wpmath0000013_14

Unit_doublet.html

  1. ( x * u 1 ) ( t ) = d x ( t ) d t (x*u_{1})(t)=\frac{dx(t)}{dt}

Unit_hyperbola.html

  1. x 2 - y 2 = 1. x^{2}-y^{2}=1.
  2. r = x 2 - y 2 . r=\sqrt{x^{2}-y^{2}}.
  3. y 2 - x 2 = 1 y^{2}-x^{2}=1
  4. r = y 2 - x 2 . r=\sqrt{y^{2}-x^{2}}.
  5. 2 . \sqrt{2}.
  6. f = x 2 - y 2 - 1 \scriptstyle f=x^{2}-y^{2}-1
  7. F = x 2 - y 2 - z 2 , \scriptstyle F=x^{2}-y^{2}-z^{2},
  8. ( e t , e - t ) . (e^{t},\ e^{-t}).
  9. A = 1 2 ( 1 1 1 - 1 ) : A=\tfrac{1}{2}\begin{pmatrix}1&1\\ 1&-1\end{pmatrix}\ :
  10. ( e t , e - t ) A = ( e t + e - t 2 , e t - e - t 2 ) = ( cosh t , sinh t ) . (e^{t},\ e^{-t})\ A=(\frac{e^{t}+e^{-t}}{2},\ \frac{e^{t}-e^{-t}}{2})=(\cosh t% ,\ \sinh t).
  11. ρ = α cosh ( n t + ϵ ) + β sinh ( n t + ϵ ) \rho=\alpha\cosh(nt+\epsilon)+\beta\sinh(nt+\epsilon)
  12. ρ ¨ = n 2 ρ ; \ddot{\rho}=n^{2}\rho\ ;
  13. x 2 - y 2 = 1 x^{2}-y^{2}=1
  14. ( x 1 , y 1 ) (x_{1},\ y_{1})
  15. ( x 2 , y 2 ) (x_{2},\ y_{2})
  16. ( x 1 x 2 + y 1 y 2 , y 1 x 2 + y 2 x 1 ) (x_{1}x_{2}+y_{1}y_{2},\ y_{1}x_{2}+y_{2}x_{1})
  17. x = cosh t x=\cosh\ t
  18. y = sinh t y=\sinh\ t
  19. ± ( cosh a + j sinh a ) \pm(\cosh a+j\sinh a)
  20. exp ( a j ) e x p ( b j ) = exp ( ( a + b ) j ) , \exp(aj)exp(bj)=\exp((a+b)j),

Unit_sphere.html

  1. ( x 1 , , x n ) (x_{1},\ldots,x_{n})
  2. x 1 2 + x 2 2 + + x n 2 = 1. x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=1.
  3. x 1 2 + x 2 2 + + x n 2 < 1 , x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}<1,
  4. x 1 2 + x 2 2 + + x n 2 1. x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}\leq 1.
  5. f ( x , y , z ) = x 2 + y 2 + z 2 = 1 f(x,y,z)=x^{2}+y^{2}+z^{2}=1
  6. V n = π n / 2 Γ ( 1 + n / 2 ) = { π n / 2 / ( n / 2 ) ! if n 0 is even , π n / 2 2 n / 2 / n ! ! if n 0 is odd , V_{n}=\frac{\pi^{n/2}}{\Gamma(1+n/2)}=\begin{cases}{\pi^{n/2}}/{(n/2)!}&% \mathrm{if~{}}n\geq 0\mathrm{~{}is~{}even,}\\ \\ {\pi^{\lfloor n/2\rfloor}2^{\lceil n/2\rceil}}/{n!!}&\mathrm{if~{}}n\geq 0% \mathrm{~{}is~{}odd,}\end{cases}
  7. A n = n V n = n π n / 2 Γ ( 1 + n / 2 ) = 2 π n / 2 Γ ( n / 2 ) , A_{n}=nV_{n}=\frac{n\pi^{n/2}}{\Gamma(1+n/2)}=\frac{2\pi^{n/2}}{\Gamma(n/2)}\,,
  8. n n
  9. n n
  10. A n A_{n}
  11. V n V_{n}
  12. 0 ( 1 / 0 ! ) π 0 0(1/0!)\pi^{0}
  13. ( 1 / 0 ! ) π 0 (1/0!)\pi^{0}
  14. 1 ( 2 1 / 1 < t h > ) π 0 < / t h > < t d > 2 < / t d > < t d > < m a t h > ( 2 1 / 1 < t h > ) π 0 < / t h > < t d > 2 < / t d > 1(2^{1}/1<th>)\pi^{0}</th><td>2</td><td><math>(2^{1}/1<th>)\pi^{0}</th><td>2% \par </td>
  15. 2 ( 1 / 1 ! ) π 1 = 2 π 2(1/1!)\pi^{1}=2\pi
  16. ( 1 / 1 ! ) π 1 = π (1/1!)\pi^{1}=\pi
  17. 3 ( 2 2 / 3 < t h > ) π 1 = 4 π < / t h > < t d > 12.57 < / t d > < t d > < m a t h > ( 2 2 / 3 < t h > ) π 1 = ( 4 / 3 ) π < / t h > < t d > 4.189 < / t d > 3(2^{2}/3<th>)\pi^{1}=4\pi</th><td>12.57</td><td><math>(2^{2}/3<th>)\pi^{1}=(4% /3)\pi</th><td>4.189\par </td>
  18. 4 ( 1 / 2 ! ) π 2 = 2 π 2 4(1/2!)\pi^{2}=2\pi^{2}
  19. ( 1 / 2 ! ) π 2 = ( 1 / 2 ) π 2 (1/2!)\pi^{2}=(1/2)\pi^{2}
  20. 5 ( 2 3 / 5 < t h > ) π 2 = ( 8 / 3 ) π 2 < / t h > < t d > 26.32 < / t d > < t d > < m a t h > ( 2 3 / 5 < t h > ) π 2 = ( 8 / 15 ) π 2 < / t h > < t d > 5.264 < / t d > 5(2^{3}/5<th>)\pi^{2}=(8/3)\pi^{2}</th><td>26.32</td><td><math>(2^{3}/5<th>)% \pi^{2}=(8/15)\pi^{2}</th><td>5.264\par </td>
  21. 6 ( 1 / 3 ! ) π 3 = π 3 6(1/3!)\pi^{3}=\pi^{3}
  22. ( 1 / 3 ! ) π 3 = ( 1 / 6 ) π 3 (1/3!)\pi^{3}=(1/6)\pi^{3}
  23. 7 ( 2 4 / 7 < t h > ) π 3 = ( 16 / 15 ) π 3 < / t h > < t d > 33.07 < / t d > < t d > < m a t h > ( 2 4 / 7 < t h > ) π 3 = ( 16 / 105 ) π 3 < / t h > < t d > 4.725 < / t d > 7(2^{4}/7<th>)\pi^{3}=(16/15)\pi^{3}</th><td>33.07</td><td><math>(2^{4}/7<th>)% \pi^{3}=(16/105)\pi^{3}</th><td>4.725\par </td>
  24. 8 ( 1 / 4 ! ) π 4 = ( 1 / 3 ) π 4 8(1/4!)\pi^{4}=(1/3)\pi^{4}
  25. ( 1 / 4 ! ) π 4 = ( 1 / 24 ) π 4 (1/4!)\pi^{4}=(1/24)\pi^{4}
  26. 9 ( 2 5 / 9 < t h > ) π 4 = ( 32 / 105 ) π 4 < / t h > < t d > 29.69 < / t d > < t d > < m a t h > ( 2 5 / 9 < t h > ) π 4 = ( 32 / 945 ) π 4 < / t h > < t d > 3.299 < / t d > 9(2^{5}/9<th>)\pi^{4}=(32/105)\pi^{4}</th><td>29.69</td><td><math>(2^{5}/9<th>% )\pi^{4}=(32/945)\pi^{4}</th><td>3.299\par </td>
  27. 10 ( 1 / 5 ! ) π 5 = ( 1 / 12 ) π 5 10(1/5!)\pi^{5}=(1/12)\pi^{5}
  28. ( 1 / 5 ! ) π 5 = ( 1 / 120 ) π 5 (1/5!)\pi^{5}=(1/120)\pi^{5}
  29. A 0 = 0 A_{0}=0
  30. A 1 = 2 A_{1}=2
  31. A 2 = 2 π A_{2}=2\pi
  32. A n = 2 π n - 2 A n - 2 A_{n}=\frac{2\pi}{n-2}A_{n-2}
  33. n > 2 n>2
  34. V 0 = 1 V_{0}=1
  35. V 1 = 2 V_{1}=2
  36. V n = 2 π n V n - 2 V_{n}=\frac{2\pi}{n}V_{n-2}
  37. n > 1 n>1
  38. V V
  39. \|\cdot\|
  40. { x V : x < 1 } . \{x\in V:\|x\|<1\}.
  41. { x V : x 1 } . \{x\in V:\|x\|\leq 1\}.
  42. { x V : x = 1 } . \{x\in V:\|x\|=1\}.
  43. p \ell^{p}
  44. p \ell^{p}
  45. C p C_{p}
  46. C 0 = C = 8 C_{0}=C_{\infty}=8
  47. C 1 = 4 2 C_{1}=4\sqrt{2}
  48. C 2 = 2 π . C_{2}=2\pi\,.
  49. x 2 - y 2 x^{2}-y^{2}

Unital.html

  1. ϕ ( I ) = I \phi(I)=I

Unknotting_number.html

  1. n n
  2. n n
  3. ( p , q ) (p,q)
  4. ( p - 1 ) ( q - 1 ) / 2 (p-1)(q-1)/2

Unknown_key-share_attack.html

  1. A A
  2. B B
  3. B B
  4. E A E\neq A

UQCR11.html

  1. \rightleftharpoons

Utilization_factor.html

  1. k u k\text{u}

Vagrant_predicate.html

  1. \exists

Valuation_effects.html

  1. Change in NFA = Current Account + Valuation Effects \begin{aligned}\displaystyle\mbox{Change in NFA}&\displaystyle=\mbox{Current % Account}~{}+\mbox{Valuation Effects}\\ \end{aligned}

Value-form.html

  1. [ e i t h e r A a m o u n t o f c o m m o d i t y B o r , C a m o u n t o f c o m m o d i t y D o r , E a m o u n t o f c o m m o d i t y F o r , G a m o u n t o f c o m m o d i t y H o r , J a m o u n t o f c o m m o d i t y K ] \begin{bmatrix}either&A&amount&of&commodity&B\\ or,&C&amount&of&commodity&D\\ or,&E&amount&of&commodity&F\\ or,&G&amount&of&commodity&H\\ or,&J&amount&of&commodity&K\\ \end{bmatrix}
  2. [ X a m o u n t o f c o m m o d i t y Y ] \begin{bmatrix}X&amount&of&commodity&Y\end{bmatrix}
  3. [ e i t h e r A a m o u n t o f c o m m o d i t y B o r , C a m o u n t o f c o m m o d i t y D o r , E a m o u n t o f c o m m o d i t y F o r , G a m o u n t o f c o m m o d i t y H o r , J a m o u n t o f c o m m o d i t y K ] \begin{bmatrix}either&A&amount&of&commodity&B\\ or,&C&amount&of&commodity&D\\ or,&E&amount&of&commodity&F\\ or,&G&amount&of&commodity&H\\ or,&J&amount&of&commodity&K\\ \end{bmatrix}
  4. [ X a m o u n t o f m o n e y ] \begin{bmatrix}X&amount&of&money\end{bmatrix}

Valya_algebra.html

  1. g ( A , B ) = - g ( B , A ) g(A,B)=-g(B,A)
  2. A , B M A,B\in M
  3. J ( g ( A 1 , A 2 ) , g ( A 3 , A 4 ) , g ( A 5 , A 6 ) ) = 0 J(g(A_{1},A_{2}),g(A_{3},A_{4}),g(A_{5},A_{6}))=0
  4. A k M A_{k}\in M
  5. J ( A , B , C ) := g ( g ( A , B ) , C ) + g ( g ( B , C ) , A ) + g ( g ( C , A ) , B ) . J(A,B,C):=g(g(A,B),C)+g(g(B,C),A)+g(g(C,A),B).
  6. g ( a A + b B , C ) = a g ( A , C ) + b g ( B , C ) g(aA+bB,C)=ag(A,C)+bg(B,C)
  7. A , B , C M A,B,C\in M
  8. a , b F a,b\in F
  9. M ( - ) M^{(-)}
  10. M ( - ) M^{(-)}
  11. α = F k ( x ) d x k , β = G k ( x ) d x k \alpha=F_{k}(x)\,dx^{k},\quad\beta=G_{k}(x)\,dx^{k}
  12. ( α , β ) 0 = d Ψ ( α , β ) + Ψ ( d α , β ) + Ψ ( α , d β ) , (\alpha,\beta)_{0}=d\Psi(\alpha,\beta)+\Psi(d\alpha,\beta)+\Psi(\alpha,d\beta),\,
  13. ( α , β ) (\alpha,\beta)
  14. α \alpha
  15. β \beta
  16. d α = d β = 0 d\alpha=d\beta=0
  17. ( α , β ) = d Ψ ( α , β ) . (\alpha,\beta)=d\Psi(\alpha,\beta).\,
  18. ( α , β ) (\alpha,\beta)

Van_der_Corput_lemma_(harmonic_analysis).html

  1. ϕ ( x ) \phi(x)\,
  2. ( a , b ) (a,b)\,
  3. | ϕ ( k ) ( x ) | 1 |\phi^{(k)}(x)|\geq 1
  4. x ( a , b ) x\in(a,b)
  5. k 2 k\geq 2
  6. k = 1 k=1\,
  7. ϕ ( x ) \phi^{\prime}(x)\,
  8. x \R x\in\R
  9. c k c_{k}\,
  10. ϕ \phi\,
  11. | a b e i λ ϕ ( x ) | c k λ - 1 / k , \Big|\int_{a}^{b}e^{i\lambda\phi(x)}\Big|\leq c_{k}\lambda^{-1/k},
  12. λ \R \lambda\in\R
  13. ϵ \epsilon\,
  14. ϕ ( x ) \phi(x)\,
  15. I \R I\subset\R
  16. | ϕ ( k ) ( x ) | 1 |\phi^{(k)}(x)|\geq 1\,
  17. x I x\in I
  18. c k c_{k}\,
  19. ϕ \phi\,
  20. ϵ 0 \epsilon\geq 0\,
  21. { x I : | ϕ ( x ) | ϵ } \{x\in I:|\phi(x)|\leq\epsilon\}
  22. c k ϵ 1 / k c_{k}\epsilon^{1/k}\,

Van_Houtum_distribution.html

  1. e i t a p a + e i t b p b + 1 - p a - p b b - a - 1 e ( a + 1 ) i t - e b i t e i t - 1 e^{ita}p_{a}+e^{itb}p_{b}+\frac{1-p_{a}-p_{b}}{b-a-1}\frac{e^{(a+1)it}-e^{bit}% }{e^{it}-1}
  2. Pr ( U = u ) = { p a if u = a ; p b if u = b 1 - p a - p b b - a - 1 if a < u < b 0 otherwise \Pr(U=u)=\begin{cases}p_{a}&\,\text{if }u=a;\\ p_{b}&\,\text{if }u=b\\ \dfrac{1-p_{a}-p_{b}}{b-a-1}&\,\text{if }a<u<b\\ 0&\,\text{otherwise}\end{cases}
  3. X X
  4. μ \mu
  5. c 2 c^{2}
  6. U U
  7. U U
  8. X X
  9. a a
  10. b b
  11. p a p_{a}
  12. p b p_{b}
  13. a \displaystyle a
  14. μ \mu
  15. c 2 c^{2}
  16. μ \mu
  17. μ \lfloor\mu\rfloor
  18. μ \lceil\mu\rceil
  19. c 2 μ 2 ( μ - μ ) ( 1 + μ - μ ) 2 + ( μ - μ ) 2 ( 1 + μ - μ ) . c^{2}\mu^{2}\geq(\mu-\lfloor\mu\rfloor)(1+\mu-\lceil\mu\rceil)^{2}+(\mu-% \lfloor\mu\rfloor)^{2}(1+\mu-\lceil\mu\rceil).

