wpmath0000006_6

Infinity-Borel_set.html

  1. X X
  2. X X
  3. X X
  4. 𝒩 i | i < ω \left\langle\mathcal{N}_{i}|i<\omega\right\rangle
  5. 𝒩 i \mathcal{N}_{i}
  6. i th i^{\mathrm{th}}
  7. i i
  8. 𝒩 i \mathcal{N}_{i}
  9. c c
  10. A c A_{c}
  11. 0 , c \left\langle 0,c\right\rangle
  12. A c A_{c}
  13. X A c X\setminus A_{c}
  14. c \vec{c}
  15. c β c_{\beta}
  16. A c β A_{c_{\beta}}
  17. 1 , c \left\langle 1,\vec{c}\right\rangle
  18. β < α A c β \bigcup_{\beta<\alpha}A_{c_{\beta}}
  19. X X
  20. X X
  21. x A L [ S , x ] ϕ ( S , x ) x\in A\iff L[S,x]\models\phi(S,x)

Infinity_(disambiguation).html

  1. \infty

Information_flow_(information_theory).html

  1. x x
  2. y y
  3. L L
  4. H H
  5. L H L\leq H
  6. L L
  7. L L
  8. H H
  9. H H
  10. L L
  11. H H
  12. H H
  13. L L
  14. l L l\in L
  15. h H h\in H
  16. L L
  17. H H
  18. e x p : τ \;\vdash exp\;:\;\tau
  19. e x p exp
  20. τ \;\tau
  21. [ s c ] C [sc]\vdash C
  22. C C
  23. s c sc
  24. [ E 1 - 2 ] e x p : h i g h h V a r s ( e x p ) e x p : l o w [E1-2]\quad\vdash exp:high\qquad\frac{h\notin Vars(exp)}{\vdash exp\;:\;low}
  25. [ C 1 - 3 ] [ s c ] 𝐬𝐤𝐢𝐩 [ s c ] h := e x p e x p : l o w [ l o w ] l := e x p [C1-3]\quad[sc]\vdash\,\textbf{skip}\qquad[sc]\vdash h\;:=\;exp\qquad\frac{% \vdash exp\;:\;low}{[low]\vdash l\;:=\;exp}
  26. [ C 4 - 5 ] [ s c ] C 1 [ s c ] C 2 [ s c ] C 1 ; C 2 e x p : s c [ s c ] C [ s c ] 𝐰𝐡𝐢𝐥𝐞 e x p 𝐝𝐨 C [C4-5]\quad\frac{[sc]\vdash C_{1}\quad[sc]\vdash C_{2}}{[sc]\vdash C_{1}\;;\;C% _{2}}\qquad\frac{\vdash exp\;:\;sc\quad[sc]\vdash C}{[sc]\vdash\,\textbf{while% }\ exp\ \,\textbf{do}\ C}
  27. [ C 6 - 7 ] e x p : s c [ s c ] C 1 [ s c ] C 2 [ s c ] 𝐢𝐟 e x p 𝐭𝐡𝐞𝐧 C 1 𝐞𝐥𝐬𝐞 C 2 [ h i g h ] C [ l o w ] C [C6-7]\quad\frac{\vdash exp\;:\;sc\quad[sc]\vdash C_{1}\quad[sc]\vdash C_{2}}{% [sc]\vdash\,\textbf{if}\ exp\ \,\textbf{then}\ C_{1}\ \,\textbf{else}\ C_{2}}% \qquad\frac{[high]\vdash C}{[low]\vdash C}
  28. [ l o w ] 𝐢𝐟 l = 42 𝐭𝐡𝐞𝐧 h := 3 𝐞𝐥𝐬𝐞 l := 0 [low]\vdash\ \,\textbf{if}\ l=42\ \,\textbf{then}\ h\;:=\;3\ \,\textbf{else}\ % l\;:=\;0
  29. l := 0 ; 𝐰𝐡𝐢𝐥𝐞 l < h 𝐝𝐨 l := l + 1 l\;:=\;0\ ;\ \,\textbf{while}\ l<h\ \,\textbf{do}\ l\;:=\;l+1
  30. h h
  31. l l
  32. \prec

Information_gain_in_decision_trees.html

  1. H H
  2. I G ( T , a ) = H ( T ) - H ( T | a ) IG(T,a)=H(T)-H(T|a)
  3. T T
  4. ( 𝐱 , y ) = ( x 1 , x 2 , x 3 , , x k , y ) (\,\textbf{x},y)=(x_{1},x_{2},x_{3},...,x_{k},y)
  5. x a v a l s ( a ) x_{a}\in vals(a)
  6. a a
  7. 𝐱 \,\textbf{x}
  8. y y
  9. a a
  10. H ( ) H()
  11. I G ( T , a ) = H ( T ) - v v a l s ( a ) | { 𝐱 T | x a = v } | | T | H ( { 𝐱 T | x a = v } ) IG(T,a)=H(T)-\sum_{v\in vals(a)}\frac{|\{\,\textbf{x}\in T|x_{a}=v\}|}{|T|}% \cdot H(\{\,\textbf{x}\in T|x_{a}=v\})

Infraparticle.html

  1. Q = d 3 x ρ ( x ) Q=\int d^{3}x\rho(\vec{x})
  2. S 2 J d S \oint_{S^{2}}\vec{J}\cdot d\vec{S}
  3. δ ψ ( x ) = i q α ( x ) ψ ( x ) \delta\psi(\vec{x})=iq\alpha(\vec{x})\psi(\vec{x})
  4. d 3 x [ α ( x ) ρ ( x ) + ϵ 0 E ( x ) α ( x ) ] \int d^{3}x\left[\alpha(\vec{x})\rho(\vec{x})+\epsilon_{0}\vec{E}(\vec{x})% \cdot\nabla\alpha(\vec{x})\right]
  5. E \vec{E}
  6. S 2 α E d S + d 3 x α [ ρ - ϵ 0 E ] . \oint_{S^{2}}\alpha\vec{E}\cdot d\vec{S}+\int d^{3}x\alpha\left[\rho-\epsilon_% {0}\nabla\cdot\vec{E}\right].
  7. lim r ϵ 0 r 2 E r ( r , θ , ϕ ) \lim_{r\rightarrow\infty}\epsilon_{0}r^{2}E_{r}(r,\theta,\phi)
  8. t = - t=-\infty
  9. t = t=\infty

Inseparable_differential_equation.html

  1. d y d x + p ( x ) y = q ( x ) \frac{dy}{dx}+p(x)y=q(x)
  2. μ = e p ( x ) d x \mu=e^{\int p(x)dx}
  3. d μ d x = ( e p ( x ) d x ) d d x ( p ( x ) d x ) \frac{d\mu}{dx}=(e^{\int p(x)dx})\frac{d}{dx}(\int p(x)dx)
  4. d μ d x = μ p ( x ) \frac{d\mu}{dx}=\mu p(x)
  5. μ d y d x + μ p ( x ) y = μ q ( x ) \mu\frac{dy}{dx}+\mu p(x)y=\mu q(x)
  6. μ d y d x + y d μ d x = μ q ( x ) \mu\frac{dy}{dx}+y\frac{d\mu}{dx}=\mu q(x)
  7. d d x ( μ y ) = μ q ( x ) \frac{d}{dx}(\mu y)=\mu q(x)
  8. μ y = μ q ( x ) d x \mu y=\int\mu q(x)dx
  9. y = μ q ( x ) d x μ y=\frac{\int\mu q(x)dx}{\mu}
  10. d y d x = x + y \frac{dy}{dx}=x+y
  11. d y d x - y = x \frac{dy}{dx}-y=x
  12. p ( x ) = - 1 p(x)=-1
  13. q ( x ) = x q(x)=x
  14. d y d x + p ( x ) y = q ( x ) \frac{dy}{dx}+p(x)y=q(x)
  15. y = μ q ( x ) d x μ . y=\frac{\int\mu q(x)dx}{\mu}.
  16. μ = e p ( x ) d x = e - 1 d x = e - x \mu=e^{\int p(x)dx}=e^{\int-1dx}=e^{-x}
  17. y = x e - x e - x y=\frac{\int xe^{-x}}{e^{-x}}
  18. y = e x ( - x e - x - e - x + C ) y=e^{x}(-xe^{-x}-e^{-x}+C)
  19. y = C e x - x - 1 y=Ce^{x}-x-1
  20. 2 y ′′ + 3 y + y = 5. 2y^{\prime\prime}+3y^{\prime}+y=5.
  21. { f } = s { f } - f ( 0 ) \mathcal{L}\{f^{\prime}\}=s\mathcal{L}\{f\}-f(0)
  22. { f ′′ } = s 2 { f } - s f ( 0 ) - f ( 0 ) \mathcal{L}\{f^{\prime\prime}\}=s^{2}\mathcal{L}\{f\}-sf(0)-f^{\prime}(0)
  23. { f ( n ) } = s n { f } - s n - 1 f ( 0 ) - - f ( n - 1 ) ( 0 ) . \mathcal{L}\left\{f^{(n)}\right\}=s^{n}\mathcal{L}\{f\}-s^{n-1}f(0)-\cdots-f^{% (n-1)}(0).
  24. y ( 0 ) = 0 y(0)=0
  25. y ( 0 ) = 0. y^{\prime}(0)=0.
  26. 2 ( s 2 Y - s 0 - 0 ) + 3 ( s Y - 0 ) + Y = 5 s . 2(s^{2}Y-s\cdot 0-0)+3(sY-0)+Y=\frac{5}{s}.
  27. ( 2 s + 1 ) ( s + 1 ) Y = 5 s (2s+1)(s+1)Y=\frac{5}{s}
  28. Y = 5 s ( 2 s + 1 ) ( s + 1 ) . Y=\frac{5}{s(2s+1)(s+1)}.

Instantaneous_phase.html

  1. ϕ ( t ) = arg [ s ( t ) ] , \phi(t)=\arg[s(t)],
  2. ϕ ( t ) = arg [ s a ( t ) ] . \phi(t)=\arg[s_{\mathrm{a}}(t)].
  3. s ( t ) = A cos ( ω t + θ ) , s(t)=A\cos(\omega t+\theta),
  4. s a ( t ) = A e j ( ω t + θ ) , s_{\mathrm{a}}(t)=Ae^{j(\omega t+\theta)},
  5. ϕ ( t ) = ω t + θ . \phi(t)=\omega t+\theta.
  6. s ( t ) = A sin ( ω t ) = A cos ( ω t - π / 2 ) , s(t)=A\sin(\omega t)=A\cos(\omega t-\pi/2),
  7. s a ( t ) = A e j ( ω t - π / 2 ) , s_{\mathrm{a}}(t)=Ae^{j(\omega t-\pi/2)},
  8. ϕ ( t ) = ω t - π / 2. \phi(t)=\omega t-\pi/2.
  9. ω ( t ) = d ϕ d t ( t ) , \omega(t)=\frac{d\phi}{dt}(t),
  10. f ( t ) = 1 2 π d ϕ d t ( t ) = ω ( t ) 2 π . f(t)=\frac{1}{2\pi}\frac{d\phi}{dt}(t)=\frac{\omega(t)}{2\pi}.
  11. ϕ ( t ) \displaystyle\phi(t)
  12. ϕ ( n T ) = ϕ [ ( n - 1 ) T ] + 2 π T f ( n T ) = ϕ [ ( n - 1 ) T ] + arg [ s a ( n T ) ] - arg [ s a ( ( n - 1 ) T ) ] Δ ϕ ( n T ) . \phi(nT)=\phi[(n-1)T]+2\pi Tf(nT)=\phi[(n-1)T]+\underbrace{\arg[s_{\mathrm{a}}% (nT)]-\arg[s_{\mathrm{a}}((n-1)T)]}_{\Delta\phi(nT)}.
  13. ϕ ( n T ) = ϕ [ ( n - 1 ) T ] + arg [ s a ( n T ) s a * ( ( n - 1 ) T ) ] , \phi(nT)=\phi[(n-1)T]+\arg[s_{\mathrm{a}}(nT)\,s_{\mathrm{a}}^{*}((n-1)T)],
  14. e i ϕ ( t ) \displaystyle e^{i\phi(t)}

Institution_(computer_science).html

  1. S i g n Sign
  2. s e n : S i g n sen:Sign\to
  3. Σ \Sigma
  4. s e n ( Σ ) sen(\Sigma)
  5. σ : Σ Σ \sigma:\Sigma\to\Sigma^{\prime}
  6. s e n ( σ ) : s e n ( Σ ) s e n ( Σ ) sen(\sigma):sen(\Sigma)\to sen(\Sigma^{\prime})
  7. s e n ( σ ) ( φ ) sen(\sigma)(\varphi)
  8. σ ( φ ) \sigma(\varphi)
  9. M o d : S i g n o p Mod:Sign^{op}\to
  10. Σ \Sigma
  11. M o d ( Σ ) Mod(\Sigma)
  12. σ : Σ Σ \sigma:\Sigma\to\Sigma^{\prime}
  13. M o d ( σ ) : M o d ( Σ ) M o d ( Σ ) Mod(\sigma):Mod(\Sigma^{\prime})\to Mod(\Sigma)
  14. M o d ( σ ) ( M ) Mod(\sigma)(M^{\prime})
  15. M | σ M^{\prime}|_{\sigma}
  16. Σ | M o d ( Σ ) | × s e n ( Σ ) {\models_{\Sigma}}\subseteq|{Mod(\Sigma)|\times sen(\Sigma)}
  17. Σ S i g n \Sigma\in Sign
  18. σ : Σ Σ \sigma:\Sigma\to\Sigma^{\prime}
  19. S i g n Sign
  20. M Σ σ ( φ ) M^{\prime}\models_{\Sigma^{\prime}}\sigma(\varphi)
  21. M | σ Σ φ M^{\prime}|_{\sigma}\models_{\Sigma}\varphi
  22. M M o d ( Σ ) M^{\prime}\in Mod(\Sigma^{\prime})
  23. φ s e n ( Σ ) \varphi\in sen(\Sigma)

Integrable_system.html

  1. 2 n 2n
  2. n n
  3. 1 1

Integral_symbol.html

  1. \displaystyle\int
  2. \iint
  3. \iiint
  4. \oint
  5. \oiint \oiint
  6. \oiiint \oiiint
  7. 0 T f ( t ) d t \int_{0}^{T}f(t)\;dt
  8. 0 T f ( t ) d t \int\limits_{0}^{T}f(t)\;\mathrm{d}t

Intelligent_agent.html

  1. f : P A f:P^{\ast}\rightarrow A

Intercept_method.html

  1. s i z e = l a r g e s i n ( H c ) = s i n ( l a t ) · s i n ( d e c ) + c o s ( l a t ) · c o s ( d e c ) · c o s ( L H A ) size=largesin(Hc)=sin(lat)·sin(dec)+cos(lat)·cos(dec)·cos(LHA)
  2. tan ( Zn ) = sin ( LHA ) sin ( lat ) cos ( LHA ) - cos ( lat ) tan ( dec ) \mathrm{tan(Zn)=\frac{sin(LHA)}{sin(lat)\cdot cos(LHA)-cos(lat)\cdot tan(dec)}}
  3. cos ( Zn ) = sin ( dec ) - sin ( lat ) sin ( Hc ) cos ( lat ) cos ( Hc ) \mathrm{cos(Zn)=\frac{sin(dec)-sin(lat)\cdot sin(Hc)}{cos(lat)\cdot cos(Hc)}}
  4. haversin ( Hc ¯ ) = haversin ( LHA ) cos ( lat ) cos ( dec ) + haversin ( lat ± dec ) \mathrm{haversin(\overline{Hc})=haversin(LHA)\cdot cos(lat)\cdot cos(dec)+% haversin(lat\pm dec)}
  5. H c ¯ \overline{Hc}
  6. H c ¯ \overline{Hc}
  7. hav ( Zn ) = cos ( lat - Hc ) - sin ( dec ) 2 cos ( lat ) cos ( Hc ) \mathrm{hav(Zn)=\frac{cos(lat-Hc)-sin(dec)}{2\cdot cos(lat)\cdot cos(Hc)}}

Interest_rate_parity.html

  1. ( 1 + i $ ) = E t ( S t + k ) S t ( 1 + i c ) (1+i_{\$})=\frac{E_{t}(S_{t+k})}{S_{t}}(1+i_{c})
  2. E t ( S t + k ) E_{t}(S_{t+k})
  3. 1 + i $ 1+i_{\$}
  4. E t ( S t + k ) S t ( 1 + i c ) \frac{E_{t}(S_{t+k})}{S_{t}}(1+i_{c})
  5. i $ = i c + Δ E t ( S t + k ) S t i_{\$}=i_{c}+\frac{{\Delta}E_{t}(S_{t+k})}{S_{t}}
  6. Δ E t ( S t + k ) {{\Delta}E_{t}(S_{t+k})}
  7. Δ E t ( S t + k ) / S t {{\Delta}E_{t}(S_{t+k})}/{S_{t}}
  8. ( 1 + i $ ) = F t S t ( 1 + i c ) (1+i_{\$})=\frac{F_{t}}{S_{t}}(1+i_{c})
  9. F t F_{t}
  10. 1 + i $ 1+i_{\$}
  11. F t S t ( 1 + i c ) \frac{F_{t}}{S_{t}}(1+i_{c})
  12. U I R P : ( 1 + i $ ) = E t ( S t + k ) S t ( 1 + i c ) UIRP:(1+i_{\$})=\frac{E_{t}(S_{t+k})}{S_{t}}(1+i_{c})
  13. C I R P : ( 1 + i $ ) = F t S t ( 1 + i c ) CIRP:(1+i_{\$})=\frac{F_{t}}{S_{t}}(1+i_{c})
  14. 1 = E t ( S t + k ) F t 1=\frac{E_{t}(S_{t+k})}{F_{t}}
  15. F t = E t ( S t + k ) {F_{t}}=E_{t}(S_{t+k})
  16. U I R P : Δ E t ( S t + k ) = E t ( S t + k ) - S t = i $ - i c UIRP:{\Delta}E_{t}(S_{t+k})=E_{t}(S_{t+k})-S_{t}=i_{\$}-i_{c}
  17. P P P : Δ E t ( S t + k ) = Δ E t ( p $ t + k ) - Δ E t ( p c t + k ) PPP:{\Delta}E_{t}(S_{t+k})={\Delta}E_{t}({p^{\$}}_{t+k})-{\Delta}E_{t}({p^{c}}% _{t+k})
  18. p p
  19. R I R P : i $ - Δ E t ( p $ t + k ) = i c - Δ E t ( p c t + k ) RIRP:i_{\$}-{\Delta}E_{t}({p^{\$}}_{t+k})=i_{c}-{\Delta}E_{t}({p^{c}}_{t+k})

Interior_product.html

  1. ι X : Ω p ( M ) Ω p - 1 ( M ) \iota_{X}\colon\Omega^{p}(M)\to\Omega^{p-1}(M)
  2. ( ι X ω ) ( X 1 , , X p - 1 ) = ω ( X , X 1 , , X p - 1 ) (\iota_{X}\omega)(X_{1},\ldots,X_{p-1})=\omega(X,X_{1},\ldots,X_{p-1})
  3. ι X α = α ( X ) = α , X \displaystyle\iota_{X}\alpha=\alpha(X)=\langle\alpha,X\rangle
  4. ι X ( β γ ) = ( ι X β ) γ + ( - 1 ) p β ( ι X γ ) . \iota_{X}(\beta\wedge\gamma)=(\iota_{X}\beta)\wedge\gamma+(-1)^{p}\beta\wedge(% \iota_{X}\gamma).
  5. ι X ι Y ω = - ι Y ι X ω \iota_{X}\iota_{Y}\omega=-\iota_{Y}\iota_{X}\omega
  6. ι X 2 = 0 \iota_{X}^{2}=0
  7. X ω = d ( ι X ω ) + ι X d ω . \mathcal{L}_{X}\omega=\mathrm{d}(\iota_{X}\omega)+\iota_{X}\mathrm{d}\omega.
  8. X X
  9. Y Y
  10. ι [ X , Y ] = [ X , ι Y ] . \iota_{[X,Y]}=\left[\mathcal{L}_{X},\iota_{Y}\right].

Interleave_sequence.html

  1. S S
  2. ( x i ) (x_{i})
  3. ( y i ) (y_{i})
  4. i = 0 , 1 , 2 , , i=0,1,2,...,
  5. S . S.
  6. x 0 , y 0 , x 1 , y 1 , . x_{0},y_{0},x_{1},y_{1},\dots.
  7. ( z i ) , i = 0 , 1 , 2 , (z_{i}),i=0,1,2,...
  8. z i := { x k if i = 2 k is even, y k if i = 2 k + 1 is odd. z_{i}:=\left\{\begin{matrix}x_{k}&\mbox{ if }~{}i=2k\mbox{ is even,}\\ y_{k}&\mbox{ if }~{}i=2k+1\mbox{ is odd.}\end{matrix}\right.
  9. ( z i ) (z_{i})
  10. ( x i ) (x_{i})
  11. ( y i ) (y_{i})

Intermediate_Jacobian.html

  1. H n ( M , R ) C = H n , 0 ( M ) H 0 , n ( M ) . H^{n}(M,{R})\otimes{C}=H^{n,0}(M)\oplus\cdots\oplus H^{0,n}(M).\,

Internal_conversion_coefficient.html

  1. α = number of de-excitations via electron emission number of de-excitations via gamma-ray emission \alpha=\frac{\mbox{number of de-excitations via electron emission}~{}}{\mbox{% number of de-excitations via gamma-ray emission}~{}}

Internal_set.html

  1. u n \langle u_{n}\rangle
  2. A n \langle A_{n}\rangle
  3. [ u n ] [u_{n}]
  4. [ A n ] * [A_{n}]\subset\;^{*}\!{\mathbb{R}}
  5. u n A n u_{n}\in A_{n}
  6. 𝒫 * ( ) {}^{*}\mathcal{P}(\mathbb{R})
  7. 𝒫 ( ) \mathcal{P}(\mathbb{R})
  8. \mathbb{R}

International_Rule_(sailing).html

  1. R metres = L + B + 1 / 2 G + 3 d + 1 / 3 S - F 2 R\mbox{ metres}~{}=\frac{L+B+1/2G+3d+1/3\sqrt{S}-F}{2}
  2. L L
  3. B B
  4. G G
  5. d d
  6. S S
  7. F F
  8. R metres = L + 0.25 G + 2 d + S - F 2.5 R\mbox{ metres}~{}=\frac{L+0.25G+2d+\sqrt{S}-F}{2.5}
  9. L L
  10. G G
  11. d d
  12. S S
  13. F F
  14. R metres = L + 2 d + S - F 2.37 R\mbox{ metres}~{}=\frac{L+2d+\sqrt{S}-F}{2.37}
  15. L L
  16. d d
  17. S S
  18. F F
  19. {}^{†}
  20. {}^{†}
  21. {}^{†}

Internet_Movie_Database.html

  1. W = R v + C m v + m W=\frac{Rv+Cm}{v+m}
  2. W W
  3. R R
  4. v v
  5. m m
  6. C C
  7. W W

Intersection_theory.html

  1. M M
  2. 2 n 2n
  3. n n
  4. M M MM
  5. λ M : H n ( M , M ) × H n ( M , M ) 𝐙 \lambda_{M}\colon H^{n}(M,\partial M)\times H^{n}(M,\partial M)\to\mathbf{Z}
  6. λ M ( a , b ) = a b , [ M ] 𝐙 \lambda_{M}(a,b)=\langle a\smile b,[M]\rangle\in\mathbf{Z}
  7. λ M ( a , b ) = ( - 1 ) n λ M ( b , a ) 𝐙 . \lambda_{M}(a,b)=(-1)^{n}\lambda_{M}(b,a)\in\mathbf{Z}.
  8. n n
  9. 2 n = 4 k 2n=4k
  10. M M
  11. n n
  12. 2 n = 4 k + 2 2n=4k+2
  13. ε ε
  14. 𝐙 / 2 𝐙 \mathbf{Z}/2\mathbf{Z}
  15. n n
  16. A A
  17. B B
  18. a a
  19. b b
  20. A A
  21. B B
  22. A A
  23. B B
  24. A A
  25. B B
  26. X X
  27. A · B A·B
  28. A B A∩B
  29. A A
  30. B B
  31. X X
  32. d i m ( A B ) = d i m A + d i m B d i m X dim(A∩B)=dimA+dimB−dimX
  33. A · B A·B
  34. A B A∩B
  35. A = B A=B
  36. A · B A·B
  37. A A
  38. X X
  39. V V
  40. W W
  41. V W V∩W
  42. V · W V·W
  43. V V
  44. W W
  45. V W V∩W
  46. V V
  47. W W
  48. V V
  49. W W
  50. V V′
  51. W W′
  52. V W V′∩W′
  53. V V′′
  54. W W′′
  55. V W V′∩W′
  56. V W V′′∩W′′
  57. r r
  58. X X
  59. f f
  60. ( r + 1 ) (r+1)
  61. Y Y
  62. k ( Y ) k(Y)
  63. y = 0 y=0
  64. 2 · ( 0 , 0 ) 2·(0,0)
  65. ( 0 , 0 ) (0,0)
  66. y = 3 y=−3
  67. X X
  68. V V
  69. W W
  70. I I
  71. J J
  72. X X
  73. Z Z
  74. V W V∩W
  75. z z
  76. Z Z
  77. V · W V·W
  78. μ ( Z ; V , W ) := i = 0 ( - 1 ) i length 𝒪 X , z Tor 𝒪 X , z i ( 𝒪 X , z / I , 𝒪 X , z / J ) \mu(Z;V,W):=\sum^{\infty}_{i=0}(-1)^{i}\,\text{length}_{\mathcal{O}_{X,z}}\,% \text{Tor}^{i}_{\mathcal{O}_{X,z}}(\mathcal{O}_{X,z}/I,\mathcal{O}_{X,z}/J)
  79. X X
  80. z z
  81. ( 𝒪 X , z / I ) 𝒪 X , z ( 𝒪 X , z / J ) = 𝒪 Z , z \left(\mathcal{O}_{X,z}/I\right)\otimes_{\mathcal{O}_{X,z}}\left(\mathcal{O}_{% X,z}/J\right)=\mathcal{O}_{Z,z}
  82. 𝒪 X , z \mathcal{O}_{X,z}
  83. V V
  84. W W
  85. μ ( Z ; V , W ) = μ ( Z ; W , V ) μ(Z;V,W)=μ(Z;W,V)
  86. V W := i μ ( Z i ; V , W ) Z i V\cdot W:=\sum_{i}\mu(Z_{i};V,W)Z_{i}
  87. V W = ∪︀ Z < s u b > i V∩W=∪︀Z<sub>i

Interval_arithmetic.html

  1. T * S = { x | there is some y in T , and some z in S , such that x = y * z } T*S=\left\{x|\,\text{there is some }y\,\text{ in }T,\,\text{ and some }z\,% \text{ in }S,\,\text{ such that }x=y*z\right\}
  2. [ a , b ] [a,b]
  3. [ c , d ] [c,d]
  4. ( - , ) (-\infty,\infty)
  5. [ a , b ] + [ c , d ] = [ a + c , b + d ] [a,b]+[c,d]=[a+c,b+d]
  6. [ a , b ] - [ c , d ] = [ a - d , b - c ] [a,b]-[c,d]=[a-d,b-c]
  7. [ a , b ] [ c , d ] = [ min ( a c , a d , b c , b d ) , max ( a c , a d , b c , b d ) ] [a,b]\cdot[c,d]=\left[\min(ac,ad,bc,bd),\max(ac,ad,bc,bd)\right]
  8. [ a , b ] [ c , d ] = [ min ( a c , a d , b c , b d ) , max ( a c , a d , b c , b d ) ] when 0 is not in [ c , d ] . \frac{[a,b]}{[c,d]}=\left[\min\left(\frac{a}{c},\frac{a}{d},\frac{b}{c},\frac{% b}{d}\right),\max\left(\frac{a}{c},\frac{a}{d},\frac{b}{c},\frac{b}{d}\right)% \right]\,\text{ when }0\,\text{ is not in }[c,d].
  9. X ( Y + Z ) X(Y+Z)
  10. X Y + X Z XY+XZ
  11. x x
  12. [ a , b ] [a,b]
  13. x x
  14. x x
  15. a a
  16. b b
  17. f f
  18. x x
  19. f f
  20. [ c , d ] [c,d]
  21. f ( x ) f(x)
  22. x [ a , b ] x\in[a,b]
  23. [ a , b ] = { x | a x b } , [a,b]=\{x\in\mathbb{R}\,|\,a\leq x\leq b\},
  24. a = - a={-\infty}
  25. b = b={\infty}
  26. [ 79.5 ; 80.5 ] / ( [ 1.795 ; 1.805 ] ) 2 = [ 24.4 ; 25.0 ] . [79{.}5;80{.}5]/([1{.}795;1{.}805])^{2}=[24{.}4;25{.}0].
  27. op {\langle\!\mathrm{op}\!\rangle}
  28. [ x 1 , x 2 ] op [ y 1 , y 2 ] = { x op y | x [ x 1 , x 2 ] and y [ y 1 , y 2 ] } [x_{1},x_{2}]{\,\langle\!\mathrm{op}\!\rangle\,}[y_{1},y_{2}]=\{x{\,\langle\!% \mathrm{op}\!\rangle\,}y\,|\,x\in[x_{1},x_{2}]\,\mbox{and}~{}\,y\in[y_{1},y_{2% }]\}
  29. [ ] [ x 1 , x 2 ] op [ y 1 , y 2 ] = [ min ( x 1 op y 1 , x 1 op y 2 , x 2 op y 1 , x 2 op y 2 ) , max ( x 1 op y 1 , x 1 op y 2 , x 2 op y 1 , x 2 op y 2 ) ] , \begin{aligned}\displaystyle[][x_{1},x_{2}]\,\langle\!\mathrm{op}\!\rangle\,[y% _{1},y_{2}]&\displaystyle=\left[\min(x_{1}{\langle\!\mathrm{op}\!\rangle}y_{1}% ,x_{1}\langle\!\mathrm{op}\!\rangle y_{2},x_{2}\langle\!\mathrm{op}\!\rangle y% _{1},x_{2}\langle\!\mathrm{op}\!\rangle y_{2}),\right.\\ &\displaystyle{}\qquad\left.\;\max(x_{1}{\langle\!\mathrm{op}\!\rangle}y_{1},x% _{1}{\langle\!\mathrm{op}\!\rangle}y_{2},x_{2}{\langle\!\mathrm{op}\!\rangle}y% _{1},x_{2}{\langle\!\mathrm{op}\!\rangle}y_{2})\right]\,\mathrm{,}\end{aligned}
  30. x op y x{\,\langle\!\mathrm{op}\!\rangle\,}y
  31. x [ x 1 , x 2 ] x\in[x_{1},x_{2}]
  32. y [ y 1 , y 2 ] y\in[y_{1},y_{2}]
  33. [ x 1 , x 2 ] + [ y 1 , y 2 ] = [ x 1 + y 1 , x 2 + y 2 ] [x_{1},x_{2}]+[y_{1},y_{2}]=[x_{1}+y_{1},x_{2}+y_{2}]
  34. [ x 1 , x 2 ] - [ y 1 , y 2 ] = [ x 1 - y 2 , x 2 - y 1 ] [x_{1},x_{2}]-[y_{1},y_{2}]=[x_{1}-y_{2},x_{2}-y_{1}]
  35. [ x 1 , x 2 ] [ y 1 , y 2 ] = [ min ( x 1 y 1 , x 1 y 2 , x 2 y 1 , x 2 y 2 ) , max ( x 1 y 1 , x 1 y 2 , x 2 y 1 , x 2 y 2 ) ] [x_{1},x_{2}]\cdot[y_{1},y_{2}]=[\min(x_{1}y_{1},x_{1}y_{2},x_{2}y_{1},x_{2}y_% {2}),\max(x_{1}y_{1},x_{1}y_{2},x_{2}y_{1},x_{2}y_{2})]
  36. [ x 1 , x 2 ] / [ y 1 , y 2 ] = [ x 1 , x 2 ] ( 1 / [ y 1 , y 2 ] ) [x_{1},x_{2}]/[y_{1},y_{2}]=[x_{1},x_{2}]\cdot(1/[y_{1},y_{2}])
  37. 1 / [ y 1 , y 2 ] = [ 1 / y 2 , 1 / y 1 ] 1/[y_{1},y_{2}]=[1/y_{2},1/y_{1}]
  38. 0 [ y 1 , y 2 ] 0\notin[y_{1},y_{2}]
  39. 1 / [ y 1 , 0 ] = [ - , 1 / y 1 ] 1/[y_{1},0]=[-\infty,1/y_{1}]
  40. 1 / [ 0 , y 2 ] = [ 1 / y 2 , ] 1/[0,y_{2}]=[1/y_{2},\infty]
  41. y 1 < 0 < y 2 y_{1}<0<y_{2}
  42. 1 / [ y 1 , y 2 ] = [ - , 1 / y 1 ] [ 1 / y 2 , ] 1/[y_{1},y_{2}]=[-\infty,1/y_{1}]\cup[1/y_{2},\infty]
  43. 1 / [ y 1 , y 2 ] = [ - , ] 1/[y_{1},y_{2}]=[-\infty,\infty]
  44. ( 1 / y 1 , 1 / y 2 ) (1/y_{1},1/y_{2})
  45. [ - , 1 / y 1 ] [-\infty,1/y_{1}]
  46. [ 1 / y 2 , ] [1/y_{2},\infty]
  47. i = 1 l [ x i 1 , x i 2 ] \textstyle\bigcup_{i=1}^{l}[x_{i1},x_{i2}]
  48. r r\in\mathbb{R}
  49. [ r , r ] [r,r]
  50. f ( a , b , x ) = a x + b f(a,b,x)=a\cdot x+b
  51. a = [ 1 , 2 ] a=[1,2]
  52. b = [ 5 , 7 ] b=[5,7]
  53. x = [ 2 , 3 ] x=[2,3]
  54. f ( a , b , x ) = ( [ 1 , 2 ] [ 2 , 3 ] ) + [ 5 , 7 ] = [ 1 2 , 2 3 ] + [ 5 , 7 ] = [ 7 , 13 ] f(a,b,x)=([1,2]\cdot[2,3])+[5,7]=[1\cdot 2,2\cdot 3]+[5,7]=[7,13]
  55. f ( a , b , x ) f(a,b,x)
  56. x x
  57. a a
  58. b b
  59. f ( [ 1 , 2 ] , [ 5 , 7 ] , x ) = ( [ 1 , 2 ] x ) + [ 5 , 7 ] = 0 [ 1 , 2 ] x = [ - 7 , - 5 ] x = [ - 7 , - 5 ] / [ 1 , 2 ] , f([1,2],[5,7],x)=([1,2]\cdot x)+[5,7]=0\Leftrightarrow[1,2]\cdot x=[-7,-5]% \Leftrightarrow x=[-7,-5]/[1,2],
  60. [ - 7 , - 2.5 ] [-7,{-2.5}]
  61. [ x 1 , x 2 ] [ y 1 , y 2 ] = [ x 1 y 1 , x 2 y 2 ] , if x 1 , y 1 0. [x_{1},x_{2}]\cdot[y_{1},y_{2}]=[x_{1}\cdot y_{1},x_{2}\cdot y_{2}],\,\text{ % if }x_{1},y_{1}\geq 0.
  62. [ - , ] [-\infty,\infty]
  63. 0 0\cdot\infty
  64. [ x ] [ x 1 , x 2 ] [x]\equiv[x_{1},x_{2}]
  65. [ ] := { [ x 1 , x 2 ] | x 1 x 2 and x 1 , x 2 { - , } } [\mathbb{R}]:=\big\{\,[x_{1},x_{2}]\,|\,x_{1}\leq x_{2}\,\text{ and }x_{1},x_{% 2}\in\mathbb{R}\cup\{-\infty,\infty\}\big\}
  66. ( [ x ] 1 , , [ x ] n ) [ ] n \big([x]_{1},\ldots,[x]_{n}\big)\in[\mathbb{R}]^{n}
  67. [ 𝐱 ] [\mathbf{x}]
  68. [ x ] [x]
  69. [ x 1 , x 1 ] [x_{1},x_{1}]
  70. f : f:\mathbb{R}\rightarrow\mathbb{R}
  71. [ x 1 , x 2 ] [x_{1},x_{2}]
  72. y 1 , y 2 [ x 1 , x 2 ] y_{1},y_{2}\in[x_{1},x_{2}]
  73. y 1 y 2 y_{1}\leq y_{2}
  74. f ( y 1 ) f ( y 2 ) f(y_{1})\leq f(y_{2})
  75. f ( y 1 ) f ( y 2 ) f(y_{1})\geq f(y_{2})
  76. [ y 1 , y 2 ] [ x 1 , x 2 ] [y_{1},y_{2}]\subseteq[x_{1},x_{2}]
  77. y 1 y_{1}
  78. y 2 y_{2}
  79. f ( [ y 1 , y 2 ] ) = [ min { f ( y 1 ) , f ( y 2 ) } , max { f ( y 1 ) , f ( y 2 ) } ] f([y_{1},y_{2}])=\left[\min\big\{f(y_{1}),f(y_{2})\big\},\max\big\{f(y_{1}),f(% y_{2})\big\}\right]
  80. a [ x 1 , x 2 ] = [ a x 1 , a x 2 ] a^{[x_{1},x_{2}]}=[a^{x_{1}},a^{x_{2}}]
  81. a > 1 a>1
  82. log a ( [ x 1 , x 2 ] ) = [ log a x 1 , log a x 2 ] \log_{a}\big({[x_{1},x_{2}]}\big)=[\log_{a}{x_{1}},\log_{a}{x_{2}}]
  83. [ x 1 , x 2 ] [x_{1},x_{2}]
  84. a > 1 a>1
  85. [ x 1 , x 2 ] n = [ x 1 n , x 2 n ] {[x_{1},x_{2}]}^{n}=[{x_{1}}^{n},{x_{2}}^{n}]
  86. n n\in\mathbb{N}
  87. x n x^{n}
  88. x [ - 1 , 1 ] x\in[-1,1]
  89. [ 0 , 1 ] [0,1]
  90. n = 2 , 4 , 6 , n=2,4,6,\ldots
  91. [ - 1 , 1 ] n [-1,1]^{n}
  92. [ - 1 , 1 ] [ - 1 , 1 ] [-1,1]\cdot\ldots\cdot[-1,1]
  93. [ - 1 , 1 ] [-1,1]
  94. x n x^{n}
  95. x < 0 x<0
  96. x > 0 x>0
  97. n n\in\mathbb{N}
  98. [ x 1 , x 2 ] n = [ x 1 n , x 2 n ] {[x_{1},x_{2}]}^{n}=[x_{1}^{n},x_{2}^{n}]
  99. x 1 0 x_{1}\geq 0
  100. [ x 1 , x 2 ] n = [ x 2 n , x 1 n ] {[x_{1},x_{2}]}^{n}=[x_{2}^{n},x_{1}^{n}]
  101. x 2 < 0 x_{2}<0
  102. [ x 1 , x 2 ] n = [ 0 , max { x 1 n , x 2 n } ] {[x_{1},x_{2}]}^{n}=[0,\max\{x_{1}^{n},x_{2}^{n}\}]
  103. x 1 , x 2 x_{1},x_{2}
  104. [ x 1 , x 2 ] [x_{1},x_{2}]
  105. ( / 1 + 2 n ) π \left({}^{1}\!\!/\!{}_{2}+{n}\right)\cdot\pi
  106. n π {n}\cdot\pi
  107. n n\in\mathbb{Z}
  108. [ - 1 , 1 ] [-1,1]
  109. f : n f:\mathbb{R}^{n}\rightarrow\mathbb{R}
  110. [ f ] : [ ] n [ ] [f]:[\mathbb{R}]^{n}\rightarrow[\mathbb{R}]
  111. f f
  112. [ f ] ( [ 𝐱 ] ) { f ( 𝐲 ) | 𝐲 [ 𝐱 ] } [f]([\mathbf{x}])\supseteq\{f(\mathbf{y})|\mathbf{y}\in[\mathbf{x}]\}
  113. [ f ] ( [ x 1 , x 2 ] ) = [ e x 1 , e x 2 ] [f]([x_{1},x_{2}])=[e^{x_{1}},e^{x_{2}}]
  114. [ g ] ( [ x 1 , x 2 ] ) = [ - , ] [g]([x_{1},x_{2}])=[{-\infty},{\infty}]
  115. [ f ] [f]
  116. f ( x 1 , , x n ) f(x_{1},\cdots,x_{n})
  117. k k
  118. k + 1 k+1
  119. f f
  120. [ f ] ( [ 𝐱 ] ) := f ( 𝐲 ) + i = 1 k 1 i ! D i f ( 𝐲 ) ( [ 𝐱 ] - 𝐲 ) i + [ r ] ( [ 𝐱 ] , [ 𝐱 ] , 𝐲 ) [f]([\mathbf{x}]):=f(\mathbf{y})+\sum_{i=1}^{k}\frac{1}{i!}\mathrm{D}^{i}f(% \mathbf{y})\cdot([\mathbf{x}]-\mathbf{y})^{i}+[r]([\mathbf{x}],[\mathbf{x}],% \mathbf{y})
  121. 𝐲 [ 𝐱 ] \mathbf{y}\in[\mathbf{x}]
  122. D i f ( 𝐲 ) \mathrm{D}^{i}f(\mathbf{y})
  123. i i
  124. f f
  125. 𝐲 \mathbf{y}
  126. [ r ] [r]
  127. r ( 𝐱 , ξ , 𝐲 ) = 1 ( k + 1 ) ! D k + 1 f ( ξ ) ( 𝐱 - 𝐲 ) k + 1 . r(\mathbf{x},\xi,\mathbf{y})=\frac{1}{(k+1)!}\mathrm{D}^{k+1}f(\xi)\cdot(% \mathbf{x}-\mathbf{y})^{k+1}.
  128. ξ \xi
  129. 𝐱 \mathbf{x}
  130. 𝐲 \mathbf{y}
  131. 𝐱 , 𝐲 [ 𝐱 ] \mathbf{x},\mathbf{y}\in[\mathbf{x}]
  132. ξ \xi
  133. [ 𝐱 ] [\mathbf{x}]
  134. 𝐲 \mathbf{y}
  135. k = 0 k=0
  136. [ J f ] ( [ 𝐱 ] ) [J_{f}](\mathbf{[x]})
  137. [ f ] ( [ 𝐱 ] ) := f ( 𝐲 ) + [ J f ] ( [ 𝐱 ] ) ( [ 𝐱 ] - 𝐲 ) [f]([\mathbf{x}]):=f(\mathbf{y})+[J_{f}](\mathbf{[x]})\cdot([\mathbf{x}]-% \mathbf{y})
  138. f ( x , y ) = x + y f(x,y)=x+y
  139. x [ 0.1 , 0.8 ] x\in[0.1,0.8]
  140. y [ 0.06 , 0.08 ] y\in[0.06,0.08]
  141. [ 0.16 , 0.88 ] [0.16,0.88]
  142. [ 0.2 , 0.9 ] [0.2,0.9]
  143. [ 0.2 , 0.9 ] [ 0.16 , 0.88 ] [0.2,0.9]\not\supseteq[0.16,0.88]
  144. f ( [ 0.1 , 0.8 ] , [ 0.06 , 0.08 ] ) f([0.1,0.8],[0.06,0.08])
  145. [ 0.1 , 0.9 ] [0.1,0.9]
  146. [ ε 1 , ε 2 ] [\varepsilon_{1},\varepsilon_{2}]
  147. f f
  148. f ( x ) = x 2 + x f(x)=x^{2}+x
  149. [ - 1 , 1 ] [-1,1]
  150. [ - 1 / 4 , 2 ] [-1/4,2]
  151. [ - 1 , 1 ] 2 + [ - 1 , 1 ] = [ 0 , 1 ] + [ - 1 , 1 ] = [ - 1 , 2 ] [-1,1]^{2}+[-1,1]=[0,1]+[-1,1]=[-1,2]
  152. h ( x , y ) = x 2 + y h(x,y)=x^{2}+y
  153. x , y [ - 1 , 1 ] x,y\in[-1,1]
  154. f f
  155. x x
  156. f ( x ) = x 2 + x f(x)=x^{2}+x
  157. f ( x ) = ( x + 1 2 ) 2 - 1 4 f(x)=\left(x+\frac{1}{2}\right)^{2}-\frac{1}{4}
  158. ( [ - 1 , 1 ] + 1 2 ) 2 - 1 4 = [ - 1 2 , 3 2 ] 2 - 1 4 = [ 0 , 9 4 ] - 1 4 = [ - 1 4 , 2 ] \left([-1,1]+\frac{1}{2}\right)^{2}-\frac{1}{4}=\left[-\frac{1}{2},\frac{3}{2}% \right]^{2}-\frac{1}{4}=\left[0,\frac{9}{4}\right]-\frac{1}{4}=\left[-\frac{1}% {4},2\right]
  159. f f
  160. x = p y = p \begin{matrix}x&=&p\\ y&=&p\end{matrix}
  161. p [ - 1 , 1 ] p\in[-1,1]
  162. ( - 1 , - 1 ) (-1,-1)
  163. ( 1 , 1 ) (1,1)
  164. [ - 1 , 1 ] × [ - 1 , 1 ] [-1,1]\times[-1,1]
  165. [ 𝐀 ] [ ] n × m [\mathbf{A}]\in[\mathbb{R}]^{n\times m}
  166. [ 𝐛 ] [ ] n [\mathbf{b}]\in[\mathbb{R}]^{n}
  167. [ 𝐱 ] [ ] m [\mathbf{x}]\in[\mathbb{R}]^{m}
  168. 𝐱 m \mathbf{x}\in\mathbb{R}^{m}
  169. ( 𝐀 , 𝐛 ) (\mathbf{A},\mathbf{b})
  170. 𝐀 [ 𝐀 ] \mathbf{A}\in[\mathbf{A}]
  171. 𝐛 [ 𝐛 ] \mathbf{b}\in[\mathbf{b}]
  172. 𝐀 𝐱 = 𝐛 \mathbf{A}\cdot\mathbf{x}=\mathbf{b}
  173. n = m n=m
  174. [ 𝐱 ] [\mathbf{x}]
  175. [ 𝐀 ] [\mathbf{A}]
  176. [ 𝐛 ] [\mathbf{b}]
  177. [ 𝐱 ] [\mathbf{x}]
  178. i i
  179. ( [ a 11 ] [ a 1 n ] [ a n 1 ] [ a n n ] ) ( x 1 x n ) = ( [ b 1 ] [ b n ] ) \begin{pmatrix}{[a_{11}]}&\cdots&{[a_{1n}]}\\ \vdots&\ddots&\vdots\\ {[a_{n1}]}&\cdots&{[a_{nn}]}\end{pmatrix}\cdot\begin{pmatrix}{x_{1}}\\ \vdots\\ {x_{n}}\end{pmatrix}=\begin{pmatrix}{[b_{1}]}\\ \vdots\\ {[b_{n}]}\end{pmatrix}
  180. x i x_{i}
  181. 1 / [ a i i ] 1/[a_{ii}]
  182. x j [ x j ] x_{j}\in[x_{j}]
  183. x j [ b i ] - k j [ a i k ] [ x k ] [ a i i ] x_{j}\in\frac{[b_{i}]-\sum\limits_{k\not=j}[a_{ik}]\cdot[x_{k}]}{[a_{ii}]}
  184. [ x j ] [x_{j}]
  185. [ x j ] [ b i ] - k j [ a i k ] [ x k ] [ a i i ] [x_{j}]\cap\frac{[b_{i}]-\sum\limits_{k\not=j}[a_{ik}]\cdot[x_{k}]}{[a_{ii}]}
  186. [ 𝐱 ] [\mathbf{x}]
  187. [ 𝐀 ] 𝐱 = [ 𝐛 ] , [\mathbf{A}]\cdot\mathbf{x}=[\mathbf{b}]\mbox{,}~{}
  188. 𝐌 \mathbf{M}
  189. ( 𝐌 [ 𝐀 ] ) 𝐱 = 𝐌 [ 𝐛 ] (\mathbf{M}\cdot[\mathbf{A}])\cdot\mathbf{x}=\mathbf{M}\cdot[\mathbf{b}]
  190. 𝐌 = 𝐀 - 1 \mathbf{M}=\mathbf{A}^{-1}
  191. 𝐀 [ 𝐀 ] \mathbf{A}\in[\mathbf{A}]
  192. 𝐌 [ 𝐀 ] \mathbf{M}\cdot[\mathbf{A}]
  193. 𝐀 [ 𝐀 ] \mathbf{A}\in[\mathbf{A}]
  194. n × n n\times n
  195. 2 n 2 2^{n^{2}}
  196. 2 n 2^{n}
  197. [ 𝐱 ] [\mathbf{x}]
  198. 𝐳 [ 𝐱 ] \mathbf{z}\in[\mathbf{x}]
  199. 𝐲 [ 𝐱 ] \mathbf{y}\in[\mathbf{x}]
  200. f ( 𝐳 ) f ( 𝐲 ) + [ J f ] ( [ 𝐱 ] ) ( 𝐳 - 𝐲 ) f(\mathbf{z})\in f(\mathbf{y})+[J_{f}](\mathbf{[x]})\cdot(\mathbf{z}-\mathbf{y})
  201. 𝐳 \mathbf{z}
  202. f ( z ) = 0 f(z)=0
  203. f ( 𝐲 ) + [ J f ] ( [ 𝐱 ] ) ( 𝐳 - 𝐲 ) = 0 f(\mathbf{y})+[J_{f}](\mathbf{[x]})\cdot(\mathbf{z}-\mathbf{y})=0
  204. 𝐳 𝐲 - [ J f ] ( [ 𝐱 ] ) - 1 f ( 𝐲 ) \mathbf{z}\in\mathbf{y}-[J_{f}](\mathbf{[x]})^{-1}\cdot f(\mathbf{y})
  205. [ J f ] ( [ 𝐱 ] ) - 1 f ( 𝐲 ) ) [J_{f}](\mathbf{[x]})^{-1}\cdot f(\mathbf{y}))
  206. [ 𝐱 ] [ ] n [\mathbf{x}]\in[\mathbb{R}]^{n}
  207. [ 𝐱 ] ( 𝐲 - [ J f ] ( [ 𝐱 ] ) - 1 f ( 𝐲 ) ) [\mathbf{x}]\cap\left(\mathbf{y}-[J_{f}](\mathbf{[x]})^{-1}\cdot f(\mathbf{y})\right)
  208. f f
  209. [ 𝐱 ] [\mathbf{x}]
  210. f ( x ) = x 2 - 2 f(x)=x^{2}-2
  211. [ x ] = [ - 2 , 2 ] [x]=[-2,2]
  212. y = 0 y=0
  213. J f ( x ) = 2 x J_{f}(x)=2\,x
  214. [ - 2 , 2 ] ( 0 - 1 2 [ - 2 , 2 ] ( 0 - 2 ) ) = [ - 2 , 2 ] ( [ - , - 0.5 ] [ 0.5 , ] ) = [ - 2 , - 0.5 ] [ 0.5 , 2 ] [-2,2]\cap\left(0-\frac{1}{2\cdot[-2,2]}(0-2)\right)=[-2,2]\cap\Big([{-\infty}% ,{-0.5}]\cup[{0.5},{\infty}]\Big)=[{-2},{-0.5}]\cup[{0.5},{2}]
  215. x [ - 2 , - 0.5 ] x\in[{-2},{-0.5}]
  216. [ 0.5 , 2 ] [{0.5},{2}]
  217. - 2 -\sqrt{2}
  218. + 2 +\sqrt{2}
  219. g ( x ) = x 2 - [ 2 , 3 ] g(x)=x^{2}-[2,3]
  220. [ - 3 , - 2 ] [ 2 , 3 ] \left[-\sqrt{3},-\sqrt{2}\right]\cup\left[\sqrt{2},\sqrt{3}\right]
  221. [ 𝐱 ] [\mathbf{x}]
  222. [ 𝐱 1 ] , , [ 𝐱 k ] , [\mathbf{x}_{1}],\dots,[\mathbf{x}_{k}]\mbox{,}~{}
  223. [ 𝐱 ] = i = 1 k [ 𝐱 i ] , \textstyle[\mathbf{x}]=\bigcup_{i=1}^{k}[\mathbf{x}_{i}]\mbox{,}~{}
  224. f ( [ 𝐱 ] ) = i = 1 k f ( [ 𝐱 i ] ) . \textstyle f([\mathbf{x}])=\bigcup_{i=1}^{k}f([\mathbf{x}_{i}])\mbox{.}~{}
  225. [ f ] ( [ 𝐱 ] ) i = 1 k [ f ] ( [ 𝐱 i ] ) \textstyle[f]([\mathbf{x}])\supseteq\bigcup_{i=1}^{k}[f]([\mathbf{x}_{i}])
  226. [ f ] ( [ 𝐱 ] ) [f]([\mathbf{x}])
  227. [ x i 1 , x i 2 ] [x_{i1},x_{i2}]
  228. [ 𝐱 ] = ( [ x 11 , x 12 ] , , [ x n 1 , x n 2 ] ) [\mathbf{x}]=([x_{11},x_{12}],\dots,[x_{n1},x_{n2}])
  229. [ x i 1 , ( x i 1 + x i 2 ) / 2 ] [x_{i1},(x_{i1}+x_{i2})/2]
  230. [ ( x i 1 + x i 2 ) / 2 , x i 2 ] [(x_{i1}+x_{i2})/2,x_{i2}]
  231. 2 r 2^{r}
  232. r r
  233. abs ( a - b ) \mathrm{abs}(a-b)
  234. [ a , b ] [a,b]
  235. f ( 𝐱 , 𝐩 ) = 0 f(\mathbf{x},\mathbf{p})=0
  236. 𝐩 [ 𝐩 ] \mathbf{p}\in[\mathbf{p}]
  237. 𝐱 \mathbf{x}
  238. { 𝐱 | 𝐩 [ 𝐩 ] , f ( 𝐱 , 𝐩 ) = 0 } \{\mathbf{x}\,|\,\exists\mathbf{p}\in[\mathbf{p}],f(\mathbf{x},\mathbf{p})=0\}
  239. x [ x ] x\in[x]
  240. x [ x ] x\not\in[x]
  241. μ [ 0 , 1 ] \mu\in[0,1]
  242. μ = 1 \mu=1
  243. μ = 0 \mu=0
  244. μ i [ 0 , 1 ] \mu_{i}\in[0,1]
  245. [ x ( 1 ) ] [ x ( 2 ) ] [ x ( k ) ] \left[x^{(1)}\right]\supset\left[x^{(2)}\right]\supset\cdots\supset\left[x^{(k% )}\right]
  246. [ x ( i ) ] [x^{(i)}]
  247. μ i \mu_{i}
  248. f ( x 1 , , x n ) f(x_{1},\cdots,x_{n})
  249. x 1 , , x n x_{1},\cdots,x_{n}
  250. [ x 1 ( 1 ) ] [ x 1 ( k ) ] , , [ x n ( 1 ) ] [ x n ( k ) ] \left[x_{1}^{(1)}\right]\supset\cdots\supset\left[x_{1}^{(k)}\right],\cdots,% \left[x_{n}^{(1)}\right]\supset\cdots\supset\left[x_{n}^{(k)}\right]
  251. [ y ( 1 ) ] [ y ( k ) ] \left[y^{(1)}\right]\supset\cdots\supset\left[y^{(k)}\right]
  252. [ y ( i ) ] \left[y^{(i)}\right]
  253. [ y ( i ) ] = f ( [ x 1 ( i ) ] , [ x n ( i ) ] ) \left[y^{(i)}\right]=f\left(\left[x_{1}^{(i)}\right],\cdots\left[x_{n}^{(i)}% \right]\right)
  254. [ y ( 1 ) ] \left[y^{(1)}\right]

