Axioms of Choice
https://axiomsofchoice.org/
2024-03-19T11:30:45+0100Axioms of Choice
https://axiomsofchoice.org/
https://axiomsofchoice.org/lib/tpl/vector/images/favicon.icotext/html2023-07-05T21:09:09+0100nikolaj (nikolaj@undisclosed.example.com)Nikolaj-K's notebook
https://axiomsofchoice.org/nikolajs_notebook?rev=1688584149&do=diff
Nikolaj-K's notebook
In this wiki I try to keep track of some of the vast amount of mathematical objects and learn about their relationships.
This is the credible and neatly interlinked (interlinked) notebook of a physicist and where the content leans towards applications, it's with an eye on stochastics, statistical physics and their computational implementation.
It also contains content for a book and ideas for a typed programming language for the formal sciences.
The pagetext/html2020-08-23T16:33:26+0100nikolaj (nikolaj@undisclosed.example.com)Hermitian matrix
https://axiomsofchoice.org/hermitian_matrix?rev=1598193206&do=diff
Hermitian matrix
Set
context $n\in\mathbb N$ definiendum $ A \in \mathrm{HermitianMatrix}(n) $ postulate $ A \in \mathrm{SquareMatrix}(n,\mathbb C) $ postulate $ A^*=A $
Discussion
Reference
Wikipedia: Hermitian matrix
Parents
Subset of
Square matrix
Context
Matrix conjugate transpose
Related
Symmetric matrixtext/html2020-05-20T14:23:48+0100nikolaj (nikolaj@undisclosed.example.com)First infinite von Neumann ordinal
https://axiomsofchoice.org/first_infinite_von_neumann_ordinal?rev=1589977428&do=diff
First infinite von Neumann ordinal
Set
definiendum $ \omega_{\mathcal N}$
...
----------
As is common, I'll also use the symbol $\mathbb N$ to denote the set theoretic object $\omega_{\mathcal N}$.
Idea
This is probably the most straightforward way to set up a countably infinite set.
Elaboration
$\omega_{\mathcal N}$$\emptyset$$\omega_{\mathcal N}$$\omega_{\mathcal N}$$\omega_{\mathcal N}$$\emptyset$$m$${\mathrm{succ}}\ m\equiv m\cup\{m\}$$0\equiv \emptyset$$1\equiv {\mathrm{succ}}\…text/html2020-01-10T23:08:53+0100nikolaj (nikolaj@undisclosed.example.com)x^x
https://axiomsofchoice.org/x_x?rev=1578694133&do=diff
x^x
Function
definiendum $ \zeta: \mathbb{C}\setminus\{???\} \to \mathbb C$ definiendum $ x\mapsto x^x$
----------
Note
Representations
$x^x={\mathrm e}^{x\log(x)}=\left({\mathrm e}^x\right)^{\log(x)}$
"todo: write down the above with an expanded $\log$ to third order"
Because of this, the local minimum of $x^x$ is that of $x\log(x)$, namely $\frac{1}{\mathrm e}\approx 0.37$, and then see
Secretary problem (Wikipedia)
Furthermore
$x^x = \sum_{n=0}^\infty \prod_{k=1}^n (1-x)\lef…text/html2019-10-06T23:37:09+0100nikolaj (nikolaj@undisclosed.example.com)Holomorphic function - [Discussion]
https://axiomsofchoice.org/holomorphic_function?rev=1570397829&do=diff
Holomorphic function
Set
context $\mathcal O\subset \mathbb C$ definiendum $f\in \mathrm{it}$ inclusion $f:\mathcal O\to\mathbb C$ for all $z_0\in\mathcal O$ postulate $\left(\lim_{z \to z_0} {f(z) - f(z_0) \over z - z_0 }\right)\in\mathbb C $
Discussion
The following discussion is an elaboration/derivation on holomorphic functions from the viewpoint of analysis on $\mathbb R^n$. The article $\mathbb R^n, n>1$$a+i\,b\in\mathbb C$$\langle a,b\rangle\in\mathbb R^…text/html2019-10-06T17:26:04+0100nikolaj (nikolaj@undisclosed.example.com)Exponential function - typo
https://axiomsofchoice.org/exponential_function?rev=1570375564&do=diff
Exponential function
Function
definition $\exp: \mathbb C\to\mathbb C$ definition $\exp(z):=\sum_{k=0}^\infty \frac{1}{k!} z^k $
----------
Discussion
Theorems
$\mathrm{e}^z = \exp(z) $
Because per definition $\mathrm{e}^z:=\exp(z\cdot \mathrm{ln}(\mathrm{e}))$.
