https://axiomsofchoice.org/ 2026-05-08T13:11:41+0200 https://axiomsofchoice.org/ https://axiomsofchoice.org/lib/tpl/vector/images/favicon.ico text/html 2015-01-15T10:09:00+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/%CE%B5-%CE%B4_function_limit?rev=1421312940&do=diff ε-δ function limit Set "todo: Rather define the set of all sequences which have a limit in the below sense this is the domain of “lim” though of as function (on all sequences it would be a partial function)" context $\langle X,d_X\rangle$ ... metric space $\langle Y,d_Y\rangle$$f:X\to Y$$\xi\in X$$\mathrm{lim}_{x\to \xi}\ f(x)\equiv y_\xi$$\varepsilon,\delta\in \mathbb R_+^*$$\forall\varepsilon.\ \exists \delta.\ \forall x.\ [\ 0<d_X(x,\xi)<\delta\ ] \Rightarrow [\ d_Y(f(x),y_\xi)<\v… text/html 2016-06-20T18:46:46+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/%CF%83-algebra?rev=1466441206&do=diff σ-algebra Set context $X$ definiendum $\Sigma$ in it postulate $\Sigma\subseteq \mathcal P(X)$ postulate $ \Sigma\ne\emptyset $ forall $E\in\Sigma$ postulate $ X\smallsetminus E \in \Sigma $ forall $A\in\mathrm{Sequence}(\Sigma)$ forall $n\in \mathbb N$ postulate $ \bigcup_{i=1}^n A_i \in \Sigma $ Ramifications Reference Wikipedia: Sigma-algebra Parents Subset of text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/%E2%84%9A_valued_function?rev=1395396676&do=diff ℚ valued function Function context $ X $ definiendum $ \mathrm{it}\equiv X \to \mathbb Q $ Discussion Parents Subset of ℝ valued function Requirements Integer text/html 2014-12-08T13:39:31+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/%E2%84%9D_valued_function?rev=1418042371&do=diff ℝ valued function Function context $ X $ ... set definiendum $ \mathrm{it}\equiv X \to \mathbb R $ Discussion Parents Subset of ℂ valued function Requirements Real number text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/%E2%84%82_valued_function?rev=1395396676&do=diff ℂ valued function Function context $ X $ definiendum $ \mathrm{it}\equiv X \to \mathbb C $ Discussion Parents Subset of Function Context Complex number text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/%E2%84%92%E1%B5%96_space?rev=1395396676&do=diff ℒᵖ space Set context $ p\in [1,\infty) $ context $ \mathbb K = \mathbb C \lor \mathbb R $ context $ \langle X,\Sigma,\mu\rangle $ ... measure space definiendum $f\in\mathcal L^p(X,\mu)$ postulate $f:X\to \mathbb K $ postulate $\left(\int_X\ |f|^p\ \text d\mu\right)^\frac{1}{p}$ ... finite Discussion Trivial remark: As explained in the notation section of the entry relation concatenation, the symbol $|f|^p$ denotes the function obtained by concatenation … text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/%E2%84%95_valued_function?rev=1395396676&do=diff ℕ valued function Function context $ X $ definiendum $ \mathrm{it}\equiv X \to \mathbb N $ Discussion Parents Subset of ℤ valued function Requirements Natural number text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/%E2%84%A4_valued_function?rev=1395396676&do=diff ℤ valued function Function context $ X $ definiendum $ \mathrm{it}\equiv X \to \mathbb Z $ Discussion Parents Subset of ℚ valued function Requirements Integer text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/2-regular_graph?rev=1395396676&do=diff 2-regular graph Set context $V,E$ ... set definiendum $ \langle V,E,\psi\rangle \in \mathrm{it}(E,V) $ inclusion $ \langle V,E,\psi\rangle $ ... undirected graph for all $ v\in V $ postulate $ d(v)=2 $ Discussion A finite 2-regular graph consists disconnected cycles. A general 2-regular graph consists disconnected cycles or infinite chains. text/html 2016-05-21T18:25:54+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/a_triangle_limit?rev=1463847954&do=diff A triangle limit Collection Consider the circle graph with 3 vertices $a,b,c$ and 3 edges. There are, up to relabeling, two directed versions of it: * The graph where all arrows go in one direction, say $h:a\to c$ $g:c\to b$ $f:b\to a$ * The graph where one of the arrows point in another direction, say$h:a\to c$$g:c\to b$$f:a\to b$$f$$g$$b$$h$$h:c\simeq a$$c$$a$${\bf{Set}}$$h$$e=\{x\in a\ |\ f(x)=g(x)\}$$h$$a\times_b c$$a\times c$$e=\{\langle x,y\rangle\in a\times_b c\,\mid\,h(y)=… text/html 2016-10-16T14:59:47+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/a_type_system_as_a_model_for_some_concepts?rev=1476622787&do=diff A type system as a model for some concepts Drawing arrows and coding functions $\blacktriangleright$ A type system as a model for some concepts $\blacktriangleright$ Foundational temp1 Guide We introduce some basic notions of Idris, namely those that can be seen as “semantics” for logic, set theory and category theory (or rather topoi, as discussed in $f:X\to Y$$i:S\to{X}$$j:X\to{T}$$(f\circ{i}):S\to Y$$(j\circ{f}):X\to T$$g$ text/html 2017-04-22T22:48:44+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/abelian_group?rev=1492894124&do=diff Abelian group Set context $\langle X,* \rangle \in \mathrm{Group}(X)$ definiendum $\langle X,* \rangle \in \text{it}$ for all $a,b\in X$ postulate $a*b=b*a$ Discussion One generally calls $X$ the group, i.e. the set where the operation “$+$” is defined on. An abelian group is also a module over the ring of integers. Reference text/html 2015-02-02T19:08:47+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/abelian_monoid?rev=1422900527&do=diff Abelian monoid Set context $ \langle M,* \rangle \in \mathrm{Monoid}(M)$ definiendum $ \langle M,* \rangle \in \text{it}$ for all $a,b\in X$ postulate $a*b=b*a$ ---------- Reference Wikipedia: Monoid ---------- Subset of Monoid text/html 2016-07-05T20:38:40+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/about?rev=1467743920&do=diff About About $\blacktriangleright$ Specifying syntax An apple pie from scratch Perspective Meta This entry explains the motivation and content for this website, the notation for the wiki and the structure of the graph. All of the content can viewed via an interactive graph, which one can also access from each entry by clicking the yellow lemon in the upper right corner. $+$$\langle a,b,c\rangle$$\langle \langle a,b\rangle,c\rangle$$\langle a,\langle b,c\rangle\rangle$$\langle a,b\rangle$$a$… text/html 2016-12-10T13:16:43+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/accelerated_cpp?rev=1481372203&do=diff Accelerated Cpp Book ### The 3. Edition * 647 pages total * has exercises * Had 4 parts with 17 chapters total: * 2-4 Basics (Math Foundations) * 5-6 Localization (robot Motion?) * 9-13 Mapping (world representation?) * 14-17 Planing and Control (future actions?)$bel_0(\neg faulty)=\frac{9}{10}$ text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/adjacency_list?rev=1395396676&do=diff Adjacency list Set context $V$ ... countable set definiendum $ \phi\in\mathrm{it} $ postulate $ \mathrm{dom}\ \phi = V $ for all $ v,u\in V $ postulate $ \phi(v)\subseteq V $ postulate $ u\in\phi(v)\implies v\in\phi(u) $ Discussion The value $\phi(v)$ denotes the set of vertices which are connected to $v$. The adjacency lists describe simple graph. Parents text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/adjacency_matrix?rev=1395396676&do=diff Adjacency matrix Set context $n\in\mathbb N$ definiendum $ A \in \mathrm{it}(n) $ postulate $ A \in \mathrm{SquareMatrix}(n,\mathbb N) $ Discussion If the indices $i,j$ label two vertices of a finite undirected graph, then the value $A_{ij}$ determines the number of edges joining them. Theorems The number $(A^n)_{ij}$ is the number of paths from $v_i$ to $v_j$. And so, for example, $\frac{1}{2}\cdot\frac{1}{3}\cdot\mathrm{tr}\,A^3$ text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/algebra_over_a_commutative_ring?rev=1395396676&do=diff Algebra over a commutative ring Set context $A$...R-module postulate $\langle A,[\cdot,\cdot]\rangle \in \mathrm{algebra}(A)$ context $[\cdot,\cdot] : A\times A\to A$ $x,y,z\in A$ $r,s\in R$ Bilinearity: postulate $[r\cdot x+s\cdot y,z]=r\cdot [x,z]+s\cdot [y,z]$ postulate $[z,r\cdot x+s\cdot y]=r\cdot [z,x]+s\cdot [z,y]$ Discussion Reference Wikipedia: Algebra (Ring theory) Parents Refinement of Module text/html 2016-10-15T14:23:44+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/an_apple_pie_from_scratch?rev=1476534224&do=diff An apple pie from scratch An apple pie from scratch $\blacktriangleright$ Outline Guide $$ \require{AMScd} \begin{CD} {\large\hbar} @>{\large{!}}>> {\large{*}} \\ @V{{\large{m}}}VV @VV{{\large\top}}V \\ {\large\heartsuit} @>>{\large{\chi}}> {\large{\Omega}} \end{CD} $$ What? As a working physicist, one regularly learns new mathematical structures and then doesn't know how they fit into the bigger picture or how it's related/… text/html 2018-03-10T19:52:29+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/analytic_function?rev=1520707949&do=diff Analytic function Set context $\mathcal O\subset \mathbb C$ definiendum $f\in \mathrm{it}$ inclusion $f:\mathcal O\to\mathbb C$ for all $c$ ... series in $\mathbb C$ "todo, roughly $\exists c.\ \forall z.\ f(z)=\sum_{n=-\infty}^\infty c_n\,z^n$ " ---------- Discussion Picture a continuous function $f:\mathbb R^2\to\mathbb R$ as a surface given by $f(x,y)$ and imagine drawing a circle of radius $1$ around the point at the origin and is parametrized by $\langle \c… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/anti-symmetric_relation?rev=1395396676&do=diff Anti-symmetric relation Set context $X$ definiendum $ R\in\mathrm{SymmetricRel}(X) $ context $ R \in \mathrm{Rel}(X) $ $ x,y\in X $ postulate $ xRy \land yRx \implies x=y $ Discussion Reference Wikipedia: Anti-symmetric relation Parents Subset of Binary relation on a set text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/antisymmetric_multilinear_functional?rev=1395396676&do=diff Antisymmetric multilinear functional Set context $X$...$\mathcal F$-vector space context $n\in \mathbb N$ definiendum $M\in \mathrm{SymMultiLin}(X^n)$ context $M\in \mathrm{MultiLin}(X^n)$ $ v_1,\dots,v_n\in X $ $ 1\le i<j\le n $ postulate $ M(v_1,\dots,v_i,\dots,v_j,\dots,v_n)=-M(v_1,\dots,v_j,\dots,v_i,\dots,v_n) $ Discussion Parents Subset of Multilinear functional text/html 2015-03-18T11:23:34+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/aoc_edits?rev=1426674214&do=diff AoC edits Meta ---------- Stuck that's in the todo list for the AxiomsOfChoice site: entries --- Natural number $\to$ First infinite von Neumann ordinal --- ==== Discussion ==== $\to$ 5- --- ==== Parents ==== $\to$ 5- graph --- Sequel of (red) --- Exposition (light red) ---------- Related nikolajs notebook text/html 2014-08-25T00:06:04+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/applicative?rev=1408917964&do=diff Applicative Haskell class Definition -- definition class Functor f => Applicative f where pure :: a -> f a (<*>) :: f (a -> b) -> f a -> f b pure id <*> v $\ \leftrightsquigarrow\ $ v pure f <*> pure x $\ \leftrightsquigarrow\ $ pure (f x) u <*> pure y $\ \leftrightsquigarrow\ $ pure ($ y) <*> u u <*> (v <*> w) $\ \leftrightsquigarrow\ $ text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/arbitrary_intersection?rev=1395396676&do=diff Arbitrary intersection Set context $S$ definiendum $x\in \bigcap S$ $X\in S$ postulate $ x\in X $ Discussion This can be viewed as a version of intersection, which however requires one more quantifier. Reference Wikipedia: Arbitrary intersection Parents Context Intersection text/html 2014-04-03T22:41:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/arbitrary_union?rev=1396557676&do=diff Arbitrary union Set context $ S\in \mathfrak U $ definiendum $ x\in \bigcup S $ range $ X\in S $ postulate $ \exists X.\ x\in X $ Discussion Reference Wikipedia: Arbitrary union Parents Element of Set universe Context* Set universe text/html 2014-11-20T12:57:52+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/archimedes_constant_%CF%80?rev=1416484672&do=diff Archimedes constant π Set "todo Requirements" definiendum $\pi:=V_2(1)$ Discussion The constant $\pi$ is defined as the volume of the disc of radius $1$, where the underlying metric space is taken to be the two dimensional Euclidean space $\mathbb E^2$. * $\pi = 3.14159\dots\approx \frac{22}{7}$ Reference Wikipedia: text/html 2015-12-16T10:24:19+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/arcus_tangens?rev=1450257859&do=diff Arcus Tangens Function definiendum $\arctan: \{z\in\mathbb C\mid |z|\le 1, z\neq \pm i\}\to\mathbb C$ definiendum $\arctan(z):=\sum_{n=0}^\infty (-1)^n\frac{1}{2n+1} z^{2n+1} $ ---------- Theorems $\frac{{\mathrm d}}{{\mathrm d}z}\arctan(z)=\frac{1}{1+z^2}$ References Wikipedia: Inverse trigonometric functions ---------- Related Exponential function text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/arithmetic_structure_of_complex_numbers?rev=1395396676&do=diff Arithmetic structure of complex numbers Set definiendum $\langle \mathbb C,+_\mathbb{C},\cdot_\mathbb{C} \rangle$ postulate $(a+ib)+_\mathbb{C}(c+id)=(a+_\mathbb{R}c)+i(b+_\mathbb{R}d)$ postulate $(a+ib)\cdot_\mathbb{C}(c+id)=(a\cdot_\mathbb{R} c-_\mathbb{R}b\cdot_\mathbb{R} d)+i(a\cdot_\mathbb{R} d +_\mathbb{R}b\cdot_\mathbb{R} c)$ As defined in complex number, the pattern with $x+iy$ denotes $\langle x,y\rangle$ with $x,y\in \mathbb R$. The operations $+_\mathbb{R}$ and $\cdo… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/arithmetic_structure_of_integers?rev=1395396676&do=diff Arithmetic structure of integers Set definiendum $\langle \mathbb Z,+_\mathbb{Z},\cdot_\mathbb{Z} \rangle$ postulate $[\langle a,b\rangle]+_\mathbb{Z}[\langle m,n\rangle]=[\langle a+m,b+n\rangle]$ postulate $[\langle a,b\rangle]\cdot_\mathbb{Z}[\langle m,n\rangle]=[\langle a\ m+b\ n,a\ n+b\ m\rangle]$ The operations $+$ and $\cdot$ on the right hand sides are these of arithmetic structure of natural numbers. Discussion We'll generally use the notation introduced in integer. W… text/html 2014-04-01T14:18:29+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/arithmetic_structure_of_natural_numbers?rev=1396354709&do=diff Arithmetic structure of natural numbers Set definiendum $\langle \mathbb N,+,\cdot \rangle$ $ m=S(k) $ postulate $n + 0 = n$ postulate $n + m = S(n) + k$ postulate $n \cdot 0 = 0$ postulate $n \cdot m = n + (n \cdot k) $ Discussion "todo: rewrite the defintion in my current notation" We'll often omit the multiplication sign. Reference text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/arithmetic_structure_of_rational_numbers?rev=1395396676&do=diff Arithmetic structure of rational numbers Set definiendum $\langle \mathbb Q,+_\mathbb{Q},\cdot_\mathbb{Q} \rangle$ postulate $[\langle a,b\rangle]+_\mathbb{Q}[\langle m,n\rangle]=[\langle a\ n+b\ m,b\ n\rangle]$ postulate $[\langle a,b\rangle]\cdot_\mathbb{Q}[\langle m,n\rangle]=[\langle a\ m,b\ n\rangle]$ The operations $+$ and $\cdot$ on the right hand sides are these of arithmetic structure of integers. Discussion We'll generally use the notation introduced in integer as w… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/arithmetic_structure_of_real_numbers?rev=1395396676&do=diff Arithmetic structure of real numbers Set definiendum $\langle \mathbb R,+_\mathbb{R},\cdot_\mathbb{R} \rangle$ postulate $ r +_\mathbb{R} s = \{q+_\mathbb{Q}p\ |\ (q\in r)\land (p\in s)\} $ postulate $ r -_\mathbb{R} s = \{q-_\mathbb{Q}p\ |\ (q\in r)\land (p\in \mathbb Q\setminus s)\} $ postulate $ -_{\mathbb R}r = \{q-_\mathbb{Q}p\ |\ (p\in \mathbb Q\setminus r)\land (q<0)\} $ postulate $ r\ge 0\land s\ge 0\implies r\cdot_\mathbb{R}s = \{q\cdot_\mathbb{Q}p\ |\ (q\in… text/html 2015-12-16T21:06:11+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/arxiv_1512.04660?rev=1450296371&do=diff arxiv_1512.04660 Paper <http://arxiv.org/abs/1512.04660> ?? - electromechanical vacuum coupling rate References ArXiv: <http://arxiv.org/abs/1512.04660> Wikipdia: Silicon nitride ---------- Related Todo papers On electronics . note text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/associative_algebra?rev=1395396676&do=diff Associative algebra Set context $A$...R-module definiendum $\langle A,[\cdot,\cdot]\rangle \in \mathrm{AssociativeAlgebra}(A)$ postulate $\langle A,[\cdot,\cdot]\rangle$ ... algebra over $A$ postulate $[\cdot,\cdot]$ ... associative Discussion Reference Wikipedia: Associative algebra Parents Subset of Algebra over a commutative ring text/html 2014-12-04T15:57:40+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/atlas?rev=1417705060&do=diff Atlas Set context $\langle M,T\rangle$ ... second-countable Hausdorff space context $n\in \mathbb N$ definiendum $A\in$ it inclusion $A\subseteq$ chart $\left(\langle M,T\rangle,n\right)$ forall $x\in M$ exists $\langle U,\varphi\rangle\in A$ postulate $x\in U$ Discussion Idea An atlas is a set of charts, so that no point $x\in M$ is left out from being mapped to $\mathbb R^n$$M$$U$$\langle U,\varphi\rangle$$A$$\bigcup_{chart\in A}\pi_1(chart… text/html 2015-04-16T19:49:40+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/automorphism?rev=1429206580&do=diff Automorphism Collection context $A\in\mathrm{Ob}_{\bf C}$ definition $A\cong A$ ---------- Reference Wikipedia: Isomorphism, Automorphism ---------- Subset of Isomorphism text/html 2015-04-17T11:03:11+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/automorphism_group?rev=1429261391&do=diff Automorphism group Set context $\bf C$ ... concrete category context $X:\mathrm{Ob}_{\bf C}$ definiendum $\mathrm{Aut}(X)\equiv\langle\!\langle X\cong X,\circ\rangle\!\rangle$ ---------- Elaboration Obviously, $\circ$ denotes the concatentaion of arrows in $\bf C$ here. Theorems The automorphisms (which for concrete $\bf C$ can always be viewed as functions) equipped with $\circ$ text/html 2017-03-13T00:13:01+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/babbys_first_hott?rev=1489360381&do=diff Babbys first HoTT Note Curry-Howard correspondence and geometric semantics syntax computation logic geometry $P$ type proposition space $x$ with $x:P$ term $x$ of type $P$ argument $x$ for proposition $P$ point $x$ of $P$ $\bf{0}$ empty type false proposition empty set ${\bf 1}$${\bf 2}$$^\star$$^\star$$x:P$$Q(x)$$x$${\large{\prod}}_{x:P} Q(x)$$s$$x$$Q(x)$$\forall (x\in P).\,Q(x)$${\large{\sum}}_{x:P} Q(x)$$\langle x,y\rangle$$x:P$$y:Q(x)$$\exists (x\in P).\,Q(x)$$s:{\large{\pr… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/ball_volume?rev=1395396676&do=diff Ball volume Set context $ p\in \mathbb N $ definiendum $V_p:\mathbb R_+\to \mathbb R_+$ definiendum $V_p(r):=\beta^p(B_0(r))$ Discussion Theorems For all $a\in \mathbb R^p$, the volume of the ball $B_a(r)$ is the same and given by $V_p(r)= \pi^{p/2}\ \Gamma(p/2+1)^{-1}\ r^p $ Reference Wikipedia: Volume of an n-ball Parents Context Lebesgue-Borel measure, Open ball text/html 2015-02-03T10:21:46+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/banach_space?rev=1422955306&do=diff Banach space Set context $V$ ... normed $F$-vector space definiendum $\mathcal V \in \mathrm{it}$ forall $v\in \mathrm{CauchySeq}(V)$ postulate $\exists v_\infty.\,\mathrm{lim}_{n\to\infty}\Vert v_n-v_\infty \Vert = 0$ ---------- Elaboration For each Cauchy sequence $(v)_{i\in\mathbb N}$, there is a limit $v_\infty\in\mathcal V$ w.r.t. the natural norm. $\Longleftrightarrow$ The space $\mathcal V$ is complete. Reference Wikipedia: text/html 2014-10-29T21:09:12+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/base_for_a_topology?rev=1414613352&do=diff Base for a topology Set context $\langle X,T\rangle$ ... topological space definiendum $B\in$ it inclusion $B\subseteq T$ for all $U\in T$ exists $C\subseteq B$ postulate $U=\bigcup C$ Discussion A base $B$ for a topology is a collection its open sets, which suffice to cover any open set. text/html 2016-10-30T17:43:32+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/bayes_algorithm?rev=1477845812&do=diff Bayes algorithm Function context $K_u:(X\times X)\to {\mathbb R}$ context $W_z:X\to {\mathbb R}$ definition $\Gamma: (X\to {\mathbb R})\to X\to {\mathbb R}$ definition $bel_{\mathrm out}[bel_{\mathrm in}](x) := N^*W_z(x)\int_A K_u(x,x')\,bel_{\mathrm in}(x'){\mathrm d}x'$ "this is the algorithm for the case where all the ingredient have these types. In practice, Coming up with an initial $bel$ is a also part of the task. $N^*$ is supposed to be the normalization of t… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/bbgky_hierarchy?rev=1395396676&do=diff BBGKY hierarchy Theorem context $\langle \mathcal M,H\rangle $ ... classical Hamilonian system range $3N\equiv \text{dim}(\mathcal M)$ range $ {\bf q} \in \mathcal M $ range $ {\bf p} \in T^*\mathcal M $ $q^i,p_i$ denote tupples of three components. context $H({\bf q},{\bf p})=\sum_{i=1}^N \left(T(p_i)+\Phi_\text{ext}(q^i)+\sum_{j<i}\Phi_\text{int}(|q^i-q^j|)\right)$ $s\in\text{range}(N)$ context $f_s$ ... symmetrized reduced distribution … text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/bernoulli_number?rev=1395396676&do=diff Bernoulli numbers Function definiendum $B:\mathbb N\to\mathbb Q$ for all $z\in \mathbb C$ postulate $\frac{z}{\mathrm{e}^z-1}=\sum_{k=0}^\infty B_k\frac{1}{k!}z^k$ Discussion Examples $B_0=1$ $B_1=-\tfrac{1}{2}$ $B_2=\frac{1}{6}$ $B_4=-\frac{1}{30}$ $B_6=-\frac{1}{42}$ Parents Subset of Infinite series, ℚ valued function text/html 2016-04-13T15:55:28+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/bessel_function?rev=1460555728&do=diff Bessel function Function definition $J_{}: ?\to ?$ definition $J_\alpha(x) := \sum_{m=0}^\infty \dfrac{1}{\Gamma(m+0+1)\,\Gamma(m+\alpha+1)} (-1)^m{\left(\dfrac{x}{2}\right)}^{2m+\alpha}$ ---------- Discussion The Bessel functions are basically the angle part of a fouriertransform of radial functions in ${\mathbb R}^n$, $\int_\text{angles}{\mathrm e}^{i\langle k,x\rangle}$. They solve $x^2 \dfrac{d^2 y}{dx^2} + x \dfrac{dy}{dx} + (x^2 - \alpha^2)y = 0$ Theorems $J_n (x) = \… text/html 2014-12-05T11:28:29+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/bijective_function?rev=1417775309&do=diff Bijective function Set context $X,Y$ ... set definiendum $ f\in \mathrm{Bijective}(X,Y) $ inclusion $ f\in \mathrm{Injective}(X,Y) $ inclusion $ f\in \mathrm{Surjective}(X,Y) $ Discussion Predicates predicate $X\approx Y\equiv \mathrm{Bijective}(X,Y)\ne\emptyset$ We also write $X$ equinumerous $Y$. predicate $X\preccurlyeq Y \equiv \exists (X'\subseteq Y).\ X'\approx X$ We also write $X$ smaller $Y$. predicate $Y$ ... countably infinite $\equiv \math… text/html 2015-02-02T19:09:21+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/binary_function?rev=1422900561&do=diff Binary function Set context $X,Y$ definiendum $ f\in \text{it}(X,Y) $ postulate $ f:X\times X\to Y $ ---------- ---------- Subset of Function text/html 2015-04-17T15:16:37+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/binary_operation?rev=1429276597&do=diff Binary operation Set context $X$ definition it $\equiv X\times X\to X$ ---------- Discussion Also called “law of composition”. It's is a binary function with which takes elements of $X$ to an element of $X$ itself. Reference Wikipedia: Binary operation ---------- Subset of text/html 2014-04-01T16:26:00+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/binary_relation?rev=1396362360&do=diff Binary relation Set context $X,Y$ definiendum $ \text{Rel}(X,Y) \equiv \mathcal{P}(X\times Y) $ Discussion The set of all binary relations on $X\times Y$. If $R$ is a binary relation, we write $x R y\equiv \langle x,y\rangle \in R$ Reference Wikipedia: Binary relation Parents Requirements Cartesian product, Power set text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/binary_relation_on_a_set?rev=1395396676&do=diff Binary relation on a set Set context $X$ definiendum $ \text{Rel}(X) \equiv \text{Rel}(X,X) $ Discussion Reference Wikipedia: Relation Parents Subset of Binary relation text/html 2016-08-21T11:33:20+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/binomial_coefficient_over_the_complex_numbers?rev=1471772000&do=diff Binomial coefficient over the complex numbers Function definition $??$ definition ${x\choose{y}} := \dfrac{1}{\Gamma(y+1)}\dfrac{\Gamma(x+1)}{\Gamma((x+1)-(y+1)+1)}$ ---------- $(1+z)^s$ Sum[Gamma[1 + x]/(Gamma[1 + x - y] Gamma[1 + y])z^y, {y, 0, \[Infinity]}] Discussion For natural numbers $n\ge{k}$ we get ${n\choose{k}} = \dfrac{1}{k!}\dfrac{n!}{(n-k)!}$ Theorems ${m\choose{m}} = 1$ ${m\choose{0}} = 1$ Pascal's identity (recursive formula) ${n\choose{k}} = {n\choose{k-1}} … text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/bipartite_adjacency_matrix?rev=1395396676&do=diff Bipartite adjacency matrix Set context $n_X,n_Y\in\mathbb N$ definiendum $ B \in \mathrm{it}(n_X,n_Y) $ postulate $ B \in \mathrm{Matrix}(n_X,n_Y,\mathbb N) $ Discussion If the indices $i,j$ label two vertices belonging to the partitions $X,Y$ of a finite bipartite graph, respectively, then the value $B_{ij}$ determines the number of edges joining them. Parents text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/bipartite_complete_graph?rev=1395396676&do=diff Bipartite complete graph Set context $V,E$ ... set definiendum $\langle V,E,\psi\rangle \in \mathrm{it}(E,V) $ postulate $\langle V,E,\psi\rangle $ ... undirected graph range $ X\cap Y=\emptyset $ range $ x\in X $ range $ y\in Y $ postulate $\exists X,Y.\ (\forall u,v.\ \{u,v\}\in\mathrm{im}(\psi)\implies (u\in X\land v\in Y)\lor (v\in X\land u\in Y)) \land (\forall x,y.\ \{x,y\}\in\mathrm{im}\ \psi) $ Discussion Let $G$ be a bipartite com… text/html 2014-04-06T14:51:45+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/bipartite_graph?rev=1396788705&do=diff Bipartite graph Set context $V$ ... set definiendum $\langle V,E\rangle \in \mathrm{it}(E,V) $ postulate $ \langle V,E\rangle $ ... undirected graph range $ X\cup Y=V $ range $ X\cap Y=\emptyset $ range $ v,w\in V $ postulate $\exists X,Y.\ \forall u,v.\ \{u,v\}\in E\implies \neg(u\in X\land v\in X)\land \neg(v\in Y\land u\in Y) $ Discussion We denote the graph $G$ with bipartition $X,Y$ by $G[X,Y]$ and call $X$ and $Y$ its parts.$G[X,Y]$$… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/bit_string?rev=1395396676&do=diff Bit string Set context $ T:\mathbb N\to\mathbb N $ definiendum $ \mathrm{it}\equiv\{0,1\}^* $ Discussion Parents Requirements Kleene star, Natural number text/html 2015-06-15T17:48:25+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/boltzmann_equation?rev=1434383305&do=diff Boltzmann equation Set context $ {\bf K}:\mathbb R^3\times\mathbb R^3\times\mathbb t\to\mathbb R $ range $ :: {\bf K}({\bf x},{\bf v},t)$ context $ S\in\mathbb N $ $i,j\in\text{range}(S)$ context $ m_i\in \mathbb R^* $ context $ I_{ij}: \mathbb R^3\times[-\tfrac{\pi}{2},\tfrac{\pi}{2}]\times[0,2\pi]\to\mathbb R $ range $ :: I_{ij}({\bf v},\vartheta,\varphi) $ The integer $S$ denote the species, $m_i$ their respective masses and $I_{ij}$ the … text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/borel_algebra?rev=1395396676&do=diff Borel algebra Set context $\langle X,\mathcal T\rangle \in \mathrm{TopSpace}(X) $ definiendum $\mathcal B(X):\equiv \sigma(\mathcal T)$ Discussion The Borel algebra of $X$ is the smallest σ-algebra formed from its open sets. Reference Wikipedia: Borel set Parents Context Topological space, Smallest generated σ-algebra text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/borel_subsets_of_the_reals?rev=1395396676&do=diff Borel subsets of the reals Set context $p\in \mathbb N$ definiendum $\mathcal B^p\equiv \mathcal B(\mathbb R^p)$ Where the corresponding σ-algebra is choosen to be the Euclidean topology. Discussion We also set $\overline{\mathcal B}^p\equiv \mathcal B(\overline{\mathbb R}^p)$, see Euclidean topology. Reference Wikipedia: Borel set Parents Subset of Euclidean topology Context Borel algebra text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/bounded_function?rev=1395396676&do=diff Bounded function Set context $ \langle X,d\rangle $ ... metric space definiendum $f\in \text{BoundedFunction}(X)$ range $x,a\in X$ range $M\in \mathbb R$ postulate $ \exists a, M.\ \forall x.\ d(f(x),a) < M$ Discussion For linear operators between normed vector spaces, being bounded for each input element and continuous coincide. text/html 2016-07-31T12:42:26+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/bounded_linear_operator?rev=1469961746&do=diff Bounded linear operator Set context $V,W$ ... normed vector spaces definiendum $A\in\mathrm{BoundedLinOp}(V,W)$ postulate $A\in\mathrm{Hom}(V,W)$ range $M\in\mathbb R, M>0$ $v\in V$ postulate $\exists M.\ \Vert Av\Vert_W\le M\Vert v\Vert_V $ ---------- Discussion A linear operator on a metrizable vector space is bounded if and only if it is continuous. text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/caloric_equation_of_state?rev=1395396676&do=diff Caloric equation of state Theorem context $ \Omega(\beta,\mu) $ ... grand potential postulate $ U = \frac{\partial}{\partial\beta}(\beta\ \Omega) + \mu\ \langle\hat N\rangle $ Discussion Parents Context Particle number expectation value, Grand potential text/html 2015-08-16T13:41:12+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/canonical_entropy?rev=1439725272&do=diff Canonical entropy Set context $ F(\beta) $ ... classical canonical free energy range $T\equiv 1/(k_B \beta)$ definiendum $S(T):=-\frac{\partial F(T)}{\partial T} $ ---------- Discussion ---------- Context Classical canonical free energy text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/canonical_injection?rev=1395396676&do=diff Canonical injection Set $X,Y$ $X\subseteq Y$ $ \{\iota\} $ $ \iota:X\to Y $ $ \iota=\text{id}_X $ Ramifications Reference Wikipedia: Inclusion map Parents Context Function Related Identity relation text/html 2014-03-30T20:20:19+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/cardinal_arithmetic_with_types?rev=1396203619&do=diff Cardinal arithmetic with types Theorem Sum, product and function type If $X$ has two terms $a,b:X$ and $Y$ has three terms $u,v,w:Y$, then $X+Y$ has five terms * $\nu_{\mathcal l}(a),\ \nu_{\mathcal l}(b),\ \nu_{\mathcal r}(u),\ \nu_{\mathcal r}(v),\ \nu_{\mathcal r}(w):X+Y$ Similarly, $X\times Y$ has six terms * $\langle a,u\rangle,\ \langle a,v\rangle,\ \langle a,w\rangle,\ \langle b,u\rangle,\ \langle b,v\rangle,\ \langle b,w\rangle:X\times Y$ and $X\to Y$ has eight terms * $a\m… text/html 2015-12-17T19:28:51+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/cartesian_closed_category?rev=1450376931&do=diff Cartesian closed category Collection definiendum ${\bf C}$ in it inclusion ${\bf C}$ ... category postulate ${\bf C}$ has a terminal object postulate For all $X,Y\in{\bf C}$, the product $X\times Y$ exists postulate For all $Y\in{\bf C}$, the functor $-\times Y$ from ${\bf C}$ to ${\bf C}$ has a right adjoint ---------- $((A\times Y)\to B)\cong(A\to B^Y)$ text/html 2014-04-06T19:10:39+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/cartesian_product?rev=1396804239&do=diff Cartesian product Set context $X,Y$ ... small set definiendum $ p\in X \times Y $ range $ x\in X$ range $ y\in Y$ postulate $ \exists x,y.\,p=\langle x,y\rangle $ Discussion $ X \times Y\subset \mathcal{P}(\mathcal{P}(X \cap Y))$ Definitions In accordance to the defintion of the n-tuple in Ordered pair, we set $ X_1\times X_2\times X_3 \equiv (X_1\times X_2)\times X_3 $ and inductively for $ X_1\times X_1\times X_3\times \ \dots\ \times X_{n-1}\time… text/html 2014-12-02T12:09:39+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/cat?rev=1417518579&do=diff Cat Category definiendum ${\bf C}\in{\bf Cat}$ postulate ${\bf C}$ ... category postulate $\mathrm{Ob}_{\bf C},\mathrm{Mor}_{\bf C} $ ... small for all ${\bf D}\in{\bf Cat}$ postulate ${\bf Cat}[{\bf C},{\bf D}]$ ... functor category $({\bf C},{\bf D})$ Discussion Elaboration ${\bf Cat}$ is the archetypical example for what is called a 2-cateogry: Each hom-class ${\bf Cat}[{\bf C},{\bf D}]$ is again a (ordinary) category. ${\bf Cat}$${\bf C}$$\equiv {\bf C}$${\b… text/html 2014-12-26T16:07:08+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/categories?rev=1419606428&do=diff Categories Meta ${\mathfrak D}_\mathrm{Cats}$ ---------- ---------- Related Category theory, Domain of discourse text/html 2014-10-28T18:09:01+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/category_._set_theory?rev=1414516141&do=diff Category . set theory Set context $\mathcal{O},M$ ... set definiendum $ \langle \mathcal{O},M,id,* \rangle \in \mathrm{it}$ definition $\mathrm{Mor}:\mathcal{O}\times\mathcal{O}\to M$ definition $\circ:{\large\prod}_{A,B,C:\mathcal{O}}\,\mathrm{Mor}(B,C)\times\mathrm{Mor}(A,B)\to\mathrm{Mor}(A,C)$ definition $id:{\large\prod}_{A:\mathcal{O}}\,\mathrm{Mor}_O(A,A)$ postulate $\mathrm{Mor}(A,B)\cap\mathrm{Mor}(U,V)\ne\emptyset\implies U=A\land V=B$ postulate … text/html 2014-12-04T22:35:35+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/category_of_f-algebras?rev=1417728935&do=diff Category of F-algebras Collection context $F$ in ${\bf C}\longrightarrow{\bf C}$ definiendum $\mathcal{A}:\mathrm{Ob}_\mathrm{it}$ postulate $\mathcal{A}$ ... $F$-algebra definiendum $\langle f\rangle:\mathrm{it}[\langle A,\alpha\rangle, \langle B,\beta\rangle]$ postulate $f\circ\alpha=\beta\circ F(f)$ Discussion The category of F-algebras and F-algebra homomorphisms. The postulate says that it can't matter if you perform the operation ($\alpha$$\beta$$f$$\alpha,\b… text/html 2014-10-29T13:39:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/category_of_open_sets?rev=1414586356&do=diff Category of open sets Set context $\langle X,\mathcal T\rangle$ ... topological space inclusion $\mathrm{Op}(X)$ ... category definition $\mathrm{Ob}_{\mathrm{Op}(X)}\equiv \mathcal T$ for all $V,U\in\mathrm{Ob}_{\mathrm{Op}(X)}$ definition $\mathrm{Op}(X)[V,U]\equiv\{i:V\to U\ |\ i(x)=x\}$ Discussion In the category of open sets, the arrows are the inclusion functions. In the case $V\subseteq U$, the hom-set $\mathrm{Op}(X)[U,V]$$\{i\}$ text/html 2016-04-14T20:11:29+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/category_theory?rev=1460657489&do=diff Category theory Note Category theory can be written down in predicate logic (see below) or formulated within type theory. More or less well, it can also be modeled within set theory (see Category . set theory). Direct axiomatization in logic The nLab gives some formal definitions for categories as theory of concatenation in first order logic. $\mathrm{Arrow}$$s$$t$$c$$s$$t$$c(f,g,h)$$g$$f$$h$$f\circ g = h$${\large\frac{}{ ss(f) \, = \, s(f) \, = \, ts(f)\,\land\,tt(f) \, = \, t(f) \, = \,… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/cauchy_principal_value?rev=1395396676&do=diff Cauchy principal value Partial function definiendum $\mathcal P\int_a^b: \mathbb R^2\times(\mathbb R\to\overline{\mathbb R})\to\overline{\mathbb R}$ $p$ ... ordered sequence of the $m$ poles of $f$ definiendum $\mathcal P\int_a^b(f):=\mathrm{lim}_{\varepsilon\to 0}\left(\int_a^{p_1-\varepsilon}f(x)\,\mathrm dx+\int_{p_1+\varepsilon}^{p_2-\varepsilon}f(x)\,\mathrm dx+\cdots+\int_{p_m+\varepsilon}^b f(x)\,\mathrm dx\right)$ Discussion The Cauchy principal value is the value of an int… text/html 2014-03-24T19:38:32+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/cauchy_sequence?rev=1395686312&do=diff Cauchy sequence Set context $\langle X,d\rangle$ $\dots$ Metric space definiendum $x\in \text{CauchySeq}(X) $ context $x\in \text{InfSequence}(X) $ range $\varepsilon\in \mathbb R,\ \varepsilon>0$ range $n,m,N\in\mathbb N$ postulate $\forall\varepsilon.