wpmath0000015_6

Normalized_compression_distance.html

  1. x x
  2. y y
  3. p p
  4. x x
  5. y y
  6. p p
  7. | p | = max { K ( x y ) , K ( y x ) } |p|=\max\{K(x\mid y),K(y\mid x)\}
  8. N I D ( x , y ) = max { K ( x y ) , K ( y x ) } max { K ( x ) , K ( y ) } , NID(x,y)=\frac{\max\{K{(x\mid y)},K{(y\mid x)}\}}{\max\{K(x),K(y)\}},
  9. K ( x y ) K(x\mid y)
  10. x x
  11. y y
  12. N I D ( x , y ) NID(x,y)
  13. K K
  14. Z ( x ) Z(x)
  15. x x
  16. N C D Z ( x , y ) = Z ( x y ) - min { Z ( x ) , Z ( y ) } max { Z ( x ) , Z ( y ) } . NCD_{Z}(x,y)=\frac{Z(xy)-\min\{Z(x),Z(y)\}}{\max\{Z(x),Z(y)\}}.
  17. N G D ( x , y ) = max { log f ( x ) , log f ( y ) } - log f ( x , y ) log N - min { log f ( x ) , log f ( y ) } , NGD(x,y)=\frac{\max\{\log f(x),\log f(y)\}-\log f(x,y)}{\log N-\min\{\log f(x)% ,\log f(y)\}},
  18. f ( x ) f(x)
  19. x x
  20. f ( x , y ) f(x,y)
  21. x x
  22. y y
  23. N N

Nu_function.html

  1. ν ( x ) 0 x t d t Γ ( t + 1 ) ν ( x , α ) 0 x α + t d t Γ ( α + t + 1 ) \begin{aligned}\displaystyle\nu(x)&\displaystyle\equiv\int_{0}^{\infty}\frac{x% ^{t}\,dt}{\Gamma(t+1)}\\ \displaystyle\nu(x,\alpha)&\displaystyle\equiv\int_{0}^{\infty}\frac{x^{\alpha% +t}\,dt}{\Gamma(\alpha+t+1)}\end{aligned}
  2. Γ ( z ) \Gamma(z)

Nuclear_forensics.html

  1. A + n B * + γ A+n\to B^{*}+\gamma
  2. A A
  3. n n
  4. B * B^{*}
  5. γ \gamma

Nuclear_transparency.html

  1. σ N \sigma_{N}
  2. σ 0 \sigma_{0}
  3. T = σ N / A σ 0 T=\sigma_{N}/A\sigma_{0}
  4. σ N \sigma_{N}
  5. σ 0 \sigma_{0}
  6. σ N = A α σ 0 \sigma_{N}=A^{\alpha}\sigma_{0}
  7. T = A α - 1 T=A^{\alpha-1}

Nucleoside-triphosphate-hexose-1-phosphate_nucleotidyltransferase.html

  1. \rightleftharpoons

Nucleoside_oxidase.html

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

Nucleoside_oxidase_(H2O2-forming).html

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

O-aminophenol_oxidase.html

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

O-phospho-L-seryl-tRNA:Cys-tRNA_synthase.html

  1. \rightleftharpoons

O-phospho-L-seryl-tRNASec:L-selenocysteinyl-tRNA_synthase.html

  1. \rightleftharpoons

O-phosphoseryl-tRNASec_kinase.html

  1. \rightleftharpoons

Octanoyl-(GcvH):protein_N-octanoyltransferase.html

  1. \rightleftharpoons

Octave_Boudouard.html

  1. \rightleftharpoons

Octic_reciprocity.html

  1. ( p | q ) 8 = ( q | p ) 8 = ( a B - b A | q ) 4 ( c D - d C | q ) 2 . (p|q)_{8}=(q|p)_{8}=(aB-bA|q)_{4}(cD-dC|q)_{2}\ .

Oersted's_law.html

  1. 𝐁 ( 𝐱 ) \mathbf{B}(\mathbf{x})\,
  2. C C\,
  3. I I\,
  4. C 𝐁 d s y m b o l = μ 0 I \oint_{C}\mathbf{B}\cdot\mathrm{d}symbol{\ell}=\mu_{0}I\,
  5. μ 0 \mu_{0}\,
  6. C C\,
  7. 𝐉 ( 𝐱 ) \mathbf{J}(\mathbf{x})\,
  8. S S\,
  9. I I\,
  10. C 𝐁 d s y m b o l = μ 0 S 𝐉 d 𝐒 \oint_{C}\mathbf{B}\cdot\mathrm{d}symbol{\ell}=\mu_{0}\iint_{S}\mathbf{J}\cdot% \mathrm{d}\mathbf{S}\,
  11. S S\,
  12. C C\,

Offshore_crane_shock_absorber.html

  1. F m = ψ m s g F_{m}=\psi m_{s}g
  2. F m F_{m}
  3. ψ \psi
  4. m s m_{s}
  5. g g
  6. ψ \psi
  7. m s m_{s}
  8. F m ψ g \frac{F_{m}}{\psi g}
  9. E k = 1 2 m s v r 2 E_{k}=\frac{1}{2}m_{s}v_{r}^{2}
  10. E k E_{k}
  11. v r v_{r}
  12. E c = 1 2 k y 2 E_{c}=\frac{1}{2}ky^{2}
  13. E c E_{c}
  14. k k
  15. y y
  16. F c = k y F_{c}=ky
  17. E c = 1 2 k ( F c k ) 2 = F c 2 2 k E_{c}=\frac{1}{2}k(\frac{F_{c}}{k})^{2}=\frac{F_{c}^{2}}{2k}
  18. E c = ( m s ψ g - m s g ) 2 2 k E_{c}=\frac{(m_{s}\psi g-m_{s}g)^{2}}{2k}
  19. E c E_{c}
  20. E k E_{k}
  21. ( m s ψ g - m s g ) 2 2 k = 1 2 m s v r 2 \frac{(m_{s}\psi g-m_{s}g)^{2}}{2k}=\frac{1}{2}m_{s}v_{r}^{2}
  22. ψ \psi
  23. ψ = 1 + v r g k m \psi=1+\frac{v_{r}}{g}\sqrt{\frac{k}{m}}
  24. ψ \psi
  25. v r v_{r}
  26. E a = η μ S m s g ( ψ - 1 ) E_{a}=\eta\mu Sm_{s}g(\psi-1)
  27. E a E_{a}
  28. μ \mu
  29. η \eta
  30. S S
  31. ( m s ψ g - m s g ) 2 2 k + η μ S m s g ( ψ - 1 ) = 1 2 m s v r 2 \frac{(m_{s}\psi g-m_{s}g)^{2}}{2k}+\eta\mu Sm_{s}g(\psi-1)=\frac{1}{2}m_{s}v_% {r}^{2}
  32. ψ = 1 + ( μ η S k ) 2 + m s k v r 2 - μ η S k m s g \psi=1+\frac{\sqrt{(\mu\eta Sk)^{2}+m_{s}kv_{r}^{2}}-\mu\eta Sk}{m_{s}g}
  33. v r = 1 2 v L + v d 2 + v c 2 v_{r}=\frac{1}{2}v_{L}+\sqrt{v_{d}^{2}+v_{c}^{2}}
  34. v L v_{L}
  35. v d v_{d}
  36. v c v_{c}
  37. v L v_{L}
  38. v d v_{d}
  39. 0
  40. 0.6 H s 0.6H_{s}
  41. H s H_{s}
  42. 1.8 + 0.3 ( H s - 3 ) 1.8+0.3(H_{s}-3)
  43. H s H_{s}
  44. v c v_{c}
  45. v d v_{d}
  46. 0
  47. 0.05 H s 0.05H_{s}
  48. 0.082 H s 2 0.082H_{s}^{2}
  49. 0.164 H s 2 0.164H_{s}^{2}

Oil_dispersants.html

  1. h t + ( h ( U + τ f ) ) - ( E h ) = R \frac{\partial h}{\partial t}+\vec{\nabla}\left(h\left(\vec{U}+\frac{\vec{\tau% }}{f}\right)\right)-\vec{\nabla}(E\vec{\nabla}h)=R
  2. U \vec{U}
  3. τ \vec{\tau}

Omega-hydroxypalmitate_O-feruloyl_transferase.html

  1. \rightleftharpoons

Omnitruncated_6-simplex_honeycomb.html

  1. A ~ 7 {\tilde{A}}_{7}
  2. 6 * {}^{*}_{6}
  3. 6 * {}^{*}_{6}
  4. 6 7 {}^{7}_{6}
  5. A ~ 7 {\tilde{A}}_{7}
  6. C ~ 4 {\tilde{C}}_{4}

Omnitruncated_7-simplex_honeycomb.html

  1. A ~ 8 {\tilde{A}}_{8}
  2. 7 * {}^{*}_{7}
  3. 7 8 {}^{8}_{7}

Omnitruncated_8-simplex_honeycomb.html

  1. A ~ 9 {\tilde{A}}_{9}
  2. 8 * {}^{*}_{8}
  3. 8 * {}^{*}_{8}
  4. 8 9 {}^{9}_{8}

On-base_plus_slugging_plus_runs_batted_in.html

  1. O P S B I = ( ( O B P + S L G ) * 1000 ) + R B I OPSBI=((OBP+SLG)*1000)+RBI\,
  2. O B P = H + B B + H B P A B + B B + S F + H B P OBP=\frac{H+BB+HBP}{AB+BB+SF+HBP}
  3. S L G = T B A B SLG=\frac{TB}{AB}
  4. O P S B I = A B * ( H + B B + H B P ) + T B * ( A B + B B + S F + H B P ) A B * ( A B + B B + S F + H B P ) * 1000 + R B I OPSBI=\frac{AB*(H+BB+HBP)+TB*(AB+BB+SF+HBP)}{AB*(AB+BB+SF+HBP)}*1000+RBI

On_the_Equilibrium_of_Planes.html

  1. A : B = C D : E C A:B=CD:EC\,

One-dimensional_Saint-Venant_equation.html

  1. u t + u u x + v u y + w u z = - p x 1 ρ + ν ( 2 u x 2 + 2 u y 2 + 2 u z 2 ) + f x , \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u% }{\partial y}+w\frac{\partial u}{\partial z}=-\frac{\partial p}{\partial x}% \frac{1}{\rho}+\nu\left(\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2% }u}{\partial y^{2}}+\frac{\partial^{2}u}{\partial z^{2}}\right)+f_{x},
  2. ν ( 2 u x 2 + 2 u y 2 + 2 u z 2 ) = 0. \nu\left(\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^% {2}}+\frac{\partial^{2}u}{\partial z^{2}}\right)=0.
  3. v u y + w u z = 0 v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}=0
  4. p = ρ g h p=\rho gh
  5. p = ρ g ( h ) . \partial p=\rho g\left(\partial h\right).
  6. - p x 1 ρ = - 1 ρ ρ g ( h ) x = - g h x . -\frac{\partial p}{\partial x}\frac{1}{\rho}=-\frac{1}{\rho}\frac{\rho g\left(% \partial h\right)}{\partial x}=-g\frac{\partial h}{\partial x}.
  7. f x = f x , g + f x , f f_{x}=f_{x,g}+f_{x,f}
  8. F g = ( sin θ ) g M F_{g}=(\sin\theta)gM
  9. sin θ = o p p h y p . \sin\theta=\frac{opp}{hyp}.
  10. sin θ = tan θ = o p p a d j = S \sin\theta=\tan\theta=\frac{opp}{adj}=S
  11. f x , g = g S . f_{x,g}=gS.
  12. f x , f = S f g . f_{x,f}=S_{f}g.
  13. u t + u u x + g h x + g ( S - S f ) = 0 \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+g\frac{\partial h% }{\partial x}+g(S-S_{f})=0
  14. ( a ) ( b ) ( c ) ( d ) ( e ) (a)\quad\ \ (b)\quad\ \ \ (c)\qquad(d)\ \ \ (e)
  15. ( S - S f ) = 0. (S-S_{f})=0.
  16. g h x + g ( S - S f ) = 0. g\frac{\partial h}{\partial x}+g(S-S_{f})=0.
  17. Q t + V w Q x = 0 \frac{\partial Q}{\partial t}+V_{w}\frac{\partial Q}{\partial x}=0
  18. V w = d Q d h T = 1 T d d h ( 1.0 n A R 2 / 3 S 1 / 2 ) V_{w}=\frac{\frac{dQ}{dh}}{T}=\frac{1}{T}\frac{d}{dh}\left(\frac{1.0}{n}AR^{2/% 3}S^{1/2}\right)
  19. V w = 5 3 V . V_{w}=\frac{5}{3}V.
  20. C = V w Δ t Δ x C=V_{w}\frac{\Delta t}{\Delta x}
  21. Q i j + 1 Q^{j+1}_{i}
  22. Q i j Q^{j}_{i}
  23. Q i j + 1 = ( 1 - C ) Q i j + C ( Q i - 1 j ) . Q^{j+1}_{i}=\left(1-C\right)Q^{j}_{i}+C\left(Q^{j}_{i-1}\right).
  24. Q i j = 9.6 m 3 / s Q^{j}_{i}=9.6\,\text{ m}^{3}/\,\text{s}
  25. Q i - 1 j = 10.3 m 3 / s Q^{j}_{i-1}=10.3\,\text{ m}^{3}/\,\text{s}
  26. Q i j + 1 = ( 1 - 0.9 ) 9.6 + 0.9 ( 10.3 ) = 10.2 m 3 / s . Q^{j+1}_{i}=\left(1-0.9\right)9.6+0.9\left(10.3\right)=10.2\,\text{ m}^{3}/\,% \text{s}.

Onsager–Machlup_function.html

  1. X X
  2. t = 0 t=0
  3. t = T t=T
  4. d X t = b ( X t ) d t + σ ( X t ) d W t dX_{t}=b(X_{t})\,dt+\sigma(X_{t})\,dW_{t}
  5. W W
  6. p ( x 1 , , x n ) = ( i = 1 n - 1 1 2 π σ ( x i ) 2 Δ t i ) exp ( - i = 1 n - 1 L ( x i , x i + 1 - x i Δ t i ) Δ t i ) p(x_{1},\ldots,x_{n})=\left(\prod^{n-1}_{i=1}\frac{1}{\sqrt{2\pi\sigma(x_{i})^% {2}\Delta t_{i}}}\right)\exp\left(-\sum^{n-1}_{i=1}L\left(x_{i},\frac{x_{i+1}-% x_{i}}{\Delta t_{i}}\right)\,\Delta t_{i}\right)
  7. L ( x , v ) = 1 2 ( v - b ( x ) σ ) 2 L(x,v)=\frac{1}{2}\left(\frac{v-b(x)}{\sigma}\right)^{2}
  8. 1 2 π σ ( x i ) 2 Δ t i \frac{1}{\sqrt{2\pi\sigma(x_{i})^{2}\Delta t_{i}}}
  9. X X
  10. X X
  11. ε ε
  12. P ( | X t - φ 1 ( t ) | ε for every t [ 0 , T ] ) P ( | X t - φ 2 ( t ) | ε for every t [ 0 , T ] ) exp ( - 0 T L ( φ 1 ( t ) , φ ˙ 1 ( t ) ) d t + 0 T L ( φ 2 ( t ) , φ ˙ 2 ( t ) ) d t ) \frac{P\left(\left|X_{t}-\varphi_{1}(t)\right|\leq\varepsilon\,\text{ for % every }t\in[0,T]\right)}{P\left(\left|X_{t}-\varphi_{2}(t)\right|\leq% \varepsilon\,\text{ for every }t\in[0,T]\right)}\to\exp\left(-\int^{T}_{0}L% \left(\varphi_{1}(t),\dot{\varphi}_{1}(t)\right)\,dt+\int^{T}_{0}L\left(% \varphi_{2}(t),\dot{\varphi}_{2}(t)\right)\,dt\right)
  13. ε 0 ε→0
  14. L L
  15. d d
  16. M M
  17. M M
  18. b b
  19. lim ε 0 P ( ρ ( X t , φ 1 ( t ) ) ε for every t [ 0 , T ] ) P ( ρ ( X t , φ 2 ( t ) ) ε for every t [ 0 , T ] ) = exp ( - 0 T L ( φ 1 ( t ) , φ ˙ 1 ( t ) ) d t + 0 T L ( φ 2 ( t ) , φ ˙ 2 ( t ) ) d t ) \lim_{\varepsilon\downarrow 0}\frac{P\left(\rho(X_{t},\varphi_{1}(t))\leq% \varepsilon\,\text{ for every }t\in[0,T]\right)}{P\left(\rho(X_{t},\varphi_{2}% (t))\leq\varepsilon\,\text{ for every }t\in[0,T]\right)}=\exp\left(-\int^{T}_{% 0}L\left(\varphi_{1}(t),\dot{\varphi}_{1}(t)\right)\,dt+\int^{T}_{0}L\left(% \varphi_{2}(t),\dot{\varphi}_{2}(t)\right)\,dt\right)
  20. ρ ρ
  21. φ ˙ 1 , φ ˙ 2 \scriptstyle\dot{\varphi}_{1},\dot{\varphi}_{2}
  22. L L
  23. L ( x , v ) = 1 2 v - b ( x ) x 2 + 1 2 div b ( x ) - 1 12 R ( x ) , L(x,v)=\tfrac{1}{2}\|v-b(x)\|_{x}^{2}+\tfrac{1}{2}\operatorname{div}\,b(x)-% \tfrac{1}{12}R(x),
  24. x x
  25. d i v b ( x ) divb(x)
  26. b b
  27. x x
  28. R ( x ) R(x)
  29. x x
  30. 𝐑 \mathbf{R}
  31. L ( x , v ) = 1 2 | v | 2 . L(x,v)=\tfrac{1}{2}|v|^{2}.
  32. 𝐑 \mathbf{R}
  33. φ : 0 , T 𝐑 φ:0,T→\mathbf{R}
  34. P φ = exp ( 0 T φ ˙ ( t ) d X t φ + 0 T 1 2 | φ ˙ ( t ) | 2 d t ) d P . P^{\varphi}=\exp\left(\int^{T}_{0}\dot{\varphi}(t)\,dX^{\varphi}_{t}+\int^{T}_% {0}\tfrac{1}{2}\left|\dot{\varphi}(t)\right|^{2}\,dt\right)\,dP.
  35. ε > 0 ε>0
  36. t 0 , T t∈0,T
  37. P ( | X t - φ ( t ) | ε for every t [ 0 , T ] ) = P ( | X t φ | ε for every t [ 0 , T ] ) = { | X t φ | ε for every t [ 0 , T ] } exp ( - 0 T φ ˙ ( t ) d X t φ - 0 T 1 2 | φ ˙ ( t ) | 2 d t ) d P φ . \begin{aligned}\displaystyle P\left(\left|X_{t}-\varphi(t)\right|\leq% \varepsilon\,\text{ for every }t\in[0,T]\right)&\displaystyle=P\left(\left|X^{% \varphi}_{t}\right|\leq\varepsilon\,\text{ for every }t\in[0,T]\right)\\ &\displaystyle=\int_{\left\{\left|X^{\varphi}_{t}\right|\leq\varepsilon\,\text% { for every }t\in[0,T]\right\}}\exp\left(-\int^{T}_{0}\dot{\varphi}(t)\,dX^{% \varphi}_{t}-\int^{T}_{0}\tfrac{1}{2}|\dot{\varphi}(t)|^{2}\,dt\right)\,dP^{% \varphi}.\end{aligned}
  38. X X
  39. P P
  40. P ( | X t - φ ( t ) | ε for every t [ 0 , T ] ) = { | X t φ | ε for every t [ 0 , T ] } exp ( - 0 T φ ˙ ( t ) d X t - 0 T 1 2 | φ ˙ ( t ) | 2 d t ) d P . P(|X_{t}-\varphi(t)|\leq\varepsilon\,\text{ for every }t\in[0,T])=\int_{\left% \{\left|X^{\varphi}_{t}\right|\leq\varepsilon\,\text{ for every }t\in[0,T]% \right\}}\exp\left(-\int^{T}_{0}\dot{\varphi}(t)\,dX_{t}-\int^{T}_{0}\tfrac{1}% {2}|\dot{\varphi}(t)|^{2}\,dt\right)\,dP.
  41. 0 T φ ˙ ( t ) d X t = φ ˙ ( T ) X T - 0 T φ ¨ ( t ) X t d t , \int^{T}_{0}\dot{\varphi}(t)\,dX_{t}=\dot{\varphi}(T)X_{T}-\int^{T}_{0}\ddot{% \varphi}(t)X_{t}\,dt,
  42. φ ¨ \scriptstyle\ddot{\varphi}
  43. φ φ
  44. ε ε
  45. t 0 , T t∈0,T
  46. ε 0 ε→0
  47. lim ε 0 P ( | X t - φ ( t ) | ε for every t [ 0 , T ] ) P ( | X t | ε for every t [ 0 , T ] ) = exp ( - 0 T 1 2 | φ ˙ ( t ) | 2 d t ) . \lim_{\varepsilon\downarrow 0}\frac{P(|X_{t}-\varphi(t)|\leq\varepsilon\,\text% { for every }t\in[0,T])}{P(|X_{t}|\leq\varepsilon\,\text{ for every }t\in[0,T]% )}=\exp\left(-\int^{T}_{0}\tfrac{1}{2}|\dot{\varphi}(t)|^{2}\,dt\right).
  48. σ σ
  49. L ( x , v ) = 1 2 | v - b ( x ) σ | 2 + 1 2 d b d x ( x ) . L(x,v)=\frac{1}{2}\left|\frac{v-b(x)}{\sigma}\right|^{2}+\frac{1}{2}\frac{db}{% dx}(x).
  50. d d
  51. σ σ
  52. L ( x , v ) = 1 2 v - b ( x ) 2 + 1 2 ( div b ) ( x ) , L(x,v)=\frac{1}{2}\|v-b(x)\|^{2}+\frac{1}{2}(\operatorname{div}\,b)(x),
  53. [ u ! ! ] [ u ! ! ] [u^{\prime}!!^{\prime}]⋅[u^{\prime}!!^{\prime}]
  54. ( div b ) ( x ) = i = 1 d 1 2 x i b i ( x ) . (\operatorname{div}\,b)(x)=\sum_{i=1}^{d}\frac{1}{2}\frac{\partial}{\partial x% _{i}}b_{i}(x).
  55. φ φ

Open-circuit_test.html

  1. 𝐈 𝟎 \mathbf{I_{0}}
  2. 𝐖 \mathbf{W}
  3. 𝐖 = 𝐕 𝟏 𝐈 𝟎 cos ϕ 0 \mathbf{W}=\mathbf{V_{1}}\mathbf{I_{0}}\cos\phi_{0}
  4. cos ϕ 0 = 𝐖 𝐕 𝟏 𝐈 𝟎 \cos\phi_{0}=\frac{\mathbf{W}}{\mathbf{V_{1}}\mathbf{I_{0}}}
  5. 𝐈 𝐦 = 𝐈 𝟎 sin ϕ 0 \mathbf{I_{m}}=\mathbf{I_{0}}\sin\phi_{0}
  6. 𝐈 𝐰 = 𝐈 𝟎 cos ϕ 0 \mathbf{I_{w}}=\mathbf{I_{0}}\cos\phi_{0}
  7. 𝐗 𝟎 \mathbf{X_{0}}
  8. 𝐑 𝟎 \mathbf{R_{0}}
  9. 𝐗 𝟎 = 𝐕 𝟏 𝐈 𝐦 \mathbf{X_{0}}=\frac{\mathbf{V_{1}}}{\mathbf{I_{m}}}
  10. 𝐑 𝟎 = 𝐕 𝟏 𝐈 𝐰 \mathbf{R_{0}}=\frac{\mathbf{V_{1}}}{\mathbf{I_{w}}}
  11. 𝐙 𝟎 = 𝐑 𝟎 2 + 𝐗 𝟎 2 \mathbf{Z_{0}}=\sqrt{\mathbf{R_{0}}^{2}+\mathbf{X_{0}}^{2}}
  12. 𝐙 𝟎 = 𝐑 𝟎 + 𝐣𝐗 𝟎 \mathbf{Z_{0}}=\mathbf{R_{0}}+\mathbf{j}\mathbf{X_{0}}
  13. 𝐘 𝟎 = 1 𝐙 𝟎 \mathbf{Y_{0}}=\frac{1}{\mathbf{Z_{0}}}
  14. 𝐆 𝟎 \mathbf{G_{0}}
  15. 𝐆 𝟎 = 𝐖 𝐕 𝟏 2 \mathbf{G_{0}}=\frac{\mathbf{W}}{\mathbf{V_{1}}^{2}}
  16. 𝐁 𝟎 = 𝐘 𝟎 2 - 𝐆 𝟎 2 \mathbf{B_{0}}=\sqrt{\mathbf{Y_{0}}^{2}-\mathbf{G_{0}}^{2}}
  17. 𝐘 𝟎 = 𝐆 𝟎 + 𝐣𝐁 𝟎 \mathbf{Y_{0}}=\mathbf{G_{0}}+\mathbf{j}\mathbf{B_{0}}
  18. 𝐖 \mathbf{W}
  19. 𝐕 𝟏 \mathbf{V_{1}}
  20. 𝐈 𝟎 \mathbf{I_{0}}
  21. 𝐈 𝐦 \mathbf{I_{m}}
  22. 𝐈 𝐰 \mathbf{I_{w}}
  23. 𝐙 𝟎 \mathbf{Z_{0}}
  24. 𝐘 𝟎 \mathbf{Y_{0}}

Operational_availability.html

  1. A o A_{o}
  2. A o = 0.9 877 h o u r s d o w n t i m e p e r y e a r A_{o}=0.9\approx 877\ hours\ down\ time\ per\ year
  3. A o = 0.99 87 h o u r s d o w n t i m e p e r y e a r A_{o}=0.99\approx 87\ hours\ down\ time\ per\ year
  4. A o = 0.999 8 h o u r s d o w n t i m e p e r y e a r A_{o}=0.999\approx 8\ hours\ down\ time\ per\ year
  5. A o = 0.9999 52 m i n u t e s d o w n t i m e p e r y e a r A_{o}=0.9999\approx 52\ minutes\ down\ time\ per\ year
  6. A o = 0.99999 5 m i n u t e s d o w n t i m e p e r y e a r A_{o}=0.99999\approx 5\ minutes\ down\ time\ per\ year
  7. A o D = ( T o t a l T i m e T o t a l T i m e + D i a g n o s t i c D o w n T i m e ) A_{o}^{D}=\left(\frac{Total\ Time}{Total\ Time+Diagnostic\ Down\ Time}\right)
  8. A o F I = ( T o t a l T i m e T o t a l T i m e + F a u l t I s o l a t i o n D o w n T i m e ) A_{o}^{FI}=\left(\frac{Total\ Time}{Total\ Time+Fault\ Isolation\ Down\ Time}\right)
  9. A o L = ( T o t a l T i m e T o t a l T i m e + L o g i s t i c s D o w n T i m e ) A_{o}^{L}=\left(\frac{Total\ Time}{Total\ Time+Logistics\ Down\ Time}\right)
  10. A o C = ( T o t a l T i m e T o t a l T i m e + C o r r e c t i v e D o w n T i m e ) A_{o}^{C}=\left(\frac{Total\ Time}{Total\ Time+Corrective\ Down\ Time}\right)
  11. A o = A o D × A o F I × A o L × A o C A_{o}=A_{o}^{D}\times A_{o}^{FI}\times A_{o}^{L}\times A_{o}^{C}

Optical_stretcher.html

  1. p = h λ = h n λ 0 p=\frac{h}{\lambda}=\frac{hn}{\lambda_{0}}

Optimal_computing_budget_allocation.html

  1. max τ 1 , τ 2 , , τ k PCS \displaystyle\max_{\tau_{1},\tau_{2},\ldots,\tau_{k}}\mathrm{PCS}
  2. i = 1 k τ i = τ \sum_{i=1}^{k}\tau_{i}=\tau
  3. Pr { C S } Pr { ( i S p E i ) ( i S ¯ p E i c ) } , \Pr\{CS\}\equiv\Pr\left\{\left(\bigcap_{i\in S_{p}}E_{i}\right)\bigcap\left(% \bigcap_{i\in\overline{S}_{p}}E_{i}^{c}\right)\right\},
  4. S p S_{p}
  5. S ¯ p \overline{S}_{p}
  6. S ¯ p = Θ \ S p \overline{S}_{p}=\Theta\backslash S_{p}
  7. E i E_{i}
  8. i i
  9. E i c E_{i}^{c}
  10. i i
  11. e 1 e_{1}
  12. e 2 e_{2}
  13. e 1 = 1 - Pr { i S ¯ p E i c } e_{1}=1-\Pr\left\{\bigcap_{i\in\overline{S}_{p}}E_{i}^{c}\right\}
  14. e 2 = 1 - Pr { i S p E i } e_{2}=1-\Pr\left\{\bigcap_{i\in S_{p}}E_{i}\right\}
  15. e 1 u b 1 = H | S ¯ p | - H i S ¯ p max j Θ , j i [ min l 1 , , H Pr { J ~ j l J ~ i l } ] e_{1}\leq ub_{1}=H\left|\overline{S}_{p}\right|-H\sum_{i\in\overline{S}_{p}}{% \max_{j\in\Theta,j\neq i}\left[\min_{l\in{1,\ldots,H}}\Pr\left\{\tilde{J}_{jl}% \leq\tilde{J}_{il}\right\}\right]}
  16. e 2 u b 2 = ( k - 1 ) i S p max j Θ , j i [ min l 1 , , H Pr { J ~ j l J ~ i l } ] , e_{2}\leq ub_{2}=(k-1)\sum_{i\in S_{p}}\max_{j\in\Theta,j\neq i}\left[\min_{l% \in{1,\ldots,H}}\Pr\left\{\tilde{J}_{jl}\leq\tilde{J}_{il}\right\}\right],
  17. H H
  18. J ~ i l \tilde{J}_{il}
  19. N o r m a l ( J ¯ i l , σ i l 2 N i ) . Normal\left(\bar{J}_{il},\frac{\sigma_{il}^{2}}{N_{i}}\right).
  20. J ¯ i l \bar{J}_{il}
  21. σ i l \sigma_{il}
  22. l l
  23. i i
  24. N i N_{i}
  25. Pr { C S } \Pr\{CS\}
  26. A P C S - M 1 - u b 1 - u b 2 . APCS{-}M\equiv 1-ub_{1}-ub_{2}.
  27. τ \tau\rightarrow\infty
  28. τ i = β i j Θ β j τ , \tau_{i}=\frac{\beta_{i}}{\sum_{j\in\Theta}\beta_{j}}\tau,
  29. h S A h\in S_{A}
  30. β h = ( σ ^ h l j h h 2 + σ ^ j h l j h h 2 / ρ h ) / δ h j h l j h h 2 ( σ ^ m l j m m 2 + σ ^ j m l j m m 2 / ρ m ) / δ m j m l j m m 2 \beta_{h}=\frac{\left(\hat{\sigma}^{2}_{hl_{j_{h}}^{h}}+\hat{\sigma}^{2}_{j_{h% }l_{j_{h}}^{h}}/\rho_{h}\right)/{\delta^{2}_{hj_{h}l_{j_{h}}^{h}}}}{\left(\hat% {\sigma}^{2}_{ml_{jm}^{m}}+\hat{\sigma}^{2}_{j_{m}l_{jm}^{m}}/\rho_{m}\right)/% {\delta^{2}_{mj_{m}l_{j_{m}}^{m}}}}
  31. d S B d\in S_{B}
  32. β d = i Θ d * σ d l d i 2 σ i l d i 2 β i 2 \beta_{d}=\sqrt{\sum_{i\in\Theta_{d}^{*}}\frac{\sigma^{2}_{dl_{d}^{i}}}{\sigma% ^{2}_{il_{d}^{i}}}\beta_{i}^{2}}
  33. δ i j l = J ¯ j l - J ¯ i l , \delta_{ijl}=\bar{J}_{jl}-\bar{J}_{il},
  34. j i arg max j Θ , j i l = 1 H Pr { J ~ j l J ~ i l } , j_{i}\equiv\arg\max_{j\in\Theta,j\neq i}\prod_{l=1}^{H}{\Pr\left\{\tilde{J}_{% jl}\leq\tilde{J}_{il}\right\}},
  35. l j i i arg min l 1 , , H Pr { J ~ j l J ~ i l } , l_{j_{i}}^{i}\equiv\arg\min_{l\in{1,\ldots,H}}\Pr\left\{\tilde{J}_{jl}\leq% \tilde{J}_{il}\right\},
  36. S A { d e s i g n h S δ h j h l j h h 2 σ ^ h l j h h 2 α h + σ ^ j h l j h h 2 α j h < min i Θ h δ i h l h i 2 σ ^ i l h i 2 α i + σ ^ h l h i 2 α h } , S_{A}\equiv\left\{design\;h\in S\mid\frac{\delta^{2}_{hj_{h}l^{h}_{j_{h}}}}{% \frac{\hat{\sigma}^{2}_{hl^{h}_{j_{h}}}}{\alpha_{h}}+\frac{\hat{\sigma}^{2}_{j% _{h}l^{h}_{j_{h}}}}{\alpha_{j_{h}}}}<\min_{i\in\Theta_{h}}\frac{\delta^{2}_{% ihl^{i}_{h}}}{\frac{\hat{\sigma}^{2}_{il^{i}_{h}}}{\alpha_{i}}+\frac{\hat{% \sigma}^{2}_{hl^{i}_{h}}}{\alpha_{h}}}\right\},
  37. S B S \ S A , S_{B}\equiv S\backslash S_{A},
  38. Θ h = i | i S , j i = h , \Theta_{h}={i|i\in S,j_{i}=h},
  39. Θ d * = h | h S A , j h = d , \Theta_{d}^{*}={h|h\in S_{A},j_{h}=d},
  40. ρ i = α j i / α i . \rho_{i}=\alpha_{j_{i}}/\alpha_{i}.

