手册

数学

物理

Discovery

Andrius Kulikauskas

  • ms@ms.lt
  • +370 607 27 665
  • My work is in the Public Domain for all to share freely.

Lietuvių kalba

Introduction E9F5FC

Understandable FFFFFF

Questions FFFFC0

Notes EEEEEE

Software

Book.Math istorija

Rodyti nežymius pakeitimus - Rodyti kodo pakeitimus

2019 rugpjūčio 17 d., 00:06 atliko AndriusKulikauskas -
Pakeista 1 eilutė iš:

Useful: Online Latex editor, Math Notation

į:

Useful: Online Latex editor, Math Notation, Math in Chinese

2019 rugpjūčio 10 d., 12:57 atliko AndriusKulikauskas -
Ištrintos 7-8 eilutės:

Math concepts I'm trying to understand all of math.

2019 rugpjūčio 10 d., 12:57 atliko AndriusKulikauskas -
Pakeistos 5-6 eilutės iš

Math notebook I share my results so far.

į:

Math research An overview of my research.

Math notebook Questions that I am investigating.

Ištrintos 11-12 eilutės:

Mathematical questions I am investigating

2019 sausio 20 d., 21:36 atliko AndriusKulikauskas -
Ištrintos 50-155 eilutės:
  • In the ways of figuring things out, I want to show how the three-cycle extends mathematical structure.

Questions I'm asking myself

  • How does Euler characteristic relate to homology, structures with holes?
  • What is the relationship between Pascal's triangle and the Grassmannian?

Philosophical questions about math:

  • How is love (and life) related to duality, reflections, transformations and other math concepts?
  • How does 1 mediate the duality of 0 and infinity? And how is that duality variously broken?
  • What are the six basic transformation? and their relation with symmetry?

Math investigations:

  • List out the results of universal hyperbolic geometry and state them in terms of symmetric functions.
  • Write an elegant combinatorics of the finite field and interpret what is F1^n.

Consider more examples, simple and sophisticated, of how things are figured out in math.

  • What is the relationship between the surface math problem and the deep way of figuring things out?
  • How do we discover the right way to figure out a math problem?
  • How do we combine several distinct ways of figuring things out?
  • How can I apply my results to figure things out in math, the biggest problems?

Consider how infinity, zero and one are defined in the various geometries. How do these concepts fit together? How do they involve the viewer and their perspective? How might that relate to Christopher Alexander's principles of life and the plane of the viewer?

One way to think of geometry is in terms of what happens at infinity. For example, do all lines (through plane) meet at infinity at a common point? Or at a circle? Or do the ends of the lines not meet? Or do they go to a circle of infinite length?

Didieji klausimai:

  • Keturias briaunainių šeimas sieti su keturiomis geometrijomis, metalogikomis, ketverybe, ženklų rūšimis, teigiamais įsakymais.
  • Briaunainių šeimų poras sieti su šešiomis daugybomis, pertvarkymais, permainomis, neigiamais įsakymais.

Questions to ask others at Math Overflow and elsewhere:

  • Intuitively, why are there four classical Lie algebras/groups?
  • Check what happens if I plug in different values into the Catalan power series.
  • What is a combinatorial interpretation of P - P(n), the generators of the Mandelbrot set, in terms of the Catalan numbers? Get help to generate the difference. What is the best software for that?
  • What is {$F_{1^n}$}?

What does projective geometry say about the existence of infinity?

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.
  • 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) ?

Tiesinė algebra

  • Kaip dauginti polar decomposed matrices?
  • Geriau suprasti 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?
  • Matematikai:
    • Suvokti matematikos prasmę ir išskirtinumą.
    • Savo pagrindines sąvokas išsakyti matematika.
      • Išsakyti Dievo šokį nuliu, begalybe ir vieniu.
        • Susipažinti su vienanariu lauku, su baigtiniais laukais ir jų kombinatorika, su F1n.
        • Apžvelgti dvilypumo pavyzdžius ir suvokti jų esmę.
        • Suprasti, kur ir kaip išsiskiria nulis ir begalybė, kaip išyra jų tolygumas.
        • Suvokti, kaip baigtiniai laukai ir vienanaris laukas vaizduoja begalybę.
        • Begalybę išreikšti geometrijomis.
      • Išsakyti keturis lygmenis.
        • Apžvelgti, įsisavinti ir atskirti keturias geometrijas.
        • Jas išsakyti simetrinėmis funkcijomis.
        • Išnagrinėti ir paklausti, kodėl yra keturios Lie grupės, algebros.
      • Išsakyti šešias permainas.
        • Rinkti ir rūšiuoti kintamųjų pavyzdžius.
        • Išsakyti šešis veiksmus vedančius iš vienos geometrijos į kitą.
        • Susieti su šešiais pertvarkymais.
      • Išsakyti aštuonis padalinimus.
        • Suprasti Bott periodiškumą.
    • Taikyti matematikos išsiaiškinimo būdus.
      • Parodyti, kaip trejybė plėtoja matematiką.
    • Aprėpti visą matematiką ir nurodyti, kaip išsiritulioja jos šakos, sąvokos, teiginiai ir uždaviniai.
      • Susieti matematiką su įvardijimo ir pagrindimo kalbomis.
  • Apžvelgti savo tyrimus matematikoje.
  • Nagrinėti algebrinės geometrijos teoremas.
  • Tirti Wildberger knygą.
  • Susieti šešis patikslinimus su šešiais kintamaisiais.
  • Išsiaiškinti matematikos svarbą. Jos išsiaiškinimo būdais išvesti jos turinį, jos šakas ir sąvokas
2019 sausio 20 d., 21:29 atliko AndriusKulikauskas -
Pridėta 35 eilutė:
2019 sausio 20 d., 21:27 atliko AndriusKulikauskas -
Pakeistos 155-165 eilutės iš
  • Išsiaiškinti matematikos svarbą. Jos išsiaiškinimo būdais išvesti jos turinį, jos šakas ir sąvokas

Matematika

  • Jeigu yra matematinis apibrėžimas (pavyzdžiui, baigtinė tiksli seka), kuris galioja vienam žmogui, tai tai galioja visiems. Šitą mintį mąsčiau bekliedėdamas, besirgdamas.
  • Matematika pagrįsta apibendrinimu, abstrahavimu.

The nature of math

  • Gramatika visada turi prasmę. Matematika yra tai, kas pavaldu logikai, bet nebūtinai turi prasmę.
  • Matematika mus moko, kaip besąlygiškumą reikšti sąlygomis, o tai įmanoma sąlygiškai.
į:
  • Išsiaiškinti matematikos svarbą. Jos išsiaiškinimo būdais išvesti jos turinį, jos šakas ir sąvokas
2019 sausio 20 d., 21:16 atliko AndriusKulikauskas -
Ištrintos 49-50 eilutės:
  • My Ph.D. thesis: Symmetric Functions of the Eigenvalues of a Matrix
2019 sausio 20 d., 21:16 atliko AndriusKulikauskas -
Pakeistos 27-28 eilutės iš
  • In particular, I want to show how the three-cycle extends mathematical structure.
į:
Ištrinta 29 eilutė:
Pakeistos 34-35 eilutės iš
į:
Pakeistos 47-54 eilutės iš
į:
Pridėtos 55-56 eilutės:
  • In the ways of figuring things out, I want to show how the three-cycle extends mathematical structure.
Ištrintos 60-64 eilutės:

Questions I need to ask others

  • Why can't the field with one element be thought of as the zero ring?
  • Are my weights for the simplexes known?
  • Is my interpretation of the -1 simplex known?
Pakeistos 65-66 eilutės iš
į:
Pakeistos 89-90 eilutės iš
  • What is F1n?
į:
  • What is {$F_{1^n}$}?
Ištrintos 126-146 eilutės:

Matematikos bendravimui

  • Bendrai
    • Surašyti savo mintis apie kintamuosius
  • Kartu su Rimvydu Krasausku
    • Mokytis homotipų tipų teorijos, kategorijų teorijos ir geometrijos.
  • Azimuth Project
    • Atsiliepti ir rašyti
  • Math Future
    • Parašyti ką veikiu, kaip tiriu nuojautas.
  • Kreiptis pagalbos Math Stack Exchange.
    • Suprasti skirtumas tarp keturių klasikinių Lie grupių-algebrų.
  • Foundations of Mathematics
    • Susipažinti su Harvey Friedman mintimis
    • Rašyti į FOM apie geometriją, taip pat "paraconsistency".
  • NLab
    • Parašyti apie -1 simpleksą.
2019 sausio 20 d., 21:10 atliko AndriusKulikauskas -
Pakeistos 48-49 eilutės iš

Math resources

į:
Pakeistos 54-59 eilutės iš

Useful links:

Other interests:

į:
2019 sausio 20 d., 18:22 atliko AndriusKulikauskas -
Pakeistos 52-64 eilutės iš

Interesting mathematicians:

Vilniaus universitetas

  • Rimvydas Krasauskas, Kompiuterinė geometrijos laboratorija
  • Hamletas Markšaitis, algebra, tipų teorija, kategorijų teorija
  • Giedrius Alkauskas - skaičių teorija, fraktalai, continued functions
  • Algirdas Javtokas mathematical psychology & economics, computational models in cognition modeling
  • Šarūnas Raudys
  • Stasys Norgėla - matematinė logika
į:
2019 sausio 19 d., 11:07 atliko AndriusKulikauskas -
Pakeista 3 eilutė iš:

数学....מאטעמאטיק

į:

数学 _ _ _ _ מאטעמאטיק

2019 sausio 19 d., 11:07 atliko AndriusKulikauskas -
Pakeista 3 eilutė iš:

数学

į:

数学....מאטעמאטיק

2019 sausio 19 d., 11:06 atliko AndriusKulikauskas -
Pridėtos 2-3 eilutės:

数学

2018 lapkričio 11 d., 17:34 atliko AndriusKulikauskas -
Pakeistos 207-211 eilutės iš
  • Matematika pagrįsta apibendrinimu, abstrahavimu.
į:
  • Matematika pagrįsta apibendrinimu, abstrahavimu.

The nature of math

  • Gramatika visada turi prasmę. Matematika yra tai, kas pavaldu logikai, bet nebūtinai turi prasmę.
  • Matematika mus moko, kaip besąlygiškumą reikšti sąlygomis, o tai įmanoma sąlygiškai.
2018 rugsėjo 11 d., 14:53 atliko AndriusKulikauskas -
Pridėtos 2-3 eilutės:

Math notebook I share my results so far.

2018 rugsėjo 01 d., 21:37 atliko AndriusKulikauskas -
Pakeistos 27-28 eilutės iš

I want to overview all of mathematics and show how its branches, concepts, questions (problems) and answers (theorems) unfold. I'm creating a map which will include the key:

į:

I want to overview all of mathematics and show how its branches, concepts, questions (problems) and answers (theorems) unfold. I'm creating a map of math which will include the key:

2018 rugsėjo 01 d., 21:30 atliko AndriusKulikauskas -
Pakeistos 27-30 eilutės iš

Areas of Math I want to overview all of mathematics and show how its branches, concepts, questions (problems) and answers (theorems) unfold.

Mathematical Concepts I am trying to organize an encyclopedia of mathematical concepts to see what they are and how they unfold.

į:

I want to overview all of mathematics and show how its branches, concepts, questions (problems) and answers (theorems) unfold. I'm creating a map which will include the key:

  • Branches of mathematics.
  • Concepts I am trying to organize an encyclopedia of mathematical concepts to see what they are and how they unfold.
  • Theorems.
2018 sausio 07 d., 16:48 atliko AndriusKulikauskas -
Pakeista 3 eilutė iš:
į:

Math concepts I'm trying to understand all of math.

2018 sausio 01 d., 08:48 atliko AndriusKulikauskas -
Pridėtos 29-30 eilutės:

Mathematical Concepts I am trying to organize an encyclopedia of mathematical concepts to see what they are and how they unfold.

2017 gruodžio 09 d., 17:35 atliko AndriusKulikauskas -
Pakeista 5 eilutė iš:
į:
2017 liepos 26 d., 17:59 atliko AndriusKulikauskas -
Pridėtos 2-3 eilutės:
2017 kovo 17 d., 12:45 atliko AndriusKulikauskas -
Pakeistos 193-199 eilutės iš
  • Išsiaiškinti matematikos svarbą. Jos išsiaiškinimo būdais išvesti jos turinį, jos šakas ir sąvokas
į:
  • Išsiaiškinti matematikos svarbą. Jos išsiaiškinimo būdais išvesti jos turinį, jos šakas ir sąvokas

Matematika

  • Jeigu yra matematinis apibrėžimas (pavyzdžiui, baigtinė tiksli seka), kuris galioja vienam žmogui, tai tai galioja visiems. Šitą mintį mąsčiau bekliedėdamas, besirgdamas.
  • Matematika pagrįsta apibendrinimu, abstrahavimu.
2016 gruodžio 13 d., 21:58 atliko AndriusKulikauskas -
Ištrintos 7-12 eilutės:
  • Conversation. Join with others for an ongoing conversation about collaborating on a science of math, especially, the "implicit math" that we think in our minds.
    • Describe and investigate the many dimensions of math, including discovery, beauty, insight, learning, humanity.
  • Show how investigation can and does methodically apply the particular ways of figuring things out, notably, in mathematics, but also more generally.
    • In particular, how does the three-cycle extend our existing mathematical language.
  • Math connections Express my philosophy's concept in terms of mathematics.
  • Show how math unfolds, how its various branches and concepts arise.
Pridėtos 9-14 eilutės:
  • Abstraction. Show how math unfolds, how its various branches and concepts arise, especially by studying the history of how math grows and documenting the ways of abstraction.
  • Describe and investigate the many dimensions of math, including discovery, beauty, insight, learning, humanity.
  • Math discovery Show how investigation can and does methodically apply the particular ways of figuring things out, notably, in mathematics, but also more generally.
  • Math connections Express my philosophy's concept in terms of mathematics and thus understand which mathematical concepts are most central.
  • Study math that is most relevant.
  • Conversation. Join with others for an ongoing conversation about collaborating on a science of math, especially, the "implicit math" that we think in our minds.
2016 lapkričio 22 d., 21:11 atliko AndriusKulikauskas -
Ištrintos 12-14 eilutės:
  • Express God's dance in terms of zero, infinity and one.
  • Express four levels of knowledge as four geometries.
  • Express six pairs of these levels as six fundamental transformations.
2016 lapkričio 22 d., 21:10 atliko AndriusKulikauskas -
Pakeista 12 eilutė iš:
  • Express my philosophy's concept in terms of mathematics.
į:
Ištrintos 17-50 eilutės:

My philosopy's concepts in terms of math

Math connections. I am expressing my philosopy's concepts in terms of mathematics. In particular, I wish to:

Understand God's dance in terms of zero, infinity and one.

  • understand God as the Center of polytopes (such as the Simplex) and Everything? as the Totality?.
  • understand the field with one element, learn about the combinatorics and geometry of finite fields and understand F1n.
  • overview examples of duality and understand their essence, how they relate one and zero, how they help map out the areas of math.
  • understand how zero and infinity get differentiated, how their symmetry gets variously broken.
  • understand how finite fields and the field with one element express infinity.
  • express infinity in terms of geometries.
  • understand the kinds of Opposites and how they are expressed in mathematics.

Describe four levels of knowledge (whether, what, how, why).

  • survey, understand and distinguish between four geometries (affine, projective, conformal, symplectic).
  • express them in terms of symmetric functions.
  • review recursive function theory and describe the four levels in terms of related concepts such as the Yates Index theorem (the triple jump).
  • understand intuitively, why there are four classical families of Lie groups/algebras and what four geometries they give rise to.

Discover six transformations? between these four levels.

  • collect and identify six ways of thinking about variables.
  • survey, specify and ground six transformations between the four geometries.
  • relate them to six way of thinking about multiplication.
  • relate them to six kinds of variables.
  • relate these six transformations to six visualizations (restructurings? in terms of sequences, hierarchies and networks).
  • relate them to six qualities of signs.
  • interpret them metalogically as six ways of relating two mental sheets.
  • relate them to six set theory axioms?.

