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数学笔记 Ask online Tensor Hom adjunction
4 logics for 4 geometries
S. J. Rapeli, Pratik Shah and A. K. Shukla. Remark on Sheffer Polynomials explains J(D), relates it to A(t) House rules:
Jesus Antonio: Key to physics is information and symmetry What is the combinatorics of convex spaces and how does that relate to orthogonal polynomials, which give different ways of looking at the geometry?
5 notions of independency
What are the transition matrices between orthogonal polynomials? https://en.wikipedia.org/wiki/Probabilistic_programming https://mathoverflow.net/questions/118857/forcinginhomotopytypetheory probabilistic programming paradigm (quantum computing) Measurement based quantum computer vs gate based quantum computer lattice surgery https://en.wikipedia.org/wiki/Toric_code topological quantum computer https://en.wikipedia.org/wiki/Oneway_quantum_computer https://www.aimath.org/WWN/convexalggeom/AIM.pdf Naturality in homotopy type theory breaks down when we try to do type theory in type theory. A special case of the universal property of identity types is related to the Yoneda lemma. Amelia: [The axiom of function extensionality is] inconsistent with many axioms of a more "computational" nature. For example, "formal Church's thesis" says that for any function N→N, there is a "program" (we call it a realizer) that realizes it. You can kinda see what goes wrong: this would be able to tell e.g. "λ x → x" and "λ x → x + 0" apart. You could imagine an assignment of realizers that sidesteps this, though, so to see that it's actually inconsistent takes slightly more work. What is the relationship between universal properties as proved by the function extensionality principle, and universal properties as given by Kan extensions? https://github.com/FrozenWinters/stlc SLTC project where Astra formalises the categorical semantics of function types in Agda. A063573 Counts the number S(n) of lambda terms at level n, in the case of a single variable.
Calculate the combinatorics of the lambdacalculus on a single variable, and if possible, on two or more variables. Is the lambdacalculus equivalent to the recursion relation for orthogonal polynomials?
Have all finite limits is equivalent to
These are the building blocks for limits Sean Carroll or me? Quantum field theory. Instead of space and time, consider in terms of particles and their interactions. Particle clock steps take us from possible interaction to possible interaction. Problem: field theory is based on Minkowski spacetime rather than on particles. Oneallmany relates questions (selection) and answers (judgement). Many is the regularity that every question is answered relevantly. Zero is "no" as a positive answer. Induction argument on truncation levels uses the level below (for identities) and the level above (which we're trying to reach). Similarly, the recurrence relation relates the level xP_n(x) with the level below and the level above.
Homotopy Type Theory
An empty type has no evidence for it, is not true. A nonempty type, as a proposition, is true. The notion of empty or nonempty is relevant for the sevensome, for describing {$\forall \wedge \exists$}. How is Yoneda lemma related to matrix row manipulation? And how might that help relate Cramer's rule to Kan extensions and the Yoneda lemma? YoungIl Choo  MeetUp The inclusion of Field in CRing has no left adjoint because it would carry Z to an initial field, which does not exist. How might an initial field relate to the field with one element? Riehl: 2.4. The category of elements A universal property for an object c ∈ C is expressed either by a contravariant functor F together with a representation C (−, c) F or by a covariant functor F together with a representation C (c, −) F. The representations define a natural characterization of the maps into (in the contravariant case) or out of (in the covariant case) the object c. Proposition 2.3.1 implies that a universal property characterizes the object c ∈ C up to isomorphism. More precisely, there is a unique isomorphism between c and any other object representing F that commutes with the chosen representations. In such contexts, the phrase “c is the universal object in C with an x” assets that x ∈ Fc is a universal element in the sense of Definition 2.3.3, i.e., x is the element of Fc that classifies the natural isomorphism that defines the