Vanna–Volga_pricing.html

  1. 𝒱 \mathcal{V}
  2. Vanna = 𝒱 S \textrm{Vanna}=\frac{\partial\mathcal{V}}{\partial S}
  3. σ \sigma
  4. Volga = 𝒱 σ \textrm{Volga}=\frac{\partial\mathcal{V}}{\partial\sigma}
  5. σ ( K ) \sigma(K)
  6. K 0 K_{0}
  7. σ 0 \sigma_{0}
  8. K c / p K_{c/p}
  9. Δ c a l l ( K c , σ 0 ) = 1 / 4 \Delta_{call}(K_{c},\sigma_{0})=1/4
  10. Δ p u t ( K p , σ 0 ) = - 1 / 4 \Delta_{put}(K_{p},\sigma_{0})=-1/4
  11. Δ c a l l / p u t ( K , σ ) \Delta_{call/put}(K,\sigma)
  12. ATM ( K 0 ) \displaystyle\textrm{ATM}(K_{0})
  13. Call ( K , σ ) \textrm{Call}(K,\sigma)
  14. X V V X^{VV}
  15. X X
  16. X VV = X B S + X v a n n a RR v a n n a w R R R R c o s t + X v o l g a BF v o l g a w B F B F c o s t X^{\rm VV}=X^{BS}+\underbrace{\frac{\textrm{X}_{vanna}}{\textrm{RR}_{vanna}}}_% {w_{RR}}{RR}_{cost}+\underbrace{\frac{\textrm{X}_{volga}}{\textrm{BF}_{volga}}% }_{w_{BF}}{BF}_{cost}
  17. X B S X^{BS}
  18. R R c o s t = [ Call ( K c , σ ( K c ) ) - Put ( K p , σ ( K p ) ) ] - [ Call ( K c , σ 0 ) - Put ( K p , σ 0 ) ] B F c o s t = 1 2 [ Call ( K c , σ ( K c ) ) + Put ( K p , σ ( K p ) ) ] - 1 2 [ Call ( K c , σ 0 ) + Put ( K p , σ 0 ) ] \begin{aligned}\displaystyle RR_{cost}&\displaystyle=\left[\textrm{Call}(K_{c}% ,\sigma(K_{c}))-\textrm{Put}(K_{p},\sigma(K_{p}))\right]-\left[\textrm{Call}(K% _{c},\sigma_{0})-\textrm{Put}(K_{p},\sigma_{0})\right]\\ \displaystyle BF_{cost}&\displaystyle=\frac{1}{2}\left[\textrm{Call}(K_{c},% \sigma(K_{c}))+\textrm{Put}(K_{p},\sigma(K_{p}))\right]-\frac{1}{2}\left[% \textrm{Call}(K_{c},\sigma_{0})+\textrm{Put}(K_{p},\sigma_{0})\right]\end{aligned}
  19. w R R w_{RR}
  20. w B F w_{BF}
  21. X i = w A T M A T M i + w R R R R i + w B F B F i i =vega, vanna, volga X_{i}=w_{ATM}\,{ATM_{i}}+w_{RR}\,{RR_{i}}+w_{BF}\,{BF_{i}}\,\,\,\,\,i\,\text{=% vega, vanna, volga}
  22. x = 𝔸 w \vec{x}=\mathbb{A}\vec{w}
  23. 𝔸 = ( A T M v e g a R R v e g a B F v e g a A T M v a n n a R R v a n n a B F v a n n a A T M v o l g a R R v o l g a B F v o l g a ) \mathbb{A}=\begin{pmatrix}ATM_{vega}&RR_{vega}&BF_{vega}\\ ATM_{vanna}&RR_{vanna}&BF_{vanna}\\ ATM_{volga}&RR_{volga}&BF_{volga}\end{pmatrix}
  24. w = ( w A T M w R R w B F ) \vec{w}=\begin{pmatrix}w_{ATM}\\ w_{RR}\\ w_{BF}\end{pmatrix}
  25. x = ( X v e g a X v a n n a X v o l g a ) \vec{x}=\begin{pmatrix}X_{vega}\\ X_{vanna}\\ X_{volga}\end{pmatrix}
  26. X VV \displaystyle X^{\rm VV}
  27. I = ( 0 R R m k t - R R B S B F m k t - B F B S ) \vec{I}=\begin{pmatrix}0\\ {RR}^{mkt}-{RR}^{BS}\\ {BF}^{mkt}-{BF}^{BS}\end{pmatrix}
  28. ( Ω v e g a Ω v a n n a Ω v o l g a ) = ( 𝔸 T ) - 1 I \begin{pmatrix}\Omega_{vega}\\ \Omega_{vanna}\\ \Omega_{volga}\end{pmatrix}=(\mathbb{A}^{T})^{-1}\vec{I}
  29. Ω i \Omega_{i}
  30. X V V X^{VV}
  31. X VV = X B S + p v a n n a X v a n n a Ω v a n n a + p v o l g a X v o l g a Ω v o l g a \begin{aligned}\displaystyle X^{\rm VV}&\displaystyle=X^{BS}+p_{vanna}X_{vanna% }\Omega_{vanna}+p_{volga}X_{volga}\Omega_{volga}\end{aligned}
  32. p v a n n a p_{vanna}
  33. p v o l g a p_{volga}
  34. B B
  35. S 0 S_{0}
  36. B = S 0 B=S_{0}
  37. p vanna \displaystyle p_{\rm vanna}
  38. γ [ 0 , 1 ] \gamma\in[0,1]
  39. γ = 0 f o r S 0 B γ = 1 f o r | S 0 - B | 0 \begin{aligned}\displaystyle\gamma=0&\displaystyle{for}\ \ S_{0}\to B\\ \displaystyle\gamma=1&\displaystyle{for}\ \ |S_{0}-B|\gg 0\end{aligned}
  40. a , b , c a,b,c
  41. γ \gamma
  42. p s u r v [ 0 , 1 ] p_{surv}\in[0,1]
  43. { B i } \{B_{i}\}
  44. p s u r v = 𝔼 [ 1 S t < B , t tod < t < t mat ] = NT ( B ) / DF ( t tod , t mat ) p_{surv}=\mathbb{E}[1_{S_{t}<B,t_{\textrm{tod}}<t<t_{\textrm{mat}}}]=\mathrm{% NT}(B)/\mathrm{DF}(t_{\textrm{tod}},t_{\textrm{mat}})
  45. NT ( B ) \mathrm{NT}(B)
  46. DF ( t tod , t mat ) \mathrm{DF}(t_{\textrm{tod}},t_{\textrm{mat}})
  47. u ( S t , t ) u(S_{t},t)
  48. u ( S t , t ) = u(S_{t},t)=
  49. { ϕ , T } \{\phi,T\}
  50. ϕ = inf { [ 0 , T ) } \phi=\textrm{inf}\{\ell\in[0,T)\}
  51. S t + > H S_{t+\ell}>H
  52. S t + < L S_{t+\ell}<L
  53. L , H L,H
  54. S t S_{t}
  55. u ( S , t ) t + 1 2 σ 2 S 2 2 u ( S , t ) S 2 + μ S u ( S , t ) S = 0 \frac{\partial u(S,t)}{\partial t}+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2% }u(S,t)}{\partial S^{2}}+\mu S\frac{\partial u(S,t)}{\partial S}=0
  56. u ( S , T ) = T u(S,T)=T
  57. T T
  58. u ( L , t ) = u ( H , t ) = t u(L,t^{\prime})=u(H,t^{\prime})=t^{\prime}
  59. H S 0 H\gg S_{0}
  60. L S 0 L\ll S_{0}
  61. μ \mu

Variable_kernel_density_estimation.html

  1. { x i } \{\vec{x}_{i}\}
  2. P ( x ) P(\vec{x})
  3. x \vec{x}
  4. P ( x ) W n h D P(\vec{x})\approx\frac{W}{nh^{D}}
  5. W = i = 1 n w i W=\sum_{i=1}^{n}w_{i}
  6. w i = K ( x - x i h ) w_{i}=K\left(\frac{\vec{x}-\vec{x}_{i}}{h}\right)
  7. x \vec{x}
  8. h = k [ n P ( x ) ] 1 / D h=\frac{k}{\left[nP(\vec{x})\right]^{1/D}}
  9. W = k D ( 2 π ) D / 2 W=k^{D}(2\pi)^{D/2}
  10. e 1 = P K 2 n h D e_{1}=\frac{P\int K^{2}}{nh^{D}}
  11. e 2 = h 2 n 2 P e_{2}=\frac{h^{2}}{n}\nabla^{2}P
  12. P ( j , x ) 1 n i = 1 , c i = j n w i P(j,\vec{x})\approx\frac{1}{n}\sum_{i=1,c_{i}=j}^{n}w_{i}
  13. R ( x ) = P ( 2 | x ) - P ( 1 | x ) = P ( 2 , x ) - P ( 1 , x ) P ( 1 , x ) + P ( 2 , x ) R(\vec{x})=P(2|\vec{x})-P(1|\vec{x})=\frac{P(2,\vec{x})-P(1,\vec{x})}{P(1,\vec% {x})+P(2,\vec{x})}
  14. j = arg min 𝑖 | b i - x | j=\arg\underset{i}{\min}|\vec{b_{i}}-\vec{x}|\,
  15. p = ( x - b j ) x R | x = b j p=(\vec{x}-\vec{b_{j}})\cdot\nabla_{\vec{x}}R|_{\vec{x}=\vec{b_{j}}}\,
  16. c = ( 3 + p / | p | ) / 2 c=(3+p/|p|)/2\,
  17. { b i } \{\vec{b_{i}}\}
  18. R ( x ) tanh p R(\vec{x})\approx\tanh p\,

Variance-stabilizing_transformation.html

  1. y = x y=\sqrt{x}\,
  2. var ( X ) = g ( μ ) , \operatorname{var}(X)=g(\mu),\,
  3. y = x 1 g ( v ) d v , y=\int^{x}\frac{1}{\sqrt{g(v)}}\,dv,
  4. X X
  5. E [ X ] = μ E[X]=\mu
  6. V a r ( X ) = σ 2 Var(X)=\sigma^{2}
  7. Y = g ( X ) Y=g(X)
  8. g g
  9. Y = g ( x ) Y=g(x)
  10. Y = g ( X ) g ( μ ) + g ( μ ) ( X - μ ) Y=g(X)\approx g(\mu)+g^{\prime}(\mu)(X-\mu)
  11. E [ Y ] = g ( μ ) E[Y]=g(\mu)
  12. V a r [ Y ] = σ 2 g ( μ ) 2 Var[Y]=\sigma^{2}g^{\prime}(\mu)^{2}
  13. X X
  14. E [ X ] = μ E[X]=\mu
  15. V a r [ X ] = k h ( μ ) Var[X]=kh(\mu)
  16. g g
  17. Y = g ( X ) Y=g(X)
  18. V a r [ Y ] k h ( μ ) g ( μ ) 2 = c o n s t a n t Var[Y]\approx kh(\mu)g^{\prime}(\mu)^{2}=constant
  19. d g d μ = C h ( μ ) \frac{dg}{d\mu}=\frac{C}{\sqrt{h(\mu)}}
  20. g ( μ ) = C d μ h ( μ ) g(\mu)=\int\frac{Cd\mu}{\sqrt{h(\mu)}}

Variation_of_information.html

  1. X X
  2. Y Y
  3. A A
  4. X = { X 1 , X 2 , . . , , X k } X=\{X_{1},X_{2},..,,X_{k}\}
  5. Y = { Y 1 , Y 2 , . . , , Y l } Y=\{Y_{1},Y_{2},..,,Y_{l}\}
  6. n = Σ i | X i | = Σ j | Y j | = | A | n=\Sigma_{i}|X_{i}|=\Sigma_{j}|Y_{j}|=|A|
  7. p i = | X i | / n p_{i}=|X_{i}|/n
  8. q j = | Y j | / n q_{j}=|Y_{j}|/n
  9. r i j = | X i Y j | / n r_{ij}=|X_{i}\cap Y_{j}|/n
  10. V I ( X ; Y ) = - i , j r i j [ log ( r i j / p i ) + log ( r i j / q j ) ] VI(X;Y)=-\sum_{i,j}r_{ij}\left[\log(r_{ij}/p_{i})+\log(r_{ij}/q_{j})\right]
  11. A A
  12. μ ( B ) := | B | / n \mu(B):=|B|/n
  13. B A B\subseteq A
  14. V I ( X ; Y ) = H ( X ) + H ( Y ) - 2 I ( X , Y ) VI(X;Y)=H(X)+H(Y)-2I(X,Y)
  15. H ( X ) H(X)
  16. X X
  17. I ( X , Y ) I(X,Y)
  18. X X
  19. Y Y
  20. A A

Vector_(mathematics_and_physics).html

  1. \nabla
  2. 2 \nabla^{2}

Velocity_obstacle.html

  1. A A
  2. B B
  3. V O A | B = { 𝐯 | t > 0 : ( 𝐯 - 𝐯 B ) t D ( 𝐱 B - 𝐱 A , r A + r B ) } VO_{A|B}=\{\mathbf{v}\,|\,\exists t>0:(\mathbf{v}-\mathbf{v}_{B})t\in D(% \mathbf{x}_{B}-\mathbf{x}_{A},r_{A}+r_{B})\}
  4. A A
  5. 𝐱 A \mathbf{x}_{A}
  6. r A r_{A}
  7. B B
  8. 𝐱 B \mathbf{x}_{B}
  9. r B r_{B}
  10. 𝐯 B \mathbf{v}_{B}
  11. D ( 𝐱 , r ) D(\mathbf{x},r)
  12. 𝐱 \mathbf{x}
  13. r r

Veneziano_amplitude.html

  1. Γ ( - 1 + 1 2 ( k 1 + k 2 ) 2 ) Γ ( - 1 + 1 2 ( k 2 + k 3 ) 2 ) Γ ( - 2 + 1 2 ( ( k 1 + k 2 ) 2 + ( k 2 + k 3 ) 2 ) ) \frac{\Gamma(-1+\frac{1}{2}(k_{1}+k_{2})^{2})\Gamma(-1+\frac{1}{2}(k_{2}+k_{3}% )^{2})}{\Gamma(-2+\frac{1}{2}((k_{1}+k_{2})^{2}+(k_{2}+k_{3})^{2}))}

Venturi_flume.html

  1. Q = C H n Q=CH^{n}
  2. Q = C H n - Q E Q=CH^{n}-Q_{E}
  3. Q E Q_{E}

Verbal_subgroup.html

  1. { x y } \{xy\}
  2. { x 2 , x y 2 x - 1 } \{x^{2},xy^{2}x^{-1}\}

Version_vector.html

  1. a a
  2. b b
  3. V a [ x ] = V b [ x ] = m a x ( V a [ x ] , V b [ x ] ) V_{a}[x]=V_{b}[x]=max(V_{a}[x],V_{b}[x])
  4. a a
  5. b b
  6. a = b a=b
  7. a b a\parallel b
  8. a < b a<b
  9. b < a b<a
  10. a < b a<b
  11. V a V_{a}
  12. V b V_{b}
  13. a < b a<b
  14. b < a b<a

Vertical_penetration.html

  1. m g h 2 = 1 2 m v 1 2 \,mgh_{2}=\frac{1}{2}mv_{1}^{2}
  2. h 2 \,h_{2}
  3. Δ h \,\Delta h
  4. h 1 = 0 \,h_{1}=0
  5. Δ h = 1 2 m v 1 2 m g \Delta h=\frac{\frac{1}{2}mv_{1}^{2}}{mg}
  6. v 1 \,v_{1}
  7. v \,v
  8. 1 2 \frac{1}{2}
  9. Δ h = v 2 2 g \Delta h=\frac{v^{2}}{2g}