Orbit_modeling.html

  1. r ( ν ) = a ( 1 - e 2 ) 1 + e cos ( ν ) r(\nu)=\frac{a(1-e^{2})}{1+e\cos(\nu)}
  2. r r
  3. a a
  4. e e
  5. ν \nu
  6. r ( ν ) = p 1 + e cos ( ν ) r(\nu)=\frac{p}{1+e\cos(\nu)}
  7. p p
  8. F = G m 1 m 2 r 2 F=G\frac{m_{1}m_{2}}{r^{2}}
  9. F F
  10. G G
  11. m 1 m_{1}
  12. m 2 m_{2}
  13. r r
  14. 𝐫 ¨ = G m 1 r 2 𝐫 ^ \ddot{\mathbf{r}}=\frac{Gm_{1}}{r^{2}}\mathbf{\hat{r}}
  15. n n
  16. i i
  17. j j
  18. 𝐫 ¨ i = j = 1 j i n G m j ( 𝐫 j - 𝐫 i ) r i j 3 \mathbf{\ddot{r}}_{i}=\sum_{\underset{j\neq i}{j=1}}^{n}{Gm_{j}(\mathbf{r}_{j}% -\mathbf{r}_{i})\over r_{ij}^{3}}
  19. 𝐫 ¨ i \mathbf{\ddot{r}}_{i}
  20. i i
  21. G G
  22. m j m_{j}
  23. j j
  24. 𝐫 i \mathbf{r}_{i}
  25. 𝐫 j \mathbf{r}_{j}
  26. i i
  27. j j
  28. r i j r_{ij}
  29. i i
  30. j j
  31. x x
  32. y y
  33. z z
  34. s y m b o l ρ symbol{\rho}
  35. 𝐫 \mathbf{r}
  36. δ 𝐫 \delta\mathbf{r}
  37. δ 𝐫 ¨ = 𝐫 ¨ - s y m b o l ρ ¨ . \ddot{\delta\mathbf{r}}=\mathbf{\ddot{r}}-symbol{\ddot{\rho}}.
  38. 𝐫 ¨ \mathbf{\ddot{r}}
  39. s y m b o l ρ ¨ symbol{\ddot{\rho}}
  40. 𝐫 \mathbf{r}
  41. s y m b o l ρ symbol{\rho}
  42. μ = G ( M + m ) \mu=G(M+m)
  43. M M
  44. m m
  45. 𝐚 per \mathbf{a}_{\,\text{per}}
  46. r r
  47. ρ \rho
  48. 𝐫 \mathbf{r}
  49. s y m b o l ρ symbol{\rho}
  50. δ 𝐫 \delta\mathbf{r}
  51. s y m b o l ρ symbol{\rho}
  52. δ 𝐫 \delta\mathbf{r}
  53. 𝐫 \mathbf{r}
  54. s y m b o l ρ ρ 3 - 𝐫 r 3 {symbol{\rho}\over\rho^{3}}-{\mathbf{r}\over r^{3}}
  55. 𝐫 0 \mathbf{r}_{0}
  56. 𝐯 0 \mathbf{v}_{0}
  57. t 0 t_{0}
  58. s = 0 s=0
  59. r 0 = ( 𝐫 0 𝐫 0 ) 1 / 2 r_{0}=(\mathbf{r}_{0}\cdot\mathbf{r}_{0})^{1/2}
  60. a = r 0 a=r_{0}
  61. b = 𝐫 0 𝐯 0 b=\mathbf{r}_{0}\cdot\mathbf{v}_{0}
  62. τ = t 0 \tau=t_{0}
  63. s y m b o l α = 𝐫 0 symbol{\alpha}=\mathbf{r}_{0}
  64. s y m b o l β = a 𝐯 0 symbol{\beta}=a\mathbf{v}_{0}
  65. V 0 V_{0}
  66. [ V 𝐫 ] 0 \Bigg[{\partial{V}\over{\partial{\mathbf{r}}}}\Bigg]_{0}
  67. 𝐏 0 \mathbf{P}_{0}
  68. α J = 2 μ r 0 - 𝐯 0 𝐯 0 - 2 V 0 \alpha_{J}=\frac{2\mu}{r_{0}}-\mathbf{v}_{0}\cdot\mathbf{v}_{0}-2V_{0}
  69. γ = μ - α J α \gamma=\mu-\alpha_{J}\alpha
  70. s y m b o l δ = - ( 𝐯 0 𝐯 0 ) 𝐫 0 + ( 𝐫 0 𝐯 0 ) 𝐯 0 + μ r 0 𝐫 0 - α J 𝐫 0 symbol{\delta}=-(\mathbf{v}_{0}\cdot\mathbf{v}_{0})\mathbf{r}_{0}+(\mathbf{r}_% {0}\cdot\mathbf{v}_{0})\mathbf{v}_{0}+\frac{\mu}{r_{0}}\mathbf{r}_{0}-\alpha_{% J}\mathbf{r}_{0}
  71. σ = 0 \sigma=0
  72. 𝐫 = s y m b o l α + s y m b o l β s c 1 + s y m b o l δ s 2 c 2 \mathbf{r}=symbol{\alpha}+symbol{\beta}sc_{1}+symbol{\delta}s^{2}c_{2}
  73. 𝐫 = s y m b o l β c 0 + s y m b o l δ s c 1 \mathbf{r^{\prime}}=symbol{\beta}c_{0}+symbol{\delta}sc_{1}
  74. 𝐱 3 = α j ( s y m b o l α - 𝐫 ) + s y m b o l δ \mathbf{x}_{3}=\alpha_{j}(symbol{\alpha}-\mathbf{r})+symbol{\delta}
  75. γ = μ - α j a \gamma=\mu-\alpha_{j}a
  76. r = a + b s c 1 + γ s 2 c 2 r=a+bsc_{1}+\gamma s^{2}c_{2}
  77. 𝐯 = 𝐫 / r \mathbf{v}=\mathbf{r^{\prime}}/r
  78. r = b c 0 + γ s c 1 r^{\prime}=bc_{0}+\gamma sc_{1}
  79. t = τ + a s + b s 2 c 2 + γ s 3 c 3 t=\tau+as+bs^{2}c_{2}+\gamma s^{3}c_{3}
  80. c 0 , c 1 , c 2 , c 3 c_{0},c_{1},c_{2},c_{3}
  81. 𝐅 = 𝐏 - V 𝐫 \mathbf{F}=\mathbf{P}-{\partial{V}\over\partial{\mathbf{r}}}
  82. 𝐐 = r 2 𝐅 + 2 𝐫 ( - V + σ ) \mathbf{Q}=r^{2}\mathbf{F}+2\mathbf{r}(-V+\sigma)
  83. α j = 2 ( - 𝐫 + r s y m b o l ω × 𝐫 ) 𝐏 \alpha^{\prime}_{j}=2(-\mathbf{r^{\prime}}+rsymbol{\omega}\times\mathbf{r})% \cdot\mathbf{P}
  84. \musymbol ϵ = 2 ( 𝐫 𝐅 ) 𝐫 - ( 𝐫 𝐅 ) 𝐫 - ( 𝐫 𝐫 ) 𝐅 \musymbol{\epsilon}^{\prime}=2(\mathbf{r^{\prime}}\cdot\mathbf{F})\mathbf{r}-(% \mathbf{r}\cdot\mathbf{F})\mathbf{r^{\prime}}-(\mathbf{r}\cdot\mathbf{r^{% \prime}})\mathbf{F}
  85. s y m b o l α = - 𝐐 s c 1 - \musymbol ϵ s 2 c 2 - α j [ s y m b o l α s 2 c 2 + 2 s y m b o l β s 3 c ¯ 3 + 1 2 s y m b o l δ s 4 c 2 2 ] symbol{\alpha}^{\prime}=-\mathbf{Q}sc_{1}-\musymbol{\epsilon}^{\prime}s^{2}c_{% 2}-\alpha^{\prime}_{j}\big[symbol{\alpha}s^{2}c_{2}+2symbol{\beta}s^{3}\bar{c}% _{3}+\frac{1}{2}symbol{\delta}s^{4}c^{2}_{2}\big]
  86. s y m b o l β = 𝐐 c 0 + \musymbol ϵ s c 1 + α j [ s y m b o l α s c 1 + s y m b o l β s 2 c ¯ 2 - s y m b o l δ s 3 ( 2 c ¯ 3 - c 1 c 2 ) ] symbol{\beta}^{\prime}=\mathbf{Q}c_{0}+\musymbol{\epsilon}^{\prime}sc_{1}+% \alpha^{\prime}_{j}\big[symbol{\alpha}sc_{1}+symbol{\beta}s^{2}\bar{c}_{2}-% symbol{\delta}s^{3}(2\bar{c}_{3}-c_{1}c_{2})\big]
  87. s y m b o l δ = 𝐐 α j s c 1 - \musymbol ϵ c 0 + α j [ - s y m b o l α c 0 + 2 α j s y m b o l β s 3 c ¯ 3 + 1 2 s y m b o l δ α j s 4 c 2 2 ] symbol{\delta}^{\prime}=\mathbf{Q}\alpha_{j}sc_{1}-\musymbol{\epsilon}^{\prime% }c_{0}+\alpha^{\prime}_{j}\big[-symbol{\alpha}c_{0}+2\alpha_{j}symbol{\beta}s^% {3}\bar{c}_{3}+\frac{1}{2}symbol{\delta}\alpha_{j}s^{4}c^{2}_{2}\big]
  88. σ = r s y m b o l ω 𝐫 × 𝐅 \sigma^{\prime}=rsymbol{\omega}\cdot\mathbf{r}\times\mathbf{F}
  89. a = - 1 r 𝐫 𝐐 s c 1 - α j [ a s 2 c 2 + 2 b s 3 c ¯ 3 + 1 2 γ s 4 c 2 2 ] a^{\prime}=-\frac{1}{r}\mathbf{r}\cdot\mathbf{Q}sc_{1}-\alpha_{j}^{\prime}\big% [as^{2}c_{2}+2bs^{3}\bar{c}_{3}+\frac{1}{2}\gamma s^{4}c^{2}_{2}\big]
  90. b = 1 r 𝐫 𝐐 c 0 + α j [ a s c 1 + b s 2 c ¯ 2 - γ s 3 ( 2 c ¯ 3 - c 1 c 2 ) ] b^{\prime}=\frac{1}{r}\mathbf{r}\cdot\mathbf{Q}c_{0}+\alpha_{j}^{\prime}\big[% asc_{1}+bs^{2}\bar{c}_{2}-\gamma s^{3}(2\bar{c}_{3}-c_{1}c_{2})\big]
  91. γ = - 1 r 𝐫 𝐐 α j s c 1 + α j [ - a c 0 + 2 b α j s 3 c ¯ 3 + 1 2 γ α j s 4 c 2 2 ] \gamma^{\prime}=-\frac{1}{r}\mathbf{r}\cdot\mathbf{Q}\alpha_{j}sc_{1}+\alpha_{% j}^{\prime}\big[-ac_{0}+2b\alpha_{j}s^{3}\bar{c}_{3}+\frac{1}{2}\gamma\alpha_{% j}s^{4}c^{2}_{2}\big]
  92. τ = 1 r 𝐫 𝐐 s 2 c 2 + α j [ a s 3 c 3 + 1 2 b s 4 c 2 2 - 2 γ s 5 ( c 5 - 4 c ¯ 5 ) ] \tau^{\prime}=\frac{1}{r}\mathbf{r}\cdot\mathbf{Q}s^{2}c_{2}+\alpha_{j}^{% \prime}\big[as^{3}c_{3}+\frac{1}{2}bs^{4}c^{2}_{2}-2\gamma s^{5}(c_{5}-4\bar{c% }_{5})\big]
  93. Δ s \Delta s
  94. s + Δ s s+\Delta s
  95. s = s + Δ s s=s+\Delta s
  96. 𝐟 = - μ R 2 𝐫 ^ + n = 2 m = 0 n 𝐟 n , m {\mathbf{f}}=-\frac{\mu}{R^{2}}\mathbf{\hat{r}}+\sum_{n=2}^{\infty}\sum_{m=0}^% {n}{\mathbf{f}}_{n,m}
  97. μ {\mu}
  98. 𝐫 ^ \mathbf{\hat{r}}
  99. R {R}
  100. 𝐟 n , m {\mathbf{f}}_{n,m}
  101. 𝐟 {\mathbf{f}}
  102. 𝐟 n , m \displaystyle\mathbf{f}_{n,m}
  103. R O R_{O}
  104. R R
  105. C n , m C_{n,m}
  106. S n , m S_{n,m}
  107. 𝐞 ^ 1 , 𝐞 ^ 2 , 𝐞 ^ 3 \mathbf{\hat{e}}_{1},\mathbf{\hat{e}}_{2},\mathbf{\hat{e}}_{3}
  108. 𝐞 ^ 1 \mathbf{\hat{e}}_{1}
  109. 𝐞 ^ 3 \mathbf{\hat{e}}_{3}
  110. 𝐞 ^ 2 = 𝐞 ^ 3 × 𝐞 ^ 1 \mathbf{\hat{e}}_{2}=\mathbf{\hat{e}}_{3}\times\mathbf{\hat{e}}_{1}
  111. A n , m A_{n,m}
  112. A n , m ( u ) = 1 n - m ( ( 2 n - 1 ) u A n - 1 , m ( u ) - ( n + m - 1 ) A n - 2 , m ( u ) ) A_{n,m}(u)=\frac{1}{n-m}((2n-1)uA_{n-1,m}(u)-(n+m-1)A_{n-2,m}(u))
  113. s λ s_{\lambda}
  114. 𝐫 ^ 𝐞 ^ 3 \mathbf{\hat{r}}\cdot\mathbf{\hat{e}}_{3}
  115. 𝒞 m , 𝒮 m \mathcal{C}_{m},\mathcal{S}_{m}
  116. 𝒞 m = 𝒞 1 𝒞 m - 1 - 𝒮 1 𝒮 m - 1 , 𝒮 m = 𝒮 1 𝒞 m - 1 + 𝒞 1 𝒮 m - 1 , 𝒮 0 = 0 , 𝒮 1 = 𝐑 𝐞 ^ 2 , 𝒞 0 = 1 , 𝒞 1 = 𝐑 𝐞 ^ 1 \mathcal{C}_{m}=\mathcal{C}_{1}\mathcal{C}_{m-1}-\mathcal{S}_{1}\mathcal{S}_{m% -1},\mathcal{S}_{m}=\mathcal{S}_{1}\mathcal{C}_{m-1}+\mathcal{C}_{1}\mathcal{S% }_{m-1},\mathcal{S}_{0}=0,\mathcal{S}_{1}=\mathbf{R}\cdot\mathbf{\hat{e}}_{2},% \mathcal{C}_{0}=1,\mathcal{C}_{1}=\mathbf{R}\cdot\mathbf{\hat{e}}_{1}
  117. 𝐟 n , m {\mathbf{f}}_{n,m}
  118. - μ R 2 𝐫 ^ -\frac{\mu}{R^{2}}\mathbf{\hat{r}}
  119. Ω ˙ MOON = - 0.00338 ( cos ( i ) ) / n \dot{\Omega}_{\mathrm{MOON}}=-0.00338(\cos(i))/n
  120. ω ˙ MOON = - 0.00169 ( 4 - 5 sin 2 ( i ) ) / n \dot{\omega}_{\mathrm{MOON}}=-0.00169(4-5\sin^{2}(i))/n
  121. Ω ˙ \dot{\Omega}
  122. ω ˙ \dot{\omega}
  123. i i
  124. n n
  125. a R - 4.5 × 10 - 6 ( 1 + r ) A / m a_{R}\approx-4.5\times 10^{-6}(1+r)A/m
  126. a R a_{R}
  127. A A
  128. m m
  129. r r
  130. r = 0 r=0
  131. r = 1 r=1
  132. r 0.4 r\approx 0.4
  133. F n = m ˙ v e = m ˙ v e-act + A e ( p e - p amb ) F_{n}=\dot{m}\;v\text{e}=\dot{m}\;v\text{e-act}+A\text{e}(p\text{e}-p\text{amb})
  134. m ˙ \dot{m}
  135. v e v\text{e}
  136. v e-act v\text{e-act}
  137. A e A\text{e}
  138. p e p\text{e}
  139. p amb p\text{amb}
  140. F D = 1 2 ρ v 2 C d A , F_{D}\,=\,\tfrac{1}{2}\,\rho\,v^{2}\,C_{d}\,A,
  141. 𝐅 D \mathbf{F}_{D}
  142. ρ \mathbf{}\rho
  143. v \mathbf{}v
  144. C d \mathbf{}C_{d}
  145. A \mathbf{}A
  146. 𝐁 = n = 1 m = 0 n 𝐁 n , m {\mathbf{B}}=\sum_{n=1}^{\infty}\sum_{m=0}^{n}{\mathbf{B}}_{n,m}
  147. 𝐁 {\mathbf{B}}
  148. 𝐁 n , m {\mathbf{B}}_{n,m}
  149. 𝐁 {\mathbf{B}}
  150. 𝐁 n , m = K n , m a n + 2 R n + m + 1 [ g n , m 𝒞 m + h n , m 𝒮 m R ( ( s λ A n , m + 1 + ( n + m + 1 ) A n , m ) 𝐫 ^ ) - A n , m + 1 𝐞 ^ 3 ] - m A n , m ( ( g n , m 𝒞 m - 1 + h n , m 𝒮 m - 1 ) 𝐞 ^ 1 + ( h n , m 𝒞 m - 1 - g n , m 𝒮 m - 1 ) 𝐞 ^ 2 ) ) \begin{aligned}\displaystyle\mathbf{B}_{n,m}&\displaystyle=\frac{K_{n,m}a^{n+2% }}{R^{n+m+1}}\left[\frac{g_{n,m}\mathcal{C}_{m}+h_{n,m}\mathcal{S}_{m}}{R}((s_% {\lambda}A_{n,m+1}+(n+m+1)A_{n,m})\mathbf{\hat{r}})-A_{n,m+1}\mathbf{\hat{e}}_% {3}\right]\\ &\displaystyle{}\quad{}-mA_{n,m}((g_{n,m}\mathcal{C}_{m-1}+h_{n,m}\mathcal{S}_% {m-1})\mathbf{\hat{e}}_{1}+(h_{n,m}\mathcal{C}_{m-1}-g_{n,m}\mathcal{S}_{m-1})% \mathbf{\hat{e}}_{2}))\end{aligned}
  151. a a
  152. R R
  153. 𝐫 ^ \mathbf{\hat{r}}
  154. g n , m g_{n,m}
  155. h n , m h_{n,m}
  156. 𝐞 ^ 1 , 𝐞 ^ 2 , 𝐞 ^ 3 \mathbf{\hat{e}}_{1},\mathbf{\hat{e}}_{2},\mathbf{\hat{e}}_{3}
  157. 𝐞 ^ 1 \mathbf{\hat{e}}_{1}
  158. 𝐞 ^ 3 \mathbf{\hat{e}}_{3}
  159. 𝐞 ^ 2 = 𝐞 ^ 3 × 𝐞 ^ 1 \mathbf{\hat{e}}_{2}=\mathbf{\hat{e}}_{3}\times\mathbf{\hat{e}}_{1}
  160. A n , m A_{n,m}
  161. A n , m ( u ) = 1 n - m ( ( 2 n - 1 ) u A n - 1 , m ( u ) - ( n + m - 1 ) A n - 2 , m ( u ) ) A_{n,m}(u)=\frac{1}{n-m}((2n-1)uA_{n-1,m}(u)-(n+m-1)A_{n-2,m}(u))
  162. K n , m K_{n,m}
  163. [ n - m n + m ] 0.5 K n - 1 , m \big[\frac{n-m}{n+m}\big]^{0.5}K_{n-1,m}
  164. n ( m + 1 ) n\geq(m+1)
  165. m = [ 1 ] m=[1\ldots\infty]
  166. [ ( n + m ) ( n - m + 1 ) ] - 0.5 K n , m - 1 [(n+m)(n-m+1)]^{-0.5}K_{n,m-1}
  167. n m n\geq m
  168. m = [ 2 ] m=[2\ldots\infty]
  169. s λ s_{\lambda}
  170. 𝐫 ^ 𝐞 ^ 3 \mathbf{\hat{r}}\cdot\mathbf{\hat{e}}_{3}
  171. 𝒞 m , 𝒮 m \mathcal{C}_{m},\mathcal{S}_{m}
  172. 𝒞 m = 𝒞 1 𝒞 m - 1 - 𝒮 1 𝒮 m - 1 , 𝒮 m = 𝒮 1 𝒞 m - 1 + 𝒞 1 𝒮 m - 1 , 𝒮 0 = 0 , 𝒮 1 = 𝐑 𝐞 ^ 2 , 𝒞 0 = 1 , 𝒞 1 = 𝐑 𝐞 ^ 1 \mathcal{C}_{m}=\mathcal{C}_{1}\mathcal{C}_{m-1}-\mathcal{S}_{1}\mathcal{S}_{m% -1},\mathcal{S}_{m}=\mathcal{S}_{1}\mathcal{C}_{m-1}+\mathcal{C}_{1}\mathcal{S% }_{m-1},\mathcal{S}_{0}=0,\mathcal{S}_{1}=\mathbf{R}\cdot\mathbf{\hat{e}}_{2},% \mathcal{C}_{0}=1,\mathcal{C}_{1}=\mathbf{R}\cdot\mathbf{\hat{e}}_{1}

Order-maintenance_problem.html

  1. O ( log n ) O(\log n)
  2. O ( log n ) O(\log n)
  3. O ( 1 ) O(1)
  4. { 1 , 2 , , u } \{1,2,\ldots,u\}
  5. u u
  6. n n
  7. u u
  8. n n
  9. Ω ( log 2 n ) \Omega(\log^{2}n)
  10. O ( log n ) O(\log n)
  11. n n
  12. σ \sigma
  13. σ = ( left , right , right ) \sigma=(\,\text{left},\,\text{right},\,\text{right})
  14. left \,\text{left}
  15. σ \sigma
  16. right \,\text{right}
  17. σ \sigma
  18. h h
  19. { 1 , 2 , , 3 h + 1 - 2 } \{1,2,\ldots,3^{h+1}-2\}
  20. c log n c\log n
  21. n n
  22. k k
  23. Ω ( k ) \Omega(k)
  24. n n
  25. Ω ( k ) \Omega(k)
  26. k k
  27. O ( k ) O(k)
  28. O ( log n ) O(\log n)
  29. log 1 / α n + 1 \log_{1/\alpha}n+1
  30. 3 log 1 / α n + 2 - 2 9 n 1 log 3 ( 1 / α ) 3^{\log_{1/\alpha}n+2}-2\leq 9n^{\frac{1}{\log_{3}(1/\alpha)}}
  31. α = 2 3 \alpha=\tfrac{2}{3}
  32. 9 n 3 9n^{3}
  33. n n
  34. w w
  35. 2 w 2^{w}
  36. n < 2 w n<2^{w}
  37. 9 2 3 w 9\cdot 2^{3w}
  38. 4 + 3 w 4+3w
  39. Ω ( log n ) \Omega(\log n)
  40. Ω ( log n ) \Omega(\log n)
  41. n n
  42. n / log n n/\log n
  43. log n \log n
  44. O ( log n ) O(\log n)
  45. O ( N / log N ) O(N/\log N)
  46. O ( log N ) O(\log N)
  47. N N
  48. N 3 n 2 N \tfrac{N}{3}\leq n\leq 2N
  49. N N
  50. O ( N / log N ) O(N/\log N)
  51. Ω ( log N ) \Omega(\log N)
  52. log N , 2 log N , , log 2 N , \log N,2\log N,\ldots,\log^{2}N,\ldots
  53. ( Y ) + log N \ell(Y)+\log N
  54. ( Y ) \ell(Y)
  55. log N - 1 \log N-1
  56. k k
  57. O ( k / log N ) O(k/\log N)
  58. O ( log N ) O(\log N)
  59. O ( k ) O(k)
  60. n 2 N n\leq 2N
  61. O ( log N ) O(\log N)
  62. O ( k ) O(k)
  63. Ω ( log N ) \Omega(\log N)
  64. O ( log N ) O(\log N)

Order_of_accuracy.html

  1. n n
  2. E E
  3. h h
  4. n n
  5. E ( h ) = C h n E(h)=Ch^{n}
  6. h h
  7. n n
  8. O ( h n ) O(h^{n})
  9. n n
  10. m m

Order_tracking_(signal_processing).html

  1. T k T_{k}
  2. ( k = 1 : K ) (k=1:K)
  3. 2 π k 2\pi k
  4. α i = 2 π i K N \alpha_{i}=2\cdot\pi\frac{iK}{N}
  5. Δ α = K N \Delta\alpha=\frac{K}{N}
  6. t ( i Δ α ) = i n t e r p o l a t i o n ( { 2 π k , T k } , α i ) t(i\Delta\alpha)=interpolation(\{2\pi k,T_{k}\},\alpha_{i})
  7. x ( i Δ α ) x(i\Delta\alpha)
  8. x ( j Δ t ) x(j\Delta t)
  9. x ( i Δ α ) = i n t e r p o l a t i o n ( { x ( j Δ t ) , j Δ t } , t ( i Δ α ) ) x(i\Delta\alpha)=interpolation(\{x(j\Delta t),j\Delta t\},t(i\Delta\alpha))
  10. X ( Ω ) = Δ t Θ n = 1 N x ( n Δ t ) e - j Ω θ ( n Δ t ) ω ( n Δ t ) X(\Omega)=\frac{\Delta t}{\Theta}\sum_{n=1}^{N}x(n\Delta t)e^{-j\Omega\theta(n% \Delta t)}\omega(n\Delta t)
  11. Ω \Omega
  12. Θ \Theta
  13. θ \theta
  14. ω \omega

Order_unit.html

  1. K X K\subseteq X
  2. X X
  3. e K e\in K
  4. K K
  5. x X x\in X
  6. λ x > 0 \lambda_{x}>0
  7. λ x e - x K \lambda_{x}e-x\in K
  8. x K λ x e x\leq_{K}\lambda_{x}e
  9. K X K\subseteq X
  10. K K
  11. core ( K ) \operatorname{core}(K)
  12. X = X=\mathbb{R}
  13. K = + = { x : x 0 } K=\mathbb{R}_{+}=\{x\in\mathbb{R}:x\geq 0\}
  14. 1 1
  15. X = n X=\mathbb{R}^{n}
  16. K = + n = { x : i = 1 , , n : x i 0 } K=\mathbb{R}^{n}_{+}=\{x\in\mathbb{R}:\forall i=1,\ldots,n:x_{i}\geq 0\}
  17. 1 = ( 1 , , 1 ) \vec{1}=(1,\ldots,1)

Ordinal_regression.html

  1. p p
  2. 1 , , K 1,...,K
  3. p p
  4. 𝐰 \mathbf{w}
  5. K K
  6. K K
  7. Pr ( y k | 𝐱 ) = σ ( θ i - 𝐰 𝐱 ) \Pr(y\leq k|\mathbf{x})=\sigma(\theta_{i}-\mathbf{w}\cdot\mathbf{x})
  8. y y
  9. k k
  10. σ σ
  11. 𝐱 \mathbf{x}
  12. σ σ
  13. σ ( θ i - 𝐰 𝐱 ) = 1 1 + e 𝐰 𝐱 - θ i \sigma(\theta_{i}-\mathbf{w}\cdot\mathbf{x})=\frac{1}{1+e^{\mathbf{w}\cdot% \mathbf{x}-\theta_{i}}}
  14. σ ( θ i - 𝐰 𝐱 ) = exp ( - exp ( θ i - 𝐰 𝐱 ) ) \sigma(\theta_{i}-\mathbf{w}\cdot\mathbf{x})=\exp(-\exp(\theta_{i}-\mathbf{w}% \cdot\mathbf{x}))
  15. y * y*
  16. y * = 𝐰 𝐱 + ε y^{*}=\mathbf{w}\cdot\mathbf{x}+\varepsilon
  17. ε ε
  18. 𝐱 \mathbf{x}
  19. y y
  20. y * y*
  21. y * y*
  22. y = { 1 if y * θ 1 , 2 if θ 1 < y * θ 2 , 3 if θ 2 < y * θ 3 K if θ K - 1 < y * . y=\begin{cases}1~{}~{}\,\text{if}~{}~{}y^{*}\leq\theta_{1},\\ 2~{}~{}\,\text{if}~{}~{}\theta_{1}<y^{*}\leq\theta_{2},\\ 3~{}~{}\,\text{if}~{}~{}\theta_{2}<y^{*}\leq\theta_{3}\\ \vdots\\ K~{}~{}\,\text{if}~{}~{}\theta_{K-1}<y^{*}.\end{cases}
  23. y = k y=k
  24. y y
  25. P ( y = k | 𝐱 ) = P ( θ k - 1 y * θ k | 𝐱 ) = P ( θ k - 1 𝐰 𝐱 + ε θ k ) = Φ ( θ k - 𝐰 𝐱 ) - Φ ( θ k - 1 - 𝐰 𝐱 ) \begin{aligned}\displaystyle P(y=k|\mathbf{x})&\displaystyle=P(\theta_{k-1}% \leq y^{*}\leq\theta_{k}|\mathbf{x})\\ &\displaystyle=P(\theta_{k-1}\leq\mathbf{w}\cdot\mathbf{x}+\varepsilon\leq% \theta_{k})\\ &\displaystyle=\Phi(\theta_{k}-\mathbf{w}\cdot\mathbf{x})-\Phi(\theta_{k-1}-% \mathbf{w}\cdot\mathbf{x})\end{aligned}
  26. Φ Φ
  27. σ σ
  28. log ( 𝐰 , θ | 𝐱 i , y i ) = k = 1 K [ y i = k ] log [ Φ ( θ k - 𝐰 𝐱 ) - Φ ( θ k - 1 - 𝐰 𝐱 ) ] \log\mathcal{L}(\mathbf{w},\mathbf{\theta}|\mathbf{x}_{i},y_{i})=\sum_{k=1}^{K% }[y_{i}=k]\log[\Phi(\theta_{k}-\mathbf{w}\cdot\mathbf{x})-\Phi(\theta_{k-1}-% \mathbf{w}\cdot\mathbf{x})]
  29. [ - F o r m u l a E r r o r - ] [-FormulaError-]

Ostrowski_numeration.html

  1. x = n = 1 b n β n x=\sum_{n=1}^{\infty}b_{n}\beta_{n}
  2. N = n = 1 k b n q n N=\sum_{n=1}^{k}b_{n}q_{n}