Interval_exchange_transformation.html

  1. n > 0 n>0
  2. π \pi
  3. 1 , , n 1,\dots,n
  4. λ = ( λ 1 , , λ n ) \lambda=(\lambda_{1},\dots,\lambda_{n})
  5. i = 1 n λ i = 1. \sum_{i=1}^{n}\lambda_{i}=1.
  6. T π , λ : [ 0 , 1 ] [ 0 , 1 ] , T_{\pi,\lambda}:[0,1]\rightarrow[0,1],
  7. ( π , λ ) (\pi,\lambda)
  8. 1 i n 1\leq i\leq n
  9. a i = 1 j < i λ j and a i = 1 j < π ( i ) λ π - 1 ( j ) . a_{i}=\sum_{1\leq j<i}\lambda_{j}\quad\,\text{and}\quad a^{\prime}_{i}=\sum_{1% \leq j<\pi(i)}\lambda_{\pi^{-1}(j)}.
  10. x [ 0 , 1 ] x\in[0,1]
  11. T π , λ ( x ) = x - a i + a i T_{\pi,\lambda}(x)=x-a_{i}+a^{\prime}_{i}
  12. x x
  13. [ a i , a i + λ i ) [a_{i},a_{i}+\lambda_{i})
  14. T π , λ T_{\pi,\lambda}
  15. [ a i , a i + λ i ) [a_{i},a_{i}+\lambda_{i})
  16. i i
  17. π ( i ) \pi(i)
  18. T π , λ T_{\pi,\lambda}
  19. [ 0 , 1 ] [0,1]
  20. T π , λ T_{\pi,\lambda}
  21. T π - 1 , λ T_{\pi^{-1},\lambda^{\prime}}
  22. λ i = λ π - 1 ( i ) \lambda^{\prime}_{i}=\lambda_{\pi^{-1}(i)}
  23. 1 i n 1\leq i\leq n
  24. n = 2 n=2
  25. π = ( 12 ) \pi=(12)
  26. T π , λ T_{\pi,\lambda}
  27. λ 1 \lambda_{1}
  28. T π , λ T_{\pi,\lambda}
  29. [ 0 , 1 ] [0,1]
  30. λ 1 \lambda_{1}
  31. λ 1 \lambda_{1}
  32. n > 2 n>2
  33. π \pi
  34. 0 < k < n 0<k<n
  35. π ( { 1 , , k } ) = { 1 , , k } \pi(\{1,\dots,k\})=\{1,\dots,k\}
  36. λ \lambda
  37. { ( t 1 , , t n ) : t i = 1 } \{(t_{1},\dots,t_{n}):\sum t_{i}=1\}
  38. T π , λ T_{\pi,\lambda}
  39. n 4 n\geq 4
  40. ( π , λ ) (\pi,\lambda)
  41. T π , λ T_{\pi,\lambda}
  42. T π , λ T_{\pi,\lambda}
  43. n n

Intervalence_charge_transfer.html

  1. 1 / 2 \scriptstyle 1/\sqrt{2}

Intraspecific_competition.html

  1. d N ( t ) d t = r N ( t ) ( 1 - N ( t ) K ) {dN(t)\over dt}=rN(t)\left(1-\frac{N(t)}{K}\right)

Introduction_to_quantum_mechanics.html

  1. h h
  2. E E
  3. f f
  4. E = n h f , where n = 1 , 2 , 3 , E=nhf,\quad\,\text{where}\quad n=1,2,3,\ldots
  5. E = h f E=hf
  6. f f
  7. h f hf
  8. h f hf
  9. φ φ
  10. φ = h f 0 . \varphi=hf_{0}.
  11. f f
  12. h f hf
  13. E K = h f - φ = h ( f - f 0 ) . E_{K}=hf-\varphi=h(f-f_{0}).
  14. f f
  15. h f hf
  16. λ λ
  17. n n
  18. λ = B ( n 2 n 2 - 4 ) n = 3 , 4 , 5 , 6 \lambda=B\left(\frac{n^{2}}{n^{2}-4}\right)\qquad\qquad n=3,4,5,6
  19. B B
  20. λ λ
  21. n n
  22. m m
  23. 1 λ = R ( 1 m 2 - 1 n 2 ) , \frac{1}{\lambda}=R\left(\frac{1}{m^{2}}-\frac{1}{n^{2}}\right),
  24. m = 2 m=2
  25. n = 3 , 4 , 5 , 6 n=3,4,5,6
  26. m = 1 m=1
  27. n > 1 n>1
  28. m = 3 m=3
  29. n > 3 n>3
  30. L L
  31. L = n h 2 π , L=n\frac{h}{2\pi},
  32. n n
  33. h h
  34. n n
  35. r r
  36. r = n 2 h 2 4 π 2 k e m e 2 r=\frac{n^{2}h^{2}}{4\pi^{2}k_{e}me^{2}}
  37. m m
  38. e e
  39. r = n 2 a 0 , r=n^{2}a_{0},\!
  40. E = - k e e 2 2 a 0 1 n 2 E=-\frac{k_{\mathrm{e}}e^{2}}{2a_{0}}\frac{1}{n^{2}}
  41. E γ = E n - E m = k e e 2 2 a 0 ( 1 m 2 - 1 n 2 ) E_{\gamma}=E_{n}-E_{m}=\frac{k_{\mathrm{e}}e^{2}}{2a_{0}}\left(\frac{1}{m^{2}}% -\frac{1}{n^{2}}\right)
  42. 1 λ = k e e 2 2 a 0 h c ( 1 m 2 - 1 n 2 ) . \frac{1}{\lambda}=\frac{k_{\mathrm{e}}e^{2}}{2a_{0}hc}\left(\frac{1}{m^{2}}-% \frac{1}{n^{2}}\right).
  43. R R
  44. R = k e e 2 2 a 0 h c . R=\frac{k_{\mathrm{e}}e^{2}}{2a_{0}hc}.
  45. p = h λ . p=\frac{h}{\lambda}.
  46. ψ \psi
  47. ψ \psi
  48. Ψ Ψ
  49. V V
  50. z z
  51. n n
  52. n n
  53. n n
  54. n = 1 , 2 , 3 n=1,2,3\ldots
  55. l l
  56. l l
  57. n 1 n−1
  58. l = 0 , 1 , , n - 1. l=0,1,\ldots,n-1.
  59. s s
  60. p p
  61. d d
  62. f f
  63. g g
  64. l −l
  65. l l
  66. m l = - l , - ( l - 1 ) , , 0 , 1 , , l . m_{l}=-l,-(l-1),\ldots,0,1,\ldots,l.
  67. m < s u b > s m<sub>s

Invariants_of_tensors.html

  1. p ( λ ) := det ( 𝐀 - λ 𝐄 ) \ p(\lambda):=\det(\mathbf{A}-\lambda\mathbf{E})
  2. 𝐄 \mathbf{E}
  3. λ \lambda\in\mathbb{C}
  4. λ \lambda
  5. p ( 𝐀 ) p(\mathbf{A})
  6. I A I_{A}
  7. λ n - 1 \lambda^{n-1}
  8. λ n \lambda^{n}
  9. I I A II_{A}
  10. λ n - 2 \lambda^{n-2}
  11. A A
  12. I A I_{A}
  13. I A = A 11 + A 22 + + A n n = tr ( 𝐀 ) \ I_{A}=A_{11}+A_{22}+\cdots+A_{nn}=\mathrm{tr}(\mathbf{A})\,
  14. ± det 𝐀 \pm\det\mathbf{A}
  15. 𝐀 \mathbf{A}
  16. I A = tr ( 𝐀 ) = A 11 + A 22 + A 33 = A 1 + A 2 + A 3 \mathrm{I}_{A}=\mathrm{tr}(\mathbf{A})=A_{11}+A_{22}+A_{33}=A_{1}+A_{2}+A_{3}\,
  17. II A = 1 2 ( ( tr 𝐀 ) 2 - tr ( 𝐀𝐀 ) ) = A 11 A 22 + A 22 A 33 + A 11 A 33 - A 12 A 21 - A 23 A 32 - A 13 A 31 \mathrm{II}_{A}=\frac{1}{2}\left((\mathrm{tr}\mathbf{A})^{2}-\mathrm{tr}(% \mathbf{A}\mathbf{A})\right)=A_{11}A_{22}+A_{22}A_{33}+A_{11}A_{33}-A_{12}A_{2% 1}-A_{23}A_{32}-A_{13}A_{31}
  18. = A 1 A 2 + A 2 A 3 + A 1 A 3 III A = det ( 𝐀 ) = A 1 A 2 A 3 \begin{aligned}&\displaystyle=A_{1}A_{2}+A_{2}A_{3}+A_{1}A_{3}\\ \displaystyle\mathrm{III}_{A}&\displaystyle=\det(\mathbf{A})=A_{1}A_{2}A_{3}% \end{aligned}
  19. A 1 A_{1}
  20. A 2 A_{2}
  21. A 3 A_{3}
  22. 𝐀 3 - I A 𝐀 2 + II A 𝐀 - III A 𝐄 = 0 \ \mathbf{A}^{3}-\mathrm{I}_{A}\mathbf{A}^{2}+\mathrm{II}_{A}\mathbf{A}-% \mathrm{III}_{A}\mathbf{E}=0
  23. 𝐀 \mathbf{A}
  24. g : R 3 R \ g:R^{3}\to R
  25. f ( 𝐀 ) = g ( I A , I I A , I I I A ) . \ f(\mathbf{A})=g(I_{A},II_{A},III_{A}).\,

Inverse_(mathematics).html

  1. y = k / x . y=k/x.

Inverse_photoemission_spectroscopy.html

  1. h ν h\nu
  2. E i E_{i}
  3. E f E_{f}
  4. E i = E f + h ν E_{i}=E_{f}+h\nu\,
  5. E i E_{i}
  6. h ν h\nu
  7. E f E_{f}

Ion-mobility_spectrometry.html

  1. v d = K E v_{d}=KE
  2. K 0 = K n n 0 = K T 0 T p p 0 K_{0}=K\frac{n}{n_{0}}=K\ \frac{T_{0}}{T}\ \frac{p}{p_{0}}
  3. K = L 2 t D U K=\frac{L^{2}}{t_{D}U}
  4. K = 3 16 2 π μ k T Q n σ K=\frac{3}{16}\sqrt{\frac{2\pi}{\mu kT}}\frac{Q}{n\sigma}
  5. R = t Δ t = L E Q 16 k T ln 2 R=\frac{t}{\Delta t}=\sqrt{\frac{LEQ}{16kT\ln 2}}

Ionic_radius.html

  1. r m {r_{m}}
  2. r x {r_{x}}
  3. d m x {d_{mx}}
  4. d m x k = r m k + r x k {d_{mx}}^{k}={r_{m}}^{k}+{r_{x}}^{k}
  5. k k
  6. k k
  7. d m x = r m + r x {d_{mx}}={r_{m}}+{r_{x}}
  8. k k
  9. k k

Ionic_strength.html

  1. I = 1 2 i = 1 n c i z i 2 I=\begin{matrix}\frac{1}{2}\end{matrix}\sum_{{\rm i}=1}^{n}c_{\rm i}z_{\rm i}^% {2}
  2. I = 1 2 i = 1 n b i z i 2 I=\begin{matrix}\frac{1}{2}\end{matrix}\sum_{{i}=1}^{n}b_{i}z_{i}^{2}

IOPS.html

  1. IOPS * TransferSizeInBytes = BytesPerSec \,\text{IOPS}*\,\text{TransferSizeInBytes}=\,\text{BytesPerSec}

IP_(complexity).html

  1. w L Pr [ V P accepts w ] 2 3 w\in L\Rightarrow\Pr[V\leftrightarrow P\,\text{ accepts }w]\geq\tfrac{2}{3}
  2. w L Pr [ V Q accepts w ] 1 3 w\not\in L\Rightarrow\Pr[V\leftrightarrow Q\,\text{ accepts }w]\leq\tfrac{1}{3}
  3. Pr [ V accepts w starting at M j ] = max P Pr [ V P accepts w starting at M j ] \Pr[V\,\text{ accepts }w\,\text{ starting at }M_{j}]=\max\nolimits_{P}\Pr\left% [V\leftrightarrow P\,\text{ accepts }w\,\text{ starting at }M_{j}\right]
  4. N M j = { 0 j = p and m p = reject 1 j = p and m p = accept max m j + 1 N M j + 1 j < p and j is odd wt-avg m j + 1 N M j + 1 j < p and j is even N_{M_{j}}=\begin{cases}0&j=p\,\text{ and }m_{p}=\,\text{reject}\\ 1&j=p\,\text{ and }m_{p}=\,\text{accept}\\ \max_{m_{j+1}}N_{M_{j+1}}&j<p\,\text{ and }j\,\text{ is odd}\\ \,\text{wt-avg}_{m_{j+1}}N_{M_{j+1}}&j<p\,\text{ and }j\,\text{ is even}\\ \end{cases}
  5. wt-avg m j + 1 N M j + 1 := m j + 1 Pr r [ V ( w , r , M j ) ] \,\text{wt-avg}_{m_{j+1}}N_{M_{j+1}}:=\sum\nolimits_{m_{j+1}}\Pr\nolimits_{r}[% V(w,r,M_{j})]
  6. N M j = m j + 1 Pr r [ V ( w , r , M j ) = m j + 1 ] N M j + 1 . N_{M_{j}}=\sum\nolimits_{m_{j+1}}\Pr\nolimits_{r}\left[V(w,r,M_{j})=m_{j+1}% \right]N_{M_{j+1}}.
  7. m j + 1 Pr r [ V ( w , r , M j ) = m j + 1 ] * Pr [ V accepts w starting at M j + 1 ] . \sum\nolimits_{m_{j+1}}\Pr\nolimits_{r}\left[V(w,r,M_{j})=m_{j+1}\right]*\Pr% \left[V\,\text{ accepts }w\,\text{ starting at }M_{j+1}\right].
  8. N M j = max m j + 1 N M j + 1 . N_{M_{j}}=\max\nolimits_{m_{j+1}}N_{M_{j+1}}.
  9. max m j + 1 * Pr [ V accepts w starting at M j + 1 ] . \max\nolimits_{m_{j+1}}*\Pr[V\,\text{ accepts }w\,\text{ starting at }M_{j+1}].
  10. max m j + 1 Pr [ V accepts w starting at M j + 1 ] Pr [ V accepts w starting at M j ] \max\nolimits_{m_{j+1}}\Pr[V\,\text{ accepts }w\,\text{ starting at }M_{j+1}]% \leq\Pr[V\,\text{ accepts w starting at }M_{j}]
  11. max m j + 1 Pr [ V accepts w starting at M j + 1 ] Pr [ V accepts w starting at M j ] \max\nolimits_{m_{j+1}}\Pr\left[V\,\text{ accepts }w\,\text{ starting at }M_{j% +1}\right]\geq\Pr\left[V\,\text{ accepts }w\,\text{ starting at }M_{j}\right]
  12. # SAT = { φ , k : φ is a CNF-formula with exactly k satisfying assignments } . \#\,\text{SAT}=\left\{\langle\varphi,k\rangle\ :\ \varphi\,\text{ is a CNF-% formula with exactly }k\,\text{ satisfying assignments}\right\}.
  13. p φ \displaystyle p_{\varphi}
  14. a 1 , , a i - 1 F a_{1},\dots,a_{i-1}\in F
  15. a 1 , , a i F a_{1},\dots,a_{i}\in F
  16. f i ( a 1 , , a i ) = a i + 1 , , a n { 0 , 1 } p ( a 1 , , a n ) . f_{i}(a_{1},\dots,a_{i})=\sum\nolimits_{a_{i+1},\dots,a_{n}\in\{0,1\}}p(a_{1},% \dots,a_{n}).
  17. f i ( r 1 , , r i - 1 , z ) f_{i}(r_{1},\dots,r_{i-1},z)
  18. f i - 1 ( r 1 , , r i - 1 ) = f i ( r 1 , , r i - 1 , 0 ) + f i ( r 1 , , r i - 1 , 1 ) f_{i-1}(r_{1},\dots,r_{i-1})=f_{i}(r_{1},\dots,r_{i-1},0)+f_{i}(r_{1},\dots,r_% {i-1},1)
  19. p ( r 1 , , r n ) p(r_{1},\dots,r_{n})
  20. f n ( r 1 , , r n ) f_{n}(r_{1},\dots,r_{n})
  21. P ~ \tilde{P}
  22. P ~ \tilde{P}
  23. f ~ 0 ( ) \tilde{f}_{0}()
  24. P ~ \tilde{P}
  25. f ~ 1 \tilde{f}_{1}
  26. f ~ 1 ( 0 ) + f ~ 1 ( 1 ) = f ~ 0 ( ) \tilde{f}_{1}(0)+\tilde{f}_{1}(1)=\tilde{f}_{0}()
  27. Pr [ f ~ 1 ( r 1 ) = f 1 ( r 1 ) ] < 1 n 2 . \Pr\left[\tilde{f}_{1}(r_{1})=f_{1}(r_{1})\right]<\tfrac{1}{n^{2}}.
  28. n / 2 n < n / n 3 n/2^{n}<n/n^{3}
  29. f ~ i - 1 ( r 1 , , r i - 1 ) f i - 1 ( r 1 , , r i - 1 ) , \tilde{f}_{i-1}(r_{1},\dots,r_{i-1})\neq f_{i-1}(r_{1},\dots,r_{i-1}),
  30. Pr [ f ~ ( r 1 , , r i ) = f i ( r 1 , , r i ) ] 1 n 2 . \Pr\left[\tilde{f}(r_{1},\dots,r_{i})=f_{i}(r_{1},\dots,r_{i})\right]\leq% \tfrac{1}{n^{2}}.
  31. P ~ \tilde{P}
  32. ψ = Q 1 x 1 Q m x m [ φ ] \psi=Q_{1}x_{1}\dots Q_{m}x_{m}[\varphi]
  33. f i ( a 1 , , a i ) = { f i ( a 1 , , a m ) = 1 Q i + 1 x i + 1 Q m x m [ φ ( a 1 , , a i ) ] is true 0 otherwise f_{i}(a_{1},\dots,a_{i})=\begin{cases}f_{i}(a_{1},\dots,a_{m})=1&Q_{i+1}x_{i+1% }\dots Q_{m}x_{m}[\varphi(a_{1},\dots,a_{i})]\,\text{ is true}\\ 0&\,\text{otherwise}\end{cases}
  34. f i ( a 1 , , a i ) = { f i + 1 ( a 1 , , a i , 0 ) f i + 1 ( a 1 , , a i , 1 ) Q i + 1 = f i + 1 ( a 1 , , a i , 0 ) * f i + 1 ( a 1 , , a i , 1 ) Q i + 1 = f_{i}(a_{1},\dots,a_{i})=\begin{cases}f_{i+1}(a_{1},\dots,a_{i},0)\cdot f_{i+1% }(a_{1},\dots,a_{i},1)&Q_{i+1}=\forall\\ f_{i+1}(a_{1},\dots,a_{i},0)*f_{i+1}(a_{1},\dots,a_{i},1)&Q_{i+1}=\exists\end{cases}
  35. ψ = Q 1 x 1 Q m x m [ φ ] \psi=Q_{1}x_{1}\dots Q_{m}x_{m}[\varphi]
  36. ψ = Q 1 x 1 R x 1 Q 2 R x 1 R x 2 Q m R x 1 R x m [ φ ] \psi^{\prime}=Q_{1}x_{1}Rx_{1}Q_{2}Rx_{1}Rx_{2}\dots Q_{m}Rx_{1}\dots Rx_{m}[\varphi]
  37. ψ = S 1 y 1 S k y k [ φ ] , where S i { , , R } , y i { x 1 , , x m } \psi^{\prime}=S_{1}y_{1}\dots S_{k}y_{k}[\varphi],\qquad\,\text{ where }S_{i}% \in\{\forall,\exists,R\},\ y_{i}\in\{x_{1},\dots,x_{m}\}
  38. f k ( x 1 , , x m ) f_{k}(x_{1},\dots,x_{m})
  39. If S i + 1 = , f i ( a 1 , , a i ) = f i + 1 ( a 1 , , a i , 0 ) f i + 1 ( a 1 , , a i , 1 ) \,\text{If }S_{i+1}=\forall,\quad f_{i}(a_{1},\dots,a_{i})=f_{i+1}(a_{1},\dots% ,a_{i},0)\cdot f_{i+1}(a_{1},\dots,a_{i},1)
  40. If S i + 1 = , f i ( a 1 , , a i ) = f i + 1 ( a 1 , , a i , 0 ) * f i + 1 ( a 1 , , a i , 1 ) \,\text{If }S_{i+1}=\exists,\quad f_{i}(a_{1},\dots,a_{i})=f_{i+1}(a_{1},\dots% ,a_{i},0)*f_{i+1}(a_{1},\dots,a_{i},1)
  41. If S i + 1 = R , f i ( a 1 , , a i , a ) = ( 1 - a ) f i + 1 ( a 1 , , a i , 0 ) + a f i + 1 ( a 1 , , a i , 1 ) \,\text{If }S_{i+1}=R,\quad f_{i}(a_{1},\dots,a_{i},a)=(1-a)f_{i+1}(a_{1},% \dots,a_{i},0)+af_{i+1}(a_{1},\dots,a_{i},1)
  42. R x 1 R x i R_{x_{1}}\dots R_{x_{i}}
  43. f 0 ( ) = { f 1 ( 0 ) f 1 ( 1 ) if S = f 1 ( 0 ) * f 1 ( 1 ) if S = . ( 1 - r ) f 1 ( 0 ) + r f 1 ( 1 ) if S = R . f_{0}()=\begin{cases}f_{1}(0)\cdot f_{1}(1)&\,\text{ if }S=\forall\\ f_{1}(0)*f_{1}(1)&\,\text{ if }S=\exists.\\ (1-r)f_{1}(0)+rf_{1}(1)&\,\text{ if }S=R.\end{cases}
  44. f i ( r 1 , , r i - 1 , z ) f_{i}(r_{1},\dots,r_{i-1},z)
  45. r 1 , , r i - 1 r_{1},\dots,r_{i-1}
  46. f i ( r 1 , , r i - 1 , 0 ) f_{i}(r_{1},\dots,r_{i-1},0)
  47. f i ( r 1 , , r i - 1 , 1 ) f_{i}(r_{1},\dots,r_{i-1},1)
  48. f i - 1 ( r 1 , , r i - 1 ) = { f i ( r 1 , , r i - 1 , 0 ) f i ( r 1 , , r i - 1 , 1 ) S = f i ( r 1 , , r i - 1 , 0 ) * f i ( r 1 , , r i - 1 , 1 ) S = . f_{i-1}(r_{1},\dots,r_{i-1})=\begin{cases}f_{i}(r_{1},\dots,r_{i-1},0)\cdot f_% {i}(r_{1},\dots,r_{i-1},1)&S=\forall\\ f_{i}(r_{1},\dots,r_{i-1},0)*f_{i}(r_{1},\dots,r_{i-1},1)&S=\exists.\end{cases}
  49. f i - 1 ( r 1 r ) = ( 1 - r ) f i ( r 1 , , r i - 1 , 0 ) + r f i ( r 1 , , r i - 1 , 1 ) if S = R . f_{i-1}(r_{1}\dots r)=(1-r)f_{i}(r_{1},\dots,r_{i-1},0)+rf_{i}(r_{1},\dots,r_{% i-1},1)\,\text{ if }S=R.
  50. f i ( r 1 , , r ) f_{i}(r_{1},\dots,r)
  51. p ( r 1 , , r m ) p(r_{1},\dots,r_{m})
  52. p ( r 1 , , r m ) = f k ( r 1 , , r m ) p(r_{1},\dots,r_{m})=f_{k}(r_{1},\dots,r_{m})
  53. P ~ \tilde{P}
  54. P ~ \tilde{P}
  55. f i - 1 ( r 1 , ) f_{i-1}(r_{1},\dots)
  56. f i ( r 1 , , 0 ) f_{i}(r_{1},\dots,0)
  57. f i ( r 1 , , 1 ) f_{i}(r_{1},\dots,1)
  58. P ~ \tilde{P}
  59. n / n 4 n/n^{4}
  60. P ~ \tilde{P}
  61. P ~ \tilde{P}
  62. w L Pr [ V P accepts w ] = 1 w\in L\Rightarrow\Pr[V\leftrightarrow P\,\text{ accepts }w]=1

Irreducible_component.html

  1. X = X 1 X 2 X=X_{1}\cup X_{2}
  2. X 1 X_{1}
  3. X 2 X_{2}
  4. X X
  5. F F
  6. F = ( G 1 F ) ( G 2 F ) F=(G_{1}\cap F)\cup(G_{2}\cap F)
  7. G 1 , G 2 G_{1},G_{2}
  8. X X
  9. F F

Isothermal_flow.html

  1. 1 / k 1/\sqrt{k}

Isotonic_regression.html

  1. x n x\in\mathbb{R}^{n}
  2. a n a\in\mathbb{R}^{n}
  3. w n w\in\mathbb{R}^{n}
  4. x i x j x_{i}\geq x_{j}
  5. G = ( N , E ) G=(N,E)
  6. x i x j x_{i}\geq x_{j}
  7. min i = 1 n w i ( x i - a i ) 2 \min\sum_{i=1}^{n}w_{i}(x_{i}-a_{i})^{2}
  8. subject to x i x j for all ( i , j ) E . \,\text{subject to }x_{i}\geq x_{j}~{}\,\text{ for all }(i,j)\in E.
  9. G = ( N , E ) G=(N,E)
  10. L p L_{p}
  11. p > 0 p>0
  12. min i = 1 n w i | x i - a i | p \min\sum_{i=1}^{n}w_{i}|x_{i}-a_{i}|^{p}
  13. subject to x i x j for all ( i , j ) E . \mathrm{subject~{}to~{}}x_{i}\geq x_{j}~{}\,\text{ for all }(i,j)\in E.
  14. x 1 x 2 x n x_{1}\leq x_{2}\leq\ldots\leq x_{n}
  15. f ( x 1 ) f ( x 2 ) f ( x n ) f(x_{1})\leq f(x_{2})\leq\ldots\leq f(x_{n})
  16. w i 0 w_{i}\geq 0
  17. g * g^{*}
  18. min g i = 1 n w i ( g ( x i ) - f ( x i ) ) 2 \min_{g}\sum_{i=1}^{n}w_{i}(g(x_{i})-f(x_{i}))^{2}
  19. g g
  20. f f

Isotopes_of_beryllium.html

  1. × 10 17 \times 10^{−}17
  2. Be 4 5 Unknown Li 3 4 + H 1 1 \mathrm{{}^{5}_{4}Be}\ \xrightarrow{\ \mathrm{Unknown}}\ \mathrm{{}^{4}_{3}Li}% +\mathrm{{}^{1}_{1}H}
  3. Be 4 6 5 zs He 2 4 + 2 H 1 1 \mathrm{{}^{6}_{4}Be}\ \xrightarrow{\ \mathrm{5zs}}\ \mathrm{{}^{4}_{2}He}+% \mathrm{2{}^{1}_{1}H}
  4. Be 4 7 + e 53.22 d - Li 3 7 \mathrm{{}^{7}_{4}Be}+\mathrm{e{}^{-}}\ \xrightarrow{\ \mathrm{53.22d}}\ % \mathrm{{}^{7}_{3}Li}
  5. Be 4 8 67 as 2 He 2 4 \mathrm{{}^{8}_{4}Be}\ \xrightarrow{\ \mathrm{67as}}\ \mathrm{2{}^{4}_{2}He}
  6. Be 4 10 1.39 Ma B 5 10 + e - \mathrm{{}^{10}_{4}Be}\ \xrightarrow{\ \mathrm{1.39Ma}}\ \mathrm{{}^{10}_{5}B}% +\mathrm{e{}^{-}}
  7. Be 4 11 13.81 s B 5 11 + e - \mathrm{{}^{11}_{4}Be}\ \xrightarrow{\ \mathrm{13.81s}}\ \mathrm{{}^{11}_{5}B}% +\mathrm{e{}^{-}}
  8. Be 4 11 13.81 s Li 3 7 + He 2 4 + e - \mathrm{{}^{11}_{4}Be}\ \xrightarrow{\ \mathrm{13.81s}}\ \mathrm{{}^{7}_{3}Li}% +\mathrm{{}^{4}_{2}He}+\mathrm{e{}^{-}}
  9. Be 4 12 21.49 ms B 5 12 + e - \mathrm{{}^{12}_{4}Be}\ \xrightarrow{\ \mathrm{21.49ms}}\ \mathrm{{}^{12}_{5}B% }+\mathrm{e{}^{-}}
  10. Be 4 12 21.49 ms B 5 11 + n 0 1 + e - \mathrm{{}^{12}_{4}Be}\ \xrightarrow{\ \mathrm{21.49ms}}\ \mathrm{{}^{11}_{5}B% }+\mathrm{{}^{1}_{0}n}+\mathrm{e{}^{-}}
  11. Be 4 13 2.7 zs Be 4 12 + n 0 1 \mathrm{{}^{13}_{4}Be}\ \xrightarrow{\ \mathrm{2.7zs}}\ \mathrm{{}^{12}_{4}Be}% +\mathrm{{}^{1}_{0}n}
  12. Be 4 14 4.84 ms B 5 13 + n 0 1 + e - \mathrm{{}^{14}_{4}Be}\ \xrightarrow{\ \mathrm{4.84ms}}\ \mathrm{{}^{13}_{5}B}% +\mathrm{{}^{1}_{0}n}+\mathrm{e{}^{-}}
  13. Be 4 14 4.84 ms B 5 14 + e - \mathrm{{}^{14}_{4}Be}\ \xrightarrow{\ \mathrm{4.84ms}}\ \mathrm{{}^{14}_{5}B}% +\mathrm{e{}^{-}}
  14. Be 4 14 4.84 ms B 5 12 + 2 n 0 1 + e - \mathrm{{}^{14}_{4}Be}\ \xrightarrow{\ \mathrm{4.84ms}}\ \mathrm{{}^{12}_{5}B}% +\mathrm{2{}^{1}_{0}n}+\mathrm{e{}^{-}}
  15. Be 4 16 < 200 n s Be 4 14 + 2 n 0 1 \mathrm{{}^{16}_{4}Be}\ \xrightarrow{\ \mathrm{<200ns}}\ \mathrm{{}^{14}_{4}Be% }+\mathrm{2{}^{1}_{0}n}

Isotopes_of_carbon.html

  1. δ 13 C = ( ( C 13 C 12 ) s a m p l e ( C 13 C 12 ) s t a n d a r d - 1 ) * 1000 o / o o \delta^{13}C=\Biggl(\frac{\bigl(\frac{{}^{13}C}{{}^{12}C}\bigr)_{sample}}{% \bigl(\frac{{}^{13}C}{{}^{12}C}\bigr)_{standard}}-1\Biggr)*1000\ ^{o}\!/\!_{oo}

