wpmath0000014_5

Georg_Scheffers.html

  1. e 1 , e 2 , e 3 e_{1},\ e_{2},\ e_{3}
  2. e 1 e 2 - e 2 e 1 = 2 e 3 , e 2 e 3 - e 3 e 2 = 2 e 1 , e 3 e 1 - e 1 e 3 = 2 e 2 . e_{1}e_{2}-e_{2}e_{1}=2e_{3},\quad e_{2}e_{3}-e_{3}e_{2}=2e_{1},\quad e_{3}e_{% 1}-e_{1}e_{3}=2e_{2}.

George_Phillips_Odom,_Jr..html

  1. Φ \Phi
  2. | A B | | B C | = | A C | | A B | = Φ \tfrac{|AB|}{|BC|}=\tfrac{|AC|}{|AB|}=\Phi
  3. | A E | | A F | = | E F | | A E | = Φ \tfrac{|AE|}{|AF|}=\tfrac{|EF|}{|AE|}=\Phi
  4. | A C | | A B | = | A B | | B C | = Φ \tfrac{|AC|}{|AB|}=\tfrac{|AB|}{|BC|}=\Phi

Geotechnical_centrifuge_modeling.html

  1. x * = x m x p x^{*}=\frac{x_{m}}{x_{p}}
  2. x * x^{*}\,
  3. x x\,
  4. σ * = σ m σ p = 1 \sigma^{\prime*}=\frac{\sigma^{\prime}_{m}}{\sigma^{\prime}_{p}}=1
  5. σ m \sigma^{\prime}_{m}
  6. σ p \sigma^{\prime}_{p}
  7. σ \sigma^{\prime}
  8. σ = σ t - u \sigma^{\prime}=\sigma^{t}-u\,
  9. σ t \sigma^{t}
  10. u u
  11. H H
  12. σ t = ρ g H \sigma^{t}=\rho gH\,
  13. ρ \rho
  14. g g
  15. ρ * = 1 \rho^{*}=1\,
  16. L * L^{*}
  17. ρ * g * H * = ( 1 ) g * L * = 1 \rho^{*}g^{*}H^{*}=(1)g^{*}L^{*}=1\,
  18. g * = 1 L * g^{*}=\frac{1}{L^{*}}
  19. a * = g * = 1 L * a^{*}=g^{*}=\frac{1}{L^{*}}
  20. L T 2 \frac{L}{T^{2}}
  21. a * = L * T * 2 a^{*}=\frac{L^{*}}{T^{*2}}
  22. 1 L * = L * T * 2 \frac{1}{L^{*}}=\frac{L^{*}}{T^{*2}}
  23. T * = L * T^{*}=L^{*}\,
  24. f * = 1 L * f^{*}=\frac{1}{L{*}}
  25. v * = L * T * = 1 v^{*}=\frac{L^{*}}{T{*}}=1
  26. T * = L * 2 T^{*}=L^{*2}\,

Geroch's_splitting_theorem.html

  1. ( M , g a b ) (M,g_{ab})
  2. ( M , g a b ) (M,g_{ab})
  3. f : M f:M\rightarrow\mathbb{R}
  4. t t\in\mathbb{R}
  5. f - 1 ( t ) f^{-1}(t)
  6. f f
  7. M M
  8. S × S\times\mathbb{R}
  9. S S
  10. M M

Gevrey_class.html

  1. | D α g ( x ) | C R k k σ k |D^{\alpha}g(x)|\leq CR^{k}k^{\sigma k}

Gibbs–Thomson_equation.html

  1. x x
  2. Δ T m ( x ) = T m B - T m ( x ) = T m B 4 σ s l H f ρ s x \Delta\,T_{m}(x)=T_{mB}-T_{m}(x)=T_{mB}\frac{4\sigma\,_{sl}}{H_{f}\rho\,_{s}x}
  3. x x
  4. d d
  5. d d
  6. c o s ϕ cos\phi\,
  7. Δ T f / Δ T m \Delta\,T_{f}/\Delta\,T_{m}
  8. x x
  9. Δ T m ( x ) = T m B - T m ( x ) = - T m B 4 σ s l c o s ϕ H f ρ s x \Delta\,T_{m}(x)=T_{mB}-T_{m}(x)=-T_{mB}\frac{4\sigma\,_{sl}cos\phi\,}{H_{f}% \rho\,_{s}x}
  10. Δ T m ( x ) = k G T x \Delta\,T_{m}(x)=\frac{k_{GT}}{x}
  11. k G T k_{GT}
  12. Δ T m ( x ) = k G T x = k g k s k i x \Delta\,T_{m}(x)=\frac{k_{GT}}{x}=\frac{k_{g}\,k_{s}\,k_{i}}{x}
  13. k g k_{g}
  14. k s k_{s}
  15. k i k_{i}
  16. p ( r 1 , r 2 ) = P - γ ρ v a p o r ( ρ l i q u i d - ρ v a p o r ) ( 1 r 1 + 1 r 2 ) p(r_{1},r_{2})=P-\frac{\gamma\,\rho\,_{vapor}}{(\rho\,_{liquid}-\rho\,_{vapor}% )}\left(\frac{1}{r_{1}}+\frac{1}{r_{2}}\right)
  17. p ( r ) p(r)
  18. r r
  19. P P
  20. r = r=\infty
  21. p e q p_{eq}
  22. γ \gamma
  23. ρ v a p o r \rho\,_{vapor}
  24. ρ l i q u i d \rho\,_{liquid}
  25. r 1 r_{1}
  26. r 2 r_{2}
  27. ln ( p ( r ) P ) = 2 γ V m o l e c u l e k B T r \ln\left(\frac{p(r)}{P}\right)=\frac{2\gamma V_{molecule}}{k_{B}Tr}
  28. ln ( P 2 P 1 ) = L R ( 1 T 1 - 1 T 2 ) . \ln\left(\frac{P_{2}}{P_{1}}\right)=\frac{L}{R}\left(\frac{1}{T_{1}}-\frac{1}{% T_{2}}\right).

Gilbert–Johnson–Keerthi_distance_algorithm.html

  1. S u p p o r t ( s h a p e , d ) Support(shape,\vec{d})
  2. s h a p e shape
  3. d \vec{d}
  4. N e a r e s t S i m p l e x ( s ) NearestSimplex(s)
  5. s s
  6. s s
  7. s s
  8. N e a r e s t S i m p l e x NearestSimplex
  9. s s
  10. N e a r e s t S i m p l e x NearestSimplex
  11. 𝐑 < s u p > n \mathbf{R}<sup>n

Giovanni_Battista_Rizza.html

  1. F F
  2. 𝐀 \mathbf{A}
  3. Γ 1 F ( X ) dX = 0 \int_{\Gamma_{1}}\mathrm{F}(\mathrm{X})\mathrm{d}\mathrm{X}=0
  4. Γ 1 Γ_{1}
  5. F F
  6. 𝐀 * \mathbf{A}^{*}
  7. 𝐀 \mathbf{A}
  8. Γ 1 F ( X ) X - Ξ dX = 2 π i s = 1 k N ( s ) u ( s ) F ( Ξ ) \int_{\Gamma_{1}}\frac{\mathrm{F}(\mathrm{X})}{\mathrm{X}-\Xi}\mathrm{d}% \mathrm{X}=2\pi i\sum^{k}_{s=1}\mathrm{N}^{(s)}u^{(s)}\mathrm{F}(\Xi)
  9. X x * x X ≡x^{*}≡x
  10. 𝐀 \mathbf{A}
  11. 𝐀 * \mathbf{A}^{*}
  12. Γ 1 Γ_{1}
  13. N ( s ) N^{(s)}
  14. Γ 1 Γ_{1}
  15. 2 n 2n
  16. n > 2 n> 2
  17. u ν = | ρ | 3 Q ( ρ ) E u \frac{\partial u}{\partial\nu}=\frac{|\nabla\rho|^{3}}{Q(\rho)}Eu
  18. u u
  19. Ω Ω
  20. ρ ρ
  21. Ω Ω
  22. Ω = { x 2 n | ρ ( x ) = 0 } , \partial\Omega=\{x\in\mathbb{R}^{2n}|\rho(x)=0\},
  23. Q ( ρ ) Q(ρ)
  24. ρ ρ
  25. E E
  26. Ω ∂Ω
  27. [ u S u p , u n ] ℂ[u^{\prime}Sup^{\prime},u^{\prime}n^{\prime}]
  28. n > 2 n> 2
  29. r r
  30. r r
  31. R [ u I s u p , u 2 , u n ] R[u^{\prime}Isup^{\prime},u^{\prime}2^{\prime},u^{\prime}n^{\prime}]
  32. 𝐀 * \mathbf{A}^{*}
  33. 𝐀 \mathbf{A}

Glas_Gaibhnenn.html

  1. 14 B .23 \tfrac{14}{B.23}

Glenn_Firebaugh.html

  1. X ¯ ) \scriptstyle\overline{X})
  2. X ¯ ) \scriptstyle\overline{X})
  3. Σ j p j f ( r j ) , \Sigma_{j}p_{j}f(r_{j}),

Glossary_of_elementary_quantum_mechanics.html

  1. | x |x\rangle
  2. | α , | β , | γ |\alpha\rangle,|\beta\rangle,|\gamma\rangle...
  3. Ψ \Psi
  4. ψ \psi
  5. ψ α ( x , t ) \psi_{\alpha}(x,t)
  6. x | α \langle x|\alpha\rangle
  7. α | = ( | α ) \langle\alpha|=(|\alpha\rangle)^{\dagger}
  8. | α |\alpha\rangle
  9. | α β | |\alpha\rangle\langle\beta|
  10. | α |\alpha\rangle
  11. | α α | |\alpha\rangle\langle\alpha|
  12. Tr ( ρ ) = 1 \operatorname{Tr}(\rho)=1
  13. ρ = ρ \rho^{\dagger}=\rho
  14. | a |a\rangle
  15. Tr ( ρ 2 ) = 1 \operatorname{Tr}(\rho^{2})=1
  16. Tr ( ρ 2 ) < 1 \operatorname{Tr}(\rho^{2})<1
  17. | α |\alpha^{\prime}\rangle
  18. α | α < \langle\alpha^{\prime}|\alpha^{\prime}\rangle<\infty
  19. | a α = | α α | α |a^{\prime}\rangle\to\alpha=\frac{|\alpha^{\prime}\rangle}{\sqrt{\langle\alpha% ^{\prime}|\alpha^{\prime}\rangle}}
  20. | a |a\rangle
  21. a | a = 1 \langle a|a\rangle=1
  22. ( n , l , m , s ) (n,l,m,s)
  23. H H
  24. r r
  25. L L
  26. L z L_{z}
  27. z z
  28. S z S_{z}
  29. H A H B H_{A}\otimes H_{B}
  30. H A H_{A}
  31. H B H_{B}
  32. Ψ \Psi
  33. ψ α ( x 0 ) = x 0 | α \psi_{\alpha}(x_{0})=\langle x_{0}|\alpha\rangle
  34. | x 0 |x_{0}\rangle
  35. H ^ \hat{H}
  36. i t | α = H ^ | α i\hbar\frac{\partial}{\partial t}|\alpha\rangle=\hat{H}|\alpha\rangle
  37. E α = H ^ | α E\alpha\rangle=\hat{H}|\alpha\rangle
  38. i t Ψ α ( 𝐫 , t ) = H ^ Ψ = ( - 2 2 m 2 + V ( 𝐫 ) ) Ψ α ( 𝐫 , t ) = - 2 2 m 2 Ψ α ( 𝐫 , t ) + V ( 𝐫 ) Ψ α ( 𝐫 , t ) i\hbar\frac{\partial}{\partial t}\Psi_{\alpha}(\mathbf{r},\,t)=\hat{H}\Psi=% \left(-\frac{\hbar^{2}}{2m}\nabla^{2}+V(\mathbf{r})\right)\Psi_{\alpha}(% \mathbf{r},\,t)=-\frac{\hbar^{2}}{2m}\nabla^{2}\Psi_{\alpha}(\mathbf{r},\,t)+V% (\mathbf{r})\Psi_{\alpha}(\mathbf{r},\,t)
  39. Ψ α ( x , t ) := x | α \Psi_{\alpha}(x,t):=\langle x|\alpha\rangle
  40. H ^ := - 2 2 m 2 + V ^ \hat{H}:=-\frac{\hbar^{2}}{2m}\nabla^{2}+\hat{V}
  41. | ψ ( 𝐫 , t ) | 2 0 |\psi(\mathbf{r},t)|^{2}\to 0
  42. | 𝐫 | + |\mathbf{r}|\to+\infty
  43. t > 0 t>0
  44. E E
  45. E < min { V ( r - ) , V ( r + ) } E<\operatorname{min}\{V(r\to-\infty),V(r\to+\infty)\}
  46. Ψ α ( x , t ) := x | α \Psi_{\alpha}(x,t):=\langle x|\alpha\rangle
  47. Ψ ~ α ( p , t ) := p | α \tilde{\Psi}_{\alpha}(p,t):=\langle p|\alpha\rangle
  48. | x |x\rangle
  49. | p |p\rangle
  50. J J
  51. J ( x , t ) = i 2 m ( ψ ψ * x - ψ x ψ ) J(x,t)=\frac{i\hbar}{2m}(\psi\frac{\partial\psi^{*}}{\partial x}-\frac{% \partial\psi}{\partial x}\psi)
  52. t | ψ ( x , t ) | 2 + 𝐉 ( 𝐱 , 𝐭 ) = 0 \frac{\partial}{\partial t}|\psi(x,t)|^{2}+\nabla\cdot\mathbf{J(x,t)}=0
  53. | ψ ( x , t ) | 2 |\psi(x,t)|^{2}
  54. x x
  55. t t
  56. | ψ ( x 0 , t ) | 2 d x |\psi(x_{0},t)|^{2}\,dx
  57. x 0 x_{0}
  58. E E
  59. E > min { V ( r - ) , V ( r + ) } E>\operatorname{min}\{V(r\to-\infty),V(r\to+\infty)\}
  60. Ψ ( x , t ) \Psi(x,t)
  61. - + | Ψ ( x , t ) | 2 d x < + \int_{-\infty}^{+\infty}|\Psi(x,t)|^{2}\,dx<+\infty
  62. V | Ψ ( 𝐫 , t ) | 2 d V < + \int_{V}|\Psi(\mathbf{r},t)|^{2}\,dV<+\infty
  63. | ψ ( x , t ) | 2 |\psi(x,t)|^{2}
  64. d d t | Ψ ( x , t ) | 2 = 0 \frac{d}{dt}|\Psi(x,t)|^{2}=0
  65. | α |\alpha\rangle
  66. | k |k\rangle
  67. | k | α | 2 |\langle k|\alpha\rangle|^{2}
  68. A A
  69. A | α = c | α A|\alpha\rangle=c|\alpha\rangle
  70. c c
  71. < M > <M>
  72. | α |\alpha
  73. M M
  74. | α |\alpha
  75. < M Align g t ; <M&gt;
  76. < M α | M | α <M>=\langle\alpha|M|\alpha\rangle
  77. ρ \rho
  78. < M Tr ( M ρ ) <M>=\operatorname{Tr}(M\rho)
  79. A = A A=A^{\dagger}
  80. α | A | α = α | A | α \langle\alpha|A|\alpha\rangle=\langle\alpha|A^{\dagger}|\alpha\rangle
  81. | α |\alpha\rangle
  82. x x ^ , p i x x\to\hat{x},\,p\to i\hbar\frac{\partial}{\partial x}

Good_cover_(algebraic_topology).html

  1. X X
  2. X X
  3. U α 1 α n = U α 1 α n - 1 U α n U_{\alpha_{1}\ldots\alpha_{n}}=U_{\alpha_{1}\ldots\alpha_{n-1}}\cap U_{\alpha_% {n}}
  4. U α 1 α n U_{\alpha_{1}\ldots\alpha_{n}}
  5. n n
  6. n \mathbb{R}^{n}
  7. S 2 S^{2}

Goodness_factor.html

  1. G = ω resistance × reluctance = ω μ σ A m A e l m l e G=\frac{\omega}{\mathrm{resistance}\times\mathrm{reluctance}}=\frac{\omega\mu% \sigma A_{\mathrm{m}}A_{\mathrm{e}}}{l_{\mathrm{m}}l_{\mathrm{e}}}
  2. G ω μ 0 p 2 ρ r g G\propto\frac{\omega\mu_{0}p^{2}}{\rho_{\mathrm{r}}g}

Good–deal_bounds.html

  1. A A
  2. ρ : p \rho:\mathcal{L}^{p}\to\mathbb{R}
  3. ρ ( X ) = inf { t : V T A T : X + t + V T A } = inf { t : X + t A - A T } \rho(X)=\inf\left\{t\in\mathbb{R}:\exists V_{T}\in A_{T}:X+t+V_{T}\in A\right% \}=\inf\left\{t\in\mathbb{R}:X+t\in A-A_{T}\right\}
  4. A T A_{T}
  5. ( - ρ ( X ) , ρ ( - X ) ) (-\rho(X),\rho(-X))
  6. A = { Z 0 : Z 0 - a . s . } A=\left\{Z\in\mathcal{L}^{0}:Z\geq 0\;\mathbb{P}-a.s.\right\}
  7. A = { Z 0 : 𝔼 [ u ( Z ) ] 𝔼 [ u ( 0 ) ] } A=\left\{Z\in\mathcal{L}^{0}:\mathbb{E}[u(Z)]\geq\mathbb{E}[u(0)]\right\}
  8. u u

Gottlieb_polynomials.html

  1. n ( x , λ ) = e - n λ k ( 1 - e λ ) k ( n k ) ( x k ) = e - n λ F 1 2 ( - n , - x ; 1 ; 1 - e λ ) \displaystyle\ell_{n}(x,\lambda)=e^{-n\lambda}\sum_{k}(1-e^{\lambda})^{k}{% \left({{n}\atop{k}}\right)}{\left({{x}\atop{k}}\right)}=e^{-n\lambda}{}_{2}F_{% 1}(-n,-x;1;1-e^{\lambda})

Gould_polynomials.html

  1. exp ( x f ( t ) ) = n G n ( x ; a , b ) t n / n ! \displaystyle\exp(xf(t))=\sum_{n}G_{n}(x;a,b)t^{n}/n!
  2. f ( t ) = k 1 1 b ( - ( b + a k ) / b k - 1 ) t k k ! f(t)=\sum_{k\geq 1}\frac{1}{b}{\left({{-(b+ak)/b}\atop{k-1}}\right)}\frac{t^{k% }}{k!}

GPS_enhancement.html

  1. 1 s 1575.42 × 10 6 = 0.63475 ns 1 ns \frac{1\ \mathrm{s}}{1575.42\times 10^{6}}=0.63475\ \mathrm{ns}\approx 1\ % \mathrm{ns}
  2. 1 s 1023 × 10 3 = 977.5 ns 1000 ns \frac{1\ \mathrm{s}}{1023\times 10^{3}}=977.5\ \mathrm{ns}\ \approx 1000\ % \mathrm{ns}
  3. ϕ ( r i , s j , t k ) \ \phi(r_{i},s_{j},t_{k})
  4. t k \ \ t_{k}
  5. ϕ \ \phi
  6. ϕ i , j , k = ϕ ( r i , s j , t k ) \ \phi_{i,j,k}=\phi(r_{i},s_{j},t_{k})
  7. Δ r , Δ s , Δ t \ \Delta^{r},\Delta^{s},\Delta^{t}
  8. α i , j , k \ \alpha_{i,j,k}
  9. Δ r , Δ s , Δ t \ \Delta^{r},\Delta^{s},\Delta^{t}
  10. Δ r ( α i , j , k ) = α i + 1 , j , k - α i , j , k \ \Delta^{r}(\alpha_{i,j,k})=\alpha_{i+1,j,k}-\alpha_{i,j,k}
  11. Δ s ( α i , j , k ) = α i , j + 1 , k - α i , j , k \ \Delta^{s}(\alpha_{i,j,k})=\alpha_{i,j+1,k}-\alpha_{i,j,k}
  12. Δ t ( α i , j , k ) = α i , j , k + 1 - α i , j , k \ \Delta^{t}(\alpha_{i,j,k})=\alpha_{i,j,k+1}-\alpha_{i,j,k}
  13. α i , j , k a n d β l , m , n \ \alpha_{i,j,k}\ and\ \beta_{l,m,n}
  14. ( a α i , j , k + b β l , m , n ) \ (a\ \alpha_{i,j,k}+b\ \beta_{l,m,n})
  15. Δ r ( a α i , j , k + b β l , m , n ) = a Δ r ( α i , j , k ) + b Δ r ( β l , m , n ) \ \Delta^{r}(a\ \alpha_{i,j,k}+b\ \beta_{l,m,n})=a\ \Delta^{r}(\alpha_{i,j,k})% +b\ \Delta^{r}(\beta_{l,m,n})
  16. Δ s ( a α i , j , k + b β l , m , n ) = a Δ s ( α i , j , k ) + b Δ s ( β l , m , n ) \ \Delta^{s}(a\ \alpha_{i,j,k}+b\ \beta_{l,m,n})=a\ \Delta^{s}(\alpha_{i,j,k})% +b\ \Delta^{s}(\beta_{l,m,n})
  17. Δ t ( a α i , j , k + b β l , m , n ) = a Δ t ( α i , j , k ) + b Δ t ( β l , m , n ) \ \Delta^{t}(a\ \alpha_{i,j,k}+b\ \beta_{l,m,n})=a\ \Delta^{t}(\alpha_{i,j,k})% +b\ \Delta^{t}(\beta_{l,m,n})
  18. Δ s ( ϕ 1 , 1 , 1 ) = ϕ 1 , 2 , 1 - ϕ 1 , 1 , 1 \ \Delta^{s}(\phi_{1,1,1})=\phi_{1,2,1}-\phi_{1,1,1}
  19. Δ r ( Δ s ( ϕ 1 , 1 , 1 ) ) \displaystyle\Delta^{r}(\Delta^{s}(\phi_{1,1,1}))
  20. t 2 \ \ t_{2}
  21. t 1 \ \ t_{1}
  22. Δ t ( Δ r ( Δ s ( ϕ 1 , 1 , 1 ) ) ) \ \Delta^{t}(\Delta^{r}(\Delta^{s}(\phi_{1,1,1})))

Gradient-like_vector_field.html

  1. X f > 0 , X\cdot f>0,
  2. f ( x ) = f ( b ) - x 1 2 - - x α 2 + x α + 1 2 + + x n 2 f(x)=f(b)-x_{1}^{2}-\cdots-x_{\alpha}^{2}+x_{\alpha+1}^{2}+\cdots+x_{n}^{2}

Gradually_varied_surface.html

  1. Σ \Sigma
  2. { A 1 , , A m } \{A_{1},\dots,A_{m}\}
  3. A 1 < < A m A_{1}<\cdots<A_{m}
  4. A i A_{i}
  5. Σ \Sigma
  6. f ( x ) = A i f(x)=A_{i}
  7. f ( y ) = A i f(y)=A_{i}
  8. f ( x ) = A i + 1 f(x)=A_{i+1}
  9. A i - 1 A_{i-1}
  10. D Σ D\subset\Sigma
  11. f : D { A 1 , , A m } f:D\rightarrow\{A_{1},\dots,A_{m}\}
  12. F F
  13. f f
  14. x x
  15. y y
  16. D D
  17. f ( x ) = A i f(x)=A_{i}
  18. f ( y ) = A j f(y)=A_{j}
  19. | i - j | d ( x , y ) |i-j|\leq d(x,y)
  20. d ( x , y ) d(x,y)
  21. x x
  22. y y

Grain_size.html

  1. ϕ = - log 2 D / D 0 , \phi=-\log_{2}{D/D_{0}},
  2. ϕ \phi
  3. D D
  4. D 0 D_{0}
  5. D = D 0 × 2 - ϕ D=D_{0}\times 2^{-\phi}\,

Granville_number.html

  1. 𝒮 \mathcal{S}
  2. 1 𝒮 1\in\mathcal{S}
  3. n , n > 1 n\in{\mathbb{N}},\;n>1
  4. n 𝒮 n\in{\mathcal{S}}
  5. d n , d < n , d 𝒮 d n \sum_{d\mid{n},\;d<n,\;d\in\mathcal{S}}d\leq{n}
  6. 𝒮 \mathcal{S}
  7. 𝒮 \mathcal{S}
  8. 𝒮 \mathcal{S}
  9. 𝒮 \mathcal{S}
  10. 𝒮 \mathcal{S}
  11. 𝒮 \mathcal{S}
  12. 𝒮 \mathcal{S}
  13. 𝒮 \mathcal{S}
  14. 𝒮 \mathcal{S}
  15. 𝒮 \mathcal{S}
  16. 𝒮 \mathcal{S}
  17. 𝒮 \mathcal{S}
  18. 𝒮 \mathcal{S}
  19. 𝒮 \mathcal{S}
  20. 𝒮 \mathcal{S}
  21. 𝒮 \mathcal{S}
  22. 𝒮 \mathcal{S}
  23. 𝒮 \mathcal{S}
  24. 𝒮 \mathcal{S}
  25. 𝒮 \mathcal{S}
  26. 𝒮 \mathcal{S}
  27. 𝒮 \mathcal{S}
  28. 𝒮 \mathcal{S}
  29. 𝒮 \mathcal{S}
  30. 𝒮 \mathcal{S}
  31. 𝒮 \mathcal{S}
  32. 𝒮 \mathcal{S}
  33. 𝒮 \mathcal{S}
  34. 𝒮 \mathcal{S}
  35. 𝒮 \mathcal{S}
  36. 𝒮 \mathcal{S}
  37. 𝒮 \mathcal{S}

Graph_energy.html

  1. λ i \lambda_{i}
  2. i = 1 , , n i=1,\ldots,n
  3. E ( G ) = i = 1 n | λ i | . E(G)=\sum_{i=1}^{n}|\lambda_{i}|.

Graphical_unitary_group_approach.html

  1. C v C_{\infty v}

Grasshoff.html

  1. Gr \mathrm{Gr}

Graver_basis.html

  1. A A
  2. G ( A ) G(A)
  3. { x n : A x = 0 , x 0 } \{x\in\mathbb{Z}^{n}:Ax=0,\ x\neq 0\}\,
  4. n \mathbb{Z}^{n}
  5. x y x\sqsubseteq y
  6. x i y i 0 x_{i}y_{i}\geq 0
  7. | x i | | y i | |x_{i}|\leq|y_{i}|
  8. A = ( 121 ) A=(121)
  9. min { f ( x ) : x n , A x = b , l x u } . \min\{f(x)\ :\ x\in\mathbb{Z}^{n},\ Ax=b,\ l\leq x\leq u\}\ .
  10. G ( A ) G(A)
  11. A A
  12. min { w x : x n , A x = b , l x u } . \min\{wx\ :\ x\in\mathbb{Z}^{n},\ Ax=b,\ l\leq x\leq u\}\ .
  13. G ( A ) G(A)
  14. A A
  15. G ( A ) G(A)
  16. min { f ( x ) : x n , A x = b , l x u } \min\{f(x)\ :\ x\in\mathbb{Z}^{n},\ Ax=b,\ l\leq x\leq u\}
  17. G ( A ) G(A)
  18. f ( x ) = i = 1 n f i ( x i ) f(x)=\sum_{i=1}^{n}f_{i}(x_{i})
  19. f ( x ) = x p f(x)=\|x\|_{p}
  20. min { w x : x + l × m × n , i x i , j , k = a j , k , j x i , j , k = b i , k , k x i , j , k = c i , j } , \min\{wx\ :\ x\in{\mathbb{Z}}_{+}^{l\times m\times n}\,,\ \sum_{i}x_{i,j,k}=a_% {j,k}\,,\ \sum_{j}x_{i,j,k}=b_{i,k}\,,\ \sum_{k}x_{i,j,k}=c_{i,j}\}\ ,
  21. a j , k a_{j,k}
  22. b i , k b_{i,k}
  23. c i , j c_{i,j}
  24. A A
  25. G ( A ) G(A)
  26. m 1 , , m d m_{1},\dots,m_{d}
  27. m 1 × × m d × n m_{1}\times\cdots\times m_{d}\times n
  28. G ( m , n ) G(m,n)
  29. A A
  30. G ( m , n ) G(m,n)
  31. G ( m , n ) G(m,n)
  32. G ( m , n ) G(m,n)
  33. ( n g ( m ) ) {n\choose g(m)}
  34. G ( m , g ( m ) ) G(m,g(m))
  35. G ( m , n ) G(m,n)
  36. G k ( m , n ) G_{k}(m,n)
  37. G ( m , n ) G(m,n)
  38. G k ( m , n ) G_{k}(m,n)
  39. G k ( m , n ) G_{k}(m,n)
  40. O ( m g ( k ) n k ) O\left(m^{g(k)}n^{k}\right)
  41. O ( n g ) O\left(n^{g}\right)
  42. G ( A ) G(A)
  43. O ( n 3 ) O\left(n^{3}\right)

Greenwood_frequency.html

  1. f G = 2.31 λ - 6 / 5 [ sec ζ Path C n 2 ( z ) v Wind ( z ) 5 / 3 d z ] 3 / 5 f_{\mathrm{G}}=2.31\,\lambda^{-6/5}\left[\sec\zeta\int_{\mathrm{Path}}C_{n}^{2% }(z)\,v_{\mathrm{Wind}}(z)^{5/3}\,dz\right]^{3/5}
  2. ζ \zeta
  3. v Wind ( z ) v_{\mathrm{Wind}}(z)
  4. C n 2 ( z ) C_{n}^{2}(z)

Greenwood_statistic.html

  1. G ( n ) = i = 1 n + 1 D i 2 , G(n)=\sum^{n+1}_{i=1}D^{2}_{i},
  2. D i D_{i}
  3. D i = 1 D_{i}=1
  4. G ( n ) = i = 1 n + 1 X i 2 T n 2 , G(n)=\frac{\sum^{n+1}_{i=1}X^{2}_{i}}{T_{n}^{2}},
  5. T n = i = 1 n + 1 X i , T_{n}=\sum^{n+1}_{i=1}X_{i},
  6. X i X_{i}
  7. G ( n ) = 1 n + 1 ( n n + 1 C v 2 + 1 ) , G(n)=\tfrac{1}{n+1}(\tfrac{n}{n+1}C_{v}^{2}+1),
  8. C v C_{v}

Griffiths_inequality.html

  1. σ = { σ j } j Λ \textstyle\sigma=\{\sigma_{j}\}_{j\in\Lambda}
  2. σ A = j A σ j \textstyle\sigma_{A}=\prod_{j\in A}\sigma_{j}
  3. H ( σ ) = - A J A σ A , H(\sigma)=-\sum_{A}J_{A}\sigma_{A}~{},
  4. Z = d μ ( σ ) e - H ( σ ) Z=\int d\mu(\sigma)e^{-H(\sigma)}
  5. = 1 Z σ ( σ ) e - H ( σ ) \langle\cdot\rangle=\frac{1}{Z}\sum_{\sigma}\cdot(\sigma)e^{-H(\sigma)}
  6. τ k = { σ k , k j , - σ k , k = j . \tau_{k}=\begin{cases}\sigma_{k},&k\neq j,\\ -\sigma_{k},&k=j.\end{cases}
  7. σ A 0 \langle\sigma_{A}\rangle\geq 0
  8. σ A σ B σ A σ B \langle\sigma_{A}\sigma_{B}\rangle\geq\langle\sigma_{A}\rangle\langle\sigma_{B}\rangle
  9. e - H ( σ ) = B k 0 J B k σ B k k ! = { k C } C B J B k B σ B k B k B ! , e^{-H(\sigma)}=\prod_{B}\sum_{k\geq 0}\frac{J_{B}^{k}\sigma_{B}^{k}}{k!}=\sum_% {\{k_{C}\}_{C}}\prod_{B}\frac{J_{B}^{k_{B}}\sigma_{B}^{k_{B}}}{k_{B}!}~{},
  10. Z σ A = d μ ( σ ) σ A e - H ( σ ) = { k C } C B J B k B k B ! d μ ( σ ) σ A σ B k B = { k C } C B J B k B k B ! d μ ( σ ) j Λ σ j n A ( j ) + n B ( j ) , \begin{aligned}\displaystyle Z\langle\sigma_{A}\rangle&\displaystyle=\int d\mu% (\sigma)\sigma_{A}e^{-H(\sigma)}=\sum_{\{k_{C}\}_{C}}\prod_{B}\frac{J_{B}^{k_{% B}}}{k_{B}!}\int d\mu(\sigma)\sigma_{A}\sigma_{B}^{k_{B}}\\ &\displaystyle=\sum_{\{k_{C}\}_{C}}\prod_{B}\frac{J_{B}^{k_{B}}}{k_{B}!}\int d% \mu(\sigma)\prod_{j\in\Lambda}\sigma_{j}^{n_{A}(j)+n_{B}(j)}~{},\end{aligned}
  11. d μ ( σ ) j σ j n ( j ) = 0 \int d\mu(\sigma)\prod_{j}\sigma_{j}^{n(j)}=0
  12. σ \sigma^{\prime}
  13. σ \sigma
  14. σ A σ B - σ A σ B = σ A ( σ B - σ B ) . \langle\sigma_{A}\sigma_{B}\rangle-\langle\sigma_{A}\rangle\langle\sigma_{B}% \rangle=\langle\langle\sigma_{A}(\sigma_{B}-\sigma^{\prime}_{B})\rangle\rangle% ~{}.
  15. σ j = τ j + τ j , σ j = τ j - τ j . \sigma_{j}=\tau_{j}+\tau_{j}^{\prime}~{},\qquad\sigma^{\prime}_{j}=\tau_{j}-% \tau_{j}^{\prime}~{}.
  16. \langle\langle\;\cdot\;\rangle\rangle
  17. τ , τ \tau,\tau^{\prime}
  18. - H ( σ ) - H ( σ ) -H(\sigma)-H(\sigma^{\prime})
  19. τ , τ \tau,\tau^{\prime}
  20. A J A ( σ A + σ A ) = A J A X A [ 1 + ( - 1 ) | X | ] τ A X τ X \begin{aligned}\displaystyle\sum_{A}J_{A}(\sigma_{A}+\sigma^{\prime}_{A})&% \displaystyle=\sum_{A}J_{A}\sum_{X\subset A}\left[1+(-1)^{|X|}\right]\tau_{A% \setminus X}\tau^{\prime}_{X}\end{aligned}
  21. τ , τ \tau,\tau^{\prime}
  22. d μ ( σ ) d μ ( σ ) d\mu(\sigma)d\mu(\sigma^{\prime})
  23. σ A \sigma_{A}
  24. σ B - σ B \sigma_{B}-\sigma^{\prime}_{B}
  25. τ , τ \tau,\tau^{\prime}
  26. σ A = X A τ A X τ X , σ B - σ B = X B [ 1 - ( - 1 ) | X | ] τ B X τ X . \begin{aligned}\displaystyle\sigma_{A}&\displaystyle=\sum_{X\subset A}\tau_{A% \setminus X}\tau^{\prime}_{X}~{},\\ \displaystyle\sigma_{B}-\sigma^{\prime}_{B}&\displaystyle=\sum_{X\subset B}% \left[1-(-1)^{|X|}\right]\tau_{B\setminus X}\tau^{\prime}_{X}~{}.\end{aligned}
  27. σ A ( σ B - σ B ) \langle\langle\sigma_{A}(\sigma_{B}-\sigma^{\prime}_{B})\rangle\rangle
  28. f h = f ( x ) e - h ( x ) d μ ( x ) / e - h ( x ) d μ ( x ) . \langle f\rangle_{h}=\int f(x)e^{-h(x)}\,d\mu(x)\Big/\int e^{-h(x)}\,d\mu(x).
  29. d μ ( x ) d μ ( y ) j = 1 n ( f j ( x ) ± f j ( y ) ) 0. \iint d\mu(x)\,d\mu(y)\prod_{j=1}^{n}(f_{j}(x)\pm f_{j}(y))\geq 0.
  30. f g h - f h g h 0. \langle fg\rangle_{h}-\langle f\rangle_{h}\langle g\rangle_{h}\geq 0.
  31. Z h = e - h ( x ) d μ ( x ) . Z_{h}=\int e^{-h(x)}\,d\mu(x).
  32. Z h 2 ( f g h - f h g h ) \displaystyle Z_{h}^{2}\left(\langle fg\rangle_{h}-\langle f\rangle_{h}\langle g% \rangle_{h}\right)
  33. f ( x ) = 1 2 ( f ( x ) + f ( y ) ) + 1 2 ( f ( x ) - f ( y ) ) . f(x)=\frac{1}{2}(f(x)+f(y))+\frac{1}{2}(f(x)-f(y)).
  34. J B σ A = σ A σ B - σ A σ B 0 \frac{\partial}{\partial J_{B}}\langle\sigma_{A}\rangle=\langle\sigma_{A}% \sigma_{B}\rangle-\langle\sigma_{A}\rangle\langle\sigma_{B}\rangle\geq 0
  35. σ A \langle\sigma_{A}\rangle
  36. J x , y | x - y | - α J_{x,y}\sim|x-y|^{-\alpha}
  37. 1 < α < 2 1<\alpha<2
  38. J x , y | x - y | - α J_{x,y}\sim|x-y|^{-\alpha}
  39. 2 < α < 4 2<\alpha<4
  40. D D
  41. J > 0 J>0
  42. β \beta
  43. D D
  44. J > 0 J>0
  45. β / 2 \beta/2
  46. 𝐬 i 𝐬 j J , 2 β σ i σ j J , β \langle\mathbf{s}_{i}\cdot\mathbf{s}_{j}\rangle_{J,2\beta}\leq\langle\sigma_{i% }\sigma_{j}\rangle_{J,\beta}
  47. β \beta
  48. β c X Y 2 β c Is ; \beta_{c}^{XY}\geq 2\beta_{c}^{\rm Is}~{};
  49. β c X Y ln ( 1 + 2 ) 0.88 . \beta_{c}^{XY}\geq\ln(1+\sqrt{2})\approx 0.88~{}.

