wpmath0000016_15

Varphi_Josephson_junction.html

  1. U U
  2. ϕ \phi
  3. 2 π 2\pi
  4. - π < ϕ + π -\pi<\phi\leq+\pi
  5. U ( ϕ ) U(\phi)
  6. ϕ = 0 \phi=0
  7. U ( ϕ ) = Φ 0 I c 2 π [ 1 - cos ( ϕ ) ] U(\phi)=\frac{\Phi_{0}I_{c}}{2\pi}[1-\cos(\phi)]
  8. Φ 0 \Phi_{0}
  9. U ( ϕ ) U(\phi)
  10. U ( ϕ ) U(\phi)
  11. ϕ 0 \phi\neq 0
  12. U ( ϕ ) U(\phi)
  13. ϕ = ± φ \phi=\pm\varphi
  14. φ \varphi
  15. 0 < φ < π 0<\varphi<\pi
  16. U ( ϕ ) = Φ 0 2 π { I c 1 [ 1 - cos ( ϕ ) ] + 1 2 I c 2 [ 1 - cos ( 2 ϕ ) ] } U(\phi)=\frac{\Phi_{0}}{2\pi}\left\{I_{c1}[1-\cos(\phi)]+\frac{1}{2}I_{c2}[1-% \cos(2\phi)]\right\}
  17. I s ( ϕ ) = I c 1 sin ( ϕ ) + I c 2 sin ( 2 ϕ ) I_{s}(\phi)=I_{c1}\sin(\phi)+I_{c2}\sin(2\phi)
  18. I s ( ϕ ) = I c sin ( ϕ - φ 0 ) I_{s}(\phi)=I_{c}\sin(\phi-\varphi_{0})
  19. ϕ = φ 0 \phi=\varphi_{0}
  20. ϕ = ± φ \phi=\pm\varphi
  21. φ 0 \varphi_{0}
  22. Φ < s u b > 1 Align l t ; Φ 0 Φ<sub>1&lt;Φ_{0}

Vecten_points.html

  1. O a , O b , O c O_{a},O_{b},O_{c}
  2. A O a , B O b AO_{a},BO_{b}
  3. C O c CO_{c}
  4. I a , I b , I c I_{a},I_{b},I_{c}
  5. A I a , B I b AI_{a},BI_{b}
  6. C I c CI_{c}
  7. X ( 485 ) X ( 486 ) X(485)X(486)
  8. A B C ABC

Vector_addition_system.html

  1. V d V\subseteq\mathbb{Z}^{d}
  2. d > 0 d>0
  3. ( Q , T ) (Q,T)
  4. T Q × d × Q T\subseteq Q\times\mathbb{Z}^{d}\times Q
  5. d > 0 d>0
  6. V d V\subseteq\mathbb{Z}^{d}
  7. u d u\in\mathbb{N}^{d}
  8. u + v u+v
  9. v V v\in V
  10. u + v d u+v\in\mathbb{N}^{d}
  11. ( Q , T ) (Q,T)
  12. ( p , u ) Q × d (p,u)\in Q\times\mathbb{N}^{d}
  13. ( q , u + v ) (q,u+v)
  14. ( p , v , q ) T (p,v,q)\in T
  15. u + v d u+v\in\mathbb{N}^{d}

Velocity_gradient.html

  1. τ = μ u y \tau=\mu\frac{\partial u}{\partial y}

Verification-based_message-passing_algorithms_in_compressed_sensing.html

  1. A x = y Ax=y
  2. A A
  3. x x
  4. y y
  5. A A
  6. G = ( V l V r , E ) G=(V_{l}\cup V_{r},E)
  7. V l V_{l}
  8. G G
  9. x x
  10. V r V_{r}
  11. y y
  12. e = ( u , v ) e=(u,v)
  13. u V l u\in V_{l}
  14. v V r v\in V_{r}
  15. A A
  16. A v , u 0 A_{v,u}\neq 0
  17. w ( e ) = A v , u w(e)=A_{v,u}
  18. A = [ 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 ] A=\left[\begin{array}[]{c c c c c c c c c c c c}0&0&1&0&0&0&0&0&1&0&1&0\\ 0&0&0&1&0&1&0&1&0&0&0&0\\ 1&0&0&0&0&1&0&0&0&0&1&0\\ 1&1&1&0&0&0&0&0&0&0&0&0\\ 0&0&0&1&0&0&1&0&0&0&0&1\\ 0&0&0&0&1&0&1&0&0&0&0&1\\ 0&0&0&0&0&0&0&1&1&1&0&0\\ 0&1&0&0&1&0&0&0&0&1&0&0\end{array}\right]
  19. x x
  20. μ v ( v i ) : V l × { 0 , 1 } \mu^{v}(v_{i}):~{}V_{l}\mapsto\mathbb{R}\times\{0,1\}
  21. v i v_{i}
  22. μ c ( c i ) : V r × + \mu^{c}(c_{i}):~{}V_{r}\mapsto\mathbb{R}\times\mathbb{Z}^{+}
  23. c i c_{i}
  24. μ i \mu_{i}
  25. i t h i^{th}
  26. V N VN
  27. V N VN^{\prime}
  28. μ c ( c ) := ( y ( c ) , d c ) c V r \mu^{c}(c):=(y(c),d_{c})~{}~{}\forall c\in V_{r}
  29. μ v ( v ) := ( 0 , 0 ) v V l \mu^{v}(v):=(0,0)~{}~{}\forall v\in V_{l}
  30. V N := VN:=\emptyset
  31. V N := { - 1 } VN^{\prime}:=\{-1\}
  32. V N V N VN^{\prime}\neq VN
  33. V N := V N VN^{\prime}:=VN
  34. c V r c\in V_{r}
  35. v a l u e := y ( c ) - v 𝒩 ( c ) μ 2 v ( v ) μ 1 v ( v ) A ( c , v ) value:=y(c)-\sum_{v\in\mathcal{N}(c)}{\mu_{2}^{v}(v)\mu_{1}^{v}(v)A(c,v)}
  36. d e g r e e := d c - v 𝒩 ( c ) μ 2 v ( v ) degree:=d_{c}-\sum_{v\in\mathcal{N}(c)}{\mu_{2}^{v}(v)}
  37. μ c ( c ) := ( v a l u e , d e g r e e ) \mu^{c}(c):=(value,degree)
  38. v V r V N v\in V_{r}\setminus VN
  39. c V r c\in V_{r}
  40. v a l u e := y ( c ) - v 𝒩 ( c ) μ 2 v ( v ) μ 1 v ( v ) A ( c , v ) value:=y(c)-\sum_{v\in\mathcal{N}(c)}{\mu_{2}^{v}(v)\mu_{1}^{v}(v)A(c,v)}
  41. d e g r e e := d c - v 𝒩 ( c ) μ 2 v ( v ) degree:=d_{c}-\sum_{v\in\mathcal{N}(c)}{\mu_{2}^{v}(v)}
  42. μ c ( c ) := ( v a l u e , d e g r e e ) \mu^{c}(c):=(value,degree)
  43. v V l V N v\in V_{l}\setminus VN
  44. c 𝒩 ( v ) : μ 1 c ( c ) = 0 \exists c\in\mathcal{N}(v):~{}\mu_{1}^{c}(c)=0
  45. μ v ( v ) := ( 0 , 1 ) \mu^{v}(v):=(0,1)
  46. μ 1 v ( v ) v V l \mu^{v}_{1}(v)~{}~{}\forall v\in V_{l}
  47. d d
  48. d - 1 d-1
  49. d t h d^{th}
  50. d - 1 d-1
  51. v v
  52. v v^{\prime}
  53. v , v v,v^{\prime}
  54. x ( v ) + x ( v ) x(v^{\prime})+x(v)
  55. v v
  56. c c
  57. v v
  58. v v^{\prime}
  59. c c
  60. x ( v ) + x ( v ) x(v)+x(v^{\prime})
  61. v v^{\prime}
  62. x ( v ) x(v^{\prime})
  63. x x
  64. x ( v ) x(v^{\prime})
  65. v v
  66. v 1 , v 2 , , v q v_{1},v_{2},...,v_{q}
  67. v = v 1 + v 2 + + v q v^{\prime}=v_{1}+v_{2}+...+v_{q}
  68. O ( 1 ) O(1)
  69. | V l | |V_{l}|
  70. O ( | V l | ) O(|V_{l}|)

Vermeil's_theorem.html

  1. R R
  2. g μ ν g_{\mu\nu}
  3. R R
  4. g μ ν g_{\mu\nu}
  5. g μ ν , g^{\mu\nu},
  6. g μ ν g_{\mu\nu}

Versor_(physics).html

  1. 𝐑 \mathbf{R}
  2. 𝐢 = ( 1 , 0 , 0 ) , \mathbf{i}=(1,0,0),
  3. 𝐣 = ( 0 , 1 , 0 ) , \mathbf{j}=(0,1,0),
  4. 𝐤 = ( 0 , 0 , 1 ) . \mathbf{k}=(0,0,1).
  5. 𝐚 = 𝐚 x + 𝐚 y + 𝐚 z = a x 𝐢 + a y 𝐣 + a z 𝐤 , \mathbf{a}=\mathbf{a}_{x}+\mathbf{a}_{y}+\mathbf{a}_{z}=a_{x}\mathbf{i}+a_{y}% \mathbf{j}+a_{z}\mathbf{k},
  6. ı ^ \hat{{\imath}}
  7. ı , \vec{\imath},
  8. ȷ , \vec{\jmath},
  9. k \vec{k}
  10. ( x ^ , y ^ , z ^ ) (\hat{{x}},\hat{{y}},\hat{{z}})
  11. ( x ^ 1 , x ^ 2 , x ^ 3 ) (\hat{{x}}_{1},\hat{{x}}_{2},\hat{{x}}_{3})
  12. ( e ^ x , e ^ y , e ^ z ) (\hat{{e}}_{x},\hat{{e}}_{y},\hat{{e}}_{z})
  13. ( e ^ 1 , e ^ 2 , e ^ 3 ) (\hat{{e}}_{1},\hat{{e}}_{2},\hat{{e}}_{3})
  14. 𝐮 ^ \hat{\mathbf{u}}
  15. 𝐮 \mathbf{u}
  16. 𝐮 \mathbf{u}
  17. 𝐮 ^ = 𝐮 𝐮 . \hat{\mathbf{u}}=\frac{\mathbf{u}}{\|\mathbf{u}\|}.
  18. 𝐮 \|\mathbf{u}\|
  19. 𝐮 \mathbf{u}

Vertical_vector_field.html

  1. π : P M \pi:P\to M
  2. d π d\pi
  3. π : P M \pi:P\to M

Very-long-chain_(3R)-3-hydroxyacyl-(acyl-carrier_protein)_dehydratase.html

  1. \rightleftharpoons

Vincent_average.html

  1. F i - 1 ( α ) = inf { t : F i ( t ) α ) } , 0 < α < 1. F_{i}^{-1}(\alpha)=\inf\{t\in\mathbb{R}:F_{i}(t)\geq\alpha)\},\quad 0<\alpha<1.
  2. F - 1 ( α ) = 𝔼 w i F i - 1 ( α ) , 0 < α < 1 , i = 1 , , n F^{-1}(\alpha)=\sum\mathbb{E}w_{i}F_{i}^{-1}(\alpha),\quad 0<\alpha<1,\quad i=% 1,\ldots,n

VingCard_Elsafe.html

  1. 2 32 4.3 × 10 9 2^{32}\approx 4.3\times 10^{9}

Vinogradov's_mean-value_theorem.html

  1. J s , k ( X ) J_{s,k}(X)
  2. k k
  3. 2 s 2s
  4. x 1 j + x 2 j + + x s j = y 1 j + y 2 j + + y s j ( 1 j k ) x_{1}^{j}+x_{2}^{j}+\cdots+x_{s}^{j}=y_{1}^{j}+y_{2}^{j}+\cdots+y_{s}^{j}\quad% (1\leq j\leq k)
  5. 1 x i , y i X , ( 1 i s ) 1\leq x_{i},y_{i}\leq X,(1\leq i\leq s)
  6. J s , k ( X ) J_{s,k}(X)
  7. J s , k ( X ) = [ 0 , 1 ) k | f k ( α ; X ) | 2 s d α J_{s,k}(X)=\int_{[0,1)^{k}}|f_{k}(\mathbf{\alpha};X)|^{2s}d\mathbf{\alpha}
  8. f k ( α ; X ) = 1 x X exp ( 2 π i ( α 1 x + + α k x k ) ) . f_{k}(\mathbf{\alpha};X)=\sum_{1\leq x\leq X}\exp(2\pi i(\alpha_{1}x+\cdots+% \alpha_{k}x^{k})).
  9. J s , k ( X ) J_{s,k}(X)
  10. J s , k ( X ) J_{s,k}(X)
  11. s s
  12. k k
  13. s s
  14. k k
  15. X s X^{s}
  16. x i = y i , ( 1 i s ) x_{i}=y_{i},(1\leq i\leq s)
  17. J s , k ( X ) X s J_{s,k}(X)\gg X^{s}
  18. J s , k X s + X 2 s - 1 2 k ( k + 1 ) . J_{s,k}\gg X^{s}+X^{2s-\frac{1}{2}k(k+1)}.
  19. ϵ > 0 \epsilon>0
  20. J s , k ( X ) X s + ϵ + X 2 s - 1 2 k ( k + 1 ) + ϵ . J_{s,k}(X)\ll X^{s+\epsilon}+X^{2s-\frac{1}{2}k(k+1)+\epsilon}.
  21. s k ( k + 1 ) s\geq k(k+1)
  22. J s , k ( X ) X 2 s - 1 2 k ( k + 1 ) + ϵ . J_{s,k}(X)\ll X^{2s-\frac{1}{2}k(k+1)+\epsilon}.
  23. s k ( k + 1 ) s\leq k(k+1)
  24. J s , k ( X ) X s + ϵ . J_{s,k}(X)\ll X^{s+\epsilon}.
  25. J s , k J_{s,k}
  26. s s
  27. k k
  28. J s , k 𝒞 ( s , k ) X 2 s - 1 2 k ( k + 1 ) , J_{s,k}\sim\mathcal{C}(s,k)X^{2s-\frac{1}{2}k(k+1)},
  29. 𝒞 ( s , k ) \mathcal{C}(s,k)
  30. s s
  31. k k
  32. s , k s,k
  33. s k 2 log ( k 2 + k ) + 1 4 k 2 + 5 4 k + 1 s\geq k^{2}\log(k^{2}+k)+\frac{1}{4}k^{2}+\frac{5}{4}k+1
  34. D ( s , k ) D(s,k)
  35. J s , k ( X ) D ( s , k ) ( log X ) 2 s X 2 s - 1 2 k ( k + 1 ) + 1 2 , J_{s,k}(X)\leq D(s,k)(\log X)^{2s}X^{2s-\frac{1}{2}k(k+1)+\frac{1}{2}},
  36. ϵ > 1 2 \epsilon>\frac{1}{2}
  37. s k s\geq k
  38. D ( s , k ) D(s,k)
  39. J s , k ( X ) D ( s , k ) X 2 s - 1 2 k ( k + 1 ) + η s , k , J_{s,k}(X)\leq D(s,k)X^{2s-\frac{1}{2}k(k+1)+\eta_{s,k}},
  40. η s , k = 1 2 k 2 ( 1 - 1 k ) [ s k ] k 2 e - s / k 2 . \eta_{s,k}=\frac{1}{2}k^{2}\left(1-\frac{1}{k}\right)^{\left[\frac{s}{k}\right% ]}\leq k^{2}e^{-s/k^{2}}.
  41. s > k 2 ( 2 log k - log ϵ ) s>k^{2}(2\log k-\log\epsilon)
  42. η s , k < ϵ \eta_{s,k}<\epsilon
  43. s s
  44. J s , k 𝒞 ( s , k ) X 2 s - 1 2 k ( k + 1 ) , J_{s,k}\sim\mathcal{C}(s,k)X^{2s-\frac{1}{2}k(k+1)},
  45. s s
  46. k k
  47. s s
  48. k 2 k\geq 2
  49. s k ( k + 1 ) s\geq k(k+1)
  50. ϵ > 0 \epsilon>0
  51. J s , k ( X ) X 2 s - 1 2 k ( k + 1 ) + ϵ . J_{s,k}(X)\ll X^{2s-\frac{1}{2}k(k+1)+\epsilon}.
  52. s s
  53. k k
  54. k 4 k\geq 4
  55. 1 s 1 4 ( k + 1 ) 2 1\leq s\leq\frac{1}{4}(k+1)^{2}
  56. ϵ > 0 \epsilon>0
  57. J s , k ( X ) X s + ϵ . J_{s,k}(X)\ll X^{s+\epsilon}.

