wpmath0000007_5

Free_entropy.html

  1. S = 1 T U + P T V - i = 1 s μ i T N i S=\frac{1}{T}U+\frac{P}{T}V-\sum_{i=1}^{s}\frac{\mu_{i}}{T}N_{i}\,
  2. U , V , { N i } ~{}~{}~{}~{}~{}U,V,\{N_{i}\}\,
  3. Φ = S - 1 T U \Phi=S-\frac{1}{T}U
  4. = - A T =-\frac{A}{T}
  5. 1 T , V , { N i } ~{}~{}~{}~{}~{}\frac{1}{T},V,\{N_{i}\}\,
  6. Ξ = Φ - P T V \Xi=\Phi-\frac{P}{T}V
  7. = - G T =-\frac{G}{T}
  8. 1 T , P T , { N i } ~{}~{}~{}~{}~{}\frac{1}{T},\frac{P}{T},\{N_{i}\}\,
  9. S S
  10. Φ \Phi
  11. Ξ \Xi
  12. U U
  13. T T
  14. P P
  15. V V
  16. A A
  17. G G
  18. N i N_{i}
  19. μ i \mu_{i}
  20. s s
  21. i i
  22. i i
  23. ψ \psi
  24. ψ \psi
  25. S = S ( U , V , { N i } ) S=S(U,V,\{N_{i}\})
  26. d S = S U d U + S V d V + i = 1 s S N i d N i dS=\frac{\partial S}{\partial U}dU+\frac{\partial S}{\partial V}dV+\sum_{i=1}^% {s}\frac{\partial S}{\partial N_{i}}dN_{i}
  27. d S = 1 T d U + P T d V + i = 1 s ( - μ i T ) d N i dS=\frac{1}{T}dU+\frac{P}{T}dV+\sum_{i=1}^{s}(-\frac{\mu_{i}}{T})dN_{i}
  28. S = U T + p V T + i = 1 s ( - μ i N T ) S=\frac{U}{T}+\frac{pV}{T}+\sum_{i=1}^{s}(-\frac{\mu_{i}N}{T})
  29. Φ = S - U T \Phi=S-\frac{U}{T}
  30. Φ = U T + P V T + i = 1 s ( - μ i N T ) - U T \Phi=\frac{U}{T}+\frac{PV}{T}+\sum_{i=1}^{s}(-\frac{\mu_{i}N}{T})-\frac{U}{T}
  31. Φ = P V T + i = 1 s ( - μ i N T ) \Phi=\frac{PV}{T}+\sum_{i=1}^{s}(-\frac{\mu_{i}N}{T})
  32. Φ \Phi
  33. d Φ = d S - 1 T d U - U d 1 T d\Phi=dS-\frac{1}{T}dU-Ud\frac{1}{T}
  34. d Φ = 1 T d U + P T d V + i = 1 s ( - μ i T ) d N i - 1 T d U - U d 1 T d\Phi=\frac{1}{T}dU+\frac{P}{T}dV+\sum_{i=1}^{s}(-\frac{\mu_{i}}{T})dN_{i}-% \frac{1}{T}dU-Ud\frac{1}{T}
  35. d Φ = - U d 1 T + P T d V + i = 1 s ( - μ i T ) d N i d\Phi=-Ud\frac{1}{T}+\frac{P}{T}dV+\sum_{i=1}^{s}(-\frac{\mu_{i}}{T})dN_{i}
  36. d Φ d\Phi
  37. Φ = Φ ( 1 T , V , { N i } ) \Phi=\Phi(\frac{1}{T},V,\{N_{i}\})
  38. d Φ = d S - T d U - U d T T 2 d\Phi=dS-\frac{TdU-UdT}{T^{2}}
  39. d Φ = d S - 1 T d U + U T 2 d T d\Phi=dS-\frac{1}{T}dU+\frac{U}{T^{2}}dT
  40. d Φ = 1 T d U + P T d V + i = 1 s ( - μ i T ) d N i - 1 T d U + U T 2 d T d\Phi=\frac{1}{T}dU+\frac{P}{T}dV+\sum_{i=1}^{s}(-\frac{\mu_{i}}{T})dN_{i}-% \frac{1}{T}dU+\frac{U}{T^{2}}dT
  41. d Φ = U T 2 d T + P T d V + i = 1 s ( - μ i T ) d N i d\Phi=\frac{U}{T^{2}}dT+\frac{P}{T}dV+\sum_{i=1}^{s}(-\frac{\mu_{i}}{T})dN_{i}
  42. Φ = Φ ( T , V , { N i } ) \Phi=\Phi(T,V,\{N_{i}\})
  43. Ξ = Φ - P V T \Xi=\Phi-\frac{PV}{T}
  44. Ξ = P V T + i = 1 s ( - μ i N T ) - P V T \Xi=\frac{PV}{T}+\sum_{i=1}^{s}(-\frac{\mu_{i}N}{T})-\frac{PV}{T}
  45. Ξ = i = 1 s ( - μ i N T ) \Xi=\sum_{i=1}^{s}(-\frac{\mu_{i}N}{T})
  46. Ξ \Xi
  47. d Ξ = d Φ - P T d V - V d P T d\Xi=d\Phi-\frac{P}{T}dV-Vd\frac{P}{T}
  48. d Ξ = - U d 2 T + P T d V + i = 1 s ( - μ i T ) d N i - P T d V - V d P T d\Xi=-Ud\frac{2}{T}+\frac{P}{T}dV+\sum_{i=1}^{s}(-\frac{\mu_{i}}{T})dN_{i}-% \frac{P}{T}dV-Vd\frac{P}{T}
  49. d Ξ = - U d 1 T - V d P T + i = 1 s ( - μ i T ) d N i d\Xi=-Ud\frac{1}{T}-Vd\frac{P}{T}+\sum_{i=1}^{s}(-\frac{\mu_{i}}{T})dN_{i}
  50. d Ξ d\Xi
  51. Ξ = Ξ ( 1 T , P T , { N i } ) \Xi=\Xi(\frac{1}{T},\frac{P}{T},\{N_{i}\})
  52. d Ξ = d Φ - T ( P d V + V d P ) - P V d T T 2 d\Xi=d\Phi-\frac{T(PdV+VdP)-PVdT}{T^{2}}
  53. d Ξ = d Φ - P T d V - V T d P + P V T 2 d T d\Xi=d\Phi-\frac{P}{T}dV-\frac{V}{T}dP+\frac{PV}{T^{2}}dT
  54. d Ξ = U T 2 d T + P T d V + i = 1 s ( - μ i T ) d N i - P T d V - V T d P + P V T 2 d T d\Xi=\frac{U}{T^{2}}dT+\frac{P}{T}dV+\sum_{i=1}^{s}(-\frac{\mu_{i}}{T})dN_{i}-% \frac{P}{T}dV-\frac{V}{T}dP+\frac{PV}{T^{2}}dT
  55. d Ξ = U + P V T 2 d T - V T d P + i = 1 s ( - μ i T ) d N i d\Xi=\frac{U+PV}{T^{2}}dT-\frac{V}{T}dP+\sum_{i=1}^{s}(-\frac{\mu_{i}}{T})dN_{i}
  56. Ξ = Ξ ( T , P , { N i } ) \Xi=\Xi(T,P,\{N_{i}\})

Free_water_clearance.html

  1. V = C o s m + C H 2 O V=C_{osm}+C_{H_{2}O}
  2. C H 2 O = V - C o s m C_{H_{2}O}=V-C_{osm}
  3. C H 2 O = V - U o s m P o s m V C_{H_{2}O}=V-\frac{U_{osm}}{P_{osm}}V
  4. C H 2 O = 4 ml/min - 140 mOsm/L 280 mOsm/L × 4 ml/min = 2 ml/min C_{H_{2}O}=4\ \mbox{ml/min}~{}-\frac{140\ \mbox{mOsm/L}~{}}{280\ \mbox{mOsm/L}% ~{}}\times 4\ \mbox{ml/min}~{}=2\ \mbox{ml/min}~{}

Frege's_theorem.html

  1. ( P ( Q R ) ) ( ( P Q ) ( P R ) ) (P\to(Q\to R))\to((P\to Q)\to(P\to R))

Freivalds'_algorithm.html

  1. 2 - k 2^{-k}
  2. r \vec{r}
  3. P = A × ( B r ) - C r \vec{P}=A\times(B\vec{r})-C\vec{r}
  4. P = ( 0 , 0 , , 0 ) T \vec{P}=(0,0,\ldots,0)^{T}
  5. A B = [ 2 3 3 4 ] [ 1 0 1 2 ] = ? [ 6 5 8 7 ] = C . AB=\begin{bmatrix}2&3\\ 3&4\end{bmatrix}\begin{bmatrix}1&0\\ 1&2\end{bmatrix}\stackrel{?}{=}\begin{bmatrix}6&5\\ 8&7\end{bmatrix}=C.
  6. r = [ 1 1 ] \vec{r}=\begin{bmatrix}1\\ 1\end{bmatrix}
  7. A × ( B r ) - C r \displaystyle A\times(B\vec{r})-C\vec{r}
  8. r = [ 1 0 ] \vec{r}=\begin{bmatrix}1\\ 0\end{bmatrix}
  9. A × ( B r ) - C r = [ 2 3 3 4 ] ( [ 1 0 1 2 ] [ 1 0 ] ) - [ 6 5 8 7 ] [ 1 0 ] = [ - 1 - 1 ] . A\times(B\vec{r})-C\vec{r}=\begin{bmatrix}2&3\\ 3&4\end{bmatrix}\left(\begin{bmatrix}1&0\\ 1&2\end{bmatrix}\begin{bmatrix}1\\ 0\end{bmatrix}\right)-\begin{bmatrix}6&5\\ 8&7\end{bmatrix}\begin{bmatrix}1\\ 0\end{bmatrix}=\begin{bmatrix}-1\\ -1\end{bmatrix}.
  10. r = [ 0 0 ] \vec{r}=\begin{bmatrix}0\\ 0\end{bmatrix}
  11. r = [ 1 1 ] \vec{r}=\begin{bmatrix}1\\ 1\end{bmatrix}
  12. P = A × ( B r ) - C r = ( A × B ) r - C r = ( A × B - C ) r = 0 \begin{aligned}\displaystyle\vec{P}&\displaystyle=A\times(B\vec{r})-C\vec{r}\\ &\displaystyle=(A\times B)\vec{r}-C\vec{r}\\ &\displaystyle=(A\times B-C)\vec{r}\\ &\displaystyle=\vec{0}\end{aligned}
  13. r \vec{r}
  14. A × B - C = 0 A\times B-C=0
  15. Pr [ P 0 ] = 0 \Pr[\vec{P}\neq 0]=0
  16. P = D × r = ( p 1 , p 2 , , p n ) T \vec{P}=D\times\vec{r}=(p_{1},p_{2},\dots,p_{n})^{T}
  17. D = A × B - C = ( d i j ) D=A\times B-C=(d_{ij})
  18. A × B C A\times B\neq C
  19. D D
  20. d i j 0 d_{ij}\neq 0
  21. p i = k = 1 n d i k r k = d i 1 r 1 + + d i j r j + + d i n r n = d i j r j + y p_{i}=\sum_{k=1}^{n}d_{ik}r_{k}=d_{i1}r_{1}+\cdots+d_{ij}r_{j}+\cdots+d_{in}r_% {n}=d_{ij}r_{j}+y
  22. y y
  23. y y
  24. Pr [ p i = 0 | y = 0 ] = Pr [ r j = 0 ] = 1 2 \Pr[p_{i}=0|y=0]=\Pr[r_{j}=0]=\frac{1}{2}
  25. Pr [ p i = 0 | y 0 ] = Pr [ r j = 1 and d i j = - y ] Pr [ r j = 1 ] = 1 2 \Pr[p_{i}=0|y\neq 0]=\Pr[r_{j}=1\and d_{ij}=-y]\leq\Pr[r_{j}=1]=\frac{1}{2}
  26. Pr [ p i = 0 ] \displaystyle\Pr[p_{i}=0]
  27. Pr [ P = 0 ] = Pr [ p 0 = 0 and p 1 = 0 and ] Pr [ p i = 0 ] 1 2 . \Pr[\vec{P}=0]=\Pr[p_{0}=0\and p_{1}=0\and\dots]\leq\Pr[p_{i}=0]\leq\frac{1}{2}.
  28. 1 2 k \frac{1}{2^{k}}

Frenkel_defect.html

  1. M g M g × {Mg}^{\times}_{Mg}
  2. O O × {O}^{\times}_{O}
  3. O i ′′ {O}^{{}^{\prime\prime}}_{i}
  4. V O {V}^{\bullet\bullet}_{O}
  5. M g M g × {Mg}^{\times}_{Mg}

Frequency_(statistics).html

  1. i i
  2. n i n_{i}
  3. f i = n i N = n i j n j . f_{i}=\frac{n_{i}}{N}=\frac{n_{i}}{\sum_{j}n_{j}}.
  4. f i f_{i}
  5. i i

Frequency_comb.html

  1. f ( n ) = f 0 + n f r f(n)=f_{0}+n\,f_{r}
  2. n n
  3. f r f_{r}
  4. f 0 f_{0}
  5. f r f_{r}
  6. f 0 f_{0}
  7. f r f_{r}
  8. f 0 f_{0}
  9. f 1 , f 2 , f 3 f_{1},f_{2},f_{3}
  10. f 4 = f 1 + f 2 - f 3 f_{4}=f_{1}+f_{2}-f_{3}
  11. f 1 , f 2 f_{1},f_{2}
  12. 2 f 1 - f 2 2f_{1}-f_{2}

Fréchet_manifold.html

  1. ϕ α β := ϕ α ϕ β - 1 | ϕ β ( U β U α ) \phi_{\alpha\beta}:=\phi_{\alpha}\circ\phi_{\beta}^{-1}|_{\phi_{\beta}(U_{% \beta}\cap U_{\alpha})}

Friendly_number.html

  1. σ - 1 ( n ) \sigma_{-\!1}(n)
  2. σ k \sigma_{k}
  3. σ k ( n ) \sigma_{k}(n)
  4. σ ( 30 ) 30 = 1 + 2 + 3 + 5 + 6 + 10 + 15 + 30 30 = 72 30 = 12 5 \tfrac{\sigma(30)}{30}=\tfrac{1+2+3+5+6+10+15+30}{30}=\tfrac{72}{30}=\tfrac{12% }{5}
  5. σ ( 140 ) 140 = 1 + 2 + 4 + 5 + 7 + 10 + 14 + 20 + 28 + 35 + 70 + 140 140 = 336 140 = 12 5 . \tfrac{\sigma(140)}{140}=\tfrac{1+2+4+5+7+10+14+20+28+35+70+140}{140}=\tfrac{3% 36}{140}=\tfrac{12}{5}.

Frobenius_theorem_(real_division_algebras).html

  1. 𝐑 \mathbf{R}
  2. 𝐂 \mathbf{C}
  3. 𝐇 \mathbf{H}
  4. 1 , 2 1,2
  5. 4 4
  6. 𝐑 \mathbf{R}
  7. 𝐂 \mathbf{C}
  8. 𝐇 \mathbf{H}
  9. 1 1
  10. 𝐑 \mathbf{R}
  11. a 0 a≤0
  12. a a
  13. D D
  14. a a
  15. 𝐑 \mathbf{R}
  16. D D
  17. 𝐑 \mathbf{R}
  18. d d
  19. D D
  20. D D
  21. d d
  22. d d
  23. z z
  24. 𝐂 \mathbf{C}
  25. Q ( z ; x ) = x 2 - 2 Re ( z ) x + | z | 2 = ( x - z ) ( x - z ¯ ) 𝐑 [ x ] . Q(z;x)=x^{2}-2\operatorname{Re}(z)x+|z|^{2}=(x-z)(x-\overline{z})\in\mathbf{R}% [x].
  26. z 𝐂 𝐑 z∈\mathbf{C}\ \mathbf{R}
  27. Q ( z ; x ) Q(z;x)
  28. 𝐑 \mathbf{R}
  29. V V
  30. a a
  31. D D
  32. D D
  33. 1 1
  34. D = 𝐑 V D=\mathbf{R}⊕V
  35. 𝐑 \mathbf{R}
  36. V V
  37. D D
  38. m m
  39. D D
  40. 𝐑 \mathbf{R}
  41. a a
  42. D D
  43. p ( x ) p(x)
  44. p ( x ) = ( x - t 1 ) ( x - t r ) ( x - z 1 ) ( x - z 1 ¯ ) ( x - z s ) ( x - z s ¯ ) , t i 𝐑 , z j 𝐂 \ 𝐑 . p(x)=(x-t_{1})\cdots(x-t_{r})(x-z_{1})(x-\overline{z_{1}})\cdots(x-z_{s})(x-% \overline{z_{s}}),\qquad t_{i}\in\mathbf{R},\quad z_{j}\in\mathbf{C}\backslash% \mathbf{R}.
  45. p ( x ) p(x)
  46. Q ( z ; x ) Q(z;x)
  47. p ( x ) = ( x - t 1 ) ( x - t r ) Q ( z 1 ; x ) Q ( z s ; x ) . p(x)=(x-t_{1})\cdots(x-t_{r})Q(z_{1};x)\cdots Q(z_{s};x).
  48. 𝐑 \mathbf{R}
  49. p ( a ) = 0 p(a)=0
  50. D D
  51. i i
  52. j j
  53. a a
  54. a a
  55. p ( x ) p(x)
  56. p ( x ) = Q ( z j ; x ) k = ( x 2 - 2 Re ( z j ) x + | z j | 2 ) k p(x)=Q(z_{j};x)^{k}=\left(x^{2}-2\operatorname{Re}(z_{j})x+|z_{j}|^{2}\right)^% {k}
  57. p ( x ) p(x)
  58. a a
  59. p ( x ) p(x)
  60. t r ( a ) tr(a)
  61. t r ( a ) = 0 tr(a)=0
  62. t r ( a ) = 0 tr(a)=0
  63. V V
  64. a a
  65. t r ( a ) = 0 tr(a)=0
  66. V V
  67. 1 1
  68. D D
  69. 𝐑 \mathbf{R}
  70. V V
  71. a , b a,b
  72. V V
  73. B ( a , b ) = ( a b b a ) / 2 B(a,b)=(−ab−ba)/2
  74. B ( a , b ) B(a,b)
  75. B ( a , a ) > 0 B(a,a)>0
  76. a 0 a≠0
  77. B B
  78. V V
  79. W W
  80. V V
  81. D D
  82. W W
  83. B −B
  84. e i 2 = - 1 , e i e j = - e j e i . e_{i}^{2}=-1,\quad e_{i}e_{j}=-e_{j}e_{i}.
  85. n = 0 n=0
  86. D D
  87. 𝐑 \mathbf{R}
  88. n = 1 n=1
  89. D D
  90. 1 1
  91. e [ u s u , u b = 1 , u p = 2 ] = 1 e[u^{\prime}su^{\prime},u^{\prime}b=1^{\prime},u^{\prime}p=2^{\prime}]=−1
  92. 𝐂 \mathbf{C}
  93. n = 2 n=2
  94. D D
  95. e 1 2 = e 2 2 = - 1 , e 1 e 2 = - e 2 e 1 , ( e 1 e 2 ) ( e 1 e 2 ) = - 1. e_{1}^{2}=e_{2}^{2}=-1,\quad e_{1}e_{2}=-e_{2}e_{1},\quad(e_{1}e_{2})(e_{1}e_{% 2})=-1.
  96. 𝐇 \mathbf{H}
  97. n > 2 n>2
  98. D D
  99. n > 2 n>2
  100. u = e < s u b > 1 e 2 e n u=e<sub>1e_{2}e_{n}

Frölicher–Nijenhuis_bracket.html

  1. Ω * ( M ) = k = 0 Ω k ( M ) . \Omega^{*}(M)=\bigoplus_{k=0}^{\infty}\Omega^{k}(M).
  2. D : Ω * ( M ) Ω * + l ( M ) D:\Omega^{*}(M)\to\Omega^{*+l}(M)
  3. D ( α β ) = D ( α ) β + ( - 1 ) deg ( α ) α D ( β ) . D(\alpha\wedge\beta)=D(\alpha)\wedge\beta+(-1)^{\ell\deg(\alpha)}\alpha\wedge D% (\beta).
  4. Der Ω * ( M ) = k = - Der k Ω * ( M ) . \mathrm{Der}\,\Omega^{*}(M)=\bigoplus_{k=-\infty}^{\infty}\mathrm{Der}_{k}\,% \Omega^{*}(M).
  5. [ D 1 , D 2 ] = D 1 D 2 - ( - 1 ) d 1 d 2 D 2 D 1 . [D_{1},D_{2}]=D_{1}\circ D_{2}-(-1)^{d_{1}d_{2}}D_{2}\circ D_{1}.
  6. i K ω ( X 1 , , X k + - 1 ) = 1 k ! ( - 1 ) ! σ S k + - 1 sign σ ω ( K ( X σ ( 1 ) , , X σ ( k ) ) , X σ ( k + 1 ) , , X σ ( k + - 1 ) ) i_{K}\,\omega(X_{1},\dots,X_{k+\ell-1})=\frac{1}{k!(\ell-1)!}\sum_{\sigma\in{S% }_{k+\ell-1}}\textrm{sign}\,\sigma\cdot\omega(K(X_{\sigma(1)},\dots,X_{\sigma(% k)}),X_{\sigma(k+1)},\dots,X_{\sigma(k+\ell-1)})
  7. K = [ d , i K ] = d i K - ( - 1 ) k - 1 i K d \mathcal{L}_{K}=[d,i_{K}]=d\,{\circ}\,i_{K}-(-1)^{k-1}i_{K}{\circ}\,d
  8. [ , ] : Ω k ( M , T M ) × Ω ( M , T M ) Ω k + ( M , T M ) : ( K , L ) [ K , L ] [\cdot,\cdot]:\Omega^{k}(M,\mathrm{T}M)\times\Omega^{\ell}(M,\mathrm{T}M)\to% \Omega^{k+\ell}(M,\mathrm{T}M):(K,L)\mapsto[K,L]
  9. [ K , L ] = [ K , L ] . \mathcal{L}_{[K,L]}=[\mathcal{L}_{K},\mathcal{L}_{L}].
  10. K = [ d , i K ] = d i K + i K d . \mathcal{L}_{K}=[d,i_{K}]=d\,{\circ}\,i_{K}+i_{K}\,{\circ}\,d.
  11. ϕ X \phi\otimes X
  12. ψ Y \psi\otimes Y
  13. [ ϕ X , ψ Y ] = ϕ ψ [ X , Y ] + ϕ X ψ Y - Y ϕ ψ X + ( - 1 ) deg ( ϕ ) ( d ϕ i X ( ψ ) Y + i Y ( ϕ ) d ψ X ) . \left.\right.[\phi\otimes X,\psi\otimes Y]=\phi\wedge\psi\otimes[X,Y]+\phi% \wedge\mathcal{L}_{X}\psi\otimes Y-\mathcal{L}_{Y}\phi\wedge\psi\otimes X+(-1)% ^{\deg(\phi)}(d\phi\wedge i_{X}(\psi)\otimes Y+i_{Y}(\phi)\wedge d\psi\otimes X).
  14. i L + K i_{L}+\mathcal{L}_{K}
  15. K \mathcal{L}_{K}
  16. [ K 1 , K 2 ] = [ K 1 , K 2 ] [\mathcal{L}_{K_{1}},\mathcal{L}_{K_{2}}]=\mathcal{L}_{[K_{1},K_{2}]}
  17. Ω ( M , T M ) \Omega(M,\mathrm{T}M)
  18. i L i_{L}
  19. [ i L 1 , i L 2 ] = i [ L 1 , L 2 ] and [i_{L_{1}},i_{L_{2}}]=i_{[L_{1},L_{2}]^{\and}}
  20. [ K , i L ] = i [ K , L ] - ( - 1 ) k l i L K [\mathcal{L}_{K},i_{L}]=i_{[K,L]}-(-1)^{kl}\mathcal{L}_{i_{L}K}

Fructosamine.html

  1. HbA1c = 0.017 × Fructosamine + 1.61 {\rm HbA1c}={\rm 0.017}\times{\rm Fructosamine}+{\rm 1.61}
  2. Fructosamine = ( HbA1c - 1.61 ) × 58.82 {\rm Fructosamine}=({\rm HbA1c}-1.61)\times{\rm 58.82}

Fuel_economy_in_automobiles.html

  1. F = d W d s Fuel economy F=\frac{dW}{ds}\propto\,\text{Fuel economy}

Fuel_saving_device.html

  1. h = 1 - 1 r v g - 1 h=1-{1\over rv^{g-1}}

Fully_normalized_subgroup.html

  1. H H
  2. G G
  3. σ \sigma
  4. H H
  5. g G g\in G
  6. x g x g - 1 x\mapsto gxg^{-1}
  7. H H
  8. σ \sigma

Function_and_Concept.html

  1. F F
  2. F F
  3. f f
  4. x y f ( x , y ) z ( f ( x , z ) y = z ) \forall x\forall yf(x,y)\rightarrow\forall z(f(x,z)\rightarrow y=z)

Fundamental_matrix_(computer_vision).html

  1. 𝐅 \mathbf{F}
  2. 𝐱 𝐅𝐱 = 0. \mathbf{x}^{\prime\top}\mathbf{Fx}=0.
  3. 𝐅 \mathbf{F}
  4. 𝐄 \mathbf{E}
  5. 𝐄 = 𝐊 𝐅 𝐊 \mathbf{E}=\mathbf{K}^{\prime\top}\;\mathbf{F}\;\mathbf{K}
  6. 𝐊 \mathbf{K}
  7. 𝐊 \mathbf{K}^{\prime}
  8. 𝐱 𝐱 \mathbf{x}\leftrightarrow\mathbf{x^{\prime}}
  9. 𝐗 \,\textbf{X}
  10. ( 𝐏 , 𝐏 ) \left(\,\textbf{P},\,\textbf{P}^{\prime}\right)
  11. 𝐱 = 𝐏 𝐗 𝐱 = 𝐏 𝐗 \begin{aligned}\displaystyle\mathbf{x}&\displaystyle=\,\textbf{P}\,\textbf{X}% \\ \displaystyle\mathbf{x^{\prime}}&\displaystyle=\,\textbf{P}^{\prime}\,\textbf{% X}\end{aligned}
  12. 𝐇 4 × 4 \,\textbf{H}_{4\times 4}
  13. 𝐗 0 = 𝐇 𝐗 \,\textbf{X}_{0}=\,\textbf{H}\,\textbf{X}
  14. 𝐏 0 = 𝐏 𝐇 - 1 𝐏 0 = 𝐏 𝐇 - 1 \begin{aligned}\displaystyle\,\textbf{P}_{0}&\displaystyle=\,\textbf{P}\,% \textbf{H}^{-1}\\ \displaystyle\,\textbf{P}_{0}^{\prime}&\displaystyle=\,\textbf{P}^{\prime}\,% \textbf{H}^{-1}\end{aligned}
  15. 𝐏 0 𝐗 0 = 𝐏 𝐇 - 1 𝐇 𝐗 = 𝐏 𝐗 = 𝐱 \,\textbf{P}_{0}\,\textbf{X}_{0}=\,\textbf{P}\,\textbf{H}^{-1}\,\textbf{H}\,% \textbf{X}=\,\textbf{P}\,\textbf{X}=\mathbf{x}
  16. 𝐏 0 \,\textbf{P}_{0}^{\prime}

Fundamental_unit_(number_theory).html

  1. K = 𝐐 ( d ) K=\mathbf{Q}(\sqrt{d})
  2. ε > 1 ε>1
  3. ε = a + b Δ 2 \varepsilon=\frac{a+b\sqrt{\Delta}}{2}
  4. x 2 - Δ y 2 = ± 4 x^{2}-\Delta y^{2}=\pm 4
  5. Δ \sqrt{\Delta}
  6. Δ \sqrt{\Delta}
  7. lim X D - ( x ) D ( x ) = 1 - j 1 odd ( 1 - 2 - j ) . \lim_{X\rightarrow\infty}\frac{D^{-}(x)}{D(x)}=1-\prod_{j\geq 1\,\text{ odd}}% \left(1-2^{-j}\right).
  8. ϵ 3 > | Δ | - 27 4 . \epsilon^{3}>\frac{|\Delta|-27}{4}.
  9. 𝐐 ( 2 3 ) \mathbf{Q}(\sqrt[3]{2})
  10. 1 + 2 3 + 2 2 3 1+\sqrt[3]{2}+\sqrt[3]{2^{2}}
  11. | Δ | - 27 4 = 20.25. \frac{|\Delta|-27}{4}=20.25.

