Numerical Analysis Homework #8
≈ 5.113832919861613
%% 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 }
% 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} }
% 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}
% 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 5.113832919861613 ≈ 5.113832919861613 def adaptive_quad (f, a_in, b_in, TOL_in, N ):
APP = 0
i = 1
TOL = dict ()
a = dict ()
h = dict ()
FA = dict ()
FC = dict ()
FB = dict ()
S = dict ()
L = dict ()
TOL[i] = 10 * TOL_in
a[i] = a_in
h[i] = (b_in - a_in)/2
FA[i] = f(a_in)
FC[i] = f(a_in + h[i])
FB[i] = f(b_in)
S[i] = h[i] * (FA[i] + 4 *FC[i] + FB[i])/3
L[i] = 1
while i > 0 :
FD = f(a[i] + h[i]/2 )
FE = f(a[i] + 3 *h[i]/2 )
S1 = h[i]*(FA[i] + 4 *FD + FC[i])/6
S2 = h[i]*(FC[i] + 4 *FE + FB[i])/6
v1 = a[i]
v2 = FA[i]
v3 = FC[i]
v4 = FB[i]
v5 = h[i]
v6 = TOL[i]
v7 = S[i]
v8 = L[i]
i -= 1
if abs (S1 + S2 - v7) < v6:
APP += S1 + S2
elif v8 >= N:
raise Exception("level exceeded" )
else :
i += 1
a[i] = v1 + v5
FA[i] = v3
FC[i] = FE
FB[i] = v4
h[i] = v5/2
TOL[i] = v6/2
S[i] = S2
L[i] = v8 + 1
i += 1
a[i] = v1
FA[i] = v2
FC[i] = FD
FB[i] = v3
h[i] = h[i-1 ]
TOL[i] = TOL[i-1 ]
S[i] = S1
L[i] = L[i-1 ]
return APP
print (adaptive_quad(
f=lambda x: ( 6 *cos(4 *x) + 4 *sin(6 *x) ) * exp(x),
a_in=0 , b_in=pi/2 ,
TOL_in=10 **(-5 ), N=10 ,
))
1980-01-01T00:00:00+00:00 image/svg+xml Matplotlib v3.7.0, https://matplotlib.org/
The integral is ≈ 0.6737815084529792
%% 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 }
% 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} }
% 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}
% 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.{6737815084529792} ≈ 0. 6737815084529792 1980-01-01T00:00:00+00:00 image/svg+xml Matplotlib v3.7.0, https://matplotlib.org/
The integral is ≈ − 5 . 724587470723463 × 10 − 17
%% 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 }
% 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} }
% 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}
% 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 {-5}.{724587470723463} \times {10}^{-17} ≈ − 5 . 724587470723463 × 10 − 17 ∫ = h 2 ( f ( a ) + f ( b ) ) − h 3 12 f ′ ′ ( ξ 1 ( x ) ) = T ( a , b ) − h 3 12 f ′ ′ ( ξ 1 ( 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 }
% 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} }
% 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}
% 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 &= \frac h 2 ( f(a) + f(b) ) - \frac{h^3}{12} f''(\xi_1(x))
\\ &= T(a,b) - \frac{h^3}{12} f''(\xi_1(x))
\end{align*} ∫ = 2 h ( f ( a ) + f ( b )) − 12 h 3 f ′′ ( ξ 1 ( x )) = T ( a , b ) − 12 h 3 f ′′ ( ξ 1 ( x )) ∫ = h 1 3 [ f ( a ) + 2 f ( a + b 2 ) + f ( b ) ] − b − a 12 h 1 2 f ′ ′ ( ξ 2 ( x ) ) = h 6 [ f ( a ) + 2 f ( a + b 2 ) + f ( b ) ] − b − a 12 h 2 4 f ′ ′ ( ξ 2 ( x ) ) h 1 = h / 2 = h 6 [ f ( a ) + 2 f ( a + b 2 ) + f ( b ) ] − h 3 48 f ′ ′ ( ξ 2 ( x ) ) h = a − b = T ( a , a + b 2 ) + T ( a + b 2 , b ) − h 3 48 f ′ ′ ( ξ 2 ( 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 }
% 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} }
% 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}
% 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
&= \frac {h_1} 3 \tbrak{ f(a) + 2 f(\frac{a+b}2) + f(b) } - \frac{b-a}{12} {h_1}^2 f''(\xi_2(x))
\\&= \frac h 6 \tbrak{ f(a) + 2 f(\frac{a+b}2) + f(b) } - \frac{b-a}{12} \frac{h^2}4 f''(\xi_2(x)) && h_1 = h/2
\\&= \frac h 6 \tbrak{ f(a) + 2 f(\frac{a+b}2) + f(b) } - \frac{h^3}{48} f''(\xi_2(x)) && h = a - b
\\&= T(a, \frac{a+b}2) + T(\frac{a+b}2, b) - \frac{h^3}{48} f''(\xi_2(x))
\end{align*} ∫ = 3 h 1 [ f ( a ) + 2 f ( 2 a + b ) + f ( b ) ] − 12 b − a h 1 2 f ′′ ( ξ 2 ( x )) = 6 h [ f ( a ) + 2 f ( 2 a + b ) + f ( b ) ] − 12 b − a 4 h 2 f ′′ ( ξ 2 ( x )) = 6 h [ f ( a ) + 2 f ( 2 a + b ) + f ( b ) ] − 48 h 3 f ′′ ( ξ 2 ( x )) = T ( a , 2 a + b ) + T ( 2 a + b , b ) − 48 h 3 f ′′ ( ξ 2 ( x )) h 1 = h /2 h = a − b f ′ ′ ( ξ 1 ) = f ′ ′ ( ξ 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 }
% 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} }
% 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}
% 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_1) = f''(\xi_2) f ′′ ( ξ 1 ) = f ′′ ( ξ 2 ) T ( a , b ) − h 3 12 f ′ ′ ( ξ 1 ( x ) ) = T ( a , a + b 2 ) + T ( a + b 2 , b ) − h 3 48 f ′ ′ ( ξ 2 ( x ) ) T ( a , b ) − ( T ( a , a + b 2 ) + T ( a + b 2 , b ) ) = 3 h 3 48 f ′ ′ ( ξ ) ∣ T ( a , b ) − ( T ( a , a + b 2 ) + T ( a + b 2 , b ) ) ∣ = 3 h 3 48 ∣ f ′ ′ ( 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 }
% 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} }