$\mathrm{e}^z \neq 0 $ $\frac{\mathrm d}{\mathrm d z}\mathrm{e}^{f(z)} = \frac{\mathrm d}{\mathrm dz}f(z)\cdot \mathrm{e}^{f(z)} $
$a,b,r,\theta\in\mathbb R$
$\exp(i\theta)=\cos(\theta)+i\sin(\theta)$ $\forall a,b.\ …text/html2019-09-28T18:11:56+0100nikolaj (nikolaj@undisclosed.example.com)Dependent product functor
https://axiomsofchoice.org/dependent_product_functor?rev=1569687116&do=diff
Dependent product functor
Functor
context ${\bf C}$ ... Cartesian closed category with all limits let $\prod_f$ ... right adjoint to the pullback functor $f^*$, where $f$ is an arrow in ${\bf C}$ context $!_X:X\to *$ ... terminal morphisms for $X\in{\bf C}$ definition $\prod_X p := {\mathrm{dom}}\prod_{!_X} p$
"todo: fmap of $\prod_X$$f^*$${\bf C}/Y$${\bf C}/X$$Y=*$$\prod_{!_X} p$${{\bf C}/*}$$\prod_X p$${\bf C}$${{\bf C}/*}$${\bf C}$$!_B:B\to{*}$$B$${\mathrm{…text/html2019-09-27T00:19:00+0100nikolaj (nikolaj@undisclosed.example.com)Magic Gaussian integral
https://axiomsofchoice.org/magic_gaussian_integral?rev=1569536340&do=diff
Magic Gaussian integral
Partial function
Discussion
The $\varepsilon$ prescription in the definition is just there so that one can evaluate the integral for certain complex matrices $A$ where it wouldn't exist otherwise. For example if $A$ has imaginary eigenvalues, then the naive integral will not be finite, while if we use $A_\varepsilon:=A-\varepsilon\,\mathrm{1}$$\mathrm{e}^{-\varepsilon\,\left\langle\phi\left|\right.\phi\right\rangle}$$I_a:=\int_{-\infty}^\infty{\mathrm e}^{-\tfrac{1}{2}…text/html2019-09-23T21:19:31+0100nikolaj (nikolaj@undisclosed.example.com)Infinite geometric series
https://axiomsofchoice.org/infinite_geometric_series?rev=1569266371&do=diff
Infinite geometric series
Function
definition $Q_\infty: \{z\in{\mathbb C}\mid \vert{z}\vert<1\}\to\mathbb C$ definition $Q_\infty(z):=\sum_{k=0}^\infty z^k $
----------
$Q_\infty(z)=\dfrac{1}{1-z}$
This can also be written as
$\sum_{k=0}^\infty\left(\dfrac{1}{1+z}\right)^k = 1+\dfrac{1}{z}$
and
$\sum_{k=0}^\infty\left(1-\dfrac{1}{z}\right)^k = z$
or, for $z>0$ and $X<1+z$ resp. $X<z/(z-1)$
$\sum_{k=0}^\infty\left(\dfrac{1}{1+z}\right)^kX^k = 1+\dfrac{1}{z}+(X-1)(z-1) \,z\dfra…text/html2019-09-03T15:33:33+0100nikolaj (nikolaj@undisclosed.example.com)Natural logarithm of real numbers
https://axiomsofchoice.org/natural_logarithm_of_real_numbers?rev=1567517613&do=diff
Natural logarithm of real numbers
Function
definiendum $\mathrm{ln}:\mathbb R_+^*\to \mathbb R$ postulate $\mathrm{ln}=\mathrm{exp}^{-1}$
----------
$\int_1^y \frac {1 } {x} {\mathrm d}x = \ln(y) $
$\int_0^{y} \frac {1 } {1+x } {\mathrm d}x = \ln(1+y) $
Log[a] == Log[b] + Integrate[1/(t+b)-1/(t+a),{t,0,Infinity}]
The function $x\mapsto\frac{x}{x-1}\log(x)$ is one without bad behaviours (singularities) on $[0,\infty)$.
----------
Subset of
Real logarithm,text/html2019-08-25T13:48:39+0100nikolaj (nikolaj@undisclosed.example.com)Diagonal construction - fix $
https://axiomsofchoice.org/diagonal_construction?rev=1566733719&do=diff
Diagonal construction
Set
context $ f:C\to \mathcal {\mathcal P}(C) $ definiendum $ x\in D_f $ inclusion $ D_f \subseteq C $ postulate $ x \notin f(x) $
Discussion
We take an arbitrary set $C$ and argue about all the functions $f:C\to \mathcal {\mathcal P}(C)$ from $C$ to the powerset ${\mathcal P}(C)$.