Fundamental_vector_field.html

  1. M M
  2. X X
  3. X X
  4. p M p\in M
  5. γ p : M \gamma_{p}:\mathbb{R}\to M
  6. γ p ( t ) = X γ p ( t ) , γ p ( 0 ) = p , \gamma_{p}^{\prime}(t)=X_{\gamma_{p}(t)},\qquad\gamma_{p}(0)=p,
  7. X X
  8. X X
  9. X X
  10. M M
  11. ϕ X : × M M \phi_{X}:\mathbb{R}\times M\to M
  12. ϕ X ( t , p ) = γ p ( t ) \phi_{X}(t,p)=\gamma_{p}(t)
  13. ( , + ) (\mathbb{R},+)
  14. M M
  15. A : × M M A:\mathbb{R}\times M\to M
  16. X X
  17. X p = d d t | t = 0 A ( t , p ) . X_{p}=\left.\frac{d}{dt}\right|_{t=0}A(t,p).
  18. \mathbb{R}
  19. M M
  20. M M
  21. X X
  22. M M
  23. G G
  24. 𝔤 \mathfrak{g}
  25. M M
  26. A : G × M M A:G\times M\to M
  27. A p : G M A_{p}:G\to M
  28. A p ( g ) = A ( g , p ) A_{p}(g)=A(g,p)
  29. A A
  30. p p
  31. X 𝔤 X\in\mathfrak{g}
  32. X # X^{\#}
  33. X X
  34. X p # = d e A p ( X ) X^{\#}_{p}=d_{e}A_{p}(X)
  35. X p # = d ( e , p ) A ( X , 0 T p M ) X^{\#}_{p}=d_{(e,p)}A\left(X,0_{T_{p}M}\right)
  36. X p # = d d t | t = 0 A ( exp ( t X ) , p ) X^{\#}_{p}=\left.\frac{d}{dt}\right|_{t=0}A\left(\exp(tX),p\right)
  37. d d
  38. 0 T p M 0_{T_{p}M}
  39. T p M T_{p}M
  40. 𝔤 Γ ( T M ) , X X # \mathfrak{g}\to\Gamma(TM),X\mapsto X^{\#}
  41. G G
  42. G G
  43. T e G T_{e}G
  44. G G
  45. \mathbb{R}
  46. ( M , ω ) (M,\omega)
  47. X H X_{H}
  48. H : M H:M\to\mathbb{R}
  49. d H = ι X H ω dH=\iota_{X_{H}}\omega
  50. ι \iota
  51. G G
  52. 𝔤 \mathfrak{g}
  53. A : G × M M A:G\times M\to M
  54. G G
  55. M M
  56. A A
  57. μ : M 𝔤 * \mu:M\to\mathfrak{g}^{*}
  58. X 𝔤 X\in\mathfrak{g}
  59. d μ X = ι X # ω , d\mu^{X}=\iota_{X^{\#}}\omega,
  60. μ X : M , p μ ( p ) , X \mu^{X}:M\to\mathbb{R},p\mapsto\langle\mu(p),X\rangle
  61. X # X^{\#}
  62. X X

Gabor_transform.html

  1. G x ( t , f ) = - e - π ( τ - t ) 2 e - j 2 π f τ x ( τ ) d τ G_{x}(t,f)=\int_{-\infty}^{\infty}e^{-\pi(\tau-t)^{2}}e^{-j2\pi f\tau}x(\tau)% \,d\tau
  2. { e - π a 2 0.00001 ; | a | 1.9143 e - π a 2 < 0.00001 ; | a | > 1.9143 \begin{cases}e^{-{\pi}a^{2}}\geq 0.00001;&\left|a\right|\leq 1.9143\\ e^{-{\pi}a^{2}}<0.00001;&\left|a\right|>1.9143\end{cases}
  3. | a | > 1.9143 \left|a\right|>1.9143
  4. G x ( t , f ) = - 1.9143 + t 1.9143 + t e - π ( τ - t ) 2 e - j 2 π f τ x ( τ ) d τ G_{x}(t,f)=\int_{-1.9143+t}^{1.9143+t}e^{-\pi(\tau-t)^{2}}e^{-j2\pi f\tau}x(% \tau)\,d\tau
  5. - π ( τ - t ) 2 {-{\pi}(\tau-t)^{2}}
  6. - π α ( τ - t ) 2 {-{\pi}\alpha(\tau-t)^{2}}
  7. x ( t ) = - - G x ( τ , f ) e j 2 π t f d f d τ x(t)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}G_{x}(\tau,f)e^{j2\pi tf}\,% df\,d\tau
  8. x ( t ) x(t)\,
  9. G x ( t , f ) = - e - π ( τ - t ) 2 e - j 2 π f τ x ( τ ) d τ G_{x}(t,f)=\int_{-\infty}^{\infty}e^{-\pi(\tau-t)^{2}}e^{-j2\pi f\tau}x(\tau)% \,d\tau
  10. a x ( t ) + b y ( t ) a\cdot x(t)+b\cdot y(t)\,
  11. a G x ( t , f ) + b G y ( t , f ) a\cdot G_{x}(t,f)+b\cdot G_{y}(t,f)\,
  12. x ( t - t 0 ) x(t-t_{0})\,
  13. G x ( t - t 0 , f ) e - j 2 π f t 0 G_{x}(t-t_{0},f)e^{-j2\pi ft_{0}}\,
  14. x ( t ) e j 2 π f 0 t x(t)e^{j2\pi f_{0}t}\,
  15. G x ( t , f - f 0 ) G_{x}(t,f-f_{0})\,
  16. - | G x ( t , f ) | 2 d f = - e - 2 π ( τ - t ) 2 | x ( τ ) | 2 d τ u - 1.9143 u + 1.9143 e - 2 π ( τ - u ) 2 | x ( τ ) | 2 d τ \int_{-\infty}^{\infty}\left|G_{x}(t,f)\right|^{2}\,df=\int_{-\infty}^{\infty}% e^{-2\pi(\tau-t)^{2}}\left|x(\tau)\right|^{2}d\tau\approx\int_{u-1.9143}^{u+1.% 9143}e^{-2\pi(\tau-u)^{2}}\left|x(\tau)\right|^{2}d\tau
  17. - - G x ( t , f ) G y * ( t , f ) d f d t = - x ( τ ) y * ( τ ) d τ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}G_{x}(t,f)G_{y}^{*}(t,f)\,df\,dt% =\int_{-\infty}^{\infty}x(\tau)y^{*}(\tau)\,d\tau
  18. { - | G x ( t , f ) | 2 d f < e - 2 π ( t - t 0 ) 2 - | G x ( t 0 , f ) | 2 d f ; if x ( t ) = 0 for t > t 0 - | G x ( t , f ) | 2 d t < e - 2 π ( f - f 0 ) 2 - | G x ( t , f 0 ) | 2 d t ; if X ( f ) = F T [ x ( t ) ] = 0 for f > f 0 \begin{cases}\displaystyle\int_{-\infty}^{\infty}\left|G_{x}(t,f)\right|^{2}df% <e^{-2\pi(t-t_{0})^{2}}\int_{-\infty}^{\infty}\left|G_{x}(t_{0},f)\right|^{2}% \,df;&\,\text{if }x(t)=0\,\text{ for }t>t_{0}\\ \displaystyle\int_{-\infty}^{\infty}\left|G_{x}(t,f)\right|^{2}\,dt<e^{-2\pi(f% -f_{0})^{2}}\int_{-\infty}^{\infty}\left|G_{x}(t,f_{0})\right|^{2}\,dt;&\,% \text{if }X(f)=FT[x(t)]=0\,\text{ for }f>f_{0}\end{cases}
  19. - G x ( t , f ) e j 2 π k t f d f = e - π ( k - 1 ) 2 t 2 x ( k t ) \int_{-\infty}^{\infty}G_{x}(t,f)e^{j2\pi ktf}\,df=e^{-\pi(k-1)^{2}t^{2}}x(kt)
  20. - G x ( t , f ) e j 2 π t f d f = x ( t ) \int_{-\infty}^{\infty}G_{x}(t,f)e^{j2\pi tf}\,df=x(t)
  21. x ( t ) = { cos ( 2 π t ) for t 0 , cos ( 4 π t ) for t > 0. x(t)=\begin{cases}\cos(2\pi t)&\,\text{for }t\leq 0,\\ \cos(4\pi t)&\,\text{for }t>0.\end{cases}
  22. y ( t ) = m = - n = - C n m g n m ( t ) y(t)=\sum_{m=-\infty}^{\infty}\sum_{n=-\infty}^{\infty}C_{nm}\cdot g_{nm}(t)
  23. g n m ( t ) = s ( t - m τ 0 ) e j Ω n t g_{nm}(t)=s(t-m\tau_{0})\cdot e^{j\Omega nt}
  24. Ω 2 π τ 0 \Omega\leq\tfrac{2\pi}{\tau_{0}}
  25. Ω = 2 π N \Omega=\tfrac{2\pi}{N}
  26. \cdot
  27. y ( k ) = m = 0 M - 1 n = 0 N - 1 C n m g n m ( k ) y(k)=\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}C_{nm}\cdot g_{nm}(k)
  28. g n m ( k ) = s ( k - m N ) e j Ω n k g_{nm}(k)=s(k-mN)\cdot e^{j\Omega nk}
  29. \cdot
  30. C n m C_{nm}
  31. Ω \Omega
  32. Ω 2 π N = 2 π N \Omega\leq\tfrac{2\pi}{N}=\tfrac{2\pi}{N^{\prime}}
  33. \cdot

Gain–bandwidth_product.html

  1. × 10 6 \times 10^{6}
  2. ω ω c \omega>>\omega_{c}
  3. A 1 ( ω ) A_{1}(\omega)
  4. A 1 ( ω ) = < m t p l > H 0 1 + ( ω ω c ) 2 A_{1}(\omega)=\frac{<mtpl>{{H_{0}}}}{{\sqrt{1+{{\left({\frac{\omega}{{{\omega_% {c}}}}}\right)}^{2}}}}}
  5. G B W P ω ω c = A 1 ( ω ) ω c o n s t . GBWP_{\omega>>{\omega_{c}}}={A_{1}}(\omega)\cdot\omega\approx const.
  6. G B W P = A 1 ( ω ) ω = < m t p l > H 0 1 + ( ω ω c ) 2 ω H 0 ( ω ω c ) 2 ω = H 0 ω c = c o n s t . GBWP={A_{1}}(\omega)\cdot\omega=\frac{<mtpl>{{H_{0}}}}{{\sqrt{1+{{\left({\frac% {\omega}{{{\omega_{c}}}}}\right)}^{2}}}}}\cdot\omega\simeq\frac{{{H_{0}}}}{{% \sqrt{{{\left({\frac{\omega}{{{\omega_{c}}}}}\right)}^{2}}}}}\cdot\omega={H_{0% }}\cdot{\omega_{c}}=const.
  7. ω = 5 ω c \omega=5\cdot\omega_{c}
  8. G B W P = < m t p l > H 0 ω c 2 + 25 ω c 2 ω c 2 5 ω c = 5 26 H 0 ω c = 0.98 H 0 ω c GBWP=\frac{<mtpl>{{H_{0}}}}{{\sqrt{\frac{{\omega_{c}^{2}+25{\omega_{c}}^{2}}}{% {\omega_{c}^{2}}}}}}\cdot 5{\omega_{c}}=\frac{5}{{\sqrt{26}}}{H_{0}}\cdot{% \omega_{c}}=0.98\cdot{H_{0}}\cdot{\omega_{c}}
  9. f < s u b > T f<sub>T

Galilei_number.html

  1. Ga = R e 2 R i = g L 3 ν 2 \mathrm{Ga}=Re^{2}Ri=\frac{g\,L^{3}}{\nu^{2}}

Gamma_process.html

  1. Γ ( t ; γ , λ ) \Gamma(t;\gamma,\lambda)
  2. ν ( x ) = γ x - 1 exp ( - λ x ) \nu(x)=\gamma x^{-1}\exp(-\lambda x)
  3. x x
  4. [ x , x + d x ] [x,x+dx]
  5. ν ( x ) d x . \nu(x)dx.
  6. γ \gamma
  7. λ \lambda
  8. μ \mu
  9. v v
  10. γ = μ 2 / v \gamma=\mu^{2}/v
  11. λ = μ / v \lambda=\mu/v
  12. t t
  13. γ t / λ \gamma t/\lambda
  14. γ t / λ 2 . \gamma t/\lambda^{2}.
  15. α Γ ( t ; γ , λ ) = Γ ( t ; γ , λ / α ) \alpha\Gamma(t;\gamma,\lambda)=\Gamma(t;\gamma,\lambda/\alpha)\,
  16. Γ ( t ; γ 1 , λ ) + Γ ( t ; γ 2 , λ ) = Γ ( t ; γ 1 + γ 2 , λ ) \Gamma(t;\gamma_{1},\lambda)+\Gamma(t;\gamma_{2},\lambda)=\Gamma(t;\gamma_{1}+% \gamma_{2},\lambda)\,
  17. 𝔼 ( X t n ) = λ - n Γ ( γ t + n ) / Γ ( γ t ) , n 0 , \mathbb{E}(X_{t}^{n})=\lambda^{-n}\Gamma(\gamma t+n)/\Gamma(\gamma t),\ \quad n% \geq 0,
  18. Γ ( z ) \Gamma(z)
  19. 𝔼 ( exp ( θ X t ) ) = ( 1 - θ / λ ) - γ t , θ < λ \mathbb{E}\Big(\exp(\theta X_{t})\Big)=(1-\theta/\lambda)^{-\gamma t},\ \quad% \theta<\lambda
  20. Corr ( X s , X t ) = s / t , s < t \operatorname{Corr}(X_{s},X_{t})=\sqrt{s/t},\ s<t
  21. X ( t ) . X(t).

Gas-phase_ion_chemistry.html

  1. A * + B A B + + e - A^{*}+B\to AB^{+\bullet}+e^{-}
  2. A + + B A + B + A^{+}+B\to A+B^{+}
  3. G * + M M + + e - + G G^{*}+M\to M^{+\bullet}+e^{-}+G
  4. G * + M M + + e - + G G^{*}+M\to M^{+\bullet}+e^{-}+G
  5. G * + M M G + + e - G^{*}+M\to MG^{+\bullet}+e^{-}
  6. A 2 + + B A + + B + A^{2+}+B\to A^{+}+B^{+}
  7. A + + B A 2 + + B + e - A^{+}+B\to A^{2+}+B+e^{-}
  8. A + + B A - + B 2 + A^{+}+B\to A^{-}+B^{2+}
  9. A - + B A + + B + 2 e - A^{-}+B\to A^{+}+B+2e^{-}

Gauge_factor.html

  1. G F = Δ R R ε = Δ ρ ρ ε + 1 + 2 ν GF=\frac{\frac{\Delta R}{R}}{\varepsilon}=\frac{\frac{\Delta\rho}{\rho}}{% \varepsilon}+1+2\nu
  2. Δ L / L o \Delta L/Lo
  3. Δ L \Delta L
  4. L o Lo
  5. Δ ρ = 0 \Delta\rho=0
  6. G F = 1 + 2 ν GF=1+2\nu
  7. Δ R R = G F ε + α θ \frac{\Delta R}{R}=GF\varepsilon+\alpha\theta

Gaussian_surface.html

  1. S d A = 4 π r 2 \int\!\!\!\!\int_{S}dA=4\pi r^{2}
  2. Φ E = E 4 π r 2 \Phi_{E}=E4\pi r^{2}
  3. Φ E = Q A ε 0 \Phi_{E}=\frac{Q_{A}}{\varepsilon_{0}}
  4. E 4 π r 2 = Q A ε 0 E = Q A 4 π ε 0 r 2 . E4\pi r^{2}=\frac{Q_{A}}{\varepsilon_{0}}\quad\Rightarrow\quad E=\frac{Q_{A}}{% 4\pi\varepsilon_{0}r^{2}}.
  5. q = λ h q=\lambda h
  6. Φ E = a E d A cos 90 + b E d A cos 90 + c E d A cos 0 = E c d A \begin{aligned}\displaystyle\Phi_{E}&\displaystyle=\int\!\!\!\!\int_{a}EdA\cos 9% 0^{\circ}+\int\!\!\!\!\int_{b}EdA\cos 90^{\circ}+\int\!\!\!\!\int_{c}EdA\cos 0% ^{\circ}\\ &\displaystyle=E\int\!\!\!\!\int_{c}dA\\ \end{aligned}
  7. c d A = 2 π r h \int\!\!\!\!\int_{c}dA=2\pi rh
  8. Φ E = E 2 π r h \Phi_{E}=E2\pi rh
  9. Φ E = q ε 0 \Phi_{E}=\frac{q}{\varepsilon_{0}}
  10. E 2 π r h = λ h ε 0 E = λ 2 π ε 0 r E2\pi rh=\frac{\lambda h}{\varepsilon_{0}}\quad\Rightarrow\quad E=\frac{% \lambda}{2\pi\varepsilon_{0}r}

Gauss–Manin_connection.html

  1. H D R k ( V ) H^{k}_{DR}(V)
  2. V λ ( x , y , z ) = x 3 + y 3 + z 3 - λ x y z = 0 V_{\lambda}(x,y,z)=x^{3}+y^{3}+z^{3}-\lambda xyz=0\;
  3. λ \lambda
  4. λ \lambda
  5. ω λ \omega_{\lambda}
  6. ω λ H d R 1 ( V λ ) . \omega_{\lambda}\in H^{1}_{dR}(V_{\lambda}).
  7. ( λ 3 - 27 ) 2 ω λ λ 2 + 3 λ 2 ω λ λ + λ ω λ = 0. (\lambda^{3}-27)\frac{\partial^{2}\omega_{\lambda}}{\partial\lambda^{2}}+3% \lambda^{2}\frac{\partial\omega_{\lambda}}{\partial\lambda}+\lambda\omega_{% \lambda}=0.

Gee-haw_whammy_diddle.html

  1. F y F_{y}
  2. F y ( t ) = Y c o s ( ω t ) F_{y}(t)=Ycos(\omega t)
  3. h ω = 2 π / T h\omega=2\pi/T
  4. Y Y
  5. F x F_{x}
  6. F x ( t ) = X c o s ( ω t + ϕ ) F_{x}(t)=Xcos(\omega t+\phi)
  7. ϕ \phi
  8. ω \omega
  9. ϕ \phi

Generalised_cost.html

  1. g = p + u ( w ) g=p+u(w)
  2. u ( w ) = τ t u(w)=\tau t
  3. g = p + u ( w ) + v ( q , w ) g=p+u(w)+v(q,w)
  4. v = 2 q v=2q
  5. 0.05 q 0.05q

Generalised_hyperbolic_distribution.html

  1. μ + δ β K λ + 1 ( δ γ ) γ K λ ( δ γ ) \mu+\frac{\delta\beta K_{\lambda+1}(\delta\gamma)}{\gamma K_{\lambda}(\delta% \gamma)}
  2. δ K λ + 1 ( δ γ ) γ K λ ( δ γ ) + β 2 δ 2 γ 2 ( K λ + 2 ( δ γ ) K λ ( δ γ ) - K λ + 1 2 ( δ γ ) K λ 2 ( δ γ ) ) \frac{\delta K_{\lambda+1}(\delta\gamma)}{\gamma K_{\lambda}(\delta\gamma)}+% \frac{\beta^{2}\delta^{2}}{\gamma^{2}}\left(\frac{K_{\lambda+2}(\delta\gamma)}% {K_{\lambda}(\delta\gamma)}-\frac{K_{\lambda+1}^{2}(\delta\gamma)}{K_{\lambda}% ^{2}(\delta\gamma)}\right)
  3. e μ z γ λ ( α 2 - ( β + z ) 2 ) λ K λ ( δ α 2 - ( β + z ) 2 ) K λ ( δ γ ) \frac{e^{\mu z}\gamma^{\lambda}}{(\sqrt{\alpha^{2}-(\beta+z)^{2}})^{\lambda}}% \frac{K_{\lambda}(\delta\sqrt{\alpha^{2}-(\beta+z)^{2}})}{K_{\lambda}(\delta% \gamma)}
  4. K λ K_{\lambda}
  5. X GH ( - ν 2 , 0 , 0 , ν , μ ) X\sim\mathrm{GH}(-\frac{\nu}{2},0,0,\sqrt{\nu},\mu)\,
  6. ν \nu
  7. X GH ( 1 , α , β , δ , μ ) X\sim\mathrm{GH}(1,\alpha,\beta,\delta,\mu)\,
  8. X GH ( - 1 / 2 , α , β , δ , μ ) X\sim\mathrm{GH}(-1/2,\alpha,\beta,\delta,\mu)\,
  9. X GH ( ? , ? , ? , ? , ? ) X\sim\mathrm{GH}(?,?,?,?,?)\,
  10. X GH ( ? , ? , ? , ? , ? ) X\sim\mathrm{GH}(?,?,?,?,?)\,
  11. X GH ( λ , α , β , 0 , μ ) X\sim\mathrm{GH}(\lambda,\alpha,\beta,0,\mu)\,

Generalized_additive_model.html

  1. g ( E ( Y ) ) = β 0 + f 1 ( x 1 ) + f 2 ( x 2 ) + + f m ( x m ) . g(\operatorname{E}(Y))=\beta_{0}+f_{1}(x_{1})+f_{2}(x_{2})+\cdots+f_{m}(x_{m})% .\,\!
  2. f ( x ) fᵢ(xᵢ)
  3. f ( x ) fᵢ(xᵢ)

Generalized_Helmholtz_theorem.html

  1. 𝐩 = ( p 1 , p 2 , , p s ) , \mathbf{p}=(p_{1},p_{2},...,p_{s}),
  2. 𝐪 = ( q 1 , q 2 , , q s ) , \mathbf{q}=(q_{1},q_{2},...,q_{s}),
  3. H ( 𝐩 , 𝐪 ; V ) = K ( 𝐩 ) + φ ( 𝐪 ; V ) H(\mathbf{p},\mathbf{q};V)=K(\mathbf{p})+\varphi(\mathbf{q};V)
  4. K = i = 1 s p i 2 2 m K=\sum_{i=1}^{s}\frac{p_{i}^{2}}{2m}
  5. φ ( 𝐪 ; V ) \varphi(\mathbf{q};V)
  6. V V
  7. t \left\langle\cdot\right\rangle_{t}
  8. E E
  9. P P
  10. T T
  11. S S
  12. E = K + φ E=K+\varphi
  13. T = 2 s K t T=\frac{2}{s}\left\langle K\right\rangle_{t}
  14. P = - φ V t P=\left\langle-\frac{\partial\varphi}{\partial V}\right\rangle_{t}
  15. S ( E , V ) = log H ( 𝐩 , 𝐪 ; V ) E d s 𝐩 d s 𝐪 . S(E,V)=\log\int_{H(\mathbf{p},\mathbf{q};V)\leq E}d^{s}\mathbf{p}d^{s}\mathbf{% q}.
  16. d S = d E + P d V T . dS=\frac{dE+PdV}{T}.
  17. T T
  18. S S

Generalized_least_squares.html

  1. { y i , x i j } i = 1.. n , j = 1.. p \{y_{i},x_{ij}\}_{i=1..n,j=1..p}
  2. Y = X β + ε , E [ ε | X ] = 0 , Var [ ε | X ] = Ω . Y=X\beta+\varepsilon,\qquad\mathrm{E}[\varepsilon|X]=0,\ \operatorname{Var}[% \varepsilon|X]=\Omega.
  3. β ^ = arg min 𝑏 ( Y - Xb ) Ω - 1 ( Y - Xb ) , \hat{\beta}=\underset{b}{\rm arg\,min}\,(Y-Xb)^{\prime}\,\Omega^{-1}(Y-Xb),
  4. β ^ = ( X Ω - 1 X ) - 1 X Ω - 1 Y . \hat{\beta}=(X^{\prime}\Omega^{-1}X)^{-1}X^{\prime}\Omega^{-1}Y.
  5. n ( β ^ - β ) 𝑑 𝒩 ( 0 , ( X Ω - 1 X ) - 1 ) . \sqrt{n}(\hat{\beta}-\beta)\ \xrightarrow{d}\ \mathcal{N}\!\left(0,\,(X^{% \prime}\,\Omega^{-1}X)^{-1}\right).
  6. ( Y * - X * b ) ( Y * - X * b ) = ( Y - X b ) Ω - 1 ( Y - X b ) . (Y^{*}-X^{*}b)^{\prime}(Y^{*}-X^{*}b)=(Y-Xb)^{\prime}\,\Omega^{-1}(Y-Xb).
  7. Ω \Omega
  8. Ω \Omega
  9. Ω ^ \widehat{\Omega}
  10. σ 2 * ( X X ) - 1 \sigma^{2}*(X^{\prime}X)^{-1}
  11. β ^ O L S = ( X X ) - 1 X y \widehat{\beta}_{OLS}=(X^{\prime}X)^{-1}X^{\prime}y
  12. u ^ j = ( Y - X b ) j \widehat{u}_{j}=(Y-Xb)_{j}
  13. Ω \Omega
  14. u ^ j \widehat{u}_{j}
  15. Ω ^ O L S \widehat{\Omega}_{OLS}
  16. Ω ^ O L S = diag ( σ ^ 1 2 , σ ^ 2 2 , , σ ^ n 2 ) . \widehat{\Omega}_{OLS}=\operatorname{diag}(\widehat{\sigma}^{2}_{1},\widehat{% \sigma}^{2}_{2},\dots,\widehat{\sigma}^{2}_{n}).
  17. β F G L S 1 \beta_{FGLS1}
  18. Ω ^ O L S \widehat{\Omega}_{OLS}
  19. β ^ F G L S 1 = ( X Ω ^ O L S - 1 X ) - 1 X Ω ^ O L S - 1 y \widehat{\beta}_{FGLS1}=(X^{\prime}\widehat{\Omega}^{-1}_{OLS}X)^{-1}X^{\prime% }\widehat{\Omega}^{-1}_{OLS}y
  20. u ^ F G L S 1 = Y - X β ^ F G L S 1 \widehat{u}_{FGLS1}=Y-X\widehat{\beta}_{FGLS1}
  21. Ω ^ F G L S 1 = diag ( σ ^ F G L S 1 , 1 2 , σ ^ F G L S 1 , 2 2 , , σ ^ F G L S 1 , n 2 ) \widehat{\Omega}_{FGLS1}=\operatorname{diag}(\widehat{\sigma}^{2}_{FGLS1,1},% \widehat{\sigma}^{2}_{FGLS1,2},\dots,\widehat{\sigma}^{2}_{FGLS1,n})
  22. β ^ F G L S 2 = ( X Ω ^ F G L S 1 - 1 X ) - 1 X Ω ^ F G L S 1 - 1 y \widehat{\beta}_{FGLS2}=(X^{\prime}\widehat{\Omega}^{-1}_{FGLS1}X)^{-1}X^{% \prime}\widehat{\Omega}^{-1}_{FGLS1}y
  23. Ω ^ \widehat{\Omega}
  24. n ( β ^ F G L S - β ) 𝑑 𝒩 ( 0 , V ) . \sqrt{n}(\hat{\beta}_{FGLS}-\beta)\ \xrightarrow{d}\ \mathcal{N}\!\left(0,\,V% \right).
  25. V = p-lim ( X Ω - 1 X / T ) V=\,\text{p-lim}(X^{\prime}\Omega^{-1}X/T)

Generalized_minimal_residual_method.html

  1. v \|v\|
  2. A x = b . Ax=b.\,
  3. K n = K n ( A , b ) = span { b , A b , A 2 b , , A n - 1 b } . K_{n}=K_{n}(A,b)=\operatorname{span}\,\{b,Ab,A^{2}b,\ldots,A^{n-1}b\}.\,
  4. q 1 , q 2 , , q n q_{1},q_{2},\ldots,q_{n}\,
  5. H ~ n \tilde{H}_{n}
  6. A Q n = Q n + 1 H ~ n . AQ_{n}=Q_{n+1}\tilde{H}_{n}.\,
  7. Q n Q_{n}
  8. A x n - b = H ~ n y n - β e 1 , \|Ax_{n}-b\|=\|\tilde{H}_{n}y_{n}-\beta e_{1}\|,\,
  9. e 1 = ( 1 , 0 , 0 , , 0 ) T e_{1}=(1,0,0,\ldots,0)^{T}\,
  10. β = b - A x 0 , \beta=\|b-Ax_{0}\|\,,
  11. x 0 x_{0}
  12. x n x_{n}
  13. r n = H ~ n y n - β e 1 . r_{n}=\tilde{H}_{n}y_{n}-\beta e_{1}.
  14. q n q_{n}
  15. y n y_{n}
  16. x n = Q n y n x_{n}=Q_{n}y_{n}
  17. r n ( 1 - λ min 2 ( 1 / 2 ( A T + A ) ) λ max ( A T A ) ) n / 2 r 0 , \|r_{n}\|\leq\left(1-\frac{\lambda_{\mathrm{min}}^{2}(1/2(A^{T}+A))}{\lambda_{% \mathrm{max}}(A^{T}A)}\right)^{n/2}\|r_{0}\|,
  18. λ min ( M ) \lambda_{\mathrm{min}}(M)
  19. λ max ( M ) \lambda_{\mathrm{max}}(M)
  20. M M
  21. r n ( 2 κ 2 ( A ) - 1 2 κ 2 ( A ) ) n / 2 r 0 . \|r_{n}\|\leq\left(\frac{2\kappa_{2}(A)-1}{2\kappa_{2}(A)}\right)^{n/2}\|r_{0}\|.
  22. κ 2 ( A ) \kappa_{2}(A)
  23. r n inf p P n p ( A ) κ 2 ( V ) inf p P n max λ σ ( A ) | p ( λ ) | r 0 , \|r_{n}\|\leq\inf_{p\in P_{n}}\|p(A)\|\leq\kappa_{2}(V)\inf_{p\in P_{n}}\max_{% \lambda\in\sigma(A)}|p(\lambda)|\|r_{0}\|,\,
  24. y n y_{n}
  25. H ~ n y n - β e 1 . \|\tilde{H}_{n}y_{n}-\beta e_{1}\|.\,
  26. H ~ n \tilde{H}_{n}
  27. R ~ n \tilde{R}_{n}
  28. Ω n H ~ n = R ~ n . \Omega_{n}\tilde{H}_{n}=\tilde{R}_{n}.
  29. R ~ n = [ R n 0 ] , \tilde{R}_{n}=\begin{bmatrix}R_{n}\\ 0\end{bmatrix},
  30. R n R_{n}
  31. H ~ n + 1 = [ H ~ n h n + 1 0 h n + 2 , n + 1 ] , \tilde{H}_{n+1}=\begin{bmatrix}\tilde{H}_{n}&h_{n+1}\\ 0&h_{n+2,n+1}\end{bmatrix},
  32. [ Ω n 0 0 1 ] H ~ n + 1 = [ R n r n + 1 0 ρ 0 σ ] \begin{bmatrix}\Omega_{n}&0\\ 0&1\end{bmatrix}\tilde{H}_{n+1}=\begin{bmatrix}R_{n}&r_{n+1}\\ 0&\rho\\ 0&\sigma\end{bmatrix}
  33. G n = [ I n 0 0 0 c n s n 0 - s n c n ] G_{n}=\begin{bmatrix}I_{n}&0&0\\ 0&c_{n}&s_{n}\\ 0&-s_{n}&c_{n}\end{bmatrix}
  34. c n = ρ ρ 2 + σ 2 and s n = σ ρ 2 + σ 2 . c_{n}=\frac{\rho}{\sqrt{\rho^{2}+\sigma^{2}}}\quad\mbox{and}~{}\quad s_{n}=% \frac{\sigma}{\sqrt{\rho^{2}+\sigma^{2}}}.
  35. Ω n + 1 = G n [ Ω n 0 0 1 ] . \Omega_{n+1}=G_{n}\begin{bmatrix}\Omega_{n}&0\\ 0&1\end{bmatrix}.
  36. Ω n + 1 H ~ n + 1 = [ R n r n + 1 0 r n + 1 , n + 1 0 0 ] with r n + 1 , n + 1 = ρ 2 + σ 2 \Omega_{n+1}\tilde{H}_{n+1}=\begin{bmatrix}R_{n}&r_{n+1}\\ 0&r_{n+1,n+1}\\ 0&0\end{bmatrix}\quad\,\text{with}\quad r_{n+1,n+1}=\sqrt{\rho^{2}+\sigma^{2}}
  37. H ~ n y n - β e 1 = Ω n ( H ~ n y n - β e 1 ) = R ~ n y n - β Ω n e 1 . \|\tilde{H}_{n}y_{n}-\beta e_{1}\|=\|\Omega_{n}(\tilde{H}_{n}y_{n}-\beta e_{1}% )\|=\|\tilde{R}_{n}y_{n}-\beta\Omega_{n}e_{1}\|.
  38. β Ω n e 1 \beta\Omega_{n}e_{1}
  39. g ~ n = [ g n γ n ] \tilde{g}_{n}=\begin{bmatrix}g_{n}\\ \gamma_{n}\end{bmatrix}
  40. H ~ n y n - β e 1 = R ~ n y n - β Ω n e 1 = [ R n 0 ] y n - [ g n γ n ] . \|\tilde{H}_{n}y_{n}-\beta e_{1}\|=\|\tilde{R}_{n}y_{n}-\beta\Omega_{n}e_{1}\|% =\left\|\begin{bmatrix}R_{n}\\ 0\end{bmatrix}y_{n}-\begin{bmatrix}g_{n}\\ \gamma_{n}\end{bmatrix}\right\|.
  41. y n = R n - 1 g n . y_{n}=R_{n}^{-1}g_{n}.
  42. g n g_{n}

Generalized_polygon.html

  1. n 3 n\geq 3
  2. P , L , I P,L,I
  3. P P
  4. L L
  5. I P × L I\subseteq P\times L
  6. 2 m < n 2\leq m<n
  7. { A 1 , A 2 } P L \{A_{1},A_{2}\}\subseteq P\cup L
  8. P , L , I P^{\prime},L^{\prime},I^{\prime}
  9. { A 1 , A 2 } P L \{A_{1},A_{2}\}\subseteq P^{\prime}\cup L^{\prime}
  10. P L P\cup L
  11. L L
  12. P P
  13. P , L , I P,L,I
  14. I I

Generalized_quadrangle.html

  1. | P | = ( s t + 1 ) ( s + 1 ) |P|=(st+1)(s+1)
  2. | B | = ( s t + 1 ) ( t + 1 ) |B|=(st+1)(t+1)
  3. ( s + t ) | s t ( s + 1 ) ( t + 1 ) (s+t)|st(s+1)(t+1)
  4. s 1 t s 2 s\neq 1\Longrightarrow t\leq s^{2}
  5. t 1 s t 2 t\neq 1\Longrightarrow s\leq t^{2}
  6. Q + ( 3 , q ) Q^{+}(3,q)
  7. Q ( 4 , q ) Q(4,q)
  8. Q - ( 5 , q ) Q^{-}(5,q)
  9. Q ( 3 , q ) : s = q , t = 1 Q(3,q):\ s=q,t=1
  10. Q ( 4 , q ) : s = q , t = q Q(4,q):\ s=q,t=q
  11. Q ( 5 , q ) : s = q , t = q 2 Q(5,q):\ s=q,t=q^{2}
  12. H ( n , q 2 ) H(n,q^{2})
  13. H ( 3 , q 2 ) : s = q 2 , t = q H(3,q^{2}):\ s=q^{2},t=q
  14. H ( 4 , q 2 ) : s = q 2 , t = q 3 H(4,q^{2}):\ s=q^{2},t=q^{3}
  15. P G ( 2 d + 1 , q ) PG(2d+1,q)
  16. d = 1 d=1
  17. W ( 3 , q ) W(3,q)
  18. s = q , t = q s=q,t=q
  19. Q ( 4 , q ) Q(4,q)
  20. W ( 3 , q ) W(3,q)
  21. q q
  22. P G ( 2 , q ) PG(2,q)
  23. π \pi
  24. P G ( 3 , q ) PG(3,q)
  25. T 2 * ( O ) T_{2}^{*}(O)
  26. π \pi
  27. π \pi
  28. π \pi
  29. θ \theta
  30. P G ( 3 , q ) PG(3,q)
  31. π = p θ \pi=p^{\theta}
  32. π \pi
  33. π \pi
  34. P G ( 3 , q ) PG(3,q)
  35. π \pi
  36. ( q , q ) (q,q)
  37. ( q , q 2 ) (q,q^{2})
  38. ( q 2 , q ) (q^{2},q)
  39. ( q 2 , q 3 ) (q^{2},q^{3})
  40. ( q 3 , q 2 ) (q^{3},q^{2})
  41. ( q - 1 , q + 1 ) (q-1,q+1)
  42. ( q + 1 , q - 1 ) (q+1,q-1)

Generalized_tree_alignment.html

  1. S S
  2. d d
  3. T T
  4. S S
  5. Σ e T d ( e ) \Sigma_{e\in T}d(e)
  6. d ( e ) d(e)
  7. e e

Generalized_Wiener_process.html

  1. a ( x , t ) d t + b ( x , t ) η d t a(x,t)dt+b(x,t)\eta\sqrt{dt}

Genocchi_number.html

  1. 2 t e t + 1 = n = 1 G n t n n ! \frac{2t}{e^{t}+1}=\sum_{n=1}^{\infty}G_{n}\frac{t^{n}}{n!}
  2. G n = 2 ( 1 - 2 n ) B n . G_{n}=2\,(1-2^{n})\,B_{n}.
  3. G n G_{n}
  4. B 1 = - 1 / 2 B_{1}=-1/2
  5. G n 1 G_{n_{1}}
  6. B 1 = 1 / 2 B_{1}=1/2
  7. G n 2 G_{n_{2}}
  8. - 2 1 + e - t \frac{-2}{1+e^{-t}}
  9. t tan ( t 2 ) = n 1 ( - 1 ) n G 2 n t 2 n ( 2 n ) ! t\tan(\frac{t}{2})=\sum_{n\geq 1}(-1)^{n}G_{2n}\frac{t^{2n}}{(2n)!}