Isotopes_of_helium.html

  1. × 10 22 \times 10^{−}22
  2. He 2 2 Unknown 2 H 1 1 \mathrm{{}^{2}_{2}He}\ \xrightarrow{\ \mathrm{Unknown}}\ \mathrm{2{}^{1}_{1}H}
  3. He 2 2 Unknown H 1 2 + e + \mathrm{{}^{2}_{2}He}\ \xrightarrow{\ \mathrm{Unknown}}\ \mathrm{{}^{2}_{1}H}+% \mathrm{e{}^{+}}
  4. He 2 5 700 ys He 2 4 + n 0 1 \mathrm{{}^{5}_{2}He}\ \xrightarrow{\ \mathrm{700ys}}\ \mathrm{{}^{4}_{2}He}+% \mathrm{{}^{1}_{0}n}
  5. He 2 6 806.7 ms Li 3 6 + e - \mathrm{{}^{6}_{2}He}\ \xrightarrow{\ \mathrm{806.7ms}}\ \mathrm{{}^{6}_{3}Li}% +\mathrm{e{}^{-}}
  6. He 2 6 806.7 ms He 2 4 + H 1 2 + e - \mathrm{{}^{6}_{2}He}\ \xrightarrow{\ \mathrm{806.7ms}}\ \mathrm{{}^{4}_{2}He}% +\mathrm{{}^{2}_{1}H}+\mathrm{e{}^{-}}
  7. He 2 7 2.9 zs He 2 6 + n 0 1 \mathrm{{}^{7}_{2}He}\ \xrightarrow{\ \mathrm{2.9zs}}\ \mathrm{{}^{6}_{2}He}+% \mathrm{{}^{1}_{0}n}
  8. He 2 8 119 ms Li 3 8 + e - \mathrm{{}^{8}_{2}He}\ \xrightarrow{\ \mathrm{119ms}}\ \mathrm{{}^{8}_{3}Li}+% \mathrm{e{}^{-}}
  9. He 2 8 119 ms Li 3 7 + n 0 1 + e - \mathrm{{}^{8}_{2}He}\ \xrightarrow{\ \mathrm{119ms}}\ \mathrm{{}^{7}_{3}Li}+% \mathrm{{}^{1}_{0}n}+\mathrm{e{}^{-}}
  10. He 2 8 119 ms He 2 5 + H 1 3 + e - \mathrm{{}^{8}_{2}He}\ \xrightarrow{\ \mathrm{119ms}}\ \mathrm{{}^{5}_{2}He}+% \mathrm{{}^{3}_{1}H}+\mathrm{e{}^{-}}
  11. He 2 9 7 zs He 2 8 + n 0 1 \mathrm{{}^{9}_{2}He}\ \xrightarrow{\ \mathrm{7zs}}\ \mathrm{{}^{8}_{2}He}+% \mathrm{{}^{1}_{0}n}
  12. He 2 10 2.7 zs He 2 8 + 2 n 0 1 \mathrm{{}^{10}_{2}He}\ \xrightarrow{\ \mathrm{2.7zs}}\ \mathrm{{}^{8}_{2}He}+% \mathrm{2{}^{1}_{0}n}

Isotopes_of_hydrogen.html

  1. 1 / 2 {1}/{2}
  2. H 1 3 12.32 y He 2 3 + e - \mathrm{{}^{3}_{1}H}\ \xrightarrow{\ \mathrm{12.32y}}\ \mathrm{{}^{3}_{2}He}+% \mathrm{e{}^{-}}
  3. H 1 4 139 ys H 1 3 + n 0 1 \mathrm{{}^{4}_{1}H}\ \xrightarrow{\ \mathrm{139ys}}\ \mathrm{{}^{3}_{1}H}+% \mathrm{{}^{1}_{0}n}
  4. H 1 5 > 910 y s H 1 3 + 2 n 0 1 \mathrm{{}^{5}_{1}H}\ \xrightarrow{\ \mathrm{>910ys}}\ \mathrm{{}^{3}_{1}H}+% \mathrm{2{}^{1}_{0}n}
  5. H 1 6 290 ys H 1 3 + 3 n 0 1 \mathrm{{}^{6}_{1}H}\ \xrightarrow{\ \mathrm{290ys}}\ \mathrm{{}^{3}_{1}H}+% \mathrm{3{}^{1}_{0}n}
  6. H 1 6 290 ys H 1 2 + 4 n 0 1 \mathrm{{}^{6}_{1}H}\ \xrightarrow{\ \mathrm{290ys}}\ \mathrm{{}^{2}_{1}H}+% \mathrm{4{}^{1}_{0}n}
  7. H 1 7 23 ys H 1 3 + 4 n 0 1 \mathrm{{}^{7}_{1}H}\ \xrightarrow{\ \mathrm{23ys}}\ \mathrm{{}^{3}_{1}H}+% \mathrm{4{}^{1}_{0}n}

Isotopes_of_lithium.html

  1. Li 3 4 91 ys He 2 3 + H 1 1 \mathrm{{}^{4}_{3}Li}\ \xrightarrow{\ \mathrm{91ys}}\ \mathrm{{}^{3}_{2}He}+% \mathrm{{}^{1}_{1}H}
  2. Li 3 5 370 ys He 2 4 + H 1 1 \mathrm{{}^{5}_{3}Li}\ \xrightarrow{\ \mathrm{370ys}}\ \mathrm{{}^{4}_{2}He}+% \mathrm{{}^{1}_{1}H}
  3. Li 3 8 840.3 ms Be 4 8 + e - \mathrm{{}^{8}_{3}Li}\ \xrightarrow{\ \mathrm{840.3ms}}\ \mathrm{{}^{8}_{4}Be}% +\mathrm{e{}^{-}}
  4. Li 3 9 178.3 ms Be 4 8 + n 0 1 + e - \mathrm{{}^{9}_{3}Li}\ \xrightarrow{\ \mathrm{178.3ms}}\ \mathrm{{}^{8}_{4}Be}% +\mathrm{{}^{1}_{0}n}+\mathrm{e{}^{-}}
  5. Li 3 9 178.3 ms Be 4 9 + e - \mathrm{{}^{9}_{3}Li}\ \xrightarrow{\ \mathrm{178.3ms}}\ \mathrm{{}^{9}_{4}Be}% +\mathrm{e{}^{-}}
  6. Li 3 10 2 zs Li 3 9 + n 0 1 \mathrm{{}^{10}_{3}Li}\ \xrightarrow{\ \mathrm{2zs}}\ \mathrm{{}^{9}_{3}Li}+% \mathrm{{}^{1}_{0}n}
  7. Li 3 11 8.75 ms Be 4 10 + n 0 1 + e - \mathrm{{}^{11}_{3}Li}\ \xrightarrow{\ \mathrm{8.75ms}}\ \mathrm{{}^{10}_{4}Be% }+\mathrm{{}^{1}_{0}n}+\mathrm{e{}^{-}}
  8. Li 3 11 8.75 ms Be 4 11 + e - \mathrm{{}^{11}_{3}Li}\ \xrightarrow{\ \mathrm{8.75ms}}\ \mathrm{{}^{11}_{4}Be% }+\mathrm{e{}^{-}}
  9. Li 3 11 8.75 ms Be 4 9 + 2 n 0 1 + e - \mathrm{{}^{11}_{3}Li}\ \xrightarrow{\ \mathrm{8.75ms}}\ \mathrm{{}^{9}_{4}Be}% +\mathrm{2{}^{1}_{0}n}+\mathrm{e{}^{-}}
  10. Li 3 11 8.75 ms Be 4 8 + 3 n 0 1 + e - \mathrm{{}^{11}_{3}Li}\ \xrightarrow{\ \mathrm{8.75ms}}\ \mathrm{{}^{8}_{4}Be}% +\mathrm{3{}^{1}_{0}n}+\mathrm{e{}^{-}}
  11. Li 3 11 8.75 ms He 4 7 + He 2 4 + e - \mathrm{{}^{11}_{3}Li}\ \xrightarrow{\ \mathrm{8.75ms}}\ \mathrm{{}^{7}_{4}He}% +\mathrm{{}^{4}_{2}He}+\mathrm{e{}^{-}}
  12. Li 3 11 8.75 ms Li 3 8 + H 1 3 + e - \mathrm{{}^{11}_{3}Li}\ \xrightarrow{\ \mathrm{8.75ms}}\ \mathrm{{}^{8}_{3}Li}% +\mathrm{{}^{3}_{1}H}+\mathrm{e{}^{-}}
  13. Li 3 11 8.75 ms Li 3 9 + H 1 2 + e - \mathrm{{}^{11}_{3}Li}\ \xrightarrow{\ \mathrm{8.75ms}}\ \mathrm{{}^{9}_{3}Li}% +\mathrm{{}^{2}_{1}H}+\mathrm{e{}^{-}}
  14. Li 3 12 < 10 n s Li 3 11 + n 0 1 \mathrm{{}^{12}_{3}Li}\ \xrightarrow{\ \mathrm{<10ns}}\ \mathrm{{}^{11}_{3}Li}% +\mathrm{{}^{1}_{0}n}

Isotopes_of_rutherfordium.html

  1. U 92 238 + 12 24 Mg 104 259 Rf + 3 0 1 n \,{}^{238}_{92}\mathrm{U}\ +\,^{24}_{12}\mathrm{Mg}\to\,^{259}_{104}\mathrm{Rf% }\ +3\,^{1}_{0}\mathrm{n}

Isotopes_of_strontium.html

  1. ν ¯ e \bar{\nu}_{e}
  2. Sr 38 89 Y 39 89 + e - + ν ¯ e \mathrm{{}^{89}_{38}Sr}\rightarrow\mathrm{{}^{89}_{39}Y}+e^{-}+\bar{\nu}_{e}
  3. Sr 38 90 Y 39 90 + e - + ν ¯ e \mathrm{{}^{90}_{38}Sr}\rightarrow\mathrm{{}^{90}_{39}Y}+e^{-}+\bar{\nu}_{e}

Isotropic_coordinates.html

  1. d s 2 = - f ( r ) 2 d t 2 + g ( r ) 2 ( d r 2 + r 2 ( d θ 2 + sin ( θ ) 2 d ϕ 2 ) ) , ds^{2}=-f(r)^{2}\,dt^{2}+g(r)^{2}\,\left(dr^{2}+r^{2}\,\left(d\theta^{2}+\sin(% \theta)^{2}\,d\phi^{2}\right)\right),
  2. - < t < , r 0 < r < r 1 , 0 < θ < π , - π < ϕ < π -\infty<t<\infty,\,r_{0}<r<r_{1},\,0<\theta<\pi,\,-\pi<\phi<\pi
  3. t \partial_{t}
  4. ϕ , sin ( ϕ ) θ + cot ( θ ) cos ( ϕ ) ϕ , cos ( ϕ ) θ - cot ( θ ) sin ( ϕ ) ϕ \partial_{\phi},\;\;\sin(\phi)\,\partial_{\theta}+\cot(\theta)\,\cos(\phi)% \partial_{\phi},\;\;\cos(\phi)\,\partial_{\theta}-\cot(\theta)\,\sin(\phi)% \partial_{\phi}
  5. X = t \vec{X}=\partial_{t}
  6. t = t 0 t=t_{0}
  7. t = t 0 , r = r 0 t=t_{0},\,r=r_{0}
  8. d σ 2 = g ( r 0 ) 2 r 0 2 ( d θ 2 + sin ( θ ) 2 d ϕ 2 ) , 0 < θ < π , - π < ϕ < π d\sigma^{2}=g(r_{0})^{2}\,r_{0}^{2}\,\left(d\theta^{2}+\sin(\theta)^{2}\,d\phi% ^{2}\right),\;0<\theta<\pi,-\pi<\phi<\pi
  9. g ( r 0 ) r g(r_{0})\,r
  10. r r
  11. ϕ = - π , π \phi=-\pi,\,\pi
  12. σ 0 = - f ( r ) d t \sigma^{0}=-f(r)\,dt
  13. σ 1 = g ( r ) d r \sigma^{1}=g(r)\,dr
  14. σ 2 = g ( r ) r d θ \sigma^{2}=g(r)\,r\,d\theta
  15. σ 3 = g ( r ) r sin ( θ ) d ϕ \sigma^{3}=g(r)\,r\,\sin(\theta)\,d\phi
  16. ω 0 1 = f d t g {\omega^{0}}_{1}=\frac{f^{\prime}\,dt}{g}
  17. ω 1 2 = - ( 1 + r g g ) d θ {\omega^{1}}_{2}=-\left(1+\frac{r\,g^{\prime}}{g}\right)\,d\theta
  18. ω 1 3 = - ( 1 + r g g ) sin ( θ ) d ϕ {\omega^{1}}_{3}=-\left(1+\frac{r\,g^{\prime}}{g}\right)\,\sin(\theta)\,d\phi
  19. ω 2 3 = - cos ( θ ) d ϕ {\omega^{2}}_{3}=-\cos(\theta)\,d\phi

Isotropic_radiator.html

  1. G \scriptstyle G
  2. I \scriptstyle I
  3. I iso \scriptstyle I\text{iso}
  4. G = I I iso G={I\over I\text{iso}}\,
  5. sin ( θ ) \sin(\theta)

Iterative_deepening_A*.html

  1. f ( n ) = g ( n ) + h ( n ) f(n)=g(n)+h(n)
  2. g ( n ) g(n)
  3. n n
  4. h ( n ) h(n)
  5. n n
  6. h h
  7. h ( n ) h * ( n ) h(n)\leq h^{*}(n)
  8. n n
  9. h * h*
  10. n n
  11. h h
  12. h ( n ) cost ( n , n ) + h ( n ) h(n)\leq\mathrm{cost}(n,n^{\prime})+h(n^{\prime})
  13. n n
  14. n n
  15. n n

Iterative_learning_control.html

  1. r ( t ) r(t)
  2. u p + 1 = u p + K * e p u_{p+1}=u_{p}+K*e_{p}
  3. u p u_{p}
  4. e p e_{p}
  5. e p e_{p}
  6. p p
  7. K K

ITU-R_468_noise_weighting.html

  1. R I T U ( f ) = 1.246332637532143 10 - 4 f ( h 1 ( f ) ) 2 + ( h 2 ( f ) ) 2 R_{ITU}(f)=\frac{1.246332637532143\cdot 10^{-4}\,f}{\sqrt{(h_{1}(f))^{2}+(h_{2% }(f))^{2}}}
  2. I T U ( f ) = 18.2 + 20 log 10 ( R I T U ( f ) ) ITU(f)=18.2+20\log_{10}\left(R_{ITU}(f)\right)
  3. h 1 ( f ) = - 4.737338981378384 10 - 24 f 6 + 2.043828333606125 10 - 15 f 4 - 1.363894795463638 10 - 7 f 2 + 1 h_{1}(f)=-4.737338981378384\cdot 10^{-24}\,f^{6}+2.043828333606125\cdot 10^{-1% 5}\,f^{4}-1.363894795463638\cdot 10^{-7}\,f^{2}+1
  4. h 2 ( f ) = 1.306612257412824 10 - 19 f 5 - 2.118150887518656 10 - 11 f 3 + 5.559488023498642 10 - 4 f h_{2}(f)=1.306612257412824\cdot 10^{-19}\,f^{5}-2.118150887518656\cdot 10^{-11% }\,f^{3}+5.559488023498642\cdot 10^{-4}\,f

J-class_yacht.html

  1. R = 0.18 L S D 3 R=\frac{0.18\cdot L\cdot\sqrt{S}}{\sqrt[3]{D}}
  2. L L
  3. S S
  4. D D
  5. R R
  6. R R
  7. L . W . L L.W.L
  8. L . O . A L.O.A

J-homomorphism.html

  1. J : π r ( SO ( q ) ) π r + q ( S q ) J\colon\pi_{r}(\mathrm{SO}(q))\to\pi_{r+q}(S^{q})\,\!
  2. S q - 1 S q - 1 S^{q-1}\rightarrow S^{q-1}
  3. S r × S q - 1 S q - 1 S^{r}\times S^{q-1}\rightarrow S^{q-1}
  4. S r + q = S r * S q - 1 S ( S q - 1 ) = S q S^{r+q}=S^{r}*S^{q-1}\rightarrow S(S^{q-1})=S^{q}
  5. J : π r ( SO ) π r S , J\colon\pi_{r}(\mathrm{SO})\to\pi_{r}^{S},\,\!

Jaccard_index.html

  1. J ( A , B ) = | A B | | A B | . J(A,B)={{|A\cap B|}\over{|A\cup B|}}.
  2. 0 J ( A , B ) 1. 0\leq J(A,B)\leq 1.
  3. d J ( A , B ) = 1 - J ( A , B ) = | A B | - | A B | | A B | . d_{J}(A,B)=1-J(A,B)={{|A\cup B|-|A\cap B|}\over|A\cup B|}.
  4. A B = ( A B ) - ( A B ) A\triangle B=(A\cup B)-(A\cap B)
  5. μ \mu
  6. X X
  7. J μ ( A , B ) = μ ( A B ) μ ( A B ) J_{\mu}(A,B)={{\mu(A\cap B)}\over{\mu(A\cup B)}}
  8. d μ ( A , B ) = 1 - J μ ( A , B ) = μ ( A B ) μ ( A B ) d_{\mu}(A,B)=1-J_{\mu}(A,B)={{\mu(A\triangle B)}\over{\mu(A\cup B)}}
  9. μ ( A B ) = 0 \mu(A\cup B)=0
  10. \infty
  11. M 11 M_{11}
  12. M 01 M_{01}
  13. M 10 M_{10}
  14. M 00 M_{00}
  15. M 11 + M 01 + M 10 + M 00 = n . M_{11}+M_{01}+M_{10}+M_{00}=n.
  16. J = M 11 M 01 + M 10 + M 11 . J={M_{11}\over M_{01}+M_{10}+M_{11}}.
  17. d J = M 01 + M 10 M 01 + M 10 + M 11 . d_{J}={M_{01}+M_{10}\over M_{01}+M_{10}+M_{11}}.
  18. 𝐱 = ( x 1 , x 2 , , x n ) \mathbf{x}=(x_{1},x_{2},\ldots,x_{n})
  19. 𝐲 = ( y 1 , y 2 , , y n ) \mathbf{y}=(y_{1},y_{2},\ldots,y_{n})
  20. x i , y i 0 x_{i},y_{i}\geq 0
  21. J ( 𝐱 , 𝐲 ) = i min ( x i , y i ) i max ( x i , y i ) , J(\mathbf{x},\mathbf{y})=\frac{\sum_{i}\min(x_{i},y_{i})}{\sum_{i}\max(x_{i},y% _{i})},
  22. d J ( 𝐱 , 𝐲 ) = 1 - J ( 𝐱 , 𝐲 ) . d_{J}(\mathbf{x},\mathbf{y})=1-J(\mathbf{x},\mathbf{y}).
  23. f f
  24. g g
  25. X X
  26. μ \mu
  27. J ( f , g ) = min ( f , g ) d μ max ( f , g ) d μ , J(f,g)=\frac{\int\min(f,g)d\mu}{\int\max(f,g)d\mu},
  28. max \max
  29. min \min
  30. d J ( f , g ) = 1 - J ( f , g ) . d_{J}(f,g)=1-J(f,g).
  31. A , B X A,B\subseteq X
  32. J μ ( A , B ) = J ( χ A , χ B ) , J_{\mu}(A,B)=J(\chi_{A},\chi_{B}),
  33. χ A \chi_{A}
  34. χ B \chi_{B}
  35. X i X_{i}
  36. , \land,\lor
  37. T s T_{s}
  38. T s ( X , Y ) = i ( X i Y i ) i ( X i Y i ) T_{s}(X,Y)=\frac{\sum_{i}(X_{i}\land Y_{i})}{\sum_{i}(X_{i}\lor Y_{i})}
  39. T d ( X , Y ) = - log 2 ( T s ( X , Y ) ) T_{d}(X,Y)=-\log_{2}(T_{s}(X,Y))
  40. 1 - T s 1-T_{s}
  41. f ( A , B ) = A B | A | 2 + | B | 2 - A B f(A,B)=\frac{A\cdot B}{|A|^{2}+|B|^{2}-A\cdot B}
  42. A B = i A i B i = i ( A i B i ) A\cdot B=\sum_{i}A_{i}B_{i}=\sum_{i}(A_{i}\land B_{i})
  43. | A | 2 = i A i 2 = i A i {|A|}^{2}=\sum_{i}A_{i}^{2}=\sum_{i}A_{i}
  44. T s T_{s}
  45. f f
  46. 1 - f 1-f
  47. 1 - T s 1-T_{s}
  48. 1 - f 1-f
  49. f f
  50. 1 - f 1-f
  51. W W
  52. A i { 0 , W i } A_{i}\in\{0,W_{i}\}
  53. J = D / ( 2 - D ) J=D/(2-D)
  54. D = 2 J / ( 1 + J ) D=2J/(1+J)

Jackscrew.html

  1. F load F in = 2 π r l \frac{F\text{load}}{F\text{in}}=\frac{2\pi r}{l}\,
  2. F load F\text{load}\,
  3. F in F\text{in}\,
  4. r r\,
  5. l l\,

Jacobi's_formula.html

  1. d d t det A ( t ) = tr ( adj ( A ( t ) ) d A ( t ) d t ) . \frac{d}{dt}\det A(t)=\mathrm{tr}\left(\mathrm{adj}(A(t))\,\frac{dA(t)}{dt}% \right)~{}.
  2. d det ( A ) = tr ( adj ( A ) d A ) . d\det(A)=\mathrm{tr}(\mathrm{adj}(A)\,dA).
  3. i j A i j B i j = tr ( A T B ) . \sum_{i}\sum_{j}A_{ij}B_{ij}=\mathrm{tr}(A^{\rm T}B).
  4. ( A B ) j k = i A j i B i k . (AB)_{jk}=\sum_{i}A_{ji}B_{ik}.\,
  5. ( A T B ) j k = i A i j B i k . (A^{\rm T}B)_{jk}=\sum_{i}A_{ij}B_{ik}.
  6. tr ( A T B ) = j ( A T B ) j j = j i A i j B i j = i j A i j B i j . \mathrm{tr}(A^{\rm T}B)=\sum_{j}(A^{\rm T}B)_{jj}=\sum_{j}\sum_{i}A_{ij}B_{ij}% =\sum_{i}\sum_{j}A_{ij}B_{ij}.\ \square
  7. d det ( A ) = tr ( adj ( A ) d A ) . d\det(A)=\mathrm{tr}(\mathrm{adj}(A)\,dA).
  8. det ( A ) = j A i j adj T ( A ) i j . \det(A)=\sum_{j}A_{ij}\mathrm{adj}^{\rm T}(A)_{ij}.
  9. det ( A ) = F ( A 11 , A 12 , , A 21 , A 22 , , A n n ) \det(A)=F\,(A_{11},A_{12},\ldots,A_{21},A_{22},\ldots,A_{nn})
  10. d det ( A ) = i j F A i j d A i j . d\det(A)=\sum_{i}\sum_{j}{\partial F\over\partial A_{ij}}\,dA_{ij}.
  11. det ( A ) A i j = k A i k adj T ( A ) i k A i j = k ( A i k adj T ( A ) i k ) A i j {\partial\det(A)\over\partial A_{ij}}={\partial\sum_{k}A_{ik}\mathrm{adj}^{\rm T% }(A)_{ik}\over\partial A_{ij}}=\sum_{k}{\partial(A_{ik}\mathrm{adj}^{\rm T}(A)% _{ik})\over\partial A_{ij}}
  12. det ( A ) A i j = k A i k A i j adj T ( A ) i k + k A i k adj T ( A ) i k A i j . {\partial\det(A)\over\partial A_{ij}}=\sum_{k}{\partial A_{ik}\over\partial A_% {ij}}\mathrm{adj}^{\rm T}(A)_{ik}+\sum_{k}A_{ik}{\partial\,\mathrm{adj}^{\rm T% }(A)_{ik}\over\partial A_{ij}}.
  13. adj T ( A ) i k A i j = 0 , {\partial\,\mathrm{adj}^{\rm T}(A)_{ik}\over\partial A_{ij}}=0,
  14. det ( A ) A i j = k adj T ( A ) i k A i k A i j . {\partial\det(A)\over\partial A_{ij}}=\sum_{k}\mathrm{adj}^{\rm T}(A)_{ik}{% \partial A_{ik}\over\partial A_{ij}}.
  15. A i k A i j = δ j k , {\partial A_{ik}\over\partial A_{ij}}=\delta_{jk},
  16. det ( A ) A i j = k adj T ( A ) i k δ j k = adj T ( A ) i j . {\partial\det(A)\over\partial A_{ij}}=\sum_{k}\mathrm{adj}^{\rm T}(A)_{ik}% \delta_{jk}=\mathrm{adj}^{\rm T}(A)_{ij}.
  17. d ( det ( A ) ) = i j adj T ( A ) i j d A i j , d(\det(A))=\sum_{i}\sum_{j}\mathrm{adj}^{\rm T}(A)_{ij}\,dA_{ij},
  18. d ( det ( A ) ) = tr ( adj ( A ) d A ) . d(\det(A))=\mathrm{tr}(\mathrm{adj}(A)\,dA).\ \square
  19. d d t det A ( t ) = ( det A ( t ) ) tr ( A ( t ) - 1 d d t A ( t ) ) \frac{d}{dt}\det A(t)=(\det A(t))\,\mathrm{tr}\left(A(t)^{-1}\,\frac{d}{dt}A(t% )\right)
  20. d d t log det A ( t ) = tr ( A ( t ) - 1 d d t A ( t ) ) . \frac{d}{dt}\log\det A(t)=\mathrm{tr}\left(A(t)^{-1}\,\frac{d}{dt}A(t)\right).
  21. d d t det e t B = tr ( B ) det e t B , \frac{d}{dt}\det e^{tB}=\mathrm{tr}\left(B\right)\det e^{tB},
  22. d d t det ( A ( t ) ) = ( det ( A ( t ) ) ) : ( d d t A ( t ) ) = tr ( adj ( A ( t ) ) d d t A ( t ) ) \frac{d}{dt}\mbox{det}~{}\left(A\left(t\right)\right)=\left(\nabla\mbox{det}~{% }\left(A\left(t\right)\right)\right):\left(\frac{d}{dt}A\left(t\right)\right)=% \mbox{tr}~{}\left(\mbox{adj}~{}\left(A\left(t\right)\right)\frac{d}{dt}A\left(% t\right)\right)

Jacobi_sum.html

  1. χ \chi
  2. ψ \psi
  3. χ \chi
  4. ψ \psi
  5. J ( χ , ψ ) = χ ( a ) ψ ( 1 - a ) , J(\chi,\psi)=\sum\chi(a)\psi(1-a),\,
  6. g g
  7. χ ψ \chi\psi
  8. χ \chi
  9. ψ \psi
  10. g ( χ ) g(\chi)
  11. g ( ψ ) g(\psi)
  12. g ( χ ψ ) g(\chi\psi)
  13. g g
  14. χ \chi
  15. ψ \psi
  16. χ ψ \chi\psi
  17. χ \chi
  18. ψ \psi
  19. g g
  20. χ \chi
  21. ψ \psi
  22. χ \chi
  23. χ \chi
  24. χ \chi
  25. χ \chi

Jacques_Philippe_Marie_Binet.html

  1. n th n\text{th}
  2. u n = u n - 1 + u n - 2 , for n > 1 , u_{n}=u_{n-1}+u_{n-2},\,\text{ for }n>1,\,
  3. u 0 = 0 u_{0}=0\,
  4. u 1 = 1 u_{1}=1\,
  5. u n = ( 1 + 5 ) n - ( 1 - 5 ) n 2 n 5 u_{n}=\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n}\sqrt{5}}

Jacques_Touchard.html

  1. C k = 1 k + 1 ( 2 k k ) , k 0 C_{k}={1\over{k+1}}{{2k}\choose{k}},\quad k\geq 0
  2. C n + 1 = k n / 2 2 n - 2 k ( n 2 k ) C k . C_{n+1}=\sum_{k\,\leq\,n/2}2^{n-2k}{n\choose 2k}C_{k}.\,
  3. C ( t ) = n 0 C n t n = 1 - 1 - 4 t 2 t C(t)=\sum_{n\geq 0}C_{n}t^{n}={{1-\sqrt{1-4t}}\over{2t}}
  4. t 1 - 2 t C ( t 2 ( 1 - 2 t ) 2 ) = C ( t ) - 1 {t\over{1-2t}}C\left({t^{2}\over(1-2t)^{2}}\right)=C(t)-1

Jensen's_Device.html

  1. Σ k = l u a k \Sigma_{k=l}^{u}a_{k}

Jet_(particle_physics).html

  1. σ i j k = i , j d x 1 d x 2 d t ^ f i 1 ( x 1 , Q 2 ) f j 2 ( x 2 , Q 2 ) d σ ^ i j k d t ^ , \sigma_{ij\rightarrow k}=\sum_{i,j}\int dx_{1}dx_{2}d\hat{t}f_{i}^{1}(x_{1},Q^% {2})f_{j}^{2}(x_{2},Q^{2})\frac{d\hat{\sigma}_{ij\rightarrow k}}{d\hat{t}},
  2. σ ^ i j k \hat{\sigma}_{ij\rightarrow k}
  3. f i a ( x , Q 2 ) f_{i}^{a}(x,Q^{2})
  4. σ ^ \hat{\sigma}
  5. P j i ( x z , Q 2 ) P_{ji}\!\left(\frac{x}{z},Q^{2}\right)
  6. ln Q 2 D i h ( x , Q 2 ) = j x 1 d z z α S 4 π P j i ( x z , Q 2 ) D j h ( z , Q 2 ) \frac{\partial}{\partial\ln Q^{2}}D_{i}^{h}(x,Q^{2})=\sum_{j}\int_{x}^{1}\frac% {dz}{z}\frac{\alpha_{S}}{4\pi}P_{ji}\!\left(\frac{x}{z},Q^{2}\right)D_{j}^{h}(% z,Q^{2})

Jet_group.html

  1. W = 𝐑 × ( 𝐑 * ) k × S 2 ( ( 𝐑 * ) k ) × × S m ( ( 𝐑 * ) k ) . W=\mathbf{R}\times(\mathbf{R}^{*})^{k}\times S^{2}((\mathbf{R}^{*})^{k})\times% \cdots\times S^{m}((\mathbf{R}^{*})^{k}).
  2. p = ( x , x ) J m ( 𝐑 n ) p=(x,x^{\prime})\in J^{m}(\mathbf{R}^{n})
  3. j k ( f p ) ( x ) = x j^{k}(f_{p})(x)=x^{\prime}
  4. ( x , x ) * ( y , y ) = ( x + y , j m f p ( y ) + y ) (x,x^{\prime})*(y,y^{\prime})=(x+y,j^{m}f_{p}(y)+y^{\prime})

John_5.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}

John_Clive_Ward.html

  1. | ψ = ( | x , y - | y , x ) \left|\psi\right\rangle=(\left|x,y\right\rangle-\left|y,x\right\rangle)
  2. | ψ = 1 2 ( | x 1 | y 2 - | y 1 | x 2 ) \left|\psi\right\rangle={1\over\sqrt{2}}(\left|x\right\rangle_{1}\left|y\right% \rangle_{2}-\left|y\right\rangle_{1}\left|x\right\rangle_{2})

John_Colson.html

  1. 9 , 8 , 7 , 6 9,8,7,6
  2. 1 1 ¯ , 1 2 ¯ , 1 3 ¯ , 1 4 ¯ 1\bar{1},1\bar{2},1\bar{3},1\bar{4}

Johnson_graph.html

  1. J ( n , k ) J(n,k)
  2. k k
  3. n n
  4. ( k - 1 ) (k-1)
  5. J ( n , 1 ) J(n,1)
  6. J ( 4 , 2 ) J(4,2)
  7. J ( 5 , 2 ) J(5,2)
  8. n n
  9. J ( n , 2 ) J(n,2)
  10. K G ( n , 2 ) KG(n,2)
  11. d d
  12. ( a d , b d , c d ) (a_{d},b_{d},c_{d})
  13. ( v , w ) (v,w)
  14. d d
  15. w w
  16. a i a_{i}
  17. d d
  18. v v
  19. b i b_{i}
  20. d + 1 d+1
  21. v v
  22. c i c_{i}
  23. d - 1 d-1
  24. v v
  25. J ( 8 , 2 ) J(8,2)
  26. J ( 8 , 2 ) J(8,2)
  27. J ( n , r ) J(n,r)~{}
  28. r ( n - r ) ~{}r(n-r)
  29. J ( n , k ) J(n,k)
  30. k k
  31. k k
  32. ( 2 k + 1 ) (2k+1)

Join_(topology).html

  1. A B A\star B
  2. ( A × B × I ) / R , (A\times B\times I)/R,\,
  3. ( a , b 1 , 0 ) ( a , b 2 , 0 ) for all a A and b 1 , b 2 B , (a,b_{1},0)\sim(a,b_{2},0)\quad\mbox{for all }~{}a\in A\mbox{ and }~{}b_{1},b_% {2}\in B,
  4. ( a 1 , b , 1 ) ( a 2 , b , 1 ) for all a 1 , a 2 A and b B . (a_{1},b,1)\sim(a_{2},b,1)\quad\mbox{for all }~{}a_{1},a_{2}\in A\mbox{ and }~% {}b\in B.
  5. A × B × { 0 } A\times B\times\{0\}
  6. A A
  7. A × B × { 1 } A\times B\times\{1\}
  8. B B
  9. A B A\star B
  10. A B C ( A ) × B A × B C ( B ) × A . A\star B\cong C(A)\times B\cup_{A\times B}C(B)\times A.
  11. A B A { b 0 } { a 0 } B \frac{A\star B}{A\star\{b_{0}\}\cup\{a_{0}\}\star B}
  12. Σ ( A B ) \Sigma(A\wedge B)
  13. A { b 0 } { a 0 } B {A\star\{b_{0}\}\cup\{a_{0}\}\star B}
  14. A B Σ ( A B ) . A\star B\simeq\Sigma(A\wedge B).
  15. S 0 S^{0}
  16. S X SX
  17. S n S^{n}
  18. S m S^{m}
  19. S n + m + 1 S^{n+m+1}

Jolly_balance.html

  1. w w
  2. w w^{\prime}
  3. w ( w - w ) \frac{w}{(w-w^{\prime})}

Joseph_Ludwig_Raabe.html

  1. a a + 1 log Γ ( t ) d t = 1 2 log 2 π + a log a - a , a 0. \int\limits_{a}^{a+1}\log\Gamma(t)\,\mathrm{d}t=\tfrac{1}{2}\log 2\pi+a\log a-% a,\quad a\geq 0.

Jump_search.html

  1. k k\in\mathbb{N}

K-ary_tree.html

  1. k h k^{h}
  2. k h + 1 - 1 k - 1 \left\lfloor\frac{k^{h+1}-1}{k-1}\right\rfloor
  3. log k ( k - 1 ) + log k ( 𝑛𝑢𝑚𝑏𝑒𝑟 _ 𝑜𝑓 _ 𝑛𝑜𝑑𝑒𝑠 ) - 1 . \left\lceil\log_{k}(k-1)+\log_{k}(\mathit{number\_of\_nodes})-1\right\rceil.
  4. log k [ ( k - 1 ) * 𝑛𝑢𝑚𝑏𝑒𝑟 _ 𝑜𝑓 _ 𝑛𝑜𝑑𝑒𝑠 + 1 ] - 1 , k 2. \log_{k}[(k-1)*\mathit{number\_of\_nodes}+1]-1,k\geq 2.
  5. k * i + 1 + c k*i+1+c
  6. i - 1 k \left\lfloor\frac{i-1}{k}\right\rfloor

K-function.html

  1. K ( z ) = ( 2 π ) ( - z + 1 ) / 2 exp [ ( z 2 ) + 0 z - 1 ln ( t ! ) d t ] . K(z)=(2\pi)^{(-z+1)/2}\exp\left[\begin{pmatrix}z\\ 2\end{pmatrix}+\int_{0}^{z-1}\ln(t!)\,dt\right].
  2. K ( z ) = exp [ ζ ( - 1 , z ) - ζ ( - 1 ) ] K(z)=\exp\left[\zeta^{\prime}(-1,z)-\zeta^{\prime}(-1)\right]
  3. ζ ( a , z ) = def [ d ζ ( s , z ) d s ] s = a . \zeta^{\prime}(a,z)\ \stackrel{\mathrm{def}}{=}\ \left[\frac{d\zeta(s,z)}{ds}% \right]_{s=a}.
  4. K ( z ) = exp ( ψ ( - 2 ) ( z ) + z 2 - z 2 - z 2 ln ( 2 π ) ) K(z)=\exp\left(\psi^{(-2)}(z)+\frac{z^{2}-z}{2}-\frac{z}{2}\ln(2\pi)\right)
  5. K ( z ) = A e ψ ( - 2 , z ) + z 2 - z 2 K(z)=Ae^{\psi(-2,z)+\frac{z^{2}-z}{2}}
  6. K ( n ) = ( Γ ( n ) ) n - 1 G ( n ) . K(n)=\frac{(\Gamma(n))^{n-1}}{G(n)}.
  7. K ( n + 1 ) = 1 1 2 2 3 3 n n . K(n+1)=1^{1}\,2^{2}\,3^{3}\cdots n^{n}.

K-homology.html

  1. C * C^{*}
  2. ( , F 0 , Γ ) (\mathcal{H},F_{0},\Gamma)
  3. ( , F 1 , Γ ) (\mathcal{H},F_{1},\Gamma)
  4. t ( , F t , Γ ) t\mapsto(\mathcal{H},F_{t},\Gamma)
  5. t [ 0 , 1 ] . t\in[0,1].
  6. K 0 ( A ) K^{0}(A)
  7. K 1 ( A ) K^{1}(A)
  8. ( , F , Γ ) (\mathcal{H},F,\Gamma)
  9. ( , - F , - Γ ) . (\mathcal{H},-F,-\Gamma).

Kaczmarz_method.html

  1. A x = b Ax=b
  2. A x = b Ax=b
  3. m m
  4. a i a_{i}
  5. i i
  6. A A
  7. x 0 x_{0}
  8. A x = b Ax=b
  9. k = 0 , 1 , k=0,1,...
  10. x k + 1 = x k + b i - a i , x k a i 2 a i ¯ x_{k+1}=x_{k}+\frac{b_{i}-\langle a_{i},x_{k}\rangle}{\lVert a_{i}\rVert^{2}}% \overline{a_{i}}
  11. i = k mod m + 1 i=k\,\bmod\,m+1
  12. a i ¯ \overline{a_{i}}
  13. a i a_{i}
  14. x k x_{k}
  15. λ k \lambda_{k}
  16. x k + 1 = x k + λ k b i - a i , x k a i 2 a i ¯ x^{k+1}=x^{k}+\lambda_{k}\frac{b_{i}-\langle a_{i},x^{k}\rangle}{\lVert a_{i}% \rVert^{2}}\overline{a_{i}}
  17. a i 2 \lVert a_{i}\rVert^{2}
  18. x k x_{k}
  19. A x = b Ax=b
  20. κ ( A ) \kappa(A)
  21. x x
  22. A x = b Ax=b
  23. x x
  24. E x k - x 2 ( 1 - κ ( A ) - 2 ) k x 0 - x 2 . E{\lVert x_{k}-x\rVert^{2}}\leq(1-\kappa(A)^{-2})^{k}\cdot{\lVert x_{0}-x% \rVert^{2}}.
  25. j = 1 m | z , a j | 2 z 2 A - 1 2 ( 1 ) \sum_{j=1}^{m}|\langle z,a_{j}\rangle|^{2}\geq\frac{\lVert z\rVert^{2}}{\lVert A% ^{-1}\rVert^{2}}\qquad\qquad\qquad\qquad(1)
  26. z n . z\in\mathbb{C}^{n}.
  27. A 2 = j = 1 m a j 2 {\lVert A\rVert^{2}}=\sum_{j=1}^{m}{\lVert a_{j}\rVert^{2}}
  28. j = 1 m < m t p l > a j 2 A 2 | z , a j a j | 2 κ ( A ) - 2 z 2 ( 2 ) \begin{aligned}\displaystyle\sum_{j=1}^{m}\frac{<}{m}tpl>{{\lVert a_{j}\rVert^% {2}}}{\lVert A\rVert^{2}}\left|\left\langle z,\frac{a_{j}}{\lVert a_{j}\rVert}% \right\rangle\right|^{2}\geq\kappa(A)^{-2}{\lVert z\rVert^{2}}\qquad\qquad% \qquad\qquad(2)\end{aligned}
  29. z n . z\in\mathbb{C}^{n}.
  30. j - t h j-th
  31. A x = b Ax=b
  32. y : y , a j = b j {y:\langle y,a_{j}\rangle=b_{j}}
  33. a j a j 2 . \frac{a_{j}}{\lVert a_{j}\rVert^{2}}.
  34. A x = b Ax=b
  35. Z = a j a j Z=\frac{a_{j}}{\lVert a_{j}\rVert}
  36. a j 2 A 2 j = 1 , , m \frac{\lVert a_{j}\rVert^{2}}{\lVert A\rVert^{2}}\qquad\qquad\qquad j=1,\cdots,m
  37. 𝔼 | z , Z | 2 κ ( A ) - 2 z 2 ( 3 ) \begin{aligned}\displaystyle\mathbb{E}|\langle z,Z\rangle|^{2}\geq\kappa(A)^{-% 2}{\lVert z\rVert^{2}}\qquad\qquad(3)\end{aligned}
  38. z n . z\in\mathbb{C}^{n}.
  39. P P
  40. A x = b Ax=b
  41. P z = z - z - x , Z Z . Pz=z-\langle z-x,Z\rangle Z.
  42. x k - x 2 {\lVert x_{k}-x\rVert^{2}}
  43. ( 1 - κ ( A ) - 2 ) . (1-\kappa(A)^{-2}).
  44. x k x_{k}
  45. x k - 1 x_{k-1}
  46. x k = P k x k - 1 , x_{k}=P_{k}x_{k-1},
  47. P 1 , P 2 , P_{1},P_{2},\cdots
  48. P . P.
  49. x k - 1 - x k x_{k-1}-x_{k}
  50. P k . P_{k}.
  51. P k P_{k}
  52. x k - x x_{k}-x
  53. x x
  54. x k - x 2 = x k - 1 - x 2 - x k - 1 - x k 2 . {\lVert x_{k}-x\rVert^{2}}={\lVert x_{k-1}-x\rVert^{2}}-{\lVert x_{k-1}-x_{k}% \rVert^{2}}.
  55. x k - 1 - x k 2 {\lVert x_{k-1}-x_{k}\rVert^{2}}
  56. x k x_{k}
  57. x k - 1 - x k = x k - 1 - x , Z k {\lVert x_{k-1}-x_{k}\rVert}=\langle x_{k-1}-x,Z_{k}\rangle
  58. Z 1 , Z 2 , Z_{1},Z_{2},\cdots
  59. Z . Z.
  60. x k - x 2 = ( 1 - | x k - 1 - x x k - 1 - x , Z k | 2 ) x k - 1 - x 2 . {\lVert x_{k}-x\rVert^{2}}=\left(1-\left|\left\langle\frac{x_{k-1}-x}{\lVert x% _{k-1}-x\rVert},Z_{k}\right\rangle\right|^{2}\right){\lVert x_{k-1}-x\rVert^{2% }}.
  61. Z 1 , , Z k - 1 Z_{1},\cdots,Z_{k-1}
  62. P 1 , , P k - 1 P_{1},\cdots,P_{k-1}
  63. x 1 , , x k - 1 x_{1},\cdots,x_{k-1}
  64. Z k Z_{k}
  65. 𝔼 Z 1 , , Z k - 1 x k - x 2 = ( 1 - 𝔼 Z 1 , , Z k - 1 | x k - 1 - x x k - 1 - x , Z k | 2 ) x k - 1 - x 2 . \mathbb{E}_{{Z_{1},\cdots,Z_{k-1}}}{\lVert x_{k}-x\rVert^{2}}=\left(1-\mathbb{% E}_{{Z_{1},\cdots,Z_{k-1}}}\left|\left\langle\frac{x_{k-1}-x}{\lVert x_{k-1}-x% \rVert},Z_{k}\right\rangle\right|^{2}\right){\lVert x_{k-1}-x\rVert^{2}}.
  66. 𝔼 Z 1 , , Z k - 1 x k - x 2 ( 1 - κ ( A ) - 2 ) x k - 1 - x 2 . \mathbb{E}_{{Z_{1},\cdots,Z_{k-1}}}{\lVert x_{k}-x\rVert^{2}}\leq(1-\kappa(A)^% {-2}){\lVert x_{k-1}-x\rVert^{2}}.
  67. 𝔼 x k - x 2 ( 1 - κ ( A ) - 2 ) 𝔼 x k - 1 - x 2 . \mathbb{E}{\lVert x_{k}-x\rVert^{2}}\leq(1-\kappa(A)^{-2})\mathbb{E}{\lVert x_% {k-1}-x\rVert^{2}}.
  68. \blacksquare

