wpmath0000014_14

Weight-of-conflict_conjecture.html

  1. Q 1 Q_{1}
  2. Q 2 Q_{2}
  3. S 1 S_{1}
  4. S 2 S_{2}
  5. Θ \Theta
  6. Q 1 ( A ) Q 2 ( A ) Q_{1}(A)\leq Q_{2}(A)
  7. W S 1 W S 2 W_{S_{1}}\geq W_{S_{2}}
  8. A Θ : Q 1 ( A ) Q 2 ( A ) W S 1 W S 2 \forall A\subseteq\Theta:Q_{1}(A)\leq Q_{2}(A)\implies W_{S_{1}}\geq W_{S_{2}}

Weighted_product_model.html

  1. P ( A K / A L ) = j = 1 n ( a K j / a L j ) w j , for K , L = 1 , 2 , 3 , , m . P(A_{K}/A_{L})=\prod_{j=1}^{n}(a_{Kj}/a_{Lj})^{w_{j}},\,\text{ for }K,L=1,2,3,% \dots,m.
  2. P ( A 1 / A 2 ) = ( 25 / 10 ) 0.20 × ( 20 / 30 ) 0.15 × ( 15 / 20 ) 0.40 × ( 30 / 30 ) 0.25 = 1.007 > 1. P(A_{1}/A_{2})=(25/10)^{0.20}\times(20/30)^{0.15}\times(15/20)^{0.40}\times(30% /30)^{0.25}=1.007>1.
  3. P ( A 1 / A 3 ) = 1.067 > 1 , and P ( A 2 / A 3 ) = 1.059 > 1. P(A_{1}/A_{3})=1.067>1,\,\text{ and }P(A_{2}/A_{3})=1.059>1.\,
  4. P ( A K ) = j = 1 n ( a K j ) w j , for K = 1 , 2 , 3 , , m . P(A_{K})=\prod_{j=1}^{n}(a_{Kj})^{w_{j}},\,\text{ for }K=1,2,3,\dots,m.

Weighted_sum_model.html

  1. A WSM-score i = j = 1 n w j a i j , for i = 1 , 2 , 3 , , m . A\text{WSM-score}_{i}=\sum_{j=1}^{n}w_{j}a_{ij},\,\text{ for }i=1,2,3,\dots,m.
  2. A WSM-score 1 = 25 × 0.20 + 20 × 0.15 + 15 × 0.40 + 30 × 0.25 = 21.50. A\text{WSM-score}_{1}=25\times 0.20+20\times 0.15+15\times 0.40+30\times 0.25=% 21.50.
  3. A WSM-score 2 = 22.00 , and A WSM-score 3 = 22.00. A\text{WSM-score}_{2}=22.00,\,\text{ and }A\text{WSM-score}_{3}=22.00.

Weighting_pattern.html

  1. u u
  2. y y
  3. x ˙ ( t ) = A ( t ) x ( t ) + B ( t ) u ( t ) \dot{x}(t)=A(t)x(t)+B(t)u(t)
  4. y ( t ) = C ( t ) x ( t ) y(t)=C(t)x(t)
  5. y ( t ) = y ( t 0 ) + t 0 t T ( t , σ ) u ( σ ) d σ y(t)=y(t_{0})+\int_{t_{0}}^{t}T(t,\sigma)u(\sigma)d\sigma
  6. T ( , ) T(\cdot,\cdot)
  7. T ( t , σ ) = C ( t ) ϕ ( t , σ ) B ( σ ) T(t,\sigma)=C(t)\phi(t,\sigma)B(\sigma)
  8. ϕ \phi
  9. T ( t , σ ) = C e A ( t - σ ) B T(t,\sigma)=Ce^{A(t-\sigma)}B
  10. e A ( t - σ ) e^{A(t-\sigma)}
  11. T ( k , l ) = C A k - l - 1 B T(k,l)=CA^{k-l-1}B

Well-structured_transition_system.html

  1. \leq
  2. X X
  3. x 0 , x 1 , x 2 , x_{0},x_{1},x_{2},\ldots
  4. X X
  5. x i x j x_{i}\leq x_{j}
  6. i < j i<j
  7. X X
  8. 𝒮 = S , , \mathcal{S}=\langle S,\rightarrow,\cdots\rangle
  9. S S
  10. S × S \rightarrow\subseteq S\times S
  11. S , , \langle S,\to,\leq\rangle
  12. S × S \leq\subseteq S\times S
  13. \leq
  14. \to
  15. s 1 s 2 s_{1}\to s_{2}
  16. ( s 1 , s 2 ) (s_{1},s_{2})\in\to
  17. t 1 t_{1}
  18. s 1 t 1 s_{1}\leq t_{1}
  19. t 2 t_{2}
  20. t 1 * t 2 t_{1}\xrightarrow{*}t_{2}
  21. t 2 t_{2}
  22. t 1 t_{1}
  23. s 2 t 2 s_{2}\leq t_{2}
  24. ( S , ) (S,\to)
  25. S = Q × D S=Q\times D
  26. Q Q
  27. D D
  28. D × D \leq\subseteq D\times D
  29. ( q , d ) ( q , d ) q = q d d (q,d)\leq(q^{\prime},d^{\prime})\Leftrightarrow q=q^{\prime}\wedge d\leq d^{\prime}
  30. \to
  31. \leq
  32. S S
  33. \leq
  34. \to
  35. s s
  36. s 0 s_{0}
  37. s s s^{\prime}\geq s
  38. s s
  39. s s
  40. s s
  41. S e S_{e}
  42. ( A , ) (A,\leq)
  43. S 1 S 2 S_{1}\subseteq S_{2}\subseteq...
  44. A A
  45. S s S_{s}
  46. S e S_{e}
  47. S s S_{s}
  48. s 0 s_{0}
  49. S e S_{e}

Wendel's_theorem.html

  1. p n , N = 2 - N + 1 k = 0 n - 1 ( N - 1 k ) . p_{n,N}=2^{-N+1}\sum_{k=0}^{n-1}{\left({{N-1}\atop{k}}\right)}.

Wente_torus.html

  1. 3 \mathbb{R}^{3}

Weyl_equation.html

  1. σ μ μ ψ = 0 \sigma^{\mu}\partial_{\mu}\psi=0
  2. I 2 1 c ψ t + σ x ψ x + σ y ψ y + σ z ψ z = 0 I_{2}\frac{1}{c}\frac{\partial\psi}{\partial t}+\sigma_{x}\frac{\partial\psi}{% \partial x}+\sigma_{y}\frac{\partial\psi}{\partial y}+\sigma_{z}\frac{\partial% \psi}{\partial z}=0
  3. σ μ = ( σ 0 , σ 1 , σ 2 , σ 3 ) = ( I 2 , σ x , σ y , σ z ) \sigma_{\mu}=(\sigma_{0},\sigma_{1},\sigma_{2},\sigma_{3})=(I_{2},\sigma_{x},% \sigma_{y},\sigma_{z})
  4. ψ = ( ψ 1 ψ 2 ) = χ e - i ( 𝐤 𝐫 - ω t ) = χ e - i ( 𝐩 𝐫 - E t ) / \psi=\begin{pmatrix}\psi_{1}\\ \psi_{2}\\ \end{pmatrix}=\chi e^{-i(\mathbf{k}\cdot\mathbf{r}-\omega t)}=\chi e^{-i(% \mathbf{p}\cdot\mathbf{r}-Et)/\hbar}
  5. χ = ( χ 1 χ 2 ) \chi=\begin{pmatrix}\chi_{1}\\ \chi_{2}\\ \end{pmatrix}
  6. | 𝐩 | = | 𝐤 | = ω / c | 𝐤 | = ω / c |\mathbf{p}|=\hbar|\mathbf{k}|=\hbar\omega/c\,\rightarrow\,|\mathbf{k}|=\omega/c
  7. σ μ μ ψ R = 0 \displaystyle\sigma^{\mu}\partial_{\mu}\psi_{R}=0
  8. 𝐩 𝐉 | 𝐩 , λ = λ | 𝐩 | | 𝐩 , λ \mathbf{p}\cdot\mathbf{J}\left|\mathbf{p},\lambda\right\rangle=\lambda|\mathbf% {p}|\left|\mathbf{p},\lambda\right\rangle
  9. λ = ± 1 / 2 \lambda=\pm 1/2
  10. = i ψ R σ μ μ ψ R \mathcal{L}=i\psi_{R}^{\dagger}\sigma^{\mu}\partial_{\mu}\psi_{R}
  11. = i ψ L σ ¯ μ μ ψ L \mathcal{L}=i\psi_{L}^{\dagger}\bar{\sigma}^{\mu}\partial_{\mu}\psi_{L}
  12. \dagger

Weyl_law.html

  1. Ω d \Omega\subset\mathbb{R}^{d}
  2. N ( x ) N(x)
  3. x x
  4. lim x N ( x ) x d / 2 = ( 2 π ) - d ω d vol ( Ω ) \lim_{x\rightarrow\infty}\frac{N(x)}{x^{d/2}}=(2\pi)^{-d}\omega_{d}\mathrm{vol% }(\Omega)
  5. ω d \omega_{d}
  6. d \mathbb{R}^{d}
  7. o ( λ d / 2 ) o(\lambda^{d/2})
  8. O ( λ ( d - 1 ) / 2 ) O(\lambda^{(d-1)/2})
  9. o ( λ ( d - 1 ) / 2 ) o(\lambda^{(d-1)/2})
  10. H = - h 2 Δ + V ( x ) H=-h^{2}\Delta+V(x)
  11. N ( λ , h ) ( 2 π h ) - d ω d { | ξ | 2 + V ( x ) < λ } d x d ξ N(\lambda,h)\sim(2\pi h)^{-d}\omega_{d}\int_{\{|\xi|^{2}+V(x)<\lambda\}}dxd\xi
  12. λ \lambda
  13. + +\infty
  14. h + 0 h\to+0
  15. N ( λ , h ) N(\lambda,h)
  16. H H
  17. λ \lambda
  18. λ \lambda
  19. N ( λ , h ) = + N(\lambda,h)=+\infty
  20. λ \lambda
  21. N ( λ ) = ( 2 π ) - d λ d / 2 vol ( Ω ) 1 4 ( 2 π ) 1 - d λ ( d - 1 ) / 2 area ( Ω ) + o ( λ ( d - 1 ) / 2 ) . N(\lambda)=(2\pi)^{-d}\lambda^{d/2}\mathrm{vol}(\Omega)\mp\frac{1}{4}(2\pi)^{1% -d}\lambda^{(d-1)/2}\mathrm{area}(\partial\Omega)+o(\lambda^{(d-1)/2}).
  22. O ( λ ( d - 1 ) / 2 log λ ) O(\lambda^{(d-1)/2}\log\lambda)
  23. O ( λ ( d - 1 ) / 2 ) O(\lambda^{(d-1)/2})
  24. o ( λ ( d - 1 ) / 2 ) o(\lambda^{(d-1)/2})
  25. 𝐑 3 \mathbf{R}^{3}

WHIS_ratio.html

  1. W H I S = α i β i WHIS=\frac{\alpha_{i}}{\beta_{i}}
  2. W H I S WHIS\equiv
  3. α i \alpha_{i}\equiv
  4. β i \beta_{i}\equiv

Whitney_topologies.html

  1. S k ( U ) = { f C ( M , N ) : ( J k f ) ( M ) U } . S^{k}(U)=\{f\in C^{\infty}(M,N):(J^{k}f)(M)\subseteq U\}.
  2. W = k = 0 W k . W=\bigcup_{k=0}^{\infty}W^{k}.
  3. dim { \R k [ x 1 , , x m ] } = i = 1 k ( m + i - 1 ) ! ( m - 1 ) ! i ! = ( ( m + k ) ! m ! k ! - 1 ) . \dim\left\{\R^{k}[x_{1},\ldots,x_{m}]\right\}=\sum_{i=1}^{k}\frac{(m+i-1)!}{(m% -1)!\cdot i!}=\left(\frac{(m+k)!}{m!\cdot k!}-1\right).
  4. B m , n k = i = 1 n \R k [ x 1 , , x m ] , dim { B m , n k } = n dim { A m k } = n ( ( m + k ) ! m ! k ! - 1 ) . B_{m,n}^{k}=\bigoplus_{i=1}^{n}\R^{k}[x_{1},\ldots,x_{m}],\implies\dim\left\{B% _{m,n}^{k}\right\}=n\dim\left\{A_{m}^{k}\right\}=n\left(\frac{(m+k)!}{m!\cdot k% !}-1\right).
  5. dim { J k ( M , N ) } = m + n + dim { B n , m k } = m + n ( ( m + k ) ! m ! k ! ) . \dim\!\left\{J^{k}(M,N)\right\}=m+n+\dim\!\left\{B_{n,m}^{k}\right\}=m+n\left(% \frac{(m+k)!}{m!\cdot k!}\right).
  6. π k : C ( M , N ) J k ( M , N ) where π k ( f ) = ( j k f ) ( M ) . \pi^{k}:C^{\infty}(M,N)\twoheadrightarrow J^{k}(M,N)\ \mbox{where}~{}\ \pi^{k}% (f)=(j^{k}f)(M).