Gross–Koblitz_formula.html

  1. τ q ( r ) = - π s p ( r ) 0 i < f Γ p ( r ( i ) / ( q - 1 ) ) \tau_{q}(r)=-\pi^{s_{p}(r)}\prod_{0\leq i<f}\Gamma_{p}(r^{(i)}/(q-1))
  2. τ q ( r ) = a q - 1 = 1 a - r ζ π Tr ( a ) \tau_{q}(r)=\sum_{a^{q-1}=1}a^{-r}\zeta_{\pi}^{\,\text{Tr}(a)}

Grothendieck_category.html

  1. V V
  2. Qcoh ( V ) \operatorname{Qcoh}(V)
  3. V V
  4. V V
  5. V V
  6. Qcoh ( V ) \operatorname{Qcoh}(V)
  7. 𝒜 \mathcal{A}
  8. 𝒜 \mathcal{A}
  9. 𝒜 \mathcal{A}
  10. 𝒜 \mathcal{A}
  11. 𝒜 \mathcal{A}
  12. 𝒜 \mathcal{A}
  13. G G
  14. 𝒜 \mathcal{A}
  15. Hom ( G , - ) \operatorname{Hom}(G,-)
  16. 𝒜 \mathcal{A}
  17. X X
  18. 𝒜 \mathcal{A}
  19. G ( I ) X G^{(I)}\rightarrow X
  20. G ( I ) G^{(I)}
  21. G G
  22. I I
  23. \mathbb{Z}
  24. R R
  25. 1 1
  26. Mod ( R ) \operatorname{Mod}(R)
  27. R R
  28. R R
  29. X X
  30. X X
  31. R R
  32. X X
  33. R R
  34. ( X , 𝒪 X ) (X,\mathcal{O}_{X})
  35. V V
  36. Qcoh ( V ) \operatorname{Qcoh}(V)
  37. V V
  38. 𝒞 \mathcal{C}
  39. 𝒜 \mathcal{A}
  40. Funct ( 𝒞 , 𝒜 ) \operatorname{Funct}(\mathcal{C},\mathcal{A})
  41. 𝒞 \mathcal{C}
  42. Add ( 𝒞 , 𝒜 ) \operatorname{Add}(\mathcal{C},\mathcal{A})
  43. 𝒞 \mathcal{C}
  44. 𝒜 \mathcal{A}
  45. 𝒜 \mathcal{A}
  46. 𝒞 \mathcal{C}
  47. 𝒜 \mathcal{A}
  48. 𝒜 / 𝒞 \mathcal{A}/\mathcal{C}
  49. / \mathbb{Q}/\mathbb{Z}
  50. 𝒜 \mathcal{A}
  51. 𝒜 \mathcal{A}
  52. 𝒜 \mathcal{A}
  53. ( U i ) (U_{i})
  54. X X
  55. i U i \sum_{i}U_{i}
  56. X X
  57. ( U i ) (U_{i})
  58. V V
  59. X X
  60. i ( U i V ) = ( i U i ) V . \sum_{i}(U_{i}\cap V)=\left(\sum_{i}U_{i}\right)\cap V.
  61. 𝒜 \mathcal{A}
  62. Mod ( R ) \operatorname{Mod}(R)
  63. R R
  64. 𝒜 \mathcal{A}
  65. 𝒜 \mathcal{A}
  66. Mod ( R ) \operatorname{Mod}(R)

Grothendieck–Ogg–Shafarevich_formula.html

  1. χ ( C , F ) = n ( 2 - 2 g ) - x X ( n + S w x ( F ) ) \chi(C,F)=n(2-2g)-\sum_{x\in X}(n+Sw_{x}(F))

Group_of_rational_points_on_the_unit_circle.html

  1. G SO ( 2 , ) . G\cong\mathrm{SO}(2,\mathbb{Q}).
  2. S 1 S^{1}
  3. SO ( 2 , ) \mathrm{SO}(2,\mathbb{R})
  4. G G 2 p 1 mod 4 G p . G\cong G_{2}\oplus\bigoplus_{p\equiv 1\bmod 4}G_{p}.
  5. a + i b c \frac{a+ib}{c}
  6. ( c / a ) 2 - ( b / a ) 2 = 1 , (c/a)^{2}-(b/a)^{2}=1,
  7. ( x , y ) × ( u , v ) = ( x u + y v , x v + y u ) , (x,y)\times(u,v)=(xu+yv,xv+yu),
  8. w 2 + x 2 - y 2 + z 2 = 0. w^{2}+x^{2}-y^{2}+z^{2}=0.
  9. ( a , b , c , d ) × ( w , x , y , z ) = ( a w - b x , a x + b w , c y + d z , c z + d y ) . (a,b,c,d)\times(w,x,y,z)=(aw-bx,ax+bw,cy+dz,cz+dy).
  10. w 2 + x 2 = 1 , w^{2}+x^{2}=1,
  11. y 2 - z 2 = 1 , y^{2}-z^{2}=1,

Grouped_data.html

  1. x ¯ \bar{x}
  2. x ¯ = f x f . \bar{x}=\frac{\sum{f\,x}}{\sum{f}}.
  3. x ¯ = f x f = 405 20 = 20.25 \bar{x}=\frac{\sum{f\,x}}{\sum{f}}=\frac{405}{20}=20.25

Groupoid_algebra.html

  1. ( G , ) (G,\cdot)
  2. K K
  3. K G KG
  4. K K
  5. G G
  6. g * h = g h g*h=g\cdot h
  7. g * h = 0 g*h=0

Growing_self-organizing_map.html

  1. G T GT
  2. D D
  3. S F SF
  4. G T = - D × ln ( S F ) GT=-D\times\ln(SF)
  5. q q^{\prime}
  6. | v - w q | | v - w q | q \left|v-w_{q^{\prime}}\right|\leq\left|v-w_{q}\right|\forall q\in\mathbb{N}
  7. v v
  8. w w
  9. q q
  10. \mathbb{N}
  11. w j ( k + 1 ) = { w j ( k ) if j \Nu k + 1 w j ( k ) + L R ( k ) × ( x k - w j ( k ) ) if j \Nu k + 1 w_{j}(k+1)=\begin{cases}w_{j}(k)&\mbox{if}~{}j\notin\Nu_{k+1}\\ w_{j}(k)+LR(k)\times(x_{k}-w_{j}(k))&\mbox{if}~{}j\in\Nu_{k+1}\end{cases}
  12. L R ( k ) LR(k)
  13. k k\in\mathbb{N}
  14. k k\to\infty
  15. w j ( k ) w_{j}(k)
  16. w j ( k + 1 ) w_{j}(k+1)
  17. j j
  18. \Nu k + 1 \Nu_{k+1}
  19. ( k + 1 ) (k+1)
  20. L R ( k ) LR(k)
  21. k k
  22. T E i > G T TE_{i}>GT
  23. T E i TE_{i}
  24. i i
  25. G T GT
  26. i i
  27. L R LR

Grubel–Lloyd_index.html

  1. G L i = ( X i + M i ) - | X i - M i | X i + M i = 1 - | X i - M i | X i + M i ; 0 G L i 1 GL_{i}=\dfrac{(X_{i}+M_{i})-\left|X_{i}-M_{i}\right|}{X_{i}+M_{i}}=1-\dfrac{% \left|X_{i}-M_{i}\right|}{X_{i}+M_{i}}\qquad;\ 0\leq GL_{i}\leq 1

Grunsky's_theorem.html

  1. | z f ( z ) f ( z ) | 1 + | z | 1 - | z | \left|{zf^{\prime}(z)\over f(z)}\right|\leq{1+|z|\over 1-|z|}
  2. | arg z f ( z ) f ( z ) | log 1 + | z | 1 - | z | . \left|\arg{zf^{\prime}(z)\over f(z)}\right|\leq\log{1+|z|\over 1-|z|}.
  3. g w ( ζ ) = ζ ( 1 - w ¯ ζ ) 2 , g_{w}(\zeta)={\zeta\over(1-\overline{w}\zeta)^{2}},
  4. g ( z ) = z + b 1 z - 1 + b 2 z - 2 + . g(z)=z+b_{1}z^{-1}+b_{2}z^{-2}+\cdots.
  5. | i = 1 n j = 1 n λ i λ j log g ( z i ) - g ( z j ) z i - z j | i = 1 n j = 1 n λ i λ j ¯ log z i z j ¯ z i z j ¯ - 1 , \left|\sum_{i=1}^{n}\sum_{j=1}^{n}\lambda_{i}\lambda_{j}\log{g(z_{i})-g(z_{j})% \over z_{i}-z_{j}}\right|\leq\sum_{i=1}^{n}\sum_{j=1}^{n}\lambda_{i}\overline{% \lambda_{j}}\log{z_{i}\overline{z_{j}}\over z_{i}\overline{z_{j}}-1},
  6. | log g ( ζ ) g ( η ) ( ζ - η ) 2 ( g ( ζ ) - g ( η ) ) 2 | log | 1 - ζ η ¯ | 2 ( | ζ | 2 - 1 ) ( | η | 2 - 1 ) . \left|\log{g^{\prime}(\zeta)g^{\prime}(\eta)(\zeta-\eta)^{2}\over(g(\zeta)-g(% \eta))^{2}}\right|\leq\log{|1-\zeta\overline{\eta}|^{2}\over(|\zeta|^{2}-1)(|% \eta|^{2}-1)}.
  7. | log ζ g ( ζ ) g ( ζ ) | | ζ | 2 + 1 | ζ | 2 - 1 . \left|\log{\zeta g^{\prime}(\zeta)\over g(\zeta)}\right|\leq{|\zeta|^{2}+1% \over|\zeta|^{2}-1}.
  8. g ( ζ ) = f ( ζ - 2 ) - 1 2 g(\zeta)=f(\zeta^{-2})^{-{1\over 2}}
  9. z = ζ - 2 . z=\zeta^{-2}.
  10. | arg z f ( z ) f ( z ) | log 1 + | z | 1 - | z | . \left|\arg{zf^{\prime}(z)\over f(z)}\right|\leq\log{1+|z|\over 1-|z|}.
  11. log 1 + | z | 1 - | z | π 2 , \log{1+|z|\over 1-|z|}\leq{\pi\over 2},
  12. | z | tanh π 4 |z|\leq\tanh{\pi\over 4}

Grunsky_matrix.html

  1. g ( z ) = f ( z - 1 ) - 1 g(z)=f(z^{-1})^{-1}
  2. g ( z ) = z + b 0 + b 1 z - 1 + b 2 z - 2 + g(z)=z+b_{0}+b_{1}z^{-1}+b_{2}z^{-2}+\cdots
  3. log g ( z ) - g ( ζ ) z - ζ = - m , n > 0 c n m z - m ζ - n \log\frac{g(z)-g(\zeta)}{z-\zeta}=-\sum_{m,n>0}c_{nm}z^{-m}\zeta^{-n}
  4. log g ( z - 1 ) - g ( ζ - 1 ) z - 1 - ζ - 1 = log f ( z ) - f ( ζ ) z - ζ - log f ( z ) z - log f ( ζ ) ζ , \log{g(z^{-1})-g(\zeta^{-1})\over z^{-1}-\zeta^{-1}}=\log{f(z)-f(\zeta)\over z% -\zeta}-\log{f(z)\over z}-\log{f(\zeta)\over\zeta},
  5. log f ( z ) - f ( ζ ) z - ζ = - m , n 0 d m n z n ζ n , \log{f(z)-f(\zeta)\over z-\zeta}=-\sum_{m,n\geq 0}d_{mn}z^{n}\zeta^{n},
  6. d m n = c m n \displaystyle{d_{mn}=c_{mn}}
  7. log f ( z ) z = n > 0 d 0 n z n \log{f(z)\over z}=\sum_{n>0}d_{0n}z^{n}
  8. d 00 = 0. \displaystyle{d_{00}=0.}
  9. | 1 m , n N c m n λ m λ n | 1 n N | λ n | 2 / n \left|\sum_{1\leq m,n\leq N}c_{mn}\lambda_{m}\lambda_{n}\right|\leq\sum_{1\leq n% \leq N}|\lambda_{n}|^{2}/n
  10. g ( z ) = z + b 0 + b 1 z - 1 + b 2 z - 2 + g(z)=z+b_{0}+b_{1}z^{-1}+b_{2}z^{-2}+\cdots
  11. z g ( z ) g ( z ) - g ( ζ ) - z z - ζ = m , n > 0 m c m n z - m ζ - n . {zg^{\prime}(z)\over g(z)-g(\zeta)}-{z\over z-\zeta}=\sum_{m,n>0}mc_{mn}z^{-m}% \zeta^{-n}.
  12. z g ( z ) g ( z ) - w = n 0 Φ n ( w ) z - n . {zg^{\prime}(z)\over g(z)-w}=\sum_{n\geq 0}\Phi_{n}(w)z^{-n}.
  13. log g ( z ) - w z = - n 1 1 n Φ n ( w ) z - n . \log{g(z)-w\over z}=-\sum_{n\geq 1}{1\over n}\Phi_{n}(w)z^{-n}.
  14. Φ n ( w ) = ( w - b 0 ) Φ n - 1 ( w ) - n b n - 0 i n - 1 b n - i Φ i ( w ) \Phi_{n}(w)=(w-b_{0})\Phi_{n-1}(w)-nb_{n}-\sum_{0\leq i\leq n-1}b_{n-i}\Phi_{i% }(w)
  15. Φ 0 ( w ) 1. \Phi_{0}(w)\equiv 1.
  16. n 0 Φ n ( g ( z ) ) ζ - n = 1 + n 1 ( z n + m 1 c n m z - m ) ζ - n , \sum_{n\geq 0}\Phi_{n}(g(z))\zeta^{-n}=1+\sum_{n\geq 1}\left(z^{n}+\sum_{m\geq 1% }c_{nm}z^{-m}\right)\zeta^{-n},
  17. Φ n ( g ( z ) ) = z n + m 1 c n m z - m . \Phi_{n}(g(z))=z^{n}+\sum_{m\geq 1}c_{nm}z^{-m}.
  18. g ( z ) = z + b 1 z - 1 + b 2 z - 2 + g(z)=z+b_{1}z^{-1}+b_{2}z^{-2}+\cdots
  19. f ( g ( z ) ) = - c n z n f(g(z))=\sum_{-\infty}^{\infty}c_{n}z^{n}
  20. n > 0 n | c n | 2 n > 0 n | c - n | 2 . \sum_{n>0}n|c_{n}|^{2}\leq\sum_{n>0}n|c_{-n}|^{2}.
  21. Ω h ( z ) d z = Ω ω = Ω d ω = Ω ( i x - y ) h d x d y = 2 i Ω z ¯ h d x d y . \int_{\partial\Omega}h(z)\,dz=\int_{\partial\Omega}\omega=\iint_{\Omega}d% \omega=\iint_{\Omega}(i\partial_{x}-\partial_{y})h\,dx\,dy=2i\iint_{\Omega}% \partial_{\overline{z}}h\,dx\,dy.
  22. A ( r ) = Ω r | f ( z ) | 2 d x d y = 1 2 i Ω r f ( z ) ¯ f ( z ) d z = 1 2 i | w | = r f ( g ( w ) ) ) ¯ f ( g ( w ) ) g ( w ) d w . A(r)=\iint_{\Omega_{r}}|f^{\prime}(z)|^{2}\,dx\,dy={1\over 2i}\int_{\partial% \Omega_{r}}\overline{f(z)}f^{\prime}(z)\,dz={1\over 2i}\int_{|w|=r}\overline{f% (g(w)))}f^{\prime}(g(w))g^{\prime}(w)\,dw.
  23. A ( r ) = π n n | c - n | 2 r 2 n . \displaystyle{A(r)=\pi\sum_{n}n|c_{-n}|^{2}r^{2n}.}
  24. n > 0 n | c n | 2 r - 2 n n > 0 n | c - n | 2 r 2 n . \sum_{n>0}n|c_{n}|^{2}r^{-2n}\leq\sum_{n>0}n|c_{-n}|^{2}r^{2n}.
  25. p ( w ) = n = 1 N n - 1 λ n Φ n ( w ) , p(w)=\sum_{n=1}^{N}n^{-1}\lambda_{n}\Phi_{n}(w),
  26. p ( g ( z ) ) = ( n = 1 N n - 1 λ n z n ) + ( m = 1 n = 1 N λ n c n m z - m ) . p(g(z))=\left(\sum_{n=1}^{N}n^{-1}\lambda_{n}z^{n}\right)+\left(\sum_{m=1}^{% \infty}\sum_{n=1}^{N}\lambda_{n}c_{nm}z^{-m}\right).
  27. m = 1 m | n = 1 N c m n λ n | 2 n = 1 N 1 n | λ n | 2 . \sum_{m=1}^{\infty}m\left|\sum_{n=1}^{N}c_{mn}\lambda_{n}\right|^{2}\leq\sum_{% n=1}^{N}{1\over n}|\lambda_{n}|^{2}.
  28. m = 1 N m | n = 1 N c m n λ n | 2 n = 1 N 1 n | λ n | 2 . \sum_{m=1}^{N}m\left|\sum_{n=1}^{N}c_{mn}\lambda_{n}\right|^{2}\leq\sum_{n=1}^% {N}{1\over n}|\lambda_{n}|^{2}.
  29. a m n = ( m n ) 1 / 2 c m n a_{mn}=(mn)^{1/2}c_{mn}
  30. A x x . \|Ax\|\leq\|x\|.
  31. | ( A x , y ) | x y . |(Ax,y)|\leq\|x\|\cdot\|y\|.
  32. x n = n - 1 / 2 λ n = y n ¯ x_{n}=n^{-1/2}\lambda_{n}=\overline{y_{n}}
  33. | m = 1 N n = 1 N c m n λ m λ n | 2 n = 1 N 1 n | λ n | 2 , \left|\sum_{m=1}^{N}\sum_{n=1}^{N}c_{mn}\lambda_{m}\lambda_{n}\right|^{2}\leq% \sum_{n=1}^{N}{1\over n}|\lambda_{n}|^{2},
  34. g ( z ) = z + b 0 + b 1 z - 1 + b 2 z - 2 + g(z)=z+b_{0}+b_{1}z^{-1}+b_{2}z^{-2}+\cdots
  35. log g ( z ) - g ( ζ ) z - ζ = - m , n 1 c m n z - m ζ - n \log{g(z)-g(\zeta)\over z-\zeta}=-\sum_{m,n\geq 1}c_{mn}z^{-m}\zeta^{-n}
  36. g ( ζ ) = ζ + a 0 + b 1 ζ - 1 + b 2 ζ - 2 + g(\zeta)=\zeta+a_{0}+b_{1}\zeta^{-1}+b_{2}\zeta^{-2}+\cdots
  37. F ( z ) = a f ( z ) \displaystyle{F(z)=af(z)}
  38. f ( z ) = z + a 2 z 2 + a 3 z 3 + f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots
  39. log g ( ζ ) - g ( η ) ζ - η = - m , n 1 c m n ζ - m η - n \log{g(\zeta)-g(\eta)\over\zeta-\eta}=-\sum_{m,n\geq 1}c_{mn}\zeta^{-m}\eta^{-n}
  40. log g ( ζ ) - f ( z ) ζ - log g ( ζ ) ζ = - m , n 1 c - m , n z m ζ - n \log{g(\zeta)-f(z)\over\zeta}-\log{g(\zeta)\over\zeta}=-\sum_{m,n\geq 1}c_{-m,% n}z^{m}\zeta^{-n}
  41. log f ( z ) - f ( w ) z - w - log f ( z ) z - log f ( w ) w = - m , n 1 c - m , - n z m w n \log{f(z)-f(w)\over z-w}-\log{f(z)\over z}-\log{f(w)\over w}=-\sum_{m,n\geq 1}% c_{-m,-n}z^{m}w^{n}
  42. | n , m 0 c m n λ m λ n | n 0 1 | n | | λ n | 2 . |\sum_{n,m\neq 0}c_{mn}\lambda_{m}\lambda_{n}|\leq\sum_{n\neq 0}{1\over|n|}|% \lambda_{n}|^{2}.
  43. f ( z - 1 ) - 1 f(z^{-1})^{-1}
  44. h ( w ) = n 1 λ n n Φ n ( w ) + n 1 λ - n n Φ - n ( a w ) . h(w)=\sum_{n\geq 1}{\lambda_{n}\over n}\Phi_{n}(w)+\sum_{n\geq 1}{\lambda_{-n}% \over n}\Phi_{-n}({a\over w}).
  45. h ( g ( ζ ) ) = n 1 λ n n ζ n + β + n 1 β n ζ - n , β n = m c n m λ m . h(g(\zeta))=\sum_{n\geq 1}{\lambda_{n}\over n}\zeta^{n}+\beta+\sum_{n\geq 1}% \beta_{n}\zeta^{-n},\,\,\,\beta_{n}=\sum_{m}c_{nm}\lambda_{m}.
  46. | h ( z ) | 2 d x d y = 1 2 i C 1 h ¯ ( z ) h ( z ) d z - 1 2 i C 2 h ¯ ( z ) h ( z ) d z , \int|h^{\prime}(z)|^{2}\,dx\,dy={1\over 2i}\int_{C_{1}}\overline{h}(z)h^{% \prime}(z)\,dz-{1\over 2i}\int_{C_{2}}\overline{h}(z)h^{\prime}(z)\,dz,
  47. 1 π | h | 2 d x d y = [ n 1 1 n | λ - n | 2 - n 1 | α n | 2 r 2 n ] + [ n 1 1 n | λ n | 2 - n 1 | β n | 2 R - 2 n ] . {1\over\pi}\iint|h^{\prime}|^{2}\,dx\,dy=\left[\sum_{n\geq 1}{1\over n}|% \lambda_{-n}|^{2}-\sum_{n\geq 1}|\alpha_{n}|^{2}r^{2n}\right]+\left[\sum_{n% \geq 1}{1\over n}|\lambda_{n}|^{2}-\sum_{n\geq 1}|\beta_{n}|^{2}R^{-2n}\right].
  48. m 0 | m | | n 0 c m n λ n | 2 m 0 1 | m | | λ m | 2 \sum_{m\neq 0}|m|\left|\sum_{n\neq 0}c_{mn}\lambda_{n}\right|^{2}\leq\sum_{m% \neq 0}{1\over|m|}|\lambda_{m}|^{2}
  49. a m n = | m n | c m n a_{mn}=\sqrt{|mn|}\cdot c_{mn}
  50. A * A = I , \displaystyle{A^{*}A=I,}
  51. J A J = A * , J A * J = A \displaystyle{JAJ=A^{*},\,\,JA^{*}J=A}
  52. A A * = J A * A J = I \displaystyle{AA^{*}=JA^{*}AJ=I}
  53. | m = 1 N n = 1 N α m α n log g ( z m ) - g ( z n ) z m - z n | 2 m = 1 N n = 1 N α m α n ¯ log 1 1 - ( z m z n ¯ ) - 1 . \left|\sum_{m=1}^{N}\sum_{n=1}^{N}\alpha_{m}\alpha_{n}\log{g(z_{m})-g(z_{n})% \over z_{m}-z_{n}}\right|^{2}\leq\sum_{m=1}^{N}\sum_{n=1}^{N}\alpha_{m}% \overline{\alpha_{n}}\log{1\over 1-(z_{m}\overline{z_{n}})^{-1}}.
  54. λ k = n = 1 N α n z n - k . \displaystyle{\lambda_{k}=\sum_{n=1}^{N}\alpha_{n}z_{n}^{-k}.}
  55. α m = 1 N n = 1 N λ n z n m . \displaystyle{\alpha_{m}={1\over N}\sum_{n=1}^{N}\lambda_{n}z_{n}^{m}.}
  56. z n = r e 2 π i n N \displaystyle{z_{n}=re^{2\pi in\over N}}
  57. | m = 1 N n = 1 N α m α n [ f ( z m ) f ( z n ) ( f ( z m ) - f ( z n ) ) 2 - 1 ( z m - z n ) 2 ] | m = 1 N n = 1 N α m α n ¯ 1 ( 1 - z m z n ¯ ) 2 . \left|\sum_{m=1}^{N}\sum_{n=1}^{N}\alpha_{m}\alpha_{n}\left[{f^{\prime}(z_{m})% f^{\prime}(z_{n})\over(f(z_{m})-f(z_{n}))^{2}}-{1\over(z_{m}-z_{n})^{2}}\right% ]\right|\leq\sum_{m=1}^{N}\sum_{n=1}^{N}\alpha_{m}\overline{\alpha_{n}}{1\over% (1-z_{m}\overline{z_{n}})^{2}}.
  58. λ k = k n = 1 N α n z n k . \displaystyle{\lambda_{k}=k\sum_{n=1}^{N}\alpha_{n}z_{n}^{k}.}
  59. α m = 1 N n = 1 N 1 n λ n z n m . \displaystyle{\alpha_{m}={1\over N}\sum_{n=1}^{N}{1\over n}\lambda_{n}z_{n}^{m% }.}
  60. z n = r e 2 π i n N \displaystyle{z_{n}=re^{2\pi in\over N}}
  61. f ( x ) = z + a 2 z 2 + a 3 z 3 + a 4 z 4 + f(x)=z+a_{2}z^{2}+a_{3}z^{3}+a_{4}z^{4}+\cdots
  62. log g ( z ) - g ( ζ ) z - ζ = - n 1 a n ( ζ - 1 ) z - n . \log{g(z)-g(\zeta)\over z-\zeta}=-\sum_{n\geq 1}a_{n}(\zeta^{-1})z^{-n}.
  63. n 1 n | a n ( w ) | 2 - log ( 1 - | w | 2 ) . \displaystyle{\sum_{n\geq 1}n|a_{n}(w)|^{2}\leq-\log(1-|w|^{2}).}
  64. m 0 b m t m = exp n 1 a n t n \displaystyle{\sum_{m\geq 0}b_{m}t^{m}=\exp\sum_{n\geq 1}a_{n}t^{n}}
  65. n 0 | b n | 2 exp n 1 n | a n | 2 . \displaystyle{\sum_{n\geq 0}|b_{n}|^{2}\leq\exp\sum_{n\geq 1}n|a_{n}|^{2}.}
  66. 1 2 π 0 2 π | e g | 2 d θ e A , {1\over 2\pi}\int_{0}^{2\pi}|e^{g}|^{2}\,d\theta\leq e^{A},
  67. b n = 1 n m = 1 n m a m b n - m \displaystyle{b_{n}={1\over n}\sum_{m=1}^{n}ma_{m}b_{n-m}}
  68. | b n | 2 1 n m 2 | a m | 2 | b n - m | 2 . \displaystyle{|b_{n}|^{2}\leq{1\over n}\sum m^{2}|a_{m}|^{2}|b_{n-m}|^{2}.}
  69. c n = 1 n m 2 | a m | 2 c n - m \displaystyle{c_{n}={1\over n}\sum m^{2}|a_{m}|^{2}c_{n-m}}
  70. | b n | 2 c n |b_{n}|^{2}\leq c_{n}
  71. | b n | 2 c n = exp m 1 m | a m | 2 . \displaystyle{\sum|b_{n}|^{2}\leq\sum c_{n}=\exp\sum_{m\geq 1}m|a_{m}|^{2}.}
  72. b n ( ζ - 1 ) z - n = exp a m ( ζ - 1 ) z - m = g ( z ) - g ( ζ ) z - ζ , \displaystyle{\sum b_{n}(\zeta^{-1})z^{-n}=\exp\sum a_{m}(\zeta^{-1})z^{-m}={g% (z)-g(\zeta)\over z-\zeta},}
  73. z / z ¯ z/\overline{z}
  74. ( T h ) ( w ) = lim ε 0 - 1 π | z - w | ε h ( z ) ( z - w ) 2 d x d y . (Th)(w)=\lim_{\varepsilon\rightarrow 0}-{1\over\pi}\iint_{|z-w|\geq\varepsilon% }{h(z)\over(z-w)^{2}}\,dx\,dy.
  75. K f ( z , w ) = f ( z ) f ( w ) ( f ( z ) - f ( w ) ) 2 . K_{f}(z,w)={f^{\prime}(z)f^{\prime}(w)\over(f(z)-f(w))^{2}}.
  76. z ¯ n \overline{z}^{n}
  77. f ( z ) f ( w ) ( f ( z ) - f ( w ) ) 2 - 1 ( z - w ) 2 = 2 z w log f ( z ) - f ( w ) z - w = - m , n 1 m n c m n z m - 1 w n - 1 {f^{\prime}(z)f^{\prime}(w)\over(f(z)-f(w))^{2}}\,-\,{1\over(z-w)^{2}}={% \partial^{2}\over\partial z\partial w}\log{f(z)-f(w)\over z-w}=-\sum_{m,n\geq 1% }mnc_{mn}z^{m-1}w^{n-1}
  78. p ( z ) = λ 1 + λ 2 z ¯ + λ 3 z ¯ 2 + + λ N z ¯ N - 1 , p(z)=\lambda_{1}+\lambda_{2}\overline{z}+\lambda_{3}\overline{z}^{2}+\cdots+% \lambda_{N}\overline{z}^{N-1},
  79. m = 1 N | n = 1 N c m n λ n | 2 = ( T f - T z ) p 2 = T f p 2 p 2 = n = 1 N 1 n | λ n | 2 . \sum_{m=1}^{N}\left|\sum_{n=1}^{N}c_{mn}\lambda_{n}\right|^{2}=\|(T_{f}-T_{z})% p\|^{2}=\|T_{f}p\|^{2}\leq\|p\|^{2}=\sum_{n=1}^{N}{1\over n}|\lambda_{n}|^{2}.
  80. ( T Ω u ) ( z ) = lim ε 0 1 π | z - w | ε u ( z ) ¯ ( z - w ) 2 d x d y (T_{\Omega}u)(z)=\lim_{\varepsilon\rightarrow 0}{1\over\pi}\iint_{|z-w|\geq% \varepsilon}{\overline{u(z)}\over(z-w)^{2}}\,\,dx\,dy
  81. f ( z ) f ( w ) ( f ( z ) - f ( w ) ) 2 - 1 ( z - w ) 2 , {f^{\prime}(z)f^{\prime}(w)\over(f(z)-f(w))^{2}}-{1\over(z-w)^{2}},
  82. ( T u , v ) = ( T v , u ) \displaystyle{(Tu,v)=(Tv,u)}
  83. ( A u , u ) = ( T u , T u ) = T u 2 0 , \displaystyle{(Au,u)=(Tu,Tu)=\|Tu\|^{2}\geq 0,}
  84. A u n = μ n u n , \displaystyle{Au_{n}=\mu_{n}u_{n},}
  85. μ n = λ n 2 \displaystyle{\mu_{n}=\lambda_{n}^{2}}
  86. T u n = λ n u n . \displaystyle{Tu_{n}=\lambda_{n}u_{n}.}
  87. T ( i u n ) = - λ n i u n T(iu_{n})=-\lambda_{n}iu_{n}
  88. 0 λ n T 1. \displaystyle{0\leq\lambda_{n}\leq\|T\|\leq 1.}
  89. T Ω < 1. \displaystyle{\|T_{\Omega}\|<1.}
  90. Δ Ω = det ( I - T Ω 2 ) = ( 1 - λ n 2 ) . \Delta_{\Omega}=\det(I-T_{\Omega}^{2})=\prod(1-\lambda_{n}^{2}).
  91. Δ Ω = - 1 12 π [ z log f D 2 + z log g D c 2 - 2 z log f ( z ) / z D 2 - 2 z log g ( z ) / z D c 2 ] . \Delta_{\Omega}=-{1\over 12\pi}\left[\|\partial_{z}\log f^{\prime}\|^{2}_{D}+% \|\partial_{z}\log g^{\prime}\|^{2}_{D^{c}}-2\|\partial_{z}\log f(z)/z\|^{2}_{% D}-2\|\partial_{z}\log g(z)/z\|^{2}_{D^{c}}\right].