Genomic_library.html

  1. N = l n ( 1 - P ) l n ( 1 - f ) N=\frac{ln(1-P)}{ln(1-f)}
  2. N N
  3. P P
  4. f f
  5. f f
  6. f = i g f=\frac{i}{g}
  7. i i
  8. g g
  9. N = l n ( 1 - P ) l n ( 1 - i g ) N=\frac{ln(1-P)}{ln(1-\frac{i}{g})}
  10. N = l n ( 1 - 0.99 ) l n [ 1 - 2.0 × 10 4 b a s e p a i r s 3.0 × 10 9 b a s e p a i r s ] N=\frac{ln(1-0.99)}{ln[1-\frac{2.0\times 10^{4}basepairs}{3.0\times 10^{9}% basepairs}]}
  11. N = - 4.61 - 6.7 × 10 - 6 N=\frac{-4.61}{-6.7\times 10^{-6}}
  12. N = 688 , 060 N=688,060

Gentzen's_consistency_proof.html

  1. α \alpha
  2. ω α = α \omega^{\alpha}=\alpha
  3. ω , ω ω , ω ω ω , \omega,\ \omega^{\omega},\ \omega^{\omega^{\omega}},\ \ldots

Geometric_topology_(object).html

  1. { M i } \{M_{i}\}
  2. ϵ i \epsilon_{i}
  3. ( 1 + ϵ i ) (1+\epsilon_{i})
  4. ϕ i : M i , [ ϵ i , ) M [ ϵ i , ) , \phi_{i}:M_{i,[\epsilon_{i},\infty)}\rightarrow M_{[\epsilon_{i},\infty)},
  5. ϵ i \epsilon_{i}
  6. M i M_{i}

George_Secor.html

  1. ( 15 14 ÷ 16 15 = 225 224 ) \left(\tfrac{15}{14}\div\tfrac{16}{15}=\tfrac{225}{224}\right)
  2. ( 3 2 ÷ ( 8 7 ) 3 = 1029 1024 ) \left(\tfrac{3}{2}\div\left(\tfrac{8}{7}\right)^{3}=\tfrac{1029}{1024}\right)
  3. ( 8 7 × 6 5 = 48 35 ) \left(\tfrac{8}{7}\times\tfrac{6}{5}=\tfrac{48}{35}\right)
  4. ( 11 8 ÷ 48 35 = 385 384 ) \left(\tfrac{11}{8}\div\tfrac{48}{35}=\tfrac{385}{384}\right)

Gianni_Bellocchi.html

  1. S ( x ; a ; b ) = { 0 x a ( x - a ) 2 × 2 ( b - a ) 2 a x c 1 - ( b - x ) 2 × 2 ( b - a ) 2 c x b 1 b x S(x;a;b)=\begin{cases}0&x\leq a\\ \frac{\left(x-a\right)^{2}\times 2}{\left(b-a\right)^{2}}&a\leq x\leq c\\ 1-\frac{\left(b-x\right)^{2}\times 2}{\left(b-a\right)^{2}}&c\leq x\leq b\\ 1&b\leq x\end{cases}

Gillespie_algorithm.html

  1. k D k_{D}
  2. k B k_{B}
  3. k D k_{D}
  4. n A n_{A}
  5. n B n_{B}
  6. k D n A n B k_{D}n_{A}n_{B}
  7. n A B n_{AB}
  8. k B n A B k_{B}n_{AB}
  9. R T O T R_{TOT}
  10. R T O T = k D n A n B + k B n A B R_{TOT}=k_{D}n_{A}n_{B}+k_{B}n_{AB}
  11. R T O T R_{TOT}
  12. 1 / R T O T 1/R_{TOT}
  13. = k D n A n B / R T O T =k_{D}n_{A}n_{B}/R_{TOT}
  14. n A n_{A}
  15. n B n_{B}
  16. n A B n_{AB}
  17. n A n_{A}
  18. n B n_{B}
  19. n A B n_{AB}
  20. n A = n B = 10 n_{A}=n_{B}=10
  21. n A B = 0 n_{AB}=0
  22. k D = 2 k_{D}=2
  23. k U = 1 k_{U}=1
  24. n A B n_{AB}

Giuseppe_Lauricella.html

  1. F A , F B , F C , F D F_{A},F_{B},F_{C},F_{D}

Glaisher's_theorem.html

  1. N N
  2. d d
  3. N = N 1 + + N k N=N_{1}+\cdots+N_{k}
  4. N i N i + 1 N_{i}\geq N_{i+1}
  5. N i N i + d - 1 + 1 , N_{i}\geq N_{i+d-1}+1,
  6. d = 2 d=2
  7. N N
  8. N N

Gliese_667.html

  1. e = r a - r p r a + r p \scriptstyle e={{r_{a}-r_{p}}\over{r_{a}+r_{p}}}
  2. h h = ( T eff T eff ) 2 * L a \begin{smallmatrix}\frac{h}{{h}_{\odot}}={\left(\frac{{{T}_{\odot}}_{\rm eff}}% {{T}_{\rm eff}}\right)^{2}}*\frac{\sqrt{L}}{a}\end{smallmatrix}
  3. h \begin{smallmatrix}{h}\end{smallmatrix}
  4. h \begin{smallmatrix}{{h}_{\odot}}\end{smallmatrix}
  5. T eff \begin{smallmatrix}{{T}_{\odot}}_{\rm eff}\end{smallmatrix}
  6. T eff \begin{smallmatrix}{{T}_{\rm eff}}\end{smallmatrix}
  7. L \begin{smallmatrix}{L}\end{smallmatrix}
  8. a \begin{smallmatrix}{a}\end{smallmatrix}

Glitch_(astronomy).html

  1. R 2.9 G M c 2 R\geq 2.9\frac{GM}{c^{2}}

Glivenko–Cantelli_theorem.html

  1. X 1 , X 2 , X_{1},X_{2},\dots
  2. \mathbb{R}
  3. F ( x ) F(x)
  4. X 1 , , X n X_{1},\dots,X_{n}
  5. F n ( x ) = 1 n i = 1 n I ( - , x ] ( X i ) , F_{n}(x)=\frac{1}{n}\sum_{i=1}^{n}I_{(-\infty,x]}(X_{i}),
  6. I C I_{C}
  7. C C
  8. x x
  9. F n ( x ) F_{n}(x)
  10. F ( x ) F(x)
  11. F n F_{n}
  12. F F
  13. F n F_{n}
  14. F F
  15. F n - F = sup x | F n ( x ) - F ( x ) | 0 \|F_{n}-F\|_{\infty}=\sup_{x\in\mathbb{R}}|F_{n}(x)-F(x)|{\longrightarrow}0
  16. X n X_{n}
  17. F n ( x ) F_{n}(x)
  18. F ( x ) = E ( 1 X 1 x ) F(x)=E(1_{X_{1}\leq x})
  19. ( - , x ] (-\infty,x]
  20. 𝒞 \mathcal{C}
  21. C 𝒞 . C\in\mathcal{C}.
  22. P n ( C ) = 1 n i = 1 n I C ( X i ) , C 𝒞 P_{n}(C)=\frac{1}{n}\sum_{i=1}^{n}I_{C}(X_{i}),C\in\mathcal{C}
  23. I C ( x ) I_{C}(x)
  24. C C
  25. P n P_{n}
  26. f P n f = S f d P n = 1 n i = 1 n f ( X i ) , f . f\mapsto P_{n}f=\int_{S}fdP_{n}=\frac{1}{n}\sum_{i=1}^{n}f(X_{i}),f\in\mathcal% {F}.
  27. \mathcal{F}
  28. 𝒞 \mathcal{C}
  29. 𝒮 \mathcal{S}
  30. 𝒞 { C : C is measurable subset of 𝒮 } {\mathcal{C}}\subset\{C:C\mbox{ is measurable subset of }~{}\mathcal{S}\}
  31. { f : 𝒮 , f is measurable } \mathcal{F}\subset\{f:\mathcal{S}\to\mathbb{R},f\mbox{ is measurable}~{}\,\}
  32. P n - P 𝒞 = sup c 𝒞 | P n ( C ) - P ( C ) | \|P_{n}-P\|_{\mathcal{C}}=\sup_{c\in{\mathcal{C}}}|P_{n}(C)-P(C)|
  33. P n - P = sup f | P n f - P ( f ) | \|P_{n}-P\|_{\mathcal{F}}=\sup_{f\in{\mathcal{F}}}|P_{n}f-P(f)|
  34. P n ( C ) P_{n}(C)
  35. P n f P_{n}f
  36. 𝔼 f = 𝒮 f d P = P ( f ) \mathbb{E}f=\int_{\mathcal{S}}fdP=P(f)
  37. 𝒞 \mathcal{C}
  38. P n - P 𝒞 0 \|P_{n}-P\|_{\mathcal{C}}\to 0
  39. n n\to\infty
  40. P n - P 𝒞 0 \|P_{n}-P\|_{\mathcal{C}}\to 0
  41. n n\to\infty
  42. 𝔼 P n - P 𝒞 0 \mathbb{E}\|P_{n}-P\|_{\mathcal{C}}\to 0
  43. n n\to\infty
  44. sup P 𝒫 ( S , A ) 𝔼 P n - P 𝒞 0 ; \sup_{P\in\mathcal{P}(S,A)}\mathbb{E}\|P_{n}-P\|_{\mathcal{C}}\to 0;
  45. sup P 𝒫 ( S , A ) 𝔼 P n - P 0. \sup_{P\in\mathcal{P}(S,A)}\mathbb{E}\|P_{n}-P\|_{\mathcal{F}}\to 0.
  46. 𝒞 \mathcal{C}
  47. S = S=\mathbb{R}
  48. 𝒞 = { ( - , t ] : t } {\mathcal{C}}=\{(-\infty,t]:t\in{\mathbb{R}}\}
  49. sup P 𝒫 ( S , A ) P n - P 𝒞 n - 1 / 2 \sup_{P\in\mathcal{P}(S,A)}\|P_{n}-P\|_{\mathcal{C}}\sim n^{-1/2}
  50. 𝒞 \mathcal{C}
  51. 𝒞 \mathcal{C}
  52. A n = { X 1 , , X n } 𝒞 A_{n}=\{X_{1},\ldots,X_{n}\}\in\mathcal{C}
  53. P ( A n ) = 0 P(A_{n})=0
  54. P n ( A n ) = 1 P_{n}(A_{n})=1
  55. P n - P 𝒞 = 1 \|P_{n}-P\|_{\mathcal{C}}=1
  56. 𝒞 \mathcal{C}

Glossary_of_machine_vision.html

  1. Q = 2 π f o P Q=\frac{2\pi f_{o}\mathcal{E}}{P}
  2. f o f_{o}
  3. \mathcal{E}
  4. P = - d E d t P=-\frac{dE}{dt}

GMROII.html

  1. ( M a r g i n % / ( 100 % - M a r g i n % ) ) * A n n u a l I n v e n t o r y T u r n s (Margin\%/(100\%-Margin\%))*AnnualInventoryTurns
  2. ( M a r g i n / C O G S ) * A n n u a l I n v e n t o r y T u r n s (Margin/COGS)*AnnualInventoryTurns
  3. M a r g i n % * ( S a l e s / A v g I n v e n t o r y C o s t ) Margin\%*(Sales/AvgInventoryCost)
  4. S e l l i n g P r i c e * G M R O I I SellingPrice*GMROII
  5. S e l l i n g P r i c e * M a r g i n % * A n n u a l I n v e n t o r y T u r n s SellingPrice*Margin\%*AnnualInventoryTurns
  6. A m e r i c a n L e v i s G M R O I I * D e n s i t y AmericanLevi^{\prime}sGMROII*Density

Golden_triangle_(mathematics).html

  1. a b = φ = 1 + 5 2 . {a\over b}=\varphi={1+\sqrt{5}\over 2}.
  2. θ = cos - 1 ( φ 2 ) = π 5 = 36 . \theta=\cos^{-1}\left({\varphi\over 2}\right)={\pi\over 5}=36^{\circ}.

Golygon.html

  1. ± 1 ± 3 ± ( n - 1 ) = 0 \pm 1\pm 3\cdots\pm(n-1)=0
  2. ± 2 ± 4 ± n = 0. \pm 2\pm 4\cdots\pm n=0.

Gompertz_function.html

  1. a a
  2. b b
  3. c c
  4. y ( t ) = a e - b e - c t y(t)=ae^{-be^{-ct}}
  5. a e b e - = a e 0 = a ae^{be^{-\infty}}=ae^{0}=a
  6. k r 1 y ( t ) k^{r}\propto\frac{1}{y(t)}
  7. r = y ( t ) y ( t ) r=\frac{y^{\prime}(t)}{y(t)}
  8. X ( t ) = K exp ( log ( X ( 0 ) K ) exp ( - α t ) ) X(t)=K\exp\left(\log\left(\frac{X(0)}{K}\right)\exp\left(-\alpha t\right)\right)
  9. lim t + X ( t ) = K \lim_{t\rightarrow+\infty}X(t)=K
  10. X ( t ) = α log ( K X ( t ) ) X ( t ) X^{\prime}(t)=\alpha\log\left(\frac{K}{X(t)}\right)X(t)
  11. X ( t ) = F ( X ( t ) ) X ( t ) , F ( X ) 0 X^{\prime}(t)=F\left(X(t)\right)X(t),F^{\prime}(X)\leq 0
  12. F ( X ) = α ( 1 - ( X K ) ν ) F ( 0 ) = α < + F(X)=\alpha\left(1-\left(\frac{X}{K}\right)^{\nu}\right)\Rightarrow F(0)=% \alpha<+\infty
  13. lim X 0 + F ( X ) = lim X 0 + α log ( K X ) = + \lim_{X\rightarrow 0^{+}}F(X)=\lim_{X\rightarrow 0^{+}}\alpha\log\left(\frac{K% }{X}\right)=+\infty
  14. X ( t ) = α log ( K X ( t ) ) X ( t ) X^{\prime}(t)=\alpha\log\left(\frac{K}{X(t)}\right)X(t)
  15. X ( t ) = α ν ( 1 - ( X ( t ) K ) 1 ν ) X ( t ) X^{\prime}(t)=\alpha\nu\left(1-\left(\frac{X(t)}{K}\right)^{\frac{1}{\nu}}% \right)X(t)
  16. ν > 0 \nu>0
  17. lim ν + ν ( 1 - x 1 / ν ) = - log ( x ) \lim_{\nu\rightarrow+\infty}\nu\left(1-x^{1/\nu}\right)=-\log\left(x\right)
  18. X ( t ) = ( ν ν + 1 ) ν K X(t)=\left(\frac{\nu}{\nu+1}\right)^{\nu}K
  19. X ( t ) = K e = K lim ν + ( ν ν + 1 ) ν X(t)=\frac{K}{e}=K\cdot\lim_{\nu\rightarrow+\infty}\left(\frac{\nu}{\nu+1}% \right)^{\nu}
  20. X C X_{C}
  21. X > X C X>X_{C}
  22. F ( X ) = max ( a , α log ( K X ) ) F(X)=\max\left(a,\alpha\log\left(\frac{K}{X}\right)\right)
  23. X C = K exp ( - a α ) . X_{C}=K\exp\left(-\frac{a}{\alpha}\right).
  24. X C X_{C}
  25. X C 10 9 X_{C}\approx 10^{9}
  26. X C 10 6 X_{C}\approx 10^{6}

Gompertz–Makeham_law_of_mortality.html

  1. h ( x ) = α e β x + λ h(x)=\alpha e^{\beta x}+\lambda
  2. Q ( u ) = α β λ - 1 λ ln ( 1 - u ) - 1 β W 0 ( α e α / λ ( 1 - u ) - ( β / λ ) λ ) Q(u)=\frac{\alpha}{\beta\lambda}-\frac{1}{\lambda}\ln(1-u)-\frac{1}{\beta}W_{0% }\left(\frac{\alpha e^{\alpha/\lambda}(1-u)^{-(\beta/\lambda)}}{\lambda}\right)

Gosper_curve.html

  1. A A
  2. A A - B - - B + A + + A A + B - A\mapsto A-B--B+A++AA+B-
  3. B + A - B B - - B - A + + A + B B\mapsto+A-BB--B-A++A+B

Gönen.html

  1. M w M_{\mathrm{w}}

Gradient-related.html

  1. { d k } \{d^{k}\}
  2. { x k } \{x^{k}\}
  3. { x k } k K \{x^{k}\}_{k\in K}
  4. { d k } k K \{d^{k}\}_{k\in K}
  5. lim sup k , k K f ( x k ) d k < 0. \limsup_{k\rightarrow\infty,k\in K}\nabla f(x^{k})^{\prime}d^{k}<0.
  6. f f
  7. k k
  8. x k x^{k}
  9. d k d^{k}

Graduate_Aptitude_Test_in_Engineering.html

  1. M ¯ \overline{M}
  2. M ¯ \overline{M}
  3. M ¯ \overline{M}
  4. M ¯ \overline{M}
  5. S = S q + ( S t - S q ) M - M q M ¯ t - M q S=S_{q}+(S_{t}-S_{q})\frac{M-M_{q}}{\overline{M}_{t}-M_{q}}
  6. M ¯ \overline{M}
  7. M ¯ \overline{M}
  8. A l l I n d i a r a n k N o . o f c a n d i d a t e s i n t h a t s u b j e c t \frac{AllIndiarank}{No.ofcandidatesinthatsubject}
  9. 10 ( a g + s g m - a S ) 10(a_{g}+s_{g}\frac{m-a}{S})

Grain_growth.html

  1. v = M σ κ v=M\sigma\kappa
  2. v v
  3. M M
  4. σ \sigma
  5. κ \kappa
  6. v = M σ 2 R v=M\sigma\frac{2}{R}
  7. R R
  8. d 2 - d 0 2 = k t d^{2}-{d_{0}}^{2}=kt\,\!
  9. k = k 0 exp ( - Q R T ) k=k_{0}\exp\left(\frac{-Q}{RT}\right)\,\!

Graph_state.html

  1. | G = ( a , b ) E U { a , b } | + V {\left|G\right\rangle}=\prod_{(a,b)\in E}U^{\{a,b\}}{\left|+\right\rangle}^{% \otimes V}
  2. U { a , b } U^{\{a,b\}}
  3. U { a , b } = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 - 1 ] U^{\{a,b\}}=\left[\begin{array}[]{cccc}{1}&{0}&{0}&{0}\\ {0}&{1}&{0}&{0}\\ {0}&{0}&{1}&{0}\\ {0}&{0}&{0}&{-1}\end{array}\right]
  4. | + = | 0 + | 1 2 {\left|+\right\rangle}=\frac{{\left|0\right\rangle}+{\left|1\right\rangle}}{% \sqrt{2}}
  5. K G ( a ) K_{G}^{(a)}
  6. K G ( a ) = σ x ( a ) b N ( a ) σ z ( b ) K_{G}^{(a)}=\sigma_{x}^{(a)}\prod_{b\in N(a)}\sigma_{z}^{(b)}
  7. ( a , b ) E (a,b)\in E
  8. σ x , y , z \sigma_{x,y,z}
  9. | G {\left|G\right\rangle}
  10. N = | V | N=\left|V\right|
  11. { K G ( a ) } a V \left\{K_{G}^{(a)}\right\}_{a\in V}
  12. K G ( a ) | G = | G K_{G}^{(a)}{\left|G\right\rangle}={\left|G\right\rangle}

Grassmann_integral.html

  1. [ a f ( θ ) + b g ( θ ) ] d θ = a f ( θ ) d θ + b g ( θ ) d θ \int[af(\theta)+bg(\theta)]\,d\theta=a\int f(\theta)\,d\theta+b\int g(\theta)% \,d\theta
  2. θ d θ = 1 \int\theta\,d\theta=1
  3. d θ = 0 \int\,d\theta=0
  4. θ f ( θ ) d θ = 0. \int\frac{\partial}{\partial\theta}f(\theta)\,d\theta=0.
  5. ( a θ + b ) d θ = a . \int(a\theta+b)\,d\theta=a.
  6. f 1 ( θ 1 ) f n ( θ n ) d θ 1 d θ n = f 1 ( θ 1 ) d θ 1 f n ( θ n ) d θ n . \int f_{1}(\theta_{1})\cdots f_{n}(\theta_{n})\,d\theta_{1}\cdots\,d\theta_{n}% =\int f_{1}(\theta_{1})\,d\theta_{1}\cdots\int f_{n}(\theta_{n})\,d\theta_{n}.
  7. θ i = θ i ( ξ j ) \theta_{i}=\theta_{i}(\xi_{j})
  8. J i j = θ i ξ j . J_{ij}=\frac{\partial\theta_{i}}{\partial\xi_{j}}.
  9. f ( θ i ) d θ = f ( θ i ( ξ j ) ) det ( J i j ) - 1 d ξ . \int f(\theta_{i})\,d\theta=\int f(\theta_{i}(\xi_{j}))\det(J_{ij})^{-1}\,d\xi.
  10. x a = x a ( y b , ξ j ) , θ i = θ i ( y b , ξ j ) , x_{a}=x_{a}(y_{b},\xi_{j})\,,\;\theta_{i}=\theta_{i}(y_{b},\xi_{j})\;,
  11. J α β = ( x a , θ i ) ( y b , ξ j ) . J_{\alpha\beta}=\frac{\partial(x_{a},\theta_{i})}{\partial(y_{b},\xi_{j})}.
  12. J = [ A B C D ] . J=\begin{bmatrix}A&B\\ C&D\end{bmatrix}.
  13. Ber + - J α β = sgn det A Ber J α β . \operatorname{Ber}_{+-}J_{\alpha\beta}=\operatorname{sgn}\,\operatorname{det}A% \,\operatorname{Ber}J_{\alpha\beta}.
  14. U f ( x a , θ i ) d ( x , θ ) = U f ( x a , θ i ) Ber + - ( x a , θ i ) ( y b , ξ j ) d ( y , ξ ) . \int_{U}f(x_{a},\theta_{i})\,d(x,\theta)=\int_{U^{\prime}}f(x_{a},\theta_{i})% \operatorname{Ber}_{+-}\,\frac{\partial(x_{a},\theta_{i})}{\partial(y_{b},\xi_% {j})}\,d(y,\xi).
  15. exp [ θ T A η ] d θ d η = det A \int\exp\left[\theta^{T}A\eta\right]\,d\theta\,d\eta=\det A
  16. A A
  17. n × n n\times n
  18. exp [ - θ T M θ ] d θ = { 2 n 2 det M , n even 0 , n odd \int\exp\left[-\theta^{T}M\theta\right]\,d\theta=\begin{cases}2^{n\over 2}% \sqrt{\det M},&n\mbox{ even}\\ 0,&n\mbox{ odd}\end{cases}
  19. M M
  20. n × n n\times n
  21. d θ = d θ 1 d θ n d\theta=d\theta_{1}\cdots\,d\theta_{n}
  22. exp [ θ T A η + θ T J + K T η ] d θ d η = det A exp [ - K T A - 1 J ] \int\exp\left[\theta^{T}A\eta+\theta^{T}J+K^{T}\eta\right]\,d\theta\,d\eta=% \det A\,\,\exp[-K^{T}A^{-1}J]
  23. A A
  24. n × n n\times n
  25. K K
  26. J J
  27. exp [ - 1 2 θ T M θ + η T θ ] d θ = { det M exp [ - 1 2 η T M - 1 η ] , n even 0 , n odd \int\exp\left[-{1\over 2}\theta^{T}M\theta+\eta^{T}\theta\right]\,d\theta=% \begin{cases}\sqrt{\det M}\exp\left[-{1\over 2}\eta^{T}M^{-1}\eta\right],&n% \mbox{ even}\\ 0,&n\mbox{ odd}\end{cases}
  28. M M
  29. n × n n\times n
  30. η \eta

Gravitational_coupling_constant.html

  1. α G = G m e 2 c = ( m e m P ) 2 1.7518 × 10 - 45 \alpha\text{G}=\frac{Gm\text{e}^{2}}{\hbar c}=\left(\frac{m\text{e}}{m\text{P}% }\right)^{2}\approx 1.7518\times 10^{-45}
  2. 4 π G = c = = ε 0 = 1 4\pi G=c=\hbar=\varepsilon_{0}=1
  3. α G = m e 2 4 π \alpha\text{G}=\frac{m\text{e}^{2}}{4\pi}
  4. α \alpha
  5. α G \alpha_{G}
  6. α G \alpha_{G}
  7. α G \alpha_{G}
  8. α α G \frac{\alpha}{\alpha_{G}}
  9. 4 π G = c = = ε 0 = 1 4\pi G=c=\hbar=\varepsilon_{0}=1
  10. α = e 2 4 π \alpha=\frac{e^{2}}{4\pi}
  11. α G = m e 2 4 π \alpha_{G}=\frac{m_{e}^{2}}{4\pi}
  12. α α G = ( e m e ) 2 \frac{\alpha}{\alpha_{G}}=\left(\frac{e}{m_{e}}\right)^{2}
  13. α \alpha
  14. α G \alpha_{G}
  15. α G \alpha_{G}
  16. α G \alpha_{G}
  17. α G \alpha_{G}
  18. α G = G m e 2 c = ( t P ω C ) 2 \alpha_{G}=\frac{Gm_{e}^{2}}{\hbar c}=\left(t_{P}\omega_{C}\right)^{2}
  19. t P t_{P}
  20. α G \alpha_{G}
  21. ω C \omega_{C}

Gravitational_energy.html

  1. m m
  2. M M
  3. F = G m M / r 2 F=GmM/r^{2}
  4. r = r 0 r=r_{0}
  5. r = r 1 r=r_{1}
  6. F = - d U d x F=-{dU\over dx}
  7. r 0 r 1 m M G r 2 d r = m M G r | r 1 r 0 = m M G r 0 - m M G r 1 = E \int_{r_{0}}^{r_{1}}{mMG\over r^{2}}dr=\left.{mMG\over r}\right|_{r_{1}}^{r_{0% }}={mMG\over r_{0}}-{mMG\over r_{1}}=E
  8. E = m M G / r 0 E={mMG/r_{0}}
  9. r 1 = r_{1}=\infty
  10. E = m M G / r 0 E={mMG/r_{0}}
  11. r 0 = 0 r_{0}=0

Gravity_(alcoholic_beverage).html

  1. S G true = ρ sample ρ water SG\text{true}={\rho\text{sample}\over\rho\text{water}}
  2. S G true = S G apparent - ρ air ρ water ( S G apparent - 1 ) SG\text{true}=SG\text{apparent}-{\rho\text{air}\over\rho\text{water}}(SG\text{% apparent}-1)
  3. p p
  4. m m
  5. n = P r e c o n S G r e c o n S G b e e r n=P_{recon}{SG_{recon}\over SG_{beer}}
  6. n n
  7. A w = ( p - n ) ( 2.0665 - 1.0665 p / 100 ) = f p n ( p - n ) A_{w}={(p-n)\over(2.0665-1.0665p/100)}=f_{pn}(p-n)
  8. f p n = 1 ( 2.0665 - 1.0665 p / 100 ) f_{pn}={1\over(2.0665-1.0665p/100)}
  9. ( p - n ) (p-n)
  10. f p n f_{pn}
  11. f p n f_{pn}
  12. f p n = 1 ( 2.0665 - 1.0665 p / 100 ) 0.48394 + 0.0024688 p + 0.00001561 p 2 f_{pn}={1\over(2.0665-1.0665p/100)}\approx 0.48394+0.0024688p+0.00001561p^{2}
  13. m m
  14. n = P ( P - 1 ( m ) + 1 - ρ EtOH ( A w ) ρ water ) n=P(P^{-1}(m)+1-\frac{\rho\text{EtOH}(A_{w})}{\rho\text{water}})
  15. P P
  16. P - 1 P^{-1}
  17. ρ EtOH ( A w ) \rho\text{EtOH}(A_{w})
  18. A w A_{w}
  19. [ p - P ( P - 1 ( m ) + 1 - ρ EtOH ( A w ) ρ water ) ] ( 2.0665 - 1.0665 p / 100 ) - A w = 0 {\left[p-P\left(P^{-1}(m)+1-\frac{\rho\text{EtOH}(A_{w})}{\rho\text{water}}% \right)\right]\over(2.0665-1.0665p/100)}-A_{w}=0
  20. A w A_{w}
  21. A w = f p m ( p - m ) A_{w}=f_{pm}(p-m)\,
  22. f p m = 0.39661 + 0.001709 p + 0.000010788 p 2 f_{pm}=0.39661+0.001709p+0.000010788p^{2}
  23. A v = A w S G beer 0.79661 A_{v}=A_{w}{SG\text{beer}\over 0.79661}
  24. A v = k ( p - m ) A_{v}=k(p-m)\,
  25. k k
  26. p 1000 ( S G - 1 ) / 4 p\approx 1000(SG-1)/4
  27. A v = 250 f p m ( O G - F G ) S G beer 0.79661 A_{v}=250f_{pm}(OG-FG){SG\text{beer}\over 0.79661}
  28. f p m f_{pm}
  29. A v = 132.715 ( O G - F G ) = ( O G - F G ) / 0.00753 A_{v}=132.715(OG-FG)=(OG-FG)/0.00753\,
  30. R D F = 100 ( p - n ) p RDF=100{(p-n)\over p}
  31. A D F = 100 ( p - m ) p 100 ( O G - F G ) ( O G - 1 ) ADF=100{(p-m)\over p}\approx 100{(OG-FG)\over(OG-1)}
  32. p t := 1000 ( S G - 1 ) p_{t}:=1000(SG-1)\,
  33. p p t / 4 = 1000 ( S G - 1 ) / 4. p\approx p_{t}/4=1000(SG-1)/4.