Viridiflorene_synthase.html

  1. \rightleftharpoons

Virtual_metrology.html

  1. S i 3 N 4 Si_{3}N_{4}

Visco-elastic_jets.html

  1. O h = η 0 ρ λ R 0 Oh=\frac{\eta_{0}}{\sqrt[]{\rho\lambda R_{0}}}
  2. γ η 0 \frac{\gamma}{\eta_{0}}
  3. D e = λ γ ρ R 0 3 De=\lambda\sqrt[]{\gamma\rho R_{0}^{3}}
  4. t r = ρ R 0 3 / γ t_{r}=\sqrt[]{\rho R_{0}^{3}/\gamma}
  5. ρ \rho
  6. η 0 \eta_{0}
  7. γ \gamma
  8. R 0 R_{0}
  9. λ \lambda
  10. R t + v R 2 z = 0. \frac{\partial\ R}{\partial t}+\frac{\partial\ vR^{2}}{\partial z}=0.
  11. ρ ( v t + v z ) = - γ κ t + 3 η s R 2 * ( R 2 v z ) z + 1 R 2 ( R 2 ( σ z z - σ r r ) ) z . \rho(\frac{\partial\ v}{\partial t}+\frac{v\partial}{\partial z})=-\gamma\frac% {\partial\kappa}{\partial t}+\frac{3\eta_{s}}{R^{2}}*\frac{\partial(R^{2}\frac% {\partial v}{\partial z})}{\partial z}+\frac{\frac{1}{R^{2}}\partial(R^{2}(% \sigma_{zz}-\sigma_{rr}))}{\partial z}.
  12. κ = 1 R ( 1 + R z 2 ) 1 2 - R z z ( 1 + R z z 2 ) 3 2 \kappa=\frac{1}{R(1+R_{z}^{2})^{\frac{1}{2}}}-\frac{R_{zz}}{(1+R_{zz}^{2})^{% \frac{3}{2}}}
  13. η s \eta_{s}
  14. η p \eta_{p}
  15. η 0 = η s + η p \eta_{0}=\eta_{s}+\eta_{p}
  16. R z R_{z}
  17. R z \frac{\partial R}{\partial z}
  18. σ z z \sigma_{zz}
  19. σ r r \sigma_{rr}
  20. σ z z \sigma_{zz}
  21. σ r r \sigma_{rr}
  22. σ z z + λ ( σ z z t + v σ z z z - 2 v z σ z z ) + α λ η p σ z z 2 = 2 η p v z \sigma_{zz}+\lambda(\frac{\partial\sigma_{zz}}{\partial t}+v\frac{\partial% \sigma_{zz}}{\partial z}-2\frac{\partial v}{\partial z}\sigma_{zz})+\frac{% \alpha\lambda}{\eta_{p}}\sigma_{zz}^{2}=2\eta_{p}\frac{\partial v}{\partial z}
  23. σ r r + λ ( σ r r t + v σ r r z + v z σ r r ) + α λ η p σ r r 2 = - η p v z \sigma_{rr}+\lambda(\frac{\partial\sigma_{rr}}{\partial t}+v\frac{\partial% \sigma_{rr}}{\partial z}+\frac{\partial v}{\partial z}\sigma_{rr})+\frac{% \alpha\lambda}{\eta_{p}}\sigma_{rr}^{2}=-\eta_{p}\frac{\partial v}{\partial z}
  24. λ \lambda
  25. α \alpha

Vitali–Carathéodory_theorem.html

  1. X ( v - u ) d μ < ε . \int_{X}(v-u)\,\mathrm{d}\mu<\varepsilon.

Vitaly_Shafranov.html

  1. β p \beta_{p}
  2. l i l_{i}

Void_(composites).html

  1. V o i d R a t i o ( e ) = V v V t - V v VoidRatio(e)=\frac{Vv}{Vt-Vv}
  2. V 1 = - 32.28 - 11.8 * l o g ( v ) V1=-32.28-11.8*log(v)
  3. V 2 = 6.35 + 2.35 * l o g ( v ) V2=6.35+2.35*log(v)

Volkenborn_integral.html

  1. f : \Z p C p f:\Z_{p}\rightarrow\mathbb{\mathbb{}}{C}_{p}
  2. p f ( x ) d x = lim n 1 p n x = 0 p n - 1 f ( x ) . \int_{\mathbb{Z}_{p}}f(x)\,{\rm d}x=\lim_{n\to\infty}\frac{1}{p^{n}}\sum_{x=0}% ^{p^{n}-1}f(x).
  3. R n = { x = i = r n - 1 b i x i | b i = 0 , , p - 1 for r < n } R_{n}=\left\{x=\sum_{i=r}^{n-1}b_{i}x^{i}|b_{i}=0,\ldots,p-1\,\text{ for }r<n\right\}
  4. K f ( x ) d x = lim n 1 p n x R n K f ( x ) . \int_{K}f(x)\,{\rm d}x=\lim_{n\to\infty}\frac{1}{p^{n}}\sum_{x\in R_{n}\cap K}% f(x).
  5. p 1 d x = 1 \int_{\mathbb{Z}_{p}}1\,{\rm d}x=1
  6. p x d x = - 1 2 \int_{\mathbb{Z}_{p}}x\,{\rm d}x=-\frac{1}{2}
  7. p x 2 d x = 1 6 \int_{\mathbb{Z}_{p}}x^{2}\,{\rm d}x=\frac{1}{6}
  8. p x k d x = B k \int_{\mathbb{Z}_{p}}x^{k}\,{\rm d}x=B_{k}
  9. p ( x k ) d x = ( - 1 ) k k + 1 \int_{\mathbb{Z}_{p}}{x\choose k}\,{\rm d}x=\frac{(-1)^{k}}{k+1}
  10. p ( 1 + a ) x d x = log ( 1 + a ) a \int_{\mathbb{Z}_{p}}(1+a)^{x}\,{\rm d}x=\frac{\log(1+a)}{a}
  11. p e a x d x = a e a - 1 \int_{\mathbb{Z}_{p}}e^{ax}\,{\rm d}x=\frac{a}{e^{a}-1}
  12. p log p ( x + u ) d u = ψ p ( x ) \int_{\mathbb{Z}_{p}}\log_{p}(x+u)\,{\rm d}u=\psi_{p}(x)
  13. log p \log_{p}
  14. ψ p \psi_{p}
  15. p f ( x + m ) d x = p f ( x ) d x + x = 0 m - 1 f ( x ) \int_{\mathbb{Z}_{p}}f(x+m)\,{\rm d}x=\int_{\mathbb{Z}_{p}}f(x)\,{\rm d}x+\sum% _{x=0}^{m-1}f^{\prime}(x)
  16. P t = p t p P^{t}=p^{t}\mathbb{Z}_{p}
  17. P t f ( x ) d x = 1 p t p f ( p t x ) d x \int_{P^{t}}f(x)\,{\rm d}x=\frac{1}{p^{t}}\int_{\mathbb{Z}_{p}}f(p^{t}x)\,{\rm d}x

Volodin_space.html

  1. X X
  2. B G L ( R ) BGL(R)
  3. X = n , σ B ( U n ( R ) σ ) X=\bigcup_{n,\sigma}B(U_{n}(R)^{\sigma})
  4. U n ( R ) G L n ( R ) U_{n}(R)\subset GL_{n}(R)
  5. σ \sigma
  6. G L n ( R ) GL_{n}(R)
  7. π 1 X \pi_{1}X
  8. St ( R ) \operatorname{St}(R)
  9. B G L ( R ) / X B G L + ( R ) BGL(R)/X\simeq BGL^{+}(R)

Von_Foerster_equation_(ecology).html

  1. n t + n a = - m ( a ) n , \frac{\partial n}{\partial t}+\frac{\partial n}{\partial a}=-m(a)n,
  2. n t = - n a \frac{\partial n}{\partial t}=-\frac{\partial n}{\partial a}
  3. n ( t , 0 ) = 0 b ( a ) n ( t , a ) d t , n(t,0)=\int_{0}^{\infty}b(a)n(t,a)\,dt,
  4. n ( 0 , a ) = f ( a ) , n(0,a)=f(a),\,

Von_Kármán_wind_turbulence_model.html

  1. Φ u g ( Ω ) = σ u 2 2 L u π 1 ( 1 + ( 1.339 L u Ω ) 2 ) 5 6 Φ v g ( Ω ) = σ v 2 2 L v π 1 + 8 3 ( 2.678 L v Ω ) 2 ( 1 + ( 2.678 L v Ω ) 2 ) 11 6 Φ w g ( Ω ) = σ w 2 2 L w π 1 + 8 3 ( 2.678 L w Ω ) 2 ( 1 + ( 2.678 L w Ω ) 2 ) 11 6 \begin{aligned}\displaystyle\Phi_{u_{g}}(\Omega)&\displaystyle=\sigma_{u}^{2}% \frac{2L_{u}}{\pi}\frac{1}{\left(1+(1.339L_{u}\Omega)^{2}\right)^{\frac{5}{6}}% }\\ \displaystyle\Phi_{v_{g}}(\Omega)&\displaystyle=\sigma_{v}^{2}\frac{2L_{v}}{% \pi}\frac{1+\frac{8}{3}(2.678L_{v}\Omega)^{2}}{\left(1+(2.678L_{v}\Omega)^{2}% \right)^{\frac{11}{6}}}\\ \displaystyle\Phi_{w_{g}}(\Omega)&\displaystyle=\sigma_{w}^{2}\frac{2L_{w}}{% \pi}\frac{1+\frac{8}{3}(2.678L_{w}\Omega)^{2}}{\left(1+(2.678L_{w}\Omega)^{2}% \right)^{\frac{11}{6}}}\end{aligned}
  2. Ω = ω V Φ i ( Ω ) = V Φ i ( ω V ) \begin{aligned}\displaystyle\Omega&\displaystyle=\frac{\omega}{V}\\ \displaystyle\Phi_{i}(\Omega)&\displaystyle=V\Phi_{i}\left(\frac{\omega}{V}% \right)\end{aligned}
  3. p g = w g y q g = w g x r g = - v g x \begin{aligned}\displaystyle p_{g}&\displaystyle=\frac{\partial w_{g}}{% \partial y}\\ \displaystyle q_{g}&\displaystyle=\frac{\partial w_{g}}{\partial x}\\ \displaystyle r_{g}&\displaystyle=-\frac{\partial v_{g}}{\partial x}\end{aligned}
  4. Φ p g ( ω ) = σ w 2 2 V L w 0.8 ( 2 π L w 4 b ) 1 3 1 + ( 4 b ω π V ) 2 Φ q g ( ω ) = ± ( ω V ) 2 1 + ( 4 b ω π V ) 2 Φ w g ( ω ) Φ r g ( ω ) = ( ω V ) 2 1 + ( 3 b ω π V ) 2 Φ v g ( ω ) \begin{aligned}\displaystyle\Phi_{p_{g}}(\omega)&\displaystyle=\frac{\sigma_{w% }^{2}}{2VL_{w}}\frac{0.8\left(\frac{2\pi L_{w}}{4b}\right)^{\frac{1}{3}}}{1+% \left(\frac{4b\omega}{\pi V}\right)^{2}}\\ \displaystyle\Phi_{q_{g}}(\omega)&\displaystyle=\frac{\pm\left(\frac{\omega}{V% }\right)^{2}}{1+\left(\frac{4b\omega}{\pi V}\right)^{2}}\Phi_{w_{g}}(\omega)\\ \displaystyle\Phi_{r_{g}}(\omega)&\displaystyle=\frac{\mp\left(\frac{\omega}{V% }\right)^{2}}{1+\left(\frac{3b\omega}{\pi V}\right)^{2}}\Phi_{v_{g}}(\omega)% \end{aligned}
  5. Φ y ( ω ) = | G ( i ω ) | 2 \Phi_{y}(\omega)=|G(i\omega)|^{2}
  6. G u g ( s ) = σ u 2 L u π V ( 1 + 0.25 L u V s ) 1 + 1.357 L u V s + 0.1987 ( L u V s ) 2 G v g ( s ) = σ v 2 L v π V ( 1 + 2.7478 2 L v V s + 0.3398 ( 2 L v V s ) 2 ) 1 + 2.9958 2 L v V s + 1.9754 ( 2 L v V s ) 2 + 0.1539 ( 2 L v V s ) 3 G w g ( s ) = σ w 2 L w π V ( 1 + 2.7478 2 L w V s + 0.3398 ( 2 L w V s ) 2 ) 1 + 2.9958 2 L w V s + 1.9754 ( 2 L w V s ) 2 + 0.1539 ( 2 L w V s ) 3 G p g ( s ) = σ w 0.8 V ( π 4 b ) 1 6 ( 2 L w ) 1 3 ( 1 + 4 b π V s ) G q g ( s ) = ± s V 1 + 4 b π V s G w g ( s ) G r g ( s ) = s V 1 + 3 b π V s G v g ( s ) \begin{aligned}\displaystyle G_{u_{g}}(s)&\displaystyle=\frac{\sigma_{u}\sqrt{% \frac{2L_{u}}{\pi V}}\left(1+0.25\frac{L_{u}}{V}s\right)}{1+1.357\frac{L_{u}}{% V}s+0.1987\left(\frac{L_{u}}{V}s\right)^{2}}\\ \displaystyle G_{v_{g}}(s)&\displaystyle=\frac{\sigma_{v}\sqrt{\frac{2L_{v}}{% \pi V}}\left(1+2.7478\frac{2L_{v}}{V}s+0.3398\left(\frac{2L_{v}}{V}s\right)^{2% }\right)}{1+2.9958\frac{2L_{v}}{V}s+1.9754\left(\frac{2L_{v}}{V}s\right)^{2}+0% .1539\left(\frac{2L_{v}}{V}s\right)^{3}}\\ \displaystyle G_{w_{g}}(s)&\displaystyle=\frac{\sigma_{w}\sqrt{\frac{2L_{w}}{% \pi V}}\left(1+2.7478\frac{2L_{w}}{V}s+0.3398\left(\frac{2L_{w}}{V}s\right)^{2% }\right)}{1+2.9958\frac{2L_{w}}{V}s+1.9754\left(\frac{2L_{w}}{V}s\right)^{2}+0% .1539\left(\frac{2L_{w}}{V}s\right)^{3}}\\ \displaystyle G_{p_{g}}(s)&\displaystyle=\sigma_{w}\sqrt{\frac{0.8}{V}}\frac{% \left(\frac{\pi}{4b}\right)^{\frac{1}{6}}}{(2L_{w})^{\frac{1}{3}}\left(1+\frac% {4b}{\pi V}s\right)}\\ \displaystyle G_{q_{g}}(s)&\displaystyle=\frac{\pm\frac{s}{V}}{1+\frac{4b}{\pi V% }s}G_{w_{g}}(s)\\ \displaystyle G_{r_{g}}(s)&\displaystyle=\frac{\mp\frac{s}{V}}{1+\frac{3b}{\pi V% }s}G_{v_{g}}(s)\end{aligned}