Very_smooth_hash.html

  1. b x 2 mod n b\equiv x^{2}\mod n
  2. p 1 = 2 , p 2 = 3 , p 3 = 5 , p_{1}=2,p_{2}=3,p_{3}=5,\dots
  3. x n * x\in\mathbb{Z}^{*}_{n}
  4. x 2 i = 0 k p i e i \textstyle x^{2}\equiv\prod_{i=0}^{k}p_{i}^{e_{i}}
  5. log n \log n
  6. s s
  7. s s
  8. n n
  9. c = 5 c=5
  10. n = 31 n=31
  11. m 1 = 35 = 5 7 m_{1}=35=5\cdot 7
  12. ( log 31 ) 5 = ˙ 7.37 (\log 31)^{5}~{}\dot{=}~{}7.37
  13. m 1 m_{1}
  14. m 2 = 55 = 5 11 m_{2}=55=5\cdot 11
  15. c = 5 c=5
  16. n = 31 n=31
  17. b 1 = 9 b_{1}=9
  18. n n
  19. c , n c,n
  20. x 1 = 3 x_{1}=3
  21. x 1 2 = b 1 x_{1}^{2}=b_{1}
  22. n n
  23. 3 2 n 3^{2}\not\geq n
  24. b 2 = 15 b_{2}=15
  25. n n
  26. x 2 = 20 x_{2}=20
  27. 20 2 = 400 15 20^{2}=400\equiv 15
  28. n n
  29. x 2 x_{2}
  30. b 2 b_{2}
  31. n n
  32. n n
  33. n n
  34. p 1 = 2 , p 2 = 3 , p_{1}=2,p_{2}=3,\ldots
  35. k k
  36. i = 1 k p i < n \textstyle\prod_{i=1}^{k}p_{i}<n
  37. m m
  38. \ell
  39. ( m 1 , , m ) (m_{1},\ldots,m_{\ell})
  40. < 2 k \ell<2^{k}
  41. m m
  42. L L
  43. l / k l/k
  44. m i = 0 m_{i}=0
  45. l < i L k l<i\leq Lk
  46. = i = 1 k l i 2 i - 1 \textstyle\ell=\sum_{i=1}^{k}l_{i}2^{i-1}
  47. i { 0 , 1 } \ell_{i}\in\{0,1\}
  48. \ell
  49. m L k + i = i m_{Lk+i}=\ell_{i}
  50. 1 i k 1\leq i\leq k
  51. x j + 1 = x j 2 i = 1 k p i m j k + i mod n x_{j+1}=x_{j}^{2}\prod_{i=1}^{k}p_{i}^{m_{jk+i}}\mod n
  52. Ω ( log n / log log n ) \Omega(\log n/\log\log n)
  53. p i p_{i}
  54. i i
  55. c c
  56. p p
  57. q q
  58. p = 2 q + 1 p=2q+1
  59. k ( log p ) c k\leq(\log p)^{c}
  60. p p
  61. e 1 , , e k e_{1},...,e_{k}
  62. 2 e 1 i = 2 k p i e i mod p 2^{e_{1}}\equiv\prod_{i=2}^{k}p_{i}^{e_{i}}\mod p
  63. | e i | < q |e_{i}|<q
  64. i = 1 , , k i=1,...,k
  65. e 1 , , e k e_{1},...,e_{k}
  66. log p \log p
  67. p p
  68. \ell
  69. 2 / 2 2^{\ell/2}
  70. 2 2^{\ell}
  71. 2 / 3 2^{\ell/3}
  72. 2 / 3 2^{\ell/3}
  73. 2 / 3 2^{\ell/3}
  74. 2 / 2 2^{\ell/2}

Vienna_rectifier.html

  1. i ¯ D = G u ¯ C G u ¯ 1 \underline{i}_{D}=G\star\underline{u}_{C}\approx G\star\underline{u}_{1}
  2. u ¯ D = u ¯ - j ω 1 L 1 1 ¯ D \underline{u}_{D}\star=\underline{u}-j\omega_{1}L_{1}\underline{1}_{D}
  3. L 1 L1
  4. u ¯ D u ¯ 1 \underline{u}_{D}\star\approx\underline{u}_{1}
  5. i D a > 0 , i D b , i D c < 0 iDa>0,iDb,iDc<0
  6. ϕ 1 = - 30 + 30 \phi_{1}=-30^{\circ}...+30^{\circ}
  7. ϕ 1 \phi_{1}
  8. i D i 1 i_{D}\approx i_{1}

View-through_rate.html

  1. V T R = 100 * V i e w t h r o u g h / I m p r e s s i o n s VTR=100*Viewthrough/Impressions
  2. C T R = 100 * C l i c k s / I m p r e s s i o n s CTR=100*Clicks/Impressions
  3. T R R = ( V i e w t h r o u g h s + C l i c k s ) / I m p r e s s i o n s TRR=(Viewthroughs+Clicks)/Impressions
  4. V C R = V i e w t h r o u g h s / T o t a l I m p r e s s i o n s VCR=Viewthroughs/TotalImpressions
  5. V - C V R = V i e w t h r o u g h C o n v e r s i o n s / V i e w t h r o u g h V i s i t s V-CVR=ViewthroughConversions/ViewthroughVisits

Vilém_Klíma.html

  1. ξ p = { sin ( π Q s Y k q 1 ν ) - e j π t Y k ν sin ( π Q s Y k q 2 ν ) ( q 1 + q 2 ) sin ( π Q s Y k ν ) e j π Q s Y k ( q 1 - 1 ) ν q 1 q 2 sin ( π Q s Y k q 1 ν ) q 1 sin ( π Q s Y k ν ) q 1 = q 2 \xi_{p}=\begin{cases}\cfrac{\sin\left(\frac{\pi}{Q_{s}}Y_{k}q_{1}\nu\right)-e^% {j\frac{\pi}{t}Y_{k}\nu}\sin\left(\frac{\pi}{Q_{s}}Y_{k}q_{2}\nu\right)}{\left% (q_{1}+q_{2}\right)\sin\left(\frac{\pi}{Q_{s}}Y_{k}\nu\right)}e^{j\frac{\pi}{Q% _{s}}Y_{k}\left(q_{1}-1\right)\nu}&q_{1}\neq q_{2}\\ \cfrac{\sin\left(\frac{\pi}{Q_{s}}Y_{k}q_{1}\nu\right)}{q_{1}\sin\left(\frac{% \pi}{Q_{s}}Y_{k}\nu\right)}&q_{1}=q_{2}\\ \end{cases}
  2. Y k = g Q s p + Q s Q b p { t = gcd ( Q s , p ) Q b = Q s t g = smallest integer for which Y k Y_{k}=\frac{gQ_{s}}{p}+\frac{Q_{s}}{Q_{b}p}\quad\begin{cases}t=\mbox{gcd}~{}(Q% _{s},p)\\ Q_{b}=\cfrac{Q_{s}}{t}\\ g=\mbox{smallest integer for which}~{}\ Y_{k}\in\mathbb{N}\\ \end{cases}
  3. q 1 = { q 2 = Q b 2 m Q b even q 2 + 1 = Q b + m 2 m Q b odd q_{1}=\begin{cases}q_{2}=\cfrac{Q_{b}}{2m}&Q_{b}\ \mbox{even}\\ q_{2}+1=\cfrac{Q_{b}+m}{2m}&Q_{b}\ \mbox{odd}\\ \end{cases}

Vincent_Lafforgue.html

  1. S L ( 3 , ) SL(3,\mathbb{R})
  2. S L ( 3 , ) SL(3,\mathbb{C})
  3. S L ( 3 , p ) SL(3,\mathbb{Q}_{p})

Virial_mass.html

  1. r vir r_{\rm vir}
  2. ρ ( < r vir ) = Δ c ρ c \rho(<r_{\rm vir})=\Delta_{c}\rho_{c}
  3. ρ ( < r ) \rho(<r)
  4. ρ c \rho_{c}
  5. ρ c \rho_{c}
  6. ρ M = Ω M ρ c \rho_{M}=\Omega_{M}\rho_{c}
  7. Ω M 0.27 \Omega_{M}\simeq 0.27
  8. Δ c \Delta_{c}
  9. Δ c = 200 \Delta_{c}=200
  10. M 200 = M ( < r 200 ) M_{200}=M(<r_{200})

Voigt-Thomson_law.html

  1. ρ ( ϑ ) = ρ 0 + Δ ρ c o s 2 ϑ \rho(\vartheta)=\rho_{0}+\Delta\rho\cdot cos^{2}\vartheta
  2. ϑ \vartheta
  3. ρ 0 \rho_{0}
  4. Δ ρ \Delta\rho
  5. ρ ( ϑ ) = ρ c o s 2 ϑ + ρ s i n 2 ϑ \rho(\vartheta)=\rho_{\parallel}\cdot cos^{2}\vartheta+\rho_{\perp}\cdot sin^{% 2}\vartheta
  6. ρ \rho_{\parallel}
  7. ρ \rho_{\perp}

Volume_(thermodynamics).html

  1. p V n pV^{n}
  2. p p
  3. V V
  4. n n
  5. n n
  6. m 3 \mathrm{m^{3}}
  7. l \mathrm{l}
  8. ft 3 \mathrm{ft}^{3}
  9. p V pV
  10. H H
  11. H = U + p V , H=U+pV,\,
  12. U U
  13. ν \nu
  14. m 3 kg \frac{\mathrm{m^{3}}}{\mathrm{kg}}
  15. ft 3 lbm \frac{\mathrm{ft^{3}}}{\mathrm{lbm}}
  16. ft 3 slug \frac{\mathrm{ft^{3}}}{\mathrm{slug}}
  17. mL g \frac{\mathrm{mL}}{\mathrm{g}}
  18. ν = V m = 1 ρ \nu=\frac{V}{m}=\frac{1}{\rho}
  19. V V
  20. m m
  21. ρ \rho
  22. ν = R ¯ T P \nu=\frac{{\bar{R}}T}{P}
  23. R ¯ {\bar{R}}
  24. T T
  25. P P
  26. V = n R T p V=\frac{nRT}{p}
  27. V 2 = V 1 × T 2 T 1 × p 1 - p w , 1 p 2 - p w , 2 V_{2}=V_{1}\times\frac{T_{2}}{T_{1}}\times\frac{p_{1}-p_{w,1}}{p_{2}-p_{w,2}}
  28. V l = 1 l × 310 K 273 K × 100 kPa - 0 kPa 100 kPa - 6.2 kPa = 1.21 l V_{l}=1\ \mathrm{l}\times\frac{310\ \mathrm{K}}{273\ \mathrm{K}}\times\frac{10% 0\ \mathrm{kPa}-0\ \mathrm{kPa}}{100\ \mathrm{kPa}-6.2\ \mathrm{kPa}}=1.21\ % \mathrm{l}
  29. V x = V t o t × P x P t o t = V t o t × n x n t o t V_{x}=V_{tot}\times\frac{P_{x}}{P_{tot}}=V_{tot}\times\frac{n_{x}}{n_{tot}}

Volume_conjecture.html

  1. K N \langle K\rangle_{N}
  2. K K
  3. N N
  4. J K , N ( q ) J_{K,N}(q)
  5. K K
  6. K N = lim q e 2 π i / N J K , N ( q ) J O , N ( q ) . \langle K\rangle_{N}=\lim_{q\to e^{2\pi i/N}}\frac{J_{K,N}(q)}{J_{O,N}(q)}.
  7. lim N 2 π log | K N | N = vol ( K ) , \lim_{N\to\infty}\frac{2\pi\log|\langle K\rangle_{N}|}{N}=\operatorname{vol}(K% ),\,
  8. vol ( K ) \operatorname{vol}(K)
  9. K K
  10. 4 1 4_{1}
  11. 5 2 5_{2}
  12. 6 1 6_{1}
  13. K K
  14. N N
  15. q = exp ( 2 π i / N ) q=\exp{(2\pi i/N)}
  16. exp i π N \exp{\frac{i\pi}{N}}
  17. lim N 2 π log K N N = vol ( S 3 \ K ) + C S ( S 3 \ K ) , \lim_{N\to\infty}\frac{2\pi\log\langle K\rangle_{N}}{N}=\operatorname{vol}(S^{% 3}\backslash K)+CS(S^{3}\backslash K),
  18. C S ( S 3 \ K ) CS(S^{3}\backslash K)

Von_Neumann–Morgenstern_utility_theorem.html

  1. L = 0.25 A + 0.75 B L=0.25A+0.75B\,
  2. L = p i A i , L=\sum p_{i}A_{i},\,
  3. p i p_{i}
  4. L M . L\prec M.
  5. L M . L\preceq M.
  6. L M . L\sim M.
  7. L M \,L\prec M\,
  8. M L \,M\prec L\,
  9. L M \,L\sim M
  10. L M \,L\preceq M\,
  11. M N \,M\preceq N\,
  12. L N \,L\preceq N\,
  13. L M N \,L\preceq M\preceq N\,
  14. p [ 0 , 1 ] \,p\in[0,1]\,
  15. p L + ( 1 - p ) N M \,pL+(1-p)N\,\sim\,M\,
  16. L M N \,L\prec M\prec N\,
  17. ε ( 0 , 1 ) \,\varepsilon\in(0,1)
  18. ( 1 - ε ) L + ε N M ε L + ( 1 - ε ) N . \,(1-\varepsilon)L+\varepsilon N\,\prec\,M\,\prec\,\varepsilon L+(1-% \varepsilon)N.\,
  19. L M \,L\prec M\,
  20. N \,N\,
  21. p ( 0 , 1 ] \,p\in(0,1]\,
  22. p L + ( 1 - p ) N p M + ( 1 - p ) N . \,pL+(1-p)N\prec pM+(1-p)N.\,
  23. Z , W , Z,\,W,
  24. p , q , r ( 0 , 1 ] p,q,r\in(0,1]
  25. r q = p , rq=p,
  26. X = q Z + ( 1 - q ) W , X=qZ+(1-q)W,
  27. p Z + ( 1 - p ) W r X + ( 1 - r ) W . pZ+(1-p)W\sim rX+(1-r)W.
  28. L M iff E ( u ( L ) ) < E ( u ( M ) ) , L\prec M\;\mathrm{iff}\;E(u(L))<E(u(M)),\,
  29. E u ( p 1 A 1 + + p n A n ) = p 1 u ( A 1 ) + + p n u ( A n ) . Eu(p_{1}A_{1}+\ldots+p_{n}A_{n})=p_{1}u(A_{1})+\cdots+p_{n}u(A_{n}).\,
  30. 20 % ( $ 10 , 000 ) + 80 % ( $ 0 ) = $ 2000 > 100 % ( $ 1000 ) 20\%(\$10,000)+80\%(\$0)=\$2000>100\%(\$1000)
  31. 20 % u ( $ 10 , 000 ) + 80 % u ( $ 0 ) < u ( $ 1000 ) 20\%u(\$10,000)+80\%u(\$0)<u(\$1000)
  32. N N
  33. p M pM
  34. 1 N 1N
  35. p M + ( 1 - p ) 0 pM+(1-p)0
  36. L M = N L\prec M=N
  37. L N L\prec N
  38. \,\sim\,

Vortex_stretching.html

  1. D ω D t = ( ω ) v , {D\vec{\omega}\over Dt}=(\vec{\omega}\cdot\vec{\nabla})\vec{v},
  2. ω \vec{\omega}
  3. ω \vec{\omega}

Vorticity_confinement.html

  1. u t + u u + P ρ = F D ( u ) - F C ( u ) \frac{\partial u}{\partial t}+u\cdot\nabla u+\nabla\frac{P}{\rho}=F_{D}(u)-F_{% C}(u)
  2. F D F_{D}
  3. F C F_{C}
  4. F D F_{D}
  5. F C F_{C}