Wiener's_attack.html

  1. e d = 1 mod φ ( N ) ed=1\bmod\varphi(N)
  2. φ ( N ) = ( p - 1 ) ( q - 1 ) \varphi(N)=(p-1)(q-1)
  3. N * \mathbb{Z}_{N}^{*}
  4. φ ( N ) \varphi(N)
  5. C M e mod φ ( N ) C\equiv M^{e}\bmod\varphi(N)
  6. C C
  7. C d ( M e ) d M ( e d ) M mod φ ( N ) C^{d}\equiv(M^{e})^{d}\equiv M^{(ed)}\equiv M\bmod\varphi(N)
  8. d < 1 3 N 1 4 d<\frac{1}{3}N^{\frac{1}{4}}
  9. e e
  10. e e^{\prime}
  11. e = e + k . φ ( N ) e^{\prime}=e+k.\varphi(N)
  12. k k
  13. e e^{\prime}
  14. e > N 3 2 e^{\prime}>N^{\frac{3}{2}}
  15. d d
  16. d p = d mod ( p - 1 ) d_{p}=d\bmod\ (p-1)
  17. d q = d mod ( q - 1 ) d_{q}=d\bmod\ (q-1)
  18. d d
  19. C C
  20. M p C d p mod p M_{p}\equiv C^{d_{p}}\bmod\ p
  21. M q C d q mod q M_{q}\equiv C^{d_{q}}\bmod\ q
  22. M M\in\mathbb{Z_{N}}
  23. M M p mod p M\equiv M_{p}\bmod\ p
  24. M M q mod q M\equiv M_{q}\bmod\ q
  25. M M
  26. M C d mod N M\equiv C^{d}\bmod\ N
  27. d mod φ ( N ) d\bmod\ \varphi(N)
  28. e d = 1 mod lcm ( p - 1 , q - 1 ) ed=1(\bmod\ \operatorname{lcm}(p-1,q-1))
  29. e d = K × lcm ( p - 1 , q - 1 ) + 1 ed=K\times\operatorname{lcm}(p-1,q-1)+1
  30. G = gcd ( p - 1 , q - 1 ) G=\gcd(p-1,q-1)
  31. e d = K G ( p - 1 ) ( q - 1 ) + 1 ed=\frac{K}{G}(p-1)(q-1)+1
  32. k = K gcd ( K , G ) k=\frac{K}{\gcd(K,G)}
  33. g = G gcd ( K , G ) g=\frac{G}{\gcd(K,G)}
  34. e d = k g ( p - 1 ) ( q - 1 ) + 1 ed=\frac{k}{g}(p-1)(q-1)+1
  35. d p q dpq
  36. e p q = k d g ( 1 - δ ) \frac{e}{pq}=\frac{k}{dg}(1-\delta)
  37. δ = p + q - 1 - g k p q \delta=\frac{p+q-1-\frac{g}{k}}{pq}
  38. e p q \frac{e}{pq}
  39. k d g \frac{k}{dg}
  40. e d > p q ed>pq
  41. G G
  42. e d g = k . ( p - 1 ) ( q - 1 ) + g edg=k.(p-1)(q-1)+g
  43. N = p q \ N=pq
  44. q < p < 2 q \ q<p<2q
  45. d < 1 3 N 1 4 d<\frac{1}{3}N^{\frac{1}{4}}
  46. N , e \left\langle N,e\right\rangle
  47. e d = 1 mod φ ( N ) ed=1(\bmod\ \varphi(N))
  48. d d
  49. N , e = 90581 , 17993 \left\langle N,e\right\rangle=\left\langle 90581,17993\right\rangle
  50. d d
  51. d d
  52. e N \frac{e}{N}
  53. e N = 17993 90581 = 1 5 + 1 29 + + 1 3 = [ 0 , 5 , 29 , 4 , 1 , 3 , 2 , 4 , 3 ] \frac{e}{N}=\frac{17993}{90581}=\cfrac{1}{5+\cfrac{1}{29+\dots+\cfrac{1}{3}}}=% \left[0,5,29,4,1,3,2,4,3\right]
  54. e N \frac{e}{N}
  55. k d \frac{k}{d}
  56. k d = 0 , 1 5 , 29 146 , 117 589 , 146 735 , 555 2794 , 1256 6323 , 5579 28086 , 17993 90581 \frac{k}{d}=0,\frac{1}{5},\frac{29}{146},\frac{117}{589},\frac{146}{735},\frac% {555}{2794},\frac{1256}{6323},\frac{5579}{28086},\frac{17993}{90581}
  57. N N
  58. 1 5 \frac{1}{5}
  59. φ ( N ) = e . d - 1 k = 17993 × 5 - 1 1 = 89964 \varphi(N)=\frac{e.d-1}{k}=\frac{17993\times 5-1}{1}=89964
  60. x 2 - ( ( N - φ ( N ) ) + 1 ) x + N = 0 x^{2}-\left(\left(N-\varphi(N)\right)+1\right)x+N=0
  61. x 2 - ( ( 90581 - 89964 ) + 1 ) x + 90581 = 0 x^{2}-\left(\left(90581-89964\right)+1\right)x+90581=0
  62. x 2 - ( 618 ) x + 90581 = 0 x^{2}-\left(618\right)x+90581=0
  63. x = 379 ; 239 x=379;239
  64. N = 90581 = 379 × 239 = p × q N=90581=379\times 239=p\times q
  65. N = 90581 N=90581
  66. d < N 1 4 3 5.7828 d<\frac{N^{\frac{1}{4}}}{3}\approx 5.7828
  67. e d = 1 mod φ ( N ) ed=1\bmod\varphi(N)
  68. k \mathit{k}
  69. e d - k φ ( N ) = 1 ed-k\varphi(N)=1
  70. | e φ ( N ) - k d | = 1 d φ ( N ) \left|\frac{e}{\varphi(N)}-\frac{k}{d}\right|=\frac{1}{d\varphi(N)}
  71. k d \frac{k}{d}
  72. e φ ( N ) \frac{e}{\varphi(N)}
  73. φ ( N ) \varphi(N)
  74. N N
  75. φ ( N ) = N - p - q + 1 \varphi(N)=N-p-q+1
  76. p + q - 1 < 3 N p+q-1<3\sqrt{N}
  77. | p + q - 1 | < 3 N \left|p+q-1\right|<3\sqrt{N}
  78. | N + 1 - φ ( N ) - 1 | < 3 N \left|N+1-\varphi(N)-1\right|<3\sqrt{N}
  79. N N
  80. φ ( N ) \varphi(N)
  81. | e N - k d | = | e d - k N N d | \left|\frac{e}{N}-\frac{k}{d}\right|=\left|\frac{ed-kN}{Nd}\right|
  82. = | e d - k φ ( N ) - k N + k φ ( N ) N d | \qquad=\left|\frac{ed-k\varphi(N)-kN+k\varphi(N)}{Nd}\right|
  83. = | 1 - k ( N - φ ( N ) ) N d | =\left|\frac{1-k(N-\varphi(N))}{Nd}\right|
  84. | 3 k N N d | = 3 k N N N d = 3 k d N \leq\left|\frac{3k\sqrt{N}}{Nd}\right|=\frac{3k\sqrt{N}}{\sqrt{N}\sqrt{N}d}=% \frac{3k}{d\sqrt{N}}
  85. k φ ( N ) = e d - 1 < e d k\varphi(N)=ed-1<ed
  86. k φ ( N ) < e d k\varphi(N)<ed
  87. e < φ ( N ) e<\varphi(N)
  88. k φ ( N ) < e d < φ ( N ) d k\varphi(N)<ed<\varphi(N)d
  89. k φ ( N ) < φ ( N ) d k\varphi(N)<\varphi(N)d
  90. k < d k<d
  91. k < d k<d
  92. d < 1 3 N 1 4 d<\frac{1}{3}N^{\frac{1}{4}}
  93. | e N - k d | 1 d N 1 4 \left|\frac{e}{N}-\frac{k}{d}\right|\leq\frac{1}{dN^{\frac{1}{4}}}
  94. d < 1 3 N 1 4 , 2 d < 3 d , d<\frac{1}{3}N^{\frac{1}{4}},2d<3d,
  95. 2 d < 3 d < N 1 4 2d<3d<N^{\frac{1}{4}}
  96. 2 d < N 1 4 , 2d<N^{\frac{1}{4}},
  97. 1 2 d > 1 N 1 4 \frac{1}{2d}>\frac{1}{N^{\frac{1}{4}}}
  98. | e N - k d | 3 k d N < 1 d 2 d = 1 2 d 2 \left|\frac{e}{N}-\frac{k}{d}\right|\leq\frac{3k}{d\sqrt{N}}<\frac{1}{d\cdot 2% d}=\frac{1}{2d^{2}}\blacksquare

Wiener_algebra.html

  1. A ( 𝐓 ) A(\mathbf{T})
  2. f A ( 𝐓 ) f∈A(\mathbf{T})
  3. f = n = - | f ^ ( n ) | , \|f\|=\sum_{n=-\infty}^{\infty}|\hat{f}(n)|,\,
  4. f ^ ( n ) = 1 2 π - π π f ( t ) e - i n t d t \hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(t)e^{-int}\,dt
  5. f f
  6. A ( 𝐓 ) A(\mathbf{T})
  7. f ( t ) g ( t ) = m f ^ ( m ) e i m t n g ^ ( n ) e i n t = n , m f ^ ( m ) g ^ ( n ) e i ( m + n ) t = n { m f ^ ( n - m ) g ^ ( m ) } e i n t , f , g A ( 𝕋 ) ; \begin{aligned}\displaystyle f(t)g(t)&\displaystyle=\sum_{m\in\mathbb{Z}}\hat{% f}(m)e^{imt}\,\cdot\,\sum_{n\in\mathbb{Z}}\hat{g}(n)e^{int}\\ &\displaystyle=\sum_{n,m\in\mathbb{Z}}\hat{f}(m)\hat{g}(n)e^{i(m+n)t}\\ &\displaystyle=\sum_{n\in\mathbb{Z}}\left\{\sum_{m\in\mathbb{Z}}\hat{f}(n-m)% \hat{g}(m)\right\}e^{int},\qquad f,g\in A(\mathbb{T});\end{aligned}
  8. f g = n | m f ^ ( n - m ) g ^ ( m ) | m | f ^ ( m ) | n | g ^ ( n ) | = f g . \|fg\|=\sum_{n\in\mathbb{Z}}\left|\sum_{m\in\mathbb{Z}}\hat{f}(n-m)\hat{g}(m)% \right|\leq\sum_{m}|\hat{f}(m)|\sum_{n}|\hat{g}(n)|=\|f\|\,\|g\|.\,
  9. A ( 𝐓 ) A(\mathbf{T})
  10. A ( 𝕋 ) C ( 𝕋 ) A(\mathbb{T})\subset C(\mathbb{T})
  11. C ( 𝐓 ) C(\mathbf{T})
  12. C 1 ( 𝕋 ) A ( 𝕋 ) . C^{1}(\mathbb{T})\subset A(\mathbb{T}).\,
  13. Lip α ( 𝕋 ) A ( 𝕋 ) C ( 𝕋 ) \mathrm{Lip}_{\alpha}(\mathbb{T})\subset A(\mathbb{T})\subset C(\mathbb{T})
  14. α > 1 / 2 \alpha>1/2
  15. f f
  16. 1 / f 1/f
  17. A ( 𝐓 ) A(\mathbf{T})
  18. M x = { f A ( 𝕋 ) f ( x ) = 0 } , x 𝕋 , M_{x}=\left\{f\in A(\mathbb{T})\,\mid\,f(x)=0\right\},\quad x\in\mathbb{T}~{},

