Juodraštis? FFFFFF

Užrašai FCFCFC

Klausimai FFFFC0

Gvildenimai CAE7FA

Pavyzdžiai? F6EEF6

Šaltiniai? EFCFE1

Duomenys? FFE6E6

Išsiaiškinimai D8F1D8

Pratimai? FF9999

Dievas man? FFECC0

Pavaizdavimai? E6E6FF

Miglos? AAAAAA

Asmeniškai? BA9696

Mieli dalyviai! Visa mano kūryba ir kartu visi šie puslapiai yra visuomenės turtas, kuriuo visi kviečiami laisvai naudotis, dalintis, visaip perkurti. - Andrius

## Mintys.Matematika istorija

2016 birželio 19 d., 14:54 atliko AndriusKulikauskas -
Pakeistos 1-6 eilutės iš
Žr. [[Book/Math]]
į:
Žr. anglų kalba: [[Book/Math]]

* [[Matematikos rūmai]]
* [[Matematikos grožis
]]
2016 birželio 19 d., 12:29 atliko AndriusKulikauskas -
Pakeistos 1-604 eilutės iš
Žr. [[http://www.selflearners.net/wiki/Math/Math | SelfLearners/Math]], [[Tensor]], [[Simplex]], [[Matematikos rūmai]], [[Matematikos grožis]]

Žiūriu Representation of Geometry paskaita. II. 0:45

>>bgcolor=#FFFFC0<<

Matematikos išsiaiškinimo būdus išryškinti. Išryškinti matematiką išreiškiančią Dievo šokį ir sandaras. Iškelti matematikos visumos pagrindus. Išmąstyti fizikos išsiaiškinimo būdus ir susieti su matematimos išsiaiškinimo būdais. Išvystyti susidomėjimą vidine matematika.
* Toliau vystyti židinio reikšmę. Išsiaiškinti, kaip suprasti dviejų takų susipynimą coxeter diagramoje. Suprasti išimtines lie grupes. Suprasti kaip klasikinės grupės ir algebros iškyla iš politipų šeimynų.
* Susieti Paskalio trikampį su aritmetikos hierarchija. Ir su homologija, Eulerio charakteristika.
* Ieškoti pagrindimo pertvarkymams aibių teorijoje ir kategorijų teorijoje.
* Suprasti Yates indekso teoriją.
* Ištirti dvejybių rūšis.
* Ištirti kintamųjų rūšis.
* Požiūrius ir permainas išreikšti kategorijų teorija.

* interpret the [[https://en.m.wikipedia.org/wiki/Simplex | binomial theorem for simplexes]].
* what does it mean that the -1 simplex is the empty set? the spirit?
* what does it mean that a point is the marked opposite for the empty set?
* how does this come up in symplicial homology?

Kokie yra matematikos pagrindai?
* Kas yra geometrija? Iš ko jinai susidaro? Iš klausimų?
** How is one dimension embedded in other dimensions?
** What is a line segment? What makes it "straight"?
** What is a circle?
** What does it mean for figures to intersect?
** Can a line intersect with itself?

Kaip kompleksiniais skaičiais išvesti ir suprasti d/dz (e^z) ?

Study the [[https://en.wikipedia.org/wiki/Finite_field | finite field GF(8)]] and relate it to the divisions of everything.

Tiesinė algebra
* Kaip dauginti polar decomposed matrices?
* Geriau suprasti [[https://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix | Eigenvector decomposition]].
** Kokios matricos turi pilną eigenvector rinkinį?
** Kokius eigenvectors ir eigenvalues turi pasukimo matricos?
** Kaip suprasti eigenvector koordinačių sistemą? Kiekviena (neišsigimusi) matrica turi naturalią koordinačių sistemą (?)
** Kaip suprasti matricą kaip lygčių sistemą?
** Palyginti matricų naudojimą Galois teorijoje.

* Kaip apsieiti be begalybės aksiomos? Tačiau su židiniu?

>><<

%width=900px% [[Attach:MatematikosSakosDidelis.png | Attach:MatematikosSakosDidelis.png]]

'''Kas yra matematika?'''

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.

Much of my work may be considered pre-mathematics, the grounding of the structures that ultimately allow for mathematics.

'''Matematikos apžvalga'''

* Add time to the diaram
* catalan numbers are related to semantics and to the generating function of the mandelbtot set
* Consider [[https://en.m.wikipedia.org/wiki/Mathematics,_Form_and_Function |Mathematics, Form and Function]] by Aleksandrov, Kolmogorov, Lavrentev
* Mathematics: Its Content, Methods and Meaning
* Wikipedia: The concept of a universal cover was first developed to define a natural domain for the analytic continuation of an analytic function. The idea of finding the maximal analytic continuation of a function in turn led to the development of the idea of Riemann surfaces. The power series defined below is generalized by the idea of a germ. The general theory of analytic continuation and its generalizations are known as sheaf theory.
* Wikipedia: Covering spaces play an important role in homotopy theory, harmonic analysis, Riemannian geometry and differential topology. In Riemannian geometry for example, ramification is a generalization of the notion of covering maps. Covering spaces are also deeply intertwined with the study of homotopy groups and, in particular, the fundamental group. An important application comes from the result that, if X is a "sufficiently good" topological space, there is a bijection between the collection of all isomorphism classes of connected coverings of X and the conjugacy classes of subgroups of the fundamental group of X.
* [[Characteristic class]] of different kinds are related to the classical linear groups.

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

svarbūs pavyzdžiai
* https://en.m.wikipedia.org/wiki/Möbius_transformation

'''Ko noriu mokytis'''

* Lie group ir algebra teorijos
** [[https://en.wikipedia.org/wiki/Coxeter_group | Finite Coxeter group properties]] žiūrėti lentelę
** [[http://math.ucr.edu/home/baez/qg-fall2008/ | Lie theory through examples]]
** [[http://arxiv.org/pdf/0902.0431v1 | Exceptional Lie groups]]
** [[http://math.ucr.edu/home/baez/octonions/ | Octonions]]
** http://phyweb.lbl.gov/~rncahn/www/liealgebras/texall.pdf
** [[http://www.nbi.dk/GroupTheory/ | Group theory]], representation theory of Symmetric group
** [[https://en.wikipedia.org/wiki/Symmetric_space | Symmetric space]], including modern classification by Huang and Leung.
** [[https://en.wikipedia.org/wiki/Amplituhedron | Amplituhedron]], [[http://www.preposterousuniverse.com/blog/2014/03/31/guest-post-jaroslav-trnka-on-the-amplituhedron/ | post by Jaroslav Trnka]] related to walks on trees? [[http://susy2013.ictp.it/video/05_Friday/2013_08_30_Arkani-Hamed_4-3.html | video]]
* Rekursyvinę funkcijų teoriją
** Yates-Index theorem
* [[http://math.ucr.edu/home/baez/ | John Baez Network Theory]]
* Kategorijų teorija
** [[https://math.berkeley.edu/~jhicks/links/SOTS/jhicks022614.pdf | Jeff Hicks Categorification]]
** [[http://arxiv.org/abs/0908.2469 | A Pre-history of n-categorical Physics]] by John Baez
** http://math.ucr.edu/home/baez/rosetta.pdf
** [[http://arxiv.org/pdf/1302.6946v3 | Category theory for scientists]]
** john baez spans in quantum theory
** [[http://www.j-paine.org/make_category_theory_intuitive.html | Make category theory intuitive]]
** http://math.ucr.edu/home/baez/qg-winter2016/
* [[https://www.msri.org/programs/276 | Geometric Representation Theory]]
** [[http://math.ucr.edu/home/baez/qg-fall2007/ | Geometric Representation Theory Seminar 2007]] John Baez and James Dolan
** [[https://www.youtube.com/watch?v=FKSbWiGMY30 | Theory X and Life]]
** [[https://www.ma.utexas.edu/users/djordan/QGpublic.pdf | Introductory Book for Geometric Representation Theory]]
** [[https://ncatlab.org/nlab/show/Borel-Weil+theorem | Borel-Weil theorem]]
* Clifford Algebra
** [[https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-visual-introduction/ | Clifford Algebra]]
** [[https://slehar.wordpress.com/2014/06/26/geometric-algebra-projective-geometry/ | Projective geometry]]
** [[https://slehar.wordpress.com/2014/07/24/geometric-algebra-conformal-geometry/ | Conformal geometry]]
** [[https://doubleconformal.wordpress.com/ | Double Conformal Mapping]]
** [[https://slehar.wordpress.com/2014/09/12/the-perceptual-origins-of-mathematics/ | The Perceptual Origin of Mathematics]]
** [[http://cns-alumni.bu.edu/~slehar/webstuff/persintro/indep.html | Stephen Lehar]]
** [[http://cns-alumni.bu.edu/~slehar/epist/epist.html | Stephan Lehar's theory of mind and brain]]
* Entropija
* Riemann-Zeta funkcijos pagrindus
* Model theory. Taylor Dupuy youtube
* [[https://ncatlab.org/nlab/show/Hegelian%20taco | Hegelian taco]]?

'''Matematikos pagrindai'''

Ieškau matematikos pagrindų. Apžvelgiu matematikos sritis ir jas išdėstau pagal tai, kaip viena nuo kitos priklauso.

Triviality: Distinguishing what is trivial from what is nontrivial. What is assumed or understood. Prerequisites for duality.
* Integers
* Rationals. Proportionality.
* Fractions. Equivalence classes. Duality of numerator and denominator, vector and covector. Confusion of numerator as answer or as amount. Identification of denominator as unit. Relation to coordinate systems. Ambiguity inherent in expression. Use of explicit coordinate (denominator) but implicit meaning.
* Linear (algebra), linear functions, linearity (derivatives)
* Matrix, array
* Scalars
* Tensors
** Tensors relate passive and active transformations, see Penrose. Tensors perhaps are related to breakdown in terms of positive and negative eigenvalues, see Orthogonal group and symmetry matrices.
** Use of units (every answer is an amount and a unit) implicitly and explicitly. d/dx is vector (unit - denominator), dX is covector (amount - numerator). "Applying the unit" (such as "tenth") is a vector field, maps from a scalar field to a scalar field, from 3 to 0.3, not necessarily Cartesian. "Dropping the unit" is the covector field, maps us from the vector field to the scalar field, from 0.3 to 3, that is necessarily Cartesian. Units and scales can be most confusing because they are knowledge that is supposed, assumed, in the background. Consider conversion of units. Implicitness and explicitness of units. Vectors/units are lists "list different units". Covectors/amounts are distributions "combine like units". Implicitness is a sign of premathematics, explicitness of postmathematics.
** Tensors are required for symmtery and invariants. And duality in general. And beauty? and the related topologies?
** Partial derivatives are explicit, total derivatives implicit - this distinction between explicit and implicit.
** Tensor symmetry: Wigner-Eckart.
* Note link to divisibility of numbers and prime decomposition.
* Rectangles, rectangular areas and volumes
* Rootedness in a world, our world. Partial world. Our relationsip with the world.

Nontrivial
* Square numbers and square roots and distances and metrics. Pythagorean theorem.
* Triangles and Geometry.
* Circles and spheres
* Real numbers
* Platonic solids
* Conic sections
* Power series
* Infinite sequences
* Worlds unto themselves. Wholeness. Total world with or without us.
* Rotations, reflections.
* Complex numbers
* Normality is a key tool for understanding a subworld unto itself.

In between
* Stitching: continuity, extension of domain, self superposition

Geometry
* [[http://math.ucr.edu/home/baez/week181.html | John Baez]]: Whenever we pick a Dynkin diagram and a field we get a geometry: An projective, Bn Cn conformal, Dn symplectic.
* [[https://en.wikipedia.org/wiki/Klein_geometry | Klein geometry]]
* [[http://arxiv.org/pdf/math/9912235.pdf | Victor Kac's paper]]: “Each of the four types W, S, H, K of simple primitive Lie algebras (L, L0) correspond to the four most important types of geometries of manifolds: all manifolds, oriented manifolds, symplectic and contact manifolds.”
* ''At its roots, geometry consists of a notion of congruence between different objects in a space. In the late 19th century, notions of congruence were typically supplied by the action of a Lie group on space. Lie groups generally act quite rigidly, and so a [[https://en.wikipedia.org/wiki/Cartan_connection | Cartan geometry]] is a generalization of this notion of congruence to allow for curvature to be present. The flat Cartan geometries — those with zero curvature — are locally equivalent to homogeneous spaces, hence geometries in the sense of Klein.''
* [[https://en.wikipedia.org/wiki/Erlangen_program | Erlangen program]]
* Lie groups play an enormous role in modern geometry, on several different levels. Felix Klein argued in his Erlangen program that one can consider various "geometries" by specifying an appropriate transformation group that leaves certain geometric properties invariant. Thus Euclidean geometry corresponds to the choice of the group E(3) of distance-preserving transformations of the Euclidean space R3, conformal geometry corresponds to enlarging the group to the conformal group, whereas in projective geometry one is interested in the properties invariant under the projective group. This idea later led to the notion of a G-structure, where G is a Lie group of "local" symmetries of a manifold.
* Start with 0 dimension: a point. Every point is the same point. Then consider embedding a point in 1 dimension. The point does not yet distinguish between the two sides because there is no orientation. A distinction comes with the arisal of a second point. But whether the second point distinguishes the two sides depends on global knowledge. So there must be a third point. This is the relationship between "persons": I, You, Other. Either the dimension is a closed curve or it is an open line. This is "global knowledge". So there is the distinction between local knowledge and global knowledge. But basically geometry is a construction of the continuum, either locally or globally. The construction takes place through infinite sequences, through completion. This completion is not relevant for all constructions.
* Tensors give the embedding of a lower dimension into a higher dimension. Taip pat tensoriai sieja erdvę ir jos papildinį, kaip kad gyvybę ir meilę. Tai vyksta vektoriais (tangent space) ir kovektoriais (normal space?) Tad geometrijos pagrindas būtų Tensors over a ring. Kovektoriai išsako idealią bazę. Tensorius susidaro iš kovektorių ir kokovektorių. Ir tie, ir tie yra tiesiniai funkcionalai. Tikai baigtinių dimensijų vektorių erdvėse kokovektoriai tolygūs vektoriams.
* A [[https://ncatlab.org/nlab/show/geometric+embedding | geometric embedding]] is the right notion of embedding or inclusion of topoi F↪E F \hookrightarrow E, i.e. of subtoposes. Notably the inclusion Sh(S)↪PSh(S) Sh(S) \hookrightarrow P of a category of sheaves into its presheaf topos or more generally the inclusion ShjE↪E Sh_j E \hookrightarrow E of sheaves in a topos E E into E E itself, is a geometric embedding. Actually every geometric embedding is of this form, up to equivalence of topoi. Another perspective is that a geometric embedding F↪E F \hookrightarrow E is the localizations of E E at the class W W or morphisms that the left adjoint E→F E \to F sends to isomorphisms in F F.
* Dflags explain how to fit a lower dimensional vector space into a higher dimensional vector space.

A finite set can be thought of as a finite dimensional vector space over the field with one element. But no such field exists!

Equivalences
* [[https://en.wikipedia.org/wiki/Twelvefold_way | Twelvefold way]] - reikėtų palyginti su aplinkybėmis, taip pat su equivalence relations, kokių gali būti.
* Category theory - Categories are helpful in making fruitful definitions
* [[https://golem.ph.utexas.edu/category/2015/02/concepts_of_sameness_part_1.html | Baez on sameness]]
* Is a set simply an equivalence class, in some sense? For example, the set is unordered but everything is labeled so that it could be ordered.
* Standard foundations - need to "label" and then "unlabel" (create an equivalence class). Why? Isn't that a lie?

Dualities. Duality arises from a symmetry between two ways of looking at something where there is no reason to choose one over the other. For example:
* '''Square roots of -i.''' There are two square roots of -1. One we call +i, the other -i, but neither should have priority over the other. Similarly, clockwise and counterclockwise rotations should not be favored. Complex conjugation is a way of asserting this. (Note that the integer +1 is naturally favored over -1. But there is no such natural favoring for i. It is purely conventional, a misleading artificial contrivance.)
* A rectangular matrix can be written out from left to right or right to left. So we have the transpose matrix.
** Normality says conjugate invariancy: gN = Ng.
* '''[[Opposite category | Opposite category]]''' Morphisms can be organized from left to right or from right to left. The opposite category turns all of the arrows around.
** Colimits and limits
** Monomorphisms ("one-to-one") and epimorphisms (forcing "onto").
** Coproducts and products
** Initial and terminal objects
** Wikipedia: In applications to logic, this then looks like a very general description of negation (that is, proofs run in the opposite direction). If we take the opposite of a lattice, we will find that meets and joins have their roles interchanged. This is an abstract form of De Morgan's laws, or of duality applied to lattices.
** Wikipedia: Reversing the direction of inequalities in a partial order. (Partial orders correspond to a certain kind of category in which Hom(A,B) can have at most one element.)
** Wikipedia: Fibrations and cofibrations are examples of dual notions in algebraic topology and homotopy theory. In this context, the duality is often called [[https://en.wikipedia.org/wiki/Eckmann%E2%80%93Hilton_duality | Eckmann–Hilton duality]].
** [[https://en.m.wikipedia.org/wiki/Adjoint | Adjoint]] bendrai ir [[https://en.wikipedia.org/wiki/Adjoint_functors | Adjoint functors]]. Wikipedia: It can be said that an adjoint functor is a way of giving the most efficient solution to some problem via a method which is formulaic. A construction is most efficient if it satisfies a universal property, and is formulaic if it defines a functor. Universal properties come in two types: initial properties and terminal properties. Since these are dual (opposite) notions, it is only necessary to discuss one of them.
* Switching of "existing" and "nonexisting", for example, edges in a graph. This underlies [[https://en.wikipedia.org/wiki/Ramsey's_theorem | Ramsey's theorem]]. Tao: "the Ramsey-type theorem, each one of which being a different formalisation of the newly gained insight in mathematics that complete disorder is impossible."
* Coordinate systems can be organized "bottom up" or "top down". This yields the duality in projective geometry.
** Root systems relate reflections (hyperplanes) and root vectors. Given a root R, reflecting across its hyperplane, every root S is taken to another root -S, and the difference between the two roots is an integer multiple of R. But this relates to the commutator sending the differences into the module based on R.
* Analysis provides lower and upper bounds on a function or phenomenon which helps define the geometry of this space.
* We can look at the operators that act or the objects they act upon. This brings to mind the two representations of the foursome.
** This is related to the duality between left and right multiplication. Examples include Polish notation.
* Faces of an object and corners of an object. (Why are they dual?)
* Coxeter groups are built from reflections. Reflections are dualities.
* Any two structures which have a nice map from one to the other have a duality in that you can start from one and go to the other.
** Galois theory: field extensions (solutions of polynomials) and groups
** Lie groups: solutions to differential equations..
Read [[https://ncatlab.org/nlab/show/duality | nLab: Duality]]. Here are examples to consider:
* [[https://en.wikipedia.org/wiki/Duality_%28projective_geometry%29 | Duality (projective geometry)]]. Interchange the role of "points" and "lines" to get a dual truth: The plane dual statement of "Two points are on a unique line" is "Two lines meet at a unique point". (Compare with the construction of an equilateral triangle and the lattice of conditions.)
* Atiyah-Singer index theorem...
* Riemann-Roch theorem
* Covectors and vectors
* Cotangent space and tangent space
* [[https://en.m.wikipedia.org/wiki/De_Rham_cohomology | de Rham cohomology]] links algebraic topology and differential topology
* [[https://en.wikipedia.org/wiki/Modular_theorem |Modularity theorem]].
* [[https://en.m.wikipedia.org/wiki/Langlands_program | Langlands program]]
* general Stokes theorem: duality between the boundary operator on chains and the exterior derivative
* [[https://en.m.wikipedia.org/wiki/Hilbert%27s_Nullstellensatz | Hilbert's Nullstellensatz]]
* Class field theory provides a one-to-one correspondence between finite abelian extensions of a fixed global field and appropriate classes of ideals of the field or open subgroups of the idele class group of the field.
* Lie's idée fixe was to develop a theory of symmetries of differential equations that would accomplish for them what Évariste Galois had done for algebraic equations: namely, to classify them in terms of group theory. Lie and other mathematicians showed that the most important equations for special functions and orthogonal polynomials tend to arise from group theoretical symmetries. In Lie's early work, the idea was to construct a theory of continuous groups, to complement the theory of discrete groups that had developed in the theory of modular forms, in the hands of Felix Klein and Henri Poincaré. The initial application that Lie had in mind was to the theory of differential equations. On the model of Galois theory and polynomial equations, the driving conception was of a theory capable of unifying, by the study of symmetry, the whole area of ordinary differential equations. However, the hope that Lie Theory would unify the entire field of ordinary differential equations was not fulfilled. Symmetry methods for ODEs continue to be studied, but do not dominate the subject. There is a differential Galois theory, but it was developed by others, such as Picard and Vessiot, and it provides a theory of quadratures, the indefinite integrals required to express solutions.
* One may ask analytic questions about algebraic numbers, and use analytic means to answer such questions; it is thus that algebraic and analytic number theory intersect. For example, one may define prime ideals (generalizations of prime numbers in the field of algebraic numbers) and ask how many prime ideals there are up to a certain size. This question can be answered by means of an examination of Dedekind zeta functions, which are generalizations of the Riemann zeta function, a key analytic object at the roots of the subject. This is an example of a general procedure in analytic number theory: deriving information about the distribution of a sequence (here, prime ideals or prime numbers) from the analytic behavior of an appropriately constructed complex-valued function.
* Meromorphic function is the quotient of two holomorphic functions, thus compares them.
* [[https://ncatlab.org/nlab/show/Isbell+duality | Isbell duality]] relates higher geometry with higher algebra.
* [[https://ncatlab.org/nlab/show/topos | Topos]] links geometry and logic.
* For integers, decomposition into primes is a "bottom up" result which states that a typical number can be compactly represented as the product of its prime components. The "top down" result is that this depends on an infinite number of exceptions ("primes") for which this compact representation does not make them more compact.
* The two facts that this method of turning rngs into rings is most efficient and formulaic can be expressed simultaneously by saying that it defines an adjoint functor. Continuing this discussion, suppose we started with the functor F, and posed the following (vague) question: is there a problem to which F is the most efficient solution? The notion that F is the most efficient solution to the problem posed by G is, in a certain rigorous sense, equivalent to the notion that G poses the most difficult problem that F solves.
* https://en.m.wikipedia.org/wiki/Coherent_duality https://en.m.wikipedia.org/wiki/Serre_duality https://en.m.wikipedia.org/wiki/Verdier_duality https://en.m.wikipedia.org/wiki/Poincaré_duality
* https://en.m.wikipedia.org/wiki/Dual_polyhedron
* a very general comment of William Lawvere is that syntax and semantics are adjoint: take C to be the set of all logical theories (axiomatizations), and D the power set of the set of all mathematical structures. For a theory T in C, let F(T) be the set of all structures that satisfy the axioms T; for a set of mathematical structures S, let G(S) be the minimal axiomatization of S. We can then say that F(T) is a subset of S if and only if T logically implies G(S): the "semantics functor" F is left adjoint to the "syntax functor" G.
* division is (in general) the attempt to invert multiplication, but many examples, such as the introduction of implication in propositional logic, or the ideal quotient for division by ring ideals, can be recognised as the attempt to provide an adjoint.
* Tensor products are adjoint to a set of homomorphisms.
* Duality - parity - išsiaiškinimo rūšis. Įvairios simetrijos - išsiaiškinimo būdų sandaros.
* In mathematics, monstrous moonshine, or moonshine theory, is a term devised by John Conway and Simon P. Norton in 1979, used to describe the unexpected connection between the monster group M and modular functions, in particular, the j function. It is now known that lying behind monstrous moonshine is a vertex operator algebra called the moonshine module or monster vertex algebra, constructed by Igor Frenkel, James Lepowsky, and Arne Meurman in 1988, having the monster group as symmetries. This vertex operator algebra is commonly interpreted as a structure underlying a conformal field theory, allowing physics to form a bridge between two mathematical areas. The conjectures made by Conway and Norton were proved by Richard Borcherds for the moonshine module in 1992 using the no-ghost theorem from string theory and the theory of vertex operator algebras and generalized Kac–Moody algebras.
[[https://en.wikipedia.org/wiki/List_of_dualities | List of dualities (Wikipedia)]]
* Alexander duality
* Alvis–Curtis duality
* Araki duality
* Beta-dual space
* Coherent duality
* De Groot dual
* Dual abelian variety
* Dual basis in a field extension
* Dual bundle
* Dual curve
* Dual (category theory)
* Dual graph
* Dual group
* Dual object
* Dual pair
* Dual polygon
* Dual polyhedron
* Dual problem
* Dual representation
* Dual q-Hahn polynomials
* Dual q-Krawtchouk polynomials
* Dual space
* Dual topology
* Dual wavelet
* Duality (optimization)
* Duality (order theory)
* Duality of stereotype spaces
* Duality (projective geometry)
* Duality theory for distributive lattices
* Dualizing complex
* Dualizing sheaf
* Esakia duality
* Fenchel's duality theorem
* Haag duality
* Hodge dual
* Jónsson–Tarski duality
* Lagrange duality
* Langlands dual
* Lefschetz duality
* Local Tate duality
* Poincaré duality
* Twisted Poincaré duality
* Poitou–Tate duality
* Pontryagin duality
* S-duality (homotopy theory)
* Schur–Weyl duality
* Serre duality
* Stone's duality
* Tannaka–Krein duality
* Verdier duality
* AGT correspondence
* A "transformation group" is a group acting as transformations of some set S. Every transformation group is the group of all permutations preserving some structure on S, and this structure is essentially unique. The bigger the transformation group, the less structure: symmetry and structure are dual, just like "entropy" and "information", or "relativity" and "invariance".