Gravity_model_of_trade.html

  1. F i j = G ( M i β 1 M j β 2 / D i j β 3 ) F_{ij}=G(M_{i}^{\beta_{1}}M_{j}^{\beta_{2}}/D_{ij}^{\beta_{3}}){\ }
  2. F i j = G ( M i β 1 M j β 2 / D i j β 3 ) η i j F_{ij}=G(M_{i}^{\beta_{1}}M_{j}^{\beta_{2}}/D_{ij}^{\beta_{3}})\eta_{ij}
  3. F i j F_{ij}
  4. i i
  5. j j
  6. M i M_{i}
  7. M j M_{j}
  8. i i
  9. j j
  10. D i j D_{ij}
  11. η \eta
  12. β 0 \beta_{0}
  13. ln ( F i j ) = β 0 + β 1 ln ( M i ) + β 2 ln ( M j ) - β 3 ln ( D i j ) + ϵ i j \ln(F_{ij})=\beta_{0}+\beta_{1}\ln(M_{i})+\beta_{2}\ln(M_{j})-\beta_{3}\ln(D_{% ij})+\epsilon_{ij}
  14. F i j F_{ij}
  15. F i j = exp [ β 0 + β 1 ln ( M i ) + β 2 ln ( M j ) - β 3 ln ( D i j ) ] η i j F_{ij}=\exp[\beta_{0}+\beta_{1}\ln(M_{i})+\beta_{2}\ln(M_{j})-\beta_{3}\ln(D_{% ij})]\eta_{ij}
  16. l n ( F i j , t ) = l n ( Y i Y j ) - a l n ( τ i j , t ) - b l n ( τ i j , t - 1 ) ln(F_{ij,t})=ln(Y_{i}Y_{j})-aln(\tau_{ij,t})-bln(\tau_{ij,t-1})

Gravity_of_Earth.html

  1. g g
  2. g g
  3. G G
  4. G G
  5. g g
  6. g h = g 0 ( r e r e + h ) 2 g_{h}=g_{0}\left(\frac{r_{\mathrm{e}}}{r_{\mathrm{e}}+h}\right)^{2}
  7. h h
  8. r r
  9. r r
  10. g ( r ) = - G M ( r ) r 2 . g(r)=-\frac{GM(r)}{r^{2}}.
  11. G G
  12. M ( r ) M(r)
  13. r r
  14. ρ ρ
  15. g ( r ) = 4 π 3 G ρ r . g(r)=\frac{4\pi}{3}G\rho r.
  16. g g
  17. d d
  18. = 𝐠 ( 𝟏 - 𝐝 / 𝐑 ) \mathbf{=g(1-d/R)}
  19. g g
  20. d d
  21. R R
  22. g ( r ) = 4 π 3 G ρ 0 r - π G ( ρ 0 - ρ 1 ) r 2 r e . g(r)=\frac{4\pi}{3}G\rho_{0}r-\pi G\left(\rho_{0}-\rho_{1}\right)\frac{r^{2}}{% r_{\mathrm{e}}}.
  23. g ϕ g_{\phi}
  24. ϕ \phi
  25. g ϕ \displaystyle g_{\phi}
  26. g ϕ = ( 9.7803267714 1 + 0.00193185138639 sin 2 ϕ 1 - 0.00669437999013 sin 2 ϕ ) m s 2 \ g_{\phi}=\left(9.7803267714~{}\frac{1+0.00193185138639\sin^{2}\phi}{\sqrt{1-% 0.00669437999013\sin^{2}\phi}}\right)\,\frac{\mathrm{m}}{\mathrm{s}^{2}}
  27. r Earth = 6.371 × 10 6 m r_{\mathrm{Earth}}=6.371\times 10^{6}\,\mathrm{m}
  28. m Earth = 5.9722 × 10 24 kg m_{\mathrm{Earth}}=5.9722\times 10^{24}\,\mathrm{kg}
  29. g 0 = G m Earth / r Earth 2 = 9.8196 m s 2 g_{0}=G\,m_{\mathrm{Earth}}/r_{\mathrm{Earth}}^{2}=9.8196\,\frac{\mathrm{m}}{% \mathrm{s}^{2}}
  30. G = 6.67384 × 10 - 11 m 3 kg s 2 . G=6.67384\times 10^{-11}\,\frac{\mathrm{m}^{3}}{\mathrm{kg}\cdot\mathrm{s}^{2}}.
  31. g h = G m Earth / ( r Earth + h ) 2 g_{h}=G\,m_{\mathrm{Earth}}/\left(r_{\mathrm{Earth}}+h\right)^{2}
  32. Δ g h = [ G m Earth / ( r Earth + h ) 2 ] - [ G m Earth / r Earth 2 ] \Delta g_{h}=\left[G\,m_{\mathrm{Earth}}/\left(r_{\mathrm{Earth}}+h\right)^{2}% \right]-\left[G\,m_{\mathrm{Earth}}/r_{\mathrm{Earth}}^{2}\right]
  33. Δ g h - G m Earth r Earth 2 × 2 h r Earth \Delta g_{h}\approx-\,\dfrac{G\,m_{\mathrm{Earth}}}{r_{\mathrm{Earth}}^{2}}% \times\dfrac{2\,h}{r_{\mathrm{Earth}}}
  34. Δ g h - 3.083 × 10 - 6 h \Delta g_{h}\approx-3.083\times 10^{-6}\,h
  35. g ϕ , h = 9.780327 ( 1 + 0.0053024 sin 2 ϕ - 0.0000058 sin 2 2 ϕ ) - 3.086 × 10 - 6 h g_{\phi,h}=9.780327\left(1+0.0053024\sin^{2}\phi-0.0000058\sin^{2}2\phi\right)% -3.086\times 10^{-6}h
  36. g ϕ , h \ g_{\phi,h}
  37. ϕ \ \phi
  38. g ϕ , h = 9.780327 [ ( 1 + 0.0053024 sin 2 ϕ - 0.0000058 sin 2 2 ϕ ) - 3.155 × 10 - 7 h ] m s 2 g_{\phi,h}=9.780327\left[\left(1+0.0053024\sin^{2}\phi-0.0000058\sin^{2}2\phi% \right)-3.155\times 10^{-7}h\right]\,\frac{\mathrm{m}}{\mathrm{s}^{2}}
  39. × 10 10 \times 10^{−}10
  40. × 10 6 \times 10^{−}6
  41. F = G m 1 m 2 r 2 = ( G m 1 r 2 ) m 2 F=G\frac{m_{1}m_{2}}{r^{2}}=\left(G\frac{m_{1}}{r^{2}}\right)m_{2}
  42. F = m 2 g F=m_{2}g\,
  43. g = G m 1 r 2 g=G\frac{m_{1}}{r^{2}}
  44. g = G m 1 r 2 = ( 6.67384 × 10 - 11 ) 5.9722 × 10 24 ( 6.371 × 10 6 ) 2 = 9.8196 m s - 2 g=G\frac{m_{1}}{r^{2}}=(6.67384\times 10^{-11})\frac{5.9722\times 10^{24}}{(6.% 371\times 10^{6})^{2}}=9.8196\mbox{m}~{}\cdot\mbox{s}~{}^{-2}

Great-circle_navigation.html

  1. tan α 1 = sin λ 12 cos ϕ 1 tan ϕ 2 - sin ϕ 1 cos λ 12 , tan α 2 = sin λ 12 - cos ϕ 2 tan ϕ 1 + sin ϕ 2 cos λ 12 , \begin{aligned}\displaystyle\tan\alpha_{1}&\displaystyle=\frac{\sin\lambda_{12% }}{\cos\phi_{1}\tan\phi_{2}-\sin\phi_{1}\cos\lambda_{12}},\\ \displaystyle\tan\alpha_{2}&\displaystyle=\frac{\sin\lambda_{12}}{-\cos\phi_{2% }\tan\phi_{1}+\sin\phi_{2}\cos\lambda_{12}},\\ \end{aligned}
  2. cos σ 12 = sin ϕ 1 sin ϕ 2 + cos ϕ 1 cos ϕ 2 cos λ 12 . \cos\sigma_{12}=\sin\phi_{1}\sin\phi_{2}+\cos\phi_{1}\cos\phi_{2}\cos\lambda_{% 12}.
  3. sin α 0 = sin α 1 cos ϕ 1 . \sin\alpha_{0}=\sin\alpha_{1}\cos\phi_{1}.
  4. tan σ 01 = tan ϕ 1 cos α 1 \tan\sigma_{01}=\frac{\tan\phi_{1}}{\cos\alpha_{1}}\qquad
  5. tan λ 01 = sin α 0 sin σ 01 cos σ 01 , λ 0 = λ 1 - λ 01 . \begin{aligned}\displaystyle\tan\lambda_{01}&\displaystyle=\frac{\sin\alpha_{0% }\sin\sigma_{01}}{\cos\sigma_{01}},\\ \displaystyle\lambda_{0}&\displaystyle=\lambda_{1}-\lambda_{01}.\end{aligned}
  6. \color w h i t e . ) sin ϕ = cos α 0 sin σ , {\color{white}.\,\qquad)}\sin\phi=\cos\alpha_{0}\sin\sigma,
  7. tan ( λ - λ 0 ) = sin α 0 sin σ cos σ , tan α = tan α 0 cos σ . \begin{aligned}\displaystyle\tan(\lambda-\lambda_{0})&\displaystyle=\frac{\sin% \alpha_{0}\sin\sigma}{\cos\sigma},\\ \displaystyle\tan\alpha&\displaystyle=\frac{\tan\alpha_{0}}{\cos\sigma}.\end{aligned}
  8. tan ϕ = cot α 0 sin ( λ - λ 0 ) . \tan\phi=\cot\alpha_{0}\sin(\lambda-\lambda_{0}).
  9. ( ϕ , λ ) (\phi,\lambda)

Great_ditrigonal_icosidodecahedron.html

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

Great_icosidodecahedron.html

  1. φ 3 = 1 + 2 φ = 2 + 5 \varphi^{3}=1+2\varphi\!=2+\sqrt{5}
  2. φ \varphi\!

Great_retrosnub_icosidodecahedron.html

  1. ξ = ( 1 + i 3 ) ( 1 2 τ + τ - 2 4 - 8 27 ) 1 3 + ( 1 - i 3 ) ( 1 2 τ - τ - 2 4 - 8 27 ) 1 3 2 \xi=\frac{\left(1+i\sqrt{3}\right)\left(\frac{1}{2\tau}+\sqrt{\frac{\tau^{-2}}% {4}-\frac{8}{27}}\right)^{\frac{1}{3}}+\left(1-i\sqrt{3}\right)\left(\frac{1}{% 2\tau}-\sqrt{\frac{\tau^{-2}}{4}-\frac{8}{27}}\right)^{\frac{1}{3}}}{2}

Green's_matrix.html

  1. x = A ( t ) x + g ( t ) x^{\prime}=A(t)x+g(t)\,
  2. x x\,
  3. A ( t ) A(t)\,
  4. n × n n\times n\,
  5. t t\,
  6. t \isin I , a t b t\isin I,a\leq t\leq b\,
  7. I I\,
  8. x 1 ( t ) , , x n ( t ) x^{1}(t),\ldots,x^{n}(t)\,
  9. n n\,
  10. x = A ( t ) x x^{\prime}=A(t)x\,
  11. X ( t ) = [ x 1 ( t ) , , x n ( t ) ] . X(t)=\left[x^{1}(t),\ldots,x^{n}(t)\right].\,
  12. X ( t ) X(t)\,
  13. n × n n\times n\,
  14. X = A X X^{\prime}=AX\,
  15. x = X y x=Xy\,
  16. x = X y + X y = A X y + X y = A x + X y . \begin{aligned}\displaystyle x^{\prime}&\displaystyle=X^{\prime}y+Xy^{\prime}% \\ &\displaystyle=AXy+Xy^{\prime}\\ &\displaystyle=Ax+Xy^{\prime}.\end{aligned}
  17. X y = g Xy^{\prime}=g\,
  18. y = c + a t X - 1 ( s ) g ( s ) d s y=c+\int_{a}^{t}X^{-1}(s)g(s)\,ds\,
  19. c c\,
  20. x = X ( t ) c + X ( t ) a t X - 1 ( s ) g ( s ) d s . x=X(t)c+X(t)\int_{a}^{t}X^{-1}(s)g(s)\,ds.\,
  21. G 0 ( t , s ) = { 0 t s b X ( t ) X - 1 ( s ) a s < t . G_{0}(t,s)=\begin{cases}0&t\leq s\leq b\\ X(t)X^{-1}(s)&a\leq s<t.\end{cases}\,
  22. x p ( t ) = a b G 0 ( t , s ) g ( s ) d s . x_{p}(t)=\int_{a}^{b}G_{0}(t,s)g(s)\,ds.\,

Greenberger–Horne–Zeilinger_state.html

  1. M > 2 M>2
  2. | GHZ = | 0 M + | 1 M 2 . |\mathrm{GHZ}\rangle=\frac{|0\rangle^{\otimes M}+|1\rangle^{\otimes M}}{\sqrt{% 2}}.
  3. | GHZ = | 000 + | 111 2 . |\mathrm{GHZ}\rangle=\frac{|000\rangle+|111\rangle}{\sqrt{2}}.
  4. Tr 3 ( ( | 000 + | 111 ) ( 000 | + 111 | ) ) = ( | 00 00 | + | 11 11 | ) 2 \mathrm{Tr}_{3}\big((|000\rangle+|111\rangle)(\langle 000|+\langle 111|)\big)=% \frac{(|00\rangle\langle 00|+|11\rangle\langle 11|)}{2}
  5. | 00 |00\rangle
  6. | 11 |11\rangle

Greenwood_function.html

  1. f = 0 x Δ f c b = A ( 10 a x - K ) f=\int_{0}^{x}\!{\Delta f_{cb}}=A(10^{ax}-K)
  2. f = 165.4 ( 10 2.1 x - 1 ) f=165.4(10^{2.1x}-1)
  3. f = 165.4 ( 10 2.1 * 10 35 - 0.88 ) = 513 Hz . f=165.4(10^{\frac{2.1*10}{35}}-0.88)=513\ \mathrm{Hz.}

GRENOUILLE.html

  1. τ p \tau_{p}
  2. G V M ( λ 0 ) ( 1 ν g ( λ 0 / 2 ) - 1 ν g ( λ 0 ) ) GVM(\lambda_{0})\equiv\left(\frac{1}{\nu_{g}(\lambda_{0}/2)}-\frac{1}{\nu_{g}(% \lambda_{0})}\right)
  3. ν g ( λ ) \nu_{g}(\lambda)
  4. λ \lambda
  5. G V M ( λ 0 ) L τ p GVM(\lambda_{0})L\gg\tau_{p}
  6. τ c \tau_{c}
  7. G V D ( λ 0 ) ( 1 ν g ( λ 0 - δ λ / 2 ) - 1 ν g ( λ 0 + δ λ / 2 ) ) GVD(\lambda_{0})\equiv\left(\frac{1}{\nu_{g}(\lambda_{0}-\delta\lambda/2)}-% \frac{1}{\nu_{g}(\lambda_{0}+\delta\lambda/2)}\right)
  8. δ λ \delta\lambda
  9. τ c G V D ( λ 0 ) L \tau_{c}\gg GVD(\lambda_{0})L
  10. G V D τ p τ c τ p L G V M GVD\frac{\tau_{p}}{\tau_{c}}\ll\frac{\tau_{p}}{L}\ll GVM
  11. τ p / τ c \tau_{p}/\tau_{c}
  12. G V M G V D T B P \frac{GVM}{GVD}\gg TBP
  13. G V D ( λ 0 ) A L τ p A G V M ( λ 0 ) L \frac{GVD(\lambda_{0})}{A}L\leq\tau_{p}\leq AGVM(\lambda_{0})L

Gross_enrolment_ratio.html

  1. 900000 1000000 = 0.90 \frac{900000}{1000000}={0.90}
  2. 0.90 * 100 % = 90 % 0.90*100\%=90\%

Group_with_operators.html

  1. Ω \Omega
  2. Ω \Omega
  3. Ω × G G : ( ω , g ) g ω \ \Omega\times G\rightarrow G:(\omega,g)\mapsto g^{\omega}
  4. ( g h ) ω = g ω h ω . \ (gh)^{\omega}=g^{\omega}h^{\omega}.
  5. ω Ω \omega\in\Omega
  6. g g ω \ g\mapsto g^{\omega}
  7. ( u ω ) ω Ω (u_{\omega})_{\omega\in\Omega}
  8. Ω \Omega
  9. Ω \Omega
  10. \to
  11. ω Ω , g G : f ( g ω ) = ( f ( g ) ) ω . \forall\omega\in\Omega,\forall g\in G:f(g^{\omega})=(f(g))^{\omega}.
  12. ω \omega
  13. Ω \Omega
  14. s S , ω Ω : s ω S . \forall s\in S,\forall\omega\in\Omega:s^{\omega}\in S.
  15. Ω \Omega
  16. Ω End 𝐆𝐫𝐩 ( G ) , \Omega\rightarrow\operatorname{End}_{\mathbf{Grp}}(G),
  17. End 𝐆𝐫𝐩 ( G ) \operatorname{End}_{\mathbf{Grp}}(G)

Grüneisen_parameter.html

  1. γ = V ( d P d E ) V = α K S C P ρ = α K T C V ρ \gamma=V\left(\frac{dP}{dE}\right)_{V}=\frac{\alpha K_{S}}{C_{P}\rho}=\frac{% \alpha K_{T}}{C_{V}\rho}
  2. C P C_{P}
  3. C V C_{V}
  4. K S K_{S}
  5. K T K_{T}
  6. d d
  7. Γ 0 = - 1 2 d Π ′′′ ( a ) a 2 + ( d - 1 ) [ Π ′′ ( a ) a - Π ( a ) ] Π ′′ ( a ) a + ( d - 1 ) Π ( a ) , \Gamma_{0}=-\frac{1}{2d}\frac{\Pi^{\prime\prime\prime}(a)a^{2}+(d-1)\left[\Pi^% {\prime\prime}(a)a-\Pi^{\prime}(a)\right]}{\Pi^{\prime\prime}(a)a+(d-1)\Pi^{% \prime}(a)},
  8. Π \Pi
  9. a a
  10. d d
  11. d = 1 d=1
  12. 10 1 2 10\frac{1}{2}
  13. m + n + 3 2 \frac{m+n+3}{2}
  14. 3 α a 2 \frac{3\alpha a}{2}
  15. d = 2 d=2
  16. 5 5
  17. m + n + 2 4 \frac{m+n+2}{4}
  18. 3 α a - 1 4 \frac{3\alpha a-1}{4}
  19. d = 3 d=3
  20. 19 6 \frac{19}{6}
  21. n + m + 1 6 \frac{n+m+1}{6}
  22. 3 α a - 2 6 \frac{3\alpha a-2}{6}
  23. d = d=\infty
  24. - 1 2 -\frac{1}{2}
  25. - 1 2 -\frac{1}{2}
  26. - 1 2 -\frac{1}{2}
  27. d d
  28. 11 d - 1 2 \frac{11}{d}-\frac{1}{2}
  29. m + n + 4 2 d - 1 2 \frac{m+n+4}{2d}-\frac{1}{2}
  30. 3 α a + 1 2 d - 1 2 \frac{3\alpha a+1}{2d}-\frac{1}{2}
  31. Π ′′′ ( a ) a > - ( d - 1 ) Π ′′ ( a ) , \Pi^{\prime\prime\prime}(a)a>-(d-1)\Pi^{\prime\prime}(a),
  32. V V
  33. i i
  34. ω i \omega_{i}
  35. γ i = - V ω i ω i V . \gamma_{i}=-\frac{V}{\omega_{i}}\frac{\partial\omega_{i}}{\partial V}.
  36. γ = α K T C V ρ \gamma=\frac{\alpha K_{T}}{C_{V}\rho}
  37. γ \gamma
  38. γ = i γ i c V , i i c V , i , \gamma=\frac{\sum_{i}\gamma_{i}c_{V,i}}{\sum_{i}c_{V,i}},
  39. c V , i c_{V,i}
  40. C V = 1 ρ V i c V , i . C_{V}=\frac{1}{\rho V}\sum_{i}c_{V,i}.
  41. C ~ V = i c V , i \tilde{C}_{V}=\sum_{i}c_{V,i}
  42. i γ i c V , i C ~ V = α K T C V ρ = α V K T C ~ V \frac{\sum_{i}\gamma_{i}c_{V,i}}{\tilde{C}_{V}}=\frac{\alpha K_{T}}{C_{V}\rho}% =\frac{\alpha VK_{T}}{\tilde{C}_{V}}
  43. i γ i c V , i = α V K T \sum_{i}\gamma_{i}c_{V,i}=\alpha VK_{T}
  44. i γ i c V , i = i [ - V ω i ω i V ] [ k B ( ω i k B T ) 2 exp ( ω i k B T ) [ exp ( ω i k B T ) - 1 ] 2 ] \sum_{i}\gamma_{i}c_{V,i}=\sum_{i}\left[-\frac{V}{\omega_{i}}\frac{\partial% \omega_{i}}{\partial V}\right]\left[k_{B}\left(\frac{\hbar\omega_{i}}{k_{B}T}% \right)^{2}\frac{\exp\left(\frac{\hbar\omega_{i}}{k_{B}T}\right)}{\left[\exp% \left(\frac{\hbar\omega_{i}}{k_{B}T}\right)-1\right]^{2}}\right]
  45. α V K T = [ 1 V ( V T ) P ] V [ - V ( P V ) T ] = - V ( V T ) P ( P V ) T \alpha VK_{T}=\left[\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{P}% \right]V\left[-V\left(\frac{\partial P}{\partial V}\right)_{T}\right]=-V\left(% \frac{\partial V}{\partial T}\right)_{P}\left(\frac{\partial P}{\partial V}% \right)_{T}
  46. ( V T ) P = T ( G P ) T = P ( G T ) P = - ( S P ) T \left(\frac{\partial V}{\partial T}\right)_{P}=\frac{\partial}{\partial T}% \left(\frac{\partial G}{\partial P}\right)_{T}=\frac{\partial}{\partial P}% \left(\frac{\partial G}{\partial T}\right)_{P}=-\left(\frac{\partial S}{% \partial P}\right)_{T}
  47. α V K T = V ( S P ) T ( P V ) T = V ( S V ) T \alpha VK_{T}=V\left(\frac{\partial S}{\partial P}\right)_{T}\left(\frac{% \partial P}{\partial V}\right)_{T}=V\left(\frac{\partial S}{\partial V}\right)% _{T}
  48. S V = V { - i k B ln [ 1 - exp ( - ω i ( V ) k B T ) ] + i 1 T ω i ( V ) exp ( ω i ( V ) k B T ) - 1 } \frac{\partial S}{\partial V}=\frac{\partial}{\partial V}\left\{-\sum_{i}k_{B}% \ln\left[1-\exp\left(-\frac{\hbar\omega_{i}(V)}{k_{B}T}\right)\right]+\sum_{i}% \frac{1}{T}\frac{\hbar\omega_{i}(V)}{\exp\left(\frac{\hbar\omega_{i}(V)}{k_{B}% T}\right)-1}\right\}
  49. V S V = - i V ω i ω i V k B ( ω i k B T ) 2 exp ( ω i k B T ) [ exp ( ω i k B T ) - 1 ] 2 = i γ i c V , i V\frac{\partial S}{\partial V}=-\sum_{i}\frac{V}{\omega_{i}}\frac{\partial% \omega_{i}}{\partial V}\;\;k_{B}\left(\frac{\hbar\omega_{i}}{k_{B}T}\right)^{2% }\frac{\exp\left(\frac{\hbar\omega_{i}}{k_{B}T}\right)}{\left[\exp\left(\frac{% \hbar\omega_{i}}{k_{B}T}\right)-1\right]^{2}}=\sum_{i}\gamma_{i}c_{V,i}
  50. γ = i γ i c V , i i c V , i = α V K T C ~ V \gamma=\dfrac{\sum_{i}\gamma_{i}c_{V,i}}{\sum_{i}c_{V,i}}=\dfrac{\alpha VK_{T}% }{\tilde{C}_{V}}

Gummel–Poon_model.html

  1. β F \beta_{\mathrm{F}}
  2. β R \beta_{\mathrm{R}}

Gustafson's_law.html

  1. S ( P ) = P - α ( P - 1 ) S(P)=P-\alpha\cdot(P-1)
  2. α \alpha
  3. ( a + b ) (a+b)
  4. a a
  5. b b
  6. a + P b . a+P\cdot b.
  7. ( a + P b ) / ( a + b ) . (a+P\cdot b)/(a+b).
  8. α = a / ( a + b ) \alpha=a/(a+b)
  9. S ( P ) = α + P ( 1 - α ) = P - α ( P - 1 ) . S(P)=\alpha+P\cdot(1-\alpha)=P-\alpha\cdot(P-1).
  10. α \alpha
  11. α \alpha

Gustavo_Colonnetti.html

  1. W W^{\ast}

Gutenberg–Richter_law.html

  1. log 10 N = a - b M \!\,\log_{10}N=a-bM
  2. N = 10 a - b M \!\,N=10^{a-bM}
  3. N \!\,N
  4. M \!\,\geq M
  5. a \!\,a
  6. b \!\,b
  7. N = N TOT 10 - b M N=N_{\mathrm{TOT}}10^{-bM}
  8. N TOT = 10 a , N_{\mathrm{TOT}}=10^{a},

Gyro_monorail.html

  1. Ω \Omega
  2. A d 2 ϕ d t 2 + H ( d θ d t + Ω ϕ ) = W h ϕ A\frac{d^{2}\phi}{dt^{2}}+H(\frac{d\theta}{dt}+\Omega\phi)=Wh\phi
  3. ( W h - H Ω ) k A J \frac{(Wh-H\Omega)k}{AJ}
  4. Ω = W h H \Omega=\frac{Wh}{H}

Gyroid.html

  1. 3 ¯ \overline{3}
  2. sin x cos y + sin y cos z + sin z cos x = 0 \sin x\cdot\cos y+\sin y\cdot\cos z+\sin z\cdot\cos x=0

Gyroradius.html

  1. r g = m v | q | B r_{g}=\frac{mv_{\perp}}{|q|B}
  2. m m
  3. v v_{\perp}
  4. q q
  5. B B
  6. ω g = | q | B m \omega_{g}=\frac{|q|B}{m}
  7. Ω g = q B m \Omega_{g}=\frac{qB}{m}
  8. f g = q B 2 π m f_{g}=\frac{qB}{2\pi m}
  9. f g , e = ( 2.8 × 10 10 Hertz / Tesla ) × B f_{g,e}=(2.8\times 10^{10}\,\mathrm{Hertz}/\mathrm{Tesla})\times B
  10. r g = m c v | q | B r_{g}=\frac{mcv_{\perp}}{|q|B}
  11. ω g = | q | B m c \omega_{g}=\frac{|q|B}{mc}
  12. c c
  13. r g / meter = 3.3 × ( m c 2 / GeV ) ( v / c ) ( | q | / e ) ( B / Tesla ) r_{g}/\mathrm{meter}=3.3\times\frac{(mc^{2}/\mathrm{GeV})(v_{\perp}/c)}{(|q|/e% )(B/\mathrm{Tesla})}
  14. c c
  15. GeV \mathrm{GeV}
  16. e e
  17. F = q ( v × B ) \vec{F}=q(\vec{v}\times\vec{B})
  18. v \vec{v}
  19. B \vec{B}
  20. r g r_{g}
  21. m v 2 r g = | q | v B \frac{mv_{\perp}^{2}}{r_{g}}=|q|v_{\perp}B
  22. r g = m v | q | B r_{g}=\frac{mv_{\perp}}{|q|B}
  23. T g = 2 π r g v T_{g}=\frac{2\pi r_{g}}{v_{\perp}}
  24. f g = 1 T g = | q | B 2 π m f_{g}=\frac{1}{T_{g}}=\frac{|q|B}{2\pi m}
  25. ω g = | q | B m \omega_{g}=\frac{|q|B}{m}

H-index.html

  1. max i min ( f ( i ) , i ) \max_{i}\min(f(i),i)
  2. h 0.54 N h\approx 0.54\sqrt{N}
  3. h / h d h/\langle h\rangle_{d}

Hagen–Poiseuille_flow_from_the_Navier–Stokes_equations.html

  1. ( ) / t = 0 \partial(...)/\partial t=0
  2. u r = u θ = 0 u_{r}=u_{\theta}=0
  3. ( ) / θ = 0 \partial(...)/\partial\theta=0
  4. u z / z = 0 \partial u_{z}/\partial z=0
  5. p / r = 0 \partial p/\partial r=0
  6. p p
  7. z z
  8. 1 r r ( r u z r ) = 1 μ p z \frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial u_{z}}{\partial r}% \right)=\frac{1}{\mu}\frac{\partial p}{\partial z}
  9. μ \mu
  10. u z = 1 4 μ p z r 2 + c 1 ln r + c 2 u_{z}=\frac{1}{4\mu}\frac{\partial p}{\partial z}r^{2}+c_{1}\ln r+c_{2}
  11. u z u_{z}
  12. r = 0 r=0
  13. c 1 = 0 c_{1}=0
  14. u z = 0 u_{z}=0
  15. r = R r=R
  16. c 2 = - 1 4 μ p z R 2 . c_{2}=-\frac{1}{4\mu}\frac{\partial p}{\partial z}R^{2}.
  17. u z = - 1 4 μ p z ( R 2 - r 2 ) . u_{z}=-\frac{1}{4\mu}\frac{\partial p}{\partial z}(R^{2}-r^{2}).
  18. r = 0 r=0
  19. u z m a x = R 2 4 μ ( - p z ) . {u_{z}}_{max}=\frac{R^{2}}{4\mu}\left(-\frac{\partial p}{\partial z}\right).
  20. u z avg = 1 π R 2 0 R u z 2 π r d r = 0.5 u z max . {u_{z}}_{\mathrm{avg}}=\frac{1}{\pi R^{2}}\int_{0}^{R}u_{z}\cdot 2\pi rdr=0.5{% u_{z}}_{\mathrm{max}}.
  21. Δ p \Delta p
  22. L L
  23. u z avg {u_{z}}_{\mathrm{avg}}
  24. - p z = Δ p L -\frac{\partial p}{\partial z}=\frac{\Delta p}{L}
  25. u z max {u_{z}}_{\mathrm{max}}
  26. u z avg {u_{z}}_{\mathrm{avg}}
  27. D = 2 R D=2R
  28. u z a v g = D 2 32 μ Δ p L . {u_{z}}_{avg}=\frac{D^{2}}{32\mu}\frac{\Delta p}{L}.
  29. Δ p = 32 μ L u z avg D 2 . \Delta p=\frac{32\mu L~{}{u_{z}}_{\mathrm{avg}}}{D^{2}}.

Half_power_point.html

  1. 20 log 10 ( 1 2 ) - 3.0103 dB 20\log_{10}\left(\tfrac{1}{\sqrt{2}}\right)\approx-3.0103\,\mathrm{dB}
  2. 10 log 10 ( 1 2 ) - 3.0103 dB 10\log_{10}\left(\tfrac{1}{2}\right)\approx-3.0103\,\mathrm{dB}
  3. log 10 2 = .3010... .3 , \log_{10}2=.3010...\approx.3,
  4. 10 log 10 2 = 3.010... 3 ; 10\log_{10}2=3.010...\approx 3;
  5. 10 log 10 r . 10\log_{10}r.
  6. G = 10 3 10 × 1 = 1.99526... 2 G=10^{\frac{3}{10}}\times 1\ =1.99526...\approx 2\,
  7. 2 10 = 1 , 024 1 , 000 = 10 3 2^{10}=1,024\approx 1,000=10^{3}
  8. 2 10 10 3 2^{10}\approx 10^{3}
  9. 10 log 10 2 3 10\log_{10}2\approx 3

Halpern–Läuchli_theorem.html

  1. T i : i d \langle T_{i}:i\in d\rangle
  2. n ω ( i < d T i ( n ) ) = C 1 C r , \bigcup_{n\in\omega}\left(\prod_{i<d}T_{i}(n)\right)=C_{1}\cup\cdots\cup C_{r},
  3. S i : i d \langle S_{i}:i\in d\rangle
  4. T i : i d \langle T_{i}:i\in d\rangle
  5. n ω ( i < d S i ( n ) ) C k for some k r . \bigcup_{n\in\omega}\left(\prod_{i<d}S_{i}(n)\right)\subset C_{k}\,\text{ for % some }k\leq r.
  6. S T i : i d d = n ω ( i < d T i ( n ) ) S^{d}_{\langle T_{i}:i\in d\rangle}=\bigcup_{n\in\omega}\left(\prod_{i<d}T_{i}% (n)\right)
  7. 𝕊 d = T i : i d S T i : i d d . \mathbb{S}^{d}=\bigcup_{\langle T_{i}:i\in d\rangle}S^{d}_{\langle T_{i}:i\in d% \rangle}.
  8. 𝕊 d \mathbb{S}^{d}

Hamburger_moment_problem.html

  1. m n = - x n d μ ( x ) ? m_{n}=\int_{-\infty}^{\infty}x^{n}\,d\mu(x)\ ?
  2. [ 0 , + ) [0,+\infty)
  3. A = ( m 0 m 1 m 2 m 1 m 2 m 3 m 2 m 3 m 4 ) A=\left(\begin{matrix}m_{0}&m_{1}&m_{2}&\cdots\\ m_{1}&m_{2}&m_{3}&\cdots\\ m_{2}&m_{3}&m_{4}&\cdots\\ \vdots&\vdots&\vdots&\ddots\\ \end{matrix}\right)
  4. j , k 0 m j + k c j c ¯ k 0 \sum_{j,k\geq 0}m_{j+k}c_{j}\bar{c}_{k}\geq 0
  5. j , k 0 m j + k c j c ¯ k = - | j 0 c j x j | 2 d μ ( x ) \sum_{j,k\geq 0}m_{j+k}c_{j}\bar{c}_{k}=\int_{-\infty}^{\infty}\left|\sum_{j% \geq 0}c_{j}x^{j}\right|^{2}\,d\mu(x)
  6. μ \mu
  7. ( , , ) (\mathcal{H},\langle,\;\rangle)
  8. [ e n + 1 ] , [ e m ] = A m , n + 1 = m m + n + 1 = [ e n ] , [ e m + 1 ] . \langle[e_{n+1}],[e_{m}]\rangle=A_{m,n+1}=m_{m+n+1}=\langle[e_{n}],[e_{m+1}]\rangle.
  9. \mathcal{H}
  10. m n = - x n d μ ( x ) . m_{n}=\int_{-\infty}^{\infty}x^{n}\,d\mu(x).
  11. T ¯ \bar{T}\,
  12. T ¯ n [ 1 ] , [ 1 ] = x n d μ ( x ) . \langle\bar{T}^{n}[1],[1]\rangle=\int x^{n}d\mu(x).
  13. T ¯ n [ 1 ] , [ 1 ] = T n [ e 0 ] , [ e 0 ] = m n . \langle\bar{T}^{n}[1],[1]\rangle=\langle T^{n}[e_{0}],[e_{0}]\rangle=m_{n}.\,
  14. Δ n = [ m 0 m 1 m 2 m n m 1 m 2 m 3 m n + 1 m 2 m 3 m 4 m n + 2 m n m n + 1 m n + 2 m 2 n ] . \Delta_{n}=\left[\begin{matrix}m_{0}&m_{1}&m_{2}&\cdots&m_{n}\\ m_{1}&m_{2}&m_{3}&\cdots&m_{n+1}\\ m_{2}&m_{3}&m_{4}&\cdots&m_{n+2}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ m_{n}&m_{n+1}&m_{n+2}&\cdots&m_{2n}\end{matrix}\right].
  15. ( , , ) (\mathcal{H},\langle,\;\rangle)
  16. T ¯ \bar{T}\,
  17. m ( t ) = n = 0 m n t n n ! , m(t)=\sum_{n=0}m_{n}\frac{t^{n}}{n!},