Wiener_amalgam_space.html

  1. X X
  2. X \|\cdot\|_{X}
  3. X X
  4. L m p L^{p}_{m}
  5. L p L^{p}
  6. m m
  7. W ( X , L p ) = { f : ( d f ( ) g ¯ ( - x ) X p m ( x ) p d x ) 1 / p < } , W(X,L^{p})=\left\{f\ :\ \left(\int_{\mathbb{R}^{d}}\|f(\cdot)\bar{g}(\cdot-x)% \|^{p}_{X}m(x)^{p}\,dx\right)^{1/p}<\infty\right\},
  8. g g
  9. x 𝕕 g ( z - x ) = 1 \sum_{x\in\mathbb{Z^{d}}}g(z-x)=1
  10. z d z\in\mathbb{R}^{d}
  11. g g

Wiener–Wintner_theorem.html

  1. lim 1 2 + 1 j = - e i j λ f ( τ j P ) \lim_{\ell\rightarrow\infty}\frac{1}{2\ell+1}\sum_{j=-\ell}^{\ell}e^{ij\lambda% }f(\tau^{j}P)

Williamson's_model_of_managerial_discretion.html

  1. U = U ( S , M , I D ) U=U(S,M,I_{D})\,
  2. Π = R - C - S \Pi=R-C-S\,
  3. Π r = Π - M \Pi_{r}=\Pi-M\,
  4. Π r Π 0 + T \Pi_{r}\geq\Pi_{0}+T\,
  5. Π D = Π - Π 0 - T \Pi_{D}=\Pi-\Pi_{0}-T\,
  6. I D = Π r - Π 0 - T I_{D}=\Pi_{r}-\Pi_{0}-T\,
  7. Π D = I D + M \Pi_{D}=I_{D}+M\,
  8. Π r = Π or Π D = I D \Pi_{r}=\Pi\,\text{ or }\Pi_{D}=I_{D}\,
  9. U = U ( S , I D ) U=U(S,I_{D})\,
  10. I D = Π - Π 0 - T I_{D}=\Pi-\Pi_{0}-T\,
  11. U = U ( S , Π - Π 0 - T ) U=U(S,\Pi-\Pi_{0}-T)\,
  12. X = f ( S , P ¯ , E ¯ ) X=f(S,\bar{P},\bar{E})\,
  13. Π = f ( X ) = f ( S , P ¯ , E ¯ ) \Pi=f(X)=f(S,\bar{P},\bar{E})\,
  14. Π D = f ( S , P ¯ , E ¯ ) - Π 0 - T ) \Pi_{D}=f(S,\bar{P},\bar{E})-\Pi_{0}-T)\,
  15. max U = U ( S , Π - Π 0 - T ) \max U=U(S,\Pi-\Pi_{0}-T)\,
  16. subject to Π Π 0 + T \,\text{subject to}\Pi\geq\Pi_{0}+T\,

Wingate_test.html

  1. P = F × d t P=\tfrac{F\times d}{t}
  2. d = r e v o l u t i o n s × d f d=revolutions\times d_{f}
  3. d f d_{f}
  4. R P P = P P B W RPP=\tfrac{PP}{BW}
  5. A F = P P - L P P P AF=\tfrac{PP-LP}{PP}
  6. i = 0 n P i \sum_{i=0}^{n}P_{i}
  7. P i P_{i}

Wins_Above_Replacement.html

  1. b W A R = ( P r u n s - A r u n s ) + ( A r u n s - R r u n s ) bWAR=(P_{runs}-A_{runs})+(A_{runs}-R_{runs})
  2. P r u n s - A r u n s P_{runs}-A_{runs}
  3. w R A A = w O B A - .320 1.25 * ( A B + B B + H B P + S F + S H ) wRAA=\tfrac{wOBA-.320}{1.25}*(AB+BB+HBP+SF+SH)
  4. w O B A = ( α 1 * u B B + α 2 * H B P + α 3 * 1 B + α 4 * 2 B + α 5 * 3 B + α 6 * H R + α 7 * S B - α 8 * C S ) ( A B + B B - I B B + H B P + S F ) wOBA={(\alpha_{1}*uBB+\alpha_{2}*HBP+\alpha_{3}*1B+\alpha_{4}*2B+\alpha_{5}*3B% +\alpha_{6}*HR+\alpha_{7}*SB-\alpha_{8}*CS)\over(AB+BB-IBB+HBP+SF)}
  5. α 1 \alpha_{1}
  6. α 8 \alpha_{8}
  7. f W A R = w R A A + U Z R + P o s i t i o n + 20 600 * P A fWAR=wRAA+UZR+Position+\tfrac{20}{600}*PA
  8. W i n s = 52.7 + 0.97 * f W A R Wins=52.7+0.97*fWAR
  9. W i n s = 52 + f W A R Wins=52+fWAR
  10. W i n s = 63.83 + 0.68 * f W A R Wins=63.83+0.68*fWAR

Wirtinger_derivatives.html

  1. x k + i y k = z k x k - i y k = u k x_{k}+iy_{k}=z_{k}\qquad x_{k}-iy_{k}=u_{k}
  2. k k
  3. n n
  4. V V
  5. x k x_{k}
  6. y q y_{q}
  7. k k
  8. q q
  9. n n
  10. d 2 V d z k d u q = 0 \frac{d^{2}V}{dz_{k}\,du_{q}}=0
  11. g ( z ) g(z)
  12. z 0 z_{0}
  13. g z ¯ ( z 0 ) = def lim r 0 1 2 π i r 2 Γ ( z 0 , r ) g ( z ) d z {\frac{\partial g}{\partial\bar{z}}(z_{0})}\overset{\mathrm{def}}{=}\lim_{r\to 0% }\frac{1}{2\pi ir^{2}}{\oint_{\Gamma(z_{0},r)}\!\!\!\!\!\!\!\!\!\!g(z){\mathrm% {d}z}}
  14. Γ ( z 0 , r ) = D ( z 0 , r ) \scriptstyle\Gamma_{(z_{0},r)}=\partial D(z_{0},r)
  15. r r
  16. g ( z ) g(z)
  17. z = z 0 z=z_{0}
  18. 2 = { ( x , y ) x , y } \mathbb{C}\equiv\mathbb{R}^{2}=\{(x,y)\mid x\in\mathbb{R},\ y\in\mathbb{R}\}
  19. z = 1 2 ( x - i y ) , z ¯ = 1 2 ( x + i y ) . \frac{\partial}{\partial z}=\frac{1}{2}\left(\frac{\partial}{\partial x}-i% \frac{\partial}{\partial y}\right),\quad\frac{\partial}{\partial\bar{z}}=\frac% {1}{2}\left(\frac{\partial}{\partial x}+i\frac{\partial}{\partial y}\right).
  20. C 1 C^{1}
  21. Ω 2 \Omega\subseteq\mathbb{R}^{2}
  22. n = 2 n = { ( 𝐱 , 𝐲 ) = ( x 1 , , x n , y 1 , , y n ) 𝐱 , 𝐲 n } \mathbb{C}^{n}=\mathbb{R}^{2n}=\left\{\left(\mathbf{x},\mathbf{y}\right)=\left% (x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}\right)\mid\mathbf{x},\mathbf{y}\in% \mathbb{R}^{n}\right\}
  23. { z 1 = 1 2 ( x 1 - i y 1 ) z n = 1 2 ( x n - i y n ) , \left\{\begin{aligned}\displaystyle\frac{\partial}{\partial z_{1}}&% \displaystyle=\frac{1}{2}\left(\frac{\partial}{\partial x_{1}}-i\frac{\partial% }{\partial y_{1}}\right)\\ &\displaystyle\qquad\qquad\vdots\\ \displaystyle\frac{\partial}{\partial z_{n}}&\displaystyle=\frac{1}{2}\left(% \frac{\partial}{\partial x_{n}}-i\frac{\partial}{\partial y_{n}}\right)\\ \end{aligned}\right.\quad,\quad
  24. { z ¯ 1 = 1 2 ( x 1 + i y 1 ) z ¯ n = 1 2 ( x n + i y n ) . \left\{\begin{aligned}\displaystyle\frac{\partial}{\partial\bar{z}_{1}}&% \displaystyle=\frac{1}{2}\left(\frac{\partial}{\partial x_{1}}+i\frac{\partial% }{\partial y_{1}}\right)\\ &\displaystyle\qquad\qquad\vdots\\ \displaystyle\frac{\partial}{\partial\bar{z}_{n}}&\displaystyle=\frac{1}{2}% \left(\frac{\partial}{\partial x_{n}}+i\frac{\partial}{\partial y_{n}}\right)% \\ \end{aligned}\right..
  25. C 1 C^{1}
  26. Ω \Omega
  27. z n z\in\mathbb{C}^{n}
  28. z ( x , y ) = ( x 1 , , x n , y 1 , , y n ) z\equiv(x,y)=(x_{1},\ldots,x_{n},y_{1},\ldots,y_{n})
  29. x x
  30. y y
  31. Ω \Omega
  32. f , g C 1 ( Ω ) \scriptstyle f,g\in C^{1}(\Omega)
  33. α , β \alpha,\beta
  34. i = 1 , , n \scriptstyle i=1,\dots,n
  35. z i ( α f + β g ) = α f z i + β g z i , z ¯ i ( α f + β g ) = α f z ¯ i + β g z ¯ i \frac{\partial}{\partial z_{i}}\left(\alpha f+\beta g\right)=\alpha\frac{% \partial f}{\partial z_{i}}+\beta\frac{\partial g}{\partial z_{i}},\quad\frac{% \partial}{\partial\bar{z}_{i}}\left(\alpha f+\beta g\right)=\alpha\frac{% \partial f}{\partial\bar{z}_{i}}+\beta\frac{\partial g}{\partial\bar{z}_{i}}
  36. f , g C 1 ( Ω ) \scriptstyle f,g\in C^{1}(\Omega)
  37. i = 1 , , n \scriptstyle i=1,\dots,n
  38. z i ( f g ) = f z i g + f g z i , z ¯ i ( f g ) = f z ¯ i g + f g z ¯ i \frac{\partial}{\partial z_{i}}\left(f\cdot g\right)=\frac{\partial f}{% \partial z_{i}}\cdot g+f\cdot\frac{\partial g}{\partial z_{i}},\quad\frac{% \partial}{\partial\bar{z}_{i}}\left(f\cdot g\right)=\frac{\partial f}{\partial% \bar{z}_{i}}\cdot g+f\cdot\frac{\partial g}{\partial\bar{z}_{i}}
  39. Ω m \Omega^{\prime}\subseteq\mathbb{C}^{m}
  40. Ω ′′ p \Omega^{\prime\prime}\subseteq\mathbb{C}^{p}
  41. g : Ω Ω g:\Omega^{\prime}\to\Omega
  42. f : Ω Ω ′′ f:\Omega\to\Omega^{\prime\prime}
  43. f , g C 1 ( Ω ) \scriptstyle f,g\in C^{1}(\Omega)
  44. g ( Ω ) Ω g(\Omega)\subseteq\Omega
  45. z ( f g ) = ( f z g ) g z + ( f z ¯ g ) g ¯ z \frac{\partial}{\partial z}\left(f\circ g\right)=\left(\frac{\partial f}{% \partial z}\circ g\right)\frac{\partial g}{\partial z}+\left(\frac{\partial f}% {\partial\bar{z}}\circ g\right)\frac{\partial\bar{g}}{\partial z}
  46. z ¯ ( f g ) = ( f z g ) g z ¯ + ( f z ¯ g ) g ¯ z ¯ \frac{\partial}{\partial\bar{z}}\left(f\circ g\right)=\left(\frac{\partial f}{% \partial z}\circ g\right)\frac{\partial g}{\partial\bar{z}}+\left(\frac{% \partial f}{\partial\bar{z}}\circ g\right)\frac{\partial\bar{g}}{\partial\bar{% z}}
  47. g C 1 ( Ω , Ω ) \scriptstyle g\in C^{1}(\Omega^{\prime},\Omega)
  48. f C 1 ( Ω , Ω ′′ ) \scriptstyle f\in C^{1}(\Omega,\Omega^{\prime\prime})
  49. i = 1 , , m \scriptstyle i=1,\dots,m
  50. z i ( f g ) = j = 1 n ( f z j g ) g j z i + j = 1 n ( f z ¯ j g ) g ¯ j z i \frac{\partial}{\partial z_{i}}\left(f\circ g\right)=\sum_{j=1}^{n}\left(\frac% {\partial f}{\partial z_{j}}\circ g\right)\frac{\partial g_{j}}{\partial z_{i}% }+\sum_{j=1}^{n}\left(\frac{\partial f}{\partial\bar{z}_{j}}\circ g\right)% \frac{\partial\bar{g}_{j}}{\partial z_{i}}
  51. z ¯ i ( f g ) = j = 1 n ( f z j g ) g j z ¯ i + j = 1 n ( f z ¯ j g ) g ¯ j z ¯ i \frac{\partial}{\partial\bar{z}_{i}}\left(f\circ g\right)=\sum_{j=1}^{n}\left(% \frac{\partial f}{\partial z_{j}}\circ g\right)\frac{\partial g_{j}}{\partial% \bar{z}_{i}}+\sum_{j=1}^{n}\left(\frac{\partial f}{\partial\bar{z}_{j}}\circ g% \right)\frac{\partial\bar{g}_{j}}{\partial\bar{z}_{i}}
  52. f C 1 ( Ω ) \scriptstyle f\in C^{1}(\Omega)
  53. i = 1 , , n \scriptstyle i=1,\dots,n
  54. f ¯ z i = f ¯ z ¯ i , f ¯ z ¯ i = f ¯ z i \frac{\overline{\partial f}}{\partial z_{i}}=\frac{\partial\bar{f}}{\partial% \bar{z}_{i}},\quad\frac{\overline{\partial f}}{\partial\bar{z}_{i}}=\frac{% \partial\bar{f}}{\partial z_{i}}
  55. d d
  56. z ¯ w \scriptstyle\partial_{\bar{z}}w
  57. L p ( Ω ) L_{p}(\Omega)
  58. w ( z ) w(z)
  59. G G
  60. z ¯ w \scriptstyle\partial_{\bar{z}}w
  61. C 1 C^{1}
  62. C 1 C^{1}