Matricos
* I thought this was the most basic object in mathematics. Note that the index set may be arbitrary, not necessarily numbers.
* Representations - A very important idea, which is that we access a deep structure (such as a division of everything) not directly, but by way of some representation. This term is used in algebra, for example, to distinguish a system (like a group) from the matrices which serve as its multiplication table.
* [[https://en.wikipedia.org/wiki/Polar_decomposition | Polar decomposition]]. Square complex matrix A can always be written as A = UP where U is a unitary matrix and P is a positive-semidefinite Hermitian matrix. The eigenvalues of U all lie on the unit circle. The real analogue of U is the orthogonal matrix, whose determinant is either +1 (rotations) or -1 (reflections). U = e^iH where H is some Hermitian matrix. P has all nonnegative eigenvalues. It is the stretching of the eigenvectors. Thus every matrix A = B*e^iC where B and C have all nonnegative real eigenvalues.
* Symmetric and [[https://en.wikipedia.org/wiki/Skew-symmetric_matrix | skew-symmetric]]. Every matrix A can be broken down as the sum of a skew-symmetric matrix 1/2*(A-AT) and a symmetric matrix 1/2*(A+AT).
* Note that a category may be thought of as a deductive system, a directive graph, and hence a matrix. My thesis was on the combinatorics of the general matrix, which apparently is all generated by the symmetric functions of the eigenvalues of a matrix. So I am interested to see if that might be of value here. I want to show how a category arises from first principles. I imagine that perspectives may be thought of as morphisms (or functors).
* LinearAlgebra - Is the study of the basic properties of matrices and their effects.
* Mano tezė. Jeigu matricą išrašome Jordan canonical form, tai didžiausi ciklai tėra dvejetukai.
* Symplectic form is related to complexification and also the linking of losition and momentum.

Difference between complex numbers and real numbers
* Quantum possibilities vs. actualities
* Cauchy's integral theorem: for complexes, derivative and integral are mirrors, but not for reals
* There is a sense in which the reals give the magnitude and the imaginaries give the rotation. The function 1/x sends x+iy to x-iy divided by x2+y2. It sends r to 1/r (across the boundary of the unit circle) and it sends theta to -theta.
* Real numbers are used for independent x, y. Imaginary number i denotes a link between two otherwise indepedent variables so that y = ix links indepedent axes by a 90 degree rotation.
* Similarly the polar decomposition of a matrix distinguishes (as for a number) the change in magnitude (scaling) and the rotation. It separates them.
* Complex numbers have two natural coordinate systems that correspond to addition (x,y) and multiplication (r,theta).
* Circle folding relates to "reflection" of the complex conjugate across an x-axis. Thinking of inverse rotation as this reflection.
* The number "i" is highly misleading in that it actually has no priority over "-i". Both are square roots of -1. Thus often (or always?) they should both be referenced - they are a "coupled" pair of numbers, not a single number. They should be referenced by a single Number "I" which is understood to have two meanings.
* The purpose of complex numbers is to define two unmarked opposites (we know them, unfortunately, as "i" and "-i", where one is marked with regard to the other, but in truth they should be both unmarked). The purpose of the real numbers is to provide that context for this unmarkedness. (Is there a simpler way to create it?)
* The quantum world is based on the two unmarked opposites ("i" and "j") as with spin 1/2 particles, "up" and "down". Symmetry breaking - the breaking of the symmetry between "i" and "-i" enforced by complex conjugation - occurs (and is defined to be) when there is a measurement, so that we collapse to the reals, where this symmetry is broken.
* The truth of the heart does not mark the opposites. The truth of the world marks one opposite with regard to the other.
* Ar teisinga? Skaičius turėtų rašyti: xr + yi pabrėžti jog tai skiritingi matai. Bet r tampa 1. Vienas matas gali būti "default" ir užtat išbrauktas. Jisai tada tampa "identity". Every answer is an amount and a unit - šis dėsnis paneigtas.
* Complex numbers: local = global. ([[https://en.wikipedia.org/wiki/Identity_theorem | Identity theorem]]). Real numbers: local != global.
* [[http://math.ucr.edu/home/baez/week201.html | Galois group of C/R]]

Symmetry
* 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).

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

Extention of a domain
* [[https://en.wikipedia.org/wiki/Analytic_continuation | Analytic continuation]] - complex numbers - dealing with divergent series.

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
* http://math.ucr.edu/home/baez/42.html
* http://math.ucr.edu/home/baez/numbers/

Lie Bracket:
* Remiasi tuo, kad summing over permutations of 1 yield 0. [x,x]=0
* Summing over permutations of 2 yields 0. [x,y]+[y,x]=0
* Summing over permutations of 3 yields 0. [x,[y,z]] + [y,[z,x]] + [z,[x,y]]=0
That's true writing out [x,y]=xy-yx and summing out you get a positive and a negative term for each permutation. But also true in the brackets directly permuting cyclically. What would it look like to sum over permutations of 4?

Lie groups and Lie algebras
* ways of breaking up an identity into two elements that are inverses of each other
* orthogonal: symmetric transposes of each other
* unitary: conjugate transposes of each other
* symplectic: antisymmetric transposes of each other
* [[https://en.m.wikipedia.org/wiki/Unitary_group#2-out-of-3_property | two out of three property]] 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 skew-symmetric (symplectic)—and these are related by the complex structure (which is the compatibility).

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.

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

Algebra
* 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?
* [[http://oeis.org/A000607 | Number of ways to partition a number into primes]].

https://en.m.wikipedia.org/wiki/Field_with_one_element

[[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).
* Gaussian binomial coefficients [[http://math.stackexchange.com/questions/214065/proving-q-binomial-identities | interpretation related to Young tableaux]]

'''Pagrindiniai matematikos dėsniai'''

Kaip [[http://www.selflearners.net/Math/DeepIdeas | 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: [[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.

'''Sąsajos tarp mano sąvokų ir matematikos'''

Dievas
* Field of one element. Roots of unity = divisions of everything?
* [[https://www.youtube.com/watch?v=1XRna0vUYdo | field of one element video]]
* Dievas išeina už savęs: 3 matai -> 2 matai -> (flip to dual) 1 matas -> 0 matas (taškas: gera širdis).
* The trivial tensor: T00. It is a zero-dimensional array, thus a scalar. An array is a perspective, and so it is having no perspective. A scalar is "spirit". Thus it is spirit with no perspective.
* Simplex, interpret the -1 face. Sometimes considered the empty set. Attention which is free. It is the center of the simplex. You can always add a new center to imagine the next simplex but in the current dimensions. Interpret (a+x)**n. It is (implicit + explicit)**n dimensions.
** [[http://www.geometrictools.com/Documentation/CentersOfSimplex.pdf | Centers of Simplex]]

Septynerybė aštuonerybė
* triangle 1 unknown 3 vertices +3 edges +1 whole

Unmarked opposite
* turinys = raiška. "Those things are which show themselves to be." buvimo pagrindas
* inner 2-cycle, kurio paprastai nebūna.
* complex numbers i=j iš kurio atsiveria 1<>-1, i<>-i. Paprastai i -> j -> i ... banguoja, o šitą sustabdžius gaunasi +1 +1 +1 +1 amžinai ir atitinkamai -1-1-1-1 amžinai.

* Bott periodicity [[http://math.ucr.edu/home/baez/week105.html | John Baez]]
* [[http://www.math.illinois.edu/K-theory/handbook/1-111-138.pdf | Max Karoubi vadovėlis apie Bott periodicity]]
* palyginti susijusias Lie grupes (ir jų ryšį su gaubliu) su požiūrių permainomis
* Bott periodicity turėtų būti susijęs su aštuonerybės sugriuvimu prieštaravimu
* susipažinti su Clifford algebra ir clock shift veiksmais
* Max Karoubi savo video paskaitoje paminėjo loop lygtį žiedams kurioje R,C,H,H' ir epsilon = +/-1 gaunasi 10 homotopy equivalences. Kodėl 10? 8+2=10? ar 6+4=10, dešimt Dievo įsakymų?
* Žiūrėk taip pat [[https://en.wikipedia.org/wiki/Hopf_fibration Hopf fibration]], [[https://en.wikipedia.org/wiki/Hopf_invariant | Hopf invariant]] ir Adam's theorem. [[https://en.wikipedia.org/wiki/Homotopy_groups_of_spheres | Homotopy group of spheres]]. [[https://en.wikipedia.org/wiki/Clifford_parallel | Clifford paralells]] ir quaternions.
* [[https://golem.ph.utexas.edu/category/2007/10/higher_clifford_algebras.html | Higher Clifford Algebras]]

[[http://math.ucr.edu/home/baez/week82.html | Clifford algebra periodicity]]
* C0 R
* C1 C
* C2 H
* C3 H + H
* C4 H(2)
* C5 C(4)
* C6 R(8)
* C7 R(8) + R(8)
* C8 R(16)
''C_{n+8} consists of 16 x 16 matrices with entries in Cn ! For a proof you might try

2) H. Blaine Lawson, Jr. and Marie-Louise Michelson, "Spin Geometry", Princeton U. Press, Princeton, 1989. or 3) Dale Husemoller, "Fibre Bundles", Springer-Verlag, Berlin, 1994. These books also describe some of the amazing consequences of this periodicity phenomenon. The topology of n-dimensional manifolds is very similar to the topology of (n+8)-dimensional manifolds in some subtle but important ways!'' Physics of fermions.

[[http://math.ucr.edu/home/baez/week61.html | Introduction to rotation groups]] '''Triality of octonions.''' ''More generally, it turns out that the representation theory of Spin(n) depends strongly on whether n is even or odd. When n is even (and bigger than 2), it turns out that Spin(n) has left-handed and right-handed spinor representations, each of dimension 2^{n/2 - 1}. When n is odd there is just one spinor representation. Of course, there is always the representation of Spin(n) coming from the vector representation of SO(n), which is n-dimensional. This leads to something very curious. If you are an ordinary 4-dimensional physicist you undoubtedly tend to think of spinors as "smaller" than vectors, since the spinor representations are 2-dimensional, while the vector representation is 3-dimensional. However, in general, when the dimension n of space (or spacetime) is even, the dimension of the spinor representations is 2^(n/2 - 1), while that of the vector representation is n, so after a while the spinor representation catches up with the vector representation and becomes bigger! This is a little bit curious, or at least it may seem so at first, but what's really curious is what happens exactly when the spinor representation catches up with the vector representation. That's when 2^(n/2 - 1) = n, or n = 8. The group Spin(8) has three 8-dimensional irreducible representations: the vector, left-handed spinor, and right-handed spinor representation. While they are not equivalent to each other, they are darn close; they are related by a symmetry of Spin(8) called "triality". And, to top it off, the octonions can be seen as a kind of spin-off of this triality symmetry... as one might have guessed, from all this 8-dimensional stuff. One can, in fact, describe the product of octonions in these terms. So now let's dig in a bit deeper and describe how this triality business works. For this, unfortunately, I will need to assume some vague familiarity with exterior algebras, Clifford algebras, and their relation to the spin group. But we will have a fair amount of fun getting our hands on a 24-dimensional beast called the Chevalley algebra, which contains the vector and spinor representations of Spin(8)!''

Gyvenimo lygtis:
* Dvasia ir sandara susieti "duality", veiksmu +2.

Požiūriai:
* Complex numbers: dvimačiai: širdies tiesa. Real numbers: pasaulio tiesa.
* Kategorijų teorija.
* Kvantoriai ir septynerybė.
* Nėra quantum frequency. Fotonai yra "požiūriai". Bosonai yra "jėgų nešėjai", "santykiai", jie "neegzistuoja".
* Ar požiūriai yra neasociatyvūs?
* Perspectives are (multidimensional) arrays. The number of array dimensions is the number of divisions of everything.

Ketverybė:
* 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.

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

Septynerybė:
* 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?

Pertvarkymai:
* Aibės struktūra primena medį nes gali būti aibės aibėse. Tačiau svarbu, kad nėra ratų.

Niekas
* Taškas yra niekas.

Aštuongubas kelias
* SetTheory - The Zermelo Frankel axioms of set theory seem to be structured by what I call the EightfoldWay.

* Topologies - Systems of constraints that may be thought of as defining worlds. Topology is the study of topologies.

Matematikos įrodymų būdai - laipsnynas

24
* 4 netroškimai: 6x4=24 ir dar 0-inis požiūris ir 7-as požiūris. Iš viso 24+2=26 kaip Monster grupėje.

Walks on trees
* Julia sets

'''Ko norėčiau išmokti matematikoje'''

Category theory
* [[http://math.ucr.edu/home/baez/qg-winter2016/ | John Baez: Category Theory]]

Foundations of Mathematics (understand how models work)
* [[http://sakharov.net/foundation.html | Foundations of Mathematics by Alexander Sakharov]]
* [[http://www.math.wisc.edu/~miller/old/m771-10/kunen770.pdf | The Foundations of Mathematics by Kenneth Kunen]]

Lie groups, Lie algebras
* Root systems: Lie algebrų rūšių pagrindas
* [[http://www.continuummechanics.org | Continuum mechanics]]

[+Matematikos išsiaiškinimo būdai+]

* [[http://www.cut-the-knot.org/m/ProblemSolving.shtml | Alexander Bogomolny sarašas]]

[+Matematikos grožis+]

'''Kas gražu matematikoje ir kodėl?'''

One milestone (or miracle) in knowing everything is to generate all mathematical objects and truths in a unified way, but especially, generate all mathematical insights in a way that builds our intuition. The relevant outlook will, I imagine, be in terms of beauty. I mean that we may think of mathematical insight as guided by the wish for beauty.

Beauty - wholeness preserving transformations
* natural generalizations
* coordinate free

[[https://en.wikipedia.org/wiki/Mathematical_beauty | Mathematical beauty]]
* Wikipedia: In the 1990s, [[https://en.wikipedia.org/wiki/J%C3%BCrgen_Schmidhuber | Jürgen Schmidhuber]] formulated a mathematical theory of observer-dependent subjective beauty based on algorithmic information theory: the most beautiful objects among subjectively comparable objects have short algorithmic descriptions (i.e., Kolmogorov complexity) relative to what the observer already knows. Schmidhuber explicitly distinguishes between beautiful and interesting. The latter corresponds to the first derivative of subjectively perceived beauty: the observer continually tries to improve the predictability and compressibility of the observations by discovering regularities such as repetitions and symmetries and fractal self-similarity. Whenever the observer's learning process (possibly a predictive artificial neural network) leads to improved data compression such that the observation sequence can be described by fewer bits than before, the temporary interestingness of the data corresponds to the compression progress, and is proportional to the observer's internal curiosity reward.
* [[https://arxiv.org/pdf/math/0702396v1.pdf | What is Good Mathematics?]] Terrence Tao.
* Alexander Bogomolny, [[http://www.cut-the-knot.org/manifesto/index.shtml | Cut the Knot Manifesto]] "The peculiar beauty of Mathematics lies in deduction, in the dependency of one fact upon another. The less expected a dependency is, the simpler the facts on which the deduction is based -- the more beautiful is the result."
* [[http://mathoverflow.net/questions/74841/an-example-of-a-beautiful-proof-that-would-be-accessible-at-the-high-school-level | Beautiful, simple proofs]]

Mathematicians are often guided by their sense of beauty. Beauty is a key to the big picture in mathematics as well as physics and philosophy.

* What is meant by beauty?
* What principles determine it?
* To what extent is beauty objective and subjective?
* How does beauty lead to mathematical insight?

Investigation: Collect examples

In this section, we collect examples of beauty in mathematics.

[[http://www-math.mit.edu/~rstan/ | Richard Stanley’s]] post at Math Overflow, [[http://mathoverflow.net/questions/49151/most-intricate-and-most-beautiful-structures-in-mathematics | Most intricate and most beautiful structures in mathematics]], produced the following list, ordered by votes received:

* The absolute Galois group of the rationals
* The natural numbers (and variations)
* Homotopy groups of spheres
* The Mandelbrot set
* The Littlewood Richardson coefficients (representations of Sn etc.)
* The class of ordinals
* The monster vertex algebra
* Classical Hopf fibration
* Exotic Lie groups (See the Scientific American article about A.Garrett Lisi’s application of E8 to physics: [[http://www.cs.virginia.edu/~robins/A_Geometric_Theory_of_Everything.pdf | A Geometric Theory of Everything]])
* The Cantor set
* The 24 dimensional packing of unit spheres with kissing number 196560 (related to the monster vertex algebra).
* The simplicial symmetric sphere spectrum
* F_un (whatever it is)
* The Grothendiek-Teichmuller tower.
* Riemann’s zeta function
* Schwartz space of functions

Below we gather more examples, both basic and advanced:

* e^2pii + 1 = 0
* more generally, Euler’s formula: e^2piix = cos(x) + i sin(x)
* the unique decomposition of natural numbers into prime numbers
* Euler’s polyhedron formula V - E + F = 2
* the classification of the Platonic solids
* the relationship between a polynomial and its graph
* binomial theorem and Pascal's triangle
* elementary proofs of the Pythagorean theorem (such as: "four times a right triangle is the difference of two squares" - the basis for infinitesimal rotation).

Investigation: Analyze examples

In this section, we analyze the examples collected above to consider:

* In what sense are they beautiful?
* What makes them beautiful?
* What are the simplest examples of beauty?
* Which examples yield the most beauty for the least drudgery?

Investigation: Look for unifying principles or contexts.

* Urs Schreiber notes that many of the beautiful structures relate to string theory.
* Relation between two completely different domains, especially dual, complementary domains.

Investigation: Compare with beauty in chess.

'''Įdomūs, prasmingi reiškiniai matematikoje'''

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.

Logika
* 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)?

'''Matematikos visuomenės'''

* [[http://archives.math.utk.edu/news.html | Electronic Newsgroups and Listservs]]
* [[https://ncatlab.org/nlab/show/math%20blogs | nLab visuomenių sąrašas]]
* [[http://www.artofproblemsolving.com/community | Art of Problem Solving: Community]]
* [[https://www.physicsforums.com | Physics forums]]
* [[http://deferentialgeometry.org | Deferential Geometry: Garrett Lisi]]

--------------------------

The Big Picture in Mathematics

Mathematics has grown as a discipline to the extent that no single person is able to overview it all. This is a research page to encourage and organize efforts to foster a perspective upon all of mathematics.

Big questions

Several big questions provide new angles on mathematics as an endeavor:

* discovery What are the ways of figuring things out in mathematics? We can study mathematics as an activity by which we create and solve mathematical problems. These techniques are much more limited than the mathematical output which they generate.
* beauty Mathematicians are guided by a sense of beauty. What is meant by beauty? What principles determine it? How does beauty lead to mathematical insight?
* organization How can mathematics be organized so as to survey the most basic concepts from which it arises and understand it in terms of its most fundamental divisions and how they relate to each other?
* education What resources are available to mathematicians that would help them most effectively learn mathematics so as to try to understand it as a whole? How might mathematicians collaborate effectively in trying to understand the big picture?
* insight What are the most fruitful insights in trying to understand mathematics? How can such insights best be stated?
* premathematics What concepts express intuitions that are prior to explicit mathematics and make it possible?
* history How can the history of mathematical discovery inform frameworks for the future development of mathematics?
* humanity What parts or aspects of mathematics are specific to the human mind, body, culture, society, and what might be more broadly meaningful to other species in the universe?