Hamilton's_principle.html

  1. 𝒮 [ 𝐪 ] = def t 1 t 2 L ( 𝐪 ( t ) , 𝐪 ˙ ( t ) , t ) d t \mathcal{S}[\mathbf{q}]\ \stackrel{\mathrm{def}}{=}\ \int_{t_{1}}^{t_{2}}L(% \mathbf{q}(t),\dot{\mathbf{q}}(t),t)\,dt
  2. L ( 𝐪 , 𝐪 ˙ , t ) L(\mathbf{q},\dot{\mathbf{q}},t)
  3. 𝒮 \mathcal{S}
  4. 𝒮 \mathcal{S}
  5. 𝒮 \mathcal{S}
  6. s y m b o l ε ( t 1 ) = s y m b o l ε ( t 2 ) = def 0 symbol\varepsilon(t_{1})=symbol\varepsilon(t_{2})\ \stackrel{\mathrm{def}}{=}\ 0
  7. δ 𝒮 \delta\mathcal{S}
  8. δ 𝒮 = t 1 t 2 [ L ( 𝐪 + s y m b o l ε , 𝐪 ˙ + s y m b o l ε ˙ ) - L ( 𝐪 , 𝐪 ˙ ) ] d t = t 1 t 2 ( s y m b o l ε L 𝐪 + s y m b o l ε ˙ L 𝐪 ˙ ) d t \delta\mathcal{S}=\int_{t_{1}}^{t_{2}}\;\left[L(\mathbf{q}+symbol\varepsilon,% \dot{\mathbf{q}}+\dot{symbol{\varepsilon}})-L(\mathbf{q},\dot{\mathbf{q}})% \right]dt=\int_{t_{1}}^{t_{2}}\;\left(symbol\varepsilon\cdot\frac{\partial L}{% \partial\mathbf{q}}+\dot{symbol{\varepsilon}}\cdot\frac{\partial L}{\partial% \dot{\mathbf{q}}}\right)\,dt
  9. δ 𝒮 = [ s y m b o l ε L 𝐪 ˙ ] t 1 t 2 + t 1 t 2 ( s y m b o l ε L 𝐪 - s y m b o l ε d d t L 𝐪 ˙ ) d t \delta\mathcal{S}=\left[symbol\varepsilon\cdot\frac{\partial L}{\partial\dot{% \mathbf{q}}}\right]_{t_{1}}^{t_{2}}+\int_{t_{1}}^{t_{2}}\;\left(symbol% \varepsilon\cdot\frac{\partial L}{\partial\mathbf{q}}-symbol\varepsilon\cdot% \frac{d}{dt}\frac{\partial L}{\partial\dot{\mathbf{q}}}\right)\,dt
  10. s y m b o l ε ( t 1 ) = s y m b o l ε ( t 2 ) = def 0 symbol\varepsilon(t_{1})=symbol\varepsilon(t_{2})\ \stackrel{\mathrm{def}}{=}\ 0
  11. δ 𝒮 = t 1 t 2 s y m b o l ε ( L 𝐪 - d d t L 𝐪 ˙ ) d t \delta\mathcal{S}=\int_{t_{1}}^{t_{2}}\;symbol\varepsilon\cdot\left(\frac{% \partial L}{\partial\mathbf{q}}-\frac{d}{dt}\frac{\partial L}{\partial\dot{% \mathbf{q}}}\right)\,dt
  12. δ 𝒮 \delta\mathcal{S}
  13. 𝒮 \mathcal{S}
  14. p k = def L q ˙ k p_{k}\ \stackrel{\mathrm{def}}{=}\ \frac{\partial L}{\partial\dot{q}_{k}}
  15. L q k = 0 d d t L q ˙ k = 0 d p k d t = 0 , \frac{\partial L}{\partial q_{k}}=0\quad\Rightarrow\quad\frac{d}{dt}\frac{% \partial L}{\partial\dot{q}_{k}}=0\quad\Rightarrow\quad\frac{dp_{k}}{dt}=0\,,
  16. L = 1 2 m v 2 = 1 2 m ( x ˙ 2 + y ˙ 2 ) L=\frac{1}{2}mv^{2}=\frac{1}{2}m\left(\dot{x}^{2}+\dot{y}^{2}\right)
  17. d d t ( L x ˙ ) - L x = 0 m x ¨ = 0 \frac{d}{dt}\left(\frac{\partial L}{\partial\dot{x}}\right)-\frac{\partial L}{% \partial x}=0\qquad\Rightarrow\qquad m\ddot{x}=0
  18. L = 1 2 m ( r ˙ 2 + r 2 φ ˙ 2 ) . L=\frac{1}{2}m\left(\dot{r}^{2}+r^{2}\dot{\varphi}^{2}\right).
  19. d d t ( L r ˙ ) - L r = 0 r ¨ - r φ ˙ 2 = 0 \frac{d}{dt}\left(\frac{\partial L}{\partial\dot{r}}\right)-\frac{\partial L}{% \partial r}=0\qquad\Rightarrow\qquad\ddot{r}-r\dot{\varphi}^{2}=0
  20. d d t ( L φ ˙ ) - L φ = 0 φ ¨ + 2 r r ˙ φ ˙ = 0. \frac{d}{dt}\left(\frac{\partial L}{\partial\dot{\varphi}}\right)-\frac{% \partial L}{\partial\varphi}=0\qquad\Rightarrow\qquad\ddot{\varphi}+\frac{2}{r% }\dot{r}\dot{\varphi}=0.
  21. r = ( a t + b ) 2 + c 2 r=\sqrt{(at+b)^{2}+c^{2}}
  22. φ = tan - 1 ( a t + b c ) + d \varphi=\tan^{-1}\left(\frac{at+b}{c}\right)+d
  23. t 1 t 2 [ δ W e + δ T - δ U ] d t = 0 \int_{t1}^{t2}\left[\delta W_{e}+\delta T-\delta U\right]dt=0
  24. δ t 1 t 2 [ T - ( U + V ) ] d t = 0. \delta\int_{t1}^{t2}\left[T-(U+V)\right]dt=0.
  25. 𝒮 0 = def 𝐩 d 𝐪 \mathcal{S}_{0}\ \stackrel{\mathrm{def}}{=}\ \int\mathbf{p}\cdot d\mathbf{q}
  26. 𝒮 \mathcal{S}

Hammar_experiment.html

  1. t 1 = A B c + v + B C + D A c 2 - v 2 + C D c - v + Δ v t_{1}=\frac{AB}{c+v}+\frac{BC+DA}{\sqrt{c^{2}-v^{2}}}+\frac{CD}{c-v+\Delta v}
  2. t 2 = A B c - v + B C + D A c 2 - v 2 + C D c + v - Δ v t_{2}=\frac{AB}{c-v}+\frac{BC+DA}{\sqrt{c^{2}-v^{2}}}+\frac{CD}{c+v-\Delta v}
  3. Δ v \Delta v
  4. Δ t 2 l Δ v c 2 \Delta t\simeq\frac{2l\Delta v}{c^{2}}
  5. Δ v < 0.074 \Delta v<0.074

Hammett_equation.html

  1. log K K 0 = σ ρ \log\frac{K}{K_{0}}=\sigma\rho
  2. log k k 0 = σ ρ . \log\frac{k}{k_{0}}=\sigma\rho.

Hamming_space.html

  1. 2 N 2^{N}
  2. 2 2 m \mathbb{Z}_{2}^{2m}
  3. 4 m \mathbb{Z}_{4}^{m}

Hann_function.html

  1. w ( n ) = 0.5 ( 1 - cos ( 2 π n N - 1 ) ) w(n)=0.5\;\left(1-\cos\left(\frac{2\pi n}{N-1}\right)\right)
  2. w ( n ) = sin 2 ( π n N - 1 ) w(n)=\sin^{2}\left(\frac{\pi n}{N-1}\right)
  3. w ( n ) = haversin ( 2 π n N - 1 ) . w(n)=\operatorname{haversin}\left(\frac{2\pi n}{N-1}\right).
  4. w r = 𝟏 [ 0 , N - 1 ] w_{r}=\mathbf{1}_{[0,N-1]}
  5. w ( n ) = 1 2 w r ( n ) - 1 4 e i2 π n N - 1 w r ( n ) - 1 4 e - i2 π n N - 1 w r ( n ) w(n)=\frac{1}{2}\,w_{r}(n)-\frac{1}{4}e^{\mathrm{i}2\pi\frac{n}{N-1}}w_{r}(n)-% \frac{1}{4}e^{-\mathrm{i}2\pi\frac{n}{N-1}}w_{r}(n)
  6. w ^ ( ω ) = 1 2 w ^ r ( ω ) - 1 4 w ^ r ( ω + 2 π N - 1 ) - 1 4 w ^ r ( ω - 2 π N - 1 ) \hat{w}(\omega)=\frac{1}{2}\hat{w}_{r}(\omega)-\frac{1}{4}\hat{w}_{r}\left(% \omega+\frac{2\pi}{N-1}\right)-\frac{1}{4}\hat{w}_{r}\left(\omega-\frac{2\pi}{% N-1}\right)
  7. w ^ r ( ω ) = e - i ω N - 1 2 sin ( N ω / 2 ) sin ( ω / 2 ) \hat{w}_{r}(\omega)=e^{-\mathrm{i}\omega\frac{N-1}{2}}\frac{\sin(N\omega/2)}{% \sin(\omega/2)}
  8. S ( τ ) = w ( t + τ ) f ( t ) d t S(\tau)=\int w(t+\tau)f(t)\,dt

Hans_Heilbronn.html

  1. ( - d ) \mathbb{Q}(\sqrt{-d})
  2. d d
  3. ( - d ) \mathbb{Q}(\sqrt{-d})
  4. d d
  5. f f
  6. k k
  7. n = 2 k n=2^{k}
  8. x x
  9. n \mathbb{Z}^{n}
  10. | f ( x ) | |f(x)|

Harmonic_division.html

  1. C A C B = D A D B . \frac{CA}{CB}=\frac{DA}{DB}.
  2. B C B D = A C A D \frac{BC}{BD}=\frac{AC}{AD}

Harmonic_map.html

  1. E ( ϕ ) = M d ϕ 2 d Vol . E(\phi)=\int_{M}\|d\phi\|^{2}\,d\operatorname{Vol}.
  2. e ( ϕ ) = 1 2 d ϕ 2 e(\phi)=\frac{1}{2}\|d\phi\|^{2}
  3. d ϕ 2 \|d\phi\|^{2}
  4. ϕ \phi
  5. T * M ϕ - 1 T N T^{*}M\otimes\phi^{-1}TN
  6. E ( ϕ ) = M e ( ϕ ) d v g = 1 2 M d ϕ 2 d v g E(\phi)=\int_{M}e(\phi)\,dv_{g}=\frac{1}{2}\int_{M}\|d\phi\|^{2}\,dv_{g}
  7. e ( ϕ ) = 1 2 trace g ϕ * h . e(\phi)=\frac{1}{2}\operatorname{trace}_{g}\phi^{*}h.
  8. e ( ϕ ) = 1 2 g i j h α β ϕ α x i ϕ β x j . e(\phi)=\frac{1}{2}g^{ij}h_{\alpha\beta}\frac{\partial\phi^{\alpha}}{\partial x% ^{i}}\frac{\partial\phi^{\beta}}{\partial x^{j}}.
  9. τ ( ϕ ) = def trace g d ϕ = 0 \tau(\phi)\ \stackrel{\,\text{def}}{=}\ \operatorname{trace}_{g}\nabla d\phi=0
  10. e ϵ ( u ) ( x ) = M d 2 ( u ( x ) , u ( y ) ) d μ x ϵ ( y ) M d 2 ( x , y ) d μ x ϵ ( y ) e_{\epsilon}(u)(x)=\frac{\int_{M}d^{2}(u(x),u(y))\,d\mu^{\epsilon}_{x}(y)}{% \int_{M}d^{2}(x,y)\,d\mu^{\epsilon}_{x}(y)}

Hartley_function.html

  1. H 0 ( A ) := log b | A | . H_{0}(A):=\mathrm{log}_{b}|A|.
  2. H 0 ( X ) = 1 1 - 0 log i = 1 | X | p i 0 = log | X | . H_{0}(X)=\frac{1}{1-0}\log\sum_{i=1}^{|X|}p_{i}^{0}=\log|X|.
  3. H ( m n ) = H ( m ) + H ( n ) H(mn)=H(m)+H(n)
  4. H ( m ) H ( m + 1 ) H(m)\leq H(m+1)
  5. H ( 2 ) = 1 H(2)=1
  6. H ( m n ) = H ( m ) + H ( n ) H(mn)=H(m)+H(n)\,
  7. H ( m ) H ( m + 1 ) H(m)\leq H(m+1)\,
  8. H ( 2 ) = 1 H(2)=1\,
  9. f ( n k ) = k f ( n ) . f(n^{k})=kf(n).\,
  10. a s b t a s + 1 . ( 1 ) a^{s}\leq b^{t}\leq a^{s+1}.\qquad(1)
  11. s log 2 a t log 2 b ( s + 1 ) log 2 a s\log_{2}a\leq t\log_{2}b\leq(s+1)\log_{2}a\,
  12. s t log 2 b log 2 a s + 1 t . \frac{s}{t}\leq\frac{\log_{2}b}{\log_{2}a}\leq\frac{s+1}{t}.
  13. f ( a s ) f ( b t ) f ( a s + 1 ) . f(a^{s})\leq f(b^{t})\leq f(a^{s+1}).\,
  14. s f ( a ) t f ( b ) ( s + 1 ) f ( a ) , sf(a)\leq tf(b)\leq(s+1)f(a),\,
  15. s t f ( b ) f ( a ) s + 1 t . \frac{s}{t}\leq\frac{f(b)}{f(a)}\leq\frac{s+1}{t}.
  16. | f ( b ) f ( a ) - log 2 ( b ) log 2 ( a ) | 1 t . \Big|\frac{f(b)}{f(a)}-\frac{\log_{2}(b)}{\log_{2}(a)}\Big|\leq\frac{1}{t}.
  17. f ( b ) f ( a ) = log 2 ( b ) log 2 ( a ) . \frac{f(b)}{f(a)}=\frac{\log_{2}(b)}{\log_{2}(a)}.
  18. f ( a ) = μ log 2 ( a ) f(a)=\mu\log_{2}(a)\,

Hasty_Pudding_cipher.html

  1. \oplus
  2. \oplus
  3. \oplus
  4. \oplus

Hausdorff_moment_problem.html

  1. m n = 0 1 x n d μ ( x ) m_{n}=\int_{0}^{1}x^{n}\,d\mu(x)\,
  2. ( - 1 ) k ( Δ k m ) n 0 (-1)^{k}(\Delta^{k}m)_{n}\geq 0
  3. ( Δ m ) n = m n + 1 - m n . (\Delta m)_{n}=m_{n+1}-m_{n}.
  4. ( - 1 ) k ( Δ k m ) n = 0 1 x n ( 1 - x ) k d μ ( x ) , (-1)^{k}(\Delta^{k}m)_{n}=\int_{0}^{1}x^{n}(1-x)^{k}d\mu(x),
  5. Δ 4 m 6 = m 6 - 4 m 7 + 6 m 8 - 4 m 9 + m 10 = x 6 ( 1 - x ) 4 d μ ( x ) 0. \Delta^{4}m_{6}=m_{6}-4m_{7}+6m_{8}-4m_{9}+m_{10}=\int x^{6}(1-x)^{4}d\mu(x)% \geq 0.

Haynes–Shockley_experiment.html

  1. j e = + μ n n E + D n n x j_{e}=+\mu_{n}nE+D_{n}\frac{\partial n}{\partial x}
  2. j p = + μ p p E - D p p x j_{p}=+\mu_{p}pE-D_{p}\frac{\partial p}{\partial x}
  3. n t = - ( n - n 0 ) τ n - j e x \frac{\partial n}{\partial t}=\frac{-(n-n_{0})}{\tau_{n}}-\frac{\partial j_{e}% }{\partial x}
  4. p t = - ( p - p 0 ) τ p - j p x \frac{\partial p}{\partial t}=\frac{-(p-p_{0})}{\tau_{p}}-\frac{\partial j_{p}% }{\partial x}
  5. p 1 = p - p 0 , n 1 = n - n 0 p_{1}=p-p_{0}\,,\quad n_{1}=n-n_{0}
  6. p 1 t = D p 2 p 1 x 2 - μ p p E x - μ p E p 1 x - p 1 τ p \frac{\partial p_{1}}{\partial t}=D_{p}\frac{\partial^{2}p_{1}}{\partial x^{2}% }-\mu_{p}p\frac{\partial E}{\partial x}-\mu_{p}E\frac{\partial p_{1}}{\partial x% }-\frac{p_{1}}{\tau_{p}}
  7. n 1 t = D n 2 n 1 x 2 + μ n n E x + μ n E n 1 x - n 1 τ n \frac{\partial n_{1}}{\partial t}=D_{n}\frac{\partial^{2}n_{1}}{\partial x^{2}% }+\mu_{n}n\frac{\partial E}{\partial x}+\mu_{n}E\frac{\partial n_{1}}{\partial x% }-\frac{n_{1}}{\tau_{n}}
  8. E x = ρ ϵ ϵ 0 = e 0 ( ( p - p 0 ) - ( n - n 0 ) ) ϵ ϵ 0 = e 0 ( p 1 - n 1 ) ϵ ϵ 0 \frac{\partial E}{\partial x}=\frac{\rho}{\epsilon\epsilon_{0}}=\frac{e_{0}((p% -p_{0})-(n-n_{0}))}{\epsilon\epsilon_{0}}=\frac{e_{0}(p_{1}-n_{1})}{\epsilon% \epsilon_{0}}
  9. p 1 = n mean + δ , n 1 = n mean - δ , p_{1}=n\text{mean}+\delta\,,\quad n_{1}=n\text{mean}-\delta\,,
  10. n mean n\text{mean}
  11. n mean t = D p 2 n mean x 2 - μ p p E x - μ p E n mean x - n mean τ p \frac{\partial n\text{mean}}{\partial t}=D_{p}\frac{\partial^{2}n\text{mean}}{% \partial x^{2}}-\mu_{p}p\frac{\partial E}{\partial x}-\mu_{p}E\frac{\partial n% \text{mean}}{\partial x}-\frac{n\text{mean}}{\tau_{p}}
  12. n mean t = D n 2 n mean x 2 + μ n n E x + μ n E n mean x - n mean τ n \frac{\partial n\text{mean}}{\partial t}=D_{n}\frac{\partial^{2}n\text{mean}}{% \partial x^{2}}+\mu_{n}n\frac{\partial E}{\partial x}+\mu_{n}E\frac{\partial n% \text{mean}}{\partial x}-\frac{n\text{mean}}{\tau_{n}}
  13. μ = e β D \mu=e\beta D
  14. n mean t = D * 2 n mean x 2 - μ * E n mean x - n mean τ * , \frac{\partial n\text{mean}}{\partial t}=D^{*}\frac{\partial^{2}n\text{mean}}{% \partial x^{2}}-\mu^{*}E\frac{\partial n\text{mean}}{\partial x}-\frac{n\text{% mean}}{\tau^{*}},
  15. D * = D n D p ( n + p ) p D p + n D n D^{*}=\frac{D_{n}D_{p}(n+p)}{pD_{p}+nD_{n}}
  16. μ * = μ n μ p ( n - p ) p μ p + n μ n \mu^{*}=\frac{\mu_{n}\mu_{p}(n-p)}{p\mu_{p}+n\mu_{n}}
  17. 1 τ * = p μ p τ p + n μ n τ n τ p τ n ( p μ p + n μ n ) . \frac{1}{\tau^{*}}=\frac{p\mu_{p}\tau_{p}+n\mu_{n}\tau_{n}}{\tau_{p}\tau_{n}(p% \mu_{p}+n\mu_{n})}.
  18. n mean ( x , t ) = A 1 4 π D * t e - t / τ * e - ( x + μ * E t - x 0 ) 2 4 D * t n\text{mean}(x,t)=A\frac{1}{\sqrt{4\pi D^{*}t}}e^{-t/\tau^{*}}e^{-\frac{(x+\mu% ^{*}Et-x_{0})^{2}}{4D^{*}t}}
  19. μ * = d E t 0 \mu^{*}=\frac{d}{Et_{0}}
  20. D * = ( μ * E ) 2 ( δ t ) 2 16 t 0 D^{*}=(\mu^{*}E)^{2}\frac{(\delta t)^{2}}{16t_{0}}

Hazard_analysis.html

  1. 1 × 10 - 5 1\times 10^{-5}
  2. 1 × 10 - 5 1\times 10^{-5}
  3. 1 × 10 - 7 1\times 10^{-7}
  4. 1 × 10 - 7 1\times 10^{-7}
  5. 1 × 10 - 9 1\times 10^{-9}
  6. 1 × 10 - 9 1\times 10^{-9}

Hazen–Williams_equation.html

  1. V = C R S = C R 0.5 S 0.5 V=C\sqrt{RS}=C\,R^{0.5}\,S^{0.5}
  2. V = k C R 0.63 S 0.54 V=k\,C\,R^{0.63}\,S^{0.54}
  3. V = k C R 0.63 S 0.54 V=k\,C\,R^{0.63}\,S^{0.54}
  4. 1 / 0.54 1/0.54
  5. V 1.85 = k 1.85 C 1.85 R 1.17 S V^{1.85}=k^{1.85}\,C^{1.85}\,R^{1.17}\,S
  6. S = V 1.85 k 1.85 C 1.85 R 1.17 S={V^{1.85}\over k^{1.85}\,C^{1.85}\,R^{1.17}}
  7. S = V 1.85 A 1.85 k 1.85 C 1.85 R 1.17 A 1.85 = Q 1.85 k 1.85 C 1.85 R 1.17 A 1.85 S={V^{1.85}A^{1.85}\over k^{1.85}\,C^{1.85}\,R^{1.17}\,A^{1.85}}={Q^{1.85}% \over k^{1.85}\,C^{1.85}\,R^{1.17}\,A^{1.85}}
  8. π d 2 / 4 \pi d^{2}/4
  9. S = 4 1.17 4 1.85 Q 1.85 π 1.85 k 1.85 C 1.85 d 1.17 d 3.70 = 4 3.02 Q 1.85 π 1.85 k 1.85 C 1.85 d 4.87 = 4 3.02 π 1.85 k 1.85 Q 1.85 C 1.85 d 4.87 = 7.916 k 1.85 Q 1.85 C 1.85 d 4.87 S={4^{1.17}\,4^{1.85}\,Q^{1.85}\over\pi^{1.85}\,k^{1.85}\,C^{1.85}\,d^{1.17}\,% d^{3.70}}={4^{3.02}\,Q^{1.85}\over\pi^{1.85}\,k^{1.85}\,C^{1.85}\,d^{4.87}}={4% ^{3.02}\over\pi^{1.85}\,k^{1.85}}{Q^{1.85}\over C^{1.85}\,d^{4.87}}={7.916% \over k^{1.85}}{Q^{1.85}\over C^{1.85}\,d^{4.87}}
  10. S psi per foot = P d L = 4.52 Q 1.85 C 1.85 d 4.87 S_{\mathrm{psi\ per\ foot}}=\frac{P_{d}}{L}=\frac{4.52\ Q^{1.85}}{C^{1.85}\ d^% {4.87}}
  11. S = h f L = 10.67 Q 1.85 C 1.85 d 4.87 S=\frac{h_{f}}{L}=\frac{10.67\ Q^{1.85}}{C^{1.85}\ d^{4.87}}

Heat_kernel.html

  1. K ( t , x , y ) = 1 ( 4 π t ) d / 2 e - | x - y | 2 / 4 t K(t,x,y)=\frac{1}{(4\pi t)^{d/2}}e^{-|x-y|^{2}/4t}\,
  2. K t ( t , x , y ) = Δ x K ( t , x , y ) \frac{\partial K}{\partial t}(t,x,y)=\Delta_{x}K(t,x,y)\,
  3. lim t 0 K ( t , x , y ) = δ ( x - y ) = δ x ( y ) \lim_{t\to 0}K(t,x,y)=\delta(x-y)=\delta_{x}(y)
  4. lim t 0 𝐑 d K ( t , x , y ) ϕ ( y ) d y = ϕ ( x ) . \lim_{t\to 0}\int_{\mathbf{R}^{d}}K(t,x,y)\phi(y)\,dy=\phi(x).
  5. K t ( t , x , y ) = Δ K ( t , x , y ) for all t > 0 and x , y Ω \frac{\partial K}{\partial t}(t,x,y)=\Delta K(t,x,y)\rm{\ \ for\ all\ }t>0\rm{% \ and\ }x,y\in\Omega
  6. lim t 0 K ( t , x , y ) = δ x ( y ) for all x , y Ω \lim_{t\to 0}K(t,x,y)=\delta_{x}(y)\rm{\ \ for\ all\ }x,y\in\Omega
  7. K ( t , x , y ) = 0 , x Ω or y Ω . K(t,x,y)=0,\quad x\in\partial\Omega\rm{\ or\ }y\in\partial\Omega.
  8. { Δ ϕ + λ ϕ = 0 in U ϕ = 0 on U . \left\{\begin{array}[]{ll}\Delta\phi+\lambda\phi=0&\mathrm{in\ }\ U\\ \phi=0&\mathrm{on\ }\ \partial U.\end{array}\right.
  9. 0 < λ 1 < λ 2 λ 3 , λ n . 0<\lambda_{1}<\lambda_{2}\leq\lambda_{3}\leq\cdots,\quad\lambda_{n}\to\infty.
  10. K ( t , x , y ) = n = 0 e - λ n t ϕ n ( x ) ϕ n ( y ) . K(t,x,y)=\sum_{n=0}^{\infty}e^{-\lambda_{n}t}\phi_{n}(x)\phi_{n}(y).
  11. T ϕ = Ω K ( t , x , y ) ϕ ( y ) d y . T\phi=\int_{\Omega}K(t,x,y)\phi(y)\,dy.
  12. T = e t Δ . T=e^{t\Delta}.

Hectogon.html

  1. A = 25 t 2 cot π 100 A=25t^{2}\cot\frac{\pi}{100}
  2. r = 1 2 t cot π 100 r=\frac{1}{2}t\cot\frac{\pi}{100}
  3. R = 1 2 t csc π 100 R=\frac{1}{2}t\csc\frac{\pi}{100}

Heisenberg_model_(quantum).html

  1. σ i { ± 1 } \sigma_{i}\in\{\pm 1\}
  2. H ^ = - J j = 1 N σ j σ j + 1 - h j = 1 N σ j \hat{H}=-J\sum_{j=1}^{N}\sigma_{j}\sigma_{j+1}-h\sum_{j=1}^{N}\sigma_{j}
  3. J J
  4. σ N + 1 = σ 1 \sigma_{N+1}=\sigma_{1}
  5. J x , J y , J_{x},J_{y},
  6. J z J_{z}
  7. H ^ = - 1 2 j = 1 N ( J x σ j x σ j + 1 x + J y σ j y σ j + 1 y + J z σ j z σ j + 1 z + h σ j z ) \hat{H}=-\frac{1}{2}\sum_{j=1}^{N}(J_{x}\sigma_{j}^{x}\sigma_{j+1}^{x}+J_{y}% \sigma_{j}^{y}\sigma_{j+1}^{y}+J_{z}\sigma_{j}^{z}\sigma_{j+1}^{z}+h\sigma_{j}% ^{z})
  8. h h
  9. s = 1 / 2 s=1/2
  10. σ x = ( 0 1 1 0 ) \sigma^{x}=\begin{pmatrix}0&1\\ 1&0\end{pmatrix}
  11. σ y = ( 0 - i i 0 ) \sigma^{y}=\begin{pmatrix}0&-i\\ i&0\end{pmatrix}
  12. σ z = ( 1 0 0 - 1 ) \sigma^{z}=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}
  13. ( 2 ) N (\mathbb{C}^{2})^{\otimes N}
  14. 2 N 2^{N}
  15. H ^ = - J z j = 1 N σ j z σ j + 1 z - g J z j = 1 N σ j x \hat{H}=-J_{z}\sum_{j=1}^{N}\sigma_{j}^{z}\sigma_{j+1}^{z}-gJ_{z}\sum_{j=1}^{N% }\sigma_{j}^{x}
  16. σ i z = j i S j x \sigma_{i}^{z}=\prod_{j\leq i}S^{x}_{j}
  17. σ i x = S i z S i + 1 z \sigma_{i}^{x}=S^{z}_{i}S^{z}_{i+1}
  18. S x S^{x}
  19. S z S^{z}
  20. H ^ = - g J z j = 1 N S j z S j + 1 z - g J z j = 1 N S j x \hat{H}=-gJ_{z}\sum_{j=1}^{N}S_{j}^{z}S_{j+1}^{z}-gJ_{z}\sum_{j=1}^{N}S_{j}^{x}
  21. g g
  22. g = 1 g=1
  23. J = J x = J y J z = Δ J=J_{x}=J_{y}\neq J_{z}=\Delta
  24. J J
  25. J J
  26. J J

Hereditary_set.html

  1. { } \{\varnothing\}
  2. \varnothing

Hermann–Mauguin_notation.html

  1. n ¯ \overline{n}
  2. 1 ¯ \overline{1}
  3. 3 ¯ \overline{3}
  4. 4 ¯ \overline{4}
  5. 5 ¯ \overline{5}
  6. 6 ¯ \overline{6}
  7. 7 ¯ \overline{7}
  8. 8 ¯ \overline{8}
  9. 2 ¯ \overline{2}
  10. 2 ¯ \overline{2}
  11. n m \tfrac{n}{m}
  12. 3 ¯ \overline{3}
  13. 4 ¯ \overline{4}
  14. 5 ¯ \overline{5}
  15. 6 ¯ \overline{6}
  16. 7 ¯ \overline{7}
  17. 8 ¯ \overline{8}
  18. 3 m \tfrac{3}{m}
  19. 6 ¯ \overline{6}
  20. 6 ¯ \overline{6}
  21. 6 ¯ \overline{6}
  22. 3 m \tfrac{3}{m}
  23. 6 ¯ m \tfrac{\bar{6}}{m}
  24. 6 ¯ \overline{6}
  25. 3 ¯ \overline{3}
  26. 3 ¯ \overline{3}
  27. 4 m \tfrac{4}{m}
  28. 4 ¯ m \tfrac{\bar{4}}{m}
  29. 4 ¯ \overline{4}
  30. 6 m \tfrac{6}{m}
  31. 3 ¯ \overline{3}
  32. 6 ¯ \overline{6}
  33. 3 ¯ \overline{3}
  34. 6 ¯ \overline{6}
  35. 6 m \tfrac{6}{m}
  36. 1 ¯ \overline{1}
  37. 2 m \tfrac{2}{m}
  38. 2 m 2 m 2 m \tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}
  39. 2 m \tfrac{2}{m}
  40. 2 m \tfrac{2}{m}
  41. 3 ¯ \overline{3}
  42. 3 ¯ \overline{3}
  43. 2 m \tfrac{2}{m}
  44. 4 ¯ \overline{4}
  45. 4 ¯ \overline{4}
  46. 4 m \tfrac{4}{m}
  47. 4 m 2 m 2 m \tfrac{4}{m}\tfrac{2}{m}\tfrac{2}{m}
  48. 6 ¯ \overline{6}
  49. 6 ¯ \overline{6}
  50. 6 m \tfrac{6}{m}
  51. 6 m 2 m 2 m \tfrac{6}{m}\tfrac{2}{m}\tfrac{2}{m}
  52. \infty
  53. \infty
  54. 2 \infty 2
  55. m \infty m
  56. 𝐧 𝐦 \mathbf{\tfrac{n}{m}}
  57. \color R e d 3 m \color{Red}{\tfrac{3}{m}}
  58. 6 ¯ \overline{6}
  59. 4 m \tfrac{4}{m}
  60. \color R e d 5 m \color{Red}{\tfrac{5}{m}}
  61. 10 ¯ \overline{10}
  62. 6 m \tfrac{6}{m}
  63. \color R e d 7 m \color{Red}{\tfrac{7}{m}}
  64. 14 ¯ \overline{14}
  65. 8 m \tfrac{8}{m}
  66. m \tfrac{\infty}{m}
  67. 𝐧 𝐦 𝟐 𝐦 𝟐 𝐦 \mathbf{\tfrac{n}{m}\tfrac{2}{m}\tfrac{2}{m}}
  68. \color R e d 3 m \color{Red}{\tfrac{3}{m}}
  69. 4 m 2 m 2 m \tfrac{4}{m}\tfrac{2}{m}\tfrac{2}{m}
  70. \color R e d 5 m \color{Red}{\tfrac{5}{m}}
  71. 6 m 2 m 2 m \tfrac{6}{m}\tfrac{2}{m}\tfrac{2}{m}
  72. \color R e d 7 m \color{Red}{\tfrac{7}{m}}
  73. 8 m 2 m 2 m \tfrac{8}{m}\tfrac{2}{m}\tfrac{2}{m}
  74. m m \tfrac{\infty}{m}m
  75. 𝐧 ¯ \mathbf{\bar{n}}
  76. 3 ¯ \bar{3}
  77. 4 ¯ \bar{4}
  78. 5 ¯ \bar{5}
  79. 6 ¯ \bar{6}
  80. 7 ¯ \bar{7}
  81. 8 ¯ \bar{8}
  82. \color R e d ¯ = m {\color{Red}{\bar{\infty}}}=\tfrac{\infty}{m}
  83. 𝐧 ¯ 𝟐 𝐦 \mathbf{\bar{n}\tfrac{2}{m}}
  84. 𝐧 ¯ 𝟐 𝐦 \mathbf{\bar{n}2m}
  85. 3 ¯ \overline{3}
  86. 2 m \tfrac{2}{m}
  87. 4 ¯ \overline{4}
  88. 5 ¯ \overline{5}
  89. 2 m \tfrac{2}{m}
  90. 6 ¯ \overline{6}
  91. 7 ¯ \overline{7}
  92. 2 m \tfrac{2}{m}
  93. 8 ¯ \overline{8}
  94. \color R e d ¯ m = m m {\color{Red}{\bar{\infty}m}}=\tfrac{\infty}{m}m
  95. n ¯ \overline{n}
  96. n ¯ \overline{n}
  97. n 2 \tfrac{n}{2}
  98. n 2 \tfrac{n}{2}
  99. 4 ¯ \overline{4}
  100. 6 ¯ \overline{6}
  101. 8 ¯ \overline{8}
  102. 4 ¯ \overline{4}
  103. 6 ¯ \overline{6}
  104. 8 ¯ \overline{8}
  105. 6 ¯ \overline{6}
  106. 6 ¯ \overline{6}
  107. 3 ¯ \overline{3}
  108. 3 ¯ \overline{3}
  109. 2 m \tfrac{2}{m}
  110. 2 m \tfrac{2}{m}
  111. 2 m \tfrac{2}{m}
  112. 3 ¯ \overline{3}
  113. n = n=\infty
  114. 2 m \tfrac{2}{m}
  115. 3 ¯ \overline{3}
  116. 4 ¯ \overline{4}
  117. 4 m \tfrac{4}{m}
  118. 3 ¯ \overline{3}
  119. 2 m \tfrac{2}{m}
  120. 4 m \tfrac{4}{m}
  121. 3 ¯ \overline{3}
  122. 2 m \tfrac{2}{m}
  123. 3 ¯ \overline{3}
  124. 2 m \tfrac{2}{m}
  125. n m \tfrac{n}{m}
  126. 2 m 2 m 2 m \tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}
  127. 4 m 2 m 2 m \tfrac{4}{m}\tfrac{2}{m}\tfrac{2}{m}
  128. 4 m \tfrac{4}{m}
  129. 4 m \tfrac{4}{m}
  130. 3 ¯ \overline{3}
  131. 2 m \tfrac{2}{m}
  132. 3 ¯ \overline{3}
  133. 4 m 2 m 2 m \tfrac{4}{m}\tfrac{2}{m}\tfrac{2}{m}
  134. 6 m 2 m 2 m \tfrac{6}{m}\tfrac{2}{m}\tfrac{2}{m}
  135. 4 m \tfrac{4}{m}
  136. 6 m \tfrac{6}{m}
  137. 3 ¯ \overline{3}
  138. 2 m \tfrac{2}{m}
  139. 3 ¯ \overline{3}
  140. 35 ¯ \overline{35}
  141. 5 ¯ \overline{5}
  142. 5 ¯ \overline{5}
  143. 53 ¯ \overline{53}
  144. \infty\infty
  145. 2 2\infty
  146. m \tfrac{\infty}{m}\infty
  147. m ¯ m\bar{\infty}
  148. m \infty\infty m
  149. 360 n \tfrac{360^{\circ}}{n}
  150. a a
  151. b b
  152. c c
  153. n n
  154. d d
  155. d d
  156. a a
  157. b b
  158. c c
  159. n n
  160. d d
  161. e e

Hermite's_identity.html

  1. k = 0 n - 1 x + k n = n x . \sum_{k=0}^{n-1}\left\lfloor x+\frac{k}{n}\right\rfloor=\lfloor nx\rfloor.
  2. x x
  3. x = x + { x } x=\lfloor x\rfloor+\{x\}
  4. k { 1 , , n } k^{\prime}\in\{1,\ldots,n\}
  5. x = x + k - 1 n x < x + k n = x + 1. \lfloor x\rfloor=\left\lfloor x+\frac{k^{\prime}-1}{n}\right\rfloor\leq x<% \left\lfloor x+\frac{k^{\prime}}{n}\right\rfloor=\lfloor x\rfloor+1.
  6. x \lfloor x\rfloor
  7. 0 = { x } + k - 1 n { x } < { x } + k n = 1. 0=\left\lfloor\{x\}+\frac{k^{\prime}-1}{n}\right\rfloor\leq\{x\}<\left\lfloor% \{x\}+\frac{k^{\prime}}{n}\right\rfloor=1.
  8. 1 - k n { x } < 1 - k - 1 n , 1-\frac{k^{\prime}}{n}\leq\{x\}<1-\frac{k^{\prime}-1}{n},
  9. n n
  10. n - k n { x } < n - k + 1. n-k^{\prime}\leq n\,\{x\}<n-k^{\prime}+1.
  11. k k^{\prime}
  12. k = 0 n - 1 x + k n = k = 0 k - 1 x + k = k n - 1 ( x + 1 ) = n x + n - k = n x + n { x } = n x + n { x } = n x . \sum_{k=0}^{n-1}\left\lfloor x+\frac{k}{n}\right\rfloor=\sum_{k=0}^{k^{\prime}% -1}\lfloor x\rfloor+\sum_{k=k^{\prime}}^{n-1}(\lfloor x\rfloor+1)=n\,\lfloor x% \rfloor+n-k^{\prime}=n\,\lfloor x\rfloor+\lfloor n\,\{x\}\rfloor=\left\lfloor n% \,\lfloor x\rfloor+n\,\{x\}\right\rfloor=\lfloor nx\rfloor.