Kahn_process_networks.html

  1. b b
  2. b b
  3. b b
  4. b b
  5. b b

Kakutani_fixed-point_theorem.html

  1. x X x\in X
  2. { x n } n \{x_{n}\}_{n\in\mathbb{N}}
  3. { y n } n \{y_{n}\}_{n\in\mathbb{N}}
  4. x n x x_{n}\to x
  5. y n y y_{n}\to y
  6. y n φ ( x n ) y_{n}\in\varphi(x_{n})
  7. n n
  8. y φ ( x ) y\in\varphi(x)
  9. φ ( x ) = { 3 / 4 0 x < 0.5 { 3 / 4 , 1 / 4 } x = 0.5 1 / 4 0.5 < x 1 \varphi(x)=\begin{cases}3/4&0\leq x<0.5\\ \{3/4,1/4\}&x=0.5\\ 1/4&0.5<x\leq 1\\ \end{cases}
  10. x = ( x - q * p * - q * ) p * + ( 1 - x - q * p * - q * ) q * x=\left(\frac{x-q^{*}}{p^{*}-q^{*}}\right)p^{*}+\left(1-\frac{x-q^{*}}{p^{*}-q% ^{*}}\right)q^{*}

Kalb–Ramond_field.html

  1. - q d x μ A μ -q\int dx^{\mu}A_{\mu}
  2. - d x μ d x ν B μ ν -\int dx^{\mu}dx^{\nu}B_{\mu\nu}

Kamioka_Liquid_Scintillator_Antineutrino_Detector.html

  1. Δ m 21 2 = 7.59 ± 0.21 10 - 5 eV 2 , tan 2 θ 12 = 0.47 - 0.05 + 0.06 \Delta m_{21}^{2}=7.59\pm 0.21\cdot 10^{-5}\,\,\text{eV}^{2},\tan^{2}\theta_{1% 2}=0.47^{+0.06}_{-0.05}

Kampyle_of_Eudoxus.html

  1. x 4 = x 2 + y 2 x^{4}=x^{2}+y^{2}
  2. r = sec 2 θ . r=\sec^{2}\theta\,.
  3. x = a sec ( t ) , y = a tan ( t ) sec ( t ) x=a\sec(t),y=a\tan(t)\sec(t)
  4. x x
  5. y y
  6. x x
  7. ( - 1 , 0 ) (-1,0)
  8. ( 1 , 0 ) (1,0)
  9. ( ± 3 / 2 , ± 3 / 2 ) (\pm\sqrt{3/2},\pm\sqrt{3}/2)
  10. x 2 - 1 2 x^{2}-\frac{1}{2}
  11. x x\to\infty
  12. y = x 2 1 - x - 2 = x 2 - 1 2 n 0 C n ( 2 x ) - 2 n y=x^{2}\sqrt{1-x^{-2}}=x^{2}-\frac{1}{2}\sum_{n\geq 0}C_{n}(2x)^{-2n}
  13. C n = 1 n + 1 ( 2 n n ) C_{n}=\frac{1}{n+1}{\left({{2n}\atop{n}}\right)}
  14. n n

Kan_extension.html

  1. 𝐀 , 𝐁 , 𝐂 \mathbf{A},\mathbf{B},\mathbf{C}
  2. X : 𝐀 𝐂 , F : 𝐀 𝐁 X\colon\mathbf{A}\to\mathbf{C},F\colon\mathbf{A}\to\mathbf{B}
  3. X X
  4. F F
  5. η \eta
  6. X X
  7. F F
  8. R : 𝐁 𝐂 R\colon\mathbf{B}\to\mathbf{C}
  9. η : R F X \eta\colon RF\to X
  10. M : 𝐁 𝐂 M\colon\mathbf{B}\to\mathbf{C}
  11. μ : M F X \mu\colon MF\to X
  12. δ : M R \delta\colon M\to R
  13. δ F \delta_{F}
  14. δ F ( a ) = δ ( F a ) : M F ( a ) R F ( a ) \delta_{F}(a)=\delta(Fa)\colon MF(a)\to RF(a)
  15. a a
  16. 𝐀 \mathbf{A}
  17. Ran F X \operatorname{Ran}_{F}X
  18. T T
  19. F , G : 𝐂 𝐃 F,G\colon\mathbf{C}\to\mathbf{D}
  20. T ( a ) : F ( a ) G ( a ) T(a)\colon F(a)\to G(a)
  21. a a
  22. 𝐂 \mathbf{C}
  23. T ( a ) T(a)
  24. T T
  25. X X
  26. F F
  27. L : 𝐁 𝐂 L\colon\mathbf{B}\to\mathbf{C}
  28. ϵ : X L F \epsilon\colon X\to LF
  29. M : 𝐁 𝐂 M\colon\mathbf{B}\to\mathbf{C}
  30. α : X M F \alpha\colon X\to MF
  31. σ : L M \sigma\colon L\to M
  32. σ F \sigma_{F}
  33. σ F ( a ) = σ ( F a ) : L F ( a ) M F ( a ) \sigma_{F}(a)=\sigma(Fa)\colon LF(a)\to MF(a)
  34. a a
  35. 𝐀 \mathbf{A}
  36. Lan F X \operatorname{Lan}_{F}X
  37. L , M L,M
  38. X X
  39. F F
  40. ϵ , α \epsilon,\alpha
  41. σ : L M \sigma\colon L\to M
  42. X : 𝐀 𝐂 X:\mathbf{A}\to\mathbf{C}
  43. F : 𝐀 𝐁 F:\mathbf{A}\to\mathbf{B}
  44. Lan F X \mathrm{Lan}_{F}X
  45. X X
  46. F F
  47. ( Lan F X ) ( b ) = lim f : F a b X ( a ) (\mathrm{Lan}_{F}X)(b)=\underrightarrow{\lim}_{f:Fa\to b}X(a)
  48. ( F b ) (F\downarrow b)
  49. F F
  50. K : 𝐌 𝐂 K:\mathbf{M}\to\mathbf{C}
  51. T : 𝐌 𝐀 T:\mathbf{M}\to\mathbf{A}
  52. 𝐂 ( K m , c ) T m \mathbf{C}(Km^{\prime},c)\cdot Tm
  53. L c = ( Lan K T ) c = m 𝐂 ( K m , c ) T m Lc=(\mathrm{Lan}_{K}T)c=\int^{m}\mathbf{C}(Km,c)\cdot Tm
  54. ( Ran K T ) c = m T m 𝐂 ( c , K m ) (\mathrm{Ran}_{K}T)c=\int_{m}Tm^{\mathbf{C}(c,Km)}
  55. F : C D F:C\to D
  56. lim F = Ran E F \mathrm{lim}F=\mathrm{Ran}_{E}F
  57. E E
  58. C C
  59. C a t Cat
  60. F F
  61. colim F = Lan E F \mathrm{colim}F=\mathrm{Lan}_{E}F
  62. F : C D F:C\to D
  63. Id : C C \mathrm{Id}:C\to C
  64. F F
  65. F F
  66. Ran F Id \mathrm{Ran}_{F}\mathrm{Id}
  67. C E C\to E
  68. F F
  69. F F

Kaplan–Meier_estimator.html

  1. t 1 t 2 t 3 t N . t_{1}\leq t_{2}\leq t_{3}\leq\cdots\leq t_{N}.
  2. i i
  3. t i t_{i}
  4. d i d_{i}
  5. n i n_{i}
  6. S ^ ( t ) = t i < t n i - d i n i . \hat{S}(t)=\prod\limits_{t_{i}<t}\frac{n_{i}-d_{i}}{n_{i}}.
  7. S ^ ( t ) = t i t n i - d i n i . \hat{S}(t)=\prod\limits_{t_{i}\leq t}\frac{n_{i}-d_{i}}{n_{i}}.
  8. S ( t ) = P [ T > t ] = 1 - P [ T t ] = 1 - F ( t ) . S(t)=P[T>t]=1-P[T\leq t]=1-F(t).\,
  9. S ^ ( t ) \scriptstyle\hat{S}(t)
  10. Var ^ ( S ^ ( t ) ) = S ^ ( t ) 2 t i t d i n i ( n i - d i ) . \widehat{\operatorname{Var}}(\widehat{S}(t))=\widehat{S}(t)^{2}\sum\limits_{t_% {i}\leq t}\frac{d_{i}}{n_{i}(n_{i}-d_{i})}.

Kaplan–Yorke_map.html

  1. x n + 1 = 2 x n ( mod 1 ) x_{n+1}=2x_{n}\ (\textrm{mod}~{}1)\,
  2. y n + 1 = α y n + cos ( 4 π x n ) y_{n+1}=\alpha y_{n}+\cos(4\pi x_{n})\,
  3. a n + 1 = 2 a n ( mod b ) a_{n+1}=2a_{n}\ (\textrm{mod}~{}b)\,
  4. x n + 1 = a n / b x_{n+1}=a_{n}/b\,
  5. y n + 1 = α y n + cos ( 4 π x n ) y_{n+1}=\alpha y_{n}+\cos(4\pi x_{n})\,
  6. a n a_{n}
  7. b b
  8. b b
  9. x n x_{n}

Karl_Georg_Christian_von_Staudt.html

  1. ( C A , B D ) + ( C A , B D ) = ( C A , B S ) . (CA,BD)+(CA,BD^{\prime})=(CA,BS).
  2. ( C A , B D ) ( C A , D D ) = ( C A , B D ) . (CA,BD)\cdot(CA,DD^{\prime})=(CA,BD^{\prime}).
  3. P P_{\infty}

Karp–Lipton_theorem.html

  1. Π 2 = Σ 2 \Pi_{2}=\Sigma_{2}\,
  2. PH = Σ 2 . \mathrm{PH}=\Sigma_{2}.\,
  3. Σ 3 \Sigma_{3}
  4. Σ 2 \Sigma_{2}
  5. Σ 2 \Sigma_{2}
  6. Σ 2 \Sigma_{2}
  7. x y ψ ( x , y ) \exists x\forall y\;\psi(x,y)
  8. ψ \psi
  9. Σ 2 \Sigma_{2}
  10. Π 1 \Pi_{1}
  11. Π 1 \Pi_{1}
  12. Π 1 \Pi_{1}
  13. Π 2 \Pi_{2}
  14. ϕ = x y ψ ( x , y ) \phi=\forall x\exists y\;\psi(x,y)
  15. ψ \psi
  16. Σ 2 \Sigma_{2}
  17. s ( x ) = y ψ ( x , y ) s(x)=\exists y\;\psi(x,y)
  18. ϕ \phi
  19. c ( x , z ) V ( c , z ) c ( s ( x ) ) \exists c\forall(x,z)\;V(c,z)\wedge c(s(x))\,
  20. Σ 2 = Π 2 . \Sigma_{2}=\Pi_{2}.
  21. P H = Σ 2 . PH=\Sigma_{2}.
  22. 𝖭𝖯 𝖯 / 𝗉𝗈𝗅𝗒 \mathsf{NP}\subseteq\mathsf{P/poly}
  23. C n C_{n}
  24. D n D_{n}
  25. Π 2 \Pi_{2}
  26. L = { z : x . y . ϕ ( x , y , z ) } L=\{z:\forall x.\exists y.\phi(x,y,z)\}\,
  27. y . ϕ ( x , y , z ) \exists y.\phi(x,y,z)
  28. D n D_{n}
  29. n = | z | n=|z|
  30. ¬ y . ϕ ( x , y , z ) \neg\exists y.\phi(x,y,z)
  31. ϕ ( x , D ( x , z ) , z ) \phi(x,D(x,z),z)\;
  32. Π 2 \Pi_{2}
  33. L L
  34. Σ 2 \Sigma_{2}
  35. Π 2 \Pi_{2}
  36. Π 2 \Pi_{2}
  37. Π 2 𝖲 2 P Σ 2 \Pi_{2}\subseteq\mathsf{S}_{2}^{P}\subseteq\Sigma_{2}
  38. 𝖭𝖯 𝖯 / 𝗉𝗈𝗅𝗒 𝖠𝖬 = 𝖬𝖠 \mathsf{NP}\subseteq\mathsf{P/poly}\implies\mathsf{AM}=\mathsf{MA}
  39. z L Pr x [ y . ϕ ( x , y , z ) ] 2 3 z\in L\implies\Pr\nolimits_{x}[\exists y.\phi(x,y,z)]\geq\tfrac{2}{3}
  40. z L Pr x [ y . ϕ ( x , y , z ) ] 1 3 z\notin L\implies\Pr\nolimits_{x}[\exists y.\phi(x,y,z)]\leq\tfrac{1}{3}
  41. y . ϕ ( x , y , z ) \exists y.\phi(x,y,z)
  42. D n D_{n}
  43. z L Pr x [ ϕ ( x , D n ( x , z ) , z ) ] 2 3 z\in L\implies\Pr\nolimits_{x}[\phi(x,D_{n}(x,z),z)]\geq\tfrac{2}{3}
  44. z L Pr x [ ϕ ( x , D n ( x , z ) , z ) ] 1 3 z\notin L\implies\Pr\nolimits_{x}[\phi(x,D_{n}(x,z),z)]\leq\tfrac{1}{3}
  45. D n D_{n}
  46. z L D . Pr x [ ϕ ( x , D ( x , z ) , z ) ] 2 3 z\in L\implies\exists D.\Pr\nolimits_{x}[\phi(x,D(x,z),z)]\geq\tfrac{2}{3}
  47. z L D . Pr x [ ϕ ( x , D ( x , z ) , z ) ] 1 3 z\notin L\implies\forall D.\Pr\nolimits_{x}[\phi(x,D(x,z),z)]\leq\tfrac{1}{3}
  48. L L
  49. L L
  50. Σ 2 \Sigma_{2}
  51. Σ 2 𝖯 / 𝗉𝗈𝗅𝗒 \Sigma_{2}\not\subseteq\mathsf{P/poly}
  52. L Σ 4 - 𝖲𝖨𝖹𝖤 ( n k ) L\in\Sigma_{4}-\mathsf{SIZE}(n^{k})
  53. 𝖲𝖠𝖳 𝖯 / 𝗉𝗈𝗅𝗒 \mathsf{SAT}\notin\mathsf{P/poly}
  54. 𝖲𝖠𝖳 𝖲𝖨𝖹𝖤 ( n k ) \mathsf{SAT}\notin\mathsf{SIZE}(n^{k})
  55. 𝖲𝖠𝖳 𝖯 / 𝗉𝗈𝗅𝗒 \mathsf{SAT}\in\mathsf{P/poly}
  56. Σ 4 = Σ 2 \Sigma_{4}=\Sigma_{2}
  57. L Σ 2 - 𝖲𝖨𝖹𝖤 ( n k ) L\in\Sigma_{2}-\mathsf{SIZE}(n^{k})
  58. L 𝖲 2 P - 𝖲𝖨𝖹𝖤 ( n k ) L\in\mathsf{S}_{2}^{P}-\mathsf{SIZE}(n^{k})
  59. 𝖲𝖨𝖹𝖤 ( n k ) \mathsf{SIZE}(n^{k})
  60. 𝖯𝖯 𝖯 / 𝗉𝗈𝗅𝗒 \mathsf{PP}\not\subseteq\mathsf{P/poly}
  61. 𝖯𝖯 𝖲𝖨𝖹𝖤 ( n k ) \mathsf{PP}\not\subseteq\mathsf{SIZE}(n^{k})
  62. 𝖯 𝖯 𝖯 / 𝗉𝗈𝗅𝗒 \mathsf{P^{\sharp P}}\subseteq\mathsf{P/poly}
  63. 𝖯 𝖯 𝖯𝖯 𝖬𝖠 \mathsf{P^{\sharp P}}\supseteq\mathsf{PP}\supseteq\mathsf{MA}
  64. 𝖯 𝖯 𝖯𝖧 Σ 2 𝖬𝖠 \mathsf{P^{\sharp P}}\supseteq\mathsf{PH}\supseteq\Sigma_{2}\supseteq\mathsf{MA}
  65. 𝖯 𝖯 = 𝖬𝖠 \mathsf{P^{\sharp P}}=\mathsf{MA}
  66. 𝖯𝖯 = Σ 2 𝖲𝖨𝖹𝖤 ( n k ) \mathsf{PP}=\Sigma_{2}\not\subseteq\mathsf{SIZE}(n^{k})
  67. S 2 P 𝖹𝖯𝖯 𝖭𝖯 S_{2}^{P}\subseteq\mathsf{ZPP}^{\mathsf{NP}}

Karush–Kuhn–Tucker_conditions.html

  1. f ( x ) f(x)
  2. g i ( x ) 0 , h j ( x ) = 0 g_{i}(x)\leq 0,h_{j}(x)=0
  3. f f
  4. g i ( i = 1 , , m ) g_{i}\ (i=1,\ldots,m)
  5. h j ( j = 1 , , l ) h_{j}\ (j=1,\ldots,l)
  6. f : n f:\mathbb{R}^{n}\rightarrow\mathbb{R}
  7. g i : n g_{i}:\,\!\mathbb{R}^{n}\rightarrow\mathbb{R}
  8. h j : n h_{j}:\,\!\mathbb{R}^{n}\rightarrow\mathbb{R}
  9. x * x^{*}
  10. x * x^{*}
  11. μ i ( i = 1 , , m ) \mu_{i}\ (i=1,\ldots,m)
  12. λ j ( j = 1 , , l ) \lambda_{j}\ (j=1,\ldots,l)
  13. f ( x * ) = i = 1 m μ i g i ( x * ) + j = 1 l λ j h j ( x * ) , \nabla f(x^{*})=\sum_{i=1}^{m}\mu_{i}\nabla g_{i}(x^{*})+\sum_{j=1}^{l}\lambda% _{j}\nabla h_{j}(x^{*}),
  14. - f ( x * ) = i = 1 m μ i g i ( x * ) + j = 1 l λ j h j ( x * ) , -\nabla f(x^{*})=\sum_{i=1}^{m}\mu_{i}\nabla g_{i}(x^{*})+\sum_{j=1}^{l}% \lambda_{j}\nabla h_{j}(x^{*}),
  15. g i ( x * ) 0 , for all i = 1 , , m g_{i}(x^{*})\leq 0,\mbox{ for all }~{}i=1,\ldots,m
  16. h j ( x * ) = 0 , for all j = 1 , , l h_{j}(x^{*})=0,\mbox{ for all }~{}j=1,\ldots,l\,\!
  17. μ i 0 , for all i = 1 , , m \mu_{i}\geq 0,\mbox{ for all }~{}i=1,\ldots,m
  18. μ i g i ( x * ) = 0 , for all i = 1 , , m . \mu_{i}g_{i}(x^{*})=0,\mbox{for all}~{}\;i=1,\ldots,m.
  19. m = 0 m=0
  20. x * x^{*}
  21. g i g_{i}
  22. h j h_{j}
  23. x * x^{*}
  24. x * x^{*}
  25. x * x^{*}
  26. x * x^{*}
  27. x * x^{*}
  28. x * x^{*}
  29. λ i \lambda_{i}
  30. μ j \mu_{j}
  31. x k x * x_{k}\to x^{*}
  32. λ i 0 λ i h i ( x k ) > 0 \lambda_{i}\neq 0\Rightarrow\lambda_{i}h_{i}(x_{k})>0
  33. μ j 0 μ j g j ( x k ) > 0 \mu_{j}\neq 0\Rightarrow\mu_{j}g_{j}(x_{k})>0
  34. x x
  35. h ( x ) = 0 h(x)=0
  36. g i ( x ) < 0 g_{i}(x)<0
  37. v 1 , , v n v_{1},\ldots,v_{n}
  38. a 1 0 , , a n 0 a_{1}\geq 0,\ldots,a_{n}\geq 0
  39. a 1 v 1 + + a n v n = 0 a_{1}v_{1}+\cdots+a_{n}v_{n}=0
  40. f f
  41. g j g_{j}
  42. h i h_{i}
  43. x * , λ * , ρ * x^{*},\lambda^{*},\rho^{*}
  44. ρ * \rho^{*}
  45. x * x^{*}
  46. ρ i > 0 \mathbb{\rho}_{i}>0
  47. s 0 s\neq 0
  48. [ g ( x * ) x , h ( x * ) x ] T s = 0 \left[\frac{\partial g(x^{*})}{\partial x},\frac{\partial h(x^{*})}{\partial x% }\right]^{T}s=0
  49. s x x 2 L ( x * , λ * , ρ * ) s > 0 s^{\prime}\nabla^{2}_{xx}L(x^{*},\lambda^{*},\rho^{*})s>0
  50. G m i n G_{min}
  51. - R ( Q ) -R(Q)
  52. G m i n R ( Q ) - C ( Q ) G_{min}\leq R(Q)-C(Q)
  53. Q 0 , Q\geq 0,
  54. ( d R / d Q ) ( 1 + μ ) - μ ( d C / d Q ) 0 , (\,\text{d}R/\,\text{d}Q)(1+\mu)-\mu(\,\text{d}C/\,\text{d}Q)\leq 0,
  55. Q 0 , Q\geq 0,
  56. Q [ ( d R / d Q ) ( 1 + μ ) - μ ( d C / d Q ) ] = 0 , Q[(\,\text{d}R/\,\text{d}Q)(1+\mu)-\mu(\,\text{d}C/\,\text{d}Q)]=0,
  57. R ( Q ) - C ( Q ) - G m i n 0 , R(Q)-C(Q)-G_{min}\geq 0,
  58. μ 0 , \mu\geq 0,
  59. μ [ R ( Q ) - C ( Q ) - G m i n ] = 0. \mu[R(Q)-C(Q)-G_{min}]=0.
  60. d R / d Q = μ 1 + μ ( d C / d Q ) . \,\text{d}R/\,\text{d}Q=\frac{\mu}{1+\mu}(\,\text{d}C/\,\text{d}Q).
  61. d R / d Q \,\text{d}R/\,\text{d}Q
  62. d C / d Q \,\text{d}C/\,\text{d}Q
  63. μ \mu
  64. μ \mu
  65. d R / d Q \,\text{d}R/\,\text{d}Q
  66. d C / d Q \,\text{d}C/\,\text{d}Q
  67. Maximize f ( x ) \,\text{Maximize }\;f(x)
  68. subject to \,\text{subject to }
  69. g i ( x ) a i , h j ( x ) = 0. g_{i}(x)\leq a_{i},h_{j}(x)=0.
  70. V ( a 1 , , a n ) = sup x f ( x ) V(a_{1},\ldots,a_{n})=\sup\limits_{x}f(x)
  71. subject to \,\text{subject to }
  72. g i ( x ) a i , h j ( x ) = 0 g_{i}(x)\leq a_{i},h_{j}(x)=0
  73. j { 1 , , l } , i { 1 , , m } . j\in\{1,\ldots,l\},i\in\{1,\ldots,m\}.
  74. { a m | for some x X , g i ( x ) a i , i { 1 , , m } . \{a\in\mathbb{R}^{m}|\,\text{for some }x\in X,g_{i}(x)\leq a_{i},i\in\{1,% \ldots,m\}.
  75. μ i \mu_{i}
  76. a i a_{i}
  77. a i a_{i}
  78. μ 0 \mu_{0}
  79. f ( x * ) \nabla f(x^{*})
  80. μ 0 f ( x * ) + i = 1 m μ i g i ( x * ) + j = 1 l λ j h j ( x * ) = 0 , \mu_{0}\nabla f(x^{*})+\sum_{i=1}^{m}\mu_{i}\nabla g_{i}(x^{*})+\sum_{j=1}^{l}% \lambda_{j}\nabla h_{j}(x^{*})=0,

Kaufmann_(Scully)_vortex.html

  1. V Θ ( r ) = Γ 2 π r r c 2 + r 2 V_{\Theta}\ (r)=\frac{\Gamma}{2\pi}\frac{r}{r_{c}^{2}+r^{2}}

Kaup–Kupershmidt_equation.html

  1. u t = u x x x x x + 10 u x x x u + 25 u x x u x + 20 u 2 u x = 1 6 ( 6 u x x x x + 60 u u x x + 45 u x 2 + 40 u 3 ) x . u_{t}=u_{xxxxx}+10u_{xxx}u+25u_{xx}u_{x}+20u^{2}u_{x}=\frac{1}{6}(6u_{xxxx}+60% uu_{xx}+45u_{x}^{2}+40u^{3})_{x}.
  2. x 3 + 2 u x + u x , \partial_{x}^{3}+2u\partial_{x}+u_{x},

KBounce.html

  1. 2 ( x - 75 ) ( y + 5 ) 2\left(x-75\right)(y+5)
  2. 2 ( x - 75 ) ( y + 5 ) + ( z 15 ) 2\left(x-75\right)(y+5)+\left(z\cdot 15\right)
  3. 68 2 = 136 68\cdot 2=136
  4. 136 / 512 = 25 % 136/512=25\%
  5. 2 ( x - 75 ) ( y + 5 ) + ( z 15 ) 2\left(x-75\right)(y+5)+\left(z\cdot 15\right)
  6. k = 1 135 2 ( 100 - ( 504 - ( k + 1 ) 504 100 ) ) ( k + 5 ) + ( ( k + 1 ) 15 ) 490921.43 \sum_{k=1}^{135}\ 2\left(100-\left(\frac{504-(k+1)}{504}\cdot 100\right)\right% )(k+5)+((k+1)\cdot 15)\approx 490921.43

Kelly_criterion.html

  1. f * = b p - q b = p ( b + 1 ) - 1 b , f^{*}=\frac{bp-q}{b}=\frac{p(b+1)-1}{b},\!
  2. f * = expected net winnings net winnings if you win f^{*}=\frac{\,\text{expected net winnings}}{\,\text{net winnings if you win}}\!
  3. f * = p - q . f^{*}=p-q.\!
  4. f * = 2 p - 1. f^{*}=2p-1.\!
  5. p p
  6. 1 1
  7. 1 + b 1+b
  8. q = 1 - p q=1-p
  9. 1 1
  10. 1 - a 1-a
  11. f * = p / a - q / b . f^{*}=p/a-q/b.\!
  12. f * = p - q f^{*}=p-q
  13. b = a = 1 b=a=1
  14. ( f * > 0 ) (f^{*}>0)
  15. p b > q a . pb>qa.\!
  16. a > 1 a>1
  17. p p
  18. a a
  19. 2 N p K ( 1 - p ) N - K W . 2^{N}p^{K}(1-p)^{N-K}W\!.
  20. Δ \Delta
  21. Δ \Delta
  22. Δ \Delta
  23. Δ \Delta
  24. ( 2 p + Δ ) K [ 2 ( 1 - p ) - Δ ] N - K W (2p+\Delta)^{K}[2(1-p)-\Delta]^{N-K}W\!
  25. Δ \Delta
  26. K ( 2 p + Δ ) K - 1 [ 2 ( 1 - p ) - Δ ] N - K W - ( N - K ) ( 2 p + Δ ) K [ 2 ( 1 - p ) - Δ ] N - K - 1 W K(2p+\Delta)^{K-1}[2(1-p)-\Delta]^{N-K}W-(N-K)(2p+\Delta)^{K}[2(1-p)-\Delta]^{% N-K-1}W\!
  27. K [ 2 ( 1 - p ) - Δ ] = ( N - K ) ( 2 p + Δ ) K[2(1-p)-\Delta]=(N-K)(2p+\Delta)\!
  28. Δ = 2 ( K N - p ) \Delta=2(\frac{K}{N}-p)\!
  29. lim N + K N = p \lim_{N\to+\infty}\frac{K}{N}=p\!
  30. Δ \Delta
  31. ( 2 K N - 1 ) W \left(2\frac{K}{N}-1\right)W\!
  32. f f
  33. 1 - f + f ( 1 + b ) = 1 + f b 1-f+f(1+b)=1+fb
  34. 1 - f a 1-fa
  35. N N
  36. p N pN
  37. q N qN
  38. C N = ( 1 + f b ) p N ( 1 - f a ) q N . C_{N}=(1+fb)^{pN}(1-fa)^{qN}.
  39. log ( C N ) / N \log(C_{N})/N
  40. C N C_{N}
  41. f f
  42. f * = p / a - q / b . f^{*}=p/a-q/b.
  43. p p
  44. p p
  45. f * f^{*}
  46. p p
  47. b b
  48. p N pN
  49. q N qN
  50. N N
  51. p k p_{k}
  52. B k B_{k}
  53. β k = B k i B i = 1 1 + Q k , \beta_{k}=\frac{B_{k}}{\sum_{i}B_{i}}=\frac{1}{1+Q_{k}},
  54. Q k Q_{k}
  55. D = 1 - t t D=1-tt
  56. t t tt
  57. D β k \frac{D}{\beta_{k}}
  58. f k f_{k}
  59. S o S^{o}
  60. f k o f^{o}_{k}
  61. S o S^{o}
  62. e r k = D β k p k = D ( 1 + Q k ) p k . er_{k}=\frac{D}{\beta_{k}}p_{k}=D(1+Q_{k})p_{k}.
  63. e r k er_{k}
  64. e r 1 er_{1}
  65. S = S=\varnothing
  66. k = 1 k=1
  67. R ( S ) = 1 R(S)=1
  68. e r k = e r 1 er_{k}=er_{1}
  69. e r k = D β k p k > R ( S ) er_{k}=\frac{D}{\beta_{k}}p_{k}>R(S)
  70. S = S { k } S=S\cup\{k\}
  71. R ( S ) R(S)
  72. R ( S ) = 1 - i S p i 1 - i S β i D R(S)=\frac{1-\sum_{i\in S}{p_{i}}}{1-\sum_{i\in S}\frac{\beta_{i}}{D}}
  73. k = k + 1 k=k+1
  74. S o = S S^{o}=S
  75. S o S^{o}
  76. S o S^{o}
  77. f k o f^{o}_{k}
  78. f k o = e r k - R ( S o ) D β k = p k - R ( S o ) D β k f^{o}_{k}=\frac{er_{k}-R(S^{o})}{\frac{D}{\beta_{k}}}=p_{k}-\frac{R(S^{o})}{% \frac{D}{\beta_{k}}}
  79. R ( S o ) = 1 - i S o f i o R(S^{o})=1-\sum_{i\in S^{o}}{f^{o}_{i}}
  80. e r k = D β k p k > R ( S ) er_{k}=\frac{D}{\beta_{k}}p_{k}>R(S)
  81. S o S^{o}
  82. f k o f^{o}_{k}
  83. G o = i S p i log 2 ( e r i ) + ( 1 - i S p i ) log 2 ( R ( S o ) ) , G^{o}=\sum_{i\in S}{p_{i}\log_{2}{(er_{i})}}+(1-\sum_{i\in S}{p_{i}})\log_{2}{% (R(S^{o}))},
  84. T d = 1 G o . T_{d}=\frac{1}{G^{o}}.
  85. p k p_{k}
  86. i p i < 1 \sum_{i}{p_{i}}<1
  87. i β i < 1 \sum_{i}{\beta_{i}}<1
  88. n n
  89. S k S_{k}
  90. r k r_{k}
  91. k = 1 , , n k=1,...,n
  92. r r
  93. u k u_{k}
  94. S k S_{k}
  95. 𝔼 [ ln ( ( 1 + r ) + k = 1 n u k ( r k - r ) ) ] \mathbb{E}\left[\ln\left((1+r)+\sum\limits_{k=1}^{n}u_{k}(r_{k}-r)\right)\right]
  96. u 0 = ( 0 , , 0 ) \vec{u_{0}}=(0,\ldots,0)
  97. 𝔼 [ ln ( 1 + r ) + k = 1 n u k ( r k - r ) 1 + r - 1 2 k = 1 n j = 1 n u k u j ( r k - r ) ( r j - r ) ( 1 + r ) 2 ] \mathbb{E}\left[\ln(1+r)+\sum\limits_{k=1}^{n}\frac{u_{k}(r_{k}-r)}{1+r}-\frac% {1}{2}\sum\limits_{k=1}^{n}\sum\limits_{j=1}^{n}u_{k}u_{j}\frac{(r_{k}-r)(r_{j% }-r)}{(1+r)^{2}}\right]
  98. u = ( 1 + r ) ( Σ ^ ) - 1 ( r ^ - r ) \vec{u^{\star}}=(1+r)(\widehat{\Sigma})^{-1}(\widehat{\vec{r}}-r)
  99. r ^ \widehat{\vec{r}}
  100. Σ ^ \widehat{\Sigma}

Kelvin_probe_force_microscope.html

  1. E = 1 2 C [ V D C + V A C sin ( ω 0 t ) ] 2 = 1 2 C [ 2 V D C V A C sin ( ω 0 t ) - 1 2 V A C 2 cos ( 2 ω 0 t ) ] E=\frac{1}{2}C[V_{DC}+V_{AC}\sin(\omega_{0}t)]^{2}=\frac{1}{2}C[2V_{DC}V_{AC}% \sin(\omega_{0}t)-\frac{1}{2}V_{AC}^{2}\cos(2\omega_{0}t)]
  2. V = ( V D C - V C P D ) + V A C sin ( ω t ) V=(V_{DC}-V_{CPD})+V_{AC}\cdot\sin(\omega t)
  3. F = 1 2 d C d z V 2 F=\frac{1}{2}\frac{dC}{dz}V^{2}
  4. F = F D C + F ω + F 2 ω F=F_{DC}+F_{\omega}+F_{2\omega}
  5. F D C = d C d z [ 1 2 ( V D C - V C P D ) 2 + 1 4 V A C 2 ] F_{DC}=\frac{dC}{dz}[\frac{1}{2}(V_{DC}-V_{CPD})^{2}+\frac{1}{4}V^{2}_{AC}]
  6. F ω = d C d z [ V D C - V C P D ] V A C sin ( ω t ) F_{\omega}=\frac{dC}{dz}[V_{DC}-V_{CPD}]V_{AC}\sin(\omega t)
  7. F 2 ω = - 1 4 d C d z V A C 2 cos ( 2 ω t ) F_{2\omega}=-\frac{1}{4}\frac{dC}{dz}V^{2}_{AC}\cos(2\omega t)

Kelvin–Voigt_material.html

  1. ε Total = ε D = ε S . \varepsilon\text{Total}=\varepsilon_{D}=\varepsilon_{S}.
  2. σ Total = σ D + σ S . \sigma\text{Total}=\sigma_{D}+\sigma_{S}.
  3. σ ( t ) = E ε ( t ) + η d ε ( t ) d t , \sigma(t)=E\varepsilon(t)+\eta\frac{d\varepsilon(t)}{dt},
  4. η \eta
  5. σ 0 \sigma_{0}
  6. σ 0 / E \sigma_{0}/E
  7. ε ( t ) = σ 0 E ( 1 - e - λ t ) , \varepsilon(t)=\frac{\sigma_{0}}{E}(1-e^{-\lambda t}),
  8. λ \lambda
  9. λ = E η \lambda=\frac{E}{\eta}
  10. t 1 t_{1}
  11. ε ( t > t 1 ) = ε ( t 1 ) e - λ ( t - t 1 ) . \varepsilon(t>t_{1})=\varepsilon(t_{1})e^{-\lambda(t-t_{1})}.
  12. E ε ( t ) σ 0 \frac{E\varepsilon(t)}{\sigma_{0}}
  13. λ t \lambda t
  14. t = 0 t=0
  15. t 1 * = λ t 1 t_{1}^{*}=\lambda t_{1}
  16. lim t ε = σ 0 E , \lim_{t\to\infty}\varepsilon=\frac{\sigma_{0}}{E},
  17. E ( ω ) = E + i η ω . E^{\star}(\omega)=E+i\eta\omega.
  18. E 1 = [ E ( ω ) ] = E , E_{1}=\Re[E(\omega)]=E,
  19. E 2 = [ E ( ω ) ] = η ω . E_{2}=\Im[E(\omega)]=\eta\omega.
  20. E 1 E_{1}
  21. E 2 E_{2}
  22. η \eta

Kepler_problem.html

  1. 𝐅 = k r 2 𝐫 ^ \mathbf{F}=\frac{k}{r^{2}}\mathbf{\hat{r}}
  2. 𝐫 ^ \mathbf{\hat{r}}
  3. V ( r ) = k r V(r)=\frac{k}{r}
  4. r r
  5. m m
  6. V ( r ) V(r)
  7. m d 2 r d t 2 - m r ω 2 = m d 2 r d t 2 - L 2 m r 3 = - d V d r m\frac{d^{2}r}{dt^{2}}-mr\omega^{2}=m\frac{d^{2}r}{dt^{2}}-\frac{L^{2}}{mr^{3}% }=-\frac{dV}{dr}
  8. ω d θ d t \omega\equiv\frac{d\theta}{dt}
  9. L = m r 2 ω L=mr^{2}\omega
  10. d V d r \frac{dV}{dr}
  11. m r ω 2 mr\omega^{2}
  12. t t
  13. θ \theta
  14. d d t = L m r 2 d d θ \frac{d}{dt}=\frac{L}{mr^{2}}\frac{d}{d\theta}
  15. L r 2 d d θ ( L m r 2 d r d θ ) - L 2 m r 3 = - d V d r \frac{L}{r^{2}}\frac{d}{d\theta}\left(\frac{L}{mr^{2}}\frac{dr}{d\theta}\right% )-\frac{L^{2}}{mr^{3}}=-\frac{dV}{dr}
  16. L r 2 d d θ ( L m r 2 d r d θ ) = - 2 L 2 m r 5 ( d r d θ ) 2 + L 2 m r 4 d 2 r d θ 2 \frac{L}{r^{2}}\frac{d}{d\theta}\left(\frac{L}{mr^{2}}\frac{dr}{d\theta}\right% )=-\frac{{2}L^{2}}{mr^{5}}\left(\frac{dr}{d\theta}\right)^{2}+\frac{L^{2}}{mr^% {4}}\frac{d^{2}r}{d\theta^{2}}
  17. u 1 r u\equiv\frac{1}{r}
  18. m r 2 L 2 \frac{mr^{2}}{L^{2}}
  19. d u d θ = - 1 r 2 d r d θ \frac{du}{d\theta}=\frac{-1}{r^{2}}\frac{dr}{d\theta}
  20. d 2 u d θ 2 = 2 r 3 ( d r d θ ) 2 - 1 r 2 d 2 r d θ 2 \frac{d^{2}u}{d\theta^{2}}=\frac{2}{r^{3}}\left(\frac{dr}{d\theta}\right)^{2}-% \frac{1}{r^{2}}\frac{d^{2}r}{d\theta^{2}}
  21. d 2 u d θ 2 + u = - m L 2 d d u V ( 1 / u ) \frac{d^{2}u}{d\theta^{2}}+u=-\frac{m}{L^{2}}\frac{d}{du}V(1/u)
  22. V ( 𝐫 ) = k r = k u V(\mathbf{r})=\frac{k}{r}=ku
  23. u ( θ ) u(\theta)
  24. d 2 u d θ 2 + u = - m L 2 d d u V ( 1 / u ) = - k m L 2 \frac{d^{2}u}{d\theta^{2}}+u=-\frac{m}{L^{2}}\frac{d}{du}V(1/u)=-\frac{km}{L^{% 2}}
  25. - k m L 2 -\frac{km}{L^{2}}
  26. u 1 r = - k m L 2 [ 1 + e cos ( θ - θ 0 ) ] u\equiv\frac{1}{r}=-\frac{km}{L^{2}}\left[1+e\cos\left(\theta-\theta_{0}\right% )\right]
  27. e e
  28. θ 0 \theta_{0}
  29. e = 0 e=0
  30. e < 1 e<1
  31. e = 1 e=1
  32. e > 1 e>1
  33. e e
  34. E E
  35. e = 1 + 2 E L 2 k 2 m e=\sqrt{1+\frac{2EL^{2}}{k^{2}m}}
  36. E < 0 E<0
  37. E = 0 E=0
  38. E > 0 E>0
  39. E = - k 2 m 2 L 2 E=-\frac{k^{2}m}{2L^{2}}

Kerala_school_of_astronomy_and_mathematics.html

  1. 1 1 - x = 1 + x + x 2 + x 3 + \frac{1}{1-x}=1+x+x^{2}+x^{3}+\dots
  2. | x | < 1 |x|<1
  3. 1 p + 2 p + + n p n p + 1 p + 1 1^{p}+2^{p}+\cdots+n^{p}\approx\frac{n^{p+1}}{p+1}
  4. sin x \sin x
  5. cos x \cos x
  6. arctan x \arctan x
  7. r arctan ( y x ) = 1 1 r y x - 1 3 r y 3 x 3 + 1 5 r y 5 x 5 - , r\arctan(\frac{y}{x})=\frac{1}{1}\cdot\frac{ry}{x}-\frac{1}{3}\cdot\frac{ry^{3% }}{x^{3}}+\frac{1}{5}\cdot\frac{ry^{5}}{x^{5}}-\cdots,
  8. y / x 1. y/x\leq 1.
  9. r sin x r = x - x x 2 ( 2 2 + 2 ) r 2 + x x 2 ( 2 2 + 2 ) r 2 x 2 ( 4 2 + 4 ) r 2 - r\sin\frac{x}{r}=x-x\cdot\frac{x^{2}}{(2^{2}+2)r^{2}}+x\cdot\frac{x^{2}}{(2^{2% }+2)r^{2}}\cdot\frac{x^{2}}{(4^{2}+4)r^{2}}-\cdot
  10. r ( 1 - cos x r ) = r x 2 ( 2 2 - 2 ) r 2 - r x 2 ( 2 2 - 2 ) r 2 x 2 ( 4 2 - 4 ) r 2 + , r(1-\cos\frac{x}{r})=r\cdot\frac{x^{2}}{(2^{2}-2)r^{2}}-r\cdot\frac{x^{2}}{(2^% {2}-2)r^{2}}\cdot\frac{x^{2}}{(4^{2}-4)r^{2}}+\cdots,
  11. r = 1 r=1
  12. sin x = x - x 3 3 ! + x 5 5 ! - x 7 7 ! + \sin x=x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\frac{x^{7}}{7!}+\cdots
  13. cos x = 1 - x 2 2 ! + x 4 4 ! - x 6 6 ! + \cos x=1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-\frac{x^{6}}{6!}+\cdots
  14. arctan x \arctan x
  15. π \pi
  16. π 4 = 1 - 1 3 + 1 5 - 1 7 + \frac{\pi}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\ldots
  17. f i ( n + 1 ) f_{i}(n+1)
  18. π 4 1 - 1 3 + 1 5 - ( - 1 ) ( n - 1 ) / 2 1 n + ( - 1 ) ( n + 1 ) / 2 f i ( n + 1 ) \frac{\pi}{4}\approx 1-\frac{1}{3}+\frac{1}{5}-\cdots(-1)^{(n-1)/2}\frac{1}{n}% +(-1)^{(n+1)/2}f_{i}(n+1)
  19. f 1 ( n ) = 1 2 n , f 2 ( n ) = n / 2 n 2 + 1 , f 3 ( n ) = ( n / 2 ) 2 + 1 ( n 2 + 5 ) n / 2 . f_{1}(n)=\frac{1}{2n},\ f_{2}(n)=\frac{n/2}{n^{2}+1},\ f_{3}(n)=\frac{(n/2)^{2% }+1}{(n^{2}+5)n/2}.
  20. 1 n 3 - n \frac{1}{n^{3}-n}
  21. π \pi
  22. π 4 = 3 4 + 1 3 3 - 3 - 1 5 3 - 5 + 1 7 3 - 7 - \frac{\pi}{4}=\frac{3}{4}+\frac{1}{3^{3}-3}-\frac{1}{5^{3}-5}+\frac{1}{7^{3}-7% }-\cdots
  23. 104348 / 33215 104348/33215
  24. π \pi
  25. 3.141592653 3.141592653
  26. π \pi