Guided_tour_puzzle_protocol.html

  1. N N
  2. N 2 N\geq 2
  3. k j s k_{js}
  4. G j G_{j}
  5. 0 j < N 0\leq j<N
  6. K s K_{s}
  7. L L
  8. N = 2 N=2
  9. L = 5 L=5
  10. x x
  11. x x
  12. L L
  13. h 0 h_{0}
  14. h 0 h_{0}
  15. h 0 = h a s h ( A x | | L | | t s | | K s ) h_{0}=hash(A_{x}\;||\;L\;||\;ts\;||\;K_{s})
  16. | | ||
  17. A x A_{x}
  18. x x
  19. t s ts
  20. h a s h hash
  21. l l
  22. S l = ( h l - 1 m o d N ) S_{l}=(h_{l-1}\;mod\;N)
  23. 0 < l L 0<l\leq L
  24. x x
  25. G j G_{j}
  26. h l h_{l}
  27. 0 < l L 0<l\leq L
  28. h l = h a s h ( h l - 1 | | l | | L | | A x | | t s | | k j s ) h_{l}=hash(h_{l-1}\;||\;l\;||\;L\;||\;A_{x}\;||\;ts\;||\;k_{js})
  29. l l
  30. l l
  31. k j s k_{js}
  32. G j G_{j}
  33. x x
  34. S 1 S_{1}
  35. h 0 h_{0}
  36. 1 1
  37. L L
  38. G S 1 G_{S_{1}}
  39. h 1 h_{1}
  40. G S 1 G_{S_{1}}
  41. h 1 h_{1}
  42. h l h_{l}
  43. x x
  44. L - 1 L-1
  45. G S 2 , G S 3 , , G S L G_{S_{2}},G_{S_{3}},...,G_{S_{L}}
  46. G S L G_{S_{L}}
  47. h L h_{L}
  48. x x
  49. h 0 , h L h_{0},\;h_{L}
  50. x x
  51. h 0 h^{\prime}_{0}
  52. h L h^{\prime}_{L}
  53. h 0 h^{\prime}_{0}
  54. h 0 h_{0}
  55. h 0 h_{0}
  56. h L h_{L}
  57. h l h_{l}
  58. h L h^{\prime}_{L}
  59. h L h_{L}
  60. k 1 s , k 2 s , , k N s k_{1s},k_{2s},\dots,k_{Ns}
  61. h 1 , h 2 , , h L h_{1},h_{2},...,h_{L}

GV-linear-code.html

  1. q 2 q\geq 2
  2. 0 δ < 1 - 1 q 0\leq\delta<1-\frac{1}{q}
  3. 0 < ε 1 - H q ( δ ) 0<\varepsilon\leq 1-H_{q}(\delta)
  4. R 1 - H q ( δ ) - ε R\geq 1-H_{q}(\delta)-\varepsilon
  5. δ \delta
  6. H q H_{q}
  7. H q ( x ) = x log q ( q - 1 ) - x log q x - ( 1 - x ) log q ( 1 - x ) . H_{q}(x)=x\log_{q}(q-1)-x\log_{q}x-(1-x)\log_{q}(1-x).
  8. G G
  9. 𝔽 q n \mathbb{F}_{q}^{n}
  10. G G
  11. d d
  12. m 𝔽 q k \ { 0 } , w t ( m G ) d m\in\mathbb{F}_{q}^{k}\backslash\left\{0\right\},wt(mG)\geq d
  13. G G
  14. d d
  15. n n
  16. d d
  17. d d
  18. n n
  19. G G
  20. G G
  21. k n kn
  22. k n kn
  23. 𝔽 q \mathbb{F}_{q}
  24. w t ( y ) wt(y)
  25. y y
  26. P \displaystyle P
  27. y y
  28. G G
  29. y = m G y=mG
  30. m 𝔽 q k m\in\mathbb{F}_{q}^{k}
  31. P = Pr random G [ there exists a vector m 𝔽 q k \ { 0 } such that w t ( m G ) < d ] P={\Pr}_{\,\text{random }G}[\,\text{there exists a vector }m\in\mathbb{F}_{q}^% {k}\backslash\{0\}\,\text{ such that }wt(mG)<d]
  32. P m 𝔽 q k \ { 0 } Pr random G [ w t ( m G ) < d ] P\leq\sum\limits_{m\in\mathbb{F}_{q}^{k}\backslash\{0\}}{{\Pr}_{\,\text{random% }G}}[wt(mG)<d]
  33. m 𝔽 q k \ { 0 } m\in\mathbb{F}_{q}^{k}\backslash\{0\}
  34. W = Pr random G [ w t ( m G ) < d ] W={\Pr}_{\,\text{random }G}[wt(mG)<d]
  35. Δ ( m 1 , m 2 ) \Delta(m_{1},m_{2})
  36. m 1 m_{1}
  37. m 2 m_{2}
  38. m m
  39. w t ( m ) = Δ ( 0 , m ) wt(m)=\Delta(0,m)
  40. W = all y 𝔽 q n s.t. Δ ( 0 , y ) d - 1 Pr random G [ m G = y ] W=\sum\limits_{\,\text{all }y\in\mathbb{F}_{q}^{n}\,\text{s.t. }\Delta(0,y)% \leq d-1}{{\Pr}_{\,\text{random }G}[mG=y]}
  41. G G
  42. m G mG
  43. 𝔽 q n \mathbb{F}_{q}^{n}
  44. Pr random G [ m G = y ] = q - n {\Pr}_{\,\text{random }G}[mG=y]=q^{-n}
  45. Vol q ( r , n ) \,\text{Vol}_{q}(r,n)
  46. r r
  47. W = Vol q ( d - 1 , n ) q n Vol q ( δ n , n ) q n q n H q ( δ ) q n W=\frac{\,\text{Vol}_{q}(d-1,n)}{q^{n}}\leq\frac{\,\text{Vol}_{q}(\delta n,n)}% {q^{n}}\leq\frac{q^{nH_{q}(\delta)}}{q^{n}}
  48. P q k W q k q n H q ( δ ) q n = q k q - n ( 1 - H q ( δ ) ) P\leq q^{k}\cdot W\leq q^{k}\frac{q^{nH_{q}(\delta)}}{q^{n}}=q^{k}q^{-n(1-H_{q% }(\delta))}
  49. k = ( 1 - H q ( δ ) - ε ) n k=(1-H_{q}(\delta)-\varepsilon)n
  50. P q - ε n P\leq q^{-\varepsilon n}
  51. q - ε n 1 q^{-\varepsilon n}\ll 1
  52. C C
  53. δ \delta
  54. R R
  55. ( 1 - H q ( δ ) - ε ) (1-H_{q}(\delta)-\varepsilon)
  56. G G
  57. k n kn
  58. 𝔽 q \mathbb{F}_{q}

György_Paál.html

  1. Ω Λ 2 / 3 \Omega_{\Lambda}\simeq 2/3

H-closed_space.html

  1. [ 0 , 1 ] [0,1]
  2. Q [ 0 , 1 ] Q\cap[0,1]

H._Pierre_Noyes.html

  1. c / e 2 = 137 \hbar c/e^{2}=137
  2. / 2 m e c \hbar/2m_{e}c
  3. ( 137 × ( 2 m e c 2 ) ) m π c 2 (137\times(2m_{e}c^{2}))\simeq m_{\pi}c^{2}
  4. 2 127 , 2^{127},
  5. / m p c . \hbar/m_{p}c.

Hadamard's_maximal_determinant_problem.html

  1. [ 1 1 1 1 1 - 1 - 1 1 1 1 - 1 - 1 1 - 1 1 - 1 ] [ 1 1 1 1 0 - 2 - 2 0 0 0 - 2 - 2 0 - 2 0 - 2 ] [ - 2 - 2 0 0 - 2 - 2 - 2 0 - 2 ] [ 1 1 0 0 1 1 1 0 1 ] \begin{bmatrix}1&1&1&1\\ 1&-1&-1&1\\ 1&1&-1&-1\\ 1&-1&1&-1\end{bmatrix}\rightarrow\left[\begin{array}[]{c|ccc}1&1&1&1\\ \hline 0&-2&-2&0\\ 0&0&-2&-2\\ 0&-2&0&-2\end{array}\right]\rightarrow\begin{bmatrix}-2&-2&0\\ 0&-2&-2\\ -2&0&-2\end{bmatrix}\rightarrow\begin{bmatrix}1&1&0\\ 0&1&1\\ 1&0&1\end{bmatrix}
  2. G = [ A B B T D ] , G=\begin{bmatrix}A&B\\ B^{\mathrm{T}}&D\end{bmatrix},
  3. R = [ W X Y Z ] R=\begin{bmatrix}W&X\\ Y&Z\end{bmatrix}
  4. r = n / s . r=\lfloor n/s\rfloor.
  5. det R ( n - 3 ) ( n - s ) / 2 ( n - 3 + 4 r ) u / 2 ( n + 1 + 4 r ) v / 2 [ 1 - u r n - 3 + 4 r - v ( r + 1 ) n + 1 + 4 r ] 1 / 2 , \det R\leq(n-3)^{(n-s)/2}(n-3+4r)^{u/2}(n+1+4r)^{v/2}\left[1-\frac{ur}{n-3+4r}% -\frac{v(r+1)}{n+1+4r}\right]^{1/2},

Hadamard_product_(matrices).html

  1. A , B A,B
  2. m × n m\times n
  3. A B A\circ B
  4. ( A B ) i , j = ( A ) i , j ( B ) i , j (A\circ B)_{i,j}=(A)_{i,j}\cdot(B)_{i,j}
  5. m × n m\times n
  6. p × q p\times q
  7. m p m\not=p
  8. n q n\not=q
  9. ( a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ) ( b 11 b 12 b 13 b 21 b 22 b 23 b 31 b 32 b 33 ) = ( a 11 b 11 a 12 b 12 a 13 b 13 a 21 b 21 a 22 b 22 a 23 b 23 a 31 b 31 a 32 b 32 a 33 b 33 ) \left(\begin{array}[]{ccc}\mathrm{a}_{11}&\mathrm{a}_{12}&\mathrm{a}_{13}\\ \mathrm{a}_{21}&\mathrm{a}_{22}&\mathrm{a}_{23}\\ \mathrm{a}_{31}&\mathrm{a}_{32}&\mathrm{a}_{33}\end{array}\right)\circ\left(% \begin{array}[]{ccc}\mathrm{b}_{11}&\mathrm{b}_{12}&\mathrm{b}_{13}\\ \mathrm{b}_{21}&\mathrm{b}_{22}&\mathrm{b}_{23}\\ \mathrm{b}_{31}&\mathrm{b}_{32}&\mathrm{b}_{33}\end{array}\right)=\left(\begin% {array}[]{ccc}\mathrm{a}_{11}\,\mathrm{b}_{11}&\mathrm{a}_{12}\,\mathrm{b}_{12% }&\mathrm{a}_{13}\,\mathrm{b}_{13}\\ \mathrm{a}_{21}\,\mathrm{b}_{21}&\mathrm{a}_{22}\,\mathrm{b}_{22}&\mathrm{a}_{% 23}\,\mathrm{b}_{23}\\ \mathrm{a}_{31}\,\mathrm{b}_{31}&\mathrm{a}_{32}\,\mathrm{b}_{32}&\mathrm{a}_{% 33}\,\mathrm{b}_{33}\end{array}\right)
  10. A B = B A , A\circ B=B\circ A,
  11. A ( B C ) = ( A B ) C , A\circ(B\circ C)=(A\circ B)\circ C,
  12. A ( B + C ) = A B + A C . A\circ(B+C)=A\circ B+A\circ C.
  13. x x
  14. y y
  15. D x D_{x}
  16. D y D_{y}
  17. x * ( A B ) y = tr ( D x * A D y B T ) x^{*}(A\circ B)y=\mathrm{tr}(D_{x}^{*}AD_{y}B^{T})
  18. x * x^{*}
  19. x x
  20. A B T AB^{T}
  21. A A
  22. B B
  23. A B T : AB^{T}:
  24. j ( A B ) i , j = ( A B T ) i , i . \sum_{j}(A\circ B)_{i,j}=(AB^{T})_{i,i}.
  25. rank ( A B ) rank ( A ) rank ( B ) \operatorname{rank}(A\circ B)\leq\operatorname{rank}(A)\operatorname{rank}(B)
  26. det ( A B ) det ( A ) det ( B ) . \det(A\circ B)\geq\det(A)\det(B).\,
  27. B = A 2 {B}={A}^{\circ 2}
  28. B i j = A i j 2 {B}_{ij}={A}^{2}_{ij}
  29. B = A 1 2 {B}={A}^{\circ\frac{1}{2}}
  30. B i j = A i j 1 2 {B}_{ij}={A}^{\frac{1}{2}}_{ij}

Hadamard_three-lines_theorem.html

  1. { x + i y : a x b } , \{x+iy:a\leq x\leq b\},
  2. M ( x ) = sup y | f ( x + i y ) | , M(x)=\sup_{y}|f(x+iy)|,\,
  3. x = t a + ( 1 - t ) b x=ta+(1-t)b
  4. 0 t 1 0\leq t\leq 1
  5. M ( x ) M ( a ) t M ( b ) 1 - t . M(x)\leq M(a)^{t}M(b)^{1-t}.\,
  6. F ( z ) F(z)
  7. F ( z ) = f ( z ) M ( a ) < m t p l > z - b b - a M ( b ) z - a a - b . F(z)=f(z)M(a)^{<}mtpl>{{z-b\over b-a}}M(b)^{{z-a\over a-b}}.
  8. F n ( z ) = F ( z ) e z 2 / n e - 1 / n F_{n}(z)=F(z)e^{z^{2}/n}e^{-1/n}
  9. g ( z ) g(z)
  10. { z : r | z | R } \{z:r\leq|z|\leq R\}
  11. f ( z ) = g ( e z ) , f(z)=g(e^{z}),\,
  12. m ( s ) = sup | z | = e s | g ( z ) | , m(s)=\sup_{|z|=e^{s}}|g(z)|,\,
  13. log m ( s ) \log\,m(s)
  14. | g h | ( | g | p ) 1 p ( | h | q ) 1 q , \int|gh|\leq\left(\int|g|^{p}\right)^{1\over p}\cdot\left(\int|h|^{q}\right)^{% 1\over q},
  15. 1 p + 1 q = 1 {1\over p}+{1\over q}=1
  16. f ( z ) = | g | p z | h | q ( 1 - z ) . f(z)=\int|g|^{pz}|h|^{q(1-z)}.

Haem_peroxidase.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons

Hahn–Exton_q-Bessel_function.html

  1. J ν ( 3 ) ( x ; q ) = x ν ( q ν + 1 ; q ) ( q ; q ) k 0 ( - 1 ) k q k ( k + 1 ) / 2 x 2 k ( q ν + 1 ; q ) k ( q ; q ) k J_{\nu}^{(3)}(x;q)=\frac{x^{\nu}(q^{\nu+1};q)_{\infty}}{(q;q)_{\infty}}\sum_{k% \geq 0}\frac{(-1)^{k}q^{k(k+1)/2}x^{2k}}{(q^{\nu+1};q)_{k}(q;q)_{k}}

Hall's_identity.html

  1. [ x , y - 1 , z ] y [ y , z - 1 , x ] z [ z , x - 1 , y ] x = 1 [x,y^{-1},z]^{y}\cdot[y,z^{-1},x]^{z}\cdot[z,x^{-1},y]^{x}=1

Hallade_method.html

  1. v e r s i n ( θ ) = 1 - c o s ( θ ) versin(\theta)=1-cos(\theta)
  2. v L 2 8 r v\approx\frac{L^{2}}{8r}
  3. O C = O A 2 - A C 2 OC=\sqrt{OA^{2}-AC^{2}}
  4. r - v = r 2 - ( L 2 ) 2 , r-v=\sqrt{r^{2}-(\frac{L}{2})^{2}},
  5. r 2 - 2 r v + v 2 = r 2 - ( L 2 ) 2 , r^{2}-2rv+v^{2}=r^{2}-(\frac{L}{2})^{2},
  6. 2 r - v = L 2 4 v , 2r-v=\frac{L^{2}}{4v},
  7. r = L 2 8 v + v 2 . r=\frac{L^{2}}{8v}+\frac{v}{2}.
  8. r L 2 8 v r\approx\frac{L^{2}}{8v}
  9. v L 2 8 r v\approx\frac{L^{2}}{8r}
  10. r L 2 8 v r\approx\frac{L^{2}}{8v}

Hamiltonian_optics.html

  1. ( q 1 ( σ ) , , q N ( σ ) ) \left(q_{1}\left(\sigma\right),\cdots,q_{N}\left(\sigma\right)\right)
  2. N N
  3. δ S = δ σ A σ B L ( q 1 , , q N , q ˙ 1 , , q ˙ N , σ ) d σ = 0 \delta S=\delta\int_{\sigma_{A}}^{\sigma_{B}}L\left(q_{1},\cdots,q_{N},\dot{q}% _{1},\cdots,\dot{q}_{N},\sigma\right)\,d\sigma=0
  4. q ˙ k = d q k / d σ \dot{q}_{k}=dq_{k}/d\sigma
  5. δ S = 0 \delta S=0
  6. L q k - d d σ L q ˙ k = 0 \frac{\partial L}{\partial q_{k}}-\frac{d}{d\sigma}\frac{\partial L}{\partial% \dot{q}_{k}}=0
  7. k = 1 , , N k=1,\cdots,N
  8. p k = L q ˙ k p_{k}=\frac{\partial L}{\partial\dot{q}_{k}}
  9. p ˙ k = L q k \dot{p}_{k}=\frac{\partial L}{\partial q_{k}}
  10. p ˙ k = d p k / d σ \dot{p}_{k}=dp_{k}/d\sigma
  11. H = k q ˙ k p k - L H=\sum_{k}{\dot{q}_{k}}p_{k}-L
  12. q i q_{i}\,
  13. q ˙ i \dot{q}_{i}
  14. H q k = - p ˙ k , H p k = q ˙ k , H σ = - L σ . \frac{\partial H}{\partial q_{k}}=-\dot{p}_{k}\,,\quad\frac{\partial H}{% \partial p_{k}}=\dot{q}_{k}\,,\quad\frac{\partial H}{\partial\sigma}=-{% \partial L\over\partial\sigma}\,.
  15. k = 1 , , N k=1,\cdots,N
  16. n = n ( x 1 , x 2 , x 3 ) n=n\left(x_{1},x_{2},x_{3}\right)
  17. s = ( x 1 ( x 3 ) , x 2 ( x 3 ) , x 3 ) s=\left(x_{1}\left(x_{3}\right),x_{2}\left(x_{3}\right),x_{3}\right)
  18. 𝐀 = ( x 1 ( x 3 A ) , x 2 ( x 3 A ) , x 3 A ) \mathbf{A}=\left(x_{1}\left(x_{3A}\right),x_{2}\left(x_{3A}\right),x_{3A}\right)
  19. 𝐁 = ( x 1 ( x 3 B ) , x 2 ( x 3 B ) , x 3 B ) \mathbf{B}=\left(x_{1}\left(x_{3B}\right),x_{2}\left(x_{3B}\right),x_{3B}\right)
  20. x 1 x_{1}
  21. x 2 x_{2}
  22. q k q_{k}
  23. x 3 x_{3}
  24. σ \sigma
  25. δ S = δ 𝐀 𝐁 n d s = δ x 3 A x 3 B n d s d x 3 d x 3 = δ x 3 A x 3 B L ( x 1 , x 2 , x ˙ 1 , x ˙ 2 , x 3 ) d x 3 = 0 \delta S=\delta\int_{\mathbf{A}}^{\mathbf{B}}n\,ds=\delta\int_{x_{3A}}^{x_{3B}% }n\frac{ds}{dx_{3}}\,dx_{3}=\delta\int_{x_{3A}}^{x_{3B}}L\left(x_{1},x_{2},% \dot{x}_{1},\dot{x}_{2},x_{3}\right)\,dx_{3}=0
  26. d s = d x 1 2 + d x 2 2 + d x 3 2 ds=\sqrt{dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}}
  27. L = n d s d x 3 = n ( x 1 , x 2 , x 3 ) 1 + x ˙ 1 2 + x ˙ 2 2 L=n\frac{ds}{dx_{3}}=n\left(x_{1},x_{2},x_{3}\right)\sqrt{1+\dot{x}_{1}^{2}+% \dot{x}_{2}^{2}}
  28. x ˙ k = d x k / d x 3 \dot{x}_{k}=dx_{k}/dx_{3}
  29. S = 𝐀 𝐁 n d s = 𝐀 𝐁 L d x 3 S=\int_{\mathbf{A}}^{\mathbf{B}}n\,ds=\int_{\mathbf{A}}^{\mathbf{B}}L\,dx_{3}
  30. L x k - d d x 3 L x ˙ k = 0 \frac{\partial L}{\partial x_{k}}-\frac{d}{dx_{3}}\frac{\partial L}{\partial% \dot{x}_{k}}=0
  31. x ˙ k = d x k / d x 3 \dot{x}_{k}=dx_{k}/dx_{3}
  32. p k = L x ˙ k p_{k}=\frac{\partial L}{\partial\dot{x}_{k}}
  33. L = n 1 + x ˙ 1 2 + x ˙ 2 2 L=n\sqrt{1+\dot{x}_{1}^{2}+\dot{x}_{2}^{2}}
  34. p k = n x ˙ k x ˙ 1 2 + x ˙ 2 2 + x ˙ 3 2 = n d x k d x 1 2 + d x 2 2 + d x 3 2 = n d x k d s p_{k}=n\frac{\dot{x}_{k}}{\sqrt{\dot{x}_{1}^{2}+\dot{x}_{2}^{2}+\dot{x}_{3}^{2% }}}=n\frac{dx_{k}}{\sqrt{dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}}}=n\frac{dx_{k}}{ds}
  35. 𝐩 = n 𝐝𝐬 d s = ( p 1 , p 2 , p 3 ) \mathbf{p}=n\frac{\mathbf{ds}}{ds}=\left(p_{1},p_{2},p_{3}\right)
  36. = ( n cos α 1 , n cos α 2 , n cos α 3 ) = n 𝐞 ^ =\left(n\cos\alpha_{1},n\cos\alpha_{2},n\cos\alpha_{3}\right)=n\mathbf{\hat{e}}
  37. 𝐞 ^ \mathbf{\hat{e}}
  38. 𝐩 = p 1 2 + p 2 2 + p 3 2 = n \|\mathbf{p}\|=\sqrt{p_{1}^{2}+p_{2}^{2}+p_{3}^{2}}=n
  39. x ˙ 3 = d x 3 / d x 3 = 1 \dot{x}_{3}=dx_{3}/dx_{3}=1
  40. L = n x ˙ 1 2 + x ˙ 2 2 + x ˙ 3 2 = x ˙ 1 n x ˙ 1 x ˙ 1 2 + x ˙ 2 2 + x ˙ 3 2 + x ˙ 2 n x ˙ 2 x ˙ 1 2 + x ˙ 2 2 + x ˙ 3 2 + n x ˙ 3 x ˙ 1 2 + x ˙ 2 2 + x ˙ 3 2 L=n\sqrt{\dot{x}_{1}^{2}+\dot{x}_{2}^{2}+\dot{x}_{3}^{2}}=\dot{x}_{1}\frac{n% \dot{x}_{1}}{\sqrt{\dot{x}_{1}^{2}+\dot{x}_{2}^{2}+\dot{x}_{3}^{2}}}+\dot{x}_{% 2}\frac{n\dot{x}_{2}}{\sqrt{\dot{x}_{1}^{2}+\dot{x}_{2}^{2}+\dot{x}_{3}^{2}}}+% \frac{n\dot{x}_{3}}{\sqrt{\dot{x}_{1}^{2}+\dot{x}_{2}^{2}+\dot{x}_{3}^{2}}}
  41. = x ˙ 1 p 1 + x ˙ 2 p 2 + x ˙ 3 p 3 = x ˙ 1 p 1 + x ˙ 2 p 2 + p 3 =\dot{x}_{1}p_{1}+\dot{x}_{2}p_{2}+\dot{x}_{3}p_{3}=\dot{x}_{1}p_{1}+\dot{x}_{% 2}p_{2}+p_{3}
  42. S = L d x 3 = 𝐩 d 𝐬 S=\int L\,dx_{3}=\int\mathbf{p}\cdot d\mathbf{s}
  43. x 1 ( x 3 ) x_{1}\left(x_{3}\right)
  44. x 2 ( x 3 ) x_{2}\left(x_{3}\right)
  45. H = x ˙ 1 p 1 + x ˙ 2 p 2 - L H=\dot{x}_{1}p_{1}+\dot{x}_{2}p_{2}-L
  46. L = x ˙ 1 p 1 + x ˙ 2 p 2 + p 3 L=\dot{x}_{1}p_{1}+\dot{x}_{2}p_{2}+p_{3}
  47. H = - p 3 = - n 2 - p 1 2 - p 2 2 H=-p_{3}=-\sqrt{n^{2}-p_{1}^{2}-p_{2}^{2}}
  48. H x k = - p ˙ k , H p k = x ˙ k \frac{\partial H}{\partial x_{k}}=-\dot{p}_{k}\,,\quad\frac{\partial H}{% \partial p_{k}}=\dot{x}_{k}
  49. x ˙ k = d x k / d x 3 \dot{x}_{k}=dx_{k}/dx_{3}
  50. p ˙ k = d p k / d x 3 \dot{p}_{k}=dp_{k}/dx_{3}
  51. x 1 x_{1}
  52. x 2 x_{2}
  53. q k q_{k}
  54. x 3 x_{3}
  55. σ \sigma
  56. n ( x 3 ) = { n A if x 3 < 0 n B if x 3 > 0 n(x_{3})=\begin{cases}n_{A}&\mbox{if }~{}x_{3}<0\\ n_{B}&\mbox{if }~{}x_{3}>0\\ \end{cases}
  57. H x k = - x k n ( x 3 ) 2 - p 1 2 - p 2 2 = 0 \frac{\partial H}{\partial x_{k}}=-\frac{\partial}{\partial x_{k}}\sqrt{n(x_{3% })^{2}-p_{1}^{2}-p_{2}^{2}}=0
  58. p ˙ k = 0 \dot{p}_{k}=0
  59. p k = Constant p_{k}=\,\text{Constant}
  60. d = 𝐩 A sin θ A = 𝐩 B sin θ B d=\|\mathbf{p}_{A}\|\sin\theta_{A}=\|\mathbf{p}_{B}\|\sin\theta_{B}
  61. 𝐩 A = n A \|\mathbf{p}_{A}\|=n_{A}
  62. 𝐩 B = n B \|\mathbf{p}_{B}\|=n_{B}
  63. n A sin θ A = n B sin θ B n_{A}\sin\theta_{A}=n_{B}\sin\theta_{B}
  64. 𝐯 = 𝐩 A - 𝐩 B \mathbf{v}=\mathbf{p}_{A}-\mathbf{p}_{B}
  65. 𝐧 = 𝐯 / 𝐯 \mathbf{n}=\mathbf{v}/\|\mathbf{v}\|
  66. 𝐧 = 𝐩 A - 𝐩 B 𝐩 A - 𝐩 B = n A 𝐢 - n B 𝐫 n A 𝐢 - n B 𝐫 \mathbf{n}=\frac{\mathbf{p}_{A}-\mathbf{p}_{B}}{\|\mathbf{p}_{A}-\mathbf{p}_{B% }\|}=\frac{n_{A}\mathbf{i}-n_{B}\mathbf{r}}{\|n_{A}\mathbf{i}-n_{B}\mathbf{r}\|}
  67. 𝐩 B \mathbf{p}_{B}
  68. 𝐩 A \mathbf{p}_{A}
  69. 𝐧 \mathbf{n}
  70. 𝐧 = 𝐢 - 𝐫 𝐢 - 𝐫 \mathbf{n}=\frac{\mathbf{i}-\mathbf{r}}{\|\mathbf{i}-\mathbf{r}\|}
  71. 𝐫 = n A n B 𝐢 + ( - ( 𝐢 𝐧 ) n A n B + Δ ) 𝐧 \mathbf{r}=\frac{n_{A}}{n_{B}}\mathbf{i}+\left(-\left(\mathbf{i}\cdot\mathbf{n% }\right)\frac{n_{A}}{n_{B}}+\sqrt{\Delta}\right)\mathbf{n}
  72. Δ = 1 - ( n A n B ) 2 ( 1 - ( 𝐢 𝐧 ) 2 ) \Delta=1-\left(\frac{n_{A}}{n_{B}}\right)^{2}\left(1-\left(\mathbf{i}\cdot% \mathbf{n}\right)^{2}\right)
  73. 𝐫 = 𝐢 - 2 ( 𝐢 𝐧 ) 𝐧 \mathbf{r}=\mathbf{i}-2\left(\mathbf{i}\cdot\mathbf{n}\right)\mathbf{n}
  74. S = L d x 3 S=\int L\,dx_{3}
  75. S x k = L x k d x 3 = d p k d x 3 d x 3 = p k \frac{\partial S}{\partial x_{k}}=\int\frac{\partial L}{\partial x_{k}}\,dx_{3% }=\int\frac{dp_{k}}{dx_{3}}\,dx_{3}=p_{k}
  76. L / x k = d p k / d x 3 \partial L/\partial x_{k}=dp_{k}/dx_{3}
  77. H / x 3 = - L / x 3 \partial H/\partial x_{3}=-\partial L/\partial x_{3}
  78. H = - p 3 H=-p_{3}
  79. S x 3 = L x 3 d x 3 = d p 3 d x 3 d x 3 = p 3 \frac{\partial S}{\partial x_{3}}=\int\frac{\partial L}{\partial x_{3}}\,dx_{3% }=\int\frac{dp_{3}}{dx_{3}}\,dx_{3}=p_{3}
  80. 𝐩 = S \mathbf{p}=\nabla S
  81. 𝐩 = S \mathbf{p}=\nabla S
  82. S = 𝐀 𝐁 𝐩 d 𝐬 = 𝐀 𝐁 S d 𝐬 = S ( 𝐁 ) - S ( 𝐀 ) S=\int_{\mathbf{A}}^{\mathbf{B}}\mathbf{p}\cdot d\mathbf{s}=\int_{\mathbf{A}}^% {\mathbf{B}}\nabla S\cdot d\mathbf{s}=S(\mathbf{B})-S(\mathbf{A})
  83. S = S d 𝐬 = 0 S=\oint\nabla S\cdot d\mathbf{s}=0
  84. S = 𝐀 𝐁 𝐩 d 𝐬 + 𝐁 𝐂 𝐩 d 𝐬 + 𝐂 𝐃 𝐩 d 𝐬 + 𝐃 𝐀 𝐩 d 𝐬 = 0 S=\int_{\mathbf{A}}^{\mathbf{B}}\mathbf{p}\cdot d\mathbf{s}+\int_{\mathbf{B}}^% {\mathbf{C}}\mathbf{p}\cdot d\mathbf{s}+\int_{\mathbf{C}}^{\mathbf{D}}\mathbf{% p}\cdot d\mathbf{s}+\int_{\mathbf{D}}^{\mathbf{A}}\mathbf{p}\cdot d\mathbf{s}=0
  85. 𝐩 d 𝐬 = 0 \mathbf{p}\cdot d\mathbf{s}=0
  86. 𝐩 d 𝐬 = n d s \mathbf{p}\cdot d\mathbf{s}=nds
  87. 𝐩 d 𝐬 = - n d s \mathbf{p}\cdot d\mathbf{s}=-n\,ds
  88. 𝐩 d 𝐬 = n d s \mathbf{p}\cdot d\mathbf{s}=n\,ds
  89. 𝐁 𝐂 n d s = 𝐀 𝐃 n d s \int_{\mathbf{B}}^{\mathbf{C}}n\,ds=\int_{\mathbf{A}}^{\mathbf{D}}n\,ds
  90. S 𝐁𝐂 = S 𝐀𝐃 S_{\mathbf{BC}}=S_{\mathbf{AD}}
  91. p 1 2 + p 3 2 = n 2 p_{1}^{2}+p_{3}^{2}=n^{2}
  92. 𝐩 = ( p 1 , p 3 ) \mathbf{p}=(p_{1},p_{3})
  93. 𝐩 = n \|\mathbf{p}\|=n
  94. 𝐩 = ( p 1 , p 2 , p 3 ) \mathbf{p}=(p_{1},p_{2},p_{3})
  95. p 1 2 + p 2 2 + p 3 2 = n 2 p_{1}^{2}+p_{2}^{2}+p_{3}^{2}=n^{2}
  96. d V = d A ( 𝐯 𝐧 ) d t dV=dA(\mathbf{v}\cdot\mathbf{n})dt
  97. d V d t = A 𝐯 𝐧 d A = V 𝐯 d V \frac{dV}{dt}=\int_{A}\mathbf{v}\cdot\mathbf{n}\,dA=\int_{V}\nabla\cdot\mathbf% {v}\,dV
  98. x 3 x_{3}
  99. ( x 1 , x 2 , p 1 , p 2 ) (x_{1},x_{2},p_{1},p_{2})
  100. 𝐯 = ( x ˙ 1 , x ˙ 2 , p ˙ 1 , p ˙ 2 ) \mathbf{v}=(\dot{x}_{1},\dot{x}_{2},\dot{p}_{1},\dot{p}_{2})
  101. x 3 x_{3}
  102. d x 1 dx_{1}
  103. x 1 x_{1}
  104. d x 2 dx_{2}
  105. x 2 x_{2}
  106. d p 1 dp_{1}
  107. p 1 p_{1}
  108. d p 2 dp_{2}
  109. p 2 p_{2}
  110. d V = d x 1 d x 2 d p 1 d p 2 dV=dx_{1}dx_{2}dp_{1}dp_{2}
  111. V V
  112. d V d x 3 = V 𝐯 d V \frac{dV}{dx_{3}}=\int_{V}\nabla\cdot\mathbf{v}\,dV
  113. 𝐯 = x ˙ 1 x 1 + x ˙ 2 x 2 + p ˙ 1 p 1 + p ˙ 2 p 2 = x 1 H p 1 + x 2 H p 2 - p 1 H x 1 - p 2 H x 2 = 0 \nabla\cdot\mathbf{v}=\frac{\partial\dot{x}_{1}}{\partial x_{1}}+\frac{% \partial\dot{x}_{2}}{\partial x_{2}}+\frac{\partial\dot{p}_{1}}{\partial p_{1}% }+\frac{\partial\dot{p}_{2}}{\partial p_{2}}=\frac{\partial}{\partial x_{1}}% \frac{\partial H}{\partial p_{1}}+\frac{\partial}{\partial x_{2}}\frac{% \partial H}{\partial p_{2}}-\frac{\partial}{\partial p_{1}}\frac{\partial H}{% \partial x_{1}}-\frac{\partial}{\partial p_{2}}\frac{\partial H}{\partial x_{2% }}=0
  114. d V / d x 3 = 0 dV/dx_{3}=0
  115. d V = d x 1 d x 2 d p 1 d p 2 = Constant dV=dx_{1}dx_{2}dp_{1}dp_{2}=\,\text{Constant}
  116. x 1 x_{1}
  117. x 2 x_{2}
  118. q k q_{k}
  119. x 3 x_{3}
  120. σ \sigma
  121. s = ( x 1 ( σ ) , x 2 ( σ ) , x 3 ( σ ) ) s=\left(x_{1}\left(\sigma\right),x_{2}\left(\sigma\right),x_{3}\left(\sigma% \right)\right)
  122. x 1 x_{1}
  123. x 2 x_{2}
  124. x 3 x_{3}
  125. q k q_{k}
  126. δ S = δ 𝐀 𝐁 n d s = δ σ A σ B n d s d σ d σ \delta S=\delta\int_{\mathbf{A}}^{\mathbf{B}}n\,ds=\delta\int_{\sigma_{A}}^{% \sigma_{B}}n\frac{ds}{d\sigma}\,d\sigma
  127. = δ σ A σ B L ( x 1 , x 2 , x 3 , x ˙ 1 , x ˙ 2 , x ˙ 3 , σ ) d σ = 0 =\delta\int_{\sigma_{A}}^{\sigma_{B}}L\left(x_{1},x_{2},x_{3},\dot{x}_{1},\dot% {x}_{2},\dot{x}_{3},\sigma\right)\,d\sigma=0
  128. L = n d s / d σ L=nds/d\sigma
  129. x ˙ k = d x k / d σ \dot{x}_{k}=dx_{k}/d\sigma
  130. L x k - d d σ L x ˙ k = 0 \frac{\partial L}{\partial x_{k}}-\frac{d}{d\sigma}\frac{\partial L}{\partial% \dot{x}_{k}}=0
  131. p k = L x ˙ k p_{k}=\frac{\partial L}{\partial\dot{x}_{k}}
  132. x 1 ( σ ) x_{1}\left(\sigma\right)
  133. x 2 ( σ ) x_{2}\left(\sigma\right)
  134. x 3 ( σ ) x_{3}\left(\sigma\right)
  135. P = x ˙ 1 p 1 + x ˙ 2 p 2 + x ˙ 3 p 3 - L P=\dot{x}_{1}p_{1}+\dot{x}_{2}p_{2}+\dot{x}_{3}p_{3}-L
  136. H x k = - p ˙ k , H p k = x ˙ k \frac{\partial H}{\partial x_{k}}=-\dot{p}_{k}\,,\quad\frac{\partial H}{% \partial p_{k}}=\dot{x}_{k}
  137. x ˙ k = d x k / d σ \dot{x}_{k}=dx_{k}/d\sigma
  138. p ˙ k = d p k / d σ \dot{p}_{k}=dp_{k}/d\sigma
  139. L = n d s d σ = n ( x 1 , x 2 , x 3 ) x ˙ 1 2 + x ˙ 2 2 + x ˙ 3 2 = L ( x 1 , x 2 , x 3 , x ˙ 1 , x ˙ 2 , x ˙ 3 ) L=n\frac{ds}{d\sigma}=n\left(x_{1},x_{2},x_{3}\right)\sqrt{\dot{x}_{1}^{2}+% \dot{x}_{2}^{2}+\dot{x}_{3}^{2}}=L\left(x_{1},x_{2},x_{3},\dot{x}_{1},\dot{x}_% {2},\dot{x}_{3}\right)
  140. p k = n x ˙ k x ˙ 1 2 + x ˙ 2 2 + x ˙ 3 2 = n d x k d x 1 2 + d x 2 2 + d x 3 2 = n d x k d s p_{k}=n\frac{\dot{x}_{k}}{\sqrt{\dot{x}_{1}^{2}+\dot{x}_{2}^{2}+\dot{x}_{3}^{2% }}}=n\frac{dx_{k}}{\sqrt{dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}}}=n\frac{dx_{k}}{ds}
  141. x ˙ k = d x k / d σ \dot{x}_{k}=dx_{k}/d\sigma
  142. L = n x ˙ 1 2 + x ˙ 2 2 + x ˙ 3 2 = x ˙ 1 n x ˙ 1 x ˙ 1 2 + x ˙ 2 2 + x ˙ 3 2 + x ˙ 2 n x ˙ 2 x ˙ 1 2 + x ˙ 2 2 + x ˙ 3 2 + x ˙ 3 n x ˙ 3 x ˙ 1 2 + x ˙ 2 2 + x ˙ 3 2 L=n\sqrt{\dot{x}_{1}^{2}+\dot{x}_{2}^{2}+\dot{x}_{3}^{2}}=\dot{x}_{1}\frac{n% \dot{x}_{1}}{\sqrt{\dot{x}_{1}^{2}+\dot{x}_{2}^{2}+\dot{x}_{3}^{2}}}+\dot{x}_{% 2}\frac{n\dot{x}_{2}}{\sqrt{\dot{x}_{1}^{2}+\dot{x}_{2}^{2}+\dot{x}_{3}^{2}}}+% \dot{x}_{3}\frac{n\dot{x}_{3}}{\sqrt{\dot{x}_{1}^{2}+\dot{x}_{2}^{2}+\dot{x}_{% 3}^{2}}}
  143. = x ˙ 1 p 1 + x ˙ 2 p 2 + x ˙ 3 p 3 =\dot{x}_{1}p_{1}+\dot{x}_{2}p_{2}+\dot{x}_{3}p_{3}
  144. P = 0 P=0
  145. p k p_{k}
  146. p 1 2 + p 2 2 + p 3 2 - n 2 ( x 1 , x 2 , x 3 ) = 0 p_{1}^{2}+p_{2}^{2}+p_{3}^{2}-n^{2}\left(x_{1},x_{2},x_{3}\right)=0
  147. P = p 1 2 + p 2 2 + p 3 2 - n 2 ( x 1 , x 2 , x 3 ) = 0 P=p_{1}^{2}+p_{2}^{2}+p_{3}^{2}-n^{2}\left(x_{1},x_{2},x_{3}\right)=0
  148. P = 0 P=0
  149. ( q 1 ( σ ) , q 2 ( σ ) , q 3 ( σ ) ) \left(q_{1}\left(\sigma\right),q_{2}\left(\sigma\right),q_{3}\left(\sigma% \right)\right)
  150. ( u 1 ( σ ) , u 2 ( σ ) , u 3 ( σ ) ) \left(u_{1}\left(\sigma\right),u_{2}\left(\sigma\right),u_{3}\left(\sigma% \right)\right)
  151. d q 1 d σ = P u 1 d u 1 d σ = - P q 1 \frac{dq_{1}}{d\sigma}=\frac{\partial P}{\partial u_{1}}\quad\quad\frac{du_{1}% }{d\sigma}=-\frac{\partial P}{\partial q_{1}}
  152. d q 2 d σ = P u 2 d u 2 d σ = - P q 2 \frac{dq_{2}}{d\sigma}=\frac{\partial P}{\partial u_{2}}\quad\quad\frac{du_{2}% }{d\sigma}=-\frac{\partial P}{\partial q_{2}}
  153. d q 3 d σ = P u 3 d u 3 d σ = - P q 3 \frac{dq_{3}}{d\sigma}=\frac{\partial P}{\partial u_{3}}\quad\quad\frac{du_{3}% }{d\sigma}=-\frac{\partial P}{\partial q_{3}}
  154. P = 𝐩 𝐩 - n 2 = 0 P=\mathbf{p}\cdot\mathbf{p}-n^{2}=0
  155. 𝐩 = u 1 q 1 + u 2 q 2 + u 3 q 3 \mathbf{p}=u_{1}\nabla q_{1}+u_{2}\nabla q_{2}+u_{3}\nabla q_{3}
  156. = u 1 q 1 q 1 q 1 + u 2 q 2 q 2 q 2 + u 3 q 3 q 3 q 3 =u_{1}\|\nabla q_{1}\|\frac{\nabla q_{1}}{\|\nabla q_{1}\|}+u_{2}\|\nabla q_{2% }\|\frac{\nabla q_{2}}{\|\nabla q_{2}\|}+u_{3}\|\nabla q_{3}\|\frac{\nabla q_{% 3}}{\|\nabla q_{3}\|}
  157. = u 1 a 1 𝐞 ^ 1 + u 2 a 2 𝐞 ^ 2 + u 3 a 3 𝐞 ^ 3 =u_{1}a_{1}\mathbf{\hat{e}}_{1}+u_{2}a_{2}\mathbf{\hat{e}}_{2}+u_{3}a_{3}% \mathbf{\hat{e}}_{3}
  158. 𝐞 ^ 1 \mathbf{\hat{e}}_{1}
  159. 𝐞 ^ 2 \mathbf{\hat{e}}_{2}
  160. 𝐞 ^ 3 \mathbf{\hat{e}}_{3}
  161. u k a k / n u_{k}a_{k}/n
  162. 𝐩 \mathbf{p}
  163. 𝐞 ^ k \mathbf{\hat{e}}_{k}