Express eight Divisions of everything, if possible, in terms of Bott periodicity.

Relate mathematics to three cognitive languages by which things come to matter (argumentation), have meaning (verbalization) and take place (narration).

  • Relate a hierarchy of six methods of proof to the prayer "Our Father" and a language of argumentation.
2016 spalio 30 d., 09:00 atliko AndriusKulikauskas -
Pakeista 1 eilutė iš:

Useful: Online Latex editor

į:

Useful: Online Latex editor, Math Notation

2016 spalio 30 d., 05:47 atliko AndriusKulikauskas -
Pakeistos 228-230 eilutės iš
  • Susieti šešis patikslinimus su šešiais kintamaisiais.
į:
  • Susieti šešis patikslinimus su šešiais kintamaisiais.
  • Išsiaiškinti matematikos svarbą. Jos išsiaiškinimo būdais išvesti jos turinį, jos šakas ir sąvokas
2016 spalio 30 d., 05:32 atliko AndriusKulikauskas -
Pakeistos 223-228 eilutės iš
  • Susieti matematiką su įvardijimo ir pagrindimo kalbomis.
į:
  • Susieti matematiką su įvardijimo ir pagrindimo kalbomis.
  • Apžvelgti savo tyrimus matematikoje.
  • Nagrinėti algebrinės geometrijos teoremas.
  • Tirti Wildberger knygą.
  • Susieti šešis patikslinimus su šešiais kintamaisiais.
2016 spalio 30 d., 05:31 atliko AndriusKulikauskas -
Pakeistos 178-223 eilutės iš
į:

Matematikos bendravimui

  • Bendrai
    • Surašyti savo mintis apie kintamuosius
  • Kartu su Rimvydu Krasausku
    • Mokytis homotipų tipų teorijos, kategorijų teorijos ir geometrijos.
  • Azimuth Project
    • Atsiliepti ir rašyti
  • Math Future
    • Parašyti ką veikiu, kaip tiriu nuojautas.
  • Kreiptis pagalbos Math Stack Exchange.
    • Suprasti skirtumas tarp keturių klasikinių Lie grupių-algebrų.
  • Foundations of Mathematics
    • Susipažinti su Harvey Friedman mintimis
    • Rašyti į FOM apie geometriją, taip pat "paraconsistency".
  • NLab
    • Parašyti apie -1 simpleksą.
  • Matematikai:
    • Suvokti matematikos prasmę ir išskirtinumą.
    • Savo pagrindines sąvokas išsakyti matematika.
      • Išsakyti Dievo šokį nuliu, begalybe ir vieniu.
        • Susipažinti su vienanariu lauku, su baigtiniais laukais ir jų kombinatorika, su F1n.
        • Apžvelgti dvilypumo pavyzdžius ir suvokti jų esmę.
        • Suprasti, kur ir kaip išsiskiria nulis ir begalybė, kaip išyra jų tolygumas.
        • Suvokti, kaip baigtiniai laukai ir vienanaris laukas vaizduoja begalybę.
        • Begalybę išreikšti geometrijomis.
      • Išsakyti keturis lygmenis.
        • Apžvelgti, įsisavinti ir atskirti keturias geometrijas.
        • Jas išsakyti simetrinėmis funkcijomis.
        • Išnagrinėti ir paklausti, kodėl yra keturios Lie grupės, algebros.
      • Išsakyti šešias permainas.
        • Rinkti ir rūšiuoti kintamųjų pavyzdžius.
        • Išsakyti šešis veiksmus vedančius iš vienos geometrijos į kitą.
        • Susieti su šešiais pertvarkymais.
      • Išsakyti aštuonis padalinimus.
        • Suprasti Bott periodiškumą.
    • Taikyti matematikos išsiaiškinimo būdus.
      • Parodyti, kaip trejybė plėtoja matematiką.
    • Aprėpti visą matematiką ir nurodyti, kaip išsiritulioja jos šakos, sąvokos, teiginiai ir uždaviniai.
      • Susieti matematiką su įvardijimo ir pagrindimo kalbomis.
2016 spalio 08 d., 17:35 atliko AndriusKulikauskas -
Pridėtos 3-6 eilutės:

Mathematics I am studying

Mathematical questions I am investigating

Ištrintos 73-119 eilutės:

Mathematics I am studying

Areas of math that I want to learn more about:

Other assorted concepts: Tetrahedron, Triality, Associativity, Unit, Matrix

Math theorems to study and master:

  • The theorem distinguishing the reals, complexes, quaternions, octonions and why there is nothing higher.
  • Fundamental Theorem of Algebra
  • Galois correspondence

Math questions I am focusing on

Math I am currently focusing on

  • understanding how Pascal's triangle relates to homology and the Gauss-Bonnet theorem and Euler's characteristic
  • understanding the different kinds of Pascal triangles and how they relate to Grassmannians
  • understanding the Gaussian binomial coefficients in terms of Coxeter generators
  • understanding the relationship between the discrete and continuous case of projective geometry
2016 spalio 03 d., 14:08 atliko AndriusKulikauskas -
Pakeista 74 eilutė iš:
į:
2016 spalio 03 d., 14:05 atliko AndriusKulikauskas -
Pakeistos 77-81 eilutės iš
į:
2016 rugsėjo 27 d., 12:16 atliko AndriusKulikauskas -
Pridėtos 136-144 eilutės:

Questions I'm asking myself

  • How does Euler characteristic relate to homology, structures with holes?
  • What is the relationship between Pascal's triangle and the Grassmannian?

Questions I need to ask others

  • Why can't the field with one element be thought of as the zero ring?
  • Are my weights for the simplexes known?
  • Is my interpretation of the -1 simplex known?
2016 rugsėjo 05 d., 08:46 atliko AndriusKulikauskas -
Pridėta 99 eilutė:
  • Galois correspondence
2016 rugsėjo 05 d., 08:45 atliko AndriusKulikauskas -
Pridėtos 95-98 eilutės:

Math theorems to study and master:

  • The theorem distinguishing the reals, complexes, quaternions, octonions and why there is nothing higher.
  • Fundamental Theorem of Algebra
2016 rugsėjo 05 d., 08:44 atliko AndriusKulikauskas -
Pakeistos 71-80 eilutės iš

Math I am currently focusing on

  • understanding how Pascal's triangle relates to homology and the Gauss-Bonnet theorem and Euler's characteristic
  • understanding the different kinds of Pascal triangles and how they relate to Grassmannians
  • understanding the Gaussian binomial coefficients in terms of Coxeter generators
  • understanding the relationship between the discrete and continuous case of projective geometry
į:

Mathematics I am studying

Pakeistos 96-97 eilutės iš
į:

Math questions I am focusing on

Math I am currently focusing on

  • understanding how Pascal's triangle relates to homology and the Gauss-Bonnet theorem and Euler's characteristic
  • understanding the different kinds of Pascal triangles and how they relate to Grassmannians
  • understanding the Gaussian binomial coefficients in terms of Coxeter generators
  • understanding the relationship between the discrete and continuous case of projective geometry

Math resources

Math communities

Ištrintos 120-121 eilutės:
2016 rugsėjo 05 d., 08:42 atliko AndriusKulikauskas -
Pakeistos 17-18 eilutės iš

I am expressing my philosopy's concepts in terms of mathematics. In particular, I wish to:

į:

Math connections. I am expressing my philosopy's concepts in terms of mathematics. In particular, I wish to:

Pridėtos 51-52 eilutės:

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. The techniques and structures that we use in our minds are much more elemental than the mathematical output which they generate.

Pakeista 58 eilutė iš:

I want to overview all of mathemtics and show how its branches, concepts, questions (problems) and answers (theorems) unfold.

į:

Areas of Math I want to overview all of mathematics and show how its branches, concepts, questions (problems) and answers (theorems) unfold.

Pakeistos 61-63 eilutės iš
į:

Other questions about the big picture

Here are questions about the big picture in mathematics:

  • 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?
  • 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?
Pakeistos 80-81 eilutės iš

Math connections with my philosophy.