W._Dale_Brownawell.html

  1. e e e^{e}
  2. e e 2 e^{e^{2}}
  3. e e

Wahlquist_fluid.html

  1. d s 2 = f ( d t - A ~ d φ ) 2 - r 0 2 ( ζ 2 + ξ 2 ) [ d ζ 2 ( 1 - k ~ 2 ζ 2 ) h ~ 1 + d ξ 2 ( 1 + k ~ 2 ξ 2 ) h ~ 2 + h ~ 1 h ~ 2 h ~ 1 - h ~ 2 d φ 2 ] ds^{2}=f(dt-\tilde{A}d\varphi)^{2}-r_{0}^{2}(\zeta^{2}+\xi^{2})[\frac{d\zeta^{% 2}}{(1-\tilde{k}^{2}\zeta^{2})\tilde{h}_{1}}+\frac{d\xi^{2}}{(1+\tilde{k}^{2}% \xi^{2})\tilde{h}_{2}}+\frac{\tilde{h}_{1}\tilde{h}_{2}}{\tilde{h}_{1}-\tilde{% h}_{2}}d\varphi^{2}]
  2. f = h ~ 1 - h ~ 2 ζ 2 + ξ 2 f=\frac{\tilde{h}_{1}-\tilde{h}_{2}}{\zeta^{2}+\xi^{2}}
  3. A ~ = r 0 ( ξ 2 h ~ 1 + ζ 2 h ~ 2 h ~ 1 - h ~ 2 - ξ A 2 ) \tilde{A}=r_{0}(\frac{\xi^{2}\tilde{h}_{1}+\zeta^{2}\tilde{h}_{2}}{\tilde{h}_{% 1}-\tilde{h}_{2}}-\xi_{A}^{2})
  4. h ~ 1 ( ζ ) = 1 + ζ 2 + ζ κ 2 [ ζ + 1 k ~ 1 - k ~ 2 ζ 2 arcsin ( k ~ ζ ) ] \tilde{h}_{1}(\zeta)=1+\zeta^{2}+\frac{\zeta}{\kappa^{2}}[\zeta_{+}\frac{1}{% \tilde{k}}\sqrt{1-\tilde{k}^{2}\zeta^{2}}\arcsin(\tilde{k}\zeta)]
  5. h ~ 2 ( ξ ) = 1 - ξ 2 - ξ κ 2 [ ξ - 1 k ~ 1 + k ~ 2 ξ 2 sinh - 1 ( k ~ ξ ) ] \tilde{h}_{2}(\xi)=1-\xi^{2}-\frac{\xi}{\kappa^{2}}[\xi_{-}\frac{1}{\tilde{k}}% \sqrt{1+\tilde{k}^{2}\xi^{2}}\sinh^{-1}(\tilde{k}\xi)]
  6. ξ A \xi_{A}
  7. h ~ 2 ( ξ A ) = 0 \tilde{h}_{2}(\xi_{A})=0
  8. μ + 3 p = μ 0 \mu+3p=\mu_{0}
  9. μ 0 \mu_{0}
  10. p = 1 2 μ 0 ( 1 - κ 2 f ) p=\frac{1}{2}\mu_{0}(1-\kappa^{2}f)
  11. μ = 1 2 μ 0 ( 3 κ 2 f - 1 ) \mu=\frac{1}{2}\mu_{0}(3\kappa^{2}f-1)

Walk-on-spheres_method.html

  1. Ω \Omega
  2. d \mathbb{R}^{d}
  3. Γ \Gamma
  4. Γ \Gamma
  5. x x
  6. Ω \Omega
  7. { Δ u ( x ) = 0 if x Ω u ( x ) = h ( x ) if x Γ . \begin{cases}\Delta u(x)=0&\mbox{if }~{}x\in\Omega\\ u(x)=h(x)&\mbox{if }~{}x\in\Gamma.\end{cases}
  8. u u
  9. x Ω x\in\Omega
  10. u ( x ) = 𝔼 x [ h ( W τ ) ] u(x)=\mathbb{E}_{x}[h(W_{\tau})]
  11. W W
  12. d d
  13. 𝔼 x [ h ( W τ ) ] 1 n i = 1 n h ( W τ i ) \mathbb{E}_{x}[h(W_{\tau})]\sim\frac{1}{n}\sum_{i=1}^{n}h(W^{i}_{\tau})
  14. ( d 1 ) (d− 1)
  15. x x
  16. W W
  17. 𝒮 0 \mathcal{S}_{0}
  18. x x
  19. 𝒮 0 \mathcal{S}_{0}
  20. ε \varepsilon
  21. ε \varepsilon
  22. Ω \Omega
  23. ε \varepsilon
  24. ε > 0 \varepsilon>0
  25. x ( 0 ) = x x^{(0)}=x
  26. d ( x ( n ) , Γ ) > ε d(x^{(n)},\Gamma)>\varepsilon
  27. r n = d ( x n , Γ ) r_{n}=d(x^{n},\Gamma)
  28. γ n \gamma_{n}
  29. x ( n + 1 ) := x ( n ) + r n γ n x^{(n+1)}:=x^{(n)}+r_{n}\gamma_{n}
  30. d ( x ( n ) , Γ ) ε d(x^{(n)},\Gamma)\leq\varepsilon
  31. x f := p Γ ( x ( n ) ) x_{f}:=p_{\Gamma}(x^{(n)})
  32. x ( n ) x^{(n)}
  33. Γ \Gamma
  34. x f x_{f}
  35. x f x_{f}
  36. Ω \Omega
  37. x x
  38. ε \varepsilon
  39. 𝒪 ( ε ) \mathcal{O}(\varepsilon)
  40. ε \varepsilon
  41. 𝒪 ( | log ( ε ) | ) \mathcal{O}(|\log(\varepsilon)|)
  42. 3 3
  43. Δ u = c u + f \Delta u=cu+f
  44. { t u ( x , t ) + 1 2 Δ x u ( x , t ) = 0 if x Ω and t < T u ( x , T ) = h ( x , T ) if x Ω ¯ u ( x , t ) = h ( x , t ) if x Γ . \begin{cases}\partial_{t}u(x,t)+\frac{1}{2}\Delta_{x}u(x,t)=0&\mbox{if }~{}x% \in\Omega\mbox{and }~{}t<T\\ u(x,T)=h(x,T)&\mbox{if }~{}x\in\bar{\Omega}\\ u(x,t)=h(x,t)&\mbox{if }~{}x\in\Gamma.\end{cases}
  45. u ( x , t ) = 𝔼 t , x ( h ( X T τ , T τ ) ) u(x,t)=\mathbb{E}_{t,x}(h(X_{T\wedge\tau},T\wedge\tau))
  46. X t = x X_{t}=x
  47. τ 0 \tau_{0}
  48. R R
  49. 𝔼 ( exp ( - s τ 0 ) ) ) = R 2 s sinh ( R 2 s ) \mathbb{E}(\exp(-s\tau_{0})))=\frac{R\sqrt{2s}}{\sinh(R\sqrt{2s})}
  50. τ \tau
  51. T T

Warp-field_experiments.html

  1. d s 2 = - c d t 2 + [ d x - v s ( t ) f ( r s ) d t ] 2 + d y 2 + d z 2 ds^{2}=-cdt^{2}+[dx-v_{s}(t)f(r_{s})dt]^{2}+dy^{2}+dz^{2}
  2. x = x s ( t ) , y = 0 , z = 0 x=x_{s}(t),y=0,z=0
  3. x s x_{s}
  4. r s = [ ( x - x s ( t ) ) 2 + y 2 + z 2 ] r_{s}=[(x-x_{s}(t))^{2}+y^{2}+z^{2}]
  5. c c
  6. v s ( t ) v_{s}(t)
  7. d x s / d t dx_{s}/dt
  8. f ( r s ) f(r_{s})
  9. f ( 0 ) = 1 f(0)=1
  10. r s > R r_{s}>R
  11. R R
  12. v s > c v_{s}>c
  13. Θ \Theta
  14. Θ = v s c x s r s d f d r s \Theta=\frac{v_{s}}{c}\frac{x_{s}}{r_{s}}\frac{df}{dr_{s}}
  15. T μ ν T^{\mu\nu}
  16. T 00 = - ( c 4 8 π G ) [ v s 2 ( t ) ρ 2 4 r s 2 c 2 ] ( d f d r s ) 2 T^{00}=-\left(\frac{c^{4}}{8\pi G}\right)\left[\frac{v_{s}^{2}(t)\rho^{2}}{4r_% {s}^{2}c^{2}}\right]\left(\frac{df}{dr_{s}}\right)^{2}
  17. G G
  18. ρ = y 2 + z 2 \rho=\sqrt{y^{2}+z^{2}}
  19. Φ \Phi
  20. γ \gamma
  21. γ = cosh [ 1 2 c 2 ( ln | 1 - ( v s c ) 2 f ( r s ) | ) ] \gamma=\cosh\left[\frac{1}{2}c^{2}\left(\ln\left|1-\left(\frac{v_{s}}{c}\right% )^{2}f(r_{s})\right|\right)\right]
  22. v s = γ v i n i v_{s}=\gamma\cdot v_{ini}
  23. v s > c v_{s}>c
  24. d s 2 = - c 2 d t 2 + a ( t ) 2 e 2 k U d X 2 + d U 2 ds^{2}=-c^{2}dt^{2}+\frac{a(t)^{2}}{e^{2kU}}dX^{2}+dU^{2}
  25. d X 2 dX^{2}
  26. d U 2 dU^{2}
  27. U = 0 U=0
  28. a ( t ) a(t)
  29. k k
  30. d X d t = c e k U a ( t ) 1 - d U 2 c 2 d t 2 \frac{dX}{dt}=\frac{ce^{kU}}{a(t)}\sqrt{1-\frac{dU^{2}}{c^{2}dt^{2}}}
  31. U U
  32. U 1 U>>1
  33. d X d t \frac{dX}{dt}
  34. γ e U \gamma\approx e^{U}
  35. Φ U \Phi\approx U
  36. d Φ d t d U d t \frac{d\Phi}{dt}\approx\frac{dU}{dt}
  37. Φ \Phi
  38. d E = - p s d V dE=-p_{s}dV
  39. d E dE
  40. ρ s d V \rho_{s}dV
  41. d V dV
  42. p s p_{s}
  43. ρ s \rho_{s}
  44. ρ s = - p s \rho_{s}=-p_{s}
  45. ρ s \rho_{s}
  46. p s p_{s}
  47. ρ s \rho_{s}
  48. p s p_{s}

WASP_(cricket_calculation_tool).html

  1. V ( b , w ) = r ( b , w ) + p ( b , w ) V ( b + 1 , w + 1 ) + ( 1 - p ( b , w ) ) V ( b + 1 , w ) V(b,w)=r(b,w)+p(b,w)V(b+1,w+1)+(1-p(b,w))V(b+1,w)

Waste_weir.html

  1. Q Q
  2. L L
  3. H H
  4. Q = K L H 3 / 2 Q=KLH^{3/2}
  5. K = 2 3 C d 2 g K=\frac{2}{3}C_{d}\sqrt{2g}
  6. 9.8 m / s e c 2 9.8m/{sec^{2}}
  7. C d C_{d}

Wave_action_(continuum_mechanics).html

  1. 𝒜 = E ω i , \mathcal{A}=\frac{E}{\omega_{i}},
  2. E E
  3. ω i \omega_{i}
  4. t 𝒜 + s y m b o l s y m b o l = 0 , \frac{\partial}{\partial t}\mathcal{A}+symbol{\nabla}\cdot symbol{\mathcal{B}}% =0,
  5. s y m b o l symbol{\mathcal{B}}
  6. s y m b o l \cdotsymbol symbol{\nabla}\cdotsymbol{\mathcal{B}}
  7. s y m b o l symbol{\mathcal{B}}
  8. t ( E ω i ) + s y m b o l [ ( s y m b o l U + s y m b o l c g ) E ω i ] = 0 , \frac{\partial}{\partial t}\left(\frac{E}{\omega_{i}}\right)+symbol{\nabla}% \cdot\left[\left(symbol{U}+symbol{c}_{g}\right)\,\frac{E}{\omega_{i}}\right]=0,
  9. 𝒜 = E ω i \mathcal{A}=\frac{E}{\omega_{i}}
  10. s y m b o l = ( s y m b o l U + s y m b o l c g ) 𝒜 , symbol{\mathcal{B}}=\left(symbol{U}+symbol{c}_{g}\right)\mathcal{A},
  11. s y m b o l c g symbol{c}_{g}
  12. s y m b o l U symbol{U}

Weak_Büchi_automaton.html

  1. q q
  2. q q^{\prime}
  3. q q
  4. q q^{\prime}
  5. q q
  6. q q^{\prime}
  7. w w
  8. F F
  9. O ( n log ( n ) ) O(n\log(n))

Weak_trace-class_operator.html

  1. = =
  2. L 1 , = { A K ( H ) : μ ( n , A ) = O ( n - 1 ) } . L_{1,\infty}=\{A\in K(H):\mu(n,A)=O(n^{-1})\}.
  3. A w = sup n 0 ( 1 + n ) μ ( n , A ) , \|A\|_{w}=\sup_{n\geq 0}(1+n)\mu(n,A),

Weakly_holomorphic_modular_form.html

  1. f ( ( a τ + b ) / ( c τ + d ) ) = ( c τ + d ) k f ( τ ) f((a\tau+b)/(c\tau+d))=(c\tau+d)^{k}f(\tau)

Wear_coefficient.html

  1. V = K P L 3 H V=K\frac{PL}{3H}
  2. H H
  3. V V
  4. P P
  5. L L
  6. K K
  7. K K
  8. K = 3 H V P L K=\frac{3HV}{PL}
  9. V V
  10. W W
  11. ρ \rho
  12. K = 3 H W P L ρ K=\frac{3HW}{PL_{\rho}}
  13. K N = 3 H V s P L s K_{N}=\frac{3HV_{s}}{PL_{s}}
  14. L s L_{s}
  15. V s V_{s}
  16. K = V A p L K=\frac{V}{A_{p}L}
  17. A p A_{p}
  18. μ = F t P \mu=\frac{F_{t}}{P}
  19. F t F_{t}
  20. V u Vu
  21. F L FL
  22. K = 3 μ H V μ P L = 3 μ V u F L V u F L K=\frac{3\mu HV}{\mu PL}=3\mu\frac{Vu}{FL}\approx\frac{Vu}{FL}
  23. K H \frac{K}{H}
  24. K = 3 g 1 d ( 1 - f v ) g 3 f v L [ 1 - e x p ( - g 3 f v L d ( 1 - f v ) ) ] K=\frac{3g_{1}d(1-f_{v})}{g_{3}f_{v}L}\left[1-exp\left(\frac{-g_{3}f_{v}L}{d(1% -f_{v})}\right)\right]
  25. g 3 g_{3}
  26. d d
  27. f v f_{v}
  28. g 1 g_{1}
  29. P P
  30. H H
  31. m A m_{A}
  32. V c V_{c}
  33. L = 0 L=0
  34. g 1 = H m A P g_{1}=\frac{Hm_{A}}{P}
  35. K = 3 H m A d ( 1 - f v ) P L g 3 f v L [ 1 - e x p ( - g 3 f v L d ( 1 - f v ) ) ] K=\frac{3Hm_{A}d(1-f_{v})}{PLg_{3}f_{v}L}\left[1-exp\left(\frac{-g_{3}f_{v}L}{% d(1-f_{v})}\right)\right]

