wpmath0000009_8

Minnesota_Starvation_Experiment.html

  1. W ( t ) = W f + K ( 24 - t ) 2 W(t)=W_{f}+K\,(24-t)^{2}
  2. t t
  3. W ( t ) W(t)
  4. t t
  5. W f W_{f}
  6. K K
  7. W ( t = 0 ) W(t=0)
  8. W i W_{i}
  9. W i = W f + K ( 24 - 0 ) 2 W_{i}=W_{f}+K\,(24-0)^{2}
  10. K K
  11. K = W i - W f 24 2 K=\frac{W_{i}-W_{f}}{24^{2}}
  12. P P
  13. P = 100 × W i - W f W i P=100\times\frac{W_{i}-W_{f}}{W_{i}}
  14. K = P 100 × 24 2 W i K=\frac{P}{100\times 24^{2}}\,W_{i}

Minus-end-directed_kinesin_ATPase.html

  1. \rightleftharpoons

Misiurewicz_point.html

  1. c c
  2. M k , n M_{k,n}\,
  3. f c ( k ) ( z c r ) = f c ( k + n ) ( z c r ) f_{c}^{(k)}(z_{cr})=f_{c}^{(k+n)}(z_{cr})\,
  4. f c ( k - 1 ) ( z c r ) f c ( k + n - 1 ) ( z c r ) f_{c}^{(k-1)}(z_{cr})\neq f_{c}^{(k+n-1)}(z_{cr})\,
  5. M k , n = c : f c ( k ) ( z c r ) = f c ( k + n ) ( z c r ) M_{k,n}=c:f_{c}^{(k)}(z_{cr})=f_{c}^{(k+n)}(z_{cr})\,
  6. z c r z_{cr}\,
  7. f c f_{c}\,
  8. k k\,
  9. n n\,
  10. f c k f_{c}^{k}
  11. k k
  12. f c f_{c}
  13. P c ( z ) = z 2 + c P_{c}(z)=z^{2}+c\,
  14. z = 0 z=0\,
  15. P c ( k ) ( 0 ) = P c ( k + n ) ( 0 ) P_{c}^{(k)}(0)=P_{c}^{(k+n)}(0)
  16. P c ( n ) = P c ( P c ( n - 1 ) ) P_{c}^{(n)}=P_{c}(P_{c}^{(n-1)})\,
  17. P c ( z ) = z 2 + c P_{c}(z)=z^{2}+c\,
  18. P c P_{c}\,
  19. P c ( 2 ) ( 0 ) = P c ( 3 ) ( 0 ) P_{c}^{(2)}(0)=P_{c}^{(3)}(0)
  20. c 2 + c = ( c 2 + c ) 2 + c \Rightarrow c^{2}+c=(c^{2}+c)^{2}+c
  21. c 4 + 2 c 3 = 0 \Rightarrow c^{4}+2c^{3}=0
  22. c c\,
  23. c c\,
  24. J c J_{c}\,
  25. c = M 2 , 2 = i c=M_{2,2}=i\,
  26. { 0 , i , i - 1 , - i , i - 1 , - i } \{0,i,i-1,-i,i-1,-i...\}\,
  27. c = M 2 , 1 = - 2 c=M_{2,1}=-2\,
  28. { 0 , - 2 , 2 , 2 , 2 , } \{0,-2,2,2,2,...\}\,
  29. z = - 2 z=-2\,
  30. c = - 2 c=-2\,
  31. c = - 2 = M 2 , 1 c=-2=M_{2,1}\,
  32. c = M 23 , 2 c=M_{23,2}\,
  33. c = - 0.77568377 + 0.13646737 * i c=-0.77568377+0.13646737*i\,
  34. M 23 , 2 M_{23,2}\,
  35. 8388611 25165824 \frac{8388611}{25165824}\,
  36. 8388613 25165824 \frac{8388613}{25165824}\,
  37. 3 * 2 23 3*2^{23}\,
  38. k = 23 k=23\,
  39. n = 2 n=2\,
  40. c = - 1.54368901269109 c=-1.54368901269109\,
  41. M 3 , 1 M_{3,1}\,
  42. 5 12 \frac{5}{12}\,
  43. 7 12 \frac{7}{12}\,
  44. k = 3 k=3\,
  45. n = 1 n=1\,
  46. c = - 0.1010... + 0.9562... * i = M 4 , 1 c=-0.1010...+0.9562...*i=M_{4,1}\,

Misner_space.html

  1. 1 , 1 / b o o s t \mathbb{R}^{1,1}/boost

Mixing_patterns.html

  1. e i j e_{ij}
  2. e i j = e j i e_{ij}=e_{ji}
  3. i j e i j = 1 , j e i j = a i , i e i j = b j \sum_{ij}{e_{ij}=1},\quad\sum_{j}{e_{ij}=a_{i}},\quad\sum_{i}{e_{ij}=b_{j}}
  4. a i a_{i}
  5. b i b_{i}
  6. i i
  7. a i = b i a_{i}=b_{i}
  8. r = i e i i - i a i b i 1 - i a i b i r=\frac{\sum_{i}{e_{ii}}-\sum_{i}{a_{i}b_{i}}}{1-\sum_{i}{a_{i}b_{i}}}
  9. r m i n = - i a i b i 1 - i a i b i r_{min}=-\frac{\sum_{i}{a_{i}b_{i}}}{1-\sum_{i}{a_{i}b_{i}}}
  10. r = 0 r=0
  11. e i j = a i b j e_{ij}=a_{i}b_{j}
  12. r = 1 r=1
  13. r = r m i n r=r_{min}
  14. - 1 r < 0 -1\leq r<0
  15. r m i n r_{min}
  16. r = 0 r=0
  17. p k ( i ) p_{k}^{(i)}
  18. i i
  19. e i j e_{ij}
  20. a i a_{i}
  21. b i b_{i}
  22. p k ( i ) p_{k}^{(i)}
  23. e i j e_{ij}
  24. p k ( i ) p_{k}^{(i)}
  25. G 0 ( i ) ( x 1 , , x n ) = k p k ( i ) x k G_{0}^{(i)}(x_{1},...,x_{n})=\sum_{k}p_{k}^{(i)}x^{k}
  26. G 1 ( i ) = 1 z i d G 0 ( i ) d x | x = 1 G_{1}^{(i)}=\frac{1}{z_{i}}\frac{dG_{0}^{(i)}}{dx}\Bigg|_{x=1}
  27. x i = j e i j x j j e i j x_{i}=\frac{\sum_{j}e_{ij}x_{j}}{\sum_{j}e_{ij}}
  28. i i
  29. r i r_{i}
  30. z i z_{i}
  31. i i
  32. H 1 ( i ) ( x ) = x G 1 ( i ) [ H 1 ( 1 ) ( x ) , , H 1 ( n ) ( x ) ] H_{1}^{(i)}(x)=xG_{1}^{(i)}[H_{1}^{(1)}(x),...,H_{1}^{(n)}(x)]
  33. i i
  34. H 0 ( i ) ( x ) = x G 0 ( i ) [ H 1 ( 1 ) ( x ) , , H 1 ( n ) ( x ) ] H_{0}^{(i)}(x)=xG_{0}^{(i)}[H_{1}^{(1)}(x),...,H_{1}^{(n)}(x)]
  35. s i s_{i}
  36. i i
  37. s i = d H 0 ( i ) d x | x = 1 = 1 + G 0 ( i ) ( 1 ) j e i j H 1 ( i ) ( 1 ) j e i j s_{i}=\frac{dH_{0}^{(i)}}{dx}\Bigg|_{x=1}=1+G_{0}^{(i)^{\prime}}(1)\frac{\sum_% {j}e_{ij}H_{1}^{(i)^{\prime}}(1)}{\sum_{j}e_{ij}}
  38. u i u_{i}
  39. i i
  40. S S
  41. S = 1 - i a i z i G 0 ( i ) ( u 1 , , u n ) S=1-\sum_{i}\frac{a_{i}}{z_{i}}G_{0}^{(i)}(u_{1},...,u_{n})
  42. r = j k j k ( e j k - q j q k ) σ q 2 r=\frac{\sum_{jk}jk(e_{jk}-q_{j}q_{k})}{\sigma^{2}_{q}}
  43. p k p_{k}
  44. q k = ( k + 1 ) p k + 1 z q_{k}=\frac{(k+1)p_{k+1}}{z}
  45. σ q \sigma_{q}
  46. q k q_{k}
  47. r = j k j k ( e j k - q j i n q k o u t ) σ i n σ o u t r=\frac{\sum_{jk}jk(e_{jk}-q_{j}^{in}q_{k}^{out})}{\sigma_{in}\sigma_{out}}
  48. p k k - τ p_{k}\sim k^{-\tau}
  49. q k q_{k}
  50. τ > 3 \tau>3
  51. p k = k - τ e - k / κ Li τ ( e - 1 / κ ) for k 1 p_{k}=\frac{k^{-\tau}\mathrm{e}^{-k/\kappa}}{\mathrm{Li}_{\tau}(\mathrm{e}^{-1% /\kappa})}\ \mathrm{for}\ k\geq 1
  52. q k ( k + 1 ) 1 - τ e - ( k + 1 ) / κ q_{k}\sim(k+1)^{1-\tau}\mathrm{e}^{-(k+1)/\kappa}
  53. κ \kappa
  54. r r
  55. κ \kappa

Modern_Arabic_mathematical_notation.html

  1. a a
  2. b b
  3. c c
  4. d d
  5. x x
  6. y y
  7. z z
  8. e e
  9. i i
  10. π \pi
  11. π \pi
  12. r r
  13. \mathbb{N}
  14. \mathbb{Z}
  15. \mathbb{Q}
  16. \mathbb{R}
  17. 𝕀 \mathbb{I}
  18. \mathbb{C}
  19. \varnothing
  20. \varnothing
  21. \in
  22. \ni
  23. \subset
  24. \supset
  25. \supset
  26. \subset
  27. 𝐒 \mathbf{S}
  28. \propto
  29. n \sqrt[n]{\,\,\,}
  30. log \log
  31. log b \log_{b}
  32. ln \ln
  33. \sum
  34. \prod
  35. \prod
  36. n ! n!
  37. 𝐏 r n {}^{n}\mathbf{P}_{r}
  38. 𝐏 ( n , r ) \mathbf{P}(n,r)
  39. 𝐂 k n {}^{n}\mathbf{C}_{k}
  40. 𝐂 ( n , k ) \mathbf{C}(n,k)
  41. ( n k ) n\choose k
  42. sin \sin
  43. cos \cos
  44. tan \tan
  45. cot \cot
  46. sec \sec
  47. csc \csc
  48. s i n - 1 sin^{-1}
  49. lim \lim
  50. 𝐟 ( x ) \mathbf{f}(x)
  51. 𝐟 ( x ) , d y d x , d 2 y d x 2 , y x \mathbf{f^{\prime}}(x),\dfrac{dy}{dx},\dfrac{d^{2}y}{dx^{2}},\dfrac{\partial{y% }}{\partial{x}}
  52. , , , \int{},\iint{},\iiint{},\oint{}
  53. z = x + i y = r ( cos φ + i sin φ ) = r e i φ = r φ z=x+iy=r(\cos{\varphi}+i\sin{\varphi})=re^{i\varphi}=r\angle{\varphi}

Modes_of_convergence.html

  1. Σ | b k | \Sigma|b_{k}|
  2. d \mathbb{R}^{d}
  3. Σ | g k | \Sigma|g_{k}|
  4. Σ | g k | \Sigma|g_{k}|
  5. Σ | g k | \Sigma|g_{k}|

Modified_Richardson_iteration.html

  1. A x = b . Ax=b.\,
  2. x ( k + 1 ) = x ( k ) + ω ( b - A x ( k ) ) , x^{(k+1)}=x^{(k)}+\omega\left(b-Ax^{(k)}\right),
  3. ω \omega
  4. x ( k ) x^{(k)}
  5. x ( k + 1 ) x ( k ) x^{(k+1)}\approx x^{(k)}
  6. x ( k ) x^{(k)}
  7. A x = b Ax=b
  8. x x
  9. e ( k ) = x ( k ) - x e^{(k)}=x^{(k)}-x
  10. e ( k + 1 ) = e ( k ) - ω A e ( k ) = ( I - ω A ) e ( k ) . e^{(k+1)}=e^{(k)}-\omega Ae^{(k)}=(I-\omega A)e^{(k)}.
  11. e ( k + 1 ) = ( I - ω A ) e ( k ) I - ω A e ( k ) , \|e^{(k+1)}\|=\|(I-\omega A)e^{(k)}\|\leq\|I-\omega A\|\|e^{(k)}\|,
  12. I - ω A < 1 \|I-\omega A\|<1
  13. A A
  14. ( λ j , v j ) (\lambda_{j},v_{j})
  15. A A
  16. 0
  17. | 1 - ω λ j | < 1 |1-\omega\lambda_{j}|<1
  18. λ j \lambda_{j}
  19. ω \omega
  20. 0 < ω < 2 / λ m a x ( A ) 0<\omega<2/\lambda_{max}(A)
  21. | 1 - ω λ j | |1-\omega\lambda_{j}|
  22. ω = 2 / ( λ m i n ( A ) + λ m a x ( A ) ) \omega=2/(\lambda_{min}(A)+\lambda_{max}(A))
  23. ω \omega
  24. e ( 0 ) e^{(0)}
  25. F ( x ) = 1 2 A ~ x - b 2 2 F(x)=\frac{1}{2}\|\tilde{A}x-b\|_{2}^{2}
  26. F ( x ) = 0 \nabla F(x)=0
  27. A ~ T A ~ x = A ~ T b ~ . \tilde{A}^{T}\tilde{A}x=\tilde{A}^{T}\tilde{b}.
  28. A = A ~ T A ~ A=\tilde{A}^{T}\tilde{A}
  29. b = A ~ T b ~ b=\tilde{A}^{T}\tilde{b}
  30. x ( k + 1 ) = x ( k ) - t F ( x ( k ) ) = x ( k ) - t ( A x ( k ) - b ) x^{(k+1)}=x^{(k)}-t\nabla F(x^{(k)})=x^{(k)}-t(Ax^{(k)}-b)
  31. t = ω t=\omega

Modular_multiplicative_inverse.html

  1. a x 1 ( mod m ) . a\,x\equiv 1\;\;(\mathop{{\rm mod}}m).
  2. m \mathbb{Z}_{m}
  3. a - 1 a^{-1}
  4. x 3 - 1 ( mod 11 ) x\equiv 3^{-1}\;\;(\mathop{{\rm mod}}11)
  5. 3 x 1 ( mod 11 ) 3x\equiv 1\;\;(\mathop{{\rm mod}}11)
  6. 11 \mathbb{Z}_{11}
  7. 3 ( 4 ) = 12 1 ( mod 11 ) 3(4)=12\equiv 1\;\;(\mathop{{\rm mod}}11)
  8. 11 \mathbb{Z}_{11}
  9. 11 \mathbb{Z}_{11}
  10. \mathbb{Z}
  11. 4 + ( 11 z ) , z 4+(11\cdot z),z\in\mathbb{Z}
  12. a x + b y = gcd ( a , b ) ax+by=\gcd(a,b)\,
  13. a x 1 ( mod m ) , ax\equiv 1\;\;(\mathop{{\rm mod}}m),
  14. a x - 1 = q m . ax-1=qm.\,
  15. a x - q m = 1 , ax-qm=1,\,
  16. a φ ( m ) 1 ( mod m ) a^{\varphi(m)}\equiv 1\;\;(\mathop{{\rm mod}}m)
  17. a φ ( m ) - 1 a - 1 ( mod m ) . a^{\varphi(m)-1}\equiv a^{-1}\;\;(\mathop{{\rm mod}}m).
  18. a - 1 a m - 2 ( mod m ) . a^{-1}\equiv a^{m-2}\;\;(\mathop{{\rm mod}}m).

Modulation_error_ratio.html

  1. MER ( dB ) = 10 log 10 ( P signal P error ) \mathrm{MER(dB)}=10\log_{10}\left({P_{\mathrm{signal}}\over P_{\mathrm{error}}% }\right)
  2. MER ( % ) = P error P signal × 100 % \mathrm{MER(\%)}=\sqrt{{P_{\mathrm{error}}\over P_{\mathrm{signal}}}}\times 100\%

Moisture_advection.html

  1. A d v ( ρ m ) = - 𝐕 ρ m Adv(\rho_{m})=-\mathbf{V}\cdot\nabla\rho_{m}\!
  2. ρ m \rho_{m}
  3. A d v ( T d ) = - 𝐕 T d Adv(T_{d})=-\mathbf{V}\cdot\nabla T_{d}\!
  4. 𝐟 = q 𝐕 \mathbf{f}=q\mathbf{V}\!
  5. 𝐅 = 0 ρ 𝐟 d z = - P 0 𝐟 g d p \mathbf{F}=\int_{0}^{\infty}\!\rho\mathbf{f}\,dz\,=-\int_{P}^{0}\!\frac{% \mathbf{f}}{g}\,dp\,
  6. ρ \rho
  7. P - E - ( 0 ρ q d z ) t = - 𝐅 P-E-\frac{\partial(\int_{0}^{\infty}\!\rho q\,dz\,)}{\partial t}=-\nabla\cdot% \mathbf{F}\!

Molar_conductivity.html

  1. Λ m = κ c \Lambda_{\mathrm{m}}=\frac{\kappa}{c}
  2. Λ m = Λ m - K c \Lambda_{\mathrm{m}}=\Lambda_{\mathrm{m}}^{\circ}-K\sqrt{c}
  3. Λ m \Lambda_{\mathrm{m}}^{\circ}
  4. Λ m = Σ i ν i λ i \Lambda_{\mathrm{m}}^{\circ}=\Sigma_{i}\nu_{i}\lambda_{i}
  5. λ i \lambda_{i}
  6. ν i \nu_{i}

Molecular_binding.html

  1. A + B A B : log K I = log ( [ A B ] [ A ] [ B ] ) = p K I A+B\rightleftharpoons AB:\log K_{I}=\log\left(\frac{[AB]}{[A][B]}\right)=pK_{I}

Molecular_replacement.html

  1. ρ ( x , y , z ) = 1 V h k l | F h k l | exp ( 2 π i ( h x + k y + l z ) + i Φ ( h k l ) ) . \rho(x,y,z)=\frac{1}{V}\sum_{h}\sum_{k}\sum_{l}|F_{hkl}|\exp(2\pi i(hx+ky+lz)+% i\Phi(hkl)).
  2. I = F F * I=F\cdot F^{*}
  3. Φ \Phi

Molecular_symmetry.html

  1. 360 n \tfrac{360^{\circ}}{n}
  2. 360 n \tfrac{360^{\circ}}{n}
  3. [ - 1 0 0 0 - 1 0 0 0 1 ] C 2 × [ 1 0 0 0 - 1 0 0 0 1 ] σ v = [ - 1 0 0 0 1 0 0 0 1 ] σ v \underbrace{\begin{bmatrix}-1&0&0\\ 0&-1&0\\ 0&0&1\\ \end{bmatrix}}_{C_{2}}\times\underbrace{\begin{bmatrix}1&0&0\\ 0&-1&0\\ 0&0&1\\ \end{bmatrix}}_{\sigma_{v}}=\underbrace{\begin{bmatrix}-1&0&0\\ 0&1&0\\ 0&0&1\\ \end{bmatrix}}_{\sigma^{\prime}_{v}}

Molecular_vibration.html

  1. Q s 1 = q 1 + q 2 + q 3 + q 4 Q_{s1}=q_{1}+q_{2}+q_{3}+q_{4}\!
  2. Q s 2 = q 1 + q 2 - q 3 - q 4 Q_{s2}=q_{1}+q_{2}-q_{3}-q_{4}\!
  3. Q s 3 = q 1 - q 2 + q 3 - q 4 Q_{s3}=q_{1}-q_{2}+q_{3}-q_{4}\!
  4. Q s 4 = q 1 - q 2 - q 3 + q 4 Q_{s4}=q_{1}-q_{2}-q_{3}+q_{4}\!
  5. q 1 - q 4 q_{1}-q_{4}
  6. Force = - k Q \mathrm{Force}=-kQ\!
  7. Force = μ d 2 Q d t 2 \mathrm{Force}=\mu\frac{d^{2}Q}{dt^{2}}
  8. μ d 2 Q d t 2 + k Q = 0 \mu\frac{d^{2}Q}{dt^{2}}+kQ=0
  9. Q ( t ) = A cos ( 2 π ν t ) ; ν = 1 2 π k μ . Q(t)=A\cos(2\pi\nu t);\ \ \nu={1\over{2\pi}}\sqrt{k\over\mu}.\!
  10. 1 μ = 1 m A + 1 m B . \frac{1}{\mu}=\frac{1}{m_{A}}+\frac{1}{m_{B}}.
  11. k = 2 V Q 2 k=\frac{\partial^{2}V}{\partial Q^{2}}
  12. E n = h ( n + 1 2 ) ν = h ( n + 1 2 ) 1 2 π k m E_{n}=h\left(n+{1\over 2}\right)\nu=h\left(n+{1\over 2}\right){1\over{2\pi}}% \sqrt{k\over m}\!
  13. h ν h\nu
  14. ν \nu
  15. Δ n = ± 1 \Delta n=\pm 1

Monetary_inflation.html

  1. M V = P T MV=PT

Moni_Naor.html

  1. ϵ \epsilon
  2. δ \delta

Monodromy_theorem.html

  1. γ : [ 0 , 1 ] . \gamma:[0,1]\to\mathbb{C}.
  2. f f
  3. U U
  4. γ ( 0 ) . \gamma(0).
  5. ( f , U ) (f,U)
  6. γ \gamma
  7. ( f t , U t ) (f_{t},U_{t})
  8. 0 t 1 0\leq t\leq 1
  9. f 0 = f f_{0}=f
  10. U 0 = U U_{0}=U
  11. t [ 0 , 1 ] , t\in[0,1],
  12. U t U_{t}
  13. γ ( t ) \gamma(t)
  14. f t : U t f_{t}:U_{t}\to\mathbb{C}
  15. t [ 0 , 1 ] t\in[0,1]
  16. ε > 0 \varepsilon>0
  17. t [ 0 , 1 ] t^{\prime}\in[0,1]
  18. | t - t | < ε |t-t^{\prime}|<\varepsilon
  19. γ ( t ) U t \gamma(t^{\prime})\in U_{t}
  20. U t U_{t}
  21. U t U_{t^{\prime}}
  22. f t f_{t}
  23. f t f_{t^{\prime}}
  24. U t U t . U_{t}\cap U_{t^{\prime}}.
  25. ( f t , U t ) (f_{t},U_{t})
  26. ( g t , V t ) (g_{t},V_{t})
  27. ( 0 t 1 ) (0\leq t\leq 1)
  28. ( f , U ) (f,U)
  29. γ , \gamma,
  30. f 1 f_{1}
  31. g 1 g_{1}
  32. U 1 V 1 . U_{1}\cap V_{1}.
  33. ( f , U ) (f,U)
  34. γ \gamma
  35. γ ( 1 ) . \gamma(1).
  36. γ \gamma
  37. γ ( 0 ) = γ ( 1 ) \gamma(0)=\gamma(1)
  38. f 0 f_{0}
  39. f 1 f_{1}
  40. γ ( 0 ) . \gamma(0).
  41. ( a , 0 ) (a,0)
  42. a > 0 a>0
  43. γ \gamma
  44. a a
  45. ( a , 0 ) (a,0)
  46. ( a , 0 ) (a,0)
  47. 2 π i 2\pi i
  48. ( a , 0 ) (a,0)
  49. a . a.
  50. ( a , 0 ) (a,0)
  51. ( - a , 0 ) (-a,0)
  52. ( - a , 0 ) (-a,0)
  53. 2 π i . 2\pi i.
  54. U U
  55. P P
  56. f : U f:U\to\mathbb{C}
  57. Q Q
  58. γ s : [ 0 , 1 ] \gamma_{s}:[0,1]\to\mathbb{C}
  59. s [ 0 , 1 ] s\in[0,1]
  60. γ s ( 0 ) = P \gamma_{s}(0)=P
  61. γ s ( 1 ) = Q \gamma_{s}(1)=Q
  62. s [ 0 , 1 ] , s\in[0,1],
  63. ( s , t ) [ 0 , 1 ] × [ 0 , 1 ] γ s ( t ) (s,t)\in[0,1]\times[0,1]\to\gamma_{s}(t)\in\mathbb{C}
  64. s [ 0 , 1 ] s\in[0,1]
  65. f f
  66. γ s , \gamma_{s},
  67. f f
  68. γ 0 \gamma_{0}
  69. γ 1 \gamma_{1}
  70. Q . Q.
  71. U U
  72. P P
  73. f : U f:U\to\mathbb{C}
  74. W W
  75. U U
  76. f f
  77. W W
  78. P , P,
  79. f f
  80. W , W,
  81. g : W g:W\to\mathbb{C}
  82. U U
  83. f . f.

Monoidal_t-norm_logic.html

  1. * *
  2. \Rightarrow
  3. x * y z x*y\leq z
  4. x ( y z ) . x\leq(y\Rightarrow z).
  5. ( x y ) = sup { z z * x y } . (x\Rightarrow y)=\sup\{z\mid z*x\leq y\}.
  6. x * ( x y ) y . x*(x\Rightarrow y)\leq y.
  7. ¬ x = ( x 0 ) . \neg x=(x\Rightarrow 0).
  8. * , *,
  9. * - *\mbox{-}~{}
  10. * - *\mbox{-}~{}
  11. * , *,
  12. \rightarrow
  13. \otimes
  14. \otimes
  15. \wedge
  16. \bot
  17. 0
  18. 0 ¯ \overline{0}
  19. ¬ \neg
  20. ¬ A A \neg A\equiv A\rightarrow\bot
  21. \leftrightarrow
  22. A B ( A B ) ( B A ) A\leftrightarrow B\equiv(A\rightarrow B)\wedge(B\rightarrow A)
  23. ( A B ) ( B A ) . (A\rightarrow B)\otimes(B\rightarrow A).
  24. \vee
  25. A B ( ( A B ) B ) ( ( B A ) A ) A\vee B\equiv((A\rightarrow B)\rightarrow B)\wedge((B\rightarrow A)\rightarrow A)
  26. \top
  27. 1 1
  28. 1 ¯ \overline{1}
  29. \top\equiv\bot\rightarrow\bot
  30. A A
  31. A B A\rightarrow B
  32. B . B.
  33. ( MTL1 ) : ( A B ) ( ( B C ) ( A C ) ) ( MTL2 ) : A B A ( MTL3 ) : A B B A ( MTL4a ) : A B A ( MTL4b ) : A B B A ( MTL4c ) : A ( A B ) A B ( MTL5a ) : ( A ( B C ) ) ( A B C ) ( MTL5b ) : ( A B C ) ( A ( B C ) ) ( MTL6 ) : ( ( A B ) C ) ( ( ( B A ) C ) C ) ( MTL7 ) : A \begin{array}[]{ll}{\rm(MTL1)}\colon&(A\rightarrow B)\rightarrow((B\rightarrow C% )\rightarrow(A\rightarrow C))\\ {\rm(MTL2)}\colon&A\otimes B\rightarrow A\\ {\rm(MTL3)}\colon&A\otimes B\rightarrow B\otimes A\\ {\rm(MTL4a)}\colon&A\wedge B\rightarrow A\\ {\rm(MTL4b)}\colon&A\wedge B\rightarrow B\wedge A\\ {\rm(MTL4c)}\colon&A\otimes(A\rightarrow B)\rightarrow A\wedge B\\ {\rm(MTL5a)}\colon&(A\rightarrow(B\rightarrow C))\rightarrow(A\otimes B% \rightarrow C)\\ {\rm(MTL5b)}\colon&(A\otimes B\rightarrow C)\rightarrow(A\rightarrow(B% \rightarrow C))\\ {\rm(MTL6)}\colon&((A\rightarrow B)\rightarrow C)\rightarrow(((B\rightarrow A)% \rightarrow C)\rightarrow C)\\ {\rm(MTL7)}\colon&\bot\rightarrow A\end{array}
  34. ( L , , , , , 0 , 1 ) (L,\wedge,\vee,\ast,\Rightarrow,0,1)
  35. ( L , , , 0 , 1 ) (L,\wedge,\vee,0,1)
  36. ( L , , 1 ) (L,\ast,1)
  37. \ast
  38. \Rightarrow
  39. z * x y z*x\leq y
  40. z x y , z\leq x\Rightarrow y,
  41. \leq
  42. ( L , , ) , (L,\wedge,\vee),
  43. L L
  44. ( x y ) ( y x ) = 1 (x\Rightarrow y)\vee(y\Rightarrow x)=1
  45. = \ast=\wedge
  46. \ast
  47. \Rightarrow
  48. \ast
  49. \wedge
  50. , \vee,
  51. \Leftrightarrow
  52. x y ( x y ) ( y x ) x\Leftrightarrow y\equiv(x\Rightarrow y)\wedge(y\Rightarrow x)
  53. \ast
  54. , \wedge,
  55. x y ( x y ) ( y x ) x\Leftrightarrow y\equiv(x\Rightarrow y)\ast(y\Rightarrow x)
  56. - x x 0 -x\equiv x\Rightarrow 0
  57. e ( p ) = e v ( p ) e ( ) = 0 e ( ) = 1 e ( A B ) = e ( A ) e ( B ) e ( A B ) = e ( A ) e ( B ) e ( A B ) = e ( A ) e ( B ) e ( A B ) = e ( A ) e ( B ) e ( A B ) = e ( A ) e ( B ) e ( ¬ A ) = e ( A ) 0 \begin{array}[]{rcl}e(p)&=&e_{\mathrm{v}}(p)\\ e(\bot)&=&0\\ e(\top)&=&1\\ e(A\otimes B)&=&e(A)\ast e(B)\\ e(A\rightarrow B)&=&e(A)\Rightarrow e(B)\\ e(A\wedge B)&=&e(A)\wedge e(B)\\ e(A\vee B)&=&e(A)\vee e(B)\\ e(A\leftrightarrow B)&=&e(A)\Leftrightarrow e(B)\\ e(\neg A)&=&e(A)\Rightarrow 0\end{array}
  58. Γ A , \Gamma\models A,
  59. \ast
  60. \ast
  61. [ 0 , 1 ] . [0,1]_{\ast}.
  62. [ 0 , 1 ] , [0,1]_{\ast},
  63. , \ast,

Monotone_cubic_interpolation.html

  1. m i m_{i}
  2. ( x k , y k ) (x_{k},y_{k})
  3. k = 1 , , n k=1,...,n
  4. Δ k = y k + 1 - y k x k + 1 - x k \Delta_{k}=\frac{y_{k+1}-y_{k}}{x_{k+1}-x_{k}}
  5. k = 1 , , n - 1 k=1,\dots,n-1
  6. m k = Δ k - 1 + Δ k 2 m_{k}=\frac{\Delta_{k-1}+\Delta_{k}}{2}
  7. k = 2 , , n - 1 k=2,\dots,n-1
  8. Δ k - 1 \Delta_{k-1}
  9. Δ k \Delta_{k}
  10. m k = 0 m_{k}=0
  11. m 1 = Δ 1 and m n = Δ n - 1 m_{1}=\Delta_{1}\quad\,\text{and}\quad m_{n}=\Delta_{n-1}
  12. k = 1 , , n - 1 k=1,\dots,n-1
  13. Δ k = 0 \Delta_{k}=0
  14. y k = y k + 1 y_{k}=y_{k+1}
  15. m k = m k + 1 = 0 , m_{k}=m_{k+1}=0,
  16. k k
  17. α k = m k / Δ k \alpha_{k}=m_{k}/\Delta_{k}
  18. β k = m k + 1 / Δ k \beta_{k}=m_{k+1}/\Delta_{k}
  19. α k \alpha_{k}
  20. β k - 1 \beta_{k-1}
  21. ( x k , y k ) (x_{k},y_{k})
  22. m k = 0 m_{k}=0
  23. ϕ ( α , β ) = α - ( 2 α + β - 3 ) 2 3 ( α + β - 2 ) \phi(\alpha,\beta)=\alpha-\frac{(2\alpha+\beta-3)^{2}}{3(\alpha+\beta-2)}
  24. α + 2 β - 3 0 \alpha+2\beta-3\leq 0
  25. 2 α + β - 3 0 2\alpha+\beta-3\leq 0
  26. ϕ ( α , β ) \phi(\alpha,\beta)
  27. ( α k , β k ) (\alpha_{k},\beta_{k})
  28. α k 2 + β k 2 > 9 \alpha_{k}^{2}+\beta_{k}^{2}>9
  29. m k = τ k α k Δ k m_{k}=\tau_{k}\alpha_{k}\Delta_{k}
  30. m k + 1 = τ k β k Δ k m_{k+1}=\tau_{k}\beta_{k}\Delta_{k}
  31. τ k = 3 α k 2 + β k 2 \tau_{k}=\frac{3}{\sqrt{\alpha_{k}^{2}+\beta_{k}^{2}}}
  32. α k 3 \alpha_{k}\leq 3
  33. β k 3 \beta_{k}\leq 3
  34. α k > 3 \alpha_{k}>3
  35. m k = 3 × Δ k m_{k}=3\times\Delta_{k}
  36. β \beta
  37. x k x_{k}
  38. y k y_{k}
  39. m k m_{k}
  40. k = 1 , , n k=1,...,n
  41. x x
  42. x x
  43. x upper x\text{upper}
  44. x x
  45. x lower x\text{lower}
  46. x k x_{k}
  47. x lower x x upper x\text{lower}\leq x\leq x\text{upper}
  48. h = x upper - x lower h=x\text{upper}-x\text{lower}
  49. t = x - x lower h t=\frac{x-x\text{lower}}{h}
  50. f interpolated ( x ) = y lower h 00 ( t ) + h m lower h 10 ( t ) + y upper h 01 ( t ) + h m upper h 11 ( t ) f\text{interpolated}(x)=y\text{lower}h_{00}(t)+hm\text{lower}h_{10}(t)+y\text{% upper}h_{01}(t)+hm\text{upper}h_{11}(t)
  51. h i i h_{ii}