Oversampled_binary_image_sensor.html

  1. λ 0 ( x ) \lambda_{0}(x)
  2. x x
  3. λ 0 ( x ) \lambda_{0}(x)
  4. R a R_{a}
  5. R a = 1.22 w f , R_{a}=1.22\,wf,
  6. w w
  7. f f
  8. λ ( x ) \lambda(x)
  9. b m = 1 b_{m}=1
  10. b m = 0 b_{m}=0
  11. s m s_{m}
  12. s m s_{m}
  13. y m y_{m}
  14. m m
  15. s m s_{m}
  16. y m y_{m}
  17. y m y_{m}
  18. s m s_{m}
  19. b m b_{m}
  20. m m
  21. y m y_{m}
  22. b m b_{m}
  23. τ \tau
  24. λ ( x ) \lambda(x)
  25. b m b_{m}

Oxalate_oxidoreductase.html

  1. \rightleftharpoons

Oxepin-CoA_hydrolase.html

  1. \rightleftharpoons

Oxygenation_index.html

  1. O I = F i O 2 * M P A W P a O 2 OI=\frac{F_{i}O_{2}*M_{PAW}}{P_{a}O_{2}}

P-cycle_protection.html

  1. 1 / ( d ¯ - 1 ) 1/(\bar{d}-1)

P-stable_group.html

  1. O p ( G ) O_{p}(G)
  2. O p ( G ) O_{p^{\prime}}(G)
  3. x N G ( P ) x\in N_{G}(P)
  4. x ¯ \bar{x}
  5. C G ( P ) C_{G}(P)
  6. [ P , x , x ] = 1 [P,x,x]=1
  7. x ¯ O n ( N G ( P ) / C G ( P ) ) \overline{x}\in O_{n}(N_{G}(P)/C_{G}(P))
  8. p ( G ) \mathcal{M}_{p}(G)
  9. O p ( M ) 1 O_{p}(M)\not=1
  10. p ( G ) \mathcal{M}_{p}(G)
  11. F * ( H ) = O p ( H ) F^{*}(H)=O_{p}(H)
  12. g H g\in H
  13. [ P , g , g ] = 1 [P,g,g]=1
  14. g C H ( P ) O p ( H / C H ( P ) ) gC_{H}(P)\in O_{p}(H/C_{H}(P))
  15. C G ( P ) P C_{G}(P)\leqslant P
  16. Z ( J 0 ( S ) ) Z(J_{0}(S))
  17. J 0 ( S ) J_{0}(S)

Pairwise_error_probability.html

  1. X X
  2. X ^ \widehat{X}
  3. P ( e ) P(e)
  4. ( X ^ ) (\widehat{X})
  5. ( X ) (X)
  6. P ( e ) 1 M x ( X X ^ | X ) P(e)\triangleq\frac{1}{M}\sum_{x}\mathbb{P}(X\neq\widehat{X}|X)
  7. M M
  8. P ( X X ^ ) P(X\to\widehat{X})
  9. X X
  10. X ^ \widehat{X}
  11. P ( e | X ) P(e|X)
  12. X ^ X \widehat{X}\neq X
  13. X X
  14. Y Y
  15. P ( e | X ) X ^ X P ( X X ^ ) P(e|X)\leq\sum_{\widehat{X}\neq X}P(X\to\widehat{X})
  16. P ( e ) = 1 M X S P ( e | X ) 1 M X S X ^ X P ( X X ^ ) P(e)=\tfrac{1}{M}\sum_{X\in S}P(e|X)\leq\tfrac{1}{M}\sum_{X\in S}\sum_{% \widehat{X}\neq X}P(X\to\widehat{X})
  17. Y = X + Z , Z i 𝒩 ( 0 , N 0 2 I n ) Y=X+Z,Z_{i}\sim\mathcal{N}(0,\tfrac{N_{0}}{2}I_{n})\,\!
  18. P ( X X ^ ) \displaystyle P(X\to\widehat{X})
  19. ( Z , X - X ^ ) (Z,X-\widehat{X})
  20. N 0 || X - X ^ || 2 / 2 N_{0}||X-\widehat{X}||^{2}/2
  21. σ 2 = 1 \sigma^{2}=1
  22. P ( X > x ) = Q ( x ) = 1 2 π x + e - t 2 2 d t P(X>x)=Q(x)=\frac{1}{\sqrt{2\pi}}\int_{x}^{+\infty}e^{-}\tfrac{t^{2}}{2}dt
  23. P ( X X ^ ) = Q ( || X - X ^ || 2 2 N 0 || X - X ^ || 2 2 ) = Q ( || X - X ^ || 2 2 . 2 N 0 || X - X ^ || 2 ) = Q ( || X - X ^ || 2 N 0 ) \begin{aligned}\displaystyle P(X\to\widehat{X})&\displaystyle=Q\bigg(\tfrac{% \tfrac{||X-\widehat{X}||^{2}}{2}}{\sqrt{\tfrac{N_{0}||X-\widehat{X}||^{2}}{2}}% }\bigg)=Q\bigg(\tfrac{||X-\widehat{X}||^{2}}{2}.\sqrt{\tfrac{2}{N_{0}||X-% \widehat{X}||^{2}}}\bigg)\\ &\displaystyle=Q\bigg(\tfrac{||X-\widehat{X}||}{\sqrt{2N_{0}}}\bigg)\end{aligned}

Palinstrophy.html

  1. 1 2 ( × ω ) , \frac{1}{2}\left(\nabla\times\omega\right),
  2. ω \omega

Pantoate_kinase.html

  1. \rightleftharpoons

Parallel_Processing_(DSP_implementation).html

  1. y ( n ) = a x ( n ) + b x ( n - 1 ) + c x ( n - 2 ) y(n)=ax(n)+bx(n-1)+cx(n-2)
  2. T s a m p l e T m + 2 T a {T_{sample}\geq T_{m}+2T_{a}}
  3. T s a m p l e T c l o c k N = T m + 2 T a 3 {T_{sample}\geq\frac{T_{clock}}{N}=\frac{T_{m}+2T_{a}}{3}}
  4. T s a m p l e T c l o c k T_{sample}\neq T_{clock}
  5. T s a m p l e = T c l o c k T_{sample}=T_{clock}
  6. H ( z ) = z - 1 1 - a * z - 1 H(z)=\frac{z^{-1}}{1-a*z^{-1}}
  7. y ( n + 1 ) = a y ( n ) + u ( n ) y(n+1)=ay(n)+u(n)
  8. y ( n + 4 ) = a 4 y ( n ) + a 3 u ( n ) + a 2 u ( n + 1 ) + a u ( n + 2 ) + u ( n + 3 ) y(n+4)=a^{4}y(n)+a^{3}u(n)+a^{2}u(n+1)+au(n+2)+u(n+3)
  9. y ( 4 k + 4 ) = a 4 y ( 4 k ) + a 3 u ( 4 k ) + a 2 u ( 4 k + 1 ) + a u ( 4 k + 2 ) + u ( 4 k + 3 ) \rightarrow y(4k+4)=a^{4}y(4k)+a^{3}u(4k)+a^{2}u(4k+1)+au(4k+2)+u(4k+3)
  10. P s e q = C t o t a l * V 0 2 * f P_{seq}=C_{total}*V_{0}^{2}*f
  11. P p a r a = ( N C t o t a l ) * ( β V 0 2 ) * f N = β 2 * P s e q P_{para}=(NC_{total})*(\beta V_{0}^{2})*\frac{f}{N}=\beta^{2}*P_{seq}
  12. N ( β V 0 - V t ) 2 = β ( V 0 - V t ) 2 N(\beta V_{0}-V_{t})^{2}=\beta(V_{0}-V_{t})^{2}

Parent_function.html

  1. y = a x 2 + b x + c , y=ax^{2}+bx+c\,,
  2. y = x 2 y=x^{2}
  3. 2 {}^{2}
  4. 2 {}^{2}
  5. 2 {}^{2}
  6. A / B {A}/{B}
  7. 2 {}^{2}
  8. 2 {}^{2}
  9. 2 {}^{2}
  10. n {}^{n}
  11. 2 {}^{2}
  12. 3 {}^{3}

Paromamine_6'-oxidase.html

  1. \rightleftharpoons

Parry_point_(triangle).html

  1. 3 ( b 2 - c 2 ) ( c 2 - a 2 ) ( a 2 - b 2 ) ( a 2 y z + b 2 z x + c 2 x y ) + ( x + y + z ) ( cyclic b 2 c 2 ( b 2 - c 2 ) ( b 2 + c 2 - 2 a 2 ) x ) = 0 \begin{aligned}&\displaystyle 3(b^{2}-c^{2})(c^{2}-a^{2})(a^{2}-b^{2})(a^{2}yz% +b^{2}zx+c^{2}xy)\\ &\displaystyle{}+(x+y+z)\left(\sum\text{cyclic}b^{2}c^{2}(b^{2}-c^{2})(b^{2}+c% ^{2}-2a^{2})x\right)=0\end{aligned}

Parshall_flume.html

  1. Q = C H a n Q=CH_{a}^{n}
  2. E = y + q 2 2 g y 2 E=y+\frac{q^{2}}{2gy^{2}}
  3. q = Q b q=\frac{Q}{b}
  4. E 1 + Δ z = E 2 + Δ z = E 3 E_{1}+\Delta z=E_{2}+\Delta z=E_{3}
  5. E c = 3 2 ( q 2 g ) 1 / 3 = 3 2 y c E_{c}=\frac{3}{2}(\frac{q^{2}}{g})^{1/3}=\frac{3}{2}y_{c}
  6. y 1 + ( Q b 1 ) 2 ( 2 g y 1 2 ) = 3 2 y c y_{1}+\frac{\left(\frac{Q}{b_{1}}\right)^{2}}{\left(2gy_{1}^{2}\right)}=\frac{% 3}{2}y_{c}
  7. 2 3 ( y 1 + ( Q b 1 ) 2 2 g y 1 2 ) = y c \frac{2}{3}(y_{1}+\frac{(\frac{Q}{b_{1}})^{2}}{2gy_{1}^{2}})=y_{c}
  8. Q = b 2 g ( 2 3 ( y 1 + ( Q b 1 ) 2 2 g y 1 2 ) ) 3 / 2 Q=b_{2}\sqrt{g}(\frac{2}{3}(y_{1}+\frac{(\frac{Q}{b_{1}})^{2}}{2gy_{1}^{2}}))^% {3/2}
  9. ( Q b 1 ) 2 2 g y 1 2 = v 2 2 g \frac{(\frac{Q}{b_{1}})^{2}}{2gy_{1}^{2}}=\frac{v^{2}}{2g}
  10. E 1 = y 1 + v 2 2 g E_{1}=y_{1}+\frac{v^{2}}{2g}
  11. Q b 2 g ( 2 3 y 1 ) 3 2 Q\approx b_{2}\sqrt{g}(\frac{2}{3}y_{1})^{\frac{3}{2}}
  12. Q 3.088 ( b 2 ) y 1.5 Q\approx 3.088(b_{2})y^{1.5}
  13. Q 1.704 ( b 2 ) y 1.5 Q\approx 1.704(b_{2})y^{1.5}

Partial_differential_algebraic_equation.html

  1. 0 = 𝐅 ( 𝐱 , 𝐲 , y i x j , 2 y i x j x k , , 𝐳 ) , 0=\mathbf{F}\left(\mathbf{x},\mathbf{y},\frac{\partial y_{i}}{\partial x_{j}},% \frac{\partial^{2}y_{i}}{\partial x_{j}\partial x_{k}},\ldots,\mathbf{z}\right),

Partial_dislocations.html

  1. | s y m b o l b 1 | 2 > \displaystyle|symbol{b_{1}}|^{2}>
  2. s y m b o l b 1 \rightarrowsymbol b 2 + s y m b o l b 3 \begin{aligned}\displaystyle symbol{b_{1}}\rightarrowsymbol{b_{2}}+symbol{b_{3% }}\end{aligned}
  3. a 2 [ 10 1 ¯ ] a 6 [ 2 1 ¯ 1 ¯ ] + a 6 [ 11 2 ¯ ] \displaystyle\frac{a}{2}[10\overline{1}]\rightarrow\frac{a}{6}[2\overline{1}% \overline{1}]+\frac{a}{6}[11\overline{2}]
  4. | a 2 1 2 + 0 2 + ( - 1 ) 2 | 2 > \displaystyle|\frac{a}{2}\sqrt{1^{2}+0^{2}+(-1)^{2}}|^{2}>
  5. a 2 ( 1 ) = \displaystyle\frac{a}{2}(1)=
  6. s y m b o l b frank = \displaystyle symbol{b}\text{frank}=

Partially_ordered_space.html

  1. X X
  2. \leq
  3. { ( x , y ) X 2 | x y } \{(x,y)\in X^{2}|x\leq y\}
  4. X 2 X^{2}
  5. X X
  6. \leq
  7. X X
  8. x , y X x,y\in X
  9. x y x\not\leq y
  10. U , V X U,V\subset X
  11. x U , y V x\in U,y\in V
  12. u v u\not\leq v
  13. u U , v V u\in U,v\in V
  14. x , y X x,y\in X
  15. x y x\not\leq y
  16. U U
  17. x x
  18. V V
  19. y y
  20. U U
  21. V V
  22. = =

Particle_deposition.html

  1. d Γ d t = k c {d\Gamma\over dt}=kc
  2. d Γ d t = k c B ( Γ ) {d\Gamma\over dt}=kcB(\Gamma)
  3. B ( Γ ) = 1 - Γ / Γ 0 B(\Gamma)=1-\Gamma/\Gamma_{0}

Partition_matroid.html

  1. B i B_{i}
  2. d i d_{i}
  3. 0 d i | B i | 0\leq d_{i}\leq|B_{i}|
  4. I I
  5. i i
  6. | I B i | d i |I\cap B_{i}|\leq d_{i}
  7. B i B_{i}
  8. B i B_{i}
  9. d i d_{i}
  10. B i B_{i}
  11. d i + 1 d_{i}+1
  12. d i \sum d_{i}
  13. U n r U{}^{r}_{n}
  14. B 1 B_{1}
  15. n n
  16. d 1 = r d_{1}=r
  17. d i = 1 d_{i}=1
  18. ( U , V ) (U,V)
  19. U U
  20. U U
  21. d i d_{i}
  22. V V
  23. G G
  24. G G
  25. G G
  26. d i = 1 d_{i}=1
  27. n n
  28. n = 0 , 1 , 2 , n=0,1,2,\dots
  29. f ( x ) = exp ( e x ( x - 1 ) + 2 x + 1 ) f(x)=\exp(e^{x}(x-1)+2x+1)
  30. d i = 1 d_{i}=1

Passive_heave_compensation.html

  1. ( m + m A ) y ¨ = - k c ( y + H cos ω t ) (m+m_{A})\ddot{y}=-k_{c}(y+H\cos\omega t)
  2. m m
  3. m A m_{A}
  4. y ¨ \ddot{y}
  5. k c k_{c}
  6. y y
  7. H H
  8. ω \omega
  9. t t
  10. A H = k c m + m A ω 2 - k c m + m A \frac{A}{H}=\frac{\frac{k_{c}}{m+m_{A}}}{\omega^{2}-\frac{k_{c}}{m+m_{A}}}
  11. ω 0 \omega_{0}
  12. ω 0 = k c m + m A \omega_{0}=\sqrt{\frac{k_{c}}{m+m_{A}}}
  13. A H = 1 ( ω ω 0 ) 2 - 1 \frac{A}{H}=\frac{1}{({\frac{\omega}{\omega_{0}}})^{2}-1}
  14. T R T_{R}
  15. T R = | 1 ( ω ω 0 ) 2 - 1 | T_{R}=\left|\frac{1}{({\frac{\omega}{\omega_{0}}})^{2}-1}\right|
  16. η P H C = 1 - T R \eta_{PHC}=1-T_{R}
  17. k c = p 0 A S ( C κ - 1 ) k_{c}=\frac{p_{0}A}{S}(C^{\kappa}-1)
  18. p 0 p_{0}
  19. A A
  20. S S
  21. C C
  22. κ \kappa
  23. p 0 A p_{0}A

Passive_sign_convention.html

  1. p = v i p=vi\,
  2. r = v / i r=v/i\,
  3. p = - v i p=-vi\,
  4. r = - v / i r=-v/i\qquad\,
  5. p = v i ( 1 ) p=vi\qquad\qquad\qquad\,\,\,(1)\,
  6. r = v / i ( 2 ) r=v/i\qquad\qquad\qquad(2)\,
  7. p = v ( - i ) = - v i p=v(-i)=-vi\,
  8. r = v / ( - i ) = - v / i r=v/(-i)=-v/i\,
  9. n p n = n v n i n = 0 \sum_{n}p_{n}=\sum_{n}v_{n}i_{n}=0\,
  10. P = - V I P=-VI\,
  11. R = - V / I R=-V/I\,

Patent_Box.html

  1. 𝐏𝐁 * ( M R - P B R M R ) \mathbf{P}\mathbf{B}*\left(\frac{MR-PBR}{MR}\right)

Patent_cliff.html

  1. S a l e s = A * Y B Sales={A}*{Y^{B}}
  2. S a l e s = A Y Sales=\frac{A}{Y}

Paul_A._Catlin.html

  1. k 4 k\leq 4
  2. k 7 k\geq 7

Paving_matroid.html

  1. r r
  2. r + 1 r+1
  3. { r , r + 1 } \{r,r+1\}
  4. r r
  5. r + 1 r+1
  6. K 4 K_{4}
  7. S ( t , k , v ) S(t,k,v)
  8. ( S , 𝒟 ) (S,\mathcal{D})
  9. S S
  10. v v
  11. 𝒟 \mathcal{D}
  12. k k
  13. S S
  14. t t
  15. S S
  16. 𝒟 \mathcal{D}
  17. 𝒟 \mathcal{D}
  18. t t
  19. S S
  20. S S
  21. d + 1 d+1
  22. d d
  23. \mathcal{F}
  24. d d
  25. \mathcal{F}
  26. d d
  27. d d
  28. \cup\mathcal{F}
  29. \mathcal{F}
  30. \mathcal{F}
  31. d d
  32. E = E=\cup\mathcal{F}
  33. \mathcal{F}
  34. I I
  35. E E
  36. | I | d |I|\leq d
  37. | I | = d + 1 |I|=d+1
  38. I I
  39. \mathcal{F}
  40. d d

Pedometric_mapping.html

  1. Z ( 𝐬 ) = m ( 𝐬 ) + ε ( 𝐬 ) + ε ′′ Z(\mathbf{s})=m(\mathbf{s})+\varepsilon^{\prime}(\mathbf{s})+\varepsilon^{% \prime\prime}
  2. m ( 𝐬 ) \textstyle{m(\mathbf{s})}
  3. ε ( 𝐬 ) \textstyle{\varepsilon^{\prime}(\mathbf{s})}
  4. ε ′′ \textstyle{\varepsilon^{\prime\prime}}
  5. 𝐬 \textstyle{\mathbf{s}}

Penrose_square_root_law.html

  1. ψ \psi
  2. 1 / N 1/\sqrt{N}
  3. P j = ( 1 2 ) 2 j ( 2 j ) ! ( j ! ) 2 . P_{j}=\left(\frac{1}{2}\right)^{2j}\frac{\left(2j\right)!}{\left(j!\right)^{2}}.
  4. ψ \psi
  5. ψ = P j 2 - 2 j ( 2 j / e ) 2 j 4 π j [ ( j / e ) j 2 π j ] 2 = 1 π j 2 π 1 N . \psi=P_{j}\sim 2^{-2j}\frac{(2j/e)^{2j}\sqrt{4\pi j}}{[(j/e)^{j}\sqrt{2\pi j}]% ^{2}}\ =\ \frac{1}{\sqrt{\pi j}}\sim\sqrt{\frac{2}{\pi}}\frac{1}{\sqrt{N}}.

Pentachlorophenol_monooxygenase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Pentalenene_oxygenase.html

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

Pentalenic_acid_synthase.html

  1. \rightleftharpoons

Pentalenolactone_D_synthase.html

  1. \rightleftharpoons

Pentalenolactone_F_synthase.html

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

Pentalenolactone_synthase.html

  1. \rightleftharpoons

Peptide-methionine_(S)-S-oxide_reductase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Peptidoglycan-N-acetylglucosamine_deacetylase.html

  1. \rightleftharpoons

Peptidyl-glutamate_4-carboxylase.html

  1. \rightleftharpoons

Perakine_reductase.html

  1. \rightleftharpoons

Perfect_thermal_contact.html

  1. A A
  2. T | = T e | A T\big|=T_{e}\big|_{A}\,
  3. - k T n | A = - k e T e n | A -k\frac{\partial T}{\partial n}\bigg|_{A}=-k_{e}\frac{\partial T_{e}}{\partial n% }\bigg|_{A}\,
  4. T , T e T,~{}T_{e}
  5. k , k e k,~{}k_{e}
  6. n n
  7. A A
  8. A A
  9. - k T n | A + k e T e n | A = q -k\frac{\partial T}{\partial n}\bigg|_{A}+k_{e}\frac{\partial T_{e}}{\partial n% }\bigg|_{A}=q\,
  10. q q

Perfectoid.html

  1. K = lim x x p K . K^{\flat}=\underleftarrow{\lim}_{x\mapsto x^{p}}K\ .

Periodic_travelling_wave.html

  1. u t = 2 u x 2 + λ ( r ) u - ω ( r ) v \frac{\partial u}{\partial t}=\frac{\partial^{2}u}{\partial x^{2}}+\lambda(r)u% -\omega(r)v
  2. v t = 2 v x 2 + ω ( r ) u + λ ( r ) v \frac{\partial v}{\partial t}=\frac{\partial^{2}v}{\partial x^{2}}+\omega(r)u+% \lambda(r)v
  3. A t = A + ( 1 + i b ) 2 A x 2 - ( 1 + i c ) | A | 2 A \frac{\partial A}{\partial t}=A+(1+ib)\frac{\partial^{2}A}{\partial x^{2}}-(1+% ic)|A|^{2}A

Perles_configuration.html

  1. 1 + φ 1+\varphi
  2. φ \varphi

Peroxyureidoacrylate::ureidoacrylate_amidohydrolase.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons
  4. \rightleftharpoons
  5. \rightleftharpoons
  6. \rightleftharpoons

Perpendicular_bisector_construction_of_a_quadrilateral.html

  1. Q Q
  2. Q 1 , Q 2 , Q 3 , Q 4 Q_{1},Q_{2},Q_{3},Q_{4}
  3. b 1 , b 2 , b 3 , b 4 b_{1},b_{2},b_{3},b_{4}
  4. Q 1 Q 2 , Q 2 Q 3 , Q 3 Q 4 , Q 4 Q 1 Q_{1}Q_{2},Q_{2}Q_{3},Q_{3}Q_{4},Q_{4}Q_{1}
  5. Q i ( 2 ) = b i + 2 b i + 3 Q_{i}^{(2)}=b_{i+2}b_{i+3}
  6. Q ( 2 ) Q^{(2)}
  7. Q ( 2 ) Q^{(2)}
  8. Q ( 3 ) Q^{(3)}
  9. Q ( i + 1 ) Q^{(i+1)}
  10. Q ( i ) Q^{(i)}
  11. Q ( 1 ) Q^{(1)}
  12. Q ( 2 ) Q^{(2)}
  13. Q ( 2 ) Q^{(2)}
  14. Q ( 3 ) Q^{(3)}
  15. Q ( 1 ) Q^{(1)}
  16. Q ( 3 ) Q^{(3)}
  17. Q ( 2 ) Q^{(2)}
  18. Q ( 4 ) Q^{(4)}
  19. Q ( i + 1 ) Q^{(i+1)}
  20. Q ( i ) Q^{(i)}
  21. α , β , γ , δ \alpha,\beta,\gamma,\delta
  22. Q ( 1 ) Q^{(1)}
  23. i i
  24. Q ( i ) Q^{(i)}
  25. Q ( i + 1 ) Q^{(i+1)}
  26. ( 1 / 4 ) ( cot ( α ) + cot ( γ ) ) ( cot ( β ) + cot ( δ ) ) . (1/4)(\cot(\alpha)+\cot(\gamma))(\cot(\beta)+\cot(\delta)).
  27. Q ( 1 ) Q^{(1)}
  28. Q ( 1 ) , Q ( 2 ) , Q^{(1)},Q^{(2)},\ldots
  29. Q ( 1 ) Q^{(1)}
  30. Q ( i ) Q^{(i)}
  31. Q ( 1 ) Q^{(1)}
  32. Q ( 1 ) , Q ( 0 ) , Q ( - 1 ) , Q^{(1)},Q^{(0)},Q^{(-1)},\ldots
  33. Q ( i ) Q^{(i)}

Perpetuant.html

  1. x 2 n - 1 - 1 ( 1 - x 2 ) ( 1 - x 3 ) ( 1 - x n ) \frac{x^{2^{n-1}-1}}{(1-x^{2})(1-x^{3})\cdots(1-x^{n})}

Petersen's_theorem.html

  1. G = ( V , E ) G=(V,E)
  2. U V U⊆V
  3. V U V−U
  4. U U
  5. G G
  6. V U V−U
  7. G G
  8. v V i deg ( v ) = 2 | E i | + m i , \sum\nolimits_{v\in V_{i}}\deg(v)=2|E_{i}|+m_{i},
  9. G i G_{i}
  10. v V i deg ( v ) = 3 | V i | , \sum\nolimits_{v\in V_{i}}\deg(v)=3|V_{i}|,
  11. G G
  12. m m
  13. U U
  14. G G
  15. 3 m 3m
  16. U U
  17. m m
  18. n n
  19. 2 n / 3656 2^{n/3656}
  20. O ( n l o g < s u p > 4 n ) O(nlog<sup>4n)

Petersen_matrix.html

  1. for all process i : j = 1 n a i j ρ j ˙ = 0 , \,\text{for all process }i:\sum_{j=1}^{n}a_{ij}\dot{\rho_{j}}=0\;,
  2. ρ j ˙ \dot{\rho_{j}}
  3. for all component j : C j t = i = 1 m a i j r i , \,\text{for all component }j:\frac{\partial C_{j}}{\partial t}=\sum_{i=1}^{m}a% _{ij}r_{i}\;,
  4. r i r_{i}
  5. A + B S A+B\rightarrow S
  6. E + S k r k f E S k cat E + P , E+S\,\overset{k_{f}}{\underset{k_{r}}{\rightleftharpoons}}\,ES\,\overset{k_{% \mathrm{cat}}}{\longrightarrow}\,E+P\;,
  7. k 1 [ A ] [ B ] k_{1}[A][B]
  8. k f [ E ] [ S ] k_{f}[E][S]
  9. k r [ E S ] k_{r}[ES]
  10. k cat [ E S ] k_{\mathrm{cat}}[ES]
  11. d [ A ] d t = - k 1 [ A ] [ B ] \frac{d[A]}{dt}=-k_{1}[A][B]
  12. d [ B ] d t = - k 1 [ A ] [ B ] \frac{d[B]}{dt}=-k_{1}[A][B]
  13. d [ S ] d t = k 1 [ A ] [ B ] - k f [ E ] [ S ] + k r [ E S ] \frac{d[S]}{dt}=k_{1}[A][B]-k_{f}[E][S]+k_{r}[ES]
  14. d [ E S ] d t = k f [ E ] [ S ] - k r [ E S ] - k cat [ E S ] \frac{d[ES]}{dt}=k_{f}[E][S]-k_{r}[ES]-k_{\mathrm{cat}}[ES]
  15. d [ P ] d t = k cat [ E S ] \frac{d[P]}{dt}=k_{\mathrm{cat}}[ES]

Petkovšek's_algorithm.html

  1. 4 ( n + 2 ) ( 2 n + 3 ) ( 2 n + 5 ) a ( n + 2 ) - 12 ( 2 n + 3 ) ( 9 n 2 + 27 n + 22 ) a ( n + 1 ) + 81 ( n + 1 ) ( 3 n + 2 ) ( 3 n + 4 ) a ( n ) = 0 , 4(n+2)(2n+3)(2n+5)a(n+2)-12(2n+3)(9n^{2}+27n+22)a(n+1)+81(n+1)(3n+2)(3n+4)a(n)% =0,
  2. Γ ( n + 1 ) Γ ( n + 3 / 2 ) ( 27 4 ) n , Γ ( n + 2 / 3 ) Γ ( n + 4 / 3 ) Γ ( n + 3 / 2 ) Γ ( n + 1 ) ( 27 4 ) n . {\frac{\Gamma\left(n+1\right)}{\Gamma\left(n+3/2\right)}\left({\frac{27}{4}}% \right)^{n}},\qquad{\frac{\Gamma\left(n+2/3\right)\Gamma\left(n+4/3\right)}{% \Gamma\left(n+3/2\right)\Gamma\left(n+1\right)}\left({\frac{27}{4}}\right)^{n}}.
  3. Γ \Gamma
  4. ( 3 n + 1 n ) {\left({{3n+1}\atop{n}}\right)}
  5. a ( n ) = k = 0 n ( n k ) 2 ( n + k k ) 2 , a(n)=\sum_{k=0}^{n}{{\left({{n}\atop{k}}\right)}^{2}{\left({{n+k}\atop{k}}% \right)}^{2}},
  6. ζ ( 3 ) \zeta(3)
  7. ( n + 2 ) 3 a ( n + 2 ) - ( 17 n 2 + 51 n + 39 ) ( 2 n + 3 ) a ( n + 1 ) + ( n + 1 ) 3 a ( n ) = 0. (n+2)^{3}a(n+2)-(17n^{2}+51n+39)(2n+3)a(n+1)+(n+1)^{3}a(n)=0.
  8. a ( n ) a(n)

