenerate matrices, they are poorly behaved. If we eliminate them, then the remaining semisimple Lie algebras are beautifully behaved. In this sense, abelian Lie algebras are poorly behaved. * Study how orthogonal and symplectic matrices are subsets of special linear matrices. In what sense are R and H subsets of C? * Find the proof and understand it: "An important property of connected Lie groups is that every open set of the identity generates the entire Lie group. Thus all there is to know about a connected Lie group is encoded near the identity." (Ruben Arenas) * Differentiating {$AA^{1}=I$} at {$A(0)=I$} we get {$A(A^{1})'+A'A^{1}=0$} and so {$A'=(A^{1})'$} for any element {$A'$} of a Lie algebra. * For any {$A$} and {$B$} in Lie algebra {$\mathfrak{g}$}, {$exp(A+B) = exp(A) + exp(B)$} if and only if {$[A,B]=0$}. * Symmetry of axes  Bn, Cn  leads, in the case of symmetry, to the equivalence of the total symmetry with the individual symmetries, so that for Dn we must divide by two the hyperoctahedral group. * the unitary matrix in the polar decomposition of an invertible matrix is uniquely defined. * In {$D_n$}, think of {$x_ix_j$} and {$x_i+x_j$} as complex conjugates. * In Lie root systems, reflections yield a geometry. They also yield an algebra of what addition of root is allowable. * Special linear group has determinant 1. In general when the determinant is +/ 1 then by Cramer's rule this means that the inverse is an integer and so can have a combinatorial interpretation as such. It means that we can have combinatorial symmetry between a matrix and its inverse  neither is distinguished. * Determinant 1 iff trace is 0. And trace is 0 makes for the links +1 with 1. It grows by adding such rows. * {$A_n$} tracefree condition is similar to working with independent variables in the the center of mass frame of a multiparticle system. (Sunil, Mukunda). In other words, the system has a center! And every subsystem has a center. * Symplectic algebras are always even dimensional whereas orthogonal algebras can be odd or even. What do odd dimensional orthogonal algebras mean? How are we to understand them? * An relates to "center of mass". How does this relate to the asymmetry of whole and center? * Išėjimas už savęs reiškiasi kaip susilankstymas, išsivertimas, užtat tėra keturi skaičiai: +0, +1, +2, +3. Šie pirmieji skaičiai yra išskirtiniai. Toliau gaunasi (didėjančio ir mažėjančio laisvumo palaikomas) bendras skaičiavimas, yra dešimts tūkstantys daiktų, kaip sako Dao De Jing. Trečias yra begalybė. * Kaip sekos lankstymą susieti su baltymų lankstymu ir pasukimu? * Fizikoje, posūkis yra viskas. Palyginti su ortogonaline grupe. * {$x_0$} is fundamentally different from {$x_i$}. The former appears in the positive form {$\pm(x_0x_1)$} but the others appear both positive and negative. * Kaip dvi skaičiavimo kryptis (conjugate) sujungti apsisukimu? * How to interpret possible expansions? For example, composition of function has a distinctive direction. Whereas a commutative product, or a set, does not. * Dots in Dynkin diagrams are figures in the geometry, and edges are invariant relationships. A point can lie on a line, a line can lie on a plane. Those relationships are invariant under the actions of the symmetries. Dots in a Dynkin diagram correspond to maximal parabolic subgroups. They are the stablizer groups of these types of figures. * G2 requires three lines to get between any two points (?) Relate this to the threecycle. Limits * Analytic limits need neighborhoods. Categorical limits need maps. Third kind of limit: Ultraproducts. Terrence Tao. Linear algebra * In solving for eigenvalues {$\lambda_i$} and eigenvectors {$v_i$} of {$M$}, make the matrix {$M\lambda I$} degenerate. Thus {$\text{det}(M\lambda I)=0$}. The matrix is degenerate when one row is a linear combination of the other rows. So the determinant is a geometrical expression for volume, for collinearity and noncollinearity. * An inner product on a vector space allows it to be broken up into vector spaces that complement each other, thus into irreducible vector spaces. * Matrices {$A=PBP^{1}$} and {$B=P^{1}AP$} have the same eigenvalues. They are simply written in terms of different coordinate systems. If v is an eigenvector of A with eigenvalue λ, then {$P^{1}v$} is an eigenvector of B with the same eigenvalue λ. * Eigenvectors are the pure dimensions into which the action of a matrix (or linear transformation) can be decomposed. Logic * Jordan algebra projections are "propositions". This makes sense with regard to the sevensome (logic) and the eightsome (the semiotic square). * Logikos prielaidos bene slypi daugybė matematikos sąvokų. Pavyzdžiui, begalybė slypi galimybėje tą patį logikos veiksnį taikyti neribotą skaičių kartų. O tai slypi tame kad logika veiksnio taikymas nekeičia loginės aplinkos. * Ar gali būti, kad aibių teorijos problemos (pavyzdžiui, sau nepriklausanti aibė) yra iš tikrųjų logikos problemos? Kaip atskirti aibių teoriją ir logiką (pavyzdžiui, axiom of choice)? * [[https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence  CurryHowardLambek correspondence]] of logic, programming and category theory * Algebra and geometry are linked by logic  intersections and unions make sense in both. * Nand gates and Nor gates (and And gates and Or gates) relate one and all. The Nand and Nor mark this relation with a negation sign. * Are Nand gates (Nor gates) related to perspectives? * Study how all logical relations derive from composition of Nand gates. * How is a Nor gate made from [[Nand]] gates? (And vice versa.) Mandelbrot set * How special is the