% 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}
% 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*} T(a,b) - \frac{h^3}{12} f''(\xi_1(x)) &= T(a, \frac{a+b}2) + T(\frac{a+b}2, b) - \frac{h^3}{48} f''(\xi_2(x))
\\ T(a,b) - ( T(a, \frac{a+b}2) + T(\frac{a+b}2, b) ) &= \frac{3 h^3}{48} f''(\xi)
\\ \mag{ T(a,b) - ( T(a, \frac{a+b}2) + T(\frac{a+b}2, b) ) } &= \frac{3 h^3}{48} \mag{ f''(\xi) }
\end{align*} T ( a , b ) − 12 h 3 f ′′ ( ξ 1 ( x )) T ( a , b ) − ( T ( a , 2 a + b ) + T ( 2 a + b , b )) ∣ ∣ T ( a , b ) − ( T ( a , 2 a + b ) + T ( 2 a + b , b )) ∣ ∣ = T ( a , 2 a + b ) + T ( 2 a + b , b ) − 48 h 3 f ′′ ( ξ 2 ( x )) = 48 3 h 3 f ′′ ( ξ ) = 48 3 h 3 ∣ f ′′ ( ξ ) ∣ ∣ ∫ − ( T ( a , a + b 2 ) + T ( a + b 2 , b ) ) ∣ = h 3 48 ∣ f ′ ′ ( 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 }
% 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} }
% 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}
% 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} }
\mag{ \int - ( T(a, \frac{a+b}2) + T(\frac{a+b}2, b) ) } = \frac{h^3}{48} \mag{ f''(\xi) }
∣ ∣ ∫ − ( T ( a , 2 a + b ) + T ( 2 a + b , b )) ∣ ∣ = 48 h 3 ∣ f ′′ ( ξ ) ∣ 3 ∣ ∫ − ( T ( a , a + b 2 ) + T ( a + b 2 , b ) ) ∣ = ∣ T ( a , b ) − ( T ( a , a + b 2 ) + T ( a + b 2 , b ) ) ∣
%% 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 }
% 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} }
% 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}
% 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 \mag{ \int - ( T(a, \frac{a+b}2) + T(\frac{a+b}2, b) ) } = \mag{ T(a,b) - ( T(a, \frac{a+b}2) + T(\frac{a+b}2, b) ) }
3 ∣ ∣ ∫ − ( T ( a , 2 a + b ) + T ( 2 a + b , b )) ∣ ∣ = ∣ ∣ T ( a , b ) − ( T ( a , 2 a + b ) + T ( 2 a + b , b )) ∣ ∣ ∫ 3 3.5 x x 2 − 4 d x = ∫ − 1 1 0.5 t + 6.5 1 4 ( 0.5 t + 6.5 ) 2 − 4 ⋅ 1 8 d t
%% 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 }
% 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} }
% 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}
% 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_{3}^{3.5} \frac{x}{\sqrt{x^2-4}} \id x
= \int_{-1}^1 \frac {0.5t + 6.5}{\sqrt{\frac 1 4 (0.5t + 6.5)^2 - 4}} \cdot \frac 1 8 \id t
∫ 3 3.5 x 2 − 4 x d x = ∫ − 1 1 4 1 ( 0.5 t + 6.5 ) 2 − 4 0.5 t + 6.5 ⋅ 8 1 d t n = 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 }
% 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} }
% 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}
% 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 = 2 n = 2 n = 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 }
% 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} }
% 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}
% 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 = 2 n = 2 ∫ − 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 }
% 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} }
% 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}
% 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 ) 1 8 0.5 ( − 3 / 3 ) + 6.5 1 4 ( 0.5 ( − 3 / 3 ) + 6.5 ) 2 − 4 + 1 8 0.5 ( 3 / 3 ) + 6.5 1 4 ( 0.5 ( 3 / 3 ) + 6.5 ) 2 − 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 }
% 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} }
% 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}
% 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 1 8 \frac {0.5(-\sqrt 3 / 3) + 6.5}{\sqrt{\frac 1 4 (0.5(-\sqrt 3 / 3) + 6.5)^2 - 4}} + \frac 1 8 \frac {0.5(\sqrt 3 / 3) + 6.5}{\sqrt{\frac 1 4 (0.5(\sqrt 3 / 3) + 6.5)^2 - 4}}
8 1 4 1 ( 0.5 ( − 3 /3 ) + 6.5 ) 2 − 4 0.5 ( − 3 /3 ) + 6.5 + 8 1 4 1 ( 0.5 ( 3 /3 ) + 6.5 ) 2 − 4 0.5 ( 3 /3 ) + 6.5 ≈ 0.6361965649627966518698075581974524572132499841757342095041156097
%% 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 }
% 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} }
% 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}
% 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.{6361965649627966518698075581974524572132499841757342095041156097} ≈ 0. 6361965649627966518698075581974524572132499841757342095041156097 0.636213
%% 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 }
% 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} }
% 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}
% 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.{636213} 0. 636213 n = 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 }
% 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} }
% 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}
% 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 = 4 n = 4 { x i }
%% 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 }
% 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} }