Hans_Georg_Feichtinger.html

  1. S 0 L 2 S 0 S_{0}\subset L^{2}\subset S_{0}^{\prime}

Hans_Volker_Klapdor-Kleingrothaus.html

  1. β \beta

Hardy_field.html

  1. lim x f ( x ) \lim_{x\rightarrow\infty}f(x)

Hardy–Littlewood_zeta-function_conjectures.html

  1. ζ ( 1 2 + i t ) \zeta\bigl(\tfrac{1}{2}+it\bigr)
  2. N ( T ) N(T)
  3. N 0 ( T ) N_{0}(T)
  4. ζ ( 1 2 + i t ) \zeta\bigl(\tfrac{1}{2}+it\bigr)
  5. ( 0 , T ] (0,T]
  6. ζ ( 1 2 + i t ) \zeta\bigl(\tfrac{1}{2}+it\bigr)
  7. ζ ( 1 2 + i t ) \zeta\bigl(\tfrac{1}{2}+it\bigr)
  8. ( T , T + H ] (T,T+H]
  9. T > 0 T>0
  10. H = T a + ε H=T^{a+\varepsilon}
  11. a > 0 a>0
  12. ε > 0 \varepsilon>0
  13. ε > 0 \varepsilon>0
  14. T 0 = T 0 ( ε ) > 0 T_{0}=T_{0}(\varepsilon)>0
  15. T T 0 T\geq T_{0}
  16. H = T 0.25 + ε H=T^{0.25+\varepsilon}
  17. ( T , T + H ] (T,T+H]
  18. ζ ( 1 2 + i t ) \zeta\bigl(\tfrac{1}{2}+it\bigr)
  19. ε > 0 \varepsilon>0
  20. T 0 = T 0 ( ε ) > 0 T_{0}=T_{0}(\varepsilon)>0
  21. c = c ( ε ) > 0 c=c(\varepsilon)>0
  22. T T 0 T\geq T_{0}
  23. H = T 0.5 + ε H=T^{0.5+\varepsilon}
  24. N 0 ( T + H ) - N 0 ( T ) c H N_{0}(T+H)-N_{0}(T)\geq cH
  25. ε > 0 \varepsilon>0
  26. T 0 = T 0 ( ε ) > 0 T_{0}=T_{0}(\varepsilon)>0
  27. c = c ( ε ) > 0 c=c(\varepsilon)>0
  28. T T 0 T\geq T_{0}
  29. H = T 0.5 + ε H=T^{0.5+\varepsilon}
  30. N ( T + H ) - N ( T ) c H log T N(T+H)-N(T)\geq cH\log T
  31. a = 0.5 a=0.5
  32. H = T 0.5 + ε H=T^{0.5+\varepsilon}

Harish-Chandra's_c-function.html

  1. 𝔞 + * \mathfrak{a}_{+}^{*}
  2. - 𝔞 + * -\mathfrak{a}_{+}^{*}
  3. c ( λ ) = c s 0 ( λ ) . c(\lambda)=c_{s_{0}}(\lambda).
  4. A ( s , λ ) ξ 0 = c s ( λ ) ξ 0 , \displaystyle A(s,\lambda)\xi_{0}=c_{s}(\lambda)\xi_{0},
  5. c s 1 s 2 ( λ ) = c s 1 ( s 2 λ ) c s 2 ( λ ) c_{s_{1}s_{2}}(\lambda)=c_{s_{1}}(s_{2}\lambda)c_{s_{2}}(\lambda)
  6. ( s 1 s 2 ) = ( s 1 ) + ( s 2 ) . \ell(s_{1}s_{2})=\ell(s_{1})+\ell(s_{2}).
  7. 𝔤 ± α \mathfrak{g}_{\pm\alpha}
  8. c s α ( λ ) = c 0 2 - i ( λ , α 0 ) Γ ( i ( λ , α 0 ) ) Γ ( 1 2 ( 1 2 m α + 1 + i ( λ , α 0 ) ) Γ ( 1 2 ( 1 2 m α + m 2 α + i ( λ , α 0 ) ) , c_{s_{\alpha}}(\lambda)=c_{0}{2^{-i(\lambda,\alpha_{0})}\Gamma(i(\lambda,% \alpha_{0}))\over\Gamma({1\over 2}({1\over 2}m_{\alpha}+1+i(\lambda,\alpha_{0}% ))\Gamma({1\over 2}({1\over 2}m_{\alpha}+m_{2\alpha}+i(\lambda,\alpha_{0}))},
  9. c 0 = 2 m α / 2 + m 2 α Γ ( 1 2 ( m α + m 2 α + 1 ) ) c_{0}=2^{m_{\alpha}/2+m_{2\alpha}}\Gamma\left({1\over 2}(m_{\alpha}+m_{2\alpha% }+1)\right)
  10. c ( λ ) = c 0 α Σ 0 + 2 - i ( λ , α 0 ) Γ ( i ( λ , α 0 ) ) Γ ( 1 2 ( 1 2 m α + 1 + i ( λ , α 0 ) ) Γ ( 1 2 ( 1 2 m α + m 2 α + i ( λ , α 0 ) ) , c(\lambda)=c_{0}\prod_{\alpha\in\Sigma_{0}^{+}}{2^{-i(\lambda,\alpha_{0})}% \Gamma(i(\lambda,\alpha_{0}))\over\Gamma({1\over 2}({1\over 2}m_{\alpha}+1+i(% \lambda,\alpha_{0}))\Gamma({1\over 2}({1\over 2}m_{\alpha}+m_{2\alpha}+i(% \lambda,\alpha_{0}))},

Harish-Chandra's_Schwartz_space.html

  1. σ ( x ) = X \sigma(x)=\|X\|
  2. ( 1 + σ ) r | D f | Ξ \frac{(1+\sigma)^{r}|Df|}{\Xi}

Harish-Chandra's_Ξ_function.html

  1. Ξ ( g ) = K a ( k g ) ρ d k , \Xi(g)=\int_{K}a(kg)^{\rho}dk,

Harish-Chandra_module.html

  1. ( 𝔤 , K ) (\mathfrak{g},K)
  2. ( π , V ) (\pi,V)
  3. π \pi
  4. φ v : G V \varphi_{v}:G\longrightarrow V
  5. φ v ( g ) = π ( g ) v \varphi_{v}(g)=\pi(g)v
  6. span { π ( k ) v : k K } \,\text{span}\{\pi(k)v:k\in K\}
  7. ( 𝔤 , K ) (\mathfrak{g},K)
  8. ( 𝔤 , K ) (\mathfrak{g},K)
  9. k v , w = v , k - 1 w \langle k\cdot v,w\rangle=\langle v,k^{-1}\cdot w\rangle
  10. Y v , w = - v , Y w \langle Y\cdot v,w\rangle=-\langle v,Y\cdot w\rangle
  11. Y 𝔤 Y\in\mathfrak{g}
  12. k K k\in K

Harish-Chandra_transform.html

  1. f P ( m ) = a - ρ N f ( n m ) d n f^{P}(m)=a^{-\rho}\int_{N}f(nm)\;dn

Hartogs–Rosenthal_theorem.html

  1. z z
  2. z ¯ \overline{z}
  3. z ¯ \overline{z}
  4. f ( z ) = g ( z ) z ¯ . f(z)=g(z)\cdot\overline{z}.
  5. f ( z ) = 1 2 π i < m t p l > C \ K f w ¯ d w d w ¯ w - z , f(z)=\frac{1}{2\pi i}\iint_{<}mtpl>{{C\backslash K}}\frac{\partial f}{\partial% \bar{w}}\frac{dw\wedge d\bar{w}}{w-z},
  6. z ¯ \bar{z}

Hartree_product.html

  1. Ψ ( 𝐱 1 , 𝐱 2 ) = χ 1 ( 𝐱 1 ) χ 2 ( 𝐱 2 ) . \Psi(\mathbf{x}_{1},\mathbf{x}_{2})=\chi_{1}(\mathbf{x}_{1})\chi_{2}(\mathbf{x% }_{2}).

Hautus_lemma.html

  1. 𝐀 M n ( ) \mathbf{A}\in M_{n}(\Re)
  2. 𝐁 M n × m ( ) \mathbf{B}\in M_{n\times m}(\Re)
  3. ( 𝐀 , 𝐁 ) (\mathbf{A},\mathbf{B})
  4. λ \lambda\in\mathbb{C}
  5. rank [ λ 𝐈 - 𝐀 , 𝐁 ] = n \operatorname{rank}[\lambda\mathbf{I}-\mathbf{A},\mathbf{B}]=n
  6. λ \lambda\in\mathbb{C}
  7. 𝐀 \mathbf{A}
  8. rank [ λ 𝐈 - 𝐀 , 𝐁 ] = n \operatorname{rank}[\lambda\mathbf{I}-\mathbf{A},\mathbf{B}]=n

Haynsworth_inertia_additivity_formula.html

  1. In ( H ) = ( π ( H ) , ν ( H ) , δ ( H ) ) \mathrm{In}(H)=\left(\pi(H),\nu(H),\delta(H)\right)\,
  2. H = [ H 11 H 12 H 12 H 22 ] H=\begin{bmatrix}H_{11}&H_{12}\\ H_{12}^{\ast}&H_{22}\end{bmatrix}
  3. In [ H 11 H 12 H 12 H 22 ] = In ( H 11 ) + In ( H / H 11 ) \mathrm{In}\begin{bmatrix}H_{11}&H_{12}\\ H_{12}^{\ast}&H_{22}\end{bmatrix}=\mathrm{In}(H_{11})+\mathrm{In}(H/H_{11})
  4. H / H 11 = H 22 - H 12 H 11 - 1 H 12 . H/H_{11}=H_{22}-H_{12}^{\ast}H_{11}^{-1}H_{12}.\,
  5. π ( H ) π ( H 11 ) + π ( H / H 11 ) \pi(H)\geq\pi(H_{11})+\pi(H/H_{11})
  6. ν ( H ) ν ( H 11 ) + ν ( H / H 11 ) \nu(H)\geq\nu(H_{11})+\nu(H/H_{11})

Heat_kernel_signature.html

  1. u 0 ( x ) u_{0}(x)
  2. h t ( x , y ) h_{t}(x,y)
  3. x x
  4. y y
  5. t t
  6. h t ( x , y ) h_{t}(x,y)
  7. h t ( x , x ) h_{t}(x,x)
  8. M M
  9. ( Δ + t ) u ( x , t ) = 0 \left(\Delta+\frac{\partial}{\partial t}\right)u(x,t)=0
  10. Δ \Delta
  11. u ( x , t ) u(x,t)
  12. x x
  13. t t
  14. u ( x , t ) = h t ( x , y ) u 0 ( y ) d y . u(x,t)=\int h_{t}(x,y)u_{0}(y)dy.
  15. h t ( x , y ) = i = 0 exp ( - λ i t ) ϕ i ( x ) ϕ i ( y ) h_{t}(x,y)=\sum_{i=0}^{\infty}\exp(-\lambda_{i}t)\phi_{i}(x)\phi_{i}(y)
  16. λ i \lambda_{i}
  17. ϕ i \phi_{i}
  18. i t h i^{th}
  19. Δ \Delta
  20. T : M N T:M\rightarrow N
  21. M M
  22. N N
  23. h t ( x , y ) = h t ( T ( x ) , T ( y ) ) h_{t}(x,y)=h_{t}(T(x),T(y))
  24. T T
  25. h t ( x , x ) = i = 0 exp ( - λ i t ) ϕ i 2 ( x ) . h_{t}(x,x)=\sum_{i=0}^{\infty}\exp(-\lambda_{i}t)\phi_{i}^{2}(x).
  26. Δ \Delta
  27. M M
  28. N N
  29. exp ( - λ i t ) \exp(-\lambda_{i}t)
  30. λ i \lambda_{i}
  31. h t ( x , x ) h_{t}(x,x)
  32. { h t 1 ( x , x ) , , h t n ( x , x ) } \{h_{t_{1}}(x,x),\ldots,h_{t_{n}}(x,x)\}
  33. t 1 , , t n t_{1},\ldots,t_{n}
  34. Δ \Delta
  35. L = A - 1 W L=A^{-1}W
  36. A A
  37. A ( i , i ) A(i,i)
  38. i i
  39. W W
  40. L L
  41. L = Φ Λ Φ T A L=\Phi\Lambda\Phi^{T}A
  42. Λ \Lambda
  43. L L
  44. Φ \Phi
  45. K t = Φ exp ( - t Λ ) Φ T . K_{t}=\Phi\exp(-t\Lambda)\Phi^{T}.
  46. k t ( i , j ) k_{t}(i,j)
  47. i i
  48. j j
  49. t t
  50. n × n n\times n
  51. n n
  52. n n
  53. x x
  54. h t ( x , x ) h_{t}(x,x)
  55. s ( x ) s(x)
  56. h t ( x , x ) = 1 4 π t + s ( x ) 12 π + O ( t ) . h_{t}(x,x)=\frac{1}{4\pi t}+\frac{s(x)}{12\pi}+O(t).
  57. x x
  58. t t
  59. ( i Δ + t ) ψ ( x , t ) = 0 \left(i\Delta+\frac{\partial}{\partial t}\right)\psi(x,t)=0
  60. ψ ( x , t ) \psi(x,t)
  61. x x
  62. p ( x ) = i = 0 f 2 ( λ i ) ϕ i 2 ( x ) p(x)=\sum_{i=0}^{\infty}f^{2}(\lambda_{i})\phi_{i}^{2}(x)
  63. f f
  64. f i ( x ) f_{i}(x)
  65. { p f 1 ( x ) , , p f n ( x ) } \{p_{f_{1}}(x),\ldots,p_{f_{n}}(x)\}
  66. x x
  67. x x
  68. s \displaystyle s

Helicopter_Cube.html

  1. 7 ! 3 6 6 ! 4 2 \frac{7!\cdot 3^{6}\cdot 6!^{4}}{2}

Helly_space.html

  1. I I = i I I i I^{I}=\prod_{i\in I}I_{i}

Hemithioacetal.html

  1. \overrightarrow{\leftarrow}

Heptacontagon.html

  1. 6 / 7 {6}/{7}
  2. 1 / 7 {1}/{7}
  3. A = 35 2 t 2 cot π 70 A=\frac{35}{2}t^{2}\cot\frac{\pi}{70}
  4. r = 1 2 t cot π 70 r=\frac{1}{2}t\cot\frac{\pi}{70}
  5. R = 1 2 t csc π 70 R=\frac{1}{2}t\csc\frac{\pi}{70}

Herbert_Wagner_(physicist).html

  1. d 2 d\leq 2

Herman_ring.html

  1. ϕ : U { ζ : 0 < r < | ζ | < 1 } \phi:U\rightarrow\{\zeta:0<r<|\zeta|<1\}
  2. θ \theta
  3. ϕ f p ϕ - 1 ( ζ ) = e 2 π i θ ζ . \phi\circ f^{\circ p}\circ\phi^{-1}(\zeta)=e^{2\pi i\theta}\zeta.
  4. f ( z ) = e 2 π i τ z 2 ( z - 4 ) 1 - 4 z f(z)=\frac{e^{2\pi i\tau}z^{2}(z-4)}{1-4z}
  5. τ = 0.6151732 \tau=0.6151732\dots
  6. ( 5 - 1 ) / 2 (\sqrt{5}-1)/2
  7. g a , b , c ( z ) = z 2 ( z - a ) z - b + c , g_{a,b,c}(z)=\frac{z^{2}(z-a)}{z-b}+c,\,
  8. a \displaystyle a
  9. h ( z ) = z 2 - 1 - e 5 π i 4 h(z)=z^{2}-1-\frac{e^{\sqrt{5}\pi i}}{4}
  10. a \displaystyle a

Hermite's_cotangent_identity.html

  1. A n , k = 1 j n j k cot ( a k - a j ) A_{n,k}=\prod_{\begin{smallmatrix}1\leq j\leq n\\ j\neq k\end{smallmatrix}}\cot(a_{k}-a_{j})
  2. cot ( z - a 1 ) cot ( z - a n ) = cos n π 2 + k = 1 n A n , k cot ( z - a k ) . \cot(z-a_{1})\cdots\cot(z-a_{n})=\cos\frac{n\pi}{2}+\sum_{k=1}^{n}A_{n,k}\cot(% z-a_{k}).
  3. cot ( z - a 1 ) cot ( z - a 2 ) = - 1 + cot ( a 1 - a 2 ) cot ( z - a 1 ) + cot ( a 2 - a 1 ) cot ( z - a 2 ) . \cot(z-a_{1})\cot(z-a_{2})=-1+\cot(a_{1}-a_{2})\cot(z-a_{1})+\cot(a_{2}-a_{1})% \cot(z-a_{2}).\,

Hermite's_problem.html

  1. x = a 0 . a 1 a 2 a 3 x=a_{0}.a_{1}a_{2}a_{3}\ldots
  2. x = n = 0 a n 10 n . x=\sum_{n=0}^{\infty}\frac{a_{n}}{10^{n}}.
  3. x = [ a 0 ; a 1 , a 2 , a 3 , ] , x=[a_{0};a_{1},a_{2},a_{3},\ldots],
  4. x = a 0 + 1 a 1 + 1 a 2 + 1 a 3 + . x=a_{0}+\cfrac{1}{a_{1}+\cfrac{1}{a_{2}+\cfrac{1}{a_{3}+\ddots}}}.

Hernandaline.html

  1. [ α ] D 20 = + 36.5 [\alpha]_{D}^{20}=+36.5

Heterogeneous_Earliest_Finish_Time.html

  1. n i n_{i}
  2. r a n k u ( n i ) = w i ¯ + max n j s u c c ( n i ) ( c i , j ¯ + r a n k u ( n j ) ) rank_{u}(n_{i})=\overline{w_{i}}+\max_{n_{j}\in succ(n_{i})}(\overline{c_{i,j}% }+rank_{u}(n_{j}))
  3. n i n_{i}
  4. i t h i^{th}
  5. w i ¯ \overline{w_{i}}
  6. s u c c ( n i ) succ(n_{i})
  7. n i n_{i}
  8. c i , j ¯ \overline{c_{i,j}}
  9. n i n_{i}
  10. n j n_{j}
  11. r a n k u ( n i ) rank_{u}(n_{i})
  12. w i ¯ \overline{w_{i}}

Heterogeneous_random_walk_in_one_dimension.html

  1. t t\rightarrow\infty
  2. τ \tau
  3. τ = e L \tau=e^{L}
  4. G i j ( t ; L ) G_{ij}(t;L)
  5. G i j ( t ; L ) G_{ij}(t;L)
  6. ψ i j ( t ) \psi_{ij}(t)
  7. ψ i j ( t ) \psi_{ij}(t)
  8. j 0 ψ i j ( t ) = 1. \sum_{j}\int_{0}^{\infty}\psi_{ij}(t)=1.
  9. ψ i I ( t ) \psi_{iI}(t)
  10. ψ i j ( t ) \psi_{ij}(t)
  11. G i j ( t ) G_{ij}(t)
  12. f ¯ ( s ) = 0 e - s t f ( t ) d t \bar{f}(s)=\int_{0}^{\infty}e^{-st}f(t)\,dt
  13. ψ i j ( t ) \psi_{ij}(t)
  14. G ¯ i j ( s ) = Γ ¯ i j ( s ) Φ ¯ ( s , L ~ ) Φ ¯ ( s , L ) Ψ ¯ i ( s ) . \bar{G}_{ij}(s)=\bar{\Gamma}_{ij}(s)\frac{\bar{\Phi}(s,\tilde{L})}{\bar{\Phi}(% s,L)}\bar{\Psi}_{i}(s).
  15. Ψ ¯ i ( s ) = j Ψ ¯ i j ( s ) \bar{\Psi}_{i}(s)=\sum_{j}\bar{\Psi}_{ij}(s)
  16. Ψ ¯ i j ( s ) = 1 - ψ ¯ i j ( s ) s . \bar{\Psi}_{ij}(s)=\frac{1-\bar{\psi}_{ij}(s)}{s}.
  17. Γ ¯ i j ( s ) = c = 0 i 1 ψ ¯ c ± 1 c ( s ) , \bar{\Gamma}_{ij}(s)=\prod_{c=0}^{i\mp 1}\bar{\psi}_{c\pm 1c}(s),
  18. Φ ¯ ( s , L ) = 1 + c = 1 [ L / 2 ] ( - 1 ) c h ¯ ( s , c ; L ) \bar{\Phi}(s,L)=1+\sum_{c=1}^{[L/2]}(-1)^{c}\bar{h}(s,c;L)
  19. h ¯ ( s , i ; L ) = c = 1 i k c = 2 + k c - 1 L - 1 - 2 ( i - c ) f ¯ k c ( s ) \bar{h}(s,i;L)=\prod_{c=1}^{i}\sum_{k_{c}=2+k_{c-1}}^{L-1-2(i-c)}\bar{f}_{k_{c% }}(s)
  20. f ¯ k j ( s ) = ψ ¯ k j k j + 1 ( s ) ψ ¯ k j + 1 k j ( s ) . \bar{f}_{k_{j}}(s)=\bar{\psi}_{k_{j}k_{j}+1}(s)\bar{\psi}_{k_{j}+1k_{j}}(s).
  21. Φ ¯ ( s ; L ) = 1 \bar{\Phi}(s;L)=1
  22. Φ ( s , L ~ ) \Phi(s,\tilde{L})
  23. Φ ¯ ( s ; L ) \bar{\Phi}(s;L)
  24. L ~ \tilde{L}
  25. L ~ \tilde{L}
  26. L ~ \tilde{L}
  27. Φ ¯ ( s ; L ~ ) = 1 \bar{\Phi}(s;\tilde{L})=1
  28. G ¯ i j ( s ; L ) \bar{G}_{ij}(s;L)
  29. G ¯ i j ( s ; L ) \bar{G}_{ij}(s;L)
  30. G ¯ i j ( s ; L ) \bar{G}_{ij}(s;L)
  31. G i j ( t ) G_{ij}(t)
  32. G i j ( t ; L ) = 0 Ψ i ( t - τ ) W i j ( τ ; L ) d τ , G_{ij}(t;L)=\int_{0}^{\infty}\Psi_{i}(t-\tau)W_{ij}(\tau;L)\,d\tau,
  33. W i j ( t ; L ) W_{ij}(t;L)
  34. W i j ( τ ; L ) = n = 0 w i j ( τ , 2 n + γ i j ; L ) . W_{ij}(\tau;L)=\sum_{n=0}^{\infty}w_{ij}(\tau,2n+\gamma_{ij};L).
  35. W i j ( t ; L ) W_{ij}(t;L)
  36. 2 n + γ i j 2n+\gamma_{ij}
  37. w i j ( τ , 2 n + γ i j ; L ) w_{ij}(\tau,2n+\gamma_{ij};L)
  38. 2 n + γ i j 2n+\gamma_{ij}
  39. W ¯ i j ( s ; L ) = W ¯ 1 L ( s ; L ) / W ¯ 1 L ~ ( s ; L ~ ) \bar{W}_{ij}(s;L)=\bar{W}_{1L}(s;L)/\bar{W}_{1\tilde{L}}(s;\tilde{L})
  40. w ¯ 1 L ( s ; L ) \bar{w}_{1L}(s;L)
  41. w i j ( τ , 2 n + γ i j ; L ) w_{ij}(\tau,2n+\gamma_{ij};L)
  42. h ¯ ( s , i ; L ) \bar{h}(s,i;L)
  43. w ¯ 1 L ( s , 2 n + γ 1 L ; L ) = δ n 0 Γ ¯ 1 L ( s ) + c = 1 [ L / 2 ] ( - 1 ) c + 1 w ¯ 1 L ( s , 2 ( n - c ) + γ 1 L ; L ) h ¯ ( s , c ; L ) . \bar{w}_{1L}(s,2n+\gamma_{1L};L)=\delta_{n0}\bar{\Gamma}_{1L}(s)+\sum_{c=1}^{[% L/2]}(-1)^{c+1}\bar{w}_{1L}(s,2(n-c)+\gamma_{1L};L)\bar{h}(s,c;L).
  44. w ¯ ^ 1 L ( s , 2 z + γ 1 L ; L ) = n = 0 w ¯ 1 L ( s , 2 n + γ 1 L ; L ) z n = Γ ¯ 1 L ( s ) [ 1 - c = 1 [ L / 2 ] ( - 1 ) c + 1 h ¯ ( s , c ; L ) z i ] - 1 . \hat{\bar{w}}_{1L}(s,2z+\gamma_{1L};L)=\sum_{n=0}^{\infty}\bar{w}_{1L}(s,2n+% \gamma_{1L};L)z^{n}=\bar{\Gamma}_{1L}(s)\big[1-\sum_{c=1}^{[L/2]}(-1)^{c+1}% \bar{h}(s,c;L)z^{i}\big]^{-1}.
  45. z = 1 z=1
  46. W ¯ 1 L ( s ; L ) \bar{W}_{1L}(s;L)
  47. w ¯ 1 L ( s , 2 n + γ 1 L ; L ) \bar{w}_{1L}(s,2n+\gamma_{1L};L)
  48. w ¯ 1 L ( s , 2 n + γ 1 L ; L ) = Γ ¯ 1 L ( s ) k 0 = a 0 , n n h ¯ ( 1 , s ; L ) k 0 c k 0 ( s ; L ) . \bar{w}_{1L}(s,2n+\gamma_{1L};L)=\bar{\Gamma}_{1L}(s)\sum_{k_{0}=a_{0,n}}^{n}% \bar{h}(1,s;L)^{k_{0}}c_{k_{0}}(s;L).
  49. c ¯ k 0 ( s ; L ) \bar{c}_{k_{0}}(s;L)
  50. L = 2 , 3 L=2,3
  51. c ¯ k 0 ( s ; L ) = c = 0 [ L / 2 ] - 1 k c = a c , n n - j = 0 c - 1 k j g ¯ k c ( s ; L ) , \bar{c}_{k_{0}}(s;L)=\prod_{c=0}^{[L/2]-1}\sum_{k_{c}=a_{c,n}}^{n-\sum_{j=0}^{% c-1}k_{j}}\bar{g}_{k_{c}}(s;L),
  52. g ¯ k i ( s ; L ) = ( k i - 1 k i ) ( - h ¯ ( s , i + 1 ; L ) h ¯ ( s , i ; L ) ) k i . \bar{g}_{k_{i}}(s;L)={\left({{k_{i-1}}\atop{k_{i}}}\right)}\left(-\frac{\bar{h% }(s,i+1;L)}{\bar{h}(s,i;L)}\right)^{k_{i}}.
  53. a i , n s a_{i,n}s
  54. a i , n = [ n - j = 0 i - 1 k j + [ L / 2 ] - 1 - i [ L / 2 ] - i ] ; i > 0 , a_{i,n}=\left[\frac{n-\sum_{j=0}^{i-1}k_{j}+[L/2]-1-i}{[L/2]-i}\right];i>0,
  55. a i , 0 = [ n + [ L / 2 ] - 1 [ L / 2 ] ] . a_{i,0}=\left[\frac{n+[L/2]-1}{[L/2]}\right].

