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数学

Discovery

Andrius Kulikauskas

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Introduction E9F5FC

Understandable FFFFFF

Questions FFFFC0

Notes EEEEEE

Software

Book.FieldWithOneElement istorija

Paslėpti nežymius pakeitimus - Rodyti galutinio teksto pakeitimus

2019 birželio 30 d., 17:38 atliko AndriusKulikauskas -
Pakeista 5 eilutė iš:
The field with one element, {$F_1$}, i a nonexistent mathematical concept (fields are supposed to contain at least two distinct elements, 0 and 1) which has spurred quite a bit of research. It suggests itself in different situations as a limiting initial case.
į:
The field with one element, {$F_1$}, is a nonexistent mathematical concept (fields are supposed to contain at least two distinct elements, 0 and 1) which has spurred quite a bit of research. It suggests itself in different situations as a limiting initial case.
2019 birželio 30 d., 17:37 atliko AndriusKulikauskas -
Pridėta 48 eilutė:
* [[https://arxiv.org/pdf/0809.1564.pdf | Cyclotomy and Analytic Geometry over {$F_1$}]]
2019 birželio 30 d., 17:36 atliko AndriusKulikauskas -
Pakeista 44 eilutė iš:
* [[http://cage.ugent.be/~kthas/Fun/ | F_un Mathematics]]
į:
* [[http://cage.ugent.be/~kthas/Fun/ | F_un Mathematics]] Lieven Le Bruyn
2019 birželio 30 d., 17:34 atliko AndriusKulikauskas -
Pakeistos 38-40 eilutės iš
* [[http://www.neverendingbooks.org/the-f_un-folklore | F_un folklore]]
* [[http://www.neverendingbooks.org/f_un-with-manin | F_un with Manin]]
* See: [[http://www.neverendingbooks.org/category/absolute | Absolute]]
į:
* [[http://www.neverendingbooks.org/the-f_un-folklore | F_un folklore]] Lieven Le Bruyn
*
[[http://www.neverendingbooks.org/f_un-with-manin | F_un with Manin]] Lieven Le Bruyn
* See Lieven Le Bruyn
: [[http://www.neverendingbooks.org/category/absolute | Absolute]]
2019 birželio 30 d., 17:33 atliko AndriusKulikauskas -
Pridėtos 38-40 eilutės:
* [[http://www.neverendingbooks.org/the-f_un-folklore | F_un folklore]]
* [[http://www.neverendingbooks.org/f_un-with-manin | F_un with Manin]]
* See: [[http://www.neverendingbooks.org/category/absolute | Absolute]]
2019 vasario 09 d., 11:51 atliko AndriusKulikauskas -
Pakeistos 82-83 eilutės iš
į:
* A field relates two groups: an additive group (the level) and a multiplicative group (the metalevel of actions). As regards the action, the zero of the additive group is the negation of action - no action taken, whereas the one of the multiplicative group is the action that has no effect. Therein lies the distinction of the level and the metalevel.
* I think that an affine geometry is not so much distinguished by its not having a zero (a zero or origin can always be defined) but by its not having a one. Perhaps a projective geometry has both a zero and an infinity and so a one is naturally available.
2019 vasario 05 d., 13:47 atliko AndriusKulikauskas -
Pridėta 62 eilutė:
* The “general linear group” in n dimensions over the field of one element is the symmetric group {$S_n$}.
2019 vasario 03 d., 22:34 atliko AndriusKulikauskas -
Pridėta 45 eilutė:
* [[https://arxiv.org/abs/math/0407093 | Projective geometry over {$F_1$} and the Gaussian binomial coefficients]], [[http://math.mit.edu/~cohn/ | Henry Cohn]], [[http://math.mit.edu/~cohn/research.html | his research]]
Ištrinta 49 eilutė:
* [[https://arxiv.org/abs/math/0407093 | Projective geometry over {$F_1$} and the Gaussian binomial coefficients]], [[http://math.mit.edu/~cohn/ | Henry Cohn]], [[http://math.mit.edu/~cohn/research.html | his research]]
2019 vasario 03 d., 21:08 atliko AndriusKulikauskas -
Ištrintos 38-39 eilutės:
[[https://www.youtube.com/watch?v=1XRna0vUYdo | field of one element video]]
Pridėta 55 eilutė:
* [[https://www.youtube.com/watch?v=1XRna0vUYdo | Video: Taylor Dupuy: The field with one element and the Riemann Hypothesis]]
2019 vasario 03 d., 17:30 atliko AndriusKulikauskas -
Ištrintos 8-11 eilutės:
I believe that the field with one element can be interpreted as an element that can be interpreted as 0, 1 and infinity. For example, 0 * infinity = 1. I think that 1 can be thought of as reflecting 0 and infinity in a duality. I think of this as modeling [[Gods Dance | God's dance]].

The field with one element can be imagined by way of Pascal's triangle. For the Gaussian binomial coefficients count the number of subspaces of a vector space over a field of characteristic q. Having q->1 yields the usual binomial coefficients. Also, Pascal's triangle counts the simplexes of a subsimplex, and variants count the parts of other infinite families of polytopes.
Pridėta 25 eilutė:
* How do finite fields deal with the issue that Lie algebras deal with: how to link countings?
Pridėtos 39-40 eilutės:
[[https://www.youtube.com/watch?v=1XRna0vUYdo | field of one element video]]
Pridėtos 73-74 eilutės:
Of course, there is no field F1 with only one element, but there is a trivial ring, and it is merely a convention that we do not call it a field. However, it is an excellent convention, because the trivial ring has no nontrivial modules (if x is an element of a module, then x = 1x = 0x = 0). Calling it a field would not help solve Puzzle 1, since F n 1 does not depend on n. I know of no direct solution to this puzzle, nor of any way to make sense of vector spaces over F1. Nevertheless, the puzzle can be solved by an indirect route: it becomes much easier to understand when it is reformulated in terms of projective geometry. That may not be surprising, if one keeps in mind that many topics, such as intersection theory, become simpler when one moves to projective geometry. (The papers [11] and [22] also shed light on this puzzle by indirect routes, but not by using projective geometry.) Cohn, page 489.
Pridėtos 78-82 eilutės:
* Limit as q->1
** The field with one element can be imagined by way of Pascal's triangle. For the Gaussian binomial coefficients count the number of subspaces of a vector space over a field of characteristic q. Having q->1 yields the usual binomial coefficients. Also, Pascal's triangle counts the simplexes of a subsimplex, and variants count the parts of other infinite families of polytopes.
* I believe that the field with one element can be interpreted as an element that can be interpreted as 0, 1 and infinity. For example, 0 * infinity = 1. I think that 1 can be thought of as reflecting 0 and infinity in a duality. I think of this as modeling [[Gods Dance | God's dance]].
Pakeistos 89-90 eilutės iš
į:
* The anharmonic group (see [[https://en.wikipedia.org/wiki/Cross-ratio | Cross-ratio]]) permutes 0,1 and infinity.
Ištrintos 99-102 eilutės:
Symmetry involves a dual point of view: for example, vertices are distinct and yet not distinguishable.

