Andrius Kulikauskas

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Lietuvių kalba

Understandable FFFFFF

Questions FFFFC0



Finite exact sequences


  • Given map T: Domain T -> Codomain T we have 0-> ker T -> Domain T -> Codomain T -> coker T -> 0. Interpretation: given a linear equation T(v)=w to solve, the kernel is the space of solutions to the homogeneous equation T(v)=0, and its dimension is the number of degrees of freedom in a solution, if it exists; the cokernel is the space of constraints that must be satisfied if the equation is to have a solution, and its dimension is the number of constraints that must be satisfied for the equation to have a solution. The dimension of the cokernel plus the dimension of the image (the rank) add up to the dimension of the target space, as the dimension of the quotient space W/T(V) is simply the dimension of the space minus the dimension of the image. dim(Domain T) - dim(ker T) + dim(coker T) = dim(Codomain T). In other words: - dim(ker T) + dim (Domain T) - dim (Codomain T) + dim (coker T) = 0.
  • Note that the foursome comes up repeatedly in the Snake Lemma.
  • Relate to Yoneda lemma

Fivesome. Five lemma and the two four-lemmas.

Mitchell's embedding theorem

Nine lemma

Roots of unity = divisions of everything?


  • Everything may be an identity map IA and as such may be defined with regard to any particular object A, that is, person or vantage point.


  • Perhaps adjunction is the division of a monad into two perspectives, free and forgetful.


  • Recursive functions - There is a jump hierarchy of recursive functions that (by the Yates index theorem) has one level be "conscious" of the level that is three levels below it, which is thus relevant for the foursome's role in consciousness.
  • Reikėtų išmokti Yates Index Theorem, jinai pakankamai trumpa ir turbūt ne tokia sudėtinga. Ir paaiškinti ką jinai turi bendro su sąmoningumu.


  • Analysis allows for work with limits.
  • Eccentricity of conic sections - there are five eccentricities (for the circle, parabola, ellipse, hyperbola, line).

Sevensome and eightsome

  • Logic is the end result of structure, see the sevensome and Greimas' semiotic square.
  • Reikėtų gerai išmokti aritmetinę hierarchiją ir bandyti ją taikyti kitur. Kaip jinai rūšiuoja sąvokas? Kaip ji siejasi su pirmos eilės, antros eilės ir kitokiomis logikomis? Kaip ji siejasi su tikrųjų skaičių ir kitokių skaičių tvėrimu? Kaip kvantoriai išsako septynerybę? Ar septynerybė išsako kvantorių ir neigimo derinius? Kaip jie susiję su požiūriais, požiūrių sudūrimu ir požiūrių grandinėmis, tad su kategorijų teorija ir požiūrių algebra?
  • triangle: 1 unknown 3 vertices +3 edges +1 whole


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Puslapis paskutinį kartą pakeistas 2017 sausio 04 d., 10:44