Andrius Kulikauskas

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Lietuvių kalba

Understandable FFFFFF

Questions FFFFC0



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.

homology - holes - what is not there - thus a topic for explicit vs. implicit math

svarbūs pavyzdžiai


  • 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?
  • Meilė (simetrija) įsteigia nemirtingumą (invariant).


  • Sieja nepažymėtą priešingybę (+) -1, 0, 1 ir pažymėtą priešingybę (x) -1, 1.

Extension of a domain

Skaičius 5

  • Golden mean is the "most" irrational of numbers (based on its continued fraction). Consider series of continued fractions... as sequence patterns...
  • Susieti most "irrationality" su "randomness". Nes ką sužinai nieko daugiau nepasako apie kas liko.

Skaičius 24

  • John Baez kalba. 24 = 6 (trikampių laukas) x 4 (kvadrato laukas).
  • 24 + 2 = 26. Dievo šokis (žmogaus trejybės naryje) veikia ant žmogaus (už šokio) tad žmogus papildo šokį dviem matais. Ir gaunasi "group action". Susiję su Monster group.
  • Monster group dydis susijęs su visatos dalelyčių skaičiumi?

Ypatingi skaičiai

Vector spaces are basic. What is basic about scalars? They make possible proportionality.

AutomataTheory - There is a qualitative hierarchy of computing devices. And they can be described in terms of matrices.


  • The mathematics of counting (and generating) objects. I think of this as the basement of mathematics, which is why I studied it.


  • studies particular structures and substructures

Neural networks

  • Very powerful and simple computational systems for which Sarunas Raudys showed a hierarchy of sophistication as learning systems.

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?

Catalan, Mandelbrot, Julia sets

  • 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).

Pagrindiniai matematikos dėsniai

Kaip matematikos pagrindus pristatyti svarbiausiais dėsniais, pavyzdžiais ir žaidimais? Kuo žaidimai yra vertingi, kaip jie suveikia? Kuriu atitinkamas mokymosi priemones, tapau drobę.

Prisiminti savo matematikos mokymo dėsnius:

  • 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

Basic division rings: 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.


Naujausi pakeitimai

Puslapis paskutinį kartą pakeistas 2016 birželio 23 d., 14:53