For any such $f$, we define $ D_f$ as the subset of $C$ containing the elements $x\in C$$ x \notin f(x) $$C=\{0,1\}$${\mathcal P}(C)=\{\{\},\{0\},\{1\},\{0,1\}\}$$f$$0…text/html2018-03-16T22:57:04+0100nikolaj (nikolaj@undisclosed.example.com)avastuttgart.png - created
https://axiomsofchoice.org/?image=avastuttgart.png&ns=&rev=1521237424&tab_details=history&mediado=diff&do=media
<img src="https://axiomsofchoice.org/_media/avastuttgart.png?w=475&h=475&t=1521237424&tok=4a39a1" alt="avastuttgart.png" />text/html2017-05-19T22:28:07+0100nikolaj (nikolaj@undisclosed.example.com)admin:logo.png
https://axiomsofchoice.org/?image=admin%3Alogo.png&ns=admin&rev=1495225687&tab_details=history&mediado=diff&do=media
<img src="https://axiomsofchoice.org/_media/admin/logo.png?w=128&h=128&t=1495225687&tok=7ca79a" alt="admin:logo.png" />text/html2017-05-01T22:13:11+0100nikolaj (nikolaj@undisclosed.example.com)nikolaj_kuntner_ist_1000x661.png - created
https://axiomsofchoice.org/?image=nikolaj_kuntner_ist_1000x661.png&ns=&rev=1493669591&tab_details=history&mediado=diff&do=media
<img src="https://axiomsofchoice.org/_media/nikolaj_kuntner_ist_1000x661.png?w=330&h=500&t=1493669591&tok=36a2ab" alt="nikolaj_kuntner_ist_1000x661.png" />text/html2016-11-11T23:03:08+0100nikolaj (nikolaj@undisclosed.example.com)important_distributions.jpg - created
https://axiomsofchoice.org/?image=important_distributions.jpg&ns=&rev=1478901788&tab_details=history&mediado=diff&do=media
<img src="https://axiomsofchoice.org/_media/important_distributions.jpg?w=500&h=500&t=1478901788&tok=5b037d" alt="important_distributions.jpg" />text/html2016-05-21T15:13:43+0100nikolaj (nikolaj@undisclosed.example.com)category_theory_a_triangle_pullback.jpg - created
https://axiomsofchoice.org/?image=category_theory_a_triangle_pullback.jpg&ns=&rev=1463836423&tab_details=history&mediado=diff&do=media
<img src="https://axiomsofchoice.org/_media/category_theory_a_triangle_pullback.jpg?w=487&h=500&t=1463836423&tok=265661" alt="category_theory_a_triangle_pullback.jpg" />text/html2016-05-21T14:38:47+0100nikolaj (nikolaj@undisclosed.example.com)category_theory_equalizer_diagram.png - created
https://axiomsofchoice.org/?image=category_theory_equalizer_diagram.png&ns=&rev=1463834327&tab_details=history&mediado=diff&do=media
<img src="https://axiomsofchoice.org/_media/category_theory_equalizer_diagram.png?w=500&h=332&t=1463834327&tok=a055b9" alt="category_theory_equalizer_diagram.png" />text/html2016-01-23T14:05:25+0100nikolaj (nikolaj@undisclosed.example.com)smug_reader.jpg - created
https://axiomsofchoice.org/?image=smug_reader.jpg&ns=&rev=1453554325&tab_details=history&mediado=diff&do=media
<img src="https://axiomsofchoice.org/_media/smug_reader.jpg?w=172&h=292&t=1453554325&tok=97e1b5" alt="smug_reader.jpg" />text/html2016-01-18T21:59:34+0100nikolaj (nikolaj@undisclosed.example.com)matura_im_mai.jpg - created
https://axiomsofchoice.org/?image=matura_im_mai.jpg&ns=&rev=1453150774&tab_details=history&mediado=diff&do=media
<img src="https://axiomsofchoice.org/_media/matura_im_mai.jpg?w=375&h=500&t=1453150774&tok=f75d3a" alt="matura_im_mai.jpg" />text/html2015-12-22T22:39:00+0100nikolaj (nikolaj@undisclosed.example.com)future.jpg - created
https://axiomsofchoice.org/?image=future.jpg&ns=&rev=1450820340&tab_details=history&mediado=diff&do=media
<img src="https://axiomsofchoice.org/_media/future.jpg?w=375&h=500&t=1450820340&tok=0a9f2c" alt="future.jpg" />