Mandelbrot set? What other comparable fractals are there? Can the Mandelbrot set be understood to encompass all of mathematics? What is a combinatorial interpretation of the Mandelbrot set? How is the Mandelbrot set related to the complex numbers and numbers (normed vector spaces) more broadly? Map of Math * Basic concept: Orientation (of a simplex). Relates to determinant, homology, etc. * Vector spaces are basic. What is basic about scalars? They make possible proportionality. * Basic concepts are the ways of figuring things out. * Basic concept  orientation = parity. * [[http://msc2010.org/mediawiki/index.php?title=MSC2010  Mathematics Subject Classification wiki]] Math * Much of my work may be considered premathematics, the grounding of the structures that ultimately allow for mathematics. * Mathematics is the study of structure. It is the study of systems, what it means to live in them, and where and how and why they fail or not. * When you get the definitions right, the theorems are easy to prove. When the theorems are hard to prove, then the definitions are not right. (Tobias Osborne) So this shows how definitions and theorems coevolve. Math Discovery  House of Knowledge * Consider how the four levels of geometrylogic bring together the four levels of analysis and the four levels of algebra, yielding the 12 topologies. And why don't the two representations of the foursome yield a third representation? * Extension of a domain  [[https://en.wikipedia.org/wiki/Analytic_continuation  Analytic continuation]]  complex numbers  dealing with divergent series. * Root systems relate two spheres  they relate two "sheets". Logic likewise relates two sheets: a sheet and a metasheet for working on a problem. Similarly, we model our attention by awareness, as Graziano pointed out. This is stepping in and stepping out. * Mathematical mind/brain unites logical language (left brain  logical reasoning) with spatial visual thinking (right brain  geometric intuition). Consider what this means as two parallel ways of thinking for the 4 logicgeometry levels in the ways of discovery. * geometry (bundle) links algebra (fiber) with analysis (base) and the latter manifold is also understood (ambiguously) algebraically as a Lie group. Algebra is a (finite) cognitive pattern that restructures the (infinite) Analysis, the model of the world. Together they are a restructuring. In the house of knowledge for mathematics, the threecycle relates analysis and algebra as structuring and restructuring. Thus this restructuring (the six restructurings) is the output of the house of knowledge and the content of mathematics, its branches, concepts, statements, problems, etc. * Analysis is the infinity of sheets, the recurring sequence of not going beyond oneself. But algebra is the single sheet which is the self that it all goes into, where all of the actions, all of the sheets coincide as one sheet, one going beyond. Thus the cardinal (of algebra) arises from the ordinal (of analysis). * The ordinal (list) is deterministic whereas the cardinal (set) is nondeterministic. (polynomial coefficient) Ordinal/List/Analysis vs. Cardinal/Set/Algebra (polynomial root)  the coefficients and roots are related by the binomial theorem, the factors are choices. * Are there 6+4 branches of math? How are the branches of math related to the ways of figuring things out? * Proceed from balance  note how additive balance precedes multiplicative ratio precedes possibly negative (directional) ratio precedes anharmonic ratio precedes crossratio  and see how balance gets variously extended, as with the anharmonic group and the Moebius transformations. And relate that to the Balance as a way of figuring things out. * In the house of knowledge for mathematics, the threecycle establishes the substructures for the symmetry group. Similarly, in physics, it establishes the scales for isolating a system (?) * How does logic come from a quadratic form? Four ways of relating level and metalevel with "and". * Terrence Tao problem solving https://books.google.lt/books/about/Solving_Mathematical_Problems_A_Personal.html?id=ZBTJWhXD05MC&redir_esc=y * Physics is measurement. A single measurement is analysis. Algebra gives the relationships between disparate measurements. But why is the reverse as in the ways of figuring things out in mathematics? * Ways of discovery in math: [[http://www.tricki.org  Tricki.org]]. [[https://gowers.wordpress.com/2010/09/24/isthetrickidead/  Overview by Timothy Gowers]]. * [[https://www.laetusinpraesens.org/musings/periodt.php  Towards a Periodic Table of Ways of Knowing in the light of metaphors of mathematics]] Anthony Judge * Associative composition yields a list (and defines a list). Consider the identity morphism composed with itself. Math outreach * [[https://gowers.wordpress.com/  Timothy Gowers' Weblog]] * [[https://www.dpmms.cam.ac.uk/~wtg10/  Timothy Gowers' webpage]] Methods of proof * How is substitution, as a method of proof, related to lamba calculus, and construction? * Does induction prove an infinite number of statements or their reassembly into one statement with infinitely many realizations? It proves the parallelness of intuitive meaningful stepped in and formal stepped out. * Mathematical induction  is it possible to treat infinitely many equations as a single equation with infinitely many instantiations? Consider NavierStokes equations. * Relate methods of proof and discovery, 3 systemic and 3 not. * Induction step by step is different than the outcome, the totality, which forgets the gradation. * Mathematical induction  is infinitely many statements that