Keraunic_level.html

  1. N g = 0.04 T d 1.25 N_{g}=0.04\,{T_{d}}^{1.25}

Kernel_density_estimation.html

  1. f ^ h ( x ) = 1 n i = 1 n K h ( x - x i ) = 1 n h i = 1 n K ( x - x i h ) , \hat{f}_{h}(x)=\frac{1}{n}\sum_{i=1}^{n}K_{h}(x-x_{i})\quad=\frac{1}{nh}\sum_{% i=1}^{n}K\Big(\frac{x-x_{i}}{h}\Big),
  2. MISE ( h ) = E ( f ^ h ( x ) - f ( x ) ) 2 d x . \operatorname{MISE}(h)=E\int(\hat{f}_{h}(x)-f(x))^{2}\,dx.
  3. AMISE ( h ) = R ( K ) n h + 1 4 m 2 ( K ) 2 h 4 R ( f ′′ ) \operatorname{AMISE}(h)=\frac{R(K)}{nh}+\frac{1}{4}m_{2}(K)^{2}h^{4}R(f^{% \prime\prime})
  4. R ( g ) = g ( x ) 2 d x R(g)=\int g(x)^{2}\,dx
  5. m 2 ( K ) = x 2 K ( x ) d x m_{2}(K)=\int x^{2}K(x)\,dx
  6. h AMISE ( h ) = - R ( K ) n h 2 + m 2 ( K ) 2 h 3 R ( f ′′ ) = 0 \frac{\partial}{\partial h}\operatorname{AMISE}(h)=-\frac{R(K)}{nh^{2}}+m_{2}(% K)^{2}h^{3}R(f^{\prime\prime})=0
  7. h AMISE = R ( K ) 1 / 5 m 2 ( K ) 2 / 5 R ( f ′′ ) 1 / 5 n 1 / 5 . h_{\operatorname{AMISE}}=\frac{R(K)^{1/5}}{m_{2}(K)^{2/5}R(f^{\prime\prime})^{% 1/5}n^{1/5}}.
  8. h = ( 4 σ ^ 5 3 n ) 1 5 1.06 σ ^ n - 1 / 5 , h=\left(\frac{4\hat{\sigma}^{5}}{3n}\right)^{\frac{1}{5}}\approx 1.06\hat{% \sigma}n^{-1/5},
  9. σ ^ \hat{\sigma}
  10. φ ^ ( t ) = 1 n j = 1 n e i t x j \hat{\varphi}(t)=\frac{1}{n}\sum_{j=1}^{n}e^{itx_{j}}
  11. φ ^ ( t ) \scriptstyle\hat{\varphi}(t)
  12. φ ^ ( t ) \scriptstyle\hat{\varphi}(t)
  13. φ ^ ( t ) \scriptstyle\hat{\varphi}(t)
  14. φ ^ ( t ) \scriptstyle\hat{\varphi}(t)
  15. f ^ ( x ) \displaystyle\hat{f}(x)

Kessler_syndrome.html

  1. Δ H \Delta{H}

Keyword_density.html

  1. ( N k r / T k n ) * 100 (Nkr/Tkn)*100
  2. ( N k r * N w p / T k n ) * 100 (Nkr*Nwp/Tkn)*100

Khovanov_homology.html

  1. L 1 , L 2 L_{1},L_{2}
  2. L 3 L_{3}
  3. λ P ( L 1 ) - λ - 1 P ( L 2 ) = ( q - q - 1 ) P ( L 3 ) . \lambda P(L_{1})-\lambda^{-1}P(L_{2})=(q-q^{-1})P(L_{3}).
  4. λ = q n , n 0 \lambda=q^{n},\ n\leq 0
  5. P n ( L ) [ q , q - 1 ] P_{n}(L)\in\mathbb{Z}[q,q^{-1}]
  6. n > 0 n>0
  7. P n ( u n k n o t ) = q n - 1 + q n - 3 + + q 1 - n P_{n}(unknot)=q^{n-1}+q^{n-3}+\cdots+q^{1-n}
  8. P 0 ( u n k n o t ) = 1 P_{0}(unknot)=1
  9. n > 0 n>0
  10. P n ( L ) P_{n}(L)
  11. s l ( n ) sl(n)
  12. P 0 ( L ) P_{0}(L)
  13. U q ( g l ( 1 | 1 ) ) U_{q}(gl(1|1))
  14. P 0 ( L ) P_{0}(L)
  15. P 1 ( L ) = 1 P_{1}(L)=1
  16. P 2 ( L ) P_{2}(L)

Kirby–Siebenmann_class.html

  1. e ( M ) H 4 ( M ; 𝐙 2 ) e(M)\in H^{4}(M;\mathbf{Z}_{2})

Klee's_measure_problem.html

  1. O ( n log n ) O(n\log n)
  2. O ( n log n ) O(n\log n)
  3. O ( n d - 1 log n ) O(n^{d-1}\log n)
  4. O ( n d - 1 ) O(n^{d-1})
  5. O ( n d / 2 log n ) O(n^{d/2}\log n)
  6. O ( n d / 2 ) O(n^{d/2})
  7. Ω ( n log n ) \Omega(n\log n)
  8. O ( n d / 2 ) O(n^{d/2})

Klein_geometry.html

  1. g . ( a H ) = ( g a ) H . g.(aH)=(ga)H.
  2. H G G / H . H\to G\to G/H.
  3. K = { k G : g - 1 k g H g G } . K=\{k\in G:g^{-1}kg\in H\;\;\forall g\in G\}.
  4. 𝔥 \mathfrak{h}
  5. 𝔤 \mathfrak{g}
  6. P n \mathbb{R}\mathrm{P}^{n}
  7. PGL ( n + 1 ) \mathrm{PGL}(n+1)
  8. P P
  9. { 0 } V 1 V n \{0\}\subset V_{1}\subset V_{n}
  10. S n S^{n}
  11. ( n + 2 ) (n+2)
  12. O ( n + 1 , 1 ) \mathrm{O}(n+1,1)
  13. P P
  14. H ( n ) H(n)
  15. \R 1 , n \R^{1,n}
  16. O ( 1 , n ) / O ( 1 ) \mathrm{O}(1,n)/\mathrm{O}(1)
  17. O ( 1 ) × O ( n ) \mathrm{O}(1)\times\mathrm{O}(n)
  18. n + 1 \mathbb{R}^{n+1}
  19. O ( n + 1 ) / O ( 1 ) \mathrm{O}(n+1)/\mathrm{O}(1)
  20. S n S^{n}
  21. O ( n + 1 ) \mathrm{O}(n+1)
  22. O ( n ) \mathrm{O}(n)
  23. A ( n ) \R n A(n)\simeq\R^{n}
  24. Aff ( n ) \R n GL ( n ) \mathrm{Aff}(n)\simeq\R^{n}\rtimes\mathrm{GL}(n)
  25. GL ( n ) \mathrm{GL}(n)
  26. E ( n ) E(n)
  27. Euc ( n ) \R n O ( n ) \mathrm{Euc}(n)\simeq\R^{n}\rtimes\mathrm{O}(n)
  28. O ( n ) \mathrm{O}(n)

Klein_paradox.html

  1. V m c 2 V\sim mc^{2}
  2. V 0 V_{0}
  3. E 0 < V 0 E_{0}<V_{0}
  4. p p
  5. ψ \psi
  6. ( σ x p + V ) ψ = E 0 ψ , V = { 0 , x < 0 V 0 , x > 0 \left(\sigma_{x}p+V\right)\psi=E_{0}\psi,\quad V=\begin{cases}0,&x<0\\ V_{0},&x>0\end{cases}
  7. σ x \sigma_{x}
  8. σ x = ( 0 1 1 0 ) \sigma_{x}=\left(\begin{matrix}0&1\\ 1&0\end{matrix}\right)
  9. ψ 1 = A e i p x ( 1 1 ) + A e - i p x ( - 1 1 ) , p = E 0 \psi_{1}=Ae^{ipx}\left(\begin{matrix}1\\ 1\end{matrix}\right)+A^{\prime}e^{-ipx}\left(\begin{matrix}-1\\ 1\end{matrix}\right),\quad p=E_{0}\,
  10. ψ 2 = B e i k x ( 1 1 ) , | k | = V 0 - E 0 \psi_{2}=Be^{ikx}\left(\begin{matrix}1\\ 1\end{matrix}\right),\quad\left|k\right|=V_{0}-E_{0}\,
  11. A A
  12. A A′
  13. B B
  14. T , R . T,R.
  15. J i = ψ i σ x ψ i , i = 1 , 2 J_{i}=\psi_{i}^{\dagger}\sigma_{x}\psi_{i},\ i=1,2\,
  16. J 1 = 2 [ | A | 2 - | A | 2 ] , J 2 = 2 | B | 2 J_{1}=2\left[\left|A\right|^{2}-\left|A^{\prime}\right|^{2}\right],\quad J_{2}% =2\left|B\right|^{2}\,
  17. R = | A | 2 | A | 2 , T = | B | 2 | A | 2 R=\frac{\left|A^{\prime}\right|^{2}}{\left|A\right|^{2}},\quad T=\frac{\left|B% \right|^{2}}{\left|A\right|^{2}}\,
  18. x = 0 x=0
  19. | A | 2 = | B | 2 \left|A\right|^{2}=\left|B\right|^{2}\,
  20. | A | 2 = 0 \left|A^{\prime}\right|^{2}=0\,
  21. + m < E < V - m +m<E<V-m

Klein_quadric.html

  1. p 12 p 34 + p 13 p 42 + p 14 p 23 = 0 p_{12}p_{34}+p_{13}p_{42}+p_{14}p_{23}=0
  2. p i j = u i v j - u j v i p_{ij}=u_{i}v_{j}-u_{j}v_{i}
  3. C C
  4. C C^{\prime}

Kloosterman_sum.html

  1. a , b , m a,b,m
  2. K ( a , b ; m ) = 0 x m - 1 , gcd ( x , m ) = 1 e 2 π i m ( a x + b x * ) . K(a,b;m)=\sum_{0\leq x\leq m-1,\ \gcd(x,m)=1}e^{\frac{2\pi i}{m}(ax+bx^{*})}.
  3. x x
  4. m m
  5. a = 0 a=0
  6. b = 0 b=0
  7. K ( a , b ; m ) K(a,b;m)
  8. a a
  9. b b
  10. m m
  11. K ( a , b ; m ) = K ( b , a ; m ) K(a,b;m)=K(b,a;m)
  12. K ( a c , b ; m ) = K ( a , b c ; m ) K(ac,b;m)=K(a,bc;m)
  13. g c d ( c , m ) = 1 gcd(c,m)=1
  14. K ( a , b ; m ) = K ( n 2 a , n 2 b ; m 1 ) K ( n 1 a , n 1 b ; m 2 ) . K(a,b;m)=K\left(n_{2}a,n_{2}b;m_{1}\right)K\left(n_{1}a,n_{1}b;m_{2}\right).
  15. p p
  16. k 1 k≥1
  17. K ( a , b ; m ) K(a,b;m)
  18. K ( a , b ; m ) K(a,b;m)
  19. K 𝐑 K⊂\mathbf{R}
  20. 𝐐 ( ζ p α + ζ p α - 1 ) \mathbf{Q}\left(\zeta_{p^{\alpha}}+\zeta_{p^{\alpha}}^{-1}\right)
  21. p p
  22. 𝐐 ( ζ 2 α - 1 + ζ 2 α - 1 - 1 ) \mathbf{Q}\left(\zeta_{2^{\alpha-1}}+\zeta_{2^{\alpha-1}}^{-1}\right)
  23. 2 < s u p > α [ u ! ! ] m 2<sup>α[u^{\prime}!!^{\prime}]m
  24. K ( a , b ; m ) = d gcd ( a , b , m ) d K ( a b d 2 , 1 ; m d ) . K(a,b;m)=\sum_{d\mid\gcd(a,b,m)}d\cdot K\left(\tfrac{ab}{d^{2}},1;\tfrac{m}{d}% \right).
  25. p p
  26. K ( a , b ; p ) K(a,b;p)
  27. g c d ( a , p ) = 1 gcd(a,p)=1
  28. K ( a , a ; p ) = m = 0 p - 1 ( m 2 - 4 a 2 p ) e 2 π i m p , K(a,a;p)=\sum_{m=0}^{p-1}\left(\frac{m^{2}-4a^{2}}{p}\right)e^{\frac{2\pi im}{% p}},
  29. ( m ) \left(\tfrac{\ell}{m}\right)
  30. k > 1 , p k>1,p
  31. g c d ( p , 2 a b ) = 1 gcd(p,2ab)=1
  32. K ( a , b ; m ) = { 2 ( m ) m Re ( ε m e 4 π i m ) ( a p ) = ( b p ) 0 otherwise K(a,b;m)=\begin{cases}2\left(\frac{\ell}{m}\right)\sqrt{m}\,\text{ Re}\left(% \varepsilon_{m}e^{\frac{4\pi i\ell}{m}}\right)&\left(\tfrac{a}{p}\right)=\left% (\tfrac{b}{p}\right)\\ 0&\,\text{otherwise}\end{cases}
  33. a b m o d m ℓ≡abmodm
  34. m m
  35. ε m = { 1 m 1 mod 4 i m 3 mod 4 \varepsilon_{m}=\begin{cases}1&m\equiv 1\mod 4\\ i&m\equiv 3\mod 4\end{cases}
  36. | K ( a , b ; m ) | τ ( m ) gcd ( a , b , m ) m . |K(a,b;m)|\leq\tau(m)\sqrt{\gcd(a,b,m)}\sqrt{m}.
  37. τ ( m ) \tau(m)
  38. m m
  39. m m
  40. p p
  41. | K ( a , b ; p ) | 2 p , |K(a,b;p)|≤2\sqrt{p},
  42. p p
  43. C C
  44. C C
  45. 1 K t 1−Kt
  46. K K
  47. n A exp ( 2 π i a n * + b n m ) , \sum\limits_{n\in A}\exp\left(2\pi i\,\frac{an^{*}+bn}{m}\right),
  48. n n
  49. A A
  50. m m
  51. A \|A\|
  52. m m
  53. n * n^{*}
  54. n n
  55. m m
  56. n n * 1 mod m nn^{*}\equiv 1(\mod m)
  57. m \sqrt{m}
  58. p p
  59. α α
  60. m ε m^{\varepsilon}
  61. exp { ( ln m ) 2 / 3 + ε } \exp\{(\ln m)^{2/3+\varepsilon}\}
  62. ε > 0 \varepsilon>0
  63. n x { a n * + b n m } , p x { a p * + b p m } , {\sum_{n\leq x}}^{\prime}\left\{\frac{an^{*}+bn}{m}\right\},{\sum_{p\leq x}}^{% \prime}\left\{\frac{ap^{*}+bp}{m}\right\},
  64. n n
  65. ( n , m ) = 1 (n,m)=1
  66. p p
  67. m m
  68. α < { a n * + b n m } β \alpha<\left\{\frac{an^{*}+bn}{m}\right\}\leq\beta
  69. n , 1 n x n,1≤n≤x
  70. m m
  71. x < m x<\sqrt{m}
  72. 0 , 11 0,11
  73. { a n * + b n m } , \biggl\{\frac{an^{*}+bn}{m}\biggr\},
  74. 1 n x , ( n , m ) = 1 , x < m 1\leq n\leq x,(n,m)=1,x<\sqrt{m}
  75. c c
  76. π ( x ; q , l ) < c x φ ( q ) ln 2 x q , \pi(x;q,l)<\frac{cx}{\varphi(q)\ln\frac{2x}{q}},
  77. π ( x ; q , l ) \pi(x;q,l)
  78. p p
  79. x x
  80. p l ( mod q ) p\equiv l\;\;(\mathop{{\rm mod}}q)
  81. G G
  82. H G H\subset G
  83. G / H G/H

KMS_state.html

  1. ρ β , μ = e - β ( H - μ N ) Tr [ e - β ( H - μ N ) ] = e - β ( H - μ N ) Z ( β , μ ) \rho_{\beta,\mu}=\frac{e^{-\beta\left(H-\mu N\right)}}{\mathrm{Tr}\left[e^{-% \beta\left(H-\mu N\right)}\right]}=\frac{e^{-\beta\left(H-\mu N\right)}}{Z(% \beta,\mu)}
  2. Z ( β , μ ) = def Tr [ e - β ( H - μ N ) ] Z(\beta,\mu)\ \stackrel{\mathrm{def}}{=}\ \mathrm{Tr}\left[e^{-\beta\left(H-% \mu N\right)}\right]
  3. α τ ( A ) = def e i H τ A e - i H τ \alpha_{\tau}(A)\ \stackrel{\mathrm{def}}{=}\ e^{iH\tau}Ae^{-iH\tau}
  4. α τ μ ( A ) = def e i ( H - μ N ) τ A e - i ( H - μ N ) τ \alpha^{\mu}_{\tau}(A)\ \stackrel{\mathrm{def}}{=}\ e^{i\left(H-\mu N\right)% \tau}Ae^{-i\left(H-\mu N\right)\tau}
  5. α τ μ ( A ) B β , μ = Tr [ ρ α τ μ ( A ) B ] = Tr [ ρ B α τ + i β μ ( A ) ] = B α τ + i β μ ( A ) β , μ \langle\alpha^{\mu}_{\tau}(A)B\rangle_{\beta,\mu}=\mathrm{Tr}\left[\rho\alpha^% {\mu}_{\tau}(A)B\right]=\mathrm{Tr}\left[\rho B\alpha^{\mu}_{\tau+i\beta}(A)% \right]=\langle B\alpha^{\mu}_{\tau+i\beta}(A)\rangle_{\beta,\mu}
  6. α z μ ( A ) B \langle\alpha^{\mu}_{z}(A)B\rangle
  7. - β < z < 0 -\beta<\Im{z}<0
  8. B α z μ ( A ) \langle B\alpha^{\mu}_{z}(A)\rangle
  9. 0 < z < β 0<\Im{z}<\beta
  10. d d z α z μ ( A ) B = i α z μ ( [ H - μ N , A ] ) B \frac{d}{dz}\langle\alpha^{\mu}_{z}(A)B\rangle=i\langle\alpha^{\mu}_{z}\left(% \left[H-\mu N,A\right]\right)B\rangle
  11. d d z B α z μ ( A ) = i B α z μ ( [ H - μ N , A ] ) \frac{d}{dz}\langle B\alpha^{\mu}_{z}(A)\rangle=i\langle B\alpha^{\mu}_{z}% \left(\left[H-\mu N,A\right]\right)\rangle
  12. α τ μ ( A ) B = B α τ + i β μ ( A ) \langle\alpha^{\mu}_{\tau}(A)B\rangle=\langle B\alpha^{\mu}_{\tau+i\beta}(A)\rangle
  13. α z μ ( A ) B \langle\alpha^{\mu}_{z}(A)B\rangle
  14. B α z μ ( A ) \langle B\alpha^{\mu}_{z}(A)\rangle
  15. α τ μ ( A ) B \langle\alpha^{\mu}_{\tau}(A)B\rangle
  16. B α τ + i β μ ( A ) \langle B\alpha^{\mu}_{\tau+i\beta}(A)\rangle

Kneser_graph.html

  1. k k
  2. n n
  3. n n
  4. n n
  5. ( n - k k ) \textstyle{\left({{n-k}\atop{k}}\right)}
  6. K G < s u b > n , k KG<sub>n,k
  7. n 3 k n≥3k
  8. K G < s u b > n , k KG<sub>n,k
  9. n < 3 k n<3k
  10. k - 1 n - 2 k + 1 \left\lceil\frac{k-1}{n-2k}\right\rceil+1
  11. K G < s u b > n , k KG<sub>n,k
  12. j = 0 , , k j=0,...,k
  13. λ j = ( - 1 ) j ( n - k - j k - j ) \lambda_{j}=(-1)^{j}{\left({{n-k-j}\atop{k-j}}\right)}
  14. ( n j ) - ( n j - 1 ) {\left({{n}\atop{j}}\right)}-{\left({{n}\atop{j-1}}\right)}
  15. j > 0 j>0
  16. j = 0 j=0
  17. α ( K G n , k ) = ( n - 1 k - 1 ) \alpha(KG_{n,k})={\left({{n-1}\atop{k-1}}\right)}
  18. k k
  19. n n
  20. ( k 1 ) (k− 1)
  21. k = 2 k=2
  22. s s
  23. k k
  24. n k n−k
  25. n n
  26. ( n - k k ) \textstyle{\left({{n-k}\atop{k}}\right)}
  27. K G < s u b > n , k KG<sub>n,k

Kodaira_vanishing_theorem.html

  1. H q ( M , K M L ) = 0 H^{q}(M,K_{M}\otimes L)=0
  2. K M L K_{M}\otimes L
  3. H q ( M , L - 1 ) H^{q}(M,L^{\otimes-1})
  4. H q ( X , L Ω X / k p ) = 0 H^{q}(X,L\otimes\Omega^{p}_{X/k})=0
  5. p + q > d p+q>d
  6. H q ( X , L - 1 Ω X / k p ) = 0 H^{q}(X,L^{\otimes-1}\otimes\Omega^{p}_{X/k})=0
  7. p + q < d p+q<d

Kolmogorov's_inequality.html

  1. Pr ( max 1 k n | S k | λ ) 1 λ 2 Var [ S n ] 1 λ 2 k = 1 n Var [ X k ] , \Pr\left(\max_{1\leq k\leq n}|S_{k}|\geq\lambda\right)\leq\frac{1}{\lambda^{2}% }\operatorname{Var}[S_{n}]\equiv\frac{1}{\lambda^{2}}\sum_{k=1}^{n}% \operatorname{Var}[X_{k}],
  2. S 1 , S 2 , , S n S_{1},S_{2},\dots,S_{n}
  3. S 0 = 0 S_{0}=0
  4. S i 0 S_{i}\geq 0
  5. i i
  6. ( Z i ) i = 0 n (Z_{i})_{i=0}^{n}
  7. Z 0 = 0 Z_{0}=0
  8. Z i + 1 = { S i + 1 if max 1 j i S j < λ Z i otherwise Z_{i+1}=\left\{\begin{array}[]{ll}S_{i+1}&\,\text{ if }\displaystyle\max_{1% \leq j\leq i}S_{j}<\lambda\\ Z_{i}&\,\text{ otherwise}\end{array}\right.
  9. i i
  10. ( Z i ) i = 0 n (Z_{i})_{i=0}^{n}
  11. S i - S i - 1 S_{i}-S_{i-1}
  12. i = 1 n E [ ( S i - S i - 1 ) 2 ] = i = 1 n E [ S i 2 - 2 S i S i - 1 + S i - 1 2 ] = i = 1 n E [ S i 2 - 2 ( S i - 1 + S i - S i - 1 ) S i - 1 + S i - 1 2 ] = i = 1 n E [ S i 2 - S i - 1 2 ] - 2 E [ S i - 1 ( S i - S i - 1 ) ] = E [ S n 2 ] - E [ S 0 2 ] = E [ S n 2 ] . \begin{aligned}\displaystyle\sum_{i=1}^{n}\,\text{E}[(S_{i}-S_{i-1})^{2}]&% \displaystyle=\sum_{i=1}^{n}\,\text{E}[S_{i}^{2}-2S_{i}S_{i-1}+S_{i-1}^{2}]\\ &\displaystyle=\sum_{i=1}^{n}\,\text{E}\left[S_{i}^{2}-2(S_{i-1}+S_{i}-S_{i-1}% )S_{i-1}+S_{i-1}^{2}\right]\\ &\displaystyle=\sum_{i=1}^{n}\,\text{E}\left[S_{i}^{2}-S_{i-1}^{2}\right]-2\,% \text{E}\left[S_{i-1}(S_{i}-S_{i-1})\right]\\ &\displaystyle=\,\text{E}[S_{n}^{2}]-\,\text{E}[S_{0}^{2}]=\,\text{E}[S_{n}^{2% }].\end{aligned}
  13. ( Z i ) i = 0 n (Z_{i})_{i=0}^{n}
  14. Pr ( max 1 i n S i λ ) = Pr [ Z n λ ] 1 λ 2 E [ Z n 2 ] = 1 λ 2 i = 1 n E [ ( Z i - Z i - 1 ) 2 ] 1 λ 2 i = 1 n E [ ( S i - S i - 1 ) 2 ] = 1 λ 2 E [ S n 2 ] = 1 λ 2 Var [ S n ] \begin{aligned}\displaystyle\,\text{Pr}\left(\max_{1\leq i\leq n}S_{i}\geq% \lambda\right)&\displaystyle=\,\text{Pr}[Z_{n}\geq\lambda]\\ &\displaystyle\leq\frac{1}{\lambda^{2}}\,\text{E}[Z_{n}^{2}]=\frac{1}{\lambda^% {2}}\sum_{i=1}^{n}\,\text{E}[(Z_{i}-Z_{i-1})^{2}]\\ &\displaystyle\leq\frac{1}{\lambda^{2}}\sum_{i=1}^{n}\,\text{E}[(S_{i}-S_{i-1}% )^{2}]=\frac{1}{\lambda^{2}}\,\text{E}[S_{n}^{2}]=\frac{1}{\lambda^{2}}\,\text% {Var}[S_{n}]\end{aligned}

Kolmogorov_microscales.html

  1. η = ( ν 3 ϵ ) 1 / 4 \eta=\left(\frac{\nu^{3}}{\epsilon}\right)^{1/4}
  2. τ η = ( ν ϵ ) 1 / 2 \tau_{\eta}=\left(\frac{\nu}{\epsilon}\right)^{1/2}
  3. u η = ( ν ϵ ) 1 / 4 u_{\eta}=\left(\nu\epsilon\right)^{1/4}
  4. ϵ \epsilon
  5. ν \nu
  6. η \eta
  7. ϵ \epsilon
  8. ν \nu
  9. τ η = ( ν / ϵ ) 1 / 2 \tau_{\eta}=(\nu/\epsilon)^{1/2}
  10. ϵ \epsilon
  11. ν \nu
  12. τ η = ( 2 < E i j E i j > ) - 1 / 2 \tau_{\eta}=(2<E_{ij}E_{ij}>)^{-1/2}
  13. τ η = ( ν / ϵ ) 1 / 2 \tau_{\eta}=(\nu/\epsilon)^{1/2}
  14. ϵ = 2 ν < E i j E i j > \epsilon=2\nu<E_{ij}E_{ij}>
  15. R e = U L / ν = ( η / τ η ) η / ν = 1 Re=UL/\nu=(\eta/\tau_{\eta})\eta/\nu=1

Koopmans'_theorem.html

  1. I I
  2. N N
  3. ϵ H \epsilon_{H}
  4. N - δ N N-\delta N
  5. N N
  6. δ N 0 \delta N\rightarrow 0
  7. N + δ N N+\delta N
  8. I I
  9. | r | |{r}|\rightarrow\infty
  10. n ( r ) e - 2 2 m e I | r | n({r})\rightarrow e^{-2\sqrt{\frac{2m_{e}}{\hbar}I}|{r}|}
  11. ϵ H = - I \epsilon_{H}=-I

Kosmotropic.html

  1. Δ G HB \Delta G_{\rm HB}
  2. Δ G HB \Delta G_{\rm HB}
  3. Δ G HB \Delta G_{\rm HB}

König's_theorem_(kinetics).html

  1. T N = 1 2 𝐯 ¯ N 𝐯 ¯ N + 1 2 𝐇 ¯ N N ω R {}^{N}T=\frac{1}{2}{{}^{N}\mathbf{\bar{v}}}\cdot{{}^{N}\mathbf{\bar{v}}}+\frac% {1}{2}{{}^{N}\!\mathbf{\bar{H}}}\cdot^{N}{\!\!\mathbf{\omega}}^{R}
  2. 𝐯 ¯ N {{}^{N}\mathbf{\bar{v}}}
  3. 𝐇 ¯ N {{}^{N}\!\mathbf{\bar{H}}}
  4. ω R N {}^{N}{\!\!\mathbf{\omega}}^{R}

Kramers–Kronig_relations.html

  1. χ ( ω ) = χ 1 ( ω ) + i χ 2 ( ω ) \chi(\omega)=\chi_{1}(\omega)+i\chi_{2}(\omega)
  2. ω \omega
  3. χ 1 ( ω ) \chi_{1}(\omega)
  4. χ 2 ( ω ) \chi_{2}(\omega)
  5. ω \omega
  6. 1 / | ω | 1/|\omega|
  7. | ω | |\omega|\rightarrow\infty
  8. χ 1 ( ω ) = 1 π 𝒫 - χ 2 ( ω ) ω - ω d ω \chi_{1}(\omega)={1\over\pi}\mathcal{P}\!\!\!\int\limits_{-\infty}^{\infty}{% \chi_{2}(\omega^{\prime})\over\omega^{\prime}-\omega}\,d\omega^{\prime}
  9. χ 2 ( ω ) = - 1 π 𝒫 - χ 1 ( ω ) ω - ω d ω , \chi_{2}(\omega)=-{1\over\pi}\mathcal{P}\!\!\!\int\limits_{-\infty}^{\infty}{% \chi_{1}(\omega^{\prime})\over\omega^{\prime}-\omega}\,d\omega^{\prime},
  10. 𝒫 \mathcal{P}
  11. χ \chi
  12. ω χ ( ω ) / ( ω - ω ) \omega^{\prime}\rightarrow\chi(\omega^{\prime})/(\omega^{\prime}-\omega)
  13. ω \omega
  14. χ ( ω ) ω - ω d ω = 0 \oint{\chi(\omega^{\prime})\over\omega^{\prime}-\omega}\,d\omega^{\prime}=0
  15. ω = ω \omega=\omega^{\prime}
  16. | ω | |\omega|
  17. χ ( ω ) \chi(\omega)
  18. 1 / | ω | 1/|\omega|
  19. 0 = χ ( ω ) ω - ω d ω = 𝒫 - χ ( ω ) ω - ω d ω - i π χ ( ω ) . 0=\oint{\chi(\omega^{\prime})\over\omega^{\prime}-\omega}\,d\omega^{\prime}=% \mathcal{P}\!\!\!\int\limits_{-\infty}^{\infty}{\chi(\omega^{\prime})\over% \omega^{\prime}-\omega}\,d\omega^{\prime}-i\pi\chi(\omega).
  20. χ ( ω ) = 1 i π 𝒫 - χ ( ω ) ω - ω d ω . \chi(\omega)={1\over i\pi}\mathcal{P}\!\!\!\int\limits_{-\infty}^{\infty}{\chi% (\omega^{\prime})\over\omega^{\prime}-\omega}\,d\omega^{\prime}.
  21. i i
  22. χ ( ω ) \chi(\omega)
  23. χ ( t - t ) \chi(t-t^{\prime})\!
  24. P ( t ) P(t)\!
  25. F ( t ) F(t^{\prime})\!
  26. t . t^{\prime}.
  27. P ( t ) P(t)\!
  28. F ( t ) F(t)
  29. χ ( t - t ) \chi(t-t^{\prime})
  30. t < t t<t^{\prime}\!
  31. χ ( ω ) \chi(\omega)\!
  32. χ ( ω ) \chi(\omega)\!
  33. ω \omega\!
  34. χ ( ω ) \chi(\omega)\!
  35. - -\infty
  36. \infty
  37. χ ( ω ) \chi(\omega)
  38. χ ( t - t ) \chi(t-t^{\prime})
  39. χ ( - ω ) = χ * ( ω ) \chi(-\omega)=\chi^{*}(\omega)
  40. χ 1 ( ω ) \chi_{1}(\omega)
  41. χ 2 ( ω ) \chi_{2}(\omega)
  42. [ 0 , ) [0,\infty)
  43. χ 1 ( ω ) \chi_{1}(\omega)
  44. ω + ω \omega^{\prime}+\omega
  45. χ 1 ( ω ) = 1 π 𝒫 - ω χ 2 ( ω ) ω 2 - ω 2 d ω + ω π 𝒫 - χ 2 ( ω ) ω 2 - ω 2 d ω . \chi_{1}(\omega)={1\over\pi}\mathcal{P}\!\!\!\int\limits_{-\infty}^{\infty}{% \omega^{\prime}\chi_{2}(\omega^{\prime})\over\omega^{\prime 2}-\omega^{2}}\,d% \omega^{\prime}+{\omega\over\pi}\mathcal{P}\!\!\!\int\limits_{-\infty}^{\infty% }{\chi_{2}(\omega^{\prime})\over\omega^{\prime 2}-\omega^{2}}\,d\omega^{\prime}.
  46. χ 2 ( ω ) \chi_{2}(\omega)
  47. χ 1 ( ω ) = 2 π 𝒫 0 ω χ 2 ( ω ) ω 2 - ω 2 d ω . \chi_{1}(\omega)={2\over\pi}\mathcal{P}\!\!\!\int\limits_{0}^{\infty}{\omega^{% \prime}\chi_{2}(\omega^{\prime})\over\omega^{\prime 2}-\omega^{2}}\,d\omega^{% \prime}.
  48. χ 2 ( ω ) = - 2 π 𝒫 0 ω χ 1 ( ω ) ω 2 - ω 2 d ω = - 2 ω π 𝒫 0 χ 1 ( ω ) ω 2 - ω 2 d ω . \chi_{2}(\omega)=-{2\over\pi}\mathcal{P}\!\!\!\int\limits_{0}^{\infty}{\omega% \chi_{1}(\omega^{\prime})\over\omega^{\prime 2}-\omega^{2}}\,d\omega^{\prime}=% -{2\omega\over\pi}\mathcal{P}\!\!\!\int\limits_{0}^{\infty}{\chi_{1}(\omega^{% \prime})\over\omega^{\prime 2}-\omega^{2}}\,d\omega^{\prime}.

Kretschmann_scalar.html

  1. K = R a b c d R a b c d K=R_{abcd}\,R^{abcd}
  2. R a b c d R_{abcd}
  3. K = 48 G 2 M 2 c 4 r 6 . K=\frac{48G^{2}M^{2}}{c^{4}r^{6}}\,.
  4. C a b c d C a b c d C_{abcd}\,C^{abcd}
  5. C a b c d C_{abcd}
  6. d d
  7. R a b c d R a b c d = C a b c d C a b c d + 4 d - 2 R a b R a b - 2 ( d - 1 ) ( d - 2 ) R 2 R_{abcd}\,R^{abcd}=C_{abcd}\,C^{abcd}+\frac{4}{d-2}R_{ab}\,R^{ab}-\frac{2}{(d-% 1)(d-2)}R^{2}
  8. R a b R^{ab}
  9. R R
  10. R a b c d R a b c d R_{abcd}\,{{}^{\star}\!R}^{abcd}
  11. R a b c d {{}^{\star}R}^{abcd}
  12. F a b F a b , F a b F a b F_{ab}\,F^{ab},\;\;F_{ab}\,{{}^{\star}\!F}^{ab}

Kruskal–Szekeres_coordinates.html

  1. ( t , r , θ , ϕ ) (t,r,\theta,\phi)
  2. T = ( r 2 G M - 1 ) 1 / 2 e r / 4 G M sinh ( t 4 G M ) T=\left(\frac{r}{2GM}-1\right)^{1/2}e^{r/4GM}\sinh\left(\frac{t}{4GM}\right)
  3. X = ( r 2 G M - 1 ) 1 / 2 e r / 4 G M cosh ( t 4 G M ) X=\left(\frac{r}{2GM}-1\right)^{1/2}e^{r/4GM}\cosh\left(\frac{t}{4GM}\right)
  4. r > 2 G M r>2GM
  5. T = ( 1 - r 2 G M ) 1 / 2 e r / 4 G M cosh ( t 4 G M ) T=\left(1-\frac{r}{2GM}\right)^{1/2}e^{r/4GM}\cosh\left(\frac{t}{4GM}\right)
  6. X = ( 1 - r 2 G M ) 1 / 2 e r / 4 G M sinh ( t 4 G M ) X=\left(1-\frac{r}{2GM}\right)^{1/2}e^{r/4GM}\sinh\left(\frac{t}{4GM}\right)
  7. 0 < r < 2 G M 0<r<2GM
  8. T 2 - X 2 = ( 1 - r 2 G M ) e r / 2 G M T^{2}-X^{2}=\left(1-\frac{r}{2GM}\right)e^{r/2GM}
  9. r 2 G M = 1 + W ( X 2 - T 2 e ) \frac{r}{2GM}=1+W\left(\frac{X^{2}-T^{2}}{e}\right)
  10. d s 2 = 32 G 3 M 3 r e - r / 2 G M ( - d T 2 + d X 2 ) + r 2 d Ω 2 , ds^{2}=\frac{32G^{3}M^{3}}{r}e^{-r/2GM}(-dT^{2}+dX^{2})+r^{2}d\Omega^{2},
  11. d Ω 2 = def d θ 2 + sin 2 θ d ϕ 2 d\Omega^{2}\ \stackrel{\mathrm{def}}{=}\ d\theta^{2}+\sin^{2}\theta\,d\phi^{2}
  12. T = \plusmn X T=\plusmn X\,
  13. T 2 - X 2 = 1 T^{2}-X^{2}=1
  14. T 2 - X 2 = 1 T^{2}-X^{2}=1
  15. - < X < -\infty<X<\infty\,
  16. - < T 2 - X 2 < 1 -\infty<T^{2}-X^{2}<1
  17. - X < T < + X -X<T<+X
  18. 2 G M < r 2GM<r
  19. | X | < T < 1 + X 2 |X|<T<\sqrt{1+X^{2}}
  20. 0 < r < 2 G M 0<r<2GM
  21. + X < T < - X +X<T<-X
  22. 2 G M < r 2GM<r
  23. - 1 + X 2 < T < - | X | -\sqrt{1+X^{2}}<T<-|X|
  24. 0 < r < 2 G M 0<r<2GM
  25. tanh ( t 4 G M ) = { T / X (in I and III) X / T (in II and IV) \tanh\left(\frac{t}{4GM}\right)=\begin{cases}T/X&\mbox{(in I and III)}\\ X/T&\mbox{(in II and IV)}\end{cases}
  26. d X = \plusmn d T dX=\plusmn dT\,
  27. d s = 0 ds=0
  28. 16 π M 2 16\pi M^{2}
  29. U = T - X U=T-X
  30. V = T + X , V=T+X,
  31. d s 2 = - 32 G 3 M 3 r e - r / 2 G M ( d U d V ) + r 2 d Ω 2 , ds^{2}=-\frac{32G^{3}M^{3}}{r}e^{-r/2GM}(dUdV)+r^{2}d\Omega^{2},
  32. U V = ( 1 - r 2 G M ) e r / 2 G M . UV=\left(1-\frac{r}{2GM}\right)e^{r/2GM}.
  33. U = constant U=\,\text{constant}
  34. V = constant V=\,\text{constant}
  35. U V = 0 UV=0
  36. U V = 1 UV=1

Krylov_subspace.html

  1. A 0 = I A^{0}=I
  2. 𝒦 r ( A , b ) = span { b , A b , A 2 b , , A r - 1 b } . \mathcal{K}_{r}(A,b)=\operatorname{span}\,\{b,Ab,A^{2}b,\ldots,A^{r-1}b\}.\,
  3. A b Ab
  4. A A
  5. A 2 b A^{2}b

Kt::V.html

  1. V d C d t = - K C ( 1 ) V\frac{dC}{dt}=-K\cdot C\qquad(1)
  2. d C d t \frac{dC}{dt}
  3. d C C = - K V d t . ( 2 a ) \int\frac{dC}{C}=\int-\frac{K}{V}\,dt.\qquad(2a)
  4. ln ( C ) = - K t V + const ( 2 b ) \ln(C)=-\frac{K\cdot t}{V}+\mbox{const}~{}\qquad(2b)
  5. C = e - K t V + c o n s t ( 2 c ) C=e^{-\frac{K\cdot t}{V}+const}\qquad(2c)
  6. C = C 0 e - K t V ( 3 ) C=C_{0}e^{-\frac{K\cdot t}{V}}\qquad(3)
  7. K t V = ln C o C ( 4 ) \frac{K\cdot t}{V}=\ln\frac{C_{o}}{C}\qquad(4)
  8. K t V = - ln ( 1 - U R R ) . ( 8 ) \frac{K\cdot t}{V}=-\ln(1-URR).\qquad(8)
  9. C B = m ˙ / K . ( 6 a ) C_{B}=\dot{m}/K.\qquad(6a)
  10. m ˙ \dot{m}
  11. K = m ˙ / C B . ( 6 b ) K=\dot{m}/C_{B}.\qquad(6b)
  12. m ˙ = C E V E t . ( 6 c ) \dot{m}=\frac{C_{E}\cdot V_{E}}{t}.\qquad(6c)
  13. K D = C E V E C B t ( 6 d ) K_{D}=\frac{C_{E}\cdot V_{E}}{C_{B}\cdot t}\qquad(6d)
  14. K t V = 7 / 3 K D V D ( 7 a ) \frac{K\cdot t}{V}=\frac{7/3\cdot K_{D}}{V_{D}}\qquad(7a)
  15. Weekly K t / V = 7 K D [ l / day ] V [ l ] . ( 7 b ) \mbox{Weekly }~{}Kt/V=\frac{7K_{D}[l/\mbox{day}~{}]}{V[l]}.\qquad(7b)
  16. C B mean = 22.817 mmol/L C_{B\mbox{ mean}~{}}=22.817\mbox{ mmol/L}~{}
  17. C D = 17.524 mmol/L C_{D}=17.524\mbox{ mmol/L}~{}
  18. V D = 3.75 L per exchange or 15 L/day V_{D}=3.75\mbox{ L per exchange or }~{}15\mbox{ L/day}~{}
  19. V B = 40.6 L V_{B}=40.6\ L

Kumaraswamy_distribution.html

  1. f ( x ; a , b ) = a b x a - 1 ( 1 - x a ) b - 1 , where x [ 0 , 1 ] , f(x;a,b)=abx^{a-1}{(1-x^{a})}^{b-1},\ \ \mbox{where}~{}\ \ x\in[0,1],
  2. F ( x ; a , b ) = 0 x f ( ξ ; a , b ) d ξ = 1 - ( 1 - x a ) b . F(x;a,b)=\int_{0}^{x}f(\xi;a,b)d\xi=1-(1-x^{a})^{b}.
  3. x = z - z min z max - z min , z min z z max . x=\frac{z-z_{\,\text{min}}}{z_{\,\text{max}}-z_{\,\text{min}}},\qquad z_{\,% \text{min}}\leq z\leq z_{\,\text{max}}.\,\!
  4. m n = b Γ ( 1 + n / a ) Γ ( b ) Γ ( 1 + b + n / a ) = b B ( 1 + n / a , b ) m_{n}=\frac{b\Gamma(1+n/a)\Gamma(b)}{\Gamma(1+b+n/a)}=bB(1+n/a,b)\,
  5. σ 2 = m 2 - m 1 2 . \sigma^{2}=m_{2}-m_{1}^{2}.
  6. α = 1 \alpha=1
  7. β = b \beta=b
  8. X a , b = Y 1 , b 1 / a , X_{a,b}=Y^{1/a}_{1,b},
  9. P { X a , b x } = 0 x a b t a - 1 ( 1 - t a ) b - 1 d t = 0 x a b ( 1 - t ) b - 1 d t = P { Y 1 , b x a } = P { Y 1 , b 1 / a x } . \operatorname{P}\{X_{a,b}\leq x\}=\int_{0}^{x}abt^{a-1}(1-t^{a})^{b-1}dt=\int_% {0}^{x^{a}}b(1-t)^{b-1}dt=\operatorname{P}\{Y_{1,b}\leq x^{a}\}=\operatorname{% P}\{Y^{1/a}_{1,b}\leq x\}.
  10. Y α , β 1 / γ Y^{1/\gamma}_{\alpha,\beta}
  11. γ > 0 \gamma>0
  12. Y α , β Y_{\alpha,\beta}
  13. α \alpha
  14. β \beta
  15. m n = Γ ( α + β ) Γ ( α + n / γ ) Γ ( α ) Γ ( α + β + n / γ ) . m_{n}=\frac{\Gamma(\alpha+\beta)\Gamma(\alpha+n/\gamma)}{\Gamma(\alpha)\Gamma(% \alpha+\beta+n/\gamma)}.
  16. α = 1 \alpha=1
  17. β = b \beta=b
  18. γ = a \gamma=a
  19. X Kumaraswamy ( 1 , 1 ) X\sim\textrm{Kumaraswamy}(1,1)\,
  20. X U ( 0 , 1 ) X\sim U(0,1)\,
  21. X U ( 0 , 1 ) X\sim U(0,1)\,
  22. ( 1 - ( 1 - X ) 1 b ) 1 a Kumaraswamy ( a , b ) {\left(1-{\left(1-X\right)}^{\tfrac{1}{b}}\right)}^{\tfrac{1}{a}}\sim\textrm{% Kumaraswamy}(a,b)\,
  23. X Beta ( 1 , b ) X\sim\textrm{Beta}(1,b)\,
  24. X Kumaraswamy ( 1 , b ) X\sim\textrm{Kumaraswamy}(1,b)\,
  25. X Beta ( a , 1 ) X\sim\textrm{Beta}(a,1)\,
  26. X Kumaraswamy ( a , 1 ) X\sim\textrm{Kumaraswamy}(a,1)\,
  27. X Kumaraswamy ( a , 1 ) X\sim\textrm{Kumaraswamy}(a,1)\,
  28. ( 1 - X ) Kumaraswamy ( 1 , a ) (1-X)\sim\textrm{Kumaraswamy}(1,a)\,
  29. X Kumaraswamy ( 1 , a ) X\sim\textrm{Kumaraswamy}(1,a)\,
  30. ( 1 - X ) Kumaraswamy ( a , 1 ) (1-X)\sim\textrm{Kumaraswamy}(a,1)\,
  31. X Kumaraswamy ( a , 1 ) X\sim\textrm{Kumaraswamy}(a,1)\,
  32. - l n ( X ) Exponential ( a ) -ln(X)\sim\textrm{Exponential}(a)\,
  33. X Kumaraswamy ( 1 , b ) X\sim\textrm{Kumaraswamy}(1,b)\,
  34. - l n ( 1 - X ) Exponential ( b ) -ln(1-X)\sim\textrm{Exponential}(b)\,
  35. X Kumaraswamy ( a , b ) X\sim\textrm{Kumaraswamy}(a,b)\,
  36. X GB1 ( a , 1 , 1 , b ) X\sim\textrm{GB1}(a,1,1,b)\,