Monotonically_normal_space.html

  1. T 1 T_{1}
  2. ( X , 𝒯 ) (X,\mathcal{T})
  3. x G x\in G
  4. μ ( x , G ) \mu(x,G)
  5. x μ ( x , G ) G x\in\mu(x,G)\subseteq G
  6. μ ( x , G ) μ ( y , H ) \mu(x,G)\cap\mu(y,H)\neq\emptyset
  7. x H x\in H
  8. y G y\in G
  9. T 1 T_{1}
  10. A , B A,B
  11. G ( A , B ) G(A,B)
  12. A G ( A , B ) G ( A , B ) - X \ B A\subseteq G(A,B)\subseteq G(A,B)^{-}\subseteq X\backslash B
  13. G ( A , B ) G ( A , B ) G(A,B)\subseteq G(A^{\prime},B^{\prime})
  14. A A A\subseteq A^{\prime}
  15. B B B^{\prime}\subseteq B
  16. G G
  17. G ~ \tilde{G}
  18. G ~ ( A , B ) = G ( A , B ) \ G ( B , A ) - \tilde{G}(A,B)=G(A,B)\backslash G(B,A)^{-}
  19. G ~ \tilde{G}
  20. G ~ ( A , B ) G ~ ( B , A ) = \begin{aligned}\displaystyle\tilde{G}(A,B)\cap\tilde{G}(B,A)=\emptyset\end{aligned}
  21. T 1 T_{1}
  22. A B - = = B A - A\cap B^{-}=\emptyset=B\cap A^{-}
  23. A G ( A , B ) G ( A , B ) - X \ B , A\subseteq G(A,B)\subseteq G(A,B)^{-}\subseteq X\backslash B,
  24. G ( A , B ) G ( A , B ) whenever A A and B B G(A,B)\subseteq G(A^{\prime},B^{\prime})\mbox{ whenever }~{}A\subseteq A^{% \prime}\mbox{ and }~{}B^{\prime}\subseteq B
  25. T 1 T_{1}
  26. p H ( p , C ) X \ C p\in H(p,C)\subseteq X\backslash C
  27. p C D p\not\in C\supseteq D
  28. H ( p , C ) H ( p , D ) H(p,C)\subseteq H(p,D)
  29. p q p\neq q
  30. H ( p , { q } ) H ( q , { p } ) = H(p,\{q\})\cap H(q,\{p\})=\emptyset

Monte_Carlo_method_in_statistical_physics.html

  1. A = P S A r e - β E r Z d r \langle A\rangle=\int_{PS}A_{\vec{r}}\frac{e^{-\beta E_{\vec{r}}}}{Z}d\vec{r}
  2. E ( r ) = E r E(\vec{r})=E_{\vec{r}}
  3. r \vec{r}
  4. r = ( q , p ) \vec{r}=\left(\vec{q},\vec{p}\right)
  5. β 1 / k b T \beta\equiv 1/k_{b}T
  6. Z = P S P ( r ) d r Z=\int_{PS}P(\vec{r})d\vec{r}
  7. 1 / N 1/\sqrt{N}
  8. A = P S A r e - β E r d r / Z \langle A\rangle=\int_{PS}A_{\vec{r}}e^{-\beta E_{\vec{r}}}d\vec{r}/Z
  9. A 1 N i = 1 N A r i e - β E r i / Z \langle A\rangle\simeq\frac{1}{N}\sum_{i=1}^{N}A_{\vec{r}_{i}}e^{-\beta E_{% \vec{r}_{i}}}/Z
  10. r i \vec{r}_{i}
  11. A A
  12. e - β E r i e^{-\beta E_{\vec{r}_{i}}}
  13. p ( r ) p(\vec{r})
  14. A A
  15. A = P S p - 1 ( r ) A r p - 1 ( r ) e - β E r / Z d r = P S p - 1 ( r ) A r * e - β E r / Z d r \langle A\rangle=\int_{PS}p^{-1}(\vec{r})\frac{A_{\vec{r}}}{p^{-1}(\vec{r})}e^% {-\beta E_{\vec{r}}}/Zd\vec{r}=\int_{PS}p^{-1}(\vec{r})A^{*}_{\vec{r}}e^{-% \beta E_{\vec{r}}}/Zd\vec{r}
  16. A r * A^{*}_{\vec{r}}
  17. p ( r ) p(\vec{r})
  18. A 1 N i = 1 N p - 1 ( r i ) A r i * e - β E r i / Z \langle A\rangle\simeq\frac{1}{N}\sum_{i=1}^{N}p^{-1}(\vec{r}_{i})A^{*}_{\vec{% r}_{i}}e^{-\beta E_{\vec{r}_{i}}}/Z
  19. r i \vec{r}_{i}
  20. p ( r ) p(\vec{r})
  21. p ( r ) p(\vec{r})
  22. p ( r ) = e - β E r Z p(\vec{r})=\frac{e^{-\beta E_{\vec{r}}}}{Z}
  23. A 1 N i = 1 N A r i * \langle A\rangle\simeq\frac{1}{N}\sum_{i=1}^{N}A^{*}_{\vec{r}_{i}}
  24. p ( r ) p(\vec{r})
  25. A r * A^{*}_{\vec{r}}
  26. r i \vec{r}_{i}
  27. p ( r ) = 1 Ω ( E r ) p(\vec{r})=\frac{1}{\Omega(E_{\vec{r}})}
  28. Ω ( E ) \Omega(E)
  29. Ω ( E ) \Omega(E)
  30. β \beta
  31. N = L 2 N=L^{2}
  32. r = ( σ 1 , σ 2 , , σ N ) \vec{r}=(\sigma_{1},\sigma_{2},...,\sigma_{N})
  33. σ i { - 1 , 1 } \sigma_{i}\in\{-1,1\}
  34. E ( r ) = i = 1 N j v i z i ( 1 - J i j σ i σ j ) E(\vec{r})=\sum_{i=1}^{N}\sum_{j\in viz_{i}}(1-J_{ij}\sigma_{i}\sigma_{j})
  35. v i z i viz_{i}
  36. M \langle M\rangle
  37. M 2 \langle M^{2}\rangle
  38. M ( r ) = i = 1 N σ i M(\vec{r})=\sum_{i=1}^{N}\sigma_{i}
  39. β = 1 / k b T \beta=1/k_{b}T
  40. p ( r ) p(\vec{r})
  41. σ i \sigma_{i}
  42. α [ 0 , 1 ] \alpha\in[0,1]
  43. Δ E = 2 σ i j v i z i σ j \Delta E=2\sigma_{i}\sum_{j\in viz_{i}}\sigma_{j}
  44. Δ M = - 2 σ i \Delta M=-2\sigma_{i}
  45. α < min ( 1 , e - β Δ E ) \alpha<\min(1,e^{-\beta\Delta E})
  46. σ i = - σ i \sigma_{i}=-\sigma_{i}
  47. E = E + Δ E E=E+\Delta E
  48. M = M + Δ M M=M+\Delta M
  49. N 2 + z N^{2+z}

Moore_reduction_procedure.html

  1. n + 1 n+1

Morrie's_law.html

  1. cos ( 20 ) cos ( 40 ) cos ( 80 ) = 1 8 . \cos(20^{\circ})\cdot\cos(40^{\circ})\cdot\cos(80^{\circ})=\frac{1}{8}.
  2. 2 n k = 0 n - 1 cos ( 2 k α ) = sin ( 2 n α ) sin ( α ) 2^{n}\cdot\prod_{k=0}^{n-1}\cos(2^{k}\alpha)=\frac{\sin(2^{n}\alpha)}{\sin(% \alpha)}
  3. sin ( 160 ) sin ( 20 ) = sin ( 180 - 20 ) sin ( 20 ) = 1 \frac{\sin(160^{\circ})}{\sin(20^{\circ})}=\frac{\sin(180^{\circ}-20^{\circ})}% {\sin(20^{\circ})}=1
  4. sin ( 180 - x ) = sin ( x ) . \sin(180^{\circ}-x)=\sin(x).
  5. sin ( 20 ) sin ( 40 ) sin ( 80 ) = 3 8 . \sin(20^{\circ})\cdot\sin(40^{\circ})\cdot\sin(80^{\circ})=\frac{\sqrt{3}\ }{8}.
  6. tan ( 20 ) tan ( 40 ) tan ( 80 ) = 3 = tan ( 60 ) . \tan(20^{\circ})\cdot\tan(40^{\circ})\cdot\tan(80^{\circ})=\sqrt{3}=\tan(60^{% \circ}).\,
  7. sin ( 2 α ) = 2 sin ( α ) cos ( α ) . \sin(2\alpha)=2\sin(\alpha)\cos(\alpha).\,
  8. cos ( α ) \cos(\alpha)
  9. cos ( α ) = sin ( 2 α ) 2 sin ( α ) . \cos(\alpha)=\frac{\sin(2\alpha)}{2\sin(\alpha)}.
  10. cos ( 2 α ) \displaystyle\cos(2\alpha)
  11. cos ( α ) cos ( 2 α ) cos ( 4 α ) cos ( 2 n - 1 α ) = sin ( 2 α ) 2 sin ( α ) sin ( 4 α ) 2 sin ( 2 α ) sin ( 8 α ) 2 sin ( 4 α ) sin ( 2 n α ) 2 sin ( 2 n - 1 α ) . \cos(\alpha)\cos(2\alpha)\cos(4\alpha)\cdots\cos(2^{n-1}\alpha)=\frac{\sin(2% \alpha)}{2\sin(\alpha)}\cdot\frac{\sin(4\alpha)}{2\sin(2\alpha)}\cdot\frac{% \sin(8\alpha)}{2\sin(4\alpha)}\cdots\frac{\sin(2^{n}\alpha)}{2\sin(2^{n-1}% \alpha)}.
  12. k = 0 n - 1 cos ( 2 k α ) = sin ( 2 n α ) 2 n sin ( α ) , \prod_{k=0}^{n-1}\cos(2^{k}\alpha)=\frac{\sin(2^{n}\alpha)}{2^{n}\sin(\alpha)},

Morse–Palais_lemma.html

  1. H x D 2 f ( 0 ) ( x , - ) H * . H\ni x\mapsto\mathrm{D}^{2}f(0)(x,-)\in H^{*}.\,
  2. f ( x ) = A φ ( x ) , φ ( x ) f(x)=\langle A\varphi(x),\varphi(x)\rangle
  3. H = G G , H=G\oplus G^{\perp},
  4. ψ ( x ) = y + z with y G , z G , \psi(x)=y+z\mbox{ with }~{}y\in G,z\in G^{\perp},
  5. f ( ψ ( x ) ) = y , y - z , z f(\psi(x))=\langle y,y\rangle-\langle z,z\rangle

Mortgage_loan.html

  1. A = P r ( 1 + r ) n ( 1 + r ) n - 1 A=P\cdot\frac{r(1+r)^{n}}{(1+r)^{n}-1}
  2. A A
  3. P P
  4. r r
  5. n n

Morton_number.html

  1. Mo = g μ c 4 Δ ρ ρ c 2 σ 3 , \mathrm{Mo}=\frac{g\mu_{c}^{4}\,\Delta\rho}{\rho_{c}^{2}\sigma^{3}},
  2. μ c \mu_{c}
  3. ρ c \rho_{c}
  4. Δ ρ \Delta\rho
  5. σ \sigma
  6. Mo = g μ c 4 ρ c σ 3 . \mathrm{Mo}=\frac{g\mu_{c}^{4}}{\rho_{c}\sigma^{3}}.
  7. Mo = We 3 Fr Re 4 . \mathrm{Mo}=\frac{\mathrm{We}^{3}}{\mathrm{Fr}\,\mathrm{Re}^{4}}.
  8. Fr = V 2 g d \mathrm{Fr}=\frac{V^{2}}{gd}

Mosco_convergence.html

  1. lim inf n F n ( x n ) F ( x ) ; \liminf_{n\to\infty}F_{n}(x_{n})\geq F(x);
  2. lim sup n F n ( x n ) F ( x ) . \limsup_{n\to\infty}F_{n}(x_{n})\leq F(x).
  3. M-lim n F n = F or F n n M F . \mathop{\,\text{M-lim}}_{n\to\infty}F_{n}=F\,\text{ or }F_{n}\xrightarrow[n\to% \infty]{\mathrm{M}}F.

Moulton_plane.html

  1. 𝔐 = P , G , I \mathfrak{M}=\langle P,G,\textrm{I}\rangle
  2. P P
  3. G G
  4. $\textrm I$
  5. P := 2 P:=\mathbb{R}^{2}\,
  6. G := ( { } ) × , G:=(\mathbb{R}\cup\{\infty\})\times\mathbb{R},
  7. \infty
  8. \not\in\mathbb{R}
  9. p I g { x = b if m = y = 1 2 m x + b if m 0 , x 0 y = m x + b if m 0 or x 0. p\,\textrm{I}\,g\iff\begin{cases}x=b&\,\text{if }m=\infty\\ y=\frac{1}{2}mx+b&\,\text{if }m\leq 0,x\leq 0\\ y=mx+b&\,\text{if }m\geq 0\,\text{ or }x\geq 0.\end{cases}
  10. P G ( 2 , F ) PG(2,F)
  11. P G ( 2 , F ) PG(2,F)
  12. P ( F 3 ) P(F^{3})

Moving_shock.html

  1. u y = W + u 1 - u 2 , \ u_{y}=W+u_{1}-u_{2},
  2. u x = W . \ u_{x}=W.
  3. p 1 = p x ; p 2 = p y ; T 1 = T x ; T 2 = T y , \ p_{1}=p_{x}\quad;\quad p_{2}=p_{y}\quad;\quad T_{1}=T_{x}\quad;\quad T_{2}=T% _{y},
  4. ρ 1 = ρ x ; ρ 2 = ρ y ; a 1 = a x ; a 2 = a y , \ \rho_{1}=\rho_{x}\quad;\quad\rho_{2}=\rho_{y}\quad;\quad a_{1}=a_{x}\quad;% \quad a_{2}=a_{y},
  5. M x = u x a x = W a 1 , \ M_{x}=\frac{u_{x}}{a_{x}}=\frac{W}{a_{1}},
  6. M y = u y a y = W + u 1 - u 2 a 2 . \ M_{y}=\frac{u_{y}}{a_{y}}=\frac{W+u_{1}-u_{2}}{a_{2}}.
  7. a 2 a 1 = 1 + 2 ( γ - 1 ) ( γ + 1 ) 2 [ γ M x 2 - 1 M x 2 - ( γ - 1 ) ] , \ \frac{a_{2}}{a_{1}}=\sqrt{1+\frac{2(\gamma-1)}{(\gamma+1)^{2}}\left[\gamma M% _{x}^{2}-\frac{1}{M_{x}^{2}}-(\gamma-1)\right]},
  8. ρ 2 ρ 1 = 1 1 - 2 γ + 1 [ 1 - 1 M x 2 ] , \ \frac{\rho_{2}}{\rho_{1}}=\frac{1}{1-\frac{2}{\gamma+1}\left[1-\frac{1}{M_{x% }^{2}}\right]},
  9. p 2 p 1 = 1 + 2 γ γ + 1 [ M x 2 - 1 ] . \ \frac{p_{2}}{p_{1}}=1+\frac{2\gamma}{\gamma+1}\left[M_{x}^{2}-1\right].
  10. u y = W - u 1 + u 2 , \ u_{y}=W-u_{1}+u_{2},
  11. M y = W - u 1 + u 2 a 2 . \ M_{y}=\frac{W-u_{1}+u_{2}}{a_{2}}.

Moving_sofa_problem.html

  1. A π / 2 1.570796327 \scriptstyle A\,\geq\,\pi/2\,\approx\,1.570796327
  2. A π / 2 + 2 / π 2.207416099 \scriptstyle A\,\geq\,\pi/2+2/\pi\,\approx\,2.207416099
  3. 2 / π \scriptstyle 2/\pi\,
  4. 2 2 2.8284 \scriptstyle 2\sqrt{2}\,\approx\,2.8284

Moyal_bracket.html

  1. ħ ħ
  2. ħ 0 ħ→0
  3. { { f , g } } \displaystyle\{\{f,g\}\}
  4. f f
  5. g g
  6. { { f , g } } ( x , p ) = 2 3 π 2 d p d p ′′ d x d x ′′ f ( x + x , p + p ) g ( x + x ′′ , p + p ′′ ) sin ( 2 ( x p ′′ - x ′′ p ) ) . \{\{f,g\}\}(x,p)={2\over\hbar^{3}\pi^{2}}\int dp^{\prime}\,dp^{\prime\prime}\,% dx^{\prime}\,dx^{\prime\prime}f(x+x^{\prime},p+p^{\prime})g(x+x^{\prime\prime}% ,p+p^{\prime\prime})\sin\left(\tfrac{2}{\hbar}(x^{\prime}p^{\prime\prime}-x^{% \prime\prime}p^{\prime})\right)~{}.
  7. f f
  8. g g
  9. [ T f , T g ] = T i { { f , g } } . [T_{f}~{},T_{g}]=T_{i\hbar\{\{f,g\}\}}.
  10. x x
  11. p p
  12. 0 , 22 π 0,22π
  13. [ T m 1 , m 2 , T n 1 , n 2 ] = 2 i sin ( 2 ( n 1 m 2 - n 2 m 1 ) ) T m 1 + n 1 , m 2 + n 2 , [T_{m_{1},m_{2}}~{},T_{n_{1},n_{2}}]=2i\sin\left(\tfrac{\hbar}{2}(n_{1}m_{2}-n% _{2}m_{1})\right)~{}T_{m_{1}+n_{1},m_{2}+n_{2}},~{}
  14. N 4 π / ħ N≡ 4π/ħ
  15. { { { f , g } } } \displaystyle\{\{\{f,g\}\}\}
  16. f f
  17. g g
  18. f g fg
  19. ħ ħ
  20. ħ ħ
  21. f ( l o g < s m a l l > ) g f(log<small>★)g

Möbius_aromaticity.html

  1. L x L_{x}
  2. L z L_{z}
  3. ψ \psi
  4. ψ ( x , 0 ) = ψ ( x , L z ) \psi(x,0)=\psi(x,L_{z})
  5. ψ ( 0 , z ) = ψ ( L x , - z ) \psi(0,z)=\psi(L_{x},-z)
  6. ϕ \phi
  7. ψ ( ϕ ) = - ψ ( ϕ + 2 π ) \psi(\phi)=-\psi(\phi+2\pi)
  8. N N
  9. | ψ λ = j = 0 N - 1 c j λ | φ j = j = 0 N - 1 e i λ ϕ j | φ j = j = 0 N - 1 e < m t p l > i λ 2 π j / N | φ j |{\psi_{\lambda}}\rangle=\sum_{j=0}^{N-1}{{c^{\lambda}_{j}}|{\varphi_{j}}}% \rangle=\sum_{j=0}^{N-1}{e^{i\lambda\phi_{j}}|{\varphi_{j}}}\rangle=\sum_{j=0}% ^{N-1}{e^{<}mtpl>{{i\lambda 2\pi j/N}}|{\varphi_{j}}}\rangle
  10. ϕ j = 2 π j / N \phi_{j}=2\pi j/N
  11. j j
  12. φ j \varphi_{j}
  13. j j
  14. c j + N λ = - c j λ c_{j+N}^{\lambda}=-c_{j}^{\lambda}
  15. λ \lambda
  16. e i λ 2 π ( j + N ) / N = - e i λ 2 π j / N e^{i\lambda 2\pi(j+N)/N}=-e^{i\lambda 2\pi j/N}
  17. e i λ 2 π = - 1 e^{i\lambda 2\pi}=-1
  18. λ k = 2 k + 1 2 k = 0 , 1 , 2 , , ( N - 1 ) \lambda_{k}=\frac{2k+1}{2}\;\;\;k=0,1,2,\ldots,(N-1)
  19. λ \lambda
  20. c j ( k ) = e i π ( 2 k + 1 ) j / N c_{j}^{(k)}=e^{i\pi(2k+1)j/N}
  21. p z p_{z}
  22. ω = π / N \omega=\pi/N
  23. β \beta^{\prime}
  24. β = β cos ( π / N ) \beta^{\prime}=\beta\cos(\pi/N)
  25. β \beta
  26. ω = 0 \omega=0
  27. C 2 C_{2}
  28. N N
  29. 1 1
  30. N N
  31. - β -\beta^{\prime}
  32. N N
  33. 𝐇 \mathbf{H}
  34. 𝐇 = ( α β 0 - β β α β 0 0 β α 0 - β 0 0 α ) \mathbf{H}=\begin{pmatrix}\alpha&\beta&0&\cdots&-\beta\\ \beta&\alpha&\beta&\cdots&0\\ 0&\beta&\alpha&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ -\beta&0&0&\cdots&\alpha\end{pmatrix}
  35. 𝐇 \mathbf{H}
  36. N × N N\times N
  37. N N
  38. E k E_{k}
  39. N N
  40. x k = α - E k β x_{k}=\frac{\alpha-E_{k}}{\beta}
  41. ( x k 1 0 - 1 1 x k 1 0 0 1 x k 0 - 1 0 0 x k ) ( c 1 ( k ) c 2 ( k ) c 3 ( k ) c N ( k ) ) = 0 \begin{pmatrix}x_{k}&1&0&\cdots&-1\\ 1&x_{k}&1&\cdots&0\\ 0&1&x_{k}&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ -1&0&0&\cdots&x_{k}\end{pmatrix}\cdot\begin{pmatrix}c_{1}^{(k)}\\ c_{2}^{(k)}\\ c_{3}^{(k)}\\ \vdots\\ c_{N}^{(k)}\\ \end{pmatrix}=0
  42. N N
  43. k = 0 k=0
  44. k = N - 1 k=N-1
  45. - 1 -1
  46. { x 0 c 1 ( 0 ) + c 2 ( 0 ) - c N ( 0 ) = 0 c j - 1 ( k ) + x k c j ( k ) + c j + 1 ( k ) = 0 c N - 1 ( N - 1 ) + x N - 1 c N ( N - 1 ) - c 1 ( N - 1 ) = 0 \begin{cases}x_{0}c_{1}^{(0)}+c_{2}^{(0)}-c_{N}^{(0)}=0\\ \vdots\\ c_{j-1}^{(k)}+x_{k}c_{j}^{(k)}+c_{j+1}^{(k)}=0\\ \vdots\\ c_{N-1}^{(N-1)}+x_{N-1}c_{N}^{(N-1)}-c_{1}^{(N-1)}=0\end{cases}
  47. x k = - 2 cos ( 2 k + 1 ) π N x_{k}=-2\cos{\frac{(2k+1)\pi}{N}}
  48. E k = α + 2 β cos ( 2 k + 1 ) π N = α + 2 β cos 2 π λ k N E_{k}=\alpha+2\beta^{\prime}\cos{\frac{(2k+1)\pi}{N}}=\alpha+2\beta^{\prime}% \cos{\frac{2\pi\lambda_{k}}{N}}

Muller_automaton.html

  1. B B
  2. Q Q
  3. \emptyset
  4. ( E j , F j ) (E_{j},F_{j})
  5. F 2 Q F\subseteq 2^{Q}
  6. F E j = and F F j F\cap E_{j}=\emptyset\and F\cap F_{j}\neq\emptyset
  7. ( E j , F j ) (E_{j},F_{j})
  8. F 2 Q F\subseteq 2^{Q}
  9. F E j = F F j = F\cap E_{j}=\emptyset\rightarrow F\cap F_{j}=\emptyset

Multi-compartment_model.html

  1. d q d t = u ( t ) - k q \frac{\mathrm{d}q}{\mathrm{d}t}=u(t)-kq
  2. C = q V C=\frac{q}{V}
  3. q ˙ 1 = q 1 k 11 + q 2 k 12 + + q n k 1 n + u 1 ( t ) \displaystyle\dot{q}_{1}=q_{1}k_{11}+q_{2}k_{12}+\cdots+q_{n}k_{1n}+u_{1}(t)
  4. 0 = i = 1 n k i j 0=\sum^{n}_{i=1}{k_{ij}}
  5. j = 1 , 2 , , n j=1,2,\dots,n
  6. 𝐪 ˙ = 𝐊𝐪 + 𝐮 \mathbf{\dot{q}}=\mathbf{Kq}+\mathbf{u}
  7. 𝐊 = [ k 11 k 12 k 1 n k 21 k 22 k 2 n k n 1 k n 2 k n n ] 𝐪 = [ q 1 q 2 q n ] 𝐮 = [ u 1 ( t ) u 2 ( t ) u n ( t ) ] \mathbf{K}=\begin{bmatrix}k_{11}&k_{12}&\cdots&k_{1n}\\ k_{21}&k_{22}&\cdots&k_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ k_{n1}&k_{n2}&\cdots&k_{nn}\\ \end{bmatrix}\mathbf{q}=\begin{bmatrix}q_{1}\\ q_{2}\\ \vdots\\ q_{n}\end{bmatrix}\mathbf{u}=\begin{bmatrix}u_{1}(t)\\ u_{2}(t)\\ \vdots\\ u_{n}(t)\end{bmatrix}
  8. [ 1 1 1 ] 𝐊 = [ 0 0 0 ] \begin{bmatrix}1&1&\cdots&1\\ \end{bmatrix}\mathbf{K}=\begin{bmatrix}0&0&\cdots&0\\ \end{bmatrix}
  9. 𝐮 = 0 \mathbf{u}=0
  10. 𝐪 = c 1 e λ 1 t 𝐯 𝟏 + c 2 e λ 2 t 𝐯 𝟐 + + c n e λ n t 𝐯 𝐧 \mathbf{q}=c_{1}e^{\lambda_{1}t}\mathbf{v_{1}}+c_{2}e^{\lambda_{2}t}\mathbf{v_% {2}}+\cdots+c_{n}e^{\lambda_{n}t}\mathbf{v_{n}}
  11. λ 1 \lambda_{1}
  12. λ 2 \lambda_{2}
  13. λ n \lambda_{n}
  14. 𝐊 \mathbf{K}
  15. 𝐯 𝟏 \mathbf{v_{1}}
  16. 𝐯 𝟐 \mathbf{v_{2}}
  17. 𝐯 𝐧 \mathbf{v_{n}}
  18. 𝐊 \mathbf{K}
  19. c 1 c_{1}
  20. c 2 c_{2}
  21. c n c_{n}
  22. λ = 0 \lambda=0
  23. [ 1 1 1 ] 𝐯 = 0 \begin{bmatrix}1&1&\cdots&1\\ \end{bmatrix}\mathbf{v}=0
  24. λ 1 \lambda_{1}
  25. 𝐪 = c 1 𝐯 𝟏 + c 2 e λ 2 t 𝐯 𝟐 + + c n e λ n t 𝐯 𝐧 \mathbf{q}=c_{1}\mathbf{v_{1}}+c_{2}e^{\lambda_{2}t}\mathbf{v_{2}}+\cdots+c_{n% }e^{\lambda_{n}t}\mathbf{v_{n}}
  26. [ 1 1 1 ] 𝐯 𝐢 = 0 \begin{bmatrix}1&1&\cdots&1\\ \end{bmatrix}\mathbf{v_{i}}=0
  27. 𝐢 = 2 , 3 , n \mathbf{i}=2,3,\dots n
  28. 𝐪 = [ 𝐯 𝟏 [ c 1 0 0 ] + 𝐯 𝟐 [ 0 c 2 0 ] + + 𝐯 𝐧 [ 0 0 c n ] ] [ 1 e λ 2 t e λ n t ] \mathbf{q}=\Bigg[\mathbf{v_{1}}\begin{bmatrix}c_{1}&0&\cdots&0\\ \end{bmatrix}+\mathbf{v_{2}}\begin{bmatrix}0&c_{2}&\cdots&0\\ \end{bmatrix}+\dots+\mathbf{v_{n}}\begin{bmatrix}0&0&\cdots&c_{n}\\ \end{bmatrix}\Bigg]\begin{bmatrix}1\\ e^{\lambda_{2}t}\\ \vdots\\ e^{\lambda_{n}t}\\ \end{bmatrix}
  29. 𝐪 = 𝐀 [ 1 e λ 2 t e λ n t ] \mathbf{q}=\mathbf{A}\begin{bmatrix}1\\ e^{\lambda_{2}t}\\ \vdots\\ e^{\lambda_{n}t}\\ \end{bmatrix}
  30. 𝐀 \mathbf{A}
  31. λ 2 \lambda_{2}
  32. λ 3 \lambda_{3}
  33. λ n \lambda_{n}
  34. [ 1 1 1 ] 𝐀 = [ a 0 0 ] \begin{bmatrix}1&1&\cdots&1\\ \end{bmatrix}\mathbf{A}=\begin{bmatrix}a&0&\cdots&0\\ \end{bmatrix}

Multi-layer_insulation.html

  1. Δ T \Delta T
  2. Q = U A Δ T . Q=UA\Delta T.
  3. ϵ 1 \epsilon_{1}
  4. ϵ 2 \epsilon_{2}
  5. U = 4 σ T 3 1 1 / ϵ 1 + 1 / ϵ 2 - 1 , U=4\sigma T^{3}\frac{1}{1/\epsilon_{1}+1/\epsilon_{2}-1},
  6. σ = 5.7 × 10 - 8 \sigma=5.7\times 10^{-8}
  7. ϵ \epsilon
  8. U = 4 σ T 3 1 N ( 2 / ϵ - 1 ) + 1 . U=4\sigma T^{3}\frac{1}{N(2/\epsilon-1)+1}.