are true  relate to natural transformation, which also relates possibly infinitely many statements. Mobius transformation * Given z1, z2, z3 in projective complex plane, there exists a unique Mobius transformation such that f(z1)=0, f(z2)=1, f(z3)=infinity. Note that there is a fourth symbol z, and they get paired: 0 and infinity, 1 and z. {$$f(z)=\frac{zz_1}{zz_3} \cdot \frac{z_2z_3}{z_2z_1} $$} * The Mobius transformation f(z) which sends f(0)=p, f(1)=r, f(infinity)=s is given by: {$$f(z)=\frac{zs(pr) + p(rs)}{z(pr) + (rs)} $$} * 1/z swaps inside and outside as conjugates, just as it swaps counterclockwise rotations i with counterclockwise rotations i. * Give a geometrical interpretation of e. * SL(2,C) models the transformation in the relationship between two wills: human's and God's. The complex variable describes the will. * The squeeze function defines area. * [[https://en.m.wikipedia.org/wiki/Möbius_transformation  W: Mobius transfomation]] important examples * SL(2,C) lines (plus infinity) become circles. Do linear equations become circular equations? What does that mean? Are SL(2,C) circular equations related to the continuum? * Intuit SL(2,C) as threedimensional in C (because adbc=1 so we lose one complex dimension  intuit that). And in what sense is that different from adbc=0 (a line? a onedimensional subspace?) * SL(2,C) is the spin relativistic group. * Investigate: In what sense do the properties of being an inverse (Cramer's rule) dictate the symmetry of SL(2,C)? Because the inverse has to maintain the same form. So why do the b, c switch sign and the a, d switch places? What imposition is made on duality? * [[https://en.wikipedia.org/wiki/Jinvariant  Jinvariant]] is related to SL(2,Z) and monstrous moonshine. * Why these three structures? How do they relate to the Moebius transformations? And how do these structures relate to the classical Lie families? * [[https://smile.amazon.com/FunctionsComplexVariableGraduateMathematics/dp/0387903283/ref=smi_www_rco2_go_smi_g3905707922?_encoding=UTF8&%2AVersion%2A=1&%2Aentries%2A=0&ie=UTF8  Functions of One Complex Variable]] John Conway * Understand the relations between U(1) and electromagnetism, SU(2) and the weak force, SU(3) and the strong force. * [[https://en.wikipedia.org/wiki/Standard_Model  Standard Model]] * [[https://en.wikipedia.org/wiki/Electroweak_interaction  Electroweak interaction]] * Mobius transformations can be composed from translations, dilations, inversions. But dilations (by complex numbers) could be understood as dilations (in positive reals), reflections, and rotations. * Reconsider what ShuHong's thesis has to say about fractions of differences, and how they relate to the Mobius group. Note that the [[https://en.wikipedia.org/wiki/M%C3%B6bius_transformation#Subgroups_of_the_M%C3%B6bius_group  Mobius transformations]] classify into types which accord with my six transformations: * reflection = circular * shear = parabolic * rotation = elliptic * dilation = hyperbolic * squeeze = internal of hyperbolic (e^t e^{t}=1) * translation = internal of parabolic Need to define "internal". Also, note that reflection is like rotation but more specific. Similarly, is translation like shear, but more specific? Analyze the Mobius group in terms of what it does to circles and lines, and analyze the transformations likewise. * Study [[https://en.wikipedia.org/wiki/SL2(R)  SL(2,R)]]. Study the [[https://en.wikipedia.org/wiki/M%C3%B6bius_transformation  Möbius group]]. Understand [[https://math.stackexchange.com/questions/646183/listofconnectedliesubgroupsofmathrmsl2mathbbc  the study of subgroups]]. * [[https://paulhus.math.grinnell.edu/SMPpaper.pdf  Jennifer Paulhus. Group Actions and Riemann Surfaces]] Multiplication * Polynomial powers are "twists" of a string. One end of the string is held up and then down. Each twist of the string allows for a new maximum or minimum. This is an interpretation of multiplication. Music * [[http://wwwpersonal.umd.umich.edu/~tmfiore/1/FioreWhatIsMathMusTheoryBasicSlides.pdf  What is Mathematical Music Theory?]] Neural networks * Very powerful and simple computational systems for which Sarunas Raudys showed a hierarchy of sophistication as learning systems. Number theory * [[http://pi.math.cornell.edu/~hatcher/TN/TNpage.html  Allen Hatcher. Topology of numbers]] * Prime numbers: "Cost function". The "cost" of a number may be thought of as the sum of all of its prime factors. What might this reveal about the primes? * [[http://oeis.org/A000607  Number of ways to partition a number into primes]]. * Prime numbers introduce (non)determinism in that when a prime divides a product, then it must divide one of the factors. * Primes are both atoms and gaps. And they elucidate the gap of all numbers with respect to infinity. * Riemann hypothesis: is the unfolding of the primes "pattern free"? Or are there one or more hidden patterns there? A zero of the Riemann zeta function indicates a pattern. So what is a pattern? And what are the limitations on patterns? * Terrence Tao: Each prime p wants to have weight ln p. Compare with [[https://en.wikipedia.org/wiki/Zipf%27s_law  Zipf's law]], [[https://en.wikipedia.org/wiki/Zipf%E2%80%93Mandelbrot_law  ZipfMandelbrot law]], [[https://en.wikipedia.org/wiki/Yule%E2%80%93Simon_distribution  YuleSimon distribution]], [[https://en.wikipedia.org/wiki/Preferential_attachment  Preferential attachment]], [[https://en.wikipedia.org/wiki/Matthew_effect  Matthew effect]], [[https://en.wikipedia.org/wiki/Pareto_principle  Pareto's principle]]. * If there is a zero in the Riemann function's zone, then there is a