Of course, there is no field F1 with only one element, but there is a trivial ring, and it is merely a convention that we do not call it a field. However, it is an excellent convention, because the trivial ring has no nontrivial modules (if x is an element of a module, then x = 1x = 0x = 0). Calling it a field would not help solve Puzzle 1, since F n 1 does not depend on n. I know of no direct solution to this puzzle, nor of any way to make sense of vector spaces over F1. Nevertheless, the puzzle can be solved by an indirect route: it becomes much easier to understand when it is reformulated in terms of projective geometry. That may not be surprising, if one keeps in mind that many topics, such as intersection theory, become simpler when one moves to projective geometry. (The papers [11] and [22] also shed light on this puzzle by indirect routes, but not by using projective geometry.) Cohn, page 489.
Pakeistos 118-129 eilutės iš
Andrius

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



[[https://www.youtube.com/watch?v=1XRna0vUYdo | field of one element video]]

The anharmonic group (see [[https://en.wikipedia.org/wiki/Cross-ratio | Cross-ratio]]) permutes 0,1 and infinity.

------------------
į:
Andrius
2019 vasario 03 d., 17:25 atliko AndriusKulikauskas -
Pakeistos 40-41 eilutės iš
* [[https://en.wikipedia.org/wiki/Field_with_one_element | Field with one element]]
į:
* [[https://en.wikipedia.org/wiki/Field_with_one_element | Wikipedia: Field with one element]]
Pakeista 43 eilutė iš:
* [[http://arxiv.org/pdf/0909.0069 |[PL] J.L. Pena, O. Lorscheid: Mapping F1-land: An overview of geometries over the field with one element]] Preprint.
į:
* [[http://arxiv.org/pdf/0909.0069 | Mapping F1-land: An overview of geometries over the field with one element]] Preprint. J.L. Pena, O. Lorscheid.
2019 vasario 03 d., 17:23 atliko AndriusKulikauskas -
Pridėtos 3-8 eilutės:
[+Field with one element, {$F_1$}+]

The field with one element, {$F_1$}, i a nonexistent mathematical concept (fields are supposed to contain at least two distinct elements, 0 and 1) which has spurred quite a bit of research. It suggests itself in different situations as a limiting initial case.

My impression is that it relates to my concept of a God who goes beyond himself into himself, who asks, Is God necessary? Would I be even if I wasn't?
Pakeista 14 eilutė iš:
į:
---------------------
Pakeista 30 eilutė iš:
į:
----------
Pakeistos 33-34 eilutės iš
Readings
į:
[+Readings+]
Ištrintos 61-70 eilutės:

Finite fields
* Lyndon words - irreducible polynomials for finite fields
* Duality of q and n in {$GL_n(F_q)$}.
* Multiset of Lyndon words - reducible and irreducible. Homogeneous symmetric functions of eigenvalues.
* Interpolation between homogeneous and elementary - between commutativity and anti-commutativity.
* Lyndon words are like prime numbers.
* Dimension of free Lie algebras = number of Lyndon words of length n
* What would be the q-theory for finite fields for matrix combinatorics?
Pridėtos 68-75 eilutės:
F1-believers base their f-unny intuition on the following two mantras :
* F1 forgets about additive data and retains only multiplicative data.
* F1-objects only acquire flesh when extended to Z (or C).

What form does the binomial theorem take in a noncommutative ring? In general one can say nothing interesting, but certain special cases work out elegantly. One of the nicest, due to Schutzenberger [18], deals with variables x, y, and q such that q commutes with x and y, and yx = qxy.

[+Ideas+]
Pakeistos 84-91 eilutės iš
F1-believers base their f-unny intuition on the following two mantras :
* F1 forgets about additive data and retains only multiplicative data.
* F1
-objects only acquire flesh when extended to Z (or C).

What form does the binomial theorem take in a noncommutative ring? In general one can say nothing interesting, but certain special cases work out elegantly. One of the nicest, due to Schutzenberger [18],
deals with variables x, y, and q such that q commutes with x and y, and yx = qxy.
į:
Finite fields
* Lyndon words - irreducible polynomials for finite fields
* Duality of q and n in {$GL_n(F_q)$}.
* Multiset of Lyndon words
- reducible and irreducible. Homogeneous symmetric functions of eigenvalues.
* Interpolation between homogeneous and elementary - between commutativity and anti-commutativity.
* Lyndon words are like prime numbers.
* Dimension of free Lie algebras = number of Lyndon words
of length n
* What would be the q-theory for finite fields for matrix combinatorics?
Pakeistos 95-112 eilutės iš
Of course, there is no field F1 with only one element, but there is a trivial ring,
and it is merely a convention that we do not call it a field. However, it is an excellent
convention, because the trivial ring has no nontrivial modules (if x is an element of
a module, then x = 1x = 0x = 0). Calling it a field would not help solve Puzzle 1,
since F
n
1
does not depend on n.
I know of no direct solution to this puzzle, nor of any way to make sense of vector
spaces over F1. Nevertheless, the puzzle can be solved by an indirect route: it becomes
much easier to understand when it is reformulated in terms of projective geometry.
That may not be surprising, if one keeps in mind that many topics, such as intersection
theory, become simpler when one moves to projective geometry. (The papers [11]
and [22] also shed light on this puzzle by indirect routes, but not by using projective
geometry.)

Cohn, page 489
į:
Of course, there is no field F1 with only one element, but there is a trivial ring, and it is merely a convention that we do not call it a field. However, it is an excellent convention, because the trivial ring has no nontrivial modules (if x is an element of a module, then x = 1x = 0x = 0). Calling it a field would not help solve Puzzle 1, since F n 1 does not depend on n. I know of no direct solution to this puzzle, nor of any way to make sense of vector spaces over F1. Nevertheless, the puzzle can be solved by an indirect route: it becomes much easier to understand when it is reformulated in terms of projective geometry. That may not be surprising, if one keeps in mind that many topics, such as intersection theory, become simpler when one moves to projective geometry. (The papers [11] and [22] also shed light on this puzzle by indirect routes, but not by using projective geometry.) Cohn, page 489.
Pakeistos 119-121 eilutės iš
I just learned of the "field of one element":

