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数学

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Andrius Kulikauskas

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数学笔记

What is Mathematical Music Theory?

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)?

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.

What is the connection between the universal grammar for games and the symmetric functions of the eigenvalues of a matrix?

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.

Study how turning the counting around relates to cycles - finite fields.

Kuom skaičius skiriasi nuo pasikartojančios veiklos - būgno mušimo?

  • A) veikla kažkada prasidėjo
  • B) kiekvienas skaičius laikomas nauju, skirtingu nuo visų kitų

briauna = skirtingumas

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.

One-dimensional 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 infinite-dimensional, which is to say, it can't be captured by a one-dimensional shadow?

{$A_n$} 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.

Composition algebra. Doubling is related to duality.

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.

Esminis pasirinkimas yra: kurią pasirinkimo sampratą rinksimės?

Kaip suvokiame {$x_i$}? Koks tai per pasirinkimas?

Kaip sekos lankstymą susieti su baltymų lankstymu ir pasukimu?

Kada pasirinkimo samprata keičiasi, visgi už visų sampratų slypi bendresnis, pirmesnis suvokimų suvokimas, taip kad renkamės pačią sampratą. Folding is the basis for substitution.

Fizikoje, posūkis yra viskas. Palyginti su ortogonaline grupe.

Bott periodicity exhibits self-folding. Note the duality with the pseudoscalar. Consider the formula n(n+1)/2 does that relate to the entries of a matrix?

{$x_0$} is fundamentally different from {$x_i$}. The former appears in the positive form {$\pm(x_0-x_1)$} 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.

Keturios pasirinkimo sampratos (apimtys) visos reikalingos norint išskirti vieną paskirą pasirinkimą.

Bott periodicity is the basis for 8-fold folding and unfolding.

Use "this" and "that" as unmarked opposites - conjugates.

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 cross-polytopes we divide the "is" into two halves, preserving the "not" intact.
  • for Dn we have simply "this" and "that" (not-this).

Root systems relate two spheres - they relate two "sheets". Logic likewise relates two sheets: a sheet and a meta-sheet for working on a problem. Similarly, we model our attention by awareness, as Graziano pointed out. This is stepping in and stepping out.

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 cross-polytope with 2 vertices
  • taking a sphere and identifying antipodal elements - this is a famous group
  • polar conjugates in projective geometry (see Wildberger)

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.

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: Yang-Mills theory.

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.

Compare trejybės ratas (for the quaternions) and Fano's plane (aštuonerybė) for the octonions.

Nobody know what E8 is the symmetry group of. (Going beyond oneself?)

What is the connection between Bott periodicity and spinors? See John Baez, The Octonions.

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.

John Baez: 24 = 6 x 4 = An x Bn

Dedekind eta function is based on 24.

Discriminant of elliptic curve.

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.

Spinor requires double rotation. Projective sphere is the opposite: half a rotation gets you back where you were.

G2 requires three lines to get between any two points (?) Relate this to the three-cycle.

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?

Algebra (of the observer) and analysis (of the observed) exhibit a duality of worldviews.

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 doe all of these hit up against the limits of matrices and the amount of symmetry in its internal folding?

Projective geometry: homogeneous coordinates add a variable Z that makes each term of maximal power N by contributing the needed power of Z.

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.

John Baez on duality in logic and physics

Four geometries

  • Affine geometry preserves lines. Projective geometry also preserves zeros. Conformal geometry preserves angles. Symplectic geometry preservers oriented area. What are all these objects?

Affine geometry

  • 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.

Duality

  • 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!

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.

Cayley-Dickson construction

  • John Baez periodic table and stablization theorem - relate to Cayley Dickson construction and its dualities.

Projective geometry

  • 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.

Conformal geometry

  • Conformal geometry preserves angles. Radial coordinates distinguishes distances r and angles theta, and makes use of the exponential {$e^{\pi i \theta}$}.

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.

Symplectic geometry

  • 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.

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.

Walks on trees

  • 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.

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.

How do symmetries of paths relate to symmetries of young diagrams

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?

Note that 2-dimensional phase space (as with a spring) is the simplest as there is no 1-dimensional phase space and there can't be (we need both position and momentum).

What is the connection between symplectic geometry and homology? See Morse theory. See 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

  • A_n points and sets
  • B_n inside: perpendicular (angles) and
  • C_n outside: line and surface area
  • D_n points and position

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.

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.

Six sextactic points.

John Baez about observables (see Nr.15) and the paper A topos for algebraic quantum theory about C* algebras within a topos and outside of it.

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?

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?

Study how Set breaks duality (the significance of initial and terminal objects).

Show why there is no n-category 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.

Category theory for me: distinguishing what observations are nontrivial - intrinsic to a subject - and what are observations are content-wise trivial or universal - not related to the subject, but simply an aspect of abstraction.

"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.

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.

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.

Is Cayley's theorem (Yoneda lemma) a contentless theorem? What makes a theorem useful as a tool for discoveries?

(Conscious) Learning from (unconscious) machine learning.

Topology - getting global invariants (which can be calculated) from local information.

Simple examples that illustrate theory.

monad = black box?

Let p: E → B be a basepoint-preserving Serre fibration with fiber F, that is, a map possessing the homotopy lifting property with respect to CW complexes. Suppose that B is path-connected. Then there is a long exact sequence of homotopy groups

Primena trejybę. Wikipedia: Homotopy groups Let p: E → B be a basepoint-preserving Serre fibration with fiber F, that is, a map possessing the homotopy lifting property with respect to CW complexes. Suppose that B is path-connected. 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_{n-1}(F) \rightarrow \cdots \rightarrow \pi_0(F) \rightarrow 0. $}

Vandermonde determinant shows invertible - basis for finite Fourier transform

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

Vector bundles: Identity and self-identity (like the ends of a regular strip or a Moebius strip). Identity of a point, identity of a fiber - self-identity under continuity. How does a fiber relate to itself? Is it flipped or not? 2 kinds of self-identity (or non-identity) allows the edge to be flipped upside down.

https://golem.ph.utexas.edu/category/2017/01/basic_category_theory_free_onl.html

http://pi.math.cornell.edu/~hatcher/AT/ATpage.html

Algebra - geometry duality. (Pullback). Morphism <-> ring homomorphism. Intrinsic and extrinsic geometry. Ambient space. Relation between two spaces. Varieties [morphisms X to Y] <-> Affine F-algebras [F-linear ring homomorphisms F[Y] to F[x]]

Algebra and geometry are linked by logic - intersections and unions make sense in both.

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.

Turing machines - inner states are "states of mind" according to Turing. How do they relate to divisions of everything?

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?

Study homology, cohomology and the Snake lemma to explain how to express a gap.

Associative composition yields a list (and defines a list). Consider the identity morphism composed with itself.

MathNotes


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Puslapis paskutinį kartą pakeistas 2019 balandžio 07 d., 22:41
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