Waldyr_Alves_Rodrigues_Jr..html

  1. π \pi

Wallis's_conical_edge.html

  1. x = v cos u , y = v sin u , z = c a 2 - b 2 cos 2 u . x=v\cos u,\quad y=v\sin u,\quad z=c\sqrt{a^{2}-b^{2}\cos^{2}u}.\,

Warburg_coefficient.html

  1. A W A_{W}
  2. Z W Z_{W}
  3. A W A_{W}
  4. σ {\sigma}
  5. Ω / s e c o n d s = Ω ( s - 1 / 2 ) {\Omega}/\sqrt{seconds}={\Omega}(s^{-1/2})
  6. A W A_{W}
  7. R R
  8. 1 / ω {1}/\sqrt{\omega}
  9. A W A_{W}
  10. A W = R T A n 2 F 2 2 ( 1 D O 1 / 2 C O b + 1 D R 1 / 2 C R b ) = R T A n 2 F 2 Θ C 2 D A_{W}={\frac{RT}{An^{2}F^{2}\sqrt{2}}}{\left(\frac{1}{D_{O}^{1/2}C_{O}^{b}}+{% \frac{1}{D_{R}^{1/2}C_{R}^{b}}}\right)}=\frac{RT}{An^{2}F^{2}\Theta C\sqrt{2D}}
  11. R R
  12. T T
  13. F F
  14. n n
  15. D D
  16. O O
  17. R R
  18. C b C^{b}
  19. O O
  20. R R
  21. A A
  22. Θ \Theta
  23. R R
  24. O O
  25. A W A_{W}

Warburg_element.html

  1. Z W = A W ω + A W j ω {Z_{W}}=\frac{A_{W}}{\sqrt{\omega}}+\frac{A_{W}}{j\sqrt{\omega}}
  2. | Z W | = 2 A W ω {|Z_{W}|}=\sqrt{2}\frac{A_{W}}{\sqrt{\omega}}

Water_activity.html

  1. a w p / p 0 a_{w}\equiv p/p_{0}
  2. a w l w x w a_{w}\equiv l_{w}x_{w}
  3. ERH = a w × 100 % \mathrm{ERH}=a_{w}\times 100\%
  4. MFSL = 10 7.91 - 8.1 a w \mathrm{MFSL}=10^{7.91-8.1a_{w}}

Watson's_lemma.html

  1. 0 < T 0<T\leq\infty
  2. ϕ ( t ) = t λ g ( t ) \phi(t)=t^{\lambda}\,g(t)
  3. g ( t ) g(t)
  4. t = 0 t=0
  5. g ( 0 ) 0 g(0)\neq 0
  6. λ > - 1 \lambda>-1
  7. | ϕ ( t ) | < K e b t t > 0 , |\phi(t)|<Ke^{bt}\ \forall t>0,
  8. K , b K,b
  9. t t
  10. 0 T | ϕ ( t ) | d t < . \int_{0}^{T}|\phi(t)|\,\mathrm{d}t<\infty.
  11. x x
  12. | 0 T e - x t ϕ ( t ) d t | < \left|\int_{0}^{T}e^{-xt}\phi(t)\,\mathrm{d}t\right|<\infty
  13. 0 T e - x t ϕ ( t ) d t n = 0 g ( n ) ( 0 ) Γ ( λ + n + 1 ) n ! x λ + n + 1 , ( x > 0 , x ) . \int_{0}^{T}e^{-xt}\phi(t)\,\mathrm{d}t\sim\ \sum_{n=0}^{\infty}\frac{g^{(n)}(% 0)\ \Gamma(\lambda+n+1)}{n!\ x^{\lambda+n+1}},\ \ (x>0,\ x\rightarrow\infty).
  14. | ϕ ( t ) | |\phi(t)|
  15. t t\to\infty
  16. g ( t ) g(t)
  17. g g
  18. 0 < T 0<T\leq\infty
  19. ϕ \phi
  20. ϕ ( t ) = t λ g ( t ) \phi(t)=t^{\lambda}g(t)
  21. λ > - 1 \lambda>-1
  22. g g
  23. [ 0 , δ ] [0,\delta]
  24. 0 < δ < T 0<\delta<T
  25. | ϕ ( t ) | K e b t |\phi(t)|\leq Ke^{bt}
  26. δ t T \delta\leq t\leq T
  27. K K
  28. b b
  29. t t
  30. x x
  31. ( 1 ) 0 T e - x t ϕ ( t ) d t = 0 δ e - x t ϕ ( t ) d t + δ T e - x t ϕ ( t ) d t (1)\quad\int_{0}^{T}e^{-xt}\phi(t)\,\mathrm{d}t=\int_{0}^{\delta}e^{-xt}\phi(t% )\,\mathrm{d}t+\int_{\delta}^{T}e^{-xt}\phi(t)\,\mathrm{d}t
  32. | 0 δ e - x t ϕ ( t ) d t | 0 δ e - x t | ϕ ( t ) | d t 0 δ | ϕ ( t ) | d t \left|\int_{0}^{\delta}e^{-xt}\phi(t)\,\mathrm{d}t\right|\leq\int_{0}^{\delta}% e^{-xt}|\phi(t)|\,\mathrm{d}t\leq\int_{0}^{\delta}|\phi(t)|\,\mathrm{d}t
  33. x 0 x\geq 0
  34. g g
  35. [ 0 , δ ] [0,\delta]
  36. λ > - 1 \lambda>-1
  37. ϕ \phi
  38. x > b x>b
  39. | δ T e - x t ϕ ( t ) d t | δ T e - x t | ϕ ( t ) | d t K δ T e ( b - x ) t d t K δ e ( b - x ) t d t = K e ( b - x ) δ x - b . \begin{aligned}\displaystyle\left|\int_{\delta}^{T}e^{-xt}\phi(t)\,\mathrm{d}t% \right|&\displaystyle\leq\int_{\delta}^{T}e^{-xt}|\phi(t)|\,\mathrm{d}t\\ &\displaystyle\leq K\int_{\delta}^{T}e^{(b-x)t}\,\mathrm{d}t\\ &\displaystyle\leq K\int_{\delta}^{\infty}e^{(b-x)t}\,\mathrm{d}t\\ &\displaystyle=K\,\frac{e^{(b-x)\delta}}{x-b}.\end{aligned}
  40. ( 1 ) (1)
  41. ( 2 ) 0 T e - x t ϕ ( t ) d t = 0 δ e - x t ϕ ( t ) d t + O ( x - 1 e - δ x ) (2)\quad\int_{0}^{T}e^{-xt}\phi(t)\,\mathrm{d}t=\int_{0}^{\delta}e^{-xt}\phi(t% )\,\mathrm{d}t+O\left(x^{-1}e^{-\delta x}\right)
  42. x x\to\infty
  43. N 0 N\geq 0
  44. g ( t ) = n = 0 N g ( n ) ( 0 ) n ! t n + g ( N + 1 ) ( t * ) ( N + 1 ) ! t N + 1 g(t)=\sum_{n=0}^{N}\frac{g^{(n)}(0)}{n!}\,t^{n}+\frac{g^{(N+1)}(t^{*})}{(N+1)!% }\,t^{N+1}
  45. 0 t δ 0\leq t\leq\delta
  46. 0 t * t 0\leq t^{*}\leq t
  47. ( 2 ) (2)
  48. ( 3 ) 0 δ e - x t ϕ ( t ) d t = 0 δ e - x t t λ g ( t ) d t = n = 0 N g ( n ) ( 0 ) n ! 0 δ t λ + n e - x t d t + 1 ( N + 1 ) ! 0 δ g ( N + 1 ) ( t * ) t λ + N + 1 e - x t d t . \begin{aligned}\displaystyle(3)\quad\int_{0}^{\delta}e^{-xt}\phi(t)\,\mathrm{d% }t&\displaystyle=\int_{0}^{\delta}e^{-xt}t^{\lambda}g(t)\,\mathrm{d}t\\ &\displaystyle=\sum_{n=0}^{N}\frac{g^{(n)}(0)}{n!}\int_{0}^{\delta}t^{\lambda+% n}e^{-xt}\,\mathrm{d}t+\frac{1}{(N+1)!}\int_{0}^{\delta}g^{(N+1)}(t^{*})\,t^{% \lambda+N+1}e^{-xt}\,\mathrm{d}t.\end{aligned}
  49. g ( N + 1 ) g^{(N+1)}
  50. [ 0 , δ ] [0,\delta]
  51. | 0 δ g ( N + 1 ) ( t * ) t λ + N + 1 e - x t d t | sup t [ 0 , δ ] | g ( N + 1 ) ( t ) | 0 δ t λ + N + 1 e - x t d t < sup t [ 0 , δ ] | g ( N + 1 ) ( t ) | 0 t λ + N + 1 e - x t d t = sup t [ 0 , δ ] | g ( N + 1 ) ( t ) | Γ ( λ + N + 2 ) x λ + N + 2 . \begin{aligned}\displaystyle\left|\int_{0}^{\delta}g^{(N+1)}(t^{*})\,t^{% \lambda+N+1}e^{-xt}\,\mathrm{d}t\right|&\displaystyle\leq\sup_{t\in[0,\delta]}% \left|g^{(N+1)}(t)\right|\int_{0}^{\delta}t^{\lambda+N+1}e^{-xt}\,\mathrm{d}t% \\ &\displaystyle<\sup_{t\in[0,\delta]}\left|g^{(N+1)}(t)\right|\int_{0}^{\infty}% t^{\lambda+N+1}e^{-xt}\,\mathrm{d}t\\ &\displaystyle=\sup_{t\in[0,\delta]}\left|g^{(N+1)}(t)\right|\,\frac{\Gamma(% \lambda+N+2)}{x^{\lambda+N+2}}.\end{aligned}
  52. 0 t a e - x t d t = Γ ( a + 1 ) x a + 1 \int_{0}^{\infty}t^{a}e^{-xt}\,\mathrm{d}t=\frac{\Gamma(a+1)}{x^{a+1}}
  53. x > 0 x>0
  54. a > - 1 a>-1
  55. Γ \Gamma
  56. ( 3 ) (3)
  57. ( 4 ) 0 δ e - x t ϕ ( t ) d t = n = 0 N g ( n ) ( 0 ) n ! 0 δ t λ + n e - x t d t + O ( x - λ - N - 2 ) (4)\quad\int_{0}^{\delta}e^{-xt}\phi(t)\,\mathrm{d}t=\sum_{n=0}^{N}\frac{g^{(n% )}(0)}{n!}\int_{0}^{\delta}t^{\lambda+n}e^{-xt}\,\mathrm{d}t+O\left(x^{-% \lambda-N-2}\right)
  58. x x\to\infty
  59. ( 4 ) (4)
  60. n n
  61. 0 δ t λ + n e - x t d t = 0 t λ + n e - x t d t - δ t λ + n e - x t d t = Γ ( λ + n + 1 ) x λ + n + 1 - δ t λ + n e - x t d t , \begin{aligned}\displaystyle\int_{0}^{\delta}t^{\lambda+n}e^{-xt}\,\mathrm{d}t% &\displaystyle=\int_{0}^{\infty}t^{\lambda+n}e^{-xt}\,\mathrm{d}t-\int_{\delta% }^{\infty}t^{\lambda+n}e^{-xt}\,\mathrm{d}t\\ &\displaystyle=\frac{\Gamma(\lambda+n+1)}{x^{\lambda+n+1}}-\int_{\delta}^{% \infty}t^{\lambda+n}e^{-xt}\,\mathrm{d}t,\end{aligned}
  62. t = s + δ t=s+\delta
  63. δ t λ + n e - x t d t = 0 ( s + δ ) λ + n e - x ( s + δ ) d s = e - δ x 0 ( s + δ ) λ + n e - x s d s e - δ x 0 ( s + δ ) λ + n e - s d s \begin{aligned}\displaystyle\int_{\delta}^{\infty}t^{\lambda+n}e^{-xt}\,% \mathrm{d}t&\displaystyle=\int_{0}^{\infty}(s+\delta)^{\lambda+n}e^{-x(s+% \delta)}\,ds\\ &\displaystyle=e^{-\delta x}\int_{0}^{\infty}(s+\delta)^{\lambda+n}e^{-xs}\,ds% \\ &\displaystyle\leq e^{-\delta x}\int_{0}^{\infty}(s+\delta)^{\lambda+n}e^{-s}% \,ds\end{aligned}
  64. x 1 x\geq 1
  65. 0 δ t λ + n e - x t d t = Γ ( λ + n + 1 ) x λ + n + 1 + O ( e - δ x ) \int_{0}^{\delta}t^{\lambda+n}e^{-xt}\,\mathrm{d}t=\frac{\Gamma(\lambda+n+1)}{% x^{\lambda+n+1}}+O\left(e^{-\delta x}\right)
  66. x x\to\infty
  67. ( 4 ) (4)
  68. 0 δ e - x t ϕ ( t ) d t = n = 0 N g ( n ) ( 0 ) Γ ( λ + n + 1 ) n ! x λ + n + 1 + O ( e - δ x ) + O ( x - λ - N - 2 ) = n = 0 N g ( n ) ( 0 ) Γ ( λ + n + 1 ) n ! x λ + n + 1 + O ( x - λ - N - 2 ) \begin{aligned}\displaystyle\int_{0}^{\delta}e^{-xt}\phi(t)\,\mathrm{d}t&% \displaystyle=\sum_{n=0}^{N}\frac{g^{(n)}(0)\ \Gamma(\lambda+n+1)}{n!\ x^{% \lambda+n+1}}+O\left(e^{-\delta x}\right)+O\left(x^{-\lambda-N-2}\right)\\ &\displaystyle=\sum_{n=0}^{N}\frac{g^{(n)}(0)\ \Gamma(\lambda+n+1)}{n!\ x^{% \lambda+n+1}}+O\left(x^{-\lambda-N-2}\right)\end{aligned}
  69. x x\to\infty
  70. ( 2 ) (2)
  71. 0 T e - x t ϕ ( t ) d t = n = 0 N g ( n ) ( 0 ) Γ ( λ + n + 1 ) n ! x λ + n + 1 + O ( x - λ - N - 2 ) + O ( x - 1 e - δ x ) = n = 0 N g ( n ) ( 0 ) Γ ( λ + n + 1 ) n ! x λ + n + 1 + O ( x - λ - N - 2 ) \begin{aligned}\displaystyle\int_{0}^{T}e^{-xt}\phi(t)\,\mathrm{d}t&% \displaystyle=\sum_{n=0}^{N}\frac{g^{(n)}(0)\ \Gamma(\lambda+n+1)}{n!\ x^{% \lambda+n+1}}+O\left(x^{-\lambda-N-2}\right)+O\left(x^{-1}e^{-\delta x}\right)% \\ &\displaystyle=\sum_{n=0}^{N}\frac{g^{(n)}(0)\ \Gamma(\lambda+n+1)}{n!\ x^{% \lambda+n+1}}+O\left(x^{-\lambda-N-2}\right)\end{aligned}
  72. x x\to\infty
  73. N 0 N\geq 0
  74. 0 T e - x t ϕ ( t ) d t n = 0 g ( n ) ( 0 ) Γ ( λ + n + 1 ) n ! x λ + n + 1 \int_{0}^{T}e^{-xt}\phi(t)\,\mathrm{d}t\sim\sum_{n=0}^{\infty}\frac{g^{(n)}(0)% \ \Gamma(\lambda+n+1)}{n!\ x^{\lambda+n+1}}
  75. x x\to\infty
  76. 0 < a < b 0<a<b
  77. F 1 1 ( a , b , x ) = Γ ( b ) Γ ( a ) Γ ( b - a ) 0 1 e x t t a - 1 ( 1 - t ) b - a - 1 d t , {}_{1}F_{1}(a,b,x)=\frac{\Gamma(b)}{\Gamma(a)\Gamma(b-a)}\int_{0}^{1}e^{xt}t^{% a-1}(1-t)^{b-a-1}\,\mathrm{d}t,
  78. Γ \Gamma
  79. t = 1 - s t=1-s
  80. F 1 1 ( a , b , x ) = Γ ( b ) Γ ( a ) Γ ( b - a ) e x 0 1 e - x s ( 1 - s ) a - 1 s b - a - 1 d s , {}_{1}F_{1}(a,b,x)=\frac{\Gamma(b)}{\Gamma(a)\Gamma(b-a)}\,e^{x}\int_{0}^{1}e^% {-xs}(1-s)^{a-1}s^{b-a-1}\,ds,
  81. λ = b - a - 1 \lambda=b-a-1
  82. g ( s ) = ( 1 - s ) a - 1 g(s)=(1-s)^{a-1}
  83. 0 1 e - x s ( 1 - s ) a - 1 s b - a - 1 d s Γ ( b - a ) x a - b as x with x > 0 , \int_{0}^{1}e^{-xs}(1-s)^{a-1}s^{b-a-1}\,ds\sim\Gamma(b-a)x^{a-b}\quad\,\text{% as }x\to\infty\,\text{ with }x>0,
  84. F 1 1 ( a , b , x ) Γ ( b ) Γ ( a ) x a - b e x as x with x > 0. {}_{1}F_{1}(a,b,x)\sim\frac{\Gamma(b)}{\Gamma(a)}\,x^{a-b}e^{x}\quad\,\text{as% }x\to\infty\,\text{ with }x>0.