It's a nonexistent mathematical concept (fields are supposed to contain at least two distinct elements, 0 and 1) which apparently has spurred quite a bit of research. It suggests itself in different situations as a limiting initial case. I hope to learn more about it and report. But my impression is that it relates to my concept of a God who goes beyond himself into himself, who asks, Is God necessary? Would I be even if I wasn't?
į:
2019 vasario 03 d., 17:18 atliko AndriusKulikauskas -
Pridėtos 15-17 eilutės:
* Learn what is known about the field with one element.
** Learn the underlying algebraic geometry.
** Learn how the field with one element relates to the [[Riemann hypothesis]].
Pakeista 19 eilutė iš:
* Learn how Weyl groups can be thought of as algebraic groups over the field with one element. The symmetric group has n! elements and the General linear group over a finite field has [n!]q elements. Relate this to Schur-Weyl duality.
į:
* Learn how Weyl groups can be thought of as algebraic groups over the field with one element. The symmetric group has n! elements and the General linear group over a finite field has {$[n!]_q$} elements. Relate this to Schur-Weyl duality.
Ištrintos 20-22 eilutės:
* Learn the underlying algebraic geometry.
* Learn how the field with one element relates to the [[Riemann hypothesis]].
* Learn what is known about the field with one element.
2019 vasario 03 d., 17:07 atliko AndriusKulikauskas -
Ištrinta 10 eilutė:
* Given a finite field, calculate the probability that a matrix is noninvertible (det=0). What happens in the limit that q->1 ?
2019 vasario 03 d., 16:52 atliko AndriusKulikauskas -
Pakeista 16 eilutė iš:
* Learn about finite fields, especially their combinatorics. Be able to contemplate {{$F_/{1^n/}$}.
į:
* Learn about finite fields, especially their combinatorics. Be able to contemplate {$F_{1^n}$}.
2019 vasario 03 d., 16:51 atliko AndriusKulikauskas -
Pakeista 16 eilutė iš:
* Learn about finite fields, especially their combinatorics. Be able to contemplate {{$F_\{1^n\}$}.
į:
* Learn about finite fields, especially their combinatorics. Be able to contemplate {{$F_/{1^n/}$}.
2019 vasario 03 d., 16:51 atliko AndriusKulikauskas -
Pakeista 16 eilutė iš:
* Learn about finite fields, especially their combinatorics. Be able to contemplate {{$F_{1^n}$}.
į:
* Learn about finite fields, especially their combinatorics. Be able to contemplate {{$F_\{1^n\}$}.
2019 vasario 03 d., 16:50 atliko AndriusKulikauskas -
Pakeistos 35-36 eilutės iš
į:
* [[https://en.wikipedia.org/wiki/Field_with_one_element | Field with one element]]
Pakeista 46 eilutė iš:
į:
* [[http://arxiv.org/pdf/0909.2522 | (Non) Commutative F-un Geometry]] Lieven Le Bruyn
Ištrinta 47 eilutė:
* [[https://arxiv.org/pdf/1801.01491.pdf | The Action of Young Subgroups on the Partition Complex]], Gregory Z. Arone, D. Lukas B. Brantnerthe. Partition complexes can be thought of as Bruhat-Tits buildings over “the field with one element”. In this heuristic picture, Young subgroups correspond to parabolic subgroups. Complementary collapse then has a neat analogous consequence for parabolic restrictions of Bruhat-Tits buildings.
Pakeistos 49-51 eilutės iš
* [[http://www-users.math.umn.edu/~dgrinber/| Darij Grinberg]], [[http://www.cip.ifi.lmu.de/~grinberg/about/rs.pdf | Research interests: Carlitz-Witt vectors and function-field symmetric functions.]]
* [[https://kconrad.math.uconn.edu/blurbs/gradnumthy/carlitz.pdf | Carlitz extensions]] Keith Conrad.
* [[http://cage.ugent.be/~kthas/Fun/library/KapranovSmirnov.pdf | ohomology Determinants and Reciprocity Laws: Number Field Case.]] M.Kapranov, A.Smirnov.
į:
** [[http://www-users.math.umn.edu/~dgrinber/| Darij Grinberg]], [[http://www.cip.ifi.lmu.de/~grinberg/about/rs.pdf | Research interests: Carlitz-Witt vectors and function-field symmetric functions.]]
Pakeistos 52-57 eilutės iš


http
://arxiv.org/pdf/0909.2522
į:
Special aspects
* [[https
://arxiv.org/pdf/1801.01491.pdf | The Action of Young Subgroups on the Partition Complex]], Gregory Z. Arone, D. Lukas B. Brantnerthe. Partition complexes can be thought of as Bruhat-Tits buildings over “the field with one element”. In this heuristic picture, Young subgroups correspond to parabolic subgroups. Complementary collapse then has a neat analogous consequence for parabolic restrictions of Bruhat-Tits buildings.
* [[https://kconrad.math.uconn.edu/blurbs/gradnumthy/carlitz.pdf | Carlitz extensions]] Keith Conrad.
* [[http://cage.ugent.be/~kthas/Fun/library/KapranovSmirnov.pdf | Cohomology Determinants and Reciprocity Laws: Number Field Case.]] M.Kapranov, A.Smirnov.
Pakeista 60 eilutė iš:
* Duality of q and n in GLn(Fq).
į:
* Duality of q and n in {$GL_n(F_q)$}.
Ištrinta 64 eilutė:
* P and NP. Field with 1 element - deterministic. Char q - nondeterministic.
Ištrintos 72-73 eilutės:
Pridėta 77 eilutė:
* P and NP. Field with 1 element - deterministic. Char q - nondeterministic.
Pakeistos 81-87 eilutės iš


F1-believers base their f-unny intuition on the following
two mantras :
F1 forgets about additive data and retains only multiplicative data.
F1-objects only acquire flesh when extended to Z (or C).
į:
F1-believers base their f-unny intuition on the following two mantras :
* F1 forgets about additive data and retains only multiplicative data.
* F1-objects only acquire flesh when extended to Z (or C).
Pakeista 132 eilutė iš:
https://en.wikipedia.org/wiki/Field_with_one_element
į:
2019 vasario 03 d., 16:16 atliko AndriusKulikauskas -
Pridėta 33 eilutė:
* [[https://sbseminar.wordpress.com/2007/08/14/the-field-with-one-element/ | The Field With One Element]] Noah Snyder about Mike Shulman's class
Pridėta 42 eilutė:
* [[https://arxiv.org/abs/0704.2030 | New Approach to Arakelov Geometry]] Nikolai Durov. A theory of generalized rings and schemes.
2019 vasario 03 d., 12:03 atliko AndriusKulikauskas -
Pakeistos 38-43 eilutės iš
į:
* Recently, there have been a few approaches to "geometry over F1", such as Borger rBo09s, Connes Consani rCC09s,rCC14s, Durov rDus, Lorscheid rLo12s, Soule rSs, Takagi rTak12s, Töen- Vaquie rTVs and Haran rH07s and rH09s. For relations between these see rPLs.