% 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}
% 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} }
\{ x_i \} { x i } { c i }
%% 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 }
% 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} }
% 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}
% 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} }
\{ c_i \} { c i } { x i }
%% 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 }
% 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} }
% 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}
% 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} }
\{ x_i \} { x i } P 4 ( x ) = x 4 − 6 7 x 2 + 3 35
%% 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 }
% 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} }
% 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}
% 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} }
P_4(x) = x^4 - \frac 6 7 x^2 + \frac 3 {35} P 4 ( x ) = x 4 − 7 6 x 2 + 35 3 x ≈ x ± 0.33998 , ± 0.86114
%% 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 }
% 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} }
% 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}
% 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} }
x \approx x \pm 0.{33998}, \pm 0.{86114} x ≈ x ± 0. 33998 , ± 0. 86114 { c i }
%% 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 }
% 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} }
% 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}
% 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} }
\{ c_i \} { c i } c i = ∫ − 1 1 ∏ j = 1 , j ≠ i n x − x j x i − x j 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 }
% 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} }
% 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}
% 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} }
c_i = \int_{-1}^1 \prod_{j=1, j\neq i}^n \frac{x-x_j}{x_i-x_j} \id x c i = ∫ − 1 1 j = 1 , j = i ∏ n x i − x j x − x j d x c 1 = ∫ − 1 1 x − x 2 x 1 − x 2 ⋅ x − x 3 x 1 − x 3 ⋅ x − x 4 x 1 − x 4 d x = ∫ − 1 1 x − 0.33998 − 0.33998 − 0.33998 ⋅ x + 0.86114 − 0.33998 + 0.86114 ⋅ x − 0.86114 − 0.33998 − 0.86114 d x ≈ 0.652148
%% 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 }
% 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} }
% 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}
% 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*} c_1
&= \int_{-1}^1 \frac {x - x_2}{x_1 - x_2} \cdot \frac {x - x_3}{x_1 - x_3} \cdot \frac {x - x_4}{x_1 - x_4} \id x
\\&= \int_{-1}^1 \frac {x - 0.33998}{-0.33998 - 0.33998} \cdot \frac {x + 0.86114}{-0.33998 + 0.86114} \cdot \frac {x - 0.86114}{-0.33998 - 0.86114} \id x
\\&\approx 0.652148
\end{align*} c 1 = ∫ − 1 1 x 1 − x 2 x − x 2 ⋅ x 1 − x 3 x − x 3 ⋅ x 1 − x 4 x − x 4 d x = ∫ − 1 1 − 0.33998 − 0.33998 x − 0.33998 ⋅ − 0.33998 + 0.86114 x + 0.86114 ⋅ − 0.33998 − 0.86114 x − 0.86114 d x ≈ 0.652148 c 2 ≈ 0.652147945660487 c 3 ≈ 0.347852054339514 c 4 ≈ 0.347852054339514
%% 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 }
% 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} }
% 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}
% 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*} c_2 &\approx 0.652147945660487
\\ c_3 &\approx 0.347852054339514
\\ c_4 &\approx 0.347852054339514
\end{align*} c 2 c 3 c 4 ≈ 0.652147945660487 ≈ 0.347852054339514 ≈ 0.347852054339514 c 1 f ( x 1 ) + c 2 f ( x 2 ) + c 3 f ( x 3 ) + c 4 f ( x 4 ) ≈ 0.636213344718202
%% 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 }
% 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} }
% 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}
% 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} }
c_1 f(x_1) + c_2 f(x_2) + c_3 f(x_3) + c_4 f(x_4) \approx 0.636213344718202 c 1 f ( x 1 ) + c 2 f ( x 2 ) + c 3 f ( x 3 ) + c 4 f ( x 4 ) ≈ 0.636213344718202 ∫ − 1 1 f ( x ) d x = a f ( − 1 ) + b f ( 1 ) + c f ′ ( − 1 ) + d f ′ ( 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 }
% 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} }
% 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}
% 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 = a f(-1) + b f(1) + c f'(-1) + d f'(1) ∫ − 1 1 f ( x ) d x = a f ( − 1 ) + b f ( 1 ) + c f ′ ( − 1 ) + d f ′ ( 1 ) f ( 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 }
% 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} }
% 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}
% 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)= f ( x ) = 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 }
% 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} }