Phase_space_formulation.html

  1. f ( x , p ) f(x,p)
  2. W ( x , p ) W(x,p)
  3. P [ a X b ] = a b - W ( x , p ) d p d x . \operatorname{P}[a\leq X\leq b]=\int_{a}^{b}\int_{-\infty}^{\infty}W(x,p)\,dp% \,dx.
  4. Â ( x , p ) Â(x,p)
  5. A ( x , p ) A(x,p)
  6. A ^ = A ( x , p ) W ( x , p ) d p d x . \langle\hat{A}\rangle=\int A(x,p)W(x,p)\,dp\,dx.
  7. W ( x , p ) W(x,p)
  8. ħ ħ
  9. ħ ħ
  10. f x g = f x g f\stackrel{\leftarrow}{\partial}_{x}g=\frac{\partial f}{\partial x}\cdot g
  11. f x g = f g x . f\stackrel{\rightarrow}{\partial}_{x}g=f\cdot\frac{\partial g}{\partial x}.
  12. f g = f exp ( i 2 ( x p - p x ) ) g f\star g=f\,\exp{\left(\tfrac{i\hbar}{2}(\stackrel{\leftarrow}{\partial}_{x}% \stackrel{\rightarrow}{\partial}_{p}-\stackrel{\leftarrow}{\partial}_{p}% \stackrel{\rightarrow}{\partial}_{x})\right)}\,g
  13. ( f g ) ( x , p ) \displaystyle(f\star g)(x,p)
  14. ( f g ) ( x , p ) = 1 π 2 2 f ( x + x , p + p ) g ( x + x ′′ , p + p ′′ ) exp ( 2 i ( x p ′′ - x ′′ p ) ) d x d p d x ′′ d p ′′ . (f\star g)(x,p)=\frac{1}{\pi^{2}\hbar^{2}}\,\int f(x+x^{\prime},p+p^{\prime})% \,g(x+x^{\prime\prime},p+p^{\prime\prime})\,\exp{\left(\tfrac{2i}{\hbar}(x^{% \prime}p^{\prime\prime}-x^{\prime\prime}p^{\prime})\right)}\,dx^{\prime}dp^{% \prime}dx^{\prime\prime}dp^{\prime\prime}~{}.
  15. exp ( - a ( x 2 + p 2 ) ) exp ( - b ( x 2 + p 2 ) ) = 1 1 + 2 a b exp ( - a + b 1 + 2 a b ( x 2 + p 2 ) ) , \exp\left(-{a}(x^{2}+p^{2})\right)~{}\star~{}\exp\left(-{b}(x^{2}+p^{2})\right% )={1\over 1+\hbar^{2}ab}\exp\left(-{a+b\over 1+\hbar^{2}ab}(x^{2}+p^{2})\right),
  16. δ ( x ) δ ( p ) = 2 h exp ( 2 i x p ) , \delta(x)~{}\star~{}\delta(p)={2\over h}\exp\left(2i{xp\over\hbar}\right),
  17. H W = E W , H\star W=E\cdot W,
  18. f t = - 1 i ( f H - H f ) , \frac{\partial f}{\partial t}=-\frac{1}{i\hbar}\left(f\star H-H\star f\right),
  19. W t = - { { W , H } } = - 2 W sin ( 2 ( x p - p x ) ) H = - { W , H } + O ( 2 ) , \frac{\partial W}{\partial t}=-\{\{W,H\}\}=-\frac{2}{\hbar}W\sin\left({{\frac{% \hbar}{2}}(\stackrel{\leftarrow}{\partial}_{x}\stackrel{\rightarrow}{\partial}% _{p}-\stackrel{\leftarrow}{\partial}_{p}\stackrel{\rightarrow}{\partial}_{x})}% \right)\ H=-\{W,H\}+O(\hbar^{2}),
  20. H = 1 2 m ω 2 x 2 + p 2 2 m . H=\frac{1}{2}m\omega^{2}x^{2}+\frac{p^{2}}{2m}.
  21. H W = ( 1 2 m ω 2 x 2 + p 2 2 m ) W = ( 1 2 m ω 2 ( x + i 2 p ) 2 + 1 2 m ( p - i 2 x ) 2 ) W = ( 1 2 m ω 2 ( x 2 - 2 4 p 2 ) + 1 2 m ( p 2 - 2 4 x 2 ) ) W + i 2 ( m ω 2 x p - p m x ) W = E W . \begin{aligned}\displaystyle H\star W&\displaystyle=\left(\frac{1}{2}m\omega^{% 2}x^{2}+\frac{p^{2}}{2m}\right)\star W\\ &\displaystyle=\left(\frac{1}{2}m\omega^{2}\left(x+\frac{i\hbar}{2}\stackrel{% \rightarrow}{\partial}_{p}\right)^{2}+\frac{1}{2m}\left(p-\frac{i\hbar}{2}% \stackrel{\rightarrow}{\partial}_{x}\right)^{2}\right)~{}W\\ &\displaystyle=\left(\frac{1}{2}m\omega^{2}\left(x^{2}-\frac{\hbar^{2}}{4}% \stackrel{\rightarrow}{\partial}_{p}^{2}\right)+\frac{1}{2m}\left(p^{2}-\frac{% \hbar^{2}}{4}\stackrel{\rightarrow}{\partial}_{x}^{2}\right)\right)~{}W\\ &\displaystyle\,\,\,\,\,+\frac{i\hbar}{2}\left(m\omega^{2}x\stackrel{% \rightarrow}{\partial}_{p}-\frac{p}{m}\stackrel{\rightarrow}{\partial}_{x}% \right)~{}W\\ &\displaystyle=E\cdot W.\end{aligned}
  22. 2 ( m ω 2 x p - p m x ) W = 0 \frac{\hbar}{2}\left(m\omega^{2}x\stackrel{\rightarrow}{\partial}_{p}-\frac{p}% {m}\stackrel{\rightarrow}{\partial}_{x}\right)\cdot W=0
  23. W ( x , p ) = F ( 1 2 m ω 2 x 2 + p 2 2 m ) F ( u ) . W(x,p)=F\left(\frac{1}{2}m\omega^{2}x^{2}+\frac{p^{2}}{2m}\right)\equiv F(u).
  24. F n ( u ) = ( - 1 ) n π L n ( 4 u ω ) e - 2 u / ω , F_{n}(u)=\frac{(-1)^{n}}{\pi\hbar}L_{n}\left(4\frac{u}{\hbar\omega}\right)e^{-% 2u/\hbar\omega}~{},
  25. E n = ω ( n + 1 2 ) . E_{n}=\hbar\omega\left(n+\frac{1}{2}\right)~{}.
  26. W ( x , p ; t = 0 ) = F ( u ) W(x,p;t=0)=F(u)
  27. W ( x , p ; t ) = W ( m ω x cos ω t - p sin ω t , p cos ω t + ω m x sin ω t ; 0 ) . W(x,p;t)=W(m\omega x\cos\omega t-p\sin\omega t,~{}p\cos\omega t+\omega mx\sin% \omega t;0)~{}.
  28. E ħ ω E≫ħω
  29. W ( 𝐱 , 𝐩 ; t ) = 1 ( π ) 3 exp ( - α 2 r 2 - p 2 α 2 2 ( 1 + ( t τ ) 2 ) + 2 t τ 𝐱 𝐩 ) , W(\mathbf{x},\mathbf{p};t)=\frac{1}{(\pi\hbar)^{3}}\exp{\left(-\alpha^{2}r^{2}% -\frac{p^{2}}{\alpha^{2}\hbar^{2}}\left(1+\left(\frac{t}{\tau}\right)^{2}% \right)+\frac{2t}{\hbar\tau}\mathbf{x}\cdot\mathbf{p}\right)}~{},
  30. W 1 ( π ) 3 exp [ - α 2 ( 𝐱 - 𝐩 t m ) 2 ] . W\longrightarrow\frac{1}{(\pi\hbar)^{3}}\exp\left[-\alpha^{2}\left(\mathbf{x}-% \frac{\mathbf{p}t}{m}\right)^{2}\right]\,.
  31. K r a d = α 2 2 2 m ( 3 2 - 1 1 + ( t / τ ) 2 ) K_{rad}=\frac{\alpha^{2}\hbar^{2}}{2m}\left(\frac{3}{2}-\frac{1}{1+(t/\tau)^{2% }}\right)
  32. K a n g = α 2 2 2 m 1 1 + ( t / τ ) 2 . K_{ang}=\frac{\alpha^{2}\hbar^{2}}{2m}\frac{1}{1+(t/\tau)^{2}}~{}.

Phasor_approach_to_fluorescence_lifetime_and_spectral_imaging.html

  1. d ( t ) = e - t / τ d(t)={{e}^{-t/\tau}}
  2. D ( ω ) = 1 1 - j ω τ D(\omega)=\frac{1}{1-j\omega\tau}
  3. τ = 1 ω Im D ( ω ) Re D ( ω ) \tau=\frac{1}{\omega}\frac{\operatorname{Im}D(\omega)}{\operatorname{Re}D(% \omega)}
  4. τ 1 , 2 = 1 ± 1 - 4 u ( u + v ) 2 ω u {{\tau}_{1,2}}=\frac{1\pm\sqrt{1-4u(u+v)}}{2\omega u}
  5. D ( ω ) = sinh ( T 2 K τ ) sinh ( 1 - j ω τ 2 K τ T ) {D}^{\prime}(\omega)=\frac{\sinh\left(\frac{T}{2K\tau}\right)}{\sinh\left(% \frac{1-j\omega\tau}{\frac{2K\tau}{T}}\right)}
  6. τ 1 , 2 = T 2 K A r c c o t h ( ± 1 - 2 u 2 - ( 4 u v + 2 u 2 ) cos ( n ω T 2 K ) ± 1 2 u sin ( n ω T 2 K ) ) {{\tau}_{1,2}}=\frac{\frac{T}{2K}}{Arccoth\left(\pm\frac{\sqrt{1-2{{u}^{2}}-% \left(4uv+2{{u}^{2}}\right)\cos\left(n\omega\frac{T}{2K}\right)}\pm 1}{2u\sin% \left(n\omega\frac{T}{2K}\right)}\right)}

Phenylacetyl-CoA_1,2-epoxidase.html

  1. \rightleftharpoons

Phenylalanine_N-monooxygenase.html

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

Pheophorbidase.html

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

Pheophorbide_a_oxygenase.html

  1. \rightleftharpoons

Phosphatidylinositol-3,4,5-trisphosphate_5-phosphatase.html

  1. \rightleftharpoons

Phosphatidylinositol-4,5-bisphosphate_4-phosphatase.html

  1. \rightleftharpoons

Phosphinothricin_acetyltransferase.html

  1. \rightleftharpoons

Phosphoethanolamine::phosphocholine_phosphatase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Phosphomethylpyrimidine_synthase.html

  1. \rightleftharpoons

Phosphonoacetaldehyde_reductase_(NADH).html

  1. \rightleftharpoons

Phosphoserine_transaminase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Photoactivated_localization_microscopy.html

  1. d = λ 2 N A d=\frac{\lambda}{2NA}
  2. λ \lambda
  3. σ \sigma
  4. σ = ( s i 2 + a 2 12 N ) ( 16 9 + 4 τ s i 2 b 2 a 2 N ) \sigma=\sqrt{\left(\frac{s_{i}^{2}+\frac{a^{2}}{12}}{N}\right)\cdot\left(\frac% {16}{9}+\frac{4\tau s_{i}^{2}b^{2}}{a^{2}N}\right)}
  5. b 2 b^{2}
  6. s i s_{i}

Photon_etc..html

  1. λ B = 2 n Λ sin ( θ + ϕ ) , \lambda_{B}=2n\Lambda\sin(\theta+\phi)\,,
  2. n n
  3. λ λ
  4. Λ Λ
  5. θ θ
  6. φ φ
  7. φ φ
  8. π π
  9. φ φ

Photon_structure_function.html

  1. e + γ e + e+γ→e+
  2. x x
  3. E E
  4. t t
  5. M M
  6. Q Q
  7. Q Q
  8. x x
  9. F 2 , B γ ( x , Q 2 ) = f B ( x ) log Q 2 / Λ 2 + F^{\gamma}_{2,B}(x,Q^{2})=f_{B}(x)\log{Q^{2}/\Lambda^{2}}+...
  10. f B ( x ) = 3 α 2 π q , q ¯ e q 4 x [ x 2 + ( 1 - x ) 2 ] f_{B}(x)=\frac{3\alpha}{2\pi}\sum_{q,\bar{q}}e_{q}^{4}x[x^{2}+(1-x)^{2}]
  11. α = 1 / 137 α=1/137
  12. x x
  13. F 2 , B γ ( x , Q 2 ) F 2 γ ( x , Q 2 ) = f ( x ) log Q 2 / Λ 2 F^{\gamma}_{2,B}(x,Q^{2})\rightarrow F^{\gamma}_{2}(x,Q^{2})=f(x)\log{Q^{2}/% \Lambda^{2}}
  14. Q Q
  15. Λ Λ
  16. x x
  17. Q Q
  18. Q Q
  19. Q Q
  20. x x
  21. h h
  22. x x
  23. W W
  24. W W
  25. α α
  26. x x
  27. x x
  28. Λ Λ
  29. Λ Λ
  30. Λ Λ
  31. x x
  32. Λ Λ
  33. α s ( M Z ) = 0.1198 ± 0.0028 ( e x ) ± 0.0040 ( t h ) \alpha_{s}(M_{Z})=0.1198\pm 0.0028(ex)\pm 0.0040(th)
  34. Λ Λ
  35. x > x>
  36. Q < s u p > 2 Align g t ; Q<sup>2&gt;

Phycoerythrobilin_synthase.html

  1. \rightleftharpoons

Phytoene_desaturase_(3,4-didehydrolycopene-forming).html

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  2. \rightleftharpoons
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Phytoene_desaturase_(lycopene-forming).html

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  2. \rightleftharpoons
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  4. \rightleftharpoons
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Phytoene_desaturase_(neurosporene-forming).html

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  2. \rightleftharpoons
  3. \rightleftharpoons
  4. \rightleftharpoons

Phytoene_desaturase_(zeta-carotene-forming).html

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

Pimelyl-(acyl-carrier_protein)_methyl_ester_esterase.html

  1. \rightleftharpoons

Pipelining_(DSP_implementation).html

  1. y ( n ) = a x ( n ) + b x ( n - 1 ) + c x ( n - 2 ) y(n)=ax(n)+bx(n-1)+cx(n-2)
  2. T s a m p l e T m + 2 T a {T_{sample}\geq T_{m}+2T_{a}}
  3. T s a m p l e T m + T a T_{sample}\geq T_{m}+T_{a}
  4. { z - 1 , , z - ( M - 1 ) } \{z^{-1},...,z^{-(M-1)}\}
  5. H ( z ) = 1 1 - a * z - 1 H(z)=\frac{1}{1-a*z^{-1}}
  6. y ( n ) = a * y ( n - 1 ) + u ( n ) y(n)=a*y(n-1)+u(n)
  7. z = a , a 1 z=a,a\leq 1
  8. z = a e ± ( 2 j π 3 ) z=ae^{\pm(\frac{2j\pi}{3})}
  9. H ( z ) = 1 + a * z - 1 + a 2 * z - 2 1 - a 3 * z - 3 H(z)=\frac{1+a*z^{-1}+a^{2}*z^{-2}}{1-a^{3}*z^{-3}}

Planetary_equilibrium_temperature.html

  1. P i n = I o ( 1 - a ) π R p 2 {P}_{in}={I_{o}}\left(1-a\right)\pi{{R}_{p}}^{2}
  2. P o u t = ϵ σ A T 4 {P}_{out}=\epsilon\sigma A{T}^{4}
  3. ϵ \epsilon
  4. A = 4 π R p 2 A=4\pi{{R}_{p}}^{2}
  5. ϵ \epsilon
  6. T e q = ( I o ( 1 - a ) 4 σ ) 1 / 4 {T}_{eq}={\left({\frac{I_{o}\left(1-a\right)}{4\sigma}}\right)}^{1/4}
  7. T e q = T ( 1 - a ) 1 / 4 R 2 D {T}_{eq}={T}_{\bigodot}{\left(1-a\right)}^{1/4}\sqrt{\frac{{R}_{\bigodot}}{2D}}
  8. P i n = P o u t {P}_{in}={P}_{out}
  9. P i n = L ( 1 - a ) ( π R p 2 4 π D 2 ) {P}_{in}={L}_{\bigodot}\left(1-a\right)\left(\frac{\pi{{R}_{p}}^{2}}{4\pi{D}^{% 2}}\right)
  10. P = σ A T 4 P=\sigma A{T}^{4}
  11. P o u t = ( σ T e q 4 ) ( 4 π R p 2 ) {P}_{out}=\left(\sigma{{T}_{eq}}^{4}\right)\left(4\pi{{R}_{p}}^{2}\right)
  12. T e q = ( L ( 1 - a ) 16 σ π D 2 ) 1 / 4 {T}_{eq}={\left({\frac{{L}_{\bigodot}\left(1-a\right)}{16\sigma\pi{D}^{2}}}% \right)}^{1/4}
  13. L = ( σ T 4 ) ( 4 π R 2 ) {L}_{\bigodot}=\left(\sigma{{T}_{\bigodot}}^{4}\right)\left(4\pi{{R}_{\bigodot% }}^{2}\right)
  14. T e q = T ( 1 - a ) 1 / 4 R 2 D {T}_{eq}={T}_{\bigodot}{\left(1-a\right)}^{1/4}\sqrt{\frac{{R}_{\bigodot}}{2D}}

Plant_seed_peroxygenase.html

  1. \rightleftharpoons

Plastoquinol::plastocyanin_reductase.html

  1. \rightleftharpoons

Plebanski_tensor.html

  1. S a b S_{ab}
  2. S a b = R a b - 1 4 R g a b . S_{ab}=R_{ab}-\frac{1}{4}Rg_{ab}.
  3. P a b = c d S [ a S b ] [ c + d ] δ [ a S b ] e [ c S d ] e - 1 6 δ [ a δ b ] [ c S e f d ] S e f . P^{ab}{}_{cd}=S^{[a}{}_{[c}S^{b]}{}_{d]}+\delta^{[a}{}_{[c}S^{b]e}S_{d]e}-% \frac{1}{6}\delta^{[a}{}_{[c}\delta^{b]}{}_{d]}S^{ef}S_{ef}.

Poisson_scatter_theorem.html

  1. K 2 K\in\mathbb{R}^{2}
  2. B 1 , , B k K B_{1},\ldots,B_{k}\in K
  3. k 2 k\geq 2
  4. λ \lambda
  5. B K B\in K
  6. λ | B | \lambda|B|
  7. | B | |B|
  8. 2 \mathbb{R}^{2}
  9. | B | |B|
  10. \mathbb{R}
  11. B 1 , , B k B_{1},\ldots,B_{k}
  12. N B 1 , , N B k N_{B_{1}},\ldots,N_{B_{k}}
  13. λ \lambda
  14. E ( N B ) / | B | = λ | B | / | B | = λ E(N_{B})/|B|=\lambda|B|/|B|=\lambda
  15. P ( one hit in B ) = λ | B | e - λ | B | λ | B | as | B | P(\,\text{ one hit in B })=\lambda|B|e^{-\lambda|B|}\rightarrow\lambda|B|\text% { as }|B|\rightarrow\infty
  16. 2 \mathbb{R}^{2}
  17. K 2 K\in\mathbb{R}^{2}
  18. B i B_{i}
  19. λ i \lambda_{i}
  20. ( λ 1 , + λ 2 + + λ k ) (\lambda_{1},+\lambda_{2}+\cdots+\lambda_{k})

Poker_Effective_Hand_Strength_(EHS)_algorithm.html

  1. E H S = H S × ( 1 - N P O T ) + ( 1 - H S ) × P P O T EHS=HS\times(1-NPOT)+(1-HS)\times PPOT
  2. E H S EHS
  3. H S HS
  4. N P O T NPOT
  5. P P O T PPOT

Polar_code_(coding_theory).html

  1. O ( n log n ) O(n\log n)

Polyamine_oxidase_(propane-1,3-diamine-forming).html

  1. \rightleftharpoons

Polyhalogen_ions.html

  1. 1 2 \tfrac{1}{2}

Polymer_electrolyte_membrane_electrolysis.html

  1. 2 H 2 O ( l ) O 2 ( g ) + 4 H + ( a q ) + 4 e - 2H_{2}O(l)\rightarrow O_{2}(g)+4H^{+}(aq)+4e^{-}
  2. 4 H + ( a q ) + 4 e - 2 H 2 ( g ) 4H^{+}(aq)+4e^{-}\rightarrow 2H_{2}(g)
  3. Δ H = Δ G elec. + T Δ S heat \Delta H=\underbrace{\Delta G}_{\textrm{elec.}}+\underbrace{T\Delta S}_{% \textrm{heat}}
  4. Δ G \Delta G
  5. T T
  6. Δ S \Delta S
  7. H 2 O ( l ) + Δ H H 2 + 1 / 2 O 2 H_{2}O_{(l)}+\Delta H\rightarrow H_{2}+\ ^{1}\!/_{2}O_{2}
  8. H 2 O ( l ) + 237.2 kJ / mol electricity + 48.6 kJ / mol heat H 2 + 1 / 2 O 2 H_{2}O_{(l)}+\underbrace{237.2\ \textrm{kJ / mol}}_{\textrm{electricity}}+% \underbrace{48.6\ \textrm{kJ / mol}}_{\textrm{heat}}\rightarrow H_{2}+\ ^{1}\!% /_{2}O_{2}
  9. V rev 0 V^{0}_{\textrm{rev}}
  10. V rev 0 = Δ G 0 n F = 237 kJ/mol 2 × 96 , 485 C/mol = 1.23 V V^{0}_{\textrm{rev}}=\frac{\Delta G^{0}}{n\cdot F}=\frac{237\ \textrm{kJ/mol}}% {2\times 96,485\ \textrm{C/mol}}=1.23V
  11. n n
  12. F F
  13. V th 0 = Δ H 0 n F = 285.9 kJ/mol 2 × 96 , 485 C/mol = 1.48 V V^{0}_{\textrm{th}}=\frac{\Delta H^{0}}{n\cdot F}=\frac{285.9\ \textrm{kJ/mol}% }{2\times 96,485\ \textrm{C/mol}}=1.48V
  14. V cell = E + V act + V trans + V ohm V_{\textrm{cell}}=E+V_{\textrm{act}}+V_{\textrm{trans}}+V_{\textrm{ohm}}
  15. V = I R V=I\cdot R
  16. Q I 2 R Q\propto I^{2}\cdot R

Polynomial_kernel.html

  1. d d
  2. K ( x , y ) = ( x 𝖳 y + c ) d K(x,y)=(x^{\mathsf{T}}y+c)^{d}
  3. x x
  4. y y
  5. c 0 c≥0
  6. c = 0 c=0
  7. a a
  8. K K
  9. φ φ
  10. K ( x , y ) = φ ( x ) , φ ( y ) K(x,y)=\langle\varphi(x),\varphi(y)\rangle
  11. φ φ
  12. d = 2 d=2
  13. K ( x , y ) = ( i = 1 n x i y i + c ) 2 = i = 1 n ( x i 2 ) ( y i 2 ) + i = 2 n j = 1 i - 1 ( 2 x i x j ) ( 2 y i y j ) + i = 1 n ( 2 c x i ) ( 2 c y i ) + c 2 K(x,y)=\left(\sum_{i=1}^{n}x_{i}y_{i}+c\right)^{2}=\sum_{i=1}^{n}\left(x_{i}^{% 2}\right)\left(y_{i}^{2}\right)+\sum_{i=2}^{n}\sum_{j=1}^{i-1}\left(\sqrt{2}x_% {i}x_{j}\right)\left(\sqrt{2}y_{i}y_{j}\right)+\sum_{i=1}^{n}\left(\sqrt{2c}x_% {i}\right)\left(\sqrt{2c}y_{i}\right)+c^{2}
  14. φ ( x ) = x n 2 , , x 1 2 , 2 x n x n - 1 , , 2 x n x 1 , 2 x n - 1 x n - 2 , , 2 x n - 1 x 1 , , 2 x 2 x 1 , 2 c x n , , 2 c x 1 , c \varphi(x)=\langle x_{n}^{2},\ldots,x_{1}^{2},\sqrt{2}x_{n}x_{n-1},\ldots,% \sqrt{2}x_{n}x_{1},\sqrt{2}x_{n-1}x_{n-2},\ldots,\sqrt{2}x_{n-1}x_{1},\ldots,% \sqrt{2}x_{2}x_{1},\sqrt{2c}x_{n},\ldots,\sqrt{2c}x_{1},c\rangle
  15. d = 2 d=2
  16. φ φ
  17. [ - F o r m u l a E r r o r - ] [-FormulaError-]

Polynomial_representations_of_cyclic_redundancy_checks.html

  1. x + 1 x+1
  2. x 4 + x + 1 x^{4}+x+1
  3. x 5 + x 3 + 1 x^{5}+x^{3}+1
  4. x 5 + x 4 + x 2 + 1 x^{5}+x^{4}+x^{2}+1
  5. x 5 + x 2 + 1 x^{5}+x^{2}+1
  6. x 6 + x + 1 x^{6}+x+1
  7. x 7 + x 3 + 1 x^{7}+x^{3}+1
  8. x 8 + x 2 + x + 1 x^{8}+x^{2}+x+1
  9. x 8 + x 5 + x 4 + 1 x^{8}+x^{5}+x^{4}+1
  10. x 8 + x 7 + x 6 + x 4 + x 2 + 1 x^{8}+x^{7}+x^{6}+x^{4}+x^{2}+1
  11. x 8 + x 4 + x 3 + x 2 + 1 x^{8}+x^{4}+x^{3}+x^{2}+1
  12. x 8 + x 7 + x 4 + x 3 + x + 1 x^{8}+x^{7}+x^{4}+x^{3}+x+1
  13. x 10 + x 9 + x 5 + x 4 + x + 1 x^{10}+x^{9}+x^{5}+x^{4}+x+1
  14. x 11 + x 9 + x 8 + x 7 + x 2 + 1 x^{11}+x^{9}+x^{8}+x^{7}+x^{2}+1
  15. x 12 + x 11 + x 3 + x 2 + x + 1 x^{12}+x^{11}+x^{3}+x^{2}+x+1
  16. x 13 + x 12 + x 11 + x 10 + x 7 + x 6 + x 5 + x 4 + x 2 + 1 x^{13}+x^{12}+x^{11}+x^{10}+x^{7}+x^{6}+x^{5}+x^{4}+x^{2}+1
  17. x 15 + x 14 + x 10 + x 8 + x 7 + x 4 + x 3 + 1 x^{15}+x^{14}+x^{10}+x^{8}+x^{7}+x^{4}+x^{3}+1
  18. x 16 + x 15 + x 2 + 1 x^{16}+x^{15}+x^{2}+1
  19. x 16 + x 12 + x 5 + 1 x^{16}+x^{12}+x^{5}+1
  20. x 16 + x 15 + x 11 + x 9 + x 8 + x 7 + x 5 + x 4 + x 2 + x + 1 x^{16}+x^{15}+x^{11}+x^{9}+x^{8}+x^{7}+x^{5}+x^{4}+x^{2}+x+1
  21. x 16 + x 13 + x 12 + x 11 + x 10 + x 8 + x 6 + x 5 + x 2 + 1 x^{16}+x^{13}+x^{12}+x^{11}+x^{10}+x^{8}+x^{6}+x^{5}+x^{2}+1
  22. x 16 + x 10 + x 8 + x 7 + x 3 + 1 x^{16}+x^{10}+x^{8}+x^{7}+x^{3}+1
  23. x 24 + x 22 + x 20 + x 19 + x 18 + x 16 + x 14 + x 13 + x 11 + x 10 + x 8 + x 7 + x 6 + x 3 + x + 1 x^{24}+x^{22}+x^{20}+x^{19}+x^{18}+x^{16}+x^{14}+x^{13}+x^{11}+x^{10}+x^{8}+x^% {7}+x^{6}+x^{3}+x+1
  24. x 24 + x 23 + x 18 + x 17 + x 14 + x 11 + x 10 + x 7 + x 6 + x 5 + x 4 + x 3 + x + 1 x^{24}+x^{23}+x^{18}+x^{17}+x^{14}+x^{11}+x^{10}+x^{7}+x^{6}+x^{5}+x^{4}+x^{3}% +x+1
  25. x 30 + x 29 + x 21 + x 20 + x 15 + x 13 + x 12 + x 11 + x 8 + x 7 + x 6 + x 2 + x + 1 x^{30}+x^{29}+x^{21}+x^{20}+x^{15}+x^{13}+x^{12}+x^{11}+x^{8}+x^{7}+x^{6}+x^{2% }+x+1
  26. x 32 + x 26 + x 23 + x 22 + x 16 + x 12 + x 11 + x 10 + x 8 + x 7 + x 5 + x 4 + x 2 + x + 1 x^{32}+x^{26}+x^{23}+x^{22}+x^{16}+x^{12}+x^{11}+x^{10}+x^{8}+x^{7}+x^{5}+x^{4% }+x^{2}+x+1
  27. x 32 + x 28 + x 27 + x 26 + x 25 + x 23 + x 22 + x 20 + x 19 + x 18 + x 14 + x 13 + x 11 + x 10 + x 9 + x 8 + x 6 + 1 x^{32}+x^{28}+x^{27}+x^{26}+x^{25}+x^{23}+x^{22}+x^{20}+x^{19}+x^{18}+x^{14}+x% ^{13}+x^{11}+x^{10}+x^{9}+x^{8}+x^{6}+1
  28. x 32 + x 30 + x 29 + x 28 + x 26 + x 20 + x 19 + x 17 + x 16 + x 15 + x 11 + x 10 + x 7 + x 6 + x 4 + x 2 + x + 1 x^{32}+x^{30}+x^{29}+x^{28}+x^{26}+x^{20}+x^{19}+x^{17}+x^{16}+x^{15}+x^{11}+x% ^{10}+x^{7}+x^{6}+x^{4}+x^{2}+x+1
  29. x 32 + x 31 + x 24 + x 22 + x 16 + x 14 + x 8 + x 7 + x 5 + x 3 + x + 1 x^{32}+x^{31}+x^{24}+x^{22}+x^{16}+x^{14}+x^{8}+x^{7}+x^{5}+x^{3}+x+1
  30. x 40 + x 26 + x 23 + x 17 + x 3 + 1 x^{40}+x^{26}+x^{23}+x^{17}+x^{3}+1
  31. x 64 + x 4 + x 3 + x + 1 x^{64}+x^{4}+x^{3}+x+1
  32. x 64 + x 62 + x 57 + x 55 + x 54 + x 53 + x 52 + x 47 + x 46 + x 45 + x 40 + x 39 + x 38 + x 37 + x 35 + x 33 + x^{64}+x^{62}+x^{57}+x^{55}+x^{54}+x^{53}+x^{52}+x^{47}+x^{46}+x^{45}+x^{40}+x% ^{39}+x^{38}+x^{37}+x^{35}+x^{33}+
  33. x 32 + x 31 + x 29 + x 27 + x 24 + x 23 + x 22 + x 21 + x 19 + x 17 + x 13 + x 12 + x 10 + x 9 + x 7 + x 4 + x + 1 x^{32}+x^{31}+x^{29}+x^{27}+x^{24}+x^{23}+x^{22}+x^{21}+x^{19}+x^{17}+x^{13}+x% ^{12}+x^{10}+x^{9}+x^{7}+x^{4}+x+1

Polyprenol_reductase.html

  1. \rightleftharpoons

Polyvinyl_alcohol_dehydrogenase_(cytochrome).html

  1. \rightleftharpoons

Position_and_momentum_space.html

  1. ψ ( 𝐫 ) = j ϕ j ψ j ( 𝐫 ) \psi(\mathbf{r})=\sum_{j}\phi_{j}\psi_{j}(\mathbf{r})
  2. ψ ( 𝐫 ) = 𝐤 - space ϕ ( 𝐤 ) ψ 𝐤 ( 𝐫 ) d 3 𝐤 \psi(\mathbf{r})=\int_{\mathbf{k}{\rm-space}}\phi(\mathbf{k})\psi_{\mathbf{k}}% (\mathbf{r}){\rm d}^{3}\mathbf{k}
  3. 𝐩 ^ = - i 𝐫 \mathbf{\hat{p}}=-i\hbar\frac{\partial}{\partial\mathbf{r}}
  4. ψ 𝐤 ( 𝐫 ) = 1 ( 2 π ) 3 e i 𝐤 𝐫 \psi_{\mathbf{k}}(\mathbf{r})=\frac{1}{(\sqrt{2\pi})^{3}}e^{i\mathbf{k}\cdot% \mathbf{r}}
  5. ψ ( 𝐫 ) = 1 ( 2 π ) 3 𝐤 - space ϕ ( 𝐤 ) e i 𝐤 𝐫 d 3 𝐤 \psi(\mathbf{r})=\frac{1}{(\sqrt{2\pi})^{3}}\int_{\mathbf{k}{\rm-space}}\phi(% \mathbf{k})e^{i\mathbf{k}\cdot\mathbf{r}}{\rm d}^{3}\mathbf{k}
  6. ϕ ( 𝐤 ) = j ψ j ϕ j ( 𝐤 ) \phi(\mathbf{k})=\sum_{j}\psi_{j}\phi_{j}(\mathbf{k})
  7. ϕ ( 𝐤 ) = 𝐫 - space ψ ( 𝐫 ) ϕ 𝐫 ( 𝐤 ) d 3 𝐫 \phi(\mathbf{k})=\int_{\mathbf{r}{\rm-space}}\psi(\mathbf{r})\phi_{\mathbf{r}}% (\mathbf{k}){\rm d}^{3}\mathbf{r}
  8. 𝐫 ^ = i p = i 𝐤 \mathbf{\hat{r}}=i\hbar\frac{\partial}{\partial p}=i\frac{\partial}{\partial% \mathbf{k}}
  9. ϕ 𝐫 ( 𝐤 ) = 1 ( 2 π ) 3 e - i 𝐤 𝐫 \phi_{\mathbf{r}}(\mathbf{k})=\frac{1}{(\sqrt{2\pi})^{3}}e^{-i\mathbf{k}\cdot% \mathbf{r}}
  10. ϕ ( 𝐤 ) = 1 ( 2 π ) 3 𝐫 - space ψ ( 𝐫 ) e - i 𝐤 𝐫 d 3 𝐫 \phi(\mathbf{k})=\frac{1}{(\sqrt{2\pi})^{3}}\int_{\mathbf{r}{\rm-space}}\psi(% \mathbf{r})e^{-i\mathbf{k}\cdot\mathbf{r}}{\rm d}^{3}\mathbf{r}