[[http://www.lanet.lv/miv/ | Matematikos žodynas anglų, lietuvių ir kitomis kalbomis]]

[[http://www.ms.lt/derlius/AndriusKulikauskasThesis.pdf | Symmetric Functions of the Eigenvalues of a Matrix
]]
į:
Žr. [[Book/Math]]
2016 birželio 18 d., 12:57 atliko AndriusKulikauskas -
Pridėta 412 eilutė:
* [[https://golem.ph.utexas.edu/category/2007/10/higher_clifford_algebras.html | Higher Clifford Algebras]]
2016 birželio 18 d., 11:53 atliko AndriusKulikauskas -
Pridėta 575 eilutė:
* [[http://archives.math.utk.edu/news.html | Electronic Newsgroups and Listservs]]
2016 birželio 17 d., 20:57 atliko AndriusKulikauskas -
Pakeista 151 eilutė iš:
* John Baez: Whenever we pick a Dynkin diagram and a field we get a geometry: An projective, Bn Cn conformal, Dn symplectic.
į:
* [[http://math.ucr.edu/home/baez/week181.html | John Baez]]: Whenever we pick a Dynkin diagram and a field we get a geometry: An projective, Bn Cn conformal, Dn symplectic.
Pakeista 153 eilutė iš:
* “Each of the four types W, S, H, K of simple primitive Lie algebras (L, L0) correspond to the four most important types of geometries of manifolds: all manifolds, oriented manifolds, symplectic and contact manifolds.”
į:
* [[http://arxiv.org/pdf/math/9912235.pdf | Victor Kac's paper]]: “Each of the four types W, S, H, K of simple primitive Lie algebras (L, L0) correspond to the four most important types of geometries of manifolds: all manifolds, oriented manifolds, symplectic and contact manifolds.”
2016 birželio 17 d., 15:32 atliko AndriusKulikauskas -
Pridėta 110 eilutė:
* [[https://ncatlab.org/nlab/show/Hegelian%20taco | Hegelian taco]]?
2016 birželio 17 d., 12:08 atliko AndriusKulikauskas -
Pridėtos 101-102 eilutės:
** [[https://slehar.wordpress.com/2014/06/26/geometric-algebra-projective-geometry/ | Projective geometry]]
** [[https://slehar.wordpress.com/2014/07/24/geometric-algebra-conformal-geometry/ | Conformal geometry]]
2016 birželio 17 d., 11:55 atliko AndriusKulikauskas -
Pakeista 42 eilutė iš:
į:
* Kaip apsieiti be begalybės aksiomos? Tačiau su židiniu?
2016 birželio 16 d., 23:35 atliko AndriusKulikauskas -
Pridėtos 99-104 eilutės:
* Clifford Algebra
** [[https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-visual-introduction/ | Clifford Algebra]]
** [[https://doubleconformal.wordpress.com/ | Double Conformal Mapping]]
** [[https://slehar.wordpress.com/2014/09/12/the-perceptual-origins-of-mathematics/ | The Perceptual Origin of Mathematics]]
** [[http://cns-alumni.bu.edu/~slehar/webstuff/persintro/indep.html | Stephen Lehar]]
** [[http://cns-alumni.bu.edu/~slehar/epist/epist.html | Stephan Lehar's theory of mind and brain]]
2016 birželio 16 d., 09:00 atliko AndriusKulikauskas -
Pridėta 144 eilutė:
* “Each of the four types W, S, H, K of simple primitive Lie algebras (L, L0) correspond to the four most important types of geometries of manifolds: all manifolds, oriented manifolds, symplectic and contact manifolds.”
2016 birželio 16 d., 06:14 atliko AndriusKulikauskas -
Pridėta 87 eilutė:
** [[https://math.berkeley.edu/~jhicks/links/SOTS/jhicks022614.pdf | Jeff Hicks Categorification]]
2016 birželio 16 d., 05:51 atliko AndriusKulikauskas -
Pakeistos 141-142 eilutės iš
* John Baez: Whenever we pick a Dynkin diagram and a field we get a geometry: An projective, Bn Cn conformal, Dn symplectic.
į:
* John Baez: Whenever we pick a Dynkin diagram and a field we get a geometry: An projective, Bn Cn conformal, Dn symplectic.
* [[https://en.wikipedia.org/wiki/Klein_geometry | Klein geometry]]
2016 birželio 16 d., 05:47 atliko AndriusKulikauskas -
Pridėta 141 eilutė:
* John Baez: Whenever we pick a Dynkin diagram and a field we get a geometry: An projective, Bn Cn conformal, Dn symplectic.
2016 birželio 15 d., 18:41 atliko AndriusKulikauskas -
Pakeista 3 eilutė iš:
Žiūriu Representation of Geometry paskaita. II. 0:38
į:
Žiūriu Representation of Geometry paskaita. II. 0:45
2016 birželio 15 d., 15:00 atliko AndriusKulikauskas -
Pakeista 3 eilutė iš:
Žiūriu Representation of Geometry paskaita. II. 0:29
į:
Žiūriu Representation of Geometry paskaita. II. 0:38
2016 birželio 15 d., 14:51 atliko AndriusKulikauskas -
Pridėta 261 eilutė:
* A "transformation group" is a group acting as transformations of some set S. Every transformation group is the group of all permutations preserving some structure on S, and this structure is essentially unique. The bigger the transformation group, the less structure: symmetry and structure are dual, just like "entropy" and "information", or "relativity" and "invariance".
2016 birželio 15 d., 09:35 atliko AndriusKulikauskas -
Pakeista 10 eilutė iš:
* Ieškoti pagrindimo pertvarkymams aibių teorijoje.
į:
* Ieškoti pagrindimo pertvarkymams aibių teorijoje ir kategorijų teorijoje.
Pridėta 14 eilutė:
* Požiūrius ir permainas išreikšti kategorijų teorija.
2016 birželio 15 d., 08:20 atliko AndriusKulikauskas -
Pridėtos 12-13 eilutės:
* Ištirti dvejybių rūšis.
* Ištirti kintamųjų rūšis.
2016 birželio 15 d., 08:18 atliko AndriusKulikauskas -
Pakeistos 7-8 eilutės iš
Matematikos išsiaiškinimo būdus išryškinti. Išryškinti matematiką išreiškiančią Dievo šokį ir sandaras. Iškelti matematikos visumos pagrindus. Išmąstyti fizikos išsiaiškinimo būdus ir susieti su matematimos išsiaiškinimo būdais.
į:
Matematikos išsiaiškinimo būdus išryškinti. Išryškinti matematiką išreiškiančią Dievo šokį ir sandaras. Iškelti matematikos visumos pagrindus. Išmąstyti fizikos išsiaiškinimo būdus ir susieti su matematimos išsiaiškinimo būdais. Išvystyti susidomėjimą vidine matematika.
* Toliau vystyti židinio reikšmę. Išsiaiškinti, kaip suprasti dviejų takų susipynimą coxeter diagramoje. Suprasti išimtines lie grupes. Suprasti kaip klasikinės grupės ir algebros iškyla iš politipų šeimynų.
* Susieti Paskalio trikampį su aritmetikos hierarchija. Ir su homologija, Eulerio charakteristika.
* Ieškoti pagrindimo pertvarkymams aibių teorijoje.
* Suprasti Yates indekso teoriją
.
Pridėta 89 eilutė:
** http://math.ucr.edu/home/baez/qg-winter2016/
2016 birželio 15 d., 08:05 atliko AndriusKulikauskas -
Pridėtos 6-7 eilutės:

Matematikos išsiaiškinimo būdus išryškinti. Išryškinti matematiką išreiškiančią Dievo šokį ir sandaras. Iškelti matematikos visumos pagrindus. Išmąstyti fizikos išsiaiškinimo būdus ir susieti su matematimos išsiaiškinimo būdais.
2016 birželio 15 d., 07:59 atliko AndriusKulikauskas -
Pridėta 79 eilutė:
** http://math.ucr.edu/home/baez/rosetta.pdf
Pridėta 82 eilutė:
** [[http://www.j-paine.org/make_category_theory_intuitive.html | Make category theory intuitive]]
2016 birželio 15 d., 00:22 atliko AndriusKulikauskas -
Pridėta 79 eilutė:
** [[http://arxiv.org/pdf/1302.6946v3 | Category theory for scientists]]
2016 birželio 14 d., 10:43 atliko AndriusKulikauskas -
Pakeista 78 eilutė iš:
** http://arxiv.org/abs/0908.2469 a lre-history of n-catevorical physics by John Baez
į:
** [[http://arxiv.org/abs/0908.2469 | A Pre-history of n-categorical Physics]] by John Baez
2016 birželio 14 d., 00:26 atliko AndriusKulikauskas -
Pridėta 78 eilutė:
** http://arxiv.org/abs/0908.2469 a lre-history of n-catevorical physics by John Baez
2016 birželio 11 d., 23:08 atliko AndriusKulikauskas -
Pridėtos 55-56 eilutės:

homology - holes - what is not there - thus a topic for explicit vs. implicit math
2016 birželio 09 d., 12:48 atliko AndriusKulikauskas -
Pridėta 54 eilutė:
* [[Characteristic class]] of different kinds are related to the classical linear groups.
2016 birželio 05 d., 11:33 atliko AndriusKulikauskas -
Pridėtos 21-22 eilutės:

Study the [[https://en.wikipedia.org/wiki/Finite_field | finite field GF(8)]] and relate it to the divisions of everything.
2016 gegužės 30 d., 23:03 atliko AndriusKulikauskas -
Pridėta 59 eilutė:
** [[https://en.wikipedia.org/wiki/Coxeter_group | Finite Coxeter group properties]] žiūrėti lentelę
2016 gegužės 30 d., 16:45 atliko AndriusKulikauskas -
Pakeista 1 eilutė iš:
Žr. [[http://www.selflearners.net/wiki/Math/Math | SelfLearners/Math]], [[Tensor]], [[Matematikos rūmai]], [[Matematikos grožis]]
į:
Žr. [[http://www.selflearners.net/wiki/Math/Math | SelfLearners/Math]], [[Tensor]], [[Simplex]], [[Matematikos rūmai]], [[Matematikos grožis]]
2016 gegužės 25 d., 16:15 atliko AndriusKulikauskas -
Pridėta 329 eilutė:
* Gaussian binomial coefficients [[http://math.stackexchange.com/questions/214065/proving-q-binomial-identities | interpretation related to Young tableaux]]
2016 gegužės 24 d., 16:19 atliko AndriusKulikauskas -
Pridėtos 135-136 eilutės:
* Is a set simply an equivalence class, in some sense? For example, the set is unordered but everything is labeled so that it could be ordered.
* Standard foundations - need to "label" and then "unlabel" (create an equivalence class). Why? Isn't that a lie?
2016 gegužės 24 d., 12:43 atliko AndriusKulikauskas -
Pakeista 326 eilutė iš:
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.
į:
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).
2016 gegužės 24 d., 12:40 atliko AndriusKulikauskas -
Pakeista 326 eilutė iš:
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.
į:
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.
2016 gegužės 24 d., 12:15 atliko AndriusKulikauskas -
Pakeista 326 eilutė iš:
We "understand" the simplex in Cartesian dimensions but think it in simplex dimensions.
į:
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.
2016 gegužės 24 d., 12:08 atliko AndriusKulikauskas -
Pridėta 326 eilutė:
We "understand" the simplex in Cartesian dimensions but think it in simplex dimensions.
2016 gegužės 24 d., 11:58 atliko AndriusKulikauskas -
Pridėtos 323-325 eilutės:

* Totally independent dimensions: Cartesian
* Totally dependent dimensions: simplex
2016 gegužės 24 d., 11:36 atliko AndriusKulikauskas -
Pridėta 353 eilutė:
** [[http://www.geometrictools.com/Documentation/CentersOfSimplex.pdf | Centers of Simplex]]
2016 gegužės 24 d., 10:59 atliko AndriusKulikauskas -
Pridėta 72 eilutė:
** john baez spans in quantum theory
2016 gegužės 24 d., 10:48 atliko AndriusKulikauskas -
Pakeista 351 eilutė iš:
* Simplex, interpret the -1 face. Sometimes considered the empty set. Attention which is free. Interpret (a+x)**n.
į:
* Simplex, interpret the -1 face. Sometimes considered the empty set. Attention which is free. It is the center of the simplex. You can always add a new center to imagine the next simplex but in the current dimensions. Interpret (a+x)**n. It is (implicit + explicit)**n dimensions.
2016 gegužės 24 d., 10:41 atliko AndriusKulikauskas -
Pakeistos 351-354 eilutės iš
* Simplex, the -1 face. Attention which is free. Interpret (a+x)**n.
į:
* Simplex, interpret the -1 face. Sometimes considered the empty set. Attention which is free. Interpret (a+x)**n.

Septynerybė aštuonerybė
* triangle 1 unknown 3 vertices +3 edges +1 whole
2016 gegužės 24 d., 10:35 atliko AndriusKulikauskas -
Pakeistos 349-352 eilutės iš
į:
* Dievas išeina už savęs: 3 matai -> 2 matai -> (flip to dual) 1 matas -> 0 matas (taškas: gera širdis).
* The trivial tensor: T00. It is a zero-dimensional array, thus a scalar. An array is a perspective, and so it is having no perspective. A scalar is "spirit". Thus it is spirit with no perspective.
* Simplex, the -1 face. Attention which is free. Interpret (a+x)**n.
Ištrintos 381-385 eilutės:

Dievas:
* Dievas išeina už savęs: 3 matai -> 2 matai -> (flip to dual) 1 matas -> 0 matas (taškas: gera širdis).
* The trivial tensor: T00. It is a zero-dimensional array, thus a scalar. An array is a perspective, and so it is having no perspective. A scalar is "spirit". Thus it is spirit with no perspective.
* Simplex, the -1 face.
2016 gegužės 24 d., 10:28 atliko AndriusKulikauskas -
Pridėta 383 eilutė:
* Simplex, the -1 face.
2016 gegužės 23 d., 11:40 atliko AndriusKulikauskas -
Pakeistos 79-80 eilutės iš
į:
* Model theory. Taylor Dupuy youtube
Pridėta 348 eilutė:
* [[https://www.youtube.com/watch?v=1XRna0vUYdo | field of one element video]]
2016 gegužės 23 d., 10:35 atliko AndriusKulikauskas -
Pridėta 263 eilutė:
* [[http://math.ucr.edu/home/baez/week201.html | Galois group of C/R]]
2016 gegužės 23 d., 10:11 atliko AndriusKulikauskas -
Pridėta 132 eilutė:
* [[https://golem.ph.utexas.edu/category/2015/02/concepts_of_sameness_part_1.html | Baez on sameness]]
2016 gegužės 22 d., 12:14 atliko AndriusKulikauskas -
Pakeistos 7-10 eilutės iš
interpret the [[https://en.m.wikipedia.org/wiki/Simplex | binomial theorem for simplexes]].
į:
* interpret the [[https://en.m.wikipedia.org/wiki/Simplex | binomial theorem for simplexes]].
* what does it mean that the -1 simplex is the empty set? the spirit?
* what does it mean that a point is the marked opposite for the empty set?
* how does this come up in symplicial homology?
2016 gegužės 22 d., 12:08 atliko AndriusKulikauskas -
Pakeista 7 eilutė iš:
interpret the binomial theorem for simplexes
į:
interpret the [[https://en.m.wikipedia.org/wiki/Simplex | binomial theorem for simplexes]].
2016 gegužės 22 d., 12:07 atliko AndriusKulikauskas -
Pridėtos 6-7 eilutės:

interpret the binomial theorem for simplexes
2016 gegužės 22 d., 09:06 atliko AndriusKulikauskas -
Pridėtos 337-344 eilutės:

Dievas
* Field of one element. Roots of unity = divisions of everything?

Unmarked opposite
* turinys = raiška. "Those things are which show themselves to be." buvimo pagrindas
* inner 2-cycle, kurio paprastai nebūna.
* complex numbers i=j iš kurio atsiveria 1<>-1, i<>-i. Paprastai i -> j -> i ... banguoja, o šitą sustabdžius gaunasi +1 +1 +1 +1 amžinai ir atitinkamai -1-1-1-1 amžinai.
2016 gegužės 21 d., 01:12 atliko AndriusKulikauskas -
Pridėta 311 eilutė:
https://en.m.wikipedia.org/wiki/Field_with_one_element
2016 gegužės 21 d., 01:11 atliko AndriusKulikauskas -
2016 gegužės 21 d., 00:44 atliko AndriusKulikauskas -
Pakeista 3 eilutė iš:
Žiūriu Representation of Geometry paskaita. 1:10
į:
Žiūriu Representation of Geometry paskaita. II. 0:29
2016 gegužės 20 d., 15:19 atliko AndriusKulikauskas -
Pakeista 3 eilutė iš:
Žiūriu Representation of Geometry paskaita. ps.57
į:
Žiūriu Representation of Geometry paskaita. 1:10
2016 gegužės 20 d., 15:04 atliko AndriusKulikauskas -
Pridėtos 2-3 eilutės:

Žiūriu Representation of Geometry paskaita. ps.57
2016 gegužės 20 d., 15:01 atliko AndriusKulikauskas -
Pridėtos 119-120 eilutės:

A finite set can be thought of as a finite dimensional vector space over the field with one element. But no such field exists!
2016 gegužės 20 d., 14:45 atliko AndriusKulikauskas -
Pridėta 118 eilutė:
* Dflags explain how to fit a lower dimensional vector space into a higher dimensional vector space.
2016 gegužės 17 d., 12:34 atliko AndriusKulikauskas -
Pakeistos 288-289 eilutės iš
* [[https://en.m.wikipedia.org/wiki/Unitary_group#2-out-of-3_property | two out of three property]]
į:
* [[https://en.m.wikipedia.org/wiki/Unitary_group#2-out-of-3_property | two out of three property]] 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 skew-symmetric (symplectic)—and these are related by the complex structure (which is the compatibility).
2016 gegužės 17 d., 12:24 atliko AndriusKulikauskas -
Pridėta 288 eilutė:
* [[https://en.m.wikipedia.org/wiki/Unitary_group#2-out-of-3_property | two out of three property]]
2016 gegužės 17 d., 12:06 atliko AndriusKulikauskas -
Pridėtos 282-288 eilutės:

Lie groups and Lie algebras
* ways of breaking up an identity into two elements that are inverses of each other
* orthogonal: symmetric transposes of each other
* unitary: conjugate transposes of each other
* symplectic: antisymmetric transposes of each other
2016 gegužės 17 d., 04:08 atliko AndriusKulikauskas -
Pakeistos 525-527 eilutės iš
[[http://www.lanet.lv/miv/ | Matematikos žodynas anglų, lietuvių ir kitomis kalbomis]]
į:
[[http://www.lanet.lv/miv/ | Matematikos žodynas anglų, lietuvių ir kitomis kalbomis]]

[[http://www.ms.lt/derlius/AndriusKulikauskasThesis.pdf | Symmetric Functions of the Eigenvalues of a Matrix
]]
2016 gegužės 16 d., 14:21 atliko AndriusKulikauskas -
Pridėta 225 eilutė:
* AGT correspondence
2016 gegužės 16 d., 10:50 atliko AndriusKulikauskas -
Pakeistos 299-301 eilutės iš
Mandelbrot, Julia sets
* http://mathforum.org/kb/message.jspa?messageID=22222
* [[http://link.springer.com/book/10.1007%2F978-3-319-10094-4 | The Problem of Catalan]]
į:
[[Catalan]], Mandelbrot, Julia sets
2016 gegužės 16 d., 10:21 atliko AndriusKulikauskas -
Pridėta 301 eilutė:
* [[http://link.springer.com/book/10.1007%2F978-3-319-10094-4 | The Problem of Catalan]]
2016 gegužės 15 d., 23:42 atliko AndriusKulikauskas -
Ištrintos 295-298 eilutės:

Mandelbrot, Julia sets
* http://mathforum.org/kb/message.jspa?messageID=22222
Pridėtos 298-301 eilutės:

Mandelbrot, Julia sets
* http://mathforum.org/kb/message.jspa?messageID=22222
Pridėtos 393-395 eilutės:

Walks on trees
* Julia sets
2016 gegužės 15 d., 23:34 atliko AndriusKulikauskas -
Pridėtos 296-299 eilutės:

Mandelbrot, Julia sets
* http://mathforum.org/kb/message.jspa?messageID=22222
2016 gegužės 15 d., 06:34 atliko AndriusKulikauskas -
Pakeista 86 eilutė iš:
** Use of units (every answer is an amount and a unit) implicitly and explicitly. d/dx is vector (unit - denominator), dX is covector (amount - numerator). "Applying the unit" (such as "tenth") is a vector field, maps from a scalar field to a scalar field, from 3 to 0.3, not necessarily Cartesian. "Dropping the unit" is the covector field, maps us from the vector field to the scalar field, from 0.3 to 3, that is necessarily Cartesian. Units and scales can be most confusing because they are knowledge that is supposed, assumed, in the background. Consider conversion of units. Implicitness and explicitness of units.
į:
** Use of units (every answer is an amount and a unit) implicitly and explicitly. d/dx is vector (unit - denominator), dX is covector (amount - numerator). "Applying the unit" (such as "tenth") is a vector field, maps from a scalar field to a scalar field, from 3 to 0.3, not necessarily Cartesian. "Dropping the unit" is the covector field, maps us from the vector field to the scalar field, from 0.3 to 3, that is necessarily Cartesian. Units and scales can be most confusing because they are knowledge that is supposed, assumed, in the background. Consider conversion of units. Implicitness and explicitness of units. Vectors/units are lists "list different units". Covectors/amounts are distributions "combine like units". Implicitness is a sign of premathematics, explicitness of postmathematics.
2016 gegužės 15 d., 05:45 atliko AndriusKulikauskas -
Pakeista 1 eilutė iš:
Žr. [[http://www.selflearners.net/wiki/Math/Math | SelfLearners/Math]], [[Tensor]], [[Matematikos rūmai]]
į:
Žr. [[http://www.selflearners.net/wiki/Math/Math | SelfLearners/Math]], [[Tensor]], [[Matematikos rūmai]], [[Matematikos grožis]]
2016 gegužės 14 d., 17:52 atliko AndriusKulikauskas -
Pridėta 89 eilutė:
** Tensor symmetry: Wigner-Eckart.
Pridėta 253 eilutė:
* 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.
2016 gegužės 14 d., 17:50 atliko AndriusKulikauskas -
Pridėta 88 eilutė:
** Partial derivatives are explicit, total derivatives implicit - this distinction between explicit and implicit.
2016 gegužės 14 d., 11:27 atliko AndriusKulikauskas -
Pakeistos 39-46 eilutės iš
* Add time to the diagram.
* Indicate math structures, the main objects of study.
* Padalinti number theory into its padalinius: analytic number theory, algebraic number theory, diophantine geometry, probabilistic number theory, arithmetic combinatorics, computational number theory. Create a region for number theory, not a separate node.
* Algebraic number theory is Galois theory.
* Class field theory is a branch of algebraic number theory.
* Lie groups depend on differential topology.
* Periodic functions, elliptic functions, modular forms, automorphic forms.
* Orthogonal polynomials.
į:
* Add time to the diaram
* catalan numbers are related to semantics and to the generating function of the mandelbtot set
2016 gegužės 13 d., 11:54 atliko AndriusKulikauskas -
Pridėta 92 eilutė:
** Use of units (every answer is an amount and a unit) implicitly and explicitly. d/dx is vector (unit - denominator), dX is covector (amount - numerator). "Applying the unit" (such as "tenth") is a vector field, maps from a scalar field to a scalar field, from 3 to 0.3, not necessarily Cartesian. "Dropping the unit" is the covector field, maps us from the vector field to the scalar field, from 0.3 to 3, that is necessarily Cartesian. Units and scales can be most confusing because they are knowledge that is supposed, assumed, in the background. Consider conversion of units. Implicitness and explicitness of units.
2016 gegužės 12 d., 18:05 atliko AndriusKulikauskas -
Pakeista 83 eilutė iš:
Triviality: Distinguishing what is trivial from what is nontrivial. What is assumed or understood.
į:
Triviality: Distinguishing what is trivial from what is nontrivial. What is assumed or understood. Prerequisites for duality.
Pridėtos 91-92 eilutės:
** Tensors relate passive and active transformations, see Penrose. Tensors perhaps are related to breakdown in terms of positive and negative eigenvalues, see Orthogonal group and symmetry matrices.
** Tensors are required for symmtery and invariants. And duality in general. And beauty? and the related topologies?
Pakeista 119 eilutė iš:
* Tensors give the embedding of a lower dimension into a higher dimension. Taip pat tensoriai sieja erdvę ir jos papildinį, kaip kad gyvybę ir meilę. Tai vyksta vektoriais (tangent space) ir kovektoriais (normal space?) Tad geometrijos pagrindas būtų Tensors over a ring. Kovektoriai išsako idealią bazę. Tensorius susidaro iš kovektorių ir kokovektorių. Ir tie, ir tie yra tiesiniai funkcionalai. Tikai baigtinių dimensijų vektorių erdvėse kokovektoriai tolygūs vektoriams. Tensors relate passive and active transformations, see Penrose. Tensors perhaps are related to breakdown in terms of positive and negative eigenvalues, see Orthogonal group and symmetry matrices.
į:
* Tensors give the embedding of a lower dimension into a higher dimension. Taip pat tensoriai sieja erdvę ir jos papildinį, kaip kad gyvybę ir meilę. Tai vyksta vektoriais (tangent space) ir kovektoriais (normal space?) Tad geometrijos pagrindas būtų Tensors over a ring. Kovektoriai išsako idealią bazę. Tensorius susidaro iš kovektorių ir kokovektorių. Ir tie, ir tie yra tiesiniai funkcionalai. Tikai baigtinių dimensijų vektorių erdvėse kokovektoriai tolygūs vektoriams.
2016 gegužės 12 d., 16:15 atliko AndriusKulikauskas -
Pakeista 117 eilutė iš:
* Tensors give the embedding of a lower dimension into a higher dimension. Taip pat tensoriai sieja erdvę ir jos papildinį, kaip kad gyvybę ir meilę. Tai vyksta vektoriais (tangent space) ir kovektoriais (normal space?) Tad geometrijos pagrindas būtų Tensors over a ring. Kovektoriai išsako idealią bazę. Tensorius susidaro iš kovektorių ir kokovektorių. Ir tie, ir tie yra tiesiniai funkcionalai. Tikai baigtinių dimensijų vektorių erdvėse kokovektoriai tolygūs vektoriams.
į:
* Tensors give the embedding of a lower dimension into a higher dimension. Taip pat tensoriai sieja erdvę ir jos papildinį, kaip kad gyvybę ir meilę. Tai vyksta vektoriais (tangent space) ir kovektoriais (normal space?) Tad geometrijos pagrindas būtų Tensors over a ring. Kovektoriai išsako idealią bazę. Tensorius susidaro iš kovektorių ir kokovektorių. Ir tie, ir tie yra tiesiniai funkcionalai. Tikai baigtinių dimensijų vektorių erdvėse kokovektoriai tolygūs vektoriams. Tensors relate passive and active transformations, see Penrose. Tensors perhaps are related to breakdown in terms of positive and negative eigenvalues, see Orthogonal group and symmetry matrices.
2016 gegužės 12 d., 14:30 atliko AndriusKulikauskas -
Pridėta 107 eilutė:
* Normality is a key tool for understanding a subworld unto itself.
2016 gegužės 12 d., 14:18 atliko AndriusKulikauskas -
Pakeistos 93-94 eilutės iš
*
į:
* Rootedness in a world, our world. Partial world. Our relationsip with the world.
Pakeista 104 eilutė iš:
* Worlds unto themselves. Wholeness.
į:
* Worlds unto themselves. Wholeness. Total world with or without us.
2016 gegužės 12 d., 14:15 atliko AndriusKulikauskas -
Pakeistos 93-94 eilutės iš
į:
*
Pakeista 96 eilutė iš:
* Squares and square roots and distances and metrics. Pythagorean theorem.
į:
* Square numbers and square roots and distances and metrics. Pythagorean theorem.
Pridėtos 103-109 eilutės:
* Infinite sequences
* Worlds unto themselves. Wholeness.
* Rotations, reflections.
* Complex numbers

