Numerical Analysis Homework #7
Approximate ∫ − 0 . 5 0 x ln ( x + 1 ) d x
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% turing machines
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\int_{{-0}.5}^0 x \ln (x + 1) \id x ∫ − 0 .5 0 x ln ( x + 1 ) d x
∫ ≈ 0.5 2 ( 0 + ( − 0 . 5 ln ( − 0 . 5 + 1 ) ) ) ≈ 0.086
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% TODO: remove this?
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\int \approx \frac{0.5}{2} \tpar{ 0 + ({-0}.5 \ln({-0}.5 + 1)) } \approx 0.{086} ∫ ≈ 2 0.5 ( 0 + ( − 0 .5 ln ( − 0 .5 + 1 )) ) ≈ 0. 086 ≈ 0.052
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% TODO: remove this?
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% turing machines
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\approx 0.{052} ≈ 0. 052 ≈ 0.034
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% TODO: remove this?
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%% computability %%
% turing machines
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\approx 0.{034} ≈ 0. 034 term is
h 3 12 f ′ ′ ( ξ )
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% TODO: remove this?
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% yoneda embedding
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% TODO: remove this?
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% turing machines
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\frac{h^3}{{12}} f''(\xi) 12 h 3 f ′′ ( ξ ) − 0 . 5 ≤ ξ ≤ 0
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% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
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% hook arrows
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%% category theory %%
% category names
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% yoneda embedding
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% extra long right-arrows
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% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
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%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
{-0}.5 \leq \xi \leq 0 − 0 .5 ≤ ξ ≤ 0 f ′ ′ ( ξ )
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% shorthands
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% f \onall X = { f(x) : x ∈ X }
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% shorthands: various brackets
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% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
f''(\xi) f ′′ ( ξ ) h 3 12 ξ + 2 ( ξ + 1 ) 2
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\frac{h^3}{{12}} \frac{\xi+2}{(\xi+1)^2} 12 h 3 ( ξ + 1 ) 2 ξ + 2 Evaluation at ξ = − 0 . 5
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\xi = {-0}.5 ξ = − 0 .5 ≈ 0.062
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\approx 0.{062} ≈ 0. 062 ≈ 0.034
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\approx 0.{034} ≈ 0. 034 ∫ ≈ 0.25 3 ( 0 − 4 ⋅ 0.25 ln ( 1 − 0.25 ) − 0.5 ln ( 1 − 0.5 ) ) ≈ 0.0528546
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\int \approx \frac{0.{25}}{3} (0 - 4 \cdot 0.{25} \text{ln}(1 - 0.{25}) - 0.5 \text{ln} (1 - 0.5)) \approx 0.{0528546} ∫ ≈ 3 0. 25 ( 0 − 4 ⋅ 0. 25 ln ( 1 − 0. 25 ) − 0.5 ln ( 1 − 0.5 )) ≈ 0. 0528546 ≈ 0.0525698
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\approx 0.{0525698} ≈ 0. 0525698 ≈ 0.0003
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\approx 0.{0003} ≈ 0. 0003 term is
h 5 90 f ( 4 ) ( ξ ) = 0. 25 5 90 ( 8 ( ξ + 1 ) 3 − 6 ξ ( ξ + 1 ) 4 )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\frac{h^5}{{90}} f^{(4)}(\xi) = \frac{0.{25}^5}{{90}} \tpar { \frac 8 {(\xi + 1)^3} - \frac{6\xi}{(\xi + 1)^4} } 90 h 5 f ( 4 ) ( ξ ) = 90 0. 25 5 ( ( ξ + 1 ) 3 8 − ( ξ + 1 ) 4 6 ξ ) − 0 . 5 ≤ ξ ≤ 0
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