Kummer–Vandiver_conjecture.html

  1. K = ( ζ p ) + K=\mathbb{Q}(\zeta_{p})^{+}
  2. ( ζ p ) \mathbb{Q}(\zeta_{p})
  3. K = ( ζ p ) + K=\mathbb{Q}(\zeta_{p})^{+}

Kuramoto_model.html

  1. ω i \omega_{i}
  2. d θ i d t = ω i + K N j = 1 N sin ( θ j - θ i ) , i = 1 N \frac{d\theta_{i}}{dt}=\omega_{i}+\frac{K}{N}\sum_{j=1}^{N}\sin(\theta_{j}-% \theta_{i}),\qquad i=1\ldots N
  3. d θ i d t = ω i + ζ i + K N j = 1 N sin ( θ j - θ i ) \frac{d\theta_{i}}{dt}=\omega_{i}+\zeta_{i}+\dfrac{K}{N}\sum_{j=1}^{N}\sin(% \theta_{j}-\theta_{i})
  4. ζ i \zeta_{i}
  5. ζ i ( t ) = 0 \langle\zeta_{i}(t)\rangle=0
  6. ζ i ( t ) ζ j ( t ) = 2 D δ i j δ ( t - t ) \langle\zeta_{i}(t)\zeta_{j}(t^{\prime})\rangle=2D\delta_{ij}\delta(t-t^{% \prime})
  7. D D
  8. r e i ψ = 1 N j = 1 N e i θ j re^{i\psi}=\frac{1}{N}\sum_{j=1}^{N}e^{i\theta_{j}}
  9. e - i θ i e^{-i\theta_{i}}
  10. d θ i d t = ω i + K r sin ( ψ - θ i ) \frac{d\theta_{i}}{dt}=\omega_{i}+Kr\sin(\psi-\theta_{i})
  11. ψ = 0 \psi=0
  12. d θ i d t = ω i - K r sin ( θ i ) \frac{d\theta_{i}}{dt}=\omega_{i}-Kr\sin(\theta_{i})
  13. ρ ( θ , ω , t ) \rho(\theta,\omega,t)
  14. - π π ρ ( θ , ω , t ) d θ = 1. \int_{-\pi}^{\pi}\rho(\theta,\omega,t)\,d\theta=1.
  15. ρ t + θ [ ρ v ] = 0 , \frac{\partial\rho}{\partial t}+\frac{\partial}{\partial\theta}[\rho v]=0,
  16. ρ t + θ [ ρ ω + ρ K r sin ( ψ - θ ) ] = 0. \frac{\partial\rho}{\partial t}+\frac{\partial}{\partial\theta}[\rho\omega+% \rho Kr\sin(\psi-\theta)]=0.
  17. θ i \theta_{i}
  18. r e i ψ = - π π e i θ - ρ ( θ , ω , t ) g ( ω ) d ω d θ . re^{i\psi}=\int_{-\pi}^{\pi}e^{i\theta}\int_{-\infty}^{\infty}\rho(\theta,% \omega,t)g(\omega)\,d\omega\,d\theta.
  19. ρ = 1 / ( 2 π ) \rho=1/(2\pi)
  20. r = 0 r=0
  21. ρ = δ ( θ - ψ - arcsin ( ω K r ) ) \rho=\delta\left(\theta-\psi-\arcsin\left(\frac{\omega}{Kr}\right)\right)
  22. ρ = normalization constant ( ω - Kr sin ( θ - ψ ) ) \rho=\frac{\rm{normalization\;constant}}{(\omega-Kr\sin(\theta-\psi))}
  23. | ω | < K r |\omega|<Kr
  24. ( q 1 , p 1 , , q N , p N ) = = 1 N ω 2 ( q 2 + p 2 ) + K 4 N , m = 1 N ( q p m - q m p ) ( q m 2 + p m 2 - q 2 - p 2 ) \mathcal{H}(q_{1},p_{1},\ldots,q_{N},p_{N})=\sum_{\ell=1}^{N}\frac{\omega_{% \ell}}{2}(q_{\ell}^{2}+p_{\ell}^{2})+\frac{K}{4N}\sum_{\ell,m=1}^{N}(q_{\ell}p% _{m}-q_{m}p_{\ell})(q_{m}^{2}+p_{m}^{2}-q_{\ell}^{2}-p_{\ell}^{2})
  25. I = ( q 2 + p 2 ) / 2 I_{\ell}=\left(q_{\ell}^{2}+p_{\ell}^{2}\right)/2
  26. ϕ = arctan2 ( q / p ) \phi_{\ell}=\mathrm{arctan2}\left(q_{\ell}/p_{\ell}\right)
  27. I I I_{\ell}\equiv I
  28. Γ ( ϕ ) = sin ( ϕ ) \Gamma(\phi)=\sin(\phi)
  29. ϕ = θ j - θ i \phi=\theta_{j}-\theta_{i}
  30. Γ ( ϕ ) = sin ( ϕ ) + a 1 sin ( 2 ϕ + b 1 ) + + a n sin ( 2 n ϕ + b n ) \Gamma(\phi)=\sin(\phi)+a_{1}\sin(2\phi+b_{1})+...+a_{n}\sin(2n\phi+b_{n})
  31. a i a_{i}
  32. b i b_{i}

Ky_Fan_inequality.html

  1. ( i = 1 n x i ) 1 / n ( i = 1 n ( 1 - x i ) ) 1 / n 1 n i = 1 n x i 1 n i = 1 n ( 1 - x i ) \frac{\bigl(\prod_{i=1}^{n}x_{i}\bigr)^{1/n}}{\bigl(\prod_{i=1}^{n}(1-x_{i})% \bigr)^{1/n}}\leq\frac{\frac{1}{n}\sum_{i=1}^{n}x_{i}}{\frac{1}{n}\sum_{i=1}^{% n}(1-x_{i})}
  2. A n := 1 n i = 1 n x i , G n = ( i = 1 n x i ) 1 / n A_{n}:=\frac{1}{n}\sum_{i=1}^{n}x_{i},\qquad G_{n}=\biggl(\prod_{i=1}^{n}x_{i}% \biggr)^{1/n}
  3. A n := 1 n i = 1 n ( 1 - x i ) , G n = ( i = 1 n ( 1 - x i ) ) 1 / n A_{n}^{\prime}:=\frac{1}{n}\sum_{i=1}^{n}(1-x_{i}),\qquad G_{n}^{\prime}=% \biggl(\prod_{i=1}^{n}(1-x_{i})\biggr)^{1/n}
  4. G n G n A n A n , \frac{G_{n}}{G_{n}^{\prime}}\leq\frac{A_{n}}{A_{n}^{\prime}},
  5. i = 1 n x i γ i i = 1 n ( 1 - x i ) γ i i = 1 n γ i x i i = 1 n γ i ( 1 - x i ) \frac{\prod_{i=1}^{n}x_{i}^{\gamma_{i}}}{\prod_{i=1}^{n}(1-x_{i})^{\gamma_{i}}% }\leq\frac{\sum_{i=1}^{n}\gamma_{i}x_{i}}{\sum_{i=1}^{n}\gamma_{i}(1-x_{i})}
  6. f ( x ) := ln x - ln ( 1 - x ) = ln x 1 - x , x ( 0 , 1 2 ] . f(x):=\ln x-\ln(1-x)=\ln\frac{x}{1-x},\qquad x\in(0,\tfrac{1}{2}].
  7. f ′′ ( x ) = - 1 x 2 + 1 ( 1 - x ) 2 < 0 , x ( 0 , 1 2 ) . f^{\prime\prime}(x)=-\frac{1}{x^{2}}+\frac{1}{(1-x)^{2}}<0,\qquad x\in(0,% \tfrac{1}{2}).
  8. ln i = 1 n x i γ i i = 1 n ( 1 - x i ) γ i = ln i = 1 n ( x i 1 - x i ) γ i = i = 1 n γ i f ( x i ) < f ( i = 1 n γ i x i ) = ln i = 1 n γ i x i i = 1 n γ i ( 1 - x i ) , \begin{aligned}\displaystyle\ln\frac{\prod_{i=1}^{n}x_{i}^{\gamma_{i}}}{\prod_% {i=1}^{n}(1-x_{i})^{\gamma_{i}}}&\displaystyle=\ln\prod_{i=1}^{n}\Bigl(\frac{x% _{i}}{1-x_{i}}\Bigr)^{\gamma_{i}}\\ &\displaystyle=\sum_{i=1}^{n}\gamma_{i}f(x_{i})\\ &\displaystyle<f\biggl(\sum_{i=1}^{n}\gamma_{i}x_{i}\biggr)\\ &\displaystyle=\ln\frac{\sum_{i=1}^{n}\gamma_{i}x_{i}}{\sum_{i=1}^{n}\gamma_{i% }(1-x_{i})},\end{aligned}

L-notation.html

  1. L n [ α , c ] L_{n}[\alpha,c]
  2. n n
  3. L n [ α , c ] = e ( c + o ( 1 ) ) ( ln n ) α ( ln ln n ) 1 - α , L_{n}[\alpha,c]=e^{(c+o(1))(\ln n)^{\alpha}(\ln\ln n)^{1-\alpha}},
  4. α \alpha
  5. 0 α 1 0\leq\alpha\leq 1
  6. e c ( ln n ) α ( ln ln n ) 1 - α e^{c(\ln n)^{\alpha}(\ln\ln n)^{1-\alpha}}
  7. e o ( 1 ) ( ln n ) α ( ln ln n ) 1 - α e^{o(1)(\ln n)^{\alpha}(\ln\ln n)^{1-\alpha}}
  8. α \alpha
  9. L n [ α , c ] = L n [ 0 , c ] = e ( c + o ( 1 ) ) ln ln n = ( ln n ) c + o ( 1 ) L_{n}[\alpha,c]=L_{n}[0,c]=e^{(c+o(1))\ln\ln n}=(\ln n)^{c+o(1)}\,
  10. α \alpha
  11. L n [ α , c ] = L n [ 1 , c ] = e ( c + o ( 1 ) ) ln n = n c + o ( 1 ) L_{n}[\alpha,c]=L_{n}[1,c]=e^{(c+o(1))\ln n}=n^{c+o(1)}\,
  12. α \alpha
  13. L n [ 1 / 3 , c ] = e ( c + o ( 1 ) ) ( ln n ) 1 / 3 ( ln ln n ) 2 / 3 L_{n}[1/3,c]=e^{(c+o(1))(\ln n)^{1/3}(\ln\ln n)^{2/3}}
  14. c = ( 64 / 9 ) 1 / 3 1.923 c=(64/9)^{1/3}\approx 1.923
  15. L n [ 1 / 2 , 1 ] = e ( 1 + o ( 1 ) ) ( ln n ) 1 / 2 ( ln ln n ) 1 / 2 . L_{n}[1/2,1]=e^{(1+o(1))(\ln n)^{1/2}(\ln\ln n)^{1/2}}.\,
  16. L n [ 1 , 1 / 2 ] = n 1 / 2 + o ( 1 ) . L_{n}[1,1/2]=n^{1/2+o(1)}.\,
  17. L n [ 0 , c ] = ( ln n ) c + o ( 1 ) L_{n}[0,c]=(\ln n)^{c+o(1)}\,
  18. c c
  19. α \alpha
  20. 1 / 2 1/2
  21. L L
  22. l l
  23. O O
  24. O O

Lagrange's_identity.html

  1. ( k = 1 n a k 2 ) ( k = 1 n b k 2 ) - ( k = 1 n a k b k ) 2 = i = 1 n - 1 j = i + 1 n ( a i b j - a j b i ) 2 ( = 1 2 i = 1 n j = 1 , j i n ( a i b j - a j b i ) 2 ) , \begin{aligned}\displaystyle\biggl(\sum_{k=1}^{n}a_{k}^{2}\biggr)\biggl(\sum_{% k=1}^{n}b_{k}^{2}\biggr)-\biggl(\sum_{k=1}^{n}a_{k}b_{k}\biggr)^{2}&% \displaystyle=\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}(a_{i}b_{j}-a_{j}b_{i})^{2}\\ &\displaystyle\biggl(=\frac{1}{2}\sum_{i=1}^{n}\sum_{j=1,j\neq i}^{n}(a_{i}b_{% j}-a_{j}b_{i})^{2}\biggr),\end{aligned}
  2. 𝐚 2 𝐛 2 - ( 𝐚 𝐛 ) 2 = 1 i < j n ( a i b j - a j b i ) 2 , \|\mathbf{a}\|^{2}\ \|\mathbf{b}\|^{2}-(\mathbf{a\cdot b})^{2}=\sum_{1\leq i<j% \leq n}\left(a_{i}b_{j}-a_{j}b_{i}\right)^{2}\ ,
  3. ( k = 1 n | a k | 2 ) ( k = 1 n | b k | 2 ) - | k = 1 n a k b k | 2 = i = 1 n - 1 j = i + 1 n | a i b ¯ j - a j b ¯ i | 2 \biggl(\sum_{k=1}^{n}|a_{k}|^{2}\biggr)\biggl(\sum_{k=1}^{n}|b_{k}|^{2}\biggr)% -\biggl|\sum_{k=1}^{n}a_{k}b_{k}\biggr|^{2}=\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}|a% _{i}\overline{b}_{j}-a_{j}\overline{b}_{i}|^{2}
  4. ( a a ) ( b b ) - ( a b ) 2 = ( a b ) ( a b ) . (a\cdot a)(b\cdot b)-(a\cdot b)^{2}=(a\wedge b)\cdot(a\wedge b).
  5. a b = ( a b ) 2 - a b 2 . \|a\wedge b\|=\sqrt{(\|a\|\ \|b\|)^{2}-\|a\cdot b\|^{2}}.
  6. | 𝐚 | 2 | 𝐛 | 2 - ( 𝐚 𝐛 ) 2 = | 𝐚 × 𝐛 | 2 |\mathbf{a}|^{2}|\mathbf{b}|^{2}-(\mathbf{a\cdot b})^{2}=|\mathbf{a\times b}|^% {2}
  7. | 𝐚 | 2 | 𝐛 | 2 ( 1 - cos 2 θ ) = | 𝐚 | 2 | 𝐛 | 2 sin 2 θ |\mathbf{a}|^{2}|\mathbf{b}|^{2}(1-\cos^{2}\theta)=|\mathbf{a}|^{2}|\mathbf{b}% |^{2}\sin^{2}\theta
  8. | 𝐚 | | 𝐛 | | sin θ | , |\mathbf{a}|\,|\mathbf{b}|\,|\sin\theta|,
  9. 𝐚 × 𝐛 = ( a 2 b 3 - a 3 b 2 ) 𝐢 + ( a 3 b 1 - a 1 b 3 ) 𝐣 + ( a 1 b 2 - a 2 b 1 ) 𝐤 \mathbf{a}\times\mathbf{b}=(a_{2}b_{3}-a_{3}b_{2})\mathbf{i}+(a_{3}b_{1}-a_{1}% b_{3})\mathbf{j}+(a_{1}b_{2}-a_{2}b_{1})\mathbf{k}
  10. | 𝐚 | 2 | 𝐛 | 2 - | 𝐚 𝐛 | 2 = | 𝐚 × 𝐛 | 2 , |\mathbf{a}|^{2}|\mathbf{b}|^{2}-|\mathbf{a}\cdot\mathbf{b}|^{2}=|\mathbf{a}% \times\mathbf{b}|^{2}\ ,
  11. p = t + 𝐯 = t + x 𝐢 + y 𝐣 + z 𝐤 . p=t+\mathbf{v}=t+x\ \mathbf{i}+y\ \mathbf{j}+z\ \mathbf{k}.
  12. p q = ( s t - 𝐯 𝐰 ) + s 𝐯 + t 𝐰 + 𝐯 × 𝐰 . pq=(st-\mathbf{v}\cdot\mathbf{w})+s\ \mathbf{v}+t\ \mathbf{w}+\mathbf{v}\times% \mathbf{w}.
  13. q ¯ = t - 𝐯 , \overline{q}=t-\mathbf{v},
  14. | q | 2 = q q ¯ = t 2 + x 2 + y 2 + z 2 . |q|^{2}=q\overline{q}=t^{2}\ +\ x^{2}+\ y^{2}\ +\ z^{2}.
  15. | p q | = | p | | q | . |pq|=|p||q|.\,
  16. p = 𝐯 , q = 𝐰 . p=\mathbf{v},\quad q=\mathbf{w}.
  17. | 𝐯𝐰 | 2 = | 𝐯 | 2 | 𝐰 | 2 , |\mathbf{v}\mathbf{w}|^{2}=|\mathbf{v}|^{2}|\mathbf{w}|^{2},\,
  18. | 𝐯𝐰 | 2 = ( 𝐯 𝐰 ) 2 + | 𝐯 × 𝐰 | 2 . |\mathbf{v}\mathbf{w}|^{2}=(\mathbf{v}\cdot\mathbf{w})^{2}+|\mathbf{v}\times% \mathbf{w}|^{2}.
  19. ( k = 1 n a k 2 ) ( k = 1 n b k 2 ) = i = 1 n j = 1 n a i 2 b j 2 = k = 1 n a k 2 b k 2 + i = 1 n - 1 j = i + 1 n a i 2 b j 2 + j = 1 n - 1 i = j + 1 n a i 2 b j 2 , \left(\sum_{k=1}^{n}a_{k}^{2}\right)\left(\sum_{k=1}^{n}b_{k}^{2}\right)=\sum_% {i=1}^{n}\sum_{j=1}^{n}a_{i}^{2}b_{j}^{2}=\sum_{k=1}^{n}a_{k}^{2}b_{k}^{2}+% \sum_{i=1}^{n-1}\sum_{j=i+1}^{n}a_{i}^{2}b_{j}^{2}+\sum_{j=1}^{n-1}\sum_{i=j+1% }^{n}a_{i}^{2}b_{j}^{2}\ ,
  20. ( k = 1 n a k b k ) 2 = k = 1 n a k 2 b k 2 + 2 i = 1 n - 1 j = i + 1 n a i b i a j b j , \left(\sum_{k=1}^{n}a_{k}b_{k}\right)^{2}=\sum_{k=1}^{n}a_{k}^{2}b_{k}^{2}+2% \sum_{i=1}^{n-1}\sum_{j=i+1}^{n}a_{i}b_{i}a_{j}b_{j}\ ,
  21. i = 1 n - 1 j = i + 1 n ( a i b j - a j b i ) 2 = i = 1 n - 1 j = i + 1 n ( a i 2 b j 2 + a j 2 b i 2 - 2 a i b j a j b i ) . \sum_{i=1}^{n-1}\sum_{j=i+1}^{n}(a_{i}b_{j}-a_{j}b_{i})^{2}=\sum_{i=1}^{n-1}% \sum_{j=i+1}^{n}(a_{i}^{2}b_{j}^{2}+a_{j}^{2}b_{i}^{2}-2a_{i}b_{j}a_{j}b_{i}).
  22. i = 1 n - 1 j = i + 1 n ( a i b j - a j b i ) 2 = i = 1 n - 1 j = i + 1 n a i 2 b j 2 + i = 1 n - 1 j = i + 1 n a j 2 b i 2 - 2 i = 1 n - 1 j = i + 1 n a i b j a j b i . \sum_{i=1}^{n-1}\sum_{j=i+1}^{n}(a_{i}b_{j}-a_{j}b_{i})^{2}=\sum_{i=1}^{n-1}% \sum_{j=i+1}^{n}a_{i}^{2}b_{j}^{2}+\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}a_{j}^{2}b_% {i}^{2}-2\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}a_{i}b_{j}a_{j}b_{i}.
  23. i = 1 n - 1 j = i + 1 n ( a i b j - a j b i ) 2 = i = 1 n - 1 j = i + 1 n a i 2 b j 2 + j = 1 n - 1 i = j + 1 n a i 2 b j 2 - 2 i = 1 n - 1 j = i + 1 n a i b i a j b j . \sum_{i=1}^{n-1}\sum_{j=i+1}^{n}(a_{i}b_{j}-a_{j}b_{i})^{2}=\sum_{i=1}^{n-1}% \sum_{j=i+1}^{n}a_{i}^{2}b_{j}^{2}+\sum_{j=1}^{n-1}\sum_{i=j+1}^{n}a_{i}^{2}b_% {j}^{2}-2\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}a_{i}b_{i}a_{j}b_{j}\ .
  24. i = 1 n - 1 j = i + 1 n a i 2 b j 2 + j = 1 n - 1 i = j + 1 n a i 2 b j 2 - 2 i = 1 n - 1 j = i + 1 n a i b i a j b j \sum_{i=1}^{n-1}\sum_{j=i+1}^{n}a_{i}^{2}b_{j}^{2}+\sum_{j=1}^{n-1}\sum_{i=j+1% }^{n}a_{i}^{2}b_{j}^{2}-2\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}a_{i}b_{i}a_{j}b_{j}
  25. a i , b i a_{i},b_{i}\in\mathbb{C}
  26. i = 1 n ( 1 - a i a ¯ i - b i b ¯ i + a i a ¯ i b i b ¯ i ) = i = 1 n ( 1 - a i a ¯ i ) i = 1 n ( 1 - b i b ¯ i ) \prod_{i=1}^{n}\left(1-a_{i}\bar{a}_{i}-b_{i}\bar{b}_{i}+a_{i}\bar{a}_{i}b_{i}% \bar{b}_{i}\right)=\prod_{i=1}^{n}\left(1-a_{i}\bar{a}_{i}\right)\prod_{i=1}^{% n}\left(1-b_{i}\bar{b}_{i}\right)
  27. ( 1 + x i ) \left(1+x_{i}\right)
  28. i = 1 n ( 1 + x i ) = 1 + i = 1 n x i + i < j n x i x j + 𝒪 3 + ( x ) , \prod_{i=1}^{n}\left(1+x_{i}\right)=1+\sum_{i=1}^{n}x_{i}+\sum_{i<j}^{n}x_{i}x% _{j}+\mathcal{O}^{3+}(x),
  29. 𝒪 3 + ( x ) \mathcal{O}^{3+}(x)
  30. x x
  31. i = 1 n ( 1 - a i a ¯ i - b i b ¯ i + a i a ¯ i b i b ¯ i ) = 1 - i = 1 n ( a i a ¯ i + b i b ¯ i ) + i = 1 n a i a ¯ i b i b ¯ i + i < j n ( a i a ¯ i a j a ¯ j + b i b ¯ i b j b ¯ j ) + i < j n ( a i a ¯ i b j b ¯ j + a j a ¯ j b i b ¯ i ) + 𝒪 5 + . \prod_{i=1}^{n}\left(1-a_{i}\bar{a}_{i}-b_{i}\bar{b}_{i}+a_{i}\bar{a}_{i}b_{i}% \bar{b}_{i}\right)=1-\sum_{i=1}^{n}\left(a_{i}\bar{a}_{i}+b_{i}\bar{b}_{i}% \right)+\sum_{i=1}^{n}a_{i}\bar{a}_{i}b_{i}\bar{b}_{i}+\sum_{i<j}^{n}\left(a_{% i}\bar{a}_{i}a_{j}\bar{a}_{j}+b_{i}\bar{b}_{i}b_{j}\bar{b}_{j}\right)+\sum_{i<% j}^{n}\left(a_{i}\bar{a}_{i}b_{j}\bar{b}_{j}+a_{j}\bar{a}_{j}b_{i}\bar{b}_{i}% \right)+\mathcal{O}^{5+}.
  32. i = 1 n ( 1 - a i a ¯ i ) i = 1 n ( 1 - b i b ¯ i ) = ( 1 - i = 1 n a i a ¯ i + i < j n a i a ¯ i a j a ¯ j + 𝒪 5 + ) ( 1 - i = 1 n b i b ¯ i + i < j n b i b ¯ i b j b ¯ j + 𝒪 5 + ) . \prod_{i=1}^{n}\left(1-a_{i}\bar{a}_{i}\right)\prod_{i=1}^{n}\left(1-b_{i}\bar% {b}_{i}\right)=\left(1-\sum_{i=1}^{n}a_{i}\bar{a}_{i}+\sum_{i<j}^{n}a_{i}\bar{% a}_{i}a_{j}\bar{a}_{j}+\mathcal{O}^{5+}\right)\left(1-\sum_{i=1}^{n}b_{i}\bar{% b}_{i}+\sum_{i<j}^{n}b_{i}\bar{b}_{i}b_{j}\bar{b}_{j}+\mathcal{O}^{5+}\right).
  33. i = 1 n ( 1 - a i a ¯ i ) i = 1 n ( 1 - b i b ¯ i ) = 1 - i = 1 n ( a i a ¯ i + b i b ¯ i ) + ( i = 1 n a i a ¯ i ) ( i = 1 n b i b ¯ i ) + i < j n ( a i a ¯ i a j a ¯ j + b i b ¯ i b j b ¯ j ) + 𝒪 5 + . \prod_{i=1}^{n}\left(1-a_{i}\bar{a}_{i}\right)\prod_{i=1}^{n}\left(1-b_{i}\bar% {b}_{i}\right)=1-\sum_{i=1}^{n}\left(a_{i}\bar{a}_{i}+b_{i}\bar{b}_{i}\right)+% \left(\sum_{i=1}^{n}a_{i}\bar{a}_{i}\right)\left(\sum_{i=1}^{n}b_{i}\bar{b}_{i% }\right)+\sum_{i<j}^{n}\left(a_{i}\bar{a}_{i}a_{j}\bar{a}_{j}+b_{i}\bar{b}_{i}% b_{j}\bar{b}_{j}\right)+\mathcal{O}^{5+}.
  34. i = 1 n a i a ¯ i b i b ¯ i + i < j n ( a i a ¯ i b j b ¯ j + a j a ¯ j b i b ¯ i ) = ( i = 1 n a i a ¯ i ) ( i = 1 n b i b ¯ i ) . \sum_{i=1}^{n}a_{i}\bar{a}_{i}b_{i}\bar{b}_{i}+\sum_{i<j}^{n}\left(a_{i}\bar{a% }_{i}b_{j}\bar{b}_{j}+a_{j}\bar{a}_{j}b_{i}\bar{b}_{i}\right)=\left(\sum_{i=1}% ^{n}a_{i}\bar{a}_{i}\right)\left(\sum_{i=1}^{n}b_{i}\bar{b}_{i}\right).
  35. ( i = 1 n x i ) ( i = 1 n x ¯ i ) = i = 1 n x i x ¯ i + i < j n ( x i x ¯ j + x ¯ i x j ) \left(\sum_{i=1}^{n}x_{i}\right)\left(\sum_{i=1}^{n}\bar{x}_{i}\right)=\sum_{i% =1}^{n}x_{i}\bar{x}_{i}+\sum_{i<j}^{n}\left(x_{i}\bar{x}_{j}+\bar{x}_{i}x_{j}\right)
  36. ( i = 1 n a i b i ) ( i = 1 n a i b i ¯ ) - i < j n ( a i b i a ¯ j b ¯ j + a ¯ i b ¯ i a j b j ) + i < j n ( a i a ¯ i b j b ¯ j + a j a ¯ j b i b ¯ i ) = ( i = 1 n a i a ¯ i ) ( i = 1 n b i b ¯ i ) . \left(\sum_{i=1}^{n}a_{i}b_{i}\right)\left(\sum_{i=1}^{n}\overline{a_{i}b_{i}}% \right)-\sum_{i<j}^{n}\left(a_{i}b_{i}\bar{a}_{j}\bar{b}_{j}+\bar{a}_{i}\bar{b% }_{i}a_{j}b_{j}\right)+\sum_{i<j}^{n}\left(a_{i}\bar{a}_{i}b_{j}\bar{b}_{j}+a_% {j}\bar{a}_{j}b_{i}\bar{b}_{i}\right)=\left(\sum_{i=1}^{n}a_{i}\bar{a}_{i}% \right)\left(\sum_{i=1}^{n}b_{i}\bar{b}_{i}\right).
  37. a i a ¯ i b j b ¯ j + a j a ¯ j b i b ¯ i - a i b i a ¯ j b ¯ j - a ¯ i b ¯ i a j b j = ( a i b ¯ j - a j b ¯ i ) ( a ¯ i b j - a ¯ j b i ) , a_{i}\bar{a}_{i}b_{j}\bar{b}_{j}+a_{j}\bar{a}_{j}b_{i}\bar{b}_{i}-a_{i}b_{i}% \bar{a}_{j}\bar{b}_{j}-\bar{a}_{i}\bar{b}_{i}a_{j}b_{j}=\left(a_{i}\bar{b}_{j}% -a_{j}\bar{b}_{i}\right)\left(\bar{a}_{i}b_{j}-\bar{a}_{j}b_{i}\right),
  38. ( i = 1 n a i b i ) ( i = 1 n a i b i ¯ ) + i < j n ( a i b ¯ j - a j b ¯ i ) ( a i b ¯ j - a j b ¯ i ¯ ) = ( i = 1 n a i a ¯ i ) ( i = 1 n b i b ¯ i ) . \left(\sum_{i=1}^{n}a_{i}b_{i}\right)\left(\sum_{i=1}^{n}\overline{a_{i}b_{i}}% \right)+\sum_{i<j}^{n}\left(a_{i}\bar{b}_{j}-a_{j}\bar{b}_{i}\right)\left(% \overline{a_{i}\bar{b}_{j}-a_{j}\bar{b}_{i}}\right)=\left(\sum_{i=1}^{n}a_{i}% \bar{a}_{i}\right)\left(\sum_{i=1}^{n}b_{i}\bar{b}_{i}\right).
  39. | i = 1 n a i b i | 2 + i < j n | a i b ¯ j - a j b ¯ i | 2 = ( i = 1 n | a i | 2 ) ( i = 1 n | b i | 2 ) . \left|\sum_{i=1}^{n}a_{i}b_{i}\right|^{2}+\sum_{i<j}^{n}\left|a_{i}\bar{b}_{j}% -a_{j}\bar{b}_{i}\right|^{2}=\left(\sum_{i=1}^{n}\left|a_{i}\right|^{2}\right)% \left(\sum_{i=1}^{n}\left|b_{i}\right|^{2}\right).