Multi-objective_optimization.html

  1. min \displaystyle\min
  2. k 2 k\geq 2
  3. X X
  4. f : X k , f ( x ) = ( f 1 ( x ) , , f k ( x ) ) T f:X\to\mathbb{R}^{k},\ f(x)=(f_{1}(x),\ldots,f_{k}(x))^{T}
  5. X X
  6. Y k Y\in\mathbb{R}^{k}
  7. x * X x^{*}\in X
  8. z * := f ( x * ) k z^{*}:=f(x^{*})\in\mathbb{R}^{k}
  9. x * x^{*}
  10. x 1 X x^{1}\in X
  11. x 2 X x^{2}\in X
  12. f i ( x 1 ) f i ( x 2 ) f_{i}(x^{1})\leq f_{i}(x^{2})
  13. i { 1 , 2 , , k } i\in\left\{{1,2,\dots,k}\right\}
  14. f j ( x 1 ) < f j ( x 2 ) f_{j}(x^{1})<f_{j}(x^{2})
  15. j { 1 , 2 , , k } j\in\left\{{1,2,\dots,k}\right\}
  16. x 1 X x^{1}\in X
  17. f ( x * ) f(x^{*})
  18. z n a d z^{nad}
  19. z i d e a l z^{ideal}
  20. z i n a d = sup x X is Pareto optimal f i ( x ) for all i = 1 , , k z^{nad}_{i}=\sup_{x\in X\,\text{ is Pareto optimal}}f_{i}(x)\,\text{ for all }% i=1,\ldots,k
  21. z i i d e a l = inf x X f i ( x ) for all i = 1 , , k . z^{ideal}_{i}=\inf_{x\in X}{f_{i}(x)}\,\text{ for all }i=1,\ldots,k.
  22. z u t o p i a n z^{utopian}
  23. z i u t o p i a n = z i i d e a l - ϵ for all i = 1 , , k , z^{utopian}_{i}=z^{ideal}_{i}-\epsilon\,\text{ for all }i=1,\ldots,k,
  24. ϵ > 0 \epsilon>0
  25. min g ( f 1 ( x ) , , f k ( x ) , θ ) s.t x X θ , \begin{array}[]{ll}\min&g(f_{1}(x),\ldots,f_{k}(x),\theta)\\ \,\text{s.t }x\in X_{\theta},\end{array}
  26. θ \theta
  27. X θ X X_{\theta}\subseteq X
  28. θ \theta
  29. g : k + 1 g:\mathbb{R}^{k+1}\mapsto\mathbb{R}
  30. min x X i = 1 k w i f i ( x ) , \min_{x\in X}\sum_{i=1}^{k}w_{i}f_{i}(x),
  31. w i > 0 w_{i}>0
  32. ϵ \epsilon
  33. min f j ( x ) s.t. x X f i ( x ) ϵ j for i { 1 , , k } { j } , \begin{array}[]{ll}\min&f_{j}(x)\\ \,\text{s.t. }&x\in X\\ &f_{i}(x)\leq\epsilon_{j}\,\text{ for }i\in\{1,\ldots,k\}\setminus\{j\},\end{array}
  34. ϵ j \epsilon_{j}
  35. f j f_{j}
  36. min max i = 1 , , k [ f i ( x ) - z ¯ i z i nad - z i utopia ] + ρ i = 1 k f i ( x ) z i n a d - z i utopian subject to x S , \begin{array}[]{ll}\min&\max_{i=1,\ldots,k}\left[\frac{f_{i}(x)-\bar{z}_{i}}{z% ^{\,\text{nad}}_{i}-z_{i}^{\,\text{utopia}}}\right]+\rho\sum_{i=1}^{k}\frac{f_% {i}(x)}{z_{i}^{nad}-z_{i}^{\,\text{utopian}}}\\ \,\text{subject to }&x\in S,\end{array}
  37. ρ i = 1 k f i ( x ) z i n a d - z i utopia \rho\sum_{i=1}^{k}\frac{f_{i}(x)}{z_{i}^{nad}-z_{i}^{\,\text{utopia}}}
  38. ρ > 0 \rho>0
  39. z nad z^{\,\text{nad}}
  40. z utopian z^{\,\text{utopian}}
  41. z ¯ \bar{z}
  42. μ P \mu_{P}
  43. σ P \sigma_{P}
  44. μ P \mu_{P}
  45. μ P - b σ P \mu_{P}-b\sigma_{P}
  46. min f ( x ) - z i d e a l s.t. x X \begin{aligned}\displaystyle\min&\displaystyle\|f(x)-z^{ideal}\|\\ \displaystyle\,\text{s.t. }&\displaystyle x\in X\end{aligned}
  47. \|\cdot\|
  48. L p L_{p}
  49. L 1 L_{1}
  50. L 2 L_{2}
  51. L L_{\infty}
  52. u : Y u\colon Y\rightarrow\mathbb{R}
  53. 𝐲 1 , 𝐲 2 Y \mathbf{y}^{1},\mathbf{y}^{2}\in Y
  54. u ( 𝐲 1 ) > u ( 𝐲 2 ) u(\mathbf{y}^{1})>u(\mathbf{y}^{2})
  55. 𝐲 1 \mathbf{y}^{1}
  56. 𝐲 2 \mathbf{y}^{2}
  57. u ( 𝐲 1 ) = u ( 𝐲 2 ) u(\mathbf{y}^{1})=u(\mathbf{y}^{2})
  58. 𝐲 1 \mathbf{y}^{1}
  59. 𝐲 2 \mathbf{y}^{2}
  60. u u
  61. max u ( 𝐟 ( 𝐱 ) ) subject to 𝐱 X , \max\;u(\mathbf{f}(\mathbf{x}))\,\text{ subject to }\mathbf{x}\in X,
  62. f 1 f_{1}
  63. f k f_{k}
  64. min f l ( 𝐱 ) s.t. f j ( 𝐱 ) 𝐲 j * , j = 1 , , l - 1 , 𝐱 X , \begin{aligned}\displaystyle\min&\displaystyle f_{l}(\mathbf{x})\\ \displaystyle\,\text{s.t. }&\displaystyle f_{j}(\mathbf{x})\leq\mathbf{y}^{*}_% {j},\;j=1,\ldots,l-1,\\ &\displaystyle\mathbf{x}\in X,\end{aligned}
  65. 𝐲 j * \mathbf{y}^{*}_{j}
  66. l = j l=j
  67. 𝐲 1 * := min { f 1 ( 𝐱 ) 𝐱 X } \mathbf{y}^{*}_{1}:=\min\{f_{1}(\mathbf{x})\mid\mathbf{x}\in X\}
  68. l l
  69. 1 1
  70. k k

Multibody_system.html

  1. 𝐌 ( 𝐪 ) 𝐪 ¨ - 𝐐 v + 𝐂 𝐪 T λ = 𝐅 , \mathbf{M(q)}\ddot{\mathbf{q}}-\mathbf{Q}_{v}+\mathbf{C_{q}}^{T}\mathbf{% \lambda}=\mathbf{F},
  2. 𝐂 ( 𝐪 , 𝐪 ˙ ) = 0 \mathbf{C}(\mathbf{q},\dot{\mathbf{q}})=0
  3. 𝐪 \mathbf{q}
  4. 𝐌 ( 𝐪 ) \mathbf{M}(\mathbf{q})
  5. 𝐂 \mathbf{C}
  6. 𝐂 𝐪 \mathbf{C_{q}}
  7. λ \mathbf{\lambda}
  8. λ \mathbf{\lambda}
  9. 𝐪 = [ 𝐮 𝚿 ] T \mathbf{q}=\left[\mathbf{u}\quad\mathbf{\Psi}\right]^{T}
  10. 𝐮 \mathbf{u}
  11. 𝚿 \mathbf{\Psi}
  12. 𝐐 v \mathbf{Q}_{v}
  13. 𝐐 v \mathbf{Q}_{v}
  14. λ i \lambda_{i}
  15. C i = 0 C_{i}=0

Multidelay_block_frequency_domain_adaptive_filter.html

  1. N N
  2. K K
  3. 𝐅 \mathbf{F}
  4. 𝐞 ¯ ( ) = 𝐅 [ 𝟎 1 x N , e ( N ) , , e ( N - N - 1 ) ] T \underline{\mathbf{e}}(\ell)=\mathbf{F}\left[\mathbf{0}_{1xN},e(\ell N),\dots,% e(\ell N-N-1)\right]^{T}
  5. 𝐱 ¯ k ( ) = diag { 𝐅 [ x ( ( - k + 1 ) N ) , , x ( ( - k - 1 ) N - 1 ) ] T } \underline{\mathbf{x}}_{k}(\ell)=\mathrm{diag}\left\{\mathbf{F}\left[x((\ell-k% +1)N),\dots,x((\ell-k-1)N-1)\right]^{T}\right\}
  6. 𝐗 ¯ ( ) = [ 𝐱 ¯ 0 ( ) , 𝐱 ¯ 1 ( ) , , 𝐱 ¯ K - 1 ( ) ] \underline{\mathbf{X}}(\ell)=\left[\underline{\mathbf{x}}_{0}(\ell),\underline% {\mathbf{x}}_{1}(\ell),\dots,\underline{\mathbf{x}}_{K-1}(\ell)\right]
  7. 𝐝 ¯ ( ) = 𝐅 [ 𝟎 1 x N , d ( N ) , , d ( N - N - 1 ) ] T \underline{\mathbf{d}}(\ell)=\mathbf{F}\left[\mathbf{0}_{1xN},d(\ell N),\dots,% d(\ell N-N-1)\right]^{T}
  8. 𝐆 1 \mathbf{G}_{1}
  9. 𝐆 2 \mathbf{G}_{2}
  10. 𝐆 1 = 𝐅 [ 𝟎 N x N 𝟎 N x N 𝟎 N x N 𝐈 N x N ] 𝐅 - 1 \mathbf{G}_{1}=\mathbf{F}\begin{bmatrix}\mathbf{0}_{NxN}&\mathbf{0}_{NxN}\\ \mathbf{0}_{NxN}&\mathbf{I}_{NxN}\\ \end{bmatrix}\mathbf{F}^{-1}
  11. 𝐆 ~ 2 = 𝐅 [ 𝐈 N x N 𝟎 N x N 𝟎 N x N 𝟎 N x N ] 𝐅 - 1 \tilde{\mathbf{G}}_{2}=\mathbf{F}\begin{bmatrix}\mathbf{I}_{NxN}&\mathbf{0}_{% NxN}\\ \mathbf{0}_{NxN}&\mathbf{0}_{NxN}\\ \end{bmatrix}\mathbf{F}^{-1}
  12. 𝐆 2 = diag { 𝐆 ~ 2 , 𝐆 ~ 2 , , 𝐆 ~ 2 } \mathbf{G}_{2}=\mathrm{diag}\left\{\tilde{\mathbf{G}}_{2},\tilde{\mathbf{G}}_{% 2},\dots,\tilde{\mathbf{G}}_{2}\right\}
  13. 𝐱 \mathbf{x}
  14. 𝐆 1 \mathbf{G}_{1}
  15. 𝐱 \mathbf{x}
  16. N N
  17. 𝐲 ¯ ^ ( ) = 𝐆 1 𝐗 ¯ ( ) 𝐡 ¯ ^ ( - 1 ) \underline{\hat{\mathbf{y}}}(\ell)=\mathbf{G}_{1}\underline{\mathbf{X}}(\ell)% \underline{\hat{\mathbf{h}}}(\ell-1)
  18. 𝐞 ¯ ( ) = 𝐝 ¯ ( ) - 𝐲 ¯ ^ ( ) \underline{\mathbf{e}}(\ell)=\underline{\mathbf{d}}(\ell)-\underline{\hat{% \mathbf{y}}}(\ell)
  19. 𝚽 𝐱𝐱 = 𝐗 ¯ ( ) 𝐗 ¯ ( ) H \mathbf{\Phi}_{\mathbf{xx}}=\underline{\mathbf{X}}(\ell)\underline{\mathbf{X}}% (\ell)^{H}
  20. 𝐡 ¯ ^ ( ) = 𝐡 ¯ ^ ( - 1 ) + μ 𝐆 2 𝚽 𝐱𝐱 - 1 ( ) 𝐗 ¯ H ( ) 𝐞 ¯ ( ) \underline{\hat{\mathbf{h}}}(\ell)=\underline{\hat{\mathbf{h}}}(\ell-1)+\mu% \mathbf{G}_{2}\mathbf{\Phi}_{\mathbf{xx}}^{-1}(\ell)\underline{\mathbf{X}}^{H}% (\ell)\underline{\mathbf{e}}(\ell)
  21. 𝚽 𝐱𝐱 = 𝐗 ¯ ( ) 𝐗 ¯ ( ) H \mathbf{\Phi}_{\mathbf{xx}}=\underline{\mathbf{X}}(\ell)\underline{\mathbf{X}}% (\ell)^{H}
  22. 𝐗 ¯ ( ) \underline{\mathbf{X}}(\ell)

Multifactor_productivity.html

  1. Labor Productivity (output per hour) = Output Labor  Inputs \textrm{Labor~{}Productivity~{}(output~{}per~{}hour)}={\textrm{Output}\over% \textrm{Labor ~{}Inputs}}
  2. Multifactor Productivity = Output ( K L E M S ) \textrm{Multifactor~{}Productivity}={\textrm{Output}\over{(KLEMS)}}
  3. M F P = d ( l n f ) d t = d ( l n Y ) d t - s L d ( l n L ) d t - s K d ( l n K ) d t MFP={{d(lnf)}\over{dt}}={{d(lnY)}\over{dt}}-{{s_{L}\cdot d(lnL)}\over{dt}}-{{s% _{K}\cdot d(lnK)}\over{dt}}
  4. f f
  5. Y Y
  6. t t
  7. s L s_{L}
  8. s K s_{K}
  9. L L
  10. K K
  11. M M
  12. S S
  13. E E

Multimodal_logic.html

  1. i , i { 1 , , n } \Box_{i},i\in\{1,\ldots,n\}
  2. i \Diamond_{i}
  3. i P \Diamond_{i}P
  4. ¬ i ¬ P \lnot\Box_{i}\lnot P
  5. \Box
  6. i \Box_{i}
  7. i α \Box_{i}\alpha
  8. α \alpha
  9. α \alpha

Multiphoton_intrapulse_interference_phase_scan.html

  1. I ( ω , τ ) = | E ( t ) g ( t - τ ) e i ω t d t | 2 I(\omega,\tau)=\left|\int{E(t)g(t-\tau)e^{i\omega t}\mathrm{d}t}\right|^{2}
  2. I ( 2 ω ) = | | E ( ω ) | 2 e i ϕ d ϕ | 2 I(2\omega)=\left|\int{|E(\omega)|^{2}e^{i\phi}\mathrm{d}\phi}\right|^{2}
  3. f ( ω ) f(\omega)
  4. Φ ( ω ) \Phi(\omega)
  5. ϕ ( ω ) = Φ ( ω ) + f ( ω ) \phi(\omega)=\Phi(\omega)+f(\omega)
  6. ϕ ( ω ) \phi(\omega)
  7. Φ ( ω ) \Phi(\omega)
  8. ω \omega
  9. I ( ω ) = | | E ( ω + Ω ) | | E ( ω - Ω ) | × exp { i [ ϕ ( ω + Ω ) + ϕ ( ω - Ω ) ] } d Ω | 2 I(\omega)=\left|\int|E(\omega+\Omega)||E(\omega-\Omega)|\times\,\text{exp}\{i[% \phi(\omega+\Omega)+\phi(\omega-\Omega)]\}\mathrm{d}\Omega\right|^{2}
  10. ϕ ( ω + Ω ) + ϕ ( ω - Ω ) = 2 ϕ 0 + ϕ ′′ ( ω ) Ω 2 + + 2 ( 2 n ) ! ϕ 2 n ( ω ) Ω 2 n \phi(\omega+\Omega)+\phi(\omega-\Omega)=2\phi 0+\phi^{\prime\prime}(\omega)% \Omega^{2}+...+\frac{2}{(2n)!}\phi^{2n^{\prime}}(\omega)\Omega^{2n}
  11. ϕ ( ω + Ω ) + ϕ ( ω - Ω ) \phi(\omega+\Omega)+\phi(\omega-\Omega)
  12. Φ ′′ ( ω ) = - f ′′ ( ω ) \Phi^{\prime\prime}(\omega)=-f^{\prime\prime}(\omega)
  13. f ( ω ) f(\omega)
  14. Φ ( ω ) \Phi(\omega)
  15. 4 ( π ) 4(\pi)
  16. ω , ω \omega,\omega
  17. ω = π c / λ S H G \omega=\pi c/\lambda_{SHG}
  18. ω m ( ω ) \omega_{m}(\omega)
  19. π \pi

Multiple_comparisons_problem.html

  1. m m
  2. m m
  3. V V
  4. S S
  5. R R
  6. U U
  7. T T
  8. m - R m-R
  9. m 0 m_{0}
  10. m - m 0 m-m_{0}
  11. m m
  12. m m
  13. m 0 m_{0}
  14. m - m 0 m-m_{0}
  15. V V
  16. S S
  17. T T
  18. U U
  19. R R
  20. m m
  21. m 0 m_{0}
  22. R R
  23. S S
  24. T T
  25. U U
  26. V V
  27. α ¯ \bar{\alpha}
  28. α ¯ = 1 - ( 1 - α { per comparison } ) k \bar{\alpha}=1-\left(1-\alpha_{\mathrm{\{per\ comparison\}}}\right)^{k}
  29. α ¯ \bar{\alpha}
  30. α ¯ k α { per comparison } , \bar{\alpha}\leq k\cdot\alpha_{\mathrm{\{per\ comparison\}}},
  31. 0.2649 = 1 - ( 1 - .05 ) 6 .05 × 6 = 0.3 0.2649=1-\left(1-.05\right)^{6}\leq.05\times 6=0.3
  32. α ¯ \bar{\alpha}
  33. α { per comparison } = α ¯ / k \alpha_{\mathrm{\{per\ comparison\}}}=\bar{\alpha}/k
  34. k k
  35. α { per comparison } \alpha_{\mathrm{\{per\ comparison\}}}
  36. α { per comparison } = 1 - ( 1 - α ¯ ) 1 k \alpha_{\mathrm{\{per\ comparison\}}}=1-{\left(1-\bar{\alpha}\right)}^{\frac{1% }{k}}
  37. i = 1 i=1
  38. i > 1 i>1
  39. α { per comparison } = α ¯ / ( k - i + 1 ) \alpha_{\mathrm{\{per\ comparison\}}}=\bar{\alpha}/(k-i+1)

Multiple_EM_for_Motif_Elicitation.html

  1. x i x_{i}
  2. x x
  3. log 10 0 := - 10 \log_{10}0:=-10
  4. x i x_{i}

Multiplication_theorem.html

  1. Γ ( z ) Γ ( z + 1 2 ) = 2 1 - 2 z π Γ ( 2 z ) . \Gamma(z)\;\Gamma\left(z+\frac{1}{2}\right)=2^{1-2z}\;\sqrt{\pi}\;\Gamma(2z).\,\!
  2. Γ ( z ) Γ ( z + 1 k ) Γ ( z + 2 k ) Γ ( z + k - 1 k ) = ( 2 π ) k - 1 2 k 1 / 2 - k z Γ ( k z ) \Gamma(z)\;\Gamma\left(z+\frac{1}{k}\right)\;\Gamma\left(z+\frac{2}{k}\right)% \cdots\Gamma\left(z+\frac{k-1}{k}\right)=(2\pi)^{\frac{k-1}{2}}\;k^{1/2-kz}\;% \Gamma(kz)\,\!
  3. k m ψ ( m - 1 ) ( k z ) = n = 0 k - 1 ψ ( m - 1 ) ( z + n k ) k^{m}\psi^{(m-1)}(kz)=\sum_{n=0}^{k-1}\psi^{(m-1)}\left(z+\frac{n}{k}\right)
  4. m > 1 m>1
  5. m = 1 m=1
  6. k [ ψ ( k z ) - log ( k ) ] = n = 0 k - 1 ψ ( z + n k ) . k\left[\psi(kz)-\log(k)\right]=\sum_{n=0}^{k-1}\psi\left(z+\frac{n}{k}\right).
  7. k s ζ ( s ) = n = 1 k ζ ( s , n k ) , k^{s}\zeta(s)=\sum_{n=1}^{k}\zeta\left(s,\frac{n}{k}\right),
  8. ζ ( s ) \zeta(s)
  9. k s ζ ( s , k z ) = n = 0 k - 1 ζ ( s , z + n k ) k^{s}\,\zeta(s,kz)=\sum_{n=0}^{k-1}\zeta\left(s,z+\frac{n}{k}\right)
  10. ζ ( s , k z ) = n = 0 ( s + n - 1 n ) ( 1 - k ) n z n ζ ( s + n , z ) . \zeta(s,kz)=\sum^{\infty}_{n=0}{s+n-1\choose n}(1-k)^{n}z^{n}\zeta(s+n,z).
  11. F ( s ; q ) = m = 1 e 2 π i m q m s = Li s ( e 2 π i q ) F(s;q)=\sum_{m=1}^{\infty}\frac{e^{2\pi imq}}{m^{s}}=\operatorname{Li}_{s}% \left(e^{2\pi iq}\right)
  12. 2 - s F ( s ; q ) = F ( s , q 2 ) + F ( s , q + 1 2 ) . 2^{-s}F(s;q)=F\left(s,\frac{q}{2}\right)+F\left(s,\frac{q+1}{2}\right).
  13. k - s F ( s ; k q ) = n = 0 k - 1 F ( s , q + n k ) . k^{-s}F(s;kq)=\sum_{n=0}^{k-1}F\left(s,q+\frac{n}{k}\right).
  14. 2 1 - s Li s ( z 2 ) = Li s ( z ) + Li s ( - z ) . 2^{1-s}\operatorname{Li}_{s}(z^{2})=\operatorname{Li}_{s}(z)+\operatorname{Li}% _{s}(-z).
  15. k 1 - s Li s ( z k ) = n = 0 k - 1 Li s ( z e i 2 π n / k ) . k^{1-s}\operatorname{Li}_{s}(z^{k})=\sum_{n=0}^{k-1}\operatorname{Li}_{s}\left% (ze^{i2\pi n/k}\right).
  16. 2 1 - n Λ n ( - z 2 ) = Λ n ( z ) + Λ n ( - z ) 2^{1-n}\Lambda_{n}(-z^{2})=\Lambda_{n}(z)+\Lambda_{n}(-z)
  17. k 1 - m B m ( k x ) = n = 0 k - 1 B m ( x + n k ) k^{1-m}B_{m}(kx)=\sum_{n=0}^{k-1}B_{m}\left(x+\frac{n}{k}\right)
  18. k - m E m ( k x ) = n = 0 k - 1 ( - 1 ) n E m ( x + n k ) for k = 1 , 3 , k^{-m}E_{m}(kx)=\sum_{n=0}^{k-1}(-1)^{n}E_{m}\left(x+\frac{n}{k}\right)\quad% \mbox{ for }~{}k=1,3,\dots
  19. k - m E m ( k x ) = - 2 m + 1 n = 0 k - 1 ( - 1 ) n B m + 1 ( x + n k ) for k = 2 , 4 , . k^{-m}E_{m}(kx)=\frac{-2}{m+1}\sum_{n=0}^{k-1}(-1)^{n}B_{m+1}\left(x+\frac{n}{% k}\right)\quad\mbox{ for }~{}k=2,4,\dots.
  20. k \mathcal{L}_{k}
  21. [ k f ] ( x ) = 1 k n = 0 k - 1 f ( x + n k ) [\mathcal{L}_{k}f](x)=\frac{1}{k}\sum_{n=0}^{k-1}f\left(\frac{x+n}{k}\right)
  22. k B m = 1 k m B m \mathcal{L}_{k}B_{m}=\frac{1}{k^{m}}B_{m}
  23. k - m < 1 k^{-m}<1
  24. f ( n ) f(n)
  25. f ( m n ) = f ( m ) f ( n ) f(mn)=f(m)f(n)
  26. g ( x ) = n = 1 f ( n ) exp ( 2 π i n x ) g(x)=\sum_{n=1}^{\infty}f(n)\exp(2\pi inx)
  27. 1 k n = 0 k - 1 g ( x + n k ) = f ( k ) g ( x ) \frac{1}{k}\sum_{n=0}^{k-1}g\left(\frac{x+n}{k}\right)=f(k)g(x)
  28. f ( n ) = n - s f(n)=n^{-s}
  29. J ν ( z ) J_{\nu}(z)
  30. λ - ν J ν ( λ z ) = n = 0 1 n ! ( ( 1 - λ 2 ) z 2 ) n J ν + n ( z ) , \lambda^{-\nu}J_{\nu}(\lambda z)=\sum_{n=0}^{\infty}\frac{1}{n!}\left(\frac{(1% -\lambda^{2})z}{2}\right)^{n}J_{\nu+n}(z),
  31. λ \lambda
  32. ν \nu

Multiplicative_distance.html

  1. μ \mu
  2. μ ( A B ) > 1. \mu(AB)>1.\,
  3. μ ( A B ) = μ ( A B ) . \mu(AB)=\mu(A^{\prime}B^{\prime}).\,
  4. μ ( A B ) < μ ( A B ) . \mu(AB)<\mu(A^{\prime}B^{\prime}).\,
  5. μ ( A B + C D ) = μ ( A B ) μ ( C D ) . \mu(AB+CD)=\mu(AB)\mu(CD).\,

Multiply-with-carry.html

  1. x n = ( a x n - r + c n - 1 ) mod b , c n = a x n - r + c n - 1 b , n r , x_{n}=(ax_{n-r}+c_{n-1})\,\bmod\,b,\ c_{n}=\left\lfloor\frac{ax_{n-r}+c_{n-1}}% {b}\right\rfloor,\ n\geq r,
  2. x n + 1 = ( a x n + c ) mod 2 32 , x_{n+1}=(ax_{n}+c)\ \bmod\,2^{32},
  3. a × x mod 2 32 a\times x\ \bmod\,2^{32}
  4. x ( a x + c ) mod 2 32 , c a x + c 2 32 . x\leftarrow(ax+c)\,\bmod\,2^{32},\ \ c\leftarrow\left\lfloor\frac{ax+c}{2^{32}% }\right\rfloor.
  5. x 0 = 1 x_{0}=1
  6. x n = ( 7 x n - 1 + 3 ) mod 10 , x_{n}=(7x_{n-1}+3)\,\bmod\,10,
  7. 10 , 03 , 24 , 31 , 10 , 03 , 24 , 31 , 10 , , 10,03,24,31,10,03,24,31,10,\ldots,
  8. 0 , 3 , 4 , 1 , 0 , 3 , 4 , 1 , 0 , 3 , 4 , 1 , 0,3,4,1,0,3,4,1,0,3,4,1,\ldots
  9. x 0 = 1 , c 0 = 3 x_{0}=1,c_{0}=3
  10. x n = ( 7 x n - 1 + c n - 1 ) mod 10 , c n = 7 x n - 1 + c n - 1 10 , x_{n}=(7x_{n-1}+c_{n-1})\,\bmod\,10,\ c_{n}=\left\lfloor\frac{7x_{n-1}+c_{n-1}% }{10}\right\rfloor,
  11. x 22 n + 20 x_{22n+20}
  12. 449275 97101 449275 9710144 449275\cdots 97101\,449275\cdots 9710144\cdots
  13. 31 69 = .4492753623188405797101 4492753623 \frac{31}{69}=.4492753623188405797101\,4492753623\ldots
  14. x n = ( a x n - r + c n - 1 ) mod b , c n = a x n - r + c n - 1 b , x_{n}=(ax_{n-r}+c_{n-1})\bmod\,b\,,\ \ c_{n}=\left\lfloor\frac{ax_{n-r}+c_{n-1% }}{b}\right\rfloor,
  15. x 0 = 34 x_{0}=34
  16. x n = 7 x n - 1 mod 69 x_{n}=7x_{n-1}\,\bmod\,69
  17. x 0 , c 0 x_{0},c_{0}
  18. x 1 , x 2 , x_{1},x_{2},\ldots
  19. x n = ( a x n - 1 + c n - 1 ) mod b , c n = a x n - 1 + c n - 1 b x_{n}=(ax_{n-1}+c_{n-1})\,\bmod b\,,\ \ c_{n}=\left\lfloor\frac{ax_{n-1}+c_{n-% 1}}{b}\right\rfloor
  20. x n = ( b - 1 ) - ( a x n - r + c n - 1 ) mod b , c n = a x n - r + c n - 1 b . x_{n}=(b-1)-(ax_{n-r}+c_{n-1})\,\bmod\,b,\ c_{n}=\left\lfloor\frac{ax_{n-r}+c_% {n-1}}{b}\right\rfloor.
  21. x n = ( b - 1 ) - ( a x n - 1024 + c n - 1 ) mod b , c n = a x n - 1024 + c n - 1 b . x_{n}=(b-1)-(ax_{n-1024}+c_{n-1})\,\bmod\,b,\ c_{n}=\left\lfloor\frac{ax_{n-10% 24}+c_{n-1}}{b}\right\rfloor.
  22. \cdot
  23. \cdot
  24. \approx
  25. 2 131104 2^{131104}

Multitaper.html

  1. 𝐗 ( t ) = [ X ( 1 , t ) , X ( 2 , t ) , , X ( p , t ) ] T \mathbf{X}(t)={[X(1,t),X(2,t),\dots,X(p,t)]}^{T}
  2. 𝐗 ( t ) \mathbf{X}(t)
  3. Δ t \Delta t
  4. f N = 1 / ( 2 Δ t ) f_{N}=1/(2\Delta t)
  5. S ^ l m ( f ) = 1 K k = 0 K - 1 S ^ k l m ( f ) . \hat{S}^{lm}(f)=\frac{1}{K}\sum_{k=0}^{K-1}\hat{S}_{k}^{lm}(f).
  6. S ^ k l m ( f ) \hat{S}_{k}^{lm}(f)
  7. 0 k K 0\leq k\leq K
  8. S ^ k l m ( f ) = 1 N Δ t [ J k l ( f ) ] * [ J k m ( f ) ] , \hat{S}_{k}^{lm}(f)=\frac{1}{N\Delta t}{[J_{k}^{l}(f)]}^{*}{[J_{k}^{m}(f)]},
  9. J k l ( f ) = t = 1 N h t , k X ( l , t ) e - i 2 π f t Δ t . J_{k}^{l}(f)=\sum_{t=1}^{N}h_{t,k}X(l,t)e^{-i2\pi ft\Delta t}.
  10. { h t , k } \{h_{t,k}\}
  11. S ^ k l m ( f ) \hat{S}_{k}^{lm}(f)
  12. 2 N W Δ t 2NW\Delta t
  13. W ( 0 , f N ) W\in(0,f_{N})

Multivariate_interpolation.html

  1. ( x i , y i , z i , ) (x_{i},y_{i},z_{i},\dots)
  2. ( x , y , z , ) (x,y,z,\dots)
  3. CINT x ( f - 1 , f 0 , f 1 , f 2 ) = 𝐛 ( x ) ( f - 1 f 0 f 1 f 2 ) \mathrm{CINT}_{x}(f_{-1},f_{0},f_{1},f_{2})=\mathbf{b}(x)\cdot\left(f_{-1}f_{0% }f_{1}f_{2}\right)
  4. 𝐛 ( x ) \mathbf{b}(x)
  5. f j f_{j}
  6. j j
  7. CR ( x ) = i = - 1 2 f i b i ( x ) \mathrm{CR}(x)=\sum_{i=-1}^{2}f_{i}b_{i}(x)
  8. CR ( x 1 , , x N ) = i 1 , , i N = - 1 2 f i 1 i N j = 1 N b i j ( x j ) \mathrm{CR}(x_{1},\dots,x_{N})=\sum_{i_{1},\dots,i_{N}=-1}^{2}f_{i_{1}\dots i_% {N}}\prod_{j=1}^{N}b_{i_{j}}(x_{j})
  9. CINT \mathrm{CINT}
  10. n n
  11. CR \mathrm{CR}
  12. n N n^{N}
  13. N N

Multivariate_probit_model.html

  1. Y Y
  2. Y * Y^{*}
  3. Y 1 Y_{1}
  4. Y 2 Y_{2}
  5. Y 1 * Y^{*}_{1}
  6. Y 2 * Y^{*}_{2}
  7. Y 1 = { 1 if Y 1 * > 0 , 0 otherwise , Y_{1}=\begin{cases}1&\,\text{if }Y^{*}_{1}>0,\\ 0&\,\text{otherwise},\end{cases}
  8. Y 2 = { 1 if Y 2 * > 0 , 0 otherwise , Y_{2}=\begin{cases}1&\,\text{if }Y^{*}_{2}>0,\\ 0&\,\text{otherwise},\end{cases}
  9. { Y 1 * = X 1 β 1 + ε 1 Y 2 * = X 2 β 2 + ε 2 \begin{cases}Y_{1}^{*}=X_{1}\beta_{1}+\varepsilon_{1}\\ Y_{2}^{*}=X_{2}\beta_{2}+\varepsilon_{2}\end{cases}
  10. [ ε 1 ε 2 ] X 𝒩 ( [ 0 0 ] , [ 1 ρ ρ 1 ] ) \begin{bmatrix}\varepsilon_{1}\\ \varepsilon_{2}\end{bmatrix}\mid X\sim\mathcal{N}\left(\begin{bmatrix}0\\ 0\end{bmatrix},\begin{bmatrix}1&\rho\\ \rho&1\end{bmatrix}\right)
  11. β 1 , β 2 , \beta_{1},\ \beta_{2},
  12. ρ \rho
  13. L ( β 1 , β 2 ) = ( \displaystyle L(\beta_{1},\beta_{2})=\Big(\prod
  14. Y 1 * Y_{1}^{*}
  15. Y 2 * Y_{2}^{*}
  16. \displaystyle\sum
  17. \displaystyle\sum
  18. Φ \Phi
  19. Y 1 Y_{1}
  20. Y 2 Y_{2}

Myosin_ATPase.html

  1. \rightleftharpoons

Myrosinase.html

  1. \rightleftharpoons

N_=_1_supersymmetry_algebra_in_1_+_1_dimensions.html

  1. 𝒩 = ( 1 , 1 ) \mathcal{N}=(1,1)
  2. Q , Q ¯ Q,\bar{Q}
  3. Z Z\,
  4. H H\,
  5. P P\,
  6. N N\,
  7. Γ \Gamma\,
  8. I I\,
  9. { Γ , Γ } = 2 I \displaystyle\{\Gamma,\Gamma\}=2I
  10. Z Z\,
  11. 2 \mathbb{Z}_{2}
  12. H , P , N , Z , I H,P,N,Z,I\,
  13. Q , Q ¯ , Γ Q,\bar{Q},\Gamma\,

Nagel_point.html

  1. csc 2 ( A / 2 ) : csc 2 ( B / 2 ) : csc 2 ( C / 2 ) \csc^{2}(A/2)\,:\,\csc^{2}(B/2)\,:\,\csc^{2}(C/2)
  2. b + c - a a : c + a - b b : a + b - c c . \frac{b+c-a}{a}\,:\,\frac{c+a-b}{b}\,:\,\frac{a+b-c}{c}.