In between
* Stitching: continuity, extension of domain, self superposition
2016 gegužės 12 d., 13:54 atliko AndriusKulikauskas -
Pridėta 88 eilutė:
* Matrix, array
Pakeistos 92-93 eilutės iš
į:
* Rectangles, rectangular areas and volumes
Pakeistos 96-97 eilutės iš
* Geometry.
į:
* Triangles and Geometry.
* Circles and spheres
2016 gegužės 12 d., 13:50 atliko AndriusKulikauskas -
Pakeista 83 eilutė iš:
Triviality: Distinguishing what is trivial from what is nontrivial.
į:
Triviality: Distinguishing what is trivial from what is nontrivial. What is assumed or understood.
Pakeista 86 eilutė iš:
* Fractions. Equivalence classes. Duality of numerator and denominator, vector and covector. Confusion of numerator as answer or as amount. Identification of denominator as unit. Relation to coordinate systems.
į:
* Fractions. Equivalence classes. Duality of numerator and denominator, vector and covector. Confusion of numerator as answer or as amount. Identification of denominator as unit. Relation to coordinate systems. Ambiguity inherent in expression. Use of explicit coordinate (denominator) but implicit meaning.
Pakeistos 90-91 eilutės iš
į:
* Note link to divisibility of numbers and prime decomposition.
Pridėtos 93-95 eilutės:
* Squares and square roots and distances and metrics. Pythagorean theorem.
* Geometry.
* Real numbers
2016 gegužės 12 d., 13:45 atliko AndriusKulikauskas -
Pridėta 86 eilutė:
* Fractions. Equivalence classes. Duality of numerator and denominator, vector and covector. Confusion of numerator as answer or as amount. Identification of denominator as unit. Relation to coordinate systems.
2016 gegužės 12 d., 12:42 atliko AndriusKulikauskas -
Pridėtos 82-93 eilutės:

Triviality: Distinguishing what is trivial from what is nontrivial.
* Integers
* Rationals. Proportionality.
* Linear (algebra), linear functions, linearity (derivatives)
* Scalars
* Tensors

Nontrivial
* Platonic solids
* Conic sections
* Power series
2016 gegužės 11 d., 12:10 atliko AndriusKulikauskas -
Pridėta 226 eilutė:
* Meilė (simetrija) įsteigia nemirtingumą (invariant).
2016 gegužės 11 d., 09:44 atliko AndriusKulikauskas -
Pakeista 66 eilutė iš:
** [[https://en.wikipedia.org/wiki/Amplituhedron | Amplituhedron]], [[http://www.preposterousuniverse.com/blog/2014/03/31/guest-post-jaroslav-trnka-on-the-amplituhedron/ | post by Jaroslav Trnka]] related to walks on trees?
į:
** [[https://en.wikipedia.org/wiki/Amplituhedron | Amplituhedron]], [[http://www.preposterousuniverse.com/blog/2014/03/31/guest-post-jaroslav-trnka-on-the-amplituhedron/ | post by Jaroslav Trnka]] related to walks on trees? [[http://susy2013.ictp.it/video/05_Friday/2013_08_30_Arkani-Hamed_4-3.html | video]]
2016 gegužės 11 d., 09:42 atliko AndriusKulikauskas -
Pakeistos 64-66 eilutės iš
** [[http://www.nbi.dk/GroupTheory/ | Group theory]]
į:
** [[http://www.nbi.dk/GroupTheory/ | Group theory]], representation theory of Symmetric group
** [[https://en.wikipedia.org/wiki/Symmetric_space | Symmetric space]], including modern classification by Huang and Leung.
** [[https://en.wikipedia.org/wiki/Amplituhedron | Amplituhedron]], [[http://www.preposterousuniverse.com/blog/2014/03/31/guest-post-jaroslav-trnka-on-the-amplituhedron/ | post by Jaroslav Trnka]] related to walks on trees?
2016 gegužės 11 d., 08:59 atliko AndriusKulikauskas -
Pakeista 327 eilutė iš:
* Perspectives are (multidimensional) arrays. The number of dimensions is the number of divisions of everything.
į:
* Perspectives are (multidimensional) arrays. The number of array dimensions is the number of divisions of everything.
2016 gegužės 11 d., 08:57 atliko AndriusKulikauskas -
Pakeistos 316-317 eilutės iš
į:
* The trivial tensor: T00. It is a zero-dimensional array, thus a scalar. An array is a perspective, and so it is having no perspective. A scalar is "spirit". Thus it is spirit with no perspective.
Pridėta 327 eilutė:
* Perspectives are (multidimensional) arrays. The number of dimensions is the number of divisions of everything.
2016 gegužės 11 d., 08:25 atliko AndriusKulikauskas -
Pridėta 205 eilutė:
* Symplectic form is related to complexification and also the linking of losition and momentum.
2016 gegužės 10 d., 17:36 atliko AndriusKulikauskas -
Pakeistos 47-48 eilutės iš
* Consider [[https://en.m.wikipedia.org/wiki/Mathematics,_Form_and_Function |Mathematics, Form and Function]]
į:
* Consider [[https://en.m.wikipedia.org/wiki/Mathematics,_Form_and_Function |Mathematics, Form and Function]] by Aleksandrov, Kolmogorov, Lavrentev
* Mathematics: Its Content, Methods and Meaning
2016 gegužės 10 d., 12:19 atliko AndriusKulikauskas -
Pridėta 62 eilutė:
** http://phyweb.lbl.gov/~rncahn/www/liealgebras/texall.pdf
2016 gegužės 10 d., 12:10 atliko AndriusKulikauskas -
Pridėta 322 eilutė:
* Ar požiūriai yra neasociatyvūs?
2016 gegužės 10 d., 11:52 atliko AndriusKulikauskas -
Pridėta 62 eilutė:
** [[http://www.nbi.dk/GroupTheory/ | Group theory]]
2016 gegužės 10 d., 11:40 atliko AndriusKulikauskas -
Pridėta 61 eilutė:
** [[http://math.ucr.edu/home/baez/octonions/ | Octonions]]
2016 gegužės 10 d., 11:08 atliko AndriusKulikauskas -
Pridėta 60 eilutė:
** [[http://arxiv.org/pdf/0902.0431v1 | Exceptional Lie groups]]
2016 gegužės 10 d., 10:53 atliko AndriusKulikauskas -
Pridėta 58 eilutė:
** [[http://math.ucr.edu/home/baez/qg-fall2008/ | Lie theory through examples]]
2016 gegužės 09 d., 23:10 atliko AndriusKulikauskas -
Pakeistos 476-478 eilutės iš
Related pages

* The QED Project had a manifesto which addressed the big picture in math.
į:
[[http://www.lanet.lv/miv/ | Matematikos žodynas anglų, lietuvių ir kitomis kalbomis]]
2016 gegužės 09 d., 15:22 atliko AndriusKulikauskas -
Pridėta 58 eilutė:
2016 gegužės 09 d., 15:06 atliko AndriusKulikauskas -
Pridėtos 271-277 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.
2016 gegužės 09 d., 13:19 atliko AndriusKulikauskas -
Pridėtos 282-297 eilutės:

[[http://math.ucr.edu/home/baez/week82.html | Clifford algebra periodicity]]
* C0 R
* C1 C
* C2 H
* C3 H + H
* C4 H(2)
* C5 C(4)
* C6 R(8)
* C7 R(8) + R(8)
* C8 R(16)
''C_{n+8} consists of 16 x 16 matrices with entries in Cn ! For a proof you might try

2) H. Blaine Lawson, Jr. and Marie-Louise Michelson, "Spin Geometry", Princeton U. Press, Princeton, 1989. or 3) Dale Husemoller, "Fibre Bundles", Springer-Verlag, Berlin, 1994. These books also describe some of the amazing consequences of this periodicity phenomenon. The topology of n-dimensional manifolds is very similar to the topology of (n+8)-dimensional manifolds in some subtle but important ways!'' Physics of fermions.

[[http://math.ucr.edu/home/baez/week61.html | Introduction to rotation groups]] '''Triality of octonions.''' ''More generally, it turns out that the representation theory of Spin(n) depends strongly on whether n is even or odd. When n is even (and bigger than 2), it turns out that Spin(n) has left-handed and right-handed spinor representations, each of dimension 2^{n/2 - 1}. When n is odd there is just one spinor representation. Of course, there is always the representation of Spin(n) coming from the vector representation of SO(n), which is n-dimensional. This leads to something very curious. If you are an ordinary 4-dimensional physicist you undoubtedly tend to think of spinors as "smaller" than vectors, since the spinor representations are 2-dimensional, while the vector representation is 3-dimensional. However, in general, when the dimension n of space (or spacetime) is even, the dimension of the spinor representations is 2^(n/2 - 1), while that of the vector representation is n, so after a while the spinor representation catches up with the vector representation and becomes bigger! This is a little bit curious, or at least it may seem so at first, but what's really curious is what happens exactly when the spinor representation catches up with the vector representation. That's when 2^(n/2 - 1) = n, or n = 8. The group Spin(8) has three 8-dimensional irreducible representations: the vector, left-handed spinor, and right-handed spinor representation. While they are not equivalent to each other, they are darn close; they are related by a symmetry of Spin(8) called "triality". And, to top it off, the octonions can be seen as a kind of spin-off of this triality symmetry... as one might have guessed, from all this 8-dimensional stuff. One can, in fact, describe the product of octonions in these terms. So now let's dig in a bit deeper and describe how this triality business works. For this, unfortunately, I will need to assume some vague familiarity with exterior algebras, Clifford algebras, and their relation to the spin group. But we will have a fair amount of fun getting our hands on a 24-dimensional beast called the Chevalley algebra, which contains the vector and spinor representations of Spin(8)!''
2016 gegužės 09 d., 13:08 atliko AndriusKulikauskas -
Pakeista 281 eilutė iš:
* Žiūrėk taip pat [[https://en.wikipedia.org/wiki/Hopf_fibration Hopf fibration]], [[https://en.wikipedia.org/wiki/Hopf_invariant | Hopf invariant]] ir Adam's theorem. [[https://en.wikipedia.org/wiki/Homotopy_groups_of_spheres | Homotopy group of spheres]].
į:
* Žiūrėk taip pat [[https://en.wikipedia.org/wiki/Hopf_fibration Hopf fibration]], [[https://en.wikipedia.org/wiki/Hopf_invariant | Hopf invariant]] ir Adam's theorem. [[https://en.wikipedia.org/wiki/Homotopy_groups_of_spheres | Homotopy group of spheres]]. [[https://en.wikipedia.org/wiki/Clifford_parallel | Clifford paralells]] ir quaternions.
2016 gegužės 09 d., 13:07 atliko AndriusKulikauskas -
Pakeista 281 eilutė iš:
* Žiūrėk taip pat [[https://en.wikipedia.org/wiki/Hopf_fibration Hopf fibration]], [[https://en.wikipedia.org/wiki/Hopf_invariant | Hopf invariant]] ir Adam's theorem.
į:
* Žiūrėk taip pat [[https://en.wikipedia.org/wiki/Hopf_fibration Hopf fibration]], [[https://en.wikipedia.org/wiki/Hopf_invariant | Hopf invariant]] ir Adam's theorem. [[https://en.wikipedia.org/wiki/Homotopy_groups_of_spheres | Homotopy group of spheres]].
2016 gegužės 09 d., 13:05 atliko AndriusKulikauskas -
Pridėta 281 eilutė:
* Žiūrėk taip pat [[https://en.wikipedia.org/wiki/Hopf_fibration Hopf fibration]], [[https://en.wikipedia.org/wiki/Hopf_invariant | Hopf invariant]] ir Adam's theorem.
2016 gegužės 09 d., 13:00 atliko AndriusKulikauskas -
Pridėtos 276-280 eilutės:
* [[http://www.math.illinois.edu/K-theory/handbook/1-111-138.pdf | Max Karoubi vadovėlis apie Bott periodicity]]
* palyginti susijusias Lie grupes (ir jų ryšį su gaubliu) su požiūrių permainomis
* Bott periodicity turėtų būti susijęs su aštuonerybės sugriuvimu prieštaravimu
* susipažinti su Clifford algebra ir clock shift veiksmais
* Max Karoubi savo video paskaitoje paminėjo loop lygtį žiedams kurioje R,C,H,H' ir epsilon = +/-1 gaunasi 10 homotopy equivalences. Kodėl 10? 8+2=10? ar 6+4=10, dešimt Dievo įsakymų?
2016 gegužės 07 d., 23:35 atliko AndriusKulikauskas -
Pakeista 1 eilutė iš:
Žr. [[http://www.selflearners.net/wiki/Math/Math | SelfLearners/Math]], [[Tensor]]
į:
Žr. [[http://www.selflearners.net/wiki/Math/Math | SelfLearners/Math]], [[Tensor]], [[Matematikos rūmai]]
2016 gegužės 06 d., 08:03 atliko AndriusKulikauskas -
Pridėtos 273-275 eilutės:

* Bott periodicity [[http://math.ucr.edu/home/baez/week105.html | John Baez]]
2016 gegužės 04 d., 11:53 atliko AndriusKulikauskas -
Pridėtos 50-52 eilutės:

svarbūs pavyzdžiai
* https://en.m.wikipedia.org/wiki/Möbius_transformation
2016 gegužės 03 d., 23:59 atliko AndriusKulikauskas -
Pakeista 342 eilutė iš:
* What is Good Mathematics? Terrence Tao.
į:
* [[https://arxiv.org/pdf/math/0702396v1.pdf | What is Good Mathematics?]] Terrence Tao.
2016 gegužės 03 d., 13:22 atliko AndriusKulikauskas -
Pridėtos 233-238 eilutės:
Lie Bracket:
* Remiasi tuo, kad summing over permutations of 1 yield 0. [x,x]=0
* Summing over permutations of 2 yields 0. [x,y]+[y,x]=0
* Summing over permutations of 3 yields 0. [x,[y,z]] + [y,[z,x]] + [z,[x,y]]=0
That's true writing out [x,y]=xy-yx and summing out you get a positive and a negative term for each permutation. But also true in the brackets directly permuting cyclically. What would it look like to sum over permutations of 4?
Pridėtos 308-310 eilutės:

24
* 4 netroškimai: 6x4=24 ir dar 0-inis požiūris ir 7-as požiūris. Iš viso 24+2=26 kaip Monster grupėje.
2016 gegužės 02 d., 21:32 atliko AndriusKulikauskas -
Pakeista 1 eilutė iš:
Žr. [[http://www.selflearners.net/wiki/Math/Math | SelfLearners/Math]]
į:
Žr. [[http://www.selflearners.net/wiki/Math/Math | SelfLearners/Math]], [[Tensor]]
2016 gegužės 02 d., 21:23 atliko AndriusKulikauskas -
Pridėta 57 eilutė:
* [[http://math.ucr.edu/home/baez/ | John Baez Network Theory]]
2016 gegužės 02 d., 21:14 atliko AndriusKulikauskas -
Pridėta 60 eilutė:
** [[https://www.youtube.com/watch?v=FKSbWiGMY30 | Theory X and Life]]
2016 gegužės 02 d., 21:12 atliko AndriusKulikauskas -
Pridėtos 55-56 eilutės:
* Rekursyvinę funkcijų teoriją
** Yates-Index theorem
Pridėtos 59-60 eilutės:
** [[http://math.ucr.edu/home/baez/qg-fall2007/ | Geometric Representation Theory Seminar 2007]] John Baez and James Dolan
** [[https://www.ma.utexas.edu/users/djordan/QGpublic.pdf | Introductory Book for Geometric Representation Theory]]
2016 gegužės 02 d., 21:07 atliko AndriusKulikauskas -
Pridėtos 50-59 eilutės:

'''Ko noriu mokytis'''

* Lie group ir algebra teorijos
* Kategorijų teorija
* [[https://www.msri.org/programs/276 | Geometric Representation Theory]]
** [[https://ncatlab.org/nlab/show/Borel-Weil+theorem | Borel-Weil theorem]]
* Entropija
* Riemann-Zeta funkcijos pagrindus
2016 gegužės 01 d., 14:03 atliko AndriusKulikauskas -
Pridėta 206 eilutė:
* Susieti most "irrationality" su "randomness". Nes ką sužinai nieko daugiau nepasako apie kas liko.
2016 gegužės 01 d., 11:18 atliko AndriusKulikauskas -
Pridėtos 197-199 eilutės:

Field
* Sieja nepažymėtą priešingybę (+) -1, 0, 1 ir pažymėtą priešingybę (x) -1, 1.
2016 balandžio 29 d., 22:56 atliko AndriusKulikauskas -
Pridėtos 195-197 eilutės:
Symmetry
* E8 is the symmetry group of itself. What is the symmetry group of?
Pridėtos 200-202 eilutės:

Skaičius 5
* Golden mean is the "most" irrational of numbers (based on its continued fraction). Consider series of continued fractions... as sequence patterns...
2016 balandžio 29 d., 15:53 atliko AndriusKulikauskas -
Pridėtos 1-2 eilutės:
Žr. [[http://www.selflearners.net/wiki/Math/Math | SelfLearners/Math]]
2016 balandžio 29 d., 13:58 atliko AndriusKulikauskas -
Pridėta 247 eilutė:
* Nėra quantum frequency. Fotonai yra "požiūriai". Bosonai yra "jėgų nešėjai", "santykiai", jie "neegzistuoja".
2016 balandžio 29 d., 13:40 atliko AndriusKulikauskas -
Pridėtos 10-11 eilutės:

Kaip kompleksiniais skaičiais išvesti ir suprasti d/dz (e^z) ?
2016 balandžio 27 d., 15:42 atliko AndriusKulikauskas -
Pridėtos 199-202 eilutės:
Ypatingi skaičiai
* http://math.ucr.edu/home/baez/42.html
* http://math.ucr.edu/home/baez/numbers/
Pakeista 402 eilutė iš:
* The QED Project had a manifesto which addressed the big picture in math.
į:
* The QED Project had a manifesto which addressed the big picture in math.
2016 balandžio 27 d., 14:11 atliko AndriusKulikauskas -
Pridėta 197 eilutė:
* Monster group dydis susijęs su visatos dalelyčių skaičiumi?
2016 balandžio 27 d., 14:10 atliko AndriusKulikauskas -
Pridėta 196 eilutė:
* 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.
2016 balandžio 27 d., 14:01 atliko AndriusKulikauskas -
Pridėtos 366-373 eilutės:

'''Matematikos visuomenės'''

* [[https://ncatlab.org/nlab/show/math%20blogs | nLab visuomenių sąrašas]]
* [[http://www.artofproblemsolving.com/community | Art of Problem Solving: Community]]
* [[https://www.physicsforums.com | Physics forums]]
* [[http://deferentialgeometry.org | Deferential Geometry: Garrett Lisi]]
2016 balandžio 27 d., 13:51 atliko AndriusKulikauskas -
Pridėtos 193-195 eilutės:

Skaičius 24
* John Baez kalba. 24 = 6 (trikampių laukas) x 4 (kvadrato laukas).
2016 balandžio 27 d., 10:49 atliko AndriusKulikauskas -
Pridėta 45 eilutė:
* Wikipedia: Covering spaces play an important role in homotopy theory, harmonic analysis, Riemannian geometry and differential topology. In Riemannian geometry for example, ramification is a generalization of the notion of covering maps. Covering spaces are also deeply intertwined with the study of homotopy groups and, in particular, the fundamental group. An important application comes from the result that, if X is a "sufficiently good" topological space, there is a bijection between the collection of all isomorphism classes of connected coverings of X and the conjugacy classes of subgroups of the fundamental group of X.
2016 balandžio 27 d., 10:46 atliko AndriusKulikauskas -
Pridėta 44 eilutė:
* Wikipedia: The concept of a universal cover was first developed to define a natural domain for the analytic continuation of an analytic function. The idea of finding the maximal analytic continuation of a function in turn led to the development of the idea of Riemann surfaces. The power series defined below is generalized by the idea of a germ. The general theory of analytic continuation and its generalizations are known as sheaf theory.
2016 balandžio 27 d., 10:28 atliko AndriusKulikauskas -
Pridėta 187 eilutė:
* Complex numbers: local = global. ([[https://en.wikipedia.org/wiki/Identity_theorem | Identity theorem]]). Real numbers: local != global.
2016 balandžio 27 d., 09:55 atliko AndriusKulikauskas -
Pridėtos 187-189 eilutės:

Extention of a domain
* [[https://en.wikipedia.org/wiki/Analytic_continuation | Analytic continuation]] - complex numbers - dealing with divergent series.
2016 balandžio 26 d., 17:53 atliko AndriusKulikauskas -
Pridėta 186 eilutė:
* Ar teisinga? Skaičius turėtų rašyti: xr + yi pabrėžti jog tai skiritingi matai. Bet r tampa 1. Vienas matas gali būti "default" ir užtat išbrauktas. Jisai tada tampa "identity". Every answer is an amount and a unit - šis dėsnis paneigtas.
2016 balandžio 26 d., 17:43 atliko AndriusKulikauskas -
Pakeista 285 eilutė iš:
* [[https://mathbeauty.wordpress.com/ | Math Beauty blog]]
į:
Ištrintos 289-321 eilutės:
'''Įdomūs, prasmingi reiškiniai matematikoje'''

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.