% 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}
% 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} }
1 1 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 }
% 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} }
% 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}
% 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} }
x x x 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 }
% 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} }
% 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}
% 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} }
x^2 x 2 x 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 }
% 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} }
% 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}
% 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} }
x^3 x 3 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 }
% 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} }
% 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}
% 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} }
1 1 ∫ − 1 1 1 d x = 2 = a + b
%% 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 }
% 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} }
% 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}
% 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 1 \id x = 2 = a + b ∫ − 1 1 1 d x = 2 = a + b b = 2 − a (I)
%% 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 }
% 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} }
% 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}
% 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} }
b = 2 - a \tag I b = 2 − a ( I ) 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 }
% 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} }
% 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}
% 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} }
x x ∫ − 1 1 x d x = 0 = − a + b + c + d
%% 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 }
% 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} }
% 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}
% 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 \id x = 0 = -a + b +c + d ∫ − 1 1 x d x = 0 = − a + b + c + d a = b + c + d a = ( 2 − a ) + c + d by (I) 2 a = 2 + c + d a = 1 + 1 2 ( c + d )
%% 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 }
% 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} }
% 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}
% 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*} a &= b + c + d
\\ a &= (2 - a) + c + d && \t{by (I)}
\\ 2a &= 2 + c + d
\\ a &= 1 + \frac 1 2 ( c + d ) \tag{II}
\end{align*} a a 2 a a = b + c + d = ( 2 − a ) + c + d = 2 + c + d = 1 + 2 1 ( c + d ) by (I) ( II ) x 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 }
% 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} }
% 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}
% 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} }
x^2 x 2 ∫ − 1 1 x 2 d x = 2 3 = a + b − 2 c + 2 d (III)
%% 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 }
% 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} }
% 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}
% 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^2 \id x = \frac 2 3 = a + b - 2c + 2d \tag{III} ∫ − 1 1 x 2 d x = 3 2 = a + b − 2 c + 2 d ( III ) 2 3 = a + ( 2 − a ) − 2 c + 3 d by (I) 2 3 = 2 − 2 c + 3 d 2 = 6 − 6 c + 9 d c = 2 3 + 3 2 d
%% 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 }
% 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} }
% 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}
% 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 2 3 &= a + (2 - a) -2c + 3d &\t{by (I)}
\\ \frac 2 3 &= 2 -2c + 3d
\\ 2 &= 6 - 6c + 9d
\\ c &= \frac 2 3 + \frac 3 2 d \tag{III}
\end{align*} 3 2 3 2 2 c = a + ( 2 − a ) − 2 c + 3 d = 2 − 2 c + 3 d = 6 − 6 c + 9 d = 3 2 + 2 3 d by (I) ( III ) x 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 }
% 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} }
% 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}
% 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} }
x^3 x 3 ∫ − 1 1 x 3 d x = 0 = − a + b + 3 c + 3 d (IV)
%% 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 }
% 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} }
% 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}
% 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^3 \id x = 0 = -a + b + 3c + 3d \tag {IV} ∫ − 1 1 x 3 d x = 0 = − a + b + 3 c + 3 d ( IV ) 0 = − a + ( 2 − a ) + 3 c + 3 d by (I) 0 = 2 − 2 a + 3 c + 3 d 0 = 2 − 2 ( 1 + 1 2 ( c + d ) ) + 3 c + 3 d by (II) 0 = 2 c + 2 d 0 = 2 ( 2 3 + 3 2 d ) + 2 d by (III) 0 = 4 3 + 5 d d = − 4 15