representation by the Yoneda lemma. In this section, we prove that the term “universal” is being used in the precise sense alluded to at the beginning of this chapter: the universal element is either initial or terminal in an appropriate category. The category in question, called the category of elements, can be constructed in a canonical way from the data of the representable functor F. The main result of this section, Proposition 2.4.8, proves that any universal property can be understood as defining an initial or terminal object, as variance dictates, completing the promise made in §2.1. Riehl, page 50: The Yoneda lemma is arguably the most important result in category theory, although it takes some time to explore the depths of the consequences of this simple statement. In §2.3, we define the notion of universal element that witnesses a universal property of some object in a locally small category. The universal element witnessing the universal property of the complete graph is an ncoloring of K n , an element of the set nColor(K n ). In §2.4, we use the Yoneda lemma to show that the pair comprised of an object characterized by a universal property and its universal element defines either an initial or a terminal object in the category of elements of the functor that it represents. This gives precise meaning to the term universal: it is a synonym for either “initial” or “terminal,” with context disambiguating between the two cases. For instance, K n is the terminal ncolored graph: the terminal object in the category of ncolored graphs and graph homomorphisms that preserve the coloring of vertices. Comma category Lawvere showed that the functors F : C → D {\displaystyle F:{\mathcal {C}}\rightarrow {\mathcal {D}}} F:{\mathcal {C}}\rightarrow {\mathcal {D}} and G : D → C {\displaystyle G:{\mathcal {D}}\rightarrow {\mathcal {C}}} G:{\mathcal {D}}\rightarrow {\mathcal {C}} are adjoint if and only if the comma categories ( F ↓ i d D ) {\displaystyle (F\downarrow id_{\mathcal {D}})} (F\downarrow id_{{\mathcal {D}}}) and ( i d C ↓ G ) {\displaystyle (id_{\mathcal {C}}\downarrow G)} (id_{{\mathcal {C}}}\downarrow G), with i d D {\displaystyle id_{\mathcal {D}}} id_{{\mathcal {D}}} and i d C {\displaystyle id_{\mathcal {C}}} id_{{\mathcal {C}}} the identity functors on D {\displaystyle {\mathcal {D}}} {\mathcal {D}} and C {\displaystyle {\mathcal {C}}} {\mathcal {C}} respectively, are isomorphic, and equivalent elements in the comma category can be projected onto the same element of C × D {\displaystyle {\mathcal {C}}\times {\mathcal {D}}} {\mathcal {C}}\times {\mathcal {D}}. This allows adjunctions to be described without involving sets, and was in fact the original motivation for introducing comma categories. Every type has a unique name. Every universe is a type with a unique name. Every term in a type should have a unique name. So why can't we have a universe of unique names for all of the terms, types and universes? And if we can, then don't we run into a paradox? Or not? https://www.youtube.com/watch?v=ylM1bPtVftk The 4th movement of Beethoven's Symphony No. 5. Conducted by Arthur Nikisch. Recorded in 1913. {$\alpha$} and {$\beta$} count ascents and descents and these are steps forwards or backwards in the unfolding of space (in time?) and so they may relate to John's picture of evolution taking us forward and backward in time. Space has 3 dimensions external to the fivesome (5+3=0)(outside the division) and time has 1 dimension internal to the fivesome (the slack inside the division). https://ww3.math.ucla.edu/dls/emilyriehl/ video about contractibility An isomorphism is a special morphism but truly it is a pair of morphisms that are inverses to each other. There may be many such pairs relating two objects but in each pair the inverses are unique with respect to each other. So it is similar to complex conjugation. Bell number interpretation of Sheffer polynomials gives a foundation for (finite) (and countable) set theory. https://math.stackexchange.com/questions/989083/iscompositionofcoveringmapscoveringmap Charlier polynomials give the trivial space wrapper (the moments are the Bell numbers). In what way are the Hermite polynomials trivial? Space wrappers reinterpret Bell numbers. Types indicate comparability which is a condition for equality. From a dream: I imagined that I was entering a spherical world full of structures, and that my perspective upon those structures was a hyperbolic geometry, expressing the Lorentz contraction, thus special relativity. Comma category Adjunctions