Logika
* 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)?

--------------------------

The Big Picture in Mathematics

Mathematics has grown as a discipline to the extent that no single person is able to overview it all. This is a research page to encourage and organize efforts to foster a perspective upon all of mathematics.

Big questions

Several big questions provide new angles on mathematics as an endeavor:

* discovery What are the ways of figuring things out in mathematics? We can study mathematics as an activity by which we create and solve mathematical problems. These techniques are much more limited than the mathematical output which they generate.
* beauty Mathematicians are guided by a sense of beauty. What is meant by beauty? What principles determine it? How does beauty lead to mathematical insight?
* organization How can mathematics be organized so as to survey the most basic concepts from which it arises and understand it in terms of its most fundamental divisions and how they relate to each other?
* education What resources are available to mathematicians that would help them most effectively learn mathematics so as to try to understand it as a whole? How might mathematicians collaborate effectively in trying to understand the big picture?
* insight What are the most fruitful insights in trying to understand mathematics? How can such insights best be stated?
* premathematics What concepts express intuitions that are prior to explicit mathematics and make it possible?
* history How can the history of mathematical discovery inform frameworks for the future development of mathematics?
* humanity What parts or aspects of mathematics are specific to the human mind, body, culture, society, and what might be more broadly meaningful to other species in the universe?

Related pages

* The QED Project had a manifesto which addressed the big picture in math.

---------------------
Pakeistos 347-378 eilutės iš
Investigation: Compare with beauty in chess.
į:
Investigation: Compare with beauty in chess.

'''Įdomūs, prasmingi reiškiniai matematikoje'''

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.

Logika
* 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)?

--------------------------

The Big Picture in Mathematics

Mathematics has grown as a discipline to the extent that no single person is able to overview it all. This is a research page to encourage and organize efforts to foster a perspective upon all of mathematics.

Big questions

Several big questions provide new angles on mathematics as an endeavor:

* discovery What are the ways of figuring things out in mathematics? We can study mathematics as an activity by which we create and solve mathematical problems. These techniques are much more limited than the mathematical output which they generate.
* beauty Mathematicians are guided by a sense of beauty. What is meant by beauty? What principles determine it? How does beauty lead to mathematical insight?
* organization How can mathematics be organized so as to survey the most basic concepts from which it arises and understand it in terms of its most fundamental divisions and how they relate to each other?
* education What resources are available to mathematicians that would help them most effectively learn mathematics so as to try to understand it as a whole? How might mathematicians collaborate effectively in trying to understand the big picture?
* insight What are the most fruitful insights in trying to understand mathematics? How can such insights best be stated?
* premathematics What concepts express intuitions that are prior to explicit mathematics and make it possible?
* history How can the history of mathematical discovery inform frameworks for the future development of mathematics?
* humanity What parts or aspects of mathematics are specific to the human mind, body, culture, society, and what might be more broadly meaningful to other species in the universe?

Related pages

* The QED Project had a manifesto which addressed the big picture in math
.
2016 balandžio 26 d., 17:31 atliko AndriusKulikauskas -
Pridėta 77 eilutė:
* Analysis provides lower and upper bounds on a function or phenomenon which helps define the geometry of this space.
2016 balandžio 26 d., 16:52 atliko AndriusKulikauskas -
Pakeista 74 eilutė iš:
* Switching of "existing" and "nonexisting", for example, edges in a graph. This underlies [[https://en.wikipedia.org/wiki/Ramsey's_theorem | Ramsey's theorem]].
į:
* Switching of "existing" and "nonexisting", for example, edges in a graph. This underlies [[https://en.wikipedia.org/wiki/Ramsey's_theorem | Ramsey's theorem]]. Tao: "the Ramsey-type theorem, each one of which being a different formalisation of the newly gained insight in mathematics that complete disorder is impossible."
2016 balandžio 26 d., 16:46 atliko AndriusKulikauskas -
Pridėta 74 eilutė:
* Switching of "existing" and "nonexisting", for example, edges in a graph. This underlies [[https://en.wikipedia.org/wiki/Ramsey's_theorem | Ramsey's theorem]].
2016 balandžio 26 d., 15:59 atliko AndriusKulikauskas -
Pridėta 286 eilutė:
* [[http://mathoverflow.net/questions/74841/an-example-of-a-beautiful-proof-that-would-be-accessible-at-the-high-school-level | Beautiful, simple proofs]]
2016 balandžio 26 d., 15:48 atliko AndriusKulikauskas -
Pridėtos 267-272 eilutės:
[+Matematikos išsiaiškinimo būdai+]

* [[http://www.cut-the-knot.org/m/ProblemSolving.shtml | Alexander Bogomolny sarašas]]

[+Matematikos grožis+]
Pakeista 285 eilutė iš:
* [[http://www.cut-the-knot.org/manifesto/index.shtml | Cut the Knot Manifesto]] "The peculiar beauty of Mathematics lies in deduction, in the dependency of one fact upon another. The less expected a dependency is, the simpler the facts on which the deduction is based -- the more beautiful is the result."
į:
* Alexander Bogomolny, [[http://www.cut-the-knot.org/manifesto/index.shtml | Cut the Knot Manifesto]] "The peculiar beauty of Mathematics lies in deduction, in the dependency of one fact upon another. The less expected a dependency is, the simpler the facts on which the deduction is based -- the more beautiful is the result."
2016 balandžio 26 d., 15:45 atliko AndriusKulikauskas -
Pridėta 279 eilutė:
* [[http://www.cut-the-knot.org/manifesto/index.shtml | Cut the Knot Manifesto]] "The peculiar beauty of Mathematics lies in deduction, in the dependency of one fact upon another. The less expected a dependency is, the simpler the facts on which the deduction is based -- the more beautiful is the result."
2016 balandžio 26 d., 15:18 atliko AndriusKulikauskas -
Pridėtos 277-278 eilutės:
* [[https://mathbeauty.wordpress.com/ | Math Beauty blog]]
* What is Good Mathematics? Terrence Tao.
2016 balandžio 26 d., 15:03 atliko AndriusKulikauskas -
Pridėtos 274-276 eilutės:

[[https://en.wikipedia.org/wiki/Mathematical_beauty | Mathematical beauty]]
* Wikipedia: In the 1990s, [[https://en.wikipedia.org/wiki/J%C3%BCrgen_Schmidhuber | Jürgen Schmidhuber]] formulated a mathematical theory of observer-dependent subjective beauty based on algorithmic information theory: the most beautiful objects among subjectively comparable objects have short algorithmic descriptions (i.e., Kolmogorov complexity) relative to what the observer already knows. Schmidhuber explicitly distinguishes between beautiful and interesting. The latter corresponds to the first derivative of subjectively perceived beauty: the observer continually tries to improve the predictability and compressibility of the observations by discovering regularities such as repetitions and symmetries and fractal self-similarity. Whenever the observer's learning process (possibly a predictive artificial neural network) leads to improved data compression such that the observation sequence can be described by fewer bits than before, the temporary interestingness of the data corresponds to the compression progress, and is proportional to the observer's internal curiosity reward.
2016 balandžio 26 d., 14:57 atliko AndriusKulikauskas -
Pridėta 73 eilutė:
** [[https://en.m.wikipedia.org/wiki/Adjoint | Adjoint]] bendrai ir [[https://en.wikipedia.org/wiki/Adjoint_functors | Adjoint functors]]. Wikipedia: It can be said that an adjoint functor is a way of giving the most efficient solution to some problem via a method which is formulaic. A construction is most efficient if it satisfies a universal property, and is formulaic if it defines a functor. Universal properties come in two types: initial properties and terminal properties. Since these are dual (opposite) notions, it is only necessary to discuss one of them.
Ištrinta 100 eilutė:
2016 balandžio 26 d., 14:54 atliko AndriusKulikauskas -
Pakeistos 70-72 eilutės iš
** In applications to logic, this then looks like a very general description of negation (that is, proofs run in the opposite direction). If we take the opposite of a lattice, we will find that meets and joins have their roles interchanged. This is an abstract form of De Morgan's laws, or of duality applied to lattices.
** Reversing the direction of inequalities in a partial order. (Partial orders correspond to a certain kind of category in which Hom(A,B) can have at most one element.)
** Fibrations and cofibrations are examples of dual notions in algebraic topology and homotopy theory. In this context, the duality is often called [[https://en.wikipedia.org/wiki/Eckmann%E2%80%93Hilton_duality | Eckmann–Hilton duality]].
į:
** Wikipedia: In applications to logic, this then looks like a very general description of negation (that is, proofs run in the opposite direction). If we take the opposite of a lattice, we will find that meets and joins have their roles interchanged. This is an abstract form of De Morgan's laws, or of duality applied to lattices.
** Wikipedia: Reversing the direction of inequalities in a partial order. (Partial orders correspond to a certain kind of category in which Hom(A,B) can have at most one element.)
** Wikipedia: Fibrations and cofibrations are examples of dual notions in algebraic topology and homotopy theory. In this context, the duality is often called [[https://en.wikipedia.org/wiki/Eckmann%E2%80%93Hilton_duality | Eckmann–Hilton duality]].
2016 balandžio 26 d., 14:53 atliko AndriusKulikauskas -
Pakeista 62 eilutė iš:
* There are two square roots of -1. One we call +i, the other -i, but neither should have priority over the other. Similarly, clockwise and counterclockwise rotations should not be favored. Complex conjugation is a way of asserting this. (Note that the integer +1 is naturally favored over -1. But there is no such natural favoring for i. It is purely conventional, a misleading artificial contrivance.)
į:
* '''Square roots of -i.''' There are two square roots of -1. One we call +i, the other -i, but neither should have priority over the other. Similarly, clockwise and counterclockwise rotations should not be favored. Complex conjugation is a way of asserting this. (Note that the integer +1 is naturally favored over -1. But there is no such natural favoring for i. It is purely conventional, a misleading artificial contrivance.)
Pakeista 65 eilutė iš:
* Morphisms can be organized from left to right or from right to left. The opposite category turns all of the arrows around.
į:
* '''[[Opposite category | Opposite category]]''' Morphisms can be organized from left to right or from right to left. The opposite category turns all of the arrows around.
Pridėta 67 eilutė:
** Monomorphisms ("one-to-one") and epimorphisms (forcing "onto").
Pridėtos 70-72 eilutės:
** In applications to logic, this then looks like a very general description of negation (that is, proofs run in the opposite direction). If we take the opposite of a lattice, we will find that meets and joins have their roles interchanged. This is an abstract form of De Morgan's laws, or of duality applied to lattices.
** Reversing the direction of inequalities in a partial order. (Partial orders correspond to a certain kind of category in which Hom(A,B) can have at most one element.)
** Fibrations and cofibrations are examples of dual notions in algebraic topology and homotopy theory. In this context, the duality is often called [[https://en.wikipedia.org/wiki/Eckmann%E2%80%93Hilton_duality | Eckmann–Hilton duality]].
Pridėta 109 eilutė:
[[https://en.wikipedia.org/wiki/List_of_dualities | List of dualities (Wikipedia)]]
Ištrinta 140 eilutė:
* Eckmann–Hilton duality
Ištrinta 149 eilutė:
* Opposite category
2016 balandžio 26 d., 14:27 atliko AndriusKulikauskas -
Pridėtos 105-157 eilutės:
* Alexander duality
* Alvis–Curtis duality
* Araki duality
* Beta-dual space
* Coherent duality
* De Groot dual
* Dual abelian variety
* Dual basis in a field extension
* Dual bundle
* Dual curve
* Dual (category theory)
* Dual graph
* Dual group
* Dual object
* Dual pair
* Dual polygon
* Dual polyhedron
* Dual problem
* Dual representation
* Dual q-Hahn polynomials
* Dual q-Krawtchouk polynomials
* Dual space
* Dual topology
* Dual wavelet
* Duality (optimization)
* Duality (order theory)
* Duality of stereotype spaces
* Duality (projective geometry)
* Duality theory for distributive lattices
* Dualizing complex
* Dualizing sheaf
* Eckmann–Hilton duality
* Esakia duality
* Fenchel's duality theorem
* Haag duality
* Hodge dual
* Jónsson–Tarski duality
* Lagrange duality
* Langlands dual
* Lefschetz duality
* Local Tate duality
* Opposite category
* Poincaré duality
* Twisted Poincaré duality
* Poitou–Tate duality
* Pontryagin duality
* S-duality (homotopy theory)
* Schur–Weyl duality
* Serre duality
* Stone's duality
* Tannaka–Krein duality
* Verdier duality
2016 balandžio 26 d., 13:13 atliko AndriusKulikauskas -
Pridėta 64 eilutė:
** Normality says conjugate invariancy: gN = Ng.
2016 balandžio 26 d., 12:59 atliko AndriusKulikauskas -
Pridėta 113 eilutė:
* Mano tezė. Jeigu matricą išrašome Jordan canonical form, tai didžiausi ciklai tėra dvejetukai.
2016 balandžio 26 d., 12:56 atliko AndriusKulikauskas -
Pakeista 69 eilutė iš:
** Root systems relate reflections (hyperplanes) and root vectors. Given a root R, reflecting across its hyperplane, every root S is taken to another root -S, and the difference between the two roots is an integer multiple of R.
į:
** Root systems relate reflections (hyperplanes) and root vectors. Given a root R, reflecting across its hyperplane, every root S is taken to another root -S, and the difference between the two roots is an integer multiple of R. But this relates to the commutator sending the differences into the module based on R.
2016 balandžio 26 d., 12:51 atliko AndriusKulikauskas -
Pakeista 69 eilutė iš:
** Root systems relate reflections (hyperplanes) and root vectors.
į:
** Root systems relate reflections (hyperplanes) and root vectors. Given a root R, reflecting across its hyperplane, every root S is taken to another root -S, and the difference between the two roots is an integer multiple of R.
2016 balandžio 25 d., 20:39 atliko AndriusKulikauskas -
Pridėta 71 eilutė:
** This is related to the duality between left and right multiplication. Examples include Polish notation.
Pridėtos 123-125 eilutės:
* The purpose of complex numbers is to define two unmarked opposites (we know them, unfortunately, as "i" and "-i", where one is marked with regard to the other, but in truth they should be both unmarked). The purpose of the real numbers is to provide that context for this unmarkedness. (Is there a simpler way to create it?)
* The quantum world is based on the two unmarked opposites ("i" and "j") as with spin 1/2 particles, "up" and "down". Symmetry breaking - the breaking of the symmetry between "i" and "-i" enforced by complex conjugation - occurs (and is defined to be) when there is a measurement, so that we collapse to the reals, where this symmetry is broken.
* The truth of the heart does not mark the opposites. The truth of the world marks one opposite with regard to the other.
2016 balandžio 25 d., 08:32 atliko AndriusKulikauskas -
Pridėtos 65-67 eilutės:
** Colimits and limits
** Coproducts and products
** Initial and terminal objects
Pridėta 69 eilutė:
** Root systems relate reflections (hyperplanes) and root vectors.
Pridėtos 72-75 eilutės:
* Coxeter groups are built from reflections. Reflections are dualities.
* Any two structures which have a nice map from one to the other have a duality in that you can start from one and go to the other.
** Galois theory: field extensions (solutions of polynomials) and groups
** Lie groups: solutions to differential equations..
Ištrintos 77-78 eilutės:
* Galois theory: field extensions (solutions of polynomials) and groups
* Lie groups: solutions to differential equations..
Ištrintos 81-82 eilutės:
* Colimits and limits
* Coproducts and products
Ištrintos 102-103 eilutės:
* Coxeter groups are built from reflections. Reflections are dualities.
* Any two structures which have a nice map from one to the other have a duality in that you can start from one and go to the other.
2016 balandžio 24 d., 23:14 atliko Andrius Kulikauskas -
Pakeista 62 eilutė iš:
* There are two square roots of -1. One we call +i, the other -i, but neither should have priority over the other. Similarly, clockwise and counterclockwise rotations should not be favored. Complex conjugation is a way of asserting this.
į:
* There are two square roots of -1. One we call +i, the other -i, but neither should have priority over the other. Similarly, clockwise and counterclockwise rotations should not be favored. Complex conjugation is a way of asserting this. (Note that the integer +1 is naturally favored over -1. But there is no such natural favoring for i. It is purely conventional, a misleading artificial contrivance.)
2016 balandžio 24 d., 22:18 atliko AndriusKulikauskas -
Pridėtos 99-100 eilutės:
* Coxeter groups are built from reflections. Reflections are dualities.
* Any two structures which have a nice map from one to the other have a duality in that you can start from one and go to the other.
2016 balandžio 24 d., 21:21 atliko AndriusKulikauskas -
Pridėta 98 eilutė:
* In mathematics, monstrous moonshine, or moonshine theory, is a term devised by John Conway and Simon P. Norton in 1979, used to describe the unexpected connection between the monster group M and modular functions, in particular, the j function. It is now known that lying behind monstrous moonshine is a vertex operator algebra called the moonshine module or monster vertex algebra, constructed by Igor Frenkel, James Lepowsky, and Arne Meurman in 1988, having the monster group as symmetries. This vertex operator algebra is commonly interpreted as a structure underlying a conformal field theory, allowing physics to form a bridge between two mathematical areas. The conjectures made by Conway and Norton were proved by Richard Borcherds for the moonshine module in 1992 using the no-ghost theorem from string theory and the theory of vertex operator algebras and generalized Kac–Moody algebras.
2016 balandžio 24 d., 19:42 atliko AndriusKulikauskas -
Pridėta 97 eilutė:
* Duality - parity - išsiaiškinimo rūšis. Įvairios simetrijos - išsiaiškinimo būdų sandaros.
2016 balandžio 24 d., 17:34 atliko AndriusKulikauskas -
Pakeistos 61-68 eilutės iš
Dualities. Read [[https://ncatlab.org/nlab/show/duality | nLab: Duality]]. Here are examples to consider:
į:
Dualities. Duality arises from a symmetry between two ways of looking at something where there is no reason to choose one over the other. For example:
* There are two square roots of -1. One we call +i, the other -i, but neither should have priority over the other. Similarly, clockwise and counterclockwise rotations should not be favored. Complex conjugation is a way of asserting this.
* A rectangular matrix can be written out from left to right or right to left. So we have the transpose matrix.
* Morphisms can be organized from left to right or from right to left. The opposite category turns all of the arrows around.
* Coordinate systems can be organized "bottom up" or "top down". This yields the duality in projective geometry.
* We can look at the operators that act or the objects they act upon. This brings to mind the two representations of the foursome.
* Faces of an object and corners of an object. (Why are they dual?)