Waveguide_filter.html

  1. 5 \scriptstyle\sqrt{5}
  2. 1 5 \frac{1}{5}
  3. 2 3 \frac{2}{3}

Weierstrass–Erdmann_condition.html

  1. J = f ( t , x , y ) d t J=\int f(t,x,y)\,dt
  2. f / x \partial f/\partial x

Weingarten_function.html

  1. U d U i 1 j 1 U i q j q U j 1 i 1 * U j q i q * d U . \int_{U_{d}}U_{i_{1}j_{1}}\cdots U_{i_{q}j_{q}}U^{*}_{j^{\prime}_{1}i^{\prime}% _{1}}\cdots U^{*}_{j^{\prime}_{q}i^{\prime}_{q}}dU.
  2. U * U^{*}
  3. U U
  4. U U^{\dagger}
  5. σ , τ S q δ i 1 i σ 1 δ i q i σ q δ j 1 j τ 1 δ j q j τ q W g ( d , σ τ - 1 ) \sum_{\sigma,\tau\in S_{q}}\delta_{i_{1}i^{\prime}_{\sigma 1}}\cdots\delta_{i_% {q}i^{\prime}_{\sigma q}}\delta_{j_{1}j^{\prime}_{\tau 1}}\cdots\delta_{j_{q}j% ^{\prime}_{\tau q}}Wg(d,\sigma\tau^{-1})
  6. W g ( d , σ ) = 1 q ! 2 λ χ λ ( 1 ) 2 χ λ ( σ ) s λ , d ( 1 ) Wg(d,\sigma)=\frac{1}{q!^{2}}\sum_{\lambda}\frac{\chi^{\lambda}(1)^{2}\chi^{% \lambda}(\sigma)}{s_{\lambda,d}(1)}
  7. W g ( , d ) = 1 \displaystyle Wg(,d)=1
  8. W g ( 1 , d ) = 1 d \displaystyle Wg(1,d)=\frac{1}{d}
  9. W g ( 2 , d ) = - 1 d ( d 2 - 1 ) \displaystyle Wg(2,d)=\frac{-1}{d(d^{2}-1)}
  10. W g ( 1 2 , d ) = 1 d 2 - 1 \displaystyle Wg(1^{2},d)=\frac{1}{d^{2}-1}
  11. W g ( 3 , d ) = 2 d ( d 2 - 1 ) ( d 2 - 4 ) \displaystyle Wg(3,d)=\frac{2}{d(d^{2}-1)(d^{2}-4)}
  12. W g ( 21 , d ) = - 1 ( d 2 - 1 ) ( d 2 - 4 ) \displaystyle Wg(21,d)=\frac{-1}{(d^{2}-1)(d^{2}-4)}
  13. W g ( 1 3 , d ) = d 2 - 2 d ( d 2 - 1 ) ( d 2 - 4 ) \displaystyle Wg(1^{3},d)=\frac{d^{2}-2}{d(d^{2}-1)(d^{2}-4)}
  14. W g ( σ , d ) = d - n - | σ | i ( - 1 ) | C i | - 1 c | C i | - 1 + O ( d - n - | σ | - 2 ) Wg(\sigma,d)=d^{-n-|\sigma|}\prod_{i}(-1)^{|C_{i}|-1}c_{|C_{i}|-1}+O(d^{-n-|% \sigma|-2})

Welch_bounds.html

  1. { x 1 , , x m } \{x_{1},\ldots,x_{m}\}
  2. n \mathbb{C}^{n}
  3. c max = max i j | x i , x j | c_{\max}=\max_{i\neq j}|\langle x_{i},x_{j}\rangle|
  4. , \langle\cdot,\cdot\rangle
  5. n \mathbb{C}^{n}
  6. k = 1 , 2 , k=1,2,\dots
  7. ( c max ) 2 k 1 m - 1 [ m ( n + k - 1 k ) - 1 ] (c_{\max})^{2k}\geq\frac{1}{m-1}\left[\frac{m}{{\left({{n+k-1}\atop{k}}\right)% }}-1\right]
  8. m n m\leq n
  9. { x i } \{x_{i}\}
  10. n \mathbb{C}^{n}
  11. c max = 0 c_{\max}=0
  12. m > n m>n
  13. k = 1 k=1
  14. m × m m\times m
  15. G G
  16. { x i } \{x_{i}\}
  17. G = [ x 1 , x 1 x 1 , x m x m , x 1 x m , x m ] G=\left[\begin{array}[]{ccc}\langle x_{1},x_{1}\rangle&\cdots&\langle x_{1},x_% {m}\rangle\\ \vdots&\ddots&\vdots\\ \langle x_{m},x_{1}\rangle&\cdots&\langle x_{m},x_{m}\rangle\end{array}\right]
  18. G G
  19. G G
  20. n n
  21. G G
  22. n n
  23. G G
  24. λ 1 , , λ r \lambda_{1},\ldots,\lambda_{r}
  25. r n r\leq n
  26. r r
  27. ( Tr G ) 2 = ( i = 1 r λ i ) 2 r i = 1 r λ i 2 n i = 1 m λ i 2 (\mathrm{Tr}\;G)^{2}=\left(\sum_{i=1}^{r}\lambda_{i}\right)^{2}\leq r\sum_{i=1% }^{r}\lambda_{i}^{2}\leq n\sum_{i=1}^{m}\lambda_{i}^{2}
  28. G G
  29. || G || 2 = i = 1 m j = 1 m | x i , x j | 2 = i = 1 m λ i 2 ||G||^{2}=\sum_{i=1}^{m}\sum_{j=1}^{m}|\langle x_{i},x_{j}\rangle|^{2}=\sum_{i% =1}^{m}\lambda_{i}^{2}
  30. i = 1 m j = 1 m | x i , x j | 2 ( Tr G ) 2 n \sum_{i=1}^{m}\sum_{j=1}^{m}|\langle x_{i},x_{j}\rangle|^{2}\geq\frac{(\mathrm% {Tr}\;G)^{2}}{n}
  31. x i x_{i}
  32. G G
  33. Tr G = m \mathrm{Tr}\;G=m
  34. i = 1 m j = 1 m | x i , x j | 2 = m + i j | x i , x j | 2 m 2 n \sum_{i=1}^{m}\sum_{j=1}^{m}|\langle x_{i},x_{j}\rangle|^{2}=m+\sum_{i\neq j}|% \langle x_{i},x_{j}\rangle|^{2}\geq\frac{m^{2}}{n}
  35. i j | x i , x j | 2 m ( m - n ) n \sum_{i\neq j}|\langle x_{i},x_{j}\rangle|^{2}\geq\frac{m(m-n)}{n}
  36. a 0 a_{\ell}\geq 0
  37. = 1 , , L \ell=1,\ldots,L
  38. 1 L = 1 L a max a \frac{1}{L}\sum_{\ell=1}^{L}a_{\ell}\leq\max a_{\ell}
  39. m ( m - 1 ) m(m-1)
  40. c max 2 c_{\max}^{2}
  41. ( c max ) 2 1 m ( m - 1 ) i j | x i , x j | 2 m - n n ( m - 1 ) (c_{\max})^{2}\geq\frac{1}{m(m-1)}\sum_{i\neq j}|\langle x_{i},x_{j}\rangle|^{% 2}\geq\frac{m-n}{n(m-1)}
  42. ( c max ) 2 m - n n ( m - 1 ) (c_{\max})^{2}\geq\frac{m-n}{n(m-1)}
  43. k = 1 k=1
  44. k = 1 k=1
  45. G G
  46. { x 1 , , x m } \{x_{1},\ldots,x_{m}\}
  47. n \mathbb{C}^{n}
  48. | x i , x j | |\langle x_{i},x_{j}\rangle|
  49. i j i\neq j
  50. { x i } \{x_{i}\}
  51. n \mathbb{C}^{n}

Welding_defect.html

  1. E α Δ T E\alpha\Delta T

Welfare_cost_of_business_cycles.html

  1. λ = 1 2 σ 2 θ \lambda=\frac{1}{2}\sigma^{2}\theta
  2. λ \lambda
  3. σ \sigma
  4. θ \theta
  5. σ \sigma
  6. σ = .032 \sigma=.032
  7. θ \theta
  8. θ \theta
  9. θ = 1 \theta=1
  10. λ = 1 2 ( .032 ) 2 = .0005 \lambda=\frac{1}{2}(.032)^{2}=.0005
  11. θ \theta
  12. λ = 1 2 ( .032 ) 2 4 = .002 \lambda=\frac{1}{2}(.032)^{2}4=.002
  13. r = ρ + θ g r=\rho+\theta g
  14. r r
  15. ρ \rho
  16. g g
  17. r r
  18. θ \theta
  19. ρ = 0 \rho=0
  20. θ M a x = r / g = .05 / .02 = 2.5 \theta^{Max}=r/g=.05/.02=2.5
  21. λ = 1 2 ( .032 ) 2 2.5 = .0013 \lambda=\frac{1}{2}(.032)^{2}2.5=.0013
  22. U U
  23. β \beta
  24. u ( . ) u(.)
  25. c t c_{t}
  26. U = t = 0 β t u ( c t ) U=\sum_{t=0}^{\infty}\beta^{t}u(c_{t})
  27. c t c e r t = A e g t c_{t}^{cert}=Ae^{gt}
  28. A A
  29. g g
  30. c t v o l = ( 1 + λ ) A e g t e - 1 2 σ 2 ϵ t c_{t}^{vol}=(1+\lambda)Ae^{gt}e^{-\frac{1}{2}\sigma^{2}}\epsilon_{t}
  31. σ \sigma
  32. ϵ \epsilon
  33. l n ( ϵ t ) ln(\epsilon_{t})
  34. e - 1 2 σ 2 ϵ t e^{-\frac{1}{2}\sigma^{2}}\epsilon_{t}
  35. λ \lambda
  36. λ \lambda
  37. t = 0 β t u ( c t c e r t ) = t = 0 β t u ( c t v o l ) \sum_{t=0}^{\infty}\beta^{t}u(c_{t}^{cert})=\sum_{t=0}^{\infty}\beta^{t}u(c_{t% }^{vol})
  38. λ \lambda
  39. u ( c t ) = c t 1 - θ - 1 1 - θ u(c_{t})=\frac{c_{t}^{1-\theta}-1}{1-\theta}
  40. λ = 1 2 σ 2 θ \lambda=\frac{1}{2}\sigma^{2}\theta
  41. u ( c t ) = l n ( c t ) u(c_{t})=ln(c_{t})
  42. θ = 1 \theta=1
  43. λ = .5 σ 2 \lambda=.5\sigma^{2}
  44. θ \theta
  45. λ \lambda

Well_equidistributed_long-period_linear.html

  1. F 2 F_{2}

Werner_state.html

  1. U U U\otimes U
  2. ρ = ( U U ) ρ ( U U ) \rho=(U\otimes U)\rho(U^{\dagger}\otimes U^{\dagger})
  3. ρ = p sym 2 d 2 + d P sym + ( 1 - p sym ) 2 d 2 - d P as , \rho=p\text{sym}\frac{2}{d^{2}+d}P\text{sym}+(1-p\text{sym})\frac{2}{d^{2}-d}P% \text{as},
  4. P sym = 1 2 ( 1 + P ) , P\text{sym}=\frac{1}{2}(1+P),
  5. P as = 1 2 ( 1 - P ) , P\text{as}=\frac{1}{2}(1-P),
  6. P = i j | i j | | j i | P=\sum_{ij}|i\rangle\langle j|\otimes|j\rangle\langle i|
  7. 1 / 2 {1}/{2}
  8. ρ = 1 d 2 - d α ( 1 - α P ) , \rho=\frac{1}{d^{2}-d\alpha}(1-\alpha P),
  9. α = ( ( 1 - 2 p sym ) d + 1 ) / ( 1 - 2 p sym + d ) . \alpha=((1-2p\text{sym})d+1)/(1-2p\text{sym}+d).
  10. U U U U\otimes U\otimes...\otimes U

Where's_George?.html

  1. 100 × [ ln ( bills entered ) + ln ( hits + 1 ) ] × [ 1 - ( days of inactivity / 100 ) ] 100\times\left[\sqrt{\ln({\rm bills\ entered})}+\ln({\rm hits}+1)\right]\times% [1-({\rm days\ of\ inactivity}/100)]

Widom_insertion_method.html

  1. 𝐁 i = ρ i a i = exp ( - ψ i k B T ) \mathbf{B}_{i}=\frac{\rho_{i}}{a_{i}}=\left\langle\exp\left(-\frac{\psi_{i}}{k% _{B}T}\right)\right\rangle
  2. 𝐁 i \mathbf{B}_{i}
  3. ρ i \rho_{i}
  4. i i
  5. a i a_{i}
  6. i i
  7. k B k_{B}
  8. T T
  9. ψ \psi
  10. i i
  11. μ i = - k B T ln ( 𝐁 i ρ i λ 3 ) \mu_{i}=-k_{B}T\ln\left(\frac{\mathbf{B}_{i}}{\rho_{i}\lambda^{3}}\right)
  12. Z = P ρ k B T = 1 - ln 𝐁 + 1 ρ 0 ρ ln 𝐁 d ρ Z=\frac{P}{\rho k_{B}T}=1-\ln\mathbf{B}+\frac{1}{\rho}\int\limits_{0}^{\rho}% \ln\mathbf{B}\,d\rho^{\prime}
  13. Z Z
  14. ρ \rho
  15. ln 𝐁 \ln\mathbf{B}
  16. ln 𝐁 = i x i ln 𝐁 i \ln\mathbf{B}=\sum_{i}{x_{i}\ln\mathbf{B}_{i}}
  17. 𝐁 i = 𝐏 i n s , i exp ( - ψ i k B T ) \mathbf{B}_{i}=\mathbf{P}_{ins,i}\left\langle\exp\left(-\frac{\psi_{i}}{k_{B}T% }\right)\right\rangle
  18. 𝐏 i n s , i \mathbf{P}_{ins,i}
  19. i i
  20. 𝐁 i = 𝐏 i n s , i exp ( - ψ i k B T ) \mathbf{B}_{i}=\mathbf{P}_{ins,i}\exp\left(-\frac{\left\langle\psi_{i}\right% \rangle}{k_{B}T}\right)