{-0}.5 \leq \xi \leq 0 − 0 .5 ≤ ξ ≤ 0 ξ = − 0 . 5
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\xi = {-0}.5 ξ = − 0 .5 0.0012
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
0.{0012} 0. 0012 ∫ − 1 1 f ( x ) d x ≈ f ( − 3 / 3 ) + f ( 3 / 3 )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\int_{{-1}}^1 f(x) \id x \approx f(-\sqrt 3 / 3) + f(\sqrt 3 / 3) ∫ − 1 1 f ( x ) d x ≈ f ( − 3 /3 ) + f ( 3 /3 ) f ( x ) = x k
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
f(x) = x^k f ( x ) = x k ∫ − 1 1 x k d x = ( 1 k + 1 x k + 1 ) − 1 1 = 1 k + 1 − ( { 1 k + 1 k odd − 1 k + 1 k even ) = ( { 0 k odd 2 k + 1 k even )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\int_{-1}^1 x^k \id x = \tpar{ \frac 1 {k+1} x^{k+1} }_{-1}^1 = \frac 1 {k+1} - \tpar{ \begin{cases} \frac 1 {k+1} & k \t{ odd} \\ -\frac 1 {k+1} & k \t{ even} \end{cases} } = \tpar{\begin{cases} 0 & k \t{ odd} \\ \frac 2 {k+1} & k \t{ even} \end{cases}} ∫ − 1 1 x k d x = ( k + 1 1 x k + 1 ) − 1 1 = k + 1 1 − ( { k + 1 1 − k + 1 1 k odd k even ) = ( { 0 k + 1 2 k odd k even ) f ( − 3 / 3 ) + f ( 3 / 3 ) = ( − 3 / 3 ) k + ( 3 / 3 ) k = ( { 0 k odd 2 ( 3 / 3 ) k k even )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
f(-\sqrt 3 / 3) + f(\sqrt 3 / 3) = (-\sqrt 3 / 3)^k + (\sqrt 3 / 3)^k = \tpar{\begin{cases} 0 & k \t{ odd} \\ 2(\sqrt 3 / 3)^k & k \t{ even} \end{cases}} f ( − 3 /3 ) + f ( 3 /3 ) = ( − 3 /3 ) k + ( 3 /3 ) k = ( { 0 2 ( 3 /3 ) k k odd k even ) k = 0 , 1 , 2 , 3
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
k = 0, 1, 2, 3 k = 0 , 1 , 2 , 3 k = 4
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
k = 4 k = 4 2 k + 1 = 2 5 ≠ 2 9 = 2 ( 3 / 3 ) 4
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\frac 2 {k+1} = \frac 2 5 \neq \frac 2 9 = 2(\sqrt 3 / 3)^4 k + 1 2 = 5 2 = 9 2 = 2 ( 3 /3 ) 4 3
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
3 3 term for Simpson’s rule is a product with f ( 4 ) ( ξ )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
f^{(4)}(\xi) f ( 4 ) ( ξ ) f
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
f f ≤ 2
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\leq 2 ≤ 2 f ( 4 ) ≡ 0
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
f^{(4)} \equiv 0 f ( 4 ) ≡ 0 ∫ 0 2 f ( x ) = 1 3 f ( 0 ) + 4 3 f ( 1 ) + 1 3 f ( 2 )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\int_0^2 f(x) = \frac 1 3 f(0) + \frac 4 3 f(1) + \frac 1 3 f(2) ∫ 0 2 f ( x ) = 3 1 f ( 0 ) + 3 4 f ( 1 ) + 3 1 f ( 2 ) Use the composite trapezoidal rule to approximate ∫ − 0 . 5 0.5 x ln ( x + 1 ) d x
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\int_{{-0}.5}^{0.5} x \ln (x + 1) \id x ∫ − 0 .5 0.5 x ln ( x + 1 ) d x n = 6
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
n = 6 n = 6
∫ − 0.5 0.5 x ln ( x + 1 ) d x ≈ 0.5 − ( − 0.5 ) 2 ⋅ 6 ( f ( − 0.5 ) + 2 ∑ j = 1 5 f ( x j ) + f ( 0.5 ) ) = 1 12 ( 0.346574 + 2 [ 0.135155 + 0.0303869 + 0 + 0.0256918 + 0.095894 ] + 0.202733 ) = 0.0936302
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\begin{align*} & \int_{-0.5}^{0.5} x \ln (x + 1) \id x
\\&\approx \frac {0.5 - (-0.5)} {2 \cdot 6} \tpar{ f(-0.5) + 2\sum_{j=1}^5 f(x_j) + f(0.5) }
\\&= \frac 1 {12} \tpar{ 0.346574 + 2\tbrak{ 0.135155 + 0.0303869 + 0 + 0.0256918 + 0.095894 } + 0.202733 }
\\&= 0.0936302
\end{align*} ∫ − 0.5 0.5 x ln ( x + 1 ) d x ≈ 2 ⋅ 6 0.5 − ( − 0.5 ) ( f ( − 0.5 ) + 2 j = 1 ∑ 5 f ( x j ) + f ( 0.5 ) ) = 12 1 ( 0.346574 + 2 [ 0.135155 + 0.0303869 + 0 + 0.0256918 + 0.095894 ] + 0.202733 ) = 0.0936302 ≈ 0.0880204
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\approx 0.{0880204} ≈ 0. 0880204 ≈ 0.0056098
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\approx 0.{0056098} ≈ 0. 0056098 ∫ − 0.5 0.5 x ln ( x + 1 ) d x ≈ h 3 ( f ( − 0.5 ) + 2 ∑ j = 1 n / 2 − 1 f ( x 2 j ) + 4 ∑ j = 1 n / 2 f ( x 2 j − 1 ) + f ( 0.5 ) ) = 1 18 ( f ( − 0.5 ) + 2 [ f ( x 2 ) + f ( x 4 ) ] + 4 [ f ( x 1 ) + f ( x 3 ) + f ( x 5 ) ] + f ( 0.5 ) ) = 1 18 ( 0.346574 + 2 [ 0.0303869 + 0.0256918 ] + 4 [ 0.135155 + 0 + 0.095894 ] + 0.202733 ) = 0.08809224444 ⋯