Read [[https://ncatlab.org/nlab/show/duality | nLab: Duality]]. Here are examples to consider:
Pridėta 96 eilutė:
* Tensor products are adjoint to a set of homomorphisms.
2016 balandžio 24 d., 17:07 atliko AndriusKulikauskas -
Pakeista 107 eilutė iš:
* The number "i" is highly misleading in that it actually has no priority over "-i". Both are square roots of -1. Thus often (or always?) they should both be referenced - they are a "coupled" pair of numbers, not a single number.
į:
* The number "i" is highly misleading in that it actually has no priority over "-i". Both are square roots of -1. Thus often (or always?) they should both be referenced - they are a "coupled" pair of numbers, not a single number. They should be referenced by a single Number "I" which is understood to have two meanings.
2016 balandžio 24 d., 17:06 atliko AndriusKulikauskas -
Pridėta 107 eilutė:
* The number "i" is highly misleading in that it actually has no priority over "-i". Both are square roots of -1. Thus often (or always?) they should both be referenced - they are a "coupled" pair of numbers, not a single number.
2016 balandžio 24 d., 16:45 atliko Andrius Kulikauskas -
Pridėtos 87-89 eilutės:
* a very general comment of William Lawvere is that syntax and semantics are adjoint: take C to be the set of all logical theories (axiomatizations), and D the power set of the set of all mathematical structures. For a theory T in C, let F(T) be the set of all structures that satisfy the axioms T; for a set of mathematical structures S, let G(S) be the minimal axiomatization of S. We can then say that F(T) is a subset of S if and only if T logically implies G(S): the "semantics functor" F is left adjoint to the "syntax functor" G.
* division is (in general) the attempt to invert multiplication, but many examples, such as the introduction of implication in propositional logic, or the ideal quotient for division by ring ideals, can be recognised as the attempt to provide an adjoint.
2016 balandžio 24 d., 16:41 atliko Andrius Kulikauskas -
Pridėtos 85-86 eilutės:
* https://en.m.wikipedia.org/wiki/Coherent_duality https://en.m.wikipedia.org/wiki/Serre_duality https://en.m.wikipedia.org/wiki/Verdier_duality https://en.m.wikipedia.org/wiki/Poincaré_duality
* https://en.m.wikipedia.org/wiki/Dual_polyhedron
2016 balandžio 24 d., 16:13 atliko Andrius Kulikauskas -
Pridėta 43 eilutė:
* Consider [[https://en.m.wikipedia.org/wiki/Mathematics,_Form_and_Function |Mathematics, Form and Function]]
2016 balandžio 24 d., 15:49 atliko AndriusKulikauskas -
Pridėta 83 eilutė:
* The two facts that this method of turning rngs into rings is most efficient and formulaic can be expressed simultaneously by saying that it defines an adjoint functor. Continuing this discussion, suppose we started with the functor F, and posed the following (vague) question: is there a problem to which F is the most efficient solution? The notion that F is the most efficient solution to the problem posed by G is, in a certain rigorous sense, equivalent to the notion that G poses the most difficult problem that F solves.
2016 balandžio 24 d., 15:41 atliko AndriusKulikauskas -
Pakeista 82 eilutė iš:
į:
2016 balandžio 24 d., 15:40 atliko Andrius Kulikauskas -
Pridėta 82 eilutė:
2016 balandžio 24 d., 13:29 atliko AndriusKulikauskas -
Pridėtos 112-114 eilutės:

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]].
2016 balandžio 24 d., 11:23 atliko AndriusKulikauskas -
Pakeistos 17-20 eilutės iš
į:
** Kaip suprasti matricą kaip lygčių sistemą?
** Palyginti matricų naudojimą Galois teorijoje.
Pakeistos 81-82 eilutės iš
į:
* For integers, decomposition into primes is a "bottom up" result which states that a typical number can be compactly represented as the product of its prime components. The "top down" result is that this depends on an infinite number of exceptions ("primes") for which this compact representation does not make them more compact.
Pridėtos 97-98 eilutės:
* Complex numbers have two natural coordinate systems that correspond to addition (x,y) and multiplication (r,theta).
* Circle folding relates to "reflection" of the complex conjugate across an x-axis. Thinking of inverse rotation as this reflection.
2016 balandžio 24 d., 11:16 atliko AndriusKulikauskas -
Pridėtos 10-16 eilutės:

Tiesinė algebra
* Kaip dauginti polar decomposed matrices?
* Geriau suprasti [[https://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix | Eigenvector decomposition]].
** Kokios matricos turi pilną eigenvector rinkinį?
** Kokius eigenvectors ir eigenvalues turi pasukimo matricos?
** Kaip suprasti eigenvector koordinačių sistemą? Kiekviena (neišsigimusi) matrica turi naturalią koordinačių sistemą (?)
2016 balandžio 24 d., 10:46 atliko AndriusKulikauskas -
Pridėtos 162-165 eilutės:

Lie groups, Lie algebras
* Root systems: Lie algebrų rūšių pagrindas
* [[http://www.continuummechanics.org | Continuum mechanics]]
2016 balandžio 23 d., 11:32 atliko AndriusKulikauskas -
Pridėtos 119-121 eilutės:

Gyvenimo lygtis:
* Dvasia ir sandara susieti "duality", veiksmu +2.
2016 balandžio 22 d., 11:12 atliko Andrius Kulikauskas -
Pridėta 71 eilutė:
* [[https://ncatlab.org/nlab/show/topos | Topos]] links geometry and logic.
2016 balandžio 22 d., 10:57 atliko Andrius Kulikauskas -
Pridėta 70 eilutė:
* [[https://ncatlab.org/nlab/show/Isbell+duality | Isbell duality]] relates higher geometry with higher algebra.
2016 balandžio 22 d., 10:41 atliko Andrius Kulikauskas -
Pridėta 45 eilutė:
* A [[https://ncatlab.org/nlab/show/geometric+embedding | geometric embedding]] is the right notion of embedding or inclusion of topoi F↪E F \hookrightarrow E, i.e. of subtoposes. Notably the inclusion Sh(S)↪PSh(S) Sh(S) \hookrightarrow P of a category of sheaves into its presheaf topos or more generally the inclusion ShjE↪E Sh_j E \hookrightarrow E of sheaves in a topos E E into E E itself, is a geometric embedding. Actually every geometric embedding is of this form, up to equivalence of topoi. Another perspective is that a geometric embedding F↪E F \hookrightarrow E is the localizations of E E at the class W W or morphisms that the left adjoint E→F E \to F sends to isomorphisms in F F.
2016 balandžio 21 d., 23:10 atliko AndriusKulikauskas -
Pridėtos 67-68 eilutės:
* One may ask analytic questions about algebraic numbers, and use analytic means to answer such questions; it is thus that algebraic and analytic number theory intersect. For example, one may define prime ideals (generalizations of prime numbers in the field of algebraic numbers) and ask how many prime ideals there are up to a certain size. This question can be answered by means of an examination of Dedekind zeta functions, which are generalizations of the Riemann zeta function, a key analytic object at the roots of the subject. This is an example of a general procedure in analytic number theory: deriving information about the distribution of a sequence (here, prime ideals or prime numbers) from the analytic behavior of an appropriately constructed complex-valued function.
* Meromorphic function is the quotient of two holomorphic functions, thus compares them.
2016 balandžio 21 d., 17:56 atliko AndriusKulikauskas -
Pridėtos 111-113 eilutės:

Dievas:
* Dievas išeina už savęs: 3 matai -> 2 matai -> (flip to dual) 1 matas -> 0 matas (taškas: gera širdis).
2016 balandžio 21 d., 09:14 atliko Andrius Kulikauskas -
Pridėta 66 eilutė:
* Lie's idée fixe was to develop a theory of symmetries of differential equations that would accomplish for them what Évariste Galois had done for algebraic equations: namely, to classify them in terms of group theory. Lie and other mathematicians showed that the most important equations for special functions and orthogonal polynomials tend to arise from group theoretical symmetries. In Lie's early work, the idea was to construct a theory of continuous groups, to complement the theory of discrete groups that had developed in the theory of modular forms, in the hands of Felix Klein and Henri Poincaré. The initial application that Lie had in mind was to the theory of differential equations. On the model of Galois theory and polynomial equations, the driving conception was of a theory capable of unifying, by the study of symmetry, the whole area of ordinary differential equations. However, the hope that Lie Theory would unify the entire field of ordinary differential equations was not fulfilled. Symmetry methods for ODEs continue to be studied, but do not dominate the subject. There is a differential Galois theory, but it was developed by others, such as Picard and Vessiot, and it provides a theory of quadratures, the indefinite integrals required to express solutions.
2016 balandžio 21 d., 09:09 atliko Andrius Kulikauskas -
Pridėta 33 eilutė:
* Orthogonal polynomials.
2016 balandžio 21 d., 08:16 atliko Andrius Kulikauskas -
Pakeistos 27-28 eilutės iš
* Padalinti number theory into its padalinius. Create a region for number theory, not a separate node.
į:
* Indicate math structures, the main objects of study.
* Padalinti number theory into its padalinius: analytic number theory, algebraic number theory, diophantine geometry, probabilistic number theory, arithmetic combinatorics, computational number theory
. Create a region for number theory, not a separate node.
Pridėta 32 eilutė:
* Periodic functions, elliptic functions, modular forms, automorphic forms.
2016 balandžio 21 d., 07:27 atliko Andrius Kulikauskas -
Pridėta 29 eilutė:
* Class field theory is a branch of algebraic number theory.
2016 balandžio 21 d., 07:19 atliko Andrius Kulikauskas -
Pridėtos 23-29 eilutės:

'''Matematikos apžvalga'''

* Add time to the diagram.
* Padalinti number theory into its padalinius. Create a region for number theory, not a separate node.
* Algebraic number theory is Galois theory.
* Lie groups depend on differential topology.
2016 balandžio 21 d., 07:15 atliko Andrius Kulikauskas -
Pridėta 54 eilutė:
* Class field theory provides a one-to-one correspondence between finite abelian extensions of a fixed global field and appropriate classes of ideals of the field or open subgroups of the idele class group of the field.
2016 balandžio 20 d., 11:16 atliko AndriusKulikauskas -
Pakeistos 220-222 eilutės iš
į:
* binomial theorem and Pascal's triangle
* elementary proofs of the Pythagorean theorem (such as: "four times a right triangle is the difference of two squares" - the basis for infinitesimal rotation).
Pakeistos 234-237 eilutės iš
Urs Schreiber notes that many of the beautiful structures relate to string theory.
į:
* Urs Schreiber notes that many of the beautiful structures relate to string theory.
* Relation between two completely different domains, especially dual, complementary domains.

Investigation: Compare with beauty in chess
.
2016 balandžio 20 d., 10:57 atliko AndriusKulikauskas -
Pakeista 193 eilutė iš:
Richard Stanley’s post at Math Overflow, [[http://mathoverflow.net/questions/49151/most-intricate-and-most-beautiful-structures-in-mathematics | Most intricate and most beautiful structures in mathematics]], produced the following list, ordered by votes received:
į:
[[http://www-math.mit.edu/~rstan/ | Richard Stanley’s]] post at Math Overflow, [[http://mathoverflow.net/questions/49151/most-intricate-and-most-beautiful-structures-in-mathematics | Most intricate and most beautiful structures in mathematics]], produced the following list, ordered by votes received:
2016 balandžio 20 d., 10:48 atliko AndriusKulikauskas -
Pakeista 203 eilutė iš:
* Exotic Lie groups (See the Scientific American article about A.Garrett Lisi’s application of E8 to physics: A Geometric Theory of Everything)
į:
* Exotic Lie groups (See the Scientific American article about A.Garrett Lisi’s application of E8 to physics: [[http://www.cs.virginia.edu/~robins/A_Geometric_Theory_of_Everything.pdf | A Geometric Theory of Everything]])
2016 balandžio 20 d., 10:47 atliko AndriusKulikauskas -
Pakeista 193 eilutė iš:
Richard Stanley’s post at Math Overflow, [http://mathoverflow.net/questions/49151/most-intricate-and-most-beautiful-structures-in-mathematics | Most intricate and most beautiful structures in mathematics]], produced the following list, ordered by votes received:
į:
Richard Stanley’s post at Math Overflow, [[http://mathoverflow.net/questions/49151/most-intricate-and-most-beautiful-structures-in-mathematics | Most intricate and most beautiful structures in mathematics]], produced the following list, ordered by votes received:
2016 balandžio 20 d., 10:47 atliko AndriusKulikauskas -
Pakeista 193 eilutė iš:
Richard Stanley’s post at Math Overflow, Most intricate and most beautiful structures in mathematics, produced the following list, ordered by votes received:
į:
Richard Stanley’s post at Math Overflow, [http://mathoverflow.net/questions/49151/most-intricate-and-most-beautiful-structures-in-mathematics | Most intricate and most beautiful structures in mathematics]], produced the following list, ordered by votes received:
2016 balandžio 20 d., 10:46 atliko AndriusKulikauskas -
Pakeistos 176-232 eilutės iš
* The QED Project had a manifesto which addressed the big picture in math.
į:
* The QED Project had a manifesto which addressed the big picture in math.

---------------------

Mathematicians are often guided by their sense of beauty. Beauty is a key to the big picture in mathematics as well as physics and philosophy.

* What is meant by beauty?
* What principles determine it?
* To what extent is beauty objective and subjective?
* How does beauty lead to mathematical insight?

Investigation: Collect examples

In this section, we collect examples of beauty in mathematics.

Richard Stanley’s post at Math Overflow, Most intricate and most beautiful structures in mathematics, produced the following list, ordered by votes received:

* The absolute Galois group of the rationals
* The natural numbers (and variations)
* Homotopy groups of spheres
* The Mandelbrot set
* The Littlewood Richardson coefficients (representations of Sn etc.)
* The class of ordinals
* The monster vertex algebra
* Classical Hopf fibration
* Exotic Lie groups (See the Scientific American article about A.Garrett Lisi’s application of E8 to physics: A Geometric Theory of Everything)
* The Cantor set
* The 24 dimensional packing of unit spheres with kissing number 196560 (related to the monster vertex algebra).
* The simplicial symmetric sphere spectrum
* F_un (whatever it is)
* The Grothendiek-Teichmuller tower.
* Riemann’s zeta function
* Schwartz space of functions

Below we gather more examples, both basic and advanced:

* e^2pii + 1 = 0
* more generally, Euler’s formula: e^2piix = cos(x) + i sin(x)
* the unique decomposition of natural numbers into prime numbers
* Euler’s polyhedron formula V - E + F = 2
* the classification of the Platonic solids
* the relationship between a polynomial and its graph

Investigation: Analyze examples

In this section, we analyze the examples collected above to consider:

* In what sense are they beautiful?
* What makes them beautiful?
* What are the simplest examples of beauty?
* Which examples yield the most beauty for the least drudgery?

Investigation: Look for unifying principles or contexts.

Urs Schreiber notes that many of the beautiful structures relate to string theory
.
2016 balandžio 20 d., 10:43 atliko AndriusKulikauskas -
Pakeistos 153-176 eilutės iš
* 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)?
į:
* 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)?

--------------------------

The Big Picture in Mathematics

Mathematics has grown as a discipline to the extent that no single person is able to overview it all. This is a research page to encourage and organize efforts to foster a perspective upon all of mathematics.

Big questions

Several big questions provide new angles on mathematics as an endeavor:

* discovery What are the ways of figuring things out in mathematics? We can study mathematics as an activity by which we create and solve mathematical problems. These techniques are much more limited than the mathematical output which they generate.
* beauty Mathematicians are guided by a sense of beauty. What is meant by beauty? What principles determine it? How does beauty lead to mathematical insight?
* organization How can mathematics be organized so as to survey the most basic concepts from which it arises and understand it in terms of its most fundamental divisions and how they relate to each other?
* education What resources are available to mathematicians that would help them most effectively learn mathematics so as to try to understand it as a whole? How might mathematicians collaborate effectively in trying to understand the big picture?
* insight What are the most fruitful insights in trying to understand mathematics? How can such insights best be stated?
* premathematics What concepts express intuitions that are prior to explicit mathematics and make it possible?
* history How can the history of mathematical discovery inform frameworks for the future development of mathematics?
* humanity What parts or aspects of mathematics are specific to the human mind, body, culture, society, and what might be more broadly meaningful to other species in the universe?

Related pages

* The QED Project had a manifesto which addressed the big picture in math.
2016 balandžio 20 d., 08:58 atliko Andrius Kulikauskas -
Pakeistos 30-31 eilutės iš
* [[https://en.m.wikipedia.org/wiki/Erlangen_program | Erlangen program]]
į:
* [[https://en.wikipedia.org/wiki/Erlangen_program | Erlangen program]]
* Lie groups play an enormous role in modern geometry, on several different levels. Felix Klein argued in his Erlangen program that one can consider various "geometries" by specifying an appropriate transformation group that leaves certain geometric properties invariant. Thus Euclidean geometry corresponds to the choice of the group E(3) of distance-preserving transformations of the Euclidean space R3, conformal geometry corresponds to enlarging the group to the conformal group, whereas in projective geometry one is interested in the properties invariant under the projective group. This idea later led to the notion of a G-structure, where G is a Lie group of "local" symmetries of a manifold.
2016 balandžio 20 d., 08:37 atliko Andrius Kulikauskas -
Pridėta 52 eilutė:
* [[https://en.m.wikipedia.org/wiki/Hilbert%27s_Nullstellensatz | Hilbert's Nullstellensatz]]
2016 balandžio 20 d., 08:23 atliko Andrius Kulikauskas -
Pridėta 30 eilutė:
* [[https://en.m.wikipedia.org/wiki/Erlangen_program | Erlangen program]]
2016 balandžio 19 d., 19:41 atliko AndriusKulikauskas -
Pridėtos 64-65 eilutės:
* Real numbers are used for independent x, y. Imaginary number i denotes a link between two otherwise indepedent variables so that y = ix links indepedent axes by a 90 degree rotation.
* Similarly the polar decomposition of a matrix distinguishes (as for a number) the change in magnitude (scaling) and the rotation. It separates them.
2016 balandžio 19 d., 19:39 atliko AndriusKulikauskas -
Pakeistos 35-36 eilutės iš
į:
* Category theory - Categories are helpful in making fruitful definitions
Pridėta 54 eilutė:
* Representations - A very important idea, which is that we access a deep structure (such as a division of everything) not directly, but by way of some representation. This term is used in algebra, for example, to distinguish a system (like a group) from the matrices which serve as its multiplication table.
Pakeistos 58-59 eilutės iš
į:
* LinearAlgebra - Is the study of the basic properties of matrices and their effects.
Pridėtos 67-77 eilutės:
AutomataTheory - There is a qualitative hierarchy of computing devices. And they can be described in terms of matrices.

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

Algebra
* 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.
Pridėta 100 eilutė:
* 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.
Pridėta 108 eilutė:
* Logic is the end result of structure, see the sevensome and Greimas' semiotic square.
Pakeistos 117-124 eilutės iš
į:
Aštuongubas kelias
* SetTheory - The Zermelo Frankel axioms of set theory seem to be structured by what I call the EightfoldWay.

* Topologies - Systems of constraints that may be thought of as defining worlds. Topology is the study of topologies.

Matematikos įrodymų būdai - laipsnynas
Pakeistos 148-172 eilutės iš
* 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)?

----------------------

Here are some branches of mathematics that are basic in this respect:

* {{Combinatorics}} - The mathematics of counting (and generating) objects. I think of this as the basement of mathematics, which is why I studied it.
* AutomataTheory - There is a qualitative hierarchy of computing devices. And they can be described in terms of matrices.
* NeuralNetworks - Very powerful and simple computational systems for which Sarunas Raudys showed a hierarchy of sophistication as learning systems.
* {{Representations}} - A very important idea, which is that we access a deep structure (such as a division of everything) not directly, but by way of some representation. This term is used in algebra, for example, to distinguish a system (like a group) from the matrices which serve as its multiplication table.
* CategoryTheory - Categories are helpful in making fruitful definitions
* {{Topologies}} - Systems of constraints that may be thought of as defining worlds. Topology is the study of topologies.
* SetTheory - The Zermelo Frankel axioms of set theory seem to be structured by what I call the EightfoldWay.
* RecursiveFunctions - 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}}
* LinearAlgebra - Is the study of the basic properties of matrices and their effects.
* {{Logic}} is the end result of structure, see the {{Sevensome}} and Greimas' SemioticSquare
* {{Algebra}} studies particular structures
į:
* 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)?
2016 balandžio 19 d., 19:33 atliko AndriusKulikauskas -
Pridėtos 18-23 eilutės:
'''Kas yra matematika?'''

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.

Much of my work may be considered pre-mathematics, the grounding of the structures that ultimately allow for mathematics.
Pridėta 29 eilutė:
* ''At its roots, geometry consists of a notion of congruence between different objects in a space. In the late 19th century, notions of congruence were typically supplied by the action of a Lie group on space. Lie groups generally act quite rigidly, and so a [[https://en.wikipedia.org/wiki/Cartan_connection | Cartan geometry]] is a generalization of this notion of congruence to allow for curvature to be present. The flat Cartan geometries — those with zero curvature — are locally equivalent to homogeneous spaces, hence geometries in the sense of Klein.''
Pakeista 36 eilutė iš:
Dualities
į:
Dualities. Read [[https://ncatlab.org/nlab/show/duality | nLab: Duality]]. Here are examples to consider:
Pridėta 52 eilutė:
* I thought this was the most basic object in mathematics. Note that the index set may be arbitrary, not necessarily numbers.
Pakeistos 55-63 eilutės iš
į:
* Note that a category may be thought of as a deductive system, a directive graph, and hence a matrix. My thesis was on the combinatorics of the general matrix, which apparently is all generated by the symmetric functions of the eigenvalues of a matrix. So I am interested to see if that might be of value here. I want to show how a category arises from first principles. I imagine that perspectives may be thought of as morphisms (or functors).

Difference between complex numbers and real numbers
* Quantum possibilities vs. actualities
* Cauchy's integral theorem: for complexes, derivative and integral are mirrors, but not for reals
* There is a sense in which the reals give the magnitude and the imaginaries give the rotation. The function 1/x sends x+iy to x-iy divided by x2+y2. It sends r to 1/r (across the boundary of the unit circle) and it sends theta to -theta.

Vector spaces are basic. What is basic about scalars? They make possible proportionality.
Pridėtos 66-67 eilutės:
Kaip [[http://www.selflearners.net/Math/DeepIdeas | matematikos pagrindus]] pristatyti svarbiausiais dėsniais, pavyzdžiais ir žaidimais? Kuo žaidimai yra vertingi, kaip jie suveikia? Kuriu atitinkamas mokymosi priemones, tapau drobę.
Pridėtos 88-91 eilutės:
Penkerybė:
* Analysis allows for work with limits.
* Eccentricity of conic sections - there are five eccentricities (for the circle, parabola, ellipse, hyperbola, line).
Pridėtos 111-118 eilutės:
'''Kas gražu matematikoje ir kodėl?'''

One milestone (or miracle) in knowing everything is to generate all mathematical objects and truths in a unified way, but especially, generate all mathematical insights in a way that builds our intuition. The relevant outlook will, I imagine, be in terms of beauty. I mean that we may think of mathematical insight as guided by the wish for beauty.

Beauty - wholeness preserving transformations
* natural generalizations
* coordinate free
Pridėta 123 eilutė:
Logika
Ištrintos 130-150 eilutės:

Difference between complex numbers and real numbers
* Quantum possibilities vs. actualities
* Cauchy's integral theorem: for complexes, derivative and integral are mirrors, but not for reals
* There is a sense in which the reals give the magnitude and the imaginaries give the rotation. The function 1/x sends x+iy to x-iy divided by x2+y2. It sends r to 1/r (across the boundary of the unit circle) and it sends theta to -theta.

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

Beauty - wholeness preserving transformations
* natural generalizations
* coordinate free

Kas yra geometrija?
* ''At its roots, geometry consists of a notion of congruence between different objects in a space. In the late 19th century, notions of congruence were typically supplied by the action of a Lie group on space. Lie groups generally act quite rigidly, and so a [[https://en.wikipedia.org/wiki/Cartan_connection | Cartan geometry]] is a generalization of this notion of congruence to allow for curvature to be present. The flat Cartan geometries — those with zero curvature — are locally equivalent to homogeneous spaces, hence geometries in the sense of Klein.''
Pakeistos 133-140 eilutės iš
Kaip [[http://www.selflearners.net/Math/DeepIdeas | matematikos pagrindus]] pristatyti svarbiausiais dėsniais, pavyzdžiais ir žaidimais? Kuo žaidimai yra vertingi, kaip jie suveikia? Kuriu atitinkamas mokymosi priemones, tapau drobę.

Mathematics is the study of {{Structure}}

{{Andrius}}: Much of my work may be considered pre-mathematics, the grounding of the structures that ultimately allow for mathematics.