%% 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 }
% 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} }
% 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}
% 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*} 0 &= -a + (2 - a) + 3c + 3d &\t{by (I)}
\\ 0 &= 2 - 2a + 3c + 3d
\\ 0 &= 2 - 2(1 + \frac 1 2 (c+d)) + 3c + 3d &\t{by (II)}
\\ 0 &= 2c + 2d
\\ 0 &= 2(\frac 2 3 + \frac 3 2 d) + 2d &\t{by (III)}
\\ 0 &= \frac 4 3 + 5d
\\ d &= -\frac 4 {15}
\end{align*} 0 0 0 0 0 0 d = − a + ( 2 − a ) + 3 c + 3 d = 2 − 2 a + 3 c + 3 d = 2 − 2 ( 1 + 2 1 ( c + d )) + 3 c + 3 d = 2 c + 2 d = 2 ( 3 2 + 2 3 d ) + 2 d = 3 4 + 5 d = − 15 4 by (I) by (II) by (III) c = 2 3 + 3 2 ( − 4 15 ) = 4 15 by (III) a = 1 + 1 2 ( c + d ) = 1 by (II) b = 2 − a = 1 by (I)
%% 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 }
% 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} }
% 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}
% 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*} c &= \frac 2 3 + \frac 3 2 (-\frac 4 {15}) = \frac 4 {15} &\t{by (III)}
\\ a &= 1 + \frac 1 2 (c + d) = 1 &\t{by (II)}
\\ b &= 2 - a = 1 &\t{by (I)}
\end{align*} c a b = 3 2 + 2 3 ( − 15 4 ) = 15 4 = 1 + 2 1 ( c + d ) = 1 = 2 − a = 1 by (III) by (II) by (I) ∫ − 1 1 f ( x ) d x = f ( − 1 ) + f ( 1 ) + 4 15 f ′ ( − 1 ) − 4 15 f ′ ( 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 }
% 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} }
% 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}
% 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 = f(-1) + f(1) + \frac 4{15} f'(-1) - \frac 4{15} f'(1)
∫ − 1 1 f ( x ) d x = f ( − 1 ) + f ( 1 ) + 15 4 f ′ ( − 1 ) − 15 4 f ′ ( 1 ) ∫ a b f ( x ) d x ≈ ∑ i = 1 n c i f ( x i ) (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 }
% 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} }
% 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}
% 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_a^b f(x) \id x \approx \sum_{i=1}^n c_i f(x_i) \tag A ∫ a b f ( x ) d x ≈ i = 1 ∑ n c i f ( x i ) ( A ) 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 }
% 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} }
% 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}
% 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 b
%% 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 }
% 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} }
% 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}
% 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} }
b b { c i }
%% 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 }
% 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} }
% 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}
% 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} }
\{ c_i \} { c i } { x i }
%% 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 }
% 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} }
% 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}
% 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} }
\{ x_i \} { x i } function of 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 }
% 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} }
% 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}
% 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 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 }
% 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} }
% 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}
% 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} }
2n 2 n f ( x ) = ∏ i = 1 n ( x − x i ) 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 }
% 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} }
% 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}
% 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) = \prod_{i=1}^n (x - x_i)^2 f ( x ) = i = 1 ∏ n ( x − x i ) 2 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 }
% 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} }
% 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}
% 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 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 }
% 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} }
% 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}
% 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 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 }
% 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} }
% 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}
% 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 zero at only a finite number of points, then every nontrivial definite integral over 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 }