Witsenhausen's_counterexample.html

  1. x 0 . x_{0}.
  2. u 1 u_{1}
  3. x 2 x_{2}
  4. u 2 u_{2}
  5. y 1 = x 1 + z y_{1}=x_{1}+z
  6. x 1 x_{1}
  7. x 0 x_{0}
  8. u 1 u_{1}
  9. x 1 = x 0 + u 1 , x_{1}=x_{0}+u_{1},
  10. x 2 = x 1 - u 2 , x_{2}=x_{1}-u_{2},
  11. y 1 = x 1 + z . y_{1}=x_{1}+z.
  12. k 2 E [ u 1 2 ] + E [ x 2 2 ] k^{2}E[u_{1}^{2}]+E[x_{2}^{2}]
  13. x 0 x_{0}
  14. z z
  15. z z
  16. x 0 x_{0}
  17. u 1 ( x 0 ) and u 2 ( y 1 ) u_{1}(x_{0})\quad\,\text{and}\quad u_{2}(y_{1})
  18. u 1 ( x 0 ) u_{1}(x_{0})
  19. u 2 ( y 1 ) u_{2}(y_{1})
  20. E ( x 1 ) = 0 E(x_{1})=0
  21. x 0 x_{0}
  22. x 0 x_{0}
  23. x 0 x_{0}
  24. k k
  25. ( k = 0.2 , σ 0 = 5 ) (k=0.2,\;\sigma_{0}=5)

Witten_conjecture.html

  1. M ¯ \overline{M}
  2. M ¯ \overline{M}
  3. M ¯ \overline{M}
  4. M ¯ \overline{M}
  5. F ( t 0 , t 1 , ) = τ 0 k 0 τ 1 k 1 i 0 t i k i k i ! = t 0 3 6 + t 1 24 + t 0 t 2 24 + t 1 2 24 + t 0 2 t 3 48 + F(t_{0},t_{1},\ldots)=\sum\langle\tau_{0}^{k_{0}}\tau_{1}^{k_{1}}\cdots\rangle% \prod_{i\geq 0}\frac{t_{i}^{k_{i}}}{k_{i}!}=\frac{t_{0}^{3}}{6}+\frac{t_{1}}{2% 4}+\frac{t_{0}t_{2}}{24}+\frac{t_{1}^{2}}{24}+\frac{t_{0}^{2}t_{3}}{48}+\cdots
  6. d 1 + + d n = 3 g - 3 + n τ d 1 , , τ d n 1 i n ( 2 d i - 1 ) ! ! λ i 2 d i + 1 = Γ G g , n 2 - | X 0 | | Aut Γ | e X 1 2 λ ( e ) \sum_{d_{1}+\cdots+d_{n}=3g-3+n}\langle\tau_{d_{1}},\ldots,\tau_{d_{n}}\rangle% \prod_{1\leq i\leq n}\frac{(2d_{i}-1)!!}{\lambda_{i}^{2d_{i}+1}}=\sum_{\Gamma% \in G_{g,n}}\frac{2^{-|X_{0}|}}{|\,\text{Aut}\Gamma|}\prod_{e\in X_{1}}\frac{2% }{\lambda(e)}
  7. log exp ( i tr X 3 / 6 ) d μ \log\int\exp(i\,\text{tr}X^{3}/6)d\mu
  8. t i = - tr Λ - 1 - 2 i 1 × 3 × 5 × × ( 2 i - 1 ) t_{i}=\frac{-\,\text{tr}\Lambda^{-1-2i}}{1\times 3\times 5\times\cdots\times(2% i-1)}
  9. d μ = c Λ exp ( - tr X 2 Λ / 2 ) d X d\mu=c_{\Lambda}\exp(-\,\text{tr}X^{2}\Lambda/2)dX
  10. X i j X k l d μ = δ i l δ j k 2 Λ i + Λ j \int X_{ij}X_{kl}d\mu=\delta_{il}\delta_{jk}\frac{2}{\Lambda_{i}+\Lambda_{j}}

Wolstenholme_prime.html

  1. ( 2 p - 1 p - 1 ) 1 ( mod p 4 ) , {2p-1\choose p-1}\equiv 1\;\;(\mathop{{\rm mod}}p^{4}),
  2. ( 2 p - 1 p - 1 ) 1 ( mod p 3 ) . {2p-1\choose p-1}\equiv 1\;\;(\mathop{{\rm mod}}p^{3}).
  3. H p - 1 0 ( mod p 3 ) , H_{p-1}\equiv 0\;\;(\mathop{{\rm mod}}p^{3})\,,
  4. H p - 1 H_{p-1}
  5. × 10 7 \times 10^{7}
  6. × 10 8 \times 10^{8}
  7. × 10 8 \times 10^{8}
  8. 10 9 10^{9}
  9. W p < m t p l ( 2 p - 1 p - 1 ) - 1 p 3 . W_{p}<mtpl>{{=}}\frac{{2p-1\choose p-1}-1}{p^{3}}.

Wonderland_model.html

  1. x ( t ) x(t)
  2. y ( t ) y(t)
  3. z ( t ) z(t)
  4. p ( t ) p(t)
  5. x , y [ 0 , ) x,y\in[0,\infty)
  6. z , p [ 0 , 1 ] z,p\in[0,1]
  7. x ( t + 1 ) \displaystyle x(t+1)
  8. γ , η , λ \ \gamma,\ \eta,\ \lambda
  9. χ , δ , ρ , ω \ \chi,\ \delta,\ \rho,\ \omega
  10. α , α 0 , α 1 , α 2 , β , β 0 , β 1 , θ \ \alpha,\ \alpha_{0},\ \alpha_{1},\ \alpha_{2},\ \beta,\beta_{0},\ \beta_{1},\ \theta
  11. b ( y , z ) b(y,z)
  12. d ( y , z ) d(y,z)
  13. f ( x , y , p ) f(x,y,p)
  14. y y
  15. b b
  16. d d
  17. y = y - ϕ ( 1 - z ) μ y \displaystyle y^{\prime}=y-\phi(1-z)^{\mu}y
  18. y ( t ) y(t)
  19. f f
  20. f ( x , y , p ) = x y p - κ e ϵ ϕ ( 1 - z ) μ y x 1 + e ϵ ϕ ( 1 - z ) μ y x \displaystyle f(x,y,p)=xyp-\kappa\frac{e^{\epsilon\phi(1-z)^{\mu}yx}}{1+e^{% \epsilon\phi(1-z)^{\mu}yx}}
  21. ϕ , μ , κ \ \phi,\ \mu,\ \kappa
  22. y ( t + 1 ) \displaystyle y(t+1)
  23. γ 0 , χ 0 , τ \ \gamma_{0},\ \chi_{0},\ \tau
  24. τ \tau
  25. τ \tau
  26. y ( t ) y(t)

Wood_method.html

  1. η 1 \eta_{1}
  2. η 2 \eta_{2}
  3. η i = K c + K i K c + K i + K i 1 + K i 2 , i = 1 , 2 \eta_{i}=\frac{K_{c}+K_{i}}{K_{c}+K_{i}+K_{i}1+K_{i}2},\quad i=1,2
  4. K i K_{i}

Woo–Lam.html

  1. A , B A,B
  2. K U x KU_{x}
  3. x x
  4. K R x KR_{x}
  5. x x
  6. N x N_{x}
  7. x x
  8. I D x ID_{x}
  9. x x
  10. E k E_{k}
  11. k k
  12. S k S_{k}
  13. k k
  14. K K
  15. | | ||
  16. 1 ) A K D C : I D A | | I D B 1)A\rightarrow KDC:ID_{A}||ID_{B}
  17. 2 ) K D C A : S K R K D C [ I D B | | K U B ] 2)KDC\rightarrow A:S_{KR_{KDC}}[ID_{B}||KU_{B}]
  18. 3 ) A B : E K U B [ N A | | I D A ] 3)A\rightarrow B:E_{KU_{B}}[N_{A}||ID_{A}]
  19. 4 ) B K D C : I D B | | I D A | | E K U K D C [ N A ] 4)B\rightarrow KDC:ID_{B}||ID_{A}||E_{KU_{KDC}}[N_{A}]
  20. 5 ) K D C B : S K R K D C [ I D A | | K U A ] | | E K U B [ S K R K D C [ N A | | K | | I D B | | I D A ] ] 5)KDC\rightarrow B:S_{KR_{KDC}}[ID_{A}||KU_{A}]||E_{KU_{B}}[S_{KR_{KDC}}[N_{A}% ||K||ID_{B}||ID_{A}]]
  21. 6 ) B A : E K U A [ S K R K D C [ N A | | K | | ] | | N B ] ] 6)B\rightarrow A:E_{KU_{A}}[S_{KR_{KDC}}[N_{A}||K||]||N_{B}]]
  22. 7 ) A B : E K [ N B ] 7)A\rightarrow B:E_{K}[N_{B}]
  23. I D A ID_{A}
  24. N A N_{A}

Word_lists_by_frequency.html

  1. N N
  2. N = 0.5 - log 2 ( Frequency of this item Frequency of most common item ) N=\left\lfloor 0.5-\log_{2}\left(\frac{\,\text{Frequency of this item}}{\,% \text{Frequency of most common item}}\right)\right\rfloor
  3. \lfloor\ldots\rfloor

WORHP.html

  1. min x \R n f ( x ) \min_{x\in\R^{n}}f(x)
  2. L ( x g ( x ) ) U L\leq\begin{pmatrix}x\\ g(x)\end{pmatrix}\leq U
  3. f : \R n \R f:\R^{n}\to\R
  4. g : \R n \R m g:\R^{n}\to\R^{m}
  5. n n
  6. m m
  7. f f
  8. g g

Work_sampling.html

  1. σ P = p q n \sigma_{P}=\sqrt{\frac{pq}{n}}
  2. n = p q σ P 2 n=\frac{pq}{{\sigma_{P}}^{2}}
  3. σ P = \sigma_{P}=
  4. p = p=
  5. q = q=
  6. n = n=