One milestone (or miracle) in knowing everything is to generate all mathematical objects and truths in a unified way, but especially, generate all mathematical insights in a way that builds our intuition. The relevant outlook will, I imagine, be in terms of beauty. I mean that we may think of mathematical insight as guided by the wish for {{Beauty}}.
į:
Pakeista 136 eilutė iš:
* {{Matrix}} - I think this is the most basic object in mathematics. Note that the index set may be arbitrary, not necessarily numbers.
į:
Pakeistos 148-155 eilutės iš
* {{Analysis}} allows for work with limits, see the {{Fivesome}}

* [[https://ncatlab.org/nlab/show/duality | nLab: Duality]]

-------------

{{Andrius}}: Note that a {{Category}} may be thought of as a deductive system, a directive graph, and hence a {{Matrix}}. My thesis was on the combinatorics of the general matrix, which apparently is all generated by the symmetric functions of the eigenvalues of a matrix. So I am interested to see if that might be of value here. I want to show how a category arises from first principles. I imagine that perspectives may be thought of as morphisms (or functors).
į:
2016 balandžio 19 d., 19:25 atliko AndriusKulikauskas -
Pridėtos 29-47 eilutės:
Dualities
* [[https://en.wikipedia.org/wiki/Duality_%28projective_geometry%29 | Duality (projective geometry)]]. Interchange the role of "points" and "lines" to get a dual truth: The plane dual statement of "Two points are on a unique line" is "Two lines meet at a unique point". (Compare with the construction of an equilateral triangle and the lattice of conditions.)
* Galois theory: field extensions (solutions of polynomials) and groups
* Lie groups: solutions to differential equations..
* Atiyah-Singer index theorem...
* Riemann-Roch theorem
* Covectors and vectors
* Cotangent space and tangent space
* Colimits and limits
* Coproducts and products
* [[https://en.m.wikipedia.org/wiki/De_Rham_cohomology | de Rham cohomology]] links algebraic topology and differential topology
* [[https://en.wikipedia.org/wiki/Modular_theorem |Modularity theorem]].
* [[https://en.m.wikipedia.org/wiki/Langlands_program | Langlands program]]
* general Stokes theorem: duality between the boundary operator on chains and the exterior derivative

Matricos
* [[https://en.wikipedia.org/wiki/Polar_decomposition | Polar decomposition]]. Square complex matrix A can always be written as A = UP where U is a unitary matrix and P is a positive-semidefinite Hermitian matrix. The eigenvalues of U all lie on the unit circle. The real analogue of U is the orthogonal matrix, whose determinant is either +1 (rotations) or -1 (reflections). U = e^iH where H is some Hermitian matrix. P has all nonnegative eigenvalues. It is the stretching of the eigenvectors. Thus every matrix A = B*e^iC where B and C have all nonnegative real eigenvalues.
* Symmetric and [[https://en.wikipedia.org/wiki/Skew-symmetric_matrix | skew-symmetric]]. Every matrix A can be broken down as the sum of a skew-symmetric matrix 1/2*(A-AT) and a symmetric matrix 1/2*(A+AT).
Pakeistos 82-83 eilutės iš
į:
Category theory
* [[http://math.ucr.edu/home/baez/qg-winter2016/ | John Baez: Category Theory]]

Foundations of Mathematics (understand how models work)
* [[http://sakharov.net/foundation.html | Foundations of Mathematics by Alexander Sakharov]]
* [[http://www.math.wisc.edu/~miller/old/m771-10/kunen770.pdf | The Foundations of Mathematics by Kenneth Kunen]]
Pakeistos 98-125 eilutės iš
* [[http://sakharov.net/foundation.html | Foundations of Mathematics by Alexander Sakharov]]
* [[http://www.math.wisc.edu/~miller/old/m771-10/kunen770.pdf | The Foundations of Mathematics by Kenneth Kunen]]

Matricos
* [[https://en.wikipedia.org/wiki/Polar_decomposition | Polar decomposition]]. Square complex matrix A can always be written as A = UP where U is a unitary matrix and P is a positive-semidefinite Hermitian matrix. The eigenvalues of U all lie on the unit circle. The real analogue of U is the orthogonal matrix, whose determinant is either +1 (rotations) or -1 (reflections). U = e^iH where H is some Hermitian matrix. P has all nonnegative eigenvalues. It is the stretching of the eigenvectors. Thus every matrix A = B*e^iC where B and C have all nonnegative real eigenvalues.
* Symmetric and [[https://en.wikipedia.org/wiki/Skew-symmetric_matrix | skew-symmetric]]. Every matrix A can be broken down as the sum of a skew-symmetric matrix 1/2*(A-AT) and a symmetric matrix 1/2*(A+AT).

Dualities
* [[https://en.wikipedia.org/wiki/Duality_%28projective_geometry%29 | Duality (projective geometry)]]. Interchange the role of "points" and "lines" to get a dual truth: The plane dual statement of "Two points are on a unique line" is "Two lines meet at a unique point". (Compare with the construction of an equilateral triangle and the lattice of conditions.)
* Galois theory: field extensions (solutions of polynomials) and groups
* Lie groups: solutions to differential equations..
* Atiyah-Singer index theorem...
* Riemann-Roch theorem
* Covectors and vectors
* Cotangent space and tangent space
* Colimits and limits
* Coproducts and products
* [[https://en.m.wikipedia.org/wiki/De_Rham_cohomology | de Rham cohomology]] links algebraic topology and differential topology
* [[https://en.wikipedia.org/wiki/Modular_theorem |Modularity theorem]].
* [[https://en.m.wikipedia.org/wiki/Langlands_program | Langlands program]]
* general Stokes theorem: duality between the boundary operator on chains and the exterior derivative
į:

Pakeistos 147-148 eilutės iš
* [[http://math.ucr.edu/home/baez/qg-winter2016/ | John Baez: Category Theory]]
* [[https://ncatlab.org/nlab/show/duality |
Duality]]
į:
* [[https://ncatlab.org/nlab/show/duality | nLab: Duality]]
2016 balandžio 19 d., 19:22 atliko AndriusKulikauskas -
Pakeistos 3-4 eilutės iš
į:
Kokie yra matematikos pagrindai?
* Kas yra geometrija? Iš ko jinai susidaro? Iš klausimų?
** How is one dimension embedded in other dimensions?
** What is a line segment? What makes it "straight"?
** What is a circle?
** What does it mean for figures to intersect?
** Can a line intersect with itself?

Pakeistos 18-25 eilutės iš
Iš ko susidaro geometrija? Iš klausimų?
* What is a line segment? What makes it "straight"?
* A circle?
* What does it mean for figures to intersect?
* Can a line intersect with itself?
* How is one dimension embedded
in other dimensions?
the same point. Then consider embedding a point in 1 dimension. The point does not yet distinguish between the two sides because there is no orientation. A distinction comes with the arisal of a second point. But whether the second point distinguishes the two sides depends on global knowledge. So there must be a third point. This is the relationship between "persons": I, You, Other. Either the dimension is a closed curve or it is an open line. This is "global knowledge". So there is the distinction between local knowledge and global knowledge. But basically geometry is a construction of the continuum, either locally or globally. The construction takes place through infinite sequences, through completion. This completion is not relevant for all constructions.
į:
'''Matematikos pagrindai'''

Ieškau matematikos pagrindų. Apžvelgiu matematikos sritis ir jas išdėstau pagal tai, kaip viena nuo kitos priklauso.

Geometry
* Start with 0 dimension: a point. Every point is the same point. Then consider embedding a point in 1 dimension. The point does not yet distinguish between the two sides because there is no orientation. A distinction comes with the arisal of a second point. But whether the second point distinguishes the two sides depends on global knowledge. So there must be a third point. This is the relationship between "persons": I, You, Other. Either the dimension is a closed curve or it is an open line. This is "global knowledge". So there is the distinction between local knowledge and global knowledge. But basically geometry is a construction of the continuum, either locally or globally. The construction takes place through infinite sequences, through completion. This completion is not relevant for all constructions.
* Tensors give the embedding of a lower dimension into a higher dimension. Taip pat tensoriai sieja erdvę ir jos papildinį, kaip kad gyvybę ir meilę. Tai vyksta vektoriais (tangent space) ir kovektoriais (normal space?) Tad geometrijos pagrindas būtų Tensors over a ring. Kovektoriai išsako idealią bazę. Tensorius susidaro iš kovektorių ir kokovektorių. Ir tie, ir tie yra tiesiniai funkcionalai. Tikai baigtinių dimensijų vektorių erdvėse kokovektoriai tolygūs vektoriams.

Equivalences
* [[https://en.wikipedia.org/wiki/Twelvefold_way | Twelvefold way]] - reikėtų palyginti su aplinkybėmis, taip pat su equivalence relations, kokių gali būti.

'''Pagrindiniai matematikos dėsniai'''

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

'''Sąsajos tarp mano sąvokų ir matematikos'''
Pakeistos 48-49 eilutės iš
[https://en.wikipedia.org/wiki/Twelvefold_way | Twelvefold way]] - reikėtų palyginti su aplinkybėmis, taip pat su equivalence relations, kokių gali būti.
į:
Ketverybė:
* Reikėtų išmokti Yates Index Theorem, jinai pakankamai trumpa ir turbūt ne tokia sudėtinga. Ir paaiškinti ką jinai turi bendro su sąmoningumu.

Septynerybė:
* 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?

Pertvarkymai:
* Aibės struktūra primena medį nes gali būti aibės aibėse. Tačiau svarbu, kad nėra ratų.

Niekas
* Taškas yra niekas.

'''Ko norėčiau išmokti matematikoje'''

'''Įdomūs, prasmingi reiškiniai matematikoje'''
Pakeistos 69-76 eilutės iš
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.

Reikėtų išmokti Yates Index Theorem, jinai pakankamai trumpa ir turbūt ne tokia sudėtinga. Ir paaiškinti ką jinai turi bendro su sąmoningumu.

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?

Tensors give the embedding of a lower dimension into a higher dimension. Taip pat tensoriai sieja erdvę ir jos papildinį, kaip kad gyvybę ir meilę. Tai vyksta vektoriais (tangent space) ir kovektoriais (normal space?) Tad geometrijos pagrindas būtų Tensors over a ring. Kovektoriai išsako idealią bazę. Tensorius susidaro iš kovektorių ir kokovektorių. Ir tie, ir tie yra tiesiniai funkcionalai. Tikai baigtinių dimensijų vektorių erdvėse kokovektoriai tolygūs vektoriams.
į:
* 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)?
Pakeistos 77-91 eilutės iš
Aibės struktūra primena medį nes gali būti aibės aibėse. Tačiau svarbu, kad nėra ratų.

Taškas yra niekas.

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)?

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
į:

2016 balandžio 19 d., 18:35 atliko AndriusKulikauskas -
Pridėtos 21-22 eilutės:

[https://en.wikipedia.org/wiki/Twelvefold_way | Twelvefold way]] - reikėtų palyginti su aplinkybėmis, taip pat su equivalence relations, kokių gali būti.
2016 balandžio 19 d., 10:11 atliko Andrius Kulikauskas -
Pridėtos 77-80 eilutės:

Beauty - wholeness preserving transformations
* natural generalizations
* coordinate free
2016 balandžio 19 d., 10:05 atliko Andrius Kulikauskas -
Pakeista 68 eilutė iš:
* general Stokes theorem
į:
* general Stokes theorem: duality between the boundary operator on chains and the exterior derivative
2016 balandžio 19 d., 09:57 atliko Andrius Kulikauskas -
Pridėta 68 eilutė:
* general Stokes theorem
2016 balandžio 16 d., 17:42 atliko AndriusKulikauskas -
2016 balandžio 16 d., 17:42 atliko AndriusKulikauskas -
Pakeistos 17-20 eilutės iš
Complex numbers: dvimačiai: širdies tiesa. Real numbers: pasaulio tiesa.
į:
Požiūriai:
* Complex numbers: dvimačiai: širdies tiesa. Real numbers: pasaulio tiesa.
* Kategorijų teorija.
* Kvantoriai ir septynerybė
.
2016 balandžio 16 d., 14:29 atliko AndriusKulikauskas -
Pakeistos 103-104 eilutės iš
[[http://math.ucr.edu/home/baez/qg-winter2016/ | John Baez: Category Theory]]
į:
* [[http://math.ucr.edu/home/baez/qg-winter2016/ | John Baez: Category Theory]]
* [[https://ncatlab.org/nlab/show/duality | Duality
]]
2016 balandžio 16 d., 14:24 atliko AndriusKulikauskas -
Pridėtos 102-103 eilutės:

[[http://math.ucr.edu/home/baez/qg-winter2016/ | John Baez: Category Theory]]
2016 balandžio 16 d., 08:07 atliko Andrius Kulikauskas -
Pakeistos 64-66 eilutės iš
į:
* [[https://en.m.wikipedia.org/wiki/Langlands_program | Langlands program]]
Ištrintos 75-78 eilutės:

2016 balandžio 16 d., 01:11 atliko Andrius Kulikauskas -
Pakeista 63 eilutė iš:
* [[https://en.m.wikipedia.org/wiki/Modular_form |Modularity theorem]].
į:
* [[https://en.wikipedia.org/wiki/Modular_theorem |Modularity theorem]].
2016 balandžio 16 d., 01:09 atliko Andrius Kulikauskas -
Pakeistos 62-63 eilutės iš
*[[https://en.m.wikipedia.org/wiki/De_Rham_cohomology | de Rham cohomology]] links algebraic topology and differential topology
į:
* [[https://en.m.wikipedia.org/wiki/De_Rham_cohomology | de Rham cohomology]] links algebraic topology and differential topology
* [[https://en.m.wikipedia.org/wiki/Modular_form |Modularity theorem]].
2016 balandžio 14 d., 12:19 atliko Andrius Kulikauskas -
Pakeista 62 eilutė iš:
*[[https://en.m.wikipedia.org/wiki/De_Rham_cohomology | de Rham cohomology]] links algebraic topology and differential topology]]
į:
*[[https://en.m.wikipedia.org/wiki/De_Rham_cohomology | de Rham cohomology]] links algebraic topology and differential topology
2016 balandžio 14 d., 12:18 atliko Andrius Kulikauskas -
Pridėta 62 eilutė:
*[[https://en.m.wikipedia.org/wiki/De_Rham_cohomology | de Rham cohomology]] links algebraic topology and differential topology]]
2016 balandžio 12 d., 11:12 atliko Andrius Kulikauskas -
Pridėta 66 eilutė:
* There is a sense in which the reals give the magnitude and the imaginaries give the rotation. The function 1/x sends x+iy to x-iy divided by x2+y2. It sends r to 1/r (across the boundary of the unit circle) and it sends theta to -theta.
2016 balandžio 11 d., 14:31 atliko Andrius Kulikauskas -
Pakeistos 63-65 eilutės iš
į:
Difference between complex numbers and real numbers
* Quantum possibilities vs. actualities
* Cauchy's integral theorem: for complexes, derivative and integral are mirrors, but not for reals
2016 balandžio 11 d., 14:20 atliko Andrius Kulikauskas -
Pridėta 57 eilutė:
* Riemann-Roch theorem
2016 balandžio 11 d., 13:04 atliko Andrius Kulikauskas -
Pakeistos 56-60 eilutės iš
* Attiyah-Index-Singer theorem...
į:
* Atiyah-Singer index theorem...
* Covectors and vectors
* Cotangent space and tangent space
* Colimits and limits
* Coproducts and products
2016 balandžio 10 d., 09:27 atliko Andrius Kulikauskas -
Pridėtos 1-6 eilutės:
>>bgcolor=#FFFFC0<<

>><<
2016 balandžio 10 d., 07:13 atliko AndriusKulikauskas -
Pridėtos 10-11 eilutės:

Complex numbers: dvimačiai: širdies tiesa. Real numbers: pasaulio tiesa.
2016 balandžio 09 d., 17:26 atliko AndriusKulikauskas -
Pridėtos 14-17 eilutės:

Reikėtų išmokti Yates Index Theorem, jinai pakankamai trumpa ir turbūt ne tokia sudėtinga. Ir paaiškinti ką jinai turi bendro su sąmoningumu.

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?
2016 balandžio 09 d., 17:04 atliko AndriusKulikauskas -
Pakeista 13 eilutė iš:
Logikos prielaidos bene slypi daugybė matematikos sąvokų. Pavyzdžiui, begalybė slypi galimybėje tą patį logikos veiksnį taikyti neribotą skaičių kartų.
į:
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.
2016 balandžio 09 d., 17:04 atliko AndriusKulikauskas -
Pridėtos 12-13 eilutės:

Logikos prielaidos bene slypi daugybė matematikos sąvokų. Pavyzdžiui, begalybė slypi galimybėje tą patį logikos veiksnį taikyti neribotą skaičių kartų.
2016 balandžio 08 d., 21:56 atliko AndriusKulikauskas -
Pridėtos 19-20 eilutės:

Taškas yra niekas.
2016 balandžio 08 d., 17:04 atliko AndriusKulikauskas -
Pakeista 20 eilutė iš:
Ar gali būti, kad aibių teorijos problemos (pavyzdžiui, sau nepriklausanti aibė) yra iš tikrųjų logikos problemos?
į:
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)?
2016 balandžio 08 d., 17:03 atliko AndriusKulikauskas -
Pridėtos 17-20 eilutės:

Aibės struktūra primena medį nes gali būti aibės aibėse. Tačiau svarbu, kad nėra ratų.

Ar gali būti, kad aibių teorijos problemos (pavyzdžiui, sau nepriklausanti aibė) yra iš tikrųjų logikos problemos?
2016 balandžio 07 d., 10:13 atliko AndriusKulikauskas -
Pakeista 1 eilutė iš:
%width=900px% Attach:MatematikosSakosDidelis.png
į:
%width=900px% [[Attach:MatematikosSakosDidelis.png | Attach:MatematikosSakosDidelis.png]]
2016 balandžio 07 d., 10:12 atliko AndriusKulikauskas -
Pridėtos 1-2 eilutės:
%width=900px% Attach:MatematikosSakosDidelis.png
2016 balandžio 02 d., 16:05 atliko AndriusKulikauskas -
Pridėtos 8-9 eilutės:

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.
2016 kovo 31 d., 13:38 atliko AndriusKulikauskas -
Pakeistos 11-12 eilutės iš
[[http://sakharov.net/foundation.html | Foundations of Mathematics by Alexander Sakharov]]
į:
* [[http://sakharov.net/foundation.html | Foundations of Mathematics by Alexander Sakharov]]
* [[http://www.math.wisc.edu/~miller/old/m771-10/kunen770.pdf | The Foundations of Mathematics by Kenneth Kunen
]]
2016 kovo 31 d., 13:37 atliko AndriusKulikauskas -
Pridėtos 10-11 eilutės:

[[http://sakharov.net/foundation.html | Foundations of Mathematics by Alexander Sakharov]]
2016 kovo 31 d., 13:03 atliko AndriusKulikauskas -
Pakeista 9 eilutė iš:
Tensors give the embedding of a lower dimension into a higher dimension. Taip pat tensoriai sieja erdvę ir jos papildinį, kaip kad gyvybę ir meilę. Tai vyksta vektoriais (tangent space) ir kovektoriais (normal space?) Tad geometrijos pagrindas būtų Tensors over a ring.
į:
Tensors give the embedding of a lower dimension into a higher dimension. Taip pat tensoriai sieja erdvę ir jos papildinį, kaip kad gyvybę ir meilę. Tai vyksta vektoriais (tangent space) ir kovektoriais (normal space?) Tad geometrijos pagrindas būtų Tensors over a ring. Kovektoriai išsako idealią bazę. Tensorius susidaro iš kovektorių ir kokovektorių. Ir tie, ir tie yra tiesiniai funkcionalai. Tikai baigtinių dimensijų vektorių erdvėse kokovektoriai tolygūs vektoriams.
2016 kovo 29 d., 15:04 atliko AndriusKulikauskas -
Pridėtos 8-9 eilutės:

Tensors give the embedding of a lower dimension into a higher dimension. Taip pat tensoriai sieja erdvę ir jos papildinį, kaip kad gyvybę ir meilę. Tai vyksta vektoriais (tangent space) ir kovektoriais (normal space?) Tad geometrijos pagrindas būtų Tensors over a ring.
2016 kovo 22 d., 13:17 atliko AndriusKulikauskas -
Pakeista 7 eilutė iš:
Start with 0 dimension: a point. Every point is the same point. Then consider embedding a point in 1 dimension. Then we know that the point distinguishes two sides of the dimension. Either the dimension is a closed curve or it is an open line. This is "global knowledge". So there is the distinction between local knowledge and global knowledge. But basically geometry is a construction of the continuum, either locally or globally. The construction takes place through infinite sequences, through completion. This completion is not relevant for all constructions.
į:
Start with 0 dimension: a point. Every point is the same point. Then consider embedding a point in 1 dimension. The point does not yet distinguish between the two sides because there is no orientation. A distinction comes with the arisal of a second point. But whether the second point distinguishes the two sides depends on global knowledge. So there must be a third point. This is the relationship between "persons": I, You, Other. Either the dimension is a closed curve or it is an open line. This is "global knowledge". So there is the distinction between local knowledge and global knowledge. But basically geometry is a construction of the continuum, either locally or globally. The construction takes place through infinite sequences, through completion. This completion is not relevant for all constructions.
2016 kovo 22 d., 13:12 atliko AndriusKulikauskas -
Pakeista 10 eilutė iš:
* every answer iš anksto amount and a unit ir tt.
į:
* every answer is an amount and a unit ir tt.
2016 kovo 22 d., 13:11 atliko AndriusKulikauskas -
Pakeista 2 eilutė iš:
* What is a line segment?
į:
* What is a line segment? What makes it "straight"?
2016 kovo 22 d., 13:10 atliko AndriusKulikauskas -
Pridėta 7 eilutė:
Start with 0 dimension: a point. Every point is the same point. Then consider embedding a point in 1 dimension. Then we know that the point distinguishes two sides of the dimension. Either the dimension is a closed curve or it is an open line. This is "global knowledge". So there is the distinction between local knowledge and global knowledge. But basically geometry is a construction of the continuum, either locally or globally. The construction takes place through infinite sequences, through completion. This completion is not relevant for all constructions.
2016 kovo 22 d., 13:00 atliko AndriusKulikauskas -
Pridėtos 1-7 eilutės:
Iš ko susidaro geometrija? Iš klausimų?
* What is a line segment?
* A circle?
* What does it mean for figures to intersect?
* Can a line intersect with itself?
* How is one dimension embedded in other dimensions?
Pridėtos 10-11 eilutės:
* combine like units
* list different units
Pakeistos 13-17 eilutės iš
į:
* 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
Ištrinta 26 eilutė:
2016 kovo 21 d., 17:34 atliko Andrius Kulikauskas -
Pakeistos 1-3 eilutės iš
Prisiminti savo matematikos mokymo dėsnius: every answer iš anksto amount and a unit ir tt.
į:
Prisiminti savo matematikos mokymo dėsnius:
*
every answer iš anksto amount and a unit ir tt.
* a right triangle is half of a rectangle
2016 kovo 21 d., 11:27 atliko Andrius Kulikauskas -
Pridėtos 1-2 eilutės:
Prisiminti savo matematikos mokymo dėsnius: every answer iš anksto amount and a unit ir tt.
2016 kovo 20 d., 23:02 atliko AndriusKulikauskas -
Pridėtos 4-9 eilutės:

Dualities
* [[https://en.wikipedia.org/wiki/Duality_%28projective_geometry%29 | Duality (projective geometry)]]. Interchange the role of "points" and "lines" to get a dual truth: The plane dual statement of "Two points are on a unique line" is "Two lines meet at a unique point". (Compare with the construction of an equilateral triangle and the lattice of conditions.)
* Galois theory: field extensions (solutions of polynomials) and groups
* Lie groups: solutions to differential equations..
* Attiyah-Index-Singer theorem...
2016 kovo 20 d., 22:22 atliko AndriusKulikauskas -
Pakeista 3 eilutė iš:
*
į:
* Symmetric and [[https://en.wikipedia.org/wiki/Skew-symmetric_matrix | skew-symmetric]]. Every matrix A can be broken down as the sum of a skew-symmetric matrix 1/2*(A-AT) and a symmetric matrix 1/2*(A+AT).
2016 kovo 20 d., 22:18 atliko AndriusKulikauskas -
Pridėtos 1-7 eilutės:
Matricos
* [[https://en.wikipedia.org/wiki/Polar_decomposition | Polar decomposition]]. Square complex matrix A can always be written as A = UP where U is a unitary matrix and P is a positive-semidefinite Hermitian matrix. The eigenvalues of U all lie on the unit circle. The real analogue of U is the orthogonal matrix, whose determinant is either +1 (rotations) or -1 (reflections). U = e^iH where H is some Hermitian matrix. P has all nonnegative eigenvalues. It is the stretching of the eigenvectors. Thus every matrix A = B*e^iC where B and C have all nonnegative real eigenvalues.
*

Pakeistos 10-11 eilutės iš
Kas yra geometrija? ''At its roots, geometry consists of a notion of congruence between different objects in a space. In the late 19th century, notions of congruence were typically supplied by the action of a Lie group on space. Lie groups generally act quite rigidly, and so a [[https://en.wikipedia.org/wiki/Cartan_connection | Cartan geometry]] is a generalization of this notion of congruence to allow for curvature to be present. The flat Cartan geometries — those with zero curvature — are locally equivalent to homogeneous spaces, hence geometries in the sense of Klein.''
į:
Kas yra geometrija?
*
''At its roots, geometry consists of a notion of congruence between different objects in a space. In the late 19th century, notions of congruence were typically supplied by the action of a Lie group on space. Lie groups generally act quite rigidly, and so a [[https://en.wikipedia.org/wiki/Cartan_connection | Cartan geometry]] is a generalization of this notion of congruence to allow for curvature to be present. The flat Cartan geometries — those with zero curvature — are locally equivalent to homogeneous spaces, hence geometries in the sense of Klein.''
2016 kovo 20 d., 22:08 atliko AndriusKulikauskas -
Pakeistos 37-72 eilutės iš
{{Andrius}}: Note that a {{Category}} may be thought of as a deductive system, a directive graph, and hence a {{Matrix}}. My thesis was on the combinatorics of the general matrix, which apparently is all generated by the symmetric functions of the eigenvalues of a matrix. So I am interested to see if that might be of value here. I want to show how a category arises from first principles. I imagine that perspectives may be thought of as morphisms (or functors).