\ \exists N.\ \forall (n,m>N).\ d(x_n,x_m)<\varepsilon$ Discussion Reference Wikipedia: Cauchy sequence Parents Context Metric space Requirements Infinite sequence text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/ceiling_function?rev=1395396676&do=diff Ceiling function Function definiendum $ \lceil \ \rceil:\mathbb{R}\to\mathbb{Z}$ definiendum $ \lceil x \rceil:=\min\,\{n\in\mathbb{Z}\mid n\ge x\} $ Discussion Parents Subset of ℤ valued function, Monotonically increasing function text/html 2014-04-08T14:48:42+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/characteristic_function?rev=1396961322&do=diff Characteristic function Set context $X$ context $A\subseteq X$ definiendum $\chi_A:X\to \{0,1\} $ definiendum $ \chi_A(x) := \begin{cases} 1 & \mathrm{if}\ x\in A\\\\ 0 & \mathrm{else} \end{cases}$ The complement $A^c$ is taken w.r.t. $X$. Discussion The symbols “$0,1$” here are essentially placeholders. One could choose any set of two distinct elements as codomain. text/html 2014-11-08T16:35:30+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/chart?rev=1415460930&do=diff Chart Set context $\langle M,T_X\rangle$ ... second-countable Hausdorff space context $n\in \mathbb N$ let $\langle \mathbb R^n,T_{\mathbb R^n}\rangle$ ... Euclidean topology definiendum $\langle U,\phi\rangle\in$ it inclusion $U\in T_M$ exists $U_{\mathbb R^n}\in T_{\mathbb R^n}$ inclusion $\phi:U\to U_{\mathbb R^n}$ inclusion $\phi$ ... homeomorphism Discussion The charts on $M$$\mathbb R^n$ text/html 2014-11-11T11:14:03+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/chiron?rev=1415700843&do=diff Chiron Framework "todo: Make a syntax entry for formal language, and join it with Chiron, Haskell, as well as logical theories." Chiron is a set theory designed for writing down, reason about and do mathematics. According to its purpose, it has an unusually broad range of basic notions. It contains a rich type system and, most notably, a facility to reason about syntactic expression. So for exmaple, there is a way to express $\equiv$$\mathcal S$$\mathcal O$$(e_1,e_2,\dots,e_n)$$n\ge 0$$e_i… text/html 2015-08-18T20:30:05+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/classical_canonical_ensemble?rev=1439922605&do=diff Classical canonical ensemble Set definiendum $ \langle \mathcal M, H,{\hat\rho}\rangle \in \mathrm{it} $ postulate $\langle \mathcal M, H,{\hat\rho},{\hat\rho}_0\rangle$ ... classical statistical ensemble postulate $\hat\rho: \mathbb R\to(\Gamma_{\mathcal M} \to\mathbb R_+) $ postulate $\hat\rho(\beta;{\bf q},{\bf p}):=\mathrm{e}^{-\beta\ H({\bf q},{\bf p})}$ ---------- Discussion Equivalence of the microcanonical and canonical ensemble For a given a Hamiltonian system… text/html 2015-08-16T13:36:18+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/classical_canonical_free_energy?rev=1439724978&do=diff Classical canonical free energy Set context $ Z(\beta) $ ... canonical partition function (classical or quantum) definiendum $F(\beta):=-\frac{1}{\beta} \mathrm{ln}(Z(\beta)) $ ---------- Discussion This mirrors the Classical microcanonical entropy. The usage of the free energy is similar to that of the cumulant-generating function. text/html 2016-03-09T11:41:26+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/classical_canonical_partition_function?rev=1457520086&do=diff Classical canonical partition function Set context $ \langle \mathcal M, \mathcal H,\pi,\pi_0,{\hat\rho},{\hat\rho}_0\rangle$ ... classical canonical ensemble context $ \mathrm{dim}(\mathcal M) = 3N $ context $ \hbar$ ... Reduced Planck's constant definiendum $Z(\beta):=\frac{1}{h^{3N}N!}\int_{\Gamma_{\mathcal M}}\ \hat\rho(\beta;{\bf q},{\bf p}) \ \mathrm d\Gamma $ ---------- Discussion This mirrors the classical microcanonical phase volume. We have $Z(\beta):=… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/classical_confined_phase_volume?rev=1395396676&do=diff Classical confined phase volume Set context $ \langle \mathcal M, \mathcal H,\pi,\pi_0,{\hat\rho},{\hat\rho}_0\rangle$ ... classical microcanonical ensemble context $ \mathrm{dim}(\mathcal M) = 3N $ context $ \hbar$ ... Reduced Planck's constant definiendum $\varphi(E):=\frac{1}{h^{3N} N!}\int_{\{\langle{\bf q},{\bf p}\rangle\in \Gamma_{\mathcal M}\ |\ H({\bf q},{\bf p})\le E\}} \mathrm d\Gamma $ Discussion Parents Context Classical microcanonical ensemble text/html 2016-03-09T16:34:56+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/classical_density_of_states?rev=1457537696&do=diff Classical density of states Set context $\varphi$ ... classical confined phase volume definiendum $D(E):=\varphi'(E) $ ---------- Discussion In bounded space, the computations involve a counting energy levels, or energy Eigenvalues $\varepsilon_r$ in the quantum case, which we index by $r$$V$$\sum_r\dots\ \ \longrightarrow\ \ (2S+1)\frac{V}{(2\pi)^3}\int\mathrm d^3k \dots $$S$$V$$E\ge 0$$\varepsilon({\bf k})=\frac{\hbar^2}{2m}{\bf k}^2$$ D(E) = 2\pi\ (S+1)\frac{V}{(2\pi)^3}\left(… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/classical_ensemble_expectation_value?rev=1395396676&do=diff Classical ensemble expectation value Set context $ \langle \mathcal M, H,\pi,\pi_0,{\hat\rho},{\hat\rho}_0\rangle$ ... classical statistical ensemble definiendum $\langle \cdot \rangle: (\Gamma_{\mathcal M}\to \mathbb R)\to \mathbb R$ definiendum $\langle F\rangle:=\int_{\Gamma_{\mathcal M}}\ F\cdot\rho\ \mathrm d\Gamma$ Discussion In the definition we must use $\rho$, i.e. the noramlized ${\hat\rho}$. Parents Context Classical probability density function text/html 2015-08-18T20:33:41+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/classical_grand_canonical_ensemble?rev=1439922821&do=diff Classical grand canonical ensemble Set range $ N\in\mathbb N $ definiendum $ (\ \langle \mathcal M_N, H_N,\pi_N,\pi_{N,0},{\hat\rho}_N,{\hat\rho}_{N,0} \rangle\ )_N \in \mathrm{it} $ postulate $\forall N.\ \langle \mathcal M_N, H_N,\pi_N,\pi_{N,0},{\hat\rho}_N,{\hat\rho}_{N,0} \rangle$ ... classical canonical ensemble A sequence of phase spaces with their partition functions, one for each particle number. "todo: the $N$th manifold is a submanifold of the $N+1$ text/html 2015-08-16T15:30:38+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/classical_hamiltonian_system?rev=1439731838&do=diff Classical Hamiltonian system Set definiendum $\langle \mathcal M, H\rangle \in \mathrm{it} $ postulate $ \mathcal M$ ... smooth manifold range $ \Gamma_{\mathcal M}\equiv \mathcal M\times T^*\mathcal M $ postulate $ H:\Gamma_{\mathcal M} \times \mathbb R \to \mathbb R $ postulate $ H $ ... differentiable in $\Gamma_{\mathcal M}$ ---------- Discussion The Hamiltonian function is related to the Lagrangian function via Legendre transformation. text/html 2015-08-18T20:33:13+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/classical_microcanonical_ensemble?rev=1439922793&do=diff Classical microcanonical ensemble Set context $ \Delta \in \mathbb R_+ $ definiendum $ \langle \mathcal M, H,{\hat\rho}\rangle \in \mathrm{it} $ postulate $\langle \mathcal M, H,{\hat\rho}\rangle$ ... classical statistical ensemble postulate $\hat\rho: \mathbb R_+\to(\Gamma_{\mathcal M} \to\mathbb R_+) $ postulate $\hat\rho(E;{\bf q},{\bf p}):=\begin{cases} 1 & \mathrm{if}\ E\le H({\bf q},{\bf p})\le E+\Delta \\\\ 0 & \mathrm{else} \end{cases}$ Discussion Referen… text/html 2015-08-16T14:26:50+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/classical_microcanonical_entropy?rev=1439728010&do=diff Classical microcanonical entropy Set context $ \Gamma(E) $ ... classical phase volume definiendum $S(E):=k_B\ \mathrm{ln}(\Gamma(E)) $ ---------- Discussion This mirrors $\rho \sim {\mathrm e}^{-H/T}$ or $H \sim -T \ln(\rho)$. ---------- Context Classical microcanonical phase volume text/html 2015-08-16T14:18:05+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/classical_microcanonical_phase_volume?rev=1439727485&do=diff Classical microcanonical phase volume Set context $ \langle \mathcal M, \mathcal H,\pi,\pi_0,{\hat\rho},{\hat\rho}_0\rangle$ ... classical microcanonical ensemble context $ \mathrm{dim}(\mathcal M) = 3N $ context $ \hbar$ ... Reduced Planck's constant definiendum $\Gamma(E):=\frac{1}{h^{3N} N!}\int_{\{\langle{\bf q},{\bf p}\rangle\in \Gamma_{\mathcal M}\ |\ E\le H({\bf q},{\bf p})\le E+\Delta\}} \mathrm d\Gamma $ ---------- Discussion Alternative definitions $\Ga… text/html 2015-08-18T20:29:01+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/classical_phase_density?rev=1439922541&do=diff Classical phase density Set context $ \langle \mathcal M, H\rangle$ ... classical Hamiltonian system definiendum $ {\hat\rho} \in \mathrm{it} $ postulate $\langle \mathcal M, H\rangle$ ... Hamiltonian system range $ \Gamma_{\mathcal M} \equiv \mathcal M\times T\mathcal M $ postulate $\hat\rho: \Gamma_{\mathcal M} \times \mathbb R \to \mathbb R_+ $ range $\hat\rho:: \hat\rho({\bf q},{\bf p},t) $ postulate $ \frac{\partial}{\partial t}{\hat\rho} =… text/html 2016-03-09T10:36:21+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/classical_probability_density_function?rev=1457516181&do=diff Classical probability density function Set context $ {\hat\rho} $ ... classical phase density definiendum $\rho:=\frac{\hat\rho}{\int_\Gamma\ \hat\rho\ \mathrm d\Gamma}$ ---------- Discussion The function $\rho$ is just the normalized version of ${\hat\rho}$. ---------- Context Classical phase density text/html 2015-08-18T20:26:50+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/classical_statistical_ensemble?rev=1439922410&do=diff Classical statistical ensemble Set context $ {\hat\rho} $ ... Classical phase density definiendum $ \langle \mathcal M, H,{\hat\rho}\rangle \in \mathrm{it} $ ...with associated Classical Hamiltonian system $\langle \mathcal M, H\rangle$. ---------- Discussion ---------- Context Classical phase density text/html 2015-04-16T18:47:51+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/closed_monoid_subset?rev=1429202871&do=diff Closed monoid subset Meta context $\langle\!\langle M,*\rangle\!\rangle$ ... monoid definiendum $S\in$ it for all $x,y\in S$ postulate $x*y\in S$ ---------- ---------- Subset of Monoid text/html 2014-07-03T11:49:55+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/colimit_._category_theory?rev=1404380995&do=diff Colimit . category theory Collection context $F:{\bf C}^{\bf D}$ context $\Delta:{\bf C}\longrightarrow{\bf C}^{\bf D}$ ... diagonal functor definiendum $\mathrm{colim}\,F$ ... initial morphism from $F$ to $\Delta$ Discussion See Limit . category theory Reference Wikipedia: Limit (category theory) Parents Context Diagonal functor Requirements Initial morphism text/html 2014-12-26T16:05:46+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/collections?rev=1419606346&do=diff Collections Meta ${\mathfrak D}_{Collections}$ ---------- ---------- Related Domain of discourse text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/column_vector?rev=1395396676&do=diff Column vector Set context $F$ ... field context $m\in \mathbb N$ definiendum $ \mathrm{it}(m,F) \equiv \mathrm{Matrix}(1,m,F) $ Discussion This set of column vectors over a field can be viewed as a vector space. Reference Wikipedia: Column_vector Parents Subset of Matrix Context Field Related Vector space text/html 2014-09-19T05:46:26+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/commutative_semiring?rev=1411098386&do=diff Commutative semiring Definition context $X$ definiendum $\langle X,+,* \rangle \in \mathrm{CommSemiring}(X)$ definiendum $\langle X,+,* \rangle \in \mathrm{SemiRing}(X)$ $a,b\in X$ postulate $a*b=b*a$ Discussion Reference Wikipedia: Semiring Parents Context Semiring text/html 2014-10-25T17:00:22+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/compact_space?rev=1414249222&do=diff Compact space Set definiendum $\langle X,\mathcal{T}\rangle \in\mathrm{it} $ inclusion $\langle X,\mathcal{T}\rangle$ ... topological space for all $\mathcal{T'}\subseteq \mathcal{T}$ for all $K\subseteq \bigcup \mathcal{T}'$ exists $\mathcal{T}''\subseteq \mathcal{T}'$ postulate $K\subseteq\bigcup \mathcal{T}''$ postulate $\mathcal{T}''$ ... finite Discussion Idea A topological space is compact if each set $K$ which is covered by some collect… text/html 2014-04-03T22:41:24+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/complement?rev=1396557684&do=diff Complement Set context $X,Y\in \mathfrak U$ definiendum $ x\in X \smallsetminus Y $ postulate $ x\in X $ postulate $ x\notin Y $ Discussion Reference Wikipedia: Complement Parents Element of Set universe Context* Set universe text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/complete_graph?rev=1395396676&do=diff Complete graph Set context $V,E$ ... set definiendum $ \langle V,E,\psi\rangle \in \mathrm{it}(E,V) $ postulate $ \langle V,E,\psi\rangle $ ... simple graph for all $ u,v \in V$ postulate $ u\neq v\implies \exists !(e\in E).\ \psi(e)=\{u,v\} $ Discussion In a complete (undirected) graph, every two distinct vertices are connected. The axiom $\{u\}\notin\mathrm{im}(\psi)$ says that there are no loops on a single vertex.$n$$n$$n$$\sum_{k=1}^{n-1}k=\frac{1}{2}n(n-… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/complete_measure_space?rev=1395396676&do=diff Complete measure space Set context $X $ definiendum $ \langle X,\Sigma,\mu\rangle $ ... complete measure space over $X$ postulate $ \langle X,\Sigma,\mu\rangle $ ... measure space $\mu(N)=0$ $N'\subseteq N $ postulate $ N'\in\Sigma $  Discussion In a complete measure space, subsets of null-sets can also be measured (and they then have zero measure as well). This notion is just introduced to prevent some pathologies. text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/complex_conjugate_of_a_complex_number?rev=1395396676&do=diff Complex conjugate of a complex number Set @#55CCEE: context postulate $\mathrm{conj}:X\to \mathbb C$ $a+i\ b\in X$ $a,b\in \mathbb R$ postulate $\mathrm{conj}(a+i\ b):= a-i\ b$ Ramifications If $z\in \mathbb C$, one writes $\overline z\equiv\mathrm{conj}(z)$. Parents Related Complex number text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/complex_conjugate_of_a_fuction?rev=1395396676&do=diff Complex conjugate of a function Set context $f:X\to\mathbb C$ definiendum $ \overline f:X\to\mathbb C$ definiendum $ \overline f(x):=\overline{f(x)} $ Ramifications Put differently $\overline f\equiv \mathrm{conj}\circ f$ Parents Context Complex conjugate of a complex number text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/complex_coordinate_space?rev=1395396676&do=diff Complex coordinate space Set $ n\in \mathbb N $ $\mathbb C^n$ Recursive definition: $ \mathbb C^1=\mathbb C $ $ \mathbb C^n=\mathbb C^{n-1}\times \mathbb C, \hspace{1cm}n\neq 1 $ Ramifications Reference Wikipedia: Several complex variables Parents Refinement of Cartesian product Related Complex number text/html 2015-04-15T14:13:05+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/complex_exponents_with_positive_real_bases?rev=1429099985&do=diff Complex exponents with positive real bases Function context $ b\in\mathbb R_+^* $ definiendum $ z\mapsto b^z :\mathbb C\to\mathbb C $ definiendum $ z\mapsto b^z := \mathrm{exp}(z\cdot \mathrm{ln}(b)) $ ---------- Discussion The identity $b^{x_1+x_2}=b^{x_1}\cdot a^{x_2}$, says that exponentiation is a (the) homomorphism between $+$ and $\cdot$. The combinatorial manifestation, e.g. formulated in for $B,X_1,X_2,\dots\in\bf{Set}$$B^{\coprod_{j\in J}X_j}\cong\prod_{j\in J} B… text/html 2015-04-03T12:15:57+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/complex_line_integral?rev=1428056157&do=diff Complex line integral Function range $\mathcal{L}$ ... continuously differentiable finite lines definiendum $\int: \mathcal{L}\to(\mathbb C\to \mathbb C)\to \mathbb K$ range $L\in \mathcal{L}$ range $\gamma: [a,b]\to L$ ... parametrization definiendum $\int_L\ f(z)\,\mathrm dz:=\int_L\ f\left(\gamma(t)\right)\cdot \gamma'(t)\, \mathrm dt$ "todo: Continuously differentiable finite lines" ---------- Theorems If $f$ is holomorphic and two curves $L_1,L_2$… text/html 2014-04-01T19:14:45+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/complex_number?rev=1396372485&do=diff Complex number Set definiendum $ \mathbb C \equiv \mathbb R^2 $ Discussion We write the complex numbers as $a+ib\equiv\langle a,b\rangle$, where $a,b\in\mathbb R$. The complex numbers are then set up as a field with $i^2=-1$, see arithmetic structure of complex numbers. We identify the real numbers within $\mathbb C$ as the set of elements of the form $\langle a,0\rangle=a+i0=a$. Reference Wikipedia: Complex number Parents text/html 2015-04-17T10:33:19+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/concrete_category?rev=1429259599&do=diff Concrete category Collection definiendum $\langle\!\langle {\bf C},F\rangle\!\rangle$ in it inclusion $\bf C$ ... locally small category inclusion $F:{\bf C}\longrightarrow{\bf Set}$ inclusion $F$ ... faithful ---------- Reference Wikipedia: Concrete category ---------- Subset of Locally small category Requirements Faithful functor text/html 2016-12-27T14:30:42+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/conditional_probability?rev=1482845442&do=diff Conditional probability Function context $f:X\times Y\to {\mathbb R}$ ... measurable in $X$ definition $p_X: X\times Y\to {\mathbb R}$ definition $p_X: N_X^*f(x,y)$ "This is really just the conditional probability when coming from a joint “probability kernel”, i.e a function $f(x,y)$. The operator $N_X^*$ maps to a function that is normalized w.r.t. $X$$p_X(x,y) = N_X^*f(x,y)$$p_X(x|y)$$N_Y^*f(x,y)$$p_X(y|x)$$|$$p_X(y|x)$$x\in X$$y\in Y$$\int{ \mathrm d}x$$p_X(x,y) = \dfrac{… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/connected_graph?rev=1395396676&do=diff Connected graph Set context $V,E$ ... set definiendum $\langle V,E,\psi\rangle \in \mathrm{it}(E,V) $ postulate $\langle V,E,\psi\rangle $ ... undirected graph range $ X\cap Y=\emptyset $ range $ X\cup Y=V $ range $ x\in X $ range $ y\in Y $ postulate $ \forall X,Y. \exists x,y.\ \{x,y\}\in\mathrm{im}\ \psi $ Discussion Parents Subset of Undirected graph text/html 2014-09-26T15:07:51+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/constant_functor?rev=1411736871&do=diff Constant functor Functor context ${\bf C},{\bf D}$ ... category context $D:\mathrm{Ob}_{\bf D}$ definiendum $\Delta_D:{\bf C}\longrightarrow{\bf D}$ definition $\Delta_DC:=D$ definition $\Delta_D(f):=\mathrm{id}_D$ Discussion Reference Wikipedia: Diagonal functor nLab: constant functor Parents Element of Functor text/html 2014-10-26T13:59:20+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/continuous_function?rev=1414328360&do=diff Continuous function Set context $\langle X,\mathcal{T}_X\rangle$ ... topological space context $\langle Y,\mathcal{T}_Y\rangle$ ... topological space definiendum $ f\in \mathrm{it} $ inclusion $ f:X\to Y $ for all $V\in \mathcal{T}_Y$ postulate $ f^{-1}(V)\in\mathcal{T}_X $ Discussion Theorems A function to $\mathbb R^n$ is continuous iff all its components are. text/html 2014-09-28T23:46:17+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/coproduct_._category_theory?rev=1411940777&do=diff Coproduct . category theory Collection context $F:\mathrm{Ob}_{{\bf C}^{\bf 2}}$ range $a,b:\mathrm{Ob}_{\bf 2}$ definition $\langle Fa+Fb, [i_a,i_b]\rangle := \mathrm{colim}\,F$ Discussion Elaboration Similar to product, but $[i_a,i_b]$ denotes a functor with a sum type domain. Reference Wikipedia: Coproduct (category theory) Parents Context Functor Refinement of Colimit . category theory, Discrete category Related Sum type text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/cosine_function?rev=1395396676&do=diff Cosine function Function definiendum $\mathrm{\cos}: \mathbb C\to\mathbb C$ definiendum $\cos(z) := \sum_{k=0}^\infty \frac{(-1)^{k}}{(2k)!}z^{2n} $ Discussion $\theta\in\mathbb R$ $\cos(\theta) = \frac{1}{2}(\mathrm e^{i\theta}+\mathrm e^{-i\theta}) $ i.e. if $\zeta:=\mathrm e^{i\theta}$, then $\zeta+\overline{\zeta}=2\cos(\theta)$. Parents Context Infinite sum of complex numbers, Factorial function Related Exponential function, Sine function text/html 2016-01-03T15:54:02+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/counit-unit_adjunction?rev=1451832842&do=diff Counit-unit adjunction Collection context $F$ in ${\bf D}\longrightarrow{\bf C}$ context $G$ in ${\bf C}\longrightarrow{\bf D}$ definiendum $\langle\varepsilon,\eta\rangle$ in $F\dashv G$ inclusion $\varepsilon, \eta$ ... my nice nats $\left(F,G\right)$ for all $X\in{\bf C}, Y\in{\bf D}$ postulate $\varepsilon_{FY}\circ F(\eta_Y)=1_{FY}$ postulate $G(\varepsilon_X)\circ \eta_{GX}=1_{GX}$ ---------- Elaboration The pair $\langle\varepsilon,\eta\… text/html 2014-10-29T20:13:06+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/countable_base_for_a_topology?rev=1414609986&do=diff Countable base for a topology Set context $\langle X,T\rangle$ ... topological space definiendum $B\in$ it postulate $B$ ... base postulate $B$ ... countable Discussion Parents Context Topological space Subset of Base for a topology, Countable set text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/countable_intersection?rev=1395396676&do=diff Countable intersection Set context $X$ context $A\in \mathrm{Sequence}(X)$ context $n\in \mathbb N\cup\{\infty\}$ definiendum $\bigcap_{j=1}^n A_j \equiv \bigcap A(\mathrm{range}(n))$ Ramifications Parents Refinement of Arbitrary intersection Related Sequence text/html 2014-10-29T19:59:22+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/countable_set?rev=1414609162&do=diff Countable set Collection definiendum $X\in$ it exists $f:\mathbb N\to X$ postulate $f$ ... surjective Discussion Parents Requirements Surjective function, Natural number text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/countable_union?rev=1395396676&do=diff Countable union Set context $A\in \mathrm{Sequence}(X) $ context $n\in \mathbb N\cup\{\infty\}$ definiendum $\bigcup_{j=1}^n A_j \equiv \bigcup A(\mathrm{range}(n))$ Discussion Parents Refinement of Indexed union Context Sequence text/html 2014-10-28T19:23:31+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/cover?rev=1414520611&do=diff Cover Set context $X$ definiendum $C$ in it postulate $f:I\to C$ postulate $X \subseteq \bigcup_{i\in I,\ f}C_i$ Discussion where we have some indexing via a set $I$. Parents Requirements Indexed union text/html 2016-12-10T13:14:32+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/cpp?rev=1481372072&do=diff Cpp Notes on programming languages $\blacktriangleright$ Cpp $\blacktriangleright$ Note Language Discussion Reference A tutorial: www.cplusplus.com/doc/tutorial Online shell: http://cpp.sh (shell) Code exmaples Q & A ---------- Related Requirements Notes on programming languages text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/cubic_graph?rev=1395396676&do=diff Cubic graph Set context $V,E$ ... set definiendum $ \langle V,E,\psi\rangle \in \mathrm{it}(E,V) $ inclusion $ \langle V,E,\psi\rangle $ ... undirected graph for all $ v\in V $ postulate $ d(v)=3 $ Discussion Weeeeeeee!!! Parents Subset of Regular graph Refinement of k-regular graph text/html 2015-04-09T16:16:08+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/cumulative_distribution_function?rev=1428588968&do=diff Cumulative distribution function Set definiendum $F\in\mathrm{CDF} $ inclusion $F:\mathbb R\to\mathbb R$ ... right-continuous function, monotonically increasing postulate $\lim_{x\to\ -\infty} F(x)=0$ postulate $\lim_{x\to\ +\infty} F(x)=1$ ---------- "$P=F'$" "Let $S:({\mathbb D}\to {\mathbb R}_{\ge 0})\to{\mathbb R}_{\ge 0}$ be linear. If $f$ with $\infty>Sf>0$, then $\bar{f}:=\frac{1}{Sf}\cdot f$ has $S\bar{f}=\frac{Sf}{Sf}=1$. So we can use such $S$ to normalize … text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/cycle_._graph_theory?rev=1395396676&do=diff Cycle . graph theory Set context $V,E$ ... set definiendum $\langle V,E,\psi\rangle \in \mathrm{it}(E,V) $ inclusion $ \langle V,E,\psi\rangle $ ... path postulate $ |V|\ge 3 $ range $ u,v\in V $ range $ a$ ... sequence in $V,\ \forall i.\ a_{i+|V|}=a_i $ range $ i\in\mathbb N$ postulate $ \exists a.\ \forall u,v.\ (\exists i.\ \{a_{i},a_{i+1}\}=\{u,v\}) \leftrightarrow (\{u,v\}\dots\mathrm{edge}) $ Discussion A path is a graph whic… text/html 2018-02-03T02:11:35+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/deformed_natural?rev=1517620295&do=diff Deformed natural Function context $p\in Q$ context $u:{\mathbb N}\to{}Q\to{\mathbb A}$ definition $[n]_u(q) := \sum_{k=1}^n \dfrac{u_k(q)}{u_k(p)}$ And clearly the denominator must be nonzero. Discussion E.g., for another sequence $a_n$ consider $u(n,q):=q^{a_n}$. In particular, consider $a_n:=n*x+d$ for some $d$. a[k_] = k x + d; Sum[q^a[k], {k, 1, n}] Limit[%, q -> 1] $x:=1, d:=0$$\lim_{q\to p}[n]_u(q) = \sum_{k=1}^n 1 = n$$(a_k)$$\sum_{k=1}^n a_k = \lim_{q\to 1}\… text/html 2019-09-28T18:11:56+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/dependent_product_functor?rev=1569687116&do=diff Dependent product functor Functor context ${\bf C}$ ... Cartesian closed category with all limits let $\prod_f$ ... right adjoint to the pullback functor $f^*$, where $f$ is an arrow in ${\bf C}$ context $!_X:X\to *$ ... terminal morphisms for $X\in{\bf C}$ definition $\prod_X p := {\mathrm{dom}}\prod_{!_X} p$ "todo: fmap of $\prod_X$$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 2015-12-22T13:47:26+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/dependent_sum_functor?rev=1450788446&do=diff Dependent sum functor Functor context ${\bf C}$ ... Cartesian closed category with all limits let $\sum_f$ ... left 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 $\sum_X p := {\mathrm{dom}}\sum_{!_X} p$ "todo: fmap of $\sum_X$$f^*$${\bf C}/Y$${\bf C}/X$$!_X:X\to *$${\mathrm{dom}}\ f^*g$$\sum_fp$$f\circ p$$\sum_X p := {\mathrm{dom}}\ p$$f:{\bf C}[X,Y]$$\sum_f:{\… text/html 2015-01-08T01:14:26+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/dependent_type_theory?rev=1420676066&do=diff Dependent type theory Framework We want dependent types together with type constructors $$\Pi,\ \Sigma,\ \to,\ \times,\ +$$ and basis types $${\bf 0},\ {\bf 1},\ {\bf 2}$$ Moreover some type definition schema alla algebraic data types and/or (higher) inductive types and/or W-types etc. For example, $\mathbb N$ is nicely inductively defined via successor function (using $\to$$+$${\bf 1}$$\large\frac{(\Gamma,x\,:\,A)\ \vdash\ Q\,:\,\mathrm{Type}}{\Gamma\ \vdash\ \Pi_{x:A}Q\,:\,\mathrm{Type}}$$\h… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/det_exp_formula?rev=1395396676&do=diff Det exp formula Theorem $ A\in\mathrm{Matrix}(n,\mathbb C) $ postulate $\mathrm{det}\left(\mathrm{exp}(A)\right)=\mathrm{e}^{\mathrm{tr}\ A}$ Discussion This is a corollary of Jacobi's formula. Reference Wikipedia: Jacobi's formula Parents Context Determinant via multilinear functionals, Exponential function text/html 2016-07-21T01:00:21+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/determinant_differentiation?rev=1469055621&do=diff Determinant differentiation Theorem context $ A,\Omega\in\mathrm{Matrix}(n,\mathbb C) $ context $ A $ ... invertible postulate $\frac{\partial}{\partial t}|_{t=0}\ \mathrm{det}(A+t\,\Omega) = \mathrm{tr}(A^{-1}\Omega)\cdot\mathrm{det}(A) $ ---------- Thus $\dfrac{\partial}{\partial t}\left|_{t=0}\right. \dfrac{\mathrm{det}(A+t\,\Omega)}{\mathrm{det}(A)} = \mathrm{tr}(A^{-1}\Omega)$ and also implies $ \mathrm{tr}(\Omega) = \dfrac{\partial}{\partial t}\left|_{t=0}\ri… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/determinant_via_multilinear_functionals?rev=1395396676&do=diff Determinant via multilinear functionals Set context $V$ ... finite dimensional $\mathcal F$-vector space definiendum $\mathrm{det}:L(V,V)\to \mathcal F$ range $n\equiv \mathrm{dim}(V)$ $M\in \mathrm{MultiLin}(V^n)$ $ v_1,\dots,v_n\in V $ $A\in L(V,V)$ postulate $ M(A\ v_1,\dots,A\ v_n) = \mathrm{det}(A)\cdot M(v_1,\dots,v_n) $ Discussion Theorems * The determinant is an invariant of linear operators on finite-dimensional vector spaces.$\mathrm{det}(AB)=\mathr… text/html 2019-08-25T13:48:39+0200 Anonymous (anonymous@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 2014-12-04T22:35:00+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/diagonal_functor?rev=1417728900&do=diff Diagonal functor Functor context ${\bf C},{\bf D}$ ... category definiendum $\Delta:{\bf C}\longrightarrow{\bf C}^{\bf D}$ definition $\Delta C:=\Delta_C$ definition $\Delta(f):=\mathrm{const}_f$ Discussion Coherence Note that for constant functors $\Delta_A,\Delta_B$, the square in the definition of a natural transformation commutes trivially. Hence any arrow $f:{\bf C}[A,B]$ text/html 2016-01-23T19:07:01+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/differentiable_function?rev=1453572421&do=diff Differentiable function Set context $X,Y$ ... Banach spaces with topology context $n\in \mathbb N$ definiendum $f\in C^n(X,Y)$ postulate $\forall(k\le n).\,D^k f$ ... continuous ---------- Theorems Let $f(0)=0\neq f'(0)$, then $\dfrac{ f(y\ f^{-1}(x)) }{y} =x+(y-1)\cdot\dfrac{f''(0)}{f'(0)^2}\cdot\dfrac{1}{2}x^2+{\mathcal O}(x^3)$ Reference Wikipedia: Differentiable function ---------- Context Fréchet derivative, Iterated function text/html 2014-06-18T15:38:49+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/directed_graph?rev=1403098729&do=diff Directed graph Set context $ V,E $ ... set definiendum $ \langle V,\langle E,\psi\rangle\rangle \in \mathrm{it}(E,V) $ postulate $ \psi $ ... function postulate $ \mathrm{dom}(\psi)=E $ postulate $ \forall (e\in E).\ \exists (u,v\in V).\ \psi(e) = \langle v,u \rangle $ Discussion Parents Subset of Graph Context Function text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/disconnected_graph?rev=1395396676&do=diff Disonnected graph Set context $V,E$ ... set definiendum $\langle V,E,\psi\rangle \in \mathrm{it} $ postulate $\langle V,E,\psi\rangle $ ... undirected graph $\setminus$ connected graph Discussion Parents Subset of Undirected graph R*Requirements Connected graph Related Connected graph text/html 2014-11-01T18:16:38+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/discrete_category?rev=1414862198&do=diff Discrete category Collection definiendum ${\bf C}$ in $\mathrm{it}$ exists $F$ ... equivalence of categories $({\bf C}, {\bf D})$ for all $f:\mathrm{Mor}_{\bf D}$ exists $A\in{\bf D}$ postulate $f=1_A$ Discussion Idea A discrete category either has no non-identity arrows or at least is equivalent to such a category.$n$${\bf n}$ text/html 2017-03-25T15:59:09+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/division_ring?rev=1490453949&do=diff Division ring Set context $X$ postulate $\langle X,+,* \rangle \in \mathrm{divisionRing}(X)$ context $\langle X,+,* \rangle \in \mathrm{unitalRing}(X)$ context $\langle X,* \rangle \in \mathrm{group}(X)$ range $a,b\in X$ postulate $\exists a,b.\ (a\neq b)$ Ramifications Discussion A division ring is essentially two compatible groups over a set $X$, one of which is necessarily commutative. Compatible in the sense of the distributive laws of a ring… text/html 2015-10-10T14:15:34+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/domain_of_discourse?rev=1444479334&do=diff Domain of discourse Meta I also use the following types to classify the bulk of other entries in the wiki and these are the entries with formal content. * ${\mathfrak D}_\mathrm{Sets}$ ... Sets * ${\mathfrak D}_{\to}$ ... Functions * ${\mathfrak D}_\mathrm{Cats}$ ... Categories * ${\mathfrak D}_{\longrightarrow}$ ... Functors * ${\mathfrak D}_{\xrightarrow{\bullet}}$ ... Natural transformations * ${\mathfrak D}_\mathrm{Tuples}$ ... Tuples, finite lists of elements of the above… text/html 2016-07-17T15:28:48+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/drastic_measures?rev=1468762128&do=diff Drastic measures Set context $F$ ... set definiendum $S$ in it postulate $S:F\to{\mathbb K}\setminus\{0\}$ postulate $S$ ... ${\mathbb K}$-linear "todo: ${\mathbb K}$-linear" ---------- Discussion “Normalization w.r.t. $S$”, $N_Sf:=(Sf)^{-1}\cdot f$, has $SN_Sf=e$ and $[N_S]=[S^{-1}]$. As $S$ is linear, $N_S(c\,f)=N_S(f)$ We'll also write $\bar{f}:=(Sf)^{-1}\cdot f$ Example 1 $F$${\mathbb N}$$Sf:=\sum_{n=0}^\infty (L_nf)(n)$$(L_n)$$L_n={\mathrm{id}}$$a$$\mat… text/html 2016-10-15T14:42:42+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/drawing_arrows_and_coding_functions?rev=1476535362&do=diff Drawing arrows and coding functions Outline $\blacktriangleright$ Drawing arrows and coding functions $\blacktriangleright$ A type system as a model for some concepts Guide Type system A main part of a type system is a collection of names that stand for so called types. We will draw a lot of simple pictures with those types “lying around$$ {\mathbb Z} $$$13 : {\mathbb Z}$$4 : {\mathbb Z}$$k:=4$$k : {\mathbb Z}$$(2+3) : {\mathbb Z}$$n:=2+3$$n : {\mathbb Z}$$g : {\mathbb Z}\to{\mathbb Z}$$g$${… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/dtime?rev=1395396676&do=diff DTIME Set context $ T:\mathbb N\to\mathbb N $ definiendum $ L\in \mathrm{\bf{DTIME}}(T(n)) $ inclusion $ L\subseteq\{0,1\}^* $ range $M\in\mathrm{TM}$ range $c\in\mathbb{N}$ postulate $\exists M,c.\ M$ decides $L$ in $c\cdot T(n)$-time Discussion The ${\bf{D}}$ stands for deterministic. Parents Requirements Turing machine as partial function Subset of Bit string text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/dual_vector_space?rev=1395396676&do=diff Dual vector space Set context $X$...$\mathcal F$-vector space definiendum $X^d \equiv \mathcal L(X,\mathcal F)$ Discussion The elements of $X^d$ are called linear functionals. Parents Context Linear operator space, Vector space text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/eigenvalue?rev=1395396676&do=diff Eigenvalue Set context $V$ ... $F$-vectorspace context $A\in\mathrm{End}(V)$ definiendum $\lambda\in\mathrm{EigenVal}(A)$ postulate $\exists v.\ A\ v=\lambda\ v$ Discussion Parents Subset of Field Context Vector space endomorphism text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/eigenvector?rev=1395396676&do=diff Eigenvector Set context $V$...$F$-vectorspace context $A\in\mathrm{End}(V)$ definiendum $\mathrm{EigenVec}(A)\equiv\{v\ |\ \exists (\lambda\in F).