Wiles'_proof_of_Fermat's_Last_Theorem.html

  1. a n + b n = c n a^{n}+b^{n}=c^{n}\!
  2. y 2 = x ( x - a n ) ( x + b n ) . y^{2}=x(x-a^{n})(x+b^{n}).\,
  3. R n T n . R_{n}\rightarrow T_{n}.
  4. E ( 𝐐 ¯ ) E(\bar{\mathbf{Q}})
  5. l n l^{n}
  6. Gal ( 𝐐 ¯ / 𝐐 ) \mathrm{Gal}(\bar{\mathbf{Q}}/\mathbf{Q})
  7. GL 2 ( 𝐙 / l n 𝐙 ) \mathrm{GL}_{2}(\mathbf{Z}/l^{n}\mathbf{Z})
  8. mod l n \mod l^{n}
  9. E ( 𝐐 ¯ ) E(\bar{\mathbf{Q}})
  10. 𝐐 ¯ \bar{\mathbf{Q}}
  11. Gal ( 𝐐 ¯ / 𝐐 ) \mathrm{Gal}(\bar{\mathbf{Q}}/\mathbf{Q})
  12. l n x = 0 l^{n}x=0
  13. ( 𝐙 / l n 𝐙 ) 2 (\mathbf{Z}/l^{n}\mathbf{Z})^{2}
  14. Gal ( 𝐐 ¯ / 𝐐 ) GL 2 ( 𝐙 / l n 𝐙 ) \mathrm{Gal}(\bar{\mathbf{Q}}/\mathbf{Q})\rightarrow\mathrm{GL}_{2}(\mathbf{Z}% /l^{n}\mathbf{Z})
  15. l n l^{n}
  16. mod l n \mod l^{n}
  17. mod l n \mod l^{n}
  18. mod l n + 1 \mod l^{n+1}
  19. mod l n \mod l^{n}

Wilkie's_theorem.html

  1. x m + 1 x n f 1 ( x 1 , , x n , e x 1 , , e x n ) = = f r ( x 1 , , x n , e x 1 , , e x n ) = 0. \exists x_{m+1}\ldots\exists x_{n}\,f_{1}(x_{1},\ldots,x_{n},e^{x_{1}},\ldots,% e^{x_{n}})=\cdots=f_{r}(x_{1},\ldots,x_{n},e^{x_{1}},\ldots,e^{x_{n}})=0.\,

Wilkinson_matrix.html

  1. [ 3 1 0 0 0 0 0 1 2 1 0 0 0 0 0 1 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 1 0 0 0 0 0 1 2 1 0 0 0 0 0 1 3 ] . \begin{bmatrix}3&1&0&0&0&0&0\\ 1&2&1&0&0&0&0\\ 0&1&1&1&0&0&0\\ 0&0&1&0&1&0&0\\ 0&0&0&1&1&1&0\\ 0&0&0&0&1&2&1\\ 0&0&0&0&0&1&3\\ \end{bmatrix}.

Williams–Landel–Ferry_equation.html

  1. log ( a T ) = - C 1 ( T - T r ) C 2 + ( T - T r ) \log(a_{T})=\frac{-C_{1}(T-T_{\mathrm{r}})}{C_{2}+(T-T_{\mathrm{r}})}
  2. 10 - 2 10^{-2}
  3. 10 2 10^{2}

Willingness_to_accept.html

  1. u ( w 0 + W T A , 1 ) = u ( w 0 , 0 ) . u(w_{0}+WTA,1)=u(w_{0},0).
  2. u ( w 0 - W T P , 0 ) = u ( w 0 , 1 ) . u(w_{0}-WTP,0)=u(w_{0},1).

Windmill_graph.html

  1. i = 0 k - 1 ( x - i ) n . \prod_{i=0}^{k-1}(x-i)^{n}.

Winter's_formula.html

  1. P C O 2 = ( 1.5 × H C O 3 - ) + 8 ± 2 P_{CO_{2}}=(1.5\times HCO_{3}^{-})+8\pm 2

Wirtinger's_representation_and_projection_theorem.html

  1. H 2 \left.\right.H_{2}
  2. L 2 \left.\right.L^{2}
  3. { z : | z | < 1 } \left.\right.\{z:|z|<1\}
  4. L 2 \left.\right.L^{2}
  5. H 2 \left.\right.H_{2}
  6. F ( z ) \left.\right.\left.F(z)\right.
  7. L 2 \left.\right.L^{2}
  8. | z | < 1 \left.\right.|z|<1
  9. | z | < 1 | F ( z ) | 2 d S < + , \iint_{|z|<1}|F(z)|^{2}\,dS<+\infty,
  10. d S \left.\right.dS
  11. f ( z ) \left.\right.f(z)
  12. H 2 L 2 H_{2}\subset L^{2}
  13. | z | < 1 | F ( z ) - f ( z ) | 2 d S \iint_{|z|<1}|F(z)-f(z)|^{2}\,dS
  14. f ( z ) = 1 π | ζ | < 1 F ( ζ ) d S ( 1 - ζ ¯ z ) 2 , | z | < 1. f(z)=\frac{1}{\pi}\iint_{|\zeta|<1}F(\zeta)\frac{dS}{(1-\overline{\zeta}z)^{2}% },\quad|z|<1.
  15. L 2 \left.\right.L^{2}
  16. H 2 \left.\right.H_{2}
  17. F ( ζ ) \left.\right.F(\zeta)
  18. f ( ζ ) \left.\right.f(\zeta)
  19. f ( z ) H 2 f(z)\in H_{2}
  20. A 0 2 \left.\right.A^{2}_{0}
  21. H 2 \left.\right.H_{2}
  22. A α 2 \left.\right.A^{2}_{\alpha}
  23. f ( z ) \left.\right.f(z)
  24. | z | < 1 \left.\right.|z|<1
  25. f A α 2 = { 1 π | z | < 1 | f ( z ) | 2 ( 1 - | z | 2 ) α - 1 d S } 1 / 2 < + for some α ( 0 , + ) , \|f\|_{A^{2}_{\alpha}}=\left\{\frac{1}{\pi}\iint_{|z|<1}|f(z)|^{2}(1-|z|^{2})^% {\alpha-1}\,dS\right\}^{1/2}<+\infty\,\text{ for some }\alpha\in(0,+\infty),
  26. A ω 2 \left.\right.A^{2}_{\omega}
  27. | z | < 1 \left.\right.|z|<1
  28. | z | < 1 \left.\right.|z|<1

Witness_(mathematics).html

  1. x ϕ ( x ) \exists x\,\phi(x)
  2. x ϕ ( x ) \exists x\,\phi(x)
  3. ϕ \phi
  4. x y ϕ ( x , y ) \forall x\exists y\,\phi(x,y)
  5. f x ϕ ( x , f ( x ) ) \exists f\forall x\,\phi(x,f(x))

Witten_zeta_function.html

  1. R 1 dim ( R ) s \sum_{R}\frac{1}{\dim(R)^{s}}
  2. ζ W ( s 1 , , s n ) = m 1 , , m r > 0 α Δ + 1 ( α , m 1 λ 1 + + m r λ r ) s α , \zeta_{W}(s_{1},\dots,s_{n})=\sum_{m_{1},\dots,m_{r}>0}\prod_{\alpha\in\Delta^% {+}}\frac{1}{(\alpha^{,}m_{1}\lambda_{1}+\cdots+m_{r}\lambda_{r})^{s_{\alpha}}},

WOBA.html

  1. w O B A = ( 0.72 * N I B B ) + ( 0.75 * H B P ) + ( 0.90 * 1 B ) + ( 0.92 * R B O E ) + ( 1.24 * 2 B ) + ( 1.56 * 3 B ) + ( 1.95 * H R ) P A wOBA=\frac{(0.72*NIBB)+(0.75*HBP)+(0.90*\mathit{1}B)+(0.92*RBOE)+(1.24*\mathit% {2}B)+(1.56*\mathit{3}B)+(1.95*HR)}{PA}

Wolff_algorithm.html

  1. N 2 N^{2}
  2. N 2 + z N^{2+z}

Wrapped_Cauchy_distribution.html

  1. f W C ( θ ; μ , γ ) = n = - γ π ( γ 2 + ( θ - μ + 2 π n ) 2 ) f_{WC}(\theta;\mu,\gamma)=\sum_{n=-\infty}^{\infty}\frac{\gamma}{\pi(\gamma^{2% }+(\theta-\mu+2\pi n)^{2})}
  2. γ \gamma
  3. μ \mu
  4. f W C ( θ ; μ , γ ) = 1 2 π n = - e i n ( θ - μ ) - | n | γ = 1 2 π sinh γ cosh γ - cos ( θ - μ ) f_{WC}(\theta;\mu,\gamma)=\frac{1}{2\pi}\sum_{n=-\infty}^{\infty}e^{in(\theta-% \mu)-|n|\gamma}=\frac{1}{2\pi}\,\,\frac{\sinh\gamma}{\cosh\gamma-\cos(\theta-% \mu)}
  5. z = e i θ z=e^{i\theta}
  6. z n = Γ e i n θ f W C ( θ ; μ , γ ) d θ = e i n μ - | n | γ . \langle z^{n}\rangle=\int_{\Gamma}e^{in\theta}\,f_{WC}(\theta;\mu,\gamma)\,d% \theta=e^{in\mu-|n|\gamma}.
  7. Γ \Gamma\,
  8. 2 π 2\pi
  9. z = e i μ - γ \langle z\rangle=e^{i\mu-\gamma}
  10. θ = Arg z = μ \langle\theta\rangle=\mathrm{Arg}\langle z\rangle=\mu
  11. R = | z | = e - γ R=|\langle z\rangle|=e^{-\gamma}
  12. z n = e i θ n z_{n}=e^{i\theta_{n}}
  13. z ¯ \overline{z}
  14. z ¯ = 1 N n = 1 N z n \overline{z}=\frac{1}{N}\sum_{n=1}^{N}z_{n}
  15. z ¯ = e i μ - γ \langle\overline{z}\rangle=e^{i\mu-\gamma}
  16. z ¯ \overline{z}
  17. μ \mu
  18. [ - π , π ) [-\pi,\pi)
  19. ( z ¯ ) (\overline{z})
  20. μ \mu
  21. z n z_{n}
  22. R ¯ 2 \overline{R}^{2}
  23. R ¯ 2 = z ¯ z * ¯ = ( 1 N n = 1 N cos θ n ) 2 + ( 1 N n = 1 N sin θ n ) 2 \overline{R}^{2}=\overline{z}\,\overline{z^{*}}=\left(\frac{1}{N}\sum_{n=1}^{N% }\cos\theta_{n}\right)^{2}+\left(\frac{1}{N}\sum_{n=1}^{N}\sin\theta_{n}\right% )^{2}
  24. R ¯ 2 = 1 N + N - 1 N e - 2 γ . \langle\overline{R}^{2}\rangle=\frac{1}{N}+\frac{N-1}{N}e^{-2\gamma}.
  25. R e 2 = N N - 1 ( R ¯ 2 - 1 N ) R_{e}^{2}=\frac{N}{N-1}\left(\overline{R}^{2}-\frac{1}{N}\right)
  26. e - 2 γ e^{-2\gamma}
  27. ln ( 1 / R e 2 ) / 2 \ln(1/R_{e}^{2})/2
  28. γ \gamma
  29. H = - Γ f W C ( θ ; μ , γ ) ln ( f W C ( θ ; μ , γ ) ) d θ H=-\int_{\Gamma}f_{WC}(\theta;\mu,\gamma)\,\ln(f_{WC}(\theta;\mu,\gamma))\,d\theta
  30. Γ \Gamma
  31. 2 π 2\pi
  32. θ \theta\,
  33. ln ( f W C ( θ ; μ , γ ) ) = c 0 + 2 m = 1 c m cos ( m θ ) \ln(f_{WC}(\theta;\mu,\gamma))=c_{0}+2\sum_{m=1}^{\infty}c_{m}\cos(m\theta)
  34. c m = 1 2 π Γ ln ( sinh γ 2 π ( cosh γ - cos θ ) ) cos ( m θ ) d θ c_{m}=\frac{1}{2\pi}\int_{\Gamma}\ln\left(\frac{\sinh\gamma}{2\pi(\cosh\gamma-% \cos\theta)}\right)\cos(m\theta)\,d\theta
  35. c 0 = ln ( 1 - e - 2 γ 2 π ) c_{0}=\ln\left(\frac{1-e^{-2\gamma}}{2\pi}\right)
  36. c m = e - m γ m for m > 0 c_{m}=\frac{e^{-m\gamma}}{m}\qquad\mathrm{for}\,m>0
  37. f W C ( θ ; μ , γ ) = 1 2 π ( 1 + 2 n = 1 ϕ n cos ( n θ ) ) f_{WC}(\theta;\mu,\gamma)=\frac{1}{2\pi}\left(1+2\sum_{n=1}^{\infty}\phi_{n}% \cos(n\theta)\right)
  38. ϕ n = e - | n | γ \phi_{n}=e^{-|n|\gamma}
  39. H = - c 0 - 2 m = 1 ϕ m c m = - ln ( 1 - e - 2 γ 2 π ) - 2 m = 1 e - 2 n γ n H=-c_{0}-2\sum_{m=1}^{\infty}\phi_{m}c_{m}=-\ln\left(\frac{1-e^{-2\gamma}}{2% \pi}\right)-2\sum_{m=1}^{\infty}\frac{e^{-2n\gamma}}{n}
  40. ( 1 - e - 2 γ ) (1-e^{-2\gamma})
  41. H = ln ( 2 π ( 1 - e - 2 γ ) ) H=\ln(2\pi(1-e^{-2\gamma}))\,