į:
Pakeistos 104-113 eilutės iš

Here are questions about the big picture in mathematics:

  • 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?
į:
Ištrinta 156 eilutė:
  • What is the significance of the center of a simplex?
2016 rugsėjo 05 d., 08:30 atliko AndriusKulikauskas -
Ištrintos 58-61 eilutės:
  • Christopher Alexander's principles of life and different kinds of opposites are relevant.
2016 rugsėjo 04 d., 22:16 atliko AndriusKulikauskas -
Ištrintos 61-63 eilutės:
Ištrintos 187-188 eilutės:

Study the finite field GF(8) and relate it to the divisions of everything.

2016 rugsėjo 04 d., 18:53 atliko AndriusKulikauskas -
Pakeista 5 eilutė iš:
  • Describe and investigate the many dimensions of math, including discovery, beauty, insight, learning, humanity.
į:
  • Describe and investigate the many dimensions of math, including discovery, beauty, insight, learning, humanity.
Pridėtos 46-48 eilutės:

Relate mathematics to three cognitive languages by which things come to matter (argumentation), have meaning (verbalization) and take place (narration).

  • Relate a hierarchy of six methods of proof to the prayer "Our Father" and a language of argumentation.
Pakeista 51 eilutė iš:

I want to learn to apply the 24 ways of figuring things out in mathematics.

į:

I have described and systematized 24 ways of figuring things out in mathematics. I now want to relate that to an overall methodology for answering mathematical questions.

Pakeistos 59-62 eilutės iš

I want to relate mathematics to three cognitive languages by which things come to matter (argumentation), have meaning (verbalization) and take place (narration).

į:
2016 rugsėjo 03 d., 18:40 atliko AndriusKulikauskas -
Pakeistos 32-33 eilutės iš
į:
  • understand intuitively, why there are four classical families of Lie groups/algebras and what four geometries they give rise to.
Pridėta 37 eilutė:
Pakeista 39 eilutė iš:
  • relate these six transformations to six visualizations (in terms of sequences, hierarchies and networks).
į:
  • relate these six transformations to six visualizations (restructurings? in terms of sequences, hierarchies and networks).
Pakeistos 42-48 eilutės iš
  • I want to study six logical . They seem to be evident in six ways of thinking about multiplication. And they are six visualizations, restructurings?. They seem to visualize six set theory axioms?.
  • relate multiplication as transformations between four geometries
  • relate multiplication to set theory axioms
  • relate multiplication to restructurings

I want to understand Bott periodicity and try to relate it to the eight divisions of everything.

į:

Express eight Divisions of everything, if possible, in terms of Bott periodicity.

2016 rugsėjo 03 d., 18:36 atliko AndriusKulikauskas -
Pakeistos 17-21 eilutės iš

I am expressing my philosopy's concepts in terms of mathematics. In particular:

God's dance in terms of zero, infinity and one.

  • understand the field with one element, learn about the combinatorics and geometry of finite fields and understand F1n.
  • overview examples of duality and understand their essence.
į:

I am expressing my philosopy's concepts in terms of mathematics. In particular, I wish to:

Understand God's dance in terms of zero, infinity and one.

  • understand God as the Center of polytopes (such as the Simplex) and Everything? as the Totality?.
  • understand the field with one element, learn about the combinatorics and geometry of finite fields and understand F1n.
  • overview examples of duality and understand their essence, how they relate one and zero, how they help map out the areas of math.
Pakeistos 26-34 eilutės iš

I want to describe four levels of knowledge (whether, what, how, why).

į:
  • understand the kinds of Opposites and how they are expressed in mathematics.

Describe four levels of knowledge (whether, what, how, why).

Pakeista 33 eilutė iš:

I want to discover six transformations between these four levels.

į:

Discover six transformations? between these four levels.

Pridėta 36 eilutė:
Pakeistos 39-40 eilutės iš
į:
  • interpret them metalogically as six ways of relating two mental sheets.
  • I want to study six logical . They seem to be evident in six ways of thinking about multiplication. And they are six visualizations, restructurings?. They seem to visualize six set theory axioms?.
2016 rugsėjo 03 d., 18:28 atliko AndriusKulikauskas -
Pridėta 5 eilutė:
  • Describe and investigate the many dimensions of math, including discovery, beauty, insight, learning, humanity.
Pridėta 12 eilutė:
  • Show how math unfolds, how its various branches and concepts arise.
Ištrintos 14-15 eilutės:
  • Show how math unfolds, how its various branches and concepts arise.
Pakeistos 17-19 eilutės iš

I want to express my philosopy's concepts in terms of mathematics. In particular:

I want to express God's dance in terms of zero, infinity and one.

į:

I am expressing my philosopy's concepts in terms of mathematics. In particular:

God's dance in terms of zero, infinity and one.

2016 rugsėjo 03 d., 18:25 atliko AndriusKulikauskas -
Pakeistos 3-8 eilutės iš

Andrius Kulikauskas: I wish to show that my philosophy is very fruitful for developing a science of math, for understanding math as a cognitive language (implicit math) which takes place in the mind, and for overviewing all of math and how its branches and concepts unfold. Here are projects that I'm working on.

I want to join with others for an ongoing conversation about math.

I want to understand the purpose of math and what distinguishes it from other languages and disciplines.

į:

Andrius Kulikauskas: I wish to show that my philosophy is very fruitful for developing a science of math, for understanding math as a cognitive language (implicit math) which takes place in the mind, and for overviewing all of math and how its branches and concepts unfold. Here are projects that I'm working on:

  • Conversation. Join with others for an ongoing conversation about collaborating on a science of math, especially, the "implicit math" that we think in our minds.
  • Show how investigation can and does methodically apply the particular ways of figuring things out, notably, in mathematics, but also more generally.
    • In particular, how does the three-cycle extend our existing mathematical language.
  • Express my philosophy's concept in terms of mathematics.
    • Express God's dance in terms of zero, infinity and one.
    • Express four levels of knowledge as four geometries.
    • Express six pairs of these levels as six fundamental transformations.
  • Understand the purpose of math and what distinguishes it from other languages and disciplines.
  • Show how math unfolds, how its various branches and concepts arise.
Pakeista 17 eilutė iš:

I want to express my philosopy's concepts in mathematics. In particular:

į:

I want to express my philosopy's concepts in terms of mathematics. In particular:

2016 rugsėjo 02 d., 20:56 atliko AndriusKulikauskas -
Pridėtos 112-119 eilutės:

Vilniaus universitetas

  • Rimvydas Krasauskas, Kompiuterinė geometrijos laboratorija
  • Hamletas Markšaitis, algebra, tipų teorija, kategorijų teorija
  • Giedrius Alkauskas - skaičių teorija, fraktalai, continued functions
  • Algirdas Javtokas mathematical psychology & economics, computational models in cognition modeling
  • Šarūnas Raudys
  • Stasys Norgėla - matematinė logika
2016 rugsėjo 01 d., 23:31 atliko AndriusKulikauskas -
Pakeistos 11-12 eilutės iš

I want to express my philosopy's concepts in mathematics. In particular, I want to:

  • express God's dance in terms of zero, infinity and one.
į:

I want to express my philosopy's concepts in mathematics. In particular:

I want to express God's dance in terms of zero, infinity and one.

Pridėtos 20-26 eilutės:
Pakeistos 37-41 eilutės iš
į:
  • I want to study six logical transformations?. They seem to be evident in six ways of thinking about multiplication. And they are six visualizations, restructurings?. They seem to visualize six set theory axioms?.
  • relate multiplication as transformations between four geometries
  • relate multiplication to set theory axioms
  • relate multiplication to restructurings
Pakeistos 43-44 eilutės iš
į:
Pakeistos 53-54 eilutės iš
į:
Pakeistos 57-65 eilutės iš
 Here are concepts in my philosophy which I am working to model with math.

Here are my ongoing investigations:

į:
Pakeistos 59-62 eilutės iš
į:
Pakeistos 64-68 eilutės iš

Philosophical questions I am currently trying to address with math

  • relate multiplication as transformations between four geometries
  • relate multiplication to set theory axioms
  • relate multiplication to restructurings
į:
Ištrintos 121-122 eilutės:

See: SelfLearners/Math

2016 rugpjūčio 27 d., 11:58 atliko AndriusKulikauskas -
Pakeistos 3-25 eilutės iš

Andrius Kulikauskas: I wish to show that my philosophy is very fruitful for developing a science of math, for understanding math as a cognitive language (implicit math) which takes place in the mind, and for overviewing all of math and how its branches and concepts unfold.