Nahm_equations.html

  1. d T 1 d z = [ T 2 , T 3 ] d T 2 d z = [ T 3 , T 1 ] d T 3 d z = [ T 1 , T 2 ] , \begin{aligned}\displaystyle\frac{dT_{1}}{dz}&\displaystyle=[T_{2},T_{3}]\\ \displaystyle\frac{dT_{2}}{dz}&\displaystyle=[T_{3},T_{1}]\\ \displaystyle\frac{dT_{3}}{dz}&\displaystyle=[T_{1},T_{2}],\end{aligned}
  2. d T i d z = 1 2 j , k ϵ i j k [ T j , T k ] = j , k ϵ i j k T j T k . \frac{dT_{i}}{dz}=\frac{1}{2}\sum_{j,k}\epsilon_{ijk}[T_{j},T_{k}]=\sum_{j,k}% \epsilon_{ijk}T_{j}T_{k}.
  3. T i * = - T i ; T^{*}_{i}=-T_{i};
  4. T i ( 2 - z ) = T i ( z ) T ; T_{i}(2-z)=T_{i}(z)^{T};\,
  5. A 0 = T 1 + i T 2 , A 1 = - 2 i T 3 , A 2 = T 1 - i T 2 A ( ζ ) = A 0 + ζ A 1 + ζ 2 A 2 , B ( ζ ) = 1 2 d A d ζ = 1 2 A 1 + ζ A 2 , \begin{aligned}&\displaystyle A_{0}=T_{1}+iT_{2},\quad A_{1}=-2iT_{3},\quad A_% {2}=T_{1}-iT_{2}\\ &\displaystyle A(\zeta)=A_{0}+\zeta A_{1}+\zeta^{2}A_{2},\quad B(\zeta)=\frac{% 1}{2}\frac{dA}{d\zeta}=\frac{1}{2}A_{1}+\zeta A_{2},\end{aligned}
  6. d A d z = [ A , B ] . \frac{dA}{dz}=[A,B].
  7. det ( λ I + A ( ζ , z ) ) = 0 , \det(\lambda I+A(\zeta,z))=0,

Nakagami_distribution.html

  1. Ω ( 1 - 1 m ( Γ ( m + 1 2 ) Γ ( m ) ) 2 ) \Omega\left(1-\frac{1}{m}\left(\frac{\Gamma(m+\frac{1}{2})}{\Gamma(m)}\right)^% {2}\right)
  2. m m
  3. Ω \Omega
  4. f ( x ; m , Ω ) = 2 m m Γ ( m ) Ω m x 2 m - 1 exp ( - m Ω x 2 ) . f(x;\,m,\Omega)=\frac{2m^{m}}{\Gamma(m)\Omega^{m}}x^{2m-1}\exp\left(-\frac{m}{% \Omega}x^{2}\right).
  5. F ( x ; m , Ω ) = P ( m , m Ω x 2 ) F(x;\,m,\Omega)=P\left(m,\frac{m}{\Omega}x^{2}\right)
  6. { x Ω f ( x ) + f ( x ) ( 2 m x 2 - 2 m Ω + Ω ) = 0 , f ( 1 ) = 2 m m e - m Ω Ω - m Γ ( m ) } \left\{x\Omega f^{\prime}(x)+f(x)\left(2mx^{2}-2m\Omega+\Omega\right)=0,f(1)=% \frac{2m^{m}e^{-\frac{m}{\Omega}}\Omega^{-m}}{\Gamma(m)}\right\}
  7. m m
  8. Ω \Omega
  9. m = E 2 [ X 2 ] Var [ X 2 ] , m=\frac{\operatorname{E}^{2}\left[X^{2}\right]}{\operatorname{Var}\left[X^{2}% \right]},
  10. Ω = E [ X 2 ] . \Omega=\operatorname{E}\left[X^{2}\right].
  11. Ω \Omega
  12. Γ ( m ) = x 2 m σ m , \Gamma(m)=\frac{x^{2m}}{\sigma^{m}},
  13. σ = x 2 m \sigma=\frac{x^{2}}{m}
  14. Y Gamma ( k , θ ) Y\,\sim\textrm{Gamma}(k,\theta)
  15. X Nakagami ( m , Ω ) X\,\sim\textrm{Nakagami}(m,\Omega)
  16. k = m k=m
  17. θ = Ω / m \theta=\Omega/m
  18. Y Y
  19. X = Y X=\sqrt{Y}\,
  20. f ( y ; m , Ω ) f(y;\,m,\Omega)
  21. k k
  22. 2 m 2m
  23. X X
  24. Y χ ( 2 m ) Y\sim\chi(2m)
  25. X = ( Ω / 2 m ) Y . X=\sqrt{(\Omega/2m)}\,Y.

Nanofluidics.html

  1. 1 r d d r ( r d ϕ d r ) = κ 2 ϕ \frac{1}{r}\frac{d}{dr}\left(r\frac{d\phi}{dr}\right)=\kappa^{2}\phi
  2. κ = 8 π n e 2 ϵ k T \kappa=\sqrt{\frac{8\pi ne^{2}}{\epsilon kT}}
  3. 1 r d d r ( r d v z d r ) = 1 η d p d z - F z η \frac{1}{r}\frac{d}{dr}\left(r\frac{dv_{z}}{dr}\right)=\frac{1}{\eta}\frac{dp}% {dz}-\frac{F_{z}}{\eta}
  4. v z ( r ) = ϵ ϕ 0 4 π η E z [ 1 - I 0 ( κ r ) I 0 ( κ a ) ] v_{z}\left(r\right)=\frac{\epsilon\phi_{0}}{4\pi\eta}E_{z}\left[1-\frac{I_{0}% \left(\kappa r\right)}{I_{0}\left(\kappa a\right)}\right]

Narayana_number.html

  1. N ( n , k ) = 1 n ( n k ) ( n k - 1 ) . N(n,k)=\frac{1}{n}{n\choose k}{n\choose k-1}.
  2. N ( n , 1 ) + N ( n , 2 ) + N ( n , 3 ) + + N ( n , n ) = C n . N(n,1)+N(n,2)+N(n,3)+\cdots+N(n,n)=C_{n}.
  3. B n B_{n}
  4. B n B_{n}
  5. S ( n , k ) S(n,k)
  6. C n C_{n}
  7. N ( n , k ) N(n,k)

Natural_exponential_family.html

  1. f X ( x | θ ) = h ( x ) exp ( η ( θ ) T ( x ) - A ( θ ) ) , f_{X}(x|\theta)=h(x)\ \exp\Big(\ \eta(\theta)T(x)-A(\theta)\ \Big)\,\!,
  2. h ( x ) h(x)
  3. A ( θ ) A(\theta)
  4. f X ( x | θ ) = h ( x ) exp ( θ x - A ( θ ) ) . f_{X}(x|\theta)=h(x)\ \exp\Big(\ \theta x-A(\theta)\ \Big)\,\!.
  5. 𝐱 𝒳 p \mathbf{x}\in\mathcal{X}\subseteq\mathbb{R}^{p}
  6. f X ( 𝐱 | s y m b o l θ ) = h ( 𝐱 ) exp ( s y m b o l θ T 𝐱 - A ( s y m b o l θ ) ) , f_{X}(\mathbf{x}|symbol\theta)=h(\mathbf{x})\ \exp\Big(symbol\theta^{\rm T}% \mathbf{x}-A(symbol\theta)\ \Big)\,\!,
  7. s y m b o l θ p . symbol\theta\in\mathbb{R}^{p}.
  8. M X ( 𝐭 ) = exp ( A ( s y m b o l θ + 𝐭 ) - A ( s y m b o l θ ) ) . M_{X}(\mathbf{t})=\exp\Big(\ A(symbol\theta+\mathbf{t})-A(symbol\theta)\ \Big)\,.
  9. K X ( 𝐭 ) = A ( s y m b o l θ + 𝐭 ) - A ( s y m b o l θ ) . K_{X}(\mathbf{t})=A(symbol\theta+\mathbf{t})-A(symbol\theta)\,.
  10. r r
  11. θ = log ( λ ) \theta=\log(\lambda)
  12. f ( k ; θ ) = 1 k ! exp ( θ k - exp ( θ ) ) , f(k;\theta)=\frac{1}{k!}\exp\Big(\ \theta\ k-\exp(\theta)\ \Big)\ ,
  13. θ \theta\in\mathbb{R}
  14. h ( k ) = 1 k ! h(k)=\frac{1}{k!}
  15. A ( θ ) = exp ( θ ) . A(\theta)=\exp(\theta)\ .
  16. K X ( t ) = A ( θ + t ) - A ( θ ) . K_{X}(t)=A(\theta+t)-A(\theta)\,.
  17. K 1 = d d θ A ( t ) . K_{1}=\frac{d}{d\theta}A(t)\,.
  18. μ 1 = κ 1 = E [ X ] = K X ( 0 ) = A ( θ ) . \mu_{1}^{\prime}=\kappa_{1}=\mathrm{E}[X]=K^{\prime}_{X}(0)=A^{\prime}(\theta)\,.
  19. Var [ X ] = μ 2 = κ 2 + κ 1 2 , \mathrm{Var}[X]=\mu_{2}^{\prime}=\kappa_{2}+\kappa_{1}^{2}\,,
  20. Var [ X ] = K X ′′ ( 0 ) = A ′′ ( θ ) . \mathrm{Var}[X]=K^{\prime\prime}_{X}(0)=A^{\prime\prime}(\theta)\,.
  21. K n = d ( n ) d θ ( n ) A ( t ) . K_{n}=\frac{d^{(n)}}{d\theta^{(n)}}A(t)\,.
  22. X 1 , , X n X_{1},\ldots,X_{n}
  23. i = 1 n X i \sum_{i=1}^{n}X_{i}\,
  24. V a r ( X ) = V ( μ ) . Var(X)=V(\mu).
  25. X N E F [ μ , V ( μ ) ] . X\sim NEF[\mu,V(\mu)].
  26. E [ X ] = A ( s y m b o l θ ) \mathrm{E}[X]=\nabla A(symbol\theta)\,
  27. Cov [ X ] = T A ( s y m b o l θ ) , \mathrm{Cov}[X]=\nabla\nabla^{\rm T}A(symbol\theta)\,,
  28. \nabla
  29. T \nabla\nabla^{\rm T}
  30. V a r ( X ) = V ( μ ) = ν 0 + ν 1 μ + ν 2 μ 2 . Var(X)=V(\mu)=\nu_{0}+\nu_{1}\mu+\nu_{2}\mu^{2}.
  31. X N ( μ , σ 2 ) X\,\sim N(\mu,\sigma^{2})
  32. V a r ( X ) = V ( μ ) = σ 2 Var(X)=V(\mu)=\sigma^{2}
  33. X P o i s ( μ ) X\,\sim Pois(\mu)
  34. V a r ( X ) = V ( μ ) = μ Var(X)=V(\mu)=\mu
  35. X G a m ( r , λ ) X\,\sim Gam(r,\lambda)
  36. μ = r λ \mu=r\lambda
  37. V a r ( X ) = V ( μ ) = μ 2 / r Var(X)=V(\mu)=\mu^{2}/r
  38. X B i n ( n , p ) X\,\sim Bin(n,p)
  39. μ = n p \mu=np
  40. V a r ( X ) = n p ( 1 - p ) Var(X)=np(1-p)
  41. V ( X ) = - n p 2 + n p = - μ 2 / n + μ . V(X)=-np^{2}+np=-\mu^{2}/n+\mu.
  42. X N e g B i n ( n , p ) X\sim NegBin(n,p)
  43. μ = n p / ( 1 - p ) \mu=np/(1-p)
  44. V ( μ ) = μ 2 / n + μ . V(\mu)=\mu^{2}/n+\mu.
  45. V ( μ ) = μ 2 / n + n V(\mu)=\mu^{2}/n+n
  46. μ > 0. \mu>0.
  47. X 1 , , X n X_{1},\ldots,X_{n}
  48. Y = i = 1 n ( X i - b ) / c Y=\sum_{i=1}^{n}(X_{i}-b)/c\,
  49. μ * = n ( μ - b ) / c \mu^{*}=n(\mu-b)/c\,
  50. V a r ( X ) = V ( μ ) = ν 0 + ν 1 μ + ν 2 μ 2 , Var(X)=V(\mu)=\nu_{0}+\nu_{1}\mu+\nu_{2}\mu^{2},
  51. V a r ( Y ) = V * ( μ * ) = ν 0 * + ν 1 * μ + ν 2 * μ 2 , Var(Y)=V^{*}(\mu^{*})=\nu^{*}_{0}+\nu^{*}_{1}\mu+\nu^{*}_{2}\mu^{2},
  52. ν 0 * = n V ( b ) / c 2 , \nu^{*}_{0}=nV(b)/c^{2}\,,
  53. ν 1 * = V ( b ) / c , \nu^{*}_{1}=V^{\prime}(b)/c\,,
  54. ν 2 * / n = ν 2 / n . \nu^{*}_{2}/n=\nu_{2}/n\,.
  55. X 1 X_{1}
  56. X 2 X_{2}
  57. Y = X 1 + X 2 Y=X_{1}+X_{2}
  58. X 1 X_{1}
  59. f ( X 1 | Y ) f(X_{1}|Y)
  60. X 1 X_{1}
  61. X 2 X_{2}
  62. f ( X 1 | Y ) f(X_{1}|Y)
  63. X | μ X|\mu

Natural_neighbor.html

  1. G ( x , y ) = i = 1 n w i f ( x i , y i ) G(x,y)=\sum^{n}_{i=1}{w_{i}f(x_{i},y_{i})}
  2. G ( x , y ) G(x,y)
  3. ( x , y ) (x,y)
  4. w i w_{i}
  5. f ( x i , y i ) f(x_{i},y_{i})
  6. ( x i , y i ) (x_{i},y_{i})
  7. w i w_{i}
  8. ( x , y ) (x,y)

Navarro–Frenk–White_profile.html

  1. ρ ( r ) = ρ 0 r R s ( 1 + r R s ) 2 \rho(r)=\frac{\rho_{0}}{\frac{r}{R_{s}}\left(1~{}+~{}\frac{r}{R_{s}}\right)^{2}}
  2. M = 0 R max 4 π r 2 ρ ( r ) d r = 4 π ρ 0 R s 3 [ ln ( R s + R max R s ) - R max R s + R max ] M=\int_{0}^{R_{\max}}4\pi r^{2}\rho(r)\,dr=4\pi\rho_{0}R_{s}^{3}\left[\ln\left% (\frac{R_{s}+R_{\max}}{R_{s}}\right)-\frac{R_{\max}}{R_{s}+R_{\max}}\right]
  3. R vir = c R s R_{\mathrm{vir}}=cR_{s}
  4. R 200 R_{200}
  5. M = 0 R vir 4 π r 2 ρ ( r ) d r = 4 π ρ 0 R s 3 [ ln ( 1 + c ) - c 1 + c ] M=\int_{0}^{R_{\mathrm{vir}}}4\pi r^{2}\rho(r)\,dr=4\pi\rho_{0}R_{s}^{3}\left[% \ln(1+c)-\frac{c}{1+c}\right]
  6. 0 R max 4 π r 2 ρ ( r ) 2 d r = 4 π 3 R s 3 ρ 0 2 [ 1 - R s 3 ( R s + R max ) 3 ] \int_{0}^{R_{\max}}4\pi r^{2}\rho(r)^{2}\,dr=\frac{4\pi}{3}R_{s}^{3}\rho_{0}^{% 2}\left[1-\frac{R_{s}^{3}}{(R_{s}+R_{\max})^{3}}\right]
  7. ρ 2 R max = R s 3 ρ 0 2 R max 3 [ 1 - R s 3 ( R s + R max ) 3 ] \langle\rho^{2}\rangle_{R_{\max}}=\frac{R_{s}^{3}\rho_{0}^{2}}{R_{\max}^{3}}% \left[1-\frac{R_{s}^{3}}{(R_{s}+R_{\max})^{3}}\right]
  8. ρ 2 R vir = ρ 0 2 c 3 [ 1 - 1 ( 1 + c ) 3 ] ρ 0 2 c 3 \langle\rho^{2}\rangle_{R_{\mathrm{vir}}}=\frac{\rho_{0}^{2}}{c^{3}}\left[1-% \frac{1}{(1+c)^{3}}\right]\approx\frac{\rho_{0}^{2}}{c^{3}}
  9. ρ 2 R s = 7 8 ρ 0 2 \langle\rho^{2}\rangle_{R_{s}}=\frac{7}{8}\rho_{0}^{2}

Navigation_function.html

  1. X X
  2. X g X X_{g}\subset X
  3. ϕ ( x ) \phi(x)
  4. ϕ ( x ) = 0 x X g \phi(x)=0\ \forall x\in X_{g}
  5. ϕ ( x ) = \phi(x)=\infty
  6. X G {X_{G}}
  7. x x
  8. x X X G x\in X\setminus{X_{G}}
  9. x x^{\prime}
  10. ϕ ( x ) < ϕ ( x ) \phi(x^{\prime})<\phi(x)
  11. J J
  12. minimize J ( x 1 : T , u 1 : T ) = T L ( x t , u t , t ) d t \,\text{minimize }J(x_{1:T},u_{1:T})=\int\limits_{T}L(x_{t},u_{t},t)dt
  13. subject to x t ˙ = f ( x t , u t ) \,\text{subject to }\dot{x_{t}}=f(x_{t},u_{t})
  14. x x
  15. u u
  16. L L
  17. x x
  18. u u
  19. f f
  20. ϕ ( x t ) = min u t U ( x t ) { L ( x t , u t ) + ϕ ( f ( x t , u t ) ) } \displaystyle\phi(x_{t})=\min_{u_{t}\in U(x_{t})}\Big\{L(x_{t},u_{t})+\phi(f(x% _{t},u_{t}))\Big\}
  21. ϕ ( x ) = 0 x X g \phi(x)=0\ \forall x\in X_{g}
  22. ϕ ( x ) = \phi(x)=\infty
  23. X G {X_{G}}
  24. x x
  25. x X X G x\in X\setminus{X_{G}}
  26. x x^{\prime}
  27. ϕ ( x ) < ϕ ( x ) \phi(x^{\prime})<\phi(x)
  28. ϕ ( x t ) = min u t U ( x t ) { L ( x t , u t ) + ϕ ( f ( x t , u t ) ) } \displaystyle\phi(x_{t})=\min_{u_{t}\in U(x_{t})}\Big\{L(x_{t},u_{t})+\phi(f(x% _{t},u_{t}))\Big\}
  29. J ( x t , u t ) J(x_{t},u_{t})
  30. f f
  31. R ( x t , u t ) R(x_{t},u_{t})
  32. P ( x t + 1 | x t , u t ) P(x_{t+1}|x_{t},u_{t})

Neferneferuaten_Tasherit.html

  1. γ \gamma
  2. α \alpha

Neferneferure.html

  1. α \alpha
  2. γ \gamma
  3. α \alpha

Negative_base.html

  1. b \scriptstyle b
  2. - r \scriptstyle-r
  3. r \scriptstyle r
  4. b \scriptstyle b
  5. b 4 \scriptstyle b^{4}
  6. b 3 \scriptstyle b^{3}
  7. b 2 \scriptstyle b^{2}
  8. b 1 \scriptstyle b^{1}
  9. b 0 \scriptstyle b^{0}
  10. - r -r
  11. a a
  12. a = i = 0 n d i ( - r ) i a=\sum_{i=0}^{n}d_{i}(-r)^{i}
  13. d k \scriptstyle d_{k}
  14. r - 1 \scriptstyle r-1
  15. d n \scriptstyle d_{n}
  16. > 0 \scriptstyle>0
  17. n = 0 \scriptstyle n=0
  18. - r \scriptstyle-r
  19. a \scriptstyle a
  20. d n d n - 1 d 1 d 0 \scriptstyle d_{n}d_{n-1}\ldots d_{1}d_{0}
  21. - r \scriptstyle-r
  22. r r
  23. 17 = 2 4 + 2 0 = ( - 2 ) 4 + ( - 2 ) 0 17=2^{4}+2^{0}=(-2)^{4}+(-2)^{0}
  24. - r \scriptstyle-r
  25. - r \scriptstyle-r
  26. - r \scriptstyle-r
  27. - r \scriptstyle-r
  28. 0 , 1 , r - 1 \scriptstyle 0,1,\ldots r-1
  29. a / b = c \scriptstyle a/b=c
  30. d d
  31. b c + d = a \scriptstyle bc+d=a
  32. 146 \displaystyle 146
  33. a = ( - r ) c + d = ( - r ) c + d - r + r = ( - r ) ( c + 1 ) + ( d + r ) \scriptstyle a=(-r)c+d=(-r)c+d-r+r=(-r)(c+1)+(d+r)
  34. | d | < r \scriptstyle|d|<r
  35. ( d + r ) \scriptstyle(d+r)
  36. r r
  37. { 0 , 1 , 2 , 3 } \scriptstyle\in\;\{0,1,2,3\}
  38. - r \scriptstyle-r
  39. 0. ( 02 ) ( - 3 ) = 1 4 = 1. ( 20 ) ( - 3 ) 0.(02)\ldots_{(-3)}=\frac{1}{4}=1.(20)\ldots_{(-3)}
  40. a r + 1 b ( r + 1 ) \frac{ar+1}{b(r+1)}

Negative_pedal_curve.html

  1. X [ x , y ] = ( x 2 - y 2 ) y - 2 x y x x y - y x X[x,y]=\frac{(x^{2}-y^{2})y^{\prime}-2xyx^{\prime}}{xy^{\prime}-yx^{\prime}}
  2. Y [ x , y ] = ( x 2 - y 2 ) x + 2 x y y x y - y x Y[x,y]=\frac{(x^{2}-y^{2})x^{\prime}+2xyy^{\prime}}{xy^{\prime}-yx^{\prime}}

Negligible_function.html

  1. μ ( x ) : \mu(x):\mathbb{N}{\rightarrow}\mathbb{R}
  2. | μ ( x ) | < 1 x c . |\mu(x)|<\frac{1}{x^{c}}.
  3. μ ( x ) : \mu(x):\mathbb{N}{\rightarrow}\mathbb{R}
  4. | μ ( x ) | < 1 poly ( x ) . |\mu(x)|<\frac{1}{\,\text{poly}(x)}.
  5. \mathbb{R}
  6. f ( x ) : f(x):\mathbb{R}{\rightarrow}\mathbb{R}
  7. x = x 0 x=x_{0}
  8. ϵ > 0 \epsilon>0
  9. δ > 0 \delta>0
  10. | x - x 0 | < δ |x-x_{0}|<\delta
  11. | f ( x ) - f ( x 0 ) | < ϵ . |f(x)-f(x_{0})|<\epsilon.
  12. x 0 = x_{0}=\infty
  13. f ( x 0 ) = 0 f(x_{0})=0
  14. μ ( x ) : \mu(x):\mathbb{R}{\rightarrow}\mathbb{R}
  15. x x
  16. ϵ > 0 \epsilon>0
  17. N ϵ N_{\epsilon}
  18. x > N ϵ x>N_{\epsilon}
  19. | μ ( x ) | < ϵ . |\mu(x)|<\epsilon\,.
  20. ϵ > 0 \epsilon>0
  21. 1 / x c 1/x^{c}
  22. c > 0 c>0
  23. 1 / p o l y ( x ) 1/poly(x)
  24. p o l y ( x ) poly(x)
  25. ϵ > 0 \epsilon>0
  26. 1 / p o l y ( x ) 1/poly(x)
  27. x x
  28. n n
  29. n n
  30. x x
  31. n n
  32. x x

Nemytskii_operator.html

  1. F ( u ) ( x ) = f ( x , u ( x ) ) . F(u)(x)=f\big(x,u(x)\big).
  2. 1 p + 1 q = 1. \frac{1}{p}+\frac{1}{q}=1.
  3. | f ( x , u ) | C | u | p - 1 + g ( x ) . \big|f(x,u)\big|\leq C|u|^{p-1}+g(x).

Nested_RAID_levels.html

  1. ( N / 2 ) S min (N/2)\cdot S_{\mathrm{min}}
  2. N N
  3. S min S_{\mathrm{min}}
  4. n n
  5. 1 1 / n = 1 1 / 3 = 2 / 3 67 % 1−1/n=1−1/3=2/3≈67\%
  6. n n
  7. r r
  8. 1 - ( 1 - r ) n - n r ( 1 - r ) n - 1 \displaystyle 1-(1-r)^{n}-nr(1-r)^{n-1}
  9. s t r i p e s / n stripes/n
  10. m 1 m−1
  11. n n
  12. ( n / s p a n s ) (n/spans)

Net_volatility.html

  1. σ N = ν L σ L - ν S σ S ν L - ν S \sigma_{N}=\frac{\nu_{L}\sigma_{L}-\nu_{S}\sigma_{S}}{\nu_{L}-\nu_{S}}\,
  2. σ N \sigma_{N}
  3. σ L \sigma_{L}
  4. ν L \nu_{L}
  5. σ S \sigma_{S}
  6. ν S \nu_{S}
  7. 14.1 % 4.3 - 18.3 % 2.3 4.3 - 2.3 = 9.27 % \frac{14.1\%\cdot 4.3-18.3\%\cdot 2.3}{4.3-2.3}=9.27\%

Network_motif.html

  1. G = ( V , E ) G=(V,E)
  2. G = ( E , V ) G′=(E′,V′)
  3. G G′
  4. G G
  5. G G G′⊆G
  6. V V V′⊆V
  7. E E ( V × V ) E′⊆E∩(V′×V′)
  8. G G G′⊆G
  9. G G′
  10. u , v E ‹u,v›∈E
  11. u , v V u,v∈V′
  12. G G′
  13. G G
  14. G G′
  15. G G
  16. G G G′↔G
  17. f : V V f:V′→V
  18. u , v E f ( u ) , f ( v ) E ‹u,v›∈E′⇔‹f(u),f(v)›∈E
  19. u , v V u,v∈V′
  20. f f
  21. G G
  22. G G′
  23. G G G″⊂G
  24. G G″
  25. G G′
  26. G G′
  27. G G
  28. G G′
  29. G G
  30. G G′
  31. G G
  32. G G
  33. Ω ( G ) Ω(G)
  34. G G
  35. N N
  36. Ω ( G ) Ω(G)
  37. G G′
  38. G G
  39. G G′
  40. G G
  41. N N
  42. 1 i N 1≤i≤N
  43. G G′
  44. G G
  45. G G′
  46. G G
  47. R ( G ) Ω ( R ) R(G)⊆Ω(R)
  48. R ( G ) = N R(G)=N
  49. Z ( G ) = F G ( G ) - μ R ( G ) σ R ( G ) Z(G^{\prime})=\frac{F_{G}(G^{\prime})-\mu_{R}(G^{\prime})}{\sigma_{R}(G^{% \prime})}
  50. R ( G ) R(G)
  51. Z ( G ) Z(G′)
  52. G G′
  53. P ( G ) = 1 N i = 1 N δ ( c ( i ) ) ; c ( i ) : F R i ( G ) F G ( G ) P(G^{\prime})=\frac{1}{N}\sum_{i=1}^{N}\delta(c(i));c(i):F_{R}^{i}(G^{\prime})% \geq F_{G}(G^{\prime})
  54. N N
  55. i i
  56. δ ( c ( i ) ) δ(c(i))
  57. c ( i ) c(i)
  58. G G′
  59. G G
  60. C G ( G ) = F G ( G ) i F G ( G i ) C_{G}(G^{\prime})=\frac{F_{G}(G^{\prime})}{\sum_{i}F_{G}(G_{i})}
  61. i i
  62. F < s u b > 1 F<sub>1

Neural_cryptography.html

  1. x i j { - 1 , 0 , + 1 } x_{ij}\in\left\{-1,0,+1\right\}
  2. w i j { - L , , 0 , , + L } w_{ij}\in\left\{-L,...,0,...,+L\right\}
  3. σ i = sgn ( j = 1 N w i j x i j ) \sigma_{i}=\operatorname{sgn}(\sum_{j=1}^{N}w_{ij}x_{ij})
  4. sgn ( x ) = { - 1 if x < 0 , 0 if x = 0 , 1 if x > 0. \operatorname{sgn}(x)=\begin{cases}-1&\,\text{if }x<0,\\ 0&\,\text{if }x=0,\\ 1&\,\text{if }x>0.\end{cases}
  5. τ = i = 1 K σ i \tau=\prod_{i=1}^{K}\sigma_{i}
  6. w i + = w i + σ i x i Θ ( σ i τ ) Θ ( τ A τ B ) w_{i}^{+}=w_{i}+\sigma_{i}x_{i}\Theta(\sigma_{i}\tau)\Theta(\tau^{A}\tau^{B})
  7. w i + = w i - σ i x i Θ ( σ i τ ) Θ ( τ A τ B ) w_{i}^{+}=w_{i}-\sigma_{i}x_{i}\Theta(\sigma_{i}\tau)\Theta(\tau^{A}\tau^{B})
  8. w i + = w i + x i Θ ( σ i τ ) Θ ( τ A τ B ) w_{i}^{+}=w_{i}+x_{i}\Theta(\sigma_{i}\tau)\Theta(\tau^{A}\tau^{B})
  9. x i j { 0 , 1 } x_{ij}\in\left\{0,1\right\}
  10. w i j { 0 , 1 } w_{ij}\in\left\{0,1\right\}
  11. σ i = θ N ( j = 1 N w i j x i j ) \sigma_{i}=\theta_{N}(\sum_{j=1}^{N}w_{ij}\oplus x_{ij})
  12. θ N ( x ) \theta_{N}(x)
  13. θ N ( x ) = { 0 if x N / 2 , 1 if x > N / 2. \theta_{N}(x)=\begin{cases}0&\,\text{if }x\leq N/2,\\ 1&\,\text{if }x>N/2.\end{cases}
  14. τ = i = 1 K σ i \tau=\bigoplus_{i=1}^{K}\sigma_{i}

Neutral_axis.html

  1. γ x y = γ z x = τ x y = τ x z = 0 \gamma_{xy}=\gamma_{zx}=\tau_{xy}=\tau_{xz}=0
  2. γ \gamma
  3. τ \tau
  4. ϵ x ( y ) = L ( y ) - L L = θ ( ρ - y ) - θ ρ θ ρ = - y θ ρ θ = - y ρ \epsilon_{x}(y)=\frac{L(y)-L}{L}=\frac{\theta\,(\rho\,-y)-\theta\rho\,}{\theta% \rho\,}=\frac{-y\theta}{\rho\theta}=\frac{-y}{\rho}
  5. ϵ x \epsilon_{x}
  6. ϵ m \epsilon_{m}
  7. ϵ m = c ρ \epsilon_{m}=\frac{c}{\rho}
  8. ρ = c ϵ m \rho=\frac{c}{\epsilon_{m}}
  9. ϵ x ( y ) = - ϵ m y c \epsilon_{x}(y)=\frac{-\epsilon_{m}y}{c}
  10. σ x = E ϵ x \sigma_{x}=E\epsilon_{x}\,
  11. E ϵ x ( y ) = - E ϵ m y c E\epsilon_{x}(y)=\frac{-E\epsilon_{m}y}{c}
  12. σ x ( y ) = - σ m y c \sigma_{x}(y)=\frac{-\sigma_{m}y}{c}
  13. σ x d A = 0 \int\sigma_{x}dA=0
  14. - σ m y c d A = 0 \int\frac{-\sigma_{m}y}{c}dA=0
  15. y d A = 0 \int ydA=0
  16. τ = ( T * Q ) ÷ ( w * I ) \tau=(T*Q)\div(w*I)