\ A\ v=\lambda\ v\}$ Discussion Parents Subset of Vector space Context Vector space endomorphism text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/elementary_volume_of_%E2%84%9D%E2%81%BF?rev=1395396676&do=diff Elementary volume of ℝⁿ Set context $p\in \mathbb N$ definiendum $\mathrm{vol}:\mathfrak J^p\to \mathrm R$ definiendum $\mathrm{vol}(]a,b]):=\prod_{i=1}^p (b_i-a_i)$ Discussion This is also called the Lebesgue pre-measure Parents Context Half-open subsets of ℝⁿ, Finite product of complex numbers text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/empty_graph?rev=1395396676&do=diff Empty graph Set context $V$ ... set definiendum $ \langle V,\emptyset,\psi\rangle = \mathrm{it}(V) $ postulate $ \psi:\emptyset\to\emptyset $ Discussion The empty graph on $V$ doesn't contain more information than the set $V$ itself. Parents Subset of Undirected graph text/html 2015-10-09T15:40:38+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/empty_set?rev=1444398038&do=diff Empty set Set definiendum $x\in\emptyset$ postulate $\bot$ ---------- Here we extend our language by the symbol $\emptyset$ (sometimes $\varnothing$ or $\{\}$ is used also), denoting $\{x\mid \bot\}$, i.e. $\emptyset\equiv\{x\mid \bot\}$ This is sensible in our set theory if the proposition $\exists! y.\,y=\{x\mid \bot\}$ holds, which really is an abbreviation for$\exists! y.\,\forall x.\,\left(x\in y\Leftrightarrow \bot\right)$$\exists! y.\ \phi(y)$$\exists y.\ \forall z.\ (\p… text/html 2014-12-04T22:34:55+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/endofunctor?rev=1417728895&do=diff Endofunctor Collection context ${\bf C}$ ... category definiendum ${\bf C}\longrightarrow{\bf C}$ Discussion Reference Parents Subset of Functor Context* Categories text/html 2015-04-15T20:52:54+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/entry_structure?rev=1429123974&do=diff Entry structure Meta Order of keywords 'Entry type' 'Formal definition' 'Comment on formalities' Elaboration Alternative definitions Universal property Discussion Theorems Examples Reference Context Requirements Subset of The 'Comment on formalities' is a plain text that adds everything that keeps the definition above to be fully formal. The $:=$$\circ$$a,b$$\alpha,\beta$$\land, \lor, \forall, \exists, \neg, (, ), =$$\frac{P}{Q}, \prod, \sum$$\dots$$a,b,\dots$${\bf C}\dots\ma… text/html 2015-11-10T18:08:51+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/epanechnikov-like_bump_._pdf?rev=1447175331&do=diff Epanechnikov-like bump . PDF Function context $x_0,d:{\mathbb R}$ definition $k_n:{\mathbb N}_{\ge 0}\to{\mathbb R}\to{\mathbb R}_{\ge 0}$ definition $k_n(x):=\begin{cases} \dfrac{1}{2d}\left(1+\dfrac{1}{2n}\right)\left(1-\left(\dfrac{x-x_0}{d}\right)^{2n}\right) &\hspace{.5cm} \mathrm{if}\hspace{.5cm} \vert x \vert\le 1 \\\\ 0 \hspace{.5cm} &\hspace{.5cm} \mathrm{else} \end{cases} $ ---------- Discussion $\lim_{n\to\infty}k_n$ is the normed $x_0$-centered rectangle of heig… text/html 2015-03-17T00:00:55+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/epimorphism?rev=1426546855&do=diff Epimorphism Collection context ${\bf C}$ ... category definiendum $f \in\mathrm{it} $ inclusion $f:{\bf C}[A,B]$ postulate $\langle B,\prod_{B}1_A\rangle$ ... pushout of $f$ along itself ---------- Discussion See Monomorphism. In ${\bf{Set}}$ the epimorphisms are the surjections. But people like to point out that in general, epis are quite different from surjections and also more difficult to classify (as opposed to monos, which mostly behave exactly like injections)… text/html 2016-05-21T18:26:19+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/equalizer_._category_theory?rev=1463847979&do=diff Equalizer . category theory Tuple context $F:(a{\overset{\rightarrow}{\rightarrow}}b)\longrightarrow{\bf C}$ definition $\langle E,\langle Fa\mapsto e,Fb\mapsto e'\rangle\rangle := \mathrm{lim}\,F$ Here $a{\overset{\rightarrow}{\rightarrow}}b$ denotes the two object category with two parallel arrows. Let $A:=Fa$, $B:=Fb$ and $f$ be one of the fmap images. Then $e:{\bf C}[E,A]$ and the other arrow must be $e\circ f$ and is hence usually ignored. ---------- ${\bf{Set}}$$f,g:A\to B$… text/html 2014-08-23T18:22:45+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/equivalence_class?rev=1408810965&do=diff Equivalence class Set context $X$ context $x\in X$ context $\sim\in\text{Equiv}(X) $ definiendum $ y\in [x]_\sim $ postulate $y\sim x$ Ramifications Discussion For some combinatorics on the number of possible equivalence classes, see Bell number (Wikipedia). Reference Wikipedia: Equivalence class Parents Context text/html 2014-12-04T22:34:51+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/equivalence_of_categories?rev=1417728891&do=diff Equivalence of categories Collection context ${\bf C},{\bf D}$ ... categories definiendum $F$ in ${\bf D}\simeq{\bf C}$ inclusion $F$ in ${\bf D}\longrightarrow{\bf C}$ exists $G$ in ${\bf C}\longrightarrow{\bf D}$ exists $\alpha$ in $FG\cong Id_{\bf C}$ exists $\beta$ in $Id_{\bf D}\cong GF$ Discussion Elaboration Here $Id_{\bf C}$ denotes the identity functor on ${\bf C}$. Motivation We want to formalize when two categories are $F$$G$$G\circ … text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/equivalence_relation?rev=1395396676&do=diff Equivalence relation Set context $X$ definiendum $ \sim \in \text{EquivRel}(X) $ context $ \sim \in \mathrm{Rel}(X) $ $x,y,z\in X$ postulate $ x\sim x $ postulate $ x\sim y \Leftrightarrow y\sim x $ postulate $ x\sim y \land y\sim z \Leftrightarrow x\sim z $ Discussion The relation $\sim$ is an equivalence relation, if it's in the intersection of all reflexive, all symmetric and all transitive relation. Hence text/html 2014-10-29T20:28:40+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/euclidean_space?rev=1414610920&do=diff Euclidean space Set context $ n\in \mathbb N $ definiendum $\mathbb E^n \equiv \langle d,\mathbb R^n\rangle $ inclusion $d:\mathbb R^n\times \mathbb R^n\to \mathbb R_+ $ postulate $d(x,y):=\left( \sum_{j=1}^n (x_j-y_j)^2 \right)^\frac{1}{2} $ Discussion Reference Wikipedia: Euclidean space Parents Subset of Metric space Context Real coordinate space text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/euclidean_topology?rev=1395396676&do=diff Euclidean topology Set context $p\in \mathbb N$ "definition via Euclidean metric" Discussion "Extension for $\overline{\mathbb R}^p$ (Extended real number line) - not sure if it deserves its own entry." Reference Wikipedia: Topological space Parents Subset of Topological space Context Real number text/html 2015-03-30T00:36:01+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/euler-lagrange_equations?rev=1427668561&do=diff Euler-Lagrange equations Set context $s\in\mathbb N $ context $L:C^2(\mathbb R^s\times\mathbb R^s\times\mathbb R,\mathbb R) $ definiendum $q\in$ it inclusion $q:C(\mathbb R,\mathbb R^s) $ let $\diamond\ q(t)$ let $\diamond\ L(x^1,\dots,x^s,v^1,\dots,v^s,t)$ range $j\in\{1,\dots,s\}$ postulate $\left(\dfrac{\mathrm d}{\mathrm dt}\dfrac{\partial L}{\partial v^j}\right)(q,q',t) - \dfrac{\partial L}{\partial x^j}(q,q',t) = 0$ ----… text/html 2015-11-13T17:07:17+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/euler_beta_function?rev=1447430837&do=diff Euler beta function Function definition $ {\mathrm B}: \{z\ |\ \mathfrak{R}(z) > 0 \}^2 \to \mathbb C$ definition $ {\mathrm B}(p,q) := \int_0^1 \tau^{p-1}(1-\tau)^{q-1}\,\mathrm d\tau $ ---------- Theorems For natural numbers * ${\large{n \choose k}}=(n+1)\cdot\dfrac{1}{{\mathrm B}(n-k+1,k+1)}$ * $\dfrac{1}{{\mathrm B}(x,y)} = \frac{x\,y}{x+y} \prod_{n=1}^\infty \left( 1 + \dfrac{x\,y}{n\,(x+y+n)}\right)$ Reference Wikipedia: Beta function ---------- Subset of ℂ valued … text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/euler_s_number?rev=1395396676&do=diff Euler's number Set definiendum $$\mathrm{e}\equiv \mathrm{exp}(1)$$ Discussion References Wikipedia: <http://en.wikipedia.org/wiki/Eulers_number> Parents Context Exponential function text/html 2019-10-06T17:26:04+0200 Anonymous (anonymous@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 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/exptime?rev=1395396676&do=diff EXPTIME Set definiendum $ \mathrm{\bf{EXPTIME}} = \bigcup_{k\in\mathbb{N}} \mathrm{\bf{DTIME}}\left((2^n)^k\right) $ Discussion Parents Requirements DTIME Subset of Bit string text/html 2015-08-25T23:14:47+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/extended_quantum_action_functional_._finite?rev=1440537287&do=diff Extended quantum action functional . finite Partial function context $ \mathbb K = \mathbb C \lor \mathbb R $ context $ m\in\mathbb N $ context $ D $ .... self-adjoint operator in $\mathbb K^m$ with well behaved inverse at least for $D+i\,\varepsilon\,\mathrm 1$ definiendum $Z:(\mathbb K^2\to\mathbb R)\to \mathbb K^4\to \mathbb K $ definiendum $Z_{\mathcal L_\mathrm{int}}(J,K,\phi,\psi):=\mathrm{e}^{i\hbar^{-1}\sum_{i=1}^m\mathcal L_\mathrm{int}\left(-i\,\hbar\fr… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/extended_real_number_line?rev=1395396676&do=diff Extended real number line Set postulate $ \overline{\mathbb R} \equiv \mathbb R\cup\{-\infty,\infty\} $ "todo: model $-\infty$ and $\infty$" Discussion Reference Wikipedia: Extended real number line Parents Context Real number text/html 2014-09-23T10:36:21+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/f-algebra?rev=1411461381&do=diff F-algebra Collection context $F$ in ${\bf C}\longrightarrow{\bf C}$ definiendum $\langle A,\alpha\rangle$ in $\text{it}$ postulate $\alpha:{\bf C}[FA,A]$ Discussion Think types $\mathrm{a}$ and $\alpha$'s of type type Algebra f a = f a -> a Example The following examples assume that ${\bf C}$ contains all the relevant ingredients (e.g. products).$+:\mathbb{N}\times\mathbb{N}\to\mathbb{N}$$\langle \mathbb{N},+\rangle$$F$$FX:=X\times X$$M$$X$$\alpha:M\times X\to X$$FX:=M\… text/html 2015-12-14T18:26:38+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/factorial_function?rev=1450113998&do=diff Factorial function Function definiendum $!: \mathbb N\to \mathbb N$ definiendum $n\mapsto n!:=\prod_{k=1}^n\ k $ ---------- Discussion Thinking of $n!=\left.\frac{{\mathrm d}^n}{{\mathrm d}x^n}\right|_{x=0}x^n$ and Fermat theory, I though there must be an expression for $n!$ which is more algebraic and indeed I found Table[Sum[(-1)^k*Binomial[n, k] (-k)^n, {k, 1, n}], {n, 1, 8}] $n$$\sum_{k=0}^n\dfrac{(-1)^k (-k)^n}{k!\,(n - k)!}=1$ text/html 2015-04-17T10:31:18+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/faithful_functor?rev=1429259478&do=diff Faithful functor Collection definiendum $F$ in it inclusion $F:{\bf C}\longrightarrow{\bf D}$ inclusion ${\bf C},{\bf D}$ ... locally small postulate $\mathrm{fmap}(F)_{X,Y}:{\bf C}[X,Y]\to{\bf D}[FX,FY]$ ... injective "Here I needed to introduc the notation $\mathrm{fmap}(F)_{X,Y}$ for the polymorphic function at $X$ and $Y$" ---------- Reference Wikipedia: text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/falling_sequence?rev=1395396676&do=diff Falling sequence Set context $X$ definiendum $A\in \mathrm{FallingSequence}(X) $ postulate $A\in \mathrm{InfSequence}(X) $ $n\in \mathbb N$ postulate $A_{n+1}\subseteq A_n $ Discussion Ramifications For falling sequences we have: $\lim_{n\to\infty}A_n=\bigcap_{n=1}^\infty A_n$. For growing sequences we have: $\lim_{n\to\infty}A_n=\bigcup_{n=1}^\infty A_n$. Predicates predicate $A_n\downarrow \hat A \equiv A\in \mathrm{FallingSequence}(X)\ \land\ \lim_{n\to\in… text/html 2014-09-28T11:51:33+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/field?rev=1411897893&do=diff Field Set context $X$ definiendum $\langle X,+,* \rangle \in \mathrm{field}(X)$ postulate $\langle X,+,* \rangle \in \mathrm{divisionRing}(X)$ postulate $\langle X,* \rangle \in \mathrm{abelianGroup}(X)$ Discussion A field is essentially two compatible abelian groups over a set $X$, one of which is necessarily commutative. Compatible in the sense of the distributive laws of a ring, which is asymmetrical with respect to $+$$*$$*$$F$$p^n$$p$ text/html 2017-03-18T17:31:39+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/finite_exponential_power?rev=1489854699&do=diff Finite exponential power Function context $ m\in{\mathbb N} definition ${\rm pexp}_n: \mathbb C\to\mathbb C$ definition ${\rm pexp}_n(z) := \left(1 + \dfrac {x} {n} \right)^n $ ---------- ${\rm pexp}_n(x) = \sum_{k=0}^n a_k(n)\dfrac {1} {k!} x^k $ with $a_k(n)=\prod_{j=1}^{k-1}\left(1-\dfrac{k-j}{n}\right)\le 1$ Elaboration $\left(x+y\right)^m=\sum_{k=0}^m \dfrac{n!}{k!\,(m-k)!} x^k y^{m-k}$ so $\left(1 + b(n)\,x \right)^n = \sum_{k=0}^n \left( b(n)^{-k}\dfrac {n!} {(n… text/html 2017-03-18T17:42:45+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/finite_exponential_series?rev=1489855365&do=diff Finite exponential series Function context $ m\in{\mathbb N} definition $\exp_n: \mathbb C\to\mathbb C$ definition $\exp_n(z):=\sum_{k=0}^n \dfrac{1}{k!} z^k $ ---------- $\frac{\mathrm d}{\mathrm d z}\mathrm{exp}_0(z) = 1 $ $\mathrm{exp}_{n}(z) = \mathrm{exp}_{n-1}(z) + \dfrac{1}{n!} z^n$ Theorems $\frac{\mathrm d}{\mathrm d z}\mathrm{exp}_0(z) = 0 $ $\frac{\mathrm d}{\mathrm d z}\mathrm{exp}_n(z) = \mathrm{exp}_{n-1}(z) = \mathrm{exp}_n(z) - \dfrac{1}{n!} z^n$ Alterna… text/html 2016-06-10T02:00:11+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/finite_geometric_series?rev=1465516811&do=diff Finite geometric series Function context $n\in{\mathbb N}$ definition $Q_n: \mathbb C\to\mathbb C$ definition $Q_n(z):=\sum_{k=0}^n z^k $ ---------- "todo: In fact" $\sum_{k=0}^n a^k\,b^{n-k} = \dfrac{1}{a-b}(a^{n+1}-b^{n+1})$ so $\sum_{k=0}^n z^k = \dfrac{1}{1-z}(1-z^{n+1})$ i.e. $\dfrac{1}{1-z} = \dfrac{1}{1-z\cdot{z^n}} \sum_{k=0}^n z^k$ i.e. $z^{n+1}-1 = (z-1)\sum_{k=0}^n z^k$ Remarks The last line can be used to show that a bunch of ugly expressions have a fa… text/html 2015-04-28T14:35:11+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/finite_product_of_complex_numbers?rev=1430224511&do=diff Finite product of complex numbers Function definition it $\equiv$ Finite sum over a monoid w.r.t $\langle\!\langle {\mathbb C},\cdot \rangle\!\rangle$ ---------- $\prod_{k=1}^n\left(1+\frac{1}{k}\right)=n+1$ ---------- Subset of Finite sum over a monoid Requirements Complex number text/html 2016-07-03T20:54:21+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/finite_sequence?rev=1467572061&do=diff Finite sequence Set context $X$ definiendum $ A\in \text{FinSequence}(X) $ range $ n\in \mathbb N $ postulate $\exists n.\ A: \mathrm{range}(n)\to X$ ---------- Reference ---------- Related Natural number range text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/finite_subset?rev=1395396676&do=diff Finite subset Set $X$ $ Z\in \text{Fin}(X) $ The set of finite subsets of $X$ $ Z\in \mathcal P(X) $ $\text{isFinite}(Z)$ Ramifications Reference Mizar: FINSUB_1 Parents Set constructor Related Finite sequence text/html 2015-06-20T16:22:12+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/finite_sum_of_complex_numbers?rev=1434810132&do=diff Finite sum of complex numbers Set context $ (z_i) \in \mathrm{FinSequence}(\mathbb C)$ context $ n=\mathrm{length}((z_i)) $ definiendum $\sum: \mathrm{FinSequence}(\mathbb C)\to \mathbb C$ definiendum $\sum_{i=1}^n\ z_i:= \begin{cases} 0 & \mathrm{if}\ n=0\\\\ \left(\sum_{i=1}^{n-1}\ z_i\right)\ +\ z_n & \mathrm{else} \end{cases}$ ---------- Theorem $\sum_{k=1}^n z^k=\frac{1}{1-z}(1-z^{n+1})$ ---------- Subset of Finite sum over a monoid text/html 2015-04-17T21:44:45+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/finite_sum_over_a_monoid?rev=1429299885&do=diff Finite sum over a monoid Function context $ \langle\!\langle M,* \rangle\!\rangle$ ... monoid let $e$ ... unit of $\langle\!\langle M,* \rangle\!\rangle$ definition $\sum^n:\prod_{S:\mathrm{Sequence}(M)}\mathrm{range}(\mathrm{length}(S))\to M$ definition $\sum_{k=1}^n\ S_k:= \begin{cases} e & \mathrm{if}\ n=0\\\\ \left(\sum_{k=1}^{n-1}\ S_k\right)\ *\ S_n & \mathrm{else} \end{cases}$ ---------- Discussion Here $\prod_{S:\mathrm{Sequence}(M)}$ is a dependent func… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/finite_undirected_graph?rev=1395396676&do=diff Finite undirected graph Set context $V,E$ ... set definiendum $ \langle V,E,\psi\rangle \in \mathrm{it}(E,V) $ postulate $\langle V,E,\psi\rangle $ ... undirected graph postulate $ E,V $ ... finite Discussion Parents Subset of Undirected graph text/html 2020-05-20T14:23:48+0200 Anonymous (anonymous@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 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/floor_function?rev=1395396676&do=diff Floor function Function definiendum $ \lfloor\ \rfloor:\mathbb{R}\to\mathbb{Z}$ definiendum $ \lfloor x \rfloor:=\max\,\{n\in\mathbb{Z}\mid n\le x\} $ Discussion Parents Subset of ℤ valued function, Monotonically decreasing function Related Ceiling function text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/fokker-planck_equation?rev=1395396676&do=diff Fokker-Planck equation Set context $ n\in\mathbb N $ context $ \mu:C(\mathbb R^n,\mathbb R^n) $ context $ \mathsf{D}:C^2(\mathbb R^n,\mathbb R^{n^2}) $ range $ ::\mu(\mathbf{x}) $ range $ ::\mathsf{D}(\mathbf{x}) $ definiendum $ f \in \mathrm{it} $ postulate $ f:C^2(\mathbb R^n\times\mathbb R,\mathbb R) $ range $ ::f(\mathbf{x},t) $ postulate $ \frac{\partial }{\partial t} f = -\mathrm{div} (\mu \cdot f) + \sum_{i=1}^{n} \s… text/html 2016-06-20T18:04:27+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/foundational_temp1?rev=1466438667&do=diff Foundational temp1 Drawing arrows and coding functions $\succ$ Foundational temp1 $\succ$ Foundational temp1b Guide (Note) Logic (Note) Specifying syntax "Remark: With the Framework entries, I go up in complexity. Then, with the entries with mathematical content, I start with abstract and go to concrete." (Framework) Intuitionistic propositional logic (Note) Predicate logic ---------- text/html 2016-04-07T17:33:02+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/foundational_temp1b?rev=1460043182&do=diff Foundational temp1b Foundational temp1 $\succ$ Foundational temp1b $\succ$ On category theory basics • Guide " Fixing framework and axiomatics After this entry, the chain of definitions start" * List types * Fix Axiomatization of Cats * Fix Axiomatization of (Grothendieck-Tarski) set theory Parents text/html 2016-04-07T17:32:37+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/foundational_temp3?rev=1460043157&do=diff Foundational temp3 On universal morphisms $\succ$ Foundational temp3 $\succ$ Foundational temp4 • Guide Empty set "... and then set theory up to" Set universe (this includes First infinite von Neumann ordinal , the existence of which is granted by the Axiom of infinity (and is part of every Set universe as defined above)) text/html 2016-04-07T17:32:28+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/foundational_temp4?rev=1460043148&do=diff Foundational temp4 Foundational temp3 $\succ$ Foundational temp4 $\succ$ Foundational temp formal power series Guide Locally small categorySetHom-functorPresheaf categoryYoneda embedding Parents Sequel of Foundational temp3 Related Foundational temp3 text/html 2016-09-23T14:49:56+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/foundational_temp_formal_power_series?rev=1474634996&do=diff Foundational temp formal power series Foundational temp4 $\succ$ Foundational temp formal power series $\succ$ Guide "formal power series Probably must come only after group/ring/etc. (objects of study of abstract algebra)" "Q: how far back can analysis be pushed? " " "$(a_n)_n,(b_n)_n\in X^{\mathbb N}$$X^{\mathbb N}\to Z$$T$$X^{\mathbb N}\to Z^{\mathbb N}$$B$$X^{\mathbb N}\times Y^{\mathbb N}\to Z^{\mathbb N}$$\langle M,\cdot\rangle$$\sum_{i=0}^\infty$$a_n,b_n,B_k^{n,m}\in M$$T(b)_k… text/html 2016-12-29T16:31:47+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/frechet_derivative?rev=1483025507&do=diff Fréchet derivative Set context $X,Y$ ... Banach spaces with topology context $\mathcal O$ ... open in $X$ definiendum $D:\mathrm{Continuous}(\mathcal O,Y)\to \mathrm{Continuous}(\mathcal O,\mathrm{BoundedLinOp}(X,Y))$ definiendum $Df:=x\mapsto J_x^f$ For $J_x f$, see Linear approximation. Discussion This definition does nothing more than emphasizing the functionality of $L_x^f$ in $f$. Reference text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/frechet_derivative_chain_rule?rev=1395396676&do=diff Fréchet derivative chain rule Theorem context $X,Y,Z$ ... Banach spaces with topology context $f\in C(X,Y)$ context $g\in C(Y,Z)$ postulate $ D(g\circ f)=(Dg)\circ f\ \cdot\ Df $ where $\circ$ denotes the concatenation of functions of $X,Y$, which is taken to bind stronger than the concatenation $\cdot$ of linear operators.$f,g: \mathbb R\to\mathbb R$$\frac{\partial}{\partial x}g(f(x))=g'(f(x))\cdot f'(x)$ text/html 2016-09-30T15:56:07+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/frobenius_matrix?rev=1475243767&do=diff Frobenius matrix Set "" ---------- Discussion Reference Wikipedia: Frobenius matrix ---------- Subset of Matrix text/html 2015-08-17T15:04:07+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/fugacity?rev=1439816647&do=diff Fugacity Function definiendum $z(\beta,\mu):=\mathrm e^{\beta\cdot\mu}$ ---------- Discussion Not to be confused with the fugacity coeffient $\phi$, which is a pressure value used to model abbaviations from the ideal gas model pressures. The unitless normalized fugacity coeffient is called thermodynamic activity $a$$z$$a_B$ text/html 2014-12-26T17:57:11+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/function?rev=1419613031&do=diff Function Set context $X,Y$ definiendum $f\in Y^X $ inclusion $f\in$ partial function$(X,Y)$ postulate $\bigcup\pi_1(f)=X $ Discussion One defines $\mathrm{dom}(f):=\bigcup\pi_1(f)$. The requirement $\mathrm{dom}(f)=X$ just says that the function must make sense for all inputs. Predicates Notice that for $f\in Y^X$ and $Y\subset Z$ we also have $f\subset Z^X$$ f\in Y^X \land \mathrm{codom}(f)=Y $$\mathrm{codom}$$ f:X\to Y $$f\dots \text{functional} \equiv \exists X,Y… text/html 2015-06-20T16:17:11+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/function_integral?rev=1434809831&do=diff Function integral Set context $\mathbb K = \overline{\mathbb R}\lor \mathbb C$ context $\langle X,\Sigma,\mu\rangle\in \mathrm{MeasureSpace}(X)$ definiendum $\int_X: (X\to \mathbb K)\to \mathbb K$ definiendum $\int_X\ f\ \mathrm d\mu:=\int_X\ (\mathrm{Re}f)^+\ \mathrm d\mu-\int_X\ (\mathrm{Re}f)^-\ \mathrm d\mu+i\ \left( \int_X\ (\mathrm{Im}f)^+\ \mathrm d\mu-\ \int_X\ (\mathrm{Im}f)^-\ \mathrm d\mu \right)$ Notice that the integral on the right hand side here is that f… text/html 2017-01-05T00:00:03+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/function_integral_on_%E2%84%9D%E2%81%BF?rev=1483570803&do=diff Function integral on ℝⁿ Set context $p\in \mathbb N$ definiendum $I^p: (\mathbb R^p\to\overline{\mathbb R})\to\overline{\mathbb R}$ definiendum $I^p(f):=\int_{\mathbb R^p}\ f\ \mathrm d\lambda^p$ Discussion Because the integral above coincides with the Lebesgue–Stieltjes integral for the monotone function $F(x):=x$, we'll also denote $I^p(f)$ by $\int_{\mathbb R^p}\ f(x)\ \mathrm dx^p$ with the argument $x\in \mathbb R^p$ of $f$ becoming a dummy index.$f:X\to \mathbb R$$f'$$ \… text/html 2014-09-17T23:12:32+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/function_type?rev=1410988352&do=diff Function type Type context $X, Y$ ... type context $\Gamma$ ... context rule ${\large\frac{\Gamma,\,x\ :\ X\ \vdash\ y\ :\ Y}{\Gamma\ \vdash\ \lambda x.\,y\ :\ X\to Y}}$ (lambda abstraction) rule ${\large\frac{\Gamma\ \vdash\ f\ :\ X\to Y\hspace{1cm}\Gamma\ \vdash\ x\ :\ X}{\Gamma\ \vdash\ f\,x\ :\ Y}}$ (function application) Discussion Parse ${\large\frac{\Gamma,\,x\ :\ X\ \vdash\ y\ :\ Y}{\Gamma\ \vdash\ \lambda x.\,y\ :\ X\to Y}}$ as ${\large\frac{(\Gamma,\,(x\ :\ … text/html 2015-12-16T13:16:13+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/functions?rev=1450268173&do=diff Functions Meta ${\mathfrak D}_\mathbb{\to}$ ---------- Discussion Many definitions are implcit Caution: Many function definitions are implicit, sometimes secretly so. E.g. defining $f(x):=\sum_{n=0}^\infty\frac{(-1)^{3n}}{n!}z^n$ is defining $f(x):=\lim_{m\to\infty}\sum_{n=0}^m\frac{(-1)^{3n}}{n!}z^n$ and a Limit definition is always a Task to find said limit. Practically speaking, uor functions are partitially defined $\to$$\prod$$A\to{B}$$\prod_{a:A}B(a)$ text/html 2016-04-09T14:31:48+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/functor?rev=1460205108&do=diff Functor Collection context ${\bf C},{\bf D}$ ... category definiendum $F$ in ${\bf C}\longrightarrow{\bf D}$ rule ${\large\frac{ A\ :\ {\bf C} }{ FA\ :\ {\bf D} }}$ rule ${\large\frac{ f\ :\ {\bf C}[A,\,B] }{ F(f)\ :\ {\bf D}[FA,\,FB] }}$ postulate $F\,1_A=1_{FA}$ postulate $F(f\circ g)=F(f)\circ F(g)$ ---------- Discussion A function $f:C\to D$ maps a set of things $C=\{a,b,c,\dots\}$ into another set of things $D=\{f(a),f(b),f(c),\dots,\dots\}$ (rema… text/html 2015-04-07T11:54:44+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/functor_._haskell?rev=1428400484&do=diff Functor . Haskell Haskell class -- definition class Functor f where fmap :: (a -> b) -> f a -> f b -- methods (<$) :: a -> f b -> f a (<$) = fmap . const fmap id $\ \leftrightsquigarrow\ $ id fmap f . fmap g $\ \leftrightsquigarrow\ $ fmap (f . g) ---------- Discussion In Haskell, the first law implies the second. text/html 2015-12-17T19:21:39+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/functor_category?rev=1450376499&do=diff Functor category Collection context ${\bf C}$ ... small category context ${\bf D}$ ... category definiendum ${\bf D}^{\bf C}$ in $\mathrm{it}$ definition $\mathrm{Ob}_{{\bf D}^{\bf C}}:={\bf C}\longrightarrow{\bf D} $ definition ${\bf D}^{\bf C}[F,G]:=F\xrightarrow{\bullet}G$ ---------- Discussion Firstly, A class of sets together with functions between them form a category. The only job of the arrows between objects here is to transfer individual elements from … text/html 2014-12-26T16:06:18+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/functors?rev=1419606378&do=diff Functors Meta ${\mathfrak D}_\mathbb{\longrightarrow}$ ---------- ---------- Related Domain of discourse text/html 2016-10-26T16:10:05+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/gamma_function?rev=1477491005&do=diff Gamma function Function definition $\Gamma: \mathbb C\setminus\{-k\ |\ k\in\mathbb N\}\to \mathbb N$ definition $\Gamma(z) := \begin{cases} \int_0^\infty\ \ t^{z-1}\ \mathrm{e}^{-t}\ \mathrm d t & \mathrm{if}\ \mathrm{Re}(z)>0 \\\\ \frac{1}{z}\Gamma(z+1) & \mathrm{else} \end{cases}$ ---------- Discussion $\Gamma(z)=\Pi(z-1)$ Theorems $n\in\mathbb N\land n\neq 0 \implies \Gamma(n)=(n-1)! $ $\Gamma(z+1) = z\cdot\Gamma(z) $ $\Gamma(z)\cdot\Gamma(1-z) =\frac{\pi}{\sin(\pi\ z)} $ … text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/general_positive_semi-definite_matrix?rev=1395396676&do=diff General positive semi-definite matrix Set context $n\in\mathbb N$ definiendum $ A \in \mathrm{it}(n) $ postulate $ A \in \mathrm{SquareMatrix}(n,\mathbb C) $ $ x \in \text{ColumnVector}(n,\mathbb C) $ postulate $ A \in \mathrm{SquareMatrix}(n,\mathbb C) $ postulate $ x^* A\ x \ge 0 $ Discussion Reference Wikipedia: Positive-definite matrix Parents Subset of Square matrix Context Column vector text/html 2018-02-15T18:59:27+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/generalized_hypergeometric_function?rev=1518717567&do=diff Generalized hypergeometric function Function definition $??$ definition ${}_pF_q[a_1,…,a_p; b_1,…,b_q](z):=\sum_{n=0}^\infty c_n z^n$ with $c_n = \dfrac{1}{n!}\dfrac{\prod_{k=1}^p a_k^{\overline{n}}}{\prod_{j=1}^q b_j^{\overline{n}}}$ ---------- Discussion Definition The coefficient can more explicitly written as $c_n = \prod_{m=0}^{n-1}\dfrac{1}{(1+m)}\dfrac{\prod_{k=1}^p(a_k+m)}{\prod_{j=1}^q(b_j+m)}$ or written down in Terms of Gamma functions. The version I chose … text/html 2016-05-24T19:58:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/glossary?rev=1464112696&do=diff Glossary Meta Literature For a personal literature list, see Literature. Predicates For a list of the predicates formally used in this wiki, as well as the locations of their definitions, see predicate library. Physical constants Here are the physical constants used in the wiki.$\hbar$$k_B$ text/html 2016-01-28T01:06:52+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/grand_canonical_entropy?rev=1453939612&do=diff Grand canonical entropy Set context $ \Omega(\beta,\mu) $ ... grand canonical free energy definiendum $S(T,\mu):=-\dfrac{\partial}{\partial T}\Omega\left(\dfrac{1}{k_B T},\mu\right) $ ---------- Discussion This mirrors the definition in canonical entropy. ---------- Context Grand canonical free energy text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/grand_canonical_expectation_value?rev=1395396676&do=diff Grand canonical expectation value Set context $ w $ ... grand canonical weight definiendum $\langle A\rangle:=\sum_{N=0}^\infty w_N\cdot \langle A_N\rangle_N$ The functional $\langle \cdot\rangle_N$ denotes the expectation in the canonical ensamble of particle number $N$. So the grand canonical expectation value $\langle \cdot\rangle$ takes sequences of observables to a real.$U=\langle H\rangle$$U$$N$$H_N$$f$$A$$A_N$$f(A)$$f(A_N)$ text/html 2015-08-16T14:02:08+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/grand_canonical_free_energy?rev=1439726528&do=diff Grand canonical free energy Set context $ \Omega(\beta,\mu) $ ... grand canonical potential definiendum $ F(\beta,\mu):=\mu\cdot\langle \hat N\rangle + \Omega(\beta,\mu) $ ---------- Discussion ---------- Context Grand potential text/html 2016-03-04T18:10:19+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/grand_canonical_partition_function?rev=1457111419&do=diff Grand canonical partition function Set context $J\in\mathbb N$ $j\in \{1,\dots,J\}$ range $N_j\in \mathbb N$ context $N_j^\text{max}\in \mathbb N \cup \{\infty\}$ context $Z_{N_j}(\beta) $ sequences in $N_j$ of canonical partition functions with length $N_j^\text{max}$ definiendum $\Xi(\beta,\mu_1,\dots,\mu_J):=\sum_{j=1}^J \sum_{{N_j}=0}^{N_j^\text{max}}\ z(\beta,\mu_j)^{N_j}\cdot Z_{N_j}(\beta) $ Here $z$ denotes the fugacity. The quantity $J$ denotes … text/html 2015-08-16T14:59:54+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/grand_canonical_weight?rev=1439729994&do=diff Grand canonical weight Set context $ \Xi(\beta,\mu) $ ... grand canonical partition function definiendum $w: \mathbb N \to {\mathbb R\times\mathbb R} \to \mathbb R$ definiendum $w_N(\beta,\mu):=\frac{1}{\Xi(\beta,\mu)}\ z(\beta,\mu)^N\cdot Z_N(\beta)$ ---------- ---------- Requirements Grand canonical partition function, Fugacity text/html 2015-08-16T14:27:07+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/grand_potential?rev=1439728027&do=diff Grand potential Set context $ \Xi(\beta,\mu) $ ... grand canonical partition function definiendum $ \Omega(\beta,\mu):=-\frac{1}{\beta}\,\mathrm{ln}(\Xi(\beta,\mu)) $ ---------- Discussion This mirrors the classical microcanonical entropy and the classical canonical free energy. Also, think $\rho \sim {\mathrm e}^{-H/T}$ or $H \sim -T \ln(\rho)$. Reference Wikipedia Grand potential ---------- Related Context Grand canonical partition function text/html 2016-09-09T09:46:21+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/graph?rev=1473407181&do=diff Graph Set context $ V,E $ ... set definiendum $ \mathrm{it}(E,V) = \mathrm{undirected\ graph}(E,V) \cup \mathrm{directed\ graph}(E,V) $ Discussion Predicates For a graph $G=\langle V,E,\psi\rangle$, we write predicate $a$ ... edge in $G \equiv a\in\mathrm{im}\ \psi$ Parents Context Undirected graph, Directed graph text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/graph_edges?rev=1395396676&do=diff Graph edges Function definiendum $ \mathrm{dom}\ E=\mathrm{graph} $ definiendum $ E(\langle V,\langle A,\psi\rangle\rangle):=\psi(A) $ Discussion Parents Context Graph text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/graph_vertices?rev=1395396676&do=diff Graph vertices Function definiendum $ \mathrm{dom}\ E=\mathrm{graph} $ definiendum $ E(\langle V,\langle A,\psi\rangle\rangle):=V $ Discussion Parents Context Graph text/html 2015-08-24T19:32:47+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/grothendieck_universe?rev=1440437567&do=diff Grothendieck universe Collection definiendum ${\mathfrak G}$ in it postulate ${\mathfrak G}$ ... set forall $X\in{\mathfrak G}$ postulate $\mathcal{P}(X) \subseteq{\mathfrak G}$ forall $x\in X$ postulate $x\in{\mathfrak G}$ exists $P\in{\mathfrak G}$ postulate $\mathcal{P}(X) \subseteq P$ forall $Y$ ... set postulate $Y \subseteq {\mathfrak G} \implies Y\ {\approx}\ {\mathfrak G}\lor Y \in {\mathfrak G} $ ---------- Discuss… text/html 2015-04-16T19:17:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/group?rev=1429204636&do=diff Group Set context $G$ definiendum $ \langle G,* \rangle \in \mathrm{it}$ inclusion $\langle G,* \rangle \in \mathrm{monoid}(G)$ let $e$ such that $\forall g.\, e*a=a*e=a$ range $g,g^{-1}\in G$ postulate $\forall g.\,\exists g^{-1}.\;(g*g^{-1}=g^{-1}*g=e)$ ---------- Alternative definitions Sharper definitions We could just define left units and left inverses and prove from the group axioms that they are already units and inverses.$\langl… text/html 2014-12-03T23:06:00+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/groupoid?rev=1417644360&do=diff Groupoid Collection definiendum ${\bf C}$ in it inclusion ${\bf C}$ ... category forall $A,B\in{\bf C}$ forall $f:{\bf C}[B,A]$ postulate $f$ ... isomorphism Discussion Reference Wikipedia: Groupoid Parents Requirements Isomorphism text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/growing_sequence?rev=1395396676&do=diff Growing sequence Set context $X$ definiendum $A\in \mathrm{GrowingSequence}(X) $ postulate $A\in \mathrm{InfSequence}(X) $ $n\in \mathbb N$ postulate $A_{n}\subseteq A_{n+1} $ Discussion Ramifications For falling sequences we have: $\lim_{n\to\infty}A_n=\bigcap_{n=1}^\infty A_n$. For growing sequences we have: $\lim_{n\to\infty}A_n=\bigcup_{n=1}^\infty A_n$. Predicates predicate $A_n\uparrow \hat A \equiv A\in \mathrm{GrowingSequence}(X)\ \land\ \lim_{n\to\in… text/html 2016-05-17T18:03:01+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/guideline?rev=1463500981&do=diff Guideline On syntax $\blacktriangleright$ Guideline Note What's different between 'An apple pie from scratch' the AoC graph itself * Book-like character and hence linear. To read it, follow the red path in the axiomsofchoice graph. * The concepts are development in a logical/mathematical fashion too, while the graph shows a not necessarily linear web of dependencies.