Hierarchical_Dirichlet_process.html

  1. j = 1 , J j=1,...J
  2. x j 1 , x j n x_{j1},...x_{jn}
  3. H H
  4. j j
  5. G j G_{j}
  6. G j | G 0 \displaystyle G_{j}|G_{0}
  7. α j \alpha_{j}
  8. G 0 G_{0}
  9. G 0 \displaystyle G_{0}
  10. α 0 \alpha_{0}
  11. H H
  12. x j i x_{ji}
  13. θ j i \theta_{ji}
  14. θ j i | G j \displaystyle\theta_{ji}|G_{j}
  15. G j G_{j}
  16. F ( θ j i ) F(\theta_{ji})
  17. G 0 G_{0}
  18. G 0 \displaystyle G_{0}
  19. θ k * , k = 1 , 2 , \theta^{*}_{k},k=1,2,...
  20. H H
  21. π 0 k \pi_{0k}
  22. G 0 G_{0}
  23. G 0 G_{0}
  24. G j G_{j}
  25. G 0 G_{0}
  26. G j \displaystyle G_{j}
  27. x j i | G j \displaystyle x_{ji}|G_{j}
  28. θ k * \theta^{*}_{k}
  29. π j k \pi_{jk}

Hierarchical_matrix.html

  1. n n
  2. O ( n ) O(n)
  3. O ( n 2 ) O(n^{2})
  4. O ( n k log ( n ) ) O(nk\,\log(n))
  5. k k
  6. log ( 1 / ϵ ) γ \log(1/\epsilon)^{\gamma}
  7. γ \gamma
  8. ϵ \epsilon
  9. O ( n k α log ( n ) β ) O(nk^{\alpha}\,\log(n)^{\beta})
  10. α , β { 1 , 2 , 3 } . \alpha,\beta\in\{1,2,3\}.
  11. I , J I,J
  12. G I × J G\in{\mathbb{R}}^{I\times J}
  13. t I , s J t\subseteq I,s\subseteq J
  14. G | t × s G|_{t\times s}
  15. k k
  16. G | t × s A B * G|_{t\times s}\approx AB^{*}
  17. A t × k , B s × k A\in{\mathbb{R}}^{t\times k},B\in{\mathbb{R}}^{s\times k}
  18. G | t × s G|_{t\times s}
  19. O ( ( # t ) ( # s ) ) O((\#t)(\#s))
  20. O ( k ( # t + # s ) ) O(k(\#t+\#s))
  21. k k
  22. G G
  23. 𝒢 [ u ] ( x ) = Ω κ ( x , y ) u ( y ) d y . {\mathcal{G}}[u](x)=\int_{\Omega}\kappa(x,y)u(y)\,dy.
  24. g i j = Ω Ω κ ( x , y ) φ i ( x ) ψ j ( y ) d y d x , g_{ij}=\int_{\Omega}\int_{\Omega}\kappa(x,y)\varphi_{i}(x)\psi_{j}(y)\,dy\,dx,
  25. ( φ i ) i I (\varphi_{i})_{i\in I}
  26. ( ψ j ) j J (\psi_{j})_{j\in J}
  27. κ \kappa
  28. κ ~ ( x , y ) = ν = 1 k κ ( x , ξ ν ) ν ( y ) , \tilde{\kappa}(x,y)=\sum_{\nu=1}^{k}\kappa(x,\xi_{\nu})\ell_{\nu}(y),
  29. ( ξ ν ) ν = 1 k (\xi_{\nu})_{\nu=1}^{k}
  30. ( ν ) ν = 1 k (\ell_{\nu})_{\nu=1}^{k}
  31. κ \kappa
  32. κ ~ \tilde{\kappa}
  33. g ~ i j = Ω Ω κ ~ ( x , y ) φ i ( x ) ψ j ( y ) d y d x = ν = 1 k Ω κ ( x , ξ ν ) φ i ( x ) d x Ω ν ( y ) ψ j ( y ) d y = ν = 1 k a i ν b j ν \tilde{g}_{ij}=\int_{\Omega}\int_{\Omega}\tilde{\kappa}(x,y)\varphi_{i}(x)\psi% _{j}(y)\,dy\,dx=\sum_{\nu=1}^{k}\int_{\Omega}\kappa(x,\xi_{\nu})\varphi_{i}(x)% \,dx\int_{\Omega}\ell_{\nu}(y)\psi_{j}(y)\,dy=\sum_{\nu=1}^{k}a_{i\nu}b_{j\nu}
  34. a i ν = Ω κ ( x , ξ ν ) φ i ( x ) d x , a_{i\nu}=\int_{\Omega}\kappa(x,\xi_{\nu})\varphi_{i}(x)\,dx,
  35. b j ν = Ω ν ( y ) ψ j ( y ) d y . b_{j\nu}=\int_{\Omega}\ell_{\nu}(y)\psi_{j}(y)\,dy.
  36. t I , s J t\subseteq I,s\subseteq J
  37. i t , j s i\in t,j\in s
  38. G | t × s A B * G|_{t\times s}\approx AB^{*}
  39. x x
  40. y y
  41. G G
  42. Z = Z + α X Y Z=Z+\alpha XY
  43. X , Y , Z X,Y,Z
  44. α \alpha
  45. Z Z
  46. O ( n k 2 log ( n ) 2 ) O(nk^{2}\,\log(n)^{2})
  47. O ( n k 2 log ( n ) 2 ) O(nk^{2}\,\log(n)^{2})
  48. O ( n k ) O(nk)
  49. k k
  50. O ( n ) . O(n).
  51. 2 {\mathcal{H}}^{2}
  52. L L^{\infty}

High-refractive-index_polymer.html

  1. n c o m p = Φ p n p + Φ o r g n o r g {n_{comp}}={\Phi_{p}}{n_{p}}+{\Phi_{org}}{n_{org}}
  2. n c o m p {n_{comp}}
  3. n p {n_{p}}
  4. n o r g {n_{org}}
  5. Φ p {\Phi_{p}}
  6. Φ o r g {\Phi_{org}}

Higher-order_sinusoidal_input_describing_function.html

  1. u ( t ) = γ sin ( ω 0 t + φ 0 ) u(t)=\gamma\sin(\omega_{0}t+\varphi_{0})
  2. y ( t ) y(t)
  3. y ( t ) = k = 0 K | H k ( ω 0 , γ ) | γ k cos ( k ( ω 0 t + φ 0 ) + H k ( ω 0 , γ ) ) y(t)=\sum\limits_{k=0}^{K}|H_{k}(\omega_{0},\gamma)|\gamma^{k}\cos\big(k(% \omega_{0}t+\varphi_{0})+\angle H_{k}(\omega_{0},\gamma)\big)
  4. U ( ω ) U(\omega)
  5. Y ( ω ) Y(\omega)
  6. | U ( ω 0 ) | = γ |U(\omega_{0})|=\gamma
  7. H k ( ω 0 , γ ) = Y ( k ω 0 , γ ) U k ( ω 0 , γ ) H_{k}(\omega_{0},\gamma)=\frac{Y(k\omega_{0},\gamma)}{U^{k}(\omega_{0},\gamma)}

Highest-weight_category.html

  1. B ( α A α ) = α ( B A α ) B\cap\left(\bigcup_{\alpha}A_{\alpha}\right)=\bigcup_{\alpha}\left(B\cap A_{% \alpha}\right)
  2. dim k Hom k ( A ( λ ) , A ( μ ) ) \dim_{k}\operatorname{Hom}_{k}(A(\lambda),A(\mu))
  3. [ A ( λ ) : S ( μ ) ] [A(\lambda):S(\mu)]
  4. 0 = F 0 ( λ ) F 1 ( λ ) I ( λ ) 0=F_{0}(\lambda)\subseteq F_{1}(\lambda)\subseteq\dots\subseteq I(\lambda)
  5. F 1 ( λ ) = A ( λ ) F_{1}(\lambda)=A(\lambda)
  6. F n ( λ ) / F n - 1 ( λ ) A ( μ ) F_{n}(\lambda)/F_{n-1}(\lambda)\cong A(\mu)
  7. i F i ( λ ) = I ( λ ) . \bigcup_{i}F_{i}(\lambda)=I(\lambda).
  8. k k
  9. n × n n\times n
  10. k k
  11. k k
  12. A A

Highly_accelerated_stress_test.html

  1. A F H A S T = A F H * A F T = e ( C o n s t a n t * ( R H s n - R H o n ) * e ( E a / k ) * ( 1 / T o - 1 / T s ) AF_{HAST}=AF_{H}*AF_{T}=e^{(Constant*(RH_{s}^{n}-RH_{o}^{n})}*e^{(E_{a}/k)*(1/% T_{o}-1/T_{s})}

Higman_group.html

  1. a - 1 b a = b 2 , b - 1 c b = c 2 , c - 1 d c = d 2 , d - 1 a d = a 2 a^{-1}ba=b^{2},\quad b^{-1}cb=c^{2},\quad c^{-1}dc=d^{2},\quad d^{-1}ad=a^{2}

Hilbert's_inequality.html

  1. | r s u r u s ¯ r - s | π r | u r | 2 . \left|\sum_{r\neq s}\dfrac{u_{r}\overline{u_{s}}}{r-s}\right|\leq\pi% \displaystyle\sum_{r}|u_{r}|^{2}.
  2. π \pi
  3. π \pi
  4. m | u m | 2 < \sum_{m}|u_{m}|^{2}<\infty\,
  5. | r s u r u s ¯ r - s | π r | u r | 2 . \left|\sum_{r\neq s}\dfrac{u_{r}\overline{u_{s}}}{r-s}\right|\leq\pi% \displaystyle\sum_{r}|u_{r}|^{2}.
  6. r s u r u ¯ s csc π ( x r - x s ) \sum_{r\neq s}u_{r}\overline{u}_{s}\csc\pi(x_{r}-x_{s})
  7. r s u r u ¯ s λ r - λ s , \sum_{r\neq s}\dfrac{u_{r}\overline{u}_{s}}{\lambda_{r}-\lambda_{s}},
  8. | r s u r u s ¯ csc π ( x r - x s ) | δ - 1 r | u r | 2 . \left|\sum_{r\neq s}u_{r}\overline{u_{s}}\csc\pi(x_{r}-x_{s})\right|\leq\delta% ^{-1}\sum_{r}|u_{r}|^{2}.
  9. | r s u r u s ¯ λ r - λ s | π τ - 1 r | u r | 2 . \left|\sum_{r\neq s}\dfrac{u_{r}\overline{u_{s}}}{\lambda_{r}-\lambda_{s}}% \right|\leq\pi\tau^{-1}\sum_{r}|u_{r}|^{2}.
  10. δ = min r , s + x r - x s , τ = min r , s + λ r - λ s , \delta={\min_{r,s}}{}_{+}\|x_{r}-x_{s}\|,\quad\tau=\min_{r,s}{}_{+}\|\lambda_{% r}-\lambda_{s}\|,\,
  11. s = min m | s - m | \|s\|=\min_{m\in\mathbb{Z}}|s-m|
  12. 0 < δ r min s + x r - x s and 0 < τ r min s + λ r - λ s , 0<\delta_{r}\leq{\min_{s}}{}_{+}\|x_{r}-x_{s}\|\quad\,\text{and}\quad 0<\tau_{% r}\leq{\min_{s}}{}_{+}\|\lambda_{r}-\lambda_{s}\|,\,
  13. | r s u r u s ¯ csc π ( x r - x s ) | 3 2 r | u r | 2 δ r - 1 . \left|\sum_{r\neq s}u_{r}\overline{u_{s}}\csc\pi(x_{r}-x_{s})\right|\leq\dfrac% {3}{2}\sum_{r}|u_{r}|^{2}\delta_{r}^{-1}.
  14. | r s u r u s ¯ λ r - λ s | 3 2 π r | u r | 2 τ r - 1 . \left|\sum_{r\neq s}\dfrac{u_{r}\overline{u_{s}}}{\lambda_{r}-\lambda_{s}}% \right|\leq\dfrac{3}{2}\pi\sum_{r}|u_{r}|^{2}\tau_{r}^{-1}.

Hilbert–Burch_theorem.html

  1. 0 R m R n R R / I 0 0\rightarrow R^{m}\rightarrow R^{n}\rightarrow R\rightarrow R/I\rightarrow 0

Hille_equation.html

  1. g g
  2. l l
  3. a a
  4. ρ \rho
  5. 1 g = ( l + π a 2 ) × ρ π a 2 \frac{1}{g}=(l+\pi\frac{a}{2})\times{}\frac{\rho}{\pi{}a^{2}}
  6. l l
  7. d d
  8. 1 g = l ρ ( π ( d 2 ) 2 ) + ρ d \frac{1}{g}=\frac{l\rho}{(\pi{}(\frac{d}{2})^{2})}+\frac{\rho}{d}

Hindley–Milner_type_system.html

  1. e = x variable | e 1 e 2 application | λ x . e abstraction | 𝚕𝚎𝚝 x = e 1 𝚒𝚗 e 2 \begin{array}[]{lrll}e&=&x&\textrm{variable}\\ &|&e_{1}\ e_{2}&\textrm{application}\\ &|&\lambda\ x\ .\ e&\textrm{abstraction}\\ &|&\mathtt{let}\ x=e_{1}\ \mathtt{in}\ e_{2}\\ \end{array}
  2. mono τ = α variable | D τ τ application poly σ = τ | α . σ quantifier \begin{array}[]{llrll}\textrm{mono}&\tau&=&\alpha&\ \textrm{variable}\\ &&|&D\ \tau\dots\tau&\ \textrm{application}\\ \textrm{poly}&\sigma&=&\tau\\ &&|&\forall\ \alpha\ .\ \sigma&\ \textrm{quantifier}\\ \\ \end{array}
  3. e 1 e 2 e_{1}e_{2}
  4. e 1 e_{1}
  5. e 2 e_{2}
  6. e 1 ( e 2 ) e_{1}(e_{2})
  7. λ x . e \lambda\ x\ .\ e
  8. x x
  9. e e
  10. 𝚏𝚞𝚗𝚌𝚝𝚒𝚘𝚗 ( x ) 𝚛𝚎𝚝𝚞𝚛𝚗 e 𝚎𝚗𝚍 \mathtt{function}\,(x)\ \mathtt{return}\ e\ \mathtt{end}
  11. 𝚕𝚎𝚝 x = e 1 𝚒𝚗 e 2 \mathtt{let}\ x=e_{1}\ \mathtt{in}\ e_{2}
  12. x x
  13. e 2 e_{2}
  14. e 1 e_{1}
  15. τ \tau
  16. 𝚒𝚗𝚝 \mathtt{int}
  17. 𝚜𝚝𝚛𝚒𝚗𝚐 \mathtt{string}
  18. 𝙼𝚊𝚙 ( 𝚂𝚎𝚝 𝚜𝚝𝚛𝚒𝚗𝚐 ) 𝚒𝚗𝚝 \mathtt{Map\ (Set\ string)\ int}
  19. { 𝙼𝚊𝚙 𝟸 , 𝚂𝚎𝚝 𝟷 , 𝚜𝚝𝚛𝚒𝚗𝚐 𝟶 , 𝚒𝚗𝚝 𝟶 } \{\mathtt{Map^{2},\ Set^{1},\ string^{0},\ int^{0}}\}
  20. D D
  21. 2 \rightarrow^{2}
  22. 𝚒𝚗𝚝 𝚜𝚝𝚛𝚒𝚗𝚐 \mathtt{int}\rightarrow\mathtt{string}
  23. α \alpha
  24. 𝚒𝚗𝚝 \mathtt{int}
  25. β \beta
  26. α α \alpha\rightarrow\alpha
  27. α \alpha
  28. α \alpha
  29. α . α α \forall\alpha.\alpha\rightarrow\alpha
  30. α . α α \forall\alpha.\alpha\rightarrow\alpha
  31. α . ( 𝚂𝚎𝚝 α ) 𝚒𝚗𝚝 \forall\alpha.(\mathtt{Set}\ \alpha)\rightarrow\mathtt{int}
  32. α . α α . α \forall\alpha.\alpha\rightarrow\forall\alpha.\alpha
  33. α 1 α n . τ \forall\alpha_{1}\dots\forall\alpha_{n}.\tau
  34. free ( α ) = { α } free ( D τ 1 τ n ) = i = 1 n free ( τ i ) free ( α . σ ) = free ( σ ) - { α } \begin{array}[]{ll}\,\text{free}(\ \alpha\ )&=\ \left\{\alpha\right\}\\ \,\text{free}(\ D\ \tau_{1}\dots\tau_{n}\ )&=\ \bigcup\limits_{i=1}^{n}{\,% \text{free}(\ \tau_{i}\ )}\\ \,\text{free}(\ \forall\ \alpha\ .\ \sigma\ )&=\ \,\text{free}(\ \sigma\ )\ -% \ \left\{\alpha\right\}\\ \end{array}
  35. α 1 α n . τ \forall\alpha_{1}\dots\forall\alpha_{n}.\tau
  36. \forall
  37. α i \alpha_{i}
  38. τ \tau
  39. α i \alpha_{i}
  40. τ \tau
  41. τ \tau
  42. α . α α \forall\alpha.\alpha\rightarrow\alpha
  43. α α \alpha\rightarrow\alpha
  44. β . β α \forall\beta.\beta\rightarrow\alpha
  45. α \alpha
  46. 𝐥𝐞𝐭 𝑏𝑎𝑟 [ α . β . α ( β α ) ] = λ x . 𝐥𝐞𝐭 𝑓𝑜𝑜 [ β . β α ] = λ y . x 𝐢𝐧 𝑓𝑜𝑜 𝐢𝐧 𝑏𝑎𝑟 \begin{array}[]{l}\,\textbf{let}\ \mathit{bar}\ [\forall\alpha.\forall\beta.% \alpha\rightarrow(\beta\rightarrow\alpha)]=\lambda\ x.\\ \quad\,\textbf{let}\ \mathit{foo}\ [\forall\beta.\beta\rightarrow\alpha]=% \lambda\ y.x\\ \quad\,\textbf{in}\ \mathit{foo}\\ \,\textbf{in}\ \mathit{bar}\end{array}
  47. 𝑓𝑜𝑜 \mathit{foo}
  48. y y
  49. x x
  50. 𝑓𝑜𝑜 \mathit{foo}
  51. 𝑓𝑜𝑜 \mathit{foo}
  52. 𝑏𝑎𝑟 \mathit{bar}
  53. α . β . α ( β α ) \forall\alpha.\forall\beta.\alpha\rightarrow(\beta\rightarrow\alpha)
  54. 𝑏𝑎𝑟 1 \mathit{bar}\ 1
  55. β . β 𝑖𝑛𝑡 \forall\beta.\beta\rightarrow\ \mathit{int}
  56. α \alpha
  57. β . β α \forall\beta.\beta\rightarrow\alpha
  58. α \alpha
  59. α \alpha
  60. Context Γ = ϵ ( 𝚎𝚖𝚙𝚝𝚢 ) | Γ , x : σ Typing = Γ e : σ \begin{array}[]{llrl}\,\text{Context}&\Gamma&=&\epsilon\ \mathtt{(empty)}\\ &&|&\Gamma,\ x:\sigma\\ \,\text{Typing}&&=&\Gamma\vdash e:\sigma\\ \\ \end{array}
  61. free ( Γ ) = x : σ Γ free ( σ ) \begin{array}[]{ll}\,\text{free}(\ \Gamma\ )&=\ \bigcup\limits_{x:\sigma\in% \Gamma}\,\text{free}(\ \sigma\ )\end{array}
  62. x : σ x:\sigma
  63. x i x_{i}
  64. σ i \sigma_{i}
  65. Γ e : σ \Gamma\ \vdash\ e:\sigma
  66. Γ \Gamma
  67. e e
  68. σ \sigma
  69. 𝑓𝑜𝑜 \mathit{foo}
  70. x : α λ y . x : β . β α x:\alpha\vdash\lambda\ y.x:\forall\beta.\beta\rightarrow\alpha
  71. α \alpha
  72. x x
  73. free ( Γ ) \,\text{free}(\ \Gamma\ )
  74. σ σ \sigma\sqsubseteq\sigma^{\prime}
  75. σ \sigma^{\prime}
  76. σ \sigma
  77. α . α α \forall\alpha.\alpha\rightarrow\alpha
  78. i n t int
  79. i n t i n t int\rightarrow int
  80. α . α α i n t i n t \forall\alpha.\alpha\rightarrow\alpha\sqsubseteq\ int\rightarrow int
  81. α . τ \forall\alpha.\tau
  82. α \alpha
  83. τ \tau
  84. β . β \forall\beta.\beta
  85. β . β β . β \forall\beta.\beta\rightarrow\forall\beta.\beta
  86. s t r i n g S e t i n t string\rightarrow Set\ int
  87. β . β β . β \forall\beta.\beta\rightarrow\forall\beta.\beta
  88. β . β β \forall\beta.\beta\rightarrow\beta
  89. α . α \forall\alpha.\alpha
  90. α . α α β . β β α . α α \forall\alpha.\alpha\rightarrow\alpha\sqsubseteq\forall\beta.\beta\rightarrow% \beta\sqsubseteq\forall\alpha.\alpha\rightarrow\alpha
  91. τ = [ α i := τ i ] τ β i free ( α 1 α n . τ ) α 1 α n . τ β 1 β m . τ \displaystyle\frac{\tau^{\prime}=\left[\alpha_{i}:=\tau_{i}\right]\tau\quad% \beta_{i}\not\in\textrm{free}(\forall\alpha_{1}...\forall\alpha_{n}.\tau)}{% \forall\alpha_{1}...\forall\alpha_{n}.\tau\sqsubseteq\forall\beta_{1}...% \forall\beta_{m}.\tau^{\prime}}
  92. α 1 α n . τ \forall\alpha_{1}\dots\forall\alpha_{n}.\tau
  93. α i \alpha_{i}
  94. τ i \tau_{i}
  95. τ \tau^{\prime}
  96. τ \tau^{\prime}
  97. α . α \forall\alpha.\alpha
  98. \sqsubseteq
  99. Predicate = σ σ | α f r e e ( Γ ) | x : α Γ Judgment = Typing Premise = Judgment | Predicate Conclusion = Judgment Rule = Premise Conclusion [ 𝙽𝚊𝚖𝚎 ] \begin{array}[]{lrl}\,\text{Predicate}&=&\sigma\sqsubseteq\sigma^{\prime}\\ &|&\alpha\not\in free(\Gamma)\\ &|&x:\alpha\in\Gamma\\ \\ \,\text{Judgment}&=&\,\text{Typing}\\ \,\text{Premise}&=&\,\text{Judgment}\ |\ \,\text{Predicate}\\ \,\text{Conclusion}&=&\,\text{Judgment}\\ \\ \,\text{Rule}&=&\displaystyle\frac{\textrm{Premise}\ \dots}{\textrm{Conclusion% }}\quad[\mathtt{Name}]\end{array}
  100. [ 𝙽𝚊𝚖𝚎 ] [\mathtt{Name}]
  101. x : σ Γ Γ x : σ [ 𝚅𝚊𝚛 ] Γ e 0 : τ τ Γ e 1 : τ Γ e 0 e 1 : τ [ 𝙰𝚙𝚙 ] Γ , x : τ e : τ Γ λ x . e : τ τ [ 𝙰𝚋𝚜 ] Γ e 0 : σ Γ , x : σ e 1 : τ Γ 𝚕𝚎𝚝 x = e 0 𝚒𝚗 e 1 : τ [ 𝙻𝚎𝚝 ] Γ e : σ σ σ Γ e : σ [ 𝙸𝚗𝚜𝚝 ] Γ e : σ α free ( Γ ) Γ e : α . σ [ 𝙶𝚎𝚗 ] \begin{array}[]{cl}\displaystyle\frac{x:\sigma\in\Gamma}{\Gamma\vdash x:\sigma% }&[\mathtt{Var}]\\ \\ \displaystyle\frac{\Gamma\vdash e_{0}:\tau\rightarrow\tau^{\prime}\quad\quad% \Gamma\vdash e_{1}:\tau}{\Gamma\vdash e_{0}\ e_{1}:\tau^{\prime}}&[\mathtt{App% }]\\ \\ \displaystyle\frac{\Gamma,\;x:\tau\vdash e:\tau^{\prime}}{\Gamma\vdash\lambda% \ x\ .\ e:\tau\rightarrow\tau^{\prime}}&[\mathtt{Abs}]\\ \\ \displaystyle\frac{\Gamma\vdash e_{0}:\sigma\quad\quad\Gamma,\,x:\sigma\vdash e% _{1}:\tau}{\Gamma\vdash\mathtt{let}\ x=e_{0}\ \mathtt{in}\ e_{1}:\tau}&[% \mathtt{Let}]\\ \\ \\ \displaystyle\frac{\Gamma\vdash e:\sigma^{\prime}\quad\sigma^{\prime}% \sqsubseteq\sigma}{\Gamma\vdash e:\sigma}&[\mathtt{Inst}]\\ \\ \displaystyle\frac{\Gamma\vdash e:\sigma\quad\alpha\notin\,\text{free}(\Gamma)% }{\Gamma\vdash e:\forall\ \alpha\ .\ \sigma}&[\mathtt{Gen}]\\ \\ \end{array}
  102. [ 𝚅𝚊𝚛 ] [\mathtt{Var}]
  103. [ 𝙰𝚙𝚙 ] [\mathtt{App}]
  104. [ 𝙰𝚋𝚜 ] [\mathtt{Abs}]
  105. [ 𝙻𝚎𝚝 ] [\mathtt{Let}]
  106. [ 𝙸𝚗𝚜𝚝 ] [\mathtt{Inst}]
  107. [ 𝙶𝚎𝚗 ] [\mathtt{Gen}]
  108. [ 𝙸𝚗𝚜𝚝 ] [\mathtt{Inst}]
  109. [ 𝙶𝚎𝚗 ] [\mathtt{Gen}]
  110. α f r e e ( Γ ) \alpha\not\in free(\ \Gamma\ )
  111. Γ i d ( n ) : i n t \Gamma\vdash id(n):int
  112. Γ = i d : α . α α , n : i n t \Gamma=id:\forall\alpha.\alpha\rightarrow\alpha,\ n:int
  113. 1 : Γ i d : α . α α [ 𝚅𝚊𝚛 ] ( i d : α . α α Γ ) 2 : Γ i d : i n t i n t [ 𝙸𝚗𝚜𝚝 ] ( 1 ) , ( α . α α i n t i n t ) 3 : Γ n : i n t [ 𝚅𝚊𝚛 ] ( n : i n t Γ ) 4 : Γ i d ( n ) : i n t [ 𝙰𝚙𝚙 ] ( 2 ) , ( 3 ) \begin{array}[]{llll}1:&\Gamma\vdash id:\forall\alpha.\alpha\rightarrow\alpha&% [\mathtt{Var}]&(id:\forall\alpha.\alpha\rightarrow\alpha\in\Gamma)\\ 2:&\Gamma\vdash id:int\rightarrow int&[\mathtt{Inst}]&(1),\ (\forall\alpha.% \alpha\rightarrow\alpha\sqsubseteq int\rightarrow int)\\ 3:&\Gamma\vdash n:int&[\mathtt{Var}]&(n:int\in\Gamma)\\ 4:&\Gamma\vdash id(n):int&[\mathtt{App}]&(2),\ (3)\\ \end{array}
  114. 𝐥𝐞𝐭 i d = λ x . x 𝐢𝐧 i d : α . α α \vdash\ \,\textbf{let}\,id=\lambda x.x\ \,\textbf{in}\ id\,:\,\forall\alpha.% \alpha\rightarrow\alpha
  115. 1 : x : α x : α [ 𝚅𝚊𝚛 ] ( x : α { x : α } ) 2 : λ x . x : α α [ 𝙰𝚋𝚜 ] ( 1 ) 3 : λ x . x : α . α α [ 𝙶𝚎𝚗 ] ( 2 ) , ( α f r e e ( ϵ ) ) 4 : i d : α . α α i d : α . α α [ 𝚅𝚊𝚛 ] ( i d : α . α α { i d : α . α α } ) 5 : 𝐥𝐞𝐭 i d = λ x . x 𝐢𝐧 i d : α . α α [ 𝙻𝚎𝚝 ] ( 3 ) , ( 4 ) \begin{array}[]{llll}1:&x:\alpha\vdash x:\alpha&[\mathtt{Var}]&(x:\alpha\in% \left\{x:\alpha\right\})\\ 2:&\vdash\lambda x.x:\alpha\rightarrow\alpha&[\mathtt{Abs}]&(1)\\ 3:&\vdash\lambda x.x:\forall\alpha.\alpha\rightarrow\alpha&[\mathtt{Gen}]&(2),% \ (\alpha\not\in free(\epsilon))\\ 4:&id:\forall\alpha.\alpha\rightarrow\alpha\vdash id:\forall\alpha.\alpha% \rightarrow\alpha&[\mathtt{Var}]&(id:\forall\alpha.\alpha\rightarrow\alpha\in% \left\{id:\forall\alpha.\alpha\rightarrow\alpha\right\})\\ 5:&\vdash\,\textbf{let}\,id=\lambda x.x\ \,\textbf{in}\ id\,:\,\forall\alpha.% \alpha\rightarrow\alpha&[\mathtt{Let}]&(3),\ (4)\\ \end{array}
  116. [ 𝙰𝚋𝚜 ] [\mathtt{Abs}]
  117. [ 𝙻𝚎𝚝 ] [\mathtt{Let}]
  118. [ 𝙰𝚋𝚜 ] [\mathtt{Abs}]
  119. λ x . e \lambda x.e
  120. Γ , x : τ e : τ \Gamma,\ x:\tau\vdash e:\tau^{\prime}
  121. [ 𝙻𝚎𝚝 ] [\mathtt{Let}]
  122. Γ , x : σ e 1 : τ \Gamma,\ x:\sigma\vdash e_{1}:\tau
  123. λ \lambda
  124. λ f . ( f true , f 0 ) \lambda f.(f\,\textrm{true},f\,\textrm{0})
  125. f f
  126. 𝐥𝐞𝐭 f = λ x . x 𝐢𝐧 ( f true , f 0 ) \,\textbf{let}\ f=\lambda x.x\,\,\textbf{in}\,(f\,\textrm{true},f\,\textrm{0})
  127. ( b o o l , i n t ) (bool,int)
  128. f f
  129. 𝐥𝐞𝐭 x = e 1 𝐢𝐧 e 2 : := ( λ x . e 2 ) e 1 \,\textbf{let}\ x=e_{1}\ \,\textbf{in}\ e_{2}\ ::=(\lambda\ x.e_{2})\ e_{1}
  130. [ 𝙸𝚗𝚜𝚝 ] [\mathtt{Inst}]
  131. [ 𝙶𝚎𝚗 ] [\mathtt{Gen}]
  132. [ 𝙸𝚗𝚜𝚝 ] [\mathtt{Inst}]
  133. [ 𝙶𝚎𝚗 ] [\mathtt{Gen}]
  134. x : σ Γ τ σ Γ x : τ [ 𝚅𝚊𝚛 ] Γ e 0 : τ τ Γ e 1 : τ Γ e 0 e 1 : τ [ 𝙰𝚙𝚙 ] Γ , x : τ e : τ Γ λ x . e : τ τ [ 𝙰𝚋𝚜 ] Γ e 0 : τ Γ , x : Γ ¯ ( τ ) e 1 : τ Γ 𝚕𝚎𝚝 x = e 0 𝚒𝚗 e 1 : τ [ 𝙻𝚎𝚝 ] \begin{array}[]{cl}\displaystyle\frac{x:\sigma\in\Gamma\quad\tau\sqsubseteq% \sigma}{\Gamma\vdash x:\tau}&[\mathtt{Var}]\\ \\ \displaystyle\frac{\Gamma\vdash e_{0}:\tau\rightarrow\tau^{\prime}\quad\quad% \Gamma\vdash e_{1}:\tau}{\Gamma\vdash e_{0}\ e_{1}:\tau^{\prime}}&[\mathtt{App% }]\\ \\ \displaystyle\frac{\Gamma,\;x:\tau\vdash e:\tau^{\prime}}{\Gamma\vdash\lambda% \ x\ .\ e:\tau\rightarrow\tau^{\prime}}&[\mathtt{Abs}]\\ \\ \displaystyle\frac{\Gamma\vdash e_{0}:\tau\quad\quad\Gamma,\,x:\bar{\Gamma}(% \tau)\vdash e_{1}:\tau^{\prime}}{\Gamma\vdash\mathtt{let}\ x=e_{0}\ \mathtt{in% }\ e_{1}:\tau^{\prime}}&[\mathtt{Let}]\end{array}
  135. Γ ¯ ( τ ) = α ^ . τ α ^ = free ( τ ) - free ( Γ ) \bar{\Gamma}(\tau)=\forall\ \hat{\alpha}\ .\ \tau\quad\quad\hat{\alpha}=% \textrm{free}(\tau)-\textrm{free}(\Gamma)
  136. [ 𝚅𝚊𝚛 ] [\mathtt{Var}]
  137. [ 𝙻𝚎𝚝 ] [\mathtt{Let}]
  138. Γ ¯ ( τ ) \bar{\Gamma}(\tau)
  139. Γ \Gamma
  140. S \vdash_{S}
  141. D \vdash_{D}
  142. Γ D e : σ Γ S e : σ \Gamma\vdash_{D}\ e:\sigma\Leftrightarrow\Gamma\vdash_{S}\ e:\sigma
  143. Γ D e : σ Γ S e : σ \Gamma\vdash_{D}\ e:\sigma\Leftarrow\Gamma\vdash_{S}\ e:\sigma
  144. Γ D e : σ Γ S e : σ \Gamma\vdash_{D}\ e:\sigma\Rightarrow\Gamma\vdash_{S}\ e:\sigma
  145. [ 𝙻𝚎𝚝 ] [\mathtt{Let}]
  146. [ 𝚅𝚊𝚛 ] [\mathtt{Var}]
  147. S \vdash_{S}
  148. D \vdash_{D}
  149. S \vdash_{S}
  150. λ x . x : α . α α \lambda\ x.x:\forall\alpha.\alpha\rightarrow\alpha
  151. S \vdash_{S}
  152. λ x . x : α α \lambda\ x.x:\alpha\rightarrow\alpha
  153. Γ D e : σ Γ S e : τ Γ ¯ ( τ ) σ \Gamma\vdash_{D}\ e:\sigma\Rightarrow\Gamma\vdash_{S}\ e:\tau\wedge\bar{\Gamma% }(\tau)\sqsubseteq\sigma
  154. S \vdash_{S}
  155. D \vdash_{D}
  156. S \vdash_{S}
  157. τ \tau
  158. [ 𝚅𝚊𝚛 ] [\mathtt{Var}]
  159. [ 𝙰𝚋𝚜 ] [\mathtt{Abs}]
  160. [ 𝙰𝚙𝚙 ] [\mathtt{App}]
  161. α f r e e ( Γ ) \alpha\not\in free(\Gamma)
  162. τ \tau
  163. [ 𝙰𝚋𝚜 ] [\mathtt{Abs}]
  164. [ 𝚅𝚊𝚛 ] [\mathtt{Var}]
  165. [ 𝙰𝚙𝚙 ] [\mathtt{App}]
  166. 𝚞𝚗𝚒𝚘𝚗 \mathtt{union}
  167. 𝚏𝚒𝚗𝚍 \mathtt{find}
  168. 𝚞𝚗𝚒𝚘𝚗 ( a , b ) \mathtt{union}(a,b)
  169. a a
  170. b b
  171. x : σ Γ τ = 𝑖𝑛𝑠𝑡 ( σ ) Γ x : τ [ 𝚅𝚊𝚛 ] Γ e 0 : τ 0 Γ e 1 : τ 1 τ = 𝑛𝑒𝑤𝑣𝑎𝑟 𝑢𝑛𝑖𝑓𝑦 ( τ 0 , τ 1 τ ) Γ e 0 e 1 : τ [ 𝙰𝚙𝚙 ] τ = 𝑛𝑒𝑤𝑣𝑎𝑟 Γ , x : τ e : τ Γ λ x . e : τ τ [ 𝙰𝚋𝚜 ] Γ e 0 : τ Γ , x : Γ ¯ ( τ ) e 1 : τ Γ 𝚕𝚎𝚝 x = e 0 𝚒𝚗 e 1 : τ [ 𝙻𝚎𝚝 ] \begin{array}[]{cl}\displaystyle\frac{x:\sigma\in\Gamma\quad\tau=\mathit{inst}% (\sigma)}{\Gamma\vdash x:\tau}&[\mathtt{Var}]\\ \\ \displaystyle\frac{\Gamma\vdash e_{0}:\tau_{0}\quad\Gamma\vdash e_{1}:\tau_{1}% \quad\tau^{\prime}=\mathit{newvar}\quad\mathit{unify}(\tau_{0},\ \tau_{1}% \rightarrow\tau^{\prime})}{\Gamma\vdash e_{0}\ e_{1}:\tau^{\prime}}&[\mathtt{% App}]\\ \\ \displaystyle\frac{\tau=\mathit{newvar}\quad\Gamma,\;x:\tau\vdash e:\tau^{% \prime}}{\Gamma\vdash\lambda\ x\ .\ e:\tau\rightarrow\tau^{\prime}}&[\mathtt{% Abs}]\\ \\ \displaystyle\frac{\Gamma\vdash e_{0}:\tau\quad\quad\Gamma,\,x:\bar{\Gamma}(% \tau)\vdash e_{1}:\tau^{\prime}}{\Gamma\vdash\mathtt{let}\ x=e_{0}\ \mathtt{in% }\ e_{1}:\tau^{\prime}}&[\mathtt{Let}]\end{array}
  172. S \vdash_{S}
  173. Γ , e \Gamma,e
  174. τ \tau
  175. i n s t ( σ ) inst(\sigma)
  176. σ \sigma
  177. n e w v a r newvar
  178. Γ ¯ ( τ ) \bar{\Gamma}(\tau)
  179. τ \tau
  180. Γ ¯ ( τ ) \bar{\Gamma}(\tau)
  181. Γ ¯ ( τ ) \bar{\Gamma}(\tau)
  182. u n i o n ( α , τ ) union(\alpha,\tau)
  183. λ x . ( x x ) \lambda\ x.(x\ x)
  184. 𝑓𝑖𝑥 : α . ( α α ) α \mathit{fix}:\forall\alpha.(\alpha\rightarrow\alpha)\rightarrow\alpha
  185. 𝚛𝚎𝚌 v = e 1 𝚒𝚗 e 2 : := 𝚕𝚎𝚝 v = 𝑓𝑖𝑥 ( λ v . e 1 ) 𝚒𝚗 e 2 \mathtt{rec}\ v=e_{1}\ \mathtt{in}\ e_{2}\ ::=\mathtt{let}\ v=\mathit{fix}(% \lambda v.e_{1})\ \mathtt{in}\ e_{2}
  186. Γ , Γ e 1 : τ 1 Γ , Γ e n : τ n Γ , Γ ′′ e : τ Γ 𝚛𝚎𝚌 v 1 = e 1 𝚊𝚗𝚍 𝚊𝚗𝚍 v n = e n 𝚒𝚗 e : τ [ 𝚁𝚎𝚌 ] \displaystyle\frac{\Gamma,\Gamma^{\prime}\vdash e_{1}:\tau_{1}\quad\dots\quad% \Gamma,\Gamma^{\prime}\vdash e_{n}:\tau_{n}\quad\Gamma,\Gamma^{\prime\prime}% \vdash e:\tau}{\Gamma\ \vdash\ \mathtt{rec}\ v_{1}=e_{1}\ \mathtt{and}\ \dots% \ \mathtt{and}\ v_{n}=e_{n}\ \mathtt{in}\ e:\tau}\quad[\mathtt{Rec}]
  187. Γ = v 1 : τ 1 , , v n : τ n \Gamma^{\prime}=v_{1}:\tau_{1},\ \dots,\ v_{n}:\tau_{n}
  188. Γ ′′ = v 1 : Γ ¯ ( τ 1 ) , , v n : Γ ¯ ( τ n ) \Gamma^{\prime\prime}=v_{1}:\bar{\Gamma}(\ \tau_{1}\ ),\ \dots,\ v_{n}:\bar{% \Gamma}(\ \tau_{n}\ )
  189. [ 𝙰𝚋𝚜 ] [\mathtt{Abs}]
  190. [ 𝙻𝚎𝚝 ] [\mathtt{Let}]
  191. 𝚒𝚗 \mathtt{in}
  192. D τ τ D\ \tau\dots\tau
  193. τ τ \tau\rightarrow\tau

