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Andrius Kulikauskas

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Book.InterpretingCayley-DicksonConstruction istorija

Rodyti nežymius pakeitimus - Rodyti galutinio teksto pakeitimus

2018 gruodžio 16 d., 10:35 atliko AndriusKulikauskas -
Pridėta 12 eilutė:
* [[https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(composition_algebras) | Hurwitz's theorem]]
2018 lapkričio 25 d., 20:52 atliko AndriusKulikauskas -
Pakeista 27 eilutė iš:
Thus conjugates are relative and they reference an internal structural distinction, whereas positive and negative terms exhibit an absolute, external distinction. This relates to observation and measurement. It also brings to mind the notion of a [[https://en.wikipedia.org/wiki/Head-driven_phrase_structure_grammar | head-driven phrase structure grammar]]. Ambiguity is maintained in the head. Everything subsequently explicilty supports the existing ambiguity.
į:
Thus conjugates are relative and they reference an internal structural distinction, whereas positive and negative terms exhibit an absolute, external distinction. This relates to observation and measurement. It also brings to mind the notion of a [[https://en.wikipedia.org/wiki/Head-driven_phrase_structure_grammar | head-driven phrase structure grammar]]. Ambiguity is maintained in the head. Everything subsequently explicitly supports the existing ambiguity.
2018 lapkričio 25 d., 20:18 atliko AndriusKulikauskas -
Pakeista 27 eilutė iš:
Thus conjugates are relative and they reference an internal structural distinction, whereas positive and negative terms exhibit an absolute, external distinction. This relates to observation and measurement. It also brings to mind the notion of a [[https://en.wikipedia.org/wiki/Head-driven_phrase_structure_grammar | head-driven phrase structure grammar]].
į:
Thus conjugates are relative and they reference an internal structural distinction, whereas positive and negative terms exhibit an absolute, external distinction. This relates to observation and measurement. It also brings to mind the notion of a [[https://en.wikipedia.org/wiki/Head-driven_phrase_structure_grammar | head-driven phrase structure grammar]]. Ambiguity is maintained in the head. Everything subsequently explicilty supports the existing ambiguity.
2018 lapkričio 25 d., 20:17 atliko AndriusKulikauskas -
Pakeista 27 eilutė iš:
Thus conjugates are relative and they reference an internal structural distinction, whereas positive and negative terms exhibit an absolute, external distinction. This relates to observation and measurement. It also recalls the notion of a head-driven grammar.
į:
Thus conjugates are relative and they reference an internal structural distinction, whereas positive and negative terms exhibit an absolute, external distinction. This relates to observation and measurement. It also brings to mind the notion of a [[https://en.wikipedia.org/wiki/Head-driven_phrase_structure_grammar | head-driven phrase structure grammar]].
2018 lapkričio 25 d., 20:15 atliko AndriusKulikauskas -
Pakeista 27 eilutė iš:
Thus conjugates are relative and they reference an internal structural distinction, whereas positive and negative terms exhibit an absolute, external distinction. This relates to observation and measurement.
į:
Thus conjugates are relative and they reference an internal structural distinction, whereas positive and negative terms exhibit an absolute, external distinction. This relates to observation and measurement. It also recalls the notion of a head-driven grammar.
2018 lapkričio 25 d., 20:14 atliko AndriusKulikauskas -
Pakeista 21 eilutė iš:
The conjugate links an undistinguished opposite (one of two conjugates) with a distinguished opposite (positive or negative).
į:
The conjugate links an undistinguished opposite (one of two conjugates) with a distinguished opposite (positive or negative). Calculating we get {$((a,b),(c,d))^* = ((a,b)^*,-(c,d))=((a^*,-b),(-c,-d))$}.
2018 lapkričio 25 d., 19:54 atliko AndriusKulikauskas -
Pakeista 29 eilutė iš:
Note that there are five dualities:
į:
Note that there are four dualities:
2018 lapkričio 25 d., 19:53 atliko AndriusKulikauskas -
Pakeista 27 eilutė iš:
Thus conjugates reference an internal structural distinction, whereas positive and negative terms exhibit an external distinction.
į:
Thus conjugates are relative and they reference an internal structural distinction, whereas positive and negative terms exhibit an absolute, external distinction. This relates to observation and measurement.
2018 lapkričio 25 d., 19:53 atliko AndriusKulikauskas -
Pakeista 27 eilutė iš:
Thus conjugacy reference an internal structural distinction, whereas positive and negative are external distinctions.
į:
Thus conjugates reference an internal structural distinction, whereas positive and negative terms exhibit an external distinction.
2018 lapkričio 25 d., 19:52 atliko AndriusKulikauskas -
Pakeista 27 eilutė iš:
Suppose that
į:
Thus conjugacy reference an internal structural distinction, whereas positive and negative are external distinctions.
2018 lapkričio 25 d., 19:50 atliko AndriusKulikauskas -
Pakeista 30 eilutė iš:
* multiplying pairs left to right or right to left
į:
* multiplying elements (or pairs) left to right or right to left
Ištrinta 31 eilutė:
* multiplying elements within a pair from left to right or right to left
2018 lapkričio 25 d., 19:50 atliko AndriusKulikauskas -
Pakeistos 23-25 eilutės iš
In general, given a term