General theories
* [[http://arxiv.org/pdf/1508.04636 | New foundations for geometry: Two non-additive languages for arithmetical geometry]] Shai Haran. This point of view of the general-linear-group suggests also that for the field with one element F we have {$"GL_n(\mathbb{F}_1)"=S_n$}, the symmetric group, which embeds as a common subgroup of all the finite group {$GL_n(\mathbb{F}_p)$}, p prime (or the "field" {$\mathbb{F}\{\pm1\}$}, with {$"GL_n(\mathbb{F}\{\pm1\})"=\{\pm1\}^n \bowtie S_n$}. (The symmetry group for {$B_n$} and {$C_n$}.)
* [[http://arxiv.org/pdf/1312.4191 | Quantum {$F_{un}$}: The q=1 Limit of Galois Field Quantum Mechanics, Projective Geometry & the Field With One Element]] Chang, Lewis, Minic, Takeuchi. Model of passing from quantum to classical mechanics
Pakeista 46 eilutė iš:
* [[http://www.cip.ifi.lmu.de/~grinberg/algebra/schur-ore.pdf | Do the symmetric functions havea function-field analogue? Draft.]] Darij Grinberg.
į:
* [[http://www.cip.ifi.lmu.de/~grinberg/algebra/schur-ore.pdf | Do the symmetric functions have a function-field analogue? Draft.]] Darij Grinberg.
Ištrintos 52-61 eilutės:


[[http://arxiv.org/pdf/1508.04636 | New foundations for geometry]] Shai Haran. This point of view of the general-linear-group suggests also that for the field with one element F we have ”GLnpFq” “ Sn, the symmetric group, which embeds as a common subgroup of all the finite group GLnpFpq, p prime (or the "field" Ft˘1u, with "GLnpFt˘1uq” “ t˘1u n ¸ Sn). (the symmetry group for Bn and Cn)

Recently, there have been a few approaches to "geometry over F1", such as
Borger rBo09s, Connes Consani rCC09s,rCC14s, Durov rDus, Lorscheid rLo12s,
Soule rSs, Takagi rTak12s, Töen- Vaquie rTVs and Haran rH07s and rH09s.
For relations between these see rPLs.

http://arxiv.org/pdf/1312.4191 model of passing from quantum to classical mechanics
2019 vasario 03 d., 11:47 atliko AndriusKulikauskas -
Pridėta 31 eilutė:
* [[http://math.ucr.edu/home/baez/week259.html | John Baez on the field with one element]]
Pridėta 36 eilutė:
* [[http://arxiv.org/pdf/0909.0069 |[PL] J.L. Pena, O. Lorscheid: Mapping F1-land: An overview of geometries over the field with one element]] Preprint.
Pakeista 41 eilutė iš:
* [[http://www.cip.ifi.lmu.de/~grinberg/algebra/schur-ore.pdf | Do the symmetric functions havea function-field analogue? Draft. Darij Grinberg.
į:
* [[http://www.cip.ifi.lmu.de/~grinberg/algebra/schur-ore.pdf | Do the symmetric functions havea function-field analogue? Draft.]] Darij Grinberg.
Pridėtos 49-62 eilutės:

[[http://arxiv.org/pdf/1508.04636 | New foundations for geometry]] Shai Haran. This point of view of the general-linear-group suggests also that for the field with one element F we have ”GLnpFq” “ Sn, the symmetric group, which embeds as a common subgroup of all the finite group GLnpFpq, p prime (or the "field" Ft˘1u, with "GLnpFt˘1uq” “ t˘1u n ¸ Sn). (the symmetry group for Bn and Cn)

Recently, there have been a few approaches to "geometry over F1", such as
Borger rBo09s, Connes Consani rCC09s,rCC14s, Durov rDus, Lorscheid rLo12s,
Soule rSs, Takagi rTak12s, Töen- Vaquie rTVs and Haran rH07s and rH09s.
For relations between these see rPLs.

http://arxiv.org/pdf/1312.4191 model of passing from quantum to classical mechanics

http://arxiv.org/pdf/0909.2522

Pakeistos 73-115 eilutės iš
Sapnavau, kad Dievas man kalba, "This is the fundamental unit of information." Ir mačiau partitions (Young tableau, French notation) kurių kiekviename langelyje buvo arba ilgas kampas arba trumpas kampas. Dabar suprantu, kad eilės išreiškė Lyndon žodžius iš dviejų raidžių. Ar tai susiję su F(2)? Su forgotten basis?

A group is
a Hopf algebra over the field with one element.

There exists (there is one=1) a unique (and only
one) vs. For any (all=infinity) vs. There is none (negation=0).

[[https://en.wikipedia.org/wiki/Orthogonal_group | Orthogonal group]] The orthogonal group is generated by reflections (two reflections give a rotation), as in a Coxeter
group,[note 1] and elements have length at most n (require at most n reflections to generate; this follows from the above classification, noting that a rotation is generated by 2 reflections, and is true more generally for indefinite orthogonal groups, by the Cartan–Dieudonné theorem). A longest element (element needing the most reflections) is reflection through the origin (the map v ↦ −v), though so are other maximal combinations of rotations (and a reflection, in odd dimension).... The analogy is stronger: Weyl groups, a class of (representations of) Coxeter groups, can be considered as simple algebraic groups over the field with one element, and there are a number of analogies between algebraic groups and vector spaces on the one hand, and Weyl groups and sets on the other.

[[https://en.wikipedia.org/wiki/Clifford_algebra | Clifford algebra]]
Clifford algebras may be thought of as quantizations (cf. Quantum group) of the exterior algebra, in the same way that the Weyl algebra is a quantization of the symmetric algebra. Weyl algebras and Clifford algebras admit a further structure of a *-algebra, and can be unified as even and odd terms of a superalgebra, as discussed in CCR and CAR algebras.

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

Zero is not a choice
. The field needs to offer another choice.

Intersection
and union do not have inverses.

intersection and union are dual. they are distributive over each other. addition and multiplication are similar but addition is not
distributive over multtiplication.

http://arxiv.org/pdf/math/0407093v1
[5] H. Cohn. Projective Geometry over F1 and the Gaussian Binomial Coefficients. arXiv: 0407
.7093v1
[math.CO], 2009. Helpful and discusses -1 dimensional subspace of Pn(F).

John Baez: http://math.ucr.edu/home/baez/week259.html

http://arxiv.org/pdf/0909.0069
[PL] J.L. Pena, O. Lorscheid: Mapping F1-land: An overview of geometries
over the field with one element, preprint, arXiv: mathAG/0909.0069.

[[http://arxiv.org/pdf/1508.04636 | New foundations for geometry]] Shai Haran.
This point of view of the general-linear-group suggests also that for the
field with one element F we have ”GLnpFq” “ Sn, the symmetric group, which
embeds as a common subgroup of all the finite group GLnpFpq, p prime (or the "field" Ft˘1u, with "GLnpFt˘1uq” “ t˘1u
n ¸ Sn). (the symmetry group for Bn and Cn)

Recently, there have been a few approaches to "geometry over F1", such as
Borger rBo09s, Connes Consani rCC09s,rCC14s, Durov rDus, Lorscheid rLo12s,
Soule rSs, Takagi rTak12s, Töen- Vaquie rTVs and Haran rH07s and rH09s.
For relations between these see rPLs.

http://arxiv.org/pdf/1312.4191 model of passing from quantum to classical mechanics

http://arxiv.org/pdf/0909.2522
į:
Interpretations of mathematical structures in terms of {$F_1$}.
* A finite set can be thought of as a finite dimensional vector space over the field with one element. But no such field exists!
* A group is a Hopf algebra over the field with one element.
* The analogy is stronger: Weyl groups,
a class of (representations of) Coxeter groups, can be considered as simple algebraic groups over the field with one element, and there are a number of analogies between algebraic groups and vector spaces on the one hand, and Weyl groups and sets on the other. [[https://en.wikipedia.org/wiki/Orthogonal_group | Orthogonal group]] The orthogonal group is generated by reflections (two reflections give a rotation), as in a Coxeter group,[note 1] and elements have length at most n (require at most n reflections to generate; this follows from the above classification, noting that a rotation is generated by 2 reflections, and is true more generally for indefinite orthogonal groups, by the Cartan–Dieudonné theorem). A longest element (element needing the most reflections) is reflection through the origin (the map v ↦ −v), though so are other maximal combinations of rotations (and a reflection, in odd dimension)....
* [[https://en.wikipedia.org/wiki/Clifford_algebra | Clifford algebra]] Clifford algebras may be thought of as quantizations (cf. Quantum group)
of the exterior algebra, in the same way that the Weyl algebra is a quantization of the symmetric algebra. Weyl algebras and Clifford algebras admit a further structure of a *-algebra, and can be unified as even and odd terms of a superalgebra, as discussed in CCR and CAR algebras.