Wouthuysen–Field_coupling.html

  1. 5.9 × 10 - 6 5.9\times 10^{-6}
  2. z 6 z\sim 6

Wozencraft_ensemble.html

  1. ε \varepsilon
  2. k k
  3. C i n 1 , C i n 2 , . . , C i n N C_{in}^{1},C_{in}^{2},..,C_{in}^{N}
  4. 1 2 \frac{1}{2}
  5. N = q k - 1 N=q^{k}-1
  6. ( 1 - ε ) N \left({1-\varepsilon}\right)N
  7. C i n i C_{in}^{i}
  8. H q - 1 ( 1 2 - ε ) \geq H_{q}^{-1}(\frac{1}{2}-\varepsilon)
  9. H q H_{q}
  10. H q ( x ) = x l o g q ( q - 1 ) - x l o g q x - ( 1 - x ) l o g q ( 1 - x ) H_{q}(x)=xlog_{q}(q-1)-xlog_{q}x-(1-x)log_{q}(1-x)
  11. α 𝔽 q k - { 0 } \alpha\in\mathbb{F}_{q^{k}}-\{0\}
  12. C i n α : 𝔽 q k 𝔽 q 2 k C_{in}^{\alpha}:\mathbb{F}_{q}^{k}\to\mathbb{F}_{q}^{2k}
  13. C i n α ( x ) = ( x , α x ) C_{in}^{\alpha}(x)=(x,\alpha x)
  14. x 𝔽 q k x\in\mathbb{F}_{q}^{k}
  15. α 𝔽 q k \alpha\in\mathbb{F}_{q^{k}}
  16. α x \alpha x
  17. 𝔽 q k \mathbb{F}_{q}^{k}
  18. 𝔽 q k \mathbb{F}_{q^{k}}
  19. 𝔽 q k \mathbb{F}_{q}^{k}
  20. C i n α ( x ) + C i n α ( y ) = ( x , α x ) + ( y , α y ) = ( x + y , α ( x + y ) ) = C i n α ( x + y ) C_{in}^{\alpha}(x)+C_{in}^{\alpha}(y)=(x,\alpha x)+(y,\alpha y)=(x+y,\alpha(x+% y))=C_{in}^{\alpha}(x+y)
  21. a F q a\in F_{q}
  22. a C i n α ( x ) = a ( x , α x ) = ( a x , α ( a x ) ) = C i n α ( a x ) aC_{in}^{\alpha}(x)=a(x,\alpha x)=\left({ax,\alpha\left({ax}\right)}\right)=C_% {in}^{\alpha}(ax)
  23. C i n α C_{in}^{\alpha}
  24. α 𝔽 q k - { 0 } \alpha\in\mathbb{F}_{q^{k}}-\{0\}
  25. 1 2 \frac{1}{2}
  26. ( 1 - ε ) N \left({1-\varepsilon}\right)N
  27. H q - 1 ( 1 2 - ε ) \geq H_{q}^{-1}(\frac{1}{2}-\varepsilon)
  28. ( 1 - ε ) N (1-\varepsilon)N
  29. H q - 1 ( 1 2 - ε ) \geq H_{q}^{-1}(\frac{1}{2}-\varepsilon)
  30. ε N \varepsilon N
  31. H q - 1 ( 1 2 - ε ) 2 k H_{q}^{-1}(\frac{1}{2}-\varepsilon)\cdot 2k
  32. H q - 1 ( 1 2 - ε ) 2 k H_{q}^{-1}(\frac{1}{2}-\varepsilon)\cdot 2k
  33. P P
  34. H q - 1 ( 1 2 - ε ) 2 k H_{q}^{-1}(\frac{1}{2}-\varepsilon)\cdot 2k
  35. H q - 1 ( 1 2 - ε ) 2 k H_{q}^{-1}(\frac{1}{2}-\varepsilon)\cdot 2k
  36. C i n α 1 C_{in}^{\alpha_{1}}
  37. C i n α 2 C_{in}^{\alpha_{2}}
  38. α 1 α 2 𝔽 q k - { 0 } \alpha_{1}\neq\alpha_{2}\in\mathbb{F}_{q^{k}}-\{0\}
  39. α 1 α 2 𝔽 q k - { 0 } \alpha_{1}\neq\alpha_{2}\in\mathbb{F}_{q^{k}}-\{0\}
  40. C i n α 1 C_{in}^{\alpha_{1}}
  41. C i n α 2 C_{in}^{\alpha_{2}}
  42. y C i n α 1 - { 0 } y\in C_{in}^{\alpha_{1}}-\{0\}
  43. y = ( y 1 , α 1 y 1 ) y=(y_{1},\alpha_{1}y_{1})
  44. y 1 𝔽 q k y_{1}\in\mathbb{F}_{q}^{k}
  45. y y
  46. y 1 0 y_{1}\neq 0
  47. y = ( y 2 , α 2 y 2 ) y=(y_{2},\alpha_{2}y_{2})
  48. y 2 𝔽 q k - { 0 } y_{2}\in\mathbb{F}_{q}^{k}-\{0\}
  49. ( y 1 , α 1 y 1 ) = ( y 2 , α 2 y 2 ) (y_{1},\alpha_{1}y_{1})=(y_{2},\alpha_{2}y_{2})
  50. y 1 = y 2 0 y_{1}=y_{2}\neq 0
  51. α 1 y 1 = α 2 y 2 \alpha_{1}y_{1}=\alpha_{2}y_{2}
  52. α 1 = α 2 \alpha_{1}=\alpha_{2}
  53. H q - 1 ( 1 2 - ε ) 2 k H_{q}^{-1}(\frac{1}{2}-\varepsilon)\cdot 2k
  54. P P
  55. P P
  56. y y
  57. w t ( y ) wt(y)
  58. y y
  59. w t ( y ) wt(y)
  60. y y
  61. y y
  62. P P
  63. y F q 2 k y\in F_{q}^{2k}
  64. S = { y | w t ( y ) S=\{y|wt(y)
  65. P | S | V o l q ( H q - 1 ( 1 2 - ε ) 2 k , 2 k ) P\leq|S|\leq Vol_{q}(H_{q}^{-1}(\frac{1}{2}-\varepsilon)\cdot 2k,2k)
  66. V o l q ( r , n ) Vol_{q}(r,n)
  67. [ q ] n [q]^{n}
  68. V o l q ( p n , n ) q H q ( p ) n Vol_{q}(pn,n)\leq q^{H_{q}(p)n}
  69. P q H q ( H q - 1 ( 1 2 - ε ) ) 2 k = q ( 1 2 - ε ) 2 k = q k q 2 ε k P\leq q^{H_{q}(H_{q}^{-1}(\frac{1}{2}-\varepsilon))\cdot 2k}=q^{(\frac{1}{2}-% \varepsilon)\cdot 2k}=\frac{q^{k}}{q^{2\varepsilon k}}
  70. k k
  71. q k q 2 ε k \frac{q^{k}}{q^{2\varepsilon k}}
  72. P P
  73. ε N \varepsilon N
  74. H q - 1 ( 1 2 - ε ) 2 k H_{q}^{-1}(\frac{1}{2}-\varepsilon)\cdot 2k
  75. N - ε N = ( 1 - ε ) N N-\varepsilon N=(1-\varepsilon)N

Wrapped_exponential_distribution.html

  1. 1 - e - λ θ 1 - e - 2 π λ \frac{1-e^{-\lambda\theta}}{1-e^{-2\pi\lambda}}
  2. arctan ( 1 / λ ) \arctan(1/\lambda)
  3. 1 - λ 1 + λ 2 1-\frac{\lambda}{\sqrt{1+\lambda^{2}}}
  4. 1 + ln ( β - 1 λ ) - β β - 1 ln ( β ) 1+\ln\left(\frac{\beta-1}{\lambda}\right)-\frac{\beta}{\beta-1}\ln(\beta)
  5. β = e 2 π λ \beta=e^{2\pi\lambda}
  6. 1 1 - i n / λ \frac{1}{1-in/\lambda}
  7. f W E ( θ ; λ ) = k = 0 λ e - λ ( θ + 2 π k ) = λ e - λ θ 1 - e - 2 π λ , f_{WE}(\theta;\lambda)=\sum_{k=0}^{\infty}\lambda e^{-\lambda(\theta+2\pi k)}=% \frac{\lambda e^{-\lambda\theta}}{1-e^{-2\pi\lambda}},
  8. 0 θ < 2 π 0\leq\theta<2\pi
  9. λ > 0 \lambda>0
  10. 0 X < 2 π 0\leq X<2\pi
  11. φ n ( λ ) = 1 1 - i n / λ \varphi_{n}(\lambda)=\frac{1}{1-in/\lambda}
  12. f W E ( θ ; λ ) = 1 2 π n = - e i n θ 1 - i n / λ . f_{WE}(\theta;\lambda)=\frac{1}{2\pi}\sum_{n=-\infty}^{\infty}\frac{e^{in% \theta}}{1-in/\lambda}.
  13. z = e i θ z=e^{i\theta}
  14. z n = Γ e i n θ f W E ( θ ; λ ) d θ = 1 1 - i n / λ , \langle z^{n}\rangle=\int_{\Gamma}e^{in\theta}\,f_{WE}(\theta;\lambda)\,d% \theta=\frac{1}{1-in/\lambda},
  15. Γ \Gamma\,
  16. 2 π 2\pi
  17. z = 1 1 - i / λ . \langle z\rangle=\frac{1}{1-i/\lambda}.
  18. θ = Arg z = arctan ( 1 / λ ) , \langle\theta\rangle=\mathrm{Arg}\langle z\rangle=\arctan(1/\lambda),
  19. R = | z | = λ 2 1 + λ 2 . R=|\langle z\rangle|=\frac{\lambda^{2}}{1+\lambda^{2}}.
  20. 0 θ < 2 π 0\leq\theta<2\pi
  21. E ( θ ) \operatorname{E}(\theta)

Write_amplification.html

  1. data written to the flash memory data written by the host = write amplification \frac{\,\text{data written to the flash memory}}{\,\text{data written by the % host}}=\,\text{write amplification}
  2. physical capacity - user capacity user capacity = over-provision \frac{\,\text{physical capacity}-\,\text{user capacity}}{\,\text{user capacity% }}=\,\text{over-provision}

YAK_(cryptography).html

  1. G G
  2. g g
  3. q q
  4. g a g^{a}
  5. g b g^{b}
  6. x R [ 0 , q - 1 ] x\in\text{R}[0,q-1]
  7. g x g^{x}
  8. x x
  9. y R [ 0 , q - 1 ] y\in\text{R}[0,q-1]
  10. g y g^{y}
  11. y y
  12. R \in\text{R}
  13. K = ( g y g b ) x + a = g ( x + a ) ( y + b ) K=(g^{y}g^{b})^{x+a}=g^{(x+a)(y+b)}
  14. K = ( g x g a ) y + b = g ( x + a ) ( y + b ) K=(g^{x}g^{a})^{y+b}=g^{(x+a)(y+b)}
  15. K K
  16. κ = H ( K ) \kappa=H(K)

Yao_graph.html

  1. k 6 k≥6
  2. k k
  3. 1 / ( cos θ - sin θ ) 1/(\cos\theta-\sin\theta)
  4. θ \theta

Yoshimine_sort.html

  1. χ i \chi_{i}
  2. χ i \chi_{i}
  3. ( p q | r s ) = χ p ( 𝐫 1 ) χ q ( 𝐫 1 ) χ r ( 𝐫 2 ) χ s ( 𝐫 2 ) 𝐫 1 - 𝐫 2 d 𝐫 1 d 𝐫 2 (pq|rs)=\int\int\frac{\chi_{p}(\mathbf{r}_{1})\;\chi_{q}(\mathbf{r}_{1})\;\chi% _{r}(\mathbf{r}_{2})\;\chi_{s}(\mathbf{r}_{2})\;}{\mid\mathbf{r}_{1}-\mathbf{r% }_{2}\mid}\;\;d\mathbf{r}_{1}\;d\mathbf{r}_{2}
  4. ( p q | r s ) (pq|rs)\in\mathbb{R}
  5. ( p q | r s ) = ( q p | r s ) ( p q | s r ) ( q p | s r ) ( r s | p q ) ( s r | p q ) ( r s | q p ) ( s r | q p ) \begin{matrix}(pq|rs)=&(qp|rs)\\ &(pq|sr)\\ &(qp|sr)\\ &(rs|pq)\\ &(sr|pq)\\ &(rs|qp)\\ &(sr|qp)\end{matrix}
  6. C 1 C_{1}
  7. C 2 C_{2}
  8. A A
  9. B B
  10. ( A A | A A ) ( B B | B B ) ( A A | B B ) ( A B | A B ) \begin{matrix}(\;A\;A\;|\;A\;A\;)\\ (\;B\;B\;|\;B\;B\;)\\ (\;A\;A\;|\;B\;B\;)\\ (\;A\;B\;|\;A\;B\;)\\ \end{matrix}