Position_of_the_Sun.html

  1. n = JD - 2451545.0 n=\mathrm{JD}-2451545.0
  2. L = 280.460 + 0.9856474 n L=280.460^{\circ}+0.9856474^{\circ}n
  3. g = 357.528 + 0.9856003 n g=357.528^{\circ}+0.9856003^{\circ}n
  4. L L
  5. g g
  6. λ = L + 1.915 sin g + 0.020 sin 2 g \lambda=L+1.915^{\circ}\sin g+0.020^{\circ}\sin 2g
  7. β = 0 \beta=0
  8. R = 1.00014 - 0.01671 cos g - 0.00014 cos 2 g R=1.00014-0.01671\cos g-0.00014\cos 2g
  9. λ \lambda
  10. β \beta
  11. R R
  12. ϵ \epsilon
  13. α = arctan ( cos ϵ tan λ ) \alpha=\arctan(\cos\epsilon\tan\lambda)
  14. α \alpha
  15. λ \lambda
  16. δ = arcsin ( sin ϵ sin λ ) \delta=\arcsin(\sin\epsilon\sin\lambda)
  17. X X
  18. Y Y
  19. Z Z
  20. X = R cos λ X=R\cos\lambda
  21. Y = R cos ϵ sin λ Y=R\cos\epsilon\sin\lambda
  22. Z = R sin ϵ sin λ Z=R\sin\epsilon\sin\lambda
  23. ϵ = 23.439 - 0.0000004 n \epsilon=23.439^{\circ}-0.0000004^{\circ}n
  24. δ = arcsin [ sin ( - 23.44 ) sin ( E L ) ] \delta_{\odot}=\arcsin\left[\sin\left(-23.44^{\circ}\right)\cdot\sin\left(EL% \right)\right]
  25. δ = - 23.44 cos [ 360 365 ( N + 10 ) ] \delta_{\odot}=-23.44^{\circ}\cdot\cos\left[\frac{360^{\circ}}{365}\cdot\left(% N+10\right)\right]
  26. δ = arcsin [ sin ( - 23.44 ) cos ( 360 365.24 ( N + 10 ) + 360 π 0.0167 sin ( 360 365.24 ( N - 2 ) ) ) ] \delta_{\odot}=\arcsin\left[\sin\left(-23.44^{\circ}\right)\cdot\cos\left(% \frac{360^{\circ}}{365.24}\left(N+10\right)+\frac{360^{\circ}}{\pi}\cdot 0.016% 7\sin\left(\frac{360^{\circ}}{365.24}\left(N-2\right)\right)\right)\right]
  27. δ = - arcsin [ 0.39779 cos ( 0.98565 ( N + 10 ) + 1.914 sin ( 0.98565 ( N - 2 ) ) ) ] \delta_{\odot}=-\arcsin\left[0.39779\cos\left(0.98565\left(N+10\right)+1.914% \sin\left(0.98565\left(N-2\right)\right)\right)\right]

Potato_paradox.html

  1. 0.99 100 0.99\cdot 100
  2. x x
  3. 0.98 ( 100 - x ) 0.98(100-x)
  4. 0.99 100 - 0.98 ( 100 - x ) = x 0.99\cdot 100-0.98(100-x)=x
  5. 99 - ( 98 - 0.98 x ) = x 99-(98-0.98x)=x
  6. 99 - 98 + 0.98 x = x 99-98+0.98x=x
  7. 1 + 0.98 x = x 1+0.98x=x
  8. x x
  9. 1 + 0.98 x - 0.98 x = x - 0.98 x 1+0.98x-0.98x=x-0.98x
  10. 1 = 0.02 x 1=0.02x
  11. 1 / 0.02 = 0.02 x / 0.02 1/0.02=0.02x/0.02
  12. 50 = x 50=x
  13. 100 - x = 100 - 50 = 50 100-x=100-50=50

Poverty_gap_index.html

  1. PGI = 1 N j = 1 q ( z - y j z ) {\rm PGI}=\frac{1}{N}\sum_{j=1}^{q}\left(\frac{z-y_{j}}{z}\right)
  2. PGI = 1 N j = 1 N ( ( z - y j ) .1 ( y j < z ) z ) {\rm PGI}=\frac{1}{N}\sum_{j=1}^{N}\left(\frac{(z-y_{j}).1(y_{j}<z)}{z}\right)
  3. N N
  4. q q
  5. z z
  6. y j y_{j}
  7. j j
  8. P 1 P_{1}
  9. P 0 P_{0}
  10. F G T α FGT_{\alpha}
  11. P 2 P_{2}
  12. P SEN P_{\,\text{SEN}}
  13. P SEN = H * G z + P G I * ( 1 - G z ) {\rm P_{\,\text{SEN}}}=H*G_{z}+PGI*(1-G_{z})
  14. H H
  15. G z G_{z}
  16. W W
  17. W = 1 N j = 1 N ln ( z y j ) {\rm W}=\frac{1}{N}\sum_{j=1}^{N}\ln\left(\frac{z}{y_{j}}\right)
  18. W W

Power_diagram.html

  1. O ( n d / 2 ) O(n^{\lceil d/2\rceil})

Power_law_scheme.html

  1. ϕ \phi\,
  2. x ( ρ u ϕ ) = x Γ ϕ x \frac{\partial}{\partial x}(\rho u\phi)\,=\frac{\partial}{\partial x}\Gamma% \frac{\partial\phi}{\partial x}
  3. Γ \Gamma
  4. ρ \rho
  5. ϕ 0 = ϕ | ( x = 0 ) \phi_{0}\,=\phi|_{(}x=0)
  6. ϕ L = ϕ | ( x = L ) \phi_{L}\,=\phi|_{(}x=L)
  7. ϕ ( x ) - ϕ 0 ϕ L - ϕ 0 = exp ( P e x L ) exp ( P e ) - 1 \frac{\phi(x)-\phi_{0}}{\phi_{L}-\phi_{0}}\,=\frac{\exp(P_{e}\frac{x}{L})}{% \exp(P_{e})-1}
  8. P e = ρ u L Γ P_{e}\,=\frac{\rho uL}{\Gamma}
  9. ϕ \phi\,
  10. ϕ \phi\,
  11. ϕ \phi\,
  12. ϕ \phi\,
  13. ρ \rho
  14. Γ \Gamma

Power_residue_symbol.html

  1. 𝒪 k \mathcal{O}_{k}
  2. ζ n 𝒪 k . \zeta_{n}\in\mathcal{O}_{k}.
  3. 𝔭 𝒪 k \mathfrak{p}\subset\mathcal{O}_{k}
  4. 𝔭 \mathfrak{p}
  5. n 𝔭 n\not\in\mathfrak{p}
  6. 𝔭 \mathfrak{p}
  7. 𝒪 k / 𝔭 : N 𝔭 = | 𝒪 k / 𝔭 | . \mathcal{O}_{k}/\mathfrak{p}\;:\;\;\;\mathrm{N}\mathfrak{p}=|\mathcal{O}_{k}/% \mathfrak{p}|.
  8. 𝔭 \mathfrak{p}
  9. 𝒪 k : \mathcal{O}_{k}:
  10. α 𝒪 k , α 𝔭 , \alpha\in\mathcal{O}_{k},\;\;\;\alpha\not\in\mathfrak{p},
  11. α N 𝔭 - 1 1 ( mod 𝔭 ) . \alpha^{\mathrm{N}\mathfrak{p}-1}\equiv 1\;\;(\mathop{{\rm mod}}\mathfrak{p}).
  12. N 𝔭 1 ( mod n ) . \mathrm{N}\mathfrak{p}\equiv 1\;\;(\mathop{{\rm mod}}n).
  13. α N 𝔭 - 1 n ζ n s ( mod 𝔭 ) \alpha^{\frac{\mathrm{N}\mathfrak{p}-1}{n}}\equiv\zeta_{n}^{s}\;\;(\mathop{{% \rm mod}}\mathfrak{p})
  14. 𝒪 k , \mathcal{O}_{k},
  15. ( α 𝔭 ) n = ζ n s α N 𝔭 - 1 n ( mod 𝔭 ) . \left(\frac{\alpha}{\mathfrak{p}}\right)_{n}=\zeta_{n}^{s}\equiv\alpha^{\frac{% \mathrm{N}\mathfrak{p}-1}{n}}\;\;(\mathop{{\rm mod}}\mathfrak{p}).
  16. ( α 𝔭 ) n = { 0 if α 𝔭 1 if α 𝔭 and there is an η 𝒪 k such that α η n ( mod 𝔭 ) ζ where ζ n = 1 and ζ 1 if α 𝔭 and there is no such η \left(\frac{\alpha}{\mathfrak{p}}\right)_{n}=\begin{cases}0&\mbox{ if }~{}% \alpha\in\mathfrak{p}\\ 1&\mbox{ if }~{}\alpha\not\in\mathfrak{p}\mbox{ and there is an }~{}\eta\in% \mathcal{O}_{k}\mbox{ such that }~{}\alpha\equiv\eta^{n}\;\;(\mathop{{\rm mod}% }\mathfrak{p})\\ \zeta\mbox{ where }~{}\zeta^{n}=1\mbox{ and }~{}\zeta\neq 1&\mbox{ if }~{}% \alpha\not\in\mathfrak{p}\mbox{ and there is no such }~{}\eta\end{cases}
  17. ( α 𝔭 ) n α N 𝔭 - 1 n ( mod 𝔭 ) . \left(\frac{\alpha}{\mathfrak{p}}\right)_{n}\equiv\alpha^{\frac{\mathrm{N}% \mathfrak{p}-1}{n}}\;\;(\mathop{{\rm mod}}\mathfrak{p}).
  18. ( α 𝔭 ) n ( β 𝔭 ) n = ( α β 𝔭 ) n \left(\frac{\alpha}{\mathfrak{p}}\right)_{n}\left(\frac{\beta}{\mathfrak{p}}% \right)_{n}=\left(\frac{\alpha\beta}{\mathfrak{p}}\right)_{n}
  19. if α β ( mod 𝔭 ) then ( α 𝔭 ) n = ( β 𝔭 ) n \mbox{if }~{}\alpha\equiv\beta\;\;(\mathop{{\rm mod}}\mathfrak{p})\mbox{ then % }~{}\left(\frac{\alpha}{\mathfrak{p}}\right)_{n}=\left(\frac{\beta}{\mathfrak{% p}}\right)_{n}
  20. ( , ) 𝔭 (\cdot,\cdot)_{\mathfrak{p}}
  21. 𝔭 \mathfrak{p}
  22. ( α 𝔭 ) n = ( π , α ) 𝔭 \left(\frac{\alpha}{\mathfrak{p}}\right)_{n}=\left({\pi,\alpha}\right)_{% \mathfrak{p}}
  23. 𝔭 \mathfrak{p}
  24. π \pi
  25. K 𝔭 K_{\mathfrak{p}}
  26. 𝔞 𝒪 k \mathfrak{a}\subset\mathcal{O}_{k}
  27. 𝔞 = 𝔭 1 𝔭 2 𝔭 g . \mathfrak{a}=\mathfrak{p}_{1}\mathfrak{p}_{2}\dots\mathfrak{p}_{g}.
  28. ( α 𝔞 ) n = ( α 𝔭 1 ) n ( α 𝔭 2 ) n ( α 𝔭 g ) n . \bigg(\frac{\alpha}{\mathfrak{a}}\bigg)_{n}=\left(\frac{\alpha}{\mathfrak{p}_{% 1}}\right)_{n}\left(\frac{\alpha}{\mathfrak{p}_{2}}\right)_{n}\dots\left(\frac% {\alpha}{\mathfrak{p}_{g}}\right)_{n}.
  29. β 𝒪 k \beta\in\mathcal{O}_{k}
  30. ( α β ) n \left(\frac{\alpha}{\beta}\right)_{n}
  31. ( α β ) n = ( α ( β ) ) n , \left(\frac{\alpha}{\beta}\right)_{n}=\left(\frac{\alpha}{(\beta)}\right)_{n},
  32. ( β ) (\beta)
  33. β . \beta.
  34. ( α 𝔞 ) n ( β 𝔞 ) n = ( α β 𝔞 ) n . \bigg(\frac{\alpha}{\mathfrak{a}}\bigg)_{n}\left(\frac{\beta}{\mathfrak{a}}% \right)_{n}=\left(\frac{\alpha\beta}{\mathfrak{a}}\right)_{n}.
  35. ( α 𝔞 ) n ( α 𝔟 ) n = ( α 𝔞 𝔟 ) n . \left(\frac{\alpha}{\mathfrak{a}}\right)_{n}\left(\frac{\alpha}{\mathfrak{b}}% \right)_{n}=\left(\frac{\alpha}{\mathfrak{ab}}\right)_{n}.
  36. If ( α 𝔞 ) n 1 then α is not an n -th power ( mod 𝔞 ) . \mbox{If }~{}\left(\frac{\alpha}{\mathfrak{a}}\right)_{n}\neq 1\mbox{ then }~{% }\alpha\mbox{ is not an }~{}n\mbox{-th power}~{}\;\;(\mathop{{\rm mod}}% \mathfrak{a}).
  37. If ( α 𝔞 ) n = 1 then α may or may not be an n -th power ( mod 𝔞 ) . \mbox{If }~{}\left(\frac{\alpha}{\mathfrak{a}}\right)_{n}=1\mbox{ then }~{}% \alpha\mbox{ may or may not be an }~{}n\mbox{-th power}~{}\;\;(\mathop{{\rm mod% }}\mathfrak{a}).
  38. ( α β ) n ( β α ) n - 1 = 𝔭 | n ( α , β ) 𝔭 \left({\frac{\alpha}{\beta}}\right)_{n}\left({\frac{\beta}{\alpha}}\right)_{n}% ^{-1}=\prod_{\mathfrak{p}|n\infty}(\alpha,\beta)_{\mathfrak{p}}
  39. α \alpha
  40. β \beta

Pólya_class.html

  1. | E ( x + i y ) | | E ( x - i y ) | |E(x+iy)|\geq|E(x-iy)|
  2. | E ( x + i y ) | |E(x+iy)|
  3. | E ( x + i y ) | > | E ( x - i y ) | |E(x+iy)|>|E(x-iy)|
  4. n 1 - Im z n | z n | 2 < \sum_{n}\frac{1-\operatorname{Im}z_{n}}{|z_{n}|^{2}}<\infty
  5. z m e a + b z + c z 2 n ( 1 - z / z n ) exp ( z Re 1 z n ) z^{m}e^{a+bz+cz^{2}}\prod_{n}\left(1-z/z_{n}\right)\exp(z\operatorname{Re}% \frac{1}{z_{n}})
  6. Im - log ( E ( z ) ) z 0 \,\text{Im}\frac{-\log(E(z))}{z}\geq 0
  7. sin ( z ) , cos ( z ) , exp ( z ) , and exp ( - z 2 ) . \sin(z),\cos(z),\exp(z),\,\text{ and }\exp(-z^{2}).
  8. z z
  9. z + i z+i
  10. e x p ( - p i z ) exp(-piz)
  11. e x p ( - p z 2 ) exp(-pz^{2})
  12. sin ( z ) , cos ( z ) , exp ( z ) , exp ( - z ) , exp ( - z 2 ) . \sin(z),\cos(z),\exp(z),\exp(-z),\exp(-z^{2}).

Pregroup_grammar.html

  1. ( A , 1 , , - l , - r , ) (A,1,\cdot,-^{l},-^{r},\leq)
  2. ( A , 1 , ) (A,1,\cdot)
  3. x l x 1 x x r 1 x^{l}\cdot x\leq 1\qquad x\cdot x^{r}\leq 1
  4. 1 x x l 1 x r x 1\leq x\cdot x^{l}\qquad 1\leq x^{r}\cdot x
  5. 1 l = 1 = 1 r 1^{l}=1=1^{r}
  6. x l r = x = x r l x^{lr}=x=x^{rl}
  7. ( x y ) l = y l x l ( x y ) r = y r x r (x\cdot y)^{l}=y^{l}\cdot x^{l}\qquad(x\cdot y)^{r}=y^{r}\cdot x^{r}
  8. x l x^{l}
  9. x r x^{r}
  10. \cdot
  11. \leq
  12. \otimes
  13. \to
  14. x y x\cdot y
  15. x y xy
  16. : :
  17. John : N Mary : N the : N N 0 l dog : N 0 c a t : N 0 \,\text{John}:N\qquad\,\text{Mary}:N\qquad\,\text{the}:N\cdot N_{0}^{l}\qquad% \,\text{dog}:N_{0}\qquad cat:N_{0}
  18. m e t : N r S N l b a r k e d : N r S a t : S r N r r N r S N l met:N^{r}\cdot S\cdot N^{l}\qquad barked:N^{r}\cdot S\qquad at:S^{r}\cdot N^{% rr}\cdot N^{r}\cdot S\cdot N^{l}
  19. T S T\leq S
  20. \leq
  21. J o h n m e t M a r y : N N r S N l N John\ met\ Mary:N\cdot N^{r}\cdot S\cdot N^{l}\cdot N
  22. N N r S N l N S N\cdot N^{r}\cdot S\cdot N^{l}\cdot N\leq S
  23. N N r S N l N S N l N S N\cdot N^{r}\cdot S\cdot N^{l}\cdot N~{}\leq~{}S\cdot N^{l}\cdot N~{}\leq~{}S
  24. N N r N\cdot N^{r}
  25. N l N N^{l}\cdot N
  26. a l l a a^{ll}\neq a
  27. N N - 1 = 1 N\cdot N^{-1}=1
  28. m n m\cdot n
  29. g ( x , b ) [ x ] g(x,b)\cdot[x]
  30. I ( J o h n : N ) = j : E I(John:N)=j:E
  31. I ( M a r y : N ) = m : E I(Mary:N)=m:E
  32. I ( t h e : N N 0 l ) = ι ( p ) [ p ] : E E 0 l I(the:N\cdot N_{0}^{l})=\iota(p)\cdot[p]:E\cdot E_{0}^{l}
  33. I ( d o g : N 0 ) = d o g : E 0 I(dog:N_{0})=dog:E_{0}
  34. I ( c a t : N 0 ) = c a t : E 0 I(cat:N_{0})=cat:E_{0}
  35. I ( m e t : N r S N l ) = [ x ] m e t ( x , y ) [ y ] : E r T E l I(met:N^{r}\cdot S\cdot N^{l})=[x]\cdot met(x,y)\cdot[y]:E^{r}\cdot T\cdot E^{l}
  36. I ( b a r k e d : N r S ) = [ x ] b a r k e d ( x ) : E r T I(barked:N^{r}\cdot S)=[x]\cdot barked(x):E^{r}\cdot T
  37. I ( a t : S r N r r N r S N l ) = [ x ] y [ y ] a t ( x , z ) [ z ] : T r E r r E r T E l I(at:S^{r}\cdot N^{rr}\cdot N^{r}\cdot S\cdot N^{l})=[x]\cdot y\cdot[y]\cdot at% (x,z)\cdot[z]:T^{r}\cdot E^{rr}\cdot E^{r}\cdot T\cdot E^{l}
  38. J o h n m e t M a r y : N ( N r S N l ) N John\ met\ Mary:N\cdot(N^{r}\cdot S\cdot N^{l})\cdot N
  39. t h e d o g b a r k e d a t t h e c a t : ( N N 0 l ) N 0 ( N r S ) ( S r N r r N r S N l ) ( N N 0 l ) N 0 the\ dog\ barked\ at\ the\ cat:(N\cdot N_{0}^{l})\cdot N_{0}\cdot(N^{r}\cdot S% )\cdot(S^{r}\cdot N^{rr}\cdot N^{r}\cdot S\cdot N^{l})\cdot(N\cdot N_{0}^{l})% \cdot N_{0}

Premnaspirodiene_oxygenase.html

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

Presqualene_diphosphate_synthase.html

  1. \rightleftharpoons

Primary-amine_oxidase.html

  1. \rightleftharpoons

Primordial_element_(algebra).html

  1. e i e_{i}
  2. i I i\in I
  3. v = i I a i ( v ) e i . v=\sum_{i\in I}a_{i}(v)e_{i}.
  4. I ( v ) = { i I a i ( v ) 0 } I(v)=\{i\in I\mid a_{i}(v)\neq 0\}
  5. I ( w ) I(w)
  6. I ( w ) I(w^{\prime})
  7. 0 w W 0\neq w^{\prime}\in W
  8. a i ( w ) = 1 a_{i}(w)=1

Product_operator_formalism.html

  1. I x I_{x}
  2. I y I_{y}
  3. I z I_{z}
  4. E / 2 E/2
  5. I z I_{z}
  6. I x ω τ I z cos ( ω τ ) I x - sin ( ω τ ) I y I_{x}\xrightarrow{\omega\tau I_{z}}\cos(\omega\tau)I_{x}-\sin(\omega\tau)I_{y}
  7. I x I_{x}
  8. I y I_{y}

Profit_model.html

  1. π = p q - ( F n + w q ) \pi=pq-(F_{n}+wq)\,
  2. π \pi
  3. π = p q - ( F n + w q ) ( 1 ) \pi=pq-(F_{n}+wq)\qquad\qquad(1)
  4. π = p q - [ F n + w x + g 0 w - g 1 w ] ( 2 ) \pi=pq-[F_{n}+wx+g_{0}w-g_{1}w]\qquad\qquad(2)
  5. w = F m + v x x ( 3 ) w=\frac{F_{m}+vx}{x}\qquad\qquad(3)
  6. π = p q - [ F + v x + g 0 w - g 1 w ] ( 4 ) \pi=pq-[F+vx+g0w-g1w]\qquad\qquad(4)

Projection_formula.html

  1. f : X Y f:X\to Y
  2. \mathcal{F}
  3. \mathcal{E}
  4. R i f * R i f * ( f * ) R^{i}f_{*}\mathcal{F}\otimes\mathcal{E}\to R^{i}f_{*}(\mathcal{F}\otimes f^{*}% \mathcal{E})

Projections_onto_convex_sets.html

  1. find x n such that x C D \,\text{find}\;x\in\mathcal{R}^{n}\quad\,\text{such that}\;x\in C\cap D
  2. x 0 x_{0}
  3. x k + 1 = 𝒫 C ( 𝒫 D ( x k ) ) . x_{k+1}=\mathcal{P}_{C}\left(\mathcal{P}_{D}(x_{k})\right).
  4. x k + 1 = 1 2 ( 𝒫 C ( x k ) + 𝒫 D ( x k ) ) x_{k+1}=\frac{1}{2}(\mathcal{P}_{C}(x_{k})+\mathcal{P}_{D}(x_{k}))
  5. E = { ( x , y ) : x C , y D } E=\{(x,y):x\in C,\;y\in D\}
  6. n × n \mathcal{R}^{n}\times\mathcal{R}^{n}
  7. F = { ( x , y ) : x b , y n , x = y } . F=\{(x,y):x\in\mathcal{R}^{b},\,y\in\mathcal{R}^{n},\;x=y\}.
  8. C D C\cap D
  9. E F E\cap F
  10. E F E\cap F
  11. ( x , y ) (x,y)
  12. ( x + y , x + y ) / 2 (x+y,x+y)/2
  13. ( x k + 1 , y k + 1 ) = 𝒫 F ( 𝒫 E ( ( x k , y k ) ) ) = 𝒫 F ( ( 𝒫 C x k , 𝒫 D y k ) ) = 1 2 ( 𝒫 C ( x k ) + 𝒫 D ( y k ) , ( 𝒫 C ( x k ) + 𝒫 D ( y k ) ) . (x_{k+1},y_{k+1})=\mathcal{P}_{F}(\mathcal{P}_{E}((x_{k},y_{k})))=\mathcal{P}_% {F}((\mathcal{P}_{C}x_{k},\mathcal{P}_{D}y_{k}))=\frac{1}{2}(\mathcal{P}_{C}(x% _{k})+\mathcal{P}_{D}(y_{k}),(\mathcal{P}_{C}(x_{k})+\mathcal{P}_{D}(y_{k})).
  14. x k + 1 = y k + 1 x_{k+1}=y_{k+1}
  15. x 0 = y 0 x_{0}=y_{0}
  16. x j = y j x_{j}=y_{j}
  17. j 0 j\geq 0
  18. x k + 1 = 1 2 ( 𝒫 C ( x k ) + 𝒫 D ( x k ) ) x_{k+1}=\frac{1}{2}(\mathcal{P}_{C}(x_{k})+\mathcal{P}_{D}(x_{k}))

Proof_of_O(log*n)_time_complexity_of_union–find.html

  1. T 1 = F (link to the root) T_{1}=\sum_{F}\,\text{(link to the root)}
  2. T 2 = F (number of links traversed where the buckets are different) T_{2}=\sum_{F}\,\text{(number of links traversed where the buckets are % different)}
  3. T 3 = F (number of links traversed where the buckets are the same). T_{3}=\sum_{F}\,\text{(number of links traversed where the buckets are the % same).}
  4. T 3 = [ B , 2 B - 1 ] u 2 B T_{3}=\sum_{[B,2^{B}-1]}\sum_{u}2^{B}
  5. T 3 B 2 B 2 n 2 B 2 n log * n . T_{3}\leq\sum_{B}2^{B}\frac{2n}{2^{B}}\leq 2n\log^{*}n.

Proper_base_change_theorem.html

  1. f : X S f:X\to S
  2. \mathcal{F}
  3. X X
  4. S = Spec A S=\operatorname{Spec}A
  5. 0 K 0 K 1 K n 0 0\to K^{0}\to K^{1}\to\cdots\to K^{n}\to 0
  6. H p ( X × S Spec - , A - ) H p ( K A - ) , p 0 H^{p}(X\times_{S}\operatorname{Spec}-,\mathcal{F}\otimes_{A}-)\to H^{p}(K^{% \bullet}\otimes_{A}-),p\geq 0
  7. A A
  8. R p f * R^{p}f_{*}\mathcal{F}
  9. \mathcal{F}
  10. p 0 p\geq 0
  11. s dim k ( s ) H p ( X s , s ) : S s\mapsto\dim_{k(s)}H^{p}(X_{s},\mathcal{F}_{s}):S\to\mathbb{Z}
  12. s χ ( s ) s\mapsto\chi(\mathcal{F}_{s})
  13. χ ( ) \chi(\mathcal{F})
  14. p 0 p\geq 0
  15. s dim k ( s ) H p ( X s , s ) s\mapsto\dim_{k(s)}H^{p}(X_{s},\mathcal{F}_{s})
  16. R p f * R^{p}f_{*}\mathcal{F}
  17. R p f * 𝒪 S k ( s ) H p ( X s , s ) R^{p}f_{*}\mathcal{F}\otimes_{\mathcal{O}_{S}}k(s)\to H^{p}(X_{s},\mathcal{F}_% {s})
  18. s S s\in S
  19. R p - 1 f * 𝒪 S k ( s ) H p - 1 ( X s , s ) R^{p-1}f_{*}\mathcal{F}\otimes_{\mathcal{O}_{S}}k(s)\to H^{p-1}(X_{s},\mathcal% {F}_{s})
  20. s S s\in S
  21. H p ( X s , s ) = 0 H^{p}(X_{s},\mathcal{F}_{s})=0
  22. s S s\in S
  23. R p - 1 f * 𝒪 S k ( s ) H p - 1 ( X s , s ) R^{p-1}f_{*}\mathcal{F}\otimes_{\mathcal{O}_{S}}k(s)\to H^{p-1}(X_{s},\mathcal% {F}_{s})
  24. s S s\in S
  25. R i f * R^{i}f_{*}\mathcal{F}
  26. \mathcal{F}
  27. \mathcal{F}
  28. X et X\text{et}
  29. H r ( X , ) H^{r}(X,\mathcal{F})
  30. \mathcal{F}

Propionate_kinase.html

  1. \rightleftharpoons

Prosolanapyrone-II_oxidase.html

  1. \rightleftharpoons

Prostamide::prostaglandin_F2alpha_synthase.html

  1. \rightleftharpoons

Protein-fructosamine_3-kinase.html

  1. \rightleftharpoons

Protein-methionine-S-oxide_reductase.html

  1. \rightleftharpoons

Protein-ribulosamine_3-kinase.html

  1. \rightleftharpoons

Protein_N-terminal_methyltransferase.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons
  4. \rightleftharpoons
  5. \rightleftharpoons
  6. \rightleftharpoons
  7. \rightleftharpoons

Protein_O-GlcNAcase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Protein_phosphatase_methylesterase-1.html

  1. \rightleftharpoons

Protoporphyrinogen_IX_dehydrogenase_(menaquinone).html

  1. \rightleftharpoons

Pseudaminic_acid_cytidylyltransferase.html

  1. \rightleftharpoons

Pseudaminic_acid_synthase.html

  1. \rightleftharpoons

Pseudobaptigenin_synthase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Pseudooxynicotine_oxidase.html

  1. \rightleftharpoons

Pseudotropine_acyltransferase.html

  1. \rightleftharpoons

PSR_J0348+0432.html

  1. 2.01 ± 0.04 M 2.01\pm 0.04M_{\odot}

PSR_J1311–3430.html

  1. M M_{\odot}
  2. M M_{\odot}
  3. M M_{\odot}
  4. M M_{\odot}

Putidaredoxin—NAD+_reductase.html

  1. \rightleftharpoons

Putrescine_aminotransferase.html

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

Pyranose_dehydrogenase_(acceptor).html

  1. \rightleftharpoons
  2. \rightleftharpoons

Pyrethroid_hydrolase.html

  1. \rightleftharpoons

Pyrimidine_oxygenase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Pyrrole-2-carboxylate_decarboxylase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Pyrrole-2-carboxylate_monooxygenase.html

  1. \rightleftharpoons

Pyruvate_dehydrogenase_(quinone).html

  1. \rightleftharpoons

Quadrant_count_ratio.html

  1. X X
  2. Y Y
  3. q = n ( Quadrant I ) + n ( Quadrant III ) - n ( Quadrant II ) - n ( Quadrant IV ) N , q=\frac{n(\,\text{Quadrant I})+n(\,\text{Quadrant III})-n(\,\text{Quadrant II}% )-n(\,\text{Quadrant IV})}{N},
  4. n(Quadrant) \,\text{n(Quadrant)}
  5. N N
  6. ( 6 + 5 ) - ( 13 + 11 ) 35 = - 0.37 \frac{(6+5)-(13+11)}{35}=-0.37

Quadratic_Frobenius_test.html

  1. B B
  2. n \sqrt{n}
  3. n \sqrt{n}\in\mathbb{Z}
  4. x n + 1 2 mod ( n , x 2 - b x - c ) x^{n+1\over 2}\,\bmod\,\big(n,x^{2}-bx-c)
  5. x n + 1 2 / n x^{n+1\over 2}\notin\mathbb{Z}\big/n\mathbb{Z}
  6. x n + 1 mod ( n , x 2 - b x - c ) x^{n+1}\,\bmod\,\big(n,x^{2}-bx-c)
  7. x n + 1 - c x^{n+1}\not\equiv-c
  8. n 2 - 1 = 2 r s n^{2}-1=2^{r}s
  9. x s 1 mod ( n , x 2 - b x - c ) x^{s}\not\equiv 1\bmod\,\big(n,x^{2}-bx-c)
  10. x 2 j s - 1 mod ( n , x 2 - b x - c ) x^{2^{j}s}\not\equiv-1\bmod\,\big(n,x^{2}-bx-c)
  11. 0 j r - 2 0\leq j\leq r-2