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\begin{align*} & \int_{-0.5}^{0.5} x \ln (x + 1) \id x
\\&\approx \frac h 3 \tpar{ f(-0.5) + 2 \sum_{j=1}^{n/2-1} f(x_{2j}) + 4\sum_{j=1}^{n/2}f(x_{2j-1}) + f(0.5) }
\\&= \frac 1 {18} \tpar{ f(-0.5) + 2 \tbrak{ f(x_2) + f(x_4) } + 4\tbrak{ f(x_1) + f(x_3) + f(x_5) } + f(0.5) }
\\&= \frac 1 {18} \tpar{ 0.346574 + 2 \tbrak{ 0.0303869 + 0.0256918 } + 4\tbrak{ 0.135155 + 0 + 0.095894 } + 0.202733 }
\\&= 0.08809224444\cdots
\end{align*} ∫ − 0.5 0.5 x ln ( x + 1 ) d x ≈ 3 h ⎝ ⎛ f ( − 0.5 ) + 2 j = 1 ∑ n /2 − 1 f ( x 2 j ) + 4 j = 1 ∑ n /2 f ( x 2 j − 1 ) + f ( 0.5 ) ⎠ ⎞ = 18 1 ( f ( − 0.5 ) + 2 [ f ( x 2 ) + f ( x 4 ) ] + 4 [ f ( x 1 ) + f ( x 3 ) + f ( x 5 ) ] + f ( 0.5 ) ) = 18 1 ( 0.346574 + 2 [ 0.0303869 + 0.0256918 ] + 4 [ 0.135155 + 0 + 0.095894 ] + 0.202733 ) = 0.08809224444 ⋯ ≈ 0.0880204
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
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% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\approx 0.{0880204} ≈ 0. 0880204 ≈ 0.000071844
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
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% draw a box outlining some math
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% f \onall X = { f(x) : x ∈ X }
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% shorthands: various brackets
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% reverse \mapsto (FIXME: make better)
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% reverse-order composition
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% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
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% hook arrows
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\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
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\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
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% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\approx 0.{000071844} ≈ 0. 000071844 term of Composite Simpson’s Rule is
b − a 180 h 4 f ( 4 ) ( μ ) = 2 180 ( 2 n ) 4 ( − 119 e 2 μ sin ( 3 μ ) − 120 e 2 μ cos ( 3 μ ) ) = 1 n 4 ( 952 45 e 2 μ sin ( 3 μ ) + 64 3 e 2 μ cos ( 3 μ ) ) (equal up to magnitude)
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
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% draw a box outlining some math
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% f \onall X = { f(x) : x ∈ X }
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% shorthands: various brackets
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% reverse \mapsto (FIXME: make better)
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% reverse-order composition
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% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
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% hook arrows
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\newcommand{\embeds}{ \hookrightarrow }
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\newcommand{\projects}{ \twoheadrightarrow }
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% derivatives: use {\ddn n x y} for (dy/dx)
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\newcommand{\dd}{ \ddn{} } % first derivative
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% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
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\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
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% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\begin{align*} \frac{b - a}{180} h^4 f^{(4)}(\mu) &= \frac{2}{180} \tpar{\frac 2 n}^4 \tpar{ -119 e^{2 \mu} \sin(3 \mu) - 120 e^{2 \mu} \cos(3 \mu) }