function that it can't mimic? Observables * [[http://math.ucr.edu/home/baez/week257.html  John Baez about observables]] (see Nr.15) and the paper [[http://arxiv.org/abs/0709.4364  A topos for algebraic quantum theory]] about C* algebras within a topos and outside of it. Octonions * The octonions can model the nonassociativity of perspectives. * Complex numbers describes rotations in twodimensions, and quaternions can be used to describe rotations in three dimensions. Is there a connection between octonions and rotations in four dimensions? * Compare trejybės ratas (for the quaternions) and Fano's plane (aštuonerybė) for the octonions. Opposites * Exercise: Find all matrices with eigenvalues 1 and 1. {$\lambda=\frac{a_{11}+a_{22} \pm \sqrt{(a_{11}+a_{22})^2  4A}}{2}$} so A=1. {$\begin{pmatrix} \pm \sqrt{1 + a_{12}a_{21}} & a_{12} \\ a_{21} & \mp \sqrt{1 + a_{12}a_{21}} \end{pmatrix}$} such as {$\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$}{$\begin{pmatrix} i & 0 \\ 0 & i \end{pmatrix}${$\begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}$} * Multiplying by {$\begin{pmatrix} x \\ y \end{pmatrix}$} yields three ways of coding opposites: {$\begin{pmatrix} iy \\ ix \end{pmatrix}$} {$\begin{pmatrix} y \\ x \end{pmatrix}$} {$\begin{pmatrix} ix \\ iy \end{pmatrix}$} where in each case two of three are applied: flipping, multiplying by i, multiplying by 1. Partial differential equations * NavierStokes equations: Reynolds number relates time symmetric (high Reynolds number) and time asymmetric (low Reynolds number) situations. Peano axioms * Equality holds for both value and type, amount and unit. Peano axiom. * Peano why can't have natural numbers have two subsets, a halfline from 0 and a full line. Perspective * Think of the nullsome as the point at infinity (the eye that sees the perspective). And think of it going beyond itself and then returning back to itself, making a big circle around. And think of that circle as a discrete line with four places. * Ker/Image  the kernel are those that can relate, that can take up the perspective * Relate to perspectives: Homology groups measure how far a chain complex is from being an exact sequence. * In studying perspective: How is homology used to prove the Brouwer fixed point theorem? * How is exactness (the image of f1 matching the kernel of f2) related to perspectives, for example, the notion of the complement? * The Axiom of Choice is based on the notions of a perspective (a bundle) in that we can assign to each set a choice. The set is fiber and the set of sets is the base. * How do observables relate to perspective? * Perspective relates intrinsic and extrinsic geometry by way of ambient space. Ambient space relates base and fiber (perspective) as a bundle. * {$K_0$} and {$K_1$} perhaps express perspectives, like God's trinity and the threecycle, the two foursomes in the eightcycle. * Normal bundles involve embedding in an extrinsic space. * The possibilities for a complex plane (extra point for sphere, point removed for cylinder) are relevant for modeling perspectives. Physics * Electrons are particles when we look at them and waves when we don't. * Quantum mechanics superposition of states may be based on borrowing of energy. When the energy debt is too high, as when it gets entangled with an observer, then the debt has to be paid. Otherwise, the debt can be restructured in complicated ways. * Lagrangian L=TV expresses slack (or antislack), makes the conversion between potential and kinetic energy as smooth as possible. Hamiltonian H=T+V expresses the totality, the love. In this sense, they are dual, as per the sevensome  the Lagrangian expresses the internal slack, and the Hamiltonian expresses the wholeness of the external frame. * Potential energy is bounded from below. * Kinetic energy is always positive, absolutely. Its relation to momentum is absolute, unconditional. Whereas potential is defined relatively and its relation to position is variable. * The wave equation is defined on phase space but in such a way that it is understood in terms of a superposition of waves for position, yielding a "blob"  a wave packet, and an analogous superposition of waves for momentum, and the two are related by the Fourier transform, by the Heisenberg uncertainty principle. * Physics abstracts from the personality of other researchers. Have a common ground, communicate not on any consensus, but based on what we find, as independent witnesses. * From dream: Space is made up of all possible curves. Physics is about the geodesics, the curves with no slack. They really are all one curve that goes through every curve and every single point. * Understanding of effect. Physics, why does it work? How can I describe it efficiently and correctly? * How many parameters do I need to describe the system? (Like an object.) Minimize constraints. It becomes complicated. Multipole is abstracting the levels of relevance. Ordering them inside the dimensions I am working with. What is the important quantity? Measures quality. How transformation leaves the object invariant. Distinguish between continuous parameters that we measure against and these quantity that we want to study. We use dimensions as a language to relate the inner structure and the outer framework. To measure momentum we need to measure two different quantities. * Weak nuclear force changes quark types. Strong nuclear force changes quark positions. Electromagnetic force distinguishes between quark properties  charge. * Exchange particles  gauge bosons. * Wave function Smolin says is ensemble, I say bosonic sharing of space and time * What