$\gg$ text/html 2016-01-26T15:13:23+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/gx_fx?rev=1453817603&do=diff g(x)^f(x) Function context $ f,g : {\mathbb R} \to {\mathbb R} $ definition $ x\mapsto g(x)^{f(x)}:??$ ---------- Theorem $\dfrac{{\mathrm d}}{{\mathrm d}x} g(x)^{f(x)} = \left[f(x)\,g'(x) + f'(x)\,g(x)\cdot \log\left(g(x)\right)\right]\cdot g(x)^{f(x)-1} $ Special cases $\dfrac{{\mathrm d}}{{\mathrm d}x} x^c = c\cdot x^{c-1}$ $\dfrac{{\mathrm d}}{{\mathrm d}x} c^x = \log(c) \cdot c^x$ Reference ---------- Element of Function text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/half-open_subsets_of_%E2%84%9D%E2%81%BF?rev=1395396676&do=diff Half-open subsets of ℝⁿ Set context $p\in \mathbb N$ definiendum $\mathfrak J^p\equiv\{\ ]a,b]\ |\ a,b\in\mathbb R^p\ \land\ a\le b\}$ Discussion This set is one which generates the $\sigma$-Algebra $\mathcal B^p$ of Borel subset of $\mathbb R^p$. We also write definiendum $\mathfrak J\equiv \mathfrak J^1$ Parents Context Real coordinate space text/html 2016-09-08T23:00:10+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/hamiltonian?rev=1473368410&do=diff Hamiltonian Function "todo" ---------- Discussion Eigenstate Given one system in isolation and a corresponding Hamiltonian with a gap (positive non-zero eigenstate), the (ratios of the) differences between eigenvalues is more relevant than the value of the eigenvalues themselves. This is because we may be able to rescale our units so that $w$$2\pi$$W\,|w\rangle = w\,|w\rangle$$W\,|u\rangle = u\,|u\rangle$$u>w$$u = w + \Delta_{wu}$${\mathrm e}^{-iw}$${\mathrm e}^{-iu}={\mathrm e}^{-iw}{\ma… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/hamiltonian_equations?rev=1395396676&do=diff Hamiltonian equations Set context $ \langle \mathcal M, H\rangle $ ... Classical Hamiltonian system definiendum $ \pi \in \mathrm{it} $ postulate $ \pi:C(\mathbb R,\Gamma_{\mathcal M}) $ postulate $ \pi'(t) = X_H(\pi(t),t) $ "todo: Hamiltonian vector field" Discussion Equivalent definitions @#55CCEE: context range $ {\bf q} \in \mathcal M $ range $ {\bf p} \in T^*\mathcal M $$ H:: H({\bf q},{\bf p},t)$$ \langle q,p \rangle \in \mathrm{it} $$ … text/html 2016-08-31T15:32:34+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/harmonic_oscillator_hamiltonian?rev=1472650354&do=diff Harmonic oscillator Hamiltonian Function $A=\kappa*\left( -\dfrac{1}{\kappa}\left(L\dfrac{\partial}{\partial x}\right)^2+\kappa\,\left(\dfrac{x-x_0}{L}\right)^2 \right)$ ---------- Discussion Remark Another “quantum harmonical oscillator” is a model which looks similar, except $x$ is an operator $x(t)$ (and one a priori more general than right multiplication by $x$ as here) and where instead of $\dfrac{\partial}{\partial x}$$\dfrac{1}{L^2}x'(t)$$\kappa$$t$$a$$x$$\kappa$$\kappa\cdot x$$x$$x… text/html 2016-10-16T15:09:33+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/hask?rev=1476623373&do=diff Hask Note "todo: objects are types arrows are extensionally identified Haskell functions identity morphism: for each object, the identity morphism is the instance of the polymorphic identity for that type The identity for a type 'a' is '(id :: a$\to$$\times$$\pi_1,\pi_2$$'(\text{undefined},\text{undefined})'$$\text{undefined}$$\forall x. f(x)=g(x)\implies f=g$$\forall h. h(f)=h(g)\implies f=g$$h$$\text{flatten}$ text/html 2016-04-11T17:29:18+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/haskell?rev=1460388558&do=diff Haskell Note Language "reserved words: case class if import let module data default in infix newtype of deriving do infixl infixr then type else instance where standard operators" "note: all type variables, as in 'a -> a' are not free, i.e. implicitly bound." text/html 2016-05-10T18:21:28+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/haskell_type_system?rev=1462897288&do=diff Haskell type system Note Haskell has a two-level type system. There kinds which are inhibited by types, and these types are further inhibited by terms. Kind and type inference can be done in the Haskell ghci-environment via ':k' and ':t', respectively.$\star$$k_1\longrightarrow k_2$$k_1,k_2$$k_1\longrightarrow k_2$$\mathrm{Int}\ :\ \star\hspace{1cm}$$[]\ :\ \star\longrightarrow\star\hspace{1cm}$$[\mathrm{Int}]\ :\ \star\hspace{1cm}$$(\to)\ \mathrm{Int}\ :\ \star\longrightarrow\star\hspace{1cm}… text/html 2014-10-25T16:07:45+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/hausdorff_space?rev=1414246065&do=diff Hausdorff space Set definiendum $\langle X,\mathcal{T}_X\rangle \in\mathrm{it} $ inclusion $\langle X,\mathcal{T}_X\rangle$ ... topological space for all $x,y\in X, x\ne y$ exists $U_x\in$ Neighbourhood$(\mathcal{T}_X,x)$, $V_y\in$ Neighbourhood$(\mathcal{T}_X,y)$ postulate $U_x\cap V_y=\emptyset$ Discussion Idea A Hausdorff space $\langle X,\mathcal{T}_X\rangle$ is one where the topology $\mathcal{T}_X$ is fine enough so that separate points also have seper… text/html 2015-10-08T13:43:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/hereditarily_finite_set?rev=1444304596&do=diff Hereditarily finite set Set definiendum $V_\omega$ in it postulate $\emptyset\in V_\omega$ for all $x\in V_\omega$ postulate ${\mathcal P}(x)\in V_\omega $ postulate $x = \emptyset\ \lor\ \exists (y\in V_\omega).\ x = {\mathcal P}(y) $ ---------- Discussion Idea This is the set of all finite sets constructable when starting with $\emptyset$. It's the smallest infinite Grothendieck universe, as well as a model of ZFC. text/html 2020-08-23T16:33:26+0200 Anonymous (anonymous@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 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/hermitian_positive_semi-definite_matrix?rev=1395396676&do=diff Hermitian positive semi-definite matrix Set context $n\in\mathbb N$ definiendum $ A \in \mathrm{it}(n) $ postulate $ A \in \mathrm{HermitianMatrix}(n) $ $ x \in \text{ColumnVector}(n,\mathbb C) $ postulate $ A \in \mathrm{SquareMatrix}(n,\mathbb C) $ postulate $ x^* A\ x \ge 0 $ Discussion Reference Wikipedia: Positive-definite matrix Parents Subset of Hermitian matrix, General positive semi-definite matrix text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/higher_moments_of_the_stretched_exponential_function?rev=1395396676&do=diff Higher moments of the stretched exponential function Set definiendum $\mathrm{it}: \mathbb R^3 \to \mathbb R$ definiendum $\langle \tau_K,\beta,n \rangle \mapsto \int_0^\infty\ \ t^{n-1}\ \mathrm{e}^{-(t/\tau_K)^\beta}\ \mathrm d t$ Discussion Theorems $ \mathrm{it}(\tau_K,\beta,n)=\frac{\tau_K^n}{\beta}\Gamma(\frac{n}{\beta}) $ Reference Wikipedia: Stretched exponential function Parents Context Function integral on ℝⁿ, Complex exponents with positive real bases Related Gamma … text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/hilbert_space?rev=1395396676&do=diff Hilbert space Set definiendum $\mathrm{Hilbert}(V)\equiv \mathrm{PreHilbert}(V)\cap \mathrm{BanachSpace}(V)$ Discussion Reference Wikipedia: Hilbert space Parents Subset of Pre-Hilbert space, Banach space text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/hilbert_space_expectation_value?rev=1395396676&do=diff Hilbert space expectation value Set context $V$...Hilbert space definiendum $\langle\cdot\rangle_{-}:\mathrm{Observable}(V)\times V\to\mathbb R$ definiendum $\langle A \rangle_{\psi}:=\langle \psi | A\ \psi \rangle$ Discussion Theorems $A\ \psi=\lambda\ \psi \implies (\langle A \rangle_{\psi}=\lambda)\,\land\,(\Delta_\psi A=0)$ Parents Context Hilbert space, Observable Related Hilbert space mean value text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/hilbert_space_mean_value?rev=1395396676&do=diff Hilbert space mean value Set context $V$...Hilbert space definiendum $\overline{\cdot}_{-}:\mathrm{Observable}(V)\times V\to\mathbb R$ definiendum $\overline{A}_{\psi}:=\frac{\langle \psi | A\ \psi \rangle}{\Vert \psi \Vert^2}$ Discussion One can rewrite this in many ways using: * $\langle \psi | A\ \psi \rangle=\langle A \rangle_\psi$ * $\Vert \psi \Vert^2=\langle \psi | \psi \rangle=\langle 1 \rangle_\psi$ For any vector $\phi$ we have... * $\Delta_\psi A = \left(… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/hilbert_transform?rev=1395396676&do=diff Hilbert transform Partial Function definiendum $H: (\mathbb C\to\mathbb C)\to(\mathbb C\to\mathbb C)$ definiendum $H(f):=y\mapsto \frac{1}{\pi}\cdot\mathcal P\int_{-\infty}^\infty\frac{f(x)}{y-x}\,\mathrm dx$ Discussion $(H(f))=-f$ The Hilbert transform commutes with the Fourier transform up to a simple factor and is an anti-self adjoint operator relative to the duality pairing between $L^p(\mathbb R)$ and the dual space $L^q(\mathbb R)$. It is also used in the Kramers–Kronig relati… text/html 2019-10-06T23:37:09+0200 Anonymous (anonymous@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 2015-02-26T11:55:35+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/hom-functor?rev=1424948135&do=diff Hom-functor Functor context ${\bf C}$ ... locally small category context $A:\mathrm{Ob}_{\bf C}$ definiendum $\mathrm{Hom}_{\bf C}(A,-)$ inclusion $\mathrm{Hom}_{\bf C}(A,-)$ in ${\bf C}\longrightarrow{\bf Set}$ definition $\mathrm{Hom}_{\bf C}(A,X):={\bf C}[A,X]$ definition $\mathrm{Hom}_{\bf C}(A,f):=g\mapsto f\circ g$ Discussion Consider some objects $A$ and $X$ in the category ${\bf C}$, then ${\bf C}[A,X]$ is defined as the type of arrows from $A$$X$… text/html 2016-04-19T18:43:05+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/hom-set_adjunction?rev=1461084185&do=diff Hom-set adjunction Collection context ${\bf C},{\bf D}$ ... small category context $F$ in ${\bf D}\longrightarrow{\bf C}$ context $G$ in ${\bf C}\longrightarrow{\bf D}$ definiendum $\Phi$ in $\mathrm{it}$ postulate $\Phi$ in $\mathrm{Hom}_{\bf C}(F-,=)\cong\mathrm{Hom}_{\bf D}(-,G=)$ Discussion Here $\mathrm{Hom}_{\bf C}(F-,=),\mathrm{Hom}_{\bf D}(-,G=)$ in ${\bf Set}^{{\bf D}\times {\bf C}}$. Observe that if $\mathrm{Hom}_{\bf C}(F-,=)\cong\mathrm{Hom}_{\b… text/html 2014-10-29T21:13:27+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/homeomorphism?rev=1414613607&do=diff Homeomorphism Set context $\langle X,\mathcal{T}_X\rangle, \langle Y,\mathcal{T}_Y\rangle$ ... topological spaces definiendum $f\in$ it inclusion $f$ ... bijection, continuous postulate $f^{-1}$ ... continuous Discussion A homeomorphism is a bijection which is continuous in both directions. Reference text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/hypercube_graph?rev=1395396676&do=diff Hypercube graph Set context $ n\in\mathbb N, n\ge 1 $ range $ V\equiv \{0,1\}^n $ definiendum $Q_n\equiv \langle V,E\rangle$ range $ k\in\mathbb N,1\le k\le n $ for all $ v,w\in V $ postulate $ \{v,w\}\in E\leftrightarrow \exists!k.\ \pi_k(v)\neq\pi_k(w) $ Discussion The Hypercube graph, also called n-cube, has vertices all n-tuples of 0's and 1's and two such vertices are connected iff they differ in one coordinate.$n$$|V|=2^n$$n$$\frac{1}{2}2^n\… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/idempotent_function?rev=1395396676&do=diff Idempotent function Set context $X$ ... set definiendum $ f\in \mathrm{it}(X) $ postulate $ f:X\to X $ postulate $ f\circ f=f $ Discussion Parents Subset of Unary operation text/html 2014-10-31T20:13:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/identity_functor?rev=1414782796&do=diff Identity functor Functor context ${\bf C}$ ... category definiendum $Id_{\bf C}:{\bf C}\longrightarrow{\bf C}$ definiendum $Id_{\bf C}A:=A$ definiendum $Id_{\bf C}(f):=f$ Discussion "endofunctor" Reference Parents Element of Functor Requirements Functor text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/identity_relation?rev=1395396676&do=diff Identity relation Set $X$ $ \langle x,y\rangle \in\text{id}_X $ $ \langle x,y\rangle \in \text{unaryOp}(X) $ $ x=y $ Ramifications Reference Mizar files: RELAT_1 Wikipedia: Equality Parents Subset of Unary operation text/html 2016-07-20T01:23:20+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/identity_type?rev=1468970600&do=diff Identity type Type In the following I write “$\dots$” for some formal necessary context declarations, but which don't help understanding the rules. Type introduction rule ${\large\frac{\dots\ \vdash\ A\,:\,\mathrm{Type}}{x,y\,:\,A\ \vdash\ Id_A(x,y)\,:\,\mathrm{Type}}}$ Under Curry-Howard we understand $Id_A(x,y)$ as the proposition $x=y$. Term introduction rule ${\large\frac{\dots\ \vdash\ \dots}{\dots\ \vdash\ refl_x\,:\,Id_A(x,x)}}$$x=x$${\large\frac{\dots,\,p\,:\,Id_A(x,y)\ \vdash\ Q(x… text/html 2016-04-18T20:20:49+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/idris?rev=1461003649&do=diff Idris Notes on programming languages $\blacktriangleright$ Idris $\blacktriangleright$ Idris syntax Note Language See also Idris syntax Discussion Reference Basics including prelude: <http://www.idris-lang.org/docs/current/> Code exmaples Q & A ---------- Related Idris syntax, Haskell, Haskell type system Requirements Notes on programming languages text/html 2016-12-25T11:34:09+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/idris_book?rev=1482662049&do=diff Idris book Idris syntax $\blacktriangleright$ Idris book $\blacktriangleright$ ... Book Discussion Reference Manning: Type-driven development with idris ---------- Related Idris syntax text/html 2017-06-06T13:08:18+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/idris_syntax?rev=1496747298&do=diff Idris syntax Idris $\blacktriangleright$ Idris syntax $\blacktriangleright$ ... Note I'll try to take all the primitive notions and write down one example with them, respectively. This list largely overlaps with Haskell. Defining types data Beagle = Bigtime | Burger | Bouncer | Baggy | Bankjob | Bugle | Babyface data Kid = Tick | Trick | Track data Parent = Donald | Dasy data Grandparent = Dagobert data Singleton = Point text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/image?rev=1395396676&do=diff Image Set context $ R\in \text{Rel}(X,Y) $ postulate $ y\in \mathrm{im}(R) $ postulate $ \exists x\ (\langle x,y \rangle \in R) $ Ramifications Reference Mizar files: RELAT_1 Wikipedia: Image Parents Context Binary relation text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/imaginary_part_of_a_complex_number?rev=1395396676&do=diff Imaginary part of a complex number Set context $z\in \mathbb C$ postulate $\mathrm{Im}(z)\equiv \frac{z-\bar z}{2i}$ Ramifications Parents Related Complex conjugate of a complex number text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/imaginary_part_of_a_function?rev=1395396676&do=diff Imaginary part of a function Set context $f:X\to \mathbb C$ definiendum $ \mathrm{Im}f:X\to\mathbb C $ definiendum $ (\mathrm{Im}f)(x):=\mathrm{Im}(f(x)) $ Discussion Parents Context Imaginary part of a complex number text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/incidence_matrix?rev=1395396676&do=diff Incidence matrix Set context $n_v,m_e\in \mathbb N$ definiendum $ M\in \mathrm{it}(n_v,m_e) $ postulate $ \mathrm{Matrix}(n_v,m_e,\{0,1,2\}) $ for all $i\in\mathrm{range}(n_v)$ postulate $\sum_{j=1}^{m_e} M_{ij}=2 $ Discussion The index $i$ in $M_{ij}$ labels the vertices and the index $j$ labels the edges. The definition says that every edge has exactly two endpoints. text/html 2014-10-28T19:23:24+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/indexed_union?rev=1414520604&do=diff Indexed union Set context $f:I\to X$ definiendum $\bigcup_{i\in I,\ f} X_i \equiv \bigcup \mathrm{im}(f)$ Discussion For $f=\mathrm{id}$, the set is indexing itself and the indexed union is just the arbitrary union $\bigcup_{i\in I} X_i = \bigcup X$. Parents Refinement of Arbitrary union Context Image text/html 2015-04-16T19:05:19+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/induced_magma_power_set_magma?rev=1429203919&do=diff Induced magma power set magma Set context $\langle\!\langle M,* \rangle\!\rangle$ ... magma definiendum $\langle\!\langle {\mathcal P}M,\star \rangle\!\rangle$ definition $\star\in$ binary operation on ${\mathcal P}M$ definition $S\star T:=\{x*y\ |\ x\in S, y\in T\}$ ---------- Discussion When talking about left cosets etc., people write $xS$ for $\{x\}S$. Reference Wikipedia: Magma ---------- text/html 2019-09-23T21:19:31+0200 Anonymous (anonymous@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 2016-04-02T14:53:07+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/infinite_product_of_complex_numbers?rev=1459601587&do=diff Infinite product of complex numbers Set context $(z_i)$ ... Sequence($\mathbb C$) definition $\prod_{i=1}^\infty z_i\equiv \mathrm{lim}_{n\to\infty}\prod_{i=1}^n z_i$ ---------- Discussion "todo: Interestingly, I think I see the nLab doesn't want to allow e.g. $\prod_{n=1}^\infty (17-n)$ to be zero. (Reference below)" I recon infinite products may arise when x is written as $111\cdots11x$$a_n$$N,M,K$$\lim_{n\to\infty}a_n=a_N\cdot\prod_{n=N}^\infty\frac{a_{n+1}}{a_n}$$\lim_{n\t… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/infinite_sequence?rev=1395396676&do=diff Infinite sequence Set context $X$ definiendum $ A\in \mathrm{InfSequence}(X) $ postulate $A: \mathbb N\to X$ Discussion Instead of $\langle n,A(n)\rangle$, we also write $A_n$ and denote the whole sequence by $(A_n)_{n\in\mathbb N}$ or just $(A_n)$. Parents Subset of Function Related Natural number text/html 2015-06-20T16:25:54+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/infinite_series?rev=1434810354&do=diff Infinite series Set context $M$ ... monoid, metric space context $ S \in \mathrm{InfSequence}(M)$ definiendum $\sum_{k=1}^\infty S_k\equiv \mathrm{lim}_{n\to\infty}\sum_{k=1}^n S_k$ ---------- Element of Monoid Context Finite sum over a monoid, Limit in a metric space text/html 2016-09-28T18:21:30+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/infinite_sum_of_complex_numbers?rev=1475079690&do=diff Infinite sum of complex numbers Set context $ (z_i) \in \mathrm{Sequence}(\mathbb C)$ definiendum $\sum_{i=1}^\infty z_i\equiv \mathrm{lim}_{n\to\infty}\sum_{i=1}^n z_i$ ---------- ---------- Element of Complex number Subset of Infinite series text/html 2014-09-28T18:00:22+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/initial_algebra?rev=1411920022&do=diff Initial algebra Collection context ${\bf A}$ ... category of $F$-algebras definition $\langle I,i\rangle$ ... initial object of ${\bf A}$ Discussion It can be shown that $\langle FI, F(i)\rangle$ is isomorphic to $\langle I,i\rangle$ and so one can view the initial algebra as fixed point of $F$. It is, then, the most generic and encompassing algebra for which an operation $\alpha:FA\to A$$F$$\langle I,i\rangle$$F$$\langle A,\alpha\rangle$$\alpha$$\{\text{Nothing}\}$$L+R$$L$$R$$L+… text/html 2014-09-28T19:47:20+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/initial_morphism?rev=1411926440&do=diff Initial morphism Collection context $X:\mathrm{Ob}_{\bf C}$ context $U$ in ${\bf D}\longrightarrow{\bf C}$ definiendum $\langle A,\phi\rangle$ in $\mathrm{it}$ inclusion $A:\mathrm{Ob}_{\bf D}$ inclusion $\phi:{\bf C}[X,U(A)]$ for all $B:\mathrm{Ob}_{\bf D}$ for all $f:{\bf C}[X,U(B)]$ range $g:{\bf D}[A,B]$ postulate $\exists_!g.\ f=U(g)\circ\phi$ Discussion For an elaboration, see terminal morphism, the dual concept. Reference text/html 2014-12-04T22:34:35+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/initial_object?rev=1417728875&do=diff Initial object Object context ${\bf C}$ ... category definiendum $I:\mathrm{Ob}_{\bf C}$ for all $X:\mathrm{Ob}_{\bf C}$ postulate $\exists_!i.\ i:{\bf C}[I,X]$ Discussion Alternative definitions The initial object of ${\bf C}$ can be characterized by the initial morphism $\langle I,\mathrm{id}_\bullet\rangle$ from $\bullet:\mathrm{Ob}_{\bf 1}$ to the (unique) functor $U$ mapping to the discrete category ${\bf 1}$, which only has a single object. Because then $U(g)=… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/injective_function?rev=1395396676&do=diff Injective function Set context $X,Y$ definiendum $ f\in \mathrm{Injective}(X,Y)$ context $ f:X\to Y $ postulate $ f(x)=f(y) \implies x=y $ Discussion Parents Context Function text/html 2015-04-17T15:31:25+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/inner_group_automorphism_group?rev=1429277485&do=diff Inner group automorphism group Set context $\langle\!\langle G,\cdot\rangle\!\rangle$ ... group definition $\mathrm{Inn}(G)\equiv\langle\!\langle \{h\mapsto g\cdot h\cdot g^{-1}\,\mid\,g\in G\},*\rangle\!\rangle$ inclusion $*$ ... pointwise function product on $G^G$ ---------- Elaboration Explicitly, pointwise function product means $(\phi*\psi)(h):=\phi(h)\cdot\psi(h)$ Reference Wikipedia: Automorphism ---------- Subset of text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/integer?rev=1395396676&do=diff Integer Set definiendum $ \mathbb Z \equiv \mathbb N\times\mathbb N\ /\ \{\langle \langle a,b\rangle,\langle m,n\rangle\rangle\ |\ a+n = b+m )\} $ with $a,b,n,m\in \mathbb N$. Discussion For $a \ge b$, we denote $\langle a,b\rangle$ by $a-b$. The structure of the non-negative integers is then that of the natural numbers. For $a < b$, we have $(b-a)>0$ and we denote $\langle a,b\rangle$ by $-(b-a)$. So if $[\langle a,b\rangle]$ is the equivalence class of $\langle a,b\rangle$ with respe… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/integral_over_a_subset?rev=1395396676&do=diff Integral over a subset Set context $\mathbb K = \overline{\mathbb R}\lor \mathbb C$ context $\langle X,\Sigma,\mu_X\rangle$ ... measure space definiendum $\int_S: \mathcal P(X)\to(X\to \mathbb K)\to \mathbb K$ range $f: X\to \mathbb K$ definiendum $\int_S\ f\ \mathrm d\mu_X:=\int_X\ f\cdot \chi_S\ \mathrm d\mu_X$ Discussion If $X=\mathbb R$, $a,b\in \mathbb R$, $a<b$ and the measure $\mu_X$ is such that single points have zero measure $\mu_X(\{a\})=\mu_X(\{b\… text/html 2014-04-03T22:41:37+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/intersection?rev=1396557697&do=diff Intersection Set context $X,Y\in\mathfrak U$ definiendum $ x\in X \cap Y $ postulate $ x\in X \cap Y \Leftrightarrow (x\in X\land x\in Y) $ Discussion $ X \cap Y $ is commutative and idempotent. The intersection and union are associative and distributive with respect to another. Reference Wikipedia: Intersection Parents Element of Set universe Context* text/html 2016-10-29T17:32:48+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/introduction_to_modern_bayesian_econometrics?rev=1477755168&do=diff Introduction To Modern Bayesian Econometrics Book On Bayes algorithm Chapter 1 - The Bayesian Algorithm Overview * 69 pages, 8 sections. * There is an introduction and then on p.10 starts a section (1.4) of 50 pages with mostly examples. * The last 10 pages are conclusion, exercises, appendix and bibliographical notes text/html 2017-01-07T01:12:05+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/intuitionistic_propositional_logic?rev=1483747925&do=diff Intuitionistic propositional logic Framework We consider a logic with the standard connectives, adopt the standard usage of brackets for separation of expressions and predicates, and we use the usual binding strengths in term construction. The atomic propositions will be denoted with capital Latin letters here, e.g. $A,B,C,P,Q,...$$\land$$\lor$$\Rightarrow$$\bot$$\Leftrightarrow$$\neg$$\top$$($$)$$R$$\mathrm{Prop}$$P : \mathrm{Prop}$$P$$\mathrm{Type}$$\mathrm{Prop}:\mathrm{Type}$$A : \mathrm… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/inverse_function?rev=1395396676&do=diff Inverse function Set context $ f\in X^Y_\text{inj} $ postulate $ f^{-1} \equiv f^\smile $ Discussion We have $\text{im}(f^{-1})=\text{dom}(f)=X,$ $\text{dom}(f^{-1})=\text{im}(f).$ Injectiveness of $f$ implies there is a left “left inverse” of the function: $f^{-1}\circ f=\text{id}$. Surjectiveness implies there is a “right inverse”, so for bijective functions the above line extens to$\text{dom}(f^{-1})=\text{im}(f)=Y$$f:X\to Y$$f^{-1}:Y\to X$ text/html 2015-03-29T19:08:59+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/involution?rev=1427648939&do=diff Involution Set context $X$ ... set definiendum $ f\in \mathrm{it}(X,X) $ postulate $ f:X\to X $ postulate $ f\circ f=\mathrm{id_X} $ ---------- ---------- Subset of Unary operation text/html 2015-04-16T20:05:54+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/isomorphism?rev=1429207554&do=diff Isomorphism Collection context $A,B\in\mathrm{Ob}_{\bf C}$ definiendum $f$ in $A\cong B$ inclusion $f:{\bf C}[A,B]$ exists $f^{-1}:{\bf C}[B,A]$ postulate $f^{-1}\circ f=\mathrm{id}_A$ postulate $f\circ f^{-1}=\mathrm{id}_B$ ---------- Reference Wikipedia: Isomorphism, Automorphism ---------- Context Categories text/html 2016-08-25T09:56:36+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/ist_._tech?rev=1472111796&do=diff IST . tech Note vip.sweep_tracker = {} for k in ['Prim_sweep', 'Scnd_sweep']: vip.sweep_tracker[k] = 0 vip.sweep_tracker[k] += 1 import numpy as np from matplotlib import cm as color_map import matplotlib.pyplot as plt #from mpl_toolkits.mplot3d import Axes3D #from matplotlib.ticker import LinearLocator, FormatStrFormatter text/html 2017-05-11T09:56:29+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/ist_._times?rev=1494489389&do=diff IST . times Note $\bullet$ DO, 22.June --- IST BBQ $\bullet$ Jahreslohnzettel Runterladen { "170517 Mi", {0935, 1350} "170516 Di", {0935, 1350} "170515 Mo", {0935, 1450} "170511 Do", {0835, 1350} "170510 Mi", {0835, 1350} "170509 Di", {0835, 1350} "170508 Mo", {0835, 1450} "170505 Fr", {0935, 1450} "170504 Do", {0935, 1450} "170503 Mi", {0835, 1350} "170502 Di", {0835, 1350} "170501 Mo", "Tag der Arbeit" "170427 Do", {0940, 1450} "170426 Mi", {0940, 1450} "170425 Di", {0940, 1450} "1704… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/iterated_function?rev=1395396676&do=diff Iterated function Set context $ f:X\to X $ context $ n\in \mathbb N, n\neq 0 $ definiendum $ f^n $ Iteratively defined as follows: definiendum $ f^1:=f $ definiendum $ f^{n}:=f\circ f^{n-1} $ Discussion Parents Subset of Unary operation Context Relation concatenation, Natural number text/html 2016-07-06T13:14:47+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/ito_integral?rev=1467803687&do=diff Itō integral Partial function context $\langle\!\langle \Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge 0},\mathbb{P} \rangle\!\rangle$ ... filtered probability space context $X_t$ ... $\mathcal{F}_t$-adapted context $H_t$ ... left-continuous, locally bounded, $\mathcal{F}_t$-adapted let $h_n=\frac{b-a}n$ let $\Delta X_{n,i} = X_{a+ih_n}-X_{a+(i-1)h_n}$ definition todo: type definition $\int_a^b H_{t-}\,{\mathrm d}X_t := {\mathrm{plim}}_{n \to \… text/html 2016-02-16T14:01:51+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/its_about_time_._note?rev=1455627711&do=diff Its about time . note Note This entry is concerned with the notion of time and is a spinoff of On physical units . note. Observation A core assumption of physics that's easy to adopt is that we are able to enumerate some moments in an ordered fashion. $t_1, t_2, t_3, ...$ Usually they are considered as point on some real axis, but I guess the only thing that really matters is that this enables us to count (other) sorts of events $t_i, t_j$$\dfrac{1}{t_j-t_i}$$h_\Psi\cdot\dfrac{{\mathrm d}… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/k-cycle?rev=1395396676&do=diff k-cycle Set context $V,E$ ... set context $k\in\mathbb N$ definiendum $\langle V,E,\psi\rangle \in \mathrm{it} $ inclusion $\langle V,E,\psi\rangle $ ... cycle postulate $ |\ E\ |=k $ Discussion The first few k-cycles are called as follows: point/henagon, line/edge/digon, triangle/trigon, sqare/tetragon/quadrilateral, pentagon, hexagon, text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/k-partite_graph?rev=1395396676&do=diff k-partite graph Set context $k\in\mathrm N$ context $V$ ... set definiendum $\langle V,E\rangle \in \mathrm{it}(E,V) $ inclusion $ \langle V,E\rangle $ ... undirected graph range $ i,j\in\{1,\dots,k\} $ range $ \bigcup_i X_i=V $ range $ \forall i,j.\ X_i\cap X_j=\emptyset $ range $ v,w\in V $ postulate $\exists X_1,\dots,X_k.\ \forall u,v.\ \{u,v\}\in E\implies \forall i.\ \neg(v\in X_i\land w\in X_i) $ Discussion The $X_… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/k-path?rev=1395396676&do=diff k-path Set context $V,E$ ... set context $k\in\mathbb N$ definiendum $\langle V,E,\psi\rangle \in \mathrm{it} $ inclusion $\langle V,E,\psi\rangle $ ... path postulate $ |\ E\ |=k $ Discussion Parents Subset of Path . graph theory text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/k-regular_graph?rev=1395396676&do=diff k-regular graph Set context $V,E$ ... set context $k\in\mathbb N$ definiendum $ \langle V,E,\psi\rangle \in \mathrm{it}(E,V) $ inclusion $ \langle V,E,\psi\rangle $ ... undirected graph for all $ v\in V $ postulate $ d(v)=k $ Discussion Parents Subset of Regular graph Context Vertex degree text/html 2014-04-08T14:44:46+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/k-tape_turing_machine?rev=1396961086&do=diff k-tape Turing machine Set context $ k\in\mathbb N $ definiendum $ \langle Q,\Gamma,\Sigma,\delta\rangle \in \mathrm{TM}_k $ inclusion $ \Sigma\subset\Gamma$ inclusion $ \delta: Q\times\Gamma^k \to Q \times \Gamma^k \times \{\mathrm{L},\mathrm{S},\mathrm{R}\}^k$ postulate $ q_\mathrm{start},q_\mathrm{halt}\in Q $ postulate $ \Box,\triangleright,0,1\in\Gamma $ postulate $ \Box\notin\Sigma $ Discussion Let's discuss the 1-tape Turing machine, i.e. $k=1$. … text/html 2016-09-09T09:31:42+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/kfv_._note?rev=1473406302&do=diff KfV . note Driver Assistent System Influence Model Quantities of interest * $n(t)$ ... (measured, known) number accidents per year $t\in{\mathbb N}$ * $n^\mathrm{max}(t)$ ... (unknown) number accidents that would happen without a particular driver assistant systems. As the system prevent some accidents, we have $n(t) < n^\mathrm{max}(t)$$n_\mathrm{prev}(t) = n^\mathrm{max}(t) - n(t)$$n^\mathrm{max}(t)$$n(t)$$n_\mathrm{prev}^\mathrm{max}(t)$$n^\mathrm{max}(t)$$p := \dfrac{ n_\mathrm{prev}… text/html 2016-09-10T18:41:21+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/kullback_leibler_divergence?rev=1473525681&do=diff Kullback–Leibler divergence Function context $p, q$ ... probability distributions definition todo: type definition $D_{\mathrm{KL}}(P\|Q) = \int_{-\infty}^\infty p(x) \, \log \left( \dfrac{p(x)}{q(x)}\right) {\rm d}x$ ---------- ---------- Requirements Cumulative distribution function, Natural logarithm of real numbers Related Classical microcanonical entropy text/html 2015-12-17T10:14:31+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/kummers_function?rev=1450343671&do=diff Kummer's function Function definition $ M: $ ?? definition $ M(a,c,z) := \lim_{b\to\infty}{}_2F_1(a,b,c,z/b) $ ---------- Theorems Reference Wikipedia: Confluent hypergeometric function ---------- Requirements Generalized hypergeometric function text/html 2016-03-08T16:22:13+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/laplace_transform?rev=1457450533&do=diff Laplace transform Function "todo" ---------- Discussion Action on monomials In this article we use $\beta=\frac{1}{T}$. "todo: clean this $\beta$ vs. $T$ Thing up. Maybe drop $T$ altogether." With $u=\beta\,E=E/T\implies dE=Tdu$ and the definition of the Gamma function, we find $\int_0^\infty \left(\frac{1}{n!}E^n\right){\mathrm e}^{-E/T}dE = T^{n+1} \frac{1}{n!} \int_0^\infty u^{(n+1)-1}{\mathrm e}^{-u}du = T^{n+1},$ so that $L[f](\beta):=\int_0^\infty f(E){\mathrm e}^{-\beta E}dE… text/html 2015-04-25T19:25:48+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/least_divisor_function?rev=1429982748&do=diff Least divisor function Function definiendum $ \mathrm{ld}:\mathbb N^+\to\{1\}\cup\mathrm{Prime\ number} $ definiendum $ \mathrm{ld}(n):=\mathrm{min}\left(\mathrm{divisors}(n)\right) $ ---------- Code Haskell divides :: Integral a => a -> a -> Bool divides d n = rem n d == 0 ld :: Integral a => a -> a ld n = ldf 2 n ldf :: Integral a => a -> a -> a ldf k n | divides k n = k | k^2 > n = n | otherwise = ldf (k+1) n $n$$n$$\mathrm{ld}(n)$$\mathrm{ld}(n)^2\le n$ text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/lebesgue-borel_measure?rev=1395396676&do=diff Lebesgue-Borel measure Set context $p\in \mathbb N$ definiendum $\beta^p\equiv\lambda^p|_{\mathcal B^p}$ Discussion Reference Wikipedia: Lebesgue measure Parents Refinement of Lebesgue measure Context Borel subsets of the reals, Restricted image text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/lebesgue_measurable_subsets_of_%E2%84%9D%E2%81%BF?rev=1395396676&do=diff Lebesgue measurable subsets of ℝⁿ Set context $p\in \mathbb N$ definiendum $\mathfrak L^p\equiv\{S\ |\ \eta^p(S)\in\overline{\mathbb R}\}$ Discussion Reference Wikipedia: Lebesgue measure Parents Subset of σ-algebra Context Lebesgue outer measure text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/lebesgue_measure?rev=1395396676&do=diff Lebesgue measure Set context $p\in \mathbb N$ definiendum $\lambda^p\equiv\eta^p|_{\mathfrak{L}^p}$ Discussion $\lambda^p:\mathfrak{L}^p\to \overline{\mathbb{R}}$ is the restriction of Lebesgue outer measure to a domain which contains only actually measurable sets. Reference Wikipedia: Lebesgue measure Parents Subset of Measure Context Lebesgue measurable subsets of ℝⁿ, Restricted image text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/lebesgue_outer_measure?rev=1395396676&do=diff Lebesgue outer measure Set context $p\in \mathbb N$ definiendum $\eta^p:\mathcal P(\mathbb R^p)\to \overline{\mathbb R}$ definiendum $\eta^p(A):=\mathrm{inf}\{\ \sum_{k=1}^\infty\lambda^p(I_k)\ |\ I\in\mathrm{Sequence}(\mathfrak J^p)\ \land\ A\subset\bigcup_{k=1}^\infty I_k\ \}$ Discussion The Lebesgue outer aims at measuring subspaces of $\mathcal P(\mathbb R^p)$ as approximated by cubes which themselves are measured via Elementary volume of ℝⁿ. Reference Wikipedia: Lebesg… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/left_module?rev=1395396676&do=diff Left module Set context $M,R$ definiendum $\langle\mathcal M,\mathcal R, *\rangle \in \mathrm{leftModule}(M,R)$ context $\mathcal M\in \mathrm{abelianGroup}(M)$ context $\mathcal R\in \mathrm{ring}(R)$ context $*:R\times M\to M$ Now denote the addition in th group $\mathcal M$ by “$+$” as usual, and the addition and multiplication in the ring $\mathcal R$ by “$\hat+$” and “$\hat*$”, respectively. $x,y\in M$ $r,s\in R$ $r*(x+y) = (r*x)+(r*y)$$(r\ \hat+\ s)… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/left_module_homomorphism?rev=1395396676&do=diff Left module homomorphism Set context $M,N$...left $\mathcal R$-module definiendum $A\in \mathrm{Hom}_{\mathcal R}(M,N)$ context $A:M\to N$ postulate $A\ (r\cdot x+s\cdot y)=r\cdot A\ x+s\cdot A\ y$ Discussion Reference Wikipedia: Module Parents Context Left module text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/leibniz_formula_for_determinants?rev=1395396676&do=diff Leibniz formula for determinants Set context $n\in \mathbb N$ context $R$ ... abelian ring definiendum $ \mathrm{det}_n:\mathrm{SquareMatrix}(n,R)\to R$ definiendum $ \mathrm{det}_n(A):=\sum_{j_1,\dots,j_n}^n\varepsilon_{j_1,\dots,j_n}\cdot \prod_{k=1}^n A_{k,j_k}$ Discussion This function concides with the implicitly defined determinant of Determinant, if the matrices are taken to be linear operators in the usual way. text/html 2016-03-07T14:01:39+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/limit_._category_theory?rev=1457355699&do=diff Limit . category theory Collection context $F:{\bf C}^{\bf D}$ context $\Delta:{\bf C}\longrightarrow{\bf C}^{\bf D}$ ... diagonal functor definiendum $\mathrm{lim}\,F$ ... terminal morphism from $\Delta$ to $F$ ---------- Elaboration A limit of a functor with image in ${\bf C}$, if it exists, is a particular terminal morphism in the functor category ${\bf C}^{\bf D}$${\bf C}^{\bf D}$$F\in {\bf C}^{\bf D}$$\phi$$F$$\phi$$\Delta$$N$${\bf C}$$\Delta(N)$${\bf C}^{\bf D}$$\De… text/html 2017-02-02T21:25:54+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/limit_in_a_metric_space?rev=1486067154&do=diff Limit in a metric space Set "todo: clean up this definition" context $\langle X,d\rangle$ ... metric space context $x$ ... infinite seqeunce in $ X $ definiendum $\mathrm{lim}_{n\to\infty}\ x_n$ range $\varepsilon\in\mathbb R$ range $ \varepsilon>0 $ range $m\in\mathbb N$ range $m\ge 0 $ range $y\equiv\mathrm{lim}_{n\to\infty}\ x_n$ postulate $ \forall\varepsilon.