\\ &= \frac 1 {n^4} \tpar{ \frac{952}{45} e^{2 \mu} \sin(3 \mu) + \frac{64}{3} e^{2 \mu} \cos(3 \mu) } &&\t{(equal up to magnitude)}
\end{align*} 180 b − a h 4 f ( 4 ) ( μ ) = 180 2 ( n 2 ) 4 ( − 119 e 2 μ sin ( 3 μ ) − 120 e 2 μ cos ( 3 μ ) ) = n 4 1 ( 45 952 e 2 μ sin ( 3 μ ) + 3 64 e 2 μ cos ( 3 μ ) ) (equal up to magnitude) μ ∈ ( 0 , 2 )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
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\renewcommand{\sf}[1]{ \mathsf{#1} }
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\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
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% draw a box outlining some math
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% f \onall X = { f(x) : x ∈ X }
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% shorthands: various brackets
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% reverse \mapsto (FIXME: make better)
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% reverse-order composition
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% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
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% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
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\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
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% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\mu \in (0, 2) μ ∈ ( 0 , 2 ) μ = 2
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
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\newcommand{\card}{ \t{cd} } % cardinality
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% draw a box outlining some math
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% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
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\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
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\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
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\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\mu = 2 μ = 2 fix μ : = 2
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
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\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
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\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
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% reverse \mapsto (FIXME: make better)
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% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
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% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
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\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
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% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\mu := 2 μ := 2 ≤ 10 − 4
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
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\renewcommand{\sf}[1]{ \mathsf{#1} }
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% draw a box outlining some math
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% f \onall X = { f(x) : x ∈ X }
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% shorthands: various brackets
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% reverse \mapsto (FIXME: make better)
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\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
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\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