  • Suvokti matematikos prasmę ir išskirtinumą.
  • Savo pagrindines sąvokas išsakyti matematika.
    • Išsakyti Dievo šokį nuliu, begalybe ir vieniu.
      • Susipažinti su vienanariu lauku, su baigtiniais laukais ir jų kombinatorika, su F1n.
      • Apžvelgti dvilypumo pavyzdžius ir suvokti jų esmę.
      • Suprasti, kur ir kaip išsiskiria nulis ir begalybė, kaip išyra jų tolygumas.
      • Suvokti, kaip baigtiniai laukai ir vienanaris laukas vaizduoja begalybę.
      • Begalybę išreikšti geometrijomis.
    • Išsakyti keturis lygmenis.
      • Apžvelgti, įsisavinti ir atskirti keturias geometrijas.
      • Jas išsakyti simetrinėmis funkcijomis.
    • Išsakyti šešias permainas.
      • Rinkti ir rūšiuoti kintamųjų pavyzdžius.
      • Išsakyti šešis veiksmus vedančius iš vienos geometrijos į kitą.
      • Susieti su šešiais pertvarkymais.
    • Išsakyti aštuonis padalinimus.
      • Suprasti Bott periodiškumą.
  • Taikyti matematikos išsiaiškinimo būdus.
    • Parodyti, kaip trejybė plėtoja matematiką.
  • Aprėpti visą matematiką ir nurodyti, kaip išsiritulioja jos šakos, sąvokos, teiginiai ir uždaviniai.
    • Susieti matematiką su įvardijimo ir pagrindimo kalbomis.
į:

Andrius Kulikauskas: I wish to show that my philosophy is very fruitful for developing a science of math, for understanding math as a cognitive language (implicit math) which takes place in the mind, and for overviewing all of math and how its branches and concepts unfold. Here are projects that I'm working on.

I want to join with others for an ongoing conversation about math.

I want to understand the purpose of math and what distinguishes it from other languages and disciplines.

My philosopy's concepts in terms of math

I want to express my philosopy's concepts in mathematics. In particular, I want to:

  • express God's dance in terms of zero, infinity and one.
  • understand the field with one element, learn about the combinatorics and geometry of finite fields and understand F1n.
  • overview examples of duality and understand their essence.
  • understand how zero and infinity get differentiated, how their symmetry gets variously broken.
  • understand how finite fields and the field with one element express infinity.
  • express infinity in terms of geometries.

I want to describe four levels of knowledge (whether, what, how, why).

  • survey, understand and distinguish between four geometries (affine, projective, conformal, symplectic).
  • express them in terms of symmetric functions.
  • review recursive function theory and describe the four levels in terms of related concepts such as the Yates Index theorem (the triple jump).

I want to discover six transformations between these four levels.

  • collect and identify six ways of thinking about variables.
  • survey, specify and ground six transformations between the four geometries.
  • relate these six transformations to six visualizations (in terms of sequences, hierarchies and networks).
  • relate them to six qualities of signs.

I want to understand Bott periodicity and try to relate it to the eight divisions of everything.

Ways of figuring things out in mathematics

I want to learn to apply the 24 ways of figuring things out in mathematics.

  • In particular, I want to show how the three-cycle extends mathematical structure.

Overview mathematics

I want to overview all of mathemtics and show how its branches, concepts, questions (problems) and answers (theorems) unfold.

I want to relate mathematics to three cognitive languages by which things come to matter (argumentation), have meaning (verbalization) and take place (narration).

2016 rugpjūčio 27 d., 11:42 atliko AndriusKulikauskas -
Pakeistos 3-4 eilutės iš

Andrius Kulikauskas: These pages are quite a mess for now. But I will organize them to make clear my interests in math.

į:

Andrius Kulikauskas: I wish to show that my philosophy is very fruitful for developing a science of math, for understanding math as a cognitive language (implicit math) which takes place in the mind, and for overviewing all of math and how its branches and concepts unfold.

Pakeista 27 eilutė iš:

I wish to show that my philosophy is very fruitful for understanding the big picture in math. I am thus developing a science of math? and a theory of implicit math. Here are concepts in my philosophy which I am working to model with math.

į:
 Here are concepts in my philosophy which I am working to model with math.
2016 rugpjūčio 27 d., 11:24 atliko AndriusKulikauskas -
Pakeistos 1-4 eilutės iš

See: SelfLearners/Math

Online Latex editor

į:

Useful: Online Latex editor

Pridėtos 105-106 eilutės:

See: SelfLearners/Math

2016 rugpjūčio 26 d., 17:02 atliko AndriusKulikauskas -
Pridėtos 6-27 eilutės:
  • Suvokti matematikos prasmę ir išskirtinumą.
  • Savo pagrindines sąvokas išsakyti matematika.
    • Išsakyti Dievo šokį nuliu, begalybe ir vieniu.
      • Susipažinti su vienanariu lauku, su baigtiniais laukais ir jų kombinatorika, su F1n.
      • Apžvelgti dvilypumo pavyzdžius ir suvokti jų esmę.
      • Suprasti, kur ir kaip išsiskiria nulis ir begalybė, kaip išyra jų tolygumas.
      • Suvokti, kaip baigtiniai laukai ir vienanaris laukas vaizduoja begalybę.
      • Begalybę išreikšti geometrijomis.
    • Išsakyti keturis lygmenis.
      • Apžvelgti, įsisavinti ir atskirti keturias geometrijas.
      • Jas išsakyti simetrinėmis funkcijomis.
    • Išsakyti šešias permainas.
      • Rinkti ir rūšiuoti kintamųjų pavyzdžius.
      • Išsakyti šešis veiksmus vedančius iš vienos geometrijos į kitą.
      • Susieti su šešiais pertvarkymais.
    • Išsakyti aštuonis padalinimus.
      • Suprasti Bott periodiškumą.
  • Taikyti matematikos išsiaiškinimo būdus.
    • Parodyti, kaip trejybė plėtoja matematiką.
  • Aprėpti visą matematiką ir nurodyti, kaip išsiritulioja jos šakos, sąvokos, teiginiai ir uždaviniai.
    • Susieti matematiką su įvardijimo ir pagrindimo kalbomis.
2016 rugpjūčio 16 d., 12:59 atliko AndriusKulikauskas -
Pridėta 10 eilutė:
  • Duality of zero and infinity by way of one.
2016 rugpjūčio 12 d., 19:30 atliko AndriusKulikauskas -
Pridėtos 86-95 eilutės:

Philosophical questions about math:

  • How is love (and life) related to duality, reflections, transformations and other math concepts?
  • How does 1 mediate the duality of 0 and infinity? And how is that duality variously broken?
  • What are the six basic transformation? and their relation with symmetry?