Wrapped_distribution.html

  1. p ( ϕ ) p(\phi)
  2. θ = ϕ mod 2 π \theta=\phi\mod 2\pi
  3. 2 π 2\pi
  4. p w ( θ ) = k = - p ( θ + 2 π k ) . p_{w}(\theta)=\sum_{k=-\infty}^{\infty}{p(\theta+2\pi k)}.
  5. 2 π 2\pi
  6. ( - π < θ π ) (-\pi<\theta\leq\pi)
  7. ln ( e i θ ) = arg ( e i θ ) = θ \ln(e^{i\theta})=\arg(e^{i\theta})=\theta
  8. ϕ \phi
  9. p ( ϕ ) p(\phi)
  10. θ \theta
  11. 2 π 2\pi
  12. [ 0 , 2 π ) [0,2\pi)
  13. ϕ \phi
  14. ϕ + 2 π a \phi+2\pi a
  15. θ = ϕ + 2 π a . \theta=\phi+2\pi a.
  16. f ( θ ) = - p ( ϕ ) f ( ϕ + 2 π a ) d ϕ . \langle f(\theta)\rangle=\int_{-\infty}^{\infty}p(\phi)f(\phi+2\pi a)d\phi.
  17. 2 π 2\pi
  18. 2 π 2\pi
  19. f ( θ ) = k = - 2 π k 2 π ( k + 1 ) p ( ϕ ) f ( ϕ + 2 π a ) d ϕ . \langle f(\theta)\rangle=\sum_{k=-\infty}^{\infty}\int_{2\pi k}^{2\pi(k+1)}p(% \phi)f(\phi+2\pi a)d\phi.
  20. θ = ϕ - 2 π k \theta^{\prime}=\phi-2\pi k
  21. f ( θ ) = 0 2 π p w ( θ ) f ( θ + 2 π a ) d θ \langle f(\theta)\rangle=\int_{0}^{2\pi}p_{w}(\theta^{\prime})f(\theta^{\prime% }+2\pi a^{\prime})d\theta^{\prime}
  22. p w ( θ ) p_{w}(\theta^{\prime})
  23. f ( θ ) f(\theta)
  24. z = e i θ z=e^{i\theta}
  25. ϕ \phi
  26. z = e i θ = e i ϕ . z=e^{i\theta}=e^{i\phi}.
  27. f ( z ) = 0 2 π p w ( θ ) f ( e i θ ) d θ \langle f(z)\rangle=\int_{0}^{2\pi}p_{w}(\theta^{\prime})f(e^{i\theta^{\prime}% })d\theta^{\prime}
  28. θ \theta
  29. f ( z ) = p w ( z ) f ( z ) d z \langle f(z)\rangle=\oint p_{w}(z)f(z)\,dz
  30. p w ( z ) p_{w}(z)
  31. p w ( θ ) d θ = p w ( z ) d z p_{w}(\theta)\,d\theta=p_{w}(z)\,dz
  32. F F
  33. p w ( θ ) = k 1 , , k F = - p ( θ + 2 π k 1 𝐞 1 + + 2 π k F 𝐞 F ) p_{w}(\vec{\theta})=\sum_{k_{1},...,k_{F}=-\infty}^{\infty}{p(\vec{\theta}+2% \pi k_{1}\mathbf{e}_{1}+\dots+2\pi k_{F}\mathbf{e}_{F})}
  34. 𝐞 k = ( 0 , , 0 , 1 , 0 , , 0 ) 𝖳 \mathbf{e}_{k}=(0,\dots,0,1,0,\dots,0)^{\mathsf{T}}
  35. k k
  36. Δ 2 π ( θ ) = k = - δ ( θ + 2 π k ) . \Delta_{2\pi}(\theta)=\sum_{k=-\infty}^{\infty}{\delta(\theta+2\pi k)}.
  37. p w ( θ ) = k = - - p ( θ ) δ ( θ - θ + 2 π k ) d θ . p_{w}(\theta)=\sum_{k=-\infty}^{\infty}\int_{-\infty}^{\infty}p(\theta^{\prime% })\delta(\theta-\theta^{\prime}+2\pi k)\,d\theta^{\prime}.
  38. p w ( θ ) = - p ( θ ) Δ 2 π ( θ - θ ) d θ . p_{w}(\theta)=\int_{-\infty}^{\infty}p(\theta^{\prime})\Delta_{2\pi}(\theta-% \theta^{\prime})\,d\theta^{\prime}.
  39. p w ( θ ) = 1 2 π - p ( θ ) n = - e i n ( θ - θ ) d θ p_{w}(\theta)=\frac{1}{2\pi}\,\int_{-\infty}^{\infty}p(\theta^{\prime})\sum_{n% =-\infty}^{\infty}e^{in(\theta-\theta^{\prime})}\,d\theta^{\prime}
  40. p w ( θ ) = 1 2 π n = - - p ( θ ) e i n ( θ - θ ) d θ p_{w}(\theta)=\frac{1}{2\pi}\,\sum_{n=-\infty}^{\infty}\int_{-\infty}^{\infty}% p(\theta^{\prime})e^{in(\theta-\theta^{\prime})}\,d\theta^{\prime}
  41. ϕ ( s ) \phi(s)
  42. p ( θ ) p(\theta)
  43. p w ( θ ) = 1 2 π n = - ϕ ( - n ) e i n θ = 1 2 π n = - ϕ ( - n ) z n p_{w}(\theta)=\frac{1}{2\pi}\,\sum_{n=-\infty}^{\infty}\phi(-n)\,e^{in\theta}=% \frac{1}{2\pi}\,\sum_{n=-\infty}^{\infty}\phi(-n)\,z^{n}
  44. p w ( z ) = 1 2 π i n = - ϕ ( - n ) z n - 1 . p_{w}(z)=\frac{1}{2\pi i}\,\sum_{n=-\infty}^{\infty}\phi(-n)\,z^{n-1}.
  45. ϕ ( m ) \phi(m)
  46. p w ( z ) p_{w}(z)
  47. z m = p w ( z ) z m d z . \langle z^{m}\rangle=\oint p_{w}(z)z^{m}\,dz.
  48. p w ( z ) p_{w}(z)
  49. z m = 1 2 π i n = - ϕ ( - n ) z m + n - 1 d z . \langle z^{m}\rangle=\frac{1}{2\pi i}\sum_{n=-\infty}^{\infty}\phi(-n)\oint z^% {m+n-1}\,dz.
  50. z m + n - 1 d z = 2 π i δ m + n \oint z^{m+n-1}\,dz=2\pi i\delta_{m+n}
  51. δ k \delta_{k}
  52. z m = ϕ ( m ) . \langle z^{m}\rangle=\phi(m).
  53. f w ( θ ) f_{w}(\theta)
  54. H = - Γ f w ( θ ) ln ( f w ( θ ) ) d θ H=-\int_{\Gamma}f_{w}(\theta)\,\ln(f_{w}(\theta))\,d\theta
  55. Γ \Gamma
  56. 2 π 2\pi
  57. ϕ ( n ) \phi(n)
  58. f w ( θ ) = 1 2 π n = - ϕ n e - i n θ f_{w}(\theta)=\frac{1}{2\pi}\sum_{n=-\infty}^{\infty}\phi_{n}e^{-in\theta}
  59. ln ( f w ( θ ) ) = m = - c m e i m θ \ln(f_{w}(\theta))=\sum_{m=-\infty}^{\infty}c_{m}e^{im\theta}
  60. c m = 1 2 π Γ ln ( f w ( θ ) ) e - i m θ d θ c_{m}=\frac{1}{2\pi}\int_{\Gamma}\ln(f_{w}(\theta))e^{-im\theta}\,d\theta
  61. H = - 1 2 π m = - n = - c m ϕ n Γ e i ( m - n ) θ d θ H=-\frac{1}{2\pi}\sum_{m=-\infty}^{\infty}\sum_{n=-\infty}^{\infty}c_{m}\phi_{% n}\int_{\Gamma}e^{i(m-n)\theta}\,d\theta
  62. H = - n = - c n ϕ n H=-\sum_{n=-\infty}^{\infty}c_{n}\phi_{n}
  63. c - m = c m c_{-m}=c_{m}
  64. ln ( f w ( θ ) ) = c 0 + 2 m = 1 c m cos ( m θ ) \ln(f_{w}(\theta))=c_{0}+2\sum_{m=1}^{\infty}c_{m}\cos(m\theta)
  65. c m = 1 2 π Γ ln ( f w ( θ ) ) cos ( m θ ) d θ c_{m}=\frac{1}{2\pi}\int_{\Gamma}\ln(f_{w}(\theta))\cos(m\theta)\,d\theta
  66. ϕ 0 = 1 \phi_{0}=1
  67. H = - c 0 - 2 n = 1 c n ϕ n H=-c_{0}-2\sum_{n=1}^{\infty}c_{n}\phi_{n}

Wrapped_Lévy_distribution.html

  1. f W L ( θ ; μ , c ) = n = - c 2 π e - c / 2 ( θ + 2 π n - μ ) ( θ + 2 π n - μ ) 3 / 2 f_{WL}(\theta;\mu,c)=\sum_{n=-\infty}^{\infty}\sqrt{\frac{c}{2\pi}}\,\frac{e^{% -c/2(\theta+2\pi n-\mu)}}{(\theta+2\pi n-\mu)^{3/2}}
  2. θ + 2 π n - μ 0 \theta+2\pi n-\mu\leq 0
  3. c c
  4. μ \mu
  5. f W L ( θ ; μ , c ) = 1 2 π n = - e - i n ( θ - μ ) - c | n | ( 1 - i sgn n ) = 1 2 π ( 1 + 2 n = 1 e - c n cos ( n ( θ - μ ) - c n ) ) f_{WL}(\theta;\mu,c)=\frac{1}{2\pi}\sum_{n=-\infty}^{\infty}e^{-in(\theta-\mu)% -\sqrt{c|n|}\,(1-i\operatorname{sgn}{n})}=\frac{1}{2\pi}\left(1+2\sum_{n=1}^{% \infty}e^{-\sqrt{cn}}\cos\left(n(\theta-\mu)-\sqrt{cn}\,\right)\right)
  6. z = e i θ z=e^{i\theta}
  7. z n = Γ e i n θ f W L ( θ ; μ , c ) d θ = e i n μ - c | n | ( 1 - i sgn ( n ) ) . \langle z^{n}\rangle=\int_{\Gamma}e^{in\theta}\,f_{WL}(\theta;\mu,c)\,d\theta=% e^{in\mu-\sqrt{c|n|}\,(1-i\operatorname{sgn}(n))}.
  8. Γ \Gamma\,
  9. 2 π 2\pi
  10. z = e i μ - c ( 1 - i ) \langle z\rangle=e^{i\mu-\sqrt{c}(1-i)}
  11. θ μ = Arg z = μ + c \theta_{\mu}=\mathrm{Arg}\langle z\rangle=\mu+\sqrt{c}
  12. R = | z | = e - c R=|\langle z\rangle|=e^{-\sqrt{c}}

X_(charge).html

  1. X = 5 ( B - L ) - 2 Y W X=5(B-L)-2Y_{W}\,

Yahalom_(protocol).html

  1. K A S K_{AS}
  2. K B S K_{BS}
  3. N A N_{A}
  4. N B N_{B}
  5. K A B K_{AB}
  6. A B : A , N A A\rightarrow B:A,N_{A}
  7. B S : B , { A , N A , N B } K B S B\rightarrow S:B,\{A,N_{A},N_{B}\}_{K_{BS}}
  8. K B S K_{BS}
  9. S A : { B , K A B , N A , N B } K A S , { A , K A B } K B S S\rightarrow A:\{B,K_{AB},N_{A},N_{B}\}_{K_{AS}},\{A,K_{AB}\}_{K_{BS}}
  10. K A B K_{AB}
  11. A B : { A , K A B } K B S , { N B } K A B A\rightarrow B:\{A,K_{AB}\}_{K_{BS}},\{N_{B}\}_{K_{AB}}
  12. N A N_{A}
  13. N B N_{B}

Yang–Mills–Higgs_equations.html

  1. D A * F A + [ Φ , D A Φ ] = 0 , D_{A}*F_{A}+[\Phi,D_{A}\Phi]=0,
  2. D A * D A Φ = 0 D_{A}*D_{A}\Phi=0
  3. lim | x | | Φ | ( x ) = 1. \lim_{|x|\rightarrow\infty}|\Phi|(x)=1.

Yao's_Millionaires'_Problem.html

  1. a a
  2. b b
  3. a b a\geq b
  4. a a
  5. b b
  6. S 0 S_{0}
  7. S 1 S_{1}
  8. i { 0 , 1 } i\in\{0,1\}
  9. S i S_{i}
  10. S ( 1 - i ) S_{(1-i)}
  11. i i
  12. a a
  13. b b
  14. d d
  15. d N d\in N
  16. K K
  17. d × 2 d\times 2
  18. k k
  19. k k
  20. u u
  21. v v
  22. 0 u < 2 k 0\leq u<2k
  23. v k v\leq k
  24. K i j l K_{ijl}
  25. l l
  26. K i j K_{ij}
  27. l = 0 l=0
  28. a i a_{i}
  29. i i
  30. a a
  31. i i
  32. 1 i d 1\leq i\leq d
  33. j v j\geq v
  34. K i 1 j K_{i1j}
  35. K i 2 j K_{i2j}
  36. a i = 1 a_{i}=1
  37. l = 1 l=1
  38. l = 2 l=2
  39. j , 0 j 2 i - 1 j,0\leq j\leq 2\cdot i-1
  40. K i l j K_{ilj}
  41. m = 2 i m=2\cdot i
  42. K i l ( m + 1 ) = 1 K_{il(m+1)}=1
  43. K i l m K_{ilm}
  44. a i a_{i}
  45. i , 1 i < d i,1\leq i<d
  46. S i S_{i}
  47. k k
  48. S d S_{d}
  49. k k
  50. S d ( k - 1 ) = 1 j = 1 d - 1 S j ( k - 1 ) j = 1 d K j 1 ( k - 1 ) S_{d(k-1)}=1\oplus\bigoplus_{j=1}^{d-1}S_{j(k-1)}\oplus\bigoplus_{j=1}^{d}K_{j% 1(k-1)}
  51. S d ( k - 2 ) = 1 j = 1 d - 1 S j ( k - 2 ) j = 1 d K j 1 ( k - 2 ) S_{d(k-2)}=1\oplus\bigoplus_{j=1}^{d-1}S_{j(k-2)}\oplus\bigoplus_{j=1}^{d}K_{j% 1(k-2)}
  52. \bigoplus
  53. l = 1 , 2 l=1,2
  54. K i j = r o t ( K i l S i , u ) K^{\prime}_{ij}=rot(K_{il}\oplus S_{i},u)
  55. r o t ( x , t ) rot(x,t)
  56. x x
  57. t t
  58. i i
  59. 0 i d 0\leq i\leq d
  60. K i l K^{\prime}_{il}
  61. l = b i + 1 l=b_{i}+1
  62. b i b_{i}
  63. i i
  64. b b
  65. N = r o l ( j = 1 d S j , u ) N=rol(\bigoplus_{j=1}^{d}S_{j},u)
  66. N N
  67. c c
  68. c c
  69. c c
  70. a b a\geq b
  71. a < b a<b
  72. N i = 1 d K i ( b i + 1 ) = r o l ( i = 1 d K i ( b i + 1 ) , u ) N\oplus\bigoplus_{i=1}^{d}K^{\prime}_{i(b_{i}+1)}=rol(\bigoplus_{i=1}^{d}K_{i(% b_{i}+1)},u)
  73. c = i = 1 d K i ( b i + 1 ) c=\bigoplus_{i=1}^{d}K_{i(b_{i}+1)}
  74. K i 1 , K i 2 K_{i1},K_{i2}
  75. K i 1 K_{i1}
  76. a i = 1 a_{i}=1
  77. K i 2 K_{i2}
  78. i > j i>j
  79. K i l K_{il}
  80. K i l K j l K_{il}\oplus K_{jl}
  81. A i l A_{il}
  82. a i > b i a_{i}>b_{i}
  83. a i > b i a_{i}>b_{i}
  84. a b a\geq b
  85. a i < b i a_{i}<b_{i}
  86. r o l ( K i ( 1 + b i ) S i , u ) rol(K_{i(1+b_{i})}\oplus S_{i},u)
  87. i i
  88. S i S_{i}
  89. O ( d 2 ) O(d^{2})
  90. O ( d 2 ) O(d^{2})
  91. O ( d 2 ) O(d^{2})
  92. O ( d 2 ) . O(d^{2}).