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% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\leq {10}^{-4} ≤ 10 − 4 1 n 4 ( 952 45 e 4 sin ( 6 ) + 64 3 e 4 cos ( 6 ) ) ≤ 10 − 4 10 4 ( 952 45 e 4 sin ( 6 ) + 64 3 e 4 cos ( 6 ) ) ≤ n 4 n ≥ ( 10 4 ( 952 45 e 4 sin ( 6 ) + 64 3 e 4 cos ( 6 ) ) ) 1 / 4 ≈ 53.1102
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\begin{align*} \frac 1 {n^4} \tpar{ \frac{{952}}{{45}} e^4 \sin(6) + \frac{{64}}{3} e^4 \cos(6) } &\leq {10}^{-4}
\\ {10}^4 \tpar{ \frac{{952}}{{45}} e^4 \sin(6) + \frac{{64}}{3} e^4 \cos(6) } &\leq n^4
\\ n &\geq \tpar{{10}^4 \tpar{ \frac{{952}}{{45}} e^4 \sin(6) + \frac{{64}}{3} e^4 \cos(6) }}^{1/4} \approx {53}.{1102}
\end{align*} n 4 1 ( 45 952 e 4 sin ( 6 ) + 3 64 e 4 cos ( 6 ) ) 10 4 ( 45 952 e 4 sin ( 6 ) + 3 64 e 4 cos ( 6 ) ) n ≤ 10 − 4 ≤ n 4 ≥ ( 10 4 ( 45 952 e 4 sin ( 6 ) + 3 64 e 4 cos ( 6 ) ) ) 1/4 ≈ 53 . 1102 n ≥ 54
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
n \geq {54} n ≥ 54 (a) ≈ 68.26894 %
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\approx {68}.{26894} \% ≈ 68 . 26894 % (b) ≈ 95.44997 %
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\approx {95}.{44997} \% ≈ 95 . 44997 % (c) ≈ 99.73002 %
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\approx {99}.{73002} \% ≈ 99 . 73002 % R n , i = R n , i − 1 + 1 4 i − 1 − 1 ( R n , i − 1 − R n − 1 , i − 1 ) ( ⋆ )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
R_{n,i} = R_{n,i-1} + \frac 1 {4^{i-1}-1} ( R_{n,i-1} - R_{n-1,i-1} ) \tag{$\star$} R n , i = R n , i − 1 + 4 i − 1 − 1 1 ( R n , i − 1 − R n − 1 , i − 1 ) ( ⋆ ) R n , 1
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
R_{n,1} R n , 1 n = n
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
n=n n = n R n , 1 = 0.75 − ( − 0.75 ) 2 ⋅ n ( f ( − 0.75 ) + 2 ∑ j = 1 n − 1 f ( − 0.75 + j 0.75 − ( − 0.75 ) n ) + f ( 0.75 ) ) = 3 4 n ( 1.03972 + 2 ∑ j = 1 n − 1 f ( 6 j − 3 4 n ) + 0.419712 )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\begin{align*} R_{n,1} &= \frac{ 0.75 - (-0.75) }{ 2 \cdot n } \tpar{ f(-0.75) + 2 \sum_{j=1}^{n-1} f\tpar{ -0.75 + j\frac{0.75-(-0.75)}{n} } + f(0.75) }
\\ &= \frac{3}{4n} \tpar{ 1.03972 + 2 \sum_{j=1}^{n-1} f\tpar{ \frac{6 j - 3}{4 n} } + 0.419712 }
\end{align*} R n , 1 = 2 ⋅ n 0.75 − ( − 0.75 ) ( f ( − 0.75 ) + 2 j = 1 ∑ n − 1 f ( − 0.75 + j n 0.75 − ( − 0.75 ) ) + f ( 0.75 ) ) = 4 n 3 ( 1.03972 + 2 j = 1 ∑ n − 1 f ( 4 n 6 j − 3 ) + 0.419712 ) R 1 , 1 ≈ 1.0945740000000002 R 2 , 1 ≈ 0.6368521118770878 R 3 , 1 ≈ 0.6026068643900597
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\begin{align*} R_{1,1} &\approx 1.0945740000000002
\\ R_{2,1} &\approx 0.6368521118770878
\\ R_{3,1} &\approx 0.6026068643900597
\end{align*} R 1 , 1 R 2 , 1 R 3 , 1 ≈ 1.0945740000000002 ≈ 0.6368521118770878 ≈ 0.6026068643900597 ( ⋆ )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
(\star) ( ⋆ ) R 3 , 3 ≈ 0.5983193574093792
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
R_{3,3} \approx 0.{5983193574093792} R 3 , 3 ≈ 0. 5983193574093792 R 2 , 1 = 1 2 ( 3 − 2 2 ) ( f ( 2 ) + 2 f ( 2.5 ) + f ( 3 ) )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
R_{2,1} = \frac 1 2 \tpar{ \frac {3-2} 2 } \tpar{ f(2) + 2 f(2.5) + f(3) } R 2 , 1 = 2 1 ( 2 3 − 2 ) ( f ( 2 ) + 2 f ( 2.5 ) + f ( 3 ) ) f ( 2.5 ) = 2 R 2 , 1 − 1 2 ( f ( 2 ) + f ( 3 ) ) = 2 R 2 , 1 − 0.44065
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\begin{align*} f(2.5) &= 2 R_{2,1} - \frac 1 2 (f(2) + f(3))
\\&= 2 R_{2,1} - 0.44065 \tag A
\end{align*} f ( 2.5 ) = 2 R 2 , 1 − 2 1 ( f ( 2 ) + f ( 3 )) = 2 R 2 , 1 − 0.44065 ( A ) R 2 , 1
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
R_{2,1} R 2 , 1 R 2 , 1
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
R_{2,1} R 2 , 1 ( n , k )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
(n, k) ( n , k ) represents R n , k
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
R_{n,k} R n , k ( 2 , 1 )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
(2,1) ( 2 , 1 ) ?