Youla–Kucera_parametrization.html

  1. P ( s ) P(s)
  2. P ( s ) P(s)
  3. { Q ( s ) 1 - P ( s ) Q ( s ) , Q ( s ) Ω } \left\{\frac{Q(s)}{1-P(s)Q(s)},Q(s)\in\Omega\right\}
  4. Q ( s ) Q(s)
  5. Q ( s ) Q(s)
  6. P ( s ) P(s)
  7. P ( s ) P(s)
  8. P ( s ) = N ( s ) M ( s ) P(s)=\frac{N(s)}{M(s)}
  9. 𝐍 ( 𝐬 ) 𝐗 ( 𝐬 ) + 𝐌 ( 𝐬 ) 𝐘 ( 𝐬 ) = 𝟏 \mathbf{N(s)X(s)}+\mathbf{M(s)Y(s)}=\mathbf{1}
  10. C ( s ) = X ( s ) Y ( s ) C(s)=\frac{X(s)}{Y(s)}
  11. { X ( s ) + M ( s ) Q ( s ) Y ( s ) - N ( s ) Q ( s ) , Q ( s ) Ω } \left\{\frac{X(s)+M(s)Q(s)}{Y(s)-N(s)Q(s)},Q(s)\in\Omega\right\}
  12. 𝐏 ( 𝐬 ) \mathbf{P(s)}
  13. 𝐏 ( 𝐬 ) = 𝐍 ( 𝐬 ) 𝐃 - 𝟏 ( 𝐬 ) \mathbf{P(s)=N(s)D^{-1}(s)}
  14. 𝐏 ( 𝐬 ) = 𝐃 ~ - 𝟏 ( 𝐬 ) 𝐍 ~ ( 𝐬 ) \mathbf{P(s)=\tilde{D}^{-1}(s)\tilde{N}(s)}
  15. [ 𝐗 𝐘 - 𝐍 ~ 𝐃 ~ ] [ 𝐃 - 𝐘 ~ 𝐍 𝐗 ~ ] = [ 𝐈 0 0 𝐈 ] \left[\begin{matrix}\mathbf{X}&\mathbf{Y}\\ -\mathbf{\tilde{N}}&{\mathbf{\tilde{D}}}\\ \end{matrix}\right]\left[\begin{matrix}\mathbf{D}&-\mathbf{\tilde{Y}}\\ \mathbf{N}&{\mathbf{\tilde{X}}}\\ \end{matrix}\right]=\left[\begin{matrix}\mathbf{I}&0\\ 0&\mathbf{I}\\ \end{matrix}\right]
  16. 𝐗 , 𝐘 , 𝐗 ~ , 𝐘 ~ \mathbf{X,Y,\tilde{X},\tilde{Y}}
  17. 𝐊 ( 𝐬 ) = ( 𝐗 - 𝚫 𝐍 ~ ) - 1 ( 𝐘 + 𝚫 𝐃 ~ ) = ( 𝐘 ~ + 𝐃 𝚫 ) ( 𝐗 ~ - 𝐍 𝚫 ) - 1 \begin{aligned}&\displaystyle\mathbf{K(s)}={{\left(\mathbf{X}-\mathbf{\Delta% \tilde{N}}\right)}^{-1}}\left(\mathbf{Y}+\mathbf{\Delta\tilde{D}}\right)\\ &\displaystyle=\left(\mathbf{\tilde{Y}}+\mathbf{D\Delta}\right){{\left(\mathbf% {\tilde{X}}-\mathbf{N\Delta}\right)}^{-1}}\end{aligned}
  18. Δ \Delta

Young's_interference_experiment.html

  1. θ f λ / d \theta_{f}\approx\lambda/d
  2. θ < s u b > f θ<sub>f

Z_N_model.html

  1. Z N Z_{N}
  2. Z N Z_{N}
  3. r r
  4. s r = exp 2 π i q N s_{r}=\exp{\frac{2\pi iq}{N}}
  5. q { 0 , 1 , , N - 1 } q\in\{0,1,\ldots,N-1\}
  6. Z N Z_{N}
  7. N N
  8. r r rr^{\prime}
  9. w ( r , r ) = k = 0 N - 1 x k ( r r ) ( s r s r * ) k w\left(r,r^{\prime}\right)=\sum_{k=0}^{N-1}x_{k}^{\left(rr^{\prime}\right)}% \left(s_{r}s_{r^{\prime}}^{*}\right)^{k}
  10. * *
  11. x k ( r r ) x_{k}^{\left(rr^{\prime}\right)}
  12. r r rr^{\prime}
  13. x k ( r r ) = x N - k ( r r ) x_{k}^{\left(rr^{\prime}\right)}=x_{N-k}^{\left(rr^{\prime}\right)}
  14. x 0 x_{0}
  15. s r ω k s r s_{r}\rightarrow\omega^{k}s_{r}
  16. s r s r * s_{r}\rightarrow s^{*}_{r}
  17. Z N Z_{N}
  18. x k 1 x_{k}^{1}
  19. x k 2 x_{k}^{2}
  20. α \alpha
  21. x n 1 = x n ( α ) x_{n}^{1}=x_{n}\left(\alpha\right)
  22. x n 2 = x n ( π - α ) x_{n}^{2}=x_{n}\left(\pi-\alpha\right)
  23. x n ( α ) = k = 0 n - 1 sin ( π k / N + α / 2 N ) sin [ π ( k + 1 ) / N - α / 2 N ] x_{n}\left(\alpha\right)=\prod_{k=0}^{n-1}\frac{\sin\left(\pi k/N+\alpha/2N% \right)}{\sin\left[\pi\left(k+1\right)/N-\alpha/2N\right]}
  24. x 0 = 1 x_{0}=1
  25. Z N Z_{N}
  26. N N\rightarrow\infty
  27. Z N Z_{N}
  28. N N
  29. x k x_{k}
  30. s r = ± 1 s_{r}=\pm 1
  31. Z N Z_{N}
  32. N = 2 N=2
  33. Z N Z_{N}
  34. Z N Z_{N}
  35. N = 3 N=3
  36. x 1 = x 2 = = x N - 1 = x c x_{1}=x_{2}=\dots=x_{N-1}=x_{c}
  37. x c x_{c}
  38. N = 4 N=4
  39. Z N Z_{N}
  40. Z N Z_{N}

Zanstra_method.html

  1. ν 0 \nu_{0}
  2. ν 0 L ν h ν d ν = 0 r 1 n p n e α B d V \int_{\nu_{0}}^{\infty}\frac{L_{\nu}}{h\nu}d\nu=\int_{0}^{r_{1}}n_{p}n_{e}% \alpha_{B}dV
  3. r 1 r_{1}
  4. n p , n e n_{p},n_{e}
  5. L ν L_{\nu}
  6. α B \alpha_{B}
  7. L ν H β ν 0 L ν h ν d ν h ν H β α H β eff α B \frac{L_{\nu_{H\beta}}}{\int_{\nu_{0}}^{\infty}\frac{L_{\nu}}{h\nu}d\nu}% \approx h\nu_{H\beta}\frac{\alpha_{H\beta}\text{eff}}{\alpha_{B}}
  8. α H β eff \alpha_{H\beta}\text{eff}
  9. ν s \nu_{s}
  10. Z = L ν s ν 0 L ν h ν d ν = h ν H β α H β eff α B F ν s F H β Z=\frac{L_{\nu_{s}}}{\int_{\nu_{0}}^{\infty}\frac{L_{\nu}}{h\nu}d\nu}=h\nu_{H% \beta}\frac{\alpha_{H\beta}\text{eff}}{\alpha_{B}}\frac{F_{\nu_{s}}}{F_{H\beta}}
  11. F ν s F_{\nu_{s}}
  12. F H β F_{H\beta}

Zariski_ring.html

  1. 𝔞 \mathfrak{a}
  2. A ^ \widehat{A}
  3. 𝔞 \mathfrak{a}
  4. A ^ \widehat{A}
  5. 𝔞 \mathfrak{a}

Zeeman_energy.html

  1. E Z e e m a n = - μ 0 V 𝐌 𝐇 E x t d V E_{Zeeman}=-\mu_{0}\int_{V}\,\textbf{M}\cdot\textbf{H}_{Ext}\,\mathrm{d}V
  2. H = - 𝐦 𝐁 H=-\textbf{m}\cdot\textbf{B}

Zeeman_slower.html

  1. M M
  2. ω = c k + δ \omega=ck+\delta
  3. γ \gamma
  4. k k
  5. I = s 0 I s I=s_{0}I_{s}
  6. I s = c γ k 3 / 12 π I_{s}=\hbar c\gamma k^{3}/12\pi
  7. a = k γ 2 M s 0 1 + s 0 + ( 2 δ / γ ) 2 \vec{a}=\frac{\hbar\vec{k}\gamma}{2M}\frac{s_{0}}{1+s_{0}+\left(2\delta^{% \prime}/\gamma\right)^{2}}
  8. v v
  9. k L v k_{L}v
  10. B B
  11. μ B / \mu^{\prime}B/\hbar
  12. μ \mu^{\prime}
  13. δ = δ + k v - μ B \delta^{\prime}=\delta+kv-\frac{\mu^{\prime}B}{\hbar}
  14. δ = 0 \delta^{\prime}=0
  15. a = η a m a x a=\eta a_{max}
  16. η = s 0 / ( 1 + s 0 ) \eta=s_{0}/(1+s_{0})
  17. a m a x = k γ 2 M a_{max}=\frac{\hbar k\gamma}{2M}
  18. z z
  19. a = η a m a x a=\eta a_{max}
  20. v ( z ) = v i 2 - 2 a z v\left(z\right)=\sqrt{v_{i}^{2}-2az}
  21. B ( z ) = k μ v - δ μ = k v i μ 1 - 2 a v i 2 z - δ μ B\left(z\right)=\frac{\hbar k}{\mu^{\prime}}v-\frac{\hbar\delta}{\mu^{\prime}}% =\frac{\hbar kv_{i}}{\mu^{\prime}}\sqrt{1-\frac{2a}{v_{i}^{2}}z}-\frac{\hbar% \delta}{\mu^{\prime}}
  22. v i v_{i}
  23. v < v i v<v_{i}
  24. v > v i v>v_{i}
  25. η \eta
  26. η 1 \eta\approx 1
  27. B ( z ) = B 0 + B a 1 - z / z 0 B(z)=B_{0}+B_{a}\sqrt{1-z/z_{0}}

Zone_diagram.html

  1. n n
  2. { p 1 , , p n } \{p_{1},\ldots,p_{n}\}
  3. p k \displaystyle{p_{k}}
  4. R k \displaystyle{R_{k}}
  5. p k \displaystyle{p_{k}}
  6. p j , j k p_{j},\,j\neq k
  7. ( R k ) k = 1 n (R_{k})_{k=1}^{n}
  8. p k p_{k}
  9. p k p_{k}
  10. p k p_{k}
  11. R k R_{k}
  12. p k p_{k}
  13. R k = { x | d ( x , p k ) d ( x , R j ) , for all j k } R_{k}=\{x\,|\,\,d(x,p_{k})\leq d(x,R_{j}),\,\,\text{for all}\,j\neq k\}
  14. d ( a , b ) \displaystyle{d(a,b)}
  15. a a
  16. b b
  17. d ( x , A ) = inf { d ( x , a ) | a A } d(x,A)=\inf\{d(x,a)\,|\,a\in A\}
  18. x x
  19. A A
  20. x = ( x 1 , x 2 ) 2 x=(x_{1},x_{2})\in\mathbb{R}^{2}
  21. ( R k ) k = 1 n (R_{k})_{k=1}^{n}
  22. R k R_{k}
  23. R j \displaystyle{R_{j}}
  24. j , j k j,\,j\neq k
  25. R j \displaystyle{R_{j}}
  26. R k \displaystyle{R_{k}}
  27. ( R k ) k = 1 n (R_{k})_{k=1}^{n}
  28. { R 1 = { x 2 | d ( x , p 1 ) d ( x , R j ) , for all j 1 } R n = { x 2 | d ( x , p n ) d ( x , R j ) , for all j n } \begin{cases}R_{1}=\{x\in\mathbb{R}^{2}\,|\,\,d(x,p_{1})\leq d(x,R_{j}),\,\,% \text{for all}\,j\neq 1\}\\ \vdots\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\vdots\\ R_{n}=\{x\in\mathbb{R}^{2}\,|\,\,d(x,p_{n})\leq d(x,R_{j}),\,\,\text{for all}% \,j\neq n\}\end{cases}
  29. ( X , d ) \displaystyle{(X,d)}
  30. K \displaystyle{K}
  31. ( P k ) k K (P_{k})_{k\in K}
  32. X \displaystyle{X}
  33. R = ( R k ) k K R=(R_{k})_{k\in K}
  34. X \displaystyle{X}
  35. k K k\in K
  36. R k = { x X | d ( x , P k ) d ( x , R j ) , for all j k } . R_{k}=\{x\in X\,|\,\,d(x,P_{k})\leq d(x,R_{j}),\,\,\text{for all}\,j\neq k\}.
  37. k K k\in K
  38. X k \displaystyle{X_{k}}
  39. X \displaystyle{X}
  40. Y = k K X k Y=\prod_{k\in K}X_{k}
  41. Dom : Y Y \,\text{Dom}:Y\to Y
  42. Dom ( R ) = S \displaystyle{\,\text{Dom}(R)=S}
  43. S = ( S k ) k K S=(S_{k})_{k\in K}
  44. S k = { x X | d ( x , P k ) d ( x , R j ) , for all j k } . S_{k}=\{x\in X\,|\,\,d(x,P_{k})\leq d(x,R_{j}),\,\,\text{for all}\,j\neq k\}.
  45. R \displaystyle{R}
  46. R = Dom ( R ) R=\displaystyle{\,\text{Dom}(R)}