% 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} }
% 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}
% 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 formula ( 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 }
% 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} }
% 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}
% 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} }
(\t A) ( A ) ∫ a b f ( x ) d x ≈ ∑ i = 1 n c i f ( x i ) = ∑ c i 0 = 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 }
% 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} }
% 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}
% 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_a^b f(x) \id x \approx \sum_{i=1}^n c_i f(x_i) = \sum c_i 0 = 0 ∫ a b f ( x ) d x ≈ i = 1 ∑ n c i f ( x i ) = ∑ c i 0 = 0 ( 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 }
% 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} }
% 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}
% 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
%% 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 }
% 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} }
% 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}
% 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 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 }
% 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} }
% 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}
% 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 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 }
% 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} }
% 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}
% 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} }
2n 2 n ( 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 }
% 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} }
% 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}
% 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} }
(\t A) ( A ) 2 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 }
% 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} }
% 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}
% 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} }
2n{-1} 2 n − 1 ∫ ≈ 0.2552526215425403 ∫ = 0.2552520000000000
%% 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 }
% 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} }
% 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}
% 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 \approx\ & 0.2552526215425403
\\\int =\ & 0.2552520000000000
\end{align*} ∫ ≈ ∫ = 0.2552526215425403 0.2552520000000000
def algo_44 (f, a, b, c, d, n, m ):
h = (b-a)/n
J1 = 0
J2 = 0
J3 = 0
for i in range (0 , n+1 ):
x = a + i*h
HX = (d(x) - c(x))/m
K1 = f(x,c(x)) + f(x,d(x))
K2 = 0
K3 = 0
for j in range (1 ,m):
y = c(x) + j*HX
Q = f(x,y)
if j % 2 == 0 :
K2 += Q
else :
K3 += Q
L = (K1 + 2 *K2 + 4 *K3) * (HX / 3 )
if i == 0 or i == n: J1 += L
elif i % 2 == 0 : J2 += L
else : J3 += L
J = h*(J1 + 2 *J2 + 4 *J3)/3
return J
print (algo_44(
f=lambda x, y: exp(y-x),
a=0 , b=0.5 ,
c=lambda x: 0 , d=lambda x: 0.5 ,
n=4 , m=4 ,
))
The approximate and actual values are
∫ ≈ 16.50864062500002 ∫ = 16.50860000000000
%% 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 }
% 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} }
% 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}
% 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 \approx\ & 16.50864062500002
\\\int =\ & 16.50860000000000
\end{align*} ∫ ≈ ∫ = 16.50864062500002 16.50860000000000 print (algo_44(
f=lambda x, y: (x**2 + y**3 ),
a=2 , b=2.2 ,
c=lambda x: x, d=lambda x: 2 *x,
n=4 , m=4 ,
))
term for Composite Simpson’s rule (under some niceness conditions) is
E = − ( d − c ) ( b − a ) 180 [ h 4 D x 4 f ( η ‾ , μ ‾ ) + k 4 D y 4 f ( η ^ , μ ^ ) ]
%% 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 }
% 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} }
% 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}
% 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} }
E = - \frac{(d-c)(b-a)}{180} \tbrak{ h^4 D^4_x f (\ol \eta, \ol \mu) + k^4 D^4_y f(\hat \eta, \hat \mu) } E = − 180 ( d − c ) ( b − a ) [ h 4 D x 4 f ( η , μ ) + k 4 D y 4 f ( η ^ , μ ^ ) ] ( η ‾ , μ ‾ ) , ( η ^ , μ ^ )
%% 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 }
% 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} }
% 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}
% 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} }
(\ol \eta, \ol \mu), (\hat \eta, \hat \mu) ( η , μ ) , ( η ^ , μ ^ ) a , b , c , d , h , 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 }