Neutral_vector.html

  1. X i X_{i}
  2. X 1 , X 2 , , X k X_{1},X_{2},\ldots,X_{k}
  3. X i X_{i}
  4. X = X 1 , , X k X=X_{1},\ldots,X_{k}
  5. i = 1 k X i = 1. \sum_{i=1}^{k}X_{i}=1.
  6. X i X_{i}
  7. X 1 X_{1}
  8. X X
  9. X 1 X_{1}
  10. X 1 X_{1}
  11. X 1 * = X 2 / ( 1 - X 1 ) , X 3 / ( 1 - X 1 ) , , X k / ( 1 - X 1 ) . X^{*}_{1}=X_{2}/(1-X_{1}),X_{3}/(1-X_{1}),\ldots,X_{k}/(1-X_{1}).
  12. X 2 X_{2}
  13. X 2 / ( 1 - X 1 ) X_{2}/(1-X_{1})
  14. X 2 / ( 1 - X 1 ) X_{2}/(1-X_{1})
  15. X 1 , 2 * = X 3 / ( 1 - X 1 - X 2 ) , X 4 / ( 1 - X 1 - X 2 ) , , X k / ( 1 - X 1 - X 2 ) . X^{*}_{1,2}=X_{3}/(1-X_{1}-X_{2}),X_{4}/(1-X_{1}-X_{2}),\ldots,X_{k}/(1-X_{1}-% X_{2}).
  16. X 2 X_{2}
  17. Y = X 2 , X 3 , , X k Y=X_{2},X_{3},\ldots,X_{k}
  18. X j X_{j}
  19. X 1 , X j - 1 X_{1},\ldots X_{j-1}
  20. X 1 , , j * = X j + 1 / ( 1 - X 1 - - X j ) , X k / ( 1 - X 1 - - X j ) . X^{*}_{1,...,j}=X_{j+1}/(1-X_{1}-\cdots-X_{j}),\ldots X_{k}/(1-X_{1}-\cdots-X_% {j}).
  21. X = ( X 1 , , X K ) Dir ( α ) X=(X_{1},\ldots,X_{K})\sim\operatorname{Dir}(\alpha)
  22. X X

Neutron_supermirror.html

  1. n θ c n\cdot\theta_{c}
  2. n n
  3. θ c \theta_{c}

Newton's_inequalities.html

  1. σ k \sigma_{k}
  2. S k = σ k ( n k ) S_{k}=\frac{\sigma_{k}}{{\left({{n}\atop{k}}\right)}}
  3. S k - 1 S k + 1 S k 2 S_{k-1}S_{k+1}\leq S_{k}^{2}

Newton's_theorem_of_revolving_orbits.html

  1. ω 2 = d θ 2 d t = k d θ 1 d t = k ω 1 . \omega_{2}=\frac{d\theta_{2}}{dt}=k\frac{d\theta_{1}}{dt}=k\omega_{1}.
  2. F 2 ( r ) - F 1 ( r ) = L 1 2 m r 3 ( 1 - k 2 ) F_{2}(r)-F_{1}(r)=\frac{L_{1}^{2}}{mr^{3}}\left(1-k^{2}\right)
  3. θ 1 ω 1 ( t ) d t . \theta_{1}\equiv\int\omega_{1}(t)\,dt.
  4. 1 r = A + B cos θ 1 \frac{1}{r}=A+B\cos\theta_{1}
  5. 1 r = A + B cos ( θ 2 k ) . \frac{1}{r}=A+B\cos\left(\frac{\theta_{2}}{k}\right).
  6. ω 1 = L 1 m r 2 ; \omega_{1}=\frac{L_{1}}{mr^{2}};
  7. 1 r = 1 b cos ( θ 1 - θ 0 ) \frac{1}{r}=\frac{1}{b}\cos\ (\theta_{1}-\theta_{0})
  8. F 2 ( r ) = μ r 3 F_{2}(r)=\frac{\mu}{r^{3}}
  9. 1 r = 1 b cos ( θ 2 - θ 0 k ) \frac{1}{r}=\frac{1}{b}\cos\ \left(\frac{\theta_{2}-\theta_{0}}{k}\right)
  10. k 2 = 1 - m μ L 1 2 k^{2}=1-\frac{m\mu}{L_{1}^{2}}
  11. 1 r = 1 b cosh ( θ 0 - θ 2 λ ) \frac{1}{r}=\frac{1}{b}\cosh\ \left(\frac{\theta_{0}-\theta_{2}}{\lambda}\right)
  12. λ 2 = m μ L 1 2 - 1 \lambda^{2}=\frac{m\mu}{L_{1}^{2}}-1
  13. 1 r = A θ 2 + ε \frac{1}{r}=A\theta_{2}+\varepsilon
  14. μ = L 1 2 m \mu=\frac{L_{1}^{2}}{m}
  15. F ( r ) = C ( r ) R r 3 F(r)=\frac{C(r)}{Rr^{3}}
  16. 1 k 2 = ( R C ) d C d r | r = R \frac{1}{k^{2}}=\left(\frac{R}{C}\right)\left.\frac{dC}{dr}\right|_{r=R}
  17. C ( r ) a r m + b r n C(r)\propto ar^{m}+br^{n}
  18. k = a + b a m + b n k=\sqrt{\frac{a+b}{am+bn}}
  19. F ( r ) = - G M m r 2 + 4 / 243 F(r)=-\frac{GMm}{r^{2+4/243}}
  20. F ( r ) = A r 2 + B r F(r)=\frac{A}{r^{2}}+Br
  21. 1 r 2 ( t ) = a r 1 ( t ) + b \frac{1}{r_{2}(t)}=\frac{a}{r_{1}(t)}+b
  22. a r 2 1 - b r 2 = g ( θ 2 k ) \frac{ar_{2}}{1-br_{2}}=g\left(\frac{\theta_{2}}{k}\right)
  23. F 2 ( r 2 ) = a 3 ( 1 - b r 2 ) 2 F 1 ( a r 2 1 - b r 2 ) + L 2 m r 3 ( 1 - k 2 ) - b L 2 m r 2 F_{2}(r_{2})=\frac{a^{3}}{\left(1-br_{2}\right)^{2}}F_{1}\left(\frac{ar_{2}}{1% -br_{2}}\right)+\frac{L^{2}}{mr^{3}}\left(1-k^{2}\right)-\frac{bL^{2}}{mr^{2}}
  24. d A 1 = 1 2 r 2 d θ 1 dA_{1}=\frac{1}{2}r^{2}d\theta_{1}
  25. d A 1 d t = 1 2 r 2 d θ 1 d t = constant \frac{dA_{1}}{dt}=\frac{1}{2}r^{2}\frac{d\theta_{1}}{dt}=\mathrm{constant}
  26. h 1 = L 1 m = r v 1 = r 2 d θ 1 d t = 2 d A 1 d t h_{1}=\frac{L_{1}}{m}=rv_{1}=r^{2}\frac{d\theta_{1}}{dt}=2\frac{dA_{1}}{dt}
  27. θ 2 ( t ) = k θ 1 ( t ) \theta_{2}(t)=k\theta_{1}(t)\,\!
  28. h 2 = 2 d A 2 d t = r 2 d θ 2 d t = k r 2 d θ 1 d t = 2 k d A 1 d t = k h 1 h_{2}=2\frac{dA_{2}}{dt}=r^{2}\frac{d\theta_{2}}{dt}=kr^{2}\frac{d\theta_{1}}{% dt}=2k\frac{dA_{1}}{dt}=kh_{1}
  29. F 2 ( r ) - F 1 ( r ) = m h 1 2 - h 2 2 r 3 F_{2}(r)-F_{1}(r)=m\frac{h_{1}^{2}-h_{2}^{2}}{r^{3}}
  30. ω 2 = d θ 2 d t = k d θ 1 d t = k ω 1 \omega_{2}=\frac{d\theta_{2}}{dt}=k\frac{d\theta_{1}}{dt}=k\omega_{1}
  31. L 2 = m r 2 ω 2 = m r 2 k ω 1 = k L 1 L_{2}=mr^{2}\omega_{2}=mr^{2}k\omega_{1}=kL_{1}\,\!
  32. m d 2 r d t 2 - m r ω 2 = m d 2 r d t 2 - L 2 m r 3 = F ( r ) m\frac{d^{2}r}{dt^{2}}-mr\omega^{2}=m\frac{d^{2}r}{dt^{2}}-\frac{L^{2}}{mr^{3}% }=F(r)
  33. m d 2 r d t 2 = F 1 ( r ) + L 1 2 m r 3 = F 2 ( r ) + L 2 2 m r 3 = F 2 ( r ) + k 2 L 1 2 m r 3 m\frac{d^{2}r}{dt^{2}}=F_{1}(r)+\frac{L_{1}^{2}}{mr^{3}}=F_{2}(r)+\frac{L_{2}^% {2}}{mr^{3}}=F_{2}(r)+\frac{k^{2}L_{1}^{2}}{mr^{3}}
  34. F 2 ( r ) = F 1 ( r ) + L 1 2 m r 3 ( 1 - k 2 ) F_{2}(r)=F_{1}(r)+\frac{L_{1}^{2}}{mr^{3}}\left(1-k^{2}\right)
  35. F ( r ) = - d V d r F(r)=-\frac{dV}{dr}
  36. - d V 2 d r = - d V 1 d r + L 1 2 m r 3 ( 1 - k 2 ) -\frac{dV_{2}}{dr}=-\frac{dV_{1}}{dr}+\frac{L_{1}^{2}}{mr^{3}}\left(1-k^{2}\right)
  37. V 2 ( r ) = V 1 ( r ) + L 1 2 2 m r 2 ( 1 - k 2 ) V_{2}(r)=V_{1}(r)+\frac{L_{1}^{2}}{2mr^{2}}\left(1-k^{2}\right)

Neyman_construction.html

  1. C C\,
  2. C C\,

Néron_model.html

  1. A R ( R ) A K ( K ) A_{R}(R)\to A_{K}(K)

Néron–Tate_height.html

  1. h L h_{L}
  2. L L
  3. A A
  4. h ^ L ( P ) = lim N h L ( N P ) N 2 \hat{h}_{L}(P)=\lim_{N\rightarrow\infty}\frac{h_{L}(NP)}{N^{2}}
  5. h ^ L ( P ) = h L ( P ) + O ( 1 ) , \hat{h}_{L}(P)=h_{L}(P)+O(1),
  6. O ( 1 ) O(1)
  7. P P
  8. L L
  9. [ - 1 ] * L = L [-1]^{*}L=L
  10. h ^ L ( P ) = lim N h L ( N P ) N \hat{h}_{L}(P)=\lim_{N\rightarrow\infty}\frac{h_{L}(NP)}{N}
  11. h ^ L ( P ) = h L ( P ) + O ( 1 ) \hat{h}_{L}(P)=h_{L}(P)+O(1)
  12. h ^ L \hat{h}_{L}
  13. L 2 = ( L [ - 1 ] * L ) ( L [ - 1 ] * L - 1 ) L^{\otimes 2}=(L\otimes[-1]^{*}L)\otimes(L\otimes[-1]^{*}L^{-1})
  14. h ^ L ( P ) = 1 2 h ^ L [ - 1 ] * L ( P ) + 1 2 h ^ L [ - 1 ] * L - 1 ( P ) is the unique quadratic function satisfying h ^ L ( P ) = h L ( P ) + O ( 1 ) and h ^ L ( 0 ) = 0. \hat{h}_{L}(P)=\frac{1}{2}\hat{h}_{L\otimes[-1]^{*}L}(P)+\frac{1}{2}\hat{h}_{L% \otimes[-1]^{*}L^{-1}}(P)\qquad\mbox{is the unique quadratic function % satisfying}~{}\qquad\hat{h}_{L}(P)=h_{L}(P)+O(1)\quad\mbox{and}~{}\quad\hat{h}% _{L}(0)=0.
  15. L L
  16. A A
  17. A A
  18. A ( K ) A(K)
  19. h ^ L \hat{h}_{L}
  20. A ( K ) A(K)\otimes\mathbb{R}
  21. h ^ \hat{h}
  22. A × A ^ A\times\hat{A}
  23. A A
  24. h ^ \hat{h}
  25. P , Q = 1 2 ( h ^ ( P + Q ) - h ^ ( P ) - h ^ ( Q ) ) . \langle P,Q\rangle=\frac{1}{2}\bigl(\hat{h}(P+Q)-\hat{h}(P)-\hat{h}(Q)\bigr).
  26. Reg ( E / K ) = det ( P i , P j ) 1 i , j r , \operatorname{Reg}(E/K)=\det\bigl(\langle P_{i},P_{j}\rangle\bigr)_{1\leq i,j% \leq r},
  27. Reg ( A / K ) = det ( P i , η j P ) 1 i , j r . \operatorname{Reg}(A/K)=\det\bigl(\langle P_{i},\eta_{j}\rangle_{P}\bigr)_{1% \leq i,j\leq r}.
  28. h ^ ( P ) c ( K ) log ( Norm K / Disc ( E / K ) ) \hat{h}(P)\geq c(K)\log(\operatorname{Norm}_{K/\mathbb{Q}}\operatorname{Disc}(% E/K))\quad
  29. E / K E/K
  30. P E ( K ) . P\in E(K).
  31. h ^ ( P ) c ( E / K ) [ K ( P ) : K ] \hat{h}(P)\geq\frac{c(E/K)}{[K(P):K]}
  32. P E ( K ¯ ) . P\in E(\bar{K}).
  33. h ^ ( P ) c ( E / K ) / [ K ( P ) : K ] 3 + ϵ \hat{h}(P)\geq c(E/K)/[K(P):K]^{3+\epsilon}
  34. h ^ ( P ) c ( E / K ) / [ K ( P ) : K ] 1 + ϵ \hat{h}(P)\geq c(E/K)/[K(P):K]^{1+\epsilon}
  35. ϕ * L = L d \phi^{*}L=L^{\otimes d}
  36. h ^ V , ϕ , L ( P ) = lim n h V , L ( ϕ ( n ) ( P ) ) d n , \hat{h}_{V,\phi,L}(P)=\lim_{n\to\infty}\frac{h_{V,L}(\phi^{(n)}(P))}{d^{n}},
  37. h ^ V , ϕ , L ( P ) = 0 P is preperiodic for ϕ . \hat{h}_{V,\phi,L}(P)=0~{}~{}\Longleftrightarrow~{}~{}P~{}{\rm is~{}% preperiodic~{}for~{}}\phi.

NGC_7052.html

  1. 370 - 150 + 260 370_{-150}^{+260}

Nilpotent_orbit.html

  1. n × n n\times n
  2. λ 1 λ 2 λ r , \lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{r},
  3. λ \lambda
  4. A = [ x y z - x ] , ( x , y , z ) ( 0 , 0 , 0 ) A=\begin{bmatrix}x&y\\ z&-x\end{bmatrix},\quad(x,y,z)\neq(0,0,0)\quad{\;}
  5. x 2 + y z = 0 , x^{2}+yz=0,
  6. 2 × 2 2\times 2
  7. y - z y-z

Ninety-One_(solitaire).html

  1. 91 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 91=1+2+3+4+5+6+7+8+9+10+11+12+13
  2. 91 = 13 * 4 + 1 * 4 + 3 * 5 + 10 * 2 = 52 + 4 + 15 + 20 91=13*4+1*4+3*5+10*2=52+4+15+20

Nitrate_reductase_(cytochrome).html

  1. \rightleftharpoons

Nitrate_reductase_(NAD(P)H).html

  1. \rightleftharpoons

Nitrate_reductase_(NADH).html

  1. \rightleftharpoons

Nitrate_reductase_(NADPH).html

  1. \rightleftharpoons

Niven's_constant.html

  1. lim n 1 n j = 1 n H ( j ) = 1 + k = 2 ( 1 - 1 ζ ( k ) ) = 1.705211 \lim_{n\to\infty}\frac{1}{n}\sum_{j=1}^{n}H(j)=1+\sum_{k=2}^{\infty}\left(1-% \frac{1}{\zeta(k)}\right)=1.705211\dots\,
  2. j = 1 n h ( j ) = n + c n + o ( n ) \sum_{j=1}^{n}h(j)=n+c\sqrt{n}+o(\sqrt{n})\,
  3. c = ζ ( 3 2 ) ζ ( 3 ) , c=\frac{\zeta(\frac{3}{2})}{\zeta(3)},\,
  4. lim n 1 n j = 1 n h ( j ) = 1. \lim_{n\to\infty}\frac{1}{n}\sum_{j=1}^{n}h(j)=1.

No-teleportation_theorem.html

  1. | ψ |\psi\rangle
  2. | ψ |\psi\rangle

No-wandering-domain_theorem.html

  1. U , f ( U ) , f ( f ( U ) ) , , f n ( U ) , U,f(U),f(f(U)),\dots,f^{n}(U),\dots
  2. f n = f f f n . f^{n}=\underbrace{f\circ f\circ\cdots\circ f}_{n}.
  3. f ( z ) = z + 2 π sin ( z ) f(z)=z+2\pi\sin(z)
  4. f ( z ) = z + 2 π sin ( z ) f(z)=z+2\pi\sin(z)

Non-chaperonin_molecular_chaperone_ATPase.html

  1. \rightleftharpoons

Non-interactive_zero-knowledge_proof.html

  1. σ Setup ( 1 k ) \sigma\leftarrow\mathrm{Setup}(1^{k})
  2. y L y\in L
  3. π Prove ( σ , y , w ) \pi\leftarrow\mathrm{Prove}(\sigma,y,w)
  4. π \pi
  5. Verify ( σ , y , π ) = accept \mathrm{Verify}(\sigma,y,\pi)=\mathrm{accept}
  6. σ \sigma
  7. ( y , w ) R σ (y,w)\in R_{\sigma}
  8. σ \sigma
  9. σ Setup ( 1 k ) \sigma\in\mathrm{Setup}(1^{k})
  10. ( y , w ) R σ (y,w)\in R_{\sigma}
  11. σ Setup ( 1 k ) \sigma\in\mathrm{Setup}(1^{k})
  12. ( y , w ) R σ (y,w)\in R_{\sigma}
  13. P r [ π Prove ( σ , y , w ) : Verify ( σ , y , π ) = accept ] = 1 Pr[\pi\leftarrow\mathrm{Prove}(\sigma,y,w):\mathrm{Verify}(\sigma,y,\pi)=% \mathrm{accept}]=1
  14. y L y\not\in L
  15. P ~ \tilde{P}
  16. ν \nu
  17. P r [ σ Setup ( 1 k ) , ( y , π ) P ~ ( σ ) : y L Verify ( σ , y , π ) = accept ] = ν ( k ) . Pr[\sigma\leftarrow\mathrm{Setup}(1^{k}),(y,\pi)\leftarrow\tilde{P}(\sigma):y% \not\in L\land\mathrm{Verify}(\sigma,y,\pi)=\mathrm{accept}]=\nu(k)\;.
  18. ( Setup , Prove , Verify ) (\mathrm{Setup},\mathrm{Prove},\mathrm{Verify})
  19. Sim = ( Sim 1 , Sim 2 ) \mathrm{Sim}=(\mathrm{Sim}_{1},\mathrm{Sim}_{2})
  20. 𝒜 \mathcal{A}
  21. P r [ σ Setup ( 1 k ) : 𝒜 Prove ( σ , . , . ) ( σ ) = 1 ] P r [ ( σ , τ ) Sim 1 : 𝒜 Sim ( σ , τ , . , . ) ( σ ) = 1 ] Pr[\sigma\leftarrow\mathrm{Setup}(1^{k}):\mathcal{A}^{{\mathrm{Prove}}(\sigma,% .,.)}(\sigma)=1]\equiv Pr[(\sigma,\tau)\leftarrow\mathrm{Sim}_{1}:\mathcal{A}^% {{\mathrm{Sim}}(\sigma,\tau,.,.)}(\sigma)=1]
  22. Sim ( σ , τ , y , w ) \mathrm{Sim}(\sigma,\tau,y,w)
  23. Sim 2 ( σ , τ , y ) \mathrm{Sim_{2}}(\sigma,\tau,y)
  24. ( y , w ) R σ (y,w)\in R_{\sigma}

Noncentral_hypergeometric_distributions.html

  1. m 1 m_{1}
  2. m 2 m_{2}
  3. N = m 1 + m 2 N=m_{1}+m_{2}
  4. n n
  5. ω 1 \omega_{1}
  6. ω 2 \omega_{2}
  7. p 1 = m 1 ω 1 m 1 ω 1 + m 2 ω 2 . p_{1}=\frac{m_{1}\omega_{1}}{m_{1}\omega_{1}+m_{2}\omega_{2}}.
  8. m 1 m_{1}
  9. m 2 m_{2}
  10. ω = ω 1 / ω 2 \omega=\omega_{1}/\omega_{2}

Nonlinear_acoustics.html

  1. B / A B/A
  2. A A
  3. B B
  4. B / A B/A
  5. β = 1 + B 2 A \beta=1+\frac{B}{2A}
  6. ρ t + ( ρ 𝐮 ) = 0 \frac{\partial\rho}{\partial t}+\nabla(\rho\,\textbf{u})=0
  7. ρ ( 𝐮 t + 𝐮 𝐮 ) + p = ( λ + 2 μ ) ( 𝐮 ) \rho\left(\frac{\partial\,\textbf{u}}{\partial t}+\,\textbf{u}\cdot\nabla\,% \textbf{u}\right)+\nabla p=(\lambda+2\mu)\nabla(\nabla\cdot\,\textbf{u})
  8. ρ = 0 ε i ρ i \rho=\sum_{0}^{\infty}\varepsilon^{i}\rho_{i}
  9. p = ε ρ 1 c 0 2 ( 1 + ε B 2 ! A ρ 1 ρ 0 + O ( ε 2 ) ) p=\varepsilon\rho_{1}c_{0}^{2}\left(1+\varepsilon\frac{B}{2!A}\frac{\rho_{1}}{% \rho_{0}}+O(\varepsilon^{2})\right)
  10. 2 p - 1 c 0 2 2 p t 2 + δ c 0 4 3 p t 3 = - β ρ 0 c 0 4 2 p 2 t 2 \,\nabla^{2}p-\frac{1}{c_{0}^{2}}\frac{\partial^{2}p}{\partial t^{2}}+\frac{% \delta}{c_{0}^{4}}\frac{\partial^{3}p}{\partial t^{3}}=-\frac{\beta}{\rho_{0}c% _{0}^{4}}\frac{\partial^{2}p^{2}}{\partial t^{2}}
  11. p p
  12. c 0 c_{0}
  13. δ \delta
  14. β \beta
  15. ρ 0 \rho_{0}
  16. δ = 1 ρ 0 ( 4 3 μ + μ B ) + k ρ 0 ( 1 c v - 1 c p ) \,\delta=\frac{1}{\rho_{0}}\left(\frac{4}{3}\mu+\mu_{B}\right)+\frac{k}{\rho_{% 0}}\left(\frac{1}{c_{v}}-\frac{1}{c_{p}}\right)
  17. μ \mu
  18. μ B \mu_{B}
  19. k k
  20. c v c_{v}
  21. c p c_{p}
  22. p z - β ρ 0 c 0 3 p p τ = δ 2 c 0 3 2 p τ 2 \frac{\partial p}{\partial z}-\frac{\beta}{\rho_{0}c_{0}^{3}}p\frac{\partial p% }{\partial\tau}=\frac{\delta}{2c_{0}^{3}}\frac{\partial^{2}p}{\partial\tau^{2}}
  23. τ = t - z / c 0 \tau=t-z/c_{0}
  24. y t + u y x = d 2 y x 2 \frac{\partial y}{\partial t^{\prime}}+u\frac{\partial y}{\partial x}=d\frac{% \partial^{2}y}{\partial x^{2}}
  25. t = z c 0 t^{\prime}=\frac{z}{c_{0}}
  26. x = - ρ 0 c 0 2 β τ x=-\frac{\rho_{0}c_{0}^{2}}{\beta}\tau
  27. d = - ρ 0 c 0 2 β 2 δ d=-\frac{\rho_{0}c_{0}}{2\beta^{2}}\delta
  28. z z
  29. ( x , y ) (x,y)
  30. 2 p z τ = c 0 2 2 p + δ 2 c 0 3 3 p τ 3 + β 2 ρ 0 c 0 3 2 p 2 τ 2 \,\frac{\partial^{2}p}{\partial z\partial\tau}=\frac{c_{0}}{2}\nabla^{2}_{% \perp}p+\frac{\delta}{2c^{3}_{0}}\frac{\partial^{3}p}{\partial\tau^{3}}+\frac{% \beta}{2\rho_{0}c^{3}_{0}}\frac{\partial^{2}p^{2}}{\partial\tau^{2}}

Nonlinear_conjugate_gradient_method.html

  1. f ( x ) \displaystyle f(x)
  2. f ( x ) = A x - b 2 \displaystyle f(x)=\|Ax-b\|^{2}
  3. f f
  4. x f = 2 A ( A x - b ) = 0 \nabla_{x}f=2A^{\top}(Ax-b)=0
  5. A A x = A b \displaystyle A^{\top}Ax=A^{\top}b
  6. x f \nabla_{x}f
  7. f ( x ) \displaystyle f(x)
  8. N N
  9. x f \nabla_{x}f
  10. Δ x 0 = - x f ( x 0 ) \Delta x_{0}=-\nabla_{x}f(x_{0})
  11. α \displaystyle\alpha
  12. f \displaystyle f
  13. α 0 := arg min α f ( x 0 + α Δ x 0 ) \displaystyle\alpha_{0}:=\arg\min_{\alpha}f(x_{0}+\alpha\Delta x_{0})
  14. x 1 = x 0 + α 0 Δ x 0 \displaystyle x_{1}=x_{0}+\alpha_{0}\Delta x_{0}
  15. Δ x 0 \displaystyle\Delta x_{0}
  16. s n \displaystyle s_{n}
  17. s 0 = Δ x 0 \displaystyle s_{0}=\Delta x_{0}
  18. Δ x n = - x f ( x n ) \Delta x_{n}=-\nabla_{x}f(x_{n})
  19. β n \displaystyle\beta_{n}
  20. s n = Δ x n + β n s n - 1 \displaystyle s_{n}=\Delta x_{n}+\beta_{n}s_{n-1}
  21. α n = arg min α f ( x n + α s n ) \displaystyle\alpha_{n}=\arg\min_{\alpha}f(x_{n}+\alpha s_{n})
  22. x n + 1 = x n + α n s n \displaystyle x_{n+1}=x_{n}+\alpha_{n}s_{n}
  23. α \displaystyle\alpha
  24. β \displaystyle\beta
  25. β n \displaystyle\beta_{n}
  26. β n F R = Δ x n Δ x n Δ x n - 1 Δ x n - 1 \beta_{n}^{FR}=\frac{\Delta x_{n}^{\top}\Delta x_{n}}{\Delta x_{n-1}^{\top}% \Delta x_{n-1}}
  27. β n P R = Δ x n ( Δ x n - Δ x n - 1 ) Δ x n - 1 Δ x n - 1 \beta_{n}^{PR}=\frac{\Delta x_{n}^{\top}(\Delta x_{n}-\Delta x_{n-1})}{\Delta x% _{n-1}^{\top}\Delta x_{n-1}}
  28. β n H S = - Δ x n ( Δ x n - Δ x n - 1 ) s n - 1 ( Δ x n - Δ x n - 1 ) \beta_{n}^{HS}=-\frac{\Delta x_{n}^{\top}(\Delta x_{n}-\Delta x_{n-1})}{s_{n-1% }^{\top}(\Delta x_{n}-\Delta x_{n-1})}
  29. β n D Y = - Δ x n Δ x n s n - 1 ( Δ x n - Δ x n - 1 ) \beta_{n}^{DY}=-\frac{\Delta x_{n}^{\top}\Delta x_{n}}{s_{n-1}^{\top}(\Delta x% _{n}-\Delta x_{n-1})}
  30. β = max { 0 , β P R } \displaystyle\beta=\max\{0,\,\beta^{PR}\}