Hinge_loss.html

  1. t = ± 1 t=±1
  2. y y
  3. y y
  4. ( y ) = max ( 0 , 1 - t y ) \ell(y)=\max(0,1-t\cdot y)
  5. y y
  6. y = 𝐰 𝐱 + b y=\mathbf{w}\cdot\mathbf{x}+b
  7. t t
  8. y y
  9. y y
  10. | y | 1 |y|\geq 1
  11. ( y ) = 0 \ell(y)=0
  12. ( y ) \ell(y)
  13. y y
  14. ( y ) = max ( 0 , 1 + max t y 𝐰 t 𝐱 - 𝐰 y 𝐱 ) \ell(y)=\max(0,1+\max_{t\neq y}\mathbf{w}_{t}\mathbf{x}-\mathbf{w}_{y}\mathbf{% x})
  15. 𝐲 \mathbf{y}
  16. φ φ
  17. Δ Δ
  18. ( 𝐲 ) \displaystyle\ell(\mathbf{y})
  19. 𝐰 \mathbf{w}
  20. y = 𝐰 𝐱 y=\mathbf{w}\cdot\mathbf{x}
  21. w i = { - t x i if t y < 1 0 otherwise \frac{\partial\ell}{\partial w_{i}}=\begin{cases}-t\cdot x_{i}&\,\text{if }t% \cdot y<1\\ 0&\,\text{otherwise}\end{cases}
  22. t y = 1 ty=1
  23. ( y ) = { 1 2 - t y if t y 0 , 1 2 ( 1 - t y ) 2 if 0 < t y 1 , 0 if 1 t y \ell(y)=\begin{cases}\frac{1}{2}-ty&\,\text{if}~{}~{}ty\leq 0,\\ \frac{1}{2}(1-ty)^{2}&\,\text{if}~{}~{}0<ty\leq 1,\\ 0&\,\text{if}~{}~{}1\leq ty\end{cases}
  24. ( y ) = 1 2 γ max ( 0 , 1 - t y ) 2 \ell(y)=\frac{1}{2\gamma}\max(0,1-ty)^{2}
  25. γ = 2 \gamma=2

History_of_the_metric_system.html

  1. 1 / 10000 {1}/{10000}
  2. 1 / 100 {1}/{100}
  3. E = m c 2 E=mc^{2}
  4. k e k_{e}
  5. k m k_{m}
  6. ϵ 0 \epsilon_{0}
  7. μ 0 \mu_{0}
  8. F m , F e F_{\mathrm{m}},F_{\mathrm{e}}
  9. I 1 , I 2 I_{\mathrm{1}},I_{\mathrm{2}}
  10. q 1 , q 2 q_{\mathrm{1}},q_{\mathrm{2}}
  11. L L
  12. r r
  13. ϵ 0 \epsilon_{0}
  14. μ 0 \mu_{0}
  15. k m , k e k_{\mathrm{m}},k_{\mathrm{e}}
  16. c c
  17. F m = 2 k m I 1 I 2 r F_{\mathrm{m}}=2k_{\mathrm{m}}\frac{I_{1}I_{2}}{r}
  18. k m = μ 0 4 π k_{\mathrm{m}}=\frac{\mu_{0}}{4\pi}
  19. k m k_{\mathrm{m}}
  20. P = V I P=VI
  21. F e = k e q 1 q 2 r 2 F_{\mathrm{e}}=k_{\mathrm{e}}\frac{q_{1}q_{2}}{r^{2}}
  22. k e = 1 4 π ϵ 0 k_{\mathrm{e}}=\frac{1}{4\pi\epsilon_{0}}
  23. k e k_{\mathrm{e}}
  24. E = Q V E=QV
  25. c 2 = 1 ϵ 0 μ 0 c^{2}=\frac{1}{\epsilon_{0}\mu_{0}}
  26. M 1 2 L 3 2 T - 1 M^{\frac{1}{2}}L^{\frac{3}{2}}T^{-1}
  27. 1 / [ u n o w r a p , u 298.257 009223 009563 ] {1}/{[u^{\prime}nowrap^{\prime},u^{\prime}298.257\u{2}009223\u{2}009563^{% \prime}]}

HO_(complexity).html

  1. i i
  2. i - 1 i-1
  3. i > 1 i>1
  4. k k
  5. k k
  6. i - 1 i-1
  7. i {}^{i}
  8. i i
  9. j i {}^{i}_{j}
  10. ϕ = X 1 i ¯ X 2 i ¯ Q X j i ¯ ψ \phi=\exists\overline{X^{i}_{1}}\forall\overline{X_{2}^{i}}\dots Q\overline{X_% {j}^{i}}\psi
  11. Q Q
  12. Q X i ¯ Q\overline{X^{i}}
  13. X i ¯ \overline{X^{i}}
  14. i i
  15. j j
  16. i i
  17. \exists
  18. i - 1 i-1
  19. exp 2 0 ( x ) = x \exp_{2}^{0}(x)=x
  20. exp 2 i + 1 ( x ) = 2 exp 2 i ( x ) \exp_{2}^{i+1}(x)=2^{\exp_{2}^{i}(x)}
  21. exp 2 i + 1 ( x ) = 2 2 2 2 2 x \exp_{2}^{i+1}(x)=2^{2^{2^{2^{\dots^{2^{x}}}}}}
  22. i i
  23. 2 2
  24. i i
  25. i i
  26. i - 1 i-1
  27. HO 0 i = NTIME ( exp 2 i - 2 ( n O ( 1 ) ) ) \,\text{HO}^{i}_{0}=\,\text{NTIME}(\exp_{2}^{i-2}(n^{O(1)}))
  28. ( i - 2 ) (i-2)
  29. n c n^{c}
  30. c c
  31. S O \exists{}SO
  32. 0 2 {}^{2}_{0}
  33. n O ( 1 ) n^{O(1)}
  34. j i {}^{i}_{j}
  35. exp 2 i - 2 ( n O ( 1 ) ) Σ j P \exp_{2}^{i-2}(n^{O(1)})^{\Sigma_{j}^{\rm P}}

Hodges'_estimator.html

  1. θ ^ n \scriptstyle\hat{\theta}_{n}
  2. n ( θ ^ n - θ ) 𝑑 L θ . \sqrt{n}(\hat{\theta}_{n}-\theta)\ \xrightarrow{d}\ L_{\theta}\ .
  3. θ ^ n H \scriptstyle\hat{\theta}^{H}_{n}
  4. θ ^ n H = { θ ^ n , if | θ ^ n | n - 1 / 4 , and 0 , if | θ ^ n | < n - 1 / 4 . \hat{\theta}_{n}^{H}=\begin{cases}\hat{\theta}_{n},&\,\text{if }|\hat{\theta}_% {n}|\geq n^{-1/4},\,\text{ and}\\ 0,&\,\text{if }|\hat{\theta}_{n}|<n^{-1/4}.\end{cases}
  5. θ ^ n \scriptstyle\hat{\theta}_{n}
  6. n α ( θ ^ n H - θ ) 𝑑 0 , when θ = 0 , \displaystyle n^{\alpha}(\hat{\theta}_{n}^{H}-\theta)\ \xrightarrow{d}\ 0,% \qquad\,\text{when }\theta=0,
  7. θ ^ n \scriptstyle\hat{\theta}_{n}
  8. θ ^ n \scriptstyle\hat{\theta}_{n}
  9. x ¯ \scriptstyle\bar{x}
  10. θ ^ n H = x ¯ 𝟏 { | x ¯ | n - 1 / 4 } \scriptstyle\hat{\theta}^{H}_{n}\;=\;\bar{x}\cdot\mathbf{1}\{|\bar{x}|\,\geq\,% n^{-1/4}\}
  11. θ ^ n H \scriptstyle\hat{\theta}_{n}^{H}

Hoek–Brown_failure_criterion.html

  1. σ 1 = σ 3 + A σ 3 + B 2 \sigma_{1}=\sigma_{3}+\sqrt{A\sigma_{3}+B^{2}}
  2. σ 1 \sigma_{1}
  3. σ 3 \sigma_{3}
  4. A , B A,B
  5. σ m \sigma_{m}
  6. τ m \tau_{m}
  7. τ m = 1 2 A ( σ m - τ m ) + B 2 \tau_{m}=\tfrac{1}{2}\sqrt{A(\sigma_{m}-\tau_{m})+B^{2}}
  8. τ m = 1 2 ( σ 1 - σ 3 ) ; σ m = 1 2 ( σ 1 + σ 3 ) . \tau_{m}=\tfrac{1}{2}(\sigma_{1}-\sigma_{3})~{};~{}~{}\sigma_{m}=\tfrac{1}{2}(% \sigma_{1}+\sigma_{3})~{}.
  9. τ m \tau_{m}
  10. τ m = 1 8 [ - A ± A 2 + 4 ( A σ m + B 2 ) ] \tau_{m}=\tfrac{1}{8}\left[-A\pm\sqrt{A^{2}+4(A\sigma_{m}+B^{2})}\right]
  11. A , B A,B
  12. C 0 C_{0}
  13. T 0 T_{0}
  14. A = C 0 2 - T 0 2 T 0 ; B = C 0 . A=\cfrac{C_{0}^{2}-T_{0}^{2}}{T_{0}}~{};~{}~{}B=C_{0}~{}.
  15. σ m = 0 \sigma_{m}=0
  16. τ m = 1 8 [ - A ± A 2 + 4 B 2 ] \tau_{m}=\tfrac{1}{8}\left[-A\pm\sqrt{A^{2}+4B^{2}}\right]
  17. τ m \tau_{m}
  18. σ m \sigma_{m}
  19. σ m - τ m \sigma_{m}-\tau_{m}

Holdover_in_synchronization_applications.html

  1. T ( t ) = T 0 + 0 t R ( t ) d t + ϵ ( t ) = T 0 + ( R 0 t + 1 2 A t 2 + ) + 0 t E t ( t ) d t + ϵ ( t ) T(t)=T_{0}+\int_{0}^{t}R(t)\,dt\ +\epsilon(t)=T_{0}+(R_{0}t+\frac{1}{2}At^{2}+% ...)+\int_{0}^{t}E_{t}(t)\,dt+\epsilon(t)
  2. T 0 T_{0}
  3. t = 0 t=0
  4. R ( t ) R(t)
  5. ϵ ( t ) \epsilon(t)
  6. R 0 R_{0}
  7. R ( t ) R(t)
  8. t = 0 t=0
  9. A A
  10. E 1 ( t ) E_{1}(t)
  11. x ( t ) = x 0 + y 0 t + D 2 t 2 + ϕ ( t ) 2 π ν n o m x(t)=x_{0}+y_{0}t+\frac{D}{2}t^{2}+\frac{\phi(t)}{2\pi\nu_{nom}}
  12. x ( t ) x(t)
  13. x 0 x_{0}
  14. t = 0 t=0
  15. y 0 y_{0}
  16. t = 0 t=0
  17. D D
  18. ϕ ( t ) \phi(t)
  19. ν n o m \nu_{nom}

Holeum.html

  1. E n E_{n}
  2. m m
  3. E n = - m c 2 α g 2 4 n 2 E_{n}=-\frac{mc^{2}\alpha_{g}^{2}}{4n^{2}}
  4. n n
  5. n = 1 , 2 , , n=1,2,...,\infty
  6. α g \alpha_{g}
  7. α g = m 2 G c = m 2 m P 2 \alpha_{g}=\frac{m^{2}G}{\hbar c}=\frac{m^{2}}{m_{P}^{2}}
  8. \hbar
  9. 2 π 2\pi
  10. c c
  11. G G
  12. m H = 2 m + E n c 2 m_{H}=2m+\frac{E_{n}}{c^{2}}
  13. r n = ( n 2 R α g 2 ) ( π 2 < m t p l > 8 ) r_{n}=\left(\frac{n^{2}R}{\alpha_{g}^{2}}\right)\left(\frac{\pi^{2}}{<}mtpl>{{% 8}}\right)
  14. R = ( 2 m G c 2 ) R=\left(\frac{2mG}{c^{2}}\right)
  15. k k
  16. m m
  17. E k = - p 2 m c 2 2 n k 2 ( 1 - p 2 6 n 2 ) 2 E_{k}=-\frac{p^{2}mc^{2}}{2n_{k}^{2}}\left(1-\frac{p^{2}}{6n^{2}}\right)^{2}
  18. p = k α g p=k\alpha_{g}
  19. k 2 k\gg 2
  20. n n
  21. k t h k^{th}
  22. n k n_{k}
  23. r k = π 2 k R n k 2 16 p 2 ( 1 - p 2 6 n 2 ) r_{k}=\frac{\pi^{2}kRn_{k}^{2}}{16p^{2}\left(1-\frac{p^{2}}{6n^{2}}\right)}
  24. M k = m k ( 1 - p 2 6 n 2 ) M_{k}=mk\left(1-\frac{p^{2}}{6n^{2}}\right)
  25. R k = k R ( 1 - p 2 6 n 2 ) R_{k}=kR\left(1-\frac{p^{2}}{6n^{2}}\right)
  26. S k = k 2 S ( 1 - p 2 6 n 2 ) S_{k}=k^{2}S\left(1-\frac{p^{2}}{6n^{2}}\right)
  27. S S
  28. n = n=\infty
  29. n k = 1 n_{k}=1
  30. k - 1 k-1
  31. n = n=\infty
  32. n k = 1 n_{k}=1
  33. p 2 6 n 2 < 1 \frac{p^{2}}{6n^{2}}<1
  34. p = k α g p=k\alpha_{g}
  35. α g = m 2 m P 2 \alpha_{g}=\frac{m^{2}}{m_{P}^{2}}
  36. m < m P ( 6 ) 1 4 ( n k ) 1 2 m<m_{P}\left(6\right)^{\frac{1}{4}}\left(\frac{n}{k}\right)^{\frac{1}{2}}
  37. n = n=\infty
  38. m < m<\infty
  39. r k R k r_{k}\leqslant R_{k}
  40. r k r_{k}
  41. R k R_{k}
  42. m m P 2 ( π n k k ) 1 2 m\geqslant\frac{m_{P}}{2}\left(\frac{\pi n_{k}}{k}\right)^{\frac{1}{2}}
  43. n k = 1 n_{k}=1
  44. m m P 2 ( π k ) 1 2 m\geqslant\frac{m_{P}}{2}\left(\frac{\pi}{k}\right)^{\frac{1}{2}}
  45. k k
  46. m m
  47. n n
  48. n k n_{k}

Holomorphic_embedding_load_flow_method.html

  1. k Y i k V k + Y i sh V i = S i * V i * \sum_{k}Y_{ik}V_{k}+Y_{i}^{\,\text{sh}}V_{i}=\frac{S_{i}^{*}}{V_{i}^{*}}
  2. k Y i k V k ( s ) + Y i sh V i ( s ) = s S i * V i * ( s * ) \sum_{k}Y_{ik}V_{k}(s)+Y_{i}^{\,\text{sh}}V_{i}(s)=s\frac{S_{i}^{*}}{V_{i}^{*}% (s^{*})}
  3. V ( s ) = n = 0 a [ n ] s n \textstyle V(s)=\sum_{n=0}^{\infty}a[n]s^{n}
  4. 1 / V ( s ) = n = 0 b [ n ] s n \textstyle 1/V(s)=\sum_{n=0}^{\infty}b[n]s^{n}
  5. k Y i k a k [ n ] + Y i sh a i [ n ] = S i * b i * [ n - 1 ] ( n = 0 , , ) \sum_{k}Y_{ik}a_{k}[n]+Y_{i}^{\,\text{sh}}a_{i}[n]=S_{i}^{*}b_{i}^{*}[n-1]% \qquad(n=0,\ldots,\infty)
  6. < v a r > s = 0 <var>s=0
  7. < v a r > N <var>N

Holomorphic_Lefschetz_fixed-point_formula.html

  1. f ( p ) = p 1 det ( 1 - A p ) = q ( - 1 ) q trace ( f * | H ¯ 0 , q ( M ) ) \sum_{f(p)=p}\frac{1}{\det(1-A_{p})}=\sum_{q}(-1)^{q}\operatorname{trace}(f^{*% }|H^{0,q}_{\overline{\partial}}(M))

Holst_action.html

  1. S = 1 2 e e I α e J β ( F α β I J - α F α β I J ) 1 2 e e I α e J β ( F α β I J - α 2 ϵ K L I J F α β K L ) S=\frac{1}{2}\int ee^{\alpha}_{\ I}e^{\beta}_{\ J}(F_{\alpha\beta}^{\ \ \ IJ}-% \alpha\ast F_{\alpha\beta}^{\ \ \ IJ})\equiv\frac{1}{2}\int ee^{\alpha}_{\ I}e% ^{\beta}_{\ J}(F_{\alpha\beta}^{\ \ \ IJ}-\frac{\alpha}{2}\epsilon^{IJ}_{\;\;% \;KL}F_{\alpha\beta}^{\ \ \ KL})
  2. e I α e^{\alpha}_{\ I}
  3. e e
  4. F α β I J F_{\alpha\beta}^{\ \ \ IJ}
  5. A α β I J A_{\alpha\beta}^{\ \ \ IJ}
  6. α \alpha
  7. α = 0 \alpha=0
  8. A α β I J A_{\alpha\beta}^{\ \ \ IJ}
  9. η I J \eta_{IJ}
  10. I , J I,J
  11. e I α e^{\alpha}_{\ I}
  12. A α β I J A_{\alpha\beta}^{\ \ \ IJ}
  13. A α β I J A_{\alpha\beta}^{\ \ \ IJ}
  14. F α β I J F_{\alpha\beta}^{\ \ \ IJ}
  15. R α β I J R_{\alpha\beta}^{\ \ \ IJ}
  16. e I α e^{\alpha}_{\ I}
  17. α = i \alpha=i
  18. α \alpha
  19. β := 1 / α \beta:=1/\alpha

Holstein–Herring_method.html

  1. ( - 2 2 m 2 + V ) ψ = E ψ , \left(-\frac{\hbar^{2}}{2m}\nabla^{2}+V\right)\psi=E\psi~{},
  2. ψ = ψ ( 𝐫 ) \psi=\psi(\mathbf{r})
  3. V V
  4. V = - e 2 4 π ε 0 ( 1 r a + 1 r b ) V=-\frac{e^{2}}{4\pi\varepsilon_{0}}\left(\frac{1}{r_{a}}+\frac{1}{r_{b}}\right)
  5. = m = e = 4 π ε 0 = 1 \hbar=m=e=4\pi\varepsilon_{0}=1
  6. ( - 1 2 2 + V ( 𝐱 ) ) ψ + = E + ψ + \left(-\frac{1}{2}\nabla^{2}+V(\,\textbf{x})\right)\psi_{+}=E_{+}\psi_{+}
  7. ( - 1 2 2 + V ( 𝐱 ) ) ψ - = E - ψ - \left(-\frac{1}{2}\nabla^{2}+V(\,\textbf{x})\right)\psi_{-}=E_{-}\psi_{-}
  8. ψ - \psi_{-}
  9. ψ + \psi_{+}
  10. ψ + 2 ψ - - ψ - 2 ψ + = - 2 Δ E ψ - ψ + . \psi_{+}\nabla^{2}\psi_{-}-\psi_{-}\nabla^{2}\psi_{+}={}-2\,\Delta E\,\psi_{-}% \psi_{+}\;.
  11. Δ E = E - - E + \Delta E=E_{-}-E_{+}
  12. ϕ A \phi_{A}
  13. ϕ B \phi_{B}
  14. ψ + = 1 2 ( ϕ A + ϕ B ) , ψ - = 1 2 ( ϕ A - ϕ B ) . \psi_{+}=\frac{1}{\sqrt{\,2}}~{}(\phi_{A}+\phi_{B})\;,\qquad\psi_{-}=\frac{1}{% \sqrt{\,2}}~{}(\phi_{A}-\phi_{B})\;.
  15. ϕ A \phi_{A}
  16. ϕ B \phi_{B}
  17. R R\rightarrow\infty
  18. ϕ A , B \phi_{A,B}
  19. ϕ A , B 0 \phi_{A,B}^{0}
  20. M M
  21. 𝐳 {\mathbf{z}}
  22. z z
  23. 𝐑 3 \mathbf{R}^{3}
  24. L L
  25. R R
  26. ψ - | M = 𝐳 ψ + | M = 0 . \left.\psi_{-}\right|_{M}=\mathbf{z}\cdot\left.\mathbf{\nabla}\psi_{+}\right|_% {M}=0\;.
  27. ϕ A | M = ϕ B | M , 𝐳 ϕ A | M = - 𝐳 ϕ B | M . \left.\phi_{A}\right|_{M}=\left.\phi_{B}\right|_{M}\;,\qquad{\mathbf{z}}\cdot% \left.\mathbf{\nabla}\phi_{A}\right|_{M}={}-\mathbf{z}\cdot\left.\mathbf{% \nabla}\phi_{B}\right|_{M}\;.
  28. L ϕ A 2 d V = R ϕ B 2 d V \int_{L}\phi_{A}^{2}~{}dV=\int_{R}\phi_{B}^{2}~{}dV
  29. 2 L ψ + ψ - d V = L ( ϕ A 2 - ϕ B 2 ) d V = 1 - 2 R ϕ A 2 d V 2\int_{L}\psi_{+}\psi_{-}~{}dV=\int_{L}(\phi_{A}^{2}-\phi_{B}^{2})~{}dV=1-2% \int_{R}\phi_{A}^{2}~{}dV
  30. L ( ψ + 2 ψ - - ψ - 2 ψ + ) d V = L ( ϕ B 2 ϕ A - ϕ A 2 ϕ B ) d V \int_{L}(\psi_{+}\nabla^{2}\psi_{-}-\psi_{-}\nabla^{2}\psi_{+})~{}dV=\int_{L}(% \phi_{B}\nabla^{2}\phi_{A}-\phi_{A}\nabla^{2}\phi_{B})~{}dV
  31. Δ E = - 2 M ϕ A ϕ A d 𝐒 1 - 2 R ϕ A 2 d V \Delta E={}-2\,\frac{\int_{M}\phi_{A}\mathbf{\nabla}\phi_{A}\bullet d{\mathbf{% S}}}{1-2\int_{R}\phi_{A}^{2}~{}dV}
  32. d 𝐒 d{\mathbf{S}}
  33. 2 p σ μ 2p\sigma_{\mu}
  34. 1 s σ g 1s\sigma_{g}
  35. Δ E = E - - E + = 4 e R e - R \Delta E=E_{-}-E_{+}=\frac{4}{e}\,R\,e^{-R}
  36. 4 / 3 4/3
  37. 4 / e 4/e
  38. ϕ A \phi_{A}
  39. H 2 H_{2}
  40. R R
  41. R R

Holtsmark_distribution.html

  1. α \alpha
  2. β \beta
  3. β \beta
  4. φ ( t ; μ , c ) = exp [ i t μ - | c t | α ] , \varphi(t;\mu,c)=\exp\left[~{}it\mu\!-\!|ct|^{\alpha}~{}\right],
  5. α \alpha
  6. μ \mu
  7. α = 3 / 2 , \alpha=3/2,
  8. φ ( t ; μ , c ) = exp [ i t μ - | c t | 3 / 2 ] . \varphi(t;\mu,c)=\exp\left[~{}it\mu\!-\!|ct|^{3/2}~{}\right].
  9. μ \mu
  10. μ \mu
  11. f ( x ) = 1 2 π - φ ( t ) e - i x t d t . f(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\varphi(t)e^{-ixt}\,dt.
  12. μ \mu
  13. f ( x ; 0 , 1 ) \displaystyle f(x;0,1)
  14. Γ ( x ) {\Gamma(x)}
  15. F n m ( ) \;{}_{m}F_{n}()

Homoassociation.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons

Homoclinic_connection.html

  1. f : M M f:M\to M
  2. M M
  3. p p
  4. W s ( f , p ) W^{s}(f,p)
  5. W u ( f , p ) W^{u}(f,p)
  6. P P
  7. V V
  8. V W s ( f , p ) W u ( f , p ) V\subseteq W^{s}(f,p)\cup W^{u}(f,p)
  9. V V
  10. p p
  11. q q
  12. V V
  13. V W s ( f , p ) W u ( f , q ) V\subseteq W^{s}(f,p)\cup W^{u}(f,q)
  14. p p
  15. q q
  16. W s ( f , p ) W^{s}(f,p)
  17. W u ( f , q ) W^{u}(f,q)
  18. p = q p=q
  19. p p
  20. \infty