OpenShot eksportuoti 30 fps nes iPhone filmuoja 30 fps Open Source Software to Thank
Shot with an iPhone XS Max. https://www.youtube.com/watch?v=tOp2rdvOmd0 Schuller on Stone's Theorem https://www.freelists.org išbandyti? https://math.stackexchange.com/questions/69698/wedgesumofcirclesandthehawaiianearring https://www.facebook.com/hackersatcambridge contact team @ hackersatcambridge.com portray mu as measuring tape portray mu as super hero measuring tape with two hands ready to hold on
Weed in cracks of cement Conjugacy ? the values of adjunction In defining the minimization operator, and in coding a list of natural numbers with a single natural number:
Minimization operator: representations of nullsome have us proceed through all levels (from true to direct, from direct to constant, from constant to significant)
Constancy  search for meaning
Significance  go beyond
In the search for constancy: take a stand (as to one), follow through (across all), reflect (supposing many) In physics, orthogonal polynomials relate what is necessary (top down) and actual (bottom up) as with string theory, questions and answers. The original spectral theorem: Look for subrepresentations such that S is a onedimensional matrix eigenvalue. Induction argument. Classical (both x, p) and quantum (x). Bald and bankrupt Eastern Europe The Screwing of the Average man: How the rich get richer and you get poorer Special relativity  causal connection  are they time like connected. Wick's theorem  are operators of the same particles  propagator connects Evolution is indicated by learnability and also by sparse communication and natural differences between hierarchies, different orders of magnitude, allowing for a natural hierarchy of niches. Not only the laws of physics are sparse but also the states in nature are sparse. Rules of physics plus configuration space plus location within that space. Source of contradiction
Selfadjoint operators
Quantum measurement projects into eigenstate. The projection operator is a mathematical statement of the collapse of the wave function. If you do it twice, then you don't get anything more. Selfadjoint operators are weighted sums of projection operators. The weights you can find from experiments by applying a projection operator. Hamiltonian is the sum of all the projections onto the energy eigenstates with the energies being the weights. Uncertainty principle  has to do with representations  representation adds a perspective  so that interferes with measuring certain things. Minimization operator mu  superhero  who clings to ledges and other such things and is stretched and blown by the wind. And the shape mu gives the shape of his body clinging to the left. https://en.wikipedia.org/wiki/%CE%9C_operator {$\mu$}operator I had a dream that i was professor anthony zee... But in a quantum superposition. Was i z or not z ? Z or not z? ..... is there a third way? Yes but there is a fourth way .... Nevermind z here is m4w! A qubit specifies the relation between affirmation and negation of probabilities. In matrix form, it provides a complex number which is the coefficient that gets multiplied to the negation (in calculating the new affirmation) and whose conjugate gets multipled to the affirmation (in calculating the new negation). In classical bits, this coefficient is simply zero. Bekaert, Boulanger. The unitary representations of the Poincare group in any spacetime dimension Five zones of scattering can be thought of as Measurement establishes a quantity with regard to boundaries  it establishes the zone within which it is  identifies with a step in the algebra  whereas analysis demarcates the boundaries. Algebra is thinking stepbystep and so it exhibits finiteness. Analysis is discovering the boundaries between steps and so exhibits continuity by discovering the critical points, as in Moore's theory. Healthy irony (verbalization) codes the analogue signal (the emotional tension between expression and meaning) into a discrete alphabet (of boxes organized in a cognitive network). Antonio
Wenbo
Oliver
https://music.163.com/#/playlist?id=797393474 Wenbo The Topos of Music: Geometric Logic of Concepts, Theory, and Performance  worse Cool Math for Hot Music  better Midori https://www.youtube.com/watch?v=OOsRMECWKAE Recuerdos de la alambra Kojin Karatani, Sabu Kohso  Architecture as Metaphor_ Language, Number, Money (1995) semijoin lattice semilattice Hatcher exercise