Normal-gamma_distribution.html

  1. E ( X ) = μ , E ( \Tau ) = α β - 1 \operatorname{E}(X)=\mu\,\!,\quad\operatorname{E}(\Tau)=\alpha\beta^{-1}
  2. ( μ , α - 1 2 β ) \left(\mu,\frac{\alpha-\frac{1}{2}}{\beta}\right)
  3. var ( X ) = β λ ( α - 1 ) , var ( \Tau ) = α β - 2 \operatorname{var}(X)=\frac{\beta}{\lambda(\alpha-1)},\quad\operatorname{var}(% \Tau)=\alpha\beta^{-2}
  4. X | T N ( μ , 1 / ( λ T ) ) , X|T\sim N(\mu,1/(\lambda T))\,\!,
  5. μ \mu
  6. λ T \lambda T
  7. 1 / ( λ T ) . 1/(\lambda T).
  8. T | α , β Gamma ( α , β ) , T|\alpha,\beta\sim\mathrm{Gamma}(\alpha,\beta)\!,
  9. ( X , T ) NormalGamma ( μ , λ , α , β ) . (X,T)\sim\mathrm{NormalGamma}(\mu,\lambda,\alpha,\beta)\!.
  10. f ( x , τ | μ , λ , α , β ) = β α λ Γ ( α ) 2 π τ α - 1 2 e - β τ e - λ τ ( x - μ ) 2 2 f(x,\tau|\mu,\lambda,\alpha,\beta)=\frac{\beta^{\alpha}\sqrt{\lambda}}{\Gamma(% \alpha)\sqrt{2\pi}}\,\tau^{\alpha-\frac{1}{2}}\,e^{-\beta\tau}\,e^{-\frac{% \lambda\tau(x-\mu)^{2}}{2}}
  11. τ \tau
  12. x x
  13. τ \tau
  14. x x
  15. ( ν , μ , σ 2 ) = ( 2 α , μ , β / ( λ α ) ) (\nu,\mu,\sigma^{2})=(2\alpha,\mu,\beta/(\lambda\alpha))
  16. α - 1 / 2 , - β - λ μ 2 / 2 , λ μ , - λ / 2 \alpha-1/2,-\beta-\lambda\mu^{2}/2,\lambda\mu,-\lambda/2
  17. ln τ , τ , τ x , τ x 2 \ln\tau,\tau,\tau x,\tau x^{2}
  18. E ( ln T ) = ψ ( α ) - ln β \operatorname{E}(\ln T)=\psi\left(\alpha\right)-\ln\beta
  19. ψ ( α ) \psi\left(\alpha\right)
  20. E ( T ) = α β \operatorname{E}(T)=\frac{\alpha}{\beta}
  21. E ( T X ) = μ α β \operatorname{E}(TX)=\mu\frac{\alpha}{\beta}
  22. E ( T X 2 ) = 1 λ + μ 2 α β \operatorname{E}(TX^{2})=\frac{1}{\lambda}+\mu^{2}\frac{\alpha}{\beta}
  23. ( X , T ) NormalGamma ( μ , λ , α , β ) , (X,T)\sim\mathrm{NormalGamma}(\mu,\lambda,\alpha,\beta),
  24. NormalGamma ( b μ , λ , α , b 2 β ) . {\rm NormalGamma}(b\mu,\lambda,\alpha,b^{2}\beta).
  25. μ \mu
  26. τ \tau
  27. x 𝒩 ( μ , τ - 1 ) x\sim\mathcal{N}(\mu,\tau^{-1})
  28. μ \mu
  29. τ \tau
  30. ( μ , τ ) (\mu,\tau)
  31. ( μ , τ ) NormalGamma ( μ 0 , λ 0 , α 0 , β 0 ) , (\mu,\tau)\sim\,\text{NormalGamma}(\mu_{0},\lambda_{0},\alpha_{0},\beta_{0}),
  32. π ( μ , τ ) τ α 0 - 1 2 exp [ - β 0 τ ] exp [ - λ 0 τ ( μ - μ 0 ) 2 2 ] . \pi(\mu,\tau)\propto\tau^{\alpha_{0}-\frac{1}{2}}\,\exp[{-\beta_{0}\tau}]\,% \exp[{-\frac{\lambda_{0}\tau(\mu-\mu_{0})^{2}}{2}}].
  33. 𝐗 \mathbf{X}
  34. n n
  35. { x 1 , , x n } \{x_{1},...,x_{n}\}
  36. μ \mu
  37. τ \tau
  38. 𝐏 ( τ , μ | 𝐗 ) 𝐋 ( 𝐗 | τ , μ ) π ( τ , μ ) \mathbf{P}(\tau,\mu|\mathbf{X})\propto\mathbf{L}(\mathbf{X}|\tau,\mu)\pi(\tau,\mu)
  39. 𝐋 \mathbf{L}
  40. 𝐋 ( 𝐗 | τ , μ ) = i = 1 n 𝐋 ( x i | τ , μ ) . \mathbf{L}(\mathbf{X}|\tau,\mu)=\prod_{i=1}^{n}\mathbf{L}(x_{i}|\tau,\mu).
  41. 𝐋 ( 𝐗 | τ , μ ) \displaystyle\mathbf{L}(\mathbf{X}|\tau,\mu)
  42. x ¯ = 1 n i = 1 n x i \bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_{i}
  43. s = 1 n i = 1 n ( x i - x ¯ ) 2 s=\frac{1}{n}\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}
  44. 𝐏 ( τ , μ | 𝐗 ) 𝐋 ( 𝐗 | τ , μ ) π ( τ , μ ) τ n / 2 exp [ - τ 2 ( n s + n ( x ¯ - μ ) 2 ) ] τ α 0 - 1 2 exp [ - β 0 τ ] exp [ - λ 0 τ ( μ - μ 0 ) 2 2 ] τ n 2 + α 0 - 1 2 exp [ - τ ( 1 2 n s + β 0 ) ] exp [ - τ 2 ( λ 0 ( μ - μ 0 ) 2 + n ( x ¯ - μ ) 2 ) ] \begin{aligned}\displaystyle\mathbf{P}(\tau,\mu|\mathbf{X})&\displaystyle% \propto\mathbf{L}(\mathbf{X}|\tau,\mu)\pi(\tau,\mu)\\ &\displaystyle\propto\tau^{n/2}\exp[\frac{-\tau}{2}\left(ns+n(\bar{x}-\mu)^{2}% \right)]\tau^{\alpha_{0}-\frac{1}{2}}\,\exp[{-\beta_{0}\tau}]\,\exp[{-\frac{% \lambda_{0}\tau(\mu-\mu_{0})^{2}}{2}}]\\ &\displaystyle\propto\tau^{\frac{n}{2}+\alpha_{0}-\frac{1}{2}}\exp[-\tau\left(% \frac{1}{2}ns+\beta_{0}\right)]\exp\left[-\frac{\tau}{2}\left(\lambda_{0}(\mu-% \mu_{0})^{2}+n(\bar{x}-\mu)^{2}\right)\right]\\ \end{aligned}
  45. λ 0 ( μ - μ 0 ) 2 + n ( x ¯ - μ ) 2 = λ 0 μ 2 - 2 λ 0 μ μ 0 + λ 0 μ 0 2 + n μ 2 - 2 n x ¯ μ + n x ¯ 2 = ( λ 0 + n ) μ 2 - 2 ( λ 0 μ 0 + n x ¯ ) μ + λ 0 μ 0 2 + n x ¯ 2 = ( λ 0 + n ) ( μ 2 - 2 λ 0 μ 0 + n x ¯ λ 0 + n μ ) + λ 0 μ 0 2 + n x ¯ 2 = ( λ 0 + n ) ( μ - λ 0 μ 0 + n x ¯ λ 0 + n ) 2 + λ 0 μ 0 2 + n x ¯ 2 - ( λ 0 μ 0 + n x ¯ ) 2 λ 0 + n = ( λ 0 + n ) ( μ - λ 0 μ 0 + n x ¯ λ 0 + n ) 2 + λ 0 n ( x ¯ - μ 0 ) 2 λ 0 + n \begin{aligned}\displaystyle\lambda_{0}(\mu-\mu_{0})^{2}+n(\bar{x}-\mu)^{2}&% \displaystyle=\lambda_{0}\mu^{2}-2\lambda_{0}\mu\mu_{0}+\lambda_{0}\mu_{0}^{2}% +n\mu^{2}-2n\bar{x}\mu+n\bar{x}^{2}\\ &\displaystyle=(\lambda_{0}+n)\mu^{2}-2(\lambda_{0}\mu_{0}+n\bar{x})\mu+% \lambda_{0}\mu_{0}^{2}+n\bar{x}^{2}\\ &\displaystyle=(\lambda_{0}+n)(\mu^{2}-2\frac{\lambda_{0}\mu_{0}+n\bar{x}}{% \lambda_{0}+n}\mu)+\lambda_{0}\mu_{0}^{2}+n\bar{x}^{2}\\ &\displaystyle=(\lambda_{0}+n)\left(\mu-\frac{\lambda_{0}\mu_{0}+n\bar{x}}{% \lambda_{0}+n}\right)^{2}+\lambda_{0}\mu_{0}^{2}+n\bar{x}^{2}-\frac{\left(% \lambda_{0}\mu_{0}+n\bar{x}\right)^{2}}{\lambda_{0}+n}\\ &\displaystyle=(\lambda_{0}+n)\left(\mu-\frac{\lambda_{0}\mu_{0}+n\bar{x}}{% \lambda_{0}+n}\right)^{2}+\frac{\lambda_{0}n(\bar{x}-\mu_{0})^{2}}{\lambda_{0}% +n}\end{aligned}
  46. 𝐏 ( τ , μ | 𝐗 ) τ n 2 + α 0 - 1 2 exp [ - τ ( 1 2 n s + β 0 ) ] exp [ - τ 2 ( ( λ 0 + n ) ( μ - λ 0 μ 0 + n x ¯ λ 0 + n ) 2 + λ 0 n ( x ¯ - μ 0 ) 2 λ 0 + n ) ] τ n 2 + α 0 - 1 2 exp [ - τ ( 1 2 n s + β 0 + λ 0 n ( x - μ 0 ) 2 2 ( λ 0 + n ) ) ] exp [ - τ 2 ( λ 0 + n ) ( μ - λ 0 μ 0 + n x ¯ λ 0 + n ) 2 ] \begin{aligned}\displaystyle\mathbf{P}(\tau,\mu|\mathbf{X})&\displaystyle% \propto\tau^{\frac{n}{2}+\alpha_{0}-\frac{1}{2}}\exp\left[-\tau\left(\frac{1}{% 2}ns+\beta_{0}\right)\right]\exp\left[-\frac{\tau}{2}\left(\left(\lambda_{0}+n% \right)\left(\mu-\frac{\lambda_{0}\mu_{0}+n\bar{x}}{\lambda_{0}+n}\right)^{2}+% \frac{\lambda_{0}n(\bar{x}-\mu_{0})^{2}}{\lambda_{0}+n}\right)\right]\\ &\displaystyle\propto\tau^{\frac{n}{2}+\alpha_{0}-\frac{1}{2}}\exp\left[-\tau% \left(\frac{1}{2}ns+\beta_{0}+\frac{\lambda_{0}n(x-\mu_{0})^{2}}{2(\lambda_{0}% +n)}\right)\right]\exp\left[-\frac{\tau}{2}\left(\lambda_{0}+n\right)\left(\mu% -\frac{\lambda_{0}\mu_{0}+n\bar{x}}{\lambda_{0}+n}\right)^{2}\right]\end{aligned}
  47. 𝐏 ( τ , μ | 𝐗 ) = NormalGamma ( λ 0 μ 0 + n x ¯ λ 0 + n , λ 0 + n , α 0 + n 2 , β 0 + 1 2 ( n s + λ 0 n ( x ¯ - μ 0 ) 2 λ 0 + n ) ) \mathbf{P}(\tau,\mu|\mathbf{X})=\,\text{NormalGamma}\left(\frac{\lambda_{0}\mu% _{0}+n\bar{x}}{\lambda_{0}+n},\lambda_{0}+n,\alpha_{0}+\frac{n}{2},\beta_{0}+% \frac{1}{2}\left(ns+\frac{\lambda_{0}n(\bar{x}-\mu_{0})^{2}}{\lambda_{0}+n}% \right)\right)
  48. 2 α 2\alpha
  49. μ \mu
  50. β α \frac{\beta}{\alpha}
  51. 2 β 2\beta
  52. λ 0 \lambda_{0}
  53. n n
  54. μ 0 \mu_{0}
  55. n μ n_{\mu}
  56. τ 0 \tau_{0}
  57. n τ n_{\tau}
  58. μ \mu
  59. τ \tau
  60. 𝐏 ( τ , μ | 𝐗 ) = NormalGamma ( μ 0 , n μ , n τ 2 , n τ 2 τ 0 ) \mathbf{P}(\tau,\mu|\mathbf{X})=\,\text{NormalGamma}(\mu_{0},n_{\mu},\frac{n_{% \tau}}{2},\frac{n_{\tau}}{2\tau_{0}})
  61. n n
  62. μ \mu
  63. s s
  64. 𝐏 ( τ , μ | 𝐗 ) = NormalGamma ( n μ μ 0 + n μ n μ + n , n μ + n , 1 2 ( n τ + n ) , 1 2 ( n τ τ 0 + n s + n μ n ( μ - μ 0 ) 2 n μ + n ) ) \mathbf{P}(\tau,\mu|\mathbf{X})=\,\text{NormalGamma}\left(\frac{n_{\mu}\mu_{0}% +n\mu}{n_{\mu}+n},n_{\mu}+n,\frac{1}{2}(n_{\tau}+n),\frac{1}{2}\left(\frac{n_{% \tau}}{\tau_{0}}+ns+\frac{n_{\mu}n(\mu-\mu_{0})^{2}}{n_{\mu}+n}\right)\right)
  65. β \beta
  66. 2 τ 0 / n τ 2\tau_{0}/n_{\tau}
  67. τ \tau
  68. α \alpha
  69. β \beta
  70. x x
  71. μ \mu
  72. 1 / ( λ τ ) 1/(\lambda\tau)

Normal_polytope.html

  1. L = v + x , y P ( x - y ) d L=v+\sum_{x,y\in P\cap\mathbb{Z}}\mathbb{Z}(x-y)\subseteq\mathbb{Z}^{d}
  2. c , z c P d x 1 , , x c P d c\in\mathbb{N},z\in cP\cap\mathbb{Z}^{d}\implies\exists x_{1},\ldots,x_{c}\in P% \cap\mathbb{Z}^{d}
  3. x 1 + + x c = z x_{1}+\cdots+x_{c}=z
  4. c , z c P L x 1 , , x c P L c\in\mathbb{N},z\in cP\cap L\implies\exists x_{1},\ldots,x_{c}\in P\cap L
  5. x 1 + + x c = z x_{1}+\cdots+x_{c}=z
  6. c c∈ℕ
  7. c d i m P - 1 c≥dimP-1
  8. { x K ( M ) n x M , n } , \{x\in K(M)\mid nx\in M,\ n\in\mathbb{N}\},
  9. C ( P ) = { λ 1 ( 𝐱 1 , 1 ) + + λ n ( 𝐱 n , 1 ) 𝐱 i P , λ i , λ i 0 } . C(P)=\{\lambda_{1}(\,\textbf{x}_{1},1)+\cdots+\lambda_{n}(\,\textbf{x}_{n},1)% \mid\,\textbf{x}_{i}\in P,\ \lambda_{i}\in\mathbb{R},\lambda_{i}\geq 0\}.
  10. n n

Notation_for_differentiation.html

  1. y = f ( x ) y=f(x)
  2. y y
  3. x x
  4. d y d x \frac{dy}{dx}
  5. x x
  6. f f
  7. x x
  8. d ( f ( x ) ) d x or d d x ( f ( x ) ) \frac{d\bigl(f(x)\bigr)}{dx}\,\text{ or }\frac{d}{dx}\bigl(f(x)\bigr)
  9. d n y d x n , d n ( f ( x ) ) d x n , or d n d x n ( f ( x ) ) \frac{d^{n}y}{dx^{n}},\quad\frac{d^{n}\bigl(f(x)\bigr)}{dx^{n}},\,\text{ or }% \frac{d^{n}}{dx^{n}}\bigl(f(x)\bigr)
  10. d ( d ( d y d x ) d x ) d x = ( d d x ) 3 ( f ( x ) ) \frac{d\Bigl(\frac{d\left(\frac{dy}{dx}\right)}{dx}\Bigr)}{dx}=\left(\frac{d}{% dx}\right)^{3}\bigl(f(x)\bigr)
  11. d 3 ( d x ) 3 ( f ( x ) ) = d 3 d x 3 ( f ( x ) ) \frac{d^{3}}{\left(dx\right)^{3}}\bigl(f(x)\bigr)=\frac{d^{3}}{dx^{3}}\bigl(f(% x)\bigr)
  12. d y d x | x = a = d y d x ( a ) . \frac{dy}{dx}\left.{\!\!\frac{}{}}\right|_{x=a}=\frac{dy}{dx}(a).
  13. d y d x = d y d u d u d x . \frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}.
  14. f f^{\prime}\;
  15. f ′′ f^{\prime\prime}\;
  16. f ′′′ f^{\prime\prime\prime}\;
  17. D f Df\;
  18. D 2 f D^{2}f\;
  19. D n f D^{n}f\;
  20. D x y D_{x}y\;
  21. D x 2 y D^{2}_{x}y\;
  22. D x n y D^{n}_{x}y\;
  23. y ˙ = d y d t , \dot{y}=\frac{dy}{dt}\,,
  24. y ¨ = d 2 y d t 2 , \ddot{y}=\frac{d^{2}y}{dt^{2}}\,,
  25. f x = d f d x f_{x}=\frac{df}{dx}
  26. f x x = d 2 f d x 2 . f_{xx}=\frac{d^{2}f}{dx^{2}}.
  27. f x = f x = x f = x f , \frac{\partial f}{\partial x}=f_{x}=\partial_{x}f=\partial^{x}f,
  28. ( T V ) S \left(\frac{\partial T}{\partial V}\right)_{S}
  29. ( T V ) P \left(\frac{\partial T}{\partial V}\right)_{P}
  30. 𝐀 = ( 𝐀 x , 𝐀 y , 𝐀 z ) \mathbf{A}=(\mathbf{A}_{x},\mathbf{A}_{y},\mathbf{A}_{z})
  31. φ \varphi
  32. φ = f ( x , y , z ) \varphi=f(x,y,z)\,
  33. = ( x , y , z ) , \nabla=\left(\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{% \partial}{\partial z}\right),
  34. grad φ \mathrm{grad\,}\varphi\,
  35. φ \varphi
  36. φ \varphi
  37. grad φ \displaystyle\operatorname{grad}\varphi
  38. div 𝐀 \mathrm{div}\,\mathbf{A}\,
  39. div 𝐀 \displaystyle\operatorname{div}\mathbf{A}
  40. div grad φ \operatorname{div}\operatorname{grad}\varphi
  41. φ \varphi
  42. div grad φ \displaystyle\operatorname{div}\operatorname{grad}\varphi
  43. Δ = 2 \Delta=\nabla^{2}
  44. curl 𝐀 \mathrm{curl}\,\mathbf{A}\,
  45. rot 𝐀 \mathrm{rot}\,\mathbf{A}\,
  46. curl 𝐀 \displaystyle\operatorname{curl}\mathbf{A}
  47. ( f g ) = f g + f g ( ϕ ψ ) = ( ϕ ) ψ + ϕ ( ψ ) . (fg)^{\prime}=f^{\prime}g+fg^{\prime}~{}~{}~{}\Longrightarrow~{}~{}~{}\nabla(% \phi\psi)=(\nabla\phi)\psi+\phi(\nabla\psi).
  48. \Box
  49. Δ \Delta

Novikov's_condition.html

  1. ( X t ) 0 t T (X_{t})_{0\leq t\leq T}
  2. ( Ω , ( t ) , ) \left(\Omega,(\mathcal{F}_{t}),\mathbb{P}\right)
  3. ( W t ) 0 t T (W_{t})_{0\leq t\leq T}
  4. 𝔼 [ e 1 2 0 T | X | t 2 d t ] < \mathbb{E}\left[e^{\frac{1}{2}\int_{0}^{T}|X|_{t}^{2}\,dt}\right]<\infty
  5. Z t = e 0 t X s d W s - 1 2 0 t X s 2 d s , 0 t T \ Z_{t}\ =e^{\int_{0}^{t}X_{s}\,dW_{s}-\frac{1}{2}\int_{0}^{t}X_{s}^{2}\,ds},% \quad 0\leq t\leq T
  6. \mathbb{P}
  7. \mathcal{F}

Nowhere-zero_flow.html

  1. e δ + ( v ) φ ( e ) = e δ - ( v ) φ ( e ) , \sum_{e\in\delta^{+}(v)}\varphi(e)=\sum_{e\in\delta^{-}(v)}\varphi(e),

Np-chart.html

  1. n p ¯ ± 3 n p ¯ ( 1 - p ¯ ) n\bar{p}\pm 3\sqrt{n\bar{p}(1-\bar{p})}
  2. n p ¯ i = j = 1 n { 1 if x i j defective 0 otherwise n\bar{p}_{i}=\sum_{j=1}^{n}\begin{cases}1&\mbox{if }~{}x_{ij}\mbox{ defective}% \\ 0&\mbox{otherwise}\end{cases}
  3. n p ¯ ± 3 n p ¯ ( 1 - p ¯ ) n\bar{p}\pm 3\sqrt{n\bar{p}(1-\bar{p})}
  4. p ¯ \bar{p}
  5. n n

NTU_method.html

  1. T h , i - T c , i \ T_{h,i}-\ T_{c,i}
  2. C h \ C_{h}
  3. C c \ C_{c}
  4. C m i n \ C_{min}
  5. q m a x = C m i n ( T h , i - T c , i ) q_{max}\ =C_{min}(T_{h,i}-T_{c,i})
  6. q m a x \ q_{max}
  7. C m i n \ C_{min}
  8. E = q q m a x E\ =\frac{q}{q_{max}}
  9. q = C h ( T h , i - T h , o ) = C c ( T c , o - T c , i ) q\ =C_{h}(T_{h,i}-T_{h,o})\ =C_{c}(T_{c,o}-T_{c,i})
  10. q = E C m i n ( T h , i - T c , i ) q\ =EC_{min}(T_{h,i}-T_{c,i})
  11. E = f ( N T U , C m i n C m a x ) \ E=f(NTU,\frac{C_{min}}{C_{max}})
  12. E \ E
  13. C r = C m i n C m a x C_{r}\ =\frac{C_{min}}{C_{max}}
  14. N T U \ NTU
  15. N T U = U A C m i n NTU\ =\frac{UA}{C_{min}}
  16. U \ U
  17. A \ A
  18. E = 1 - exp [ - N T U ( 1 + C r ) ] 1 + C r E\ =\frac{1-\exp[-NTU(1+C_{r})]}{1+C_{r}}
  19. E = 1 - exp [ - N T U ( 1 - C r ) ] 1 - C r exp [ - N T U ( 1 - C r ) ] E\ =\frac{1-\exp[-NTU(1-C_{r})]}{1-C_{r}\exp[-NTU(1-C_{r})]}
  20. C r = 1 C_{r}\ =1
  21. E = N T U 1 + N T U E\ =\frac{NTU}{1+NTU}
  22. C r = 0 C_{r}\ =0
  23. E = 1 - exp [ - N T U ] E\ =1-\exp[-NTU]

Nuclear_structure.html

  1. Z Z
  2. N N
  3. E B = a V A - a S A 2 / 3 - a C Z 2 A 1 / 3 - a A ( N - Z ) 2 A - δ ( A , Z ) E_{B}=a_{V}A-a_{S}A^{2/3}-a_{C}\frac{Z^{2}}{A^{1/3}}-a_{A}\frac{(N-Z)^{2}}{A}-% \delta(A,Z)
  4. A = Z + N A=Z+N
  5. A A
  6. A 2 / 3 A^{2/3}
  7. Z 2 Z^{2}
  8. ( N - Z ) 2 (N-Z)^{2}
  9. δ ( A , Z ) \delta(A,Z)
  10. a V , a S , a C , a A a_{V},a_{S},a_{C},a_{A}

Nucleate_boiling.html

  1. N u b = C f c ( R e b , P r L ) N{{u}_{b}}={{C}_{fc}}\left(R{{e}_{b}},P{{r}_{L}}\right)
  2. N u b = ( q A ) D b ( T s - T s a t ) k L N{{u}_{b}}=\frac{\left(\frac{q}{A}\right){{D}_{b}}}{\left({{T}_{s}}-{{T}_{sat}% }\right){{k}_{L}}}
  3. D b D_{b}
  4. T s - T s a t {{T}_{s}}-{{T}_{sat}}
  5. k L k_{L}
  6. P r L Pr_{L}
  7. R e b Re_{b}
  8. R e b = D b G b μ L R{{e}_{b}}=\frac{{{D}_{b}}{{G}_{b}}}{{{\mu}_{L}}}
  9. G b G_{b}
  10. μ L {{\mu}_{L}}
  11. q A = μ L h f g [ g ( ρ L - ρ v ) σ ] 1 2 [ c p L ( T s - T s a t ) C s f h f g P r L n ] 3 \frac{q}{A}={{\mu}_{L}}{{h}_{fg}}{{\left[\frac{g\left({{\rho}_{L}}-{{\rho}_{v}% }\right)}{\sigma}\right]}^{{}^{1}\!\!\diagup\!\!{}_{2}\;}}{{\left[\frac{{{c}_{% pL}}\left({{T}_{s}}-{{T}_{sat}}\right)}{{{C}_{sf}}{{h}_{fg}}Pr_{L}^{n}}\right]% }^{3}}
  12. c p L c_{pL}
  13. C s f C_{sf}
  14. C s f C_{sf}
  15. C s f C_{sf}
  16. C s f C_{sf}
  17. C C l 4 CCl_{4}

Nucleic_acid_thermodynamics.html

  1. K = [ A ] [ B ] [ A B ] K=\frac{[A][B]}{[AB]}
  2. Δ G = - R T ln [ A ] [ B ] [ A B ] \Delta G^{\circ}=-RT\ln\frac{[A][B]}{[AB]}
  3. T m = - Δ G R ln [ A B ] i n i t i a l 2 T_{m}=-\frac{\Delta G^{\circ}}{R\ln\frac{[AB]_{initial}}{2}}
  4. T m = Δ H Δ S - R ln [ A B ] i n i t i a l 2 T_{m}=\frac{\Delta H^{\circ}}{\Delta S^{\circ}-R\ln\frac{[AB]_{initial}}{2}}
  5. T m = Δ H Δ S + R ln ( [ A ] t o t a l - [ B ] t o t a l / 2 ) T_{m}=\frac{\Delta H^{\circ}}{\Delta S^{\circ}+R\ln([A]_{total}-[B]_{total}/2)}
  6. Δ G 37 ( total ) = Δ G 37 ( initiations ) + i = 1 10 n i Δ G 37 ( i ) \Delta G_{37}^{\circ}(\mathrm{total})=\Delta G_{37}^{\circ}(\mathrm{% initiations})+\sum_{i=1}^{10}n_{i}\Delta G_{37}^{\circ}(i)
  7. Δ G ( total ) = Δ H total - T Δ S total \Delta G^{\circ}(\mathrm{total})=\Delta H_{\mathrm{total}}^{\circ}-T\Delta S_{% \mathrm{total}}^{\circ}
  8. Δ H \Delta H
  9. Δ S \Delta S
  10. Δ G \Delta G

Nucleotidyltransferase.html

  1. \rightleftharpoons

Numerical_continuation.html

  1. F ( 𝐮 , λ ) = 0 F(\mathbf{u},\lambda)=0
  2. λ \lambda
  3. λ \lambda
  4. F ( , λ ) F(\ast,\lambda)
  5. F F
  6. F = 0 F=0
  7. F ( 𝐮 ) = 0 F(\mathbf{u})=0
  8. 𝐮 \mathbf{u}
  9. F ( 𝐮 ) F(\mathbf{u})
  10. 𝐮 = F ( 𝐮 , λ ) \mathbf{u}^{\prime}=F(\mathbf{u},\lambda)
  11. 𝐮 \mathbf{u}
  12. T T
  13. 𝐮 = F ( 𝐮 , λ ) , 𝐮 ( 0 ) = 𝐮 ( T ) \mathbf{u}^{\prime}=F(\mathbf{u},\lambda),\,\mathbf{u}(0)=\mathbf{u}(T)
  14. 𝐮 ( t ) = 𝐮 ( t + Ω ) \mathbf{u}(t)=\mathbf{u}(t+\Omega)
  15. Ω \Omega
  16. 𝐮 = 𝐅 ( 𝐮 , λ ) , 𝐮 ( 0 ) = 𝐮 ( T + N . Ω ) \mathbf{u}^{\prime}=\mathbf{F}(\mathbf{u},\lambda),\,\mathbf{u}(0)=\mathbf{u}(% T+N.\Omega)
  17. N N
  18. T T
  19. 𝐮 = T 𝐅 ( 𝐮 , λ ) , 𝐮 ( 0 ) = 𝐮 ( 1 ) + N . Ω \mathbf{u}^{\prime}=T\mathbf{F}(\mathbf{u},\lambda),\,\mathbf{u}(0)=\mathbf{u}% (1)+N.\Omega
  20. 𝐮 0 ( t ) \mathbf{u}_{0}(t)
  21. λ 0 \lambda_{0}
  22. λ \lambda
  23. < 𝐮 ( 0 ) - 𝐮 0 ( 0 ) , 𝐅 ( 𝐮 0 ( 0 ) , λ 0 ) 0 <\mathbf{u}(0)-\mathbf{u}_{0}(0),\mathbf{F}(\mathbf{u}_{0}(0),\lambda_{0})>=0
  24. 𝐮 \mathbf{u}
  25. 0 1 < 𝐮 ( t ) - 𝐮 0 ( t ) , 𝐅 ( 𝐮 0 ( t ) , λ 0 ) > d t = 0 \int_{0}^{1}<\mathbf{u}(t)-\mathbf{u}_{0}(t),\mathbf{F}(\mathbf{u}_{0}(t),% \lambda_{0})>\,dt=0
  26. Γ ( 𝐮 0 , λ 0 ) \Gamma(\mathbf{u}_{0},\lambda_{0})
  27. F F
  28. ( 𝐮 , λ ) (\mathbf{u},\lambda)
  29. F ( 𝐮 , λ ) = 0 F(\mathbf{u},\lambda)=0
  30. ( 𝐮 0 , λ 0 ) (\mathbf{u}_{0},\lambda_{0})
  31. ( 𝐮 ( s ) , λ ( s ) ) (\mathbf{u}(s),\lambda(s))
  32. ( 𝐮 ( 0 ) , λ ( 0 ) ) = ( 𝐮 0 , λ 0 ) , ( 𝐮 ( 1 ) , λ ( 1 ) ) = ( 𝐮 , λ ) (\mathbf{u}(0),\lambda(0))=(\mathbf{u}_{0},\lambda_{0}),\,(\mathbf{u}(1),% \lambda(1))=(\mathbf{u},\lambda)
  33. F ( 𝐮 ( s ) , λ ( s ) ) = 0 F(\mathbf{u}(s),\lambda(s))=0
  34. ( 𝐮 0 , λ 0 ) (\mathbf{u}_{0},\lambda_{0})
  35. F ( 𝐮 0 , λ 0 ) = 0 F(\mathbf{u}_{0},\lambda_{0})=0
  36. Γ ( 𝐮 0 , λ 0 ) \Gamma(\mathbf{u}_{0},\lambda_{0})
  37. F F
  38. ( 𝐮 , λ ) (\mathbf{u},\lambda)
  39. F F
  40. ( n ) (n)
  41. ( 𝐮 0 , λ 0 ) (\mathbf{u}_{0},\lambda_{0})
  42. F F
  43. ( 𝐮 , λ ) (\mathbf{u},\lambda)
  44. F F
  45. Γ \Gamma
  46. Φ \Phi
  47. J = [ F x F λ ] J=\left[\begin{array}[]{cc}F_{x}&F_{\lambda}\\ \end{array}\right]\,
  48. Ψ \Psi
  49. J J
  50. F ( x , λ ) = 0 F(x,\lambda)=0
  51. [ ( I - Ψ Ψ T ) F ( x + Φ ξ + η ) Ψ T F ( x + Φ ξ + η ) ] = 0 , \left[\begin{array}[]{l}(I-\Psi\Psi^{T})F(x+\Phi\xi+\eta)\\ \Psi^{T}F(x+\Phi\xi+\eta)\\ \end{array}\right]=0,
  52. η \eta
  53. J J
  54. ( Φ T η = 0 ) (\Phi^{T}\,\eta=0)
  55. ξ \xi
  56. η \eta
  57. η ( ξ ) \eta(\xi)
  58. η ( 0 ) = 0 \eta(0)=0
  59. ( I - Ψ Ψ T ) F ( x + Φ ξ + η ( ξ ) ) = 0 ) (I-\Psi\Psi^{T})F(x+\Phi\xi+\eta(\xi))=0)
  60. η ( ξ ) \eta(\xi)
  61. λ \lambda
  62. λ 0 \lambda_{0}
  63. 𝐮 0 \mathbf{u}_{0}
  64. F ( 𝐮 , λ 0 ) = 0 F(\mathbf{u},\lambda_{0})=0
  65. λ \lambda
  66. λ + Δ λ \lambda+\Delta\lambda
  67. Δ λ \Delta\lambda
  68. λ \lambda
  69. Δ s \Delta s
  70. F ( u , λ ) = 0 u ˙ 0 * ( u - u 0 ) + λ ˙ 0 ( λ - λ 0 ) = Δ s \begin{array}[]{l}F(u,\lambda)=0\\ \dot{u}^{*}_{0}(u-u_{0})+\dot{\lambda}_{0}(\lambda-\lambda_{0})=\Delta s\\ \end{array}\,
  71. ( u ˙ 0 , λ ˙ 0 ) (\dot{u}_{0},\dot{\lambda}_{0})\,
  72. ( u 0 , λ 0 ) (u_{0},\lambda_{0})\,
  73. [ F u F λ u ˙ * λ ˙ ] \left[\begin{array}[]{cc}F_{u}&F_{\lambda}\\ \dot{u}^{*}&\dot{\lambda}\\ \end{array}\right]\,
  74. λ \lambda
  75. λ \lambda
  76. ( n ) (n)

Nyström_method.html

  1. n n
  2. O ( n 3 ) O(n^{3})
  3. n n
  4. a b h ( x ) d x k = 1 n w k h ( x k ) \int_{a}^{b}h(x)\;\mathrm{d}x\approx\sum_{k=1}^{n}w_{k}h(x_{k})
  5. w k w_{k}
  6. x k x_{k}
  7. f ( x ) = λ u ( x ) - a b K ( x , x ) f ( x ) d x f(x)=\lambda u(x)-\int_{a}^{b}K(x,x^{\prime})f(x^{\prime})\;\mathrm{d}x^{\prime}
  8. f ( x ) λ u ( x ) - k = 1 n w k K ( x , x k ) f ( x k ) f(x)\approx\lambda u(x)-\sum_{k=1}^{n}w_{k}K(x,x_{k})f(x_{k})

Observed_information.html

  1. X 1 , , X n X_{1},\ldots,X_{n}
  2. θ \theta
  3. X 1 , , X n X_{1},\ldots,X_{n}
  4. ( θ | X 1 , , X n ) = i = 1 n log f ( X i | θ ) \ell(\theta|X_{1},\ldots,X_{n})=\sum_{i=1}^{n}\log f(X_{i}|\theta)
  5. θ * \theta^{*}
  6. 𝒥 ( θ * ) = - ( θ ) | θ = θ * \mathcal{J}(\theta^{*})=-\left.\nabla\nabla^{\top}\ell(\theta)\right|_{\theta=% \theta^{*}}
  7. = - ( 2 θ 1 2 2 θ 1 θ 2 2 θ 1 θ n 2 θ 2 θ 1 2 θ 2 2 2 θ 2 θ n 2 θ n θ 1 2 θ n θ 2 2 θ n 2 ) ( θ ) | θ = θ * =-\left.\left(\begin{array}[]{cccc}\tfrac{\partial^{2}}{\partial\theta_{1}^{2}% }&\tfrac{\partial^{2}}{\partial\theta_{1}\partial\theta_{2}}&\cdots&\tfrac{% \partial^{2}}{\partial\theta_{1}\partial\theta_{n}}\\ \tfrac{\partial^{2}}{\partial\theta_{2}\partial\theta_{1}}&\tfrac{\partial^{2}% }{\partial\theta_{2}^{2}}&\cdots&\tfrac{\partial^{2}}{\partial\theta_{2}% \partial\theta_{n}}\\ \vdots&\vdots&\ddots&\vdots\\ \tfrac{\partial^{2}}{\partial\theta_{n}\partial\theta_{1}}&\tfrac{\partial^{2}% }{\partial\theta_{n}\partial\theta_{2}}&\cdots&\tfrac{\partial^{2}}{\partial% \theta_{n}^{2}}\\ \end{array}\right)\ell(\theta)\right|_{\theta=\theta^{*}}
  8. ( θ ) \mathcal{I}(\theta)
  9. X X
  10. θ \theta
  11. ( θ ) = E ( 𝒥 ( θ ) ) \mathcal{I}(\theta)=\mathrm{E}(\mathcal{J}(\theta))

Occurrences_of_Grandi's_series.html

  1. f ( x ) = 1 - x + x 2 - x 3 + = 1 1 + x . f(x)=1-x+x^{2}-x^{3}+\cdots=\frac{1}{1+x}.
  2. f ( x ) = π 2 sinh π sinh x . f(x)=\frac{\pi}{2\sinh\pi}\sinh x.
  3. ( - 1 ) n - 1 ( 1 n - 1 n 3 + 1 n 5 - ) = ( - 1 ) n - 1 n 1 + n 2 . (-1)^{n-1}\left(\frac{1}{n}-\frac{1}{n^{3}}+\frac{1}{n^{5}}-\cdots\right)=(-1)% ^{n-1}\frac{n}{1+n^{2}}.
  4. 2 π 0 π f ( x ) sin x d x = 1 2 sinh π ( cosh x sin x - sinh x cos x ) | 0 π = 1 2 . \frac{2}{\pi}\int_{0}^{\pi}f(x)\sin x\;dx=\frac{1}{2\sinh\pi}\left.(\cosh x% \sin x-\sinh x\cos x)\right|_{0}^{\pi}=\frac{1}{2}.
  5. cos x + cos 2 x + cos 3 x + = k = 1 cos ( k x ) . \cos x+\cos 2x+\cos 3x+\cdots=\sum_{k=1}^{\infty}\cos(kx).
  6. 1 + 2 k = 1 cos ( k x ) = 0 ? 1+2\sum_{k=1}^{\infty}\cos(kx)=0?
  7. η ( z ) = 1 - 1 2 z + 1 3 z - 1 4 z + = n = 1 ( - 1 ) n - 1 n z , \eta(z)=1-\frac{1}{2^{z}}+\frac{1}{3^{z}}-\frac{1}{4^{z}}+\cdots=\sum_{n=1}^{% \infty}\frac{(-1)^{n-1}}{n^{z}},
  8. η ( z ) = 1 + 1 2 z + 1 3 z + 1 4 z + - 2 2 z ( 1 + 1 2 z + ) = ( 1 - 2 2 z ) ζ ( z ) , \begin{array}[]{rcl}\eta(z)&=&\displaystyle 1+\frac{1}{2^{z}}+\frac{1}{3^{z}}+% \frac{1}{4^{z}}+\cdots-\frac{2}{2^{z}}\left(1+\frac{1}{2^{z}}+\cdots\right)\\ &=&\displaystyle\left(1-\frac{2}{2^{z}}\right)\zeta(z),\end{array}