Hermitian_symmetric_space.html

  1. 𝔥 \mathfrak{h}
  2. 𝔥 = 𝔨 𝔪 , \displaystyle{\mathfrak{h}=\mathfrak{k}\oplus\mathfrak{m},}
  3. 𝔨 \mathfrak{k}
  4. 𝔪 \mathfrak{m}
  5. 𝔨 \mathfrak{k}
  6. 𝔥 \mathfrak{h}
  7. 𝔥 \mathfrak{h}
  8. 𝔥 \mathfrak{h}
  9. 𝔨 \mathfrak{k}
  10. 𝔪 \mathfrak{m}
  11. 𝔪 \mathfrak{m}
  12. 𝔥 \mathfrak{h}
  13. 𝔨 \mathfrak{k}
  14. 𝔪 \mathfrak{m}
  15. 𝔥 \mathfrak{h}
  16. ( [ [ A , B ] , C ] , D ) = ( [ A , B ] , [ C , D ] ) = ( [ [ C , D ] , B ] , A ) . \displaystyle{([[A,B],C],D)=([A,B],[C,D])=([[C,D],B],A).}
  17. [ J A , J B ] = [ A , B ] . \displaystyle{[JA,JB]=[A,B].}
  18. 𝔥 \mathfrak{h}
  19. 𝔨 \mathfrak{k}
  20. 𝔥 \mathfrak{h}
  21. 𝔥 \mathfrak{h}
  22. δ ( X ) = [ T + A , X ] , \displaystyle{\delta(X)=[T+A,X],}
  23. 𝔨 \mathfrak{k}
  24. 𝔪 \mathfrak{m}
  25. 𝔨 \mathfrak{k}
  26. 𝔨 \mathfrak{k}
  27. 𝔥 \mathfrak{h}
  28. 𝔨 \mathfrak{k}
  29. 𝔪 \mathfrak{m}
  30. 𝔨 \mathfrak{k}
  31. 𝔩 \mathfrak{l}
  32. 𝔪 1 \mathfrak{m}_{1}
  33. 𝔪 \mathfrak{m}
  34. 𝔩 = 𝔨 𝔪 1 , 𝔪 1 = 𝔩 𝔪 . \displaystyle{\mathfrak{l}=\mathfrak{k}\oplus\mathfrak{m}_{1},\,\,\,\ % \mathfrak{m}_{1}=\mathfrak{l}\cap\mathfrak{m}.}
  35. 𝔤 \mathfrak{g}
  36. 𝔥 \mathfrak{h}
  37. 𝔥 = i = 1 N 𝔥 i , \displaystyle{\mathfrak{h}=\oplus_{i=1}^{N}\mathfrak{h}_{i},}
  38. 𝔥 1 \mathfrak{h}_{1}
  39. 𝔨 \mathfrak{k}
  40. 𝔪 \mathfrak{m}
  41. 𝔥 1 \mathfrak{h}_{1}
  42. 𝔪 \mathfrak{m}
  43. 𝔥 = 𝔨 𝔪 , \mathfrak{h}=\mathfrak{k}\oplus\mathfrak{m},
  44. ( 𝔪 , J ) (\mathfrak{m},J)
  45. 𝔤 = 𝔪 + 𝔩 𝔪 - \mathfrak{g}=\mathfrak{m}_{+}\oplus\mathfrak{l}\oplus\mathfrak{m}_{-}
  46. 𝔪 = 𝔪 - 𝔪 + \mathfrak{m}\otimes\mathbb{C}=\mathfrak{m}_{-}\oplus\mathfrak{m}_{+}
  47. 𝔩 = 𝔨 \mathfrak{l}=\mathfrak{k}\otimes\mathbb{C}
  48. 𝔪 + 𝔩 \mathfrak{m}^{+}\oplus\mathfrak{l}
  49. 𝔪 ± \mathfrak{m}_{\pm}
  50. 𝔪 ± \mathfrak{m}_{\pm}
  51. 𝔪 ± \mathfrak{m}_{\pm}
  52. 𝔪 \mathfrak{m}_{\mathbb{C}}
  53. 𝔪 \mathfrak{m}
  54. 𝔪 + \mathfrak{m}_{+}
  55. 𝔪 - \mathfrak{m}_{-}
  56. 𝔪 + \mathfrak{m}_{+}
  57. SL ( p + q , ) \mathrm{SL}(p+q,\mathbb{C})
  58. SU ( p + q ) \mathrm{SU}(p+q)\,
  59. S ( U ( p ) × U ( q ) ) \mathrm{S}(\mathrm{U}(p)\times\mathrm{U}(q))
  60. p + q \mathbb{C}^{p+q}
  61. SO ( 2 n , ) \mathrm{SO}(2n,\mathbb{C})
  62. SO ( 2 n ) \mathrm{SO}(2n)\,
  63. U ( n ) \mathrm{U}(n)\,
  64. 1 2 n ( n - 1 ) \tfrac{1}{2}n(n-1)
  65. [ 1 2 n ] [\tfrac{1}{2}n]
  66. 2 n \mathbb{R}^{2n}
  67. Sp ( 2 n , ) \mathrm{Sp}(2n,\mathbb{C})\,
  68. Sp ( n ) \mathrm{Sp}(n)\,
  69. U ( n ) \mathrm{U}(n)\,
  70. 1 2 n ( n + 1 ) \tfrac{1}{2}n(n+1)
  71. n \mathbb{H}^{n}
  72. SO ( n + 2 , ) \mathrm{SO}(n+2,\mathbb{C})
  73. SO ( n + 2 ) \mathrm{SO}(n+2)\,
  74. SO ( n ) × SO ( 2 ) \mathrm{SO}(n)\times\mathrm{SO}(2)
  75. n + 2 \mathbb{R}^{n+2}
  76. E 6 E_{6}^{\mathbb{C}}\,
  77. E 6 E_{6}\,
  78. SO ( 10 ) × SO ( 2 ) \mathrm{SO}(10)\times\mathrm{SO}(2)
  79. ( 𝕆 ) P 2 (\mathbb{C}\otimes\mathbb{O})P^{2}
  80. 𝕆 P 2 \mathbb{O}P^{2}
  81. E 7 E_{7}^{\mathbb{C}}
  82. E 7 E_{7}\,
  83. E 6 × SO ( 2 ) E_{6}\times\mathrm{SO}(2)
  84. ( 𝕆 ) P 2 (\mathbb{H}\otimes\mathbb{O})P^{2}
  85. ( 𝕆 ) P 2 (\mathbb{C}\otimes\mathbb{O})P^{2}
  86. S = ( - α I p 0 0 α I q ) S=\begin{pmatrix}-\alpha I_{p}&0\\ 0&\alpha I_{q}\end{pmatrix}
  87. J n = ( 0 I n - I n 0 ) J_{n}=\begin{pmatrix}0&I_{n}\\ -I_{n}&0\end{pmatrix}
  88. S = ( I p 0 0 - I 2 ) S=\begin{pmatrix}I_{p}&0\\ 0&-I_{2}\end{pmatrix}
  89. P ( p , q ) = ( A p p B p q 0 D q q ) \displaystyle{P(p,q)=\begin{pmatrix}A_{pp}&B_{pq}\\ 0&D_{qq}\end{pmatrix}}
  90. A = ( 0 J - J 0 ) . \displaystyle{A=\begin{pmatrix}0&J\\ -J&0\end{pmatrix}.}
  91. 𝔥 * = 𝔨 i 𝔪 𝔤 . \mathfrak{h}^{*}=\mathfrak{k}\oplus i\mathfrak{m}\subset\mathfrak{g}.
  92. i 𝔪 i\mathfrak{m}
  93. 𝔞 \mathfrak{a}
  94. 𝔞 \mathfrak{a}
  95. 𝔞 \mathfrak{a}
  96. 𝔞 * = i 𝔞 \mathfrak{a}^{*}=i\mathfrak{a}
  97. 𝔞 * \mathfrak{a}^{*}
  98. H * = K A * K . \displaystyle{H^{*}=KA^{*}K.}
  99. i 𝔪 i\mathfrak{m}
  100. 𝔪 \mathfrak{m}
  101. i 𝔪 i\mathfrak{m}
  102. 𝔪 \mathfrak{m}
  103. 𝔞 \mathfrak{a}
  104. 𝔞 \mathfrak{a}
  105. 𝔪 \mathfrak{m}
  106. 𝔪 \mathfrak{m}
  107. 𝔨 \mathfrak{k}
  108. 𝔞 \mathfrak{a}
  109. 𝔪 \mathfrak{m}
  110. 𝔞 \mathfrak{a}
  111. 𝔞 \mathfrak{a}
  112. 𝔞 \mathfrak{a}
  113. H = K A K , H = K exp 𝔪 \displaystyle{H=KAK,\,\,\,H=K\cdot\exp\mathfrak{m}}
  114. M = { σ ( g ) g - 1 : g H } , \displaystyle{M=\{\sigma(g)g^{-1}:g\in H\},}
  115. 𝔥 \mathfrak{h}
  116. 𝔥 \mathfrak{h}
  117. 𝔥 \mathfrak{h}
  118. dim 𝔨 - dim 𝔨 a = dim 𝔪 - dim 𝔪 a , \displaystyle{\mathrm{dim}\,\mathfrak{k}-\mathrm{dim}\,\mathfrak{k}_{a}=% \mathrm{dim}\,\mathfrak{m}-\mathrm{dim}\,\mathfrak{m}_{a},}
  119. 𝔨 a \mathfrak{k}_{a}
  120. 𝔪 a \mathfrak{m}_{a}
  121. 𝔨 a \mathfrak{k}_{a}
  122. 𝔨 \mathfrak{k}
  123. 𝔨 a \mathfrak{k}_{a}^{\perp}
  124. 𝔨 a \mathfrak{k}_{a}^{\perp}
  125. 𝔪 a \mathfrak{m}_{a}
  126. 𝔞 \mathfrak{a}
  127. 𝔞 \mathfrak{a}
  128. 𝔞 \mathfrak{a}
  129. 𝔱 \mathfrak{t}
  130. 𝔤 α \mathfrak{g}_{\alpha}
  131. 𝔤 \mathfrak{g}
  132. 𝔨 \mathfrak{k}_{\mathbb{C}}
  133. 𝔪 \mathfrak{m}_{\mathbb{C}}
  134. 𝔨 \mathfrak{k}_{\mathbb{C}}
  135. 𝔪 \mathfrak{m}_{\mathbb{C}}
  136. 𝔪 + \mathfrak{m}_{+}
  137. 𝔪 - \mathfrak{m}_{-}
  138. 𝔤 α \mathfrak{g}_{\alpha}
  139. X α = E α + E - α , Y α = i ( E α - E - α ) \displaystyle{X_{\alpha}=E_{\alpha}+E_{-\alpha},\,\,\,Y_{\alpha}=i(E_{\alpha}-% E_{-\alpha})}
  140. 𝔥 \mathfrak{h}
  141. 𝔥 \mathfrak{h}
  142. 𝔪 - \mathfrak{m}_{-}
  143. X i = E ψ i + E - ψ i \displaystyle{X_{i}=E_{\psi_{i}}+E_{-\psi_{i}}}
  144. 𝔪 \mathfrak{m}
  145. 𝔞 \mathfrak{a}
  146. [ c α E α + c α ¯ E - α , E ψ i + E - ψ i ] = 0 \displaystyle{[\sum c_{\alpha}E_{\alpha}+\overline{c_{\alpha}}E_{-\alpha},E_{% \psi_{i}}+E_{-\psi_{i}}]=0}
  147. 𝔞 \mathfrak{a}
  148. σ ( α β - β ¯ α ¯ ) = ( α - β β ¯ α ¯ ) \displaystyle{\sigma\begin{pmatrix}\alpha&\beta\\ -\overline{\beta}&\overline{\alpha}\end{pmatrix}=\begin{pmatrix}\alpha&-\beta% \\ \overline{\beta}&\overline{\alpha}\end{pmatrix}}
  149. SU ( 1 , 1 ) / 𝐓 = { z : | z | < 1 } B + / 𝐓 = SL ( 2 , ) / B = { } , \displaystyle{\mathrm{SU}(1,1)/\mathbf{T}=\{z:|z|<1\}\,\,\,\subset\,\,\,B_{+}/% \mathbf{T}_{\mathbb{C}}=\mathbb{C}\,\,\,\subset\,\,\,\mathrm{SL}(2,\mathbb{C})% /B=\mathbb{C}\cup\{\infty\},}
  150. ( 1 z 0 1 ) = exp ( 0 z 0 0 ) . \displaystyle{\begin{pmatrix}1&z\\ 0&1\end{pmatrix}=\exp\begin{pmatrix}0&z\\ 0&0\end{pmatrix}.}
  151. exp 𝔪 + \exp\mathfrak{m}_{+}
  152. 𝔪 + \mathfrak{m}_{+}
  153. 𝔪 + \mathfrak{m}_{+}
  154. 𝐗 = exp ( 𝔪 + ) K exp ( 𝔪 - ) = exp ( 𝔪 + ) P \mathbf{X}=\exp(\mathfrak{m}_{+})\cdot K_{\mathbb{C}}\cdot\exp(\mathfrak{m}_{-% })=\exp(\mathfrak{m}_{+})\cdot P
  155. exp 𝔪 + \exp\mathfrak{m}_{+}
  156. M ± = exp 𝔪 ± M_{\pm}=\exp\mathfrak{m}_{\pm}
  157. [ 𝔪 + , 𝔪 - ] 𝔨 [\mathfrak{m}_{+},\mathfrak{m}_{-}]\subset\mathfrak{k}_{\mathfrak{C}}
  158. [ 𝔪 , 𝔪 ] 𝔨 [\mathfrak{m},\mathfrak{m}]\subset\mathfrak{k}
  159. 𝔪 + \mathfrak{m}_{+}
  160. 𝔪 - \mathfrak{m}_{-}
  161. 𝔪 - \mathfrak{m}_{-}
  162. Y = Ad ( X ) Y = Y + [ X , Y ] + 1 2 [ X , [ X , Y ] ] 𝔪 + 𝔨 𝔪 - , \displaystyle{Y=\mathrm{Ad}(X)\cdot Y=Y+[X,Y]+{1\over 2}[X,[X,Y]]\in\mathfrak{% m}_{+}\oplus\mathfrak{k}_{\mathbb{C}}\oplus\mathfrak{m}_{-},}
  163. 𝔪 - \mathfrak{m}_{-}
  164. μ ( X , Y ) = Ad ( p - 1 ) X + Y = Ad ( p - 1 ) ( X Ad ( p ) Y ) , \displaystyle{\mu^{\prime}(X,Y)=\mathrm{Ad}(p^{-1})X+Y=\mathrm{Ad}(p^{-1})(X% \oplus\mathrm{Ad}(p)Y),}
  165. 𝔪 + \mathfrak{m}_{+}
  166. | z | 2 < 1 2 ( 1 + ( z z ) 2 ) < 1 |z|^{2}<{1\over 2}(1+(z\cdot z)^{2})<1
  167. g = ( A B C D ) \displaystyle{g=\begin{pmatrix}A&B\\ C&D\end{pmatrix}}
  168. g ( Z ) = ( A Z + B ) ( C Z + D ) - 1 . \displaystyle{g(Z)=(AZ+B)(CZ+D)^{-1}.}
  169. ( 0 a - a 0 ) \begin{pmatrix}0&a\\ -a&0\end{pmatrix}
  170. 𝔤 \mathfrak{g}
  171. 𝔤 \mathfrak{g}
  172. 𝔤 \mathfrak{g}
  173. 𝔤 \mathfrak{g}
  174. 𝔤 \mathfrak{g}
  175. 𝔤 \mathfrak{g}
  176. 𝔤 \mathfrak{g}
  177. 𝔤 \mathfrak{g}
  178. S S
  179. D D

Herzog–Schönheim_conjecture.html

  1. G G
  2. A = { a 1 G 1 , , a k G k } A=\{a_{1}G_{1},\ \ldots,\ a_{k}G_{k}\}
  3. G 1 , , G k G_{1},\ldots,G_{k}
  4. G G
  5. A A
  6. G G
  7. k > 1 k>1
  8. [ G : G 1 ] , , [ G : G k ] [G:G_{1}],\ldots,[G:G_{k}]
  9. H H
  10. G G
  11. k = [ G : H ] < k=[G:H]<\infty
  12. G G
  13. k k
  14. H H
  15. G 1 , , G k G_{1},\ldots,G_{k}
  16. G G
  17. G 1 , , G k G_{1},\ldots,G_{k}
  18. G G
  19. [ G : i = 1 k G i ] | i = 1 k [ G : G i ] \bigg[G:\bigcap_{i=1}^{k}G_{i}\bigg]\ \bigg|\ \prod_{i=1}^{k}[G:G_{i}]
  20. P ( [ G : i = 1 k G i ] ) = i = 1 k P ( [ G : G i ] ) , P\bigg(\bigg[G:\bigcap_{i=1}^{k}G_{i}\bigg]\ \bigg)=\bigcup_{i=1}^{k}P([G:G_{i% }]),
  21. P ( n ) P(n)
  22. n n
  23. G G
  24. \Z \Z
  25. G G

Heteroclinic_orbit.html

  1. x ˙ = f ( x ) \dot{x}=f(x)
  2. x = x 0 x=x_{0}
  3. x = x 1 x=x_{1}
  4. ϕ ( t ) \phi(t)
  5. x 0 x_{0}
  6. x 1 x_{1}
  7. ϕ ( t ) x 0 as t - \phi(t)\rightarrow x_{0}\quad\mathrm{as}\quad t\rightarrow-\infty
  8. ϕ ( t ) x 1 as t + \phi(t)\rightarrow x_{1}\quad\mathrm{as}\quad t\rightarrow+\infty
  9. x 1 x_{1}
  10. x 0 x_{0}
  11. S = { 1 , 2 , , M } S=\{1,2,\ldots,M\}
  12. σ = { ( , s - 1 , s 0 , s 1 , ) : s k S k } \sigma=\{(\ldots,s_{-1},s_{0},s_{1},\ldots):s_{k}\in S\;\forall k\in\mathbb{Z}\}
  13. p ω s 1 s 2 s n q ω p^{\omega}s_{1}s_{2}\cdots s_{n}q^{\omega}
  14. p = t 1 t 2 t k p=t_{1}t_{2}\cdots t_{k}
  15. t i S t_{i}\in S
  16. q = r 1 r 2 r m q=r_{1}r_{2}\cdots r_{m}
  17. r i S r_{i}\in S
  18. p ω p^{\omega}
  19. p ω s 1 s 2 s n p ω p^{\omega}s_{1}s_{2}\cdots s_{n}p^{\omega}
  20. s 1 s 2 s n s_{1}s_{2}\cdots s_{n}
  21. p ω p^{\omega}

Hidden_subgroup_problem.html

  1. 1 | G | x G | x | f ( x ) \frac{1}{\sqrt{|G|}}\sum_{x\in G}{|x\rangle\otimes|f(x)\rangle}

Hierarchy_(mathematics).html

  1. n n n\leq n^{\prime}
  2. m m
  3. n + m = n n+m=n^{\prime}
  4. n n^{\prime}
  5. n n
  6. n n^{\prime}
  7. n n
  8. m m
  9. n + m = n n+m=n^{\prime}
  10. m = ( n - n ) m=(n^{\prime}-n)

High-intensity_focused_ultrasound.html

  1. I = I o e - 2 α z I=I_{o}{e}^{-2\alpha\mathrm{z}}
  2. I o I_{o}
  3. α \alpha
  4. - I z = 2 α I = Q \frac{-\partial I}{\partial\mathrm{z}}=2\alpha I=Q
  5. 𝐶𝐸𝑀 = t o t f R T reference - T d t \mathit{CEM}=\int_{t_{o}}^{t_{f}}R^{T_{\mathrm{reference}}-T}dt

High-resolution_transmission_electron_microscopy.html

  1. C T F ( u ) = A ( u ) E ( u ) sin ( χ ( u ) ) CTF(u)=A(u)E(u)\sin(\chi(u))
  2. χ ( u ) = π 2 C s λ 3 u 4 - π Δ f λ u 2 \chi(u)=\frac{\pi}{2}C_{s}\lambda^{3}u^{4}-\pi\Delta f\lambda u^{2}
  3. E ( u ) = E s ( u ) E c ( u ) E d ( u ) E v ( u ) E D ( u ) , E(u)=E_{s}(u)E_{c}(u)E_{d}(u)E_{v}(u)E_{D}(u),\,
  4. E s ( u ) = exp [ - ( π α λ ) 2 ( δ \Chi ( u ) δ u ) 2 ] = exp [ - ( π α λ ) 2 ( C s λ 3 u 3 + Δ f λ u ) 2 ] , E_{s}(u)=\exp\left[-\left(\frac{\pi\alpha}{\lambda}\right)^{2}\left(\frac{% \delta\Chi(u)}{\delta u}\right)^{2}\right]=\exp\left[-\left(\frac{\pi\alpha}{% \lambda}\right)^{2}(C_{s}\lambda^{3}u^{3}+\Delta f\lambda u)^{2}\right],
  5. E c ( u ) = exp [ - 1 2 ( π λ δ ) 2 u 4 ] , E_{c}(u)=\exp\left[-\frac{1}{2}\left(\pi\lambda\delta\right)^{2}u^{4}\right],
  6. δ = C c 4 ( Δ I obj I obj ) 2 + ( Δ E V acc ) 2 + ( Δ V acc V acc ) 2 , \delta=C_{c}\sqrt{4\left(\frac{\Delta I\text{obj}}{I\text{obj}}\right)^{2}+% \left(\frac{\Delta E}{V\text{acc}}\right)^{2}+\left(\frac{\Delta V\text{acc}}{% V\text{acc}}\right)^{2}},
  7. Δ I obj / I obj \Delta I\text{obj}/I\text{obj}
  8. Δ V acc / V acc \Delta V\text{acc}/V\text{acc}
  9. Δ E / V acc \Delta E/V\text{acc}
  10. Δ f Scherzer = - 1.2 C s λ \Delta f\text{Scherzer}=-1.2\sqrt{C_{s}\lambda}\,
  11. u res ( Scherzer ) = 0.6 λ 3 / 4 C s 1 / 4 , u\text{res}(\,\text{Scherzer})=0.6\lambda^{3/4}C_{s}^{1/4},
  12. Δ f Gabor = 0.56 Δ f Scherzer \Delta f\text{Gabor}=0.56\Delta f\text{Scherzer}
  13. Δ f Lichte = - 0.75 C s ( u max λ ) 2 , \Delta f\text{Lichte}=-0.75C_{s}(u_{\max}\lambda)^{2},

High_availability.html

  1. c c
  2. x x
  3. c := - log 10 x c:=\lfloor-\log_{10}x\rfloor

Higher-order_derivative_test.html

  1. f f
  2. I \R , c I I\subset\R,\;c\in I
  3. n 1 n\geq 1
  4. f ( c ) = = f ( n ) ( c ) = 0 and f ( n + 1 ) ( c ) 0 f^{\prime}(c)=\cdots=f^{(n)}(c)=0\quad\,\text{and}\quad f^{(n+1)}(c)\,\not=0
  5. f ( n + 1 ) ( c ) < 0 c f^{(n+1)}(c)<0\Rightarrow c
  6. f ( n + 1 ) ( c ) > 0 c f^{(n+1)}(c)>0\Rightarrow c
  7. f ( n + 1 ) ( c ) < 0 c f^{(n+1)}(c)<0\Rightarrow c
  8. f ( n + 1 ) ( c ) > 0 c f^{(n+1)}(c)>0\Rightarrow c
  9. f f
  10. x 8 x^{8}

Higher-order_differential_cryptanalysis.html

  1. n n
  2. n n
  3. f : 𝔽 2 n 𝔽 2 n f:\mathbb{F}^{n}_{2}\to\mathbb{F}^{n}_{2}
  4. α \alpha
  5. β \beta
  6. α \alpha
  7. β \beta
  8. f ( m α ) f ( m ) = β f(m\oplus\alpha)\oplus f(m)=\beta
  9. m 𝔽 2 n m\in\mathbb{F}^{n}_{2}
  10. f : 𝔽 2 n 𝔽 2 n f:\mathbb{F}^{n}_{2}\to\mathbb{F}^{n}_{2}
  11. α \alpha
  12. Δ α f ( x ) := f ( x α ) f ( x ) \Delta_{\alpha}f(x):=f(x\oplus\alpha)\oplus f(x)
  13. i i
  14. ( α 1 , α 2 , , α i ) (\alpha_{1},\alpha_{2},\dots,\alpha_{i})
  15. Δ α 1 , α 2 , , α i ( i ) f ( x ) := Δ α i ( Δ α 1 , α 2 , , α i - 1 i - 1 f ( x ) ) \Delta^{(i)}_{\alpha_{1},\alpha_{2},\dots,\alpha_{i}}f(x):=\Delta_{\alpha_{i}}% \left(\Delta^{i-1}_{\alpha_{1},\alpha_{2},\dots,\alpha_{i-1}}f(x)\right)
  16. Δ α 1 , α 2 ( 2 ) f ( x ) = f ( x ) f ( x α 1 ) f ( x α 2 ) f ( x α 1 α 2 ) \Delta^{(2)}_{\alpha_{1},\alpha_{2}}f(x)=f(x)\oplus f(x\oplus\alpha_{1})\oplus f% (x\oplus\alpha_{2})\oplus f(x\oplus\alpha_{1}\oplus\alpha_{2})

Higman's_lemma.html

  1. w 1 , w 2 , w_{1},w_{2},\ldots
  2. i < j i<j
  3. w i w_{i}
  4. w j w_{j}

Hilbert's_theorem.html

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

Hilbert_curve.html

  1. 2 2
  2. H n H_{n}
  3. n n
  4. H n H_{n}
  5. 2 n - 1 2 n \textstyle 2^{n}-{1\over 2^{n}}
  6. n n
  7. n 2 - 1 n^{2}-1

Hirzebruch_surface.html

  1. O ( 0 ) + O ( - n ) . O(0)+O(-n).
  2. [ 0 1 1 - n ] , \begin{bmatrix}0&1\\ 1&-n\end{bmatrix},

Histogram_equalization.html

  1. p x ( i ) = p ( x = i ) = n i n , 0 i < L \ p_{x}(i)=p(x=i)=\frac{n_{i}}{n},\quad 0\leq i<L
  2. p x ( i ) p_{x}(i)
  3. c d f x ( i ) = j = 0 i p x ( j ) \ cdf_{x}(i)=\sum_{j=0}^{i}p_{x}(j)
  4. c d f y ( i ) = i K \ cdf_{y}(i)=iK
  5. c d f y ( y ) = c d f y ( T ( k ) ) = c d f x ( k ) \ cdf_{y}(y^{\prime})=cdf_{y}(T(k))=cdf_{x}(k)
  6. y = y ( max { x } - min { x } ) + min { x } \ y^{\prime}=y\cdot(\max\{x\}-\min\{x\})+\min\{x\}
  7. [ 52 55 61 66 70 61 64 73 63 59 55 90 109 85 69 72 62 59 68 113 144 104 66 73 63 58 71 122 154 106 70 69 67 61 68 104 126 88 68 70 79 65 60 70 77 68 58 75 85 71 64 59 55 61 65 83 87 79 69 68 65 76 78 94 ] \begin{bmatrix}52&55&61&66&70&61&64&73\\ 63&59&55&90&109&85&69&72\\ 62&59&68&113&144&104&66&73\\ 63&58&71&122&154&106&70&69\\ 67&61&68&104&126&88&68&70\\ 79&65&60&70&77&68&58&75\\ 85&71&64&59&55&61&65&83\\ 87&79&69&68&65&76&78&94\end{bmatrix}
  8. [ 0 , 255 ] [0,255]
  9. h ( v ) = round ( c d f ( v ) - c d f m i n ( M × N ) - c d f m i n × ( L - 1 ) ) h(v)=\mathrm{round}\left(\frac{cdf(v)-cdf_{min}}{(M\times N)-cdf_{min}}\times(% L-1)\right)
  10. h ( v ) = round ( c d f ( v ) - c d f m i n ( M × N ) - c d f m i n × ( L - 2 ) ) + 1 h(v)=\mathrm{round}\left(\frac{cdf(v)-cdf_{min}}{(M\times N)-cdf_{min}}\times(% L-2)\right)+1
  11. h ( v ) = round ( c d f ( v ) - 1 63 × 255 ) h(v)=\mathrm{round}\left(\frac{cdf(v)-1}{63}\times 255\right)
  12. h ( 78 ) = round ( 46 - 1 63 × 255 ) = round ( 0.714286 × 255 ) = 182 h(78)=\mathrm{round}\left(\frac{46-1}{63}\times 255\right)=\mathrm{round}\left% (0.714286\times 255\right)=182
  13. [ 0 12 53 93 146 53 73 166 65 32 12 215 235 202 130 158 57 32 117 239 251 227 93 166 65 20 154 243 255 231 146 130 97 53 117 227 247 210 117 146 190 85 36 146 178 117 20 170 202 154 73 32 12 53 85 194 206 190 130 117 85 174 182 219 ] \begin{bmatrix}0&12&53&93&146&53&73&166\\ 65&32&12&215&235&202&130&158\\ 57&32&117&239&251&227&93&166\\ 65&20&154&243&255&231&146&130\\ 97&53&117&227&247&210&117&146\\ 190&85&36&146&178&117&20&170\\ 202&154&73&32&12&53&85&194\\ 206&190&130&117&85&174&182&219\end{bmatrix}