{$$(a_1,a_2,a_3,\cdots, a_{2^{n}}) \in \bigoplus_{i=1}^{2^n}A$$} we have
į:
In general, given a term {$(a_1,a_2,a_3,\cdots, a_{2^{n}}) \in \bigoplus_{i=1}^{2^n}A$} we have
2018 lapkričio 25 d., 19:49 atliko AndriusKulikauskas -
Pakeistos 21-29 eilutės iš
The conjugate links an umarked opposite (a conjugate) with a marked opposite (positive or negative).
į:
The conjugate links an undistinguished opposite (one of two conjugates) with a distinguished opposite (positive or negative).

In general, given a term

{$$(a_1,a_2,a_3,\cdots, a_{2^{n}}) \in \bigoplus_{i=1}^{2^n}A$$} we have

{$$(a_1,a_2,a_3,\cdots, a_{2^{n}})^*=({a_1}^*,-a_2,-a_3,\cdots,-a_{2^{n}})$$}

Suppose that
2018 lapkričio 25 d., 14:10 atliko AndriusKulikauskas -
Pridėtos 20-21 eilutės:

The conjugate links an umarked opposite (a conjugate) with a marked opposite (positive or negative).
2018 lapkričio 25 d., 13:47 atliko AndriusKulikauskas -
Pakeista 21 eilutė iš:
Note that there are four dualities:
į:
Note that there are five dualities:
Pridėta 24 eilutė:
* multiplying elements within a pair from left to right or right to left
2018 lapkričio 25 d., 10:42 atliko AndriusKulikauskas -
Pridėtos 20-26 eilutės:

Note that there are four dualities:
* multiplying pairs left to right or right to left
* left and right positions within a pair
* conjugates
* positive and negative
2018 lapkričio 25 d., 10:40 atliko AndriusKulikauskas -
Pakeistos 13-20 eilutės iš
{$(a,b)(c,d)=(ac-d^{*}b,bc^{*}+da)$}
į:
Given a division algebra {$A$} with conjugates, define a new division algebra {$A\oplus A$} with the following product on the ordered pairs:

{$(a,b)(c,d)=(ac-d^{*}b,bc^{*}+da)$}

Define the conjugate as follows:

{$(a,b)^* = (a^*,-b)$}
2018 lapkričio 25 d., 10:26 atliko AndriusKulikauskas -
Pakeistos 9-13 eilutės iš
[[https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction | Wikipedia: Cayley-Dickson construction]]
į:
Readings
*
[[https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction | Wikipedia: Cayley-Dickson construction]]
* [[http://people.maths.ox.ac.uk/mcgerty/ClassicalGroups.pdf | The Classical Groups]] by Kevin McGerty.

{$(a,b)(c,d)=(ac-d^{*}b,bc^{*}+da)$}
2018 lapkričio 25 d., 10:22 atliko AndriusKulikauskas -
Pridėtos 8-9 eilutės:

[[https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction | Wikipedia: Cayley-Dickson construction]]
2018 lapkričio 25 d., 10:21 atliko AndriusKulikauskas -
Pridėtos 1-7 eilutės:
>>bgcolor=#E9F5FC<<
-------------
See: [[Math notebook]]

'''Interpret the Cayley-Dickson construction.'''
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>><<

InterpretingCayley-DicksonConstruction


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