Octonion_algebra.html

  1. N ( x y ) = N ( x ) N ( y ) N(xy)=N(x)N(y)
  2. H 1 ( F , G 2 ) H^{1}(F,G_{2})

Odd_greedy_expansion.html

  1. x y = 1 2 a i + 1 , \frac{x}{y}=\sum\frac{1}{2a_{i}+1},
  2. 8 77 = 1 10 + 1 257 + 1 197890 = 1 11 + 1 77 , \frac{8}{77}=\frac{1}{10}+\frac{1}{257}+\frac{1}{197890}=\frac{1}{11}+\frac{1}% {77},

Odd_number_theorem.html

  1. M : ( u , v ) ( u , v ) M:(u,v)\mapsto(u^{\prime},v^{\prime})\,
  2. ( u , v ) (u,v)\,
  3. V : ( s , w ) V:(s,w)\,
  4. V 0 : ( s 0 , w 0 ) V_{0}:(s_{0},w_{0})\,
  5. D = δ V = 0 | ( s 0 , w 0 ) D=\delta V=0|_{(s_{0},w_{0})}
  6. χ = index D = constant \chi=\sum\,\text{index}_{D}=\,\text{constant}\,
  7. n + n_{+}\,
  8. n - n_{-}\,
  9. χ = n + - n - = 1 \chi=n_{+}-n_{-}=1\,
  10. N = n + + n - = 2 n - + 1 N=n_{+}+n_{-}=2n_{-}+1\,

Odd–even_sort.html

  1. a 1 , , a n a_{1},...,a_{n}
  2. a i a_{i}
  3. e e
  4. e e
  5. n - e n-e
  6. n - e + 1 n-e+1
  7. e - 1 e-1
  8. n - 1 n-1
  9. ( n - 1 ) - ( e - 1 ) = n - e (n-1)-(e-1)=n-e
  10. n - e + 2 n-e+2
  11. i i
  12. n - e + i + 1 n-e+i+1
  13. e e
  14. n - e + ( e - 1 ) + 1 = n n-e+(e-1)+1=n
  15. n n

Oil_immersion.html

  1. δ = λ 2 N A \delta=\frac{\lambda}{\mathrm{2NA}}
  2. NA = n sin α 0 \mathrm{NA}=n\sin\alpha_{0}\;

Oja's_rule.html

  1. 𝐰 \mathbf{w}
  2. y y
  3. 𝐱 \mathbf{x}
  4. Δ 𝐰 = 𝐰 n + 1 - 𝐰 n = η y n ( 𝐱 n - y n 𝐰 n ) , \,\Delta\mathbf{w}~{}=~{}\mathbf{w}_{n+1}-\mathbf{w}_{n}~{}=~{}\eta\,y_{n}(% \mathbf{x}_{n}-y_{n}\mathbf{w}_{n}),
  5. η η
  6. n n
  7. d 𝐰 d t = η y ( t ) ( 𝐱 ( t ) - y ( t ) 𝐰 ( t ) ) . \,\frac{d\mathbf{w}}{dt}~{}=~{}\eta\,y(t)(\mathbf{x}(t)-y(t)\mathbf{w}(t)).
  8. Δ 𝐰 = η y ( 𝐱 n ) 𝐱 n \,\Delta\mathbf{w}~{}=~{}\eta\,y(\mathbf{x}_{n})\mathbf{x}_{n}
  9. n n
  10. w i ( n + 1 ) = w i + η y ( 𝐱 ) x i \,w_{i}(n+1)~{}=~{}w_{i}+\eta\,y(\mathbf{x})x_{i}
  11. 𝐱 \mathbf{x}
  12. w i ( n + 1 ) = w i + η y ( 𝐱 ) x i ( j = 1 m [ w j + η y ( 𝐱 ) x j ] p ) 1 / p \,w_{i}(n+1)~{}=~{}\frac{w_{i}+\eta\,y(\mathbf{x})x_{i}}{\left(\sum_{j=1}^{m}[% w_{j}+\eta\,y(\mathbf{x})x_{j}]^{p}\right)^{1/p}}
  13. p = 2 p=2
  14. | η | 1 |\eta|\ll 1
  15. w i ( n + 1 ) = w i ( j w j p ) 1 / p + η ( y x i ( j w j p ) 1 / p - w i j y x j w j ( j w j p ) ( 1 + 1 / p ) ) + O ( η 2 ) \,w_{i}(n+1)~{}=~{}\frac{w_{i}}{\left(\sum_{j}w_{j}^{p}\right)^{1/p}}~{}+~{}% \eta\left(\frac{yx_{i}}{\left(\sum_{j}w_{j}^{p}\right)^{1/p}}-\frac{w_{i}\sum_% {j}yx_{j}w_{j}}{\left(\sum_{j}w_{j}^{p}\right)^{(1+1/p)}}\right)~{}+~{}O(\eta^% {2})
  16. η η
  17. y ( 𝐱 ) = j = 1 m x j w j \,y(\mathbf{x})~{}=~{}\sum_{j=1}^{m}x_{j}w_{j}
  18. 1 1
  19. | 𝐰 | = ( j = 1 m w j p ) 1 / p = 1 \,|\mathbf{w}|~{}=~{}\left(\sum_{j=1}^{m}w_{j}^{p}\right)^{1/p}~{}=~{}1
  20. w i ( n + 1 ) = w i + η y ( x i - w i y ) \,w_{i}(n+1)~{}=~{}w_{i}+\eta\,y(x_{i}-w_{i}y)
  21. 𝐱 \mathbf{x}
  22. 𝐱 = j a j 𝐪 j \mathbf{x}~{}=~{}\sum_{j}a_{j}\mathbf{q}_{j}
  23. lim n σ 2 ( n ) = λ 1 \lim_{n\rightarrow\infty}\sigma^{2}(n)~{}=~{}\lambda_{1}
  24. η η
  25. n = 1 η ( n ) = , n = 1 η ( n ) p < , p > 1 \sum_{n=1}^{\infty}\eta(n)=\infty,~{}~{}~{}\sum_{n=1}^{\infty}\eta(n)^{p}<% \infty,~{}~{}~{}p>1
  26. y ( 𝐱 ( n ) ) y(\mathbf{x}(n))
  27. 𝐱 \mathbf{x}
  28. 𝐰 \mathbf{w}
  29. E ( 𝐰 ) = - h 𝐰 - b 𝐰 𝐕𝐰 - c 𝐰 𝐱 y E(\mathbf{w})=-h\mathbf{w}-b\mathbf{w}^{\top}\mathbf{V}\mathbf{w}-c\mathbf{w}^% {\top}\mathbf{x}y
  30. 𝐰 n + 1 = 𝐰 n + η ( h + b ( 𝐕 + 𝐕 ) 𝐰 n + c 𝐱 n + 1 y n + 1 ) \mathbf{w}_{n+1}=\mathbf{w}_{n}+\eta(h+b(\mathbf{V}+\mathbf{V}^{\top})\mathbf{% w}_{n}+c\mathbf{x}_{n+1}y_{n+1})
  31. y { - 1 , 1 } y\in\{-1,1\}
  32. b b\in\mathbb{R}
  33. c > 0 c>0
  34. h h\in\mathbb{R}
  35. 𝐕 { 0 , 1 } D × D \mathbf{V}\in\{0,1\}^{D\times D}
  36. h = 0 h=0
  37. b = 0 b=0
  38. c = 1 c=1
  39. h = 0 h=0
  40. b = - 0.5 b=-0.5
  41. c = 1 c=1
  42. 𝐕 = 𝐈 \mathbf{V}=\mathbf{I}
  43. 𝐈 \mathbf{I}
  44. 𝐰 n + 1 = 𝐰 n + η ( 2 b 𝐰 n + 𝐱 n + 1 y n + 1 ) \mathbf{w}_{n+1}=\mathbf{w}_{n}+\eta(2b\mathbf{w}_{n}+\mathbf{x}_{n+1}y_{n+1})
  45. Δ w i j x i y j - ϵ ( c pre * k w i k y k ) ( c post * y j ) , \Delta w_{ij}~{}\propto~{}\langle x_{i}y_{j}\rangle-\epsilon\left\langle\left(% c_{\mathrm{pre}}*\sum_{k}w_{ik}y_{k}\right)\cdot\left(c_{\mathrm{post}}*y_{j}% \right)\right\rangle,
  46. i i
  47. j j
  48. x x
  49. y y
  50. ε ε
  51. Δ w = C x w - w C y . \Delta w~{}=~{}Cx\cdot w-w\cdot Cy.

Okapi_BM25.html

  1. Q Q
  2. q 1 , , q n q_{1},...,q_{n}
  3. D D
  4. score ( D , Q ) = i = 1 n IDF ( q i ) f ( q i , D ) ( k 1 + 1 ) f ( q i , D ) + k 1 ( 1 - b + b | D | avgdl ) , \,\text{score}(D,Q)=\sum_{i=1}^{n}\,\text{IDF}(q_{i})\cdot\frac{f(q_{i},D)% \cdot(k_{1}+1)}{f(q_{i},D)+k_{1}\cdot(1-b+b\cdot\frac{|D|}{\,\text{avgdl}})},
  5. f ( q i , D ) f(q_{i},D)
  6. q i q_{i}
  7. D D
  8. | D | |D|
  9. D D
  10. a v g d l avgdl
  11. k 1 k_{1}
  12. b b
  13. k 1 [ 1.2 , 2.0 ] k_{1}\in[1.2,2.0]
  14. b = 0.75 b=0.75
  15. IDF ( q i ) \,\text{IDF}(q_{i})
  16. q i q_{i}
  17. IDF ( q i ) = log N - n ( q i ) + 0.5 n ( q i ) + 0.5 , \,\text{IDF}(q_{i})=\log\frac{N-n(q_{i})+0.5}{n(q_{i})+0.5},
  18. N N
  19. n ( q i ) n(q_{i})
  20. q i q_{i}
  21. ϵ \epsilon
  22. q q
  23. n ( q ) n(q)
  24. D D
  25. n ( q ) N \frac{n(q)}{N}
  26. N N
  27. D D
  28. q q
  29. - log n ( q ) N = log N n ( q ) . -\log\frac{n(q)}{N}=\log\frac{N}{n(q)}.
  30. q 1 q_{1}
  31. q 2 q_{2}
  32. q 1 q_{1}
  33. q 2 q_{2}
  34. D D
  35. n ( q 1 ) N n ( q 2 ) N , \frac{n(q_{1})}{N}\cdot\frac{n(q_{2})}{N},
  36. i = 1 2 log N n ( q i ) . \sum_{i=1}^{2}\log\frac{N}{n(q_{i})}.
  37. b b
  38. b = 1 b=1
  39. b = 0 b=0
  40. δ \delta
  41. 1.0 1.0
  42. score ( D , Q ) = i = 1 n IDF ( q i ) [ f ( q i , D ) ( k 1 + 1 ) f ( q i , D ) + k 1 ( 1 - b + b | D | avgdl ) + δ ] \,\text{score}(D,Q)=\sum_{i=1}^{n}\,\text{IDF}(q_{i})\cdot\left[\frac{f(q_{i},% D)\cdot(k_{1}+1)}{f(q_{i},D)+k_{1}\cdot(1-b+b\cdot\frac{|D|}{\,\text{avgdl}})}% +\delta\right]

Oleg_Lupanov.html

  1. C ( f ) 2 n n + o ( 2 n n ) . C(f)\leq\frac{2^{n}}{n}+o\left(\frac{2^{n}}{n}\right).

Oloid.html

  1. A = 4 π r 2 \!A=4\pi r^{2}
  2. 2 3 ( 2 E ( 3 4 ) + K ( 3 4 ) ) r 3 \frac{2}{3}\left(2E\left(\frac{3}{4}\right)+K\left(\frac{3}{4}\right)\right)r^% {3}
  3. V 3.0524184684 r 3 \!V\approx 3.0524184684r^{3}
  4. Δ h = r ( 2 2 - 3 3 8 ) 0.0576 r \Delta h=r(\frac{\sqrt{2}}{2}-{3}\frac{\sqrt{3}}{8})\approx 0.0576r
  5. l = 3 r \!l=\sqrt{3}r

On_the_Equilibrium_of_Heterogeneous_Substances.html

  1. δ Q T \int\frac{\delta Q}{T}
  2. δ Q T \int\frac{\delta Q}{T}

Operational_calculus.html

  1. F ( p ) F(p)
  2. p p
  3. F F
  4. p p
  5. F ( p ) F(p)
  6. F F
  7. H ( t ) H(t)
  8. p y = H ( t ) py=H(t)
  9. y = p - 1 H = 0 t H ( u ) d u = t H ( t ) . y=p^{-1}H=\int_{0}^{t}H(u)du=tH(t).
  10. p - 1 p^{-1}
  11. p - n p^{-n}
  12. n n
  13. p - n H ( t ) = t n n ! H ( t ) . p^{-n}H(t)=\frac{t^{n}}{n!}H(t).
  14. p p - a H ( t ) = 1 1 - a p H ( t ) \frac{p}{p-a}H(t)=\frac{1}{1-\frac{a}{p}}H(t)
  15. 1 1 - a p H ( t ) = n = 0 a n p - n H ( t ) = n = 0 a n t n n ! H ( t ) = e a t H ( t ) . \frac{1}{1-\frac{a}{p}}H(t)=\sum_{n=0}^{\infty}a^{n}p^{-n}H(t)=\sum_{n=0}^{% \infty}\frac{a^{n}t^{n}}{n!}H(t)=e^{at}H(t).
  16. p p
  17. H ( t ) H(t)
  18. 1 F ( p ) = n = 0 a n p - n \frac{1}{F(p)}=\sum_{n=0}^{\infty}a_{n}p^{-n}
  19. 1 F ( p ) H ( t ) = n = 0 a n t n n ! H ( t ) . \frac{1}{F(p)}H(t)=\sum_{n=0}^{\infty}a_{n}\frac{t^{n}}{n!}H(t).
  20. p p
  21. e < s u p > a p f ( t ) = f ( t + a ) e<sup>apf(t)=f(t+a)

Operational_transformation.html

  1. T ( o p 1 , o p 2 ) T(op_{1},op_{2})
  2. T - 1 ( o p 1 , o p 2 ) T^{-1}(op_{1},op_{2})
  3. p 1 , c 1 , s i d 1 p_{1},c_{1},sid_{1}
  4. p 2 , c 2 , s i d 2 p_{2},c_{2},sid_{2}
  5. p 1 < p 2 p_{1}<p_{2}
  6. p 1 , c 1 , s i d 1 p_{1},c_{1},sid_{1}
  7. p 1 = p 2 p_{1}=p_{2}
  8. s i d 1 < s i d 2 sid_{1}<sid_{2}
  9. p 1 , c 1 , s i d 1 p_{1},c_{1},sid_{1}
  10. p 1 + 1 , c 1 , s i d 1 p_{1}+1,c_{1},sid_{1}
  11. T - 1 T^{-1}
  12. p 1 , c 1 , s i d 1 p_{1},c_{1},sid_{1}
  13. p 2 , c 2 , s i d 2 p_{2},c_{2},sid_{2}
  14. p 1 < p 2 p_{1}<p_{2}
  15. p 1 , c 1 , s i d 1 p_{1},c_{1},sid_{1}
  16. p 1 = p 2 p_{1}=p_{2}
  17. s i d 1 < s i d 2 sid_{1}<sid_{2}
  18. p 1 , c 1 , s i d 1 p_{1},c_{1},sid_{1}
  19. p 1 - 1 , c 1 , s i d 1 p_{1}-1,c_{1},sid_{1}
  20. o p 1 op_{1}
  21. o p 2 op_{2}
  22. o p 1 T ( o p 2 , o p 1 ) o p 2 T ( o p 1 , o p 2 ) op_{1}\circ T(op_{2},op_{1})\equiv op_{2}\circ T(op_{1},op_{2})
  23. o p i o p j op_{i}\circ op_{j}
  24. o p i op_{i}
  25. o p j op_{j}
  26. \equiv
  27. o p 1 , o p 2 op_{1},op_{2}
  28. o p 3 op_{3}
  29. T ( o p 3 , o p 1 T ( o p 2 , o p 1 ) ) = T ( o p 3 , o p 2 T ( o p 1 , o p 2 ) ) T(op_{3},op_{1}\circ T(op_{2},op_{1}))=T(op_{3},op_{2}\circ T(op_{1},op_{2}))
  30. o p 3 op_{3}
  31. o p 2 op_{2}
  32. T ( o p 1 , o p 2 ) T(op_{1},op_{2})
  33. o p 3 op_{3}
  34. o p 1 op_{1}
  35. T ( o p 2 , o p 1 ) T(op_{2},op_{1})
  36. o p 1 op_{1}
  37. o p 2 op_{2}
  38. o p o p ¯ op\circ\overline{op}
  39. S o p o p ¯ = S S\circ op\circ\overline{op}=S
  40. o p o p ¯ op\circ\overline{op}
  41. o p o p ¯ op\circ\overline{op}
  42. T ( o p x , o p o p ¯ ) = o p x T(op_{x},op\circ\overline{op})=op_{x}
  43. o p x op_{x}
  44. o p o p ¯ op\circ\overline{op}
  45. o p x op_{x}
  46. o p x op_{x}
  47. o p o p ¯ op\circ\overline{op}
  48. o p 1 op_{1}
  49. o p 2 op_{2}
  50. o p 1 ¯ = T ( o p 1 ¯ , T ( o p 2 , o p 1 ) ) \overline{op_{1}}^{\prime}=T(\overline{op_{1}},T(op_{2},op_{1}))
  51. o p 1 ¯ = T ( o p 1 , o p 2 ) ¯ \overline{op_{1}^{\prime}}=\overline{T(op_{1},op_{2})}
  52. o p 1 ¯ = o p 1 ¯ \overline{op_{1}}^{\prime}=\overline{op_{1}^{\prime}}
  53. o p 1 ¯ \overline{op_{1}}^{\prime}
  54. o p 1 ¯ \overline{op_{1}^{\prime}}
  55. o p 1 ¯ \overline{op_{1}}
  56. o p 2 op_{2}
  57. o p 1 op_{1}

Operator_system.html

  1. 𝒜 \mathcal{A}
  2. 𝒜 \mathcal{M}\subseteq\mathcal{A}
  3. S := + * + 1 S:=\mathcal{M}+\mathcal{M}^{*}+\mathbb{C}1

Opial_property.html

  1. lim inf n x n - x 0 < lim inf n x n - x . \liminf_{n\to\infty}\|x_{n}-x_{0}\|<\liminf_{n\to\infty}\|x_{n}-x\|.
  2. lim inf n x n - x lim inf n x n - x 0 x = x 0 . \liminf_{n\to\infty}\|x_{n}-x\|\leq\liminf_{n\to\infty}\|x_{n}-x_{0}\|\implies x% =x_{0}.
  3. lim inf n x n - x 0 < lim inf n x n - x , \liminf_{n\to\infty}\|x_{n}-x_{0}\|<\liminf_{n\to\infty}\|x_{n}-x\|,
  4. lim inf n x n - x lim inf n x n - x 0 x = x 0 . \liminf_{n\to\infty}\|x_{n}-x\|\leq\liminf_{n\to\infty}\|x_{n}-x_{0}\|\implies x% =x_{0}.
  5. 1 + r lim inf n x n - x 1+r\leq\liminf_{n\to\infty}\|x_{n}-x\|
  6. lim inf n x n 1. \liminf_{n\to\infty}\|x_{n}\|\geq 1.

Optical_correlator.html

  1. c ( x , y ) c(x,y)
  2. i ( x , y ) i(x,y)
  3. h ( x , y ) h(x,y)
  4. c ( x , y ) = i ( x , y ) h * ( - x , - y ) c(x,y)=i(x,y)\otimes h^{*}(-x,-y)
  5. C ( ξ , η ) = I ( ξ , η ) H * ( - ξ , - η ) C(\xi,\eta)=I(\xi,\eta)H^{*}(-\xi,-\eta)
  6. f f
  7. f f
  8. f f
  9. r ( x , y ) r(x,y)
  10. H ( ξ , η ) = R ( ξ , η ) H(\xi,\eta)=R(\xi,\eta)
  11. H ( ξ , η ) = R ( ξ , η ) | R ( ξ , η ) | H(\xi,\eta)=\frac{R(\xi,\eta)}{\left|R(\xi,\eta)\right|}

Optical_DPSK_demodulator.html

  1. ± π / 4 \pm\pi/4

Optical_properties_of_water_and_ice.html

  1. n 2 - 1 n 2 + 2 ( 1 / ρ ¯ ) = a 0 + a 1 ρ ¯ + a 2 T ¯ + a 3 λ ¯ 2 T ¯ + a 4 λ ¯ 2 + a 5 λ ¯ 2 - λ ¯ 𝑈𝑉 2 + a 6 λ ¯ 2 - λ ¯ 𝐼𝑅 2 + a 7 ρ ¯ 2 \frac{n^{2}-1}{n^{2}+2}(1/\overline{\rho})=a_{0}+a_{1}\overline{\rho}+a_{2}% \overline{T}+a_{3}{\overline{\lambda}}^{2}\overline{T}+\frac{a_{4}}{{\overline% {\lambda}}^{2}}+\frac{a_{5}}{{\overline{\lambda}}^{2}-{\overline{\lambda}}_{% \mathit{UV}}^{2}}+\frac{a_{6}}{{\overline{\lambda}}^{2}-{\overline{\lambda}}_{% \mathit{IR}}^{2}}+a_{7}{\overline{\rho}}^{2}
  2. T ¯ = T T * , \overline{T}=\frac{T}{T^{\,\text{*}}},\,\text{ }
  3. ρ ¯ = ρ ρ * , and \overline{\rho}=\frac{\rho}{\rho^{\,\text{*}}},\,\text{ and }
  4. λ ¯ = λ λ * \overline{\lambda}=\frac{\lambda}{\lambda^{\,\text{*}}}
  5. a 0 a_{0}
  6. a 1 a_{1}
  7. a 2 a_{2}
  8. a 3 a_{3}
  9. a 4 a_{4}
  10. a 5 a_{5}
  11. a 6 a_{6}
  12. a 7 a_{7}
  13. T * T^{*}
  14. ρ * \rho^{*}
  15. λ * \lambda^{*}
  16. λ ¯ IR \overline{\lambda}_{\,\text{IR}}
  17. λ ¯ UV \overline{\lambda}_{\,\text{UV}}
  18. λ \lambda
  19. ρ \rho

Optimal_matching.html

  1. S = ( s 1 , s 2 , s 3 , s T ) S=(s_{1},s_{2},s_{3},\ldots s_{T})
  2. s i s_{i}
  3. 𝐒 {\mathbf{S}}
  4. a i : 𝐒 𝐒 a_{i}:{\mathbf{S}}\rightarrow{\mathbf{S}}
  5. s s
  6. a s Ins ( s 1 , s 2 , s 3 , s T ) = ( s 1 , s 2 , s 3 , , s , s T ) a^{\rm Ins}_{s^{\prime}}(s_{1},s_{2},s_{3},\ldots s_{T})=(s_{1},s_{2},s_{3},% \ldots,s^{\prime},\ldots s_{T})
  7. a s 2 Del ( s 1 , s 2 , s 3 , s T ) = ( s 1 , s 3 , s T ) a^{\rm Del}_{s_{2}}(s_{1},s_{2},s_{3},\ldots s_{T})=(s_{1},s_{3},\ldots s_{T})
  8. s 1 s_{1}
  9. s 1 s^{\prime}_{1}
  10. a s 1 , s 1 Sub ( s 1 , s 2 , s 3 , s T ) = ( s 1 , s 2 , s 3 , s T ) a^{\rm Sub}_{s_{1},s^{\prime}_{1}}(s_{1},s_{2},s_{3},\ldots s_{T})=(s^{\prime}% _{1},s_{2},s_{3},\ldots s_{T})
  11. c ( a i ) 𝐑 0 + c(a_{i})\in{\mathbf{R}}^{+}_{0}
  12. S 1 S_{1}
  13. S 2 S_{2}
  14. S 2 S_{2}
  15. S 1 S_{1}
  16. A = a 1 , a 2 , a n A={a_{1},a_{2},\ldots a_{n}}
  17. A A
  18. S 1 S_{1}
  19. S 2 S_{2}
  20. S 2 = a 1 a 2 a n ( S 1 ) S_{2}=a_{1}\circ a_{2}\circ\ldots\circ a_{n}(S_{1})
  21. a 1 a 2 a_{1}\circ a_{2}
  22. c ( A ) = i = 1 n c ( a i ) c(A)=\sum_{i=1}^{n}c(a_{i})
  23. A A
  24. S 1 S_{1}
  25. S 2 S_{2}
  26. d ( S 1 , S 2 ) = min A { c ( A ) such that S 2 = A ( S 1 ) } d(S_{1},S_{2})=\min_{A}\left\{c(A)~{}{\rm such~{}that}~{}S_{2}=A(S_{1})\right\}
  27. S 1 S_{1}
  28. S 2 S_{2}
  29. d ( S 1 , S 2 ) d(S_{1},S_{2})
  30. d ( S 1 , S 2 ) = 0 d(S_{1},S_{2})=0
  31. S 1 = S 2 S_{1}=S_{2}
  32. c ( a Ins ) = c ( a Del ) c(a^{\rm Ins})=c(a^{\rm Del})
  33. c ( a i ) c(a_{i})

Option_(finance).html

  1. d C = Δ d S + Γ d S 2 2 + κ d σ + θ d t dC=\Delta dS+\Gamma\frac{dS^{2}}{2}+\kappa d\sigma+\theta dt\,
  2. Δ \Delta
  3. Γ \Gamma
  4. κ \kappa
  5. θ \theta
  6. d S dS
  7. d σ d\sigma
  8. d t dt
  9. d S dS
  10. d σ d\sigma
  11. d t dt
  12. - Δ -\Delta
  13. Π \Pi
  14. d Π = Δ d S + Γ d S 2 2 + κ d σ + θ d t = Γ d S 2 2 + κ d σ + θ d t d\Pi=\Delta dS+\Gamma\frac{dS^{2}}{2}+\kappa d\sigma+\theta dt=\Gamma\frac{dS^% {2}}{2}+\kappa d\sigma+\theta dt\,
  15. Δ \Delta
  16. Γ \Gamma
  17. κ \kappa
  18. θ \theta
  19. d C = ( 0.439 0.5 ) + ( 0.0631 0.5 2 2 ) + ( 9.6 - 0.015 ) + ( - 0.022 1 ) = 0.0614 dC=(0.439\cdot 0.5)+\left(0.0631\cdot\frac{0.5^{2}}{2}\right)+(9.6\cdot-0.015)% +(-0.022\cdot 1)=0.0614

Order_of_integration.html

  1. k = 0 b k 2 < , \sum_{k=0}^{\infty}\mid{b_{k}}^{2}\mid<\infty,
  2. b b
  3. ( 1 - L ) d X t (1-L)^{d}X_{t}
  4. L L
  5. 1 - L 1-L
  6. ( 1 - L ) X t = X t - X t - 1 = Δ X . (1-L)X_{t}=X_{t}-X_{t-1}=\Delta X.
  7. X t X_{t}
  8. Z t = k = 0 t X k Z_{t}=\sum_{k=0}^{t}X_{k}
  9. Z t = X t , \triangle Z_{t}=X_{t},
  10. X t I ( d - 1 ) . X_{t}\sim I(d-1).\,

Ordinal_optimization.html

  1. a b a\lor b
  2. a b a\land b
  3. a b a\lor b
  4. a b a\land b
  5. \lor
  6. \land
  7. \lor
  8. \lor
  9. \land
  10. \land
  11. A = a 1 a n \bigvee A=a_{1}\lor\cdots\lor a_{n}
  12. A = a 1 a n \bigwedge A=a_{1}\land\cdots\land a_{n}
  13. A = { a 1 , , a n } A=\{a_{1},\ldots,a_{n}\}
  14. = 0 \bigvee\varnothing=0
  15. = 1 \bigwedge\varnothing=1
  16. ( A B ) = ( A ) ( B ) \bigvee\left(A\cup B\right)=\left(\bigvee A\right)\vee\left(\bigvee B\right)
  17. ( A B ) = ( A ) ( B ) \bigwedge\left(A\cup B\right)=\left(\bigwedge A\right)\wedge\left(\bigwedge B\right)
  18. ( A ) = ( A ) ( ) = ( A ) 0 = A \bigvee\left(A\cup\emptyset\right)=\left(\bigvee A\right)\vee\left(\bigvee% \emptyset\right)=\left(\bigvee A\right)\vee 0=\bigvee A
  19. ( A ) = ( A ) ( ) = ( A ) 1 = A \bigwedge\left(A\cup\emptyset\right)=\left(\bigwedge A\right)\wedge\left(% \bigwedge\emptyset\right)=\left(\bigwedge A\right)\wedge 1=\bigwedge A
  20. A = A A\cup\emptyset=A

Oregon_State_University_Radiation_Center.html

  1. n c m 2 * s \frac{n}{cm^{2}*s}

Oren–Nayar_reflectance_model.html

  1. σ 2 \sigma^{2}
  2. σ \sigma
  3. [ 0 , ) [0,\infty)
  4. E 0 E_{0}
  5. L r L_{r}
  6. L r = ρ π cos θ i ( A + ( B max [ 0 , cos ( ϕ i - ϕ r ) ] sin α tan β ) ) E 0 L_{r}=\frac{\rho}{\pi}\cdot\cos\theta_{i}\cdot(A+(B\cdot\max[0,\cos(\phi_{i}-% \phi_{r})]\cdot\sin\alpha\cdot\tan\beta))\cdot E_{0}
  7. A = 1 - 0.5 σ 2 σ 2 + 0.57 A=1-0.5\frac{\sigma^{2}}{\sigma^{2}+0.57}
  8. B = 0.45 σ 2 σ 2 + 0.09 B=0.45\frac{\sigma^{2}}{\sigma^{2}+0.09}
  9. α = max ( θ i , θ r ) \alpha=\max(\theta_{i},\theta_{r})
  10. β = min ( θ i , θ r ) \beta=\min(\theta_{i},\theta_{r})
  11. ρ \rho
  12. σ \sigma
  13. σ = 0 \sigma=0
  14. A = 1 A=1
  15. B = 0 B=0
  16. L r = ρ π cos θ i E 0 L_{r}=\frac{\rho}{\pi}\cdot\cos\theta_{i}\cdot E_{0}
  17. σ \sigma

Orientation_character.html

  1. π \pi
  2. ω : π { ± 1 } \omega\colon\pi\to\left\{\pm 1\right\}
  3. π = π 1 M \pi=\pi_{1}M
  4. ω \omega
  5. π \pi
  6. - 1 -1
  7. ω \omega
  8. 𝐙 [ π ] \mathbf{Z}[\pi]
  9. g ω ( g ) g - 1 g\mapsto\omega(g)g^{-1}
  10. ± g - 1 \pm g^{-1}
  11. g g
  12. 𝐙 [ π ] ω \mathbf{Z}[\pi]^{\omega}

Orientation_entanglement.html

  1. X = ( x 1 x 2 - i x 3 x 2 + i x 3 - x 1 ) X=\left(\begin{matrix}x_{1}&x_{2}-ix_{3}\\ x_{2}+ix_{3}&-x_{1}\end{matrix}\right)
  2. X M X M + X\mapsto MXM^{+}
  3. det ( MXM + ) = det ( X ) \rm{det}(MXM^{+})=\rm{det}(X)
  4. M X M + MXM^{+}

Oriented_coloring.html

  1. χ o ( G ) \scriptstyle\chi_{o}(G)

Oriented_projective_geometry.html

  1. 𝐑 * n \mathbf{R}_{*}^{n}
  2. 𝐑 n \mathbf{R}^{n}
  3. 𝐓 1 \mathbf{T}^{1}
  4. ( x , w ) 𝐑 * 2 (x,w)\in\mathbf{R}^{2}_{*}
  5. ( x , w ) ( a x , a w ) (x,w)\sim(ax,aw)\,
  6. a > 0 a>0
  7. 𝐓 2 \mathbf{T}^{2}
  8. ( x , y , w ) 𝐑 * 3 (x,y,w)\in\mathbf{R}^{3}_{*}
  9. ( x , y , w ) ( a x , a y , a w ) (x,y,w)\sim(ax,ay,aw)\,
  10. a > 0 a>0
  11. 𝐓 1 \mathbf{T}^{1}
  12. 𝐑 \mathbf{R}
  13. 𝐓 2 \mathbf{T}^{2}
  14. 𝐑 2 \mathbf{R}^{2}
  15. 𝐓 \mathbf{T}
  16. p = ( p x , p y , p w ) p=(p_{x},p_{y},p_{w})
  17. q = ( q x , q y , q w ) q=(q_{x},q_{y},q_{w})
  18. 𝐓 2 \mathbf{T}^{2}
  19. 𝐓 1 \mathbf{T}^{1}
  20. ( ( p x q w - q x p w ) 2 + ( p y q w - q y p w ) 2 , sign ( p w q w ) ( p w q w ) 2 ) . ((p_{x}q_{w}-q_{x}p_{w})^{2}+(p_{y}q_{w}-q_{y}p_{w})^{2},\mathrm{sign}(p_{w}q_% {w})(p_{w}q_{w})^{2})\,.