\,\exists m.\,\forall (n\ge m).\,d(x_n,y)<\varepsilon $$\… text/html 2014-04-11T16:42:13+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/linear_approximation?rev=1397227333&do=diff Linear approximation Set context $X,Y$ ... Banach spaces with topology context $\mathcal O$ ... open in $X$ context $x\in\mathcal O$ context $f:\mathcal O\to Y$ definiendum $J_x^f$ postulate $J_x^f$ ... bounded linear operator from $X$ to $Y$ postulate $\mathrm{lim}_{h\to 0}\ \Vert f(x+h)-f(x)-J_x^f(h)\Vert / \Vert h\Vert\ =\ 0$ Discussion The approximation of the value of $f$$x$$f(x+d)\sim f(x)+J_x^f(d)$$J_x^f$$x$$f$$X=\mathbb R^n,Y=\mathbb R^m… text/html 2017-01-17T01:06:57+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/linear_first-order_ode_system?rev=1484611617&do=diff Linear first-order ODE system Set context $ A:\mathbb R\to\mathrm{Matrix}(n,\mathbb R) $ context $ b:\mathbb R\to\mathbb R^n $ definiendum $ y \in \mathrm{it} $ postulate $ y:C^k(\mathbb R,\mathbb R^n) $ postulate $ y'(t)=A(t)\ y(t)+b(t) $ ---------- Theorems There exists a matrix $S(t,s)$ such that the solution of the equation above is of the form $y(t)=S(t,0)\ y_0+\int_0^t\ S(t,s)\ b(s)\ \mathrm ds$ We don't know $S(t,s)$ in general, but $S(t,0)=\lim_{n\… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/linear_operator_algebra?rev=1395396676&do=diff Linear operator algebra Set context $X$...left $\mathcal R$-module definiendum $\langle \mathrm{Hom}(X,X),+,\cdot,*,\rangle \in L(X,X)$ context $\langle \mathrm{Hom}(X,X),+,\cdot\rangle \in \mathcal L(X,X)$ context $*:\mathrm{Hom}(X,X)\times \mathrm{Hom}(X,X)\to \mathrm{Hom}(X,X)$ $ v\in M $ $A,B \in \mathrm{Hom}(X,Y)$ postulate $(A*B)v = A(B v) $ Discussion Theorem: A linear operator $A:X\to X$ is bijective if it has an inverse in $L(X,X)$. Parents Subse… text/html 2015-10-29T12:55:23+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/linear_operator_space?rev=1446119723&do=diff Linear operator space Set context $X,Y$...left $\mathcal R$-module definiendum $\langle\mathrm{Hom}(X,Y),+,\cdot \rangle \in \mathcal L(X,Y)$ context $+:\mathrm{Hom}(X,Y)\times \mathrm{Hom}(X,Y)\to \mathrm{Hom}(M,N)$ context $\cdot : \mathcal R\times\mathrm{Hom}(X,Y)\to\mathrm{Hom}(X,Y)$ $ v\in M $ $r,s \in \mathcal R$ $A,B \in \mathrm{Hom}(X,Y)$ postulate $(r \cdot A+s \cdot B)\ v = r\ (A\ v) + s\ (B\ v) $ ---------- Discussion A linear operator $A:X\t… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/linear_representation_of_functions?rev=1395396676&do=diff Linearization representation Theorem $X,Y$ ... Banach spaces with topology $f\in C(X,Y)$ postulate $ f(x+y) = f(x) + \int_0^1\ Df(x+t\ y)\cdot y\ \mathrm d t $ Discussion $ f(x) = f(0) + \int_0^1\ Df(t\ x)\cdot x\ \mathrm d t $ Reference Parents Context Fréchet derivative, Function integral text/html 2016-05-24T20:13:02+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/literature?rev=1464113582&do=diff Literature Literature $\blacktriangleright$ Todo books / Todo papers Meta "[work] - Mathematical theory of transport processes in gases / J.H. Ferziger and H.G. Kaper - Plasma kinetics in atmospheric gases / M. Capitelli " physics "- Noise theory and applications to physics / Philipe Refregier text/html 2016-04-13T11:58:53+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/locally_convex_topological_vector_space?rev=1460541533&do=diff Locally convex topological vector space Set "a vector space together with a family of seminorms" Discussion This induces a norm. Makes the Gateaux derivative possible, which is possibly non-linear and more general than the Frechet derivative Reference Wikipedia: Locally convext topological v ector space Gateaux derivative ---------- text/html 2016-09-15T01:42:20+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/locally_finite_topology_subset?rev=1473896540&do=diff Locally finite topology subset Set context $\langle X,\mathcal T\rangle$ ... topological space definiendum ${\mathcal C} in it inclusion ${\mathcal C}\subset \mathcal T$ for all $x\in X$ exists $V\in \mathcal T$ postulate $x\in V$ postulate $\{U\in {\mathcal C}\,|\,U\cap V\neq\emptyset\}$ ... finite ---------- Idea Like many properties, this is a notion of smallness. It's not about the smallness of a subset $U$$X$${\mathcal C}$$U$$X$$V\in{\mathc… text/html 2015-04-17T10:42:06+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/locally_small_category?rev=1429260126&do=diff Locally small category Collection context ${\mathfrak U}_\mathrm{Sets}$ ... set universe definiendum $\bf C$ in $\mathrm{it}$ inclusion $\bf C$ ... category postulate $ {\bf C}[A,B] $ ... ${\mathfrak U}_\mathrm{Sets}$-small set ---------- ---------- Subset of Categories Requirements Set universe Requirements* Categories text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/logarithmic_integral_function?rev=1395396676&do=diff Logarithmic integral function Function definiendum $ \mathrm{li}: \mathbb R_+\to \mathbb R_+$ definiendum $ \mathrm{li}(x) := \int_0^x \frac{1}{\ln(t)}\mathrm d t $ Discussion Reference Wikipedia: Logarithmic integral function Parents Subset of ℝ valued function Requirements Cauchy principal value text/html 2016-07-16T13:34:03+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/logic?rev=1468668843&do=diff Logic Note To speak about foundations of mathematics, we need some formal logic. Here, a logic is a framework comprising a syntactic language and agreed upon derivation $\text{rules}$, with sentences that typically represent statements of reasoning about certain things.$${\large\frac{foo}{bar}}(\text{foobar rule})$$$\text{foobar rule}$$foo$$bar$ text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/loop?rev=1395396676&do=diff Loop Set context $X$ postulate $ \langle X,* \rangle \in \text{Loop}(X)$ context $\langle X,* \rangle \in \mathrm{Quasigroup}(X)$ range $e,a\in X$ postulate $\exists e.\ \forall a.\ (a*e=e*a=a) $ Here we used infix notation for “$*$”. Ramifications Discussion The binary operation is often called multiplication. The axioms $*\in \mathrm{binaryOp}(X)$ above means that a monoid is closed with respect to the multiplication. $X$$*$ text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/loopless_graph?rev=1395396676&do=diff Loopless graph Set context $V,E$ ... set definiendum $ \langle V,E,\psi\rangle \in \mathrm{it}(E,V) $ postulate $\langle V,E,\psi\rangle $ ... undirected graph for all $u\in V$ postulate $\{u,u\} $ ... not an edge Discussion Parents Subset of Undirected graph text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/macroscopic_observables_from_kinetic_theory?rev=1395396676&do=diff Macroscopic observables from kinetic theory Set context $ f $ ... one-particle reduced distribution function range $ N \equiv \mathrm{dim}(\mathcal M) $ context $ q,m\in \mathbb R^*$ In terms of the phase space probability density, $f=f_1$ and $N$ is the number of described particles in the system with mass $m$$q$$ \langle n,\rho,u,c,\Gamma,j,J,{\mathrm p},{\mathrm P},V,C,e,E,q,Q,T \rangle \in \mathrm{it} $$ :: A({\bf v}) $$ \langle A \rangle({\bf x},t) \equiv \int\ … text/html 2019-09-27T00:19:00+0200 Anonymous (anonymous@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 2015-04-12T17:48:51+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/magma?rev=1428853731&do=diff Magma Set context $M$ ... set definiendum $ \langle\!\langle M,* \rangle\!\rangle \in$ magma inclusion $* \in$ binary operation (M) ---------- Elaboration The binary operation is often called multiplication. The axiom '$* \in$ binary operation (M)' above means that a magma is closed with respect to the multiplication. $M$$*$ text/html 2015-04-01T10:37:48+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/martin-loef_type_theory?rev=1427877468&do=diff Martin-Löf type theory Framework A Dependent type theory with Identity type. Or rather it's an umbrella term for several closely related such theories. ---------- Reference Wikipedia: Intuitionistic type theory ---------- Requirements Identity type text/html 2015-11-09T10:04:31+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/matlab?rev=1447059871&do=diff Matlab Note ---------- Language Basic commands In the command window (interactive mode): who whos clear foo clear all more on more on more off Functions exp sin log imag class % checks type of argument Reference Mathworks Constants and test matrices, text/html 2016-09-30T15:56:36+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/matrix?rev=1475243796&do=diff Matrix Set context $X$ ... set context $n,m\in \mathbb N$ definiendum $ A\in \mathrm{Matrix}(n,m,X) $ postulate $$ A\in\mathrm{FinSequence}(\mathrm{FinSequence}(X)) $$ range $ 1\ge i\ge m$ postulate $\mathrm{length}(A)=n$ postulate $\mathrm{length}(A_i)=m$ ---------- Discussion We write $A_{ij}\equiv (A_i)_j$ Reference Wikipedia: Matrix_(mathematics) Mathematica: MatrixOperations Idris: contrib/Data/Matrix.idr Haskell: package/matrix-0.… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/matrix_conjugate_transpose?rev=1395396676&do=diff Matrix conjugate transpose Set context $n,k\in\mathbb N$ definiendum $ {\cdot}^*: \mathrm{Matrix}(n,k,\mathbb C) \to \mathrm{Matrix}(k,n,\mathbb C) $ postulate $ (A^*)_{ij}=\overline{A_{ji}} $ Discussion Reference Wikipedia: Conjugate transpose Parents Context Matrix text/html 2017-07-04T22:08:15+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/matrix_eigenvalue?rev=1499198895&do=diff Matrix eigenvalue Set context $A\in\mathrm{SquareMatrix}(n,\mathbb C)$ definiendum $\lambda\in\mathrm{EigenVal}(A) $ postulate $\mathrm{det}(\lambda\cdot I_n-A)=0$ ---------- Parents Subset of Eigenvalue Context Leibniz formula for determinants text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/matrix_exponential?rev=1395396676&do=diff Matrix exponential Set context $ n\in\mathbb N $ definiendum $\mathrm{exp}: \text{SquareMatrix}(n,\mathbb C)\to\text{SquareMatrix}(n,\mathbb C)$ definiendum $\mathrm{exp}(A):=\sum_{k=0}^\infty \frac{1}{k!} A^k $ Discussion Theorems $[A,B]=0\implies \mathrm{exp}(A+B)=\mathrm{exp}(A)\cdot\mathrm{exp}(B)$ References Wikipedia: Matrix exponential, Exponential map Parents Context Square matrix text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/matrix_product?rev=1395396676&do=diff Matrix product Set context $R$ ... ring context $m,n,k\in \mathbb N$ definiendum $ *: \mathrm{Matrix}(m,n,R)\times \mathrm{Matrix}(n,k,R)\to \mathrm{Matrix}(m,k,R) $ postulate $ (A*B)_{ij}=\sum_{l=1}^m A_{il}\cdot B_{lj} $ Discussion For square matrices, the matrix product is associative. And also for general matrices, we still have $(A*B)*'C=A*''(B*'''C)$, where the four binary functions are the matrix products for the suitable dimensions. text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/matrix_ring?rev=1395396676&do=diff Matrix ring Set context $n\in \mathbb N$ context $R$ ... ring definiendum $ \langle \mathrm{SquareMatrix}(n,R),* \rangle \in \mathrm{it}(n,R) $ postulate $ * $ ... matrix product w.r.t. $n\times n$ square matrices Discussion Reference Wikipedia: Matrix ring Parents Subset of Associative algebra Context Square matrix, Matrix product text/html 2016-10-02T16:16:49+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/matrix_transpose?rev=1475417809&do=diff Matrix transpose Function context $X$ context $n,k\in\mathbb N$ definition ${\cdot}^{\ \mathrm{T}}: \mathrm{Matrix}(n,k,X) \to \mathrm{Matrix}(k,n,X) $ postulate $(A^\mathrm{T})_{ij}=A_{ji} $ ---------- Discussion Code mm = {{a, b, c}, {x, y, z}}; tm = Transpose[mm]; mm // MatrixForm tm // MatrixForm --$ idris -p contrib --$ :l this_module.idr import Data.Matrix mm : Matrix 3 2 Nat mm = [[3,5],[1,5],[2,1]] tm : Matrix 2 3 Nat tm = transpose mm text/html 2014-12-04T14:21:21+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/maximal_extension_in_a_set?rev=1417699281&do=diff Maximal extension in a set Set context $A$ ... set context $a\in A$ definiendum $\mathrm{max}(a,A)\equiv\bigcup\{b\mid b\in A\land a\subseteq b\}$ Discussion Idea Given $a\in A$, the maximal extension $\mathrm{max}(a,A)$ is the largest set in $A$ which encompasses $a$. Predicate predicate $x$ maximal in $X \equiv \mathrm{max}(x,X)=x$ Reference Parents Element of text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/maximal_vertex_degree?rev=1395396676&do=diff Maximal vertex degree Function context $ V,E $ ... set definiendum $ \Delta : \mathrm{undirectedGraph}(V,E) \to \mathbb N $ definiendum $ \Delta(G) := \mathrm{max}(\mathrm{im}\ d_G) $ Discussion Parents Context Vertex degree text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/maximum_function?rev=1395396676&do=diff Maximum function Set context $X$ context $\le$ ... non-strict partial order over $X$ definiendum $\mathrm{max}:X\times X\to X $ definiendum $ \mathrm{max}(x,y) := \begin{cases} x & \mathrm{if}\ y\le x\\\\ y & \mathrm{else} \end{cases}$ Discussion Parents Context Non-strict partial order Element of Binary operation text/html 2014-09-15T20:19:34+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/maybe?rev=1410805174&do=diff Maybe Haskell type data Maybe a = Nothing | Just a deriving (Eq, Ord) instance Monad Maybe where (Just x) >>= k = k x Nothing >>= _ = Nothing return = Just fail _ = Nothing -- utility (Just _) >> k = k Nothing >> _ = Nothing instance Functor Maybe where fmap _ Nothing = Nothing fmap f (Just a) = Just (f a) text/html 2016-03-09T10:34:43+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/means_._note?rev=1457516083&do=diff Means . Note Note context $S$ ... set context $G$ ... group context $w:S\to G$ context $I:(S\to G)\to G$ definiendum $M:(S\to G)\to G$ definiendum $\langle f\rangle:=I(f\cdot w)\cdot I(w)^{-1}$ Here $(f\cdot w)(s):=f(s)*w(s)$ where $*$ is the group operation. Real functions E.g. $\langle f\rangle_{[a,b]}:=\dfrac{\int_a^bf(x)\,{\mathrm dx}}{b-a}$ where $[a,b]\subseteq{\mathbb R}$ and $w(x):=1$. Minus twelve For $z\in(0,1)$, we find $\sum_{k=0}^\inft… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/measurable_function?rev=1395396676&do=diff Measurable function Set context $ \langle X,\Sigma_X\rangle\in \mathrm{MeasurableSpace}(X) $ context $ \langle Y,\Sigma_Y\rangle\in \mathrm{MeasurableSpace}(Y) $ postulate $ f\in \mathrm{Measurable}(X,Y) $ context $ f:X\to Y $ $y\in \Sigma_Y$ postulate $ f^{-1}(y)\in\Sigma_X $ Discussion This is very similar to the definition of continuous function. People write $f:\langle X,\Sigma_X\rangle\to\langle Y,\Sigma_Y\rangle$ to point out the function is measura… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/measurable_numerical_function?rev=1395396676&do=diff Measurable numerical function Set context $ \langle X,\Sigma_X\rangle\in \mathrm{MeasurableSpace}(X) $ postulate $\mathcal M \equiv\mathrm{Measurable}(X,\overline{\mathbb R}) $ Where for $\mathbb R$ we choose the Borel subsets of the reals. Discussion Parents Subset of Measurable function Context Borel subsets of the reals text/html 2016-06-20T18:46:55+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/measurable_space?rev=1466441215&do=diff Measurable space Set context $ X $ definiendum $ \langle X,\Sigma\rangle\in \mathrm{MeasurableSpace}(X) $ postulate $ \Sigma \in \mathrm{SigmaAlgebra}(X) $ Discussion Every σ-algebra gives us a measurable space. Predicates We call a set $X$ measurable if there is a sigma-algebra over it. Reference Wikipedia: Sigma-algebra Parents Equivalent to text/html 2015-04-01T10:30:50+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/measure?rev=1427877050&do=diff Measure Set context $\Sigma \in \mathrm{SigmaAlgebra}(X) $ definiendum $ \mu\in \mathrm{measure}(\Sigma)$ context $E\in \Sigma$ context $S\in \mathrm{Sequence}(\Sigma)$ inclusion $\mu:\Sigma\to \overline{\mathbb R} $ postulate $ \mu(E)\ge 0$ postulate $ \mu(\emptyset)=0 $ postulate $ \mu\left(\bigcup_{j=1}^\infty S_j\right)=\sum_{j=1}^\infty \mu(S_j) $ ---------- Reference Wikipedia: Measure ---------- Context σ-algebra, Extended real nu… text/html 2016-06-20T18:37:13+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/measure_space?rev=1466440633&do=diff Measure space Set context $X $ definiendum $ \langle\!\langle X,\Sigma,\mu \rangle\!\rangle$ in it postulate $ \Sigma \in \mathrm{SigmaAlgebra}(X) $ postulate $ \mu\in \mathrm{Measure}(\Sigma) $ Discussion A measure space is a measurable space together with a fixed measure. Reference Wikipedia: Measure Parents Equivalent to Measure Context σ-algebra, Extended real number line, Infinite series text/html 2016-09-24T14:52:03+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/mechanics?rev=1474721523&do=diff Mechanics Foundational temp formal power series $\succ \dots \succ$ Mechanics $\succ$ $\succ \dots \succ$ Pendulum "tmp entry on the mechanics intro part" Guide ---------- Sequel of Foundational temp formal power series Related Classical Hamiltonian system, Euler-Lagrange equations text/html 2014-11-10T21:12:48+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/mere_proposition?rev=1415650368&do=diff Mere proposition Type $isProp(A):={\large\Pi}_{x,y:A}\,Id_A(x,y)$ Discussion In HoTT, a type is a proposition iff all its terms are equal. Roughly, this means it has zero or one term. Reference Parents Requirements Identity type text/html 2014-11-20T13:00:19+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/metric?rev=1416484819&do=diff Metric Set context $ X $ ... set definiendum $d\in$ it postulate $d:X\times X \to \mathbb{R}_+$ postulate $ d(x,y)=0 \implies x=y$ postulate $ d(x,y)=d(y,x)$ postulate $ d(x,y)\le d(x,p)+d(p,y)$ Discussion Reference Wikipedia: Metric Parents Subset of Binary function Equivalent to Metric space Context Non-negative real number text/html 2015-01-21T19:41:10+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/metric_space?rev=1421865670&do=diff Metric space Set context $X$ ... set definiendum $\langle X,d\rangle\in\mathrm{it}$ postulate $d$ ... metric $X$ ---------- We can reconstruct the set underlying a metric via $\text{dom}(\text{dom}(d))=\text{dom}(X\times X)=X$, so the set of metrics and the set of metric spaces over $X$ are in bijection. Reference Wikipedia: text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/microcanonical_inverse_temperature?rev=1395396676&do=diff Microcanonical inverse temperature Set context $ S(E) $ ... classical microcanonical entropy definiendum $\beta(E):=\frac{\partial S(E)}{\partial E} $ Discussion Parents Context Classical microcanonical entropy text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/microcanonical_temperature?rev=1395396676&do=diff Microcanonical temperature Set context $ \beta $ ... microcanonical inverse temperature definiendum $T:=1/\beta$ Discussion Parents Context Microcanonical inverse temperature text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/minimal_vertex_degree?rev=1395396676&do=diff Minimal vertex degree Function context $ V,E $ ... set definiendum $ \delta : \mathrm{undirectedGraph}(V,E) \to \mathbb N $ definiendum $ \delta(G) := \mathrm{min}(\mathrm{im}\ d_G) $ Discussion Parents Context Vertex degree text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/minimum_function?rev=1395396676&do=diff Minimum function Set context $X$ context $\le$ ... non-strict partial order over $X$ definiendum $\mathrm{max}:X\times X\to X $ definiendum $ \mathrm{max}(x,y) := \begin{cases} x & \mathrm{if}\ x\le y\\\\ y & \mathrm{else} \end{cases}$ Discussion Parents Context Non-strict partial order Element of Binary operation Related Maximum function text/html 2018-06-16T00:51:42+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/modal_logic?rev=1529103102&do=diff Modal Logic Framework "Modal operators $\Box_R$ ... necessarily $\Diamond_R=\neg\Box_R\neg$ ... possibly Semantics $\langle W,R\rangle$ ... “Kripke frame” where $W$ ... (type of) worlds $R\subseteq W\times W$ ... binary accessibility relation $wRv$ ... “$v$ is accessible from $w$” $P(w)$ ... predicate $(\Box_R P)(w)$ ... $\forall v.\, wRv\rightarrow P(v)$ $(\Diamond_R P)(w)$ ... $\exists v.\, wRv\land P(v)$ The modal operators are similar to $\forall$$\exists$$P(w)$$\forall w.\,P(w)$$(\Box… text/html 2018-05-12T23:47:38+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/module?rev=1526161658&do=diff Module Set context $M,R$ postulate $\langle\mathcal M,\mathcal R, *\rangle \in \mathrm{module}(\mathcal M,\mathcal R)$ context $\langle\mathcal M,\mathcal R, *\rangle \in \mathrm{leftModule}(\mathcal M,\mathcal R)$ context $\mathcal M\in \mathrm{abelianGroup}(M)$ Now denote the multiplication in the ring $\mathcal R$ by “$\ \hat*\ $”. $r,s\in R$ postulate $r*s=s*r$ Discussion A module is a left module with a commutative ring acting on the group. One gene… text/html 2015-02-26T20:27:00+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/moduli_space?rev=1424978820&do=diff Moduli space Note Basically, a parameter space. Point is that it's internal to the theory. ---------- Related Space and quantity "what's a better related topic here?" text/html 2015-10-05T16:23:07+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/monad?rev=1444054987&do=diff Monad Collection context ${\bf C}$ ... category definiendum $\langle T,\eta,\mu\rangle$ in $\mathrm{it}$ inclusion $T$ in ${\bf C}^{\bf C}$ inclusion $\eta:1_{\bf C}\xrightarrow{\bullet}T$ inclusion $\mu:TT\xrightarrow{\bullet}T$ postulate $\mu\circ T\mu=\mu\circ\mu T$ postulate $\mu\circ T\eta=\mu\circ\eta T=1_T$ ---------- Discussion A monad is functor together with two natural transformations that fulfill some algebraic relations. $\mu_X\circ T(… text/html 2015-01-30T12:51:19+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/monad_._haskell?rev=1422618679&do=diff Monad . Haskell Haskell class -- definition class Monad m where return :: a -> m a (>>=) :: m a -> (a -> m b) -> m b -- utility functions (>=>) :: (a -> m b) -> (b -> m c) -> (a -> m c) (f >=> g) a = f a >>= g fmap :: (a -> b) -> m a -> m b fmap f ma = ma >>= (return . f) join :: m (m a) -> m a join mma = mma >>= id (>>) :: m a -> m b -> m b ma >> mb = ma >>= (const mb) $\ \leftrightsquigarrow\ $$\ \leftrightsquigarrow\ $$\ \leftrightsq… text/html 2014-08-24T02:47:40+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/monadplus?rev=1408841260&do=diff MonadPlus Haskell class > Discussion Postulates Associated methods "todo" Reference Parents Subset of Monad . Haskell text/html 2015-04-12T17:48:48+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/monoid?rev=1428853728&do=diff Monoid Set context $M$ ... set definiendum $ \langle\!\langle M,*\rangle\!\rangle \in$ it inclusion $*$ ... binary operation exists $e$ postulate $e$ ... unit element $\langle\!\langle M,*\rangle\!\rangle$ postulate $(a*b)*c=a*(b*c)$ ---------- Discussion The binary operation is often called multiplication and $e$ is called the $M$$*$$*$ text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/monoid_kernel?rev=1395396676&do=diff Monoid kernel Set context $ X $ context $ M $ ... monoid context $ f:X\to M $ definiendum $ x\in\mathrm{ker}(f) $ postulate $ f(x)=e $ Discussion $\mathrm{ker}(f)$ is $\mathrm{sol}(f)$ w.r.t. the unit of the monoid. Reference Wikipedia: Solution set Parents Subset of Solution set Context Monoid text/html 2015-03-17T14:08:53+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/monomorphism?rev=1426597733&do=diff Monomorphism Collection context ${\bf C}$ ... category definiendum $f \in\mathrm{it} $ inclusion $f:{\bf C}[A,B]$ postulate $\langle A,\prod_{A}1_A\rangle$ ... pullback of $f$ along itself ---------- Equivalent definitions The arrow $f:{\bf C}[A,B]$ is mono of for all $g,h:{\bf C}[C,A]$ holds $f\circ g=f\circ h\implies g=h$. In ${\bf{Set}}$, this can be rewritten as the definition of injections: $f(x)=g(y)\implies x=y$${\bf{Set}}$$A\times_BA$$\langle a,d\rangle\in… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/monotonically_decreasing_function?rev=1395396676&do=diff Monotonically decreasing function Function context $ X,Y $ context $ \le_X,\le_Y $ ... non-strict partial order definiendum $ f\in\mathrm{it} $ inclusion $f:X\to Y$ for all $x,y\in X$ postulate $ x\ge_X y \implies f(x)\ge_Y f(y) $ Discussion Parents Subset of Function Context Non-strict partial order text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/monotonically_increasing_function?rev=1395396676&do=diff Monotonically increasing function Function context $ X,Y $ context $ \le_X,\le_Y $ ... non-strict partial order definiendum $ f\in\mathrm{it} $ inclusion $f:X\to Y$ for all $x,y\in X$ postulate $ x\le_X y \implies f(x)\le_Y f(y) $ Discussion Parents Subset of Function Context Non-strict partial order Related Monotonically decreasing function text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/multi-index_power?rev=1395396676&do=diff Multi-index power Set context $ G $ ... group context $ g \in \text{Sequence}(G) $ context $ \alpha \in \text{Sequence}(\mathbb N) $ context $ \mathrm{length}(g)=\mathrm{length}(\alpha) $ definiendum $ \langle g,\alpha\rangle \mapsto g^\alpha := \prod_{i=1}^{\mathrm{length}(\alpha)} g_i^{\alpha_i} $ We also write $|\gamma|=\sum_i^{\mathrm{length}(\gamma)} \gamma_i $. Discussion In most cases, the base sequence is understood. E.g. if $\gamma=\langle 3,1,… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/multilinear_functional?rev=1395396676&do=diff Multilinear functional Set context $X$...$\mathcal F$-vector space context $n\in \mathbb N$ definiendum $M\in \mathrm{MultiLin}(X^n)$ context $ M:X^n \to \mathcal F$ $X^n$ being the cartesian product of $n$ instances of the vector space $X$. $ a,b\in \mathcal F $ $ v_1,\dots,v_n,w\in X $ $ 1\le j\le n $ postulate $ M(v_1,\dots,a\cdot v_j+b\cdot w,\dots,v_n)=a\ M(v_1,\dots,v_j,\dots,v_n)+b\ M(v_1,\dots,w,\dots,v_n) $ Discussion Parents Subset of Function… text/html 2014-12-04T22:34:21+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/my_equivalence_of_categories?rev=1417728861&do=diff My equivalence of categories Collection context $F$ in ${\bf D}\longrightarrow{\bf C}$ context $G$ in ${\bf C}\longrightarrow{\bf D}$ definiendum $\langle\alpha,\beta\rangle$ in $F\simeq G$ inclusion $\alpha, \beta$ ... my nice nats $\left(F,G\right)$ inclusion $\alpha,\beta$ ... natural isomorphism Discussion Elaboration $\alpha$ in $FG\cong Id_{\bf C}$ $\beta$ in $Id_{\bf D}\cong GF$. Note the two different symbols $\cong$$\simeq$ text/html 2014-12-04T16:29:54+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/my_nice_nats?rev=1417706994&do=diff My nice nats Collection context $F$ in ${\bf D}\longrightarrow{\bf C}$ context $G$ in ${\bf C}\longrightarrow{\bf D}$ definiendum $\langle\alpha,\beta\rangle$ in it inclusion $\alpha:FG\xrightarrow{\bullet}1_{\bf C}$ inclusion $\beta:1_{\bf D}\xrightarrow{\bullet}GF$ Discussion That silly name ... I made it up. The natural transformation $\beta:1_{\bf D}\xrightarrow{\bullet}GF$ squeezes every set $X\in {\bf D}$ into a set $GFX\in {\bf D}$ (although this need … text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/n-cube?rev=1395396676&do=diff k-regular graph Set context $n\in\mathbb N, n\ge 1$ definiendum $ Q_n\equiv\langle V,E \rangle $ postulate $ V=\{0,1\}^n $ for all $ v,w\in V $ range $ k\in\mathbb N, 1\le k\ne n $ postulate $ \{v,w\}\in E \leftrightarrow \exists! k.\ \pi_k(v)\neq \pi_k(w) $ Discussion The n-cube $Q_n$ is the graph with vertices being n-tuples which are connected exactly if they differ by one coordinate. $V(Q_2)=\{\langle 0,0\rangle,\langle 0,1\rangle,\langle 1,0\rang… text/html 2014-12-04T16:32:31+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/natural_isomorphism?rev=1417707151&do=diff Natural isomorphism Collection context $F,G$ in ${\bf C}\longrightarrow{\bf D}$ definiendum $\eta$ in $F\cong G$ inclusion $\eta$ in $F\xrightarrow{\bullet}G$ for all $A:\mathrm{Ob}_{\bf C}$ postulate $\eta_A$ ... isomorphsim Discussion Reference Wikipedia: Natural transformation Parents Context Functor Requirements Natural transformation, Isomorphism text/html 2016-07-22T15:08:40+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/natural_logarithm_of_complex_numbers?rev=1469192920&do=diff Natural logarithm of complex numbers Set definiendum $\mathrm{ln}:\mathbb C\setminus {]-\infty,0]} \to \mathbb R$ postulate $\mathrm{ln}(z):=\mathrm{ln}|z|+i\,\arg(z)$ ---------- "todo: Complex argument" Limits $\lim_{x\to 0}x\ln(x)=0$ Differentiation and integrals $\int \ln(x^n)\,{\mathrm d}x^n=\int \left(x^n\right)'\ln(x^n)\,{\mathrm d}x=x^n\left(\ln(x^n)-1\right)$ Series At least around $z=0$ (I think for $|z|<1$) $\ln{\left(\frac{1}{1-z}\right)} = \sum_{n=1}^\infty \frac… text/html 2019-09-03T15:33:33+0200 Anonymous (anonymous@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 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/natural_number_range?rev=1395396676&do=diff Natural number range Set definiendum $ \mathrm{range}: \mathbb N\cup\{\infty\}\to\mathcal P(\mathbb N)$ definiendum $ \mathrm{range}(n):= \begin{cases} \{k\ |\ 1\leq k\leq n\} & \mathrm{if}\ n\in \mathbb N\\\\ \mathbb N & \mathrm{if}\ n=\infty \end{cases}$ Discussion Parents Related Natural number Element of Function text/html 2015-02-18T20:36:47+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/natural_numbers?rev=1424288207&do=diff Natural numbers Framework We can set up ${\mathbb N}$ via Peano axioms. In this case induction is a seperate (schema) of axioms. If one has a notion of the set/type of Rational numbers ${\mathbb Q}$, one may consider definiendum $ {\mathbb N}$ inclusion $ {\mathbb N}\subseteq {\mathbb Q}$ postulate $ 0\in{\mathbb N}$ for all $ n\in{\mathbb N}$ postulate $ (n+1)\in{\mathbb N} $$ n = 0\ \lor\ \exists (k\in{\mathbb N}).\ n = k+1 $$0,1,2,\dots$${\mathbb N}\equiv\{0,1,2… text/html 2016-04-09T15:00:12+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/natural_transformation?rev=1460206812&do=diff Natural transformation Collection context $F,G$ in ${\bf C}\longrightarrow{\bf D}$ definiendum $\eta$ in $F\xrightarrow{\bullet}G$ inclusion $\eta:{\large\prod}_{(A:\mathrm{Ob}_{\bf C})}F\,A\to G\,A$ postulate $\eta\circ F(\,f)=G(\,f)\circ\eta$ Here, in the postulate, I've left the components ($\eta_A,\eta_B$ etc.) implicit. Discussion Idea Natural transformation form a collection of arrows within a single category which are compatible with the (structure preserving… text/html 2014-12-26T16:06:39+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/natural_transformations?rev=1419606399&do=diff Natural transformations Meta ${\mathfrak D}_{\xrightarrow{\bullet}}$ ---------- ---------- Related Domain of discourse text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/navier-stokes_equations?rev=1395396676&do=diff Navier–Stokes equations Set context $ n\in\mathbb N $ context $ \rho: \mathbb R^n\times\mathbb R\to\mathbb R^n $ context $ p \in C(\mathbb R^n\times\mathbb R,\mathbb R) $ context $ \boldsymbol{\mathsf{T}} \in C(\mathbb R^n\times\mathbb R\times\mathbb R^n,\mathbb R^{n^2}) $ context $ \mathbf{f} \in C(\mathbb R^n\times\mathbb R,\mathbb R^n) $ range $ ::\rho(\mathbf{x},t) $ range $ ::p(\mathbf{x},t) $ range $ ::\boldsymbol{\… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/negative_part_of_a_function?rev=1395396676&do=diff Negative part of a function Set context $f:X\to \overline{\mathbb R}$ definiendum $f^-:X\to \overline{\mathbb R}$ definiendum $ f^-(x) := \mathrm{max}(-f(x),0)$ Discussion Notice that $f^-(x)\ge 0$. Parents Context Maximum function, Extended real number line text/html 2014-10-25T14:47:37+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/neighbourhood?rev=1414241257&do=diff Neighbourhood Set context $\langle X,\mathcal{T}_X\rangle$ ... topological space context $p\in X$ definiendum $ U_p\in\mathrm{it} $ postulate $ \exists(\mathcal{O}\in\mathcal{T}_X).\ \mathcal{O}\subseteq U_p $ Discussion A neighbourhood of $p$ is a reasonably big set surrounding $p$. Predicates Consider $X$ together with a topology, then locally euclidean space$X$$\mathbb R^n$$X$$ \equiv \forall(x\in X).\ \exists(U_x\in\mathrm{Neighbourhood}(x)),\ f.\ f\in\mathrm… text/html 2016-07-22T15:13:54+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/niemand_sequence?rev=1469193234&do=diff Niemand seqeunce Note Let's define a Niemand sequence $(a_{n,N})$ as a sequence (in $N$) of sequences (in $n$). The map $(a_{n,N}) \mapsto (S_N):=\sum_{n=0}^N a_{n,N}$ removes the $n$-index. And then $(S_N) \mapsto \sum_{n=0}^N S_N$ removes the second. Examples a{n,N} constant in N For $a_{n,N}$ constant in $N$, the series $\sum_{n=0}^N a_{n,N}$ is just the sequence of partial sums.