Homomorphic_signatures_for_network_coding.html

  1. G = ( V , E ) G=(V,E)
  2. V V
  3. E E
  4. s V s\in V
  5. D D
  6. T V T\subseteq V
  7. W / 𝔽 p W/\mathbb{F}_{p}
  8. d d
  9. p p
  10. w 1 , , w k W w_{1},\ldots,w_{k}\in W
  11. v 1 , , v k v_{1},\ldots,v_{k}
  12. v i = ( 0 , , 0 , 1 , , 0 , w i 1 , , w i d ) v_{i}=(0,\ldots,0,1,\ldots,0,w_{i_{1}},\ldots,w{i_{d}})
  13. w i j w_{i_{j}}
  14. j j
  15. w i w_{i}
  16. ( i - 1 ) (i-1)
  17. v i v_{i}
  18. v i v_{i}
  19. 𝔽 p k + d \mathbb{F}_{p}^{k+d}
  20. V V
  21. e E e\in E
  22. y ( e ) y(e)
  23. v = i n ( e ) v=in(e)
  24. y ( e ) = f : out ( f ) = v ( m e ( f ) y ( f ) ) y(e)=\sum_{f:\mathrm{out}(f)=v}(m_{e}(f)y(f))
  25. m e ( f ) 𝔽 p m_{e}(f)\in\mathbb{F}_{p}
  26. k k
  27. k k
  28. w i w_{i}
  29. y ( e ) y(e)
  30. y ( e ) = 1 i k ( g i ( e ) v i ) y(e)=\sum_{1\leq i\leq k}(g_{i}(e)v_{i})
  31. V V
  32. g ( e ) = ( g 1 ( e ) , , g k ( e ) ) g(e)=(g_{1}(e),\ldots,g_{k}(e))
  33. y ( e ) y(e)
  34. g ( e 1 ) , , g ( e k ) g(e_{1}),\ldots,g(e_{k})
  35. e i e_{i}
  36. t T t\in T
  37. t t
  38. G t G_{t}
  39. G t G_{t}
  40. y 1 , , y k y_{1},\ldots,y_{k}
  41. w 1 , , w k w_{1},\ldots,w_{k}
  42. [ y y 2 y k ] = G t [ w 1 w 2 w k ] \begin{bmatrix}y^{\prime}\\ y_{2}^{\prime}\\ \vdots\\ y_{k}^{\prime}\end{bmatrix}=G_{t}\begin{bmatrix}w_{1}\\ w_{2}\\ \vdots\\ w_{k}\end{bmatrix}
  43. y i y_{i}^{\prime}
  44. k k
  45. y i y_{i}
  46. t T t\in T
  47. k k
  48. y 1 , , y k y_{1},\ldots,y_{k}
  49. v i v_{i}
  50. y i = ( α i 1 , , α i k , a i 1 , , a i d ) y_{i}=(\alpha_{i_{1}},\ldots,\alpha_{i_{k}},a_{i_{1}},\ldots,a_{i_{d}})
  51. y i = 1 j k ( α i j v j ) . y_{i}=\sum_{1\leq j\leq k}(\alpha_{ij}v_{j}).
  52. v i v_{i}
  53. H : V G H:V\longrightarrow G
  54. H H
  55. x x
  56. y y
  57. H ( x ) = H ( y ) H(x)=H(y)
  58. H H
  59. H ( x + y ) = H ( x ) + H ( y ) H(x+y)=H(x)+H(y)
  60. H ( v i ) H(v_{i})
  61. y = 1 i k ( α i v i ) y=\sum_{1\leq i\leq k}(\alpha_{i}v_{i})
  62. H ( y ) = 1 i k ( α i H ( v i ) ) H(y)=\sum_{1\leq i\leq k}(\alpha_{i}H(v_{i}))
  63. H H
  64. H H
  65. H H
  66. 𝔽 q \mathbb{F}_{q}
  67. q q
  68. E E
  69. 𝔽 q \mathbb{F}_{q}
  70. y 2 = x 3 + a x + b , y^{2}=x^{3}+ax+b,\,
  71. a , b 𝔽 q a,b\in\mathbb{F}_{q}
  72. 4 a 3 + 27 b 2 0 4a^{3}+27b^{2}\not=0
  73. K 𝔽 q K\supseteq\mathbb{F}_{q}
  74. E ( K ) = { ( x , y ) | y 2 = x 3 + a x + b } { O } E(K)=\{(x,y)|y^{2}=x^{3}+ax+b\}\bigcup\{O\}
  75. E E
  76. E E
  77. E / 𝔽 q E/\mathbb{F}_{q}
  78. 𝔽 ¯ q \mathbb{\bar{F}}_{q}
  79. 𝔽 q \mathbb{F}_{q}
  80. m m
  81. 𝔽 q \mathbb{F}_{q}
  82. m m
  83. E [ m ] = P E ( 𝔽 ¯ q ) : m P = O E[m]={P\in E(\mathbb{\bar{F}}_{q}):mP=O}
  84. E / 𝔽 q E/\mathbb{F}_{q}
  85. gcd ( m , q ) = 1 \gcd(m,q)=1
  86. E [ m ] ( / m ) * ( / m ) E[m]\cong(\mathbb{Z}/m\mathbb{Z})*(\mathbb{Z}/m\mathbb{Z})
  87. e m : E [ m ] * E [ m ] μ m ( 𝔽 q ) e_{m}:E[m]*E[m]\rightarrow\mu_{m}(\mathbb{F}_{q})
  88. e m ( P + R , Q ) = e m ( P , Q ) e m ( R , Q ) and e m ( P , Q + R ) = e m ( P , Q ) e m ( P , R ) e_{m}(P+R,Q)=e_{m}(P,Q)e_{m}(R,Q)\,\text{ and }e_{m}(P,Q+R)=e_{m}(P,Q)e_{m}(P,R)
  89. e m ( P , Q ) = 1 e_{m}(P,Q)=1
  90. Q = O Q=O
  91. e m ( P , P ) = 1 e_{m}(P,P)=1
  92. e m e_{m}
  93. p p
  94. q q
  95. V / 𝔽 p V/\mathbb{F}_{p}
  96. D D
  97. E / 𝔽 q E/\mathbb{F}_{q}
  98. P 1 , , P D E [ p ] P_{1},\ldots,P_{D}\in E[p]
  99. h : V E [ p ] h:V\longrightarrow E[p]
  100. h ( u 1 , , u D ) = 1 i D ( u i P i ) h(u_{1},\ldots,u_{D})=\sum_{1\leq i\leq D}(u_{i}P_{i})
  101. h h
  102. V V
  103. E [ p ] E[p]
  104. s 1 , , s D s_{1},\ldots,s_{D}
  105. 𝔽 p \mathbb{F}_{p}
  106. Q Q
  107. e p ( P i , Q ) 1 e_{p}(P_{i},Q)\not=1
  108. ( P i , s i Q ) (P_{i},s_{i}Q)
  109. 1 i D 1\leq i\leq D
  110. v = u 1 , , u D v=u_{1},\ldots,u_{D}
  111. σ ( v ) = 1 i D ( u i s i P i ) \sigma(v)=\sum_{1\leq i\leq D}(u_{i}s_{i}P_{i})
  112. v = u 1 , , u D v=u_{1},\ldots,u_{D}
  113. σ \sigma
  114. e p ( σ , Q ) \displaystyle e_{p}(\sigma,Q)
  115. σ ( v i ) \sigma(v_{i})
  116. 1 i k 1\leq i\leq k
  117. v i , σ ( v i ) v_{i},\sigma(v_{i})
  118. e e
  119. y ( e ) = f E : out ( f ) = in ( e ) ( m e ( f ) y ( f ) ) y(e)=\sum_{f\in E:\mathrm{out}(f)=\mathrm{in}(e)}(m_{e}(f)y(f))
  120. σ ( y ( e ) ) = f E : out ( f ) = in ( e ) ( m e ( f ) σ ( y ( f ) ) ) \sigma(y(e))=\sum_{f\in E:\mathrm{out}(f)=\mathrm{in}(e)}(m_{e}(f)\sigma(y(f)))
  121. E E
  122. 𝔽 q \mathbb{F}_{q}
  123. 2 log q 2\log q
  124. l o g ( p ) log(p)
  125. p p
  126. q q
  127. h ( e ) h(e)
  128. O ( d i n log p log 1 + ϵ q ) O(d_{in}\log p\log^{1+\epsilon}q)
  129. d i n d_{in}
  130. i n ( e ) in(e)
  131. O ( ( d + k ) log 2 + ϵ q ) O((d+k)\log^{2+\epsilon}q)
  132. ( P 1 , , P r ) (P_{1},\ldots,P_{r})
  133. E [ p ] E[p]
  134. a = ( a 1 , , a r ) 𝔽 p r a=(a_{1},\ldots,a_{r})\in\mathbb{F}_{p}^{r}
  135. b = ( b 1 , , b r ) 𝔽 p r b=(b_{1},\ldots,b_{r})\in\mathbb{F}_{p}^{r}
  136. a b a\not=b
  137. 1 i r ( a i P i ) = 1 j r ( b j P j ) . \sum_{1\leq i\leq r}(a_{i}P_{i})=\sum_{1\leq j\leq r}(b_{j}P_{j}).
  138. p p
  139. r = 2 r=2
  140. x P + y Q = u P + v Q xP+yQ=uP+vQ
  141. ( x - u ) P + ( y - v ) Q = 0 (x-u)P+(y-v)Q=0
  142. x u x\not=u
  143. y v y\not=v
  144. x = u x=u
  145. ( y - v ) Q = 0 (y-v)Q=0
  146. Q Q
  147. p p
  148. y - u 0 mod p y-u\equiv 0\bmod p
  149. y = v y=v
  150. 𝔽 p \mathbb{F}_{p}
  151. ( x , y ) (x,y)
  152. ( u , v ) (u,v)
  153. 𝔽 2 \mathbb{F}_{2}
  154. Q = - ( x - u ) ( y - v ) - 1 P Q=-(x-u)(y-v)^{-1}P
  155. p p
  156. P 1 = P P_{1}=P
  157. P 2 = Q P_{2}=Q
  158. P i = O P_{i}=O
  159. i i
  160. r = 2 r=2
  161. P 1 = r 1 P P_{1}=r_{1}P
  162. P i = r i Q P_{i}=r_{i}Q
  163. r i r_{i}
  164. 𝔽 p \mathbb{F}_{p}
  165. Q Q
  166. a r 1 P + 2 i r ( b i r i Q ) = 0. ar_{1}P+\sum_{2\leq i\leq r}(b_{i}r_{i}Q)=0.
  167. 2 i r b i r i 0 mod p \sum_{2\leq i\leq r}b_{i}r_{i}\not\equiv 0\bmod p
  168. r i r_{i}
  169. b i b_{i}
  170. 2 i r 2\leq i\leq r
  171. r i r_{i}
  172. 2 i r ( b i r i ) = 0 \sum_{2\leq i\leq r}(b_{i}r_{i})=0
  173. 1 p 1\over p
  174. Q Q

Homotopy_type_theory.html

  1. A A
  2. A A
  3. a a
  4. a a
  5. a : A a:A
  6. a A a\in A
  7. x : A B ( x ) x:A\ \vdash\ B(x)
  8. B A B\to A
  9. Id A ( a , b ) \mathrm{Id}_{A}(a,b)
  10. p : Id A ( a , b ) p:\mathrm{Id}_{A}(a,b)
  11. p : a b p:a\mapsto b
  12. α : Id Id A ( a , b ) ( p , q ) \alpha:\mathrm{Id}_{\mathrm{Id}_{A}(a,b)}(p,q)
  13. α : p q \alpha:p\Rightarrow q
  14. a = b a=b
  15. a a
  16. b b
  17. a a
  18. b b
  19. a a
  20. b b
  21. a a
  22. a = a a=a
  23. a = b a=b
  24. b = a b=a
  25. a = b a=b
  26. b = c b=c
  27. a = c a=c
  28. a = b a=b
  29. P ( a ) P(a)
  30. a = b a=b
  31. P ( b ) P(b)
  32. P ( a ) P(a)
  33. P ( b ) P(b)
  34. a a
  35. b b
  36. a a
  37. b b
  38. a = b a=b
  39. a a
  40. a = a a=a
  41. A A
  42. B B
  43. A B A\simeq B
  44. f : A B f:A\to B
  45. I d B B ( f g , i d B ) Id_{B\rightarrow B}(f\circ g,id_{B})
  46. I d A A ( h f , i d A ) Id_{A\rightarrow A}(h\circ f,id_{A})
  47. g g
  48. h h
  49. y : B , I d B ( f ( g ( y ) ) , y ) \forall y:B,Id_{B}(f(g(y)),y)
  50. x : A , I d A ( h ( f ( x ) ) , x ) \forall x:A,Id_{A}(h(f(x)),x)
  51. ( A = B ) ( A B ) (A=B)\simeq(A\simeq B)
  52. A = B A=B
  53. I d U ( A , B ) Id_{U}(A,B)
  54. U U
  55. A A
  56. B B
  57. ( A = B ) ( A B ) (A=B)\to(A\simeq B)

Hopf_lemma.html

  1. L u = a i j ( x ) 2 u x i x j + b i ( x ) u x i + c ( x ) u , x Ω . Lu=a_{ij}(x)\frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}+b_{i}(x)\frac{% \partial u}{\partial x_{i}}+c(x)u,\qquad x\in\Omega.
  2. Ω \Omega
  3. n \mathbb{R}^{n}
  4. L u = 0 Lu=0
  5. Ω \Omega
  6. Ω ¯ \overline{\Omega}
  7. Ω \partial\Omega
  8. x 0 Ω x_{0}\in\partial\Omega
  9. u ν ( x 0 ) 0 , \frac{\partial u}{\partial\nu}(x_{0})\geq 0,
  10. / ν \partial/\partial\nu
  11. u ( x ) u(x)
  12. x x
  13. x 0 x_{0}
  14. Ω \Omega
  15. L L
  16. u ν ( x 0 ) > 0. \frac{\partial u}{\partial\nu}(x_{0})>0.
  17. Ω \Omega
  18. 2 \mathbb{R}^{2}
  19. L L
  20. u u
  21. C 2 ( Ω ) C 1 ( Ω ¯ ) C^{2}(\Omega)\cap C^{1}(\overline{\Omega})
  22. L u 0 , in Ω . Lu\geq 0,\qquad\textrm{in~{}}\Omega.
  23. x 0 Ω x_{0}\in\partial\Omega
  24. 0 u ( x 0 ) = max x Ω ¯ u ( x ) 0\leq u(x_{0})=\max_{x\in\overline{\Omega}}u(x)
  25. Ω \Omega
  26. C 2 C^{2}
  27. x 0 x_{0}
  28. c 0 c\leq 0
  29. u u
  30. u ν ( x 0 ) > 0 \frac{\partial u}{\partial\nu}(x_{0})>0
  31. ν \nu
  32. Ω \Omega
  33. B Ω B\subset\Omega
  34. x 0 B x_{0}\in\partial B
  35. c c
  36. u ( x 0 ) = 0 u(x_{0})=0

Hopf_theorem.html

  1. f , g : M S n f,g:M\to S^{n}
  2. deg ( f ) = deg ( g ) \deg(f)=\deg(g)

Hopkins–Levitzki_theorem.html

  1. F i = J i - 1 M / J i M F_{i}=J^{i-1}M/J^{i}M
  2. F i F_{i}
  3. R / J R/J
  4. F i F_{i}
  5. F i F_{i}
  6. R / J R/J
  7. R / J R/J
  8. F i F_{i}
  9. F i F_{i}
  10. F i F_{i}

Horvitz–Thompson_estimator.html

  1. Y i , i = 1 , 2 , , n Y_{i},i=1,2,\ldots,n
  2. π i \pi_{i}
  3. Y ^ H T = i = 1 n π i - 1 Y i , \hat{Y}_{HT}=\sum_{i=1}^{n}\pi_{i}^{-1}Y_{i},
  4. μ ^ H T = N - 1 Y ^ H T = N - 1 i = 1 n π i - 1 Y i . \hat{\mu}_{HT}=N^{-1}\hat{Y}_{HT}=N^{-1}\sum_{i=1}^{n}\pi_{i}^{-1}Y_{i}.
  5. π i \pi_{i}
  6. π i - 1 Y i \pi_{i}^{-1}Y_{i}
  7. π \pi
  8. μ \mu
  9. μ ^ H T \hat{\mu}_{HT}

Humbert_polynomials.html

  1. ( 1 - m x t + t m ) - λ = n = 0 π n , m λ ( x ) t n \displaystyle(1-mxt+t^{m})^{-\lambda}=\sum^{\infty}_{n=0}\pi^{\lambda}_{n,m}(x% )t^{n}

Humbert_series.html

  1. ( q ) n = q ( q + 1 ) ( q + n - 1 ) = Γ ( q + n ) Γ ( q ) , (q)_{n}=q\,(q+1)\cdots(q+n-1)=\frac{\Gamma(q+n)}{\Gamma(q)}~{},
  2. q q
  3. q = 0 , - 1 , - 2 , q=0,-1,-2,\ldots
  4. Φ 1 ( a , b , c ; x , y ) = Γ ( c ) Γ ( a ) Γ ( c - a ) 0 1 t a - 1 ( 1 - t ) c - a - 1 ( 1 - x t ) - b e y t d t , \real c > \real a > 0 . \Phi_{1}(a,b,c;x,y)=\frac{\Gamma(c)}{\Gamma(a)\Gamma(c-a)}\int_{0}^{1}t^{a-1}(% 1-t)^{c-a-1}(1-xt)^{-b}e^{yt}\,\mathrm{d}t,\quad\real\,c>\real\,a>0~{}.
  5. Φ 2 ( b 1 , b 2 , c ; x , y ) = F 1 ( - , b 1 , b 2 , c ; x , y ) = m , n = 0 ( b 1 ) m ( b 2 ) n ( c ) m + n m ! n ! x m y n , \Phi_{2}(b_{1},b_{2},c;x,y)=F_{1}(-,b_{1},b_{2},c;x,y)=\sum_{m,n=0}^{\infty}% \frac{(b_{1})_{m}(b_{2})_{n}}{(c)_{m+n}\,m!\,n!}\,x^{m}y^{n}~{},
  6. Φ 3 ( b , c ; x , y ) = Φ 2 ( b , - , c ; x , y ) = F 1 ( - , b , - , c ; x , y ) = m , n = 0 ( b ) m ( c ) m + n m ! n ! x m y n , \Phi_{3}(b,c;x,y)=\Phi_{2}(b,-,c;x,y)=F_{1}(-,b,-,c;x,y)=\sum_{m,n=0}^{\infty}% \frac{(b)_{m}}{(c)_{m+n}\,m!\,n!}\,x^{m}y^{n}~{},
  7. Ψ 2 ( a , c 1 , c 2 ; x , y ) = Ψ 1 ( a , - , c 1 , c 2 ; x , y ) = F 2 ( a , - , - , c 1 , c 2 ; x , y ) = F 4 ( a , - , c 1 , c 2 ; x , y ) = m , n = 0 ( a ) m + n ( c 1 ) m ( c 2 ) n m ! n ! x m y n , \Psi_{2}(a,c_{1},c_{2};x,y)=\Psi_{1}(a,-,c_{1},c_{2};x,y)=F_{2}(a,-,-,c_{1},c_% {2};x,y)=F_{4}(a,-,c_{1},c_{2};x,y)=\sum_{m,n=0}^{\infty}\frac{(a)_{m+n}}{(c_{% 1})_{m}(c_{2})_{n}\,m!\,n!}\,x^{m}y^{n}~{},

Hund's_cases.html

  1. 𝐋 \mathbf{L}
  2. 𝐒 \mathbf{S}
  3. 𝐉 a = 𝐋 + 𝐒 \mathbf{J}_{a}=\mathbf{L}+\mathbf{S}
  4. 𝐉 \mathbf{J}
  5. 𝐍 = 𝐉 - 𝐒 \mathbf{N}=\mathbf{J}-\mathbf{S}
  6. 𝐑 = 𝐍 - 𝐋 \mathbf{R}=\mathbf{N}-\mathbf{L}
  7. 𝐋 \mathbf{L}
  8. 𝐋 \mathbf{L}
  9. 𝐒 \mathbf{S}
  10. 𝐉 \mathbf{J}
  11. 𝐋 \mathbf{L}
  12. 𝐒 \mathbf{S}
  13. 𝐋 \mathbf{L}
  14. 𝐋 \mathbf{L}
  15. 𝐒 \mathbf{S}
  16. Λ \Lambda
  17. Σ \Sigma
  18. s y m b o l Ω symbol\Omega
  19. Ω = Λ + Σ \Omega=\Lambda+\Sigma
  20. 𝐑 \mathbf{R}
  21. 𝐉 = s y m b o l Ω + 𝐑 \mathbf{J}=symbol\Omega+\mathbf{R}
  22. 𝐋 \mathbf{L}
  23. 𝐒 \mathbf{S}
  24. s y m b o l Ω symbol\Omega
  25. 𝐑 \mathbf{R}
  26. 𝐉 \mathbf{J}
  27. Λ \Lambda
  28. S S
  29. Σ \Sigma
  30. J J
  31. Ω \Omega
  32. H r o t = B 𝐑 2 = B ( 𝐉 - 𝐋 - 𝐒 ) 2 H_{rot}=B\mathbf{R}^{2}=B(\mathbf{J}-\mathbf{L}-\mathbf{S})^{2}
  33. B B
  34. 2 S + 1 2S+1
  35. B J ( J + 1 ) BJ(J+1)
  36. J = Ω J=\Omega
  37. Λ = 0 \Lambda=0
  38. 𝐍 = s y m b o l λ + 𝐑 \mathbf{N}=symbol\lambda+\mathbf{R}
  39. 𝐉 = 𝐍 + 𝐒 \mathbf{J}=\mathbf{N}+\mathbf{S}
  40. 𝐋 \mathbf{L}
  41. Λ \Lambda
  42. N N
  43. S S
  44. J J
  45. H r o t = B 𝐑 2 = B ( 𝐍 - 𝐋 ) 2 H_{rot}=B\mathbf{R}^{2}=B(\mathbf{N}-\mathbf{L})^{2}
  46. B B
  47. B N ( N + 1 ) BN(N+1)
  48. N = Λ N=\Lambda
  49. Λ \Lambda
  50. Σ \Sigma
  51. 𝐋 \mathbf{L}
  52. 𝐒 \mathbf{S}
  53. 𝐉 a \mathbf{J}_{a}
  54. Ω \Omega
  55. 𝐉 = s y m b o l Ω + 𝐑 \mathbf{J}=symbol\Omega+\mathbf{R}
  56. J a J_{a}
  57. J J
  58. Ω \Omega
  59. 𝐋 \mathbf{L}
  60. 𝐑 \mathbf{R}
  61. 𝐋 \mathbf{L}
  62. 𝐍 \mathbf{N}
  63. 𝐋 \mathbf{L}
  64. 𝐑 \mathbf{R}
  65. 𝐉 \mathbf{J}
  66. 𝐍 \mathbf{N}
  67. 𝐒 \mathbf{S}
  68. L L
  69. R R
  70. N N
  71. S S
  72. J J
  73. R R
  74. H r o t = B 𝐑 2 = B R ( R + 1 ) H_{rot}=B\mathbf{R}^{2}=BR(R+1)
  75. 𝐉 a \mathbf{J}_{a}
  76. 𝐉 \mathbf{J}
  77. 𝐉 a \mathbf{J}_{a}
  78. 𝐑 \mathbf{R}
  79. J a J_{a}
  80. R R
  81. J J
  82. R R
  83. H r o t = B 𝐑 2 = B R ( R + 1 ) H_{rot}=B\mathbf{R}^{2}=BR(R+1)

Hydraulic_jumps_in_rectangular_channels.html

  1. q = Q w q=\frac{Q}{w}
  2. F r F_{r}
  3. F x = Δ ( m a x ) \sum F_{x}=\Delta(ma_{x})
  4. F = Δ m × d y d t \sum F=\Delta m\times\frac{dy}{dt}
  5. Δ m v = Δ m × d y d t \Delta mv=\Delta m\times\frac{dy}{dt}
  6. F x = Δ ( m v x ) \sum F_{x}=\Delta(mv_{x})
  7. Δ ( m v ) ( w ) = ρ ( Q ) ( v 2 - v 1 ) \Delta(mv)(w)=\rho(Q)(v_{2}-v_{1})
  8. Δ ( m v ) = ρ q ( v 2 - v 1 ) \Delta(mv)=\rho q(v_{2}-v_{1})
  9. F = P 1 - P 2 - F d where P = γ × y 2 2 \sum F=P_{1}-P_{2}-F_{d}\qquad\,\text{where }P={\gamma\times y^{2}\over 2}
  10. γ × y 1 2 2 - γ × y 2 2 2 - F d = ρ q ( v 2 - v 1 ) {\gamma\times y_{1}^{2}\over 2}-{\gamma\times y_{2}^{2}\over 2}-F_{d}=\rho q(v% _{2}-v_{1})
  11. y 1 2 2 - y 2 2 2 - F d γ = ρ × q × ( v 2 - v 1 ) γ {y_{1}^{2}\over 2}-{y_{2}^{2}\over 2}-{F_{d}\over\gamma}={\rho\times q\times(v% _{2}-v_{1})\over\gamma}
  12. 1 g = ρ γ \frac{1}{g}={\rho\over\gamma}
  13. y 1 2 2 + q v 1 g = y 2 2 2 + q v 2 g {y_{1}^{2}\over 2}+{qv_{1}\over g}={y_{2}^{2}\over 2}+{qv_{2}\over g}
  14. v = q y v=\frac{q}{y}
  15. M = y 1 2 2 + q 2 g y 1 = y 2 2 2 + q 2 g y 2 M={y_{1}^{2}\over 2}+{q^{2}\over gy_{1}}={y_{2}^{2}\over 2}+{q^{2}\over gy_{2}}
  16. M = y 1 2 2 + q 2 g y 1 = y 2 2 2 + q 2 g y 2 M=\frac{y_{1}^{2}}{2}+\frac{q^{2}}{gy_{1}}=\frac{y_{2}^{2}}{2}+\frac{q^{2}}{gy% _{2}}
  17. q 2 g ( 1 y 1 - 1 y 2 ) = 1 2 ( y z 2 - y 1 2 ) \frac{q^{2}}{g}\left(\frac{1}{y_{1}}-\frac{1}{y_{2}}\right)=\frac{1}{2}\left(y% _{z}^{2}-y_{1}^{2}\right)
  18. q 2 g ( y 2 - y 1 y 1 y 2 ) = 1 2 ( y 2 - y 1 ) ( y 2 + y 1 ) \frac{q^{2}}{g}\left(\frac{y_{2}-y_{1}}{y_{1}y_{2}}\right)=\frac{1}{2}(y_{2}-y% _{1})(y_{2}+y_{1})
  19. q 2 g ( 1 y 1 y 2 ) = 1 2 ( y 2 + y 1 ) where q 1 2 = y 1 2 v 1 2 = y 2 2 v 2 2 \frac{q^{2}}{g}\left(\frac{1}{y_{1}y_{2}}\right)=\frac{1}{2}(y_{2}+y_{1})% \qquad\,\text{where }q_{1}^{2}=y_{1}^{2}v_{1}^{2}=y_{2}^{2}v_{2}^{2}
  20. v 1 2 g ( 1 y 1 y 2 ) = 1 2 y 1 2 ( y 2 + y 1 ) recall F r 1 2 = v 1 2 g y 1 \frac{v_{1}^{2}}{g}\left(\frac{1}{y_{1}y_{2}}\right)=\frac{1}{2y_{1}^{2}}(y_{2% }+y_{1})\qquad\,\text{recall }Fr_{1}^{2}=\frac{v_{1}^{2}}{gy_{1}}
  21. F r 1 2 = y 2 2 2 y 1 2 + y 2 2 y 1 Fr_{1}^{2}=\frac{y_{2}^{2}}{2y_{1}^{2}}+\frac{y_{2}}{2y_{1}}
  22. F r 1 2 = x 2 2 + x 2 Fr_{1}^{2}=\frac{x^{2}}{2}+\frac{x}{2}
  23. 0 = x 2 2 + x 2 - F r 1 2 a = 1 / 2 , b = 1 / 2 , c = F r 1 2 \Rightarrow 0=\frac{x^{2}}{2}+\frac{x}{2}-Fr_{1}^{2}\qquad a=1/2,\;b=1/2,\;c=% Fr_{1}^{2}
  24. 4 2 \tfrac{\sqrt{4}}{2}
  25. x = - 1 2 ± ( 1 / 2 ) 2 + 4 ( 1 / 2 ) ( F r 1 2 ) 2 ( 1 / 2 ) = - 1 2 ( 1 / 4 ) + 2 ( F r 1 2 ) x=\frac{-\tfrac{1}{2}\pm\sqrt{(1/2)^{2}+4(1/2)(Fr_{1}^{2})}}{2(1/2)}=-\frac{1}% {2}\sqrt{(1/4)+2(Fr_{1}^{2})}
  26. y 2 y 1 = 1 2 ( 1 + 8 F r 1 2 - 1 ) \frac{y_{2}}{y_{1}}=\frac{1}{2}\left(\sqrt{1+8Fr_{1}^{2}}-1\right)
  27. ( 1 ) y 2 = y 1 2 ( 1 + 8 F r 1 2 - 1 ) (1)\quad y_{2}=\frac{y_{1}}{2}\left(\sqrt{1+8Fr_{1}^{2}}-1\right)
  28. ( 2 ) F r 1 = v ( g y 1 ) 1 / 2 = q y 1 ( g y 1 ) 1 / 2 (2)\quad Fr_{1}=\frac{v}{(gy_{1})^{1/2}}=\frac{q}{y_{1}(gy_{1})^{1/2}}
  29. ( 3 ) F r 1 2 = q 2 g y 1 3 = ( 10 ) 2 32.2 * ( 0.24 3 ) = 224.65 (3)\quad Fr_{1}^{2}=\frac{q^{2}}{gy_{1}^{3}}=\frac{(10)^{2}}{32.2*(0.24^{3})}=% 224.65
  30. ( 4 ) y 2 = 0.24 2 1 + 8 ( 224.65 ) - 1 = 4.97 f t (4)\quad y_{2}=\frac{0.24}{2}\sqrt{1+8(224.65)}-1=4.97\;ft
  31. ( 5 ) M 1 = ( 0.24 ) 2 2 + q 2 g * ( 0.24 ) = 13 f t 2 (5)\quad M_{1}=\frac{(0.24)^{2}}{2}+\frac{q^{2}}{g*(0.24)}=13\;ft^{2}
  32. ( 6 ) M 2 = ( 4.97 ) 2 2 + q 2 g * ( 4.97 ) = 13 f t 2 (6)\quad M_{2}=\frac{(4.97)^{2}}{2}+\frac{q^{2}}{g*(4.97)}=13\;ft^{2}
  33. Δ E = E 1 - E 2 = ( y 1 + q 2 2 g y 1 2 ) - ( y 2 + q 2 2 g y 2 2 ) = ( y 2 - y 1 ) 3 4 y 1 y 2 \Delta E=E_{1}-E_{2}=\left(y_{1}+\frac{q^{2}}{2gy_{1}^{2}}\right)-\left(y_{2}+% \frac{q^{2}}{2gy_{2}^{2}}\right)=\frac{(y_{2}-y_{1})^{3}}{4y_{1}y_{2}}
  34. E 2 E 1 = ( 8 F r 1 2 + 1 ) 3 / 2 - 4 F r 1 2 + 1 8 F r 1 2 ( 2 + F r 1 2 ) {E_{2}\over E_{1}}={(8Fr_{1}^{2}+1)^{3/2}-4Fr_{1}^{2}+1\over 8Fr_{1}^{2}(2+Fr_% {1}^{2})}
  35. ( 1 ) F r 1 = v g y = 10 [ m / s ] 9.81 [ m / s 2 ] * 0.5 [ m ] = 4.5 (1)\;Fr_{1}=\frac{v}{\sqrt{gy}}=\frac{10[m/s]}{\sqrt{9.81[m/s^{2}]*0.5[m]}}=4.5
  36. ( 2 ) F r 1 > 1 (supercritical flow) (2)\;Fr_{1}>1\quad\,\text{(supercritical flow)}
  37. ( 3 ) y 2 = y 1 2 ( 1 + 8 F r 1 2 - 1 ) = 0.5 2 ( 1 + 8 * 4.5 2 - 1 ) = 2.94 m (3)\;y_{2}=\frac{y_{1}}{2}\left(\sqrt{1+8Fr_{1}^{2}}-1\right)=\frac{0.5}{2}% \left(\sqrt{1+8*4.5^{2}}-1\right)=2.94\;m
  38. ( 4 ) Δ E = ( y 2 - y 1 ) 3 4 y 1 y 2 = ( 2.94 - 0.5 ) 3 4 * 2.94 * 0.5 = 2.47 m (4)\;\Delta E=\frac{(y_{2}-y_{1})^{3}}{4y_{1}y_{2}}=\frac{(2.94-0.5)^{3}}{4*2.% 94*0.5}=2.47\;m
  39. ( 5 ) E 2 E 1 = ( 8 F r 1 2 + 1 ) 3 / 2 - 4 F r 1 2 + 1 8 F r 1 2 ( 2 + F 1 2 ) = ( 8 * 4.5 2 + 1 ) 3 / 2 - 4 * 4.5 2 + 1 8 * 4.5 2 ( 2 + 4.5 2 ) = 0.55 (5)\;\frac{E_{2}}{E_{1}}=\frac{(8Fr_{1}^{2}+1)^{3/2}-4Fr_{1}^{2}+1}{8Fr_{1}^{2% }(2+F_{1}^{2})}=\frac{(8*4.5^{2}+1)^{3/2}-4*4.5^{2}+1}{8*4.5^{2}(2+4.5^{2})}=0% .55
  40. ( 6 ) % efficiency = 0.55 * 100 = 55 % efficient (6)\;\%\,\text{ efficiency}=0.55*100=55\%\,\text{ efficient}\,
  41. ( 1 ) L = 220 × y 1 × tanh F r 1 - 1 22 (1)\;L=220\times y_{1}\times\tanh\frac{Fr_{1}-1}{22}
  42. ( 2 ) L = 220 * 0.5 * tanh ( 4.5 - 1 22 ) (2)\;L=220*0.5*\tanh\left(\frac{4.5-1}{22}\right)
  43. ( 3 ) tanh ( z ) = e z - e - z e z + e - z if necessary, this is an alternate way to calculate tanh ( z ) (3)\;\tanh(z)=\frac{e^{z}-e^{-z}}{e^{z}+e^{-z}}\qquad\,\text{if necessary, % this is an alternate way to calculate }\tanh(z)
  44. ( 4 ) z = 4.5 - 1 22 = 0.1591 (4)\;z=\frac{4.5-1}{22}=0.1591
  45. ( 5 ) tanh ( z ) = 0.1578 (5)\;\tanh(z)=0.1578
  46. ( 6 ) L = 220 * 0.5 * 0.1578 = 17.4 m (6)\;L=220*0.5*0.1578=17.4\;m
  47. ( 1 ) h j = y 2 - y 1 (1)\;h_{j}=y_{2}-y_{1}
  48. ( 2 ) y 2 = y 1 2 ( 1 + 8 F r 1 2 - 1 ) (2)\;y_{2}=\frac{y_{1}}{2}\left(\sqrt{1+8Fr_{1}^{2}}-1\right)
  49. ( 3 ) h j = y 1 2 ( 1 + 8 F r 1 2 - 1 ) - y 1 (3)\;h_{j}=\frac{y_{1}}{2}\left(\sqrt{1+8Fr_{1}^{2}}-1\right)-y_{1}
  50. ( 4 ) h j = y 1 1 + 8 F r 1 2 - 3 y 1 2 (4)\;h_{j}=\frac{y_{1}\sqrt{1+8Fr_{1}^{2}}-3y_{1}}{2}
  51. ( 5 ) h j = y 1 1 + 8 F r 1 2 - 3 y 1 2 (5)\;h_{j}=\frac{y_{1}\sqrt{1+8Fr_{1}^{2}}-3y_{1}}{2}
  52. ( 6 ) h j = 0.5 1 + 8 ( 4.5 ) 2 - ( 3 * 0.5 ) 2 (6)\;h_{j}=\frac{0.5\sqrt{1+8(4.5)^{2}}-(3*0.5)}{2}
  53. ( 7 ) h j = 2.4 m (7)\;h_{j}=2.4\;m