is the relationship between spin (and alignment to a particular axis or coordinate system) and the alignment of magnets? * Which state is which amongst "one" and "another" is maintained until it is unnecessary  this is quantum entanglement. * Massless particles acquire mass through symmetry breaking: [[https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_theory  YangMills theory]]. * [[https://www.math.columbia.edu/~woit/wordpress/?p=5927  Geometric unity]] I tend to agree with Roger Penrose that spin has been one of the great mysteries in quantum mechanics. As best as I can recall, he said it was one of two primary mysteries in a talk at NYU back in the late 1990’s. ... understand spin and I think we'll understand entanglement a lot better. Projective geometry * Any projective space is of the form {$\mathbb{KP}^n$} for some skew field {$\mathbb{K}$} except when n=2. Then it holds if the Axiom of Desargues holds in the space, but there are also exceptional spaces. This fact may relate to the fact that semisimple Lie algebras are related to {$\mathbb{R}$}, {$\mathbb{C}$} or {$\mathbb{H}$}  the classical Lie algebras  with the exceptions being very limited. * If we have the real plane and think of it projectively, then there is a line at infinity, encircling infinity, where the points on that line have orientation (given by the orientation of intersecting lines). But if we think of the plane as the complex numbers, then perhaps instead of a line it is sufficient to consider a single complex number as the point of infinity, in that it comes with a sense of angle. * Why is projective geometry related to lines, sphere, projection, point at infinity? * Harmonic pencil: Look for what it would mean for a ratio to be {$i$} and the product to be 1. The answer is {$e^{\frac{\pi}{4}i}$} and {$e^{\frac{3\pi}{4}i}$}, or {$e^{\frac{\pi}{4}i}$} and {$e^{\frac{3\pi}{4}i}$} * {$\mathrm{Aut}_{H\circ f}(\mathbb{P}^1_\mathbb{C})\simeq \mathrm{PSL}(2,\mathbb{C})$} * {$\mathrm{Aut}_{H\circ f}(\mathbb{U})\simeq \mathrm{PSL}(2,\mathbb{R})$} * {$\mathrm{Aut}_{H\circ f}(\mathbb{C})\simeq \mathrm{P}\Delta(2,\mathbb{R}) \equiv$} uppertriangular elements of {$\mathrm{PSL}(2,\mathbb{C})$} * Desargues theorem in geometry corresponds to the associative property in algebra. * A projective space is best understood from one dimension higher. And it is understood in terms of breaking down into smaller affine spaces. And these dimensions higher and lower are related to extending the chain of dimensions as given by Lie algebras. And the reversal of counting is related to the reversal of the orientation of lines etc. as a line is considered as a circle. * Homogeneous coordinates add a variable Z that makes each term of maximal power N by contributing the needed power of Z. * Spinor requires double rotation. Projective sphere is the opposite: half a rotation gets you back where you were. * Sextactic points on a simple closed curve. Quadratic form * Moebius transformation involves complex numbers and 2 x 2 matrix. Whereas Clifford algebra involves quadratic form  as a lens for perspectives? Think of a linear form (proportion) as a plain sheet of glass, and a quadratic form as a convex and/or concave lens. Quantifiers * Study existential and universal quantifiers as adjunctions and as the basis for the arithmetical hierarchy. * "For all" and "there exists" are adjoints presumably because they are on opposite sides of a negation wall that distinguishes the internal structure and external relationships. (That wall also distinguishes external context and internal structure.) (And algorithms?) So study that wall, for example, with regard to recursion theory. Randomness * Random phenomena organize themselves around a critical boundary. Real line * Real line models separation (by cutting) and connectedness (by continuity). The separating cuts become locations (points) in their own right. Restructuring * The six ways of restructuring relate the unconscious's continuum (large number) with the conscious's discrete (small number) and express the relations between the two. Consider these relations in the house of knowledge for mathematics, and how they relate to the six geometrical transformations. * Bundle. Geometry relates analysis (continuum) and algebra (discrete) as a restructuring. When the discrete grows large does it become a continuum? * Bundle = restructuring (base = continuum > fiber = discrete). A number that is "large enough" can essentially model the continuum. Riemann Surface * [[https://en.wikipedia.org/wiki/Riemann_surface  Riemann surface]] * [[http://www.math.tifr.res.in/~pablo/download/book/book.html  Riemann Surfaces Book]], Pablo Arés Gastesi Rotation * Usually multiplication by i is identified with a rotation of 90 degrees. However, we can instead identify it with a rotation of 180 degrees if we consider, as in the case of spinors, that the first time around it adds a sign of 1, and it needs to go around twice in order to establish a sign of +1. This is the definition that makes spin composition work in three dimensions, for the quaternions. The usual geometric interpretation of complex numbers is then a particular reinterpretation that is possible but not canonical. Rotation gets you on the other side of the page. * If we think of rotation by i as relating two dimensions, then {$i^2$} takes a dimension to its negative. So that is helpful when we are thinking of the "extra" distinguished dimensions (1). And if that dimension is attributed to a line, then this interpretation reflects along that line. But when we compare rotations as such, then we are not