$f$$x_0,x_1$$h(N):=\dfrac{x_1-x_0}{N}$$a_{n,N}:=h(N)\,f\left(x_0+n\,h(N)\right)$$\sum_{n=0}^N a_{n,N}$$\… text/html 2025-12-23T03:33:58+0200 Anonymous (anonymous@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 2016-04-09T15:48:31+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/non-negative_extended_real_number_line?rev=1460209711&do=diff Non-negative extended real number line Set postulate $ x\in\overline{\mathbb R}_+ $ postulate $ x\in\overline{\mathbb R} \land x\ge 0 $ Ramifications Reference Wikipedia: Extended real number line Parents Set Subset of Extended real number line text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/non-negative_rational_number?rev=1395396676&do=diff Non-negative rational number Set definiendum $ r \in\mathbb Q_+ $ context $ r \in \mathbb Q $ postulate $ r \ge 0 $ Discussion Reference Wikipedia: Rational number Parents Refinement of Non-negative real number Subset of Rational number text/html 2014-11-20T13:00:54+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/non-negative_real_number?rev=1416484854&do=diff Non-negative real number Set definiendum $ r \in\mathbb R_+ $ postulate $ r \ge 0 $ Discussion Reference Wikipedia: Real number Parents Subset of Real number text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/non-strict_partial_order?rev=1395396676&do=diff Non-strict partial order Set context $X$ definiendum $ \le\ \in\ \mathrm{it} $ The relation $\le$ is an order relation if it's in the intersection of all reflexive, all anti-symmetric and all transitive relation. Hence context $ \le\ \in\ \mathrm{Rel}(X) $ $ x,y,z \in X $ postulate $ x \le x $ postulate $ x\le y\ \land\ y\le x \implies (x=y) $$ x \le y\ \land\ y \le z \Leftrightarrow x\le z $$x\le y\ \equiv\ \le(x,y)$ text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/non-zero_integer?rev=1395396676&do=diff Non-zero integer Set definiendum $ \mathbb Z^* \equiv \mathbb Z\setminus \{0\} $ Discussion Parents Subset of Integer text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/non-zero_natural_number?rev=1395396676&do=diff Non-zero natural number Set definiendum $ \mathbb N^* \equiv \mathbb N\setminus \{0\} $ Discussion Parents Refinement of Non-zero integer Subset of Natural number text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/non-zero_rational_number?rev=1395396676&do=diff Non-zero rational number Set definiendum $ \mathbb Q^* \equiv \mathbb Q\setminus \{0\} $ Discussion Parents Subset of Rational number text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/non-zero_real_number?rev=1395396676&do=diff Non-zero real number Set definiendum $ \mathbb R^* \equiv \mathbb R\setminus \{0\} $ Discussion Parents Subset of Real number text/html 2016-05-01T16:01:56+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/norm?rev=1462111316&do=diff Norm Set context $F$ ... subfield of $\mathbb{C}$ context $V$ ... $F$-vector space definiendum $p\in \mathrm{Norm}(V)$ postulate $p:V\to \mathbb R $ $v,w\in V$ postulate $p(v+w) \le p(v)+p(w)$ postulate $p(v)=0 \implies v=0$ $\lambda\in F$ postulate $p(\lambda\cdot v) = |\lambda|\cdot p(v)$ ---------- Discussion $ p(v)\ge 0 $ The last axiom $\ p(v)=0 \implies v=0\ $ isn't part of seminorm. Reference Wikipedia: text/html 2015-11-30T11:06:52+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/normal_distribution?rev=1448878012&do=diff Normal distribution Function definition $f: {\mathbb C}\times{\mathbb C}\times{\mathbb C}\setminus\{0\}\to{\mathbb C}$ definition $f(x, \mu, \sigma) = \dfrac{1}{\sigma\sqrt{2\pi} }{\mathrm e}^{ -\dfrac{(x-\mu)^2}{2\sigma^2}}$ ---------- Discussion People also like to write ${\mathcal N}(\mu, \sigma) := \lambda x.\,f(x,\mu, \sigma)$ Theorems Reference Wikipedia: Normal_distribution ---------- Requirements Exponential function text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/normal_matrix?rev=1395396676&do=diff Normal matrix Set context $n\in\mathbb N$ definiendum $ A \in \mathrm{NormalMatrix}(n) $ postulate $ A \in \mathrm{SquareMatrix}(n,\mathbb C) $ postulate $ A^*\ A = A\ A^* $ Discussion Reference Wikipedia: Normal matrix Parents Subset of Square matrix Context Matrix conjugate transpose text/html 2015-12-25T17:19:39+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/normalized_fox-wright_function?rev=1451060379&do=diff Normalized Fox-Wright function Function definition $??$ definition ${}_p\Psi_q^*[\langle a_1, A_1\rangle,…,\langle a_p, A_p\rangle; \langle b_1, B_1\rangle,…,\langle a_q, A_q\rangle](z):= \sum_{n=0}^\infty c_n z^n$ with $c_n = \dfrac{1}{n!}\dfrac{\prod_{m=1}^p \Gamma(a_m+A_m\cdot{n})\, /\, \Gamma(a_m)}{\prod_{j=1}^q \Gamma(b_j+B_m\cdot{n})\, /\, \Gamma(b_j)}$ ---------- Discussion Elaboration/Motivation The coefficients of ${}_p\Psi_q^*$ relate very similarly to each ot… text/html 2014-09-28T11:50:59+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/normed_vector_space?rev=1411897859&do=diff Normed vector space Set context $V$ ... $F$-vector space definiendum $\langle V,\Vert\cdot\Vert\rangle$ ... normed $F$-vector space postulate $\Vert\cdot\Vert\in\mathrm{Norm}(V)$ Discussion Reference Wikipedia: Normed vector space Parents Subset of Vector space Context Norm text/html 2016-07-22T23:52:20+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/notes_on_physical_theories_._note?rev=1469224340&do=diff notes on physical theories . note this is a temporary entry to link all the physics notes to, so I find them later, when I develop the theories proper. ---------- "<https://en.wikipedia.org/wiki/List_of_quantum-mechanical_systems_with_analytical_solutions>" "todo:" Set up mechanics (QM and classical) with a maximum number of units and all unit conversions as functions, $c$$[c]=m/s$$\tau_c(x) := \tfrac{1}{c}x $$x\in{\mathbb R}$$\tau_c(x) := \tfrac{1}{c}||\,x\,||$$x\in{\mathcal B}$$\tau_c(\… text/html 2016-06-24T12:25:55+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/notes_on_programming_languages?rev=1466763955&do=diff Notes on programming languages Note ---------- Haskell Idris Python Matlab "todo: figure out a common scheme for all the above pages. I guess 'Basic commands', 'Syntax' and 'Type system' should each get their own entry. (Some of them already do)" ---------- Requirements text/html 2015-10-10T20:45:19+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/np?rev=1444502719&do=diff NP Set definiendum $ L\in \mathrm{\bf{NP}} $ inclusion $ L\subseteq\{0,1\}^* $ range $ M $ ... polynomial time 2-tape Turing machine range $ p:\mathbb{N}\to\mathbb{N} $ ... polynomial range $ x,u\in\{0,1\}^* $ ... bit string postulate $\exists M,p.\ \forall x.\ x\in L\Leftrightarrow \exists u.\ \left(M(x,u) = 1\land \left|u\right|=p(\left|x\right|)\right) $ ---------- Discussion "todo: Requirements Polynomial time Turing machine" The definition… text/html 2016-01-18T22:08:47+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/observable?rev=1453151327&do=diff Observable Set context $V$...Hilbert space definiendum $\mathrm{Observable}(V)\equiv\mathrm{SelfAdjoint}(V)\cap\mathrm{End}(V)$ ---------- Discussion Observables are the linear self-adjoint operators. * $\langle\psi|A\ \phi\rangle\in \mathbb C$ is called transition amplitude. * $\frac{|\langle\psi|A\ \phi\rangle|^2}{\Vert\psi\Vert^2\Vert\psi\Vert^2}\ge 0$ is called transition probability. * $\langle \psi | A\ \psi \rangle\in \mathbb R$ is called expectation value. * … text/html 2015-03-27T20:14:07+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/ode_system?rev=1427483647&do=diff ODE system Set context $ d,n,m,k\in\mathbb N $ context $ F:\mathbb R\times\mathbb R^{d\,(1+n)}\to \mathbb R^m $ definiendum $ y \in $ it postulate $ y:C^k(\mathbb R,\mathbb R^d) $ postulate $ F(t,y(t),y'(t),y''(t),\dots,y^{(n)}(t))=0 $ ---------- Reference Wikipedia: Ordinary differential equation ---------- Subset of PDE system text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/offset_logarithmic_integral?rev=1395396676&do=diff Offset logarithmic integral Function definiendum $ \mathrm{Li}: \mathbb R_+\to \mathbb R_+$ definiendum $ \mathrm{Li}(x) := \mathrm{li}(x)-\mathrm{li}(2) $ Discussion Alternative definition $ \mathrm{Li}(x) := \int_2^x \frac{1}{\ln(t)}\mathrm d t $ Theorems $ \mathrm{Li}(x)\approx\pi(x) $ "todo: Prime counting-function" Reference Wikipedia: Logarithmic integral function Parents Requirements Logarithmic integral function text/html 2016-04-07T17:38:37+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/on_category_theory_basics?rev=1460043517&do=diff On category theory basics Foundational temp1b $\blacktriangleright$ On category theory basics $\blacktriangleright$ On universal morphisms Guide Opposite category From every category, we can obtain another one by simply flipping the arguments of concatenation. IsomorphismGroupoidAutomorphism ---------- Functor As far as objects $A,B\in{\bf C}$ are concerned, functors $F$ in ${\bf C}\longrightarrow{\bf D}$ are plain functions, acting via $A\mapsto FA$${\bf C}[A,B]\mapsto{\bf C}[FA,FB]$$F$ text/html 2016-03-08T12:09:53+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/on_electrodynamics_._note?rev=1457435393&do=diff On electrodynamics . note Axioms * Differential geometry ${\mathrm d} F = 0$ ${\mathrm d} {\star} F = c^F {\cdot}j$ * Vectorial Form see On physical units . note ... References Wikipedia: Maxwell's_equations ---------- Related Notes on physical theories . note, PDE system text/html 2016-07-23T15:23:35+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/on_electronics_._note?rev=1469280215&do=diff On electronics . note Model Circuit elements on the microscopic level Hierarchy of devices *ordered hierarchy of viewpoints on devices such as resistors (electrical resistance)* On units Here I want to work out more natural units for stuff like capacitance, to understand the formulas describing different circuit device behaviors. See also $[q_x]$$\frac{1}{[q_x]}$$I$$[I]=\dfrac{[q_x]}{[t]}$$U$$[U]=\dfrac{1}{[q_x][t]}$$I=f_G(U)$$f_G$$I=G\cdot U$$[G]=[q_x]^2$$[q_x]^2$$[S]$$[\Omega]$$-1$$G\lef… text/html 2016-07-20T00:05:28+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/on_mathematical_theories?rev=1468965928&do=diff On mathematical theories Perspective $\blacktriangleright$ On mathematical theories $\blacktriangleright$ On reading Note Entries such as Logic discuss the framework for writing down theories and their models, formalizing mathematical entities and making definitions. Much of the process isn't captured by formalities though, especially things that relate to motivations and conceptural understandings.$\log$ text/html 2015-11-08T14:46:25+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/on_phenomenological_thermodynamics_._note?rev=1446990385&do=diff On phenomenological thermodynamics . Note Axioms The theory in which state space an internal energy $U(S,V,\{N_i\})$ is part of the axioms. (In statistical physics, $U(S,V,\{N_i\})$ is only derived, see e.g. Statistical internal energy.) Immediate definitions $U, S, V, \{N_i\}$ are all the extensive quantities. Legendre transformation of $U$$S$$F$$U$$T$$S$$T:=\frac{\partial U}{\partial S}$$U$$V$$H$$U$$-p$$V$$p:=-\frac{\partial U}{\partial V}$$F$$H$$V$$S$$G$$\mu_i:=\frac{\partial U}{\parti… text/html 2016-09-28T17:17:45+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/on_physical_units_._note?rev=1475075865&do=diff On physical units . note Perspective $\blacktriangleright$ On physical units . note $\blacktriangleright$ Its about time . note $\blacktriangleright$ On electronics . note Note Abstract notes on unit-typing Here are some ideas capturing units as a type system. We'll use it to restrict operations between structures that exhibit the same algebraic rules. E.g. a positions $x_1, x_2, \cdot$$T_1, T_2, \cdot$$T_1+T_2$$T_1+x_1$$U$$E$$\dfrac{e:E}{[e]:U}$$\dfrac{a=b}{[a]=[b]}$$[-]$$E$$U$$\dfrac{… text/html 2016-08-07T16:21:53+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/on_reading?rev=1470579713&do=diff On reading On mathematical theories $\blacktriangleright$ On reading $\blacktriangleright$ On syntax Note One strategy in reading and taking apart the content of a physics or math text is to try and classify the parts of it into the above three points and sub-points. For each piece of text one is currently presented with, one can try and classify:$F$ text/html 2016-07-19T23:42:01+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/on_syntax?rev=1468964521&do=diff On syntax On reading $\blacktriangleright$ On syntax $\blacktriangleright$ Guideline Note An apple pie from scratch. Where do we start? Mathematics is there to be used. It's a little like a game and a defining feature of it is that it always involves some collection of rules. There are, however, many many different ways to play that game, and so we inevitably need to specify how we're going to play. Every explanatory math text must a mix of formal and informal explanations. Since the mathema… text/html 2016-04-07T17:32:46+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/on_universal_morphisms?rev=1460043166&do=diff On universal morphisms On category theory basics $\succ$ On universal morphisms $\succ$ Foundational temp3 Guide Terminal morphismInitial morphism Motivation: Terminal and initial morphisms (the opposite notion, where you just flip some arrows) are called universal morphisms. On the technical side, they are a very important concept with a ridiculously broad range of examples, because you can translate many $G$${\bf C}\longrightarrow{\bf D}$$Y_i$${\bf D}$$\langle A_{Y_i},\eta_{Y_i}\rangle$$Y_… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/open_ball?rev=1395396676&do=diff Open ball Set context $\langle d,X\rangle$ ... metric space context $a\in X$ definiendum $B_a:\mathbb R_+^*\to \mathcal P(X)$ definiendum $B_a(r):= \{x\ |\ d(x,a)<r \}$ Discussion Reference Wikipedia: Ball (Mathematics) Parents Context Metric text/html 2016-09-14T15:09:01+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/open_cover?rev=1473858541&do=diff Open cover Set context $\langle X,\mathcal T\rangle$ ... topological space definiendum $C$ in it inclusion $C$ ... cover(X) postulate $C\subseteq \mathcal T$ ---------- ---------- Subset of Cover Context Topological space Requirements* Cover text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/open_subsets_of_%E2%84%9D%E2%81%BF?rev=1395396676&do=diff Open subsets of ℝⁿ Set context $p\in \mathbb N$ definiendum $\mathfrak J_o^p\equiv\{\ ]a,b[\ |\ a,b\in\mathbb R^p\ \land\ a< b\}$ Discussion We write definiendum $\mathfrak J_o\equiv \mathfrak J_o^1$ Parents Context Real coordinate space text/html 2014-12-04T22:34:13+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/opposite_category?rev=1417728853&do=diff Opposite category Category context ${\bf C}$ ... category definiendum ${\bf C}^\mathrm{op}$ definition $\mathrm{Ob}_{{\bf C}^\mathrm{op}}:=\mathrm{Ob}_{\bf C} $ definition ${\bf C}^\mathrm{op}[A,B]:={\bf C}[B,A]$ definition $f\,\circ_{{\bf C}^\mathrm{op}}\,g\ :=\ g\,\circ_{{\bf C}}\,f$ Discussion Reference Wikipedia: Opposite category Parents Context* Categories Element of Categories text/html 2016-10-16T16:31:44+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/optimization_set?rev=1476628304&do=diff Optimization set Set context $ B $ context $ \langle Y, \le \rangle $ ... Non-strict partially ordered set context $ r:B\to Y $ definition $ O_r := \{\beta\in B\mid \forall(b\in B).\,r(\beta)\le{r(b)}\}$ ---------- "todo #tag If p are parameters and c_p(x) curves with x_min(c_p)=f(p) known, try to find x_min(c') by fitting c_p to c'. Now what's p here. Is there a scheme so that we can extend the list p to have guaranteed that there are parameters so that eventua… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/order-reflecting_function?rev=1395396676&do=diff Order-reflecting function Set context $X,Y$ context $ \le_X,\le_Y $ context ...non-strict partial order definiendum $ f\in\mathrm{it} $ $f:X\to Y$ $x,y\in X$ postulate $ f(x)\le_Y f(y)\implies x\le_X y $ Discussion Reference Wikipedia: Monotone function Parents Refinement of Function Context Non-strict partial order text/html 2015-06-30T16:14:49+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/order_structure_of_real_numbers?rev=1435673689&do=diff Order structure of real numbers Set definiendum $\langle \mathbb R,\le \rangle$ We define the total order over the real numbers (in the Dedekind cut model) via $r<s \equiv r\subset s$, i.e. postulate $s \subseteq r \Leftrightarrow s\ge r$ ---------- Theorems Inequality of arithmetic and geometric means (AM-GM inequality): $\frac{1}{n}\sum_{k=1}^n x_k \ge \left(\prod_{k=1}^n x_k\right)^\frac{1}{n}$ Reference text/html 2014-12-08T10:15:49+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/ordered_pair?rev=1418030149&do=diff Ordered pair Set context $ X,Y $ definiendum $ \langle X,Y \rangle \equiv \{\{X\},\{X,Y\}\} $ Discussion The ordered pair is a model of $ \langle x,y \rangle = \langle z,u \rangle\ \Leftrightarrow\ x=z \land y=u $ See also Wikipedia: Ordered pair, Kuratowski defintion. In calculations, only this property of it should be used. Now Let $x_i$ be indexed sets and define $p \equiv \langle x_1,x_2\rangle$. Canonical projection: Using arbitrary union and arbitrary intersection, we c… text/html 2014-12-05T11:07:01+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/ordinal_number?rev=1417774021&do=diff Ordinal number Set definiendum $\alpha\in \mathrm{Ord}$ inclusion $\alpha$...transitive for all $\beta,\gamma\in\alpha$ postulate $ (\beta\in\gamma)\ \lor\ (\gamma\in\beta)\ \lor\ (\beta=\gamma) $ Discussion The second requiement says that the ordinal admits a set theoretical constuction of a certain order relation for all its elements. The first requirement means $\ \forall (\beta \in \alpha).\ \beta \subseteq \alpha\ $$\mathrm{Ord}$$\in$$<$$\beta<\gamma\equiv \bet… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/orthogonal_basis?rev=1395396676&do=diff Orthogonal basis Set context $V$...pre-Hilbert space definiendum $B\in \mathrm{OrthogonalBasis}(V)$ context $B\in \mathrm{Basis}(V)$ $v,w\in B$ postulate $v\ne w\implies \langle v|w\rangle = 0 $ Discussion Parents Subset of Vector space basis Context Pre-Hilbert space text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/orthonormal_basis?rev=1395396676&do=diff Orthonormal basis Set context $V$...pre-Hilbert space definiendum $B\in \mathrm{OrthonormalBasis}(V)$ context $B\in \mathrm{Basis}(V)$ $v,w\in B$ postulate $v\ne w\implies \langle v|w\rangle = 0 $ postulate $\Vert v\Vert = 1 $ Discussion Parents Subset of Orthogonal basis text/html 2016-11-27T15:20:14+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/outline?rev=1480256414&do=diff Outline An apple pie from scratch $\succ$ Outline $\succ$ Drawing arrows and coding functions Guide Here is an outline structure (work in progress): Concepts in the incarnation of a programming language * Assume knowledge of integers $-2,-1,0,1,2,3$ with binary operations $+,-,*$. * More on that languages syntax and examples (which are instances of relevant concepts to be discussed later)$\to$$\to$$\to$ text/html 2015-03-18T17:16:40+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/over_category?rev=1426695400&do=diff Overcategory Category context ${\bf C}$ ... category context $T\in{\bf C}$ definiendum ${\bf C}/T$ definition $\mathrm{Ob}_{{\bf C}/T}:=$ all arrows $f$, such that there is an object $S\in{\bf C}$, such that $f:{\bf C}[S,T]$ definition ${\bf C}/T[f,g]:=$ all arrows $h$, such that $h\circ g = f$ ---------- Elaboration Given a (target) object $T$${\bf C}$${\bf C}/T$$T$ text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/p?rev=1395396676&do=diff P Set definiendum $ \mathrm{\bf{P}} = \bigcup_{k\in\mathbb{N}} \mathrm{\bf{DTIME}}(n^k) $ Discussion Parents Requirements DTIME Subset of Bit string text/html 2014-04-01T16:25:38+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/partial_function?rev=1396362338&do=diff Partial function Set context $X,Y$ definiendum $ f\in\text{PartialFunction}(X,Y) $ context $ f \in \text{Rel}(X,Y) $ postulate $ \langle x,a\rangle\in f \land \langle x,b\rangle\in f \Rightarrow a=b $ Discussion The definition can be written as $\{\langle x,a\rangle,\langle x,b\rangle\}\subseteq f \Rightarrow a=b.$ It says that each argument $x$ for the function can result in only one value. (functionality) Reference Wikipedia: text/html 2015-08-16T13:28:21+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/particle_number_expectation_value?rev=1439724501&do=diff Particle number expectation value Set context $ w $ ... grand canonical weight definiendum $ \langle\hat N\rangle(\beta,\mu) := \sum_{N=0}^\infty w_N(\beta,\mu)\cdot N $ ---------- Discussion The notation “$\langle\hat N\rangle$” is chosen for the function because we can also introduce the sequence of observables $\hat N$ defined to give us the particle number of each canonical ensemble, i.e. $\hat N_N=N$$\hat N$$\hat N$$N$$\Omega(\beta,\mu)$$ \langle\hat N\rangle = - \frac{\pa… text/html 2015-08-19T19:51:12+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/particle_potential_._note?rev=1440006672&do=diff Note Morse potential Manipulate[ Plot[ (1 - Exp[-a (r - R)])^2, {r, 0, 5}, PlotRange -> {-1, 2}] , {{a, 2}, 0, 10}, {{R, 2}, 0, 10}] Reference Wikipedia: Morse potential ---------- Related Classical Hamiltonian system text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/path_._graph_theory?rev=1395396676&do=diff Path . graph theory Set context $V,E$ ... set definiendum $\langle V,E,\psi\rangle \in \mathrm{it}(E,V) $ postulate $\langle V,E,\psi\rangle $ ... simple graph range $ u,v\in V $ range $ a$ ... sequence in $V$ range $ i\in\mathbb N$ postulate $d(v)\neq 0$  postulate $ \exists a.\ \forall u,v.\ (\exists i.\ \{a_{i},a_{i+1}\}=\{u,v\}) \leftrightarrow (\{u,v\}\dots\mathrm{edge}) $ Discussion A path is a graph which can fully be describe… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/path_length?rev=1395396676&do=diff Path length Function context $V,E$ ... set definiendum $ \mathrm{dom}\ \mathrm{length} = \mathrm{Path}(V,E) $ definiendum $ \mathrm{length}(G):=|E| $ Discussion Parents Context Path . graph theory text/html 2015-03-27T20:14:34+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/pde_system?rev=1427483674&do=diff PDE system Set context $ D,d,n,m,k\in\mathbb N $ context $ \alpha $ ... sequence of length $n$ of multi-indices context $ F:\mathbb R^D\times\mathbb R^{d\ \cdot\ (1+n)}\to \mathbb R^m $ definiendum $ u \in \mathrm{it} $ postulate $u:C^k(\mathbb R^D,\mathbb R^d) $ postulate $ F(x,u(x),u^{(\alpha_1)}(x),u^{(\alpha_2)}(x),\dots,u^{(\alpha_n)}(x))=0 $ ---------- Reference Wikipedia: Partial differential equation ---------- Refinement of Monoid kernel … text/html 2016-09-24T15:18:51+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/pendulum?rev=1474723131&do=diff Pendulum Mechanics $\succ \dots \succ$ Pendulum $\succ \dots \succ$ $\dots$ Guide ---------- app brainstorming - hanging pendulum, projections to AR axes - magic stick, tracking, projection of $x(t)$, also $x'(t)$ and $x''(t)$ + fixed color scheme - cube of pendulum with side projections - replay of happenings - introducing time axes text/html 2016-07-11T02:08:26+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/perspective?rev=1468195706&do=diff Perspective About $\blacktriangleright$ Perspective $\blacktriangleright$ On reading notes on physical theories . note On physical units . note Note (This is a perspective on mathematical content as it currently develops in my head, shaped by the stuff I learn.) Three major chunks of “doing physics” are learning, computing and modeling. Of course, whatever you're focusing on at the moment, the other two are usually part of it too. $A$$X$$f:\omega\to A$$d$$A$$h'(t)=3\,h(t)$$\sum_{k=0}^n\fr… text/html 2015-01-12T18:19:54+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/pi_function?rev=1421083194&do=diff Pi function Function definiendum $\Pi: \mathbb C\setminus\{-k\ |\ k\in\mathbb N^*\}\to \mathbb N$ definiendum $\Pi(z) := \begin{cases} \int_0^\infty\ \ t^{z}\ \mathrm{e}^{-t}\ \mathrm d t & \mathrm{if}\ \mathrm{Re}(z)>0 \\\\ \frac{1}{z+1}\Pi(z+1) & \mathrm{else} \end{cases}$ Discussion $\Pi(z)=\Gamma(z+1)$ Theorems $n\in\mathbb N \implies \Pi(n)=n! $ $\Pi(z)\cdot \Pi(-z)=\frac{\tau\ z/2}{\sin(\tau\ z/2)} $ Reference Wikipedia: Gamma function Parents Context Function integral o… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/pointed_set?rev=1395396676&do=diff Pointed set Set context $ X $ definiendum $ \langle X,x_0 \rangle \in \mathrm{it} $ postulate $x_0\in X$ Discussion Reference Wikipedia: Pointed set Parents Context Ordered pair text/html 2015-04-17T15:30:20+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/pointwise_function_product?rev=1429277420&do=diff Pointwise function product Set context $S$ ... set context $\langle\!\langle M,* \rangle\!\rangle$ ... magma definition $\star\in$ binary operation on $M^S$ definition $(f\star g)(s):=f(s)*g(s)$ ---------- Discussion Extends to groups, etc. "the following could be phrased more explicitly." Note that $M^S$${\mathrm{Hom}}_{\bf{Set}}(S,M)$$F$$F$ text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/pole_of_a_complex_function?rev=1395396676&do=diff Pole of a complex function Set context $\mathcal O$ ... open subset of $\mathbb C$ context $f:\mathcal O\to \mathbb C$ definiendum $a\in\mathrm{it}$ range $U$ ... open subset of $\mathbb O$ range $g:\mathcal O\to \mathbb C$ range $n\in\mathbb N, n>0$ postulate $\exists U,g,n.\ \left(z\in U\right)\land \left(f\ \mathrm{holomorphic\ on}\ U\setminus\{z\}\right)\land \left(g\ \mathrm{holomorphic\ on}\ U\right)\land \left(f(z)=\frac{g(z)}{(z-a)… text/html 2016-05-30T16:41:53+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/polylogarithm?rev=1464619313&do=diff Polylogarithm Function "Context: s" definiendum $ \mathrm{Li}_s: {\mathbb C} \to {\mathbb C}$ definiendum $ \mathrm{Li}_s(z) := \begin{cases} \sum_{n=0}^\infty\, n^{-s} z^n&\hspace{.5cm} \mathrm{if}\hspace{.5cm} |z|<1,\hspace{.5cm} \\\\ \text{analytic continuation}\hspace{.5cm} &\hspace{.5cm} \mathrm{else} \end{cases}$ "todo “$\text{analytic continuation}$”" ---------- Theorems Representations $\mathrm{Li}_s(z) = z\dfrac{\int_0^\infty\frac{x^{s}}{e^x-z}\frac{{\mathrm d}x}{x}}{\i… text/html 2016-10-11T13:13:52+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/polynomial_function?rev=1476184432&do=diff Polynomial function Set definition $??$ definition $??$ ---------- Discussion Simply quadratic equation $q(x)=x\,x-a\,x$ $q(x)=0 \implies (x=0\lor x=a)$ A small deviation $p(x)=x\,x-a\,x+b$ $p(x)=0 \implies x=a\cdot\dfrac{1+\epsilon\sqrt{1-\left(\frac{2}{a}\right)^2\cdot b}}{2}, \epsilon=\pm 1$ $\lim_{b\to 0}\sqrt{1-\left(\frac{2}{a}\right)^2\cdot b}=1$ Series[(1 + \[Epsilon] Sqrt[1 - d])/2, {d, 0, 5}] Reference Wikipedia: ---------- Subset of Analytic function Requ… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/poset?rev=1395396676&do=diff Poset Set context $X$ definiendum $\langle X,\le \rangle \in\ \mathrm{Poset}(X) $ context $\le$ ... non-strict partial order Discussion "todo <http://i.imgur.com/xMvIkYr.png>" Reference Wikipedia: Poset Parents Equivalent to Non-strict partial order text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/positive_function_integral?rev=1395396676&do=diff Positive function integral Set context $M \in \mathrm{MeasureSpace}(X)$ postulate $\int_X:\mathcal M^+\to \mathbb R_+$ $ f\uparrow u_n$ $u_n\in \mathcal T^+$ postulate $\int_X\ f\ \mathrm d\mu:=\mathrm{lim}_{n\to \infty}\int_X\ u_n\ \mathrm d\mu$ Notice that the integral on the right hand side here is that for positive real step functions. Discussion Monotone convergence theorem: If $f_n$ is a growing sequence in $\mathcal M^+$, we have$\int_X\left(\mathrm{lim}_{n\to… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/positive_measurable_numerical_function?rev=1395396676&do=diff Positive measurable numerical function Set context $ \langle X,\Sigma_X\rangle\in \mathrm{MeasurableSpace}(X) $ postulate $f\in \mathcal M^+$ context $f\in \mathrm{Measurable}(X,\overline{\mathbb R})$ $x\in X$ postulate $f(x)\ge 0$ Discussion For the definition of the integral, it's crucial to know that for every $f\in \mathcal M^+$, there is a sequence $u_n$ with elements in the step functions $\mathcal T^+$$u_n\uparrow f$ text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/positive_part_of_a_function?rev=1395396676&do=diff Positive part of a function Set context $f:X\to \overline{\mathbb R}$ definiendum $f^+:X\to \overline{\mathbb R}$ definiendum $ f^+(x) := \mathrm{max}(f(x),0)$ Discussion Parents Context Maximum function, Extended real number line text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/positive_real_step_function?rev=1395396676&do=diff Positive real step function Set context $\langle X,\Sigma\rangle\in\mathrm{MeasurableSpace}(X)$ postulate $f\in \mathcal T^+$ context $f\in \mathcal T$ context ...Real step function $x\in X$ postulate $f(x)\ge 0$ Discussion Parents Subset of Real step function text/html 2015-10-08T20:35:53+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/power_set?rev=1444329353&do=diff Power set Set context $X$ ... set definiendum $ Y \in \mathcal{P}(X) $ postulate $ Y\subseteq X $ ---------- Here we define $\mathcal{P}(X) \equiv \{Y\mid Y\subseteq X\}$ which is sensible in our set theory if, for each set $X$, we have $\exists! P.\,P = \{Y\mid Y\subseteq X\}$ or, more formally, $\forall X.\,\exists! P.\,P = \{Y\mid Y\subseteq X\}$ which is short for $\forall X.\,\exists! P.\,\forall Y.\,\left(Y\in P\Leftrightarrow Y\subseteq X\right)$ Discussion T… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/pre-hilbert_space?rev=1395396676&do=diff Pre-Hilbert space Set context $V$ definiendum $\langle\mathcal V,\langle\cdot|\cdot\rangle\rangle \in \mathrm{PreHilbert}(V)$ context $\mathcal V \in \mathrm{VectorSpace}(V,\mathbb C)$ context $\langle\cdot|\cdot\rangle:V\times V\to \mathbb C$ $u,v,w\in V$ $a,b\in \mathbb C$ postulate $\overline{\langle v|w \rangle}=\langle w|v \rangle$ postulate $v \ne 0 \Rightarrow \langle v|v \rangle > 0 $ postulate $v = 0 \Rightarrow \langle v|v \rangle = 0 $ … text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/pre-image?rev=1395396676&do=diff Pre-image Set Pre-image of the function $f$ w.r.t the codomain subset $Z$. context $ Z $ context $ f\in X^Y $ postulate $ x\in f^{-1}(Y) $ postulate $ f(x)\in Z $ Ramifications Reference Wikipedia: Image Parents Context Function text/html 2014-12-04T13:59:54+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/predicate_library?rev=1417697994&do=diff Predicate library Meta This is a list of predicates together with the entries in which they are defined. So for example “$f$...functional” is a property which, according to the list below, is defined in the entry Function. Note that the list excludes certain predicates if they are named after the exact name of an entry. This is the case for some $V$$x$$\mathrm{isfoo}$$P$$x$$P(x)$$x$$x$$\neg P(x)$$<, \le, >, \ge$ text/html 2016-05-01T15:56:25+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/predicate_logic?rev=1462110985&do=diff Predicate logic Framework "todo: better exposition of the concepts I mention in this first paragraph.." We presuppose knowledge of the weaker theories. Informally, a formula is any logical expression (e.g. propositions or predicates with or without free variables) which can denote a truth value, while terms or variables themselves do not. On the meta-level, one must understand typing (terms vs. predicates) and $\forall, \exists$$\int_{-2}^7t^n\,\mathrm dt$$\int_{-2}^7r^n\,\mathrm dr$$\forall… text/html 2015-02-07T00:22:43+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/presheaf_._topology?rev=1423264963&do=diff Presheaf . topology Collection context $\langle X,\mathcal T\rangle$ ... topological space context ${\bf C}$ ... category definiendum ${\bf C}^{\mathrm{Op}(X)^{\mathrm{op}}}$ Discussion Motivation via fibre bundles Let $\langle X,\mathcal T\rangle$ be a topological space. A fibre bundle $Fib\to E\to X$ is given by a projection map $p:E\to X$ together with, for each open set $U\in \mathcal T$$E$$U\times Fib$$X\to E$$\sigma$$p\circ\sigma=\mathrm{id}_X$$E=X\times Fib$$\… text/html 2015-02-21T12:10:47+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/presheaf_category?rev=1424517047&do=diff Presheaf category Category context ${\bf C}$ ... small category definiendum ${\bf Set}^{{\bf C}^\mathrm{op}}$ ---------- The co- and contravariant hom-functors $\mathrm{Hom}(B,-)$ and $\mathrm{Hom}(-,B)$ are maybe the most natural functors. While forgetful functors are other examples of covariant set-valued functors, covariant functors very often have to do with function spaces. (Once we pass from presheaves to sheaves by adding some more text/html 2016-05-24T19:44:01+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/prime_enumeration?rev=1464111841&do=diff Prime enumeration Function definition $p : {\mathbb N}_+\to{\mathbb P}$ definition $p(1):=2$ definition $p(n+1):={\mathrm{min}}({\mathbb P}\setminus{X_n})$ with $X_n\equiv\{m\in{\mathbb P}\mid m\le p(n)\}$ ---------- Code primes0 :: [Integer] primes0 = filter prime0 [2..] prime0 :: Integer -> Bool prime0 n | n < 1 = error "not a positive integer" | n == 1 = False | otherwise = ld n == n text/html 2016-05-24T19:28:53+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/prime_number?rev=1464110933&do=diff Prime number Set definiendum $ p\in\mathrm{it} $ inclusion $p\in\mathbb{N}\setminus\{0,1\}$ range $1<n<p$ postulate $\nexists n.\, n\mid p$ ---------- ---------- Subset of Natural numbers Related Prime enumeration text/html 2016-04-05T15:05:46+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/primitive_notions_._wat?rev=1459861546&do=diff Primitive notions . WAT WAT Thoughts Some necessary ingrediences in the end of the HoTT book. (In an old version, it was around p.44 and at the very end.) A year ago I asked a relevant related question here <http://cstheory.stackexchange.com/questions/27400/minimal-specification-of-martin-l%C3%B6f-type-theory> Apple pie Think about what would be a good order of introduction (using WAT) (see $\to, \times$$\to$$\lambda.$${\mathbb N}$$0, 1, 2, 3, 4, 5, 6, 7, 8, 9$$+, *$${\mathbb N}\to{\math… text/html 2016-10-31T20:03:40+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/probabilistic_robotics_._book?rev=1477940620&do=diff Probabilistic Robotics . Book Book The 3. Edition * 647 pages total * has exercises * Had 4 parts with 17 chapters total: * 2-4 Basics (Math Foundations) * 5-6 Localization (robot Motion?) * 9-13 Mapping (world representation?) * 14-17 Planing and Control (future actions?)