Quadratic_Jordan_algebra.html

  1. L ( a ) : x a x L(a):x\mapsto ax
  2. Q ( a ) = 2 L ( a ) 2 - L ( a 2 ) . \displaystyle{Q(a)=2L(a)^{2}-L(a^{2}).}
  3. ( Q ( a ) - 1 a ) a = a Q ( a ) - 1 a = L ( a ) Q ( a ) - 1 a = Q ( a ) - 1 a 2 = 1. \displaystyle{(Q(a)^{-1}a)a=aQ(a)^{-1}a=L(a)Q(a)^{-1}a=Q(a)^{-1}a^{2}=1.}
  4. [ L ( a ) , L ( a 2 ) ] = 0 \displaystyle{[L(a),L(a^{2})]=0}
  5. 2 L ( a b ) L ( a ) + L ( a 2 ) L ( b ) = 2 L ( a ) L ( b ) L ( a ) + L ( a 2 b ) . \displaystyle{2L(ab)L(a)+L(a^{2})L(b)=2L(a)L(b)L(a)+L(a^{2}b).}
  6. Q ( a ) L ( a - 1 ) = L ( a ) . \displaystyle{Q(a)L(a^{-1})=L(a).}
  7. c = D c ( Q ( a ) a - 1 ) = 2 Q ( a , c ) a - 1 + Q ( a ) D c ( a - 1 ) , \displaystyle{c=D_{c}(Q(a)a^{-1})=2Q(a,c)a^{-1}+Q(a)D_{c}(a^{-1}),}
  8. Q ( a , c ) = 1 2 ( Q ( a + c ) - Q ( a ) - Q ( c ) ) = L ( a ) L ( c ) + L ( c ) L ( a ) - L ( a c ) . \displaystyle{Q(a,c)={1\over 2}(Q(a+c)-Q(a)-Q(c))=L(a)L(c)+L(c)L(a)-L(ac).}
  9. Q ( a , c ) a - 1 = ( L ( a ) L ( c ) + L ( c ) L ( a ) - L ( a c ) ) a - 1 = c . \displaystyle{Q(a,c)a^{-1}=(L(a)L(c)+L(c)L(a)-L(ac))a^{-1}=c.}
  10. c = 2 c + Q ( a ) D c ( a - 1 ) , \displaystyle{c=2c+Q(a)D_{c}(a^{-1}),}
  11. ( Q ( a ) b ) ( Q ( a - 1 ) b - 1 ) = 1. \displaystyle{(Q(a)b)(Q(a^{-1})b^{-1})=1.}
  12. ( Q ( a ) b ) - 1 = Q ( a - 1 ) b - 1 . \displaystyle{(Q(a)b)^{-1}=Q(a^{-1})b^{-1}.}
  13. - Q ( Q ( a ) b ) - 1 Q ( a ) c = - Q ( a ) - 1 Q ( b ) - 1 c . \displaystyle{-Q(Q(a)b)^{-1}Q(a)c=-Q(a)^{-1}Q(b)^{-1}c.}
  14. Q ( a , b ) a - 1 = b . \displaystyle{Q(a,b)a^{-1}=b.}
  15. 0 = Q ( c , b ) a - 1 + Q ( a , b ) D c ( a - 1 ) = Q ( c , b ) a - 1 - Q ( a , b ) Q ( a ) - 1 c . \displaystyle{0=Q(c,b)a^{-1}+Q(a,b)D_{c}(a^{-1})=Q(c,b)a^{-1}-Q(a,b)Q(a)^{-1}c.}
  16. Q ( a ) Q ( b , c ) a = Q ( a ) Q ( a - 1 , b ) Q ( a ) c = 1 2 Q ( a ) [ Q ( a - 1 + b ) - Q ( a - 1 ) - Q ( b ) ] Q ( a ) c = Q ( Q ( a ) b , a ) c . \displaystyle{Q(a)Q(b,c)a=Q(a)Q(a^{-1},b)Q(a)c=\frac{1}{2}Q(a)[Q(a^{-1}+b)-Q(a% ^{-1})-Q(b)]Q(a)c=Q(Q(a)b,a)c.}
  17. Q ( a ) Q ( b , c ) a = Q ( Q ( a ) b , a ) c . \displaystyle{Q(a)Q(b,c)a=Q(Q(a)b,a)c.}
  18. Q ( a ) Q ( b , c ) a = Q ( a , Q ( a ) c ) b . \displaystyle{Q(a)Q(b,c)a=Q(a,Q(a)c)b.}
  19. R ( x , y ) R(x,y)
  20. R ( x , y ) z = 2 Q ( x , z ) y R(x,y)z=2Q(x,z)y
  21. Q ( a ) R ( b , a ) c = R ( a , b ) Q ( a ) c , \displaystyle{Q(a)R(b,a)c=R(a,b)Q(a)c,}
  22. c ( a 2 b ) + 2 a ( a c ) b ) = a 2 ( c b ) + 2 ( a c ) ( a b ) . \displaystyle{c(a^{2}b)+2a(ac)b)=a^{2}(cb)+2(ac)(ab).}
  23. c ( ( a d ) b ) + d ( ( a c ) b ) + a ( ( d c ) b ) = ( c d ) ( a b ) + ( d a ) ( c b ) + ( a c ) ( d b ) . \displaystyle{c((ad)b)+d((ac)b)+a((dc)b)=(cd)(ab)+(da)(cb)+(ac)(db).}
  24. L ( c ) L ( b ) L ( a ) + L ( ( a c ) b ) + L ( a ) L ( b ) L ( c ) = L ( a b ) L ( c ) + L ( c b ) L ( a ) + L ( a c ) L ( b ) . \displaystyle{L(c)L(b)L(a)+L((ac)b)+L(a)L(b)L(c)=L(ab)L(c)+L(cb)L(a)+L(ac)L(b).}
  25. Q ( a ) = 2 L ( a ) 2 - L ( a 2 ) . \displaystyle{Q(a)=2L(a)^{2}-L(a^{2}).}
  26. Q ( a , b ) = ½ ( Q ( a = b ) Q ( a ) Q ( b ) ) Q(a,b)=½(Q(a=b)−Q(a)−Q(b))
  27. Q ( a , a ) = Q ( a ) Q(a,a)=Q(a)
  28. Q ( a , b ) = L ( a ) L ( b ) + L ( b ) L ( a ) - L ( a b ) . \displaystyle{Q(a,b)=L(a)L(b)+L(b)L(a)-L(ab).}
  29. Q Q ( a ) , L ( a ) = 00 QQ(a),L(a)=00
  30. R ( a , b ) = 2 L L ( a ) , L ( b ) + 22 L ( a b ) R(a,b)=2LL(a),L(b)+22L(ab)
  31. Q ( a ) R ( b , a ) = 2 [ Q ( a ) L ( b ) , L ( a ) ] + 2 Q ( a ) L ( a b ) = 2 [ Q ( a b , a ) , L ( a ) ] + 2 [ L ( a ) , L ( b ) ] Q ( a ) + 2 Q ( a ) L ( a b ) . \displaystyle{Q(a)R(b,a)=2[Q(a)L(b),L(a)]+2Q(a)L(ab)=2[Q(ab,a),L(a)]+2[L(a),L(% b)]Q(a)+2Q(a)L(ab).}
  32. Q ( a ) R ( b , a ) = 2 [ L ( a ) , L ( b ) ] Q ( a ) + 2 L ( a b ) Q ( a ) = R ( a , b ) Q ( a ) . \displaystyle{Q(a)R(b,a)=2[L(a),L(b)]Q(a)+2L(ab)Q(a)=R(a,b)Q(a).}
  33. 2 Q ( a ) Q ( b , c ) a = 2 Q ( Q ( a ) c , a ) b . \displaystyle{2Q(a)Q(b,c)a=2Q(Q(a)c,a)b.}
  34. Q ( a ) R ( b , a ) c = 2 Q ( Q ( a ) b , a ) c . \displaystyle{Q(a)R(b,a)c=2Q(Q(a)b,a)c.}
  35. Q ( Q ( a ) b ) = Q ( a ) Q ( b ) Q ( a ) Q(Q(a)b)=Q(a)Q(b)Q(a)
  36. Q ( c ) L ( x ) + L ( x ) Q ( c ) = 2 Q ( c x , c ) Q(c)L(x)+L(x)Q(c)=2Q(cx,c)
  37. Q ( c , y ) L ( x ) + L ( x ) Q ( c , y ) = Q ( y x , c ) + Q ( c x , y ) . \displaystyle{Q(c,y)L(x)+L(x)Q(c,y)=Q(yx,c)+Q(cx,y).}
  38. [ L ( x ) , R ( c , d ) ] = R ( x c , d ) - R ( c , x d ) . \displaystyle{[L(x),R(c,d)]=R(xc,d)-R(c,xd).}
  39. [ T , R ( c , d ) ] = R ( D c , d ) + R ( c , D d ) . \displaystyle{[T,R(c,d)]=R(Dc,d)+R(c,Dd).}
  40. [ R ( a , b ) , R ( x , y ) ] a = - R ( R ( x , y ) a , b ) a + R ( a , R ( y , x ) b ) a . \displaystyle{[R(a,b),R(x,y)]a=-R(R(x,y)a,b)a+R(a,R(y,x)b)a.}
  41. 2 Q ( Q ( a ) b ) = 2 R ( a , b ) Q ( Q ( a ) b , a ) - R ( b , a ) Q ( a ) R ( a , b ) + 2 Q ( a ) Q ( b ) Q ( a ) = 2 Q ( a ) Q ( b ) Q ( a ) . \displaystyle{2Q(Q(a)b)=2R(a,b)Q(Q(a)b,a)-R(b,a)Q(a)R(a,b)+2Q(a)Q(b)Q(a)=2Q(a)% Q(b)Q(a).}
  42. Q ( a ) R ( b , a ) = R ( a , b ) Q ( a ) = 2 Q ( Q ( a ) b , a ) . \displaystyle{Q(a)R(b,a)=R(a,b)Q(a)=2Q(Q(a)b,a).}
  43. 2 Q ( a ) Q ( b , c ) a = 2 Q ( Q ( a ) c , a ) b . \displaystyle{2Q(a)Q(b,c)a=2Q(Q(a)c,a)b.}
  44. Q ( a ) R ( b , a ) c = 2 Q ( Q ( a ) b , a ) c . \displaystyle{Q(a)R(b,a)c=2Q(Q(a)b,a)c.}
  45. L ( a ) = 1 2 R ( a , 1 ) . \displaystyle{L(a)=\frac{1}{2}R(a,1).}
  46. R ( a , 1 ) = R ( 1 , a ) = 2 Q ( a , 1 ) . \displaystyle{R(a,1)=R(1,a)=2Q(a,1).}
  47. L ( a ) = Q ( a , 1 ) , L ( 1 ) = Q ( 1 , 1 ) = I . \displaystyle{L(a)=Q(a,1),\,\,\,L(1)=Q(1,1)=I.}
  48. R ( a , b ) = 2 Q ( Q ( a ) b , a ) Q ( a ) - 1 = 2 Q ( a ) Q ( b , a - 1 ) . \displaystyle{R(a,b)=2Q(Q(a)b,a)Q(a)^{-1}=2Q(a)Q(b,a^{-1}).}
  49. R ( a , b ) = 2 Q ( a , b - 1 ) Q ( b ) . \displaystyle{R(a,b)=2Q(a,b^{-1})Q(b).}
  50. a b = L ( a ) b = 1 2 R ( a , 1 ) b = Q ( a , b ) 1 , \displaystyle{a\circ b=L(a)b=\frac{1}{2}R(a,1)b=Q(a,b)1,}
  51. a b = b a . \displaystyle{a\circ b=b\circ a.}
  52. [ L ( a ) , L ( a 2 ) ] = 0. \displaystyle{[L(a),L(a^{2})]=0.}
  53. Q ( Q ( a ) b ) = Q ( a ) Q ( b ) Q ( a ) , \displaystyle{Q(Q(a)b)=Q(a)Q(b)Q(a),}
  54. Q ( a ) = 2 Q ( a , 1 ) 2 - Q ( a 2 , 1 ) = 2 L ( a ) 2 - L ( a 2 ) . \displaystyle{Q(a)=2Q(a,1)^{2}-Q(a^{2},1)=2L(a)^{2}-L(a^{2}).}
  55. Q ( a ) L ( a ) = L ( a ) Q ( a ) , \displaystyle{Q(a)L(a)=L(a)Q(a),}
  56. Q ( a ) = 2 L ( a ) 2 - L ( a 2 ) , \displaystyle{Q(a)=2L(a)^{2}-L(a^{2}),}
  57. Q ( a , b ) = L ( a ) L ( b ) + L ( b ) L ( a ) - L ( a b ) . \displaystyle{Q(a,b)=L(a)L(b)+L(b)L(a)-L(ab).}
  58. 1 2 R ( a , b ) = L ( a ) L ( b ) - L ( b ) L ( a ) + L ( a b ) , \displaystyle{\frac{1}{2}R(a,b)=L(a)L(b)-L(b)L(a)+L(ab),}
  59. Q ( a , b ) = 2 L ( a ) L ( b ) - 1 2 R ( a , b ) . \displaystyle{Q(a,b)=2L(a)L(b)-\frac{1}{2}R(a,b).}
  60. R ( a , b ) Q ( a ) = Q ( a ) R ( b , a ) = 2 Q ( Q ( a ) b , a ) , \displaystyle{R(a,b)Q(a)=Q(a)R(b,a)=2Q(Q(a)b,a),}
  61. Q ( Q ( a ) b ) = Q ( a ) Q ( b ) Q ( a ) \displaystyle{Q(Q(a)b)=Q(a)Q(b)Q(a)}
  62. 2 Q ( Q ( a ) b , b ) - L ( a ) Q ( b ) L ( a ) = Q ( a ) Q ( b ) + Q ( b ) Q ( a ) - Q ( a b ) . \displaystyle{2Q(Q(a)b,b)-L(a)Q(b)L(a)=Q(a)Q(b)+Q(b)Q(a)-Q(ab).}
  63. R ( Q ( a ) b , b ) = R ( a , Q ( b ) a ) . \displaystyle{R(Q(a)b,b)=R(a,Q(b)a).}
  64. 2 L ( Q ( a ) b ) L ( b ) - Q ( Q ( a ) b , b ) = 2 L ( a ) L ( Q ( b ) a ) - Q ( a , Q ( b ) a ) . \displaystyle{2L(Q(a)b)L(b)-Q(Q(a)b,b)=2L(a)L(Q(b)a)-Q(a,Q(b)a).}
  65. [ L ( Q ( a ) b ) + L ( b ) Q ( a ) ] L ( b ) = L ( a ) [ L ( Q ( b ) a ) + Q ( b ) L ( a ) ] . \displaystyle{[L(Q(a)b)+L(b)Q(a)]L(b)=L(a)[L(Q(b)a)+Q(b)L(a)].}
  66. ½ L ( a ) R ( b , a ) L ( b ) ½L(a)R(b,a)L(b)
  67. L a ( b 2 , a ) Q ( b ) = Q ( b ) L b ( a 2 , b ) . \displaystyle{L_{a}(b^{2,a})Q(b)=Q(b)L_{b}(a^{2,b}).}
  68. R ( Q ( b ) a , a ) Q ( b ) = Q ( b ) R ( Q ( a ) b , b ) = R ( b , Q ( a ) b ) Q ( b ) , \displaystyle{R(Q(b)a,a)Q(b)=Q(b)R(Q(a)b,b)=R(b,Q(a)b)Q(b),}
  69. R ( a , b ) c = 2 Q ( a , c ) b R(a,b)c=2Q(a,c)b
  70. 2 Q ( a , c ) = Q ( a + c ) Q ( a ) Q ( c ) 2Q(a,c)=Q(a+c)−Q(a)−Q(c)
  71. Q ( Q ( a ) b ) = Q ( a ) Q ( b ) Q ( a ) , \displaystyle{Q(Q(a)b)=Q(a)Q(b)Q(a),}
  72. R ( a , b ) Q ( a ) = Q ( a ) R ( b , a ) = 2 Q ( Q ( a ) b , a ) , \displaystyle{R(a,b)Q(a)=Q(a)R(b,a)=2Q(Q(a)b,a),}
  73. R ( Q ( a ) b , b ) = R ( a , Q ( b ) a ) . \displaystyle{R(Q(a)b,b)=R(a,Q(b)a).}

Quadratic_set.html

  1. 𝔓 = ( 𝒫 , 𝒢 , ) \mathfrak{P}=({\mathcal{P}},{\mathcal{G}},\in)
  2. 𝒬 {\mathcal{Q}}
  3. 𝒫 {\mathcal{P}}
  4. g g
  5. 𝒢 {\mathcal{G}}
  6. 𝒬 {\mathcal{Q}}
  7. 𝒬 {\mathcal{Q}}
  8. g g
  9. | g 𝒬 | = 0 , | g 𝒬 | = 1 |g\cap{\mathcal{Q}}|=0,\ |g\cap{\mathcal{Q}}|=1
  10. | g 𝒬 | = 2 |g\cap{\mathcal{Q}}|=2
  11. P 𝒬 P\in{\mathcal{Q}}
  12. 𝒬 P {\mathcal{Q}}_{P}
  13. P P
  14. 𝒫 {\mathcal{P}}
  15. 𝒬 {\mathcal{Q}}
  16. P P
  17. 𝒬 P {\mathcal{Q}}_{P}
  18. 𝔓 n \mathfrak{P}_{n}
  19. n 3 n\geq 3
  20. 𝒬 {\mathcal{Q}}
  21. 𝔓 n \mathfrak{P}_{n}
  22. 𝒬 {\mathcal{Q}}
  23. 2 \geq 2
  24. 𝔓 \mathfrak{P}
  25. 2 \geq 2
  26. 𝒪 \mathcal{O}
  27. 𝔬 \mathfrak{o}
  28. 𝔬 \mathfrak{o}
  29. 𝔬 \mathfrak{o}
  30. g g
  31. g 𝔬 = { P } g\cap\mathfrak{o}=\{P\}
  32. g g
  33. | g 𝔬 | = 0 |g\cap\mathfrak{o}|=0
  34. | g 𝔬 | = 1 |g\cap\mathfrak{o}|=1
  35. | g 𝔬 | = 2 |g\cap\mathfrak{o}|=2
  36. 𝔓 \mathfrak{P}
  37. n n
  38. 𝔬 \mathfrak{o}
  39. | 𝔬 | = n + 1 |\mathfrak{o}|=n+1
  40. 𝔬 \mathfrak{o}
  41. 𝔓 \mathfrak{P}
  42. 𝔓 \mathfrak{P}
  43. 𝒪 \mathcal{O}
  44. 𝒪 \mathcal{O}
  45. g g
  46. | g 𝒪 | = 0 , | g 𝒪 | = 1 |g\cap{\mathcal{O}}|=0,\ |g\cap{\mathcal{O}}|=1
  47. | g 𝒪 | = 2 |g\cap{\mathcal{O}}|=2
  48. P 𝒪 P\in{\mathcal{O}}
  49. 𝒪 P {\mathcal{O}}_{P}
  50. P P
  51. P P
  52. n n
  53. K K
  54. | K | < |K|<\infty
  55. 𝔓 n ( K ) \mathfrak{P}_{n}(K)
  56. n = 2 n=2
  57. n = 3 n=3
  58. | K | < , c h a r K 2 |K|<\infty,\ charK\neq 2
  59. 𝔓 n ( K ) \mathfrak{P}_{n}(K)
  60. c h a r K = 2 charK=2

Quadratically_closed_field.html

  1. F 5 2 n F_{5^{2^{n}}}
  2. F 5 2 n F_{5^{2^{n}}}

Quantifier_rank.html

  1. q r ( φ ) = 0 qr(\varphi)=0
  2. q r ( φ 1 φ 2 ) = q r ( φ 1 φ 2 ) = m a x ( q r ( φ 1 ) , q r ( φ 2 ) ) qr(\varphi_{1}\land\varphi_{2})=qr(\varphi_{1}\lor\varphi_{2})=max(qr(\varphi_% {1}),qr(\varphi_{2}))
  3. q r ( ¬ φ ) = q r ( φ ) qr(\lnot\varphi)=qr(\varphi)
  4. q r ( x φ ) = q r ( φ ) + 1 qr(\exists_{x}\varphi)=qr(\varphi)+1
  5. q r ( φ ) n qr(\varphi)\leq n

Quantized_state_systems_method.html

  1. A A
  2. B B
  3. e ( t ) \vec{e}(t)
  4. | e ( t ) | | V | | ( Λ ) - 1 Λ | | V - 1 | Δ Q + | V | | ( Λ ) - 1 V - 1 B | Δ u \left|\vec{e}(t)\right|\leq\left|V\right|\ \left|\Re\left(\Lambda\right)^{-1}% \Lambda\right|\ \left|V^{-1}\right|\ \Delta\vec{Q}+\left|V\right|\ \left|\Re% \left(\Lambda\right)^{-1}V^{-1}B\right|\ \Delta\vec{u}
  5. Δ Q \Delta\vec{Q}
  6. Δ u \Delta\vec{u}
  7. V Λ V - 1 = A V\Lambda V^{-1}=A
  8. A A
  9. | | \left|\,\cdot\,\right|
  10. x ˙ ( t ) = f ( x ( t ) , t ) , x ( t 0 ) = x 0 . \dot{x}(t)=f(x(t),t),\quad x(t_{0})=x_{0}.
  11. x ˙ ( t ) = f ( q ( t ) , t ) , q ( t 0 ) = x 0 . \dot{x}(t)=f(q(t),t),\quad q(t_{0})=x_{0}.
  12. x x
  13. q q
  14. q ( t ) = { x ( t ) if | x ( t ) - q ( t - ) | Δ Q q ( t - ) otherwise q(t)=\begin{cases}x(t)&\,\text{if }\left|x(t)-q(t^{-})\right|\geq\Delta Q\\ q(t^{-})&\,\text{otherwise}\end{cases}
  15. Δ Q \Delta Q
  16. x ( t ) x(t)
  17. q ( t - ) q(t^{-})
  18. q ( t ) q(t)
  19. k th k\text{th}
  20. q k ( t ) q_{k}(t)
  21. x k ( t ) x_{k}(t)
  22. x ( t ) \vec{x}(t)
  23. q ( t ) \vec{q}(t)
  24. x ( t ) = f ( q ( t ) , t ) \vec{x}(t)=f(\vec{q}(t),t)
  25. q ( t ) q(t)
  26. x ( t ) x(t)
  27. q ( t ) q(t)
  28. t t

Quantum-optical_spectroscopy.html

  1. B ^ \hat{B}^{\dagger}
  2. B ^ \hat{B}
  3. Δ [ B ] J B K \Delta\langle\left[B^{\dagger}\right]^{J}\,B^{K}\rangle
  4. ( J + K ) (J+K)
  5. X ^ \hat{X}^{\dagger}
  6. X ^ \hat{X}
  7. H ^ lm = - B ^ X ^ + h . c . , \hat{H}_{\mathrm{lm}}=-\sum\mathcal{F}\,\hat{B}\hat{X}^{\dagger}+\mathrm{h.c.}\,,
  8. X ^ \hat{X}^{\dagger}
  9. B ^ \hat{B}
  10. \mathcal{F}
  11. B ^ \hat{B}
  12. Δ [ X ^ ] J X ^ K = η J + K 2 Δ [ B ] J B K , \Delta\langle\left[\hat{X}^{\dagger}\right]^{J}\hat{X}^{K}\rangle=\eta^{\frac{% J+K}{2}}\Delta\langle\left[B^{\dagger}\right]^{J}B^{K}\rangle\,,
  13. B ^ \langle\hat{B}\rangle
  14. X ^ \langle\hat{X}\rangle
  15. Δ X ^ X ^ \Delta\langle\hat{X}^{\dagger}\hat{X}\rangle
  16. Δ X ^ X ^ \Delta\langle\hat{X}\,\hat{X}\rangle
  17. Δ B ^ B ^ \Delta\langle\hat{B}^{\dagger}\hat{B}\rangle
  18. Δ B ^ B ^ \Delta\langle\hat{B}\hat{B}\rangle

Quantum_cylindrical_quadrupole.html

  1. i t ψ ( x , t ) = - 2 2 m 2 x 2 ψ ( x , t ) + V ( x ) ψ ( x , t ) , \mathrm{i}\hbar\frac{\partial}{\partial t}\psi(x,t)=-\frac{\hbar^{2}}{2m}\frac% {\partial^{2}}{\partial x^{2}}\psi(x,t)+V(x)\psi(x,t),
  2. \hbar
  3. m m
  4. i \mathrm{i}
  5. t t
  6. 𝐕 quad = λ d 2 C o s [ 2 ϕ ] 4 π ϵ 0 s 2 \mathbf{V}_{\mathrm{quad}}=\frac{\lambda d^{2}Cos[2\phi]}{4\pi\epsilon_{0}s^{2}}
  7. 𝐕 quad = Q λ d 2 C o s [ 2 ϕ ] 4 π ϵ 0 s 2 \mathbf{V}_{\mathrm{quad}}=\frac{Q\lambda d^{2}Cos[2\phi]}{4\pi\epsilon_{0}s^{% 2}}
  8. E ψ ( x ) = - 2 2 m s s ( s s ) ψ ( s , ϕ ) - 2 2 m s 2 2 ϕ 2 ψ ( s , ϕ ) + Q λ d 2 C o s [ 2 ϕ ] 4 π ϵ 0 s 2 ψ ( s , ϕ ) , E\psi(x)=-\frac{\hbar^{2}}{2ms}\frac{\partial}{\partial s}(s\frac{\partial}{% \partial s})\psi(s,\phi)-\frac{\hbar^{2}}{2ms^{2}}\frac{\partial^{2}}{\partial% \phi^{2}}\psi(s,\phi)+\frac{Q\lambda d^{2}Cos[2\phi]}{4\pi\epsilon_{0}s^{2}}% \psi(s,\phi),
  9. x = k s x=ks
  10. 1 x x ( x x ) S ( x ) + ( 1 - ν 2 x 2 ) S ( x ) = 0 \frac{1}{x}\frac{\partial}{\partial x}(x\frac{\partial}{\partial x})S(x)+(1-% \frac{\nu^{2}}{x^{2}})S(x)=0
  11. 2 ϕ 2 Φ ( ϕ ) + ( ν 2 - λ q m d 2 2 π ϵ 0 C o s [ 2 ϕ ] ) Φ [ ϕ ] = 0 \frac{\partial^{2}}{\partial\phi^{2}}\Phi(\phi)+(\nu^{2}-\frac{\lambda qmd^{2}% }{2\pi\epsilon_{0}\hbar}Cos[2\phi])\Phi[\phi]=0
  12. ν 2 \nu^{2}
  13. λ q m d 2 2 π ϵ 0 \frac{\lambda qmd^{2}}{2\pi\epsilon_{0}\hbar}
  14. d 2 y d x 2 + [ a - 2 q cos ( 2 x ) ] y = 0. \frac{d^{2}y}{dx^{2}}+[a-2q\cos(2x)]y=0.
  15. C ( a , q , x ) C(a,q,x)
  16. S ( a , q , x ) S(a,q,x)
  17. a n a_{n}
  18. b n b_{n}

Quasi-harmonic_approximation.html

  1. F ( T , V ) = U ( V ) + E Z P ( V ) - T S ( T , V ) F(T,V)=U(V)+E_{ZP}(V)-TS(T,V)
  2. E Z P ( V ) = 1 N 𝐤 , i 1 2 h ν 𝐤 , i ( V ) E_{ZP}(V)=\frac{1}{N}\sum_{\mathbf{k},i}\frac{1}{2}h\nu_{\mathbf{k},i}(V)
  3. S ( V ) = - 1 N 𝐤 , i k B ln [ 1 - exp ( - h ν 𝐤 , i ( V ) k B T ) ] + 1 N 𝐤 , i k B h ν 𝐤 , i ( V ) k B T [ exp ( - h ν 𝐤 , i ( V ) k B T ) - 1 ] - 1 S(V)=-\frac{1}{N}\sum_{\mathbf{k},i}k_{B}\ln\left[1-\exp\left(-\frac{h\nu_{% \mathbf{k},i}(V)}{k_{B}T}\right)\right]+\frac{1}{N}\sum_{\mathbf{k},i}k_{B}% \frac{h\nu_{\mathbf{k},i}(V)}{k_{B}T}\left[\exp\left(-\frac{h\nu_{\mathbf{k},i% }(V)}{k_{B}T}\right)-1\right]^{-1}
  4. G ( T , P ) = min V [ U ( V ) + E Z P ( V ) - T S ( T , V ) + P V ] G(T,P)=\min_{V}\left[U(V)+E_{ZP}(V)-TS(T,V)+PV\right]
  5. α V = 1 V ( V T ) P \alpha_{V}=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{P}
  6. γ i = - ln ν i ln V \gamma_{i}=-\frac{\partial\ln\nu_{i}}{\partial\ln V}

Quaternionic_structure.html

  1. 𝐐 \mathbf{Q}
  2. ( G , Q , q ) (G,Q,q)
  3. G G
  4. 2 2
  5. 1 −1
  6. Q Q
  7. 1 1
  8. q q
  9. G × G Q G×G→Q
  10. 1. q ( a , ( - 1 ) a ) = 1 , 2. q ( a , b ) = q ( a , c ) q ( a , b c ) = 1 , 3. q ( a , b ) = q ( c , d ) x | q ( a , b ) = q ( a , x ) , q ( c , d ) = q ( c , x ) . \begin{aligned}\displaystyle\,\text{1.}&\displaystyle q(a,(-1)a)=1,\\ \displaystyle\,\text{2.}&\displaystyle q(a,b)=q(a,c)\Leftrightarrow q(a,bc)=1,% \\ \displaystyle\,\text{3.}&\displaystyle q(a,b)=q(c,d)\Rightarrow\exists x|q(a,b% )=q(a,x),q(c,d)=q(c,x)\end{aligned}.
  11. F F
  12. Q Q
  13. G G
  14. F < s u p > / F 2 F<sup>∗/F^{∗2}

QUICK_scheme.html

  1. 6 / 8 {6}/{8}
  2. 3 / 8 {3}/{8}
  3. 1 / 8 {1}/{8}
  4. d ( ρ u ϕ ) d x = d d x ( r d ϕ d x ) . {d(\rho u\phi)\over dx}=\frac{d}{dx}\left(r\frac{d\phi}{dx}\right).
  5. d ( ρ u ) d x = 0. {d(\rho u)\over dx}=0.
  6. ( ρ u A ϕ ) e - ( ρ u A ϕ ) w = ( r A ϕ x ) e - ( r A ϕ x ) w (\rho uA\phi)_{e}-(\rho uA\phi)_{w}=(rA\frac{\partial\phi}{\partial x})_{e}-(% rA\frac{\partial\phi}{\partial x})_{w}

Quinate::shikimate_dehydrogenase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Quinate_dehydrogenase_(quinone).html

  1. \rightleftharpoons

Quinolinate_synthase.html

  1. \rightleftharpoons

Quorum-quenching_N-acyl-homoserine_lactonase.html

  1. \rightleftharpoons

Quotient_filter.html

  1. α \alpha
  2. α = n / m \alpha=n/m
  3. 1 - e - α / 2 r 2 - r 1-e^{-\alpha/2^{r}}\leq 2^{-r}

Rademacher_system.html

  1. { t r n ( t ) = sgn ( sin 2 n + 1 π t ) ; t [ 0 , 1 ] , n 𝒩 } . \{t\mapsto r_{n}(t)=\operatorname{sgn}(\sin 2^{n+1}\pi t);t\in[0,1],n\in% \mathcal{N}\}.

Radial_basis_function_kernel.html

  1. K ( 𝐱 , 𝐱 ) = exp ( - || 𝐱 - 𝐱 || 2 2 σ 2 ) K(\mathbf{x},\mathbf{x^{\prime}})=\exp\left(-\frac{||\mathbf{x}-\mathbf{x^{% \prime}}||^{2}}{2\sigma^{2}}\right)
  2. || 𝐱 - 𝐱 || 2 \textstyle||\mathbf{x}-\mathbf{x^{\prime}}||^{2}
  3. σ \sigma
  4. γ = 1 2 σ 2 \textstyle\gamma=\tfrac{1}{2\sigma^{2}}
  5. K ( 𝐱 , 𝐱 ) = exp ( - γ || 𝐱 - 𝐱 || 2 ) K(\mathbf{x},\mathbf{x^{\prime}})=\exp(-\gamma||\mathbf{x}-\mathbf{x^{\prime}}% ||^{2})
  6. 𝐱 = 𝐱 \mathbf{x}=\mathbf{x}
  7. σ = 1 \sigma=1
  8. exp ( - 1 2 || 𝐱 - 𝐱 || 2 ) = j = 0 ( 𝐱 𝐱 ) j j ! exp ( - 1 2 || 𝐱 || 2 ) exp ( - 1 2 || 𝐱 || 2 ) \exp\left(-\frac{1}{2}||\mathbf{x}-\mathbf{x^{\prime}}||^{2}\right)=\sum_{j=0}% ^{\infty}\frac{(\mathbf{x}^{\top}\mathbf{x^{\prime}})^{j}}{j!}\exp\left(-\frac% {1}{2}||\mathbf{x}||^{2}\right)\exp\left(-\frac{1}{2}||\mathbf{x^{\prime}}||^{% 2}\right)
  9. z ( 𝐱 ) z ( 𝐱 ) φ ( 𝐱 ) φ ( 𝐱 ) = K ( 𝐱 , 𝐱 ) z(\mathbf{x})z(\mathbf{x^{\prime}})\approx\varphi(\mathbf{x})\varphi(\mathbf{x% ^{\prime}})=K(\mathbf{x},\mathbf{x^{\prime}})
  10. φ \textstyle\varphi

Radiant_energy_density.html

  1. w e = Q e V , w_{\mathrm{e}}=\frac{\partial Q_{\mathrm{e}}}{\partial V},
  2. E e = c w e , E_{\mathrm{e}}=cw_{\mathrm{e}},
  3. L e = c 4 π w e . L_{\mathrm{e}}=\frac{c}{4\pi}w_{\mathrm{e}}.
  4. M e = π L e = c 4 w e . M_{\mathrm{e}}=\pi L_{\mathrm{e}}=\frac{c}{4}w_{\mathrm{e}}.