comparing lines, but rather rotations. In this case if we perform an entire rotation, then we flip the rotations for that other dimension that we have rotated about. So this means that the relation between rotations as such is very different than the relation with the isolated distinguised dimension. The relations between rotations is such is, I think, given by {$A_n$} whereas the distinguished dimension is an extra dimension which gets represented, I think, in terms of {$B_n$}, as the short root. * This is the difference between thinking of the negative dimension as "explicitly" written out, or thinking of it as simply as one of two "implicit" states that we switch between. * Rotation relates one dimension to two others. How does this rotation work in higher dimensions? To what extent does multiplication of rotation (through three dimensional half turns) work in higher dimensions and how does it break down? Schroedinger's equation * Schroedinger's equation relates position and momentum through complex number i. Slack in one (as given by derivative) is given by the value of the other times the ratio of the Hamiltonian over Planck's constant (the available quantum slack as given by the energy). * {$\frac{\text{d}}{\text{dt}}\Psi=i\frac{H}{\hbar}\Psi$} * In statistics, the probability that a system at a given temperature T is in a given microstate is proportional to {$e^{\frac{H}{kT}}$}, so here we likewise have the quantumization factor {${\frac{H}{kT}}$}. Set theory * The Pairing axiom can be defined from other Zermelo Fraenkel axioms. (How?) And what does that mean for my identification of the axioms with the ways of figuring things out? Six operations * Six operations (six modeling methods) 6=4+2 relates 2 perspectives (internal (tensor) & external (Hom)) by 4 scopes (functors). The six 3+3 specifications define a gap for a perspective, thus relate three perspectives. Algebra and analysis come together in one perspective. Together, in the House of Knowledge, these all make for consciousness. Sixsome * A circle (through polarity) defines triplets of points, and triplets of lines, thus sixsomes. The center of a circle is perhaps a fourth point (with every triplet) much like the identity is related to the threecycle? Symmetric functions * Relate the monomial, forgotten, Schur symmetric functions of eigenvalues with the matrices {$(IA)^{1}$} and {$e^A$}. * Exercise: Get the eigenvalues for a generic matrix: 2x2, 3x3, etc. * Exercise: Look for a method to find the eigenvalues for a generic matrix. Express the solving of the equation as a way of relating the elementary functions. Are they related to the inverse Kostka matrix? And the impossibility of a combinatorial solution? And the nondeterminism issue, P vs NP? * Think again about the combinatorial intepretation of {$K^{1}K=I$}. * What do the constraints on Lie groups and Lie algebras say about symmetric functions of eigenvalues. * Vandermonde determinant shows invertible  basis for finite Fourier transform * What is the connection between the universal grammar for games and the symmetric functions of the eigenvalues of a matrix? * How do symmetries of paths relate to symmetries of young diagrams Symmetry * [[https://theoreticalatlas.wordpress.com/2014/01/10/categorifyingglobalvslocalsymmetry/  Global vs. Local symmetry]] * Algorithmic symmetry: Algorithms fold the list of effective algorithms upon itself. * Ockham's razor gets us to focus on the structures which are most basic in that they generate the richest symmetries  the rich symmetries tend solutions towards Ockham's razor. * Analytic symmetry is related to an infinite matrix, which is what we have with the classical Lie algebras. It is also the relation that an infinite sequence (and the function it models) can have with itself. It is thus the symmetry inherent in a recurrence relation, which is the content of a threecycle, what it is relating, a self to itself. * Analytic symmetry is governed by the characteristic polynomial which can't be solved in degree 5. * Representations of the symmetric group. Symmetric  homogeneous  bosons  vectors. Antisymmetric  elementary  fermion  covectors. Euclidean space allows reflection to define inside and outside nonproblematically, thus antisymmetricity. Free vector space. Schur functions combine symmetric and antisymmetric in rows and columns. * E8 is the symmetry group of itself. What is the symmetry group of? * Love (symmetry) establishes immortality (invariant). * Note that x and y axes are separated by 90 degrees. This is the grounds for the degree four of i, the trigonometric functions, the CauchyRiemann equations, etc. * Try to express the symmetries of an object, like a polyhedron, in terms of bundle conceptions. * Symmetry: indistinguishable change, thus a lie, a nontruth, what is hidden. Hidden change, the revealing of hidden change. Symplectic geometry * Notion of derivative is related to tangency, to tangent vector, is related to the normal of the normal, is related to duality, is related to the slack modeled by symplectic geometry. * Symplectic geometry defines slack. It defines motion as oriented area. * Brouwer Fixed Point Theorem holds on a disk with boundary. He also showed that a reversible T which preserves area on the disk without boundary has a fixed point. (Conjugated through a translation.) (So area preservation is equivalent to having a boundary.) This relates perspectives and symplectic geometry. * Symplectic maps map loops to loops with the same area. Area of a closed curve is given by differential forms. There has to be an energy in the background, the