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
? ? R 1 , 1 = 1 2 ⋅ 1 1 ⋅ ( f ( 2 ) + f ( 3 ) ) = 0.44065 R 2 , 2 = R 2 , 1 + 1 3 ( R 2 , 1 − R 1 , 1 ) ≈ 4 3 R 2 , 1 − 0.146883 R 3 , 2 = R 3 , 1 + 1 3 ( R 3 , 1 − R 2 , 1 ) ≈ 0.582493 − 1 3 R 2 , 1 R 3 , 3 = R 3 , 2 + 1 15 ( R 3 , 2 − R 2 , 2 ) = 16 15 R 3 , 2 − 4 45 R 2 , 1 + 0.195844 = − 4 9 ( R 2 , 1 − 1.83863 )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\begin{align*} R_{1,1} &= \frac 1 2 \cdot \frac 1 1 \cdot (f(2) + f(3)) = 0.44065
\\ R_{2,2} &= R_{2,1} + \frac 1 3 ( R_{2,1} - R_{1,1} ) \approx \frac 4 3 R_{2,1} - 0.146883
\\ R_{3,2} &= R_{3,1} + \frac 1 3 ( R_{3,1} - R_{2,1} ) \approx 0.582493 - \frac 1 3 R_{2,1}
\\ R_{3,3} &= R_{3,2} + \frac 1 {15} ( R_{3,2} - R_{2,2} ) = \frac{16}{15} R_{3,2} - \frac{4}{45} R_{2,1} + 0.195844 = -\frac 4 9 (R_{2,1} - 1.83863)
\end{align*} R 1 , 1 R 2 , 2 R 3 , 2 R 3 , 3 = 2 1 ⋅ 1 1 ⋅ ( f ( 2 ) + f ( 3 )) = 0.44065 = R 2 , 1 + 3 1 ( R 2 , 1 − R 1 , 1 ) ≈ 3 4 R 2 , 1 − 0.146883 = R 3 , 1 + 3 1 ( R 3 , 1 − R 2 , 1 ) ≈ 0.582493 − 3 1 R 2 , 1 = R 3 , 2 + 15 1 ( R 3 , 2 − R 2 , 2 ) = 15 16 R 3 , 2 − 45 4 R 2 , 1 + 0.195844 = − 9 4 ( R 2 , 1 − 1.83863 ) R 3 , 3 = 0.43662
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
R_{3,3} = 0.{43662} R 3 , 3 = 0. 43662 R 2 , 1 ≈ 0.856237
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
R_{2,1} \approx 0.{856237} R 2 , 1 ≈ 0. 856237 ( A )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
(A) ( A ) f ( 2.5 ) ≈ 1.271824
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
f(2.5) \approx 1.{271824} f ( 2.5 ) ≈ 1. 271824 ≈ 0.8427007901827961
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\approx 0.8427007901827961 ≈ 0.8427007901827961 [ φ ] : = { 1 φ 0 ¬ φ
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
[\varphi] := \begin{cases} 1 & \varphi \\ 0 & \neg \varphi \end{cases} [ φ ] := { 1 0 φ ¬ φ ∑ i = 1 2 k − 2 f ( a + ( i − 1 2 ) h k − 1 ) + ∑ i = 1 2 k − 2 − 1 f ( a + i h k − 1 ) = ∑ i = 2 2 k − 1 [ 2 ∣ i ] f ( a + ( i 2 − 1 2 ) h k − 1 ) + ∑ i = 2 2 k − 1 − 2 [ 2 ∣ i ] f ( a + i 2 h k − 1 ) reindex sums = ∑ i = 1 2 k − 1 − 1 [ 2 ∣ ( i + 1 ) ] f ( a + ( i + 1 2 − 1 2 ) h k − 1 ) + ∑ i = 2 2 k − 1 − 2 [ 2 ∣ i ] f ( a + i 2 h k − 1 ) reindex left sum = ∑ i = 1 2 k − 1 − 1 [ 2 ∣ ( i + 1 ) ] f ( a + i 2 h k − 1 ) + ∑ i = 2 2 k − 1 − 2 [ 2 ∣ i ] f ( a + i 2 h k − 1 ) simplify = ∑ i = 1 2 k − 1 − 1 [ 2 ∣ ( i + 1 ) ] f ( a + i 2 h k − 1 ) + ∑ i = 1 2 k − 1 − 1 [ 2 ∣ i ] f ( a + i 2 h k − 1 ) extend right sum index bounds = ∑ i = 1 2 k − 1 − 1 [ 2 ∣ ( i + 1 ) ] f ( a + i 2 h k − 1 ) + [ 2 ∣ i ] f ( a + i 2 h k − 1 ) combine like-indexed sums = ∑ i = 1 2 k − 1 − 1 f ( a + i 2 h k − 1 ) simplify
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\begin{align*} & \sum_{i=1}^{2^{k-2}} f(a + (i - \frac 1 2)h_{k-1}) + \sum_{i=1}^{2^{k-2}-1} f(a+ih_{k-1})
\\&= \sum_{i=2}^{2^{k-1}} [2 \mid i] f(a + (\frac i 2 - \frac 1 2)h_{k-1}) + \sum_{i=2}^{2^{k-1}-2} [2 \mid i]f(a+ \frac i 2 h_{k-1}) &&\t{reindex sums}
\\&= \sum_{i=1}^{2^{k-1}-1} [2 \mid (i+1)] f(a + (\frac {i+1} 2 - \frac 1 2)h_{k-1}) + \sum_{i=2}^{2^{k-1}-2} [2 \mid i]f(a+ \frac i 2 h_{k-1}) &&\t{reindex left sum}
\\&= \sum_{i=1}^{2^{k-1}-1} [2 \mid (i+1)] f(a + \frac i 2 h_{k-1}) + \sum_{i=2}^{2^{k-1}-2} [2 \mid i]f(a+ \frac i 2 h_{k-1}) &&\t{simplify}
\\&= \sum_{i=1}^{2^{k-1}-1} [2 \mid (i+1)] f(a + \frac i 2 h_{k-1}) + \sum_{i=1}^{2^{k-1}-1} [2 \mid i]f(a+ \frac i 2 h_{k-1}) &&\t{extend right sum index bounds}
\\&= \sum_{i=1}^{2^{k-1}-1} [2 \mid (i+1)] f(a + \frac i 2 h_{k-1}) + [2 \mid i]f(a+ \frac i 2 h_{k-1}) &&\t{combine like-indexed sums}
\\&= \sum_{i=1}^{2^{k-1}-1} f(a + \frac i 2 h_{k-1}) &&\t{simplify}
\end{align*} i = 1 ∑ 2 k − 2 f ( a + ( i − 2 1 ) h k − 1 ) + i = 1 ∑ 2 k − 2 − 1 f ( a + i h k − 1 ) = i = 2 ∑ 2 k − 1 [ 2 ∣ i ] f ( a + ( 2 i − 2 1 ) h k − 1 ) + i = 2 ∑ 2 k − 1 − 2 [ 2 ∣ i ] f ( a + 2 i h k − 1 ) = i = 1 ∑ 2 k − 1 − 1 [ 2 ∣ ( i + 1 )] f ( a + ( 2 i + 1 − 2 1 ) h k − 1 ) + i = 2 ∑ 2 k − 1 − 2 [ 2 ∣ i ] f ( a + 2 i h k − 1 ) = i = 1 ∑ 2 k − 1 − 1 [ 2 ∣ ( i + 1 )] f ( a + 2 i h k − 1 ) + i = 2 ∑ 2 k − 1 − 2 [ 2 ∣ i ] f ( a + 2 i h k − 1 ) = i = 1 ∑ 2 k − 1 − 1 [ 2 ∣ ( i + 1 )] f ( a + 2 i h k − 1 ) + i = 1 ∑ 2 k − 1 − 1 [ 2 ∣ i ] f ( a + 2 i h k − 1 ) = i = 1 ∑ 2 k − 1 − 1 [ 2 ∣ ( i + 1 )] f ( a + 2 i h k − 1 ) + [ 2 ∣ i ] f ( a + 2 i h k − 1 ) = i = 1 ∑ 2 k − 1 − 1 f ( a + 2 i h k − 1 ) reindex sums reindex left sum simplify extend right sum index bounds combine like-indexed sums simplify 