https://axiomsofchoice.org/ 2026-04-15T23:04:06+0200 https://axiomsofchoice.org/ https://axiomsofchoice.org/lib/tpl/vector/images/favicon.ico text/html 2025-12-23T03:33:58+0200 nikolaj (nikolaj@undisclosed.example.com) https://axiomsofchoice.org/nikolajs_notebook?rev=1766457238&do=diff Nikolaj-K's notebook This wiki is inactive since 2016 or so, but here I kept 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 scienc… text/html 2020-08-23T16:33:26+0200 nikolaj (nikolaj@undisclosed.example.com) 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 matrix text/html 2020-05-20T14:23:48+0200 nikolaj (nikolaj@undisclosed.example.com) 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$$1\equiv {\mathrm{succ}}\… text/html 2020-01-10T23:08:53+0200 nikolaj (nikolaj@undisclosed.example.com) https://axiomsofchoice.org/x_x?rev=1578694133&do=diff x^x Function definiendum $ \zeta: \mathbb{C}\setminus\{???\} \to \mathbb C$ definiendum $ x\mapsto x^x$ ---------- Note Representations $x^x={\mathrm e}^{x\log(x)}=\left({\mathrm e}^x\right)^{\log(x)}$ "todo: write down the above with an expanded $\log$ to third order" Because of this, the local minimum of $x^x$ is that of $x\log(x)$, namely $\frac{1}{\mathrm e}\approx 0.37$, and then see Secretary problem (Wikipedia) Furthermore $x^x = \sum_{n=0}^\infty \prod_{k=1}^n (1-x)\lef… text/html 2019-10-06T23:37:09+0200 nikolaj (nikolaj@undisclosed.example.com) https://axiomsofchoice.org/holomorphic_function?rev=1570397829&do=diff Holomorphic function Set context $\mathcal O\subset \mathbb C$ definiendum $f\in \mathrm{it}$ inclusion $f:\mathcal O\to\mathbb C$ for all $z_0\in\mathcal O$ postulate $\left(\lim_{z \to z_0} {f(z) - f(z_0) \over z - z_0 }\right)\in\mathbb C $ Discussion The following discussion is an elaboration/derivation on holomorphic functions from the viewpoint of analysis on $\mathbb R^n$. The article $\mathbb R^n, n>1$$a+i\,b\in\mathbb C$$\langle a,b\rangle\in\mathbb R^… text/html 2019-10-06T17:26:04+0200 nikolaj (nikolaj@undisclosed.example.com) https://axiomsofchoice.org/exponential_function?rev=1570375564&do=diff Exponential function Function definition $\exp: \mathbb C\to\mathbb C$ definition $\exp(z):=\sum_{k=0}^\infty \frac{1}{k!} z^k $ ---------- Discussion Theorems $\mathrm{e}^z = \exp(z) $ Because per definition $\mathrm{e}^z:=\exp(z\cdot \mathrm{ln}(\mathrm{e}))$. $\mathrm{e}^z \neq 0 $ $\frac{\mathrm d}{\mathrm d z}\mathrm{e}^{f(z)} = \frac{\mathrm d}{\mathrm dz}f(z)\cdot \mathrm{e}^{f(z)} $ $a,b,r,\theta\in\mathbb R$ $\exp(i\theta)=\cos(\theta)+i\sin(\theta)$ $\forall a,b.\ … text/html 2019-09-28T18:11:56+0200 nikolaj (nikolaj@undisclosed.example.com) https://axiomsofchoice.org/dependent_product_functor?rev=1569687116&do=diff Dependent product functor Functor context ${\bf C}$ ... Cartesian closed category with all limits let $\prod_f$ ... right adjoint to the pullback functor $f^*$, where $f$ is an arrow in ${\bf C}$ context $!_X:X\to *$ ... terminal morphisms for $X\in{\bf C}$ definition $\prod_X p := {\mathrm{dom}}\prod_{!_X} p$ "todo: fmap of $\prod_X$$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/html 2019-09-27T00:19:00+0200 nikolaj (nikolaj@undisclosed.example.com) 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/html 2019-09-23T21:19:31+0200 nikolaj (nikolaj@undisclosed.example.com) 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/html 2019-09-03T15:33:33+0200 nikolaj (nikolaj@undisclosed.example.com) 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/html 2019-08-25T13:48:39+0200 nikolaj (nikolaj@undisclosed.example.com) 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/html 2018-03-16T22:57:04+0200 nikolaj (nikolaj@undisclosed.example.com) 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&amp;tok=4a39a1" alt="avastuttgart.png" /> text/html 2017-05-19T22:28:07+0200 nikolaj (nikolaj@undisclosed.example.com) 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&amp;tok=7ca79a" alt="admin:logo.png" /> text/html 2017-05-01T22:13:11+0200 nikolaj (nikolaj@undisclosed.example.com) 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&amp;tok=36a2ab" alt="nikolaj_kuntner_ist_1000x661.png" /> text/html 2016-11-11T23:03:08+0200 nikolaj (nikolaj@undisclosed.example.com) 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&amp;tok=5b037d" alt="important_distributions.jpg" /> text/html 2016-05-21T15:13:43+0200 nikolaj (nikolaj@undisclosed.example.com) 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&amp;tok=265661" alt="category_theory_a_triangle_pullback.jpg" /> text/html 2016-05-21T14:38:47+0200 nikolaj (nikolaj@undisclosed.example.com) 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&amp;tok=a055b9" alt="category_theory_equalizer_diagram.png" /> text/html 2016-01-23T14:05:25+0200 nikolaj (nikolaj@undisclosed.example.com) 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&amp;tok=97e1b5" alt="smug_reader.jpg" /> text/html 2016-01-18T21:59:34+0200 nikolaj (nikolaj@undisclosed.example.com) 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&amp;tok=f75d3a" alt="matura_im_mai.jpg" /> text/html 2015-12-22T22:39:00+0200 nikolaj (nikolaj@undisclosed.example.com) 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&amp;tok=0a9f2c" alt="future.jpg" />