Zwanzig_projection_operator.html

  1. ( f 1 , f 2 ) = d Γ ρ 0 ( Γ ) f 1 ( Γ ) f 2 ( Γ ) , \left(f_{1},f_{2}\right)=\int d\Gamma\rho_{0}\left(\Gamma\right)f_{1}\left(% \Gamma\right)f_{2}\left(\Gamma\right),
  2. ρ 0 ( Γ ) = δ ( H ( Γ ) - E ) d Γ δ ( H ( Γ ) - E ) , \rho_{0}\left(\Gamma\right)=\frac{\delta\left(H\left(\Gamma\right)-E\right)}{% \int d\Gamma^{\prime}\delta\left(H\left(\Gamma^{\prime}\right)-E\right)},
  3. G ( A ( Γ ) ) = d a G ( a ) δ ( A ( Γ ) - a ) . G(A\left(\Gamma\right))=\int daG\left(a\right)\delta\left(A\left(\Gamma\right)% -a\right).
  4. f ( Γ ) = F ( A ( Γ ) ) + R ( Γ ) , f\left(\Gamma\right)=F\left(A\left(\Gamma\right)\right)+R\left(\Gamma\right),
  5. d Γ ρ 0 ( Γ ) f ( Γ ) δ ( A ( Γ ) - a ) = d Γ ρ 0 ( Γ ) F ( A ( Γ ) ) δ ( A ( Γ ) - a ) = F ( a ) d Γ ρ 0 ( Γ ) δ ( A ( Γ ) - a ) . \int d\Gamma\rho_{0}\left(\Gamma\right)f\left(\Gamma\right)\delta\left(A\left(% \Gamma\right)-a\right)=\int d\Gamma\rho_{0}\left(\Gamma\right)F\left(A\left(% \Gamma\right)\right)\delta\left(A\left(\Gamma\right)-a\right)=F\left(a\right)% \int d\Gamma\rho_{0}\left(\Gamma\right)\delta\left(A\left(\Gamma\right)-a% \right).
  6. P f ( Γ ) = F ( A ( Γ ) ) = d Γ ρ 0 ( Γ ) f ( Γ ) δ ( A ( Γ ) - A ( Γ ) ) d Γ ρ 0 ( Γ ) δ ( A ( Γ ) - A ( Γ ) ) . P\cdot f\left(\Gamma\right)=F\left(A\left(\Gamma\right)\right)=\frac{\int d% \Gamma^{\prime}\rho_{0}\left(\Gamma^{\prime}\right)f\left(\Gamma^{\prime}% \right)\delta\left(A\left(\Gamma^{\prime}\right)-A\left(\Gamma\right)\right)}{% \int d\Gamma^{\prime}\rho_{0}\left(\Gamma^{\prime}\right)\delta\left(A\left(% \Gamma^{\prime}\right)-A\left(\Gamma\right)\right)}.
  7. ρ ( Γ , t ) = ρ 0 ( Γ ) σ ( Γ , t ) \rho(\Gamma,t)=\rho_{0}(\Gamma)\sigma(\Gamma,t)
  8. σ ( Γ , t ) \sigma(\Gamma,t)
  9. ρ ( Γ , t ) \rho(\Gamma,t)
  10. i t σ ( Γ , t ) = L σ ( Γ , t ) . i\frac{\partial}{\partial t}\sigma(\Gamma,t)=L\sigma(\Gamma,t).
  11. ρ 1 = P σ \rho_{1}=P\sigma
  12. ρ 2 = ( 1 - P ) σ \rho_{2}=(1-P)\sigma
  13. i t ρ 1 = P L ρ 1 + P L ρ 2 , i\frac{\partial}{\partial t}\rho_{1}=PL\rho_{1}+PL\rho_{2},
  14. i t ρ 2 = ( 1 - P ) L ρ 2 + ( 1 - P ) L ρ 1 . i\frac{\partial}{\partial t}\rho_{2}=\left(1-P\right)L\rho_{2}+\left(1-P\right% )L\rho_{1}.
  15. ρ 2 \rho_{2}
  16. ρ 2 ( Γ , t ) \rho_{2}(\Gamma,t)
  17. ρ 1 \rho_{1}
  18. p ( A ( Γ ) , t ) = p 0 ( A ( Γ ) ) ρ 1 ( Γ , t ) , p(A(\Gamma),t)=p_{0}(A(\Gamma))\rho_{1}(\Gamma,t),
  19. p 0 ( a ) p_{0}(a)
  20. 1 = P + Q , U = U P + P U Q + Q U Q , = U n = U n P + m = 1 n U n - m P ( U Q ) m + Q ( U Q ) n . \begin{aligned}\displaystyle 1&\displaystyle=P+Q,\\ \displaystyle U&\displaystyle=UP+PUQ+QUQ,\\ \displaystyle...&\displaystyle=...\\ \displaystyle U^{n}&\displaystyle=U^{n}P+\sum_{m=1}^{n}U^{n-m}P\left(UQ\right)% ^{m}+Q\left(UQ\right)^{n}.\end{aligned}
  21. e i t L = e i t L P + i 0 t d s e i ( t - s ) L P L Q e i s L Q + Q e i t L Q . e^{itL}=e^{itL}P+i\int_{0}^{t}dse^{i\left(t-s\right)L}PLQe^{isLQ}+Qe^{itLQ}.
  22. d A d t ( Γ , t ) = V + K + R , V = e i t L P A ˙ ( Γ , 0 ) , K = i 0 t d s e i ( t - s ) L P L Q e i s L Q A ˙ ( Γ , 0 ) = i 0 t d s e i ( t - s ) L P L R ( s ) , R = Q e i t L Q A ˙ ( Γ , 0 ) . \begin{aligned}\displaystyle\frac{dA}{dt}\left(\Gamma,t\right)&\displaystyle=V% +K+R,\\ \displaystyle V&\displaystyle=e^{itL}P\dot{A}\left(\Gamma,0\right),\\ \displaystyle K&\displaystyle=i\int_{0}^{t}dse^{i\left(t-s\right)L}PLQe^{isLQ}% \dot{A}\left(\Gamma,0\right)=i\int_{0}^{t}dse^{i\left(t-s\right)L}PLR\left(s% \right),\\ \displaystyle R&\displaystyle=Qe^{itLQ}\dot{A}\left(\Gamma,0\right).\end{aligned}
  23. P f ( Γ ) = n ( f , Φ n ) Φ n ( A ( Γ ) ) . P\cdot f\left(\Gamma\right)=\sum_{n}\left(f,\Phi_{n}\right)\Phi_{n}\left(A% \left(\Gamma\right)\right).

Zyablov_bound.html

  1. R R
  2. δ \delta
  3. R R
  4. C o u t C_{out}
  5. δ \delta
  6. ( max 0 r ( 1 - H q ( δ + ε ) ) ) r ( 1 - δ H q - 1 ( 1 - r ) - ε ) \mathcal{R}\geq(\max\limits_{0\leq r\leq(1-H_{q}(\delta+\varepsilon))})r(1-{% \delta\over{H_{q}^{-1}(1-r)-\varepsilon}})
  7. r r
  8. C i n C_{in}
  9. C o u t C_{out}
  10. C i n C_{in}
  11. C o u t C_{out}
  12. R R
  13. C o u t C_{out}
  14. δ \delta
  15. 1 - R {1-R}
  16. C o u t C i n C_{out}\circ C_{in}
  17. C i n C_{in}
  18. C i n C_{in}
  19. r r
  20. 0
  21. δ i n \delta_{in}
  22. 0
  23. C i n C_{in}
  24. r r
  25. δ i n H q - 1 ( 1 - r ) - ε , ε \delta_{in}\geq H_{q}^{-1}(1-r)-\varepsilon,\varepsilon
  26. 0
  27. C o u t C i n C_{out}\circ C_{in}
  28. r R rR
  29. δ = ( 1 - R ) ( H q - 1 ( 1 - r ) - ε ) \delta=(1-R)(H_{q}^{-1}(1-r)-\varepsilon)
  30. R R
  31. δ , r \delta,r
  32. R = ( 1 - δ H - 1 ( 1 - r ) - ε ) R=(1-\frac{\delta}{H^{-1}(1-r)-\varepsilon})
  33. max 0 r 1 - H q ( δ + ε ) r ( 1 - δ H q - 1 ( 1 - r ) - ε ) \mathcal{R}\geq{\max\limits_{0\leq r\leq{1-H_{q}(\delta+\varepsilon)}}}r\left(% 1-{\delta\over{H_{q}^{-1}(1-r)-\varepsilon}}\right)
  34. r r
  35. R R
  36. 0
  37. δ \delta
  38. 0
  39. R R
  40. 0
  41. [ N , K ] Q [N,K]_{Q}
  42. N = Q - 1 N=Q-1
  43. 𝔽 Q * \mathbb{F}_{Q}^{*}
  44. Q = q k Q=q^{k}
  45. k = θ ( l o g N ) k=\theta(logN)
  46. C i n C_{in}
  47. q O ( k n ) q^{O(kn)}
  48. k = r n k=rn
  49. q O ( k n ) = q O ( k 2 ) = N O ( l o g N ) q^{O(kn)}=q^{O(k^{2})}=N^{O(logN)}
  50. n N O ( l o g n N ) nN^{O(lognN)}
  51. C i n C_{in}
  52. q O ( n ) q^{O(n)}
  53. ( n N ) O ( 1 ) (nN)^{O(1)}