Hydroxyl_value.html

  1. H V = [ [ ( 56.1 ) ( N ) ( V B - V a c e t ) ] / W a c e t ] + A V HV=[[(56.1)(N)(V_{B}-V_{acet})]/W_{acet}]+AV

Hyperbolic_absolute_risk_aversion.html

  1. T ( W ) T(W)
  2. A ( W ) A(W)
  3. T ( W ) = 1 A ( W ) = W 1 - γ + b a , T(W)=\frac{1}{A(W)}=\frac{W}{1-\gamma}+\frac{b}{a},
  4. U ( W ) = 1 - γ γ ( a W 1 - γ + b ) γ U(W)=\frac{1-\gamma}{\gamma}\left(\frac{aW}{1-\gamma}+b\right)^{\gamma}
  5. a > 0 a>0
  6. b + a W 1 - γ > 0. b+\frac{aW}{1-\gamma}>0.
  7. γ < 1 \gamma<1
  8. γ > 1 \gamma>1
  9. γ \gamma
  10. γ \gamma
  11. U ( W ) = log ( W + b ) U(W)=\,\text{log}(W+b)
  12. A ( W ) < 0 A^{\prime}(W)<0
  13. γ \gamma
  14. γ \gamma
  15. γ \gamma
  16. R ( W ) > 0 R^{\prime}(W)>0
  17. R ( W ) < 0 R^{\prime}(W)<0
  18. R ( W ) = 0 R^{\prime}(W)=0
  19. γ 1 \gamma\neq 1
  20. γ = 1 \gamma=1
  21. γ = 2 \gamma=2
  22. γ \gamma
  23. γ < 1 \gamma<1
  24. a = 1 - γ a=1-\gamma
  25. a = 1 a=1
  26. γ \gamma

Hyperoperation.html

  1. a m b = a m - 1 ( a m - 1 ( a m - 1 ( ( a m - 1 ( a m - 1 a ) ) ) ) ) b copies of a \begin{matrix}a\uparrow^{m}b&=&\underbrace{a\uparrow^{m-1}(a\uparrow^{m-1}(a% \uparrow^{m-1}(...(a\uparrow^{m-1}(a\uparrow^{m-1}a))...)))}\\ &&b\mbox{ copies of }~{}a\end{matrix}
  2. a m b = a m - 1 ( a m ( b - 1 ) ) a\uparrow^{m}b=a\uparrow^{m-1}(a\uparrow^{m}(b-1))
  3. H n ( a , b ) ( 0 ) 3 0 H_{n}(a,b)(\mathbb{N}_{0})^{3}\rightarrow\mathbb{N}_{0}\,\!
  4. H n ( a , b ) = { b + 1 if n = 0 a if n = 1 , b = 0 0 if n = 2 , b = 0 1 if n 3 , b = 0 H n - 1 ( a , H n ( a , b - 1 ) ) otherwise H_{n}(a,b)=\begin{cases}b+1&\,\text{if }n=0\\ a&\,\text{if }n=1,b=0\\ 0&\,\text{if }n=2,b=0\\ 1&\,\text{if }n\geq 3,b=0\\ H_{n-1}(a,H_{n}(a,b-1))&\,\text{otherwise}\end{cases}\,\!
  5. H 0 ( a , b ) = b + 1 , H_{0}(a,b)=b+1\,\!,
  6. H 1 ( a , b ) = a + b , H_{1}(a,b)=a+b\,\!,
  7. H 2 ( a , b ) = a b , H_{2}(a,b)=a\cdot b\,\!,
  8. H 3 ( a , b ) = a b , H_{3}(a,b)=a^{b}\,\!,
  9. H 4 ( a , b ) = a b , H_{4}(a,b)=a\uparrow\uparrow{b}\,\!,
  10. H 5 ( a , b ) = a b , H_{5}(a,b)=a\uparrow\uparrow\uparrow{b}\,\!,
  11. H n ( a , b ) = a n - 2 b for n 3 , H_{n}(a,b)=a\uparrow^{n-2}b\,\text{ for }n\geq 3\,\!,
  12. H n ( a , b ) = a n - 2 b for n 0. H_{n}(a,b)=a\uparrow^{n-2}b\,\text{ for }n\geq 0.\,\!
  13. a + b = ( a + ( b - 1 ) ) + 1 a+b=(a+(b-1))+1
  14. a b = a + ( a ( b - 1 ) ) a\cdot b=a+(a\cdot(b-1))
  15. a b = a ( a ( b - 1 ) ) a^{b}=a\cdot(a^{(b-1)})
  16. a b = a a ( b - 1 ) a\uparrow\uparrow b=a^{a\uparrow\uparrow{(b-1)}}
  17. b + 1 b+1
  18. 1 + 1 + 1 + 1 + + 1 b copies of 1 {1+{\underbrace{1+1+1+\cdots+1}\atop{b\mbox{ copies of 1}~{}}}}
  19. a + b a+b
  20. a + 1 + 1 + 1 + + 1 b copies of 1 {a+{\underbrace{1+1+1+\cdots+1}\atop{b\mbox{ copies of 1}~{}}}}
  21. a b a\cdot b
  22. a + a + a + + a b copies of a {{\underbrace{a+a+a+\cdots+a}}\atop{b\mbox{ copies of }~{}a}}
  23. a b a^{b}
  24. a b a\uparrow b
  25. a a a a a b copies of a {{\underbrace{a\cdot a\cdot a\cdot a\cdot\ldots\cdot a}}\atop{b\mbox{ copies % of }~{}a}}
  26. a b {}^{b}a
  27. a b a\uparrow\uparrow b
  28. a ( a ( a a ) ) ) b copies of a {{\underbrace{a\uparrow(a\uparrow(a\uparrow\cdots\uparrow a))...)}}\atop{b% \mbox{ copies of }~{}a}}
  29. a b a\uparrow\uparrow\uparrow b
  30. a 3 b a\uparrow^{3}b
  31. a ( a ( a a ) ) ) b copies of a {{\underbrace{a\uparrow\uparrow(a\uparrow\uparrow(a\uparrow\uparrow\cdots% \uparrow\uparrow a))...)}}\atop{b\mbox{ copies of }~{}a}}
  32. a b a\uparrow\uparrow\uparrow\uparrow b
  33. a 4 b a\uparrow^{4}b
  34. a 3 ( a 3 ( a 3 3 a ) ) ) b copies of a {{\underbrace{a\uparrow^{3}(a\uparrow^{3}(a\uparrow^{3}\cdots\uparrow^{3}a))..% .)}}\atop{b\mbox{ copies of }~{}a}}
  35. 1 a \frac{1}{a}
  36. ϕ ( a , b , n ) \phi(a,b,n)\,\!
  37. G ( n , a , b ) = H n ( a , b ) G(n,a,b)=H_{n}(a,b)
  38. ϕ ( a , b , n ) \phi(a,b,n)\,\!
  39. ϕ \phi\,\!
  40. ϕ ( a , b , n ) \phi(a,b,n)\,\!
  41. ϕ \phi\,\!
  42. ϕ ( a , b , 3 ) = a ( b + 1 ) \phi(a,b,3)=a\uparrow\uparrow(b+1)\,\!
  43. ϕ ( a , b , 3 ) \phi(a,b,3)\,\!
  44. a a a a^{a^{\cdot^{\cdot^{\cdot^{a}}}}}\,\!
  45. a b a\uparrow\uparrow b\,\!
  46. H n ( a , b ) H_{n}(a,b)
  47. a n - 2 b a\uparrow^{n-2}b\,\!
  48. G ( n , a , b ) G(n,a,b)\,\!
  49. ϕ ( a , b , n - 1 ) for 1 n 3 ϕ ( a , b - 1 , n - 1 ) for n 4 \begin{matrix}\phi(a,b,n-1)\ \,\text{ for }1\leq n\leq 3\\ \phi(a,b-1,n-1)\ \,\text{ for }n\geq 4\end{matrix}\,\!
  50. A ( n , b - 3 ) + 3 for a = 2 A(n,b-3)+3\ \,\text{for }a=2\,\!
  51. a n - 1 b a\otimes^{n-1}b\,\!
  52. a n b a{\,\begin{array}[]{|c|}\hline{\!n\!}\\ \hline\end{array}\,}b\,\!
  53. a b ( n ) a{}^{(n)}b\,\!
  54. a b ( n ) a{}_{(n)}b\,\!
  55. a O n - 1 b aO_{n-1}b\,\!
  56. a [ n ] b a[n]b\,\!
  57. a b ( n - 2 ) a\to b\to(n-2)
  58. { a , b , n - 2 } \{a,b,n-2\}
  59. ϕ ( a , b , n ) \phi(a,b,n)
  60. ϕ \phi
  61. ϕ ( a , 0 , n ) = a \phi(a,0,n)=a
  62. F 0 ( a , b ) = a + b F_{0}(a,b)=a+b
  63. F 1 ( a , b ) = a b F_{1}(a,b)=a\cdot b
  64. F 2 ( a , b ) = a b F_{2}(a,b)=a^{b}
  65. F 3 ( a , b ) = a ( b + 1 ) F_{3}(a,b)=a\uparrow\uparrow(b+1)
  66. F 4 ( a , b ) = ( x a ( x + 1 ) ) b ( a ) F_{4}(a,b)=(x\mapsto a\uparrow\uparrow(x+1))^{b}(a)
  67. A ( 0 , b ) = 2 b + 1 A(0,b)=2b+1
  68. a = 2 a=2
  69. F n ( a , 0 ) = 0 F_{n}(a,0)=0
  70. F 0 ( a , b ) = b + 1 F_{0}(a,b)=b+1
  71. F 1 ( a , b ) = a + b F_{1}(a,b)=a+b
  72. F 2 ( a , b ) = a b = e ln ( a ) + ln ( b ) F_{2}(a,b)=a\cdot b=e^{\ln(a)+\ln(b)}
  73. F 3 ( a , b ) = a b F_{3}(a,b)=a^{b}
  74. F 4 ( a , b ) = a ( b - 1 ) F_{4}(a,b)=a\uparrow\uparrow(b-1)
  75. F 5 ( a , b ) = ( x a ( x - 1 ) ) b ( 0 ) = 0 if a > 0 F_{5}(a,b)=(x\mapsto a\uparrow\uparrow(x-1))^{b}(0)=0\,\text{ if }a>0
  76. a + b = ( a + ( b - 1 ) ) + 1 a+b=(a+(b-1))+1
  77. a b = ( a ( b - 1 ) ) + a a\cdot b=(a\cdot(b-1))+a
  78. a b = ( a ( b - 1 ) ) a a^{b}=(a^{(b-1)})\cdot a
  79. a ( n + 1 ) b = ( a ( n + 1 ) ( b - 1 ) ) ( n ) a a_{(n+1)}b=(a_{(n+1)}(b-1))_{(n)}a
  80. a ( 1 ) b = a + b a ( 2 ) 0 = 0 a ( n ) 1 = a for n > 2 \begin{array}[]{lcll}a_{(1)}b&=&a+b\\ a_{(2)}0&=&0\\ a_{(n)}1&=&a&\,\text{for }n>2\\ \end{array}
  81. α O 0 β = α + β α O γ β = sup η < β , ξ < γ ( α O γ η ) O ξ α \begin{array}[]{lcl}\alpha O_{0}\beta&=&\alpha+\beta\\ \alpha O_{\gamma}\beta&=&\sup\limits_{\eta<\beta,\xi<\gamma}(\alpha O_{\gamma}% \eta)O_{\xi}\alpha\end{array}
  82. a O n b = a ( n + 1 ) b aO_{n}b=a_{(n+1)}b
  83. α ( 4 ) ( 1 + β ) = α ( α β ) . \alpha_{(4)}(1+\beta)=\alpha^{(\alpha^{\beta})}.
  84. α ( 1 + 2 γ + 1 ) β α ( 1 + 2 γ ) ( 1 + 3 α β ) . \alpha_{(1+2\gamma+1)}\beta\leq\alpha_{(1+2\gamma)}(1+3\alpha\beta).
  85. F 0 ( a , b ) = a + 1 F_{0}(a,b)=a+1
  86. F 1 ( a , b ) = a + b F_{1}(a,b)=a+b
  87. F 2 ( a , b ) = a b F_{2}(a,b)=a\cdot b
  88. F 3 ( a , b ) = a b F_{3}(a,b)=a^{b}
  89. F 4 ( a , b ) = a ( a ( b - 1 ) ) F_{4}(a,b)=a^{(a^{(b-1)})}
  90. F 5 ( a , b ) = ( x x x ( a - 1 ) ) b - 1 ( a ) F_{5}(a,b)=(x\mapsto x^{x^{(a-1)}})^{b-1}(a)
  91. F n + 1 ( a , b ) = exp ( F n ( ln ( a ) , ln ( b ) ) ) F_{n+1}(a,b)=\exp(F_{n}(\ln(a),\ln(b)))
  92. F 0 ( a , b ) = ln ( e a + e b ) F_{0}(a,b)=\ln(e^{a}+e^{b})
  93. F 1 ( a , b ) = a + b F_{1}(a,b)=a+b
  94. F 2 ( a , b ) = a b = e ln ( a ) + ln ( b ) F_{2}(a,b)=a\cdot b=e^{\ln(a)+\ln(b)}
  95. F 3 ( a , b ) = a ln ( b ) = e ln ( a ) ln ( b ) F_{3}(a,b)=a^{\ln(b)}=e^{\ln(a)\ln(b)}
  96. F 4 ( a , b ) = e e ln ( ln ( a ) ) ln ( ln ( b ) ) F_{4}(a,b)=e^{e^{\ln(\ln(a))\ln(\ln(b))}}

Hyperparameter_optimization.html

  1. C { 10 , 100 , 1000 } C\in\{10,100,1000\}
  2. γ { 0.1 , 0.2 , 0.5 , 1.0 } \gamma\in\{0.1,0.2,0.5,1.0\}

Hysteresis_(economics).html

  1. U t = U t - 1 + e t , U_{t}=U_{t-1}+e_{t},
  2. U t U_{t}
  3. e t e_{t}
  4. E t - 1 ( U t + τ ) = U t - 1 E_{t-1}(U_{t+\tau})=U_{t-1}
  5. τ = 0 , 1 , \tau=0,1,\ldots
  6. E t - 1 E_{t-1}
  7. e t e_{t}
  8. τ \tau
  9. E t - 1 ( U t + τ ) E_{t-1}(U_{t+\tau})
  10. e t e_{t}
  11. U t U_{t}
  12. E t - 1 U t + τ E_{t-1}U_{t+\tau}
  13. τ \tau

IBM_Award.html

  1. ( plyr PTS - plyr FGA + plyr REB + plyr AST + plyr STL + plyr BLK - plyr PF - plyr TO + ( team wins × 10 ) ) × 250 team PTS - team FGA + team REB + team AST + team STL + team BLK - team PF - team TO \frac{(\mathrm{plyr\ PTS}-\mathrm{plyr\ FGA}+\mathrm{plyr\ REB}+\mathrm{plyr\ % AST}+\mathrm{plyr\ STL}+\mathrm{plyr\ BLK}-\mathrm{plyr\ PF}-\mathrm{plyr\ TO}% +(\mathrm{team\ wins}\times 10))\times 250}{\mathrm{team\ PTS}-\mathrm{team\ % FGA}+\mathrm{team\ REB}+\mathrm{team\ AST}+\mathrm{team\ STL}+\mathrm{team\ % BLK}-\mathrm{team\ PF}-\mathrm{team\ TO}}

ICARUS_(experiment).html

  1. A 40 r + ν K 40 + e - . {}^{40}Ar+\nu\rightarrow{}^{40}K+e^{-}\,.

Ideal_lattice_cryptography.html

  1. [ x ] / f \mathbb{Z}[x]/\langle f\rangle
  2. f f
  3. n n
  4. R R
  5. n \mathbb{Z}^{n}
  6. \mathbb{Z}
  7. n n
  8. σ \sigma
  9. R R
  10. σ ( R ) \sigma(R)
  11. n n
  12. n \mathbb{R}^{n}
  13. R R
  14. σ \sigma
  15. σ ( I ) \sigma(I)
  16. I I
  17. R . R.
  18. f [ x ] f\in\mathbb{Z}[x]
  19. n n
  20. [ x ] / f \mathbb{Z}[x]/\langle f\rangle
  21. { ( g mod f ) : g [ x ] } \{(g\ \bmod\ \ f):g\in\mathbb{Z}[x]\}
  22. [ x ] / f \mathbb{Z}[x]/\langle f\rangle
  23. n \mathbb{Z}^{n}
  24. I [ x ] / f I\subseteq\mathbb{Z}[x]/\langle f\rangle
  25. ( I ) n \mathcal{L}(I)\subseteq\mathbb{Z}^{n}
  26. ( B ) n \mathcal{L}(B)\subseteq\mathbb{Z}^{n}
  27. B = { g mod f : g I } B=\{g\ \bmod\ f:g\in I\}
  28. f f
  29. n n
  30. I [ x ] / f I\subseteq\mathbb{Z}[x]/\langle f\rangle
  31. f f
  32. f f
  33. g f \lVert g\rVert_{f}
  34. g \lVert g\rVert_{\infty}
  35. g g
  36. [ x ] / f \mathbb{Z}[x]/\langle f\rangle
  37. n \mathbb{Z}^{n}
  38. I I
  39. [ x ] / f \mathbb{Z}[x]/\langle f\rangle
  40. f f
  41. n n
  42. n \mathbb{Z}^{n}
  43. n n
  44. B ( n , n ) B\in\mathbb{Z}^{(n,n)}
  45. 𝐪 \,\textbf{q}
  46. B B
  47. 𝐪 \,\textbf{q}
  48. B B
  49. A = adj ( B ) A={\rm adj}(B)
  50. d = det ( B ) d=\det(B)
  51. z = B ( n , n ) z=B_{(n,n)}
  52. P = A M B mod d P=AMB\bmod\ d
  53. c = P ( , n ) c=P_{(\centerdot,n)}
  54. z c i z\mid c_{i}
  55. i = 1 , , n i=1,\dots,n
  56. q ( c / z ) mod ( d / z ) q^{\ast}\equiv\ (c/z)\bmod\ (d/z)
  57. q 0 mod z q^{\ast}\equiv 0\bmod\ z
  58. B q 0 mod ( d / z ) Bq^{\ast}\equiv 0\bmod\ (d/z)
  59. q = B q / d q=Bq^{\ast}/d
  60. M = ( 0 . . . 0 . . I n - 1 . 0 ) M=\begin{pmatrix}0&.&.&.&0\\ &&&&.\\ &&&&.\\ I_{n-1}&&&&.\\ &&&&0\end{pmatrix}
  61. n = 2 n=2
  62. k { 0 , ± 1 } k\in\mathbb{Z}\setminus\{0,\pm 1\}
  63. B 1 = ( k 0 0 1 ) B_{1}=\begin{pmatrix}k&0\\ 0&1\end{pmatrix}
  64. B 2 = ( 1 0 0 k ) B_{2}=\begin{pmatrix}1&0\\ 0&k\end{pmatrix}
  65. B 2 B_{2}
  66. k = 2 k=2
  67. d = 2 d=2
  68. A = ( 2 0 0 1 ) , A=\begin{pmatrix}2&0\\ 0&1\end{pmatrix},
  69. M = ( 0 0 1 0 ) M=\begin{pmatrix}0&0\\ 1&0\end{pmatrix}
  70. P = A M B mod d P=AMB\bmod d
  71. P = ( 0 0 1 0 ) . P=\begin{pmatrix}0&0\\ 1&0\end{pmatrix}.
  72. P P
  73. B B
  74. [ x ] / ( x n - 1 ) \mathbb{Z}[x]/(x^{n}-1)
  75. O ~ ( n ) \tilde{O}(n)
  76. [ x ] / f ( x ) \mathbb{Z}[x]/f(x)
  77. R = p [ x ] / f R=\mathbb{Z}_{p}[x]/\langle f\rangle
  78. f p [ x ] f\in\mathbb{Z}_{p}[x]
  79. n n
  80. p p
  81. n 2 n^{2}
  82. m m
  83. a 1 , , a m R a_{1},\dots,a_{m}\in R
  84. m m
  85. m m
  86. h = ( a 1 , , a m ) R m h=(a_{1},\ldots,a_{m})\in R^{m}
  87. D m D^{m}
  88. D D
  89. R R
  90. R R
  91. b = ( b 1 , , b m ) D m b=(b_{1},...,b_{m})\in D^{m}
  92. h ( b ) = i = 1 m α i b i h(b)=\sum_{i=1}^{m}\alpha_{i}\centerdot b_{i}
  93. O ( m n log p ) = O ( n log n ) O(mn\log p)=O(n\log n)
  94. α i b i \alpha_{i}\centerdot b_{i}
  95. O ( n log n log log n ) O(n\log n\log\log n)
  96. f f
  97. m m
  98. O ( n log n log log n ) O(n\log n\log\log n)
  99. b b D m b\neq b^{\prime}\in D^{m}
  100. h ( b ) = h ( b ) h(b)=h(b^{\prime})
  101. h R m h\in R^{m}
  102. [ x ] / f \mathbb{Z}[x]/\langle f\rangle
  103. q , n , m , d q,n,m,d
  104. n m n\mid m
  105. n \in\mathbb{Z}^{n}
  106. m / n m/n
  107. a 1 , , a m / n a_{1},...,a_{m/n}
  108. q n \mathbb{Z}_{q}^{n}
  109. f A : { 0 , , d - 1 } m q n f_{A}:\{0,...,d-1\}^{m}\longrightarrow\mathbb{Z}_{q}^{n}
  110. f A ( y ) = [ F a 1 | | F a m / n ] y mod q f_{A}(y)=[F\ast a_{1}|...|F\ast a_{m/n}]y\bmod\ q
  111. n , m , q , d n,m,q,d
  112. n \mathbb{Z}^{n}
  113. A A
  114. A ( i ) = F a ( i ) A^{(i)}=F\ast a^{(i)}
  115. Λ q ( [ F a 1 | | F a m / n ] ) \Lambda_{q}^{\perp}([F\ast a_{1}|...|F\ast a_{m/n}])
  116. n n
  117. O ( n ) ) O(\sqrt{n}))
  118. f ( x ) = x n + f n x n - 1 + + f 1 [ x ] f(x)=x^{n}+f_{n}x^{n-1}+...+f_{1}\in\mathbb{Z}[x]
  119. ( - 1 , 0 , , 0 ) (-1,0,...,0)
  120. [ 𝐅 𝐮 ] 𝐯 n \lVert[\,\textbf{F}\ast\,\textbf{u}]\,\textbf{v}\rVert\leq{\sqrt{n}}
  121. x n - 1 x^{n}-1
  122. ( - 1 , 0 , , 0 ) (-1,0,...,0)
  123. ( x - 1 ) ( x n - 1 + x n - 2 + + x + 1 ) (x-1)(x^{n-1}+x^{n-2}+\cdots+x+1)
  124. ( - 1 , 0 , , 0 ) (-1,0,...,0)
  125. ( 1 , , 1 ) n (1,...,1)\in\mathbb{Z}^{n}
  126. n + 1 n+1
  127. ( 1 , 0 , , 0 ) n (1,0,...,0)\in\mathbb{Z}^{n}
  128. n n
  129. O ~ ( n ) \tilde{O}(n)
  130. [ x ] / f \mathbb{Z}[x]/\langle f\rangle
  131. f f
  132. 1 n 1^{n}
  133. f f\in\mathbb{Z}
  134. n n
  135. p ( ϕ n ) 3 p\longleftarrow(\phi n)^{3}
  136. m log n m\longleftarrow\lceil\log n\rceil
  137. R p [ x ] / f R\longleftarrow\mathbb{Z}_{p}[x]/\langle f\rangle
  138. i i
  139. D K i DK_{i}
  140. D L i DL_{i}
  141. D K i = { y ^ R m DK_{i}=\{\hat{y}\in R^{m}
  142. y ^ 5 i p 1 / m } \lVert\hat{y}\rVert_{\infty}\leq 5ip^{1/m}\}
  143. D L i = { y ^ R m DL_{i}=\{\hat{y}\in R^{m}
  144. y ^ 5 i n ϕ p 1 / m } \lVert\hat{y}\rVert_{\infty}\leq 5in\phi p^{1/m}\}
  145. h R , m h\in\mathcal{H}_{R,m}
  146. r { 0 , 1 } log 2 n r\in\{0,1\}^{\lfloor\log^{2}n\rfloor}
  147. r = 0 log 2 n r=0^{\lfloor\log^{2}n\rfloor}
  148. j = log 2 n j=\lfloor\log^{2}n\rfloor
  149. j j
  150. r r
  151. k ^ , l ^ \hat{k},\hat{l}
  152. D K j DK_{j}
  153. D L j DL_{j}
  154. ( k ^ , l ^ ) (\hat{k},\hat{l})
  155. ( h , h ( k ^ ) , h ( l ^ ) ) (h,h(\hat{k}),h(\hat{l}))
  156. z R z\in R
  157. z 1 \lVert z\rVert_{\infty}\leq 1
  158. ( k ^ , l ^ ) (\hat{k},\hat{l})
  159. s ^ k ^ z + l ^ \hat{s}\longleftarrow\hat{k}z+\hat{l}
  160. z z
  161. s ^ \hat{s}
  162. ( h , h ( k ^ ) , h ( l ^ ) ) (h,h(\hat{k}),h(\hat{l}))
  163. s ^ 10 ϕ p 1 / m n log 2 n \lVert\hat{s}\rVert_{\infty}\leq 10\phi p^{1/m}n\log^{2}n
  164. s ^ = k ^ z + l ^ \hat{s}=\hat{k}z+\hat{l}
  165. O ~ ( m ) \tilde{O}(m)
  166. q \mathbb{Z}_{q}
  167. ( 1 , 0 , , 0 ) n (1,0,\dots,0)\in\mathbb{Z}^{n}
  168. n n
  169. x n + 1 x^{n}+1
  170. q q
  171. 2 n 2n
  172. q - 1 q-1
  173. 𝐖 q n × n \,\textbf{W}\in\mathbb{Z}^{n\times n}_{q}
  174. q \mathbb{Z}_{q}
  175. a ~ ( 1 ) , , a ~ ( m / n ) \tilde{a}^{(1)},...,\tilde{a}^{(m/n)}
  176. m / n m/n
  177. q n \mathbb{Z}^{n}_{q}
  178. y { 0 , , d - 1 } m y\in\{0,...,d-1\}^{m}
  179. 𝐖 f A ( y ) mod q \,\textbf{W}^{\centerdot}f_{A}(y)\bmod\ q
  180. 𝐀 = [ 𝐅 α ( 1 ) , , 𝐅 α ( m / n ) ] \,\textbf{A}=[\,\textbf{F}\ast\alpha^{(1)},\ldots,\,\textbf{F}\ast\alpha^{(m/n% )}]
  181. α ( i ) = 𝐖 - 1 a ~ ( i ) m o d q \alpha^{(i)}=\,\textbf{W}^{-1}\tilde{a}^{(i)}\ mod\ q
  182. 𝐖 - 1 \,\textbf{W}^{-1}
  183. a ~ q n \tilde{a}\in\mathbb{Z}^{n}_{q}
  184. α q n \alpha\in\mathbb{Z}^{n}_{q}
  185. 𝐖 f A ( y ) = 𝐖 f A ( y ) ( m o d q ) \,\textbf{W}^{\centerdot}f_{A}(y)=\,\textbf{W}^{\centerdot}f_{A}(y^{\prime})\ % (mod\ q)
  186. f A ( y ) = f A ( y ) ( m o d q ) f_{A}(y)=f_{A}(y^{\prime})\ (mod\ q)
  187. f A f_{A}
  188. n , m , q , d n,m,q,d
  189. n n
  190. q q
  191. 2 n ( q - 1 ) 2n\mid(q-1)
  192. n m n\mid m
  193. m / n m/n
  194. a ~ 1 , , a ~ m / n \tilde{a}_{1},...,\tilde{a}_{m/n}
  195. q n \mathbb{Z}_{q}^{n}
  196. m / n m/n
  197. y ( 1 ) , , y ( m / n ) { 0 , , d - 1 } n y^{(1)},\dots,y^{(m/n)}\in\{0,\dots,d-1\}^{n}
  198. i = 1 m / n a ~ ( i ) ( 𝐖 y ( i ) ) q n \sum_{i=1}^{m/n}\tilde{a}^{(i)}\odot(\,\textbf{W}y^{(i)})\in\mathbb{Z}_{q}^{n}
  199. \odot
  200. f ( x ) = x n + 1 [ x ] f(x)=x^{n}+1\in\mathbb{Z}[x]
  201. n n
  202. f ( x ) f(x)
  203. f ( x ) f(x)
  204. R = [ x ] / f ( x ) R=\mathbb{Z}[x]/\langle f(x)\rangle
  205. f ( x ) f(x)
  206. R R
  207. f ( x ) f(x)
  208. n n
  209. q 1 mod 2 n q\equiv 1\bmod 2n
  210. n n
  211. R q = R / q = q [ x ] / f ( x ) R_{q}=R/\langle q\rangle=\mathbb{Z}_{q}[x]/\langle f(x)\rangle
  212. f ( x ) f(x)
  213. q q
  214. R q R_{q}
  215. n n
  216. { 0 , , q - 1 } \{0,\dots,q-1\}
  217. s = s ( x ) R q s=s(x)\in R_{q}
  218. ( a , b a s ) R q × R q (a,b\approx a\centerdot s)\in R_{q}\times R_{q}
  219. a s a\centerdot s
  220. R R
  221. R R
  222. s R q s\in R_{q}
  223. s s
  224. K K
  225. n n
  226. α = α ( n ) ( 0 , 1 ) \alpha=\alpha(n)\in(0,1)
  227. q = q ( n ) 2 q=q(n)\geq 2
  228. α q ω ( l o g n ) \alpha\centerdot q\geq\omega(\sqrt{logn})
  229. K K
  230. D G S γ DGS_{\gamma}
  231. 𝒪 K \mathcal{O}_{K}
  232. L W E q , Ψ α LWE_{q,\Psi\leq\alpha}
  233. γ = η ϵ ( I ) ω ( l o g n ) / α \gamma=\eta_{\epsilon}(I)\centerdot\omega(\sqrt{logn})/\alpha
  234. O ~ ( n 2 ) \tilde{O}(n^{2})
  235. O ~ ( n ) \tilde{O}(n)
  236. O ~ ( 1 ) \tilde{O}(1)
  237. Ω ~ ( n ) \tilde{\Omega}(n)
  238. O ~ ( n ) \tilde{O}(n)
  239. O ~ ( n 2 ) \tilde{O}(n^{2})
  240. ε = ( 𝖪𝖾𝗒𝖦𝖾𝗇 , 𝖤𝗇𝖼𝗋𝗒𝗉𝗍 , 𝖣𝖾𝖼𝗋𝗒𝗉𝗍 , 𝖤𝗏𝖺𝗅 ) \varepsilon=(\mathsf{KeyGen},\mathsf{Encrypt},\mathsf{Decrypt},\mathsf{Eval})
  241. 𝒞 \mathcal{C}
  242. C 𝒞 C\in\mathcal{C}
  243. P K , S K 𝖪𝖾𝗒𝖦𝖾𝗇 ( 1 λ ) PK,SK\leftarrow\mathsf{KeyGen}(1^{\lambda})
  244. y = 𝖤𝗇𝖼𝗋𝗒𝗉𝗍 ( P K , x ) y=\mathsf{Encrypt}(PK,x)
  245. y = 𝖤𝗏𝖺𝗅 ( P K , C , y ) y^{\prime}=\mathsf{Eval}(PK,C,y)
  246. 𝖣𝖾𝖼𝗋𝗒𝗉𝗍 ( S K , y ) = C ( x ) \mathsf{Decrypt}(SK,y^{\prime})=C(x)
  247. ε \varepsilon
  248. p o l y ( λ ) poly(\lambda)
  249. λ \lambda