Yao's_test.html

  1. P P
  2. S = { S k } k S=\{S_{k}\}_{k}
  3. S k S_{k}
  4. P ( k ) P(k)
  5. k k
  6. μ k \mu_{k}
  7. S k S_{k}
  8. P C P_{C}
  9. C = { C k } C=\{C_{k}\}
  10. P C ( k ) P_{C}(k)
  11. p k , S C p_{k,S}^{C}
  12. s s
  13. S k S_{k}
  14. μ ( s ) \mu(s)
  15. C k ( s ) = 1 C_{k}(s)=1
  16. p k , S C = 𝒫 [ C k ( s ) = 1 | s S k with probability μ k ( s ) ] p_{k,S}^{C}={\mathcal{P}}[C_{k}(s)=1|s\in S_{k}\,\text{ with probability }\mu_% {k}(s)]
  17. p k , U C p_{k,U}^{C}
  18. C k ( s ) = 1 C_{k}(s)=1
  19. s s
  20. P ( k ) P(k)
  21. { 0 , 1 } P ( k ) \{0,1\}^{P(k)}
  22. S S
  23. C C
  24. k k
  25. Q Q
  26. | p k , S C - p k , U C | < 1 Q ( k ) |p_{k,S}^{C}-p_{k,U}^{C}|<\frac{1}{Q(k)}

Yaw_(rotation).html

  1. θ \theta
  2. ψ \psi
  3. ω \omega
  4. d ω d t = 2 k ( a - b ) I β - 2 k ( a 2 + b 2 ) V I ω \frac{d\omega}{dt}=2k\frac{(a-b)}{I}\beta-2k\frac{(a^{2}+b^{2})}{VI}\omega
  5. d β d t = - 4 k M V β + ( 1 - 2 k ) ( b - a ) M V 2 ω \frac{d\beta}{dt}=-\frac{4k}{MV}\beta+(1-2k)\frac{(b-a)}{MV^{2}}\omega
  6. d β d t \frac{d\beta}{dt}
  7. β \beta
  8. ( b > a ) (b>a)
  9. V 2 = 2 k ( a + b ) 2 M ( a - b ) V^{2}=\frac{2k(a+b)^{2}}{M(a-b)}

Year-on-Year_Inflation-Indexed_Swap.html

  1. T i T_{i}
  2. N ϕ i K N{\phi_{i}}K
  3. N ψ i [ I ( T i ) I ( T i - 1 ) - 1 ] N{\psi_{i}}[\frac{I(T_{i})}{I(T_{i-1})}-1]
  4. ϕ i \phi_{i}
  5. ψ i \psi_{i}
  6. T 0 T_{0}
  7. T i T_{i}
  8. T M T_{M}
  9. I ( T 0 ) I(T_{0})
  10. T 0 T_{0}
  11. I ( T i ) I(T_{i})
  12. T i T_{i}
  13. I ( T M ) I(T_{M})
  14. T M T_{M}

Yigu_yanduan.html

  1. 300 - 3 x 300-3x
  2. ( 300 - 3 x ) * ( 300 - 3 x ) = 900 - 1800 x + 9 x 2 (300-3x)*(300-3x)=900-1800x+9x^{2}
  3. - 3 x 2 - 1800 x + 37200 -3x^{2}-1800x+37200
  4. x + 35 x+35
  5. π 3 \pi\approx 3
  6. 3 ( x + 35 ) 2 = 3 x 2 + 210 x + 3675 3(x+35)^{2}=3x^{2}+210x+3675
  7. 3 x 2 + 210 x + 3675 - 4 x 6000 3x^{2}+210x+3675-4x6000
  8. 3 x 2 + 210 x - 20325 3x^{2}+210x-20325
  9. ( L - W ) 2 (L-W)^{2}
  10. ( L - W ) 2 (L-W)^{2}
  11. L 2 + W 2 L^{2}+W^{2}
  12. 7225 - x 2 7225-x^{2}
  13. 2 ( 7225 - x 2 ) 2(7225-x^{2})
  14. 2 ( 7225 - x 2 ) 2(7225-x^{2})
  15. 3 x 2 + 210 x - 20325 3x^{2}+210x-20325
  16. 5 x 2 + 210 x - 34775 5x^{2}+210x-34775
  17. x 2 + 76 x + 1444 x^{2}+76x+1444
  18. x 2 + 10 x x^{2}+10x
  19. 1.96 x 2 + 19.6 x 1.96x^{2}+19.6x
  20. x 2 + 76 x + 1444 x^{2}+76x+1444
  21. 1.96 x 2 + 19.6 x = 1.96x^{2}+19.6x=
  22. - 0.96 x 2 + 56.4 x + 1444 -0.96x^{2}+56.4x+1444
  23. - 0.96 x 2 + 56.4 x + 1444 -0.96x^{2}+56.4x+1444
  24. - 0.96 x 2 + 56.4 x - 810 -0.96x^{2}+56.4x-810

Yukawa–Tsuno_equation.html

  1. log k X k 0 = ρ ( σ + r ( σ + - σ ) ) \log\frac{k_{X}}{k_{0}}=\rho(\sigma+r(\sigma^{+}-\sigma))
  2. log k X k 0 = ρ ( σ + r ( σ - - σ ) ) \log\frac{k_{X}}{k_{0}}=\rho(\sigma+r(\sigma^{-}-\sigma))
  3. σ = σ R + σ I \sigma=\sigma_{R}+\sigma_{I}
  4. G ( R ) = σ + - σ \ G(R)=\sigma^{+}-\sigma
  5. G ( R ) = ( σ I + + σ R + ) - ( σ I + σ R ) G(R)=(\sigma^{+}_{I}+\sigma^{+}_{R})-(\sigma_{I}+\sigma_{R})
  6. G ( R ) = σ R + - σ R G(R)=\sigma^{+}_{R}-\sigma_{R}
  7. σ = σ + r G ( R ) = σ + r ( σ + - σ ) \ \sigma^{^{\prime}}=\sigma+rG(R)=\sigma+r(\sigma^{+}-\sigma)
  8. log ( k X k 0 ) = ρ σ \log(\frac{k_{X}}{k_{0}})=\rho\sigma^{^{\prime}}
  9. log ( k X k 0 ) = ρ ( σ + r ( σ + - σ ) ) \log(\frac{k_{X}}{k_{0}})=\rho(\sigma+r(\sigma^{+}-\sigma))
  10. σ + - σ \sigma^{+}-\sigma
  11. r = 0 r=0
  12. r > 0 r>0
  13. r < 0 r<0
  14. < V A R > r <VAR>r

Z-scan_technique.html

  1. n 2 n_{2}
  2. 0.1 < S < 0.5 0.1<S<0.5
  3. ± z 0 \pm z_{0}
  4. z 0 z_{0}
  5. z 0 = π W 0 2 λ z_{0}=\frac{\pi W_{0}^{2}}{\lambda}
  6. L L
  7. L < z 0 L<z_{0}

Zero-Coupon_Inflation-Indexed_Swap.html

  1. T M T_{M}
  2. N [ ( 1 + K ) M - 1 ] N[(1+K)^{M}-1]
  3. N [ I ( T M ) I ( T 0 ) - 1 ] N[\frac{I(T_{M})}{I(T_{0})}-1]
  4. T 0 T_{0}
  5. T M T_{M}
  6. I ( T 0 ) I(T_{0})
  7. T 0 T_{0}
  8. I ( T M ) I(T_{M})
  9. T M T_{M}

Zeta_function_(operator).html

  1. 𝒪 \mathcal{O}
  2. ζ 𝒪 ( s ) = tr 𝒪 - s \zeta_{\mathcal{O}}(s)=\operatorname{tr}\;\mathcal{O}^{-s}
  3. λ i \lambda_{i}
  4. 𝒪 \mathcal{O}
  5. ζ 𝒪 ( s ) = λ i λ i - s \zeta_{\mathcal{O}}(s)=\sum_{\lambda_{i}}\lambda_{i}^{-s}
  6. det 𝒪 := e - ζ 𝒪 ( 0 ) . \det\mathcal{O}:=e^{-\zeta^{\prime}_{\mathcal{O}}(0)}\;.

Zig-zag_product.html

  1. G , H G,H
  2. G H G\circ H
  3. G G
  4. H H
  5. H H
  6. G G
  7. G H G\circ H
  8. G G
  9. H H
  10. G G
  11. D D
  12. [ N ] [N]
  13. Rot G \mathrm{Rot}_{G}
  14. H H
  15. d d
  16. [ D ] [D]
  17. Rot H \mathrm{Rot}_{H}
  18. G H G\circ H
  19. d 2 d^{2}
  20. [ N ] × [ D ] [N]\times[D]
  21. Rot G H \mathrm{Rot}_{G\circ H}
  22. Rot G H ( ( v , a ) , ( i , j ) ) \mathrm{Rot}_{G\circ H}((v,a),(i,j))
  23. ( a , i ) = Rot H ( a , i ) (a^{\prime},i^{\prime})=\mathrm{Rot}_{H}(a,i)
  24. ( w , b ) = Rot G ( v , a ) (w,b^{\prime})=\mathrm{Rot}_{G}(v,a^{\prime})
  25. ( b , j ) = Rot H ( b , j ) (b,j^{\prime})=\mathrm{Rot}_{H}(b^{\prime},j)
  26. ( ( w , b ) , ( j , i ) ) ((w,b),(j^{\prime},i^{\prime}))
  27. G G
  28. d 2 d^{2}
  29. G G
  30. H H
  31. G G
  32. G G
  33. H H
  34. H H
  35. G G
  36. ( N , D , λ ) (N,D,\lambda)
  37. D D
  38. N N
  39. λ \lambda
  40. G 1 G_{1}
  41. ( N 1 , D 1 , λ 1 ) (N_{1},D_{1},\lambda_{1})
  42. G 2 G_{2}
  43. ( D 1 , D 2 , λ 2 ) (D_{1},D_{2},\lambda_{2})
  44. G z H G\circ{z}H
  45. ( N 1 D 1 , D 2 2 , f ( λ 1 , λ 2 ) ) (N_{1}\cdot D_{1},D_{2}^{2},f(\lambda_{1},\lambda_{2}))
  46. f ( λ 1 , λ 2 ) < λ 1 + λ 2 + λ 2 2 f(\lambda_{1},\lambda_{2})<\lambda_{1}+\lambda_{2}+\lambda_{2}^{2}
  47. G H G\circ H
  48. G G
  49. G G
  50. D D
  51. [ N ] [N]
  52. H H
  53. d d
  54. [ D ] [D]
  55. S [ N ] S\subseteq[N]
  56. G G
  57. G | S H = G H | S × D G|_{S}\circ H=G\circ H|_{S\times D}
  58. G | S G|_{S}
  59. G G
  60. S S
  61. S S
  62. G G
  63. S S

Zoltán_Füredi.html

  1. O ( n log n ) O(n\log n)

Zonal_spherical_harmonics.html

  1. Z ( ) ( θ , ϕ ) = P ( cos θ ) Z^{(\ell)}(\theta,\phi)=P_{\ell}(\cos\theta)
  2. Z 𝐱 ( ) ( 𝐲 ) Z^{(\ell)}_{\mathbf{x}}(\mathbf{y})
  3. Z ( ) ( θ , ϕ ) . Z^{(\ell)}(\theta,\phi).
  4. Z 𝐱 ( ) Z^{(\ell)}_{\mathbf{x}}
  5. P P ( 𝐱 ) P\mapsto P(\mathbf{x})
  6. Y ( 𝐱 ) = S n - 1 Z 𝐱 ( ) ( 𝐲 ) Y ( 𝐲 ) d Ω ( y ) Y(\mathbf{x})=\int_{S^{n-1}}Z^{(\ell)}_{\mathbf{x}}(\mathbf{y})Y(\mathbf{y})\,% d\Omega(y)
  7. 1 ω n - 1 1 - r 2 | 𝐱 - r 𝐲 | n = k = 0 r k Z 𝐱 ( k ) ( 𝐲 ) , \frac{1}{\omega_{n-1}}\frac{1-r^{2}}{|\mathbf{x}-r\mathbf{y}|^{n}}=\sum_{k=0}^% {\infty}r^{k}Z^{(k)}_{\mathbf{x}}(\mathbf{y}),
  8. ω n - 1 \omega_{n-1}
  9. 1 | 𝐱 - 𝐲 | n - 2 = k = 0 c n , k | 𝐱 | k | 𝐲 | n + k - 2 Z 𝐱 / | 𝐱 | ( k ) ( 𝐲 / | 𝐲 | ) \frac{1}{|\mathbf{x}-\mathbf{y}|^{n-2}}=\sum_{k=0}^{\infty}c_{n,k}\frac{|% \mathbf{x}|^{k}}{|\mathbf{y}|^{n+k-2}}Z_{\mathbf{x}/|\mathbf{x}|}^{(k)}(% \mathbf{y}/|\mathbf{y}|)
  10. c n , k = 1 ω n - 1 2 k + n - 2 ( n - 2 ) . c_{n,k}=\frac{1}{\omega_{n-1}}\frac{2k+n-2}{(n-2)}.
  11. Z 𝐱 ( ) ( 𝐲 ) = 1 c n , C ( α ) ( 𝐱 𝐲 ) Z^{(\ell)}_{\mathbf{x}}(\mathbf{y})=\frac{1}{c_{n,\ell}}C_{\ell}^{(\alpha)}(% \mathbf{x}\cdot\mathbf{y})
  12. C ( α ) C_{\ell}^{(\alpha)}
  13. Z R 𝐱 ( ) ( R 𝐲 ) = Z 𝐱 ( ) ( 𝐲 ) Z^{(\ell)}_{R\mathbf{x}}(R\mathbf{y})=Z^{(\ell)}_{\mathbf{x}}(\mathbf{y})
  14. Z 𝐱 ( ) ( 𝐲 ) = k = 1 d Y k ( 𝐱 ) Y k ( 𝐲 ) ¯ . Z^{(\ell)}_{\mathbf{x}}(\mathbf{y})=\sum_{k=1}^{d}Y_{k}(\mathbf{x})\overline{Y% _{k}(\mathbf{y})}.
  15. Z 𝐱 ( ) ( 𝐱 ) = ω n - 1 - 1 dim 𝐇 . Z^{(\ell)}_{\mathbf{x}}(\mathbf{x})=\omega_{n-1}^{-1}\dim\mathbf{H}_{\ell}.

Zygosity.html

  1. H o H_{o}
  2. H e H_{e}
  3. H o = i = 1 n ( 1 if a i 1 a i 2 ) n H_{o}=\frac{\sum\limits_{i=1}^{n}{(1\ \textrm{if}\ a_{i1}\neq a_{i2})}}{n}
  4. n n
  5. a i 1 , a i 2 a_{i1},a_{i2}
  6. i i
  7. H e = 1 - i = 1 m ( f i ) 2 H_{e}=1-\sum\limits_{i=1}^{m}{(f_{i})^{2}}
  8. m m
  9. f i f_{i}
  10. i t h i^{th}

±1-sequence.html

  1. x 1 , x 2 , . . \textstyle\langle x_{1},x_{2},..\rangle
  2. | i = 1 k x i d | > C \left|\sum_{i=1}^{k}x_{i\cdot d}\right|>C
  3. x j x_{j}
  4. | j = 1 N - v x j x j + v | 1 \left|\sum_{j=1}^{N-v}x_{j}x_{j+v}\right|\leq 1\,
  5. 1 v < N 1\leq v<N

Łojasiewicz_inequality.html

  1. dist ( x , Z ) α C | f ( x ) | . \operatorname{dist}(x,Z)^{\alpha}\leq C|f(x)|.\,
  2. | f ( x ) - f ( p ) | θ c | f ( x ) | . |f(x)-f(p)|^{\theta}\leq c|\nabla f(x)|.\,

Łoś–Tarski_preservation_theorem.html

  1. T T
  2. L L
  3. Φ ( x ¯ ) \Phi(\bar{x})
  4. L L
  5. x ¯ \bar{x}
  6. A A
  7. B B
  8. T T
  9. A B A\subseteq B
  10. a ¯ \bar{a}
  11. A A
  12. B Φ ( a ¯ ) B\models\bigwedge\Phi(\bar{a})
  13. A Φ ( a ¯ ) A\models\bigwedge\Phi(\bar{a})
  14. Φ \Phi
  15. T T
  16. Φ \Phi
  17. T T
  18. Ψ ( x ¯ ) \Psi(\bar{x})
  19. 1 \forall_{1}
  20. L L
  21. 1 \forall_{1}
  22. x ¯ [ ψ ( x ¯ ) ] \forall\bar{x}[\psi(\bar{x})]
  23. ψ ( x ¯ ) \psi(\bar{x})

Π01_class.html

  1. Π 1 0 \Pi^{0}_{1}
  2. Π 1 0 \Pi^{0}_{1}
  3. 2 ω 2^{\omega}

Ψ₀(Ωω).html

  1. Π 1 1 \Pi_{1}^{1}
  2. Ω 0 = 0 \Omega_{0}=0
  3. Ω n = n \Omega_{n}=\aleph_{n}
  4. C i ( α ) C_{i}(\alpha)
  5. Ω n \Omega_{n}
  6. Ω i \Omega_{i}
  7. Ψ j ( ξ ) \Psi_{j}(\xi)
  8. ξ C i ( α ) \xi\in C_{i}(\alpha)
  9. ξ < α \xi<\alpha
  10. Ψ i ( α ) \Psi_{i}(\alpha)
  11. C i ( α ) C_{i}(\alpha)