Hamiltonian. Symplectic space can associate area to a small loop (small triangle). * Consider how simplexes (in differential geometry and Stokes theorem) relate to symplectic geometry. What is the relevant polytope for symplectic geometry? Coordinate systems? Or crosspolytopes? * Heisenberg uncertainty principle  the slack in the vacuum  that allows the borrowing of energy on brief scales. Compare with coupling (of position and momentum, for example). * Force (and acceleration) is a second derivative  this is because of the duality, the coupling, needed between, say, momentum and position. Is this coupling similar to adjoint functors? * Kinectic energy is possibly zero and builds up from there. It can be expressed in an absolute sense. Potential is possibly infinite and subtracts from that. So it must be expressed in a relative sense. So they are coming at energy from opposite directions. * Note that symplectic geometry, in preserving areas, seems to only care about the extremal points. In what sense is that true in phase space? When and how could it not be true? What is the difference between having finitely many extremal points (convex polytope? and what is the significance of extremal points vs. extremal edges vs. extremal faces etc.? and homology/cohomology?), infinitely many (as with a fractal boundary) or all extremal points? Can symplectic geometry be considered a study of the outside and conformal geometry a study of the inside? * Lagrangian: (Force) Change in potential energy = (mass x acceleration) Change in kinetic energy. Potential energy is based on position and not time. Kinetic energy is based on time and not position. the threesome. * Symplectic  basis for coupling  coupling of electric and magnetic fields  is what is responsible for the periodic nature of waves  the higher the frequency, the higher the energy, the tighter the coupling  the coupling is across the entire universe. The coupling models looseness  slack. This brings to mind the vacillation between knowing and not knowing. * Note that 2dimensional phase space (as with a spring) is the simplest as there is no 1dimensional phase space and there can't be (we need both position and momentum). * What is the connection between symplectic geometry and homology? See [[https://en.wikipedia.org/wiki/Morse_theory  Morse theory]]. See [[https://people.ucsc.edu/~alee150/sympl.html  Floer theory]]. * Symplectic geometry is the geometry of the "outside" (using quaternions) whereas conformal geometry is the geometry of the "inside" (using complex numbers). Then what do real numbers capture the geometry of? And is there a geometry of the line vs. a geometry of the circle? And is one of them "spinorial"? * Benet linkage  keturgrandinis  lygiagretainis, antilygiagretainis * Closed curve in plane must have at least four critical points of curvature. This reflects the fourfold aspect of turning around, rotating. Caustic  a phenomenon in symplectic geometry. Tensors * Tensors stay free of a coordinate system and work with all of them. * Tensor: function in all possible coordinate spaces such that it (its values) obey certain transformation rules. * There is a duality between a tensor and its expression under a particular basis. They are interchangeable. * Tensors are invariant under linear transformations but their components do change. * So a tensor is a bringing together of components, which can be either covariant or contravariant. Is this stepping out and stepping in? Is a tensor a division of everything and each component a perspective? Theorems * "Some Fundamental Theorems in Mathematics" (Knill, 2018) https://arxiv.org/abs/1807.08416 Topology * Klein bottle (twisted torus) exhibits two kinds of duality: twist (inside/outside or not) and hole. * [[https://www.amazon.com/DifferentialAlgebraicTopologyGraduateMathematics/dp/0387906134  Bott & Tu. Differential Forms in Algebraic Topology.]] * What if there is a handle (a torus) inside a sphere? How to classify that? * Understand classification of closed surfaces: Sphere = 0. Projective plane = 1/3. Klein bottle = 2/3. Torus = 1. * [[https://www.maths.ed.ac.uk/~v1ranick/papers/botttu.pdf  Differential Forms in Algebraic Topology]], Bott & Tu * Topology  getting global invariants (which can be calculated) from local information. Topos * [[https://pdfs.semanticscholar.org/a4fb/637410e874b97d323d145c4ebed1c0b03074.pdf  Colin McLarty. The Uses and Abuses of the History of Topos Theory]] * [[https://johncarlosbaez.wordpress.com/2020/01/05/topostheorypart1/  John Baez. Topos Theory (Part 1)]] sheaves, elementary topoi, Grothendieck topoi and geometric morphisms. * [[https://johncarlosbaez.wordpress.com/2020/01/07/topostheorypart2/  Part 2]] turning presheaves into bundles and vice versa; turning sheaves into etale spaces and vice versa. * [[https://johncarlosbaez.wordpress.com/2020/01/13/topostheorypart3/  Part 3]] sheafification; the adjunction between presheaves and bundles. * [[https://johncarlosbaez.wordpress.com/2020/01/21/topostheorypart4/  Part 4]] direct and inverse images of sheaves. * [[https://johncarlosbaez.wordpress.com/2020/01/28/topostheorypart5/  Part 5]] why presheaf categories are elementary topoi: colimits and limits in presheaf categories. * [[https://johncarlosbaez.wordpress.com/2020/02/11/topostheorypart6/  Part 6]] why presheaf categories are elementary topoi: cartesian closed categories and why presheaf categories are cartesian closed. * [[https://johncarlosbaez.wordpress.com/2020/02/18/topostheorypart7/  Part 7]] subobject classifiers * [[https://johncarlosbaez.wordpress.com/2020/02/27/topostheorypart8/  Part 8]] an