Math investigations:

  • List out the results of universal hyperbolic geometry and state them in terms of symmetric functions.
  • Write an elegant combinatorics of the finite field and interpret what is F1^n.
2016 rugpjūčio 12 d., 11:14 atliko AndriusKulikauskas -
Pridėtos 102-103 eilutės:
  • What is the significance of the center of a simplex?
  • Intuitively, why are there four classical Lie algebras/groups?
Pakeistos 106-107 eilutės iš
  • Intuitively, why are there four classical Lie algebras/groups?
  • Is the -1 simplex the center of any simplex?
į:
2016 rugpjūčio 10 d., 11:38 atliko AndriusKulikauskas -
Pakeista 103 eilutė iš:
  • What is a combinatorial interpretation of P - P(n), the generators of the Mandelbrot set, in terms of the Catalan numbers? Get help to generate the difference.
į:
  • What is a combinatorial interpretation of P - P(n), the generators of the Mandelbrot set, in terms of the Catalan numbers? Get help to generate the difference. What is the best software for that?
2016 rugpjūčio 10 d., 11:37 atliko AndriusKulikauskas -
Pakeistos 102-104 eilutės iš
  • What is a combinatorial interpretation of P - P(n), the generators of the Mandelbrot set, in terms of the Catalan numbers?
į:
  • Check what happens if I plug in different values into the Catalan power series.
  • What is a combinatorial interpretation of P - P(n), the generators of the Mandelbrot set, in terms of the Catalan numbers? Get help to generate the difference.
  • Intuitively, why are there four classical Lie algebras/groups?
  • Is the -1 simplex the center of any simplex?
  • What is F1n?
2016 rugpjūčio 10 d., 10:32 atliko AndriusKulikauskas -
Pakeista 102 eilutė iš:
į:
  • What is a combinatorial interpretation of P - P(n), the generators of the Mandelbrot set, in terms of the Catalan numbers?
2016 rugpjūčio 08 d., 10:37 atliko AndriusKulikauskas -
Pakeistos 101-103 eilutės iš

Questions to ask others:

  • What is geometry?
  • Does the Catalan generating function on the complex plane yield the Mandelbrot set?
į:

Questions to ask others at Math Overflow and elsewhere:

2016 rugpjūčio 08 d., 02:06 atliko AndriusKulikauskas -
Pridėtos 2-3 eilutės:

Online Latex editor

2016 liepos 21 d., 15:36 atliko AndriusKulikauskas -
Pridėtos 90-91 eilutės:

Consider how infinity, zero and one are defined in the various geometries. How do these concepts fit together? How do they involve the viewer and their perspective? How might that relate to Christopher Alexander's principles of life and the plane of the viewer?

2016 liepos 21 d., 11:12 atliko AndriusKulikauskas -
Pridėta 89 eilutė:
  • How can I apply my results to figure things out in math, the biggest problems?
2016 liepos 21 d., 11:11 atliko AndriusKulikauskas -
Pridėtos 84-88 eilutės:

Consider more examples, simple and sophisticated, of how things are figured out in math.

  • What is the relationship between the surface math problem and the deep way of figuring things out?
  • How do we discover the right way to figure out a math problem?
  • How do we combine several distinct ways of figuring things out?
2016 liepos 10 d., 21:31 atliko AndriusKulikauskas -
Pakeista 47 eilutė iš:
į:
Pakeista 49 eilutė iš:
į:
2016 liepos 10 d., 21:30 atliko AndriusKulikauskas -
Pakeista 48 eilutė iš:
į:
Pridėta 50 eilutė:
Pakeistos 55-66 eilutės iš
į:

Other assorted concepts: Tetrahedron, Triality, Associativity, Unit, Matrix

Ištrintos 105-106 eilutės:
Ištrintos 130-141 eilutės:
2016 liepos 10 d., 16:23 atliko AndriusKulikauskas -
Ištrinta 26 eilutė:
  • correctly interpreting the hypercubes
2016 liepos 07 d., 16:19 atliko AndriusKulikauskas -
Pridėtos 86-87 eilutės:

One way to think of geometry is in terms of what happens at infinity. For example, do all lines (through plane) meet at infinity at a common point? Or at a circle? Or do the ends of the lines not meet? Or do they go to a circle of infinite length?

2016 birželio 25 d., 12:18 atliko AndriusKulikauskas -
Pridėta 73 eilutė:
2016 birželio 25 d., 11:04 atliko AndriusKulikauskas -
Pridėta 40 eilutė:
  • Geometric representation theory
2016 birželio 25 d., 10:49 atliko AndriusKulikauskas -
Pridėta 71 eilutė:
2016 birželio 23 d., 14:58 atliko AndriusKulikauskas -
Pridėtos 33-34 eilutės:

Math connections with my philosophy.

Pakeistos 134-333 eilutės iš

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. (Identity theorem). Real numbers: local != global.
  • 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

  • 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
  • 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?

  • 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 interpretation related to Young tableaux

Pagrindiniai matematikos dėsniai

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

Prisiminti savo matematikos mokymo dėsnius:

  • every answer is an amount and a unit ir tt.
  • combine like units
  • list different units
  • a right triangle is half of a rectangle
  • a triangle is the sum of two right triangles
  • four times a right triangle is the difference of two squares
  • extending the domain
  • purposes of families of functions

Basic division rings: John Baez 59

  • The real numbers are not of characteristic 2,
  • so the complex numbers don't equal their own conjugates,
  • so the quaternions aren't commutative,
  • so the octonions aren't associative,
  • so the hexadecanions aren't a division algebra.

Sąsajos tarp mano sąvokų ir matematikos

Dievas

  • Field of one element. Roots of unity = divisions of everything?
  • 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.
    • 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.

Padalinimų ratas

  • Bott periodicity John Baez
  • 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, Hopf invariant ir Adam's theorem. Homotopy group of spheres. Clifford paralells ir quaternions.
  • Higher Clifford Algebras

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.

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.

Aplinkybės

  • 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
į:
2016 birželio 23 d., 14:08 atliko AndriusKulikauskas -
Pridėta 50 eilutė:
Pakeistos 53-55 eilutės iš
į:
Pakeistos 134-242 eilutės iš

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 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

  • Clifford Algebra
    • Clifford Algebra
    • Projective geometry
    • Conformal geometry
    • Double Conformal Mapping
    • The Perceptual Origin of Mathematics
    • Stephen Lehar
    • Stephan Lehar's theory of mind and brain
  • Entropija
  • Riemann-Zeta funkcijos pagrindus
  • Model theory. Taylor Dupuy youtube
  • 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

  • John Baez: Whenever we pick a Dynkin diagram and a field we get a geometry: An projective, Bn Cn conformal, Dn symplectic.
  • Klein geometry
  • 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 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.
  • 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 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

  • Twelvefold way - reikėtų palyginti su aplinkybėmis, taip pat su equivalence relations, kokių gali būti.
  • Category theory - Categories are helpful in making fruitful definitions
  • 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?

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.
  • 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 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.
į:
2016 birželio 23 d., 13:57 atliko AndriusKulikauskas -
Pakeistos 44-47 eilutės iš
į:
Pakeistos 154-181 eilutės iš
  • Lie group ir algebra teorijos
    • Finite Coxeter group properties žiūrėti lentelę
    • pradedant tensoriais
    • Lie theory through examples
    • http://link.springer.com/book/10.1007%2F978-1-4612-0979-9
    • Exceptional Lie groups
    • Octonions
    • http://phyweb.lbl.gov/~rncahn/www/liealgebras/texall.pdf
    • Group theory, representation theory of Symmetric group
    • Symmetric space, including modern classification by Huang and Leung.
    • Amplituhedron, post by Jaroslav Trnka related to walks on trees? video
  • Rekursyvinę funkcijų teoriją
    • Yates-Index theorem
  • John Baez Network Theory
  • Kategorijų teorija
    • Jeff Hicks Categorification
    • A Pre-history of n-categorical Physics by John Baez
    • http://math.ucr.edu/home/baez/rosetta.pdf
    • Category theory for scientists
    • john baez spans in quantum theory
    • Make category theory intuitive
    • http://math.ucr.edu/home/baez/qg-winter2016/
    • Categorification
  • Geometric Representation Theory
    • Geometric Representation Theory Seminar 2007 John Baez and James Dolan
    • Theory X and Life
    • Introductory Book for Geometric Representation Theory
    • Borel-Weil theorem
į:
Ištrintos 425-441 eilutės:

Ko norėčiau išmokti matematikoje

Category theory

  • John Baez: Category Theory

Foundations of Mathematics (understand how models work)

  • Foundations of Mathematics by Alexander Sakharov
  • The Foundations of Mathematics by Kenneth Kunen

Lie groups, Lie algebras

  • Root systems: Lie algebrų rūšių pagrindas
  • Continuum mechanics

Matematikos išsiaiškinimo būdai

  • Alexander Bogomolny sarašas
2016 birželio 23 d., 12:49 atliko AndriusKulikauskas -
Ištrintos 466-473 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)?
2016 birželio 23 d., 12:47 atliko AndriusKulikauskas -
Ištrintos 466-540 eilutės:

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

Mathematical beauty

  • Wikipedia: In the 1990s, 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.[17][18][19] 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.
  • Math Beauty blog and Readings
  • What is Good Mathematics? Terrence Tao.
  • Alexander Bogomolny, 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."
  • 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.