Radó–Kneser–Choquet_theorem.html

  1. F f ( r e i θ ) = 1 2 π 0 2 π f ( φ ) 1 - r 2 1 - 2 r cos ( θ - φ ) + r 2 d φ , \displaystyle{F_{f}(re^{i\theta})={1\over 2\pi}\int_{0}^{2\pi}f(\varphi)\cdot{% 1-r^{2}\over 1-2r\cos(\theta-\varphi)+r^{2}}\,d\varphi,}
  2. J f ( a ) = | z F f ( a ) | 2 - | z ¯ F f ( a ) | 2 . \displaystyle{J_{f}(a)=|\partial_{z}F_{f}(a)|^{2}-|\partial_{\overline{z}}F_{f% }(a)|^{2}.}
  3. F f g = F f g . \displaystyle{F_{f\circ g}=F_{f}\circ g.}
  4. ( F f g ) z = [ ( F f ) ζ g ] g z , ( F f g ) z ¯ = [ ( F f ) ζ ¯ g ] g z ¯ . \displaystyle{(F_{f}\circ g)_{z}=[(F_{f})_{\zeta}\circ g]\cdot g_{z},\,\,(F_{f% }\circ g)_{\overline{z}}=[(F_{f})_{\overline{\zeta}}\circ g]\cdot\overline{g_{% z}}.}
  5. J f g ( a ) = J f ( 0 ) | g z ( a ) | 2 . \displaystyle{J_{f\circ g}(a)=J_{f}(0)\cdot|g_{z}(a)|^{2}.}
  6. J f ( 0 ) = | a 1 | 2 - | a - 1 | 2 , \displaystyle{J_{f}(0)=|a_{1}|^{2}-|a_{-1}|^{2},}
  7. a n = 1 2 π 0 2 π f ( e i θ ) e - i n θ d θ . \displaystyle{a_{n}={1\over 2\pi}\int_{0}^{2\pi}f(e^{i\theta})e^{-in\theta}\,d% \theta.}
  8. | a 1 | 2 - | a - 1 | 2 = 1 4 π 2 0 2 π 0 2 π [ f ( e i θ ) f ( e i φ ) ¯ ( e i ( θ - φ ) - e - i ( θ - φ ) ) ] d θ d φ . \displaystyle{|a_{1}|^{2}-|a_{-1}|^{2}={1\over 4\pi^{2}}\int_{0}^{2\pi}\int_{0% }^{2\pi}\Re[f(e^{i\theta})\overline{f(e^{i\varphi})}(e^{i(\theta-\varphi)}-e^{% -i(\theta-\varphi)})]\,d\theta\,d\varphi.}
  9. f ( e i θ ) = e i h ( θ ) , \displaystyle{f(e^{i\theta})=e^{ih(\theta)},}
  10. h ( θ + 2 π ) = h ( θ ) + 2 π , \displaystyle{h(\theta+2\pi)=h(\theta)+2\pi},
  11. | a 1 | 2 - | a - 1 | 2 = 1 2 π 2 0 2 π 0 2 π sin ( θ - φ ) sin ( h ( θ ) - h ( φ ) ) d θ d φ = 1 2 π 2 0 2 π 0 2 π sin ( θ ) sin ( h ( θ + φ ) - h ( φ ) ) d θ d φ . \displaystyle{|a_{1}|^{2}-|a_{-1}|^{2}={1\over 2\pi^{2}}\int_{0}^{2\pi}\int_{0% }^{2\pi}\sin(\theta-\varphi)\sin(h(\theta)-h(\varphi))\,d\theta\,d\varphi={1% \over 2\pi^{2}}\int_{0}^{2\pi}\int_{0}^{2\pi}\sin(\theta)\sin(h(\theta+\varphi% )-h(\varphi))\,d\theta\,d\varphi.}
  12. | a 1 | 2 - | a - 1 | 2 = 1 2 π 2 0 π sin θ 0 π R ( θ , φ ) d φ d θ , \displaystyle{|a_{1}|^{2}-|a_{-1}|^{2}={1\over 2\pi^{2}}\int_{0}^{\pi}\sin% \theta\int_{0}^{\pi}R(\theta,\varphi)\,d\varphi\,d\theta,}
  13. R ( θ , φ ) = sin ( h ( φ + θ ) - h ( φ ) ) + sin ( h ( φ + 2 π ) - h ( φ + θ + π ) ) + sin ( h ( φ + θ + π ) - h ( φ + π ) ) + sin ( h ( φ + π ) - h ( φ + θ ) ) . R(\theta,\varphi)=\sin(h(\varphi+\theta)-h(\varphi))+\sin(h(\varphi+2\pi)-h(% \varphi+\theta+\pi))+\sin(h(\varphi+\theta+\pi)-h(\varphi+\pi))+\sin(h(\varphi% +\pi)-h(\varphi+\theta)).

Railway_electrification_in_the_Soviet_Union.html

  1. 162 / 3 16{2}/{3}
  2. η m e a n = i p i t i η i i p i t i \eta_{mean}=\frac{\textstyle\sum_{i}p_{i}t_{i}\eta_{i}}{\textstyle\sum_{i}p_{i% }t_{i}}
  3. p i p_{i}
  4. η i \eta_{i}
  5. t i t_{i}

Random_walk_closeness_centrality.html

  1. m j k m_{jk}
  2. M ( i , j ) = a i j j = 1 n a i j M(i,j)=\frac{a_{ij}}{\sum_{j=1}^{n}a_{ij}}
  3. a i j a_{ij}
  4. C i R W C = n j = 1 n H ( j , i ) C_{i}^{RWC}=\frac{n}{\sum_{j=1}^{n}H(j,i)}
  5. H ( i , j ) = r = 1 r P ( i , j , r ) H(i,j)=\sum_{r=1}^{\infty}rP(i,j,r)
  6. M - j M_{-j}
  7. M - j r - 1 M_{-j}^{r-1}
  8. P ( i , j , r ) = k j ( ( M - j r - 1 ) ) i k m k j P(i,j,r)=\sum_{k\neq j}((M_{-j}^{r-1}))_{ik}m_{kj}
  9. H ( i , j ) = r = 1 r k j ( ( M - j r - 1 ) ) i k m k j H(i,j)=\sum_{r=1}^{\infty}r\sum_{k\neq j}((M_{-j}^{r-1}))_{ik}m_{kj}
  10. H ( i , j ) = k j ( ( I - M - j ) - 2 ) i k m k j H(i,j)=\sum_{k\neq j}((I-M_{-j})^{-2})_{ik}m_{kj}
  11. H ( . , j ) = ( I - M - j ) - 1 e H(.,j)=(I-M_{-j})^{-1}e
  12. H ( . , j ) H(.,j)
  13. C i R W B = j i k r j k C_{i}^{RWB}=\sum_{j\neq i\neq k}r_{jk}
  14. r j k r_{jk}

Range_query_(data_structures).html

  1. q f ( A , i , j ) q_{f}(A,i,j)
  2. A = [ a 1 , a 2 , . . , a n ] A=[a_{1},a_{2},..,a_{n}]
  3. S S
  4. A [ 1 , n ] A[1,n]
  5. 1 i j n 1\leq i\leq j\leq n
  6. f f
  7. S S
  8. f ( A [ i , j ] ) = f ( a i , , a j ) f(A[i,j])=f(a_{i},\ldots,a_{j})
  9. f = s u m f=sum
  10. A [ 1 , n ] A[1,n]
  11. s u m ( A , i , j ) sum(A,i,j)
  12. s u m ( A [ i , j ] ) = ( a i + + a j ) sum(A[i,j])=(a_{i}+\ldots+a_{j})
  13. 1 i j n 1\leq i\leq j\leq n
  14. O ( n ) O(n)
  15. i i
  16. A A
  17. B B
  18. B [ i ] B[i]
  19. i i
  20. A A
  21. 0 i n 0\leq i\leq n
  22. s u m ( A [ i , j ] ) = B [ j ] - B [ i - 1 ] sum(A[i,j])=B[j]-B[i-1]
  23. f f
  24. f - 1 f^{-1}
  25. f - 1 f^{-1}
  26. c c
  27. θ ( c n ) \theta(c\cdot n)
  28. f f
  29. θ ( α c ( n ) ) \theta(\alpha_{c}(n))
  30. α k \alpha_{k}
  31. f { max , min } f\in\{\max,\min\}
  32. f = min f=\min
  33. min ( A [ 1.. n ] ) \min(A[1..n])
  34. A [ 1.. n ] A[1..n]
  35. min ( A , i , j ) \min(A,i,j)
  36. O ( 1 ) O(1)
  37. O ( n ) O(n)
  38. T A T_{A}
  39. A [ 1 , n ] A[1,n]
  40. a i = m i n { a 1 , a 2 , , a n } a_{i}=min\{a_{1},a_{2},\ldots,a_{n}\}
  41. A [ 1 , i - 1 ] A[1,i-1]
  42. A [ i + 1 , n ] A[i+1,n]
  43. m i n ( A , i , j ) min(A,i,j)
  44. T A T_{A}
  45. a i a_{i}
  46. a j a_{j}
  47. O ( n ) O(n)
  48. A = [ 4 , 5 , 6 , 7 , 4 , ] A=[4,5,6,7,4,]
  49. A [ 1 , n ] A[1,n]
  50. A [ 1 , n ] A[1,n]
  51. O ( n 2 - 2 ϵ ) O(n^{2-2\epsilon})
  52. O ( n ϵ log n ) O(n^{\epsilon}\log n)
  53. 0 ϵ 1 / 2 0\leq\epsilon\leq 1/2
  54. O ( n 2 log log n / log n ) O(n^{2}\log\log n/\log n)
  55. O ( 1 ) O(1)
  56. Ω ( log n log ( S w / n ) ) \Omega\left(\frac{\log n}{\log(Sw/n)}\right)
  57. S S
  58. m e d i a n ( A , i , j ) median(A,i,j)
  59. A [ i , j ] A[i,j]
  60. m e d i a n ( A , i , j ) median(A,i,j)
  61. A [ i , j ] A[i,j]
  62. j - i 2 \frac{j-i}{2}
  63. O ( n log k + k log n ) O(n\log k+k\log n)
  64. O ( n log k ) O(n\log k)
  65. r r
  66. A [ i , j ] A[i,j]
  67. r = j - i 2 r=\frac{j-i}{2}

Rank_of_a_partition.html

  1. N ( m , n ) = N ( - m , n ) N(m,n)=N(-m,n)
  2. N ( m , q , n ) = N ( q - m , q , n ) N(m,q,n)=N(q-m,q,n)
  3. N ( m , q , n ) = r = - N ( m + r q , n ) N(m,q,n)=\sum_{r=-\infty}^{\infty}N(m+rq,n)
  4. n = 0 p ( n ) x n = k = 1 1 ( 1 - x k ) \sum_{n=0}^{\infty}p(n)x^{n}=\prod_{k=1}^{\infty}\frac{1}{(1-x^{k})}
  5. m = - n = 0 N ( m , n ) z m q n = 1 + n = 1 q n 2 k = 1 n ( 1 - z q k ) ( 1 - z - 1 q k ) \sum_{m=-\infty}^{\infty}\sum_{n=0}^{\infty}N(m,n)z^{m}q^{n}=1+\sum_{n=1}^{% \infty}\frac{q^{n^{2}}}{\prod_{k=1}^{n}(1-zq^{k})(1-z^{-1}q^{k})}
  6. n = 0 Q ( n ) x n = k = 0 1 ( 1 - x 2 k - 1 ) \sum_{n=0}^{\infty}Q(n)x^{n}=\prod_{k=0}^{\infty}\frac{1}{(1-x^{2k-1})}
  7. m , n = 0 Q ( m , n ) z m q n = 1 + s = 1 q s ( s + 1 ) / 2 ( 1 - z q ) ( 1 - z q 2 ) ( 1 - z q s ) \sum_{m,n=0}^{\infty}Q(m,n)z^{m}q^{n}=1+\sum_{s=1}^{\infty}\frac{q^{s(s+1)/2}}% {(1-zq)(1-zq^{2})\cdots(1-zq^{s})}

Ranked_voting_system.html

  1. N ! 2 ( ln 2 ) N + 1 \frac{N!}{2(\ln 2)^{N+1}}
  2. n = 1 N - 1 N ! n ! = ( e - 1 ) N ! - 1 = floor ( ( e - 1 ) N ! - 1 ) \sum_{n=1}^{N-1}\frac{N!}{n!}=\lfloor(e-1)N!-1\rfloor=\mathrm{floor}\left((e-1% )N!-1\right)

Ratio_estimator.html

  1. R = μ ¯ y / μ ¯ x R=\bar{\mu}_{y}/\bar{\mu}_{x}\,
  2. θ y = R θ x \theta_{y}=R\theta_{x}\,
  3. r = y ¯ x ¯ = i = 1 n y i = 1 n x r=\frac{\bar{y}}{\bar{x}}=\frac{\sum_{i=1}^{n}y}{\sum_{i=1}^{n}x}
  4. E ( x y ) = E ( x 1 y ) = E ( x ) E ( 1 y ) E ( x ) 1 E ( y ) = E ( x ) E ( y ) E\left(\frac{x}{y}\right)=E\left(x\frac{1}{y}\right)=E(x)E\left(\frac{1}{y}% \right)\geq E(x)\frac{1}{E(y)}=\frac{E(x)}{E(y)}
  5. r corr = r - s [ y / x ] x m x r_{\mathrm{corr}}=r-\frac{s_{[y/x]x}}{m_{x}}
  6. r corr = r + 1 n ( 1 - n - 1 N - 1 ) r s x 2 - ρ s x s y m x 2 r_{\mathrm{corr}}=r+\frac{1}{n}(1-\frac{n-1}{N-1})\frac{rs_{x}^{2}-\rho s_{x}s% _{y}}{m_{x}^{2}}
  7. r corr = r - N - n N ( r s x 2 - ρ s x s y ) n m x 2 r_{\mathrm{corr}}=r-\frac{N-n}{N}\frac{(rs_{x}^{2}-\rho s_{x}s_{y})}{nm_{x}^{2}}
  8. r corr = r [ 1 + 1 n ( 1 m x - s x y m x m y ) + 1 n 2 ( 2 m x 2 - s x y m x m y [ 2 + 3 m x ] + s x 2 y m x 2 m y ) ] r_{\mathrm{corr}}=r\left[1+\frac{1}{n}\left(\frac{1}{m_{x}}-\frac{s_{xy}}{m_{x% }m_{y}}\right)+\frac{1}{n^{2}}\left(\frac{2}{m_{x}^{2}}-\frac{s_{xy}}{m_{x}m_{% y}}\left[2+\frac{3}{m_{x}}\right]+\frac{s_{x^{2}y}}{m_{x}^{2}m_{y}}\right)\right]
  9. θ = 1 n - 1 N \theta=\frac{1}{n}-\frac{1}{N}
  10. c x 2 = s x 2 m x 2 c_{x}^{2}=\frac{s_{x}^{2}}{m_{x}^{2}}
  11. c x y = s x y m x m y c_{xy}=\frac{s_{xy}}{m_{x}m_{y}}
  12. r corr = r + N - 1 N m y - r m x n - 1 r_{\mathrm{corr}}=r+\frac{N-1}{N}\frac{m_{y}-rm_{x}}{n-1}
  13. r corr = r 1 + θ c x y 1 + θ c x 2 r_{\mathrm{corr}}=r\frac{1+\theta c_{xy}}{1+\theta c_{x}^{2}}
  14. r corr = r ( 1 + θ ( c x y - c x 2 ) ) r_{\mathrm{corr}}=r\left(1+\theta\left(c_{xy}-c_{x}^{2}\right)\right)
  15. r corr = r 1 + θ ( c x 2 - c x y ) r_{\mathrm{corr}}=\frac{r}{1+\theta(c_{x}^{2}-c_{xy})}
  16. r corr = r ( 1 + θ c x y ) ( 1 - θ c x 2 ) r_{\mathrm{corr}}=r(1+\theta c_{xy})(1-\theta c_{x}^{2})
  17. r corr = r ( 1 - θ c x 2 ) 1 - θ c x y r_{\mathrm{corr}}=\frac{r(1-\theta c_{x}^{2})}{1-\theta c_{xy}}
  18. r corr = r ( 1 + θ c x y ) ( 1 + θ c x 2 ) r_{\mathrm{corr}}=\frac{r}{(1+\theta c_{xy})(1+\theta c_{x}^{2})}
  19. r corr = r [ 1 - 2 n 2 m x ( 1 m x - s x y m x m y ) ( 1 + 13 2 n + 8 n m x ) ] r_{\mathrm{corr}}=r\left[1-\frac{2}{n^{2}m_{x}}\left(\frac{1}{m_{x}}-\frac{s_{% xy}}{m_{x}m_{y}}\right)\left(1+\frac{13}{2n}+\frac{8}{nm_{x}}\right)\right]
  20. r corr = r + c x 2 m y m x - s x y m x 2 r_{\mathrm{corr}}=r+c_{x}^{2}\frac{m_{y}}{m_{x}}-\frac{s_{xy}}{m_{x}^{2}}
  21. r corr = n r - n - 1 n i j = 1 n r i r_{\mathrm{corr}}=nr-\frac{n-1}{n}\sum_{i\neq j=1}^{n}r_{i}
  22. r corr = g r - g - 1 g i = 1 g r i r_{\mathrm{corr}}=gr-\frac{g-1}{g}\sum_{i=1}^{g}r_{i}
  23. r corr = g g + 1 r - 1 g ( g - 1 ) i = 1 g r i r_{\mathrm{corr}}=\frac{g}{g+1}r-\frac{1}{g(g-1)}\sum_{i=1}^{g}r_{i}
  24. r corr = r ¯ + n n - 1 m y - r ¯ m x m x r_{\mathrm{corr}}=\bar{r}+\frac{n}{n-1}\frac{m_{y}-\bar{r}m_{x}}{m_{x}}
  25. r corr = r g ¯ + g ( m y - r g ¯ m x ) m x r_{\mathrm{corr}}=\bar{r_{g}}+\frac{g(m_{y}-\bar{r_{g}}m_{x})}{m_{x}}
  26. r ¯ \bar{r}
  27. r g ¯ = r i g \bar{r_{g}}=\sum\frac{r_{i}^{{}^{\prime}}}{g}
  28. τ y = r τ x \tau_{y}=r\tau_{x}
  29. var ( r ) = 1 s x 2 + m x 2 [ ( s y 2 - s x 2 [ y 2 / x 2 ] ) - ( s x [ y / x ] ) 2 + 2 m y s x [ y / x ] - s x 2 m x 2 ( m y - s x [ y / x ] 2 ) ] \operatorname{var}(r)=\frac{1}{s_{x}^{2}+m_{x}^{2}}\left[(s_{y}^{2}-s_{x^{2}[y% ^{2}/x^{2}]})-(s_{x[y/x]})^{2}+2m_{y}s_{x[y/x]}-\frac{s_{x}^{2}}{m_{x}^{2}}(m_% {y}-s_{x[y/x]}^{2})\right]
  30. var ( r ) = N - n N 1 m x 2 i = 1 n ( y i - r x i ) n - 1 \operatorname{var}(r)=\frac{N-n}{N}\frac{1}{m_{x}^{2}}\frac{\sum_{i=1}^{n}(y_{% i}-rx_{i})}{n-1}
  31. var ( r ) = 1 n ( 1 - n - 1 N - 1 ) r 2 s x 2 + s y 2 - 2 r ρ s x s y m x 2 \operatorname{var}(r)=\frac{1}{n}(1-\frac{n-1}{N-1})\frac{r^{2}s_{x}^{2}+s_{y}% ^{2}-2r\rho s_{x}s_{y}}{m_{x}^{2}}
  32. var ( r ) = 1 n [ s y 2 m x 2 + m y 2 s x 2 m x 4 - 2 m y s x y m x 3 ] \operatorname{var}(r)=\frac{1}{n}\left[\frac{s_{y}^{2}}{m_{x}^{2}}+\frac{m_{y}% ^{2}s_{x}^{2}}{m_{x}^{4}}-\frac{2m_{y}s_{xy}}{m_{x}^{3}}\right]
  33. var ( r ) = r 2 [ 1 n ( 1 m x + 1 m y - 2 s x y m x m y ) + 1 n 2 ( 6 m x 2 + 3 m x m y + s x y [ 4 m y 2 - 8 m x m y - 16 m x 2 m y + 5 s x y m x 2 m y 2 ] + 4 s x 2 y m x 2 m y - 2 s x y 2 m x m y 2 ) ] \operatorname{var}(r)=r^{2}\left[\frac{1}{n}\left(\frac{1}{m_{x}}+\frac{1}{m_{% y}}-\frac{2s_{xy}}{m_{x}m_{y}}\right)+\frac{1}{n^{2}}\left(\frac{6}{m_{x}^{2}}% +\frac{3}{m_{x}m_{y}}+s_{xy}\left[\frac{4}{m_{y}^{2}}-\frac{8}{m_{x}m_{y}}-% \frac{16}{m_{x}^{2}m_{y}}+\frac{5s_{xy}}{m_{x}^{2}m_{y}^{2}}\right]+\frac{4s_{% x^{2}y}}{m_{x}^{2}m_{y}}-\frac{2s_{xy^{2}}}{m_{x}m_{y}^{2}}\right)\right]
  34. var ( r ) = 1 n ( n - 1 ) i j n ( r i - r J ) 2 \operatorname{var}(r)=\frac{1}{n(n-1)}\sum_{i\neq j}^{n}(r_{i}-r_{J})^{2}
  35. var ( τ y ) = τ y 2 var ( r ) \operatorname{var}(\tau_{y})=\tau_{y}^{2}\operatorname{var}(r)
  36. var ( y ¯ ) = m x 2 var ( r ) = N - n N 1 m x 2 i = 1 n ( y i - r x i ) n - 1 = N - n N ( s y 2 + r 2 s x 2 - 2 r ρ s x s y ) n \operatorname{var}(\bar{y})=m_{x}^{2}\operatorname{var}(r)=\frac{N-n}{N}\frac{% 1}{m_{x}^{2}}\frac{\sum_{i=1}^{n}(y_{i}-rx_{i})}{n-1}=\frac{N-n}{N}\frac{(s_{y% }^{2}+r^{2}s_{x}^{2}-2r\rho s_{x}s_{y})}{n}
  37. γ = ( m y ω n m x m y ω 2 + m x 2 m y ) ( 6 + 1 n m x [ 44 + 1 1 + ω 2 m y / m x ] ) \gamma=\left(\frac{m_{y}\omega}{\sqrt{nm_{x}m_{y}\omega^{2}+m_{x}^{2}m_{y}}}% \right)\left(6+\frac{1}{nm_{x}}\left[44+\frac{1}{1+\omega^{2}m_{y}/m_{x}}% \right]\right)
  38. ω = 1 - m x cov ( x , y ) \omega=1-m_{x}\operatorname{cov}(x,y)\,
  39. r = y i x i r=\frac{\sum y_{i}}{\sum x_{i}}
  40. P = x i ( N - 1 n - 1 ) X P=\frac{\sum x_{i}}{{N-1\choose n-1}X}

Rational_reconstruction_(mathematics).html

  1. r s \frac{r}{s}
  2. n = r × s - 1 ( mod m ) n=r\times s^{-1}\;\;(\mathop{{\rm mod}}m)
  3. v = ( m , 0 ) v=(m,0)
  4. w = ( n , 1 ) w=(n,1)
  5. N \leq N
  6. q = v 1 w 1 q=\left\lfloor{\frac{v_{1}}{w_{1}}}\right\rfloor
  7. w 1 N w_{1}\leq N
  8. w 2 < 0 w_{2}<0
  9. w 2 < D w_{2}<D
  10. gcd ( w 1 , w 2 ) = 1 \gcd(w_{1},w_{2})=1
  11. r s \frac{r}{s}
  12. r = w 1 r=w_{1}
  13. s = w 2 s=w_{2}

Rational_set.html

  1. N N
  2. RAT ( N ) \mathrm{RAT}(N)
  3. N N
  4. A , B RAT ( N ) A,B\in\mathrm{RAT}(N)
  5. A B RAT ( N ) A\cup B\in\mathrm{RAT}(N)
  6. A , B RAT ( N ) A,B\in\mathrm{RAT}(N)
  7. A × B = { a × b a A , b B } RAT ( N ) A\times B=\{a\times b\mid a\in A,b\in B\}\in\mathrm{RAT}(N)
  8. A RAT ( N ) A\in\mathrm{RAT}(N)
  9. A * = i = 1 A i RAT ( N ) A^{*}=\bigcup_{i=1}^{\infty}A^{i}\in\mathrm{RAT}(N)
  10. A 1 = A A^{1}=A
  11. A n + 1 = A n × A A^{n+1}=A^{n}\times A
  12. N N
  13. N N
  14. A A
  15. A * A^{*}
  16. A A
  17. A * A^{*}
  18. \mathbb{N}
  19. k \mathbb{N}^{k}
  20. N N
  21. RAT ( N ) \mathrm{RAT}(N)
  22. N = { a } * × { b , c } * N=\{a\}^{*}\times\{b,c\}^{*}
  23. U = { ( a n , b n c m ) n , m } U=\{(a^{n},b^{n}c^{m})\mid n,m\in\mathbb{N}\}
  24. V = { ( a n , b m c n ) n , m } V=\{(a^{n},b^{m}c^{n})\mid n,m\in\mathbb{N}\}
  25. W = U V = { ( a n , b n c n ) n } W=U\cap V=\{(a^{n},b^{n}c^{n})\mid n\in\mathbb{N}\}
  26. { b n m n n } \{b^{n}m^{n}\mid n\in\mathbb{N}\}
  27. N N
  28. M M
  29. ϕ : N M \phi:N\rightarrow M
  30. S RAT ( N ) S\in\mathrm{RAT}(N)
  31. ϕ ( S ) = { ϕ ( x ) x S } RAT ( M ) \phi(S)=\{\phi(x)\mid x\in S\}\in\mathrm{RAT}(M)

Rayleigh_mixture_distribution.html

  1. f ( x ; σ ) = x σ 2 e - x 2 / 2 σ 2 , x 0 , f(x;\sigma)=\frac{x}{\sigma^{2}}e^{-x^{2}/2\sigma^{2}},\quad x\geq 0,
  2. f ( x ; σ , n ) = 0 r e - r 2 / 2 σ 2 σ 2 τ ( x , r ; n ) d r , f(x;\sigma,n)=\int_{0}^{\infty}\frac{re^{-r^{2}/2\sigma^{2}}}{\sigma^{2}}\tau(% x,r;n)\,\mathrm{d}r,
  3. τ ( x , r ; n ) \tau(x,r;n)