Web_Mercator.html

  1. x = 128 π 2 zoom level ( λ + π ) pixels y = 128 π 2 zoom level ( π - ln [ tan ( π 4 + φ 2 ) ] ) pixels \begin{aligned}\displaystyle x&\displaystyle=\frac{128}{\pi}2^{\,\text{zoom % level}}(\lambda+\pi)\,\text{ pixels}\\ \displaystyle y&\displaystyle=\frac{128}{\pi}2^{\,\text{zoom level}}(\pi-\ln% \left[\tan\left(\frac{\pi}{4}+\frac{\varphi}{2}\right)\right])\,\text{ pixels}% \end{aligned}
  2. φ max = 180 π [ 2 arctan ( e π ) - π 2 ] \displaystyle\varphi^{\circ}_{\,\text{max}}=\frac{180}{\pi}[2\arctan(e^{\pi})-% \frac{\pi}{2}]

Webbed_space.html

  1. ( D i ) (D_{i})
  2. X = i D i X=\cup_{i}D_{i}
  3. D i D_{i}
  4. ( D i j ) (D_{ij})
  5. D i j ( 1 2 ) D i D_{ij}\subseteq(\frac{1}{2})D_{i}
  6. j D i j \cup_{j}D_{ij}
  7. D i D_{i}
  8. D i D_{i}
  9. D i D_{i}
  10. ( x n ) (x_{n})
  11. x 1 x_{1}
  12. x 2 x_{2}
  13. Σ n x n \Sigma_{n}x_{n}
  14. β ( X * , X ) \beta(X^{*},X)
  15. β ( X * , X ) \beta(X^{*},X)

Weber_problem.html

  1. i = 1 n w i x i - y , \sum_{i=1}^{n}w_{i}\,\|x_{i}-y\|,
  2. i = 1 n w i x i - y j x i - y 2 \sum_{i=1}^{n}\frac{w_{i}}{\|x_{i}-y_{j}\|}\|x_{i}-y\|^{2}
  3. y j + 1 = ( i = 1 n w i x i x i - y j ) / ( i = 1 n w i x i - y j ) . \left.y_{j+1}=\left(\sum_{i=1}^{n}\frac{w_{i}x_{i}}{\|x_{i}-y_{j}\|}\right)% \right/\left(\sum_{i=1}^{n}\frac{w_{i}}{\|x_{i}-y_{j}\|}\right).

Weighted_correlation_network_analysis.html

  1. s i j s_{ij}
  2. s i j u n s i g n e d = | c o r ( x i , x j ) | s^{unsigned}_{ij}=|cor(x_{i},x_{j})|
  3. x i x_{i}
  4. x j x_{j}
  5. x i x_{i}
  6. x j x_{j}
  7. s i j s i g n e d = 0.5 + 0.5 c o r ( x i , x j ) s^{signed}_{ij}=0.5+0.5cor(x_{i},x_{j})
  8. s i j u n s i g n e d s^{unsigned}_{ij}
  9. s i j s i g n e d s^{signed}_{ij}
  10. c o r ( x i , x j ) = - 1 cor(x_{i},x_{j})=-1
  11. A = [ a i j ] A=[a_{ij}]
  12. A A
  13. S = [ s i j ] S=[s_{ij}]
  14. S S
  15. s i j > τ s_{ij}>\tau
  16. a i j = ( s i j ) β a_{ij}=(s_{ij})^{\beta}
  17. β \beta
  18. β = 6 \beta=6
  19. β = 12 \beta=12
  20. β \beta
  21. β \beta
  22. l o g ( a i j ) = β l o g ( s i j ) log(a_{ij})=\beta log(s_{ij})
  23. β \beta
  24. k M E i = c o r ( x i , M E ) kME_{i}=cor(x_{i},ME)
  25. Z s u m m a r y Zsummary

Weil–Brezin_Map.html

  1. N N
  2. x , y , t a , b , c = x + a , y + b , t + c + x b . \langle x,y,t\rangle\langle a,b,c\rangle=\langle x+a,y+b,t+c+xb\rangle.
  3. Γ \Gamma
  4. N N
  5. Γ \Gamma
  6. N N
  7. Γ \ N \Gamma\backslash N
  8. μ = d x d y d t \mu=dx\wedge dy\wedge dt
  9. L 2 ( Γ \ N ) = n H n L^{2}(\Gamma\backslash N)=\oplus_{n\in\mathbb{Z}}H_{n}
  10. H n = { f L 2 ( Γ \ N ) f ( Γ x , y , t + s ) = exp ( 2 π i n s ) f ( Γ x , y , t ) } H_{n}=\{f\in L^{2}(\Gamma\backslash N)\mid f(\Gamma\langle x,y,t+s\rangle)=% \exp(2\pi ins)f(\Gamma\langle x,y,t\rangle)\}
  11. W : L 2 ( ) H 1 W:L^{2}(\mathbb{R})\to H_{1}
  12. W ( ψ ) ( Γ x , y , t ) = l ψ ( x + l ) e 2 π i l y e 2 π i t W(\psi)(\Gamma\langle x,y,t\rangle)=\sum_{l\in\mathbb{Z}}\psi(x+l)e^{2\pi ily}% e^{2\pi it}
  13. ψ \psi
  14. W - 1 : H 1 L 2 ( ) W^{-1}:H_{1}\to L^{2}(\mathbb{R})
  15. ( W - 1 f ) ( x ) = 0 1 f ( Γ x , y , 0 ) d y (W^{-1}f)(x)=\int_{0}^{1}f(\Gamma\langle x,y,0\rangle)dy
  16. f f
  17. H 1 H_{1}
  18. λ 0 \lambda\neq 0
  19. U λ U_{\lambda}
  20. N N
  21. L 2 ( ) L^{2}(\mathbb{R})
  22. ( U λ ( a , b , c ) ψ ) ( x ) = e 2 π i λ ( c + b x ) ψ ( x + a ) (U_{\lambda}(\langle a,b,c\rangle)\psi)(x)=e^{2\pi i\lambda(c+bx)}\psi(x+a)
  23. U λ ( a , 0 , 0 ) U λ ( 0 , b , 0 ) = e 2 π i λ a b U λ ( 0 , b , 0 ) U λ ( a , 0 , 0 ) U_{\lambda}(\langle a,0,0\rangle)U_{\lambda}(\langle 0,b,0\rangle)=e^{2\pi i% \lambda ab}U_{\lambda}(\langle 0,b,0\rangle)U_{\lambda}(\langle a,0,0\rangle)
  24. U = U 1 U=U_{1}
  25. N N
  26. L 2 ( ) L^{2}(\mathbb{R})
  27. R R
  28. N N
  29. H 1 L 2 ( Γ \ N ) H_{1}\subset L^{2}(\Gamma\backslash N)
  30. W U ( a , b , c ) = R ( a , b , c ) W WU(\langle a,b,c\rangle)=R(\langle a,b,c\rangle)W
  31. U U
  32. L 2 ( ) L^{2}(\mathbb{R})
  33. R R
  34. H 1 H_{1}
  35. J : N N J:N\to N
  36. J ( x , y , t ) = y , - x , t - x y J(\langle x,y,t\rangle)=\langle y,-x,t-xy\rangle
  37. J * : H 1 H 1 J^{*}:H_{1}\to H_{1}
  38. = W - 1 J * W \mathcal{F}=W^{-1}J^{*}W
  39. L 2 ( ) L^{2}(\mathbb{R})
  40. W W
  41. J * J^{*}
  42. ψ \psi
  43. l ψ ( l ) = W ( ψ ) ( Γ 0 , 0 , 0 ) ) = ( J * W ( ψ ) ) ( Γ 0 , 0 , 0 ) ) = W ( ψ ^ ) ( Γ 0 , 0 , 0 ) ) = l ψ ^ ( l ) \sum_{l}\psi(l)=W(\psi)(\Gamma\langle 0,0,0)\rangle)=(J^{*}W(\psi))(\Gamma% \langle 0,0,0)\rangle)=W(\hat{\psi})(\Gamma\langle 0,0,0)\rangle)=\sum_{l}\hat% {\psi}(l)
  44. n 0 n\neq 0
  45. H n L 2 ( Γ \ N ) H_{n}\subset L^{2}(\Gamma\backslash N)
  46. H n = m = 0 | n | - 1 H n , m H_{n}=\oplus_{m=0}^{|n|-1}H_{n,m}
  47. H n , m = { f H n f ( Γ x , y + 1 n , t ) = e 2 π i m / n f ( Γ x , y , t ) } H_{n,m}=\{f\in H_{n}\mid f(\Gamma\langle x,y+{1\over n},t\rangle)=e^{2\pi im/n% }f(\Gamma\langle x,y,t\rangle)\}
  48. L ( 0 , 1 / n , 0 ) L(\langle 0,1/n,0\rangle)
  49. H n H_{n}
  50. H n , 0 , , H n , | n | - 1 H_{n,0},...,H_{n,|n|-1}
  51. L ( m / n , 0 , 0 ) L(\langle m/n,0,0\rangle)
  52. H n H_{n}
  53. L ( m / n , 0 , 0 ) : H n , 0 H n , m L(\langle m/n,0,0\rangle):H_{n,0}\to H_{n,m}
  54. n 0 n\neq 0
  55. m = 0 , , | n | - 1 m=0,...,|n|-1
  56. W n , m : L 2 ( ) H n , m W_{n,m}:L^{2}(\mathbb{R})\to H_{n,m}
  57. W n , m ( ψ ) ( Γ x , y , t ) = l ψ ( x + l + m n ) e 2 π i ( n l + m ) y e 2 π i n t W_{n,m}(\psi)(\Gamma\langle x,y,t\rangle)=\sum_{l\in\mathbb{Z}}\psi(x+l+{m% \over n})e^{2\pi i(nl+m)y}e^{2\pi int}
  58. ψ \psi
  59. W n , m = L ( m / n , 0 , 0 ) W n , 0 . W_{n,m}=L(\langle m/n,0,0\rangle)\circ W_{n,0}.
  60. W n , m - 1 : H n , m L 2 ( ) W_{n,m}^{-1}:H_{n,m}\to L^{2}(\mathbb{R})
  61. ( W n , m - 1 f ) ( x ) = 0 1 e - 2 π i m y f ( Γ x - m n , y , 0 ) d y (W_{n,m}^{-1}f)(x)=\int_{0}^{1}e^{-2\pi imy}f(\Gamma\langle x-{m\over n},y,0% \rangle)dy
  62. f f
  63. H n , m H_{n,m}
  64. U n U_{n}
  65. H n , m H_{n,m}
  66. W n , m W_{n,m}
  67. W n , m U n ( a , b , c ) = R ( a , b , c ) W n , m W_{n,m}U_{n}(\langle a,b,c\rangle)=R(\langle a,b,c\rangle)W_{n,m}
  68. m , m m,m^{\prime}
  69. ( W n , m - 1 J * W n , m ψ ) ( x ) = e 2 π i m m / n ψ ^ ( n x ) (W_{n,m^{\prime}}^{-1}J^{*}W_{n,m}\psi)(x)=e^{2\pi im^{\prime}m/n}\hat{\psi}(nx)
  70. n > 0 n>0
  71. ϕ n ( x ) = ( 2 n ) 1 / 4 e - π n x 2 \phi_{n}(x)=(2n)^{1/4}e^{-\pi nx^{2}}
  72. K n K_{n}
  73. H n H_{n}
  74. { s y m b o l e n , 0 , , s y m b o l e n , n - 1 } \{symbol{e}_{n,0},...,symbol{e}_{n,n-1}\}
  75. s y m b o l e n , m = W n , m ( ϕ n ) H n , m . symbol{e}_{n,m}=W_{n,m}(\phi_{n})\in H_{n,m}.
  76. L ( 1 / n , 0 , 0 ) L(\langle 1/n,0,0\rangle)
  77. L ( 0 , 1 / n , 0 ) L(\langle 0,1/n,0\rangle)
  78. K n K_{n}
  79. J * J^{*}
  80. K n K_{n}
  81. J * s y m b o l e n , m = 1 n m e 2 π i m m / n s y m b o l e n , m . J^{*}symbol{e}_{n,m}={1\over\sqrt{n}}\sum_{m^{\prime}}e^{2\pi im^{\prime}m/n}% symbol{e}_{n,m^{\prime}}.
  82. 𝔫 \mathfrak{n}
  83. N N
  84. 𝔫 \mathfrak{n}
  85. X , Y , T X,Y,T
  86. N N
  87. X ( x , y , t ) = x , X(x,y,t)={\partial\over\partial x},
  88. Y ( x , y , t ) = y + x t , Y(x,y,t)={\partial\over\partial y}+x{\partial\over\partial t},
  89. T ( x , y , t ) = t . T(x,y,t)={\partial\over\partial t}.
  90. Γ \ N \Gamma\backslash N
  91. V - i = X - i Y V_{-i}=X-iY
  92. n > 0 n>0
  93. V - i V_{-i}
  94. C ( Γ \ N ) H n , m C^{\infty}(\Gamma\backslash N)\cap H_{n,m}
  95. s y m b o l e n , m symbol{e}_{n,m}
  96. ker ( V - i : C ( Γ \ N ) H n H n ) = { K n , n > 0 , n = 0 \ker(V_{-i}:C^{\infty}(\Gamma\backslash N)\cap H_{n}\to H_{n})=\left\{\begin{% array}[]{lr}K_{n},&n>0\\ \mathbb{C},&n=0\end{array}\right.
  97. n n
  98. Γ \ N \Gamma\backslash N
  99. n 0 K n \oplus_{n\geq 0}K_{n}
  100. K 0 = K_{0}=\mathbb{C}
  101. [ x 1 , x 2 2 , x 3 3 ] / ( x 3 6 + x 1 4 x 2 2 + x 2 6 ) \mathbb{C}[x_{1},x_{2}^{2},x_{3}^{3}]/(x_{3}^{6}+x_{1}^{4}x_{2}^{2}+x_{2}^{6})
  102. ϑ ( z ; τ ) = l = - exp ( π i l 2 τ + 2 π i l z ) \vartheta(z;\tau)=\sum_{l=-\infty}^{\infty}\exp(\pi il^{2}\tau+2\pi ilz)
  103. ϑ ( n ( x + i y ) ; n i ) = ( 2 n ) - 1 / 4 e π n y 2 s y m b o l e n , 0 ( Γ y , x , 0 ) \vartheta(n(x+iy);ni)=(2n)^{-1/4}e^{\pi ny^{2}}symbol{e}_{n,0}(\Gamma\langle y% ,x,0\rangle)
  104. f f
  105. \mathbb{C}
  106. n n
  107. τ \tau
  108. Im ( τ ) > 0 \mathrm{Im}(\tau)>0
  109. [ b a ] [^{a}_{b}]
  110. f ( z + 1 ) = exp ( π i a ) f ( z ) f(z+1)=\exp(\pi ia)f(z)
  111. f ( z + τ ) = exp ( π i b ) exp ( - π i n ( 2 z + τ ) ) f ( z ) f(z+\tau)=\exp(\pi ib)\exp(-\pi in(2z+\tau))f(z)
  112. n n
  113. τ \tau
  114. [ b a ] [^{a}_{b}]
  115. Θ n [ b a ] ( τ , A ) \Theta_{n}[^{a}_{b}](\tau,A)
  116. dim Θ n [ b a ] ( τ , A ) = n \dim\Theta_{n}[^{a}_{b}](\tau,A)=n
  117. Θ n [ 0 0 ] ( i , A ) \Theta_{n}[^{0}_{0}](i,A)
  118. θ n , m ( z ) = l exp [ - π n ( l + m n ) 2 + 2 π i ( l n + m ) z ) ] \theta_{n,m}(z)=\sum_{l\in\mathbb{Z}}\exp[-\pi n(l+{m\over n})^{2}+2\pi i(ln+m% )z)]
  119. θ n , m ( x + i y ) = ( 2 n ) - 1 / 4 e π n y 2 s y m b o l e n , m ( Γ y , x , 0 ) \theta_{n,m}(x+iy)=(2n)^{-1/4}e^{\pi ny^{2}}symbol{e}_{n,m}(\Gamma\langle y,x,% 0\rangle)