History_of_entropy.html

  1. Q T \frac{Q}{T}
  2. Q ( 1 T 2 - 1 T 1 ) Q\left(\frac{1}{T_{2}}-\frac{1}{T_{1}}\right)
  3. S = Q T S=\frac{Q}{T}
  4. Δ S = S final - S initial \Delta S=S_{\rm final}-S_{\rm initial}\,
  5. Δ S = ( Q T 2 - Q T 1 ) \Delta S=\left(\frac{Q}{T_{2}}-\frac{Q}{T_{1}}\right)
  6. Δ S = Q ( 1 T 2 - 1 T 1 ) \Delta S=Q\left(\frac{1}{T_{2}}-\frac{1}{T_{1}}\right)
  7. δ Q T = - N \int\frac{\delta Q}{T}=-N
  8. δ Q T = 0 \int\frac{\delta Q}{T}=0
  9. δ Q T 0 \int\frac{\delta Q}{T}\geq 0
  10. S S
  11. S = k B ln Ω S=k_{\rm B}\ln\Omega\!
  12. Ω Ω
  13. H = - K i = 1 k p ( i ) log p ( i ) , H=-K\sum_{i=1}^{k}p(i)\log p(i),

History_of_gravitational_theory.html

  1. Force of gravity mass of object 1 × mass of object 2 distance from centers 2 {\rm Force\,of\,gravity}\propto\frac{\rm mass\,of\,object\,1\,\times\,mass\,of% \,object\,2}{\rm distance\,from\,centers^{2}}
  2. \propto

History_of_mass_spectrometry.html

  1. [ M + H ] + [M+H]^{+}

Hoffman–Singleton_graph.html

  1. ( x - 7 ) ( x - 2 ) 28 ( x + 3 ) 21 (x-7)(x-2)^{28}(x+3)^{21}

HOL_Light.html

  1. t = t \cfrac{\qquad}{\vdash t=t}
  2. Γ s = t Δ t = u Γ Δ s = u \cfrac{\Gamma\vdash s=t\qquad\Delta\vdash t=u}{\Gamma\cup\Delta\vdash s=u}
  3. Γ f = g Δ x = y Γ Δ f ( x ) = g ( y ) \cfrac{\Gamma\vdash f=g\qquad\Delta\vdash x=y}{\Gamma\cup\Delta\vdash f(x)=g(y)}
  4. Γ s = t Γ ( λ x . s ) = ( λ x . t ) \cfrac{\Gamma\vdash s=t}{\Gamma\vdash(\lambda x.s)=(\lambda x.t)}
  5. x x
  6. Γ \Gamma
  7. ( λ x . t ) x = t \cfrac{\qquad}{\vdash(\lambda x.t)x=t}
  8. { p } p \cfrac{\qquad}{\{p\}\vdash p}
  9. p p
  10. p p
  11. Γ p = q Δ p Γ Δ q \cfrac{\Gamma\vdash p=q\qquad\Delta\vdash p}{\Gamma\cup\Delta\vdash q}
  12. Γ p Δ q ( Γ - { q } ) ( Δ - { p } ) p = q \cfrac{\Gamma\vdash p\qquad\Delta\vdash q}{(\Gamma-\{q\})\cup(\Delta-\{p\})% \vdash p=q}
  13. Γ [ x 1 , , x n ] p [ x 1 , , x n ] Γ [ t 1 , , t n ] p [ t 1 , , t n ] \cfrac{\Gamma[x_{1},\ldots,x_{n}]\vdash p[x_{1},\ldots,x_{n}]}{\Gamma[t_{1},% \ldots,t_{n}]\vdash p[t_{1},\ldots,t_{n}]}
  14. Γ [ α 1 , , α n ] p [ α 1 , , α n ] Γ [ τ 1 , , τ n ] p [ τ 1 , , τ n ] \cfrac{\Gamma[\alpha_{1},\ldots,\alpha_{n}]\vdash p[\alpha_{1},\ldots,\alpha_{% n}]}{\Gamma[\tau_{1},\ldots,\tau_{n}]\vdash p[\tau_{1},\ldots,\tau_{n}]}

Holditch's_theorem.html

  1. π p q \pi pq
  2. π p q \pi pq

Holland's_schema_theorem.html

  1. o ( H ) o(H)
  2. δ ( H ) \delta(H)
  3. E ( m ( H , t + 1 ) ) m ( H , t ) f ( H ) a t [ 1 - p ] . \operatorname{E}(m(H,t+1))\geq{m(H,t)f(H)\over a_{t}}[1-p].
  4. m ( H , t ) m(H,t)
  5. H H
  6. t t
  7. f ( H ) f(H)
  8. H H
  9. a t a_{t}
  10. t t
  11. p p
  12. H H
  13. p = δ ( H ) l - 1 p c + o ( H ) p m p={\delta(H)\over l-1}p_{c}+o(H)p_{m}
  14. o ( H ) o(H)
  15. l l
  16. p m p_{m}
  17. p c p_{c}
  18. δ ( H ) \delta(H)
  19. H H
  20. H H

Holomorphic_vector_bundle.html

  1. X X
  2. E E
  3. π : E X π:E→X
  4. ϕ U : π - 1 ( U ) U × 𝐂 k \phi_{U}:\pi^{-1}(U)\to U\times\mathbf{C}^{k}
  5. t U V : U V GL k ( 𝐂 ) t_{UV}:U\cap V\to\mathrm{GL}_{k}(\mathbf{C})
  6. E E
  7. U U
  8. X X
  9. 𝒪 ( E ) \mathcal{O}(E)
  10. E E
  11. 𝐂 ¯ \underline{\mathbf{C}}
  12. 𝒪 X \mathcal{O}_{X}
  13. X X
  14. X p , q \mathcal{E}_{X}^{p,q}
  15. ( p , q ) (p,q)
  16. ( p , q ) (p,q)
  17. E E
  18. p , q ( E ) X p , q E . \mathcal{E}^{p,q}(E)\triangleq\mathcal{E}_{X}^{p,q}\otimes E.
  19. ¯ : p , q ( E ) p , q + 1 ( E ) . \overline{\partial}:\mathcal{E}^{p,q}(E)\to\mathcal{E}^{p,q+1}(E).
  20. E E
  21. E E
  22. 𝒪 ( E ) \mathcal{O}(E)
  23. H 0 ( X , 𝒪 ( E ) ) = Γ ( X , 𝒪 ( E ) ) , H^{0}(X,\mathcal{O}(E))=\Gamma(X,\mathcal{O}(E)),
  24. E E
  25. H 1 ( X , 𝒪 ( E ) ) H^{1}(X,\mathcal{O}(E))
  26. X X
  27. E E
  28. 0 E F X × 𝐂 0 0→E→F→X×\mathbf{C}→0
  29. P i c ( X ) Pic(X)
  30. X X
  31. H 1 ( X , 𝒪 X * ) H^{1}(X,\mathcal{O}_{X}^{*})
  32. p s = ¯ s p\nabla s=\bar{\partial}s
  33. X s , t = X s , t + s , X t X\cdot\langle s,t\rangle=\langle\nabla_{X}s,t\rangle+\langle s,\nabla_{X}t\rangle
  34. X s \nabla_{X}s
  35. s \nabla s
  36. h i j = e i , e j h_{ij}=\langle e_{i},e_{j}\rangle
  37. h i k ( ω u ) j k = h i j \sum h_{ik}\,{(\omega_{u})}^{k}_{j}=\partial h_{ij}
  38. ω u = h - 1 h . \omega_{u}=h^{-1}\partial h.
  39. ω u = g - 1 d g + g ω u g - 1 , \omega_{u^{\prime}}=g^{-1}dg+g\omega_{u}g^{-1},
  40. ω ¯ T = ¯ h h - 1 {\overline{\omega}}^{T}=\overline{\partial}h\cdot h^{-1}
  41. d e i , e j = h i j + ¯ h i j = ω i k e k , e j + e i , ω j k e k = e i , e j + e i , e j . d\langle e_{i},e_{j}\rangle=\partial h_{ij}+\overline{\partial}h_{ij}=\langle{% \omega}^{k}_{i}e_{k},e_{j}\rangle+\langle e_{i},{\omega}^{k}_{j}e_{k}\rangle=% \langle\nabla e_{i},e_{j}\rangle+\langle e_{i},\nabla e_{j}\rangle.
  42. s \nabla s
  43. ¯ s \bar{\partial}s
  44. Ω = d ω + ω ω \Omega=d\omega+\omega\wedge\omega
  45. p = ¯ p\nabla=\bar{\partial}
  46. Ω = ¯ ω . \Omega=\bar{\partial}\omega.

Holomorphically_separable.html

  1. X X
  2. X X
  3. f 𝒪 ( X ) f\in\mathcal{O}(X)
  4. n \mathbb{C}^{n}
  5. n \mathbb{C}^{n}

Holor.html

  1. v i = σ x i x i v i , v^{i^{\prime}}=\sigma{{\partial x^{i^{\prime}}}\over{\partial x^{i}}}v^{i},
  2. σ = 1 \sigma=1

Homeostatic_model_assessment.html

  1. HOMA-IR = Glucose × Insulin 22.5 \,\text{HOMA-IR}=\frac{\,\text{Glucose}\times\,\text{Insulin}}{22.5}
  2. HOMA-IR = Glucose × Insulin 405 \,\text{HOMA-IR}=\frac{\,\text{Glucose}\times\,\text{Insulin}}{405}
  3. HOMA- β = 20 × Insulin Glucose - 3.5 % \,\text{HOMA-}\beta=\frac{20\times\,\text{Insulin}}{\,\text{Glucose}-3.5}\%
  4. HOMA- β = 360 × Insulin Glucose - 63 % \,\text{HOMA-}\beta=\frac{360\times\,\text{Insulin}}{\,\text{Glucose}-63}\%

Homeotopy.html

  1. π k \pi_{k}
  2. X X
  3. π k ( X ) \pi_{k}(X)
  4. S k X . S^{k}\to X.
  5. X X
  6. X X X\to X
  7. Homeo ( X ) . {\rm Homeo}(X).
  8. Homeo ( X ) {\rm Homeo}(X)
  9. X X
  10. H M E k ( X ) = π k ( Homeo ( X ) ) . HME_{k}(X)=\pi_{k}({\rm Homeo}(X)).
  11. H M E 0 ( X ) = π 0 ( Homeo ( X ) ) = M C G * ( X ) HME_{0}(X)=\pi_{0}({\rm Homeo}(X))=MCG^{*}(X)
  12. X . X.
  13. Homeo ( X ) {\rm Homeo}(X)
  14. π 0 . \pi_{0}.
  15. X X
  16. H M E 0 ( X ) = Out ( π 1 ( X ) ) , HME_{0}(X)={\rm Out}(\pi_{1}(X)),

Homestake_experiment.html

  1. ν e + 37 Cl 37 Ar + e - . \mathrm{\nu_{e}+\ ^{37}Cl\longrightarrow\ ^{37}Ar+e^{-}.}

Homoclinic_orbit.html

  1. x ˙ = f ( x ) \dot{x}=f(x)
  2. x = x 0 x=x_{0}
  3. Φ ( t ) \Phi(t)
  4. Φ ( t ) x 0 as t ± \Phi(t)\rightarrow x_{0}\quad\mathrm{as}\quad t\rightarrow\pm\infty
  5. f : M M f:M\rightarrow M
  6. M M
  7. x x
  8. p p
  9. lim n ± f n ( x ) = p . \lim_{n\rightarrow\pm\infty}f^{n}(x)=p.
  10. S = { 1 , 2 , , M } S=\{1,2,\ldots,M\}
  11. σ = { ( , s - 1 , s 0 , s 1 , ) : s k S k } \sigma=\{(\ldots,s_{-1},s_{0},s_{1},\ldots):s_{k}\in S\;\forall k\in\mathbb{Z}\}
  12. p ω s 1 s 2 s n q ω p^{\omega}s_{1}s_{2}\cdots s_{n}q^{\omega}
  13. p = t 1 t 2 t k p=t_{1}t_{2}\cdots t_{k}
  14. t i S t_{i}\in S
  15. q = r 1 r 2 r m q=r_{1}r_{2}\cdots r_{m}
  16. r i S r_{i}\in S
  17. p ω p^{\omega}
  18. p ω s 1 s 2 s n p ω p^{\omega}s_{1}s_{2}\cdots s_{n}p^{\omega}
  19. s 1 s 2 s n s_{1}s_{2}\cdots s_{n}
  20. p ω p^{\omega}

Homomorphic_encryption.html

  1. ( x ) \mathcal{E}(x)
  2. m m
  3. e e
  4. x x
  5. ( x ) = x e mod m \mathcal{E}(x)=x^{e}\;\bmod\;m
  6. ( x 1 ) ( x 2 ) = x 1 e x 2 e mod m = ( x 1 x 2 ) e mod m = ( x 1 x 2 ) \mathcal{E}(x_{1})\cdot\mathcal{E}(x_{2})=x_{1}^{e}x_{2}^{e}\;\bmod\;m=(x_{1}x% _{2})^{e}\;\bmod\;m=\mathcal{E}(x_{1}\cdot x_{2})
  7. G G
  8. q q
  9. g g
  10. ( G , q , g , h ) (G,q,g,h)
  11. h = g x h=g^{x}
  12. x x
  13. m m
  14. ( m ) = ( g r , m h r ) \mathcal{E}(m)=(g^{r},m\cdot h^{r})
  15. r { 0 , , q - 1 } r\in\{0,\ldots,q-1\}
  16. ( m 1 ) ( m 2 ) = ( g r 1 , m 1 h r 1 ) ( g r 2 , m 2 h r 2 ) = ( g r 1 + r 2 , ( m 1 m 2 ) h r 1 + r 2 ) = ( m 1 m 2 ) . \begin{aligned}&\displaystyle\mathcal{E}(m_{1})\cdot\mathcal{E}(m_{2})=(g^{r_{% 1}},m_{1}\cdot h^{r_{1}})(g^{r_{2}},m_{2}\cdot h^{r_{2}})\\ \displaystyle=&\displaystyle(g^{r_{1}+r_{2}},(m_{1}\cdot m_{2})h^{r_{1}+r_{2}}% )=\mathcal{E}(m_{1}\cdot m_{2}).\end{aligned}
  17. ( b ) = x b r 2 mod m \mathcal{E}(b)=x^{b}r^{2}\;\bmod\;m
  18. r { 0 , , m - 1 } r\in\{0,\ldots,m-1\}
  19. ( b 1 ) ( b 2 ) = x b 1 r 1 2 x b 2 r 2 2 = x b 1 + b 2 ( r 1 r 2 ) 2 = ( b 1 b 2 ) \mathcal{E}(b_{1})\cdot\mathcal{E}(b_{2})=x^{b_{1}}r_{1}^{2}x^{b_{2}}r_{2}^{2}% =x^{b_{1}+b_{2}}(r_{1}r_{2})^{2}=\mathcal{E}(b_{1}\oplus b_{2})
  20. \oplus
  21. ( x ) = g x r c mod m \mathcal{E}(x)=g^{x}r^{c}\;\bmod\;m
  22. r { 0 , , m - 1 } r\in\{0,\ldots,m-1\}
  23. ( x 1 ) ( x 2 ) = ( g x 1 r 1 c ) ( g x 2 r 2 c ) = g x 1 + x 2 ( r 1 r 2 ) c = ( x 1 + x 2 mod c ) \mathcal{E}(x_{1})\cdot\mathcal{E}(x_{2})=(g^{x_{1}}r_{1}^{c})(g^{x_{2}}r_{2}^% {c})=g^{x_{1}+x_{2}}(r_{1}r_{2})^{c}=\mathcal{E}(x_{1}+x_{2}\;\bmod\;c)
  24. ( x ) = g x r m mod m 2 \mathcal{E}(x)=g^{x}r^{m}\;\bmod\;m^{2}
  25. r { 0 , , m - 1 } r\in\{0,\ldots,m-1\}
  26. ( x 1 ) ( x 2 ) = ( g x 1 r 1 m ) ( g x 2 r 2 m ) = g x 1 + x 2 ( r 1 r 2 ) m = ( x 1 + x 2 mod m 2 ) \mathcal{E}(x_{1})\cdot\mathcal{E}(x_{2})=(g^{x_{1}}r_{1}^{m})(g^{x_{2}}r_{2}^% {m})=g^{x_{1}+x_{2}}(r_{1}r_{2})^{m}=\mathcal{E}(x_{1}+x_{2}\;\bmod\;m^{2})
  27. T T
  28. k k
  29. T p o l y l o g ( k ) T\cdot polylog(k)

Homotopy_extension_property.html

  1. X X\,\!
  2. A X A\subset X
  3. ( X , A ) (X,A)\,\!
  4. f t : A Y f_{t}\colon A\rightarrow Y
  5. F 0 : X Y F_{0}\colon X\rightarrow Y
  6. F 0 | A = f 0 F_{0}|_{A}=f_{0}
  7. F 0 F_{0}
  8. F t : X Y F_{t}\colon X\rightarrow Y
  9. F t | A = f t F_{t}|_{A}=f_{t}
  10. ( X , A ) (X,A)\,\!
  11. G : ( ( X × { 0 } ) ( A × I ) ) Y G\colon((X\times\{0\})\cup(A\times I))\rightarrow Y
  12. G : X × I Y G^{\prime}\colon X\times I\rightarrow Y
  13. G G\,\!
  14. G G^{\prime}\,\!
  15. Y Y\,\!
  16. ( X , A ) (X,A)\,\!
  17. Y Y\,\!
  18. f ~ \tilde{f}
  19. f ~ : X Y I \tilde{f}\colon X\to Y^{I}
  20. f ~ : X × I Y \tilde{f}\colon X\times I\to Y
  21. X X\,\!
  22. A A\,\!
  23. X X\,\!
  24. ( X , A ) (X,A)\,\!
  25. ( X , A ) (X,A)\,\!
  26. ( X × { 0 } A × I ) (X\times\{0\}\cup A\times I)
  27. X × I . X\times I.
  28. ( X , A ) \mathbf{\mathit{(X,A)}}
  29. i : A X i:A\to X
  30. i : Y Z i:Y\to Z
  31. Y \mathbf{\mathit{Y}}
  32. i \mathbf{\mathit{i}}

Hoop_Conjecture.html

  1. c = 2 π r s c=2\pi r_{s}\,\!
  2. c c\,\!
  3. r s r_{s}\,\!

Hoover_index.html

  1. N N
  2. A A
  3. E i E_{i}
  4. A i A_{i}
  5. E total E\text{total}
  6. N N
  7. A total A\text{total}
  8. N N
  9. H H
  10. H = 1 2 i = 1 N | E i E total - A i A total | . H={\frac{1}{2}}\sum_{i=1}^{N}\left|{\frac{{E}_{i}}{{E}\text{total}}}-{\frac{{A% }_{i}}{{A}\text{total}}}\right|.
  11. T s T_{s}
  12. T s = 1 2 i = 1 N ln E i A i ( E i E total - A i A total ) . T_{s}={\frac{1}{2}}\sum_{i=1}^{N}\ln{\frac{{E}_{i}}{{A}_{i}}}\left({\frac{{E}_% {i}}{{E}\text{total}}}-{\frac{{A}_{i}}{{A}\text{total}}}\right).

Hopf_bifurcation.html

  1. d z d t = z ( ( λ + i ) + b | z | 2 ) , \frac{dz}{dt}=z((\lambda+i)+b|z|^{2}),
  2. b = α + i β . b=\alpha+i\beta.\,
  3. z ( t ) = r e i ω t z(t)=re^{i\omega t}\,
  4. r = - λ / α and ω = 1 + β r 2 . r=\sqrt{-\lambda/\alpha}\,\text{ and }\omega=1+\beta r^{2}.\,
  5. d X d t = A + X 2 Y - ( B + 1 ) X \frac{dX}{dt}=A+X^{2}Y-(B+1)X
  6. d Y d t = B X - X 2 Y . \frac{dY}{dt}=BX-X^{2}Y.
  7. d x d t = - x + a y + x 2 y , d y d t = b - a y - x 2 y . \frac{dx}{dt}=-x+ay+x^{2}y,~{}~{}\frac{dy}{dt}=b-ay-x^{2}y.
  8. J 0 J_{0}
  9. Z e Z_{e}
  10. J 0 J_{0}
  11. ± i β \pm i\beta
  12. p 0 , p 1 , , p k p_{0},p_{1},\dots,p_{k}
  13. P P
  14. p i ( μ ) = c i , 0 μ k - i + c i , 1 μ k - i - 2 + c i , 2 μ k - i - 4 + p_{i}(\mu)=c_{i,0}\mu^{k-i}+c_{i,1}\mu^{k-i-2}+c_{i,2}\mu^{k-i-4}+\cdots
  15. c i , 0 c_{i,0}
  16. i i
  17. { 1 , , k } \{1,\dots,k\}
  18. c i , 0 c_{i,0}
  19. c k , 0 c_{k,0}
  20. c i , 0 c_{i,0}
  21. i i
  22. { 0 , , k - 2 } \{0,\dots,k-2\}
  23. c k - 1 , 0 = 0 c_{k-1,0}=0
  24. c k - 2 , 1 < 0 c_{k-2,1}<0
  25. { d x d t = μ ( 1 - y 2 ) x - y , d y d t = x . \left\{\begin{array}[]{l}\dfrac{dx}{dt}=\mu(1-y^{2})x-y,\\ \dfrac{dy}{dt}=x.\end{array}\right.
  26. J = ( - μ ( - 1 + y 2 ) - 2 μ y x - 1 1 0 ) . J=\begin{pmatrix}-\mu(-1+y^{2})&-2\mu yx-1\\ 1&0\end{pmatrix}.
  27. λ \lambda
  28. P ( λ ) = λ 2 - μ λ + 1. P(\lambda)=\lambda^{2}-\mu\lambda+1.
  29. a 0 = 1 , a 1 = - μ , a 2 = 1 a_{0}=1,a_{1}=-\mu,a_{2}=1
  30. p 0 ( λ ) = a 0 λ 2 - a 2 p 1 ( λ ) = a 1 λ \begin{array}[]{l}p_{0}(\lambda)=a_{0}\lambda^{2}-a_{2}\\ p_{1}(\lambda)=a_{1}\lambda\end{array}
  31. i = 0 , 1 i=0,1
  32. p i ( μ ) = c i , 0 μ k - i + c i , 1 μ k - i - 2 + c i , 2 μ k - i - 4 + p_{i}(\mu)=c_{i,0}\mu^{k-i}+c_{i,1}\mu^{k-i-2}+c_{i,2}\mu^{k-i-4}+\cdots
  33. c 0 , 0 = 1 > 0 , c 1 , 0 = - μ = 0 , c 0 , 1 = - 1 < 0. c_{0,0}=1>0,c_{1,0}=-\mu=0,c_{0,1}=-1<0.

Hopf_conjecture.html

  1. k - ( rank G - rank H ) 5. k-(\operatorname{rank}G-\operatorname{rank}H)\leq 5.
  2. ( - 1 ) k χ ( M 2 k ) 0. (-1)^{k}\chi(M^{2k})\geq 0.

Hopf_invariant.html

  1. η : S 3 S 2 \eta\colon S^{3}\to S^{2}
  2. η \eta
  3. η - 1 ( x ) , η - 1 ( y ) S 3 \eta^{-1}(x),\eta^{-1}(y)\subset S^{3}
  4. x y S 2 x\neq y\in S^{2}
  5. π 3 ( S 2 ) \pi_{3}(S^{2})
  6. η \eta
  7. π i ( S n ) \pi_{i}(S^{n})\otimes\mathbb{Q}
  8. n n
  9. 2 n - 1 2n-1
  10. ϕ : S 2 n - 1 S n \phi\colon S^{2n-1}\to S^{n}
  11. n > 1 n>1
  12. C ϕ = S n ϕ D 2 n , C_{\phi}=S^{n}\cup_{\phi}D^{2n},
  13. D 2 n D^{2n}
  14. 2 n 2n
  15. S n S^{n}
  16. ϕ \phi
  17. C cell * ( C ϕ ) C^{*}_{\mathrm{cell}}(C_{\phi})
  18. n n
  19. n n
  20. \mathbb{Z}
  21. n n
  22. 2 n 2n
  23. n > 1 n>1
  24. H cell i ( C ϕ ) = { i = 0 , n , 2 n , 0 otherwise . H^{i}_{\mathrm{cell}}(C_{\phi})=\begin{cases}\mathbb{Z}&i=0,n,2n,\\ 0&\mbox{otherwise}~{}.\end{cases}
  25. H n ( C ϕ ) = α H^{n}(C_{\phi})=\langle\alpha\rangle
  26. H 2 n ( C ϕ ) = β . H^{2n}(C_{\phi})=\langle\beta\rangle.
  27. α α \alpha\smile\alpha
  28. H * ( C ϕ ) = [ α , β ] / β β = α β = 0 , α α = h ( ϕ ) β . H^{*}(C_{\phi})=\mathbb{Z}[\alpha,\beta]/\langle\beta\smile\beta=\alpha\smile% \beta=0,\alpha\smile\alpha=h(\phi)\beta\rangle.
  29. h ( ϕ ) h(\phi)
  30. ϕ \phi
  31. h : π 2 n - 1 ( S n ) h\colon\pi_{2n-1}(S^{n})\to\mathbb{Z}
  32. n n
  33. h h
  34. 2 2\mathbb{Z}
  35. 1 1
  36. n = 1 , 2 , 4 , 8 n=1,2,4,8
  37. 𝔸 = , , , 𝕆 \mathbb{A}=\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}
  38. S ( 𝔸 2 ) 𝔸 1 S(\mathbb{A}^{2})\to\mathbb{PA}^{1}
  39. V V
  40. V V^{\infty}
  41. V k V\cong\mathbb{R}^{k}
  42. V S k V^{\infty}\cong S^{k}
  43. k k
  44. ( X , x 0 ) (X,x_{0})
  45. V V^{\infty}
  46. V X V^{\infty}\wedge X
  47. F : V X V Y F\colon V^{\infty}\wedge X\to V^{\infty}\wedge Y
  48. F F
  49. h ( F ) { X , Y Y } 2 h(F)\in\{X,Y\wedge Y\}_{\mathbb{Z}_{2}}
  50. 2 \mathbb{Z}_{2}
  51. X X
  52. Y Y Y\wedge Y
  53. V V
  54. k k
  55. 2 \mathbb{Z}_{2}
  56. X X
  57. Y Y Y\wedge Y
  58. Δ X : X X X \Delta_{X}\colon X\to X\wedge X
  59. I I
  60. h ( F ) := ( F F ) ( I Δ X ) - ( I Δ Y ) ( I F ) . h(F):=(F\wedge F)(I\wedge\Delta_{X})-(I\wedge\Delta_{Y})(I\wedge F).
  61. V V X V^{\infty}\wedge V^{\infty}\wedge X
  62. V V Y Y V^{\infty}\wedge V^{\infty}\wedge Y\wedge Y
  63. 2 \mathbb{Z}_{2}
  64. h V ( F ) h_{V}(F)
  65. V V

Horn–Schunck_method.html

  1. E = [ ( I x u + I y v + I t ) 2 + α 2 ( u 2 + v 2 ) ] d x d y E=\iint\left[(I_{x}u+I_{y}v+I_{t})^{2}+\alpha^{2}(\lVert\nabla u\rVert^{2}+% \lVert\nabla v\rVert^{2})\right]{{\rm d}x{\rm d}y}
  2. I x I_{x}
  3. I y I_{y}
  4. I t I_{t}
  5. V = [ u ( x , y ) , v ( x , y ) ] \vec{V}=[u(x,y),v(x,y)]^{\top}
  6. α \alpha
  7. α \alpha
  8. L u - x L u x - y L u y = 0 \frac{\partial L}{\partial u}-\frac{\partial}{\partial x}\frac{\partial L}{% \partial u_{x}}-\frac{\partial}{\partial y}\frac{\partial L}{\partial u_{y}}=0
  9. L v - x L v x - y L v y = 0 \frac{\partial L}{\partial v}-\frac{\partial}{\partial x}\frac{\partial L}{% \partial v_{x}}-\frac{\partial}{\partial y}\frac{\partial L}{\partial v_{y}}=0
  10. L L
  11. I x ( I x u + I y v + I t ) - α 2 Δ u = 0 I_{x}(I_{x}u+I_{y}v+I_{t})-\alpha^{2}\Delta u=0
  12. I y ( I x u + I y v + I t ) - α 2 Δ v = 0 I_{y}(I_{x}u+I_{y}v+I_{t})-\alpha^{2}\Delta v=0
  13. Δ = 2 x 2 + 2 y 2 \Delta=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}
  14. Δ u ( x , y ) = u ¯ ( x , y ) - u ( x , y ) \Delta u(x,y)=\overline{u}(x,y)-u(x,y)
  15. u ¯ ( x , y ) \overline{u}(x,y)
  16. u u
  17. ( I x 2 + α 2 ) u + I x I y v = α 2 u ¯ - I x I t (I_{x}^{2}+\alpha^{2})u+I_{x}I_{y}v=\alpha^{2}\overline{u}-I_{x}I_{t}
  18. I x I y u + ( I y 2 + α 2 ) v = α 2 v ¯ - I y I t I_{x}I_{y}u+(I_{y}^{2}+\alpha^{2})v=\alpha^{2}\overline{v}-I_{y}I_{t}
  19. u u
  20. v v
  21. u k + 1 = u ¯ k - I x ( I x u ¯ k + I y v ¯ k + I t ) α 2 + I x 2 + I y 2 u^{k+1}=\overline{u}^{k}-\frac{I_{x}(I_{x}\overline{u}^{k}+I_{y}\overline{v}^{% k}+I_{t})}{\alpha^{2}+I_{x}^{2}+I_{y}^{2}}
  22. v k + 1 = v ¯ k - I y ( I x u ¯ k + I y v ¯ k + I t ) α 2 + I x 2 + I y 2 v^{k+1}=\overline{v}^{k}-\frac{I_{y}(I_{x}\overline{u}^{k}+I_{y}\overline{v}^{% k}+I_{t})}{\alpha^{2}+I_{x}^{2}+I_{y}^{2}}

Horseshoe_lemma.html

  1. A A^{\prime}
  2. A ′′ A^{\prime\prime}
  3. A A^{\prime}
  4. A ′′ A^{\prime\prime}
  5. A A
  6. A A^{\prime}
  7. A ′′ A^{\prime\prime}
  8. A A
  9. A A^{\prime}
  10. A ′′ A^{\prime\prime}
  11. 𝒜 \mathcal{A}
  12. 𝒜 \mathcal{A}
  13. A A^{\prime}
  14. A ′′ A^{\prime\prime}
  15. A A
  16. P n = P n P n ′′ P_{n}=P^{\prime}_{n}\oplus P^{\prime\prime}_{n}
  17. 𝒜 \mathcal{A}
  18. A A

Hotelling's_lemma.html

  1. y ( p ) y(p)
  2. p p
  3. y ( p ) = π ( p ) p y(p)=\frac{\partial\pi(p)}{\partial p}
  4. π \pi
  5. p > 0 p>0
  6. y * ( p ) y^{*}(p)
  7. π ( p * ) - p * y * ( p ) \pi(p^{*})-p^{*}y^{*}(p)
  8. p * p^{*}
  9. π ( p ) p - y = 0 \frac{\partial\pi(p)}{\partial p}-y=0
  10. y ( p ) = π ( p ) p y(p)=\frac{\partial\pi(p)}{\partial p}