Lambda_lifting.html

  1. O ( n 2 ) O(n^{2})
  2. f x = y f = λ x . y f\ x=y\equiv f=\lambda x.y
  3. f F V ( E ) ( let f : f = E in L ( λ f . L ) E ) f\not\in FV(E)\to(\operatorname{let}f:f=E\operatorname{in}L\equiv(\lambda f.L)% \ E)
  4. x F V ( E ) ( let v , , w , x : E and F in L let v , , w : E in let x : F in L ) x\not\in FV(E)\to(\operatorname{let}v,...,w,x:E\and F\operatorname{in}L\equiv% \operatorname{let}v,...,w:E\operatorname{in}\operatorname{let}x:F\operatorname% {in}L)
  5. \equiv
  6. _ \_
  7. L [ G := S ] L[G:=S]
  8. lambda - lift - op [ S , L , P ] = P [ L := lambda - lift [ S , L ] ] \operatorname{lambda-lift-op}[S,L,P]=P[L:=\operatorname{lambda-lift}[S,L]]
  9. lambda - lift [ S , L ] let V : de - lambda [ G = S ] in L [ S := G ] \operatorname{lambda-lift}[S,L]\equiv\operatorname{let}V:\operatorname{de-% lambda}[G=S]\operatorname{in}L[S:=G]
  10. G = make - call [ V , FV [ S ] ] G=\operatorname{make-call}[V,\operatorname{FV}[S]]
  11. V vars [ let F in L ] V\not\in\operatorname{vars}[\operatorname{let}F\operatorname{in}L]
  12. vars [ E ] \operatorname{vars}[E]
  13. F = t r u e F=true
  14. L = λ f . ( λ x . f ( x x ) ) ( λ x . f ( x x ) ) L=\lambda f.(\lambda x.f\ (x\ x))\ (\lambda x.f\ (x\ x))
  15. S = λ x . f ( x x ) S=\lambda x.f\ (x\ x)
  16. G = p f G=p\ f
  17. de - lambda [ p f = λ x . f ( x x ) ] p f x = f ( x x ) \operatorname{de-lambda}[p\ f=\lambda x.f\ (x\ x)]\equiv p\ f\ x=f\ (x\ x)
  18. lambda - lift [ λ x . f ( x x ) , let t r u e in λ f . ( λ x . f ( x x ) ) ( λ x . f ( x x ) ) ] let p f x = f ( x x ) in λ f . ( p f ) ( p f ) \operatorname{lambda-lift}[\lambda x.f\ (x\ x),\operatorname{let}true% \operatorname{in}\lambda f.(\lambda x.f\ (x\ x))\ (\lambda x.f\ (x\ x))]\equiv% \operatorname{let}p\ f\ x=f\ (x\ x)\operatorname{in}\lambda f.(p\ f)\ (p\ f)
  19. X V make - call [ H , V ] make - call [ H , V ¬ { X } ] X X\in V\to\operatorname{make-call}[H,V]\equiv\operatorname{make-call}[H,V\cap% \neg\{X\}]\ X
  20. make - call [ H , { } ] H \operatorname{make-call}[H,\{\}]\equiv H
  21. S = λ x . f ( x x ) S=\lambda x.f\ (x\ x)
  22. FV ( S ) = { f } \operatorname{FV}(S)=\{f\}
  23. G make - call [ p , FV [ S ] ] make - call [ p , { f } ] make - call [ p , { } ] f p f G\equiv\operatorname{make-call}[p,\operatorname{FV}[S]]\equiv\operatorname{% make-call}[p,\{f\}]\equiv\operatorname{make-call}[p,\{\}]\ f\equiv p\ f
  24. lambda - lift [ ( λ V . E ) S , L ] let V : de - lambda [ G = S ] in L [ ( λ V . E ) S := E [ V := G ] ] \operatorname{lambda-lift}[(\lambda V.E)\ S,L]\equiv\operatorname{let}V:% \operatorname{de-lambda}[G=S]\operatorname{in}L[(\lambda V.E)\ S:=E[V:=G]]
  25. G = make - call [ V , FV [ S ] ] G=\operatorname{make-call}[V,\operatorname{FV}[S]]
  26. V = x V=x
  27. E = f ( x x ) E=f\ (x\ x)
  28. S = ( λ x . f ( x x ) ) S=(\lambda x.f\ (x\ x))
  29. L = λ f . ( λ x . f ( x x ) ) ( λ x . f ( x x ) ) L=\lambda f.(\lambda x.f\ (x\ x))\ (\lambda x.f\ (x\ x))
  30. G = x f G=x\ f
  31. E [ V := G ] = f ( x x ) [ x := x f ] = f ( ( x f ) ( x f ) ) E[V:=G]=f\ (x\ x)[x:=x\ f]=f\ ((x\ f)\ (x\ f))
  32. L [ ( λ V . E ) F := E [ V := G ] ] = L [ ( λ x . f ( x x ) ) ( λ x . f ( x x ) ) := f ( ( x f ) ( x f ) ) ] = λ f . f ( ( x f ) ( x f ) ) L[(\lambda V.E)\ F:=E[V:=G]]=L[(\lambda x.f\ (x\ x))\ (\lambda x.f\ (x\ x)):=f% \ ((x\ f)\ (x\ f))]=\lambda f.f\ ((x\ f)\ (x\ f))
  33. de - lambda [ x f = λ y . f ( y y ) ] x f y = f ( y y ) \operatorname{de-lambda}[x\ f=\lambda y.f\ (y\ y)]\equiv x\ f\ y=f\ (y\ y)
  34. lambda - lift [ ( λ x . f ( x x ) ) ( λ x . f ( x x ) ) , λ f . ( λ x . f ( x x ) ) ( λ x . f ( x x ) ) ] let x f y = f ( y y ) in λ f . ( x f ) ( x f ) \operatorname{lambda-lift}[(\lambda x.f\ (x\ x))\ (\lambda x.f\ (x\ x)),% \lambda f.(\lambda x.f\ (x\ x))\ (\lambda x.f\ (x\ x))]\equiv\operatorname{let% }x\ f\ y=f\ (y\ y)\operatorname{in}\lambda f.(x\ f)\ (x\ f)
  35. let M in N \operatorname{let}M\operatorname{in}N
  36. lambda - lift - tran [ λ f . ( λ x . f ( x x ) ) ( λ x . f ( x x ) ) ] let p f x = f ( x x ) and q p f = ( p f ) ( p f ) in q p \operatorname{lambda-lift-tran}[\lambda f.(\lambda x.f\ (x\ x))\ (\lambda x.f% \ (x\ x))]\equiv\operatorname{let}p\ f\ x=f\ (x\ x)\and q\ p\ f=(p\ f)\ (p\ f)% \operatorname{in}q\ p
  37. de - let [ lambda - lift - tran [ λ f . ( λ x . f ( x x ) ) ( λ x . f ( x x ) ) ] ] ( λ p . ( λ q . q p ) λ p . λ f . ( p f ) ( p f ) ) λ f . λ x . f ( x x ) \operatorname{de-let}[\operatorname{lambda-lift-tran}[\lambda f.(\lambda x.f\ % (x\ x))\ (\lambda x.f\ (x\ x))]]\equiv(\lambda p.(\lambda q.q\ p)\ \lambda p.% \lambda f.(p\ f)\ (p\ f))\ \lambda f.\lambda x.f\ (x\ x)
  38. lambda - lift - tran [ L ] = drop - params - tran [ merge - let [ lambda - apply [ L ] ] ] \operatorname{lambda-lift-tran}[L]=\operatorname{drop-params-tran}[% \operatorname{merge-let}[\operatorname{lambda-apply}[L]]]
  39. lambda - apply [ L ] = lambda - process [ lift - choice [ L ] , L ] \operatorname{lambda-apply}[L]=\operatorname{lambda-process}[\operatorname{% lift-choice}[L],L]
  40. lambda - process [ none , L ] = L \operatorname{lambda-process}[\operatorname{none},L]=L
  41. lambda - process [ S , L ] = lambda - apply [ lambda - lift [ S , L ] ] \operatorname{lambda-process}[S,L]=\operatorname{lambda-apply}[\operatorname{% lambda-lift}[S,L]]
  42. merge - let [ let V : E in let W : F in G ] = merge - let [ let V , W : E and F in G ] \operatorname{merge-let}[\operatorname{let}V:E\operatorname{in}\operatorname{% let}W:F\operatorname{in}G]=\operatorname{merge-let}[\operatorname{let}V,W:E% \and F\operatorname{in}G]
  43. merge - let [ E ] = E \operatorname{merge-let}[E]=E
  44. lambda - free [ λ F . X ] = false \operatorname{lambda-free}[\lambda F.X]=\operatorname{false}
  45. lambda - free [ V ] = true \operatorname{lambda-free}[V]=\operatorname{true}
  46. lambda - free [ M N ] = lambda - free [ M ] and lambda - free [ N ] \operatorname{lambda-free}[M\ N]=\operatorname{lambda-free}[M]\and% \operatorname{lambda-free}[N]
  47. λ x 1 . λ x n . X \lambda x_{1}.\ ...\ \lambda x_{n}.X
  48. lambda - anon [ λ F . X ] = lambda - free [ X ] lambda - anon [ X ] \operatorname{lambda-anon}[\lambda F.X]=\operatorname{lambda-free}[X]% \operatorname{lambda-anon}[X]
  49. lambda - anon [ V ] = false \operatorname{lambda-anon}[V]=\operatorname{false}
  50. lambda - anon [ M N ] = false \operatorname{lambda-anon}[M\ N]=\operatorname{false}
  51. lambda - anon [ X ] lift - choice [ X ] = X \operatorname{lambda-anon}[X]\to\operatorname{lift-choice}[X]=X
  52. lift - choice [ λ F . X ] = lift - choice [ X ] \operatorname{lift-choice}[\lambda F.X]=\operatorname{lift-choice}[X]
  53. lift - choice [ M ] none lift - choice [ M N ] = lift - choice [ M ] \operatorname{lift-choice}[M]\neq\operatorname{none}\to\operatorname{lift-% choice}[M\ N]=\operatorname{lift-choice}[M]
  54. lift - choice [ M N ] = lift - choice [ N ] \operatorname{lift-choice}[M\ N]=\operatorname{lift-choice}[N]
  55. lift - choice [ V ] = none \operatorname{lift-choice}[V]=\operatorname{none}
  56. λ f . ( λ x . f ( x x ) ) ( λ y . f ( y y ) ) \lambda f.(\lambda x.f\ (x\ x))\ (\lambda y.f\ (y\ y))
  57. λ x . f ( x x ) \lambda x.f\ (x\ x)
  58. lift - choice [ λ f . ( λ x . f ( x x ) ) ( λ y . f ( y y ) ) ] \operatorname{lift-choice}[\lambda f.(\lambda x.f\ (x\ x))\ (\lambda y.f\ (y\ % y))]
  59. lift - choice [ ( λ x . f ( x x ) ) ( λ y . f ( y y ) ) ] \operatorname{lift-choice}[(\lambda x.f\ (x\ x))\ (\lambda y.f\ (y\ y))]
  60. lift - choice [ λ x . f ( x x ) ] \operatorname{lift-choice}[\lambda x.f\ (x\ x)]
  61. λ x . f ( x x ) \lambda x.f\ (x\ x)
  62. λ f . ( p f ) ( p f ) \lambda f.(p\ f)\ (p\ f)
  63. λ f . ( p f ) ( p f ) \lambda f.(p\ f)\ (p\ f)
  64. lift - choice [ λ f . ( p f ) ( p f ) ] \operatorname{lift-choice}[\lambda f.(p\ f)\ (p\ f)]
  65. λ f . ( p f ) ( p f ) \lambda f.(p\ f)\ (p\ f)
  66. ( λ F . M ) N (\lambda F.M)\ N
  67. lambda - named [ ( λ F . M ) N ] = lambda - free [ M ] and lambda - anon [ N ] \operatorname{lambda-named}[(\lambda F.M)\ N]=\operatorname{lambda-free}[M]% \and\operatorname{lambda-anon}[N]
  68. lambda - named [ λ F . X ] = false \operatorname{lambda-named}[\lambda F.X]=\operatorname{false}
  69. lambda - named [ V ] = false \operatorname{lambda-named}[V]=\operatorname{false}
  70. lambda - named [ X ] lambda - anon [ X ] lift - choice [ X ] = X \operatorname{lambda-named}[X]\operatorname{lambda-anon}[X]\to\operatorname{% lift-choice}[X]=X
  71. lift - choice [ λ F . X ] = lift - choice [ X ] \operatorname{lift-choice}[\lambda F.X]=\operatorname{lift-choice}[X]
  72. lift - choice [ M ] none lift - choice [ M N ] = lift - choice [ M ] \operatorname{lift-choice}[M]\neq\operatorname{none}\to\operatorname{lift-% choice}[M\ N]=\operatorname{lift-choice}[M]
  73. lift - choice [ M N ] = lift - choice [ N ] \operatorname{lift-choice}[M\ N]=\operatorname{lift-choice}[N]
  74. lift - choice [ V ] = none \operatorname{lift-choice}[V]=\operatorname{none}
  75. λ f . ( λ x . f ( x x ) ) ( λ y . f ( y y ) ) \lambda f.(\lambda x.f\ (x\ x))\ (\lambda y.f\ (y\ y))
  76. ( λ x . f ( x x ) ) ( λ y . f ( y y ) ) (\lambda x.f\ (x\ x))\ (\lambda y.f\ (y\ y))
  77. lift - choice [ λ f . ( λ x . f ( x x ) ) ( λ y . f ( y y ) ) ] \operatorname{lift-choice}[\lambda f.(\lambda x.f\ (x\ x))\ (\lambda y.f\ (y\ % y))]
  78. lift - choice [ ( λ x . f ( x x ) ) ( λ y . f ( y y ) ) ] \operatorname{lift-choice}[(\lambda x.f\ (x\ x))\ (\lambda y.f\ (y\ y))]
  79. ( λ x . f ( x x ) ) ( λ y . f ( y y ) ) (\lambda x.f\ (x\ x))\ (\lambda y.f\ (y\ y))
  80. λ f . f ( ( x f ) ( x f ) ) \lambda f.f\ ((x\ f)\ (x\ f))
  81. λ f . f ( ( x f ) ( x f ) ) \lambda f.f\ ((x\ f)\ (x\ f))
  82. lift - choice [ λ f . f ( ( x f ) ( x f ) ) ] \operatorname{lift-choice}[\lambda f.f\ ((x\ f)\ (x\ f))]
  83. λ f . f ( ( x f ) ( x f ) ) \lambda f.f\ ((x\ f)\ (x\ f))
  84. λ f . ( λ x . f ( x x ) ) ( λ x . f ( x x ) ) \lambda f.(\lambda x.f\ (x\ x))\ (\lambda x.f\ (x\ x))
  85. let x f y = f ( y y ) and q x f = f ( ( x f ) ( x f ) ) in q x \operatorname{let}x\ f\ y=f\ (y\ y)\and q\ x\ f=f\ ((x\ f)\ (x\ f))% \operatorname{in}q\ x
  86. let x f y = f ( y y ) and q f = f ( ( x f ) ( x f ) ) in q \operatorname{let}x\ f\ y=f\ (y\ y)\and q\ f=f\ ((x\ f)\ (x\ f))\operatorname{% in}q
  87. ( λ x . ( λ q . q ) λ f . f ( x f ) ( x f ) ) λ f . λ y . f ( y y ) (\lambda x.(\lambda q.q)\ \lambda f.f\ (x\ f)\ (x\ f))\ \lambda f.\lambda y.f% \ (y\ y)
  88. λ f . ( λ x . f ( x x ) ) ( λ x . f ( x x ) ) \lambda f.(\lambda x.f\ (x\ x))\ (\lambda x.f\ (x\ x))
  89. ( λ x . f ( x x ) ) ( λ x . f ( x x ) ) (\lambda x.f\ (x\ x))\ (\lambda x.f\ (x\ x))
  90. x f x\ f
  91. { x , f } \{x,f\}
  92. ( λ f . ( λ x . f ( x x ) ) ( λ x . f ( x x ) ) ) (\lambda f.(\lambda x.f\ (x\ x))(\lambda x.f\ (x\ x)))
  93. [ ( λ x . f ( x x ) ) ( λ x . f ( x x ) ) := f ( ( x f ) ( x f ) ) ] [(\lambda x.f\ (x\ x))\ (\lambda x.f\ (x\ x)):=f\ ((x\ f)\ (x\ f))]
  94. x f = λ y . f ( y y ) x\ f=\lambda y.f\ (y\ y)
  95. { x , f , p } \{x,f,p\}
  96. λ f . f ( ( x f ) ( x f ) ) \lambda f.f\ ((x\ f)\ (x\ f))
  97. x f y = f ( y y ) x\ f\ y=f\ (y\ y)
  98. λ f . f ( ( x f ) ( x f ) ) \lambda f.f\ ((x\ f)\ (x\ f))
  99. q x q\ x
  100. { x , f , p } \{x,f,p\}
  101. λ f . f ( ( x f ) ( x f ) ) [ λ f . f ( ( x f ) ( x f ) ) := q x ] \lambda f.f\ ((x\ f)\ (x\ f))[\lambda f.f\ ((x\ f)\ (x\ f)):=q\ x]
  102. x f y = f ( y y ) and q x = λ f . f ( ( x f ) ( x f ) ) x\ f\ y=f\ (y\ y)\and q\ x=\lambda f.f\ ((x\ f)\ (x\ f))
  103. { x , f , p , q } \{x,f,p,q\}
  104. q x q\ x
  105. x f y = f ( y y ) and q x f = f ( ( x f ) ( x f ) ) x\ f\ y=f\ (y\ y)\and q\ x\ f=f\ ((x\ f)\ (x\ f))
  106. { x , f , p , q } \{x,f,p,q\}
  107. let p f x = f ( x x ) and q p f = ( p f ) ( p f ) in q p \operatorname{let}p\ f\ x=f\ (x\ x)\and q\ p\ f=(p\ f)\ (p\ f)\operatorname{in% }q\ p
  108. let p f x = f ( x x ) and q f = ( p f ) ( p f ) in q \operatorname{let}p\ f\ x=f\ (x\ x)\and q\ f=(p\ f)\ (p\ f)\operatorname{in}q
  109. ( λ p . ( λ q . q ) λ f . ( p f ) ( p f ) ) λ f . λ x . f ( x x ) (\lambda p.(\lambda q.q)\ \lambda f.(p\ f)\ (p\ f))\ \lambda f.\lambda x.f\ (x% \ x)
  110. λ f . ( λ x . f ( x x ) ) ( λ x . f ( x x ) ) \lambda f.(\lambda x.f\ (x\ x))\ (\lambda x.f\ (x\ x))
  111. λ x . f ( x x ) \lambda x.f\ (x\ x)
  112. p f p\ f
  113. { x , f } \{x,f\}
  114. ( λ f . ( λ x . f ( x x ) ) ( λ x . f ( x x ) ) ) [ λ x . f ( x x ) := p f ] (\lambda f.(\lambda x.f\ (x\ x))(\lambda x.f\ (x\ x)))[\lambda x.f\ (x\ x):=p% \ f]
  115. p f = λ x . f ( x x ) p\ f=\lambda x.f\ (x\ x)
  116. { x , f , p } \{x,f,p\}
  117. λ f . ( p f ) ( p f ) \lambda f.(p\ f)\ (p\ f)
  118. p f x = f ( x x ) p\ f\ x=f\ (x\ x)
  119. λ f . ( p f ) ( p f ) \lambda f.(p\ f)\ (p\ f)
  120. q p q\ p
  121. { x , f , p } \{x,f,p\}
  122. λ f . ( p f ) ( p f ) [ λ f . ( p f ) ( p f ) := q p ] \lambda f.(p\ f)\ (p\ f)[\lambda f.(p\ f)\ (p\ f):=q\ p]
  123. p f x = f ( x x ) and q p = λ f . ( p f ) ( p f ) p\ f\ x=f\ (x\ x)\and q\ p=\lambda f.(p\ f)\ (p\ f)
  124. { x , f , p , q } \{x,f,p,q\}
  125. q p q\ p
  126. p f x = f ( x x ) and q p f = ( p f ) ( p f ) p\ f\ x=f\ (x\ x)\and q\ p\ f=(p\ f)\ (p\ f)
  127. { x , f , p , q } \{x,f,p,q\}
  128. λ x . f ( x x ) \lambda x.f\ (x\ x)
  129. λ f . ( p f ) ( p f ) \lambda f.(p\ f)\ (p\ f)
  130. p f x = f ( x x ) p\ f\ x=f(x\ x)
  131. λ f . ( p f ) ( p f ) \lambda f.(p\ f)\ (p\ f)
  132. q p q\ p
  133. q p f = ( p f ) ( p f ) q\ p\ f=(p\ f)\ (p\ f)
  134. let p f x = f ( x x ) and q p f = ( p f ) ( p f ) in q p \operatorname{let}p\ f\ x=f\ (x\ x)\and q\ p\ f=(p\ f)\ (p\ f)\operatorname{in% }q\ p
  135. K K
  136. λ f . ( λ x . f ( x x ) ) ( λ x . f ( x x ) ) K \lambda f.(\lambda x.f\ (x\ x))\ (\lambda x.f\ (x\ x))\ K
  137. let p f x = f ( x x ) and q p f = ( p f ) ( p f ) in q p K \operatorname{let}p\ f\ x=f\ (x\ x)\and q\ p\ f=(p\ f)\ (p\ f)\operatorname{in% }q\ p\ \ K
  138. ( λ x . K ( x x ) ) ( λ x . K ( x x ) ) (\lambda x.K\ (x\ x))\ (\lambda x.K\ (x\ x))
  139. let p f x = f ( x x ) and q p f = ( p f ) ( p f ) in p K ( p K ) \operatorname{let}p\ f\ x=f\ (x\ x)\and q\ p\ f=(p\ f)\ (p\ f)\operatorname{in% }p\ K\ (p\ K)
  140. K ( ( λ x . K ( x x ) ) ( λ x . K ( x x ) ) ) K\ ((\lambda x.K\ (x\ x))\ (\lambda x.K\ (x\ x)))
  141. let p f x = f ( x x ) and q p f = p f ( p f ) in K ( p K ( p K ) ) \operatorname{let}p\ f\ x=f\ (x\ x)\and q\ p\ f=p\ f\ (p\ f)\operatorname{in}K% \ (p\ K\ (p\ K))
  142. ( λ x . K ( x x ) ) ( λ x . K ( x x ) ) = K ( ( λ x . K ( x x ) ) ( λ x . K ( x x ) ) ) ) (\lambda x.K\ (x\ x))\ (\lambda x.K\ (x\ x))=K\ ((\lambda x.K\ (x\ x))\ (% \lambda x.K\ (x\ x))))
  143. p K ( p K ) = K ( p K ( p K ) ) p\ K\ (p\ K)=K\ (p\ K\ (p\ K))
  144. lambda - drop - op [ L , P , X ] = P [ L := drop - params - tran [ sink - test [ L , X ] ] ] \operatorname{lambda-drop-op}[L,P,X]=P[L:=\operatorname{drop-params-tran}[% \operatorname{sink-test}[L,X]]]
  145. lambda - drop - tran [ L , X ] = drop - params - tran [ sink - tran [ de - let [ L , X ] ] ] \operatorname{lambda-drop-tran}[L,X]=\operatorname{drop-params-tran}[% \operatorname{sink-tran}[\operatorname{de-let}[L,X]]]
  146. sink - tran [ ( λ N . B ) Y , X ] = sink - test [ ( λ N . sink - tran [ B ] ) sink - tran [ Y ] , X ] \operatorname{sink-tran}[(\lambda N.B)\ Y,X]=\operatorname{sink-test}[(\lambda N% .\operatorname{sink-tran}[B])\ \operatorname{sink-tran}[Y],X]
  147. sink - tran [ λ N . B , X ] = λ N . sink - tran [ B , X ] \operatorname{sink-tran}[\lambda N.B,X]=\lambda N.\operatorname{sink-tran}[B,X]
  148. sink - tran [ M N , X ] = sink - tran [ M , X ] sink - tran [ M , X ] \operatorname{sink-tran}[M\ N,X]=\operatorname{sink-tran}[M,X]\ \operatorname{% sink-tran}[M,X]
  149. sink - tran [ V , X ] = V \operatorname{sink-tran}[V,X]=V
  150. E FV [ G ] and E FV [ H ] sink [ ( λ E . G H ) Y , X ] = G H E\not\in\operatorname{FV}[G]\and E\not\in\operatorname{FV}[H]\to\operatorname{% sink}[(\lambda E.G\ H)\ Y,X]=G\ H
  151. E FV [ G ] and E FV [ H ] sink [ ( λ E . G H ) Y , X ] = sink - test [ G sink - test [ ( λ E . H ) Y , X ] ] E\not\in\operatorname{FV}[G]\and E\in\operatorname{FV}[H]\to\operatorname{sink% }[(\lambda E.G\ H)\ Y,X]=\operatorname{sink-test}[G\ \operatorname{sink-test}[% (\lambda E.H)\ Y,X]]
  152. E FV [ G ] and E FV [ H ] sink [ ( λ E . G H ) Y , X ] = ( sink - test [ ( λ E . G ) Y , X ] ) H E\in\operatorname{FV}[G]\and E\not\in\operatorname{FV}[H]\to\operatorname{sink% }[(\lambda E.G\ H)\ Y,X]=(\operatorname{sink-test}[(\lambda E.G)\ Y,X])\ H
  153. E FV [ G ] and E FV [ H ] sink [ ( λ E . G H ) Y , X ] = ( λ E . G H ) Y E\in\operatorname{FV}[G]\and E\in\operatorname{FV}[H]\to\operatorname{sink}[(% \lambda E.G\ H)\ Y,X]=(\lambda E.G\ H)\ Y
  154. V W sink [ ( λ V . λ W . E ) Y , X ] = λ W . sink - test [ ( λ V . E ) Y , X ] V\neq W\to\operatorname{sink}[(\lambda V.\lambda W.E)\ Y,X]=\lambda W.% \operatorname{sink-test}[(\lambda V.E)\ Y,X]
  155. E V sink [ ( λ E . V ) Y , X ] = V E\neq V\to\operatorname{sink}[(\lambda E.V)\ Y,X]=V
  156. E = V sink [ ( λ E . V ) Y , X ] = Y E=V\to\operatorname{sink}[(\lambda E.V)\ Y,X]=Y
  157. L X sink - test [ L , X ] = L L\in X\to\operatorname{sink-test}[L,X]=L
  158. L X sink - test [ L , X ] = sink [ L , X ] L\not\in X\to\operatorname{sink-test}[L,X]=\operatorname{sink}[L,X]
  159. sink - tran [ de - let [ let p f x = f ( x x ) and q p f = ( p f ) ( p f ) in q p ] ] \operatorname{sink-tran}[\operatorname{de-let}[\operatorname{let}p\ f\ x=f\ (x% \ x)\and q\ p\ f=(p\ f)\ (p\ f)\operatorname{in}q\ p]]
  160. sink - tran [ ( λ p . ( λ q . q p ) ( λ p . λ f . ( p f ) ( p f ) ) ) ( λ f . λ x . f ( x x ) ) ] \operatorname{sink-tran}[(\lambda p.(\lambda q.q\ p)\ (\lambda p.\lambda f.(p% \ f)\ (p\ f)))\ (\lambda f.\lambda x.f\ (x\ x))]
  161. sink [ ( λ p . sink [ ( λ q . q p ) ( λ p . λ f . ( p f ) ( p f ) ) ] ) ( λ f . λ x . f ( x x ) ) ] \operatorname{sink}[(\lambda p.\operatorname{sink}[(\lambda q.q\ p)\ (\lambda p% .\lambda f.(p\ f)\ (p\ f))])\ (\lambda f.\lambda x.f\ (x\ x))]
  162. sink [ ( λ q . q p ) ( λ p . λ f . ( p f ) ( p f ) ) ] \operatorname{sink}[(\lambda q.q\ p)\ (\lambda p.\lambda f.(p\ f)\ (p\ f))]
  163. E FV [ G ] and E FV [ H ] sink [ ( λ E . G H ) Y , X ] E\in\operatorname{FV}[G]\and E\not\in\operatorname{FV}[H]\to\operatorname{sink% }[(\lambda E.G\ H)\ Y,X]
  164. E = q , G = q , H = p , Y = ( λ p . λ f . ( p f ) ( p f ) ) , X = { } E=q,G=q,H=p,Y=(\lambda p.\lambda f.(p\ f)\ (p\ f)),X=\{\}
  165. ( sink [ ( λ E . G ) Y , X ] ) H (\operatorname{sink}[(\lambda E.G)\ Y,X])\ H
  166. ( sink [ ( λ q . q ) ( λ p . λ f . ( p f ) ( p f ) ) , X ] ) p (\operatorname{sink}[(\lambda q.q)\ (\lambda p.\lambda f.(p\ f)\ (p\ f)),X])\ p
  167. sink [ ( λ p . sink [ ( λ q . q ) ( λ p . λ f . ( p f ) ( p f ) ) ] p ) ( λ f . λ x . f ( x x ) ) ] \operatorname{sink}[(\lambda p.\operatorname{sink}[(\lambda q.q)\ (\lambda p.% \lambda f.(p\ f)\ (p\ f))]\ p)\ (\lambda f.\lambda x.f\ (x\ x))]
  168. sink [ ( λ q . q ) ( λ p . λ f . ( p f ) ( p f ) ) ] \operatorname{sink}[(\lambda q.q)\ (\lambda p.\lambda f.(p\ f)\ (p\ f))]
  169. E = V sink [ ( λ E . V ) Y , X ] E=V\to\operatorname{sink}[(\lambda E.V)\ Y,X]
  170. E = q , V = q , Y = ( λ p . λ f . ( p f ) ( p f ) ) , X = { } E=q,V=q,Y=(\lambda p.\lambda f.(p\ f)\ (p\ f)),X=\{\}
  171. Y Y
  172. ( λ p . λ f . ( p f ) ( p f ) ) (\lambda p.\lambda f.(p\ f)\ (p\ f))
  173. sink [ ( λ p . ( λ p . λ f . ( p f ) ( p f ) ) p ) ( λ f . λ x . f ( x x ) ) ] \operatorname{sink}[(\lambda p.(\lambda p.\lambda f.(p\ f)\ (p\ f))\ p)\ (% \lambda f.\lambda x.f\ (x\ x))]
  174. E FV [ G ] and E FV [ H ] sink [ ( λ E . G H ) Y , X ] E\not\in\operatorname{FV}[G]\and E\in\operatorname{FV}[H]\to\operatorname{sink% }[(\lambda E.G\ H)\ Y,X]
  175. E = p , G = ( λ p . λ f . ( p f ) ( p f ) ) , H = p , Y = ( λ f . λ x . f ( x x ) ) E=p,G=(\lambda p.\lambda f.(p\ f)\ (p\ f)),H=p,Y=(\lambda f.\lambda x.f\ (x\ x))
  176. sink [ G sink [ ( λ E . H ) Y , X ] ] \operatorname{sink}[G\ \operatorname{sink}[(\lambda E.H)\ Y,X]]
  177. sink [ ( λ p . λ f . ( p f ) ( p f ) ) sink - test [ ( λ p . p ) ( λ f . λ x . f ( x x ) ) , X ] ] \operatorname{sink}[(\lambda p.\lambda f.(p\ f)\ (p\ f))\ \operatorname{sink-% test}[(\lambda p.p)\ (\lambda f.\lambda x.f\ (x\ x)),X]]
  178. sink [ ( λ p . p ) ( λ f . λ x . f ( x x ) ) , X ] \operatorname{sink}[(\lambda p.p)\ (\lambda f.\lambda x.f\ (x\ x)),X]
  179. E = V sink [ ( λ E . V ) Y , X ] E=V\to\operatorname{sink}[(\lambda E.V)\ Y,X]
  180. E = p , V = p , Y = ( λ f . λ x . f ( x x ) ) , X = { } E=p,V=p,Y=(\lambda f.\lambda x.f\ (x\ x)),X=\{\}
  181. Y Y
  182. ( λ f . λ x . f ( x x ) ) (\lambda f.\lambda x.f\ (x\ x))
  183. sink [ ( λ p . λ f . ( p f ) ( p f ) ) ( λ f . λ x . f ( x x ) ) ] \operatorname{sink}[(\lambda p.\lambda f.(p\ f)\ (p\ f))\ (\lambda f.\lambda x% .f\ (x\ x))]
  184. V W sink [ ( λ V . λ W . E ) Y , X ] V\neq W\to\operatorname{sink}[(\lambda V.\lambda W.E)\ Y,X]
  185. V = p , W = f , E = ( p f ) ( p f ) , Y = ( λ f . λ x . f ( x x ) ) V=p,W=f,E=(p\ f)\ (p\ f),Y=(\lambda f.\lambda x.f\ (x\ x))
  186. λ W . sink [ ( λ V . E ) Y , X ] \lambda W.\operatorname{sink}[(\lambda V.E)\ Y,X]
  187. λ f . sink [ ( λ p . ( p f ) ( p f ) ) ( λ f . λ x . f ( x x ) ) , X ] \lambda f.\operatorname{sink}[(\lambda p.(p\ f)\ (p\ f))\ (\lambda f.\lambda x% .f\ (x\ x)),X]
  188. sink [ ( λ p . ( p f ) ( p f ) ) ( λ f . λ x . f ( x x ) ) , X ] \operatorname{sink}[(\lambda p.(p\ f)\ (p\ f))\ (\lambda f.\lambda x.f\ (x\ x)% ),X]
  189. E FV [ G ] and E FV [ H ] sink [ ( λ E . G H ) Y , X ] E\in\operatorname{FV}[G]\and E\in\operatorname{FV}[H]\to\operatorname{sink}[(% \lambda E.G\ H)\ Y,X]
  190. E = p , G = ( p f ) , H = ( p f ) , Y = ( λ f . λ x . f ( x x ) ) E=p,G=(p\ f),H=(p\ f),Y=(\lambda f.\lambda x.f\ (x\ x))
  191. ( λ E . G H ) Y (\lambda E.G\ H)\ Y
  192. ( λ p . ( p f ) ( p f ) ) ( λ f . λ x . f ( x x ) ) (\lambda p.(p\ f)\ (p\ f))\ (\lambda f.\lambda x.f\ (x\ x))
  193. λ f . ( λ p . ( p f ) ( p f ) ) ( λ f . λ x . f ( x x ) ) \lambda f.(\lambda p.(p\ f)\ (p\ f))\ (\lambda f.\lambda x.f\ (x\ x))
  194. λ m , p , q . ( λ g . λ n . ( n ( g m p n ) ( g q p n ) ) ) λ x . λ o . λ y . o x y \lambda m,p,q.(\lambda g.\lambda n.(n\ (g\ m\ p\ n)\ (g\ q\ p\ n)))\ \lambda x% .\lambda o.\lambda y.o\ x\ y
  195. drop - params - tran [ λ m , p , q . ( λ g . λ n . n ( g m p n ) ( g q p n ) ) λ x . λ o . λ y . o x y \operatorname{drop-params-tran}[\lambda m,p,q.(\lambda g.\lambda n.n\ (g\ m\ p% \ n)\ (g\ q\ p\ n))\ \lambda x.\lambda o.\lambda y.o\ x\ y
  196. λ m , p , q . ( λ g . λ n . n ( g m n ) ( g q n ) ) λ x . λ y . p x y \equiv\lambda m,p,q.(\lambda g.\lambda n.n\ (g\ m\ n)\ (g\ q\ n))\ \lambda x.% \lambda y.p\ x\ y
  197. drop - params - tran [ λ f . ( λ p . ( p f ) ( p f ) ) ( λ f . λ x . f ( x x ) ) ] \operatorname{drop-params-tran}[\lambda f.(\lambda p.(p\ f)\ (p\ f))\ (\lambda f% .\lambda x.f\ (x\ x))]
  198. λ f . ( λ p . p p ) ( λ x . f ( x x ) ) \equiv\lambda f.(\lambda p.p\ p)\ (\lambda x.f\ (x\ x))
  199. drop - params - tran [ L ] ( drop - params [ L , D , F V [ L ] , [ ] ] ) \operatorname{drop-params-tran}[L]\equiv(\operatorname{drop-params}[L,D,FV[L],% []])
  200. build - param - list [ L , D , V , _ ] \operatorname{build-param-list}[L,D,V,\_]
  201. λ m , p , q . ( λ g . λ n . ( n ( g m p n ) ( g q p n ) ) ) λ x . λ o . λ y . o x y \lambda m,p,q.(\lambda g.\lambda n.(n\ (g\ m\ p\ n)\ (g\ q\ p\ n)))\ \lambda x% .\lambda o.\lambda y.o\ x\ y
  202. D [ g ] = [ [ x , false , _ ] , [ o , _ , p ] , [ y , _ , n ] ] D[g]=[[x,\operatorname{false},\_],[o,\_,p],[y,\_,n]]
  203. ( λ N . S ) L (\lambda N.S)\ L
  204. build - param - lists [ ( λ N . S ) L , D , V , R ] build - param - lists [ S , D , V , R ] and build - list [ L , D , V , D [ N ] ] \operatorname{build-param-lists}[(\lambda N.S)\ L,D,V,R]\equiv\operatorname{% build-param-lists}[S,D,V,R]\and\operatorname{build-list}[L,D,V,D[N]]
  205. build - param - lists [ λ N . S , D , V , R ] build - param - lists [ S , D , V , R ] \operatorname{build-param-lists}[\lambda N.S,D,V,R]\equiv\operatorname{build-% param-lists}[S,D,V,R]
  206. build - list [ λ P . B , D , V , [ X , _ , _ ] : : L ] build - list [ B , D , V , L ] \operatorname{build-list}[\lambda P.B,D,V,[X,\_,\_]::L]\equiv\operatorname{% build-list}[B,D,V,L]
  207. build - list [ B , D , V , [ ] ] build - param - lists [ B , D , V , _ ] \operatorname{build-list}[B,D,V,[]]\equiv\operatorname{build-param-lists}[B,D,% V,\_]
  208. build - param - lists [ N , D , V , D [ N ] ] \operatorname{build-param-lists}[N,D,V,D[N]]
  209. build - param - lists [ E P , D , V , R ] build - param - lists [ E , D , V , T ] and build - param - lists [ P , D , V , K ] \operatorname{build-param-lists}[E\ P,D,V,R]\equiv\operatorname{build-param-% lists}[E,D,V,T]\and\operatorname{build-param-lists}[P,D,V,K]
  210. and T = [ F , S , A ] : : R and ( S ( equate [ A , P ] and V [ F ] = A ) ) and D [ F ] = K \and T=[F,S,A]::R\and(S\implies(\operatorname{equate}[A,P]\and V[F]=A))\and D[% F]=K
  211. equate [ A , N ] A = N ( def [ V [ N ] ] and A = V [ N ] ) \operatorname{equate}[A,N]\equiv A=N(\operatorname{def}[V[N]]\and A=V[N])
  212. equate [ A , E ] A = E \operatorname{equate}[A,E]\equiv A=E
  213. ask [ S ] S { X : X = S } \operatorname{ask}[S]\equiv S\in\{X:X=S\}
  214. def [ F ] | { X : X = F } | \operatorname{def}[F]\equiv|\{X:X=F\}|
  215. build - param - list [ let V : E in L , D , V , _ ] build - param - list [ E , D , V , _ ] and build - param - list [ L , D , V , _ ] \operatorname{build-param-list}[\operatorname{let}V:E\operatorname{in}L,D,V,\_% ]\equiv\operatorname{build-param-list}[E,D,V,\_]\and\operatorname{build-param-% list}[L,D,V,\_]
  216. build - param - lists [ E and F , D , V , _ ] build - param - lists [ E , D , V , _ ] and build - param - lists [ F , D , V , _ ] \operatorname{build-param-lists}[E\and F,D,V,\_]\equiv\operatorname{build-% param-lists}[E,D,V,\_]\and\operatorname{build-param-lists}[F,D,V,\_]
  217. λ m , p , q . ( λ g . λ n . ( n ( g m p n ) ( g q p n ) ) ) λ x . λ o . λ y . o x y \lambda m,p,q.(\lambda g.\lambda n.(n\ (g\ m\ p\ n)\ (g\ q\ p\ n)))\ \lambda x% .\lambda o.\lambda y.o\ x\ y
  218. D [ g ] = [ [ x , false , _ ] , [ o , true , p ] , [ y , true , n ] ] D[g]=[[x,\operatorname{false},\_],[o,\operatorname{true},p],[y,\operatorname{% true},n]]
  219. λ m , p , q . ( λ g . λ n . ( n ( g m n ) ( g q n ) ) ) λ x . λ y . p x y \lambda m,p,q.(\lambda g.\lambda n.(n\ (g\ m\ n)\ (g\ q\ n)))\ \lambda x.% \lambda y.p\ x\ y
  220. λ m , p , q . ( λ g . λ n . ( n ( g m p n ) ( g q p n ) ) ) λ x . λ o . λ y . o x y \lambda m,p,q.(\lambda g.\lambda n.(n\ (g\ m\ p\ n)\ (g\ q\ p\ n)))\ \lambda x% .\lambda o.\lambda y.o\ x\ y
  221. build - param - list [ λ m , p , q . ( λ g . λ n . ( n ( g m p n ) ( g q p n ) ) ) λ x . λ o . λ y . o x y , D , V , _ ] \operatorname{build-param-list}[\lambda m,p,q.(\lambda g.\lambda n.(n\ (g\ m\ % p\ n)\ (g\ q\ p\ n)))\ \lambda x.\lambda o.\lambda y.o\ x\ y,D,V,\_]
  222. build - param - list [ λ m , p , q . ( λ g . λ n . ( n ( g m p n ) ( g q p n ) ) ) λ x . λ o . λ y . o x y , D , V , _ ] \operatorname{build-param-list}[\lambda m,p,q.(\lambda g.\lambda n.(n\ (g\ m\ % p\ n)\ (g\ q\ p\ n)))\ \lambda x.\lambda o.\lambda y.o\ x\ y,D,V,\_]
  223. build - param - list [ ( λ g . λ n . ( n ( g m p n ) ( g q p n ) ) ) λ x . λ o . λ y . o x y , D , V , _ ] \operatorname{build-param-list}[(\lambda g.\lambda n.(n\ (g\ m\ p\ n)\ (g\ q\ % p\ n)))\ \lambda x.\lambda o.\lambda y.o\ x\ y,D,V,\_]
  224. build - param - lists [ n ( g m p n ) ( g q p n ) , D , V , R ] and build - list [ λ x . λ o . λ y . o x y , D , V , D [ g ] ] \operatorname{build-param-lists}[n\ (g\ m\ p\ n)\ (g\ q\ p\ n),D,V,R]\and% \operatorname{build-list}[\lambda x.\lambda o.\lambda y.o\ x\ y,D,V,D[g]]
  225. build - list [ λ x . λ o . λ y . o x y , D , V , D [ g ] ] \operatorname{build-list}[\lambda x.\lambda o.\lambda y.o\ x\ y,D,V,D[g]]
  226. build - list [ λ x . λ o . λ y . o x y , D , V , D [ g ] ] and D [ g ] = L 1 \operatorname{build-list}[\lambda x.\lambda o.\lambda y.o\ x\ y,D,V,D[g]]\and D% [g]=L_{1}
  227. build - list [ λ o . λ y . o x y , D , V , L 1 ] and D [ g ] = [ x , _ , _ ] : : L 1 \operatorname{build-list}[\lambda o.\lambda y.o\ x\ y,D,V,L_{1}]\and D[g]=[x,% \_,\_]::L_{1}
  228. build - list [ λ y . o x y , D , V , L 2 ] and D [ g ] = [ x , _ , _ ] : : [ o , _ , _ ] : : L 2 \operatorname{build-list}[\lambda y.o\ x\ y,D,V,L_{2}]\and D[g]=[x,\_,\_]::[o,% \_,\_]::L_{2}
  229. build - list [ o x y , D , V , L 3 ] and D [ g ] = [ x , _ , _ ] : : [ o , _ , _ ] : : [ y , _ , _ ] : : L 3 \operatorname{build-list}[o\ x\ y,D,V,L_{3}]\and D[g]=[x,\_,\_]::[o,\_,\_]::[y% ,\_,\_]::L_{3}
  230. build - param - lists [ o x y , D , V , [ ] ] and D [ g ] = [ x , _ , _ ] : : [ o , _ , _ ] : : [ y , _ , _ ] : : [ ] \operatorname{build-param-lists}[o\ x\ y,D,V,[]]\and D[g]=[x,\_,\_]::[o,\_,\_]% ::[y,\_,\_]::[]
  231. build - param - lists [ n ( g m p n ) ( g q p n ) , D , V , R ] \operatorname{build-param-lists}[n\ (g\ m\ p\ n)\ (g\ q\ p\ n),D,V,R]
  232. build - param - lists [ n ( g m p n ) ( g q p n ) , D , V , R ] \operatorname{build-param-lists}[n\ (g\ m\ p\ n)\ (g\ q\ p\ n),D,V,R]
  233. build - param - lists [ n ( g m p n ) , D , V , T 1 ] and build - param - lists [ g q p n , D , V , K 1 ] \operatorname{build-param-lists}[n\ (g\ m\ p\ n),D,V,T_{1}]\and\operatorname{% build-param-lists}[g\ q\ p\ n,D,V,K_{1}]
  234. and ( ( T 1 = [ F 1 , S 1 , A 1 ] : : R \and((T_{1}=[F_{1},S_{1},A_{1}]::R
  235. and ( S 1 ( equate [ A 1 , g q p n ] and V [ F 1 ] = g q p n ) ) and D [ F 1 ] = K 1 ) \and(S_{1}\implies(\operatorname{equate}[A_{1},g\ q\ p\ n]\and V[F_{1}]=g\ q\ % p\ n))\and D[F_{1}]=K_{1})
  236. build - param - lists [ n , D , V , T 2 ] and build - param - lists [ g m p n , D , V , K 2 ] and build - param - lists [ g q p n , D , V , K 1 ] \operatorname{build-param-lists}[n,D,V,T_{2}]\and\operatorname{build-param-% lists}[g\ m\ p\ n,D,V,K_{2}]\and\operatorname{build-param-lists}[g\ q\ p\ n,D,% V,K_{1}]
  237. and ( ( T 2 = [ F 2 , S 2 , A 2 ] : : [ F 1 , S 1 , A 1 ] : : R \and((T_{2}=[F_{2},S_{2},A_{2}]::[F_{1},S_{1},A_{1}]::R
  238. and ( S 1 ( equate [ A 1 , g q p n ] and V [ F 1 ] = g q p n ) ) and D [ F 1 ] = K 1 ) \and(S_{1}\implies(\operatorname{equate}[A_{1},g\ q\ p\ n]\and V[F_{1}]=g\ q\ % p\ n))\and D[F_{1}]=K_{1})
  239. and ( S 2 ( equate [ A 2 , g m p n ] and V [ F 2 ] = g m p n ) ) and D [ F 2 ] = K 2 ) \and(S_{2}\implies(\operatorname{equate}[A_{2},g\ m\ p\ n]\and V[F_{2}]=g\ m\ % p\ n))\and D[F_{2}]=K_{2})
  240. build - param - lists [ g m p n , D , V , K 2 ] and build - param - lists [ g q p n , D , V , K 1 ] \operatorname{build-param-lists}[g\ m\ p\ n,D,V,K_{2}]\and\operatorname{build-% param-lists}[g\ q\ p\ n,D,V,K_{1}]
  241. and ( ( D [ n ] = [ F 2 , S 2 , A 2 ] : : [ F 1 , S 1 , A 1 ] : : R \and((D[n]=[F_{2},S_{2},A_{2}]::[F_{1},S_{1},A_{1}]::R
  242. and ( S 1 ( equate [ A 1 , g q p n ] and V [ F 1 ] = g q p n ) ) and D [ F 1 ] = K 1 ) \and(S_{1}\implies(\operatorname{equate}[A_{1},g\ q\ p\ n]\and V[F_{1}]=g\ q\ % p\ n))\and D[F_{1}]=K_{1})
  243. and ( S 2 ( equate [ A 2 , g m p n ] and V [ F 2 ] = g m p n ) ) and D [ F 2 ] = K 2 ) \and(S_{2}\implies(\operatorname{equate}[A_{2},g\ m\ p\ n]\and V[F_{2}]=g\ m\ % p\ n))\and D[F_{2}]=K_{2})
  244. D [ n ] = [ _ , _ , g m p n ] : : [ _ , _ , g q p n ] : : R D[n]=[\_,\_,g\ m\ p\ n]::[\_,\_,g\ q\ p\ n]::R
  245. build - param - lists [ g m p n , D , V , K 2 ] \operatorname{build-param-lists}[g\ m\ p\ n,D,V,K_{2}]
  246. build - param - lists [ g m p n , D , V , K 2 ] \operatorname{build-param-lists}[g\ m\ p\ n,D,V,K_{2}]