Well-separated_pair_decomposition.html

  1. S d S\subset\mathbb{R}^{d}
  2. ( A i , B i ) (A_{i},B_{i})
  3. p , q S p,q\in S
  4. A , B A,B
  5. d \mathbb{R}^{d}
  6. R ( X ) R(X)
  7. X X
  8. s > 0 s>0
  9. A A
  10. B B
  11. R ( A ) R(A)
  12. R ( B ) R(B)
  13. ρ \rho
  14. s ρ s\rho
  15. S S
  16. ( A 1 , B 1 ) , ( A 2 , B 2 ) , , ( A m , B m ) (A_{1},B_{1}),(A_{2},B_{2}),\ldots,(A_{m},B_{m})
  17. S S
  18. p , q S p,q\in S
  19. i i
  20. 1 i m 1\leq i\leq m
  21. p A i p\in A_{i}
  22. q B i q\in B_{i}
  23. q A i q\in A_{i}
  24. p B i p\in B_{i}
  25. O ( s d n ) O(s^{d}n)
  26. O ( n lg n ) O(n\lg n)
  27. O ( n lg n ) O(n\lg n)
  28. O ( n lg n + k ) O(n\lg n+k)
  29. O ( n lg n ) O(n\lg n)
  30. ( 1 - ϵ ) (1-\epsilon)
  31. O ( n lg n ) O(n\lg n)
  32. O ( n lg n ) O(n\lg n)

Wells_graph.html

  1. { 5 , 4 , 1 , 1 ; 1 , 1 , 4 , 5 } \{5,4,1,1;1,1,4,5\}
  2. 5 1 5 8 1 1 0 ( - 5 ) 8 ( - 3 ) 5 5^{1}\sqrt{5}^{8}1^{1}0(-\sqrt{5})^{8}(-3)^{5}

Wen-Hao_Zhang.html

  1. L p Lp
  2. V m Vm
  3. H g C l 2 HgCl2

Weyl_distance_function.html

  1. Σ ( W , S ) Σ(W,S)
  2. Σ ( W , S ) Σ(W,S)
  3. C 0 , C 1 , , C n . C_{0},C_{1},\dots,C_{n}.
  4. C i - 1 , C i C_{i-1},C_{i}
  5. s i s_{i}
  6. C 0 C_{0}
  7. C n C_{n}
  8. δ ( C 0 , C n ) = s 1 s 2 s n . \delta(C_{0},C_{n})=s_{1}s_{2}\cdots s_{n}.
  9. C 0 C_{0}
  10. C n C_{n}
  11. C 0 , , C n C_{0},\dots,C_{n}
  12. C 0 C_{0}
  13. C n C_{n}
  14. δ ( C 0 , C n ) = s 1 s 2 s n \delta(C_{0},C_{n})=s_{1}s_{2}\cdots s_{n}
  15. s i s_{i}
  16. C 0 C_{0}
  17. C n C_{n}
  18. d ( C 0 , C n ) d(C_{0},C_{n})
  19. δ ( C 0 , C n ) \delta(C_{0},C_{n})
  20. d ( C 0 , C n ) = ( δ ( C 0 , C n ) ) d(C_{0},C_{n})=\ell(\delta(C_{0},C_{n}))
  21. δ ( C , D ) = 1 \delta(C,D)=1
  22. C = D C=D
  23. d ( C , D ) = 0 d(C,D)=0
  24. C = D C=D
  25. δ ( C , D ) = δ ( D , C ) - 1 \delta(C,D)=\delta(D,C)^{-1}
  26. d ( C , D ) = d ( D , C ) d(C,D)=d(D,C)
  27. δ ( C , C ) = s S \delta(C^{\prime},C)=s\in S
  28. δ ( C , D ) = w \delta(C,D)=w
  29. δ ( C , D ) \delta(C^{\prime},D)
  30. ( s w ) = ( w ) + 1 \ell(sw)=\ell(w)+1
  31. δ ( C , D ) = s w \delta(C^{\prime},D)=sw
  32. δ ( C , D ) = w \delta(C,D)=w
  33. s S s\in S
  34. C C^{\prime}
  35. δ ( C , C ) = s \delta(C^{\prime},C)=s
  36. δ ( C , D ) = s w \delta(C^{\prime},D)=sw
  37. δ : C × C W \delta:C\times C\to W
  38. δ δ

Weyl−Lewis−Papapetrou_coordinates.html

  1. d s 2 = - e 2 ν d t 2 + ρ 2 B 2 e - 2 ν ( d ϕ - ω d t ) 2 + e 2 ( λ - ν ) ( d ρ 2 + d z 2 ) ds^{2}=-e^{2\nu}dt^{2}+\rho^{2}B^{2}e^{-2\nu}(d\phi-\omega dt)^{2}+e^{2(% \lambda-\nu)}(d\rho^{2}+dz^{2})

Weyr_canonical_form.html

  1. λ \lambda
  2. n × n n\times n
  3. W W
  4. n 1 + n 2 + + n r = n n_{1}+n_{2}+\cdots+n_{r}=n
  5. n n
  6. n 1 n 2 n r 1 n_{1}\geq n_{2}\geq\cdots\geq n_{r}\geq 1
  7. W W
  8. r × r r\times r
  9. ( W i j ) (W_{ij})
  10. ( i , j ) (i,j)
  11. W i j W_{ij}
  12. n i × n j n_{i}\times n_{j}
  13. W i i W_{ii}
  14. n i × n i n_{i}\times n_{i}
  15. λ I \lambda I
  16. i = 1 , , r i=1,\ldots,r
  17. W i , i + 1 W_{i,i+1}
  18. n i × n i + 1 n_{i}\times n_{i+1}
  19. i = 1 , , r - 1 i=1,\ldots,r-1
  20. W i j = 0 W_{ij}=0
  21. j i , i + 1 j\neq i,i+1
  22. W W
  23. ( n 1 , n 2 , , n r ) (n_{1},n_{2},\ldots,n_{r})
  24. W = W=
  25. = [ W 11 W 12 W 22 W 23 W 33 W 34 W 44 ] =\begin{bmatrix}W_{11}&W_{12}&&\\ &W_{22}&W_{23}&\\ &&W_{33}&W_{34}\\ &&&W_{44}\\ \end{bmatrix}
  26. n = 10 n=10
  27. n 1 = 4 , n 2 = 2 , n 3 = 2 , n 4 = 1 n_{1}=4,n_{2}=2,n_{3}=2,n_{4}=1
  28. W W
  29. ( 4 , 2 , 2 , 1 ) (4,2,2,1)
  30. W 11 = [ λ 0 0 0 0 λ 0 0 0 0 λ 0 0 0 0 λ ] = λ I 4 , W 22 = [ λ 0 0 λ ] = λ I 2 , W 33 = [ λ 0 0 λ ] = λ I 2 , W 44 = [ λ ] = λ I 1 W_{11}=\begin{bmatrix}\lambda&0&0&0\\ 0&\lambda&0&0\\ 0&0&\lambda&0\\ 0&0&0&\lambda\\ \end{bmatrix}=\lambda I_{4},\quad W_{22}=\begin{bmatrix}\lambda&0\\ 0&\lambda&\\ \end{bmatrix}=\lambda I_{2},\quad W_{33}=\begin{bmatrix}\lambda&0\\ 0&\lambda&\\ \end{bmatrix}=\lambda I_{2},\quad W_{44}=\begin{bmatrix}\lambda\\ \end{bmatrix}=\lambda I_{1}
  31. W 12 = [ 1 0 0 1 0 0 0 0 ] , W 23 = [ 1 0 0 1 ] , W 34 = [ 1 0 ] . W_{12}=\begin{bmatrix}1&0\\ 0&1\\ 0&0\\ 0&0\\ \end{bmatrix},\quad W_{23}=\begin{bmatrix}1&0\\ 0&1\\ \end{bmatrix},\quad W_{34}=\begin{bmatrix}1\\ 0\\ \end{bmatrix}.
  32. W W
  33. λ 1 , , λ k \lambda_{1},\ldots,\lambda_{k}
  34. W W
  35. W W
  36. W W
  37. W = [ W 1 W 2 W k ] W=\begin{bmatrix}W_{1}&&&\\ &W_{2}&&\\ &&\ddots&\\ &&&W_{k}\\ \end{bmatrix}
  38. W i W_{i}
  39. λ i \lambda_{i}
  40. i = 1 , , k i=1,\ldots,k
  41. A A
  42. W W
  43. W W
  44. A A
  45. A A
  46. n n
  47. A A
  48. λ 1 , λ 2 , , λ k \lambda_{1},\lambda_{2},\ldots,\lambda_{k}
  49. A A
  50. A = [ λ 1 I + N 1 λ 2 I + N 2 λ k I + N k ] = [ λ 1 I λ 2 I λ k I ] + [ N 1 N 2 N k ] = D + N A=\begin{bmatrix}\lambda_{1}I+N_{1}&&&\\ &\lambda_{2}I+N_{2}&&\\ &&\ddots&\\ &&&\lambda_{k}I+N_{k}\\ \end{bmatrix}=\begin{bmatrix}\lambda_{1}I&&&\\ &\lambda_{2}I&&\\ &&\ddots&\\ &&&\lambda_{k}I\\ \end{bmatrix}+\begin{bmatrix}N_{1}&&&\\ &N_{2}&&\\ &&\ddots&\\ &&&N_{k}\\ \end{bmatrix}=D+N
  51. D D
  52. N N
  53. A A
  54. N i N_{i}
  55. A A
  56. n n
  57. F F
  58. C C
  59. W W
  60. W = C - 1 A C W=C^{-1}AC
  61. A 1 = A A_{1}=A
  62. A 1 A_{1}
  63. A 1 A_{1}
  64. n n
  65. F n F^{n}
  66. P 1 P_{1}
  67. P 1 - 1 A 1 P 1 = [ 0 B 2 0 A 2 ] P_{1}^{-1}A_{1}P_{1}=\begin{bmatrix}0&B_{2}\\ 0&A_{2}\end{bmatrix}
  68. A 2 A_{2}
  69. n n
  70. ( A 1 ) (A_{1})
  71. A 2 A_{2}
  72. A 2 A_{2}
  73. A 2 A_{2}
  74. A 2 A_{2}
  75. n n
  76. ( A 1 ) (A_{1})
  77. P 2 P_{2}
  78. P 2 - 1 A 2 P 2 = [ 0 B 3 0 A 3 ] P_{2}^{-1}A_{2}P_{2}=\begin{bmatrix}0&B_{3}\\ 0&A_{3}\end{bmatrix}
  79. A 2 A_{2}
  80. n n
  81. ( A 1 ) (A_{1})
  82. ( A 2 ) (A_{2})
  83. A 1 , A 2 , A 3 , A_{1},A_{2},A_{3},\ldots
  84. P 1 , P 2 , P 3 , P_{1},P_{2},P_{3},\ldots
  85. A r A_{r}
  86. A A
  87. ( n 1 , n 2 , , n r ) (n_{1},n_{2},\ldots,n_{r})
  88. n i n_{i}
  89. ( A i ) (A_{i})
  90. P = P 1 [ I 0 0 P 2 ] [ I 0 0 P 3 ] [ I 0 0 P r ] P=P_{1}\begin{bmatrix}I&0\\ 0&P_{2}\end{bmatrix}\begin{bmatrix}I&0\\ 0&P_{3}\end{bmatrix}\cdots\begin{bmatrix}I&0\\ 0&P_{r}\end{bmatrix}
  91. I I
  92. X = P - 1 A P X=P^{-1}AP
  93. X X
  94. X = [ 0 X 12 X 13 X 1 , r - 1 X 1 r 0 X 23 X 2 , r - 1 X 2 r 0 X r - 1 , r 0 ] X=\begin{bmatrix}0&X_{12}&X_{13}&\cdots&X_{1,r-1}&X_{1r}\\ &0&X_{23}&\cdots&X_{2,r-1}&X_{2r}\\ &&&\ddots&\\ &&&\cdots&0&X_{r-1,r}\\ &&&&&0\end{bmatrix}
  95. Y r - 1 Y_{r-1}
  96. Y r - 1 X r , r - 1 Y_{r-1}X_{r,r-1}
  97. I r , r - 1 = [ I O ] I_{r,r-1}=\begin{bmatrix}I\\ O\end{bmatrix}
  98. Q 1 = Q_{1}=
  99. ( I , I , , Y r - 1 - 1 , I ) (I,I,\ldots,Y_{r-1}^{-1},I)
  100. Q 1 - 1 X Q 1 Q_{1}^{-1}XQ_{1}
  101. ( r , r - 1 ) (r,r-1)
  102. I r , r - 1 I_{r,r-1}
  103. R 1 R_{1}
  104. R 1 - 1 Q 1 - 1 X Q 1 R 1 R_{1}^{-1}Q_{1}^{-1}XQ_{1}R_{1}
  105. I r , r - 1 I_{r,r-1}
  106. 0
  107. r - 1 r-1
  108. ( r - 1 , r - 2 ) (r-1,r-2)
  109. I r - 1 , r - 2 I_{r-1,r-2}
  110. Q 2 Q_{2}
  111. R 2 R_{2}
  112. r - 2 , r - 3 , , 3 , 2 r-2,r-3,\ldots,3,2
  113. Q 3 , R 3 , , Q r - 2 , R r - 2 , Q r - 1 Q_{3},R_{3},\ldots,Q_{r-2},R_{r-2},Q_{r-1}
  114. W W
  115. C = P 1 diag ( I , P 2 ) diag ( I , P r - 1 ) Q 1 R 1 Q 2 R r - 2 Q r - 1 C=P_{1}\,\text{diag}(I,P_{2})\cdots\,\text{diag}(I,P_{r-1})Q_{1}R_{1}Q_{2}% \cdots R_{r-2}Q_{r-1}
  116. W = C - 1 A C W=C^{-1}AC
  117. n × n n\times n
  118. n n