example of a presheaf topos * How do sheaves relate to gradation of symmetric functions? * [[https://www.oliviacaramello.com/Videos/Videos.htm  V: Olivia Caramello]] Triality * [[http://math.ucr.edu/home/baez/octonions/conway_smith/  Baez's review of Conway and Smith]]: The octonions can be described not only as the vector representation of {$\text{Spin}(8)$}, but also the lefthanded spinor representation and the righthanded spinor representation. This fact is called 'triality'. It has many amazing spinoffs, including structures like the exceptional Lie groups and the exceptional Jordan algebra, and the fact that supersymmetric string theory works best in 10dimensional spacetime — fundamentally because {$8 + 2 = 10$}. * The triality suggests that the Father and the Son are oppositely handed spinor representations and the vector representation is the Spirit? Or the unconscious and conscious are oppositely handed and the vector representation is consciousness? * Yu. I. Manin. Cubic Forms: Algebra, Geometry, Arithmetic. Related to ternary operations and triality. * Triality: C at the center, three legs: quaternions, evendimensional reals, odddimensional reals. Fold, fuse, link. Universal hyperbolic geometry * The circle maps every point to a line and vice versa. * Quadrance (distance squared) is more correct than distance because quadrance makes positive distance and negative distance equivalent. Spread (absolute value of the sine of angle) is more correct than angle because the value of the spread is the same for all angles at an intersection, which is to say, for both theta and pi minus theta. In this way, quadrance and spread eliminate false distinctions and the problems they cause. * [[https://www.youtube.com/watch?v=LaTTqgchO2o  How Chromogeometry transcends Klein's Erlangen Program for Planar Geometries N J Wildberger]] * Try to express projective geometry (or universal hyperbolic geometry) in terms of matrices and thus symmetric functions. What then is algebraic geometry and how do polynomials get involved? What is analytic geometry and in what sense does it go beyond matrices? How do all of these hit up against the limits of matrices and the amount of symmetry in its internal folding? * Attach:GeometryFormulas.png * Attach:QuadrupleFormulas.png Variables * Variables are where the direction is reversed as regards the four relationships between level and metalevel. Variables are, on the one hand, the conclusion of math's "brain", but on the other hand, they are the start of math's "mind". * Organic variation, variables. * Homogeneous coordinates are what let us go between a simplex (when Z=1) and the coordinate system (when Z is free). Compare with the kinds of variables. Videos * http://people.math.harvard.edu/~knill/media/index.html Walks on trees * Walks from A to B in category theory are morphisms and they get mapped to the morphisms from A to B. Relate this to walks on trees. * Parentheses establish a tree structure. What are walks on these trees? How do they relate to associativity and to walks in categories from one object to another? * Walks On Trees are perhaps important as they combine both unification, as the tree has a root, and completion, as given by the walk. In college, I asked God what kind of mathematics might be relevant to knowing everything, and I understood him to say that walks on trees where the trees are made of the elements of the threesome. Yoneda Lemma * B_>C ..... How>What * External relations > Internal logic .... (Not What=Why) Hom C > Hom B (Not How=Whether) * What does the Yoneda Lemma say concretely about the category of Graphs, Groups, Lists, etc.? * What would the category of Lists look like? And what would the Yoneda Lemma look like if the functor mapped into the category of Lists? * Analyze the commutative diagram proving the Yoneda lemma by thinking it, on the one level, as a statement about four sets, but on another level, as a statement about functors and a natural transformation between them. Is there an ambiguity here? * Eduardo's Yoneda Lemma diagram is the foursome. * Loss of info from How to What is equal to the Loss of info from "Why for What" to "Why for How". * How: inner logic. What: external view. * Yoneda lemma  relates to exponentiation and logarithm * Category is a collection of examples that satisfy certain conditions. That is why you can have many examples that are essentially identical. And it's essential for the meaning of the Yoneda Lemma, for example. Because two different examples may have the same structure but one example may draw richer meaning from one domain and the other may not have that richer meaning or may have other meaning from another domain. * Whether (objects), what (morphisms), how (functors), why (natural transformations). Important for defining the same thing, equivalence. If they satisfy the same reason why, then they are the same. * Representable functors  based on arrows from the same object. * Is Cayley's theorem (Yoneda lemma) a contentless theorem? What makes a theorem useful as a tool for discoveries? * Show why there is no ncategory theory because it folds up into the foursome. * Understand the Yoneda lemma. Relate it to the four ways of looking at a triangle. * The Yoneda Lemma: the Why of the external relationships leads to the Whether of the objects in the set. The latter are considered as a set of truth statements. Žodynas. Lietuvių kalba: * sphere  sfera * trace  pėdsakas * semisimple  puspaprastis, puspaprastė * conjugate  sujungtinis * transpose  transponuota matrica, transponavimas 
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