---------

What is a [{{Morphism}} #] in the context of GOS?

{{Andrius}}: Morphism is one of the SixMethodsOfProof. It is proof by matching analogous structures. Morphism is the strongest method of proof in that it is least contingent on the details of a system.

{{HelmutLeitner}}: Andrius, please give examples for [{{Morphism}} #]. Its important that at least one example is outside of GOS and outside of abstract mathematics.

{{Andrius}}: See the example of the [http://www.davros.org/science/roadparadox.html Road Network Paradox]. The road networks are compared to a system of springs. There is a spring that links two weak springs and thus reduces the load that the system can carry. Cutting that spring improves the system. Similarly, removing a road can improve traffic. This is a morphism because it is mapping a problem from one domain into an analogous domain where it may be easier to understand, at least the relevant aspect. In general, metaphorical thinking is based on morphism, mappings, transformations.

{{HelmutLeitner}}: ok, but how can you consider such a method of transformation a proof?

{{Andrius}}: It's the essence of a proof, the heart of a proof. What is lacking? Perhaps more rigor and care is needed to make the mapping explicit. (Or in the above example, to make clear what exactly carries over from one domain to the other domain.) But what makes it a proof (sloppy or careful) is the mapping. That is where the argument stands. If the mapping is valid, then the argument is proven. Whether or not a mapping (or any mapping) can be validated (and accepted) depends also on the pragmatic context. It's possible to not accept any proof, and it's possible for all proofs to be wrong. But that doesn't mean that they are not proofs. But perhaps I didn't understand you? Or perhaps another word might be better? And yet I'm using ''proof'' in the same spirit as mathematicians do in their work. They don't spell out all the details, they make sure the main idea is compelling and when it is, the details clarify themselves and are worked out further. They don't write their papers this way, but this is how they think through their conclusions. So, for example, the above analogy makes clear that adding a road ''could'' make traffic worse. It also suggests ''why'' traffic is worse (the extra spring moves the weight to the weak springs; the extra road diverts traffic to the less efficient roads). Or vice versa. We've moved the problem from one domain to another one where possibly we may have better intuition. What is actually proven depends on the actual morphism. And note that formalism can make the proof more rigorous (more applicable regarding the details), but it can't make it stronger (clearer as to whether it is inherently valid or invalid). Yes, a proof can be stitched together, but that doesn't convince us that it will hold together. Morphism is the strongest type of proof because it is cut from whole cloth. It is all or nothing - either it applies or it doesn't.

Here are some examples of different ways of proving the same thing: [{{SelfEducation/methods}} of proof]. Morphism is the ''strongest'' (and convincing) proof, then induction, and substitution (algebraic manipulation) or examination of cases are much more fragile, much more contingent. A good morphism may be wrong in one case, but then it will be right about another, perhaps more important case. A good morphism can change what questions we choose to ask, it can have us redefine our terms and change our focus.

Here is another example: '''A right triangle is half of a rectangle.''' This is a morphism! I mean, this is to say that a ''right triangle'' is a concept that bridges ''the world of rectangles'' and ''the world of triangles''. Consequently, the shape of the right triangle is given by:
* the size of an acute angle (the world of triangles)
* the ratio of any two sides (the world of rectangles)
This ''defines'' six maps between these two worlds! and they are the trigonometric functions. We don't need to know any more details - this is the heart of trigonometry, and enough for us to start to apply it practically. In fact, this morphism tells us the kinds of geometries for which we can have a ''trigonometry''.

{{HelmutLeitner}}: I think the [http://www.davros.org/science/roadparadox.html Road Network Paradox] doesn't hold. While the symmetrical standard situation produces 83x6000=498000 time units, A maybe counterintuitive traffic A-&gt;C-&gt;B-&gt;D actually reduces the total time and so improves average time as expected (e. g. 5600 x 80.8 + 400 x 111.2 = 496960 time units). So either the mechanical example also doesn't hold or the morphism is wrong.

{{HelmutLeitner}}: The paradox is artifically constructed because people are assumed to have to go in the wrong direction. It doesn't hold, when some people go the long way on the good roads. Then average travel time decreases. Check the numbers I've given.

{{Andrius}}: Helmut, I think your alternate route is clever. (And they may not have taken that into account.) I can't take the time to think through the numbers. But is your formula suggesting that some people will travel 111.2 minutes and others will travel 80.8 minutes? They are specifically ruling that out by assuming that people will switch routes until everybody's travel time is the same. Also, from a common sense point of view, is the introduction of this clever long route an "improvement" over the original solution. That would be hard to believe (though, perhaps possible). Finally, the point that got me to believe the paradox is that the shortcut is much shorter than the long roads. Taking the shortcut makes for a 30km trip rather than a 50km trip. So people have a reason to sit through two traffic jams instead of just one. And that makes the traffic jams much worse.

{{HelmutLeitner}}: Arguments about that will take longer than checking the numbers. If there are multiple routes from some A to some B then it is improbable that the times of drivers will ever be equal, this is only because the sample model is symmetrical, so the condition - if it exists, doesn't make sense. In real life not even on a single route drivers will experience equal times. Drivers can't compare and can't optimize. Another example: I have four routes to my work, the shortest is 8 min but sometimes jammed to 30 min, so I often take other routes, the longest route guarantees me about 12-15 min. So if I have to be in time and I have 15+ minutes, I'll take the longest route. If I have enough time I take the shortest route, because it is statistically fastest. But I also take other routes regularly - because otherwise I have no experience about their qualities. -- But all such considerations are outside of the abstract model of the paradox. There is no guarantee that the model corresponds to reality, a model is a simplification. -- In effect they introduce a one-way-road and its quite clear that a one-way-road in the wrong direction will not be helpful - but this wouldn't be entertaining. If they talk about a street I have the right to assume that people may use it in either direction. I have the right to assume that they talk about the real world because they strive for a - very doubtful - real world explanation for "they add streets and traffic doesn't get better" which may have quite different reasons (e. g. that good streets attract traffic and that traffic increases). -- Also the mathematical model is doubtful: If we have a street E-&gt;G with length L and capacity C (T=L+F/C) and reinterpret it cut in to pieces as E-&gt;F-&gt;G with the same length (L/2 + L/2) and capacity, the calculated time increases (T=L+2*F/C). This has no influence onto all the aspects we discussed but it does not correspond to traffic or flow reality. Of course changes in the mathematic calculations can change the situations but can't be reflected in the morphism to the physical spring model!

{{Andrius}}: Helmut, you're right that reality is complex, as you say. But here are their assumptions: %gray%We can see that if traffic is split unevenly, one route will be faster than the other. People using the slower route will realize that they could do better by changing to the other, and no doubt will do so. The only stable situation is where half the traffic goes each way, because then anyone changing to the other route will find they are on a slower route, and thus will change back.%gray% For modeling purposes, those may or may not be good assumptions, and in your case they would at least require averaging. For example, in economics often such assumptions are made (perfect knowledge), occasionally they are not (imperfect knowledge), and there are cases where either model is appropriate. I think it's fair to allow that they are not trying to model all road systems, just some road systems. Here is the website of [http://homepage.ruhr-uni-bochum.de/Dietrich.Braess/ Dietrich Braess] who wrote the original paper in 1968, and which has been generated a lot of research, including observations of this behavior in computer networks. And [http://www.crowddynamics.com/Myriad/Braess%20Paradox.htm here's another explanation].

{{HelmutLeitner}}: Andrius, I do not doubt that there are assumptions that are consistent with their findings. I ''do'' doubt that they say something about real traffic or physical systems. Morphism - the way they use it - just seems to provide insight. Can you provide a better example of a morphism that actually proofs something?
į:
{{Andrius}}: Note that a {{Category}} may be thought of as a deductive system, a directive graph, and hence a {{Matrix}}. My thesis was on the combinatorics of the general matrix, which apparently is all generated by the symmetric functions of the eigenvalues of a matrix. So I am interested to see if that might be of value here. I want to show how a category arises from first principles. I imagine that perspectives may be thought of as morphisms (or functors).
2016 kovo 14 d., 11:57 atliko AndriusKulikauskas -
Pridėtos 2-3 eilutės:

Kas yra geometrija? ''At its roots, geometry consists of a notion of congruence between different objects in a space. In the late 19th century, notions of congruence were typically supplied by the action of a Lie group on space. Lie groups generally act quite rigidly, and so a [[https://en.wikipedia.org/wiki/Cartan_connection | Cartan geometry]] is a generalization of this notion of congruence to allow for curvature to be present. The flat Cartan geometries — those with zero curvature — are locally equivalent to homogeneous spaces, hence geometries in the sense of Klein.''
2016 kovo 14 d., 08:37 atliko AndriusKulikauskas -
Pridėtos 1-8 eilutės:
Vector spaces are basic. What is basic about scalars? They make possible proportionality.

----------------------
2014 birželio 06 d., 11:22 atliko Andrius Kulikauskas -
Pakeistos 27-62 eilutės iš
{{Andrius}}: Note that a {{Category}} may be thought of as a deductive system, a directive graph, and hence a {{Matrix}}. My thesis was on the combinatorics of the general matrix, which apparently is all generated by the symmetric functions of the eigenvalues of a matrix. So I am interested to see if that might be of value here. I want to show how a category arises from first principles. I imagine that perspectives may be thought of as morphisms (or functors).
į:
{{Andrius}}: Note that a {{Category}} may be thought of as a deductive system, a directive graph, and hence a {{Matrix}}. My thesis was on the combinatorics of the general matrix, which apparently is all generated by the symmetric functions of the eigenvalues of a matrix. So I am interested to see if that might be of value here. I want to show how a category arises from first principles. I imagine that perspectives may be thought of as morphisms (or functors).

---------

What is a [{{Morphism}} #] in the context of GOS?

{{Andrius}}: Morphism is one of the SixMethodsOfProof. It is proof by matching analogous structures. Morphism is the strongest method of proof in that it is least contingent on the details of a system.

{{HelmutLeitner}}: Andrius, please give examples for [{{Morphism}} #]. Its important that at least one example is outside of GOS and outside of abstract mathematics.

{{Andrius}}: See the example of the [http://www.davros.org/science/roadparadox.html Road Network Paradox]. The road networks are compared to a system of springs. There is a spring that links two weak springs and thus reduces the load that the system can carry. Cutting that spring improves the system. Similarly, removing a road can improve traffic. This is a morphism because it is mapping a problem from one domain into an analogous domain where it may be easier to understand, at least the relevant aspect. In general, metaphorical thinking is based on morphism, mappings, transformations.

{{HelmutLeitner}}: ok, but how can you consider such a method of transformation a proof?

{{Andrius}}: It's the essence of a proof, the heart of a proof. What is lacking? Perhaps more rigor and care is needed to make the mapping explicit. (Or in the above example, to make clear what exactly carries over from one domain to the other domain.) But what makes it a proof (sloppy or careful) is the mapping. That is where the argument stands. If the mapping is valid, then the argument is proven. Whether or not a mapping (or any mapping) can be validated (and accepted) depends also on the pragmatic context. It's possible to not accept any proof, and it's possible for all proofs to be wrong. But that doesn't mean that they are not proofs. But perhaps I didn't understand you? Or perhaps another word might be better? And yet I'm using ''proof'' in the same spirit as mathematicians do in their work. They don't spell out all the details, they make sure the main idea is compelling and when it is, the details clarify themselves and are worked out further. They don't write their papers this way, but this is how they think through their conclusions. So, for example, the above analogy makes clear that adding a road ''could'' make traffic worse. It also suggests ''why'' traffic is worse (the extra spring moves the weight to the weak springs; the extra road diverts traffic to the less efficient roads). Or vice versa. We've moved the problem from one domain to another one where possibly we may have better intuition. What is actually proven depends on the actual morphism. And note that formalism can make the proof more rigorous (more applicable regarding the details), but it can't make it stronger (clearer as to whether it is inherently valid or invalid). Yes, a proof can be stitched together, but that doesn't convince us that it will hold together. Morphism is the strongest type of proof because it is cut from whole cloth. It is all or nothing - either it applies or it doesn't.

Here are some examples of different ways of proving the same thing: [{{SelfEducation/methods}} of proof]. Morphism is the ''strongest'' (and convincing) proof, then induction, and substitution (algebraic manipulation) or examination of cases are much more fragile, much more contingent. A good morphism may be wrong in one case, but then it will be right about another, perhaps more important case. A good morphism can change what questions we choose to ask, it can have us redefine our terms and change our focus.

Here is another example: '''A right triangle is half of a rectangle.''' This is a morphism! I mean, this is to say that a ''right triangle'' is a concept that bridges ''the world of rectangles'' and ''the world of triangles''. Consequently, the shape of the right triangle is given by:
* the size of an acute angle (the world of triangles)
* the ratio of any two sides (the world of rectangles)
This ''defines'' six maps between these two worlds! and they are the trigonometric functions. We don't need to know any more details - this is the heart of trigonometry, and enough for us to start to apply it practically. In fact, this morphism tells us the kinds of geometries for which we can have a ''trigonometry''.

{{HelmutLeitner}}: I think the [http://www.davros.org/science/roadparadox.html Road Network Paradox] doesn't hold. While the symmetrical standard situation produces 83x6000=498000 time units, A maybe counterintuitive traffic A-&gt;C-&gt;B-&gt;D actually reduces the total time and so improves average time as expected (e. g. 5600 x 80.8 + 400 x 111.2 = 496960 time units). So either the mechanical example also doesn't hold or the morphism is wrong.

{{HelmutLeitner}}: The paradox is artifically constructed because people are assumed to have to go in the wrong direction. It doesn't hold, when some people go the long way on the good roads. Then average travel time decreases. Check the numbers I've given.

{{Andrius}}: Helmut, I think your alternate route is clever. (And they may not have taken that into account.) I can't take the time to think through the numbers. But is your formula suggesting that some people will travel 111.2 minutes and others will travel 80.8 minutes? They are specifically ruling that out by assuming that people will switch routes until everybody's travel time is the same. Also, from a common sense point of view, is the introduction of this clever long route an "improvement" over the original solution. That would be hard to believe (though, perhaps possible). Finally, the point that got me to believe the paradox is that the shortcut is much shorter than the long roads. Taking the shortcut makes for a 30km trip rather than a 50km trip. So people have a reason to sit through two traffic jams instead of just one. And that makes the traffic jams much worse.

{{HelmutLeitner}}: Arguments about that will take longer than checking the numbers. If there are multiple routes from some A to some B then it is improbable that the times of drivers will ever be equal, this is only because the sample model is symmetrical, so the condition - if it exists, doesn't make sense. In real life not even on a single route drivers will experience equal times. Drivers can't compare and can't optimize. Another example: I have four routes to my work, the shortest is 8 min but sometimes jammed to 30 min, so I often take other routes, the longest route guarantees me about 12-15 min. So if I have to be in time and I have 15+ minutes, I'll take the longest route. If I have enough time I take the shortest route, because it is statistically fastest. But I also take other routes regularly - because otherwise I have no experience about their qualities. -- But all such considerations are outside of the abstract model of the paradox. There is no guarantee that the model corresponds to reality, a model is a simplification. -- In effect they introduce a one-way-road and its quite clear that a one-way-road in the wrong direction will not be helpful - but this wouldn't be entertaining. If they talk about a street I have the right to assume that people may use it in either direction. I have the right to assume that they talk about the real world because they strive for a - very doubtful - real world explanation for "they add streets and traffic doesn't get better" which may have quite different reasons (e. g. that good streets attract traffic and that traffic increases). -- Also the mathematical model is doubtful: If we have a street E-&gt;G with length L and capacity C (T=L+F/C) and reinterpret it cut in to pieces as E-&gt;F-&gt;G with the same length (L/2 + L/2) and capacity, the calculated time increases (T=L+2*F/C). This has no influence onto all the aspects we discussed but it does not correspond to traffic or flow reality. Of course changes in the mathematic calculations can change the situations but can't be reflected in the morphism to the physical spring model!

{{Andrius}}: Helmut, you're right that reality is complex, as you say. But here are their assumptions: %gray%We can see that if traffic is split unevenly, one route will be faster than the other. People using the slower route will realize that they could do better by changing to the other, and no doubt will do so. The only stable situation is where half the traffic goes each way, because then anyone changing to the other route will find they are on a slower route, and thus will change back.%gray% For modeling purposes, those may or may not be good assumptions, and in your case they would at least require averaging. For example, in economics often such assumptions are made (perfect knowledge), occasionally they are not (imperfect knowledge), and there are cases where either model is appropriate. I think it's fair to allow that they are not trying to model all road systems, just some road systems. Here is the website of [http://homepage.ruhr-uni-bochum.de/Dietrich.Braess/ Dietrich Braess] who wrote the original paper in 1968, and which has been generated a lot of research, including observations of this behavior in computer networks. And [http://www.crowddynamics.com/Myriad/Braess%20Paradox.htm here's another explanation].

{{HelmutLeitner}}: Andrius, I do not doubt that there are assumptions that are consistent with their findings. I ''do'' doubt that they say something about real traffic or physical systems. Morphism - the way they use it - just seems to provide insight. Can you provide a better example of a morphism that actually proofs something?
2014 birželio 05 d., 12:59 atliko Andrius Kulikauskas -
Pakeistos 23-27 eilutės iš
* {{Analysis}} allows for work with limits, see the {{Fivesome}}
į:
* {{Analysis}} allows for work with limits, see the {{Fivesome}}

-------------

{{Andrius}}: Note that a {{Category}} may be thought of as a deductive system, a directive graph, and hence a {{Matrix}}. My thesis was on the combinatorics of the general matrix, which apparently is all generated by the symmetric functions of the eigenvalues of a matrix. So I am interested to see if that might be of value here. I want to show how a category arises from first principles. I imagine that perspectives may be thought of as morphisms (or functors).
2014 birželio 05 d., 12:58 atliko Andrius Kulikauskas -
Pakeistos 1-23 eilutės iš
Kaip [[http://www.selflearners.net/Math/DeepIdeas | matematikos pagrindus]] pristatyti svarbiausiais dėsniais, pavyzdžiais ir žaidimais? Kuo žaidimai yra vertingi, kaip jie suveikia? Kuriu atitinkamas mokymosi priemones, tapau drobę.
į:
Kaip [[http://www.selflearners.net/Math/DeepIdeas | matematikos pagrindus]] pristatyti svarbiausiais dėsniais, pavyzdžiais ir žaidimais? Kuo žaidimai yra vertingi, kaip jie suveikia? Kuriu atitinkamas mokymosi priemones, tapau drobę.

Mathematics is the study of {{Structure}}

{{Andrius}}: Much of my work may be considered pre-mathematics, the grounding of the structures that ultimately allow for mathematics.

One milestone (or miracle) in knowing everything is to generate all mathematical objects and truths in a unified way, but especially, generate all mathematical insights in a way that builds our intuition. The relevant outlook will, I imagine, be in terms of beauty. I mean that we may think of mathematical insight as guided by the wish for {{Beauty}}.

Here are some branches of mathematics that are basic in this respect:

* {{Matrix}} - I think this is the most basic object in mathematics. Note that the index set may be arbitrary, not necessarily numbers.
* {{Combinatorics}} - The mathematics of counting (and generating) objects. I think of this as the basement of mathematics, which is why I studied it.
* AutomataTheory - There is a qualitative hierarchy of computing devices. And they can be described in terms of matrices.
* NeuralNetworks - Very powerful and simple computational systems for which Sarunas Raudys showed a hierarchy of sophistication as learning systems.
* {{Representations}} - A very important idea, which is that we access a deep structure (such as a division of everything) not directly, but by way of some representation. This term is used in algebra, for example, to distinguish a system (like a group) from the matrices which serve as its multiplication table.
* CategoryTheory - Categories are helpful in making fruitful definitions
* {{Topologies}} - Systems of constraints that may be thought of as defining worlds. Topology is the study of topologies.
* SetTheory - The Zermelo Frankel axioms of set theory seem to be structured by what I call the EightfoldWay.
* RecursiveFunctions - 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}}
* LinearAlgebra - Is the study of the basic properties of matrices and their effects.
* {{Logic}} is the end result of structure, see the {{Sevensome}} and Greimas' SemioticSquare
* {{Algebra}} studies particular structures
* {{Analysis}} allows for work with limits, see the {{Fivesome}}
2012 sausio 10 d., 20:31 atliko Andrius Kulikauskas -
Pridėta 1 eilutė:
Kaip [[http://www.selflearners.net/Math/DeepIdeas | matematikos pagrindus]] pristatyti svarbiausiais dėsniais, pavyzdžiais ir žaidimais? Kuo žaidimai yra vertingi, kaip jie suveikia? Kuriu atitinkamas mokymosi priemones, tapau drobę.

#### Matematika

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 Puslapis paskutinį kartą pakeistas 2016 birželio 19 d., 14:54