Whisking_in_animals.html

  1. θ \theta
  2. ϕ \phi
  3. ζ \zeta

Whitney_inequality.html

  1. f C ( [ a , b ] ) f\in C([a,b])
  2. P n P_{n}
  3. n \leq n
  4. E n ( f ) [ a , b ] := inf P n f - P n C ( [ a , b ] ) E_{n}(f)_{[a,b]}:=\inf_{P_{n}}{\|f-P_{n}\|_{C([a,b])}}
  5. k k
  6. f C ( [ a , b ] ) f\in C([a,b])
  7. ω k ( t ) := ω k ( t ; f ; [ a , b ] ) := sup h [ 0 , t ] Δ h k ( f ; ) C ( [ a , b - k h ] ) for t [ 0 , ( b - a ) / k ] , \omega_{k}(t):=\omega_{k}(t;f;[a,b]):=\sup_{h\in[0,t]}\|\Delta_{h}^{k}(f;\cdot% )\|_{C([a,b-kh])}\quad\,\text{ for }\quad t\in[0,(b-a)/k],
  8. ω k ( t ) := ω k ( ( b - a ) / k ) for t > ( b - a ) / k , \omega_{k}(t):=\omega_{k}((b-a)/k)\quad\,\text{ for}\quad t>(b-a)/k,
  9. Δ h k \Delta_{h}^{k}
  10. k k
  11. f C ( [ a , b ] ) f\in C([a,b])
  12. E k - 1 ( f ) [ a , b ] W k ω k ( b - a k ; f ; [ a , b ] ) E_{k-1}(f)_{[a,b]}\leq W_{k}\omega_{k}\left(\frac{b-a}{k};f;[a,b]\right)
  13. W k W_{k}
  14. k k
  15. W ( k ) W(k)
  16. W k W_{k}
  17. x 0 := a , h := b - a k , x j := x + 0 + j h , F ( x ) = a x f ( u ) d u , x_{0}:=a,\quad h:=\frac{b-a}{k},\quad x_{j}:=x+0+jh,\quad F(x)=\int_{a}^{x}f(u% )\,du,
  18. G ( x ) := F ( x ) - L ( x ; F ; x 0 , , x k ) , g ( x ) := G ( x ) , G(x):=F(x)-L(x;F;x_{0},\ldots,x_{k}),\quad g(x):=G^{\prime}(x),
  19. ω k ( t ) := ω k ( t ; f ; [ a , b ] ) ω k ( t ; g ; [ a , b ] ) \omega_{k}(t):=\omega_{k}(t;f;[a,b])\equiv\omega_{k}(t;g;[a,b])
  20. L ( x ; F ; x 0 , , x k ) L(x;F;x_{0},\ldots,x_{k})
  21. F F
  22. { x 0 , , x k } \{x_{0},\ldots,x_{k}\}
  23. x [ a , b ] x\in[a,b]
  24. δ \delta
  25. ( x + k δ ) [ a , b ] (x+k\delta)\in[a,b]
  26. 0 1 Δ t δ k ( g ; x ) d t = ( - 1 ) k g ( x ) + j = 1 k ( - 1 ) k - j ( k j ) 0 1 g ( x + j t δ ) d t \int_{0}^{1}\Delta_{t\delta}^{k}(g;x)\,dt=(-1)^{k}g(x)+\sum_{j=1}^{k}(-1)^{k-j% }{\left({{k}\atop{j}}\right)}\int_{0}^{1}g(x+jt\delta)\,dt
  27. = ( - 1 ) k g ( x ) + j = 1 k ( - 1 ) k - j ( k j ) 1 j δ ( G ( x + j δ ) - G ( x ) ) , =(-1)^{k}g(x)+\sum_{j=1}^{k}{(-1)^{k-j}{\left({{k}\atop{j}}\right)}\frac{1}{j% \delta}(G(x+j\delta)-G(x))},
  28. | g ( x ) | 0 1 | Δ t δ k ( g ; x ) | d t + 2 | δ | G C ( [ a , b ] ) j = 1 k ( k j ) 1 j ω k ( | δ | ) + 1 | δ | 2 k + 1 G C ( [ a , b ] ) |g(x)|\leq\int_{0}^{1}|\Delta_{t\delta}^{k}(g;x)|\,dt+\frac{2}{|\delta|}\|G\|_% {C([a,b])}\sum_{j=1}^{k}{\left({{k}\atop{j}}\right)}\frac{1}{j}\leq\omega_{k}(% |\delta|)+\frac{1}{|\delta|}2^{k+1}\|G\|_{C([a,b])}
  29. G C ( [ a , b ] ) h ω k ( h ) \|G\|_{C([a,b])}\leq h\omega_{k}(h)
  30. E k - 1 ( f ) [ a , b ] g C ( [ a , b ] ) ω k ( | δ | ) + 1 | δ | h 2 k + 1 ω k ( h ) . E_{k-1}(f)_{[a,b]}\leq\|g\|_{C([a,b])}\leq\omega_{k}(|\delta|)+\frac{1}{|% \delta|}h2^{k+1}\omega_{k}(h).
  31. δ \delta
  32. h | δ | h / 2 h\geq|\delta|\geq h/2
  33. W ( 1 ) = 1 / 2 W(1)=1/2
  34. W ( 2 ) 1 W(2)\leq 1
  35. W ( k ) 1 W(k)\leq 1
  36. k k
  37. W ( 2 ) = 1 2 , 8 15 W ( 3 ) 0.7 W ( 4 ) 3.3 W ( 5 ) 10.4 W(2)=\frac{1}{2},\quad\frac{8}{15}\leq W(3)\leq 0.7\quad W(4)\leq 3.3\quad W(5% )\leq 10.4
  38. W ( k ) 1 2 , k W(k)\geq\frac{1}{2},\quad k\in\mathbb{N}
  39. W ( k ) = O ( k 2 k ) W(k)=O(k^{2k})
  40. W ( k ) ( k + 1 ) k k W(k)\leq(k+1)k^{k}
  41. W ( k ) = O ( k ln k ) W(k)=O(k\ln k)
  42. W ( k ) = O ( k ) W(k)=O(k)
  43. W ( k ) 1 W(k)\leq 1
  44. k k
  45. W ( k ) 6 W(k)\leq 6
  46. k k
  47. W ( k ) 2 W(k)\leq 2
  48. k 82000 k\leq 82000
  49. W ( k ) 2 + 1 e 2 W(k)\leq 2+\frac{1}{e^{2}}
  50. k k

Wichmann-Hill.html

  1. 6.95 * 10 12 6.95*10^{12}

Widening_(computer_science).html

  1. \infty

Wiener_connector.html

  1. d ( u , v ) d(u,v)
  2. u u
  3. v v
  4. S S
  5. W ( S ) W(S)
  6. W ( S ) = ( u , v ) S d ( u , v ) W(S)=\sum_{(u,v)\in S}d(u,v)
  7. V V
  8. E E
  9. Q V Q\subseteq V
  10. H V H\subseteq V
  11. arg min H W ( H Q ) \operatorname*{arg\,min}_{H}W(H\cup Q)
  12. H H
  13. H H
  14. 2 O ( n ) 2^{O(n)}
  15. O ( q ( m log n + n log 2 n ) ) O(q(m\log n+n\log^{2}n))

Wiener–Lévy_theorem.html

  1. f f
  2. 1 / f 1/f
  3. F ( θ ) = k = - c k e i k θ , θ [ 0 , 2 π ] F(\theta)=\sum\limits_{k=-\infty}^{\infty}{{c_{k}}{e^{ik\theta}}},\theta\in[0,% 2\pi]
  4. F = k = - | < m t p l > c k | . T h e v a l u e s o f < m a t h > F ( θ ) \left\|F\right\|=\sum\limits_{k=-\infty}^{\infty}{\left|<mtpl>{{c_{k}}}\right|% }.Thevaluesof<math>F(\theta)
  5. C C
  6. H ( t ) H(t)
  7. C C
  8. H [ F ( θ ) ] H[F(\theta)]
  9. H ( θ ) = l n ( θ ) H(\theta)=ln(\theta)
  10. F ( θ ) = k = 0 p k e i k θ , ( k = 0 p k = 1 ) F(\theta)=\sum\limits_{k=0}^{\infty}{{p_{k}}{e^{ik\theta}}},(\sum\limits_{k=0}% ^{\infty}{{p_{k}}=1})
  11. F ( θ ) F(\theta)
  12. F ( θ ) F(\theta)
  13. H [ F ( θ ) ] = l n ( k = 0 p k e i k θ ) = k = 0 c k e i k θ H[F(\theta)]=ln(\sum\limits_{k=0}^{\infty}{{p_{k}}{e^{ik\theta}}})=\sum\limits% _{k=0}^{\infty}{{c_{k}}{e^{ik\theta}}}
  14. H = k = 0 | < m t p l Align g t ; c k | . T h e s t a t i s t i c a l a p p l i c a t i o n o f t h i s e x a m p l e c a n b e f o u n d i n d i s c r e t e p s e u d o [ [ c o m p o u n d P o i s s o n d i s t r i b u t i o n | c o m p o u n d P o i s s o n d i s t r i b u t i o n ] ] a n d [ [ Z e r o - i n f l a t e d m o d e l | Z e r o - i n f l a t e d m o d e l ] ] . = = S e e a l s o = = * [ [ W i e n e r s t h e o r e m ( d i s a m b i g u a t i o n ) | W i e n e r s t h e o r e m ] ] = = R e f e r e n c e s = = R e f l i s t D E F A U L T S O R T : W i e n e r - L e v y t h e o r e m " [ C a t e g o r y : T h e o r e m s i n F o u r i e r a n a l y s i s C a t e g o r y : T h e o r e m s i n F o u r i e r a n a l y s i s ] < r e f e r e n c e s > < / r e f e r e n c e s > \left\|H\right\|=\sum\limits_{k=0}^{\infty}{\left|<mtpl&gt;{{c_{k}}}\right|}.% \par Thestatisticalapplicationofthisexamplecanbefoundindiscretepseudo[[% compound_{P}oisson_{d}istribution|compoundPoissondistribution]]and[[Zero-% inflated_{m}odel|Zero-inflatedmodel]].\par ==Seealso==\par *[[Wiener^{\prime}s% _{t}heorem_{(}disambiguation)|Wiener^{\prime}stheorem]]\par ==References==% \par {{Reflist}}{{DEFAULTSORT:Wiener-Levytheorem}}"\par [Category:Theorems_{i}% n_{F}ourier_{a}nalysisCategory:TheoremsinFourieranalysis]\par <references></references>

With_high_probability.html

  1. 1 - 1 / n 1-1/n

Women's_Flat_Track_Derby_Association_Rankings.html

  1. Total Game Points earned = Win/Loss Factor × Opponent Strength × 100 \,\text{Total Game Points earned}=\,\text{Win/Loss Factor}\times\,\text{% Opponent Strength}\times 100

Worst-case_distance.html

  1. s s
  2. Θ \Theta
  3. f f
  4. μ \mu
  5. σ \sigma
  6. f f
  7. W C D = f - μ σ WCD={f-\mu\over\sigma}
  8. C ^ p k \hat{C}_{pk}
  9. C p k = f - μ 3 σ {C}_{pk}={f-{\mu}\over 3{\sigma}}

X_+_Y_sorting.html

  1. X X
  2. Y Y
  3. ( x , y ) X × Y (x,y)\in X\times Y
  4. x + y x+y
  5. O ( n m l o g ( n m ) ) O(nmlog(nm))
  6. n n
  7. m m
  8. O ( n ² l o g n ² ) = O ( n ² l o g n ) O(n²logn²)=O(n²logn)
  9. m = n m=n
  10. n × m n×m
  11. O ( n ² ) O(n²)
  12. O ( n ² l o g n ) O(n²logn)
  13. w w
  14. O ( n l o g n ) O(nlogn)
  15. x x
  16. y y

Y_and_H_transforms.html

  1. Y Y
  2. 𝐇 \mathbf{H}
  3. ν ν
  4. f ( r ) f(r)
  5. Y Y
  6. ν ν
  7. F ( k ) = 0 f ( r ) Y ν ( k r ) k r d r F(k)=\int_{0}^{\infty}f(r)Y_{\nu}(kr)\sqrt{kr}\,dr
  8. 𝐇 \mathbf{H}
  9. F ( k ) F(k)
  10. 𝐇 \mathbf{H}
  11. ν ν
  12. f ( r ) = 0 F ( k ) 𝐇 ν ( k r ) k r d k f(r)=\int_{0}^{\infty}F(k)\mathbf{H}_{\nu}(kr)\sqrt{kr}\,dk
  13. Y Y
  14. 𝐇 \mathbf{H}
  15. Y Y
  16. 𝐇 \mathbf{H}
  17. Y Y
  18. 𝐇 \mathbf{H}

YaDICs.html

  1. S S D ( μ , , ) = 1 | Ω F | x i Ω F ( ( x i ) - ( T μ ( x i ) ) ) 2 SSD(\mu,\mathcal{I_{F}},\mathcal{I_{M}})=\dfrac{1}{\left|\Omega_{F}\right|}% \sum_{x_{i}\in\Omega_{F}}\left(\mathcal{I_{F}}(x_{i})-\mathcal{I_{M}}({T}_{\mu% }(x_{i}))\right)^{2}
  2. \mathcal{I_{F}}
  3. \mathcal{I_{M}}
  4. Ω F \Omega_{F}
  5. | Ω F | \left|\Omega_{F}\right|
  6. T μ {T}_{\mu}
  7. T μ ( x ) = x + { Φ ( x ) } t { μ } T_{\mu}(x)=x+\left\{\Phi(x)\right\}^{t}\left\{\mu\right\}
  8. N C C ( μ , , ) = x i Ω F ( ( x i ) - ¯ ) ( ( T μ ( x i ) ) - ¯ ) x i Ω F ( ( x i ) - ¯ ) 2 x i Ω F ( ( T μ ( x i ) ) - ¯ ) 2 NCC(\mu,\mathcal{I_{F}},\mathcal{I_{M}})=\dfrac{\sum_{x_{i}\in\Omega_{F}}\left% (\mathcal{I_{F}}(x_{i})-\overline{\mathcal{I_{F}}}\right)\left(\mathcal{I_{M}}% ({T}_{\mu}(x_{i}))-\overline{\mathcal{I_{M}}}\right)}{\sqrt{\sum_{x_{i}\in% \Omega_{F}}\left(\mathcal{I_{F}}(x_{i})-\overline{\mathcal{I_{F}}}\right)^{2}% \sum_{x_{i}\in\Omega_{F}}\left(\mathcal{I_{M}}({T}_{\mu}(x_{i}))-\overline{% \mathcal{I_{M}}}\right)^{2}}}
  9. ¯ \overline{\mathcal{I_{F}}}
  10. ¯ \overline{\mathcal{I_{M}}}
  11. S S D ( μ , , ) μ = 2 | Ω F | x i Ω F ( ( x i ) - ( T μ ( x i ) ) ) ( T μ ( x i ) μ = 2 | Ω F | x i Ω F ( ( x i ) - ( T μ ( x i ) ) ) ( T μ ( x i ) μ ) t ( T μ ( x i ) ) x \begin{array}[]{lcl}\dfrac{\partial SSD(\mu,\mathcal{I_{F}},\mathcal{I_{M}})}{% \partial\mu}&=&\dfrac{2}{\left|\Omega_{F}\right|}\sum_{x_{i}\in\Omega_{F}}% \left(\mathcal{I_{F}}(x_{i})-\mathcal{I_{M}}({T}_{\mu}(x_{i}))\right)\dfrac{% \partial\mathcal{I_{M}}({T}_{\mu}(x_{i})}{\partial\mu}\\ &=&\dfrac{2}{\left|\Omega_{F}\right|}\sum_{x_{i}\in\Omega_{F}}\left(\mathcal{I% _{F}}(x_{i})-\mathcal{I_{M}}({T}_{\mu}(x_{i}))\right)\left(\dfrac{\partial{T}_% {\mu}(x_{i})}{\partial\mu}\right)^{t}\dfrac{\partial\mathcal{I_{M}}({T}_{\mu}(% x_{i}))}{\partial x}\\ \end{array}
  12. μ k + 1 = μ k + α k d k \mu_{k+1}=\mu_{k}+\alpha_{k}d_{k}
  13. d k = - γ k 𝒞 ( μ , , ) μ d_{k}=-\gamma_{k}\dfrac{\partial\mathcal{C}(\mu,\mathcal{I_{F}},\mathcal{I_{M}% })}{\partial\mu}
  14. γ k \gamma_{k}
  15. γ k \gamma_{k}
  16. \Longrightarrow
  17. γ k = [ 𝒞 ( μ , , ) μ 𝒞 ( μ , , ) μ t ] - 1 \gamma_{k}=\left[\dfrac{\partial\mathcal{C}(\mu,\mathcal{I_{F}},\mathcal{I_{M}% })}{\partial\mu}\dfrac{\partial\mathcal{C}(\mu,\mathcal{I_{F}},\mathcal{I_{M}}% )}{\partial\mu}^{t}\right]^{-1}
  18. \Longrightarrow

YCgCo.html

  1. [ Y C g C o ] = [ 1 / 4 1 / 2 1 / 4 - 1 / 4 1 / 2 - 1 / 4 1 / 2 0 - 1 / 2 ] [ R G B ] \begin{bmatrix}Y\\ Cg\\ Co\end{bmatrix}=\begin{bmatrix}1/4&1/2&1/4\\ -1/4&1/2&-1/4\\ 1/2&0&-1/2\end{bmatrix}\cdot\begin{bmatrix}R\\ G\\ B\end{bmatrix}
  2. [ R G B ] = [ 1 - 1 1 1 1 0 1 - 1 - 1 ] [ Y C g C o ] \begin{bmatrix}R\\ G\\ B\end{bmatrix}=\begin{bmatrix}1&-1&1\\ 1&1&0\\ 1&-1&-1\end{bmatrix}\cdot\begin{bmatrix}Y\\ Cg\\ Co\end{bmatrix}

Yoneda_product.html

  1. Ext n ( M , N ) Ext m ( L , M ) Ext n + m ( L , N ) \operatorname{Ext}^{n}(M,N)\otimes\operatorname{Ext}^{m}(L,M)\to\operatorname{% Ext}^{n+m}(L,N)
  2. Hom ( M , N ) Hom ( L , M ) Hom ( L , N ) , f g f g . \operatorname{Hom}(M,N)\otimes\operatorname{Hom}(L,M)\to\operatorname{Hom}(L,N% ),\,f\otimes g\mapsto f\circ g.