% 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} }
% 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}
% 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, b, c, d, h, k a , b , c , d , h , k E = − 0.25 180 [ ( 0.5 n ) 4 D x 4 f ( η ‾ , μ ‾ ) + ( 0.5 n ) 4 D y 4 f ( η ^ , μ ^ ) ] = − 1 1440 ⋅ n 4 [ D x 4 f ( η ‾ , μ ‾ ) + D y 4 f ( η ^ , μ ^ ) ]
%% 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 }
% 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} }
% 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}
% 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*} E = & - \frac{0.25}{180} \tbrak{ \tpar{\frac{0.5}n}^4 D^4_x f (\ol \eta, \ol \mu) + \tpar{\frac{0.5}n}^4 D^4_y f(\hat \eta, \hat \mu) }
\\&= - \frac1{1440 \cdot n^4} \tbrak{ D^4_x f (\ol \eta, \ol \mu) + D^4_y f(\hat \eta, \hat \mu) }
\end{align*} E = − 180 0.25 [ ( n 0.5 ) 4 D x 4 f ( η , μ ) + ( n 0.5 ) 4 D y 4 f ( η ^ , μ ^ ) ] = − 1440 ⋅ n 4 1 [ D x 4 f ( η , μ ) + D y 4 f ( η ^ , μ ^ ) ] E = − 1 1440 ⋅ n 4 [ [ e y − x ] η ‾ , μ ‾ + [ e y − x ] η ^ , μ ^ ] = − 1 1440 ⋅ n 4 [ e η ‾ − μ ‾ + e η ^ − μ ^ ]
%% 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 }
% 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} }
% 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}
% 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*} E = & - \frac1{1440 \cdot n^4} \tbrak{ \tbrak{e^{y-x}}_{\ol \eta, \ol \mu} + \tbrak{e^{y-x}}_{\hat \eta, \hat \mu} }
\\&= - \frac1{1440 \cdot n^4} \tbrak{ e^{\ol \eta - \ol \mu} + e^{\hat \eta - \hat \mu} }
\end{align*} E = − 1440 ⋅ n 4 1 [ [ e y − x ] η , μ + [ e y − x ] η ^ , μ ^ ] = − 1440 ⋅ n 4 1 [ e η − μ + e η ^ − μ ^ ] R
%% 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 }
% 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} }
% 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}
% 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 R 0 ≤ η ‾ , μ ‾ , η ^ , μ ^ ≤ 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 }
% 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} }
% 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}
% 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 \leq \ol \eta, \ol \mu, \hat \eta, \hat \mu \leq 0.5 0 ≤ η , μ , η ^ , μ ^ ≤ 0.5 e a − b
%% 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 }
% 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} }
% 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}
% 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} }
e^{a-b} e a − b 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 }
% 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} }
% 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}
% 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 b
%% 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 }
% 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} }
% 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}
% 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} }
b b least , so our maximization of ∣ E ∣
%% 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 }
% 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} }
% 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}
% 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} }
\mag E ∣ E ∣ η ‾ = 0.5 μ ‾ = 0 η ^ = 0.5 μ ^ = 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 }
% 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} }
% 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}
% 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*} \ol \eta &= 0.5
\\ \ol \mu &= 0
\\ \hat \eta &= 0.5
\\ \hat \mu &= 0
\end{align*} η μ η ^ μ ^ = 0.5 = 0 = 0.5 = 0 maximum
∣ E ∣ = 1 1440 ⋅ n 4 ( 2 e 1 2 ) = e 1220 ⋅ n 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 }
% 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} }
% 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}
% 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} }
\mag E = \frac 1 {1440 \cdot n^4} \tpar { 2 e^{\frac 1 2} } = \frac {\sqrt e}{1220 \cdot n^4} ∣ E ∣ = 1440 ⋅ n 4 1 ( 2 e 2 1 ) = 1220 ⋅ n 4 e M = 10 − 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 }
% 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} }
% 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}
% 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} }
M = {10}^{-6} M = 10 − 6 e 1220 ⋅ n 4 ≤ M e 1220 M ≤ n 4 ( e 1220 M ) 1 4 ≤ 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 }
% 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} }
% 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}
% 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{\sqrt e}{1220 \cdot n^4} \leq M