Questions about beauty

  • 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
  • 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.

2016 birželio 23 d., 12:35 atliko AndriusKulikauskas -
Ištrintos 20-29 eilutės:

Here are questions about the big picture in mathematics:

  • 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?
Ištrinta 44 eilutė:
Pridėtos 50-59 eilutės:

Here are questions about the big picture in mathematics:

  • 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?
Ištrintos 549-556 eilutės:

Matematikos visuomenės

  • Electronic Newsgroups and Listservs
  • nLab visuomenių sąrašas
  • Art of Problem Solving: Community
  • Physics forums
  • Deferential Geometry: Garrett Lisi
2016 birželio 23 d., 12:34 atliko AndriusKulikauskas -
Pridėtos 21-30 eilutės:

Here are questions about the big picture in mathematics:

  • 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?
Ištrintos 558-577 eilutės:

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?
2016 birželio 23 d., 12:31 atliko AndriusKulikauskas -
Pakeistos 59-60 eilutės iš
į:
  • Matematikos žodynas anglų, lietuvių ir kitomis kalbomis
Pakeistos 63-64 eilutės iš
į:
  • My Ph.D. thesis: Symmetric Functions of the Eigenvalues of a Matrix
Ištrintos 569-571 eilutės:

Matematikos žodynas anglų, lietuvių ir kitomis kalbomis

Symmetric Functions of the Eigenvalues of a Matrix

2016 birželio 23 d., 00:04 atliko AndriusKulikauskas -
Ištrintos 62-63 eilutės:

Žiūriu Representation of Geometry paskaita. II. 0:45

Pridėtos 64-67 eilutės:

Didieji klausimai:

  • Keturias briaunainių šeimas sieti su keturiomis geometrijomis, metalogikomis, ketverybe, ženklų rūšimis, teigiamais įsakymais.
  • Briaunainių šeimų poras sieti su šešiomis daugybomis, pertvarkymais, permainomis, neigiamais įsakymais.
2016 birželio 23 d., 00:01 atliko AndriusKulikauskas -
Pakeistos 67-68 eilutės iš

What is geometry?

į:

Questions to ask others:

  • What is geometry?
  • Does the Catalan generating function on the complex plane yield the Mandelbrot set?
Pakeistos 82-83 eilutės iš
  • interpret the binomial theorem for simplexes.
  • what does it mean that the -1 simplex is the empty set? the spirit?
į:
2016 birželio 22 d., 19:05 atliko AndriusKulikauskas -
Pridėta 43 eilutė:
2016 birželio 22 d., 17:20 atliko AndriusKulikauskas -
Pridėtos 65-66 eilutės:

What is geometry?

2016 birželio 22 d., 17:14 atliko AndriusKulikauskas -
Pridėta 37 eilutė:
2016 birželio 22 d., 11:59 atliko AndriusKulikauskas -
Pridėta 154 eilutė:
  • Categorification
2016 birželio 22 d., 11:38 atliko AndriusKulikauskas -
Pridėtos 64-65 eilutės:

What does projective geometry say about the existence of infinity?

2016 birželio 22 d., 10:55 atliko AndriusKulikauskas -
Pakeista 28 eilutė iš:
  • understanding how Pascal's triangle relates to homology and the Gauss-Bonnet theorem and Euler's characteristic
į:
  • understanding how Pascal's triangle relates to homology and the Gauss-Bonnet theorem and Euler's characteristic
2016 birželio 22 d., 10:13 atliko AndriusKulikauskas -
Pridėtos 52-53 eilutės:
2016 birželio 22 d., 10:12 atliko AndriusKulikauskas -
Pakeista 51 eilutė iš:
  • [[Lou Kauffman]
į:
2016 birželio 22 d., 10:11 atliko AndriusKulikauskas -
Pridėta 51 eilutė:
  • [[Lou Kauffman]
2016 birželio 21 d., 23:51 atliko AndriusKulikauskas -
Pakeista 14 eilutė iš:
į:
Pridėtos 20-24 eilutės:

Philosophical questions I am currently trying to address with math

  • relate multiplication as transformations between four geometries
  • relate multiplication to set theory axioms
  • relate multiplication to restructurings
2016 birželio 21 d., 23:41 atliko AndriusKulikauskas -
Pakeista 23 eilutė iš:
  • understanding how Pascal's triangle relates to homology and the Gauss-Bonnet theorem
į:
  • understanding how Pascal's triangle relates to homology and the Gauss-Bonnet theorem and Euler's characteristic
2016 birželio 21 d., 23:39 atliko AndriusKulikauskas -
Pridėtos 20-26 eilutės:

Math I am currently focusing on

  • correctly interpreting the hypercubes
  • understanding how Pascal's triangle relates to homology and the Gauss-Bonnet theorem
  • understanding the different kinds of Pascal triangles and how they relate to Grassmannians
  • understanding the Gaussian binomial coefficients in terms of Coxeter generators
  • understanding the relationship between the discrete and continuous case of projective geometry
2016 birželio 21 d., 09:05 atliko AndriusKulikauskas -
Pridėtos 14-16 eilutės:
Ištrinta 17 eilutė:
Pridėta 19 eilutė:
  • Christopher Alexander's principles of life and different kinds of opposites are relevant.
2016 birželio 21 d., 08:59 atliko AndriusKulikauskas -
Pridėta 14 eilutė:
  • I think it's important to look at the various dualities.
Pakeistos 211-314 eilutės iš

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? 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 Eckmann–Hilton duality.
    • Adjoint bendrai ir 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 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 nLab: Duality. Here are examples to consider:

  • 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
  • de Rham cohomology links algebraic topology and differential topology
  • Modularity theorem.
  • Langlands program
  • general Stokes theorem: duality between the boundary operator on chains and the exterior derivative
  • 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.[79] 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.
  • Isbell duality relates higher geometry with higher algebra.
  • 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[2] 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.

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
  • Spanier–Whitehead 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".
į:
2016 birželio 21 d., 08:35 atliko AndriusKulikauskas -
Pridėtos 25-26 eilutės:
2016 birželio 19 d., 14:37 atliko AndriusKulikauskas -
Pakeistos 1-2 eilutės iš

See: Math Discovery, Math Beauty, Tensor, Simplex, SelfLearners/Math

į:

See: SelfLearners/Math

Pridėta 13 eilutė:
Pridėta 15 eilutė:
2016 birželio 19 d., 14:35 atliko AndriusKulikauskas -
Pakeistos 5-7 eilutės iš

Implicit math

Concepts in my philosophy that I wish to model with math.

į:

I wish to show that my philosophy is very fruitful for understanding the big picture in math. I am thus developing a science of math? and a theory of implicit math. Here are concepts in my philosophy which I am working to model with math.

Pakeistos 9-11 eilutės iš

I'm trying to understand the big picture in math.

į:

Here are my ongoing investigations:

2016 birželio 19 d., 14:32 atliko AndriusKulikauskas -
Pakeistos 6-7 eilutės iš
į:

Concepts in my philosophy that I wish to model with math.

Ištrintos 23-26 eilutės:

Concepts in my philosophy that I wish to model with math.

2016 birželio 19 d., 14:28 atliko AndriusKulikauskas -
Pridėtos 34-36 eilutės:

Other interests:

2016 birželio 19 d., 14:25 atliko AndriusKulikauskas -
Pakeista 3 eilutė iš:

These pages are quite a mess for now. But I will organize them to make clear my interests in math.

į:

Andrius Kulikauskas: These pages are quite a mess for now. But I will organize them to make clear my interests in math.

2016 birželio 19 d., 14:23 atliko AndriusKulikauskas -
Pakeista 15 eilutė iš:
į:
2016 birželio 19 d., 14:11 atliko AndriusKulikauskas -
Pakeista 8 eilutė iš:

I'm trying to understand the big picture in math.

į:

I'm trying to understand the big picture in math.

2016 birželio 19 d., 14:10 atliko AndriusKulikauskas -
Pridėta 22 eilutė:
2016 birželio 19 d., 14:09 atliko AndriusKulikauskas -
Pridėtos 30-32 eilutės:

Useful links:

2016 birželio 19 d., 14:07 atliko AndriusKulikauskas -
Pridėta 12 eilutė:
2016 birželio 19 d., 14:06 atliko AndriusKulikauskas -
Pridėtos 19-21 eilutės:

Concepts in my philosophy that I wish to model with math.