Yuri_Kovchegov.html

  1. - 1 4 G μ ν a G a μ ν -\frac{1}{4}G^{a}_{\mu\nu}G^{\mu\nu}_{a}

Zak_transform.html

  1. Z a [ f ] ( t , w ) = a k = - f ( a t + a k ) e - 2 π k w i Z_{a}[f](t,w)=\sqrt{a}\sum_{k=-\infty}^{\infty}f(at+ak)e^{-2\pi kwi}
  2. Z [ f ] ( t , w ) = k = - f ( t + k ) e - 2 π k w i Z[f](t,w)=\sum_{k=-\infty}^{\infty}f(t+k)e^{-2\pi kwi}
  3. Z [ f ] ( t , ν ) = k = - f ( t + k ) e - k ν i Z[f](t,\nu)=\sum_{k=-\infty}^{\infty}f(t+k)e^{-k\nu i}
  4. Z T [ f ] ( t , w ) = T k = - f ( t + k T ) e - 2 π k w T i Z_{T}[f](t,w)=\sqrt{T}\sum_{k=-\infty}^{\infty}f(t+kT)e^{-2\pi kwTi}
  5. ϕ ( t ) = { 1 , 0 t < 1 0 , otherwise \phi(t)=\begin{cases}1,&0\leq t<1\\ 0,&\,\text{otherwise}\end{cases}
  6. Z [ ϕ ] ( t , w ) = e - 2 π - t w i Z[\phi](t,w)=e^{-2\pi\lceil-t\rceil wi}
  7. - t \lceil-t\rceil
  8. - t -t
  9. Z [ a f + b g ] ( t , w ) = a Z [ f ] ( t , w ) + b Z [ g ] ( t , w ) Z[af+bg](t,w)=aZ[f](t,w)+bZ[g](t,w)
  10. Z [ f ] ( t , w + 1 ) = Z [ f ] ( t , w ) Z[f](t,w+1)=Z[f](t,w)
  11. Z [ f ] ( t + 1 , w ) = e 2 π w i Z [ f ] ( t , w ) Z[f](t+1,w)=e^{2\pi wi}Z[f](t,w)
  12. Z [ f ¯ ] ( t , w ) = Z [ f ] ¯ ( t , - w ) Z[\bar{f}](t,w)=\overline{Z[f]}(t,-w)
  13. Z [ f ] ( t , w ) = Z [ f ] ( - t , - w ) Z[f](t,w)=Z[f](-t,-w)
  14. Z [ f ] ( t , w ) = - Z [ f ] ( - t , - w ) Z[f](t,w)=-Z[f](-t,-w)
  15. \star
  16. Z [ f g ] ( t , w ) = Z [ f ] ( t , w ) Z [ g ] ( t , w ) Z[f\star g](t,w)=Z[f](t,w)\star Z[g](t,w)
  17. f ( t ) = 0 1 Z [ f ] ( t , w ) d w . f(t)=\int_{0}^{1}Z[f](t,w)\,dw.
  18. Z [ f ] ( n , w ) = k = - f ( n + k ) e - 2 π k w i . Z[f](n,w)=\sum_{k=-\infty}^{\infty}f(n+k)e^{-2\pi kwi}.
  19. f ( n ) = 0 1 Z [ f ] ( n , w ) d w . f(n)=\int_{0}^{1}Z[f](n,w)\,dw.

Zassenhaus_algorithm.html

  1. V V
  2. U U
  3. W W
  4. V V
  5. U = u 1 , , u n U=\langle u_{1},\ldots,u_{n}\rangle
  6. W = w 1 , , w k . W=\langle w_{1},\ldots,w_{k}\rangle.
  7. B 1 , , B m B_{1},\ldots,B_{m}
  8. u i u_{i}
  9. w i w_{i}
  10. u i = j = 1 m a i , j B j u_{i}=\sum_{j=1}^{m}a_{i,j}B_{j}
  11. w i = j = 1 m b i , j B j . w_{i}=\sum_{j=1}^{m}b_{i,j}B_{j}.
  12. U + W U+W
  13. U W U\cap W
  14. ( ( n + k ) × ( 2 m ) ) ((n+k)\times(2m))
  15. ( a 1 , 1 a 1 , 2 a 1 , m a 1 , 1 a 1 , 2 a 1 , m a n , 1 a n , 2 a n , m a n , 1 a n , 2 a n , m b 1 , 1 b 1 , 2 b 1 , m 0 0 0 b k , 1 b k , 2 b k , m 0 0 0 ) \begin{pmatrix}a_{1,1}&a_{1,2}&\cdots&a_{1,m}&a_{1,1}&a_{1,2}&\cdots&a_{1,m}\\ \vdots&\vdots&&\vdots&\vdots&\vdots&&\vdots\\ a_{n,1}&a_{n,2}&\cdots&a_{n,m}&a_{n,1}&a_{n,2}&\cdots&a_{n,m}\\ b_{1,1}&b_{1,2}&\cdots&b_{1,m}&0&0&\cdots&0\\ \vdots&\vdots&&\vdots&\vdots&\vdots&&\vdots\\ b_{k,1}&b_{k,2}&\cdots&b_{k,m}&0&0&\cdots&0\end{pmatrix}
  16. ( c 1 , 1 c 1 , 2 c 1 , m * * * c q , 1 c q , 2 c q , m * * * 0 0 0 d 1 , 1 d 1 , 2 d 1 , m 0 0 0 d l , 1 d l , 2 d l , m 0 0 0 0 0 0 0 0 0 0 0 0 ) \begin{pmatrix}c_{1,1}&c_{1,2}&\cdots&c_{1,m}&*&*&\cdots&*\\ \vdots&\vdots&&\vdots&\vdots&\vdots&&\vdots\\ c_{q,1}&c_{q,2}&\cdots&c_{q,m}&*&*&\cdots&*\\ 0&0&\cdots&0&d_{1,1}&d_{1,2}&\cdots&d_{1,m}\\ \vdots&\vdots&&\vdots&\vdots&\vdots&&\vdots\\ 0&0&\cdots&0&d_{l,1}&d_{l,2}&\cdots&d_{l,m}\\ 0&0&\cdots&0&0&0&\cdots&0\\ \vdots&\vdots&&\vdots&\vdots&\vdots&&\vdots\\ 0&0&\cdots&0&0&0&\cdots&0\end{pmatrix}
  17. * *
  18. ( c p , 1 , c p , 2 , , c p , m ) (c_{p,1},c_{p,2},\ldots,c_{p,m})
  19. p { 1 , , q } p\in\{1,\ldots,q\}
  20. ( d p , 1 , , d p , m ) (d_{p,1},\ldots,d_{p,m})
  21. p { 1 , , l } p\in\{1,\ldots,l\}
  22. ( y 1 , , y q ) (y_{1},\ldots,y_{q})
  23. y i := j = 1 m c i , j B j y_{i}:=\sum_{j=1}^{m}c_{i,j}B_{j}
  24. U + W U+W
  25. ( z 1 , , z l ) (z_{1},\ldots,z_{l})
  26. z i := j = 1 m d i , j B j z_{i}:=\sum_{j=1}^{m}d_{i,j}B_{j}
  27. U W U\cap W
  28. π 1 : V × V V , ( a , b ) a \pi_{1}:V\times V\to V,(a,b)\mapsto a
  29. H := { ( u , u ) u U } + { ( w , 0 ) w W } V × V . H:=\{(u,u)\mid u\in U\}+\{(w,0)\mid w\in W\}\leq V\times V.
  30. π 1 ( H ) = U + W \pi_{1}(H)=U+W
  31. H ( 0 × V ) = 0 × ( U W ) H\cap(0\times V)=0\times(U\cap W)
  32. H ( 0 × V ) H\cap(0\times V)
  33. π 1 | H {\pi_{1}|}_{H}
  34. H H
  35. dim ( H ) = dim ( U + W ) + dim ( U W ) \dim(H)=\dim(U+W)+\dim(U\cap W)
  36. H H
  37. m m
  38. y i y_{i}
  39. U + W U+W
  40. ( 0 , z i ) (0,z_{i})
  41. z i 0 z_{i}\neq 0
  42. H ( 0 × V ) H\cap(0\times V)
  43. ( y i , * ) (y_{i},*)
  44. ( 0 , z i ) (0,z_{i})
  45. H H
  46. dim ( U W ) \dim(U\cap W)
  47. z i z_{i}
  48. z i z_{i}
  49. U W U\cap W
  50. U = ( 1 - 1 0 1 ) , ( 0 0 1 - 1 ) U=\left\langle\begin{pmatrix}1\\ -1\\ 0\\ 1\end{pmatrix},\begin{pmatrix}0\\ 0\\ 1\\ -1\end{pmatrix}\right\rangle
  51. W = ( 5 0 - 3 3 ) , ( 0 5 - 3 - 2 ) W=\left\langle\begin{pmatrix}5\\ 0\\ -3\\ 3\end{pmatrix},\begin{pmatrix}0\\ 5\\ -3\\ -2\end{pmatrix}\right\rangle
  52. 4 \mathbb{R}^{4}
  53. ( 2 + 2 ) × ( 2 4 ) (2+2)\times(2\cdot 4)
  54. ( 1 - 1 0 1 1 - 1 0 1 0 0 1 - 1 0 0 1 - 1 5 0 - 3 3 0 0 0 0 0 5 - 3 - 2 0 0 0 0 ) . \begin{pmatrix}1&-1&0&1&&1&-1&0&1\\ 0&0&1&-1&&0&0&1&-1\\ \\ 5&0&-3&3&&0&0&0&0\\ 0&5&-3&-2&&0&0&0&0\end{pmatrix}.
  55. ( 1 0 0 0 * * * * 0 1 0 - 1 * * * * 0 0 1 - 1 * * * * 0 0 0 0 1 - 1 0 1 ) \begin{pmatrix}1&0&0&0&&*&*&*&*\\ 0&1&0&-1&&*&*&*&*\\ 0&0&1&-1&&*&*&*&*\\ \\ 0&0&0&0&&1&-1&0&1\end{pmatrix}
  56. * *
  57. ( ( 1 0 0 0 ) , ( 0 1 0 - 1 ) , ( 0 0 1 - 1 ) ) \left(\begin{pmatrix}1\\ 0\\ 0\\ 0\end{pmatrix},\begin{pmatrix}0\\ 1\\ 0\\ -1\end{pmatrix},\begin{pmatrix}0\\ 0\\ 1\\ -1\end{pmatrix}\right)
  58. U + W U+W
  59. ( ( 1 - 1 0 1 ) ) \left(\begin{pmatrix}1\\ -1\\ 0\\ 1\end{pmatrix}\right)
  60. U W U\cap W