Osborne IV 40:00 what is needed for a relativistic quantum field theory. Brody. Quantum Mechanics and Riemann Hypothesis. Brown. Topology and Groupoids. Think of 1cell as the center (of all things), the spirit. And think of 0cell not simply as a point but as a 0dimensional open arc (the point shell) with regard to that center (the spirit). The point shells are glued onto the spirit, and similarly, open arcs are glued onto point shells, and so on, inductively. https://www.thphys.uniheidelberg.de/~floerchinger/categories/ Quantum Field Theory https://bookstore.ams.org/surv149/12 https://ncatlab.org/nlab/show/cellular+approximation+theorem#applications Any functor {$F$} can be thought of as {$F:J\rightarrow C$} where {$J$} is the shape, the index set (how) and {$C$} is the image (what). And for any functor we can ask if it has a limit (an object {$L$} with maps {$\psi_X:L\rightarrow F(X)$} for all objects {$X$} of {$J$}, such that for all {$f:X\rightarrow Y$} in {$J$} we have the analogous morphisms commute in {$C$}, and that is universal as such). So that limit is Why. And is the colimit Whether? Or is whether simply the object with its identity morphism? Compare with the Yoneda lemma. Relate walks on trees to covering groups. What do conjugates (paths) mean? What is the homotopy group? https://en.wikipedia.org/wiki/Koopman%E2%80%93von_Neumann_classical_mechanics https://www.amazon.com/QuantumChallengeFoundationsMechanicsAstronomy/dp/076372470X In what sense are Feynman diagrams relativistic given that they have directions for time and for space? Instead of thinking of speed of light, think of a clock that doesn't tick, so that t=0 always. And this is the case for the quantum harmonic osciallator and for the particleclocks with no steps. One {$\exists x$}, all {$\forall x$}, many {$\neg\exists x \wedge \neg\forall x$}. Masaki Kashiwara, Pierre Schapira. Categories and Sheaves. 2006 Gerald B. Folland
Bohm Pilot Wave, Thomas Spencer Relative invariance  more global than another Relate the threecycle (taking a stand, following through, reflecting) to Kan extension as defined by filtering objects and morphisms, acting on them, and then taking the colimit or limit on that filter. Alan Turing, Cybernetics and the Secrets of Life My approach to special relativity lets me work in units in my own frame. https://en.wikipedia.org/wiki/Algorithmic_information_theory Gregory Chaitin = Shannon + Turing = CompressionDecompression as understanding. https://en.wikipedia.org/wiki/Cristian_S._Calude Philosophy of computation https://www.youtube.com/watch?v=xP5iIeKXE8 Life in life Thinking about the expansion of the universe as a reduction of density, by which the mass of particles becomes ever less important, by which we have an increase of entropy (becoming less deliberate). And we can reverse this by starting with an increase in entropy and arriving at the expansion of the universe. Relate Ellerman's heteromorphism and comma category. Consider how the understanding of Yoneda lemma in terms of a left Kan extension, and in particular, the factoring, relates to the push down automata. San Francisco Meet Up interests: Dependently typed programming languages. Language aspects of category theory. Functional programming. Topos, lambda calculus. Is type theory advantageous? Modeling infinitesimals. Kan extensions are a framework for universality. Consider example 6.1.3, the Yoneda Lemma. Think in which ways the universality of limits, colimits, adjunctions, etc. is captured by Kan extensions. All of these universal properties can be thought of in terms of initial or terminal objects in the appropriate categories, such as the category of cones, or the comma category for the universal mapping property for adjunctions. So consider the relevant categories. How do they relate to the classification of adjoint strings? Dan Shiebler. Kan Extensions in Data Science and Machine Learning Matematika išplaukia iš (poreikių tenkinimo) algoritmų taikymo, vedančio iš duotybių į bendrybes. O tos bendrybės įkūnija, išreiškia tam tikrus prieštaravimus, juos paverčia sąvokomis, kurias galima mąstyti toliau. Pavyzdžiui, apskritimas iškyla iš begalinės simetrijos visom kryptim, arba iš virve aprėpto ploto maksimalizavimo.