\\ \frac{\sqrt e}{1220 M} \leq n^4
\\ \tpar{ \frac{\sqrt e}{1220 M} }^{\frac 1 4} \leq n
\end{align*} 1220 ⋅ n 4 e ≤ M 1220 M e ≤ n 4 ( 1220 M e ) 4 1 ≤ n n ≥ ≈ 6.063
%% 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 }
% 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} }
% 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}
% 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\ \approx 6.063 n ≥ ≈ 6.063 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 }
% 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} }
% 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}
% 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 ≥ 7
%% 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 }
% 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} }
% 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}
% 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 7 n ≥ 7 term of the approximation for
∫ 2 2.2 ∫ x 2 x x 2 + y 3 d y 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 }
% 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} }
% 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}
% 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_2^{2.2} \int_x^{2x} x^2 + y^3 \id y \id x ∫ 2 2.2 ∫ x 2 x x 2 + y 3 d y d x formula for rectangular regions, but this region of integration is not rectangular. We’ll make the simplifying assumption that the error of an approximation over a non-rectangular region is bounded by the error over the smallest enclosing rectangular region, and accordingly make use of the error term for
∫ 2 2.2 ∫ 2 4.4 x 2 + y 3 d y 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 }
% 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} }
% 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}
% 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_2^{2.2} \int_2^{4.4} x^2 + y^3 \id y \id x ∫ 2 2.2 ∫ 2 4.4 x 2 + y 3 d y d x term for the approximation for this integral is
E = − ( d − c ) ( b − a ) 180 [ h 4 D x 4 f ( η ‾ , μ ‾ ) + k 4 D y 4 f ( η ^ , μ ^ ) ] = − 11 4500 [ ( 0.2 n ) 4 D x 4 f ( η ‾ , μ ‾ ) + ( 2.4 n ) 4 D y 4 f ( η ^ , μ ^ ) ] = − 11 2812500 ⋅ n 4 [ D x 4 f ( η ‾ , μ ‾ ) + 20736 D y 4 f ( η ^ , μ ^ ) ] = − 11 2812500 ⋅ n 4 [ 0 + 20736 ⋅ 0 ] = 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 }
% 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} }
% 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}
% 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*} E &= -\frac{(d-c)(b-a)}{180} \tbrak{ h^4 D^4_x f (\ol \eta, \ol \mu) + k^4 D^4_y f(\hat \eta, \hat \mu) }
\\ &= -\frac{11}{4500} \tbrak{ \tpar{\frac {0.2} n}^4 D^4_x f (\ol \eta, \ol \mu) + \tpar{\frac{2.4}n}^4 D^4_y f(\hat \eta, \hat \mu) }
\\ &= -\frac{11}{2812500 \cdot n^4} \tbrak{ D^4_x f (\ol \eta, \ol \mu) + 20736 D^4_y f(\hat \eta, \hat \mu) }
\\ &= -\frac{11}{2812500 \cdot n^4} \tbrak{ 0 + 20736 \cdot 0 }
\\ &= 0
\end{align*} E = − 180 ( d − c ) ( b − a ) [ h 4 D x 4 f ( η , μ ) + k 4 D y 4 f ( η ^ , μ ^ ) ] = − 4500 11 [ ( n 0.2 ) 4 D x 4 f ( η , μ ) + ( n 2.4 ) 4 D y 4 f ( η ^ , μ ^ ) ] = − 2812500 ⋅ n 4 11 [ D x 4 f ( η , μ ) + 20736 D y 4 f ( η ^ , μ ^ ) ] = − 2812500 ⋅ n 4 11 [ 0 + 20736 ⋅ 0 ] = 0 n , m
%% 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 }
% 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} }
% 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}
% 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, m n , m ( x ‾ , y ‾ ) ≈ ( 0.38063 , 0.38225 )
%% 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 }
% 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} }
% 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}
% 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} }
(\ol x, \ol y) \approx (0.38063,\ 0.38225) ( x , y ) ≈ ( 0.38063 , 0.38225 ) ( x ‾ , y ‾ ) ≈ ( 0.38164 , 0.38164 )
%% 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 }
% 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} }
% 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}
% 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} }
(\ol x, \ol y) \approx (0.38164,\ 0.38164) ( x , y ) ≈ ( 0.38164 , 0.38164 ) sigma = lambda x, y: exp(-(x**2 + y**2 ))
n = m = 14
xmin = 0 ; xmax = 1 ; ymin = lambda x: 0 ; ymax = lambda x: sqrt(1 - x**2 )
x_num = algo_44(
f=lambda x, y: x * sigma(x, y),
a=xmin, b=xmax, c=ymin, d=ymax, n=n, m=m,
)
x_den = algo_44(
f=lambda x, y: sigma(x, y),
a=xmin, b=xmax, c=ymin, d=ymax, n=n, m=m,
)
y_num = algo_44(
f=lambda x, y: y * sigma(x, y),
a=xmin, b=xmax, c=ymin, d=ymax, n=n, m=m,
)
y_den = algo_44(
f=lambda x, y: sigma(x, y),
a=xmin, b=xmax, c=ymin, d=ymax, n=n, m=m,
)
print ("x:" , x_num / x_den)
print ("y:" , y_num / y_den)