Cognitive ideas regarding {$F_1$}.
* Zero is not
a choice. The field needs to offer another choice.
* There exists (there is one=1) a unique (and only one) vs. For any (all=infinity) vs
. There is none (negation=0).
* Fields are "complete" mathematical structures (having all of
the operations) but thus inevitably having a "gap" by which 0 and 1 are distinct. This is the quintessential gap and the prime numbers are likewise gaps in the factorization of numbers.
* Intersection and union do not have inverses.
* Intersection and union are dual. they are
distributive over each other. addition and multiplication are similar but addition is not distributive over multtiplication.

Pakeista 104 eilutė iš:
a module, then x = 1x = 0x = 0). Calling itj a field would not help solve Puzzle 1,
į:
a module, then x = 1x = 0x = 0). Calling it a field would not help solve Puzzle 1,
2019 vasario 03 d., 11:15 atliko AndriusKulikauskas -
Pridėta 39 eilutė:
* [[http://www.cip.ifi.lmu.de/~grinberg/algebra/schur-ore.pdf | Do the symmetric functions havea function-field analogue? Draft. Darij Grinberg.
2019 vasario 03 d., 11:13 atliko AndriusKulikauskas -
Pakeista 16 eilutė iš:
* Learn about finite fields, especially their combinatorics. Be able to contemplate F1n.
į:
* Learn about finite fields, especially their combinatorics. Be able to contemplate {{$F_{1^n}$}.
2019 vasario 03 d., 11:12 atliko AndriusKulikauskas -
Pridėtos 29-36 eilutės:

Introductory
* [[https://cameroncounts.wordpress.com/2011/07/20/the-field-with-one-element/ | The field with one element]]. Peter Cameron.
* [[http://www.neverendingbooks.org/DATA3/ncg.pdf | Geometry and the Absolute Point]]

Overview
* [[http://cage.ugent.be/~kthas/Fun/ | F_un Mathematics]]
Pakeistos 39-43 eilutės iš
* [[http://www-users.math.umn.edu/~dgrinber/| Darij Grinberg]], http://www.cip.ifi.lmu.de/~grinberg/about/rs.pdf, See: Combinatorics and the field with one element. Witt vectors - p-adic integers
* http
://cage.ugent.be/~kthas/Fun/library/KapranovSmirnov.pdf
* https://cameroncounts.wordpress.com/2011/07/20/the-field-with-one-element/
* http:
//cage.ugent.be/~kthas/Fun/
* http
://arxiv.org/abs/0911.3537
į:
* [[http://www-users.math.umn.edu/~dgrinber/| Darij Grinberg]], [[http://www.cip.ifi.lmu.de/~grinberg/about/rs.pdf | Research interests: Carlitz-Witt vectors and function-field symmetric functions.]]
* [[https
://kconrad.math.uconn.edu/blurbs/gradnumthy/carlitz.pdf | Carlitz extensions]] Keith Conrad.
* [[http:
//cage.ugent.be/~kthas/Fun/library/KapranovSmirnov.pdf | ohomology Determinants and Reciprocity Laws: Number Field Case.]] M.Kapranov, A.Smirnov.
* [[http://arxiv.org/abs/0911.3537 | Characteristic one, entropy and the absolute point]]. Alain Connes, Caterina Consani.
2019 vasario 03 d., 10:46 atliko AndriusKulikauskas -
Pakeista 30 eilutė iš:
* https://arxiv.org/pdf/1801.01491.pdf Partition complexes can be thought of as Bruhat-Tits buildings over “the field with one element”. In this heuristic picture, Young subgroups correspond to parabolic subgroups. Complementary collapse then has a neat analogous consequence for parabolic restrictions of Bruhat-Tits buildings.
į:
* [[https://arxiv.org/pdf/1801.01491.pdf | The Action of Young Subgroups on the Partition Complex]], Gregory Z. Arone, D. Lukas B. Brantnerthe. Partition complexes can be thought of as Bruhat-Tits buildings over “the field with one element”. In this heuristic picture, Young subgroups correspond to parabolic subgroups. Complementary collapse then has a neat analogous consequence for parabolic restrictions of Bruhat-Tits buildings.
2019 vasario 03 d., 10:22 atliko AndriusKulikauskas -
Pakeistos 32-37 eilutės iš
į:
* http://cage.ugent.be/~kthas/Fun/library/KapranovSmirnov.pdf
* https://cameroncounts.wordpress.com/2011/07/20/the-field-with-one-element/
* http://cage.ugent.be/~kthas/Fun/
* http://arxiv.org/abs/0911.3537
Ištrintos 56-64 eilutės:

http://cage.ugent.be/~kthas/Fun/library/KapranovSmirnov.pdf

https://cameroncounts.wordpress.com/2011/07/20/the-field-with-one-element/

http://cage.ugent.be/~kthas/Fun/


http://arxiv.org/abs/0911.3537
2019 sausio 19 d., 14:02 atliko AndriusKulikauskas -
Pakeistos 30-31 eilutės iš
* https://arxiv.org/pdf/1801.01491.pdf Partition complexes can be thought of as Bruhat-Tits buildings over “the field with one element”. In this heuristic picture, Young subgroups correspond to parabolic subgroups. Complementary
collapse then has a neat analogous consequence for parabolic restrictions of Bruhat-Tits buildings.
į:
* https://arxiv.org/pdf/1801.01491.pdf Partition complexes can be thought of as Bruhat-Tits buildings over “the field with one element”. In this heuristic picture, Young subgroups correspond to parabolic subgroups. Complementary collapse then has a neat analogous consequence for parabolic restrictions of Bruhat-Tits buildings.
2019 sausio 19 d., 14:02 atliko AndriusKulikauskas -
Pridėta 28 eilutė:
Readings
Pakeistos 30-33 eilutės iš
į:
* https://arxiv.org/pdf/1801.01491.pdf Partition complexes can be thought of as Bruhat-Tits buildings over “the field with one element”. In this heuristic picture, Young subgroups correspond to parabolic subgroups. Complementary
collapse then has a neat analogous consequence for parabolic restrictions of Bruhat-Tits buildings.
* [[http://www-users.math.umn.edu/~dgrinber/| Darij Grinberg]], http://www.cip.ifi.lmu.de/~grinberg/about/rs.pdf, See: Combinatorics and the field with one element. Witt vectors - p-adic integers
Ištrintos 156-165 eilutės:

Partition complexes can be thought of as Bruhat-Tits buildings over “the field with one element”.
In this heuristic picture, Young subgroups correspond to parabolic subgroups. Complementary
collapse then has a neat analogous consequence for parabolic restrictions of Bruhat-Tits buildings. https://arxiv.org/pdf/1801.01491.pdf

Darij Grinberg
http://www-users.math.umn.edu/~dgrinber/
http://www.cip.ifi.lmu.de/~grinberg/about/rs.pdf
See: Combinatorics and the field with one element.
Witt vectors - p-adic integers
2019 sausio 19 d., 14:01 atliko AndriusKulikauskas -
Pridėtos 27-28 eilutės:

* [[https://arxiv.org/abs/math/0407093 | Projective geometry over {$F_1$} and the Gaussian binomial coefficients]], [[http://math.mit.edu/~cohn/ | Henry Cohn]], [[http://math.mit.edu/~cohn/research.html | his research]]
2018 gruodžio 09 d., 17:08 atliko AndriusKulikauskas -
Pridėtos 22-24 eilutės:

Investigate
* Does 1-1=0 in the field with one element?
2018 lapkričio 11 d., 16:56 atliko AndriusKulikauskas -
Pakeistos 151-157 eilutės iš
collapse then has a neat analogous consequence for parabolic restrictions of Bruhat-Tits buildings. https://arxiv.org/pdf/1801.01491.pdf
į:
collapse then has a neat analogous consequence for parabolic restrictions of Bruhat-Tits buildings. https://arxiv.org/pdf/1801.01491.pdf

Darij Grinberg
http://www-users.math.umn.edu/~dgrinber/
http://www.cip.ifi.lmu.de/~grinberg/about/rs.pdf
See: Combinatorics and the field with one element.
Witt vectors - p-adic integers
2018 lapkričio 08 d., 14:01 atliko AndriusKulikauskas -
Pridėta 12 eilutė:
* How the [[https://en.wikipedia.org/wiki/Hall_algebra | Hall algebra]] and Hall polynomials give rise to the symmetric functions and, in particular, Schur functions, when q goes to 0.
2018 rugsėjo 14 d., 11:12 atliko AndriusKulikauskas -
Pakeista 1 eilutė iš:
See: [[Math]], [[Binomial theorem]]
į:
See: [[Math]], [[Binomial theorem]], [[Finite fields]]
2018 rugsėjo 13 d., 16:40 atliko AndriusKulikauskas -
Ištrintos 4-7 eilutės:
* Learn what is known about the field with one element. Learn the underlying algebraic geometry.
* Learn about finite fields, especially their combinatorics. Be able to contemplate F1n.
* Learn how the field with one element relates to the [[Riemann hypothesis]].
Pakeistos 9-18 eilutės iš
Find q-analogues for each of the four interpretations of the binomial theorem. See how what they count relates to a finite field of characteristic q. Then see what happens to each of them, and their relationship, when q->1.

Consider relationship between the two starting points for [[https://en.wikipedia.org/wiki/Homography | homographies]] and projective spaces. One based on fields and one without fields. Consider that as a relationship q->1.

Given a finite field, calculate the probability that a matrix is noninvertible (det=0). What happens in the limit that q->1 ?

Relate Cayley's theorem to the field with one element.

Study:
* How Weyl groups can be thought of as algebraic groups over the field with one element. The symmetric group has n! elements and
the General linear group over a finite field has [n!]q elements. Relate this to Schur-Weyl duality.
į:
Study
*
Find q-analogues for each of the four interpretations of the binomial theorem. See how what they count relates to a finite field of characteristic q. Then see what happens to each of them, and their relationship, when q->1.
* Given a finite field, calculate the probability that a matrix is noninvertible (det=0). What happens in the limit that q->1 ?


Learn
* Learn about finite fields, especially their combinatorics
. Be able to contemplate F1n.
* Learn how Weyl groups can be thought of as algebraic groups over the field with one element. The symmetric group has n! elements and the General linear group over a finite field has [n!]q elements. Relate this to Schur-Weyl duality.
* Learn about the relationship between
the two starting points for [[https://en.wikipedia.org/wiki/Homography | homographies]] and projective spaces. One based on fields and one without fields. Consider that as a relationship q->1.
* Learn the underlying algebraic geometry.
* Learn how the field with one element relates to the [[Riemann hypothesis]].
* Learn what is known about the field with one element.
2018 rugsėjo 13 d., 16:14 atliko AndriusKulikauskas -
Pakeistos 1-2 eilutės iš
See: [[Math]]
į:
See: [[Math]], [[Binomial theorem]]

I believe that the field with one element can be interpreted as an element that can be interpreted as 0, 1 and infinity. For example, 0 * infinity = 1. I think that 1 can be thought of as reflecting 0 and infinity in a duality. I think of this as modeling [[Gods Dance | God's dance]].

* Learn what is known about the field with one element. Learn the underlying algebraic geometry.
* Learn about finite fields, especially their combinatorics. Be able to contemplate F1n.
* Learn how the field with one element relates to the [[Riemann hypothesis]].

The field with one element can be imagined by way of Pascal's triangle. For the Gaussian binomial coefficients count the number of subspaces of a vector space over a field of characteristic q. Having q->1 yields the usual binomial coefficients. Also, Pascal's triangle counts the simplexes of a subsimplex, and variants count the parts of other infinite families of polytopes.
Pridėtos 12-13 eilutės:

Find q-analogues for each of the four interpretations of the binomial theorem. See how what they count relates to a finite field of characteristic q. Then see what happens to each of them, and their relationship, when q->1.
2018 vasario 15 d., 22:19 atliko AndriusKulikauskas -
Pridėtos 15-24 eilutės:

Finite fields
* Lyndon words - irreducible polynomials for finite fields
* Duality of q and n in GLn(Fq).
* Multiset of Lyndon words - reducible and irreducible. Homogeneous symmetric functions of eigenvalues.
* Interpolation between homogeneous and elementary - between commutativity and anti-commutativity.
* Lyndon words are like prime numbers.
* Dimension of free Lie algebras = number of Lyndon words of length n
* P and NP. Field with 1 element - deterministic. Char q - nondeterministic.
* What would be the q-theory for finite fields for matrix combinatorics?
2018 sausio 15 d., 18:05 atliko AndriusKulikauskas -
Pakeista 16 eilutė iš:
Sapnavau, kad Dievas man kalba, "This is the fundamental unit of information." Ir mačiau partitions (Young tableau, French notation) kurių kiekviename langelyje buvo arba ilgas kampas arba trumpas kampas. Dabar suprantu, kad eilės išreiškė Lyndon žodžius iš dviejų raidžių.
į:
Sapnavau, kad Dievas man kalba, "This is the fundamental unit of information." Ir mačiau partitions (Young tableau, French notation) kurių kiekviename langelyje buvo arba ilgas kampas arba trumpas kampas. Dabar suprantu, kad eilės išreiškė Lyndon žodžius iš dviejų raidžių. Ar tai susiję su F(2)? Su forgotten basis?
2018 sausio 15 d., 17:57 atliko AndriusKulikauskas -
Pakeista 16 eilutė iš:
į:
Sapnavau, kad Dievas man kalba, "This is the fundamental unit of information." Ir mačiau partitions (Young tableau, French notation) kurių kiekviename langelyje buvo arba ilgas kampas arba trumpas kampas. Dabar suprantu, kad eilės išreiškė Lyndon žodžius iš dviejų raidžių.
2018 sausio 15 d., 13:58 atliko AndriusKulikauskas -
Pakeistos 127-132 eilutės iš
------------------
į:
------------------