$t=0, t=1, t=2, \dots$$\Delta t$$1$$t_{a:b} = \cup_{i=a}^b \{t_i\} =\{t_{a}, t_{a+1}, t_{a+2}, \dots t_{b-1}, t_{b-1}\}$$x_{t}$$z_{t}$$u_{t}$$(t-1,t]$$x_{t}$$p(x_{t} \,|\,x_{t-1}, u_t)$$p(x_{t} \,|\,z_{1:t-1… text/html 2015-03-31T20:08:17+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/probability_space?rev=1427825297&do=diff Probability space Set context $X $ ... set definiendum $ \langle X,\Sigma,P\rangle\in \mathrm{ProbabilitySpace}(X) $ inclusion $ \langle X,\Sigma,P\rangle $ ... measure space postulate $ P(X)=1 $ ---------- Reference Wikipedia: Probability space ---------- Subset of Measure space text/html 2016-11-11T23:03:33+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/probability_theory_._note?rev=1478901813&do=diff Probability theory . note Note ---------- Relations between distributions p(x) Univariate Distribution Relations - Lawrence M. Leemis, Jaquelyn T. McQueston Old notes: it is roghly... * Binomial (discrete grid, no interaction, values between 0 and inf) (parameter N and p is total tries/contribution and per try chance for one contribution) text/html 2014-09-28T23:46:57+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/product_._category_theory?rev=1411940817&do=diff Product . category theory Collection context $F:\mathrm{Ob}_{{\bf C}^{\bf 2}}$ range $a,b:\mathrm{Ob}_{\bf 2},\ a\neq b$ definition $\langle Fa\times Fb, \pi\rangle := \mathrm{lim}\,F$ Discussion Elaboration ${\bf C}^{\bf 2}$ is the functor category with objects being functors in ${\bf 2}\longrightarrow{\bf C}$ and the morphisms are natural transformations, i.e. families of ${\bf 2}$-indexed arrows in ${\bf C}$. So $\pi:\prod_{x:\mathrm{Ob}_{\bf 2}}\left(Fa\times Fb\right)\… text/html 2014-04-03T16:23:48+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/product_type?rev=1396535028&do=diff Product type Type context $X, Y$ ... type rule ${\large\frac{a\ :\ X\hspace{1cm}b\ :\ Y}{\langle a,b\rangle\ :\ X\times Y}}(\times\mathcal I)$ rule ${\large\frac{p :\ X\times Y}{\pi_{\mathcal l}(p)\ :\ X\hspace{1cm}\pi_{\mathcal r}(p)\ :\ Y}}(\times\mathcal E)$ rule ${\large\frac{a\ :\ X\hspace{1cm}b\ :\ Y}{\pi_{\mathcal l}(\langle a,b\rangle)\ \leftrightsquigarrow\ a \hspace{1cm}\pi_{\mathcal r}(\langle a,b\rangle)\ \leftrightsquigarrow\ b}}(\times\beta)$ rule ${\… text/html 2014-11-11T11:13:55+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/proof_theory?rev=1415700835&do=diff Proof theory Framework We can use logic to reason about logical derivations. The object language contains formulae $foo$, $bar$, etc. and we use $foo\vdash bar$, which one might reads as “if $foo$ is provable, then $bar$ also provable.” In the proof theoretic logic, we use a variable which represents a collection of object language formulae, called $\Gamma$$\land$${\large\frac{}{\phi\vdash\phi}}(identity)$${\large\frac{\Gamma,\Psi\vdash\vartheta}{\Gamma,\alpha,\Psi\vdash\vartheta}}(weake… text/html 2015-10-01T19:13:11+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/propositions?rev=1443719591&do=diff Propositions Meta ${\mathfrak D}_\mathrm{Propositions}$ ---------- "todo: make some entries for the set theory axioms." ----- Related Logic, Domain of discourse text/html 2015-03-16T21:06:26+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/pullback_._category_theory?rev=1426536386&do=diff Pullback . category theory Collection context $F:({a\rightarrow z\leftarrow b})\longrightarrow{\bf C}$ definition $\langle Fa\times_{Fz} Fb, \pi\rangle := \mathrm{lim}\,F$ Here we consider a functor $F$ from the category ${a\rightarrow z\leftarrow b}$, consisting of three object and two non-identity arrows $f_a$ and $f_b$, to a category ${\bf C}$. ---------- Universal property For readability, let's write $A\equiv{Fa}, B\equiv{Fb}, Z\equiv{Fz}, \alpha\equiv{f_a}$$\beta\equiv{f_… text/html 2015-03-18T19:59:20+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/pullback_functor?rev=1426705160&do=diff Pullback functor Functor context $A:\mathrm{Ob}_{\bf C}$ context $f...$ definiendum $f^*$ definition $...$ todo: <https://proofwiki.org/wiki/Definition:Pullback_Functor> ---------- Also called base change functor (particularly in geometry) or inverse image functor (particularly in ${\bf{Set}}$). Reference Wikipedia: Pullback functor ---------- text/html 2017-06-12T17:04:13+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/python?rev=1497279853&do=diff Python Notes on programming languages $\blacktriangleright$ Python $\blacktriangleright$ ... Note Language tmp (build-in functions) "not any all" "reversed(seq)" "mydict = dict(mykey1 = 3 ,mykey2 = 'hi' )" "isinstance(3, int) isinstance('3', int) callableI(f) iter(o)" "input()" Terminal args text/html 2016-03-03T13:33:14+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/quantum_canonical_partition_function?rev=1457008394&do=diff Quantum canonical partition function Set context $ H $ ... Hamiltonian "todo: Hamiltonian" definiendum $Z(\beta):=\mathrm{tr}(\mathrm e^{-\beta H}) $ "todo: trace" ---------- Discussion Generally, material physics of finite (i.e. non-zero) temperature derives its macroscopic relations from small scale considerations. All observables are essentially determined by the relation between the possible microscopic states and their energy, which makes evaluation of the partition func… text/html 2016-07-22T18:36:03+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/quantum_integer?rev=1469205363&do=diff Quantum integer Set context $f:{\mathbb N}\to{\mathbb R}$ definiendum $[n]_q \in \mathrm{it}$ inclusion $[n]_q:{\mathbb N}\to{\mathbb C}^*\to{\mathbb R}$ definition $[n]_q:=q^{-f(n)/2}\frac{1-q^n}{1-q}$ Discussion These are $q$-deformations of integers, so that arithmetic coincides at $q=1$. $[n]_{q} = q^{-f(n)/2}q^{-1}\sum_{k=1}^n q^k = n+\tfrac{n}{2}(n-1-f(n))\cdot(q-1)+\mathcal{O}\left((q-1)^2\right)$ In fact this doesn't require $n$ to be an integer. The case $f… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/quasigroup?rev=1395396676&do=diff Quasigroup Set context $X$ postulate $ \langle X,* \rangle \in \text{Quasigroup}(X)$ context $\langle X,* \rangle \in \mathrm{Magma}(X)$ range $a,b,x,y\in X$ postulate $ \forall a.\ \forall b.\ \exists x.\ a*x=b $ postulate $ \forall a.\ \forall b.\ \exists y.\ y*a=b $ Here we used infix notation for “$*$”. Ramifications Discussion The binary operation is often called multiplication. The axioms $*\in \mathrm{binaryOp}(X)$$X$$*$ text/html 2015-01-31T13:45:27+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/questions?rev=1422708327&do=diff Questions Meta Theta and partition function ---------- Related Todo text/html 2014-12-27T00:17:55+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/quotient_set?rev=1419635875&do=diff Quotient set Set context $X$ ... set context $ \sim\in\text{Equiv}(X) $ definiendum $ Y\in X/\sim $ postulate $ \exists x.\ Y=[x]_\sim $ ---------- Reference Wikipedia: Equivalence class ---------- Context Equivalence class text/html 2014-12-26T16:58:34+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/rational_number?rev=1419609514&do=diff Rational number Set definiendum $ \mathbb Q \equiv \mathbb Z\times (\mathbb Z\setminus \{0\})\ /\ \{\langle \langle a,b\rangle,\langle m,n\rangle\rangle\ |\ a\ n - b\ m = 0 )\} $ with $a,b,n,m\in \mathbb Z$ Discussion Reference Wikipedia: Rational number Parents Subset of Real number Context Arithmetic structure of integers, Quotient set Related Rational numbers text/html 2016-04-23T13:54:29+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/rational_numbers?rev=1461412469&do=diff Rational numbers Framework $\dots,\,-\frac{4}{3},\,0,\,\frac{1}{17},\,1,\,7.528,\,9001,\dots$ The rational numbers ${\mathbb{Q}}$ can be defined as the field of characteristic 0 which has no proper sub-field. In less primitive notions, it's the field of fractions for the integral domain of natural numbers. The second order theory of rationals (see the note below) describes a countable collection.$m$$\dfrac{1}{1-x}=\dfrac{1}{1-x\cdot x^{m}}\sum_{k=0}^m x^k$$\dfrac{1}{y}=\dfrac{1}{1-(1-y)\cdot(… text/html 2015-08-15T20:34:35+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/reaction_rate_equation?rev=1439663675&do=diff Reaction rate equation Set context $R,J\in \mathbb N$ context $ \nu^-,\nu^+\in\mathrm{Matrix}(R,J,\mathbb Q) $ context $ k\in \mathbb R^R $ definiendum $ [A] \in \mathrm{it} $ $j\in \text{range}(J)$ postulate $ [A]:C(\mathbb R,\mathbb R^J) $ range $ ::[A](t) $ postulate $ \frac{\partial}{\partial t}[A]_j=\sum_{r=1}^R k_r\cdot(\nu_{rj}^+-\nu_{rj}^-)\cdot\prod_{i=1}^J [A]_i^{\nu_{ri}^-} $ ---------- The quantities $R$ and $J$ denote the numbe… text/html 2014-04-04T16:01:15+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/real_coordinate_space?rev=1396620075&do=diff Real coordinate space Set context $ n\in \mathbb N $ definiendum $\mathbb R^n$ Discussion Explicitly, $ \mathbb R^1=\mathbb R $ $ \mathbb R^p=\mathbb R^{p-1}\times \mathbb R, \hspace{1cm}$ range $1\le i\le p$ predicate $ a< b \equiv \forall i.\ a_i< b_i$ predicate $ a> b \equiv \forall i.\ a_i> b_i$ predicate $ a\le b \equiv \forall i.\ a_i\le b_i$ predicate $ a\ge b \equiv \forall i.\ a_i\ge b_i$ "todo: define the following in a seperate entry… text/html 2016-05-31T20:42:47+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/real_function_derivative?rev=1464720167&do=diff Real function derivative Definition context $X,Y\in \mathfrak J_o^p$ context $f:X\to Y$ ... continuous let $X_\exists\equiv\{x\in X\ |\ \mathrm{lim}_{h\to 0}\frac{f(x+h)-f(x)}{h}\in Y\}$ definiendum $f':X_\mathrm{\exists}\to Y$ definiendum $f(x) := \mathrm{lim}_{h\to 0}\dfrac{f(x+h)-f(x)}{h} $ ---------- Discussion To constructing $f'$ from $f$, we must restrict the domain $X$ to $X_\exists$ where, per definition, the required limit exists. We could alternat… text/html 2016-02-10T14:59:18+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/real_logarithm?rev=1455112758&do=diff Real logarithm Set context $ x,b\in\mathbb R_+^* $ context $ b\neq 1 $ definiendum $\log_b(x):\mathbb R_+^*\to \mathbb R_+^*$ definiendum $\log_b(x):= y$ postulate $b^y=x$ ---------- The logarithm function is that of the Dimension Consider $\log_r(r^n/r^1) = \log_r(r^{n-1}) = n - 1 = \log_r(r^n) - \log_r(r^1)$ vs. ${\mathrm {dim}}({\mathbb R}^n/{\mathbb R}^1) = {\mathrm {dim}}({\mathbb R}^{n-1}) = n - 1 = {\mathrm {dim}}({\mathbb R}^n)-{\mathrm {dim}}({\math… text/html 2015-09-30T22:45:43+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/real_number?rev=1443645943&do=diff Real number Set definiendum $ r \in \mathbb R $ for all $x,y\in \mathbb Q$ for all $ r\subset \mathbb Q, r\neq \emptyset $ postulate $ y\in r\implies x\in r $ postulate $ \neg\ \exists (b\in r).\ \forall (a\in r).\ a<_{\mathbb Q}b $ ---------- Discussion Remark: We distinguish between “$\subset$” and “$\subseteq$”, i.e. the above definition implies $ r\neq \mathbb Q $. Each real number is modeled as a Dedekind cut of $ \mathbb Q $$r$$<_{\mathbb Q}$$s<r\equiv… text/html 2016-01-19T22:18:29+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/real_numbers?rev=1453238309&do=diff Real numbers Framework $\dots,\,-\frac{\pi}{2},\,0,\,1,\dots$ ${\mathbb R}$ ---------- Discussion "axiomatics and cardinality" Specker sequence Choose a programming language and index all Turing machines (or executable program) by $i$. Let $h(i,s)$ be $0$ or $1$, depending on whether the machine with index $i$$s$$h(i,s)$$s$$n$$ s=n-i $$a_n = \sum_{i+s=n} \dfrac {h(i,s)} {2^i} $ text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/real_part_of_a_complex_number?rev=1395396676&do=diff Real part of a complex number Set context $z\in \mathbb C$ postulate $\mathrm{Re}(z)\equiv \frac{z+\bar z}{2}$ Ramifications Parents Related Complex conjugate of a complex number text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/real_part_of_a_function?rev=1395396676&do=diff Real part of a function Set context $f:X\to \mathbb C$ definiendum $ \mathrm{Re}f:X\to\mathbb C $ definiendum $ (\mathrm{Re}f)(x):=\mathrm{Re}(f(x)) $ Discussion Parents Context Real part of a complex number text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/real_step_function?rev=1395396676&do=diff Real step function Set context $\langle X,\Sigma\rangle\in\mathrm{MeasurableSpace}(X)$ postulate $f\in \mathcal T$ context $f\in\mathrm{Measurable}(X,\mathbb R)$ Where we consider the Borel algebra over $\mathbb R$ postulate $\mathrm{im}(f)$ ... finite Discussion These functions can be written as $f=\sum_{j=1}^n\alpha_j\cdot\chi_{E_n}$ with $\alpha_j$'s real numbers and $E_n$'s in the measurable algebra. text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/reduced_distribution_function?rev=1395396676&do=diff Reduced distribution function Set context $ \rho $ ... Classical probability density function range $ N \equiv \mathrm{dim}(\mathcal M) $ $s < N$ definiendum $\bar f_s(q^1,p_1,q^2,p_2,\dots,q^s,p_s):=\int\ \rho\ \ \mathrm d q^{s+1} \mathrm d p_{s+1}\cdots \mathrm d q^N \mathrm d p_N$ Discussion We also set $f_N:=\rho$, but that's just introduction of different notation. Parents Context Classical probability density function text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/reflexive_relation?rev=1395396676&do=diff Reflexive relation Set context $ X $ definiendum $ R\in\mathrm{ReflexiveRel}(X) $ context $ R \in \mathrm{Rel}(X) $ $ x\inX $ postulate $ xRx $ Discussion Parents Subset of Total relation text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/regular_graph?rev=1395396676&do=diff Regular graph Set context $V,E$ ... set definiendum $ \langle V,E,\psi\rangle \in \mathrm{it}(E,V) $ inclusion $ \langle V,E,\psi\rangle $ ... undirected graph range $ k\in \mathbb N $ range $ v\in V $ postulate $ \exists k.\ \forall v.\ d(v)=k $ Discussion Parents Subset of Undirected graph Context Vertex degree text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/relation_concatenation?rev=1395396676&do=diff Relation concatenation Set context $ R \in \text{Rel}(X,U) $ context $ S \in \text{Rel}(V,Y) $ definiendum $ \langle x,y \rangle \in S\circ R $ postulate $ \exists m.\ \langle x,m \rangle \in R \land \langle m,y \rangle \in S $ Discussion Concatenations/compositions are associative. A mayority of uses of the relation concatenation is when the relation is functional, i.e. one composes functions alla $g:X\to Y,\ \ f:Y\to Z$$f\circ g:X\to Z$$(f\circ g)(x):=f(g(x))$$f:… text/html 2014-07-11T20:38:31+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/relation_domain?rev=1405103911&do=diff Relation domain Set context $ R $ $ c\in R $ postulate $ \forall c.\ \exists a,b.\ c=\langle a,b\rangle $ definiendum $ x\in \mathrm{dom}(R) $ postulate $ \exists y.\ \langle x,y \rangle\in R $ Discussion Reference Wikipedia: Domain of a function Parents Context Ordered pair text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/restricted_image?rev=1395396676&do=diff Restricted image Set context $ f:X\to Y $ context $ S\subseteq X $ postulate $ f(S)\equiv \mathrm{im}(f|_S) $ Discussion The notation $f(S)$ is overloading $f$, as $S$ is not actually in the domain $X$ of $f$. Reference Wikipedia: Image Parents Subset of Image Context Restricted relation text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/restricted_relation?rev=1395396676&do=diff Restricted relation Set context $X,Y,Z$ context $R\in \text{Rel}(X,Y)$ postulate $ \langle x,y\rangle \in R_{|Z}$ postulate $ \langle x,y\rangle \in R $ postulate $ x \in Z $ Ramifications Reference Mizar files: RELAT_1 Wikipedia: Wikipedia: Binary relation Parents Subset of Binary relation text/html 2016-08-30T20:38:12+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/retarded_propagator_._time-independent_hamiltonian?rev=1472582292&do=diff Retarded propagator . time-independent Hamiltonian Set context $\mathcal H$ ... Hilbert space context $H\in\mathrm{Observable}(\mathcal H)$ definiendum $P:\mathbb{R}\times\mathbb{R}\to L(\mathcal H,\mathcal H)$ definiendum $P(t,s):=\exp\left(i\,(t-s)\frac{1}{\hbar}H\right)$ ---------- Discussion Let $N\equiv \mathrm{dim}(V)$ and let $i,j,k,n$ range from $1$ to $N$. Let $\{|E_n\rangle\}$ as well as $\{|Q_n\rangle\}$ be two orthonormal sets of basis vectors where in p… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/retraction?rev=1395396676&do=diff Retraction Set $X,S$ $S\subseteq X$ $ \{\rho\} $ $ \rho:X\to S $ $ \rho_{|S}=\text{id}_S $ Ramifications Reference Wikipedia: Deformation retract Parents Related Identity relation text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/reversed_relation?rev=1395396676&do=diff Reversed relation Set context $ R\in\text{Rel}(X,Y) $ definiendum $ \langle x,y \rangle \in R^\smile $ postulate $ \langle y,x \rangle \in R $ Ramifications Reference Mizar files: RELAT_1 Parents Context Binary relation text/html 2016-06-03T22:54:23+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/riemann_zeta_function?rev=1464987263&do=diff Riemann zeta function Function definition $ \zeta: \mathbb{C}\setminus\{1\} \to \mathbb C$ definition $ \zeta(s) := \begin{cases} \sum_{n=1}^\infty\, n^{-s} &\hspace{.5cm} \mathrm{if}\hspace{.5cm} \mathfrak{R}(s)>1 \\\\ \text{analytic continuation}\hspace{.5cm} &\hspace{.5cm} \mathrm{else} \end{cases}$ “$\text{analytic continuation}$” ---------- Theorems Euler product: $\zeta(s)=\prod_{p\in\text{primes}}\frac{1}{1-p^{-s}}$ Representations $\zeta(s) = \dfrac{\int_0^\infty\frac{x… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/right-continuous_function?rev=1395396676&do=diff Right-continuous function Set context $p\in\mathbb N$ definiendum $f\in\mathrm{RightContinuous}(\mathbb R^p,\mathbb R) $ postulate $f:\mathbb R^p\to\mathbb R$ range $\varepsilon,\delta\in \mathbb R_+^*$ $y\in\mathbb R^p$ postulate $\forall y.\ \forall\varepsilon.\ \exists \delta.\ \forall x.\ (x\ge y\ \land\ \Vert x-y \Vert < \delta) \implies |f(x)-f(y)|<\varepsilon$ Discussion Reference Wikipedia: Continuous function Parents Subset of Continuous functio… text/html 2017-05-04T10:38:13+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/ring?rev=1493887093&do=diff Ring Set context $\langle X,+ \rangle \in \mathrm{AbelianGroup}(X)$ definiendum $\langle X,+,* \rangle\in\mathrm{it}$ for all $a,b,c\in X$ postulate $(a*b)*c=a*(b*c)$ postulate $a*(b+c)=(a*b)+(a*c)$ postulate $(b+c)*a=(b*a)+(c*a)$ Discussion One might call the commutative group operation “$+$” the addition and the other one “$*$” the multiplication. In a unital ring, the latter has an identity too.$X$$+$$*$ text/html 2016-10-11T12:53:44+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/sagittarius?rev=1476183224&do=diff Sagittarius Note ---------- brainstorming Orbital mechanics ??pipe terethi ---------- Related Mechanics, Euler-Lagrange equations text/html 2014-10-29T21:06:52+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/second-countable_hausdorff_space?rev=1414613212&do=diff Second-countable Hausdorff space Set context $X$ definiendum $\langle X,T\rangle\in$ it inclusion $\langle X,T\rangle$ ... Second-countable space inclusion $\langle X,T\rangle$ ... Hausdorff space Discussion Parents Subset of Hausdorff space, Second-countable space text/html 2014-12-04T14:24:06+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/second-countable_space?rev=1417699446&do=diff Second-countable space Set context $X$ ... set definiendum $\langle X,T\rangle\in$ it inclusion $\langle X,T\rangle$ ... topological space exists $B\subseteq T$ postulate $B$ ... countable base Discussion Parents Subset of Topological space Requirements Countable base for a topology text/html 2014-03-21T11:20:05+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/self-adjoint_operator?rev=1395397205&do=diff Self-adjoint operator Set context $V$...Hilbert space definiendum $A\in \mathrm{it}$ inclusion $A:V\to V$ for all $v,w\in V$ postulate $\langle v\,\left|\right.\,A\,w\rangle = \langle A\,v\,\left|\right.\,w\rangle$ Discussion Reference Wikipedia: Self-adjoint operator Parents Context Hilbert space text/html 2016-01-20T12:52:45+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/semigroup?rev=1453290765&do=diff Semigroup Set context $S$ ... set definiendum $\langle\!\langle S,* \rangle\!\rangle \in $ semigroup(S) inclusion $\langle\!\langle S,* \rangle\!\rangle\in $ magma(S) postulate $(a*b)*c=a*(b*c)$ ---------- Discussion The binary operation is often called multiplication. The axioms $*\in \mathrm{binaryOp}(S)$ above means that a magma is closed with respect to the multiplication. $S$$*$ text/html 2016-05-01T16:01:41+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/seminorm?rev=1462111301&do=diff Seminorm Set context $F$ ... subfield of $\mathbb{C}$ context $V$ ... $F$-vector space definiendum $p\in \mathrm{SemiNorm}(V)$ postulate $p:V\to \mathbb R $ $v,w\in V$ postulate $p(v+w) \le p(v)+p(w)$ $\lambda\in F$ postulate $p(\lambda\cdot v) = |\lambda|\cdot p(v)$ ---------- Discussion A Norm is a seminorm with the adition axiom $p(v)=0 \implies v=0$ (which I also write as $p(!0)=0$.) text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/semiring?rev=1395396676&do=diff Semiring Set context $X$ definiendum $\langle X,+,* \rangle \in \mathrm{Semiring}(X)$ context $\langle X,+ \rangle \in \mathrm{AbelianMonoid}(X)$ $a,b,c\in X$ postulate $(a*b)*c=a*(b*c)$ postulate $a*(b+c)=(a*b)+(a*c)$ postulate $(b+c)*a=(b*a)+(c*a)$ Discussion Reference Wikipedia: Semiring Parents Context Abelian monoid text/html 2014-12-11T17:27:29+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/seperated_presheaf?rev=1418315249&do=diff Seperated presheaf Collection context $\langle X,\mathcal T_X\rangle$ ... topological space definiendum $F$ in it inclusion $F$ in ${\bf Set}^{\mathrm{Op}(X)^{\mathrm{op}}}$ for all $U\in \mathcal T_X$ for all $s,t\in FU$ for all $C_U$ ... open cover$(U)$ postulate $\left(\forall (V\in C_U).\ s|_V=t|_V\right) \implies s=t$ Discussion Here we use the notation discussed in presheaf. I.e. in the last line, $s|_V=t|_V$$F(i)(s)=F(i)(t)$$i:V\to U$$FU$… text/html 2016-07-03T19:10:03+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/sequence?rev=1467565803&do=diff Sequence Set context $X$ definiendum $ \mathrm{Sequence}(X)\equiv\mathrm{FinSequence}(X)\cup\mathrm{InfSequence}(X) $ ---------- ---------- Context Finite sequence, Infinite sequence text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/sequence_begin?rev=1395396676&do=diff Sequence begin Function context $ X $ ... set definiendum $ \mathrm{first}: \mathrm{Sequence}(X)\to X $ definiendum $ \mathrm{first}(S):= \pi_1(S) $ Discussion Parents Subset of Surjective function Context Sequence length text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/sequence_end?rev=1395396676&do=diff Sequence end Function context $ X $ ... set definiendum $ \mathrm{last}: \mathrm{FiniteSequence}(X)\to X $ definiendum $ \mathrm{last}(S):= \pi_{\mathrm{length}(S)}(S) $ Discussion Parents Subset of Surjective function Context Sequence length text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/sequence_intersection?rev=1395396676&do=diff Sequence intersection Set context $X$ context $A$ ... sequence over $X$ context $n\in \mathbb N$ definiendum ${\bigcap_{k=n+1}^\infty}A_k\equiv\bigcap A(\mathbb N\smallsetminus\mathrm{range}(n))$ Discussion Parents Context Infinite sequence, Restricted image, Arbitrary intersection text/html 2014-12-02T16:37:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/sequence_length?rev=1417534636&do=diff Sequence length Set context $ X $ definiendum $ \mathrm{length}: \mathrm{Sequence}(X)\to\mathbb N\cup\{\infty\} $ definiendum $ \mathrm{length}(S):= \left|\mathrm{dom}(S)\right|$ Discussion Parents Subset of Injective function Context Sequence Refinement of Set cardinality text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/sequence_reversion?rev=1395396676&do=diff Sequence reversion Function context $ X $ ... set definiendum $ \mathrm{rev}: \mathrm{FiniteSequence}(X)\to\mathrm{FiniteSequence}(X) $ postulate $ \pi_n(\mathrm{length}(S))=\pi_{\mathrm{length}(S)+1-n}(S)$ Discussion Parents Subset of Idempotent function Context Sequence length text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/sequence_union?rev=1395396676&do=diff Sequence union Set context $X$ context $A\in \mathrm{InfSequence}(X)$ context $n\in \mathbb N$ definiendum ${\bigcup_{k=n+1}^\infty}A_k\equiv\bigcup A(\mathbb N\smallsetminus\mathrm{range}(n))$ Discussion Parents Context Infinite sequence, Restricted image, Arbitrary union text/html 2014-12-05T15:22:15+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/set?rev=1417789335&do=diff Set Category context ${\mathfrak U}_\mathrm{Sets}$ ... set universe definiendum ${\bf Set}$ postulate $\mathrm{Ob}_{\bf Set}:={\mathfrak U}_\mathrm{Sets}$ inclusion ${\bf Set}$ ... locally ${\mathfrak U}_\mathrm{Sets}$-small category postulate $f\in{\bf Set}[X,Y]\ \ \Leftrightarrow\ \ f:X\to Y$ Discussion Here '$f:X\to Y$' on the right denoty a function in the type theory sense, where domain and codomain are sets. Note that ${\bf Set}$$\mathrm{Ob}_{\bf{Set}}$$f:… text/html 2016-01-16T18:27:56+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/set_._hott?rev=1452965276&do=diff Set . HoTT Type ${\mathrm{isSet}}(A):={\large\Pi}_{x,y:A}\,isProp(Id_A(x,y))$ Discussion Elaboration See the last lines of univalence axiom. Alternative definitions ${\mathrm{isSet}}(A):={\large\Pi}_{x,y:A}\,{\large\Pi}_{p,q: {\mathrm {Id}}_A(x,y)}\,{\mathrm {Id}}_{{\mathrm {Id}}_A(x,y)}(p,q)$ ${\mathrm{isSet}}(A):={\large\Pi}(x,y:A)\,{\large\Pi}(p,q: {\mathrm {Id}}_A(x,y))\,{\mathrm {Id}}_{{\mathrm {Id}}_A(x,y)}(p,q)$ Reference Parents Requirements Mere proposition text/html 2014-12-05T11:02:40+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/set_cardinality?rev=1417773760&do=diff Set cardinality Set context $X$ ... set definiendum $ \left|X\right|\equiv\mathrm{inf}\{\alpha\ |\ \alpha\in\mathrm{Ord}\ \land\ \alpha\approx X\} $ Discussion Reference Wikipedia: Cardinal numbers, Cardinal assignment, Von Neumann cardinal assignment Parents Context Non-strict partial order, Bijective function Element of Ordinal number text/html 2014-10-25T19:13:57+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/set_limes_inferior?rev=1414257237&do=diff Set limes inferior Set context $A\in \text{Seq}(X)$ definition $\underset{n\to\infty}{\liminf}A_n\equiv{\bigcap_{n=1}^\infty}\left({\bigcup_{k=n}^\infty}A_n\right)$ Ramifications We have that $\underset{n\to\infty}{\limsup}A_n\subseteq \underset{n\to\infty}{\liminf}A_n,$ see set limes superior. If moreover $\underset{n\to\infty}{\limsup}A_n=\underset{n\to\infty}{\liminf}A_n,$ then we call it $\underset{n\to\infty}{\lim}A_n$ and say $A$ is convergent. Reference Wikipedia:… text/html 2014-10-25T19:13:17+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/set_limes_superior?rev=1414257197&do=diff Set limes superior Set context $A\in \text{Seq}(X)$ definition $\underset{n\to\infty}{\limsup}A_n\equiv{\bigcup_{n=1}^\infty}\left({\bigcap_{k=n}^\infty}A_k\right)$ Ramifications We have that $\underset{n\to\infty}{\limsup}A_n\subseteq \underset{n\to\infty}{\liminf}A_n,$ see Set limes inferior. If moreover $\underset{n\to\infty}{\limsup}A_n=\underset{n\to\infty}{\liminf}A_n,$ then we call it $\underset{n\to\infty}{\lim}A_n$ and say $A$ is convergent. Reference Wikipedia:… text/html 2016-05-24T19:19:21+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/set_of_divisors_function?rev=1464110361&do=diff Set of divisors function Function definiendum $ \mathrm{divisors}:\mathbb N^+\to\mathcal{P}(\mathbb N) $ definiendum $ \mathrm{divisors}(n):=\{a\ |\ \exists (b\in\mathbb N).\ a\cdot b=n\} $ "make this into a set" ---------- Code A related Boolean function is divides :: Integral a => a -> a -> Bool divides d n = rem n d == 0 ---------- Subset of Function Requirements text/html 2015-10-08T13:53:44+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/set_theory?rev=1444305224&do=diff Set theory Framework This wiki is a place where I can look up definitions of mathematical objects and also see their generalizations and special cases. Set theory is the most commonly chosen way to set up mathematical foundations, and accordingly most of the entries in the wiki specify mathematical sets.$x,y,z,u,X,Y,Z,s,p,U,\dots$$z_1,z_2,z_3,\dots$$\phi,\psi,\dots$$\in$$ x \notin X \equiv \neg (x \in X) $$X$$\forall (x\in X).\ \phi(x) \equiv \forall x.\ (x\in X \Rightarrow \phi(x)) $$ \exists… text/html 2015-08-25T22:49:58+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/set_universe?rev=1440535798&do=diff Set universe Collection definiendum ${\mathfrak U}_\mathrm{Sets}$ in it postulate ${\mathfrak U}_\mathrm{Sets}$ ... Grothendieck universe postulate $\omega_{\mathcal N}\subseteq {\mathfrak U}_\mathrm{Sets}$ ---------- Discussion A set universe ${\mathfrak U}_\mathrm{Sets}$ is a Grothendieck universe containing all sets generated by the first infinite von Neumann ordinal $\omega_{\mathcal N}$. It contains a model for the natural numbers, their powerset, the powersets of those… text/html 2015-10-01T19:12:24+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/sets?rev=1443719544&do=diff Sets Meta ${\mathfrak D}_\mathrm{Sets}$ ---------- ---------- Related Set theory, Domain of discourse text/html 2014-10-29T14:32:31+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/sheaf?rev=1414589551&do=diff Sheaf Collection context $\langle X,\mathcal T\rangle$ ... topological space definiendum $F$ in it inclusion $F$ ... seperated presheaf for all $U\in \mathcal T$ for all $C_U$ ... open cover$(U)$ for all $S:\prod_{V\in C} FV$ postulate $\left(\forall(V,W\in C_U).\ S(V)|_{V\cap W}=S(W)|_{V\cap W}\right) \implies \exists (s\in FU).\ \forall V.\ S(V)=s|_V$ Discussion As in seperated presheaf, $t|_V$ denotes the image of $t$ under $F(i)$$i:V\to W$$V… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/simple_graph?rev=1395396676&do=diff Simple graph Set context $V,E$ ... set definiendum $\langle V,E,\psi\rangle \in \mathrm{it}(E,V) $ postulate $\langle V,E,\psi\rangle $ ... loopless graph postulate $ \psi $ ... injective Discussion Two vertices $\{u,v\}$ of a a simple graph are connected by at most one edge. Parents Subset of Loopless graph text/html 2015-11-04T16:26:45+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/sine_function?rev=1446650805&do=diff Sine function Function definiendum $\mathrm{\sin}: \mathbb C\to\mathbb C$ definiendum $\sin(z) := \sum_{k=0}^\infty \frac{(-1)^{k}}{(2k+1)!}z^{2n+1} $ ---------- $\theta\in\mathbb R$ $\sin(\theta) = \frac{1}{2i}(\mathrm e^{i\theta}-\mathrm e^{-i\theta}) $ i.e. if $\zeta:=\mathrm e^{i\theta}$, then $\zeta_\theta-\overline{\zeta_\theta}=2i\sin(\theta)$. Theorem * From $\sum_{n=a}^{b}{\mathrm e}^{2kn}=\sum_{n=a}^{b}\left({\mathrm e}^{2k}\right)^n=\dots$ we get $\sum_{n=a}^{b}\s… text/html 2014-12-05T01:03:38+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/singleton?rev=1417737818&do=diff Singleton Set context $ X $ ... set definition $ \{X\} \equiv \{X,X\} $ Discussion Reference Wikipedia: Singleton Parents Context* Sets Requirements Unordered pair text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/singular_values_of_a_matrix?rev=1395396676&do=diff Singular values of a matrix Set context $ A \in \mathrm{SquareMatrix}(n,\mathbb C) $ definiendum $ \sqrt{\lambda} \in \mathrm{SingularVal}(A) $ postulate $ \lambda \in \mathrm{EigenVal}(A^*A) $ Discussion Note that $A^*A$ is always Hermitian positive semi-definite matrix. Singular values make sense for more general operators too. Reference Wikipedia: Singular value, Matrix norm Parents text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/skew-hermitian_matrix?rev=1395396676&do=diff Skew-Hermitian matrix Set context $n\in\mathbb N$ definiendum $ A \in \mathrm{Skew-HermitianMatrix}(n) $ postulate $ A \in \mathrm{SquareMatrix}(n,\mathbb C) $ postulate $ A^*=-A $ Discussion Reference Wikipedia: Skew-Hermitian matrix Parents Subset of Square matrix text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/skew-symmetric_matrix?rev=1395396676&do=diff Skew-symmetric matrix Set context $R$ ... ring context $n\in\mathbb N$ definiendum $ A \in \mathrm{SymmetricMatrix}(n,R) $ postulate $ A^\mathrm{T}=-A $ Discussion Reference Wikipedia: Skew-symmetric matrix Parents Subset of Matrix Context Matrix transpose, Matrix ring text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/smallest_generated_%CF%83-algebra?rev=1395396676&do=diff Smallest generated σ-algebra Set context $X$ postulate $\sigma(A):\equiv \bigcap \{C\ |\ A\subseteq C\land C\subseteq \mathrm{SigmaAlgebra}(X)\} $ Discussion The smallest σ-algebra over $X$ generated by, and hence containing, $A$. Reference Wikipedia: σ-Operator (german) Parents Element of σ-algebra text/html 2014-12-18T18:44:09+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/smooth_atlas?rev=1418924649&do=diff Smooth atlas Set context $\langle M,T\rangle$ ... second-countable Hausdorff space context $n\in \mathbb N$ definiendum $A\in$ it inclusion $A\subseteq$ atlas ($\langle M,T\rangle,n$) forall $\langle V,\phi\rangle,\langle W,\psi\rangle\in A$ postulate $\phi\circ\psi^{-1}$ ... smooth A priori “$\phi\circ\psi^{-1}$” in the postulate doesn't make sense as their domains/codomains will not much. Here, really, one must choose functions with appropriately rest… text/html 2014-12-04T16:39:49+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/smooth_function?rev=1417707589&do=diff Smooth function Set context $X,Y$ ... Banach spaces with topology definiendum $f\in C^\infty(X,Y)$ for all $k\in \mathbb N$ postulate $f\in C^k(X,Y)$ postulate $\lim_{m\to\infty}D^mf$ ... continuous Discussion Formalities Here $D^mf$ denotes the $m$th component of the sequence $Df, D^2f, D^3f,\dots$. Reference Wikipedia: text/html 2015-03-01T15:55:42+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/smooth_function_of_a_linear_operator?rev=1425221742&do=diff Smooth function of a linear operator Set ---------- Expansion Let $A(z),B(z)$ be function with expansion around $a,b$, respectively. $A(D)\,B(X)=\sum_{k=0}^\infty A^{(k)}(a) \dfrac{1}{k!} (D-a)^k \cdot \sum_{j=0}^\infty B^{(j)}(b) \dfrac{1}{j!} (X-b)^j $ Truncation at $k=j$ $D$$D^{n+1}X^{n}=0$$\mathbb C$$\mathbb C$$a$$a$$a=0$$D=Y\frac{\partial}{\partial X}$${j \choose k}=\frac{1}{k!}\frac{j!}{(j-k)!