Ornstein–Zernike_equation.html

  1. h ( r 12 ) = g ( r 12 ) - 1 h(r_{12})=g(r_{12})-1\,
  2. r 12 r_{12}
  3. g ( r 12 ) g(r_{12})
  4. c ( r 12 ) c(r_{12})
  5. h ( r 12 ) = c ( r 12 ) + ρ d 𝐫 3 c ( r 13 ) h ( r 23 ) h(r_{12})=c(r_{12})+\rho\int d\mathbf{r}_{3}c(r_{13})h(r_{23})\,
  6. c ( r ) c(r)
  7. h ( r ) h(r)
  8. 𝐫 i j := 𝐫 i - 𝐫 j \mathbf{r}_{ij}:=\mathbf{r}_{i}-\mathbf{r}_{j}
  9. i , j = 1 , 2 , 3 i,j=1,2,3
  10. h ( 𝐫 12 ) = c ( 𝐫 12 ) + ρ d 𝐫 3 c ( 𝐫 12 - 𝐫 32 ) h ( - 𝐫 32 ) = c ( 𝐫 12 ) + ρ d 𝐫 32 c ( 𝐫 12 - 𝐫 32 ) h ( 𝐫 32 ) = c ( 𝐫 12 ) + ρ ( c * h ) ( 𝐫 12 ) h(\mathbf{r}_{12})=c(\mathbf{r}_{12})+\rho\int d\mathbf{r}_{3}c(\mathbf{r}_{12% }-\mathbf{r}_{32})h(-\mathbf{r}_{32})=c(\mathbf{r}_{12})+\rho\int d\mathbf{r}_% {32}c(\mathbf{r}_{12}-\mathbf{r}_{32})h(\mathbf{r}_{32})=c(\mathbf{r}_{12})+% \rho(c\,*\,h)(\mathbf{r}_{12})\,
  11. h ( 𝐫 ) h(\mathbf{r})
  12. c ( 𝐫 ) c(\mathbf{r})
  13. H ^ ( 𝐤 ) \hat{H}(\mathbf{k})
  14. C ^ ( 𝐤 ) \hat{C}(\mathbf{k})
  15. H ^ ( 𝐤 ) = C ^ ( 𝐤 ) + ρ H ^ ( 𝐤 ) C ^ ( 𝐤 ) , \hat{H}(\mathbf{k})=\hat{C}(\mathbf{k})+\rho\hat{H}(\mathbf{k})\hat{C}(\mathbf% {k}),\,
  16. C ^ ( 𝐤 ) = H ^ ( 𝐤 ) 1 + ρ H ^ ( 𝐤 ) H ^ ( 𝐤 ) = C ^ ( 𝐤 ) 1 - ρ C ^ ( 𝐤 ) . \hat{C}(\mathbf{k})=\frac{\hat{H}(\mathbf{k})}{1+\rho\hat{H}(\mathbf{k})}\,\,% \,\,\,\,\,\hat{H}(\mathbf{k})=\frac{\hat{C}(\mathbf{k})}{1-\rho\hat{C}(\mathbf% {k})}.\,
  17. h ( r ) h(r)
  18. c ( r ) c(r)
  19. c ( r ) c(r)
  20. h ( r ) h(r)
  21. u ( r ) u(r)

Orr–Sommerfeld_equation.html

  1. 𝐮 = ( U ( z ) + u ( x , z , t ) , 0 , w ( x , z , t ) ) \mathbf{u}=\left(U(z)+u^{\prime}(x,z,t),0,w^{\prime}(x,z,t)\right)
  2. ( U ( z ) , 0 , 0 ) (U(z),0,0)
  3. 𝐮 exp ( i α ( x - c t ) ) \mathbf{u}^{\prime}\propto\exp(i\alpha(x-ct))
  4. μ i α ρ ( d 2 d z 2 - α 2 ) 2 φ = ( U - c ) ( d 2 d z 2 - α 2 ) φ - U ′′ φ \frac{\mu}{i\alpha\rho}\left({d^{2}\over dz^{2}}-\alpha^{2}\right)^{2}\varphi=% (U-c)\left({d^{2}\over dz^{2}}-\alpha^{2}\right)\varphi-U^{\prime\prime}\varphi
  5. μ \mu
  6. ρ \rho
  7. φ \varphi
  8. U 0 U_{0}
  9. h h
  10. 1 i α R e ( d 2 d z 2 - α 2 ) 2 φ = ( U - c ) ( d 2 d z 2 - α 2 ) φ - U ′′ φ {1\over i\alpha\,Re}\left({d^{2}\over dz^{2}}-\alpha^{2}\right)^{2}\varphi=(U-% c)\left({d^{2}\over dz^{2}}-\alpha^{2}\right)\varphi-U^{\prime\prime}\varphi
  11. R e = ρ U 0 h μ Re=\frac{\rho U_{0}h}{\mu}
  12. z = z 1 z=z_{1}
  13. z = z 2 z=z_{2}
  14. α φ = d φ d z = 0 \alpha\varphi={d\varphi\over dz}=0
  15. z = z 1 z=z_{1}
  16. z = z 2 , z=z_{2},
  17. φ \varphi
  18. α φ = d φ d x = 0 \alpha\varphi={d\varphi\over dx}=0
  19. z = z 1 z=z_{1}
  20. z = z 2 , z=z_{2},
  21. φ \varphi
  22. c c
  23. φ \varphi
  24. c c
  25. U U
  26. c c
  27. α \alpha
  28. R e > R e c = 5772.22 Re>Re_{c}=5772.22
  29. R e = R e c Re=Re_{c}
  30. α c = 1.02056 \alpha_{c}=1.02056
  31. c r = 0.264002 c_{r}=0.264002
  32. Im ( α c ) \,\text{Im}(\alpha{c})
  33. α \alpha
  34. λ = - i α c \lambda=-i\alpha{c}
  35. φ = d φ d z = 0 , \varphi={d\varphi\over dz}=0,
  36. z = 0 z=0
  37. d 2 φ d z 2 + α 2 φ = 0 \frac{d^{2}\varphi}{dz^{2}}+\alpha^{2}\varphi=0
  38. Ω d 3 φ d z 3 + i α R e [ ( c - U ( z 2 = 1 ) ) d φ d z + φ ] - i α R e ( 1 F r + α 2 W e ) φ c - U ( z 2 = 1 ) = 0 , \Omega\equiv\frac{d^{3}\varphi}{dz^{3}}+i\alpha Re\left[\left(c-U\left(z_{2}=1% \right)\right)\frac{d\varphi}{dz}+\varphi\right]-i\alpha Re\left(\frac{1}{Fr}+% \frac{\alpha^{2}}{We}\right)\frac{\varphi}{c-U\left(z_{2}=1\right)}=0,
  39. z = 1 \,z=1
  40. F r = U 0 2 g h , W e = ρ u 0 2 h σ Fr=\frac{U_{0}^{2}}{gh},\,\,\ We=\frac{\rho u_{0}^{2}h}{\sigma}
  41. U ( z ) = z U\left(z\right)=z
  42. χ 1 ( z ) = sinh ( α z ) , χ 2 ( z ) = cosh ( α z ) \chi_{1}\left(z\right)=\sinh\left(\alpha z\right),\qquad\chi_{2}\left(z\right)% =\cosh\left(\alpha z\right)
  43. χ 3 ( z ) = 1 α z sinh [ α ( z - ξ ) ] A i [ e i π / 6 ( α R e ) 1 / 3 ( ξ - c - i α R e ) ] d ξ , \chi_{3}\left(z\right)=\frac{1}{\alpha}\int_{\infty}^{z}\sinh\left[\alpha\left% (z-\xi\right)\right]Ai\left[e^{i\pi/6}\left(\alpha Re\right)^{1/3}\left(\xi-c-% \frac{i\alpha}{Re}\right)\right]d\xi,
  44. χ 4 ( z ) = 1 α z sinh [ α ( z - ξ ) ] A i [ e 5 i π / 6 ( α R e ) 1 / 3 ( ξ - c - i α R e ) ] d ξ , \chi_{4}\left(z\right)=\frac{1}{\alpha}\int_{\infty}^{z}\sinh\left[\alpha\left% (z-\xi\right)\right]Ai\left[e^{5i\pi/6}\left(\alpha Re\right)^{1/3}\left(\xi-c% -\frac{i\alpha}{Re}\right)\right]d\xi,
  45. A i ( ) Ai\left(\cdot\right)
  46. φ = i = 1 4 c i χ i ( z ) \varphi=\sum_{i=1}^{4}c_{i}\chi_{i}\left(z\right)
  47. c i c_{i}
  48. | χ 1 ( 0 ) χ 2 ( 0 ) χ 3 ( 0 ) χ 4 ( 0 ) χ 1 ( 0 ) χ 2 ( 0 ) χ 3 ( 0 ) χ 4 ( 0 ) Ω 1 ( 1 ) Ω 2 ( 1 ) Ω 3 ( 1 ) Ω 4 ( 1 ) χ 1 ′′ ( 1 ) + α 2 χ 1 ( 1 ) χ 2 ′′ ( 1 ) + α 2 χ 2 ( 1 ) χ 3 ′′ ( 1 ) + α 2 χ 3 ( 1 ) χ 4 ′′ ( 1 ) + α 2 χ 4 ( 1 ) | = 0 \left|\begin{array}[]{cccc}\chi_{1}\left(0\right)&\chi_{2}\left(0\right)&\chi_% {3}\left(0\right)&\chi_{4}\left(0\right)\\ \chi_{1}^{\prime}\left(0\right)&\chi_{2}^{\prime}\left(0\right)&\chi_{3}^{% \prime}\left(0\right)&\chi_{4}^{\prime}\left(0\right)\\ \Omega_{1}\left(1\right)&\Omega_{2}\left(1\right)&\Omega_{3}\left(1\right)&% \Omega_{4}\left(1\right)\\ \chi_{1}^{\prime\prime}\left(1\right)+\alpha^{2}\chi_{1}\left(1\right)&\chi_{2% }^{\prime\prime}\left(1\right)+\alpha^{2}\chi_{2}\left(1\right)&\chi_{3}^{% \prime\prime}\left(1\right)+\alpha^{2}\chi_{3}\left(1\right)&\chi_{4}^{\prime% \prime}\left(1\right)+\alpha^{2}\chi_{4}\left(1\right)\end{array}\right|=0
  49. α \alpha
  50. α c i \alpha c_{\,\text{i}}
  51. α c \alpha c

Oscillation_theory.html

  1. F ( x , y , y , , y ( n - 1 ) ) = y ( n ) x [ 0 , + ) F(x,y,y^{\prime},\ \dots,\ y^{(n-1)})=y^{(n)}\quad x\in[0,+\infty)
  2. y ′′ + y = 0 y^{\prime\prime}+y=0

Ostriker–Peebles_criterion.html

  1. T W > 0.15 \frac{T}{W}>0.15

Outliers_ratio.html

  1. [ MOS - 2 σ , MOS + 2 σ ] [\,\text{MOS}-2\sigma,\,\text{MOS}+2\sigma]\,

Outline_of_energy_storage.html

  1. E E
  2. B B
  3. E = 1 2 B 2 = 1 2 L i 2 E=\frac{1}{2}B^{2}=\frac{1}{2}Li^{2}
  4. L L
  5. i i

Overdetermined_system.html

  1. 2 x 1 + x 2 = - 1 2x_{1}+x_{2}=-1
  2. - 3 x 1 + x 2 = - 2 -3x_{1}+x_{2}=-2
  3. - x 1 + x 2 = 1 -x_{1}+x_{2}=1
  4. [ 2 1 - 3 1 - 1 1 ] [ X 1 X 2 ] = [ - 1 - 2 1 ] \begin{bmatrix}2&1\\ -3&1\\ -1&1\\ \end{bmatrix}\begin{bmatrix}X_{1}\\ X_{2}\\ \end{bmatrix}=\begin{bmatrix}-1\\ -2\\ 1\\ \end{bmatrix}
  5. A x = b , Ax=b,
  6. min x A x - b , \min_{x}\|Ax-b\|,
  7. x = ( A T A ) - 1 A T b , x=(A^{\mathrm{T}}A)^{-1}A^{\mathrm{T}}b,
  8. T {\mathrm{T}}
  9. ( A T A ) - 1 (A^{\mathrm{T}}A)^{-1}
  10. ( x - 1 ) ( x - 2 ) = 0 , ( x - 1 ) ( x - 3 ) = 0 (x-1)(x-2)=0,(x-1)(x-3)=0
  11. x = 1 , x=1,

Overlap–add_method.html

  1. x [ n ] x[n]
  2. h [ n ] h[n]
  3. y [ n ] = x [ n ] * h [ n ] = def m = - h [ m ] x [ n - m ] = m = 1 M h [ m ] x [ n - m ] , \begin{aligned}\displaystyle y[n]=x[n]*h[n]\ \stackrel{\mathrm{def}}{=}\ \sum_% {m=-\infty}^{\infty}h[m]\cdot x[n-m]=\sum_{m=1}^{M}h[m]\cdot x[n-m],\end{aligned}
  4. x [ n ] x[n]
  5. x k [ n ] = def { x [ n + k L ] n = 1 , 2 , , L 0 otherwise , x_{k}[n]\ \stackrel{\mathrm{def}}{=}\begin{cases}x[n+kL]&n=1,2,\ldots,L\\ 0&\textrm{otherwise},\end{cases}
  6. x [ n ] = k x k [ n - k L ] , x[n]=\sum_{k}x_{k}[n-kL],\,
  7. y [ n ] = ( k x k [ n - k L ] ) * h [ n ] = k ( x k [ n - k L ] * h [ n ] ) = k y k [ n - k L ] , \begin{aligned}\displaystyle y[n]=\left(\sum_{k}x_{k}[n-kL]\right)*h[n]&% \displaystyle=\sum_{k}\left(x_{k}[n-kL]*h[n]\right)\\ &\displaystyle=\sum_{k}y_{k}[n-kL],\end{aligned}
  8. y k [ n ] = def x k [ n ] * h [ n ] y_{k}[n]\ \stackrel{\mathrm{def}}{=}\ x_{k}[n]*h[n]\,
  9. N L + M - 1 , N\geq L+M-1,\,
  10. N N\,
  11. x k [ n ] x_{k}[n]\,
  12. h [ n ] h[n]\,
  13. y k [ n ] = IFFT ( FFT ( x k [ n ] ) FFT ( h [ n ] ) ) y_{k}[n]=\textrm{IFFT}\left(\textrm{FFT}\left(x_{k}[n]\right)\cdot\textrm{FFT}% \left(h[n]\right)\right)
  14. N N
  15. x [ n ] x[n]
  16. y k [ n ] y_{k}[n]
  17. x k [ n ] x_{k}[n]
  18. h [ n ] h[n]
  19. y k [ n ] y_{k}[n]
  20. y k [ n ] y_{k}[n]
  21. L L
  22. N N
  23. C = N 2 log 2 N C=\frac{N}{2}\log_{2}N
  24. C O A = N x N - M + 1 N ( log 2 N + 1 ) C_{OA}=\left\lceil\frac{N_{x}}{N-M+1}\right\rceil N\left(\log_{2}N+1\right)\,
  25. C O A C_{OA}
  26. M L M_{L}
  27. N N
  28. C O A ( N ) = C O A ( 2 m ) C_{OA}\left(N\right)=C_{OA}\left(2^{m}\right)
  29. m m
  30. log 2 ( M ) m log 2 ( N x ) \log_{2}\left(M\right)\leq m\leq\log_{2}\left(N_{x}\right)
  31. N N
  32. N N
  33. x [ n ] x[n]
  34. L = N - M + 1 L=N-M+1
  35. x [ n ] x[n]
  36. h [ n ] h[n]
  37. C S = N x ( log 2 N x + 1 ) C_{S}=N_{x}\left(\log_{2}N_{x}+1\right)\,
  38. O ( N x log 2 N ) O\left(N_{x}\log_{2}N\right)
  39. O ( N x log 2 N x ) O\left(N_{x}\log_{2}N_{x}\right)
  40. N x N_{x}
  41. M M

Overtime_rate.html

  1. Overtime Rate = Overtime Hours Regular Hours (defined) \textstyle{\mbox{Overtime Rate }~{}=\frac{\sum{\mbox{Overtime Hours}~{}}}{\sum% {\mbox{Regular Hours (defined)}~{}}}}

Oxygen_balance.html

  1. O B % = - 1600 M o l . w t . o f c o m p o u n d × ( 2 X + ( Y / 2 ) + M - Z ) OB\%=\frac{-1600}{Mol.wt.ofcompound}\times(2X+(Y/2)+M-Z)
  2. O B % = - 1600 227.1 × ( 14 + 2.5 - 6 ) OB\%=\frac{-1600}{227.1}\times(14+2.5-6)

Oxygen_enhancement_ratio.html

  1. O E R = R a d i a t i o n d o s e i n h y p o x i a R a d i a t i o n d o s e i n a i r OER=\frac{Radiation\,dose\,in\,hypoxia}{Radiation\,dose\,in\,air}

Oxygen_evolution.html

  1. \longrightarrow

P-wave_modulus.html

  1. M M
  2. σ z z = M ϵ z z \sigma_{zz}=M\epsilon_{zz}
  3. ϵ * * \epsilon_{**}
  4. M = ρ V P 2 M=\rho V_{\mathrm{P}}^{2}

Padé_table.html

  1. f ( z ) = c 0 + c 1 z + c 2 z 2 + = l = 0 c l z l , f(z)=c_{0}+c_{1}z+c_{2}z^{2}+\cdots=\sum_{l=0}^{\infty}c_{l}z^{l},
  2. R m , n ( z ) = P m ( z ) Q n ( z ) = a 0 + a 1 z + a 2 z 2 + + a m z m b 0 + b 1 z + b 2 z 2 + + b n z n R_{m,n}(z)=\frac{P_{m}(z)}{Q_{n}(z)}=\frac{a_{0}+a_{1}z+a_{2}z^{2}+\cdots+a_{m% }z^{m}}{b_{0}+b_{1}z+b_{2}z^{2}+\cdots+b_{n}z^{n}}
  3. f ( z ) l = 0 m + n c l z l = : f a p x ( z ) f(z)\approx\sum_{l=0}^{m+n}c_{l}z^{l}=:f_{apx}(z)
  4. Q n ( z ) f a p x ( z ) = P m ( z ) Q_{n}(z)f_{apx}(z)=P_{m}(z)
  5. Q n ( z ) ( c 0 + c 1 z + c 2 z 2 + + c m + n z m + n ) = P m ( z ) Q_{n}(z)\left(c_{0}+c_{1}z+c_{2}z^{2}+\cdots+c_{m+n}z^{m+n}\right)=P_{m}(z)
  6. D m , n = | c m c m - 1 c m - n + 2 c m - n + 1 c m + 1 c m c m - n + 3 c m - n + 2 c m + n - 2 c m + n - 3 c m c m - 1 c m + n - 1 c m + n - 2 c m + 1 c m | D_{m,n}=\left|\begin{matrix}c_{m}&c_{m-1}&\ldots&c_{m-n+2}&c_{m-n+1}\\ c_{m+1}&c_{m}&\ldots&c_{m-n+3}&c_{m-n+2}\\ \vdots&\vdots&&\vdots&\vdots\\ c_{m+n-2}&c_{m+n-3}&\ldots&c_{m}&c_{m-1}\\ c_{m+n-1}&c_{m+n-2}&\ldots&c_{m+1}&c_{m}\\ \end{matrix}\right|
  7. f ( z ) = b 0 + a 1 z 1 - a 2 z 1 - a 3 z 1 - a 4 z 1 - . f(z)=b_{0}+\cfrac{a_{1}z}{1-\cfrac{a_{2}z}{1-\cfrac{a_{3}z}{1-\cfrac{a_{4}z}{1% -\ddots}}}}.
  8. 1 1 \frac{1}{1}
  9. 1 1 - z \frac{1}{1-z}
  10. 1 1 - z + 1 2 z 2 \frac{1}{1-z+{\scriptstyle\frac{1}{2}}z^{2}}
  11. 1 1 - z + 1 2 z 2 - 1 6 z 3 \frac{1}{1-z+{\scriptstyle\frac{1}{2}}z^{2}-{\scriptstyle\frac{1}{6}}z^{3}}
  12. 1 + z 1 \frac{1+z}{1}
  13. 1 + 1 2 z 1 - 1 2 z \frac{1+{\scriptstyle\frac{1}{2}}z}{1-{\scriptstyle\frac{1}{2}}z}
  14. 1 + 1 3 z 1 - 2 3 z + 1 6 z 2 \frac{1+{\scriptstyle\frac{1}{3}}z}{1-{\scriptstyle\frac{2}{3}}z+{\scriptstyle% \frac{1}{6}}z^{2}}
  15. 1 + 1 4 z 1 - 3 4 z + 1 4 z 2 - 1 24 z 3 \frac{1+{\scriptstyle\frac{1}{4}}z}{1-{\scriptstyle\frac{3}{4}}z+{\scriptstyle% \frac{1}{4}}z^{2}-{\scriptstyle\frac{1}{24}}z^{3}}
  16. 1 + z + 1 2 z 2 1 \frac{1+z+{\scriptstyle\frac{1}{2}}z^{2}}{1}
  17. 1 + 2 3 z + 1 6 z 2 1 - 1 3 z \frac{1+{\scriptstyle\frac{2}{3}}z+{\scriptstyle\frac{1}{6}}z^{2}}{1-{% \scriptstyle\frac{1}{3}}z}
  18. 1 + 1 2 z + 1 12 z 2 1 - 1 2 z + 1 12 z 2 \frac{1+{\scriptstyle\frac{1}{2}}z+{\scriptstyle\frac{1}{12}}z^{2}}{1-{% \scriptstyle\frac{1}{2}}z+{\scriptstyle\frac{1}{12}}z^{2}}
  19. 1 + 2 5 z + 1 20 z 2 1 - 3 5 z + 3 20 z 2 - 1 60 z 3 \frac{1+{\scriptstyle\frac{2}{5}}z+{\scriptstyle\frac{1}{20}}z^{2}}{1-{% \scriptstyle\frac{3}{5}}z+{\scriptstyle\frac{3}{20}}z^{2}-{\scriptstyle\frac{1% }{60}}z^{3}}
  20. 1 + z + 1 2 z 2 + 1 6 z 3 1 \frac{1+z+{\scriptstyle\frac{1}{2}}z^{2}+{\scriptstyle\frac{1}{6}}z^{3}}{1}
  21. 1 + 3 4 z + 1 4 z 2 + 1 24 z 3 1 - 1 4 z \frac{1+{\scriptstyle\frac{3}{4}}z+{\scriptstyle\frac{1}{4}}z^{2}+{% \scriptstyle\frac{1}{24}}z^{3}}{1-{\scriptstyle\frac{1}{4}}z}
  22. 1 + 3 5 z + 3 20 z 2 + 1 60 z 3 1 - 2 5 z + 1 20 z 2 \frac{1+{\scriptstyle\frac{3}{5}}z+{\scriptstyle\frac{3}{20}}z^{2}+{% \scriptstyle\frac{1}{60}}z^{3}}{1-{\scriptstyle\frac{2}{5}}z+{\scriptstyle% \frac{1}{20}}z^{2}}
  23. 1 + 1 2 z + 1 10 z 2 + 1 120 z 3 1 - 1 2 z + 1 10 z 2 - 1 120 z 3 \frac{1+{\scriptstyle\frac{1}{2}}z+{\scriptstyle\frac{1}{10}}z^{2}+{% \scriptstyle\frac{1}{120}}z^{3}}{1-{\scriptstyle\frac{1}{2}}z+{\scriptstyle% \frac{1}{10}}z^{2}-{\scriptstyle\frac{1}{120}}z^{3}}
  24. 1 + z + 1 2 z 2 + 1 6 z 3 + 1 24 z 4 1 \frac{1+z+{\scriptstyle\frac{1}{2}}z^{2}+{\scriptstyle\frac{1}{6}}z^{3}+{% \scriptstyle\frac{1}{24}}z^{4}}{1}
  25. 1 + 4 5 z + 3 10 z 2 + 1 15 z 3 + 1 120 z 4 1 - 1 5 z \frac{1+{\scriptstyle\frac{4}{5}}z+{\scriptstyle\frac{3}{10}}z^{2}+{% \scriptstyle\frac{1}{15}}z^{3}+{\scriptstyle\frac{1}{120}}z^{4}}{1-{% \scriptstyle\frac{1}{5}}z}
  26. 1 + 2 3 z + 1 5 z 2 + 1 30 z 3 + 1 360 z 4 1 - 1 3 z + 1 30 z 2 \frac{1+{\scriptstyle\frac{2}{3}}z+{\scriptstyle\frac{1}{5}}z^{2}+{% \scriptstyle\frac{1}{30}}z^{3}+{\scriptstyle\frac{1}{360}}z^{4}}{1-{% \scriptstyle\frac{1}{3}}z+{\scriptstyle\frac{1}{30}}z^{2}}
  27. 1 + 4 7 z + 1 7 z 2 + 2 105 z 3 + 1 840 z 4 1 - 3 7 z + 1 14 z 2 - 1 210 z 3 \frac{1+{\scriptstyle\frac{4}{7}}z+{\scriptstyle\frac{1}{7}}z^{2}+{% \scriptstyle\frac{2}{105}}z^{3}+{\scriptstyle\frac{1}{840}}z^{4}}{1-{% \scriptstyle\frac{3}{7}}z+{\scriptstyle\frac{1}{14}}z^{2}-{\scriptstyle\frac{1% }{210}}z^{3}}
  28. F 1 1 {}_{1}F_{1}
  29. R m , n = F 1 1 ( - m ; - m - n ; z ) F 1 1 ( - n ; - m - n ; - z ) R_{m,n}=\frac{{}_{1}F_{1}(-m;-m-n;z)}{{}_{1}F_{1}(-n;-m-n;-z)}
  30. e z = 1 + z 1 - 1 2 z 1 + 1 6 z 1 - 1 6 z 1 + 1 10 z 1 - 1 10 z 1 + - . e^{z}=1+\cfrac{z}{1-\cfrac{\frac{1}{2}z}{1+\cfrac{\frac{1}{6}z}{1-\cfrac{\frac% {1}{6}z}{1+\cfrac{\frac{1}{10}z}{1-\cfrac{\frac{1}{10}z}{1+-\ddots}}}}}}.
  31. e z = 1 e - z ; e^{z}=\frac{1}{e^{-z}};
  32. e z = 1 1 - z 1 + 1 2 z 1 - 1 6 z 1 + 1 6 z 1 - 1 10 z 1 + 1 10 z 1 - + . e^{z}=\cfrac{1}{1-\cfrac{z}{1+\cfrac{\frac{1}{2}z}{1-\cfrac{\frac{1}{6}z}{1+% \cfrac{\frac{1}{6}z}{1-\cfrac{\frac{1}{10}z}{1+\cfrac{\frac{1}{10}z}{1-+\ddots% }}}}}}}.
  33. L ( z ) = c 0 + n = 1 c n k = 1 n ( z - β k ) L(z)=c_{0}+\sum_{n=1}^{\infty}c_{n}\prod_{k=1}^{n}(z-\beta_{k})
  34. a 0 + a 1 ( z - β 1 ) 1 - a 2 ( z - β 2 ) 1 - a 3 ( z - β 3 ) 1 - . a_{0}+\cfrac{a_{1}(z-\beta_{1})}{1-\cfrac{a_{2}(z-\beta_{2})}{1-\cfrac{a_{3}(z% -\beta_{3})}{1-\ddots}}}.

Pair-instability_supernova.html

  1. E = m c 2 E=mc^{2}

Pairing-based_cryptography.html

  1. e : G 1 × G 2 G T e:G_{1}\times G_{2}\to G_{T}
  2. G 1 , G 2 G_{1},G_{2}
  3. q q
  4. G T G_{T}
  5. q q
  6. e : G 1 × G 2 G T e:G_{1}\times G_{2}\rightarrow G_{T}
  7. a , b F q * , P G 1 , Q G 2 : e ( a P , b Q ) = e ( P , Q ) a b \forall a,b\in F_{q}^{*},\ \forall P\in G_{1},Q\in G_{2}:\ e\left(aP,bQ\right)% =e\left(P,Q\right)^{ab}
  8. e ( P , Q ) 1 e\left(P,Q\right)\neq 1
  9. e e
  10. G 1 = G 2 G_{1}=G_{2}
  11. G 1 = G 2 G_{1}=G_{2}
  12. G 1 G 2 G_{1}\neq G_{2}
  13. ϕ : G 2 G 1 \phi:G_{2}\to G_{1}
  14. G 1 G 2 G_{1}\neq G_{2}
  15. G 1 G_{1}
  16. G 2 G_{2}

Paley–Zygmund_inequality.html

  1. E [ Z ] = E [ Z 1 { Z θ E [ Z ] } ] + E [ Z 1 { Z > θ E [ Z ] } ] . \operatorname{E}[Z]=\operatorname{E}[Z\,\mathbf{1}_{\{Z\leq\theta\operatorname% {E}[Z]\}}]+\operatorname{E}[Z\,\mathbf{1}_{\{Z>\theta\operatorname{E}[Z]\}}].
  2. θ E [ Z ] \theta\operatorname{E}[Z]
  3. E [ Z 2 ] 1 / 2 P ( Z > θ E [ Z ] ) 1 / 2 \operatorname{E}[Z^{2}]^{1/2}\operatorname{P}(Z>\theta\operatorname{E}[Z])^{1/2}
  4. P ( Z > θ E [ Z ] ) ( 1 - θ ) 2 E [ Z ] 2 Var Z + E [ Z ] 2 . \operatorname{P}(Z>\theta\operatorname{E}[Z])\geq\frac{(1-\theta)^{2}\,% \operatorname{E}[Z]^{2}}{\operatorname{Var}Z+\operatorname{E}[Z]^{2}}.
  5. E [ Z - θ E [ Z ] ] E [ ( Z - θ E [ Z ] ) 𝟏 { Z > θ E [ Z ] } ] E [ ( Z - θ E [ Z ] ) 2 ] 1 / 2 P ( Z > θ E [ Z ] ) 1 / 2 \operatorname{E}[Z-\theta\operatorname{E}[Z]]\leq\operatorname{E}[(Z-\theta% \operatorname{E}[Z])\mathbf{1}_{\{Z>\theta\operatorname{E}[Z]\}}]\leq% \operatorname{E}[(Z-\theta\operatorname{E}[Z])^{2}]^{1/2}\operatorname{P}(Z>% \theta\operatorname{E}[Z])^{1/2}
  6. P ( Z > θ E [ Z ] ) ( 1 - θ ) 2 E [ Z ] 2 E [ ( Z - θ E [ Z ] ) 2 ] = ( 1 - θ ) 2 E [ Z ] 2 Var Z + ( 1 - θ ) 2 E [ Z ] 2 . \operatorname{P}(Z>\theta\operatorname{E}[Z])\geq\frac{(1-\theta)^{2}% \operatorname{E}[Z]^{2}}{\operatorname{E}[(Z-\theta\operatorname{E}[Z])^{2}]}=% \frac{(1-\theta)^{2}\operatorname{E}[Z]^{2}}{\operatorname{Var}Z+(1-\theta)^{2% }\operatorname{E}[Z]^{2}}.