Partition complexes can be thought of as Bruhat-Tits buildings over “the field with one element”.
In this heuristic picture, Young subgroups correspond to parabolic subgroups. Complementary
collapse then has a neat analogous consequence for parabolic restrictions of Bruhat-Tits buildings. https://arxiv.org/pdf/1801.01491.pdf
2017 lapkričio 11 d., 21:08 atliko AndriusKulikauskas -
Pakeistos 9-11 eilutės iš
Relate Cayley's theorem to the field with one element

>>bgcolor=#EEEEEE<<
į:
Relate Cayley's theorem to the field with one element.

Study:
* How Weyl groups can be thought of as algebraic groups over the field with one element. The symmetric group has n! elements and the General linear group over a finite field has [n!]q elements. Relate this to Schur-Weyl duality.

>><<
2017 spalio 04 d., 10:20 atliko AndriusKulikauskas -
Pridėtos 8-9 eilutės:

Relate Cayley's theorem to the field with one element
2017 spalio 04 d., 09:55 atliko AndriusKulikauskas -
Pridėtos 117-118 eilutės:

The anharmonic group (see [[https://en.wikipedia.org/wiki/Cross-ratio | Cross-ratio]]) permutes 0,1 and infinity.
2016 gruodžio 03 d., 10:41 atliko AndriusKulikauskas -
Pridėtos 10-11 eilutės:

A group is a Hopf algebra over the field with one element.
2016 rugsėjo 27 d., 20:02 atliko AndriusKulikauskas -
Pridėtos 10-11 eilutės:

There exists (there is one=1) a unique (and only one) vs. For any (all=infinity) vs. There is none (negation=0).
2016 rugsėjo 13 d., 11:27 atliko AndriusKulikauskas -
Pridėtos 12-13 eilutės:

[[https://en.wikipedia.org/wiki/Clifford_algebra | Clifford algebra]] Clifford algebras may be thought of as quantizations (cf. Quantum group) of the exterior algebra, in the same way that the Weyl algebra is a quantization of the symmetric algebra. Weyl algebras and Clifford algebras admit a further structure of a *-algebra, and can be unified as even and odd terms of a superalgebra, as discussed in CCR and CAR algebras.
2016 rugsėjo 13 d., 11:21 atliko AndriusKulikauskas -
Pridėtos 10-11 eilutės:

[[https://en.wikipedia.org/wiki/Orthogonal_group | Orthogonal group]] The orthogonal group is generated by reflections (two reflections give a rotation), as in a Coxeter group,[note 1] and elements have length at most n (require at most n reflections to generate; this follows from the above classification, noting that a rotation is generated by 2 reflections, and is true more generally for indefinite orthogonal groups, by the Cartan–Dieudonné theorem). A longest element (element needing the most reflections) is reflection through the origin (the map v ↦ −v), though so are other maximal combinations of rotations (and a reflection, in odd dimension).... The analogy is stronger: Weyl groups, a class of (representations of) Coxeter groups, can be considered as simple algebraic groups over the field with one element, and there are a number of analogies between algebraic groups and vector spaces on the one hand, and Weyl groups and sets on the other.
2016 rugpjūčio 30 d., 22:21 atliko AndriusKulikauskas -
Pridėtos 6-7 eilutės:

Given a finite field, calculate the probability that a matrix is noninvertible (det=0). What happens in the limit that q->1 ?
2016 rugpjūčio 15 d., 11:17 atliko AndriusKulikauskas -
Pridėtos 29-30 eilutės:

John Baez: http://math.ucr.edu/home/baez/week259.html
2016 rugpjūčio 15 d., 11:15 atliko AndriusKulikauskas -
Pakeista 28 eilutė iš:
[math.CO], 2009.
į:
[math.CO], 2009. Helpful and discusses -1 dimensional subspace of Pn(F).
2016 liepos 15 d., 12:50 atliko AndriusKulikauskas -
Pridėtos 15-16 eilutės:

http://arxiv.org/abs/0911.3537
2016 liepos 15 d., 12:45 atliko AndriusKulikauskas -
Pridėtos 8-14 eilutės:

http://cage.ugent.be/~kthas/Fun/library/KapranovSmirnov.pdf

https://cameroncounts.wordpress.com/2011/07/20/the-field-with-one-element/

http://cage.ugent.be/~kthas/Fun/
2016 liepos 10 d., 16:43 atliko AndriusKulikauskas -
Pakeista 25 eilutė iš:
http://arxiv.org/pdf/1508.04636
į:
[[http://arxiv.org/pdf/1508.04636 | New foundations for geometry]] Shai Haran.
2016 birželio 23 d., 15:35 atliko AndriusKulikauskas -
Pridėtos 94-95 eilutės:

[[https://www.youtube.com/watch?v=1XRna0vUYdo | field of one element video]]
2016 birželio 23 d., 14:06 atliko AndriusKulikauskas -
Pridėtos 8-9 eilutės:

A finite set can be thought of as a finite dimensional vector space over the field with one element. But no such field exists!
2016 birželio 23 d., 10:21 atliko AndriusKulikauskas -
Pridėtos 10-11 eilutės:

Intersection and union do not have inverses.
2016 birželio 23 d., 10:16 atliko AndriusKulikauskas -
Pridėtos 10-11 eilutės:

intersection and union are dual. they are distributive over each other. addition and multiplication are similar but addition is not distributive over multtiplication.
2016 birželio 23 d., 09:54 atliko AndriusKulikauskas -
Pridėtos 8-9 eilutės:

Zero is not a choice. The field needs to offer another choice.
2016 birželio 23 d., 01:28 atliko AndriusKulikauskas -
Pakeista 46 eilutė iš:
a module, then x = 1x = 0x = 0). Calling it a field would not help solve Puzzle 1,
į:
a module, then x = 1x = 0x = 0). Calling itj a field would not help solve Puzzle 1,
Pridėta 59 eilutė:
Cohn, page 489
2016 birželio 23 d., 01:27 atliko AndriusKulikauskas -
Pridėtos 42-57 eilutės:

Of course, there is no field F1 with only one element, but there is a trivial ring,
and it is merely a convention that we do not call it a field. However, it is an excellent
convention, because the trivial ring has no nontrivial modules (if x is an element of
a module, then x = 1x = 0x = 0). Calling it a field would not help solve Puzzle 1,
since F
n
1
does not depend on n.
I know of no direct solution to this puzzle, nor of any way to make sense of vector
spaces over F1. Nevertheless, the puzzle can be solved by an indirect route: it becomes
much easier to understand when it is reformulated in terms of projective geometry.
That may not be surprising, if one keeps in mind that many topics, such as intersection
theory, become simpler when one moves to projective geometry. (The papers [11]
and [22] also shed light on this puzzle by indirect routes, but not by using projective
geometry.)
2016 birželio 23 d., 01:24 atliko AndriusKulikauskas -
Pridėtos 36-38 eilutės:

What form does the binomial theorem take in a noncommutative ring? In general one can say nothing interesting, but certain special cases work out elegantly. One of the nicest, due to Schutzenberger [18],
deals with variables x, y, and q such that q commutes with x and y, and yx = qxy.
2016 birželio 23 d., 01:21 atliko AndriusKulikauskas -
Pakeista 9 eilutė iš:
http://arxiv.org/pdf/0407.7093
į:
http://arxiv.org/pdf/math/0407093v1
2016 birželio 23 d., 01:18 atliko AndriusKulikauskas -
Pakeista 9 eilutė iš:
http://arxiv.org/pdf/0407.7093v1
į:
http://arxiv.org/pdf/0407.7093
2016 birželio 23 d., 01:18 atliko AndriusKulikauskas -
Pridėtos 8-11 eilutės:

http://arxiv.org/pdf/0407.7093v1
[5] H. Cohn. Projective Geometry over F1 and the Gaussian Binomial Coefficients. arXiv: 0407.7093v1
[math.CO], 2009.
2016 birželio 23 d., 01:10 atliko AndriusKulikauskas -
Pridėtos 8-11 eilutės:

http://arxiv.org/pdf/0909.0069
[PL] J.L. Pena, O. Lorscheid: Mapping F1-land: An overview of geometries
over the field with one element, preprint, arXiv: mathAG/0909.0069.
2016 birželio 23 d., 01:07 atliko AndriusKulikauskas -
Pridėtos 14-18 eilutės:

Recently, there have been a few approaches to "geometry over F1", such as
Borger rBo09s, Connes Consani rCC09s,rCC14s, Durov rDus, Lorscheid rLo12s,
Soule rSs, Takagi rTak12s, Töen- Vaquie rTVs and Haran rH07s and rH09s.
For relations between these see rPLs.
2016 birželio 23 d., 01:05 atliko AndriusKulikauskas -
Pakeistos 9-13 eilutės iš
http://arxiv.org/pdf/1508.04636
į:
http://arxiv.org/pdf/1508.04636
This point of view of the general-linear-group suggests also that for the
field with one element F we have ”GLnpFq” “ Sn, the symmetric group, which
embeds as a common subgroup of all the finite group GLnpFpq, p prime (or the "field" Ft˘1u, with "GLnpFt˘1uq” “ t˘1u
n ¸ Sn). (the symmetry group for Bn and Cn)
2016 birželio 23 d., 01:00 atliko AndriusKulikauskas -
Pridėtos 14-19 eilutės:

F1-believers base their f-unny intuition on the following
two mantras :
• F1 forgets about additive data and retains only multiplicative data.
• F1-objects only acquire flesh when extended to Z (or C).
2016 birželio 23 d., 00:56 atliko AndriusKulikauskas -
Pakeista 11 eilutė iš:
http://arxiv.org/pdf/1312.4191
į:
http://arxiv.org/pdf/1312.4191 model of passing from quantum to classical mechanics
2016 birželio 22 d., 00:22 atliko AndriusKulikauskas -
Pridėtos 12-13 eilutės:

http://arxiv.org/pdf/0909.2522
2016 birželio 22 d., 00:13 atliko AndriusKulikauskas -
Pridėtos 10-11 eilutės:

http://arxiv.org/pdf/1312.4191
2016 birželio 22 d., 00:10 atliko AndriusKulikauskas -
Pridėtos 8-9 eilutės:

http://arxiv.org/pdf/1508.04636
2016 birželio 20 d., 01:28 atliko AndriusKulikauskas -
Pridėtos 3-6 eilutės:
>>bgcolor=#FFFFC0<<

Consider relationship between the two starting points for [[https://en.wikipedia.org/wiki/Homography | homographies]] and projective spaces. One based on fields and one without fields. Consider that as a relationship q->1.
Pakeistos 9-10 eilutės iš
Consider relationship between the two starting points for [[https://en.wikipedia.org/wiki/Homography | homographies]] and projective spaces. One based on fields and one without fields. Consider that as a relationship q->1.
į:
Symmetry involves a dual point of view: for example, vertices are distinct and yet not distinguishable.
2016 birželio 19 d., 22:39 atliko AndriusKulikauskas -
Pridėtos 4-9 eilutės:

Consider relationship between the two starting points for [[https://en.wikipedia.org/wiki/Homography | homographies]] and projective spaces. One based on fields and one without fields. Consider that as a relationship q->1.



--------------------
2016 birželio 19 d., 14:08 atliko AndriusKulikauskas -
Pridėtos 1-29 eilutės:
See: [[Math]]

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Dear Harvey,

Thank you for your invitations in your letter below and also earlier, "...I am trying to get a dialog going on the FOM and in these other forums as to "what foundations of mathematics are, ought to be, and what purpose they serve"."
http://www.cs.nyu.edu/pipermail/fom/2016-April/019724.html

You mentioned, in my words, that you are looking for an issue that working mathematicians are grappling with where the classical ZFC foundations are not satisfactory or sufficient. Would the "field with one element" be such issue for you?
https://ncatlab.org/nlab/show/field+with+one+element

Jacques Tits raised this issue in 1957 and it has yet to be resolved despite substantial interest, conferences, and long papers. Would that count as a "problem" for the Foundations of Mathematics? It seems that in the history of math it is very easy to simply say "that is not real math" as was the case with the rational numbers, imaginary numbers, infinitesimals, infinite series, etc.

The issue is that there are many instances where a combinatorial interpretation makes sense in terms of a finite field Fq of characteristic q, which is all the more insightful when q=1. For example, the Gaussian binomial coefficients can be interpreted as counting the number of k-dimensional subspaces of an n-dimensional vector space over a finite field Fq. When q=1, then we get the usual binomial coefficients which count the subsets of size k of a set of size n. So this suggests an important way of thinking about sets. However, F1 would be a field with one element, which means that 0=1. But if 0 and 1 are not distinct, then none of the usual properties of a field make sense. Nobody has figured out a convincing interpretation for F1. And yet the concept seems to be pervasive, meaningful and fruitful.

If there was an alternate foundations of mathematics which yielded a helpful, meaningful, fruitful interpretation of F1, would that count in its favor? And if it could do everything that FOM can do, then might it be preferable, at least for some? But especially if that interpretation was shown not to make sense in other FOMs?

I am curious what you and others think.

Andrius

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I just learned of the "field of one element":
https://en.wikipedia.org/wiki/Field_with_one_element
It's a nonexistent mathematical concept (fields are supposed to contain at least two distinct elements, 0 and 1) which apparently has spurred quite a bit of research. It suggests itself in different situations as a limiting initial case. I hope to learn more about it and report. But my impression is that it relates to my concept of a God who goes beyond himself into himself, who asks, Is God necessary? Would I be even if I wasn't?

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FieldWithOneElement


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