Mathematics is described in terms of set theory. The category of graphs {$\textrm{Set}^{A}$} where {$A$} is the category with two objects, edges E and vertices V, and two nontrivial morphisms target {$t:E\rightarrow V$} and source {$s:E\rightarrow V$}. Similarly, all mathematical structures and their structure preserving morphisms should have a similar expression in terms of sets and their relationshps. Work out various examples. Then study the role of {$\textrm{Set}^{X}$} in the Yoneda Lemma. SL(2,C) character variety related to hyperbolic geometry. SL2(C) character varieties Universal enveloping algebra is an abstraction where the generators are free and thus yield infinite generators. Whereas the Lie algebra may be in terms of concrete matrices and the underlying generators, when understood not in terms of the Lie bracket but in terms of matrix multiplication, may have relations such as {$x^2=0$}, {$h^2=1$}. Information is what you learn. What you learn grows at the boundary, has the shape of the boundary. A shape can be thought of as being created by integrating over these boundaries as they increase. TaiDanae Bradley: Information is on the Boundary
Prove that the matrix made up of eigenvectors diagonalizes a matrix. In special relativity, think of distance squared over time as surface area per time, the difference beween the surface areas of two spheres, one expanding with velocity v, and the other with velocity c. For John: How could we get negative energy? Consider how to get imaginary square roots. For example, if a speed is greater than the speed of light, then the relationship between time and position is multiplied by an imaginary number. Quaternions, Dirac equation: Pauli matrices are the threecycle for learning and they are extended by a fourth dimension of nonlearning (what is absolutely true or false) for the foursome. https://en.wikipedia.org/wiki/Lebesgue_covering_dimension Covering spaces with repetition yield the spaces they cover. Enveloping algebra (important for adjunctions) is related to Hochschild cohomology. Think of combinatorial quantum physics operator (raising and lowering operator) as a derived functor. Lifting a path is like inverting a functor. How is that related to adjunctions? Adjunction is conditional inversion. Unclear whether the empty space is path connected. Think of a universal covering space as expressing the unfolding of a space, thus expressing eternal life. Relate triangulated categories (with squiggles {$X\rightsquigarrow W = X\rightarrow TW$}) to monads with likewise squiggles. Samwel Kongere vaizdo įrašai Nafsi Afrika Acrobats  Pyramid of Peace Research/Notes
Relate {$F_1$} with the basis element 1 in a Clifford algebra. The house of knowledge for mathematics describes 4 representations (properties) of everything (onesome, totality), which through their unity establish, define space as algebraic, consisting of enumerated dimensions:
(Relate this to the binomial theorem.) And it describes 4 representation of the nullsome (center), which through their unity establish, define a point as analytic. This describes four choices:
Meanings are variously related by adjunctions. They enrich the meaning and extend the context.
4 levels of knowledge is sufficient (in the chain complexes). The house of knowledge describes those 4 levels. It relates the analytical view of a point with the algebraic view of a space. Consider the Zig Zag Lemma as applying the threecycle to set up four levels of knowledge, 4 x 3 = 12 circumstances. Counterquestions
Enveloping algebra (important for adjunctions) is related to Hochschild cohomology which is a special case of the functor Ext.
Think of combinatorial quantum physics operator (raising and lowering operator) as a derived functor.
Counterquestions
Potential energy source {$\frac{y}{2}$} (adding free cell) is in balance with kinetic energy source {$i\frac{\partial}{\partial y}$} (halflink) A chain complex is loose and has slack, which is the basis for homology. An exact sequence is tight and has no slack. A division of everything is tight and has no slack.
Fivesome
John Baez, Michael Shulman. Lectures on nCategories and Cohomology.
Note that there is only one empty set. But there could also be many empty sets. And all can be thought of as an empty set. Can the search for constancy be considered a search for emptiness? Emily Riehl, Dominic Verity. Elements of ∞Category Theory Foursome For C and D categories we have
Foursome
A functor between ordinary categories (1categories) can be:
A functor {$F:C→D$} is essentially surjective if it is surjective on objects “up to isomorphism”: If for every object {$y$} of {$D$}, there exists an object {$x$} of {$C$} and an isomorphism {$F(x)≅y$} in D. A functor F:C→D can be:
This formalism captures the intuition of how “stuff”, “structure”, and “properties” are expected to be related:
Yoneda lemma lets you go from natural isomorphism of homsets to natural isomorphism of functors. Are the doubts and counterquestions related to electromagnetism, U(1) and the related gauge theory? Observing symmetry requires breaking symmetry. https://www.masterclass.com/classes/terencetaoteachesmathematicalthinking Is the associativity diagram for monoidal categories an example of the fivesome? Involution is square root of permutation. Compare with spin as square root of geometry. Math Discovery
Local and global quantum are linked by experiments, by "the complicated interplay between infrared and ultraviolet affects", by a conspiracy of IR/UV mixing.
Walks