Zemor's_decoding_algorithm.html

  1. d d
  2. G G
  3. G G
  4. ( V , E ) \left(V,E\right)
  5. V V
  6. E E
  7. V V
  8. A A
  9. \cup
  10. B B
  11. A A
  12. \cap
  13. B B
  14. \emptyset
  15. A A
  16. B B
  17. n n
  18. | A | = | B | = n |A|=|B|=n
  19. E E
  20. N N
  21. n d nd
  22. E E
  23. A A
  24. B B
  25. E ( v ) E(v)
  26. v v
  27. V V
  28. E ( v ) E(v)
  29. v V v\in V
  30. 𝔽 = G F ( 2 ) \mathbb{F}=GF(2)
  31. x = ( x e ) , e E x=(x_{e}),e\in E
  32. 𝔽 N \mathbb{F}^{N}
  33. E ( v ) E(v)
  34. ( x ) v (x)_{v}
  35. A A
  36. B B
  37. x 𝔽 N x\in\mathbb{F}^{N}
  38. n n
  39. ( x ) v 𝔽 d \left(x\right)_{v}\in\mathbb{F}^{d}
  40. v v
  41. A A
  42. C C
  43. C o C_{o}
  44. [ d , r o d , δ ] [d,r_{o}d,\delta]
  45. q q
  46. 2 2
  47. v V v\in V
  48. v ( 1 ) , v ( 2 ) , , v ( d ) v(1),v(2),\ldots,v(d)
  49. d d
  50. E E
  51. v v
  52. x e x_{e}
  53. e e
  54. E E
  55. C C
  56. x = ( x 1 , x 2 , , x N ) x=\left(x_{1},x_{2},\ldots,x_{N}\right)
  57. { 0 , 1 } N \{0,1\}^{N}
  58. v v
  59. V V
  60. ( x v ( 1 ) , x v ( 2 ) , , x v ( d ) ) \left(x_{v(1)},x_{v(2)},\ldots,x_{v(d)}\right)
  61. C o C_{o}
  62. E E
  63. 2 2
  64. V V
  65. V V
  66. E E
  67. d d
  68. G G
  69. C C
  70. ( G , C o ) \left(G,C_{o}\right)
  71. G G
  72. C o C_{o}
  73. ( G , C o ) \left(G,C_{o}\right)
  74. v v
  75. v ( 1 ) , v ( 2 ) , , v ( d ) v(1),v(2),\ldots,v(d)
  76. C C
  77. x v C o x_{v}\in C_{o}
  78. v A , B v\in A,B
  79. C C
  80. [ N , K , D ] [N,K,D]
  81. 𝔽 \mathbb{F}
  82. C o C_{o}
  83. C C
  84. C = { c 𝔽 N : ( c ) v C o } C=\{c\in\mathbb{F}^{N}:(c)_{v}\in C_{o}\}
  85. v V v\in V
  86. ( x ) v = ( x e 1 , x e 2 , x e 3 , x e 4 ) C o (x)_{v}=\left(x_{e1},x_{e2},x_{e3},x_{e4}\right)\in C_{o}
  87. G G
  88. C C
  89. G G
  90. λ \lambda
  91. G G
  92. d d
  93. ( K N ) 2 r o - 1 \left(\dfrac{K}{N}\right)\geq 2r_{o}-1
  94. R R
  95. m m
  96. n n
  97. ( n , k ) \left(n,k\right)
  98. r = ( k n ) r=\left(\dfrac{k}{n}\right)
  99. k 1 - ( 1 - r ) m k\geq 1-\left(1-r\right)m
  100. R R
  101. K / N K/N
  102. S S
  103. n n
  104. ( n m ) S \left(\dfrac{n}{m}\right)S
  105. m m
  106. ( n ) S \left(n\right)S
  107. ( n - k ) (n-k)
  108. ( n - k ) S \left(n-k\right)S
  109. ( K N ) ( ( n m ) S - ( n - k ) S ( n m ) S ) \left(\dfrac{K}{N}\right)\geq\left(\dfrac{(\dfrac{n}{m})S-(n-k)S}{(\dfrac{n}{m% })S}\right)
  110. 1 - m ( n - k n ) \geq 1-m\left(\dfrac{n-k}{n}\right)
  111. 1 - m ( 1 - r ) \geq 1-m\left(1-r\right)
  112. m m
  113. 2 2
  114. r = r o r=r_{o}
  115. ( K N ) 2 r o - 1 \left(\dfrac{K}{N}\right)\geq 2r_{o}-1
  116. D N ( ( δ - ( λ d ) ) ( 1 - ( λ d ) ) ) 2 D\geq N\left(\dfrac{(\delta-(\dfrac{\lambda}{d}))}{(1-(\dfrac{\lambda}{d})})% \right)^{2}
  117. = N ( δ 2 - O ( λ d ) ) =N\left(\delta^{2}-O\left(\dfrac{\lambda}{d}\right)\right)
  118. ( 1 ) \rightarrow(1)
  119. S S
  120. r r
  121. d d
  122. δ \delta
  123. B B
  124. d d
  125. λ \lambda
  126. C ( B , S ) C(B,S)
  127. 2 r o - 1 2r_{o}-1
  128. ( ( δ - ( λ d ) 1 - ( λ d ) ) ) 2 \left(\left(\dfrac{\delta-\left(\dfrac{\lambda}{d}\right)}{1-\left(\dfrac{% \lambda}{d}\right)}\right)\right)^{2}
  129. B B
  130. d d
  131. G G
  132. C ( B , S ) C(B,S)
  133. ( d n 2 ) \left(\dfrac{dn}{2}\right)
  134. n n
  135. X X
  136. G G
  137. γ n \gamma n
  138. X X
  139. G G
  140. ( d n 2 ) ( γ 2 + ( λ d ) γ ( 1 - γ ) ) \left(\dfrac{dn}{2}\right)\left(\gamma^{2}+(\dfrac{\lambda}{d})\gamma\left(1-% \gamma\right)\right)
  141. ( d n 2 ) ( γ 2 + ( λ d ) γ ( 1 - γ ) ) \left(\dfrac{dn}{2}\right)\left(\gamma^{2}+\left(\dfrac{\lambda}{d}\right)% \gamma\left(1-\gamma\right)\right)
  142. γ n \gamma n
  143. ( ( 2 n d 2 ) ( γ 2 + ( λ d ) γ ( 1 - γ ) ) γ n ) \left(\dfrac{(\dfrac{2nd}{2})\left(\gamma^{2}+(\dfrac{\lambda}{d})\gamma\left(% 1-\gamma\right)\right)}{\gamma n}\right)
  144. = d ( γ + ( λ d ) ( 1 - γ ) ) =d\left(\gamma+(\dfrac{\lambda}{d})\left(1-\gamma\right)\right)
  145. ( 2 ) \rightarrow(2)
  146. d ( γ + ( λ d ) ( 1 - γ ) ) < γ d d\left(\gamma+(\dfrac{\lambda}{d})\left(1-\gamma\right)\right)<\gamma d
  147. ( γ 2 + ( λ d ) γ ( 1 - γ ) ) \left(\gamma^{2}+(\dfrac{\lambda}{d})\gamma\left(1-\gamma\right)\right)
  148. C ( B , S ) C(B,S)
  149. ( 2 ) (2)
  150. γ < ( 1 - ( λ d ) δ - ( λ d ) ) \gamma<\left(\dfrac{1-(\dfrac{\lambda}{d})}{\delta-(\dfrac{\lambda}{d})}\right)
  151. C ( B , S ) C(B,S)
  152. ( δ - ( λ d ) 1 - ( λ d ) ) 2 \left(\dfrac{\delta-(\dfrac{\lambda}{d})}{1-(\dfrac{\lambda}{d})}\right)^{2}
  153. G G
  154. λ / d \lambda/d
  155. 1 1
  156. d d
  157. d - 1 d-1
  158. d d
  159. G G
  160. λ ( G ) 2 d - 1 \lambda(G)\leq 2\sqrt{d-1}
  161. G G
  162. d d
  163. λ \lambda
  164. G G
  165. ( 1 ) (1)
  166. 0
  167. d d
  168. A A
  169. B B
  170. G G
  171. C o C_{o}
  172. A A
  173. C o C_{o}
  174. B B
  175. A A
  176. B B
  177. 𝔻 : 𝔽 d C o \mathbb{D}:\mathbb{F}^{d}\rightarrow C_{o}
  178. C o C_{o}
  179. ( d 2 ) \left(\dfrac{d}{2}\right)
  180. w = ( w e ) , e E w=(w_{e}),e\in E
  181. z z
  182. G G
  183. A A
  184. E E
  185. v A E v \cup_{v\in A}E_{v}
  186. B B
  187. E E
  188. v B E v \cup_{v\in B}E_{v}
  189. w { 0 , 1 } N w\in\{0,1\}^{N}
  190. N = d n N=dn
  191. E v E_{v}
  192. v A v\in A
  193. v A v\in A
  194. ( w v ( 1 ) , w v ( 2 ) , , w v ( d ) ) \left(w_{v(1)},w_{v(2)},\ldots,w_{v(d)}\right)
  195. C o C_{o}
  196. E v E_{v}
  197. v A v\in A
  198. n n
  199. w w
  200. z z
  201. z z
  202. A A
  203. B B
  204. B B
  205. A A
  206. B B
  207. d = n d=n
  208. G G
  209. C C
  210. C o C_{o}
  211. m m
  212. ( ( log n ) log ( 2 - α ) ) \left(\dfrac{(\log{n})}{\log(2-\alpha)}\right)
  213. ( 1 2 ) . α N δ ( ( δ 2 ) - ( λ d ) ) = ( ( 1 4 ) . α N ( δ 2 - O ( λ d ) ) (\dfrac{1}{2}).\alpha N\delta\left((\dfrac{\delta}{2})-(\dfrac{\lambda}{d})% \right)=\left((\dfrac{1}{4}).\alpha N(\delta^{2}-O(\dfrac{\lambda}{d})\right)
  214. α < 1 \alpha<1
  215. O ( N log N ) O(N\log{N})
  216. O ( log N ) O(\log{N})
  217. α N δ 2 ( 1 - ϵ ) / 4 \alpha N\delta^{2}(1-\epsilon)/4
  218. G G
  219. α < 1 \alpha<1
  220. ( α δ o 2 4 ) ( 1 - ) N (\dfrac{\alpha\delta_{o}^{2}}{4})(1-\in)N
  221. O ( log n ) O(\log{n})
  222. O O
  223. α \alpha
  224. n n
  225. w w
  226. 1 1
  227. w = w 0 w=w^{0}
  228. w 1 , w 2 , , w t w^{1},w^{2},\ldots,w^{t}
  229. t t
  230. X i = e E | x e i = 1 X^{i}={e\in E|x_{e}^{i}=1}
  231. S i = v V i | E v X i + 1 ! = S^{i}={v\in V^{i}|E_{v}\cap X^{i+1}!=\emptyset}
  232. S i S^{i}
  233. i t h i^{th}
  234. S 1 < S 0 S^{1}<S^{0}
  235. S 0 > S 1 > S 2 > S^{0}>S^{1}>S^{2}>\cdots
  236. | S i + 1 | ( 1 2 - α ) | S i | |S_{i+1}|<=(\dfrac{1}{2-\alpha})|S_{i}|
  237. α < 1 \alpha<1
  238. | S i | < n |S_{i}|<n
  239. l o g 2 - α n log_{2-\alpha}n
  240. | S i | = n ( 1 ( 2 - α ) i ) = O ( n ) \sum|S_{i}|=n\sum(\dfrac{1}{(2-\alpha)^{i}})=O(n)
  241. i t h i^{th}
  242. O ( | S i | ) O(|S_{i}|)
  243. m m
  244. [ ( log n ) / ( log ( 2 - α ) ) ] [(\log{n})/(\log(2-\alpha))]

Zero-truncated_Poisson_distribution.html

  1. g ( k ; λ ) = P ( X = k k > 0 ) = f ( k ; λ ) 1 - f ( 0 ; λ ) = λ k e - λ k ! ( 1 - e - λ ) = λ k ( e λ - 1 ) k ! g(k;\lambda)=P(X=k\mid k>0)=\frac{f(k;\lambda)}{1-f(0;\lambda)}=\frac{\lambda^% {k}e^{-\lambda}}{k!\left(1-e^{-\lambda}\right)}=\frac{\lambda^{k}}{(e^{\lambda% }-1)k!}
  2. E [ X ] = λ 1 - e - λ = λ e λ e λ - 1 \operatorname{E}[X]=\frac{\lambda}{1-e^{-\lambda}}=\frac{\lambda e^{\lambda}}{% e^{\lambda}-1}
  3. Var [ X ] = λ 1 - e - λ - λ 2 e - λ ( 1 - e - λ ) 2 = λ e λ e λ - 1 [ 1 - λ e λ - 1 ] \operatorname{Var}[X]=\frac{\lambda}{1-e^{-\lambda}}-\frac{\lambda^{2}e^{-% \lambda}}{(1-e^{-\lambda})^{2}}=\frac{\lambda e^{\lambda}}{e^{\lambda}-1}\left% [1-\frac{\lambda}{e^{\lambda}-1}\right]

Zero_lag_exponential_moving_average.html

  1. 𝐿𝑎𝑔 = ( P e r i o d - 1 ) / 2 \,\textit{Lag}={(Period-1)/2}
  2. 𝐸𝑚𝑎𝐷𝑎𝑡𝑎 = D a t a - D a t a ( L a g d a y s a g o ) \,\textit{EmaData}={Data-Data(Lagdaysago)}
  3. 𝑍𝐿𝐸𝑀𝐴 = E M A ( E m a D a t a , P e r i o d ) \,\textit{ZLEMA}={EMA(EmaData,Period)}

Zeta-carotene_isomerase.html

  1. \rightleftharpoons

Zhao_Youqin's_π_algorithm.html

  1. π \pi
  2. π \pi
  3. \ell
  4. d = r 2 - ( 2 ) 2 d=\sqrt{r^{2}-\left(\frac{\ell}{2}\right)^{2}}
  5. e = r - d = r - r 2 - ( 2 ) 2 . e=r-d=r-\sqrt{r^{2}-\left(\frac{\ell}{2}\right)^{2}}.
  6. 2 \ell_{2}
  7. 2 = ( 2 ) 2 + e 2 \ell_{2}=\sqrt{\left(\frac{\ell}{2}\right)^{2}+e^{2}}
  8. 2 = 1 2 2 + 4 ( r - 1 2 4 r 2 - 2 ) 2 \ell_{2}=\frac{1}{2}\sqrt{\ell^{2}+4\left(r-\frac{1}{2}\sqrt{4r^{2}-\ell^{2}}% \right)^{2}}
  9. l 3 l_{3}
  10. 3 = 1 2 2 2 + 4 ( r - 1 2 4 r 2 - 2 2 ) 2 \ell_{3}=\frac{1}{2}\sqrt{\ell_{2}^{2}+4\left(r-\frac{1}{2}\sqrt{4r^{2}-\ell_{% 2}^{2}}\right)^{2}}
  11. n + 1 = 1 2 n 2 + 4 ( r - 1 2 4 r 2 - n 2 ) 2 \ell_{n+1}=\frac{1}{2}\sqrt{\ell_{n}^{2}+4\left(r-\frac{1}{2}\sqrt{4r^{2}-\ell% _{n}^{2}}\right)^{2}}
  12. π = 3.141592. \pi=3.141592.\,
  13. π \pi
  14. 22 7 \frac{22}{7}
  15. 355 113 \frac{355}{113}
  16. π \pi
  17. π \pi

Zingiberene_synthase.html

  1. \rightleftharpoons

Zone_axis.html

  1. u v w \langle{uvw\rangle}\,
  2. [ u v w ] [uvw]\,
  3. { h k l } \{hkl\}\,
  4. ( h k l ) (hkl)\,

Zospeum_tholussum.html

  1. p p

Łukasiewicz–Moisil_algebra.html

  1. 1 , , n - 1 \nabla_{1},\ldots,\nabla_{n-1}
  2. ( A , , , ¬ , j J , 0 , 1 ) (A,\vee,\wedge,\neg,\nabla_{j\in J},0,1)
  3. j J n \nabla^{n}_{j\in J}
  4. j ( x y ) = ( j x ) ( j y ) \nabla_{j}(x\vee y)=(\nabla_{j}\;x)\vee(\nabla_{j}\;y)
  5. j x ¬ j x = 1 \nabla_{j}\;x\vee\neg\nabla_{j}\;x=1
  6. j ( k x ) = k x \nabla_{j}(\nabla_{k}\;x)=\nabla_{k}\;x
  7. j ¬ x = ¬ n - j x \nabla_{j}\neg x=\neg\nabla_{n-j}\;x
  8. 1 x 2 x n - 1 x \nabla_{1}\;x\leq\nabla_{2}\;x\cdots\leq\nabla_{n-1}\;x
  9. j x = j y \nabla_{j}\;x=\nabla_{j}\;y
  10. j ( x y ) = ( j x ) ( j y ) \nabla_{j}(x\wedge y)=(\nabla_{j}\;x)\wedge(\nabla_{j}\;y)
  11. j x ¬ j x = 0 \nabla_{j}\;x\wedge\neg\nabla_{j}\;x=0
  12. j 0 = 0 \nabla_{j}\;0=0
  13. j 1 = 1 \nabla_{j}\;1=1
  14. j \nabla_{j}
  15. n \mathcal{L}_{n}
  16. ¬ x = 1 - x \neg x=1-x
  17. x y = min { x , y } x\wedge y=\min\{x,y\}
  18. x y = max { x , y } x\vee y=\max\{x,y\}
  19. j ( i n - 1 ) = { 0 if i + j < n 1 if i + j n i { 0 } J , j J . \nabla_{j}\left(\frac{i}{n-1}\right)=\;\begin{cases}0&\mbox{if }~{}i+j<n\\ 1&\mbox{if }~{}i+j\geq n\\ \end{cases}\quad i\in\{0\}\cup J,\;j\in J.
  20. n \mathcal{L}_{n}
  21. n \mathcal{L}_{n}
  22. x y = def y j J ( ¬ j x ) ( j y ) x\Rightarrow y\;\overset{\mathrm{def}}{=}\;y\vee\bigwedge_{j\in J}(\neg\nabla_% {j}\;x)\vee(\nabla_{j}\;y)