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Type theory - Axioms of Choice

https://axiomsofchoice.org/type_theory

Http:/ www.cs.cmu.edu/ rwh/plbook/book.pdf. Improvements of the human condition. Last modified: 2016/04/14 11:58 by.

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Linear approximation - Axioms of Choice

https://axiomsofchoice.org/linear_approximation

X,Y$ Banach spaces with topology. Mathcal O$ open in $X$. X in mathcal O$. F: mathcal O to Y$. J x f$ bounded linear operator from $X$ to $Y$. Mathrm{lim} {h to 0} Vert f(x h)-f(x)-J x f(h) Vert / Vert h Vert = 0$. The approximation of the value of $f$ at a point $x$ is $f(x d) sim f(x) J x f(d)$. The operators $J x f$ are actually both functional in $x$ and in $f$ and this is how we can define general differentiation operators, see Fréchet derivative. Mathfrak J o$, and then if. The field $ mathbb C$ is...

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Limit . category theory - Axioms of Choice

https://axiomsofchoice.org/limit_._category_theory

Limit . category theory. F:{ bf C} { bf D}$. Delta:{ bf C} longrightarrow{ bf C} { bf D}$ diagonal functor. Mathrm{lim} ,F$ terminal morphism from $ Delta$ to $F$. Firstly, keep in mind that in a functor category ${ bf C} { bf D}$, the main data of a terminal morphism to $F in { bf C} { bf D}$ is a natural transformation $ phi$ from some functor to $F$. Loads of other natural transformations factor through this $ phi$. We now restate the above definition using a little more prose: The diagonal functor.

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k-tape Turing machine - Axioms of Choice

https://axiomsofchoice.org/k-tape_turing_machine

K in mathbb N $. Langle Q, Gamma, Sigma, delta rangle in mathrm{TM} k $. Delta: Q times Gamma k to Q times Gamma k times { mathrm{L}, mathrm{S}, mathrm{R} } k$. Q mathrm{start},q mathrm{halt} in Q $. Box, triangleright,0,1 in Gamma $. Box notin Sigma $. Let's discuss the 1-tape Turing machine, i.e. $k=1$. The general case is then clear. Triangleright underset{q mathrm{start} {a} b c d cdots v w x y z Box Box Box Box Box cdots $. Triangleright a underset{q j}{b} c d cdots $. The conventions in the pic (Sn...

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Domain of discourse - Axioms of Choice

https://axiomsofchoice.org/domain_of_discourse

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 sorts. Mathfrak D} mathrm{Collections}$ Collections. There are also entries of type Type.

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Magic Gaussian integral - Axioms of Choice

https://axiomsofchoice.org/magic_gaussian_integral

M in mathbb N $. A $ self-adjoint $n times n$ square matrix over $ mathbb C$. A varepsilon $ $n times n$ square matrix over $ mathbb C$. Lim { varepsilon to 0}A varepsilon = A $. Z: ( mathbb C m to mathbb C) to mathbb C m to mathbb C$. Z f(J):= lim { varepsilon to 0} int {- infty} infty ,f( phi) , mathrm e { frac{1}{2} left langle phi left ,A varepsilon , right phi right rangle i , left langle phi left right.J right rangle} prod {i=1} m mathrm d phi i $. Int {- infty} infty mathrm e {- tfrac{1}{2}a , phi...

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Second-countable Hausdorff space - Axioms of Choice

https://axiomsofchoice.org/second-countable_hausdorff_space

Langle X,T rangle in$ it. Langle X,T rangle$ Second-countable space. Langle X,T rangle$ Hausdorff space. Improvements of the human condition. Second-countable hausdorff space.txt. Last modified: 2014/10/29 21:06 by.

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Monad - Axioms of Choice

https://axiomsofchoice.org/monad

Langle T, eta, mu rangle$ in $ mathrm{it}$. T$ in ${ bf C} { bf C}$. Eta:1 { bf C} xrightarrow{ bullet}T$. Mu circ T mu= mu circ mu T$. Mu circ T eta= mu circ eta T=1 T$. A monad is functor together with two natural transformations that fulfill some algebraic relations. Written out in component form, the postulates read. Mu X circ T( mu X)= mu X circ mu {TX}$. Mu X circ T( eta X)= mu X circ eta {TX}=1 {TX}$. Perspective, they are the arrow images of the functor applied to the co-unit components. Eta:1 { ...

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Itō integral - Axioms of Choice

https://axiomsofchoice.org/ito_integral

Langle Omega, mathcal{F},( mathcal{F} t) {t ge 0}, mathbb{P} rangle! Rangle$ filtered probability space. X t$ $ mathcal{F} t$-adapted. H t$ left-continuous, locally bounded, $ mathcal{F} t$-adapted. Delta X {n,i} = X {a ih n}-X {a (i-1)h n}$. Int a b H {t-} ,{ mathrm d}X t := { mathrm{plim} {n to infty} sum {i=1} n H {a (i-1)h n} Delta X {n,i}$. Bullet$ filtered probability space (${F} t$ being a filter meas with passing time, more and more of the algebra is available). Trivially, $H {a (i-1)h}$ is.