Pakeista 23 eilutė iš:
į:
2016 birželio 19 d., 14:02 atliko AndriusKulikauskas -
Pridėtos 23-25 eilutės:

Interesting mathematicians:

2016 birželio 19 d., 13:55 atliko AndriusKulikauskas -
Pridėtos 7-9 eilutės:

I'm trying to understand the big picture in math.

2016 birželio 19 d., 13:51 atliko AndriusKulikauskas -
Pridėta 13 eilutė:
2016 birželio 19 d., 13:50 atliko AndriusKulikauskas -
Pridėta 18 eilutė:
2016 birželio 19 d., 13:49 atliko AndriusKulikauskas -
Pridėta 11 eilutė:
2016 birželio 19 d., 13:46 atliko AndriusKulikauskas -
Pridėta 9 eilutė:
2016 birželio 19 d., 13:42 atliko AndriusKulikauskas -
Pridėta 14 eilutė:
2016 birželio 19 d., 13:23 atliko AndriusKulikauskas -
Pridėtos 12-14 eilutės:

Assorted concepts:

2016 birželio 19 d., 13:22 atliko AndriusKulikauskas -
Pridėta 9 eilutė:
2016 birželio 19 d., 13:21 atliko AndriusKulikauskas -
Pakeista 9 eilutė iš:
į:
2016 birželio 19 d., 13:19 atliko AndriusKulikauskas -
Pakeistos 8-10 eilutės iš
į:

Areas of math that I want to learn more about:

2016 birželio 19 d., 13:15 atliko AndriusKulikauskas -
Pridėtos 4-6 eilutės:
2016 birželio 19 d., 13:10 atliko AndriusKulikauskas -
Pridėtos 2-5 eilutės:

These pages are quite a mess for now. But I will organize them to make clear my interests in math.

Circle folding

2016 birželio 19 d., 13:02 atliko AndriusKulikauskas -
Pakeista 1 eilutė iš:

See: SelfLearners/Math, Tensor, Simplex, Math Discovery, Math Beauty

į:

See: Math Discovery, Math Beauty, Tensor, Simplex, SelfLearners/Math

2016 birželio 19 d., 13:01 atliko AndriusKulikauskas -
Pakeista 1 eilutė iš:

See: SelfLearners/Math, Tensor, Simplex, Math Discovery, Mathematical Beauty

į:

See: SelfLearners/Math, Tensor, Simplex, Math Discovery, Math Beauty

2016 birželio 19 d., 13:00 atliko AndriusKulikauskas -
Pakeista 1 eilutė iš:
į:

See: SelfLearners/Math, Tensor, Simplex, Math Discovery, Mathematical Beauty

2016 birželio 19 d., 12:32 atliko AndriusKulikauskas -
Pakeista 47 eilutė iš:
į:
2016 birželio 19 d., 12:29 atliko AndriusKulikauskas -
Pridėtos 1-606 eilutės:

Žr. SelfLearners/Math, Tensor, Simplex, Matematikos rūmai?, Matematikos grožis?

Žiūriu Representation of Geometry paskaita. II. 0:45

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 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 finite field GF(8) and relate it to the divisions of everything.

Tiesinė algebra

  • Kaip dauginti polar decomposed matrices?
  • Geriau suprasti 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?

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 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
    • Finite Coxeter group properties žiūrėti lentelę
    • pradedant tensoriais
    • Lie theory through examples
    • http://link.springer.com/book/10.1007%2F978-1-4612-0979-9
    • Exceptional Lie groups
    • Octonions
    • http://phyweb.lbl.gov/~rncahn/www/liealgebras/texall.pdf
    • Group theory, representation theory of Symmetric group
    • Symmetric space, including modern classification by Huang and Leung.
    • Amplituhedron, post by Jaroslav Trnka related to walks on trees? video
  • Rekursyvinę funkcijų teoriją
    • Yates-Index theorem
  • John Baez Network Theory
  • Kategorijų teorija
    • Jeff Hicks Categorification
    • A Pre-history of n-categorical Physics by John Baez
    • http://math.ucr.edu/home/baez/rosetta.pdf
    • Category theory for scientists
    • john baez spans in quantum theory
    • Make category theory intuitive
    • http://math.ucr.edu/home/baez/qg-winter2016/
  • Geometric Representation Theory
    • Geometric Representation Theory Seminar 2007 John Baez and James Dolan
    • Theory X and Life
    • Introductory Book for Geometric Representation Theory
    • Borel-Weil theorem
  • Clifford Algebra
    • Clifford Algebra
    • Projective geometry
    • Conformal geometry
    • Double Conformal Mapping
    • The Perceptual Origin of Mathematics
    • Stephen Lehar
    • Stephan Lehar's theory of mind and brain
  • Entropija
  • Riemann-Zeta funkcijos pagrindus
  • Model theory. Taylor Dupuy youtube
  • 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

  • John Baez: Whenever we pick a Dynkin diagram and a field we get a geometry: An projective, Bn Cn conformal, Dn symplectic.
  • Klein geometry
  • 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 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.
  • 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 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

  • Twelvefold way - reikėtų palyginti su aplinkybėmis, taip pat su equivalence relations, kokių gali būti.
  • Category theory - Categories are helpful in making fruitful definitions
  • 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? 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 Eckmann–Hilton duality.
    • Adjoint bendrai ir 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 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 nLab: Duality. Here are examples to consider:

  • 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
  • de Rham cohomology links algebraic topology and differential topology
  • Modularity theorem.
  • Langlands program
  • general Stokes theorem: duality between the boundary operator on chains and the exterior derivative
  • 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.[79] 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.
  • Isbell duality relates higher geometry with higher algebra.
  • 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[2] 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.

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
  • Spanier–Whitehead 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.
  • 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 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. (Identity theorem). Real numbers: local != global.
  • 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

  • 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
  • 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?

  • 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 interpretation related to Young tableaux

Pagrindiniai matematikos dėsniai

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

Prisiminti savo matematikos mokymo dėsnius:

  • every answer is an amount and a unit ir tt.
  • combine like units
  • list different units
  • a right triangle is half of a rectangle
  • a triangle is the sum of two right triangles
  • four times a right triangle is the difference of two squares
  • extending the domain
  • purposes of families of functions

Basic division rings: John Baez 59

  • The real numbers are not of characteristic 2,
  • so the complex numbers don't equal their own conjugates,
  • so the quaternions aren't commutative,
  • so the octonions aren't associative,
  • so the hexadecanions aren't a division algebra.

Sąsajos tarp mano sąvokų ir matematikos

Dievas

  • Field of one element. Roots of unity = divisions of everything?
  • 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.
    • 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.

Padalinimų ratas

  • Bott periodicity John Baez
  • 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, Hopf invariant ir Adam's theorem. Homotopy group of spheres. Clifford paralells ir quaternions.
  • Higher Clifford Algebras

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.

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.

Aplinkybės

  • 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

  • John Baez: Category Theory

Foundations of Mathematics (understand how models work)

  • Foundations of Mathematics by Alexander Sakharov
  • The Foundations of Mathematics by Kenneth Kunen

Lie groups, Lie algebras

  • Root systems: Lie algebrų rūšių pagrindas
  • Continuum mechanics

Matematikos išsiaiškinimo būdai

  • 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

Mathematical beauty

  • Wikipedia: In the 1990s, 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.[17][18][19] 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.
  • Math Beauty blog and Readings
  • What is Good Mathematics? Terrence Tao.
  • Alexander Bogomolny, 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."
  • 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.

Questions about beauty

  • 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
  • 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

  • Electronic Newsgroups and Listservs
  • nLab visuomenių sąrašas
  • Art of Problem Solving: Community
  • Physics forums
  • 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?

Matematikos žodynas anglų, lietuvių ir kitomis kalbomis

Symmetric Functions of the Eigenvalues of a Matrix


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Puslapis paskutinį kartą pakeistas 2019 rugpjūčio 17 d., 00:06
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