}$$A=\exp$$A^{(k)}(0)=1$$(u+v)^j=\sum_{k=0}^j {j \choose k} u^k v^{j-k}$$A(D)\,B(X) =\sum… text/html 2014-12-04T15:43:23+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/smooth_manifold?rev=1417704203&do=diff Smooth manifold Set context $\langle M,T\rangle$ ... second-countable Hausdorff space context $n\in \mathbb N$ definiendum $\langle M,A\rangle\in$ it postulate $A$ maximal in smooth atlas($\langle M,T\rangle,n$) Discussion Elaboration Effectively, a smooth manifold would be given by providing any atlas. But then, due to the redundancy of some charts on small open sets, different atlases give rise to equivalent mathematical objects and so a smooth manifold is defined… text/html 2016-05-20T08:58:37+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/solution_set?rev=1463727517&do=diff Solution set Set context $ X $ context $ \langle Y,y_0 \rangle $ ... pointed set context $ f:X\to Y $ definition $ S := \{x\mid f(x)=y_0\}$ ---------- $S=f^{-1}(\{y_0\})$ where $f^{-1}:{\mathcal P}Y\to{\mathcal P}X$. Reference Wikipedia: Solution set ---------- Context Pointed set Related Equalizer . category theory text/html 2015-04-20T19:59:08+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/space_and_quantity?rev=1429552748&do=diff Space and quantity Note Spaces Presheaves can be understood as spaces in the following way: Topoi can be viewed as universes of things. The archetypical topos is ${\bf{Set}}$ and all topoi have many of it's nice features, e.g. local Cartesian closure and existence of small (co-)limits. Any presheaf category ${\bf{Set}}^{\bf{C}^{op}}$${\bf{Set}}\cong{\bf{Set}}^{\bf{1}}$${\bf{C}}$$U\in{\bf{C}}$$\mathrm{Hom}_{\bf{C}}(-,U)\in{\bf{Set}}^{{\bf{C}}^{op}}$$\beta:{\bf{C}}[U,W]$$\alpha\in\mathrm{Hom}_… text/html 2016-07-18T01:37:52+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/specifying_syntax?rev=1468798672&do=diff Specifying syntax About $\blacktriangleright$ Specifying syntax $\blacktriangleright$ Type theory Logic Note Introduction In this entry, we elaborate on the $\frac{foo}{bar}$ scheme to specify well formed syntactic expressions. In a type theory, in particular, we reason about systems of syntactical constructions. * The declaration of variables is completely formalized: If $\sigma$$x$$x:\sigma$$A$$B$$C$$$\frac{A\hspace{.5cm}B}{C}$$$\mathrm{Type}$$\sigma$$\tau$$\mathrm{Type}$$\sigma\to\t… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/spectral_norm_of_a_matrix?rev=1395396676&do=diff Spectral norm of a matrix Set context $ n\in\mathbb N $ definiendum $ \Vert\cdot\Vert: \text{SquareMatrix}(n,\mathbb C)\to \mathbb R_+ $ postulate $ \Vert A \Vert := \text{max}(\text{SingularVal}(A)) $ Discussion Reference Wikipedia: Singular value, Matrix norm Parents Subset of Norm Context Singular values of a matrix text/html 2016-09-30T15:54:05+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/square_matrix?rev=1475243645&do=diff Square matrix Set context $X$ context $n\in \mathbb N$ definiendum $ \mathrm{Matrix}(n,X)\equiv \mathrm{Matrix}(n,n,X) $ ---------- Discussion We write $A_{i,j}\equiv (A_i)_j$ ---------- Subset of Matrix text/html 2015-08-16T14:02:43+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/statistical_internal_energy?rev=1439726563&do=diff Statistical internal energy Set context $ \langle \mathcal M, H,\pi,\pi_0,{\hat\rho},{\hat\rho}_0\rangle$ ... classical statistical ensemble definiendum $U\equiv\langle H\rangle$ ---------- Discussion ---------- Context Classical ensemble expectation value text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/step_function_integral?rev=1395396676&do=diff Step function integral Set context $\langle X,\Sigma,\mu\rangle\in \mathrm{MeasureSpace}(X)$ postulate $\int_X:\mathcal T^+\to \mathbb R_+$ $ f\equiv \sum_{j=1}^n\alpha_j\cdot\chi_{E_n} \in \mathcal T^+$ postulate $\int_X\ f\ \mathrm d\mu:=\sum_{j=1}^n\alpha_j\cdot\mu(E_n) $ Discussion Reference Parents Context Positive real step function, Characteristic function, Finite sum of complex numbers text/html 2016-09-09T10:29:17+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/stochastic_recurrence_relation?rev=1473409757&do=diff Stochastic recurrence relation Set "thoughts on a formalization of the notion of sampling and repeated generation of sample paths (as opposed to the notion of random variables, also governed by a probability distribution)" Fix a collection of “$x_a, x_b, \dots x_j$$x_k$$k>j$$X_k := X_{k-1} + \mu(X_{k-1})\,{\mathrm d}t + \sigma(X_{k-1})\,{\mathrm d}{\mathcal W}_{{\mathrm d}t}$$\mu, \sigma$${\mathrm d}t$${\mathrm d}{\mathcal W}_{{\mathrm d}t}$${\mathrm d}t$${\mathrm d}X_t := \mu(X_t)\,{\mathrm… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/strict_partial_order?rev=1395396676&do=diff Strict partial order Set context $X$ definiendum $ <\ \in\ \text{StrictPartOrd}(X) $ context $ <\ \in\ \mathrm{Rel}(X) $ $ x,y,z\in X $ postulate $ x \nless x $ postulate $ x<y\land y<z \implies x<z $ Here we use infix notation: $x<y\ \equiv\ <(x,y)$. Discussion A strict partial order is automatically anti-symmetric. Reference Wikipedia: Order theory, Poset Parents Subset of text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/strictly_positive_rational_number?rev=1395396676&do=diff Strictly positive rational number Set definiendum $ \mathbb Q_+^* \equiv \mathbb Q_+ \cap \mathbb Q_* $ Ramifications Reference ProofWiki: Strictly Positive Parents Subset of Non-negative rational number, Non-zero rational number text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/strictly_positive_real_number?rev=1395396676&do=diff Strictly positive real number Set definiendum $ \mathbb R_+^* \equiv \mathbb R_+ \cap \mathbb R_* $ Ramifications Reference ProofWiki: Strictly Positive Parents Subset of Non-negative real number, Non-zero real number text/html 2016-03-04T10:22:24+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/students_t_distribution?rev=1457083344&do=diff Student's t-Distribution Function definition $f:$ ?? definition $f(t, \nu) := \dfrac{1}{\nu^{\frac{1}{2}}{\mathrm B}(\frac{1}{2},\frac{\nu}{2})}\cdot\left(1+\dfrac{t^2}{\nu}\right)^{-\frac{\nu+1}{2}}$ "todo" ---------- Discussion Relation to the Normal distribution The limit is clear using $\lim_{n\to\infty}\left(1+\dfrac{x}{n}\right)^{a\,n+b}=\lim_{n\to\infty}\left(1+\dfrac{a\,x}{n}\right)^n={\mathrm e}^{a\,x}$. Here the normalization: Plot[Sqrt[n/(2*Pi)]*Beta[1/2,n/2],{n,-… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/subfield?rev=1395396676&do=diff Subfield Set context $\langle F,+,*\rangle \in \mathrm{field}(F) $ definiendum $\langle \hat F,\hat +,\hat*\rangle\in\mathrm{subfield}(\langle F,+,*\rangle) $ postulate $ 0,1\in \hat F $ $x,y\in \hat F$ postulate $ x+y,\ x*y,\ x-y,\ x*y^{-1}\in \hat F $ postulate $ x\ \hat +\ y,\ x\ \hat *\ y,\ x\ \hat -\ y,\ x*y^{\hat{-1}} \in \hat F $ Discussion Reference Wikipedia: Field Parents Context Field text/html 2015-04-16T22:10:14+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/submonoid?rev=1429215014&do=diff Submonoid Meta context $\langle\!\langle M,*\rangle\!\rangle$ ... monoid definiendum $\langle\!\langle S,*\rangle\!\rangle\in$ it inclusion $S\in$ closed monoid subset of $\langle\!\langle M,*\rangle\!\rangle$ exists $e\in S$ postulate $e$ ... unit of $\langle\!\langle M,*\rangle\!\rangle$ ---------- ---------- Subset of closed monoid subset text/html 2016-05-29T17:16:08+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/subobject_classifier?rev=1464534968&do=diff Subobject classifier Collection context ${\bf C}$ ... category with terminal object $*$ and all pullbacks definiendum $\top$ in it inclusion $\top:{\bf C}[*,\Omega]$ for all $m_S:{\bf C}[S,X]$ ... monomorphism postulate There is a unique arrow $\chi_S$, so that the mono $m_S$ is the pullback of $\top$$\chi_S$$m_S$$S$$X$$\chi_S$$X$$\Omega$$\top$$$ \require{AMScd} \begin{CD} S @>{!_S}>> * \\ @V{m_S}VV @VV{\top}V \\ … text/html 2014-11-11T17:55:50+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/subset_._hott?rev=1415724950&do=diff Subset . HoTT Note Let $A$ be a set and let $P(x)$ be a dependent type over $A$, which is a mere proposition. The latter means that for a fixed $a:A$, the type $P(a)$ is either empty or has a unique term. Viewed as fibration, the fibre $P(a)$ over $a$ has a “thickness” of zero or one. Like a characteristic function, this specifies a subset of $A$$\star_a$$P(a)$$\langle a,\star_a\rangle:\Sigma_{x:A}P(x)$$\{x\,|\,P(x)\}\equiv \Sigma_{x:A}P(x)$$a\in\{x\,|\,P(x)\}\equiv P(a)$$\{x\,|\,Q(x)\}\subset… text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/subset_complement?rev=1395396676&do=diff Subset complement Set context $Y$ context $X\subset Y$ definiendum $ X^c \equiv Y\smallsetminus X $ Discussion Notice that the notation $ X^c $ doesn't actually denote the bigger set $Y$. It is supposed to be stated or known from the context. Reference Wikipedia: Complement Parents Context text/html 2014-11-05T21:49:03+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/subsingleton?rev=1415220543&do=diff Subsingleton Collection definiendum $u$ in it for all $x,y\in u$ postulate $x=y$ Discussion In classical (non-constructive) set theory, a subsingleton is the empty set or a singleton. Reference Parents Requirements Set universe text/html 2014-12-05T10:51:11+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/successor_set?rev=1417773071&do=diff Successor set Set context $X$ definiendum $ {\mathrm{succ}}\ X \equiv X \cup \{X\} $ Discussion Theorems $ X\in {\mathrm{succ}}\ X $ $ ({\mathrm{succ}}\ X={\mathrm{succ}}\ Y)\Rightarrow (X=Y) $ $ (Y\in {\mathrm{succ}}\ X)\Leftrightarrow (Y=X\lor Y=\{X\}) $ $ X\neq {\mathrm{succ}}\ X $ Reference Wikipedia: Successor ordinal ProofWiki: Successor set Parents Requirements Singleton, Union text/html 2015-10-07T16:50:40+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/sum_type?rev=1444229440&do=diff Sum type Type context $X, Y, Z$ ... type rule ${\large\frac{a\ :\ X}{\nu_{\mathcal l}(a)\ :\ X+Y}}(+\mathcal l \mathcal I)$ rule ${\large\frac{b\ :\ Y}{\nu_{\mathcal r}(b)\ :\ X+Y}}(+\mathcal r \mathcal I)$ rule ${\large\frac{\Gamma,\ x\ :\ X\ \vdash\ f\ :\ Z\hspace{1cm}\Gamma,\ y\ :\ Y\ \vdash\ g\ :\ Z\hspace{1cm}c\ :\ X+Y}{\Gamma\ \vdash\ \left[\mathrm{\bf if}\,c\ \text{matches} \ \nu_{\mathcal l}(x)\ \text{do} \ f(x),\ \mathrm{\bf elseif}\, c\ \text{matches} \ \nu_{… text/html 2014-10-29T19:53:56+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/surjective_function?rev=1414608836&do=diff Surjective function Set context $X,Y$ definiendum $f\in$ it inclusion $f:X\to Y $ postulate $\text{im}(f)=Y $ Discussion A function can only be or not be surjective w.r.t. to a stated codomain. A function is always surjective w.r.t. it's own image. See Function for further discussion. text/html 2014-03-27T10:55:56+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/symmetric_difference?rev=1395914156&do=diff Symmetric difference Set context $ X,Y $ definiendum $ X \triangle Y \equiv (X \smallsetminus Y) \cup (Y \smallsetminus X) $ Discussion Taken as 2-ary operation, $ X \triangle Y $ is commutative. The symmetric difference is commutative, associative and distributive w.r.t. intersection. The power set of any set becomes an abelian group under the operation of symmetric difference, with the empty set as the neutral element of the group and every element in this group being its own … text/html 2014-03-21T11:11:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/symmetric_matrix?rev=1395396676&do=diff Symmetric matrix Set context $X$ context $n\in\mathbb N$ definiendum $ A \in \mathrm{SymmetricMatrix}(n,X) $ postulate $ A^\mathrm{T}=A $ Discussion Reference Wikipedia: Symmetric matrix Parents Subset of Matrix Context Matrix transpose text/html 2014-03-21T11:11:17+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/symmetric_multilinear_functional?rev=1395396677&do=diff Symmetric multilinear functional Set context $X$...$\mathcal F$-vector space context $n\in \mathbb N$ definiendum $M\in \mathrm{SymMultiLin}(X^n)$ context $M\in \mathrm{MultiLin}(X^n)$ $ v_1,\dots,v_n\in X $ $ 1\le i<j\le n $ postulate $ M(v_1,\dots,v_i,\dots,v_j,\dots,v_n)=M(v_1,\dots,v_j,\dots,v_i,\dots,v_n) $ Discussion Parents Subset of Multilinear functional text/html 2014-03-21T11:11:17+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/symmetric_relation?rev=1395396677&do=diff Symmetric relation Set context $X$ definiendum $ R\in\mathrm{SymmetricRel}(X) $ context $ R \in \mathrm{Rel}(X) $ $ x,y\in X $ postulate $ xRy \Leftrightarrow yRx $ Discussion Parents Subset of Binary relation on a set text/html 2014-03-21T11:11:17+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/symmetrized_reduced_distribution_function?rev=1395396677&do=diff Symmetrized reduced distribution function Set context $ \bar f_s $ ... Reduced distribution function definiendum $f_s:=\frac{1}{s!}\sum_\pi \bar f_s$ where $\sum_\pi$ is the sum over argument-permutations, e.g. $\sum_\pi\ g(a,b,c)\equiv g(a,b,c)+g(c,a,b)+g(b,c,a)+g(a,c,b)+g(b,a,c)+g(c,b,a)$. Discussion Relevant for discussions of kinetics on the intermediate level $f_1$ and $f_2$$f_3$$\bar f_s$$f_s=\bar f_s$ text/html 2015-06-30T10:47:19+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/tangens?rev=1435654039&do=diff Tangens Function "todo" $\tan(x):=\dfrac{\sin(x)}{\cos(x)}$ $\tan(x)=\sum_{n=1}^\infty (-1)^{n-1}\frac{2^{2n} (2^{2n}-1)}{(2n)!}B_{2n}\,x^{2n-1}$ with $B_k$ ... Bernoulli numbers ----- Theorems $\left(\dfrac{\tan(x)}{x}\right)^m=1+m\dfrac{1}{3}x^2+m\left(\dfrac{m}{18}+\dfrac{7}{90}\right)x^4+{\mathcal O}(x^6)$ References Wikipedia: Trigonometric_functions#tangent ---------- Related Exponential function, Arcus Tangens text/html 2014-03-21T11:11:17+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/taylor_s_formula?rev=1395396677&do=diff Taylor's formula Theorem $k,n\in \mathbb N,\ k>n$ $f\in C^k(\mathbb R^n,\mathbb R)$ postulate $f(x) = \sum_{|\alpha|\le k} \frac{1}{\alpha !} f^{(\alpha)}(0)\ x^\alpha + R_k(x) $ with postulate $ R_k(x) = \sum_{|\alpha|=k+1} \frac{k+1}{\alpha !} \left( \int_0^1\ (1-s)^k\ F^{(\alpha)}(s\ x)\ \mathrm ds \right)\ x^\alpha $ where we use multi-index notation for $\alpha \in \mathrm{FinSequence}(\mathbb N)$, see Multi-index power. Discussion $f\in C^\infty(\mathbb R,\mathbb R)$, $a… text/html 2015-12-14T10:11:00+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/temp_links_._note?rev=1450084260&do=diff temp links . note Note ---------- <http://ist.ac.at/research/physical_sciences> <https://en.wikipedia.org/wiki/Teledyne_Technologies> ---------- misc: www.mathcurve.com Reference Nikolajs notebook text/html 2015-04-16T18:51:52+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/template_entry?rev=1429203112&do=diff Template entry Meta rule basic logical rule context contex definiendum definiendum definition definition inclusion subset specification let local definition range domain specification for all text/html 2016-07-17T15:47:23+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/tensor_product?rev=1468763243&do=diff Tensor product Set "todo definition as set using equivalence relations point out universal property write down rule of it - the ones we actually need when doing e.g. quantum information" ---------- Reference Wikipedia: Tensor product ---------- Requirements text/html 2014-09-28T19:46:56+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/terminal_morphism?rev=1411926416&do=diff Terminal morphism Collection context $Z:\mathrm{Ob}_{\bf C}$ context $F$ in ${\bf D}\longrightarrow{\bf C}$ definiendum $\langle B,\phi\rangle$ in $\mathrm{it}$ inclusion $A:\mathrm{Ob}_{\bf D}$ inclusion $\phi:{\bf C}[FB,Z]$ for all $B:\mathrm{Ob}_{\bf D}$ for all $\psi:{\bf C}[FA,Z]$ range $f:{\bf D}[A,B]$ postulate $\exists_!f.\ \psi=\phi\circ F(f)$ Discussion Terminology Note that the name “terminal morphisms$\langle B,\phi\… text/html 2014-12-04T22:34:05+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/terminal_object?rev=1417728845&do=diff Terminal object Object context ${\bf C}$ ... category definiendum $T$ inclusion $T:\mathrm{Ob}_{\bf C}$ for all $X:\mathrm{Ob}_{\bf C}$ postulate $\exists!i.\ i:{\bf C}[X,T]$ Discussion See Initial object for the definition of the dual, as well as a characterization as universal morphism. Reference Wikipedia: Initial and terminal objects Parents text/html 2015-01-31T13:49:31+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/theta_and_partition_function?rev=1422708571&do=diff Theta and partition function Note ---------- Related Quantum canonical partition function text/html 2015-09-19T11:46:01+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/thin_category?rev=1442655961&do=diff Thin category Collection definiendum ${\bf C}$ in it inclusion $\bf C$ ... category for all $A,B:\mathrm{Ob}_{\bf C}$ for all $f,g:{\bf C}[A,B]$ postulate $f=g$ ---------- Discussion Miau! Reference nLab: Thin category ---------- Refinement of Locally small category text/html 2017-03-17T14:37:51+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/todo?rev=1489757871&do=diff Todo Meta d = 3; a = 5; r = 1/3; f[z_] = ((z + d)^a - d^a)^r; Series[f[z], {z, 0, 6}] "IMPORTANT: formal power series Tensor product ε-δ function limit" Theormodyanmics atm. there are only some notes under On phenomenological thermodynamics . Note. topoi "subobject classifier Base change functor " Topics To solve $f(x)=x$ via fixedpoint iteration, one may consider the sequence $f^n(x_\text{guess})$$g(x):=\dfrac{h(x)}{h(f(x))}\cdot f(x)$$g^n(x_\text{guess})$$h(x):=1+x$$\neg$${\ma… text/html 2016-11-05T17:00:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/todo_books?rev=1478361616&do=diff Todo books Meta Mathematics Number theory Title Author An Introduction to the Theory of Numbers G. H. Hardy, E. M. Writgth Geometry Title Author Topology and Geometry Glen E. Bredon Differential Geometry: Manifolds, Curves and Surfaces text/html 2015-12-16T19:57:14+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/todo_papers?rev=1450292234&do=diff Todo papers Meta ---------- Related Todo, Literature text/html 2015-02-26T20:24:40+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/topological_space?rev=1424978680&do=diff Topological space Set definiendum $\langle X,\mathcal T\rangle \in \mathrm{it} $ postulate $X,\emptyset\in \mathcal T$ for all $S\subseteq \mathcal T$ postulate $\bigcup S\in \mathcal T$ postulate $S$ ... finite $\Rightarrow \bigcap S\in \mathcal T$ ---------- We call $\mathcal T$ the topology and its elements the open (sub-)sets of $X$. A comment on the intersection axiom requiring finiteness: A major motivation for topological spaces is $\mathbb R^n$$(-\tfrac{1}{n… text/html 2016-03-11T15:14:24+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/total_derivative?rev=1457705664&do=diff Total derivative Function definition $\dfrac{{\mathrm d}}{{\mathrm d}t}:\left(X_1\times\cdots \times X_n\times{\mathbb R}\to{\mathbb R}\right)\to \left(\left({\mathbb R}\to X_1\times\cdots \times X_n\right)\times {\mathbb R}\right)\to {\mathbb R}$ let $\diamond\ f(x^1,\dots,x^n,t)$ definition $\left(\dfrac{{\mathrm d}}{{\mathrm d}t}f\right)\left(\langle t\mapsto\langle r^1(t),\dots,r^n(t)\rangle,t\rangle\right):=\sum_{j=1}^n \dfrac{\partial f}{\partial x^j}(\langle r^1(t),… text/html 2014-03-21T11:11:17+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/total_order?rev=1395396677&do=diff Total order Set context $X$ definiendum $ \le\ \in\ \mathrm{it} $ The relation $\le$ is an order relation if it's in the intersection of all total, all anti-symmetric and all transitive relation. Hence context $ \le\ \in\ \mathrm{Rel}(X) $ $ x,y,z \in X $ postulate $ x \le y\ \lor\ y \le x $ postulate $ x\le y\ \land\ y\le x \implies (x=y) $ postulate $ x \le y\ \land\ y \le z \Leftrightarrow x\le z $ text/html 2014-03-21T11:11:17+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/total_relation?rev=1395396677&do=diff Total relation Set context $X$ definiendum $ R\in\mathrm{TotalRel}(X) $ context $ R\ \in\ \mathrm{Rel}(X) $ $x,y\in X$ postulate $ xRy\ \lor\ yRx $ Discussion Wikipedia: Total relation Parents Subset of Binary relation on a set text/html 2014-03-21T11:11:17+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/trace_of_square_matrices?rev=1395396677&do=diff Trace of square matrices Set context $n\in \mathbb N$ context $M$ ... abelian monoid definiendum $ \mathrm{tr}:\mathrm{SquareMatrix}(n,M)\to M$ definiendum $ \mathrm{tr}(A):=\sum_{k=1}^n A_{kk}$ Discussion Reference Wikipedia: Trace (linear algebra) Parents Context Abelian monoid, Square matrix, Finite sum over a monoid text/html 2014-03-21T11:11:17+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/transitive_relation?rev=1395396677&do=diff Transitive relation Set context $X$ definiendum $ R\in\mathrm{TransitiveRel}(X) $ context $ R \in \mathrm{Rel}(X) $ $ x,y,z\in X $ postulate definiendum $ xRy\land yRz \implies xRz $ Discussion Parents Subset of Binary relation on a set text/html 2014-03-21T11:11:17+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/transport_coefficients?rev=1395396677&do=diff Transport coefficients Set context $ n,u,Q,T,\mathrm{P},K $ ... density, mean velocity, heat flux, pressure tensor, temperature, external force definiendum $ \langle D,\kappa,p,\eta,\zeta \rangle $ These are matrices or numbers, and they might even depend on any other quantities. postulate $ u = - D\ \frac{\nabla n}{n} $ postulate $ Q = - \kappa\ \nabla T $$ \Lambda_{ik} = \frac{1}{2}\left( \frac{\partial}{\partial x_k}u_i + \frac{\partial}{\partial x_i}u_k \right) $$ P =… text/html 2014-03-21T11:11:17+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/trichotomous_relation?rev=1395396677&do=diff Trichotomous relation Set context $X$ definiendum $ R\in\mathrm{TrichotomousRel}(X) $ $ x,y\in X $ postulate $ xRy \lor yRx\lor x=y $ Discussion Parents Subset of Binary relation on a set text/html 2014-03-21T11:11:17+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/trivial_graph?rev=1395396677&do=diff Trivial graph Set context $v,E$ ... set definiendum $ \mathrm{it} = \mathrm{undirectedGraph}(E,\{v\}) $ Discussion The trivial graphs are the graphs with one vertex, namely $v$, and a bunch of loops on it, labeled by the elements of $E$. Parents Subset of Finite undirected graph text/html 2014-12-26T17:56:37+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/tuples?rev=1419612997&do=diff Tuples Meta ${\mathfrak D}_{Tuples}$ ---------- ---------- Related Domain of discourse text/html 2015-01-30T11:49:33+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/turing_machine_as_partial_function?rev=1422614973&do=diff Turing machine as partial function Partial Function context $M\equiv\langle Q,\Gamma,\Sigma,\delta\rangle \in \mathrm{TM} $ definiendum $ \mathrm{run}_M:\Gamma^*\times\mathbb{N}^3\times Q \to \Gamma^*\times\mathbb{N}\times\mathbb{N}\times Q $ definiendum $ \mathrm{run}_M(v,t,s,h,q) := \begin{cases} \left\langle v,t,1,1,q_\mathrm{start} \right\rangle & \mathrm{if}\hspace{0.5cm} q=q_\mathrm{halt} \\\\ \mathrm{run}_M\left(\mathrm{nextTape}(v,h,q),t+1,\mathrm{maxSpace}(v,s,h,q),\ma… text/html 2015-08-19T19:50:31+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/two-body_problem?rev=1440006631&do=diff Two-body problem Set definiendum $\langle \mathbb R^{2\times 3}, H\rangle \in \mathrm{it} $ ... Classical Hamiltonian system postulate $ H({\bf r}_1,{\bf r}_2,{\bf p}_1,{\bf p}_2) = \frac{1}{m_1}\frac{1}{2}{\bf p}_1^2 + \frac{1}{m_2}\frac{1}{2}{\bf p}_2^2 + V(|{\bf r}_1-{\bf r}_2|) $ ---------- Discussion Equations of motion in suitable coordinates range $ M \equiv m_1+m_2 $ range $ \mu \equiv m_1 m_2/M $ The following choice of coordinates eliminates the singl… text/html 2014-11-11T11:24:39+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/type?rev=1415701479&do=diff Type Type $\mathrm{Type}$ ... the type of all other types under consideration. See Type theory for a longer discussion. Parents Related Type theory text/html 2015-12-07T18:39:28+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/type_equivalence?rev=1449509968&do=diff Type equivalence Type $(A\simeq B) \equiv \Sigma_{f:A\to B}\ isequiv(f)$ where $isequiv(f) \equiv \left(\Sigma_{g:B\to A}(f\circ g\sim id_B)\right) \times \left(\Sigma_{g:B\to A}(h\circ f\sim id_A)\right)$ where $(k\sim l)\equiv \Pi_{x:C} \left(k(x)=_{P(x)}l(x)\right)$ where $P:C\to U$ and $k,l: \Pi_{x:C}P(x)$. Discussion We see how equivalence of types $(A\simeq B)$ eventually comes down to equality of terms. (The univalence axiom then says that to find those term-equalities, one ca… text/html 2017-04-14T23:36:52+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/type_theory?rev=1492205812&do=diff Type theory Note ---------- Reference ---------- Related About, Specifying syntax text/html 2014-03-21T11:11:17+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/unary_operation?rev=1395396677&do=diff Unary operation Set context $X$ ... set definiendum $ f\in \text{it}(X) $ inclusion $ f:X\to X $ Discussion Reference Wikipedia: Unary operation Parents Subset of Binary relation on a set, Function text/html 2014-06-18T15:40:32+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/undirected_graph?rev=1403098832&do=diff Undirected graph Set context $V,E$ ... set definiendum $ \langle V,\langle E,\psi\rangle\rangle \in \mathrm{it}(E,V) $ postulate $ \psi $ ... function postulate $ \mathrm{dom}(\psi)=E $ postulate $ \forall (e\in E).\ \exists (u,v\in V).\ \psi(e) = \{v,u\} $ Discussion In the above definition, the set $E=\{a,b,\dots\}$ in $\langle E,\psi\rangle$ is any set whos elements then each label an edge, e.g. $\psi(a)=\{v,w\}$. Instead, one can also define a graph using a… text/html 2014-12-05T00:40:39+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/union?rev=1417736439&do=diff Union Set context $X,Y$ ... set definiendum $ x\in X \cup Y $ postulate $ x\in X\lor x\in Y $ Discussion The 2-ary set construction $ X \cup Y $ is commutative and idempotent. Notice that $ X \cup \emptyset = X$. The union and intersection are associative and distributive with respect to another. Reference Wikipedia: text/html 2015-04-12T17:48:43+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/unit_element?rev=1428853723&do=diff Unit element Set context $\langle M,* \rangle$ ... magma definiendum $e \in$ it postulate $e*a=a*e=e$ ---------- Reference Wikipedia: Magma ---------- Requirements Magma text/html 2014-03-21T11:11:17+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/unit_matrix?rev=1395396677&do=diff Unit matrix Set context $n\in \mathbb N$ context $R$ ... ring definiendum $ (I_n)_{ij} := \begin{cases} e & \mathrm{if}\ i=j\\\\ 0 & \mathrm{else} \end{cases}$ postulate $ I_n \in \mathrm{SquareMatrix}(n,R) $ Discussion Reference Wikipedia: Unit matrix Parents Element of Matrix ring text/html 2014-03-21T11:11:17+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/unital_associative_algebra?rev=1395396677&do=diff Unital associative algebra Set context $A$...R-module definiendum $\langle A,[\cdot,\cdot]\rangle \in \mathrm{UnitalAssociativeAlgebra}(A)$ context $\langle A,[\cdot,\cdot]\rangle \in \mathrm{Algebra}(A)$ context $\langle A,[\cdot,\cdot]\rangle$...monoid Discussion Reference Wikipedia: Associative algebra Parents Subset of Associative algebra text/html 2015-02-02T19:08:20+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/unital_ring?rev=1422900500&do=diff Unital ring Set context $\langle X,+,* \rangle \in \mathrm{ring}(X)$ definiendum $\langle X,+,* \rangle \in \mathrm{it}$ postulate $\langle X,* \rangle \in \mathrm{monoid}(X)$ ---------- The second requirement implies that there is an identiy for the binary operation $*$. One generally (also) calls $X$ the unital ring, i.e. the set where the operations “$+$”$*$ text/html 2014-03-21T11:11:17+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/unitary_matrix?rev=1395396677&do=diff Unitary matrix Set context $n\in\mathbb N$ definiendum $ U \in \mathrm{UnitaryMatrix}(n) $ postulate $ U \in \mathrm{SquareMatrix}(n,\mathbb C) $ postulate $ U^*\ U = U\ U^* = I_n $ Discussion Reference Wikipedia: Unitary matrix Parents Subset of Normal matrix Context Unit matrix text/html 2016-07-18T01:02:43+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/univalence_axiom?rev=1468796563&do=diff Univalence axiom Note Axiom: $Id_U(A,B)\simeq(A\simeq B)$ ... is inhabited Discussion LHS: Here $A,B:U$ are some types in a universe. The identity type $Id_U(A,B)$ corresponds (in the groupoid pov) to the space of paths from $A$ to $B$ in $U$. RHS: Here the type equivalence $(A\simeq B)$ is defined in terms of functions running in opposite directions, $A\to B$$B\to A$$Id_U(A,A)$$refl_A$$id_A:A\to A$$(A\simeq B)\to Id_U(A,B)$$0, S$$+$$SS0+SSS0=_{PA}SSSSS0$$1\equiv S0,\,2\equiv S1,\,etc.$$2+… text/html 2014-08-18T14:56:47+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/universal_turing_machine?rev=1408366607&do=diff Universal Turing machine Set $ M\in U\mathrm{TM} $ $ M\subset\mathrm{TM}_2$ $ x,\alpha $ ... bit string $ M_\alpha $ ... Turing machine $ \forall M'.\ \exists\alpha.\ \forall x.\ M[x,\alpha]=M_\alpha[x] $ Discussion Turing showed that there indeed exist such universal Turing machines which are capable of running any other Turing machine $M$ when given in machine code format: We ecode the original Turing machine $M_\alpha$$\alpha$$\triangleright\ \alpha\ \Box\ \Box\ \dots$$\mathrm{\a… text/html 2015-10-09T15:52:25+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/unordered_pair?rev=1444398745&do=diff Unordered pair Set context $ X,Y$ ... set definiendum $ x\in \{X,Y\} $ postulate $ x = X \lor x = Y $ ---------- $\{X,Y\} \equiv \{x \mid x = X \lor x = Y\}$ Discussion $\{X,X\} = \{x \mid x = X \lor x = X\} = \{x \mid x = X\} = \{X\}$ Reference Wikipedia: Axiom of pairing ---------- Context Sets text/html 2015-11-05T15:52:46+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/various_physical_scales?rev=1446735166&do=diff Various physical scales Note Order Unit Variable $1-10^3$ $\mathrm{\mu V}$ input offset voltage $2-5$ $\mathrm{°C/mm}$ realistic temperature gradient over circuit $-2$ $\mathrm{mV/°C}$ realistic offset voltage per Celsius over circuit References text/html 2016-12-29T16:42:28+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/vector_derivative?rev=1483026148&do=diff Vector derivative Function postulate $x$ "todo" ---------- Discussion ${\bf x}(t)=\sum_{i=1}^nx^i(t){\bf e}_i(t)$ $\dfrac{\partial }{\partial t}{\bf x}(t)=\sum_{i=1}^n\left(\left(\dfrac{\partial }{\partial t}x^i(t)\right){\bf e}_i(t)+x^i(t)\dfrac{\partial }{\partial t}{\bf e}_i(t)\right)$ <https://en.wikipedia.org/wiki/Fictitious_force> Reference Wikipedia: <https://en.wikipedia.org/wiki/Fictitious_force> ---------- Context Fréchet derivative text/html 2014-08-30T18:51:35+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/vector_space?rev=1409417495&do=diff Vector space Set context $V,F$ definiendum $\langle\mathcal V,\mathcal F, *\rangle \in \mathrm{vectorSpace}(V,F)$ context $\langle\mathcal V,\mathcal F, *\rangle \in \mathrm{module}(V,F)$ context $\mathcal F\in \mathrm{field}(F)$ Ramifications Discussion A vector space is a $F$-module over $V$, where $F$ is a field, not just a ring. One speaks of an $F$-vector space over $V$. Here $F$ and $V$ are just sets. text/html 2014-12-02T16:37:47+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/vector_space_basis?rev=1417534667&do=diff Vector space basis Set context $V$...$\ \mathcal F$-vector space definiendum $B\in \mathrm{basis}(V)$ context $B\subset V$ $B'\subseteq B$ $B'$...finite range $n\equiv\left|B'\right|$ $v_1,\dots,v_n\in B'$ $c_1,\dots c_n\in \mathcal F$ $x\in V$ postulate $\sum_{k=1}^n c_k\cdot v_k=0\ \Rightarrow\ \forall j.\ c_j=0$ All finite subsets of the base are linearly independed. It's maybe more clear when written in the contrapositive: $\exists j.\ c_j\ne 0\ \Ri… text/html 2014-03-21T11:11:17+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/vector_space_dimension?rev=1395396677&do=diff Vector space dimension Set context $V$...$\ \mathcal F$-vector space definiendum $\mathrm{dim}(V)\equiv \mathrm{card}(B)$ context $B\in\mathrm{basis}(V)$ Discussion For finite vector spaces, the basis cardinality $\mathrm{dim}(V)$ is the only invariant w.r.t. vector space isomorphisms. The Zero vector space has an empty base. Its vector space dimension is zero. text/html 2014-03-21T11:11:17+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/vector_space_endomorphism?rev=1395396677&do=diff Vector space endomorphism Set context $V,W$...$F$-vector space definiendum $\mathrm{End}(V)\equiv\mathrm{Hom}(V,V)$ Discussion Parents Subset of Vector space homomorphism text/html 2014-03-21T11:11:17+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/vector_space_homomorphism?rev=1395396677&do=diff Vector space homomorphism Set context $V,W$...$F$-vector space definiendum $A\in \mathrm{Hom}(V,W)$ context $A:V\to W$ postulate $A\ (r\cdot x+s\cdot y)=r\cdot A\ x+s\cdot A\ y$ Discussion Parents Subset of Left module homomorphism Context Vector space text/html 2014-12-02T16:38:25+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/vertex_degree?rev=1417534705&do=diff Vertex degree Function context $G=\langle V,E,\psi\rangle$ ... undirected graph definiendum $ \mathrm{dom}\ d = V $ range $ u,v\in V, u\neq v $ range $ e\in E $ definiendum $ d(v):=\left|\{e\ |\ \exists u.\ \psi(e)=\{u,v\}\}+2\ \mathrm{card} \{e\ |\ \psi(e)=\{v,v\}\}\right| $ Discussion The value $d_G(v)$ is the number of neighbours of $v$ with loops counted twice. It is also equal to the sum over the $v$$v$$\left| \{v\ |\ d(v)...\mathrm{odd}\} \right|$$\s… text/html 2014-03-21T11:11:17+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/vertex_neighbours?rev=1395396677&do=diff Vertex neighbours Function context $G=\langle V,E,\psi\rangle$ ... undirected graph definiendum $ \mathrm{dom}\ N_G = V $ definiendum $ N_G(v):=\{u\ |\ \{v,u\}\dots\mathrm{edge\ in}\ G\} $ Discussion Parents Context Undirected graph text/html 2016-04-13T16:23:07+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/wat?rev=1460557387&do=diff WAT WAT Here I write down some syntax ideas for WAT - Writing And Typing, a programming language with a dependent type system that can very easily be parsed to the standard mathematical syntax you wouldd encounter in physics papers. "In the best case, the type both makes for an interesting mathematical structure (an $\infty$$H|\psi\rangle$$\sqrt{3}+1$$\Pi,\Sigma$ text/html 2020-01-10T23:08:53+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/x_x?rev=1578694133&do=diff x^x Function definiendum $ \zeta: \mathbb{C}\setminus\{???\} \to \mathbb C$ definiendum $ x\mapsto x^x$ ---------- Note Representations $x^x={\mathrm e}^{x\log(x)}=\left({\mathrm e}^x\right)^{\log(x)}$ "todo: write down the above with an expanded $\log$ to third order" Because of this, the local minimum of $x^x$ is that of $x\log(x)$, namely $\frac{1}{\mathrm e}\approx 0.37$, and then see Secretary problem (Wikipedia) Furthermore $x^x = \sum_{n=0}^\infty \prod_{k=1}^n (1-x)\lef… text/html 2015-03-14T01:13:09+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/yoneda_embedding?rev=1426291989&do=diff Yoneda embedding Functor context ${\bf C}$ ... small category definiendum $\mathcal y$ inclusion $\mathcal y$ in ${\bf C}\longrightarrow{\bf Set}^{{\bf C}^\mathrm{op}}$ definition $\mathcal y\,A:=\mathrm{Hom}_{\bf C}(-,A)$ definition $\mathcal y(\,f):=g\mapsto f\circ g$ Discussion Elaboration on the action on arrows Consider a general arrow $f:{\bf C}[A,B]$. In the presheaf category ${\bf Set}^{{\bf C}^\mathrm{op}}$, the image arrow $\mathcal y(\,f)$$\mathrm{nat… text/html 2014-12-04T22:33:11+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/zero_object?rev=1417728791&do=diff Zero object Collection context ${\bf C}$ ... category definiendum $Z$ postulate $Z$ ... initial object postulate $Z$ ... terminal object Discussion Reference Wikipedia: Initial and terminal objects Parents Context* Categories Requirements Initial object, Terminal object Element of* Initial morphism, Terminal morphism text/html 2015-02-07T23:47:16+0200 Anonymous (anonymous@undisclosed.example.com) https://axiomsofchoice.org/zeta_functions?rev=1423349236&do=diff Zeta functions Note A bit on encodings and basic operations Field algebra Let $S$ be the space/statespace/system (maybe with parts/states/aspects $s_0, s_1, s_2, s_3, s_4,\dots$) traceless part $S \equiv 1 + T \equiv 1 - t$ or similar ... here $1$ is the neutral/constant thing in the theory and $T$ resp. $t$ is what's really interesting about $S$$1$$T$$\dfrac{1}{S}$$X$$Q(t):=\dfrac{1}{1-t}=\sum_{n=0}^\infty t^n$$Q(t)=1+t+{\mathcal O}(t^2) \approx 1-T$$t$$1$$S\,Q(t)=2\,Q(t)-1$$\log(S)=\l…