Łukaszyk–Karmowski_metric.html

  1. D ( X , Y ) = - - | x - y | f ( x ) g ( y ) d x d y D(X,Y)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}|x-y|f(x)g(y)\,dx\,dy
  2. D δ δ ( X , Y ) = - - | x - y | δ ( x - μ x ) δ ( y - μ y ) d x d y = | μ x - μ y | D_{\delta\delta}(X,Y)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}|x-y|% \delta(x-\mu_{x})\delta(y-\mu_{y})\,dx\,dy=|\mu_{x}-\mu_{y}|
  3. μ x \mu_{x}
  4. μ y \mu_{y}
  5. D δ δ ( X , X ) = | μ x - μ x | = 0. D_{\delta\delta}(X,X)=|\mu_{x}-\mu_{x}|=0.
  6. D ( X , X ) > 0. D\left(X,X\right)>0.\,
  7. D ( X , Z ) = - - | x - z | f ( x ) h ( z ) d x d z = - - | x - z | f ( x ) h ( z ) d x d z - g ( y ) d y \displaystyle{}D(X,Z)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}|x-z|f(x)h% (z)\,dx\,dz\ =\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}|x-z|f(x)h(z)\,dx% \,dz\int_{-\infty}^{\infty}g(y)dy
  8. D ( X , Z ) D ( X , Y ) + D ( Y , Z ) . D(X,Z)\leq D(X,Y)+D(Y,Z).\,
  9. σ = 0 , σ = 0.2 , σ = 0.4 , σ = 0.6 , σ = 0.8 , σ = 1 \sigma=0,\sigma=0.2,\sigma=0.4,\sigma=0.6,\sigma=0.8,\sigma=1
  10. m x y = | μ x - μ y | m_{xy}=|\mu_{x}-\mu_{y}|
  11. - - | x - y | f ( x , y ) d x d y . \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}|x-y|f(x,y)\,dx\,dy.
  12. D N N ( X , Y ) = μ x y + 2 σ π exp ( - μ x y 2 4 σ 2 ) - μ x y erfc ( μ x y 2 σ ) , D_{NN}(X,Y)=\mu_{xy}+\frac{2\sigma}{\sqrt{\pi}}\operatorname{exp}\left(-\frac{% \mu_{xy}^{2}}{4\sigma^{2}}\right)-\mu_{xy}\operatorname{erfc}\left(\frac{\mu_{% xy}}{2\sigma}\right),
  13. μ x y = | μ x - μ y | , \mu_{xy}=\left|\mu_{x}-\mu_{y}\right|,
  14. D N N ( X , Y ) D_{NN}(X,Y)
  15. lim μ x y 0 D N N ( X , Y ) = D N N ( X , X ) = 2 σ π . \lim_{\mu_{xy}\to 0}D_{NN}(X,Y)=D_{NN}(X,X)=\frac{2\sigma}{\sqrt{\pi}}.
  16. D R R ( X , Y ) = { 24 3 σ 3 - μ x y 3 + 6 3 σ μ x y 2 36 σ 2 , μ x y < 2 3 σ , μ x y , μ x y 2 3 σ . D_{RR}(X,Y)=\begin{cases}\frac{24\sqrt{3}\sigma^{3}-\mu_{xy}^{3}+6\sqrt{3}% \sigma\mu_{xy}^{2}}{36\sigma^{2}},&\mu_{xy}<2\sqrt{3}\sigma,\\ \mu_{xy},&\mu_{xy}\geq 2\sqrt{3}\sigma.\end{cases}
  17. D R R ( X , X ) = 2 σ 3 . D_{RR}(X,X)=\frac{2\sigma}{\sqrt{3}}.
  18. D ( X , Y ) = i j | x i - y j | P ( X = x i ) P ( Y = y j ) . D(X,Y)=\sum_{i}\sum_{j}|x_{i}-y_{j}|P(X=x_{i})P(Y=y_{j}).\,
  19. D P P ( X , Y ) = x = 0 n y = 0 n | x - y | λ x x λ y y e - ( λ x + λ y ) x ! y ! . D_{PP}(X,Y)=\sum_{x=0}^{n}\sum_{y=0}^{n}|x-y|\frac{{\lambda_{x}}^{x}{\lambda_{% y}}^{y}e^{-(\lambda_{x}+\lambda_{y})}}{x!y!}.
  20. | x - y | |x-y|
  21. D ( 𝐗 , 𝐘 ) = Ω Ω d ( 𝐱 , 𝐲 ) F ( 𝐱 ) G ( 𝐲 ) d Ω x d Ω y . D(\mathbf{X},\mathbf{Y})=\int_{\Omega}\int_{\Omega}d(\mathbf{x},\mathbf{y})F(% \mathbf{x})G(\mathbf{y})\,d\Omega_{x}\,d\Omega_{y}.
  22. D ( 𝐗 , 𝐘 ) = Ω Ω i = 1 2 | x i - y i | 2 F ( x 1 , x 2 ) G ( y 1 , y 2 ) d x 1 d x 2 d y 1 d y 2 . D(\mathbf{X},\mathbf{Y})=\int_{\Omega}\int_{\Omega}\sqrt{\sum_{i=1}^{2}|x_{i}-% y_{i}|^{2}}F(x_{1},x_{2})G(y_{1},y_{2})\,dx_{1}\,dx_{2}\,dy_{1}\,dy_{2}.
  23. D ( 𝐗 , 𝐘 ) = Ω Ω d ( 𝐱 , 𝐲 ) F ( 𝐱 , 𝐲 ) d Ω x d Ω y . D(\mathbf{X},\mathbf{Y})=\int_{\Omega}\int_{\Omega}d(\mathbf{x},\mathbf{y})F(% \mathbf{x},\mathbf{y})\,d\Omega_{x}\,d\Omega_{y}.
  24. D * * ( p ) ( 𝐗 , 𝐘 ) = ( i D * * ( X i , Y i ) p ) 1 p D_{**}^{(p)}(\mathbf{X},\mathbf{Y})=\left({\sum_{i}{D_{**}(X_{i},Y_{i})}^{p}}% \right)^{\frac{1}{p}}
  25. D * * ( X i , Y i ) D_{**}(X_{i},Y_{i})\,
  26. X i X_{i}
  27. Y i Y_{i}
  28. 𝐗 , 𝐘 D * * ( p ) ( 𝐗 , 𝐘 ) = 0 𝐗 = 𝐘 \forall{\mathbf{X},\mathbf{Y}}\ D_{**}^{(p)}(\mathbf{X},\mathbf{Y})=0\ % \nLeftrightarrow\ \mathbf{X}=\mathbf{Y}\,
  29. D * * ( p ) ( 𝐗 , 𝐗 ) = 0 i D * * ( X i , X i ) = 0 D_{**}^{(p)}(\mathbf{X},\mathbf{X})=0\Leftrightarrow\ \forall{i}\ D_{**}(X_{i}% ,X_{i})=0
  30. X i D * * ( X i , X i ) > 0 \exists\ X_{i}\ D_{**}(X_{i},X_{i})>0
  31. i D * * ( X i , Y i ) > 0 \forall\ i\ D_{**}(X_{i},Y_{i})>0\,
  32. i D * * ( X i , Y i ) = D * * ( Y i , X i ) \forall\ i\ D_{**}(X_{i},Y_{i})=D_{**}(Y_{i},X_{i})
  33. 𝐗 , 𝐘 , 𝐙 D * * ( p ) ( 𝐗 , 𝐙 ) D * * ( p ) ( 𝐗 , 𝐘 ) + D * * ( p ) ( 𝐘 , 𝐙 ) \forall\ \mathbf{X},\mathbf{Y},\mathbf{Z}\ D_{**}^{(p)}(\mathbf{X},\mathbf{Z})% \leq D_{**}^{(p)}(\mathbf{X},\mathbf{Y})+D_{**}^{(p)}(\mathbf{Y},\mathbf{Z})
  34. ( i D * * ( X i , Y i ) p ) 1 p + ( i D * * ( Y i , Z i ) p ) 1 p \displaystyle{}\left({\sum_{i}{D_{**}(X_{i},Y_{i})}^{p}}\right)^{\frac{1}{p}}+% \left({\sum_{i}{D_{**}(Y_{i},Z_{i})}^{p}}\right)^{\frac{1}{p}}\ \geq
  35. d P = | ψ ( x , y , z ) | 2 d V . dP=|\psi(x,y,z)|^{2}dV.\,
  36. ( 0 ξ L ) (0\leq\xi\leq L)
  37. ψ m ( x ) = 2 L sin ( m π x L ) , \psi_{m}(x)=\sqrt{\frac{2}{L}}\sin{\left(\frac{m\pi x}{L}\right)},\,
  38. ξ ( 0 , L ) \xi\in(0,L)\,
  39. D ( X , ξ ) = 0 L | x - ξ | | ψ m ( x ) | 2 d x = \displaystyle{}D(X,\xi)=\int\limits_{0}^{L}|x-\xi||\psi_{m}(x)|^{2}dx=
  40. d ( 0 , 0.2 L ) + D ( 0.2 L , X ) 0.2 L + 0.3171 L = 0.517 L D ( 0 , X ) = 0.5 L = d ( 0 , 0.5 L ) . d(0,0.2L)+D(0.2L,X)\approx 0.2L+0.3171L=0.517L\neq D(0,X)=0.5L=d(0,0.5L).\,
  41. ψ m ( x ) = 2 L sin ( m π x L ) , \psi_{m}(x)=\sqrt{\frac{2}{L}}\sin{\left(\frac{m\pi x}{L}\right)},\,
  42. ψ n ( y ) = 2 L sin ( n π y L ) , \psi_{n}(y)=\sqrt{\frac{2}{L}}\sin{\left(\frac{n\pi y}{L}\right)},\,
  43. D ( X , Y ) = 0 L 0 L | x - y | | ψ m ( x ) | 2 | ψ n ( y ) | 2 d x d y \displaystyle{}D(X,Y)=\int\limits_{0}^{L}\int\limits_{0}^{L}|x-y||\psi_{m}(x)|% ^{2}|\psi_{n}(y)|^{2}\,dx\,dy
  44. min ( D ( X , Y ) ) = L ( 4 π 2 - 15 12 π 2 ) 0.2067 L . \min(D(X,Y))=L\left(\frac{4\pi^{2}-15}{12\pi^{2}}\right)\approx 0.2067L.\,

ΔP.html

  1. Δ p = ρ g h f \Delta p=\rho\cdot g\cdot h_{f}
  2. Δ p = f L D ρ V 2 2 \Delta p=f\cdot\frac{L}{D}\cdot\frac{\rho V^{2}}{2}
  3. C d y n = < m t p l > V T P I P - P E E P C_{dyn}=\frac{<}{m}tpl>{{V_{T}}}{{PIP-PEEP}}
  4. C s t a t = < m t p l > V T P p l a t - P E E P C_{stat}=\frac{<}{m}tpl>{{V_{T}}}{{P_{plat}-PEEP}}

Ε-net_(computational_geometry).html

  1. O ( d ε log d ε ) ; O\left(\frac{d}{\varepsilon}\log\frac{d}{\varepsilon}\right);
  2. P P
  3. X X
  4. ε \varepsilon
  5. H 2 X H\subseteq 2^{X}
  6. X X
  7. S X S\subseteq X
  8. h H h\in H
  9. P ( h ) ε S h . P(h)\geq\varepsilon\quad\Longrightarrow\quad S\cap h\neq\varnothing.
  10. S S
  11. ε \varepsilon
  12. ε \varepsilon
  13. H H
  14. S X S\subseteq X
  15. h H h\in H
  16. | P ( h ) - | S h | | S | | < ε . \left|P(h)-\frac{|S\cap h|}{|S|}\right|<\varepsilon.

Λ-ring.html

  1. Λ 2 ( V W ) Λ 2 ( V ) ( Λ 1 ( V ) Λ 1 ( W ) ) Λ 2 ( W ) \Lambda^{2}(V\oplus W)\cong\Lambda^{2}(V)\oplus\left(\Lambda^{1}(V)\otimes% \Lambda^{1}(W)\right)\oplus\Lambda^{2}(W)
  2. λ 2 ( x + y ) = λ 2 ( x ) + λ 1 ( x ) λ 1 ( y ) + λ 2 ( y ) \lambda^{2}(x+y)=\lambda^{2}(x)+\lambda^{1}(x)\lambda^{1}(y)+\lambda^{2}(y)
  3. Λ 1 ( V W ) Λ 1 ( V ) Λ 1 ( W ) \Lambda^{1}(V\otimes W)\cong\Lambda^{1}(V)\otimes\Lambda^{1}(W)
  4. λ 1 ( x y ) = λ 1 ( x ) λ 1 ( y ) \lambda^{1}(xy)=\lambda^{1}(x)\lambda^{1}(y)
  5. m λ m ( x ) t m = i ( 1 + t x i ) \displaystyle\sum_{m}\lambda^{m}(x)t^{m}=\prod_{i}(1+tx_{i})
  6. m P m ( λ 1 ( x ) , , λ m ( x ) , λ 1 ( y ) , , λ m ( y ) ) t m = i , j ( 1 + t x i y j ) \displaystyle\sum_{m}P_{m}(\lambda^{1}(x),\cdots,\lambda^{m}(x),\lambda^{1}(y)% ,\cdots,\lambda^{m}(y))t^{m}=\prod_{i,j}(1+tx_{i}y_{j})
  7. m P m , n ( λ 1 ( x ) , , λ m n ( x ) ) t m = i 1 < i 2 < < i n ( 1 + t x i 1 x i 2 x i n ) \displaystyle\sum_{m}P_{m,n}(\lambda^{1}(x),\cdots,\lambda^{mn}(x))t^{m}=\prod% _{i_{1}<i_{2}<\cdots<i_{n}}(1+tx_{i_{1}}x_{i_{2}}\cdots x_{i_{n}})

Τ-additivity.html

  1. 𝒢 Σ \mathcal{G}\subseteq\Sigma
  2. 𝒢 \mathcal{G}
  3. μ ( 𝒢 ) = sup G 𝒢 μ ( G ) \mu\left(\bigcup\mathcal{G}\right)=\sup_{G\in\mathcal{G}}\mu(G)

Continuous_dual_q-Hahn_polynomials.html

  1. p n ( x ; a , b , c | q ) = ( a b , a c ; q ) n a n * 3 Φ 2 ( q - n , a e i θ , a e - i θ ; a b , a c | q ; q ) p_{n}(x;a,b,c|q)=\frac{(ab,ac;q)_{n}}{a^{n}}*_{3}\Phi_{2}(q^{-}n,ae^{i\theta},% ae^{-i\theta};ab,ac|q;q)
  2. x = c o s ( θ ) x=cos(\theta)

Q-Meixner–Pollaczek_polynomials.html

  1. P n ( x ; a | q ) = a - n e i n ϕ P_{n}(x;a|q)=a^{-n}e^{in\phi}
  2. a 2 ; q n ( q ; q ) n \frac{a^{2};q_{n}}{(q;q)_{n}}
  3. Φ 2 3 ( q - n , a e i ( θ + 2 ϕ ) , a e - i θ ; a 2 , 0 | q ; q ) {}_{3}\Phi_{2}(q^{-}n,ae^{i(\theta+2\phi)},ae^{-i\theta};a^{2},0|q;q)