Hölder's_theorem.html

  1. P ( x ; y 0 , y 1 , , y n ) P(x;\;y_{0},\;y_{1},\ldots,\;y_{n})
  2. P ( x ; Γ ( x ) , Γ ( x ) , , Γ ( n ) ( x ) ) 0. \,P\left(x;\;\Gamma(x),\;\Gamma^{\prime}(x),\;\ldots\;,\;\Gamma^{(n)}(x)\right% )\equiv 0.\!
  3. y 0 , y 1 , , y n , y_{0},\;y_{1},\ldots,\;y_{n},
  4. y 0 , y 1 , , y n , y_{0},\;y_{1},\ldots,\;y_{n},
  5. P ( x ; y 0 , y 1 , , y n ) = ( a 0 , a 1 , , a n ) A ( a 0 , a 1 , , a n ) ( x ) ( y 0 ) a 0 ( y 1 ) a 1 ( y n ) a n \,P(x;\;y_{0},\;y_{1},\ldots,\;y_{n})=\sum_{(a_{0},\;a_{1},\ldots,\;a_{n})}A_{% (a_{0},\;a_{1},\ldots,\;a_{n})}(x)\cdot(y_{0})^{a_{0}}\cdot(y_{1})^{a_{1}}% \cdot\ldots\cdot(y_{n})^{a_{n}}\!
  6. ( a 0 , a 1 , , a n ) \,(a_{0},\;a_{1},\ldots,\;a_{n})\,
  7. A ( a 0 , a 1 , , a n ) ( x ) \,A_{(a_{0},\;a_{1},\ldots,\;a_{n})}(x)\,
  8. A ( a 0 , a 1 , , a n ) ( x ) \,A_{(a_{0},\;a_{1},\ldots,\;a_{n})}(x)\,
  9. P ( x ; y 0 , y 1 , y 2 ) = x 2 y 2 + x y 1 + ( x 2 - ν 2 ) y 0 \,P(x;\;y_{0},\;y_{1},\;y_{2})=x^{2}y_{2}+xy_{1}+(x^{2}-\nu^{2})y_{0}\,
  10. A ( 0 , 0 , 1 ) ( x ) = x 2 \,A_{(0,\,0,\;1)}(x)=x^{2}\,
  11. A ( 0 , 1 , 0 ) ( x ) = x \,A_{(0,\;1,\;0)}(x)=x\,
  12. A ( 1 , 0 , 0 ) ( x ) = ( x 2 - ν 2 ) \,A_{(1,\;0,\;0)}(x)=(x^{2}-\nu^{2})\,
  13. P ( z ; f , f , f ′′ ) = x 2 f ′′ + x f + ( x 2 - ν 2 ) f = 0 \,P(z;\;f,\;f^{\prime},\;f^{\prime\prime})=x^{2}f^{\prime\prime}+xf^{\prime}+(% x^{2}-\nu^{2})f=0\,
  14. f = J ν ( x ) \,f=J_{\nu}(x)\,
  15. f = Y ν ( x ) \,f=Y_{\nu}(x)\,
  16. P ( x ; J ν ( x ) , J ν ( x ) , J ν ′′ ( x ) ) 0. \,P\left(x;\;J_{\nu}(x),\;J_{\nu}^{\prime}(x),\;J_{\nu}^{\prime\prime}(x)% \right)\equiv 0.\!
  17. J ν ( x ) \,J_{\nu}(x)\,
  18. Y ν ( x ) \,Y_{\nu}(x)\,
  19. P ( x ; Γ ( x ) , Γ ( x ) , , Γ ( n ) ( x ) ) 0. \,P\left(x;\;\Gamma(x),\;\Gamma^{\prime}(x),\;\ldots\;,\;\Gamma^{(n)}(x)\right% )\equiv 0.\!
  20. P ( x ; y 0 , y 1 , , y n ) = ( a 0 , a 1 , , a n ) A ( a 0 , a 1 , , a n ) ( x ) ( y 0 ) a 0 ( y 1 ) a 1 ( y n ) a n \,P(x;\;y_{0},\;y_{1},\ldots,\;y_{n})=\sum_{(a_{0},\;a_{1},\ldots,\;a_{n})}A_{% (a_{0},\;a_{1},\ldots,\;a_{n})}(x)\cdot(y_{0})^{a_{0}}\cdot(y_{1})^{a_{1}}% \cdot\ldots\cdot(y_{n})^{a_{n}}\!
  21. A ( a 0 , a 1 , , a n ) \,A_{(a_{0},\;a_{1},\ldots,\;a_{n})}\,
  22. P ( x + 1 ; Γ ( x + 1 ) , Γ ( 1 ) ( x + 1 ) , , Γ ( n ) ( x + 1 ) ) = = P ( x + 1 ; x Γ ( x ) , [ x Γ ( x ) ] ( 1 ) , [ x Γ ( x ) ] ( 2 ) , , [ x Γ ( x ) ] ( n ) ) = P ( x + 1 ; x Γ ( x ) , x Γ ( 1 ) ( x ) + Γ ( x ) , x Γ ( 2 ) ( x ) + 2 Γ ( 1 ) ( x ) , , x Γ ( n ) ( x ) + n Γ ( n - 1 ) ( x ) ) \begin{aligned}&\displaystyle P\left(x+1;\;\Gamma(x+1),\;\Gamma^{(1)}(x+1),% \ldots,\;\Gamma^{(n)}(x+1)\right)=\\ &\displaystyle\;\;\;\;\;\;\;=P\left(x+1;\;x\Gamma(x),\;\left[x\Gamma(x)\right]% ^{(1)},\;\left[x\Gamma(x)\right]^{(2)},\ldots,\left[x\Gamma(x)\right]^{(n)}% \right)\\ &\displaystyle\;\;\;\;\;\;\;=P\left(x+1;\;x\Gamma(x),\;x\Gamma^{(1)}(x)+\Gamma% (x),\;x\Gamma^{(2)}(x)+2\Gamma^{(1)}(x),\ldots,\;x\Gamma^{(n)}(x)+n\Gamma^{(n-% 1)}(x)\right)\end{aligned}
  23. Q ( x ; y 0 , y 1 , , y n ) = P ( x + 1 ; x y 0 , x y 1 + y 0 , x y 2 + 2 y 1 , x y 3 + 3 y 2 , , x y n + n y ( n - 1 ) ) \begin{aligned}\displaystyle Q(x;\;y_{0},\;y_{1},\ldots,\;y_{n})=P\left(x+1;\;% xy_{0},\;xy_{1}+y_{0},\;xy_{2}+2y_{1},\;xy_{3}+3y_{2},\ldots,\;xy_{n}+ny_{(n-1% )}\right)\end{aligned}
  24. Q ( x ; Γ ( x ) , Γ ( x ) , , Γ ( n ) ( x ) ) = 0 \,Q\left(x;\;\Gamma(x),\;\Gamma^{\prime}(x),\;\ldots\;,\;\Gamma^{(n)}(x)\right% )=0\,
  25. x a 0 + a 1 + + a n A ( h 0 , h 1 , , h n ) ( x + 1 ) ( y 0 ) h 0 ( y 1 ) h 1 ( y n ) h n x^{a_{0}+a_{1}+\ldots+a_{n}}A_{(h_{0},\;h_{1},\ldots,\;h_{n})}(x+1)\cdot(y_{0}% )^{h_{0}}\cdot(y_{1})^{h_{1}}\cdot\ldots\cdot(y_{n})^{h_{n}}\!
  26. ( h 0 , h 1 , , h n ) \,(h_{0},\;h_{1},\ldots,\;h_{n})\,
  27. Q ( x ; y 0 , y 1 , , y n ) \displaystyle Q(x;\;y_{0},\;y_{1},\ldots,\;y_{n})
  28. R ( x ) A ( h 0 , , h n ) ( x ) ( y 0 ) h 0 ( y n ) h n = x h 0 + + h n A ( h 0 , , h n ) ( x + 1 ) ( y 0 ) h 0 ( y n ) h n R ( x ) A ( h 0 , , h n ) ( x ) = x h 0 + + h n A ( h 0 , , h n ) ( x + 1 ) \begin{aligned}\displaystyle R(x)A_{(h_{0},\ldots,\;h_{n})}(x)\cdot(y_{0})^{h_% {0}}\cdot\ldots(y_{n})^{h_{n}}&\displaystyle=x^{h_{0}+\ldots+h_{n}}A_{(h_{0},% \ldots,\;h_{n})}(x+1)\cdot(y_{0})^{h_{0}}\cdot\ldots(y_{n})^{h_{n}}\\ \displaystyle R(x)A_{(h_{0},\ldots,\;h_{n})}(x)&\displaystyle=x^{h_{0}+\ldots+% h_{n}}A_{(h_{0},\ldots,\;h_{n})}(x+1)\end{aligned}\!
  29. γ 0 \,\gamma\neq 0\,
  30. P ( γ + 1 ; γ y 0 , γ y 1 + y 0 , γ y 2 + 2 y 1 , γ y 3 + 3 y 2 , , γ y n + n y n - 1 ) = 0 \,P\left(\gamma+1;\;\gamma y_{0},\;\gamma y_{1}+y_{0},\;\gamma y_{2}+2y_{1},\;% \gamma y_{3}+3y_{2},\ldots,\;\gamma y_{n}+ny_{n-1}\right)=0
  31. ( z - ( γ + 1 ) ) \,(z-(\gamma+1))\,
  32. R ( x ) = x n \,R(x)=x^{n}\,
  33. γ = 0 \,\gamma=0\,
  34. P ( γ + 1 ; γ y 0 , γ y 1 + y , γ y 2 + 2 y 1 , γ y 3 + 3 y 2 , , γ y n + n y n - 1 ) = P ( 1 ; 0 , y 0 , 2 y 1 , 3 y 2 , , n y n - 1 ) = P ( 1 ; 0 , z 1 , z 2 , z 3 , , z n - 1 ) = 0. \,\begin{aligned}\displaystyle P\left(\gamma+1;\;\gamma y_{0},\;\gamma y_{1}+y% ,\;\gamma y_{2}+2y_{1},\;\gamma y_{3}+3y_{2},\ldots,\;\gamma y_{n}+ny_{n-1}% \right)&\displaystyle=P\left(1;\;0,\;y_{0},\;2y_{1},\;3y_{2},\ldots,\;ny_{n-1}% \right)\\ &\displaystyle=P\left(1;\;0,\;z_{1},\;z_{2},\;z_{3},\ldots,\;z_{n-1}\right)\\ &\displaystyle=0.\end{aligned}
  35. P ( 1 ; 0 , z 1 , z 2 , z 3 , , z n - 1 ) = 0 \,P\left(1;\;0,\;z_{1},\;z_{2},\;z_{3},\ldots,\;z_{n-1}\right)=0\,
  36. P ( x + 1 ; 0 , x y 1 + y 0 , x y 2 + 2 y 1 , x y 3 ( x ) + 3 y 2 , , x y n + n y ( n - 1 ) ) = R ( x ) P ( x ; 0 , z 1 , , z n ) \begin{aligned}\displaystyle P\left(x+1;\;0,\;xy_{1}+y_{0},\;xy_{2}+2y_{1},\;% xy_{3}(x)+3y_{2},\ldots,\;xy_{n}+ny_{(n-1)}\right)&\displaystyle=R(x)P(x;\;0,% \;z_{1},\ldots,\;z_{n})\\ \end{aligned}
  37. P ( m ; 0 , z 1 , z 2 , z 3 , , z n - 1 ) = 0 P\left(m;\;0,\;z_{1},\;z_{2},\;z_{3},\ldots,\;z_{n-1}\right)=0
  38. y 0 \,y_{0}\,

Hölder_condition.html

  1. | f ( x ) - f ( y ) | C || x - y || α |f(x)-f(y)|\leq C||x-y||^{\alpha}
  2. C k , α ( Ω ¯ ) C^{k,\alpha}(\overline{\Omega})
  3. f C k , α = f C k + max | β | = k | D β f | C 0 , α \|f\|_{C^{k,\alpha}}=\|f\|_{C^{k}}+\max_{|\beta|=k}\left|D^{\beta}f\right|_{C^% {0,\alpha}}
  4. f C k = max | β | k sup x Ω | D β f ( x ) | . \|f\|_{C^{k}}=\max_{|\beta|\leq k}\sup_{x\in\Omega}\left|D^{\beta}f(x)\right|.
  5. | f | 0 , α |f|_{0,\alpha}
  6. f k , α \|f\|_{k,\alpha}
  7. | f | 0 , α , Ω |f|_{0,\alpha,\Omega}\;
  8. f k , α , Ω \|f\|_{k,\alpha,\Omega}
  9. C k , α ( Ω ¯ ) C^{k,\alpha}(\overline{\Omega})
  10. C k , α \|\cdot\|_{C^{k,\alpha}}
  11. | f | 0 , α , Ω diam ( Ω ) β - α | f | 0 , β , Ω |f|_{0,\alpha,\Omega}\leq\mathrm{diam}(\Omega)^{\beta-\alpha}|f|_{0,\beta,\Omega}
  12. | u n - u | 0 , α = | u n | 0 , α 0 , |u_{n}-u|_{0,\alpha}=|u_{n}|_{0,\alpha}\to 0,
  13. | u n ( x ) - u n ( y ) | | x - y | α ( | u n ( x ) - u n ( y ) | | x - y | β ) α / β | u n ( x ) - u n ( y ) | 1 - α / β | u n | 0 , β β / α ( 2 u n ) 1 - α / β = o ( 1 ) . \frac{|u_{n}(x)-u_{n}(y)|}{|x-y|^{\alpha}}\leq\left(\frac{|u_{n}(x)-u_{n}(y)|}% {|x-y|^{\beta}}\right)^{\alpha/\beta}|u_{n}(x)-u_{n}(y)|^{1-\alpha/\beta}\leq|% u_{n}|_{0,\beta}^{\beta/\alpha}\,\left(2\|u_{n}\|_{\infty}\right)^{1-\alpha/% \beta}=o(1).
  14. C 0 , α ( Ω ¯ ) C^{0,\alpha}(\overline{\Omega})
  15. [ 0 , ) [0,\infty)
  16. u x , r = 1 | B r | B r ( x ) u ( y ) d y u_{x,r}=\frac{1}{|B_{r}|}\int_{B_{r}(x)}u(y)dy
  17. B r ( x ) | u ( y ) - u x , r | 2 d y C r n + 2 α , \int_{B_{r}(x)}|u(y)-u_{x,r}|^{2}dy\leq Cr^{n+2\alpha},
  18. w ( u , x 0 , r ) = sup B r ( x 0 ) u - inf B r ( x 0 ) u w(u,x_{0},r)=\sup_{B_{r}(x_{0})}u-\inf_{B_{r}(x_{0})}u
  19. w ( u , x 0 , r 2 ) λ w ( u , x 0 , r ) w(u,x_{0},\tfrac{r}{2})\leq\lambda w(u,x_{0},r)
  20. u C 0 , γ ( 𝐑 n ) C u W 1 , p ( 𝐑 n ) \|u\|_{C^{0,\gamma}(\mathbf{R}^{n})}\leq C\|u\|_{W^{1,p}(\mathbf{R}^{n})}
  21. f - f k , X = O ( k - α 1 - α ) . \|f-f_{k}\|_{\infty,X}=O\left(k^{-\frac{\alpha}{1-\alpha}}\right).
  22. f * ( x ) := inf y X { f ( y ) + C | x - y | α } . f^{*}(x):=\inf_{y\in X}\left\{f(y)+C|x-y|^{\alpha}\right\}.

Ho–Lee_model.html

  1. d r t = θ t d t + σ d W t dr_{t}=\theta_{t}\,dt+\sigma\,dW_{t}
  2. θ t \theta_{t}

HSTCP.html

  1. a ( w ) / w a(w)/w
  2. ( 1 - b ( w ) ) w (1-b(w))w

Hua's_lemma.html

  1. ε \varepsilon
  2. f ( α ) = x = 1 N exp ( 2 π i P ( x ) α ) , f(\alpha)=\sum_{x=1}^{N}\exp(2\pi iP(x)\alpha),
  3. 0 1 | f ( α ) | λ d α P , ε N μ ( λ ) \int_{0}^{1}|f(\alpha)|^{\lambda}d\alpha\ll_{P,\varepsilon}N^{\mu(\lambda)}
  4. ( λ , μ ( λ ) ) (\lambda,\mu(\lambda))
  5. ( 2 ν , 2 ν - ν + ε ) , ν = 1 , , k . (2^{\nu},2^{\nu}-\nu+\varepsilon),\quad\nu=1,\ldots,k.

Huang's_algorithm.html

  1. w w
  2. w w

Hubble–Reynolds_law.html

  1. I ( R ) = I 0 ( 1 + R / R H ) 2 I(R)=\frac{I_{0}}{(1+R/R_{H})^{2}}
  2. I ( R ) I(R)
  3. R R
  4. I 0 I_{0}
  5. R H R_{H}

Human_Poverty_Index.html

  1. [ 1 3 ( P 1 α + P 2 α + P 3 α ) ] 1 α \left[\frac{1}{3}\left(P_{1}^{\alpha}+P_{2}^{\alpha}+P_{3}^{\alpha}\right)% \right]^{\frac{1}{\alpha}}
  2. P 1 P_{1}
  3. P 2 P_{2}
  4. P 3 P_{3}
  5. α \alpha
  6. [ 1 4 ( P 1 α + P 2 α + P 3 α + P 4 α ) ] 1 α \left[\frac{1}{4}\left(P_{1}^{\alpha}+P_{2}^{\alpha}+P_{3}^{\alpha}+P_{4}^{% \alpha}\right)\right]^{\frac{1}{\alpha}}
  7. P 1 P_{1}
  8. P 2 P_{2}
  9. P 3 P_{3}
  10. P 4 P_{4}
  11. α \alpha

Hunt_process.html

  1. { F t } t 0 \{F_{t}\}_{t\geq 0}

Hunting_oscillation.html

  1. 1 R = 2 k y r d \frac{1}{R}=\frac{2ky}{rd}
  2. | d 2 y d x 2 | 1 R \left|\frac{\operatorname{d}^{2}y}{\operatorname{d}x^{2}}\right|\approx\frac{1% }{R}
  3. d 2 y d x 2 = - ( 2 k r d ) y \frac{\operatorname{d}^{2}y}{\operatorname{d}x^{2}}=-\left(\frac{2k}{rd}\right)y
  4. λ = 2 π r d 2 k \lambda=2\pi\sqrt{\frac{rd}{2k}}
  5. ( θ ) \left(\theta\right)
  6. θ = d y d x \theta=\frac{\operatorname{d}y}{\operatorname{d}x}
  7. d θ d x \displaystyle\frac{\operatorname{d}\theta}{\operatorname{d}x}
  8. d d t = U d d x \frac{\operatorname{d}}{\operatorname{d}t}=U\frac{\operatorname{d}}{% \operatorname{d}x}
  9. d 2 θ d t 2 = - U 2 ( 2 k r d ) θ \frac{\operatorname{d}^{2}\theta}{\operatorname{d}t^{2}}=-U^{2}\left(\frac{2k}% {rd}\right)\theta
  10. F d = C d 2 θ d t 2 Fd=C\frac{\operatorname{d}^{2}\theta}{\operatorname{d}t^{2}}
  11. F = - C U 2 ( 2 k r d 2 ) θ F=-CU^{2}\left(\frac{2k}{rd^{2}}\right)\theta
  12. F = μ W 2 F=\mu\frac{W}{2}
  13. μ \mu
  14. θ U 2 = μ W r d 2 4 C k \theta U^{2}=\mu W\frac{rd^{2}}{4Ck}
  15. ω = d θ d t \omega=\frac{\operatorname{d}\theta}{\operatorname{d}t}
  16. ω d d θ = d d t \omega\frac{\operatorname{d}}{\operatorname{d}\theta}=\frac{\operatorname{d}}{% \operatorname{d}t}
  17. ω \omega
  18. ω d ω d θ = - U 2 ( 2 k r d ) θ \omega\frac{\operatorname{d}\omega}{\operatorname{d}\theta}=-U^{2}\left(\frac{% 2k}{rd}\right)\theta
  19. 1 2 ω 2 = - 1 2 U 2 ( 2 k r d ) θ 2 \frac{1}{2}\omega^{2}=-\frac{1}{2}U^{2}\left(\frac{2k}{rd}\right)\theta^{2}
  20. 1 2 C ω 2 = - 1 2 C U 2 ( 2 k r d ) θ 2 \frac{1}{2}C\omega^{2}=-\frac{1}{2}CU^{2}\left(\frac{2k}{rd}\right)\theta^{2}
  21. d c o s ( θ ) = d ( 1 + 1 2 θ 2 ) \frac{d}{cos(\theta)}=d\left(1+\frac{1}{2}\theta^{2}\right)
  22. 1 2 ( d + 1 2 d θ 2 - d ) \frac{1}{2}\left(d+\frac{1}{2}d\theta^{2}-d\right)
  23. h = 1 4 k d θ 2 h=\frac{1}{4}kd\theta^{2}
  24. E = 1 4 W k d θ 2 E=\frac{1}{4}Wkd\theta^{2}
  25. V = U r ( r + k y ) V=\frac{U}{r}\left(r+ky\right)
  26. 1 4 m ( U 2 + 2 U 2 k y r + U 2 k 2 y 2 r 2 ) \frac{1}{4}m\left(U^{2}+2U^{2}\frac{ky}{r}+U^{2}\frac{k^{2}y^{2}}{r^{2}}\right)
  27. 1 4 m ( U 2 - 2 U 2 k y r + U 2 k 2 y 2 r 2 ) \frac{1}{4}m\left(U^{2}-2U^{2}\frac{ky}{r}+U^{2}\frac{k^{2}y^{2}}{r^{2}}\right)
  28. δ E = 1 2 m ( U k y r ) 2 \delta E=\frac{1}{2}m\left(\frac{Uky}{r}\right)^{2}
  29. 2 U k y r d \displaystyle 2U\frac{ky}{rd}
  30. ω 2 = - U 2 ( 2 k r d ) θ 2 \omega^{2}=-U^{2}\left(\frac{2k}{rd}\right)\theta^{2}
  31. δ E = - 1 8 U 2 m d 2 ( 2 k r d ) θ 2 \delta E=-\frac{1}{8}U^{2}md^{2}\left(\frac{2k}{rd}\right)\theta^{2}
  32. T = 1 2 U 2 ( C + m d 2 4 ) ( 2 k r d ) θ 2 T=\frac{1}{2}U^{2}\left(C+\frac{md^{2}}{4}\right)\left(\frac{2k}{rd}\right)% \theta^{2}
  33. W k d 2 = U 2 2 k r d ( C + m d 2 4 ) \frac{Wkd}{2}=U^{2}\frac{2k}{rd}\left(C+\frac{md^{2}}{4}\right)
  34. U 2 = W r d 2 4 C + m d 2 U^{2}=\frac{Wrd^{2}}{4C+md^{2}}
  35. U 2 = W a r d 2 k ( 4 C + m d 2 ) U^{2}=\frac{Ward^{2}}{k(4C+md^{2})}
  36. d 2 y d x 2 \frac{\operatorname{d}^{2}y}{\operatorname{d}x^{2}}
  37. 1 R = 2 k y r d \frac{1}{R}=\frac{2ky}{rd}

Hückel_method.html

  1. E k = α + 2 β cos k π ( n + 1 ) E_{k}=\alpha+2\beta\cos\frac{k\pi}{(n+1)}
  2. E k = α + 2 β cos 2 k π n E_{k}=\alpha+2\beta\cos\frac{2k\pi}{n}
  3. Δ E = - 4 β sin π 2 ( n + 1 ) \Delta E=-4\beta\sin\frac{\pi}{2(n+1)}
  4. π \pi
  5. Ψ \Psi\,
  6. ϕ \phi\,
  7. c c\,
  8. Ψ = c 1 ϕ 1 + c 2 ϕ 2 \ \Psi=c_{1}\phi_{1}+c_{2}\phi_{2}
  9. H Ψ = E Ψ \ H\Psi=E\Psi
  10. H H\,
  11. E E\,
  12. H c 1 ϕ 1 + H c 2 ϕ 2 = E c 1 ϕ 1 + E c 2 ϕ 2 Hc_{1}\phi_{1}+Hc_{2}\phi_{2}=Ec_{1}\phi_{1}+Ec_{2}\phi_{2}\,
  13. ϕ 1 \phi_{1}\,
  14. c 1 ( H 11 - E S 11 ) + c 2 ( H 12 - E S 12 ) = 0 c_{1}(H_{11}-ES_{11})+c_{2}(H_{12}-ES_{12})=0\,
  15. ϕ 2 \phi_{2}\,
  16. c 1 ( H 21 - E S 21 ) + c 2 ( H 22 - E S 22 ) = 0 c_{1}(H_{21}-ES_{21})+c_{2}(H_{22}-ES_{22})=0\,
  17. [ c 1 ( H 11 - E S 11 ) + c 2 ( H 12 - E S 12 ) c 1 ( H 21 - E S 21 ) + c 2 ( H 22 - E S 22 ) ] = 0 \begin{bmatrix}c_{1}(H_{11}-ES_{11})+c_{2}(H_{12}-ES_{12})\\ c_{1}(H_{21}-ES_{21})+c_{2}(H_{22}-ES_{22})\\ \end{bmatrix}=0
  18. [ H 11 - E S 11 H 12 - E S 12 H 21 - E S 21 H 22 - E S 22 ] × [ c 1 c 2 ] = 0 \begin{bmatrix}H_{11}-ES_{11}&H_{12}-ES_{12}\\ H_{21}-ES_{21}&H_{22}-ES_{22}\\ \end{bmatrix}\times\begin{bmatrix}c_{1}\\ c_{2}\\ \end{bmatrix}=0
  19. H i j = ϕ i H ϕ j d v H_{ij}=\int\phi_{i}H\phi_{j}\mathrm{d}v\,
  20. S i j = ϕ i ϕ j d v S_{ij}=\int\phi_{i}\phi_{j}\mathrm{d}v\,
  21. H i i H_{ii}\,
  22. H i j H_{ij}\,
  23. S i j = δ i j S_{ij}=\delta_{ij}\,
  24. H i j H_{ij}\,
  25. H 11 = H 22 = α H_{11}=H_{22}=\alpha\,
  26. H 12 = H 21 = β H_{12}=H_{21}=\beta\,
  27. S 11 = S 22 = 1 S_{11}=S_{22}=1\,
  28. S 12 = S 21 = 0 S_{12}=S_{21}=0\,
  29. [ α - E β β α - E ] × [ c 1 c 2 ] = 0 \begin{bmatrix}\alpha-E&\beta\\ \beta&\alpha-E\\ \end{bmatrix}\times\begin{bmatrix}c_{1}\\ c_{2}\\ \end{bmatrix}=0
  30. β \beta
  31. [ α - E β 1 1 α - E β ] × [ c 1 c 2 ] = 0 \begin{bmatrix}\frac{\alpha-E}{\beta}&1\\ 1&\frac{\alpha-E}{\beta}\\ \end{bmatrix}\times\begin{bmatrix}c_{1}\\ c_{2}\\ \end{bmatrix}=0
  32. x x
  33. α - E β \frac{\alpha-E}{\beta}
  34. [ x 1 1 x ] × [ c 1 c 2 ] = 0 \begin{bmatrix}x&1\\ 1&x\\ \end{bmatrix}\times\begin{bmatrix}c_{1}\\ c_{2}\\ \end{bmatrix}=0
  35. x = α - E β x=\frac{\alpha-E}{\beta}\,
  36. x β = α - E x\beta=\alpha-E\,
  37. E = α - x β E=\alpha-x\beta\,
  38. c 2 = - x c 1 c_{2}=-xc_{1}\,
  39. c 1 = - x c 2 c_{1}=-xc_{2}\,
  40. | x 1 1 x | = 0 \begin{vmatrix}x&1\\ 1&x\\ \end{vmatrix}=0
  41. x 2 - 1 = 0 x^{2}-1=0\,
  42. x 2 = 1 x^{2}=1\,
  43. x = ± 1 x=\pm 1\,
  44. E = α - x β E=\alpha-x\beta
  45. E = α - ± 1 × β E=\alpha-\pm 1\times\beta
  46. E = α β E=\alpha\mp\beta
  47. c 2 = - x c 1 c_{2}=-xc_{1}\,
  48. c 1 = - x c 2 c_{1}=-xc_{2}\,
  49. c 2 = - ± 1 × c 1 c_{2}=-\pm 1\times c_{1}\,
  50. c 2 = c 1 c_{2}=\mp c_{1}\,
  51. Ψ = c 1 ( ϕ 1 ϕ 2 ) \Psi=c_{1}(\phi_{1}\mp\phi_{2})\,
  52. c 1 = 1 2 , c_{1}=\frac{1}{\sqrt{2}},
  53. Ψ = 1 2 ( ϕ 1 ϕ 2 ) = ϕ 1 ϕ 2 2 \Psi=\frac{1}{\sqrt{2}}(\phi_{1}\mp\phi_{2})=\frac{\phi_{1}\mp\phi_{2}}{\sqrt{% 2}}\,
  54. α + β \alpha+\beta
  55. Ψ = 1 2 ( ϕ 1 + ϕ 2 ) \Psi=\frac{1}{\sqrt{2}}(\phi_{1}+\phi_{2})\,
  56. α - β \alpha-\beta
  57. Ψ = 1 2 ( ϕ 1 - ϕ 2 ) \Psi=\frac{1}{\sqrt{2}}(\phi_{1}-\phi_{2})\,
  58. Ψ \Psi\,
  59. ϕ \phi\,
  60. c c\,
  61. Ψ = c 1 ϕ 1 + c 2 ϕ 2 + c 3 ϕ 3 + c 4 ϕ 4 \ \Psi=c_{1}\phi_{1}+c_{2}\phi_{2}+c_{3}\phi_{3}+c_{4}\phi_{4}
  62. [ α - E β 0 0 β α - E β 0 0 β α - E β 0 0 β α - E ] × [ c 1 c 2 c 3 c 4 ] = 0 \begin{bmatrix}\alpha-E&\beta&0&0\\ \beta&\alpha-E&\beta&0\\ 0&\beta&\alpha-E&\beta\\ 0&0&\beta&\alpha-E\\ \end{bmatrix}\times\begin{bmatrix}c_{1}\\ c_{2}\\ c_{3}\\ c_{4}\\ \end{bmatrix}=0
  63. ( α - E ) ( α + β - E ) - β 2 = 0 (\alpha-E)(\alpha+\beta-E)-\beta^{2}=0\,
  64. E ± = α + 1 ± 5 2 β E\pm=\alpha+\frac{1\pm\sqrt{5}}{2}\beta

Hybrid_algorithm_(constraint_satisfaction).html

  1. b b
  2. b b
  3. b b
  4. b b
  5. b b
  6. b b
  7. b b
  8. b b
  9. b b
  10. b b
  11. b b
  12. b b
  13. b b

Hybrid_solar_cell.html

  1. E A A - E A D > U D E_{A}^{A}-E_{A}^{D}>U_{D}

Hydraulic_cylinder.html

  1. F = P A F=P\cdot A
  2. P = F p A p - A r P=\frac{F_{p}}{A_{p}-A_{r}}

Hyperbolic_Dehn_surgery.html

  1. M ( u 1 , u 2 , , u n ) M(u_{1},u_{2},\dots,u_{n})
  2. u i = p i / q i u_{i}=p_{i}/q_{i}
  3. p i p_{i}
  4. q i q_{i}
  5. u i u_{i}
  6. \infty
  7. M ( , , ) M(\infty,\dots,\infty)
  8. M ( u 1 , u 2 , , u n ) M(u_{1},u_{2},\dots,u_{n})
  9. E i E_{i}
  10. M ( u 1 , u 2 , , u n ) M(u_{1},u_{2},\dots,u_{n})
  11. p i 2 + q i 2 p_{i}^{2}+q_{i}^{2}\rightarrow\infty
  12. p i / q i p_{i}/q_{i}
  13. u i u_{i}
  14. ω ω \omega^{\omega}

Hyperbolic_equilibrium_point.html

  1. [ x n + 1 y n + 1 ] = [ 1 1 1 2 ] [ x n y n ] modulo 1 \begin{bmatrix}x_{n+1}\\ y_{n+1}\end{bmatrix}=\begin{bmatrix}1&1\\ 1&2\end{bmatrix}\begin{bmatrix}x_{n}\\ y_{n}\end{bmatrix}\quad\,\text{modulo }1
  2. λ 1 = 3 + 5 2 > 1 \lambda_{1}=\frac{3+\sqrt{5}}{2}>1
  3. λ 2 = 3 - 5 2 < 1 \lambda_{2}=\frac{3-\sqrt{5}}{2}<1
  4. d x d t = y , \frac{dx}{dt}=y,
  5. d y d t = - x - x 3 - α y , α 0 \frac{dy}{dt}=-x-x^{3}-\alpha y,~{}\alpha\neq 0
  6. J ( 0 , 0 ) = ( 0 1 - 1 - α ) J(0,0)=\begin{pmatrix}0&1\\ -1&-\alpha\end{pmatrix}
  7. - α ± α 2 - 4 2 \frac{-\alpha\pm\sqrt{\alpha^{2}-4}}{2}

Hyperboloid_model.html

  1. Q ( x 0 , x 1 , , x n ) = x 0 2 - x 1 2 - - x n 2 . Q(x_{0},x_{1},\ldots,x_{n})=x_{0}^{2}-x_{1}^{2}-\ldots-x_{n}^{2}.
  2. B ( u , v ) = ( Q ( u + v ) - Q ( u ) - Q ( v ) ) / 2. B(u,v)=(Q(u+v)-Q(u)-Q(v))/2.
  3. B ( ( x 0 , x 1 , , x n ) , ( y 0 , y 1 , , y n ) ) = x 0 y 0 - x 1 y 1 - - x n y n . B((x_{0},x_{1},\ldots,x_{n}),(y_{0},y_{1},\ldots,y_{n}))=x_{0}y_{0}-x_{1}y_{1}% -\ldots-x_{n}y_{n}.
  4. d ( u , v ) = arcosh ( B ( u , v ) ) , d(u,v)=\operatorname{arcosh}(B(u,v)),
  5. a r c o s h arcosh
  6. ( 1 0 0 0 A 0 ) \begin{pmatrix}1&0&\ldots&0\\ 0&&&\\ \vdots&&A&\\ 0&&&\\ \end{pmatrix}
  7. A A
  8. n = SO + ( 1 , n ) / SO ( n ) . \mathbb{H}^{n}=\mathrm{SO}^{+}(1,n)/\mathrm{SO}(n).
  9. , \langle\cdot,\cdot\rangle
  10. n \mathbb{R}^{n}
  11. x n x\in\mathbb{R}^{n}
  12. ( x , 1 + x , x ) n + 1 , (x,\sqrt{1+\langle x,x\rangle})\in\mathbb{R}^{n+1},
  13. ( x , 1 - x , x ) n + 1 (x,\sqrt{1-\langle x,x\rangle})\in\mathbb{R}^{n+1}
  14. cosh A + α sinh A , \cosh A+\alpha\sinh A,