  247. build - param - lists [ g m p , D , V , T 3 ] and build - param - lists [ n , D , V , K 3 ] \operatorname{build-param-lists}[g\ m\ p,D,V,T_{3}]\and\operatorname{build-% param-lists}[n,D,V,K_{3}]
  248. and ( ( T 3 = [ F 3 , S 3 , A 3 ] : : K 2 \and((T_{3}=[F_{3},S_{3},A_{3}]::K_{2}
  249. and ( S 3 ( equate [ A 3 , n ] and V [ F 3 ] = n ) ) and D [ F 3 ] = D [ n ] ) \and(S_{3}\implies(\operatorname{equate}[A_{3},n]\and V[F_{3}]=n))\and D[F_{3}% ]=D[n])
  250. build - param - lists [ g m , D , V , T 4 ] and build - param - lists [ p , D , V , K 4 ] \operatorname{build-param-lists}[g\ m,D,V,T_{4}]\and\operatorname{build-param-% lists}[p,D,V,K_{4}]
  251. and T 4 = [ _ , S 4 , A 4 ] : : [ _ , S 3 , A 3 ] : : K 2 \and T_{4}=[\_,S_{4},A_{4}]::[\_,S_{3},A_{3}]::K_{2}
  252. and ( S 3 ( equate [ A 3 , n ] and V [ F 3 ] = n ) ) and D [ F 3 ] = D [ n ] ) \and(S_{3}\implies(\operatorname{equate}[A_{3},n]\and V[F_{3}]=n))\and D[F_{3}% ]=D[n])
  253. and ( S 4 ( equate [ A 4 , p ] and V [ F 4 ] = p ) ) and D [ F 4 ] = D [ p ] \and(S_{4}\implies(\operatorname{equate}[A_{4},p]\and V[F_{4}]=p))\and D[F_{4}% ]=D[p]
  254. build - param - lists [ g , D , V , T 5 ] and build - param - lists [ m , D , V , K 5 ] \operatorname{build-param-lists}[g,D,V,T_{5}]\and\operatorname{build-param-% lists}[m,D,V,K_{5}]
  255. and T 5 = [ F 5 , S 5 , A 5 ] : : [ F 4 , S 4 , A 4 ] : : [ F 3 , S 3 , A 3 ] : : K 2 \and T_{5}=[F_{5},S_{5},A_{5}]::[F_{4},S_{4},A_{4}]::[F_{3},S_{3},A_{3}]::K_{2}
  256. and ( S 3 ( equate [ A 3 , n ] and V [ F 3 ] = n ) ) and D [ F 3 ] = D [ n ] ) \and(S_{3}\implies(\operatorname{equate}[A_{3},n]\and V[F_{3}]=n))\and D[F_{3}% ]=D[n])
  257. and ( S 4 ( equate [ A 4 , p ] and V [ F 4 ] = p ) ) and D [ F 4 ] = D [ p ] \and(S_{4}\implies(\operatorname{equate}[A_{4},p]\and V[F_{4}]=p))\and D[F_{4}% ]=D[p]
  258. and ( S 5 ( equate [ A 5 , m ] and V [ F 5 ] = m ) ) and D [ F 5 ] = D [ m ] \and(S_{5}\implies(\operatorname{equate}[A_{5},m]\and V[F_{5}]=m))\and D[F_{5}% ]=D[m]
  259. D [ g ] = [ x , S 5 , A 5 ] : : [ o , S 4 , A 4 ] : : [ y , S 3 , A 3 ] : : K 2 D[g]=[x,S_{5},A_{5}]::[o,S_{4},A_{4}]::[y,S_{3},A_{3}]::K_{2}
  260. and ( S 3 ( equate [ A 3 , n ] and V [ y ] = n ) ) and D [ y ] = D [ n ] ) \and(S_{3}\implies(\operatorname{equate}[A_{3},n]\and V[y]=n))\and D[y]=D[n])
  261. and ( S 4 ( equate [ A 4 , p ] and V [ o ] = p ) ) and D [ o ] = D [ p ] \and(S_{4}\implies(\operatorname{equate}[A_{4},p]\and V[o]=p))\and D[o]=D[p]
  262. and ( S 5 ( equate [ A 5 , m ] and V [ x ] = m ) ) and D [ x ] = D [ m ] \and(S_{5}\implies(\operatorname{equate}[A_{5},m]\and V[x]=m))\and D[x]=D[m]
  263. build - param - lists [ g q p n , D , V , K 1 ] \operatorname{build-param-lists}[g\ q\ p\ n,D,V,K_{1}]
  264. build - param - lists [ g q p n , D , V , K 1 ] \operatorname{build-param-lists}[g\ q\ p\ n,D,V,K_{1}]
  265. build - param - lists [ g q p , D , V , T 6 ] and build - param - lists [ n , D , V , K 6 ] \operatorname{build-param-lists}[g\ q\ p,D,V,T_{6}]\and\operatorname{build-% param-lists}[n,D,V,K_{6}]
  266. and ( ( T 6 = [ F 6 , S 6 , A 6 ] : : K 1 \and((T_{6}=[F_{6},S_{6},A_{6}]::K_{1}
  267. and ( S 6 ( equate [ A 6 , n ] and V [ F 6 ] = n ) ) and D [ F 6 ] = D [ n ] ) \and(S_{6}\implies(\operatorname{equate}[A_{6},n]\and V[F_{6}]=n))\and D[F_{6}% ]=D[n])
  268. build - param - lists [ g q , D , V , T 7 ] and build - param - lists [ p , D , V , K 7 ] \operatorname{build-param-lists}[g\ q,D,V,T_{7}]\and\operatorname{build-param-% lists}[p,D,V,K_{7}]
  269. and T 7 = [ _ , S 7 , A 7 ] : : [ _ , S 6 , A 6 ] : : K 1 \and T_{7}=[\_,S_{7},A_{7}]::[\_,S_{6},A_{6}]::K_{1}
  270. and ( S 6 ( equate [ A 6 , n ] and V [ F 6 ] = n ) ) and D [ F 6 ] = D [ n ] ) \and(S_{6}\implies(\operatorname{equate}[A_{6},n]\and V[F_{6}]=n))\and D[F_{6}% ]=D[n])
  271. and ( S 7 ( equate [ A 7 , p ] and V [ F 7 ] = p ) ) and D [ F 7 ] = D [ p ] \and(S_{7}\implies(\operatorname{equate}[A_{7},p]\and V[F_{7}]=p))\and D[F_{7}% ]=D[p]
  272. build - param - lists [ g , D , V , T 8 ] and build - param - lists [ m , D , V , K 8 ] \operatorname{build-param-lists}[g,D,V,T_{8}]\and\operatorname{build-param-% lists}[m,D,V,K_{8}]
  273. and T 8 = [ F 8 , S 8 , A 8 ] : : [ F 7 , S 7 , A 7 ] : : [ F 6 , S 6 , A 6 ] : : K 1 \and T_{8}=[F_{8},S_{8},A_{8}]::[F_{7},S_{7},A_{7}]::[F_{6},S_{6},A_{6}]::K_{1}
  274. and ( S 6 ( equate [ A 6 , n ] and V [ F 6 ] = n ) ) and D [ F 6 ] = D [ n ] ) \and(S_{6}\implies(\operatorname{equate}[A_{6},n]\and V[F_{6}]=n))\and D[F_{6}% ]=D[n])
  275. and ( S 7 ( equate [ A 7 , p ] and V [ F 7 ] = p ) ) and D [ F 7 ] = D [ p ] \and(S_{7}\implies(\operatorname{equate}[A_{7},p]\and V[F_{7}]=p))\and D[F_{7}% ]=D[p]
  276. and ( S 8 ( equate [ A 8 , q ] and V [ F 8 ] = q ) ) and D [ F 8 ] = D [ q ] \and(S_{8}\implies(\operatorname{equate}[A_{8},q]\and V[F_{8}]=q))\and D[F_{8}% ]=D[q]
  277. D [ g ] = [ x , S 8 , A 8 ] : : [ o , S 6 , A 7 ] : : [ y , S 6 , A 6 ] : : K 1 D[g]=[x,S_{8},A_{8}]::[o,S_{6},A_{7}]::[y,S_{6},A_{6}]::K_{1}
  278. and ( S 6 ( equate [ A 6 , n ] and V [ y ] = n ) ) and D [ y ] = D [ n ] ) \and(S_{6}\implies(\operatorname{equate}[A_{6},n]\and V[y]=n))\and D[y]=D[n])
  279. and ( S 7 ( equate [ A 7 , p ] and V [ o ] = p ) ) and D [ o ] = D [ p ] \and(S_{7}\implies(\operatorname{equate}[A_{7},p]\and V[o]=p))\and D[o]=D[p]
  280. and ( S 8 ( equate [ A 8 , q ] and V [ x ] = q ) ) and D [ x ] = D [ q ] \and(S_{8}\implies(\operatorname{equate}[A_{8},q]\and V[x]=q))\and D[x]=D[q]
  281. V [ n ] , V [ p ] , V [ q ] , V [ m ] V[n],V[p],V[q],V[m]
  282. equate [ A , N ] A = N ( def [ V [ N ] ] and A = V [ N ] ) A = N \operatorname{equate}[A,N]\equiv A=N(\operatorname{def}[V[N]]\and A=V[N])% \equiv A=N
  283. D [ g ] = [ x , S 5 , A 5 ] : : [ o , S 4 , A 4 ] : : [ y , S 3 , A 3 ] : : K 2 D[g]=[x,S_{5},A_{5}]::[o,S_{4},A_{4}]::[y,S_{3},A_{3}]::K_{2}
  284. and S 3 A 3 = n \and S_{3}\implies A_{3}=n
  285. and S 4 A 4 = p \and S_{4}\implies A_{4}=p
  286. and S 5 A 5 = m \and S_{5}\implies A_{5}=m
  287. D [ g ] = [ x , S 8 , A 8 ] : : [ o , S 6 , A 7 ] : : [ y , S 6 , A 6 ] : : K 1 D[g]=[x,S_{8},A_{8}]::[o,S_{6},A_{7}]::[y,S_{6},A_{6}]::K_{1}
  288. and S 6 A 6 = n \and S_{6}\implies A_{6}=n
  289. and S 7 A 7 = p \and S_{7}\implies A_{7}=p
  290. and S 8 A 8 = q \and S_{8}\implies A_{8}=q
  291. D [ g ] D[g]
  292. S 5 = S 8 , A 5 = A 8 , S 4 = S 7 , A 4 = A 7 , S 3 = S 6 , A 3 = A 6 S_{5}=S_{8},A_{5}=A_{8},S_{4}=S_{7},A_{4}=A_{7},S_{3}=S_{6},A_{3}=A_{6}
  293. S 3 S_{3}
  294. n = A 3 = A 6 = n n=A_{3}=A_{6}=n
  295. S 3 S_{3}
  296. S 3 = _ S_{3}=\_
  297. S 4 S_{4}
  298. p = A 4 = A 7 = p p=A_{4}=A_{7}=p
  299. S 5 S_{5}
  300. m = A 5 = A 8 = q m=A_{5}=A_{8}=q
  301. S 5 S_{5}
  302. D [ g ] = [ x , false , _ ] : : [ o , _ , p ] : : [ y , _ , n ] : : _ D[g]=[x,\operatorname{false},\_]::[o,\_,p]::[y,\_,n]::\_
  303. build - param - lists [ o x y , D , V , L ] \operatorname{build-param-lists}[o\ x\ y,D,V,L]
  304. build - param - lists [ o x y , D , V , L ] \operatorname{build-param-lists}[o\ x\ y,D,V,L]
  305. build - param - lists [ o x , D , V , T 9 ] and build - param - lists [ y , D , V , K 9 ] \operatorname{build-param-lists}[o\ x,D,V,T_{9}]\and\operatorname{build-param-% lists}[y,D,V,K_{9}]
  306. and T 9 = [ F 9 , S 9 , A 9 ] : : L \and T_{9}=[F_{9},S_{9},A_{9}]::L
  307. and ( S 9 ( equate [ A 9 , y ] and V [ F 9 ] = A 9 ) and K 9 = D [ F 9 ] \and(S_{9}\implies(\operatorname{equate}[A_{9},y]\and V[F_{9}]=A_{9})\and K_{9% }=D[F_{9}]
  308. build - param - lists [ o , D , V , T 10 ] and build - param - lists [ x , D , V , K 10 ] and build - param - lists [ y , D , V , K 10 ] \operatorname{build-param-lists}[o,D,V,T_{10}]\and\operatorname{build-param-% lists}[x,D,V,K_{10}]\and\operatorname{build-param-lists}[y,D,V,K_{10}]
  309. and T 10 = [ F 10 , S 10 , A 10 ] : : [ F 9 , S 9 , A 9 ] : : L \and T_{10}=[F_{10},S_{10},A_{10}]::[F_{9},S_{9},A_{9}]::L
  310. and ( S 9 ( equate [ A 9 , y ] and V [ F 9 ] = A 9 ) and K 9 = D [ F 9 ] \and(S_{9}\implies(\operatorname{equate}[A_{9},y]\and V[F_{9}]=A_{9})\and K_{9% }=D[F_{9}]
  311. and ( S 10 ( equate [ A 10 , y ] and V [ F 10 ] = A 10 ) and K 10 = D [ F 10 ] \and(S_{10}\implies(\operatorname{equate}[A_{10},y]\and V[F_{10}]=A_{10})\and K% _{10}=D[F_{10}]
  312. and D [ o ] = [ F 10 , S 10 , A 10 ] : : [ F 9 , S 9 , A 9 ] : : L \and D[o]=[F_{10},S_{10},A_{10}]::[F_{9},S_{9},A_{9}]::L
  313. and ( S 9 ( equate [ A 9 , y ] and V [ F 9 ] = A 9 ) and K 9 = D [ F 9 ] \and(S_{9}\implies(\operatorname{equate}[A_{9},y]\and V[F_{9}]=A_{9})\and K_{9% }=D[F_{9}]
  314. and ( S 10 ( equate [ A 10 , y ] and V [ F 10 ] = A 10 ) and K 10 = D [ F 10 ] \and(S_{10}\implies(\operatorname{equate}[A_{10},y]\and V[F_{10}]=A_{10})\and K% _{10}=D[F_{10}]
  315. D [ o ] = [ _ , _ , x ] : : [ _ , _ , y ] : : _ D[o]=[\_,\_,x]::[\_,\_,y]::\_
  316. D [ g ] = [ [ x , false , _ ] , [ o , true , p ] , [ y , true , n ] ] D[g]=[[x,\operatorname{false},\_],[o,\operatorname{true},p],[y,\operatorname{% true},n]]
  317. D [ n ] = [ [ _ , _ , ( g m p n ) ] , [ _ , _ , ( g q p n ) ] ] D[n]=[[\_,\_,(g\ m\ p\ n)],[\_,\_,(g\ q\ p\ n)]]
  318. D [ m ] = _ D[m]=\_
  319. D [ p ] = _ D[p]=\_
  320. D [ q ] = _ D[q]=\_
  321. λ f . ( ( λ p . f ( p p f ) ) ( λ q . λ x . x ( q q x ) ) \lambda f.((\lambda p.f\ (p\ p\ f))\ (\lambda q.\lambda x.x\ (q\ q\ x))
  322. D [ p ] = [ [ q , _ , p ] , [ x , _ , f ] ] D[p]=[[q,\_,p],[x,\_,f]]
  323. λ f . ( ( λ q . f ( q q ) ) ( λ q . f ( q q ) ) \lambda f.((\lambda q.f\ (q\ q))\ (\lambda q.f\ (q\ q))
  324. λ f . ( ( λ p . f ( p p f ) ) ( λ q . λ x . x ( q q x ) ) \lambda f.((\lambda p.f\ (p\ p\ f))\ (\lambda q.\lambda x.x\ (q\ q\ x))
  325. build - param - list [ λ f . ( ( λ p . f ( p p f ) ) ( λ q . λ x . x ( q q x ) ) , D , V , _ ] \operatorname{build-param-list}[\lambda f.((\lambda p.f\ (p\ p\ f))\ (\lambda q% .\lambda x.x\ (q\ q\ x)),D,V,\_]
  326. build - param - list [ λ f . ( ( λ p . f ( p p f ) ) ( λ q . λ x . x ( q q x ) ) , D , V , _ ] \operatorname{build-param-list}[\lambda f.((\lambda p.f\ (p\ p\ f))\ (\lambda q% .\lambda x.x\ (q\ q\ x)),D,V,\_]
  327. build - param - list [ ( λ p . f ( p p f ) ) ( λ q . λ x . x ( q q x ) ) , D , V , _ ] \operatorname{build-param-list}[(\lambda p.f\ (p\ p\ f))\ (\lambda q.\lambda x% .x\ (q\ q\ x)),D,V,\_]
  328. build - param - lists [ f ( p p f ) , D , _ ] and build - list [ λ q . λ x . x ( q q x ) , D , D [ p ] ] \operatorname{build-param-lists}[f\ (p\ p\ f),D,\_]\and\operatorname{build-% list}[\lambda q.\lambda x.x\ (q\ q\ x),D,D[p]]
  329. build - param - lists [ f ( p p f ) , D , _ ] and build - list [ λ q . λ x . x ( q q x ) , D , D [ p ] ] \operatorname{build-param-lists}[f\ (p\ p\ f),D,\_]\and\operatorname{build-% list}[\lambda q.\lambda x.x\ (q\ q\ x),D,D[p]]
  330. build - list [ λ q . λ x . x ( q q x ) , D , D [ p ] ] \operatorname{build-list}[\lambda q.\lambda x.x\ (q\ q\ x),D,D[p]]
  331. build - list [ λ q . λ x . x ( q q x ) , D , D [ p ] ] and D [ p ] = L 1 \operatorname{build-list}[\lambda q.\lambda x.x\ (q\ q\ x),D,D[p]]\and D[p]=L_% {1}
  332. build - list [ λ x . x ( q q x ) , D , L 2 ] and D [ p ] = [ q , _ , _ ] : : L 2 \operatorname{build-list}[\lambda x.x\ (q\ q\ x),D,L_{2}]\and D[p]=[q,\_,\_]::% L_{2}
  333. build - list [ x ( q q x ) , D , L 3 ] and D [ p ] = [ q , _ , _ ] : : [ x , _ , _ ] : : L 3 \operatorname{build-list}[x\ (q\ q\ x),D,L_{3}]\and D[p]=[q,\_,\_]::[x,\_,\_]:% :L_{3}
  334. build - param - lists [ x ( q q x ) , D , [ ] ] and D [ p ] = [ q , _ , _ ] : : [ x , _ , _ ] : : [ ] \operatorname{build-param-lists}[x\ (q\ q\ x),D,[]]\and D[p]=[q,\_,\_]::[x,\_,% \_]::[]
  335. build - param - lists [ λ p . f ( p p f ) , D , V , T 1 ] \operatorname{build-param-lists}[\lambda p.f\ (p\ p\ f),D,V,T_{1}]
  336. build - param - lists [ λ p . f ( p p f ) , D , V , T 1 ] \operatorname{build-param-lists}[\lambda p.f\ (p\ p\ f),D,V,T_{1}]
  337. build - param - lists [ f ( p p f ) , D , V , T 1 ] \operatorname{build-param-lists}[f\ (p\ p\ f),D,V,T_{1}]
  338. build - param - lists [ f , D , V , T 2 ] and build - param - lists [ p p f , D , V , K 2 ] \operatorname{build-param-lists}[f,D,V,T_{2}]\and\operatorname{build-param-% lists}[p\ p\ f,D,V,K_{2}]
  339. and T 2 = [ F 2 , S 2 , A 2 ] : : [ F 1 , S 1 , A 1 ] : : _ \and T_{2}=[F_{2},S_{2},A_{2}]::[F_{1},S_{1},A_{1}]::\_
  340. and ( S 2 ( equate [ A 2 , p p f ] and V [ F 2 ] = A 2 ) ) and D [ F 2 ] = K 2 \and(S_{2}\implies(\operatorname{equate}[A_{2},p\ p\ f]\and V[F_{2}]=A_{2}))% \and D[F_{2}]=K_{2}
  341. build - param - lists [ p p f , D , V , K 2 ] \operatorname{build-param-lists}[p\ p\ f,D,V,K_{2}]
  342. and D [ f ] = [ F 2 , S 2 , A 2 ] : : [ F 1 , S 1 , A 1 ] : : _ \and D[f]=[F_{2},S_{2},A_{2}]::[F_{1},S_{1},A_{1}]::\_
  343. and ( S 2 ( equate [ A 2 , p p f ] and V [ F 2 ] = A 2 ) ) and D [ F 2 ] = K 2 \and(S_{2}\implies(\operatorname{equate}[A_{2},p\ p\ f]\and V[F_{2}]=A_{2}))% \and D[F_{2}]=K_{2}
  344. build - param - lists [ p p , D , V , T 3 ] and build - param - lists [ f , D , V , K 3 ] \operatorname{build-param-lists}[p\ p,D,V,T_{3}]\and\operatorname{build-param-% lists}[f,D,V,K_{3}]
  345. and T 3 = [ F 3 , S 3 , A 3 ] : : K 2 \and T_{3}=[F_{3},S_{3},A_{3}]::K_{2}
  346. and ( S 3 ( equate [ A 3 , f ] and V [ F 3 ] = A 3 ) ) and D [ F 3 ] = K 3 \and(S_{3}\implies(\operatorname{equate}[A_{3},f]\and V[F_{3}]=A_{3}))\and D[F% _{3}]=K_{3}
  347. build - param - lists [ p p , D , V , T 3 ] \operatorname{build-param-lists}[p\ p,D,V,T_{3}]
  348. and T 3 = [ F 3 , S 3 , A 3 ] : : K 2 \and T_{3}=[F_{3},S_{3},A_{3}]::K_{2}
  349. and ( S 2 ( equate [ A 2 , p p f ] and V [ F 2 ] = A 2 ) ) and D [ F 2 ] = K 2 \and(S_{2}\implies(\operatorname{equate}[A_{2},p\ p\ f]\and V[F_{2}]=A_{2}))% \and D[F_{2}]=K_{2}
  350. and ( S 3 ( equate [ A 3 , f ] and V [ F 3 ] = A 3 ) ) and D [ F 3 ] = D [ f ] \and(S_{3}\implies(\operatorname{equate}[A_{3},f]\and V[F_{3}]=A_{3}))\and D[F% _{3}]=D[f]
  351. build - param - lists [ p , D , V , T 4 ] and build - param - lists [ p , D , V , K 4 ] \operatorname{build-param-lists}[p,D,V,T_{4}]\and\operatorname{build-param-% lists}[p,D,V,K_{4}]
  352. and T 4 = [ F 4 , S 4 , A 4 ] : : [ F 3 , S 3 , A 3 ] : : K 2 \and T_{4}=[F_{4},S_{4},A_{4}]::[F_{3},S_{3},A_{3}]::K_{2}
  353. and ( S 3 ( equate [ A 3 , f ] and V [ F 3 ] = A 3 ) ) and D [ F 3 ] = D [ f ] \and(S_{3}\implies(\operatorname{equate}[A_{3},f]\and V[F_{3}]=A_{3}))\and D[F% _{3}]=D[f]
  354. and ( S 4 ( equate [ A 4 , p ] and V [ F 4 ] = A 4 ) ) and D [ F 4 ] = K 4 \and(S_{4}\implies(\operatorname{equate}[A_{4},p]\and V[F_{4}]=A_{4}))\and D[F% _{4}]=K_{4}
  355. D [ p ] = [ F 4 , S 4 , A 4 ] : : [ F 3 , S 3 , A 3 ] : : K 2 D[p]=[F_{4},S_{4},A_{4}]::[F_{3},S_{3},A_{3}]::K_{2}
  356. and ( S 3 ( equate [ A 3 , f ] and V [ F 3 ] = A 3 ) ) and D [ F 3 ] = D [ f ] \and(S_{3}\implies(\operatorname{equate}[A_{3},f]\and V[F_{3}]=A_{3}))\and D[F% _{3}]=D[f]
  357. and ( S 4 ( equate [ A 4 , p ] and V [ F 4 ] = A 4 ) ) and D [ F 4 ] = D [ p ] \and(S_{4}\implies(\operatorname{equate}[A_{4},p]\and V[F_{4}]=A_{4}))\and D[F% _{4}]=D[p]
  358. build - param - lists [ x ( q q x ) ) , D , V , _ ] \operatorname{build-param-lists}[x\ (q\ q\ x)),D,V,\_]
  359. build - param - lists [ λ q . λ x . x ( q q x ) ) , D , V , _ ] \operatorname{build-param-lists}[\lambda q.\lambda x.x\ (q\ q\ x)),D,V,\_]
  360. build - param - lists [ x ( q q x ) ) , D , V , K 1 ] \operatorname{build-param-lists}[x\ (q\ q\ x)),D,V,K_{1}]
  361. build - param - lists [ x , D , V , T 5 ] and build - param - lists [ q q x , D , V , K 5 ] \operatorname{build-param-lists}[x,D,V,T_{5}]\and\operatorname{build-param-% lists}[q\ q\ x,D,V,K_{5}]
  362. and T 5 = [ F 5 , S 5 , A 5 ] : : _ \and T_{5}=[F_{5},S_{5},A_{5}]::\_
  363. and ( S 5 ( equate [ A 5 , q q x ] and V [ F 5 ] = A 5 ) ) and D [ F 5 ] = K 5 \and(S_{5}\implies(\operatorname{equate}[A_{5},q\ q\ x]\and V[F_{5}]=A_{5}))% \and D[F_{5}]=K_{5}
  364. build - param - lists [ q q x , D , V , K 5 ] \operatorname{build-param-lists}[q\ q\ x,D,V,K_{5}]
  365. and D [ x ] = [ F 5 , S 5 , A 5 ] : : _ \and D[x]=[F_{5},S_{5},A_{5}]::\_
  366. and ( S 5 ( equate [ A 5 , q q x ] and V [ F 5 ] = A 5 ) ) and D [ F 5 ] = K 5 \and(S_{5}\implies(\operatorname{equate}[A_{5},q\ q\ x]\and V[F_{5}]=A_{5}))% \and D[F_{5}]=K_{5}
  367. build - param - lists [ q q , D , V , T 6 ] and build - param - lists [ x , D , V , K 6 ] \operatorname{build-param-lists}[q\ q,D,V,T_{6}]\and\operatorname{build-param-% lists}[x,D,V,K_{6}]
  368. and T 6 = [ F 6 , S 6 , A 6 ] : : K 5 \and T_{6}=[F_{6},S_{6},A_{6}]::K_{5}
  369. and ( S 6 ( equate [ A 6 , x ] and V [ F 6 ] = A 6 ) ) and D [ F 6 ] = K 6 \and(S_{6}\implies(\operatorname{equate}[A_{6},x]\and V[F_{6}]=A_{6}))\and D[F% _{6}]=K_{6}
  370. build - param - lists [ q q , D , V , T 6 ] \operatorname{build-param-lists}[q\ q,D,V,T_{6}]
  371. and T 6 = [ F 6 , S 6 , A 6 ] : : K 5 \and T_{6}=[F_{6},S_{6},A_{6}]::K_{5}
  372. and ( S 6 ( equate [ A 6 , x ] and V [ F 6 ] = A 6 ) ) and D [ F 6 ] = D [ x ] \and(S_{6}\implies(\operatorname{equate}[A_{6},x]\and V[F_{6}]=A_{6}))\and D[F% _{6}]=D[x]
  373. build - param - lists [ q , D , V , T 7 ] and build - param - lists [ q , D , V , K 7 ] \operatorname{build-param-lists}[q,D,V,T_{7}]\and\operatorname{build-param-% lists}[q,D,V,K_{7}]
  374. and T 7 = [ F 7 , S 7 , A 7 ] : : [ F 6 , S 6 , A 6 ] : : K 5 \and T_{7}=[F_{7},S_{7},A_{7}]::[F_{6},S_{6},A_{6}]::K_{5}
  375. and ( S 6 ( equate [ A 6 , x ] and V [ F 6 ] = A 6 ) ) and D [ F 6 ] = D [ x ] \and(S_{6}\implies(\operatorname{equate}[A_{6},x]\and V[F_{6}]=A_{6}))\and D[F% _{6}]=D[x]
  376. and ( S 7 ( equate [ A 7 , q ] and V [ F 7 ] = A 7 ) ) and D [ F 7 ] = K 7 \and(S_{7}\implies(\operatorname{equate}[A_{7},q]\and V[F_{7}]=A_{7}))\and D[F% _{7}]=K_{7}
  377. build - param - lists [ q , D , V , K 7 ] \operatorname{build-param-lists}[q,D,V,K_{7}]
  378. and D [ q ] = [ F 7 , S 7 , A 7 ] : : [ F 6 , S 6 , A 6 ] : : K 5 \and D[q]=[F_{7},S_{7},A_{7}]::[F_{6},S_{6},A_{6}]::K_{5}
  379. and ( S 6 ( equate [ A 6 , x ] and V [ F 6 ] = A 6 ) ) and D [ F 6 ] = D [ x ] \and(S_{6}\implies(\operatorname{equate}[A_{6},x]\and V[F_{6}]=A_{6}))\and D[F% _{6}]=D[x]
  380. and ( S 7 ( equate [ A 7 , q ] and V [ F 7 ] = A 7 ) ) and D [ F 7 ] = D [ q ] \and(S_{7}\implies(\operatorname{equate}[A_{7},q]\and V[F_{7}]=A_{7}))\and D[F% _{7}]=D[q]
  381. D [ p ] = [ q , _ , _ ] : : [ x , _ , _ ] : : L 3 D[p]=[q,\_,\_]::[x,\_,\_]::L_{3}
  382. D [ p ] = [ F 4 , S 4 , A 4 ] : : [ F 3 , S 3 , A 3 ] : : K 2 D[p]=[F_{4},S_{4},A_{4}]::[F_{3},S_{3},A_{3}]::K_{2}
  383. and ( S 3 ( equate [ A 3 , f ] and V [ F 3 ] = A 3 ) ) and D [ F 3 ] = D [ f ] \and(S_{3}\implies(\operatorname{equate}[A_{3},f]\and V[F_{3}]=A_{3}))\and D[F% _{3}]=D[f]
  384. and ( S 4 ( equate [ A 4 , p ] and V [ F 4 ] = A 4 ) ) and D [ F 4 ] = D [ p ] \and(S_{4}\implies(\operatorname{equate}[A_{4},p]\and V[F_{4}]=A_{4}))\and D[F% _{4}]=D[p]
  385. D [ q ] = [ F 7 , S 7 , A 7 ] : : [ F 6 , S 6 , A 6 ] : : K 5 D[q]=[F_{7},S_{7},A_{7}]::[F_{6},S_{6},A_{6}]::K_{5}
  386. ( S 6 ( equate [ A 6 , x ] and V [ F 6 ] = A 6 ) ) and D [ F 6 ] = D [ x ] (S_{6}\implies(\operatorname{equate}[A_{6},x]\and V[F_{6}]=A_{6}))\and D[F_{6}% ]=D[x]
  387. ( S 7 ( equate [ A 7 , q ] and V [ F 7 ] = A 7 ) ) and D [ F 7 ] = D [ q ] (S_{7}\implies(\operatorname{equate}[A_{7},q]\and V[F_{7}]=A_{7}))\and D[F_{7}% ]=D[q]
  388. D [ p ] D[p]
  389. F 4 = q F_{4}=q
  390. F 3 = x F_{3}=x
  391. D [ p ] = [ q , S 4 , A 4 ] : : [ x , S 3 , A 3 ] : : K 2 D[p]=[q,S_{4},A_{4}]::[x,S_{3},A_{3}]::K_{2}
  392. ( S 3 ( equate [ A 3 , f ] and V [ x ] = A 3 ) ) and D [ x ] = D [ f ] (S_{3}\implies(\operatorname{equate}[A_{3},f]\and V[x]=A_{3}))\and D[x]=D[f]
  393. ( S 4 ( equate [ A 4 , p ] and V [ q ] = A 4 ) ) and D [ q ] = D [ p ] (S_{4}\implies(\operatorname{equate}[A_{4},p]\and V[q]=A_{4}))\and D[q]=D[p]
  394. D [ q ] = D [ p ] D[q]=D[p]
  395. D [ p ] = [ F 7 , S 7 , A 7 ] : : [ F 6 , S 6 , A 6 ] : : K 5 D[p]=[F_{7},S_{7},A_{7}]::[F_{6},S_{6},A_{6}]::K_{5}
  396. ( S 6 ( equate [ A 6 , x ] and V [ F 6 ] = A 6 ) ) and D [ F 6 ] = D [ x ] (S_{6}\implies(\operatorname{equate}[A_{6},x]\and V[F_{6}]=A_{6}))\and D[F_{6}% ]=D[x]
  397. ( S 7 ( equate [ A 7 , q ] and V [ F 7 ] = A 7 ) ) and D [ F 7 ] = D [ q ] (S_{7}\implies(\operatorname{equate}[A_{7},q]\and V[F_{7}]=A_{7}))\and D[F_{7}% ]=D[q]
  398. F 7 = q , F 6 = x , A 3 = A 6 , A 4 = A 7 , S 3 = S 6 , S 4 = S 7 F_{7}=q,F_{6}=x,A_{3}=A_{6},A_{4}=A_{7},S_{3}=S_{6},S_{4}=S_{7}
  399. V [ x ] = A 3 V[x]=A_{3}
  400. V [ q ] = A 4 V[q]=A_{4}
  401. S 3 A 3 = f S_{3}\implies A_{3}=f
  402. S 3 ( A 3 = x A 3 = v [ x ] ) S_{3}\implies(A_{3}=xA_{3}=v[x])
  403. S 3 A 3 = f S_{3}\implies A_{3}=f
  404. S 4 A 4 = p S_{4}\implies A_{4}=p
  405. S 4 ( A 4 = q A 4 = v [ q ] ) S_{4}\implies(A_{4}=qA_{4}=v[q])
  406. S 4 A 4 = p S_{4}\implies A_{4}=p
  407. D [ p ] = [ q , _ , p ] : : [ x , _ , f ] : : _ D[p]=[q,\_,p]::[x,\_,f]::\_
  408. drop - params [ ( λ N . S ) L , D , V , R ] ( λ N . drop - params [ S , D , F , R ] ) drop - formal [ D [ N ] , L , F ] \operatorname{drop-params}[(\lambda N.S)\ L,D,V,R]\equiv(\lambda N.% \operatorname{drop-params}[S,D,F,R])\ \operatorname{drop-formal}[D[N],L,F]
  409. F = F V [ ( λ N . S ) L ] F=FV[(\lambda N.S)\ L]
  410. drop - params [ λ N . S , D , V , R ] ( λ N . drop - params [ S , D , F , R ] ) \operatorname{drop-params}[\lambda N.S,D,V,R]\equiv(\lambda N.\operatorname{% drop-params}[S,D,F,R])
  411. F = F V [ λ N . S ] F=FV[\lambda N.S]
  412. drop - params [ N , D , V , D [ N ] ] N \operatorname{drop-params}[N,D,V,D[N]]\equiv N
  413. ( def [ F ] and ask [ S ] and F V [ A ] V ) drop - params [ E P , D , V , R ] drop - params [ E , D , V , [ F , S , A ] : : R ] (\operatorname{def}[F]\and\operatorname{ask}[S]\and FV[A]\subset V)\to% \operatorname{drop-params}[E\ P,D,V,R]\equiv\operatorname{drop-params}[E,D,V,[% F,S,A]::R]
  414. ¬ ( def [ F ] and ask [ S ] and F V [ A ] V ) drop - params [ E P , D , V , R ] drop - params [ E , D , V , [ F , S , A ] : : R ] drop - params [ P , D , V , _ ] \neg(\operatorname{def}[F]\and\operatorname{ask}[S]\and FV[A]\subset V)\to% \operatorname{drop-params}[E\ P,D,V,R]\equiv\operatorname{drop-params}[E,D,V,[% F,S,A]::R]\ \operatorname{drop-params}[P,D,V,\_]
  415. drop - params [ let V : E in L ] let V : drop - params [ E , D , F V [ E ] , [ ] ] in drop - params [ L , D , F V [ L ] , [ ] ] \operatorname{drop-params}[\operatorname{let}V:E\operatorname{in}L]\equiv% \operatorname{let}V:\operatorname{drop-params}[E,D,FV[E],[]]\operatorname{in}% \operatorname{drop-params}[L,D,FV[L],[]]
  416. drop - params [ E and F , D , V , _ ] drop - params [ E , D , V , _ ] and drop - params [ F , D , V , _ ] \operatorname{drop-params}[E\and F,D,V,\_]\equiv\operatorname{drop-params}[E,D% ,V,\_]\and\operatorname{drop-params}[F,D,V,\_]
  417. λ g . λ n . n ( g m p n ) ( g q p n ) \lambda g.\lambda n.n\ (g\ m\ p\ n)\ (g\ q\ p\ n)
  418. drop - param [ λ g . λ n . n ( g m p n ) ( g q p n ) , D , { p , q , m } , _ ] \operatorname{drop-param}[\lambda g.\lambda n.n\ (g\ m\ p\ n)\ (g\ q\ p\ n),D,% \{p,q,m\},\_]
  419. λ g . drop - param [ λ n . n ( g m p n ) ( g q p n ) , D , { p , q , m } , _ ] \lambda g.\operatorname{drop-param}[\lambda n.n\ (g\ m\ p\ n)\ (g\ q\ p\ n),D,% \{p,q,m\},\_]
  420. ¬ ( def [ F 1 ] and ) \neg(\operatorname{def}[F_{1}]\and...)
  421. λ g . λ n . drop - param [ n ( g m p n ) , D , { p , q , m } , [ F 1 , S 1 , A 1 ] : : _ ] drop - param [ ( g q p n ) , D , { p , q , m } , _ ] \lambda g.\lambda n.\operatorname{drop-param}[n\ (g\ m\ p\ n),D,\{p,q,m\},[F_{% 1},S_{1},A_{1}]::\_]\ \operatorname{drop-param}[(g\ q\ p\ n),D,\{p,q,m\},\_]
  422. ¬ ( def [ F 2 ] and ) \neg(\operatorname{def}[F_{2}]\and...)
  423. λ g . λ n . drop - param [ n , D , { p , q , m } , [ F 2 , S 2 , A 2 ] : : [ F 1 , S 1 , A 1 ] : : _ ] drop - param [ ( g m p n ) , D , { p , q , m } , _ ] drop - param [ ( g q p n ) , D , { p , q , m } , _ ] \lambda g.\lambda n.\operatorname{drop-param}[n\ ,D,\{p,q,m\},[F_{2},S_{2},A_{% 2}]::[F_{1},S_{1},A_{1}]::\_]\ \operatorname{drop-param}[(g\ m\ p\ n),D,\{p,q,% m\},\_]\ \operatorname{drop-param}[(g\ q\ p\ n),D,\{p,q,m\},\_]
  424. D [ n ] = [ F 2 , S 2 , A 2 ] : : [ F 1 , S 1 , A 1 ] : : _ ] D[n]=[F_{2},S_{2},A_{2}]::[F_{1},S_{1},A_{1}]::\_]
  425. λ g . λ n . n drop - param [ ( g m p n ) , D , { p , q , m } , _ ] drop - param [ ( g q p n ) , D , { p , q , m } , _ ] \lambda g.\lambda n.n\ \operatorname{drop-param}[(g\ m\ p\ n),D,\{p,q,m\},\_]% \ \operatorname{drop-param}[(g\ q\ p\ n),D,\{p,q,m\},\_]
  426. D [ n ] = [ [ _ , _ , ( g m p n ) ] , [ _ , _ , ( g q p n ) ] ] D[n]=[[\_,\_,(g\ m\ p\ n)],[\_,\_,(g\ q\ p\ n)]]
  427. F 1 = _ F_{1}=\_
  428. F 2 = _ F_{2}=\_
  429. def [ F 1 ] = false \operatorname{def}[F_{1}]=\operatorname{false}
  430. def [ F 2 ] = false \operatorname{def}[F_{2}]=\operatorname{false}
  431. drop - param [ ( g m p n ) , D , { p , q , m } , _ ] \operatorname{drop-param}[(g\ m\ p\ n),D,\{p,q,m\},\_]
  432. V = { p , q , m } V=\{p,q,m\}
  433. drop - param [ ( g m p n ) , D , V , _ ] \operatorname{drop-param}[(g\ m\ p\ n),D,V,\_]
  434. F V ( A 1 ) { p , q , m } FV(A_{1})\not\subset\{p,q,m\}
  435. n { p , q , m } n\not\subset\{p,q,m\}
  436. drop - params [ g m p , D , V , [ F 1 , S 1 , A 1 ] : : _ ] drop - params [ n , D , V , _ ] \operatorname{drop-params}[g\ m\ p,D,V,[F_{1},S_{1},A_{1}]::\_]\ \operatorname% {drop-params}[n,D,V,\_]
  437. def [ F 2 ] and ask [ S 2 ] and F V [ A 2 ] V \operatorname{def}[F_{2}]\and\operatorname{ask}[S_{2}]\and FV[A_{2}]\subset V
  438. def [ y ] and ask [ _ ] and F V [ p ] { p , q , m } \operatorname{def}[y]\and\operatorname{ask}[\_]\and FV[p]\subset\{p,q,m\}
  439. drop - params [ g m , D , V , [ F 2 , S 2 , A 2 ] : : [ F 1 , S 1 , A 1 ] : : _ ] drop - params [ n , D , V , _ ] \operatorname{drop-params}[g\ m,D,V,[F_{2},S_{2},A_{2}]::[F_{1},S_{1},A_{1}]::% \_]\ \operatorname{drop-params}[n,D,V,\_]
  440. ¬ ask [ S 3 ] \neg\operatorname{ask}[S_{3}]
  441. ¬ ask [ false ] \neg\operatorname{ask}[\operatorname{false}]
  442. drop - params [ g , D , V , [ F 3 , S 3 , A 3 ] : : [ F 2 , S 2 , A 2 ] : : [ F 1 , S 1 , A 1 ] : : _ ] drop - params [ m , D , V , _ ] drop - params [ n , D , V , _ ] \operatorname{drop-params}[g,D,V,[F_{3},S_{3},A_{3}]::[F_{2},S_{2},A_{2}]::[F_% {1},S_{1},A_{1}]::\_]\ \operatorname{drop-params}[m,D,V,\_]\ \operatorname{% drop-params}[n,D,V,\_]
  443. D [ g ] = [ [ x , false , _ ] , [ o , _ , p ] , [ y , _ , n ] ] D[g]=[[x,\operatorname{false},\_],[o,\_,p],[y,\_,n]]
  444. = [ F 3 , S 3 , A 3 ] : : [ F 2 , S 2 , A 2 ] : : [ F 1 , S 1 , A 1 ] : : _ ] =[F_{3},S_{3},A_{3}]::[F_{2},S_{2},A_{2}]::[F_{1},S_{1},A_{1}]::\_]
  445. F 3 = x , S 3 = false , A 3 = _ F_{3}=x,S_{3}=\operatorname{false},A_{3}=\_
  446. F 2 = o , S 2 = _ , A 2 = p F_{2}=o,S_{2}=\_,A_{2}=p
  447. F 1 = y , S 1 = _ , A 1 = n F_{1}=y,S_{1}=\_,A_{1}=n
  448. g m n g\ m\ n
  449. drop - param [ ( g q p n ) , D , { p , q , m } , _ ] \operatorname{drop-param}[(g\ q\ p\ n),D,\{p,q,m\},\_]
  450. drop - param [ ( g q p n ) , D , V , _ ] \operatorname{drop-param}[(g\ q\ p\ n),D,V,\_]
  451. F V ( A 4 ) V FV(A_{4})\not\subset V
  452. n { p , q , m } n\not\subset\{p,q,m\}
  453. drop - params [ g q p , D , V , [ F 4 , S 4 , A 4 ] : : _ ] drop - params [ n , D , V , _ ] \operatorname{drop-params}[g\ q\ p,D,V,[F_{4},S_{4},A_{4}]::\_]\ \operatorname% {drop-params}[n,D,V,\_]
  454. def [ F 5 ] and ask [ S 5 ] and F V [ A 5 ] V \operatorname{def}[F_{5}]\and\operatorname{ask}[S_{5}]\and FV[A_{5}]\subset V
  455. def [ o ] and ask [ _ ] and p { p , q , m } ) \operatorname{def}[o]\and\operatorname{ask}[\_]\and p\subset\{p,q,m\})
  456. drop - params [ g q , D , V , [ F 5 , S 5 , A 5 ] : : [ F 4 , S 4 , A 4 ] : : _ ] drop - params [ n , D , V , _ ] \operatorname{drop-params}[g\ q,D,V,[F_{5},S_{5},A_{5}]::[F_{4},S_{4},A_{4}]::% \_]\ \operatorname{drop-params}[n,D,V,\_]
  457. ¬ ask [ S - 6 ] \neg\operatorname{ask}[S-6]
  458. ¬ ask [ false ] \neg\operatorname{ask}[\operatorname{false}]
  459. drop - params [ g , D , V , [ F 6 , S 6 , A 6 ] : : [ F 5 , S 5 , A 5 ] : : [ F 4 , S 4 , A 4 ] : : _ ] drop - params [ m , D , V , _ ] drop - params [ n , D , V , _ ] \operatorname{drop-params}[g,D,V,[F_{6},S_{6},A_{6}]::[F_{5},S_{5},A_{5}]::[F_% {4},S_{4},A_{4}]::\_]\ \operatorname{drop-params}[m,D,V,\_]\ \operatorname{% drop-params}[n,D,V,\_]
  460. D [ g ] = [ [ x , false , _ ] , [ o , _ , p ] , [ y , _ , n ] ] D[g]=[[x,\operatorname{false},\_],[o,\_,p],[y,\_,n]]
  461. = [ F 6 , S 6 , A 6 ] : : [ F 5 , S 5 , A 5 ] : : [ F 4 , S 4 , A 4 ] : : _ ] =[F_{6},S_{6},A_{6}]::[F_{5},S_{5},A_{5}]::[F_{4},S_{4},A_{4}]::\_]
  462. F 6 = x , S 6 = false , A 6 = _ F_{6}=x,S_{6}=\operatorname{false},A_{6}=\_
  463. F 5 = o , S 5 = _ , A 5 = p F_{5}=o,S_{5}=\_,A_{5}=p
  464. F 4 = y , S 4 = _ , A 4 = n F_{4}=y,S_{4}=\_,A_{4}=n
  465. g q n g\ q\ n
  466. ( ask [ S ] and F V [ A ] V ) drop - formal [ [ F , S , A ] : : Z , λ F . Y , V ] drop - formal [ [ F , S , A ] : : Z , Y [ F := A ] , L ] (\operatorname{ask}[S]\and FV[A]\subset V)\to\operatorname{drop-formal}[[F,S,A% ]::Z,\lambda F.Y,V]\equiv\operatorname{drop-formal}[[F,S,A]::Z,Y[F:=A],L]
  467. ¬ ( ask [ S ] and F V [ A ] V ) drop - formal [ [ F , S , A ] : : Z , λ F . Y , V ] λ F . drop - formal [ [ F , S , A ] : : Z , Y , V ] \neg(\operatorname{ask}[S]\and FV[A]\subset V)\to\operatorname{drop-formal}[[F% ,S,A]::Z,\lambda F.Y,V]\equiv\lambda F.\operatorname{drop-formal}[[F,S,A]::Z,Y% ,V]
  468. drop - formal [ Z , Y , V ] Y \operatorname{drop-formal}[Z,Y,V]\equiv Y
  469. false \operatorname{false}
  470. drop - formal [ D , λ x . λ o . λ y . o x y , F ] \operatorname{drop-formal}[D,\lambda x.\lambda o.\lambda y.o\ x\ y,F]
  471. true and { p } F \operatorname{true}\and\{p\}\subset F
  472. λ x . drop - formal [ D , λ o . λ y . o x y , F ] \lambda x.\operatorname{drop-formal}[D,\lambda o.\lambda y.o\ x\ y,F]
  473. ¬ ( true and { n } F \neg(\operatorname{true}\and\{n\}\subset F
  474. λ x . drop - formal [ D , ( λ y . o x y ) [ o := p ] , F ] \lambda x.\operatorname{drop-formal}[D,(\lambda y.o\ x\ y)[o:=p],F]
  475. λ x . λ y . drop - formal [ D , p x y , F ] \lambda x.\lambda y.\operatorname{drop-formal}[D,p\ x\ y,F]
  476. λ x . λ y . p x y \lambda x.\lambda y.p\ x\ y
  477. let p f x = f ( x x ) and q p f = ( p f ) ( p f ) in q p \operatorname{let}p\ f\ x=f\ (x\ x)\and q\ p\ f=(p\ f)\ (p\ f)\operatorname{in% }q\ p
  478. let p f x = f ( x x ) and q p f = ( p f ) ( p f ) in q p \operatorname{let}p\ f\ x=f\ (x\ x)\and q\ p\ f=(p\ f)\ (p\ f)\operatorname{in% }q\ p
  479. let p = λ f . λ x . f ( x x ) and q = λ p . λ f . ( p f ) ( p f ) in q p \operatorname{let}p=\lambda f.\lambda x.f\ (x\ x)\and q=\lambda p.\lambda f.(p% \ f)\ (p\ f)\operatorname{in}q\ p
  480. ( λ q . ( λ p . q p ) ( λ f . λ x . f ( x x ) ) ) ( λ p . λ f . ( p f ) ( p f ) ) (\lambda q.(\lambda p.q\ p)\ (\lambda f.\lambda x.f\ (x\ x)))\ (\lambda p.% \lambda f.(p\ f)\ (p\ f))
  481. ( λ p . λ f . ( p f ) ( p f ) ) ( λ f . λ x . f ( x x ) ) (\lambda p.\lambda f.(p\ f)\ (p\ f))\ (\lambda f.\lambda x.f\ (x\ x))
  482. λ f . ( λ p . ( p f ) ( p f ) ) ( λ f . λ x . f ( x x ) ) \lambda f.(\lambda p.(p\ f)\ (p\ f))\ (\lambda f.\lambda x.f\ (x\ x))
  483. λ f . ( λ p . p p ) ( λ x . f ( x x ) ) \lambda f.(\lambda p.p\ p)\ (\lambda x.f\ (x\ x))
  484. λ f . ( λ x . f ( x x ) ) ( λ x . f ( x x ) ) \lambda f.(\lambda x.f\ (x\ x))\ (\lambda x.f\ (x\ x))
  485. λ f . ( λ x . f ( x x ) ) ( λ x . f ( x x ) ) \lambda f.(\lambda x.f\ (x\ x))\ (\lambda x.f\ (x\ x))