Rayleigh–Bénard_convection.html

  1. Ra L = g β ν α ( T b - T u ) L 3 \mathrm{Ra}_{L}=\frac{g\beta}{\nu\alpha}(T_{b}-T_{u})L^{3}
  2. 27 / 4 {27}/{4}

Rayleigh–Plesset_equation.html

  1. P B ( t ) - P ( t ) ρ L = R d 2 R d t 2 + 3 2 ( d R d t ) 2 + 4 ν L R d R d t + 2 S ρ L R \frac{P_{B}(t)-P_{\infty}(t)}{\rho_{L}}=R\frac{d^{2}R}{dt^{2}}+\frac{3}{2}% \left(\frac{dR}{dt}\right)^{2}+\frac{4\nu_{L}}{R}\frac{dR}{dt}+\frac{2S}{\rho_% {L}R}
  2. P B ( t ) P_{B}(t)
  3. P ( t ) P_{\infty}(t)
  4. ρ L \rho_{L}
  5. R ( t ) R(t)
  6. ν L \nu_{L}
  7. S S
  8. P B ( t ) P_{B}(t)
  9. P ( t ) P_{\infty}(t)
  10. R ( t ) R(t)
  11. R ( t ) R(t)
  12. t t
  13. T B ( t ) T_{B}(t)
  14. P B ( t ) P_{B}(t)
  15. ρ L \rho_{L}
  16. μ L \mu_{L}
  17. T T_{\infty}
  18. P ( t ) P_{\infty}(t)
  19. T T_{\infty}
  20. r r
  21. P ( r , t ) P(r,t)
  22. T ( r , t ) T(r,t)
  23. u ( r , t ) u(r,t)
  24. r R ( t ) r\geq R(t)
  25. u ( r , t ) u(r,t)
  26. F ( t ) F(t)
  27. u ( r , t ) = F ( t ) r 2 u(r,t)=\frac{F(t)}{r^{2}}
  28. u ( R , t ) = d R d t = F ( t ) R 2 u(R,t)=\frac{dR}{dt}=\frac{F(t)}{R^{2}}
  29. F ( t ) = R 2 d R / d t F(t)=R^{2}dR/dt
  30. d m V d t = ρ V d V d t = ρ V d ( 4 π R 3 / 3 ) d t = 4 π ρ V R 2 d R d t \frac{dm_{V}}{dt}=\rho_{V}\frac{dV}{dt}=\rho_{V}\frac{d(4\pi R^{3}/3)}{dt}=4% \pi\rho_{V}R^{2}\frac{dR}{dt}
  31. V V
  32. u L u_{L}
  33. r = R r=R
  34. d m L d t = ρ L A u L = ρ L ( 4 π R 2 ) u L \frac{dm_{L}}{dt}=\rho_{L}Au_{L}=\rho_{L}(4\pi R^{2})u_{L}
  35. A A
  36. d m v / d t = d m L / d t dm_{v}/dt=dm_{L}/dt
  37. u L = ( ρ V / ρ L ) d R / d t u_{L}=(\rho_{V}/\rho_{L})dR/dt
  38. u ( R , t ) = d R d t - u L = d R d t - ρ V ρ L d R d t = ( 1 - ρ V ρ L ) d R d t u(R,t)=\frac{dR}{dt}-u_{L}=\frac{dR}{dt}-\frac{\rho_{V}}{\rho_{L}}\frac{dR}{dt% }=\left(1-\frac{\rho_{V}}{\rho_{L}}\right)\frac{dR}{dt}
  39. F ( t ) = ( 1 - ρ V ρ L ) R 2 d R d t F(t)=\left(1-\frac{\rho_{V}}{\rho_{L}}\right)R^{2}\frac{dR}{dt}
  40. ρ L ρ V \rho_{L}\gg\rho_{V}
  41. F ( t ) F(t)
  42. F ( t ) = R 2 d R / d t F(t)=R^{2}dR/dt
  43. u ( r , t ) = F ( t ) r 2 = R 2 r 2 d R d t u(r,t)=\frac{F(t)}{r^{2}}=\frac{R^{2}}{r^{2}}\frac{dR}{dt}
  44. ρ L ( u t + u u r ) = - P r + μ L [ 1 r 2 r ( r 2 u r ) - 2 u r 2 ] \rho_{L}\left(\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial r}% \right)=-\frac{\partial P}{\partial r}+\mu_{L}\left[\frac{1}{r^{2}}\frac{% \partial}{\partial r}\left(r^{2}\frac{\partial u}{\partial r}\right)-\frac{2u}% {r^{2}}\right]
  45. ν L = μ L / ρ L \nu_{L}=\mu_{L}/\rho_{L}
  46. - 1 ρ L P r = u t + u u r - ν L [ 1 r 2 r ( r 2 u r ) - 2 u r 2 ] -\frac{1}{\rho_{L}}\frac{\partial P}{\partial r}=\frac{\partial u}{\partial t}% +u\frac{\partial u}{\partial r}-\nu_{L}\left[\frac{1}{r^{2}}\frac{\partial}{% \partial r}\left(r^{2}\frac{\partial u}{\partial r}\right)-\frac{2u}{r^{2}}\right]
  47. u ( r , t ) u(r,t)
  48. - 1 ρ L P r = 2 R r 2 ( d R d t ) 2 + R 2 r 2 d 2 R d t 2 - 2 R 4 r 5 ( d R d t ) 2 = 1 r 2 ( 2 R ( d R d t ) 2 + R 2 d 2 R d t 2 ) - 2 R 4 r 5 ( d R d t ) 2 -\frac{1}{\rho_{L}}\frac{\partial P}{\partial r}=\frac{2R}{r^{2}}\left(\frac{% dR}{dt}\right)^{2}+\frac{R^{2}}{r^{2}}\frac{d^{2}R}{dt^{2}}-\frac{2R^{4}}{r^{5% }}\left(\frac{dR}{dt}\right)^{2}=\frac{1}{r^{2}}\left(2R\left(\frac{dR}{dt}% \right)^{2}+R^{2}\frac{d^{2}R}{dt^{2}}\right)-\frac{2R^{4}}{r^{5}}\left(\frac{% dR}{dt}\right)^{2}
  49. r = R r=R
  50. r r\rightarrow\infty
  51. - 1 ρ L P ( R ) P d P = R [ 1 r 2 ( 2 R ( d R d t ) 2 + R 2 d 2 R d t 2 ) - 2 R 4 r 5 ( d R d t ) 2 ] d r -\frac{1}{\rho_{L}}\int_{P(R)}^{P_{\infty}}dP=\int_{R}^{\infty}\left[\frac{1}{% r^{2}}\left(2R\left(\frac{dR}{dt}\right)^{2}+R^{2}\frac{d^{2}R}{dt^{2}}\right)% -\frac{2R^{4}}{r^{5}}\left(\frac{dR}{dt}\right)^{2}\right]dr
  52. P ( R ) - P ρ L = [ - 1 r ( 2 R ( d R d t ) 2 + R 2 d 2 R d t 2 ) + R 4 2 r 4 ( d R d t ) 2 ] R = R d 2 R d t 2 + 3 2 ( d R d t ) 2 {\frac{P(R)-P_{\infty}}{\rho_{L}}=\left[-\frac{1}{r}\left(2R\left(\frac{dR}{dt% }\right)^{2}+R^{2}\frac{d^{2}R}{dt^{2}}\right)+\frac{R^{4}}{2r^{4}}\left(\frac% {dR}{dt}\right)^{2}\right]_{R}^{\infty}=R\frac{d^{2}R}{dt^{2}}+\frac{3}{2}% \left(\frac{dR}{dt}\right)^{2}}
  53. σ r r \sigma_{rr}
  54. σ r r = - P + 2 μ L u r \sigma_{rr}=-P+2\mu_{L}\frac{\partial u}{\partial r}
  55. σ r r ( R ) + P B - 2 S R = - P ( R ) + 2 μ L u r | r = R + P B - 2 S R = - P ( R ) + 2 μ L r ( R 2 r 2 d R d t ) r = R + P B - 2 S R = - P ( R ) - 4 μ L R d R d t + P B - 2 S R \begin{aligned}\displaystyle\sigma_{rr}(R)+P_{B}-\frac{2S}{R}&\displaystyle=-P% (R)+\left.2\mu_{L}\frac{\partial u}{\partial r}\right|_{r=R}+P_{B}-\frac{2S}{R% }\\ &\displaystyle=-P(R)+2\mu_{L}\frac{\partial}{\partial r}\left(\frac{R^{2}}{r^{% 2}}\frac{dR}{dt}\right)_{r=R}+P_{B}-\frac{2S}{R}\\ &\displaystyle=-P(R)-\frac{4\mu_{L}}{R}\frac{dR}{dt}+P_{B}-\frac{2S}{R}\\ \end{aligned}
  56. S S
  57. P ( R ) = P B - 4 μ L R d R d t - 2 S R P(R)=P_{B}-\frac{4\mu_{L}}{R}\frac{dR}{dt}-\frac{2S}{R}
  58. P ( R ) - P ρ L = P B - P ρ L - 4 μ L ρ L R d R d t - 2 S ρ L R = R d 2 R d t 2 + 3 2 ( d R d t ) 2 \frac{P(R)-P_{\infty}}{\rho_{L}}=\frac{P_{B}-P_{\infty}}{\rho_{L}}-\frac{4\mu_% {L}}{\rho_{L}R}\frac{dR}{dt}-\frac{2S}{\rho_{L}R}=R\frac{d^{2}R}{dt^{2}}+\frac% {3}{2}\left(\frac{dR}{dt}\right)^{2}
  59. ν L = μ L / ρ L \nu_{L}=\mu_{L}/\rho_{L}
  60. P B ( t ) - P ( t ) ρ L = R d 2 R d t 2 + 3 2 ( d R d t ) 2 + 4 ν L R d R d t + 2 S ρ L R \frac{P_{B}(t)-P_{\infty}(t)}{\rho_{L}}=R\frac{d^{2}R}{dt^{2}}+\frac{3}{2}% \left(\frac{dR}{dt}\right)^{2}+\frac{4\nu_{L}}{R}\frac{dR}{dt}+\frac{2S}{\rho_% {L}R}
  61. P B ( t ) - P ( t ) ρ L = R R ¨ + 3 2 ( R ˙ ) 2 + 4 ν L R ˙ R + 2 S ρ L R \frac{P_{B}(t)-P_{\infty}(t)}{\rho_{L}}=R\ddot{R}+\frac{3}{2}(\dot{R})^{2}+% \frac{4\nu_{L}\dot{R}}{R}+\frac{2S}{\rho_{L}R}
  62. P B - P = 2 S R P_{B}-P_{\infty}=\frac{2S}{R}

Reaction_field_method.html

  1. U A B = q A q B [ 1 r A B + ( ε R F - 1 ) r A B 2 ( 2 ε R F + 1 ) r c 3 ] U_{AB}=q_{A}q_{B}\left[\frac{1}{r_{AB}}+\frac{(\varepsilon_{RF}-1)r_{AB}^{2}}{% (2\varepsilon_{RF}+1)r_{c}^{3}}\right]
  2. r c r_{c}
  3. E R F = 2 ( ε R F - 1 ) 2 ε R F + 1 M r c 3 E_{RF}=\frac{2(\varepsilon_{RF}-1)}{2\varepsilon_{RF}+1}\frac{\vec{M}}{r_{c}^{% 3}}
  4. M = μ i \vec{M}=\sum\mu_{i}
  5. i i
  6. - 1 / 2 μ i E R F -1/2\mu_{i}\cdot E_{RF}
  7. i i
  8. μ i × E R F \mu_{i}\times E_{RF}
  9. r t r_{t}
  10. f ( r ) f(r)
  11. r t = .95 r c r_{t}=.95r_{c}
  12. ε R F \varepsilon_{RF}
  13. ϵ \epsilon

Reaction–diffusion–advection_equation.html

  1. u ( s y m b o l x , t ) u(symbol{x},t)
  2. s y m b o l x n symbol{x}\in\mathbb{R}^{n}
  3. u t + ( s y m b o l v u - D u ) = f , \frac{\partial u}{\partial t}+\nabla\cdot\left(symbol{v}u-D\nabla u\right)=f,
  4. s y m b o l v n symbol{v}\in\mathbb{R}^{n}
  5. f f
  6. f , D , f,D,
  7. s y m b o l v symbol{v}

Rebound_attack.html

  1. E E
  2. E = E f w E i n E b w E=E_{fw}\circ E_{in}\circ E_{bw}
  3. E i n E_{in}
  4. E f w E_{fw}
  5. E b w E_{bw}
  6. S : { 0 , 1 } 8 { 0 , 1 } 8 S:\{0,1\}^{8}\to\{0,1\}^{8}
  7. ( a , b ) { 0 , 1 } 8 (a,b)\in\{0,1\}^{8}
  8. x x
  9. S ( x ) S ( x a ) = b S(x)\oplus S(x\oplus a)=b
  10. a a
  11. b b
  12. E b w E_{bw}
  13. E i n E_{in}
  14. E f w E_{fw}

Rebreather_diving.html

  1. V O 2 V_{O_{2}}
  2. V E V_{E}
  3. K E K_{E}
  4. K E = V E V O 2 K_{E}=\frac{V_{E}}{V_{O_{2}}}
  5. d F O 2 l o o p dF_{O_{2}loop}
  6. V l o o p * d F O 2 l o o p = ( Q f e e d * F O 2 f e e d - V O 2 - ( Q f e e d - V O 2 ) * F O 2 l o o p ) d t V_{loop}*dF_{O_{2}loop}=(Q_{feed}*F_{O_{2}feed}-V_{O_{2}}-(Q_{feed}-V_{O_{2}})% *F_{O_{2}loop})dt
  7. V l o o p V_{loop}
  8. Q f e e d Q_{feed}
  9. F O 2 f e e d F_{O_{2}feed}
  10. V O 2 V_{O_{2}}
  11. d F O 2 l o o p d t = ( Q f e e d * F O 2 f e e d - V O 2 ( t ) - ( Q f e e d - V O 2 ) * F O 2 l o o p ( t ) ) V l o o p \frac{dF_{O_{2}loop}}{dt}=\frac{(Q_{feed}*F_{O_{2}feed}-V_{O_{2}}(t)-(Q_{feed}% -V_{O_{2}})*F_{O_{2}loop}(t))}{V_{loop}}
  12. F O 2 l o o p ( t ) = Q f e e d * F O 2 f e e d - V O 2 Q f e e d - V O 2 + ( F O 2 l o o p s t a r t - Q f e e d * F O 2 f e e d - V O 2 Q f e e d - V O 2 ) * e - Q f e e d - V O 2 V l o o p t F_{O_{2}loop}(t)=\frac{Q_{feed}*F_{O_{2}feed}-V_{O_{2}}}{Q_{feed}-V_{O_{2}}}+(% F_{O_{2}loop}^{start}-\frac{Q_{feed}*F_{O_{2}feed}-V_{O_{2}}}{Q_{feed}-V_{O_{2% }}})*e^{-\frac{Q_{feed}-V_{O_{2}}}{V_{loop}}t}
  13. F O 2 l o o p F_{O_{2}loop}
  14. F O 2 l o o p = ( Q f e e d * F O 2 f e e d - V O 2 ) ( Q f e e d - V O 2 ) F_{O_{2}loop}=\frac{(Q_{feed}*F_{O_{2}feed}-V_{O_{2}})}{(Q_{feed}-V_{O_{2}})}
  15. Q f e e d Q_{feed}
  16. V O 2 V_{O_{2}}
  17. F O 2 f e e d F_{O_{2}feed}
  18. d F O 2 l o o p dF_{O_{2}loop}
  19. V l o o p * d F O 2 l o o p = ( ( Q d u m p + V O 2 ) * F O 2 f e e d - V O 2 - Q d u m p * F O 2 l o o p ) d t V_{loop}*dF_{O_{2}loop}=((Q_{dump}+V_{O_{2}})*F_{O_{2}feed}-V_{O_{2}}-Q_{dump}% *F_{O_{2}loop})dt
  20. V l o o p V_{loop}
  21. F O 2 l o o p F_{O_{2}loop}
  22. Q d u m p Q_{dump}
  23. V O 2 V_{O_{2}}
  24. F O 2 f e e d F_{O_{2}feed}
  25. d F O 2 l o o p d t = ( ( Q d u m p + V O 2 ) * F O 2 f e e d ( t ) - V O 2 - Q d u m p * F O 2 l o o p ( t ) ) V l o o p \frac{dF_{O_{2}loop}}{dt}=\frac{((Q_{dump}+V_{O_{2}})*F_{O_{2}feed}(t)-V_{O_{2% }}-Q_{dump}*F_{O_{2}loop}(t))}{V_{loop}}
  26. F O 2 l o o p ( t ) = ( Q d u m p + V O 2 ) * F O 2 f e e d - V O 2 Q d u m p + ( F O 2 l o o p s t a r t - ( Q d u m p + V O 2 ) * F O 2 f e e d - V O 2 Q d u m p ) * e - Q d u m p V l o o p t F_{O_{2}loop}(t)=\frac{(Q_{dump}+V_{O_{2}})*F_{O_{2}feed}-V_{O_{2}}}{Q_{dump}}% +(F_{O_{2}loop}^{start}-\frac{(Q_{dump}+V_{O_{2}})*F_{O_{2}feed}-V_{O_{2}}}{Q_% {dump}})*e^{-\frac{Q_{dump}}{V_{loop}}t}
  27. F O 2 b a g ( t ) = ( Q d u m p + V O 2 ) * F O 2 f e e d - V O 2 Q d u m p F_{O_{2}bag}(t)=\frac{(Q_{dump}+V_{O_{2}})*F_{O_{2}feed}-V_{O_{2}}}{Q_{dump}}
  28. F O 2 l o o p F_{O_{2}loop}
  29. F O 2 l o o p = ( Q d u m p + V O 2 ) F O 2 f e e d - V O 2 Q d u m p F_{O_{2}loop}=\frac{(Q_{dump}+V_{O_{2}})F_{O_{2}feed}-V_{O_{2}}}{Q_{dump}}
  30. Q d u m p Q_{dump}
  31. V O 2 V_{O_{2}}
  32. F O 2 f e e d F_{O_{2}feed}
  33. P a m b P_{amb}
  34. Q d u m p = P a m b * K b e l l o w s * V E Q_{dump}=P_{amb}*K_{bellows}*V_{E}
  35. K b e l l o w s K_{bellows}
  36. V E V_{E}
  37. Q d u m p = P a m b * K b e l l o w s * K E * V O 2 Q_{dump}=P_{amb}*K_{bellows}*K_{E}*V_{O_{2}}
  38. F O 2 l o o p = ( P a m b * K b e l l o w s * K E * V O 2 + V O 2 ) F O 2 f e e d - V O 2 P a m b * K b e l l o w s * K E * V O 2 F_{O_{2}loop}=\frac{(P_{amb}*K_{bellows}*K_{E}*V_{O_{2}}+V_{O_{2}})F_{O_{2}% feed}-V_{O_{2}}}{P_{amb}*K_{bellows}*K_{E}*V_{O_{2}}}
  39. F O 2 l o o p = ( P a m b * K b e l l o w s * K E + 1 ) F O 2 f e e d - 1 P a m b * K b e l l o w s * K E F_{O_{2}loop}=\frac{(P_{amb}*K_{bellows}*K_{E}+1)F_{O_{2}feed}-1}{P_{amb}*K_{% bellows}*K_{E}}

Rec._2020.html

  1. E = { 4.5 E 0 E < β α E 0.45 - ( α - 1 ) β E 1 E^{\prime}=\begin{cases}4.5E&0\leq E<\beta\\ \alpha\,\!E^{0.45}-(\alpha\,\!-1)&\beta\,\!\leq E\leq 1\end{cases}

Reciprocal_distribution.html

  1. f ( x ; a , b ) = 1 x [ log e ( b ) - log e ( a ) ] for a x b and a > 0. f(x;a,b)=\frac{1}{x[\log_{e}(b)-\log_{e}(a)]}\quad\,\text{ for }a\leq x\leq b% \,\text{ and }a>0.
  2. a a
  3. b b
  4. log e \log_{e}
  5. F ( x ; a , b ) = log e ( x ) - log e ( a ) log e ( b ) - log e ( a ) for a x b . F(x;a,b)=\frac{\log_{e}(x)-\log_{e}(a)}{\log_{e}(b)-\log_{e}(a)}\quad\,\text{ % for }a\leq x\leq b.
  6. { x f ( x ) + f ( x ) = 0 , f ( 1 ) = 1 log ( b ) - log ( a ) } \left\{\begin{array}[]{l}xf^{\prime}(x)+f(x)=0,\\ f(1)=\frac{1}{\log(b)-\log(a)}\end{array}\right\}

Recognizable_set.html

  1. N N
  2. S N S\subseteq N
  3. M M
  4. ϕ \phi
  5. N N
  6. M M
  7. S = ϕ - 1 ( ϕ ( S ) ) S=\phi^{-1}(\phi(S))
  8. T T
  9. M M
  10. M M
  11. S S
  12. T T
  13. N S N\setminus S
  14. M T M\setminus T
  15. A A
  16. A * A^{*}
  17. A A
  18. A A
  19. A * A^{*}
  20. \mathbb{N}
  21. N N
  22. REC ( N ) \mathrm{REC}(N)
  23. N N
  24. N N
  25. N N
  26. REC ( N ) \mathrm{REC}(N)
  27. N = 2 N=\mathbb{N}^{2}
  28. S = { ( 1 , 1 ) } S=\{(1,1)\}
  29. S * = { ( i , i ) i } S^{*}=\{(i,i)\mid i\in\mathbb{N}\}
  30. N N
  31. M M
  32. ϕ : N M \phi:N\rightarrow M
  33. S REC ( M ) S\in\mathrm{REC}(M)
  34. ϕ - 1 ( S ) = { x ϕ ( x ) S } REC ( M ) \phi^{-1}(S)=\{x\mid\phi(x)\in S\}\in\mathrm{REC}(M)

Rectifier_(neural_networks).html

  1. f ( x ) = max ( 0 , x ) f(x)=\max(0,x)
  2. f ( x ) = ln ( 1 + e x ) f(x)=\ln(1+e^{x})
  3. f ( x ) = e x / ( e x + 1 ) = 1 / ( 1 + e - x ) f^{\prime}(x)=e^{x}/(e^{x}+1)=1/(1+e^{-x})
  4. f ( x ) = max ( 0 , x + 𝒩 ( 0 , σ ( x ) ) ) f(x)=\max(0,x+\mathcal{N}(0,\sigma(x)))
  5. f ( x ) = { x if x > 0 0.01 x otherwise f(x)=\begin{cases}x&\mbox{if }~{}x>0\\ 0.01x&\mbox{otherwise}\end{cases}

Recursion_(disambiguation).html

  1. a n a_{n}

RecycleUnits.html

  1. P P
  2. U U
  3. u U u\in U
  4. n P n\in P
  5. p p
  6. n n
  7. l l
  8. u u
  9. p = = l p==l
  10. n n
  11. u u
  12. ¬ p = = l \neg p==l
  13. n n
  14. u u
  15. n n
  16. n n
  17. n n
  18. n n
  19. n n
  20. x x
  21. x x
  22. n n
  23. x x
  24. l l
  25. r r
  26. p p
  27. n n
  28. l l
  29. r r
  30. p l . C l a u s e p\in l.Clause
  31. p r . C l a u s e p\in r.Clause
  32. n n
  33. l l
  34. r r
  35. p p
  36. p l . C l a u s e p\in l.Clause
  37. p r . C l a u s e p\notin r.Clause
  38. n n
  39. r r
  40. l l
  41. p r . C l a u s e p\in r.Clause
  42. p l . C l a u s e p\notin l.Clause
  43. n n
  44. l l
  45. r r
  46. x { l , r } x\in\{l,r\}
  47. y { l , r } { x } y\in\{l,r\}\setminus\{x\}
  48. n n
  49. x x
  50. y y
  51. C 8 C_{8}
  52. ( 1 ) ( 2 ) ( 1 ) C 1 ( 1 , 3 ) C 2 ( - 1 , 2 , 5 ) C 3 ( 2 , 3 , 5 ) C 4 ( 1 , - 2 ) C 7 ( 1 , 3 , 5 ) ( 4 ) C 5 ( - 1 , 4 ) C 6 ( - 1 , - 4 ) \color r e d C 8 ( - 1 ) C 9 ( 3 , 5 ) (1)\cfrac{(2)\cfrac{(1)\cfrac{C_{1}(1,3)\qquad C_{2}(-1,2,5)}{C_{3}(2,3,5)}% \qquad C_{4}(1,-2)}{C_{7}(1,3,5)}\qquad(4)\cfrac{C_{5}(-1,4)\qquad C_{6}(-1,-4% )}{\color{red}C_{8}(-1)}}{C_{9}(3,5)}
  53. C 3 C_{3}
  54. ( 1 ) ( 2 ) \color r e d ( 1 ) C 1 ( 1 , 3 ) C 2 ( - 1 , 2 , 5 ) C 3 ( 2 , 3 , 5 ) C 4 ( 1 , - 2 ) C 7 ( 1 , 3 , 5 ) ( 4 ) C 5 ( - 1 , 4 ) C 6 ( - 1 , - 4 ) C 8 ( - 1 ) C 9 ( 3 , 5 ) (1)\cfrac{(2)\cfrac{{\color{red}(1)}\cfrac{C_{1}(1,3)\qquad C_{2}(-1,2,5)}{C_{% 3}(2,3,5)}\qquad C_{4}(1,-2)}{C_{7}(1,3,5)}\qquad(4)\cfrac{C_{5}(-1,4)\qquad C% _{6}(-1,-4)}{C_{8}(-1)}}{C_{9}(3,5)}
  55. C 8 C_{8}
  56. C 8 * {C_{8}}^{*}
  57. C 8 C_{8}
  58. ( 1 ) ( 2 ) ( 1 ) C 1 ( 1 , 3 ) \color r e d C 8 * C 3 ( 2 , 3 , 5 ) C 4 ( 1 , - 2 ) C 7 ( 1 , 3 , 5 ) ( 4 ) C 5 ( - 1 , 4 ) C 6 ( - 1 , - 4 ) C 8 ( - 1 ) C 9 ( 3 , 5 ) (1)\cfrac{(2)\cfrac{(1)\cfrac{C_{1}(1,3)\qquad{\color{red}{C_{8}}^{*}}}{C_{3}(% 2,3,5)}\qquad C_{4}(1,-2)}{C_{7}(1,3,5)}\qquad(4)\cfrac{C_{5}(-1,4)\qquad C_{6% }(-1,-4)}{C_{8}(-1)}}{C_{9}(3,5)}
  59. C 3 C_{3}
  60. C 1 C_{1}
  61. C 8 C_{8}
  62. C 3 C_{3}
  63. C 3 C_{3}
  64. ( 1 ) ( 2 ) ( 1 ) C 1 ( 1 , 3 ) C 8 * C 3 \color r e d ( 3 ) C 4 ( 1 , - 2 ) C 7 ( 1 , 3 , 5 ) ( 4 ) C 5 ( - 1 , 4 ) C 6 ( - 1 , - 4 ) C 8 ( - 1 ) C 9 ( 3 , 5 ) (1)\cfrac{(2)\cfrac{(1)\cfrac{C_{1}(1,3)\qquad{C_{8}}^{*}}{C_{3}{\color{red}(3% )}}\qquad C_{4}(1,-2)}{C_{7}(1,3,5)}\qquad(4)\cfrac{C_{5}(-1,4)\qquad C_{6}(-1% ,-4)}{C_{8}(-1)}}{C_{9}(3,5)}
  65. C 7 C_{7}
  66. C 7 C_{7}
  67. C 7 C_{7}
  68. C 4 C_{4}
  69. C 4 C_{4}
  70. ( 1 ) ( 1 ) C 1 ( 1 , 3 ) C 8 * C 3 ( 3 ) C 7 \color r e d ( 3 ) ( 4 ) C 5 ( - 1 , 4 ) C 6 ( - 1 , - 4 ) C 8 ( - 1 ) C 9 ( 3 , 5 ) (1)\cfrac{\cfrac{(1)\cfrac{C_{1}(1,3)\qquad{C_{8}}^{*}}{C_{3}(3)}}{C_{7}{% \color{red}(3)}}\qquad(4)\cfrac{C_{5}(-1,4)\qquad C_{6}(-1,-4)}{C_{8}(-1)}}{C_% {9}(3,5)}
  71. C 9 C_{9}
  72. ( 1 ) C 1 ( 1 , 3 ) ( 4 ) C 5 ( - 1 , 4 ) C 6 ( - 1 , - 4 ) C 8 ( - 1 ) C 3 ( 3 ) C 7 ( 3 ) C 9 \color r e d ( 3 ) \cfrac{\cfrac{(1)\cfrac{C_{1}(1,3)\qquad(4)\cfrac{C_{5}(-1,4)\qquad C_{6}(-1,-% 4)}{C_{8}(-1)}}{C_{3}(3)}}{C_{7}(3)}}{C_{9}{\color{red}(3)}}
  73. C 8 * {C_{8}}^{*}
  74. C 8 C_{8}

Regression-kriging.html

  1. Z ( 𝐬 ) = m ( 𝐬 ) + ε ( 𝐬 ) + ε ′′ Z(\mathbf{s})=m(\mathbf{s})+\varepsilon^{\prime}(\mathbf{s})+\varepsilon^{% \prime\prime}
  2. z ^ ( 𝐬 0 ) = m ^ ( 𝐬 0 ) + e ^ ( 𝐬 0 ) = k = 0 p β ^ k q k ( 𝐬 0 ) + i = 1 n λ i e ( 𝐬 i ) \hat{z}(\mathbf{s}_{0})=\hat{m}(\mathbf{s}_{0})+\hat{e}(\mathbf{s}_{0})=\sum% \limits_{k=0}^{p}{\hat{\beta}_{k}\cdot q_{k}(\mathbf{s}_{0})}+\sum\limits_{i=1% }^{n}\lambda_{i}\cdot e(\mathbf{s}_{i})
  3. m ^ ( 𝐬 0 ) \hat{m}(\mathbf{s}_{0})
  4. e ^ ( 𝐬 0 ) \hat{e}(\mathbf{s}_{0})
  5. β ^ k \hat{\beta}_{k}
  6. β ^ 0 \hat{\beta}_{0}
  7. λ i \lambda_{i}
  8. e ( 𝐬 i ) e(\mathbf{s}_{i})
  9. 𝐬 i {\mathbf{s}}_{i}
  10. β ^ k \hat{\beta}_{k}
  11. β ^ 𝙶𝙻𝚂 = ( 𝐪 𝐓 𝐂 - 𝟏 𝐪 ) - 𝟏 𝐪 𝐓 𝐂 - 𝟏 𝐳 \mathbf{\hat{\beta}}_{\mathtt{GLS}}=\left(\mathbf{q}^{\mathbf{T}}\cdot\mathbf{% C}^{-\mathbf{1}}\cdot\mathbf{q}\right)^{-\mathbf{1}}\cdot\mathbf{q}^{\mathbf{T% }}\cdot\mathbf{C}^{-\mathbf{1}}\cdot\mathbf{z}
  12. β ^ 𝙶𝙻𝚂 \mathbf{\hat{\beta}}_{\mathtt{GLS}}
  13. 𝐂 \mathbf{C}
  14. 𝐪 {\mathbf{q}}
  15. 𝐳 \mathbf{z}
  16. z ^ 𝚁𝙺 ( 𝐬 0 ) = 𝐪 𝟎 𝐓 β ^ 𝙶𝙻𝚂 + λ 𝟎 𝐓 ( 𝐳 - 𝐪 β ^ 𝙶𝙻𝚂 ) \hat{z}_{\mathtt{RK}}(\mathbf{s}_{0})=\mathbf{q}_{\mathbf{0}}^{\mathbf{T}}% \cdot\mathbf{\hat{\beta}}_{\mathtt{GLS}}+\mathbf{\lambda}_{\mathbf{0}}^{% \mathbf{T}}\cdot(\mathbf{z}-\mathbf{q}\cdot\mathbf{\hat{\beta}}_{\mathtt{GLS}})
  17. z ^ ( 𝐬 0 ) \hat{z}({\mathbf{s}}_{0})
  18. 𝐬 0 {\mathbf{s}}_{0}
  19. 𝐪 𝟎 {\mathbf{q}}_{\mathbf{0}}
  20. p + 1 p+1
  21. λ 𝟎 \mathbf{\lambda}_{\mathbf{0}}
  22. n n
  23. σ ^ 𝚁𝙺 2 ( 𝐬 0 ) = ( C 0 + C 1 ) - 𝐜 𝟎 𝐓 𝐂 𝟏 𝐜 𝟎 + ( 𝐪 𝟎 - 𝐪 𝐓 𝐂 - 𝟏 𝐜 𝟎 ) 𝐓 ( 𝐪 𝐓 𝐂 - 𝟏 𝐪 ) - 𝟏 ( 𝐪 𝟎 - 𝐪 𝐓 𝐂 - 𝟏 𝐜 𝟎 ) \hat{\sigma}_{\mathtt{RK}}^{2}(\mathbf{s}_{0})=(C_{0}+C_{1})-\mathbf{c}_{% \mathbf{0}}^{\mathbf{T}}\cdot\mathbf{C}^{\mathbf{1}}\cdot\mathbf{c}_{\mathbf{0% }}+\left(\mathbf{q}_{\mathbf{0}}-\mathbf{q}^{\mathbf{T}}\cdot\mathbf{C}^{-% \mathbf{1}}\cdot\mathbf{c}_{\mathbf{0}}\right)^{\mathbf{T}}\cdot\left(\mathbf{% q}^{\mathbf{T}}\cdot\mathbf{C}^{-\mathbf{1}}\cdot\mathbf{q}\right)^{\mathbf{-1% }}\cdot\left(\mathbf{q}_{\mathbf{0}}-\mathbf{q}^{\mathbf{T}}\cdot\mathbf{C}^{-% \mathbf{1}}\cdot\mathbf{c}_{\mathbf{0}}\right)
  24. C 0 + C 1 C_{0}+C_{1}
  25. 𝐜 0 {\mathbf{c}}_{0}
  26. 𝐂 \mathbf{C}
  27. z ^ 𝙺𝙴𝙳 ( 𝐬 0 ) = i = 1 n w i 𝙺𝙴𝙳 ( 𝐬 0 ) z ( 𝐬 i ) \hat{z}_{\mathtt{KED}}(\mathbf{s}_{0})=\sum\limits_{i=1}^{n}w_{i}^{\mathtt{KED% }}(\mathbf{s}_{0})\cdot z(\mathbf{s}_{i})
  28. i = 1 n w i 𝙺𝙴𝙳 ( 𝐬 0 ) q k ( 𝐬 i ) = q k ( 𝐬 0 ) \sum\limits_{i=1}^{n}w_{i}^{\mathtt{KED}}(\mathbf{s}_{0})\cdot q_{k}(\mathbf{s% }_{i})=q_{k}(\mathbf{s}_{0})
  29. k = 1 , , p k=1,\ldots,p
  30. z ^ 𝙺𝙴𝙳 ( 𝐬 0 ) = δ 𝟎 𝐓 𝐳 \hat{z}_{\mathtt{KED}}(\mathbf{s}_{0})=\mathbf{\delta}_{\mathbf{0}}^{\mathbf{T% }}\cdot\mathbf{z}
  31. z z
  32. q k q_{k}
  33. ( 𝐬 0 ) ({\mathbf{s}}_{0})
  34. δ 𝟎 {\mathbf{\delta}}_{\mathbf{0}}
  35. w i 𝙺𝙴𝙳 w_{i}^{\mathtt{KED}}
  36. p p
  37. 𝐳 \mathbf{z}
  38. n n
  39. λ 𝟎 𝙺𝙴𝙳 = { w 1 𝙺𝙴𝙳 ( 𝐬 0 ) , , w n 𝙺𝙴𝙳 ( 𝐬 0 ) , φ 0 ( 𝐬 0 ) , , φ p ( 𝐬 0 ) } 𝐓 = 𝐂 𝙺𝙴𝙳 - 1 𝐜 𝟎 𝙺𝙴𝙳 \mathbf{\lambda}_{\mathbf{0}}^{\mathtt{KED}}=\left\{w_{1}^{\mathtt{KED}}(% \mathbf{s}_{0}),\ldots,w_{n}^{\mathtt{KED}}(\mathbf{s}_{0}),\varphi_{0}(% \mathbf{s}_{0}),\ldots,\varphi_{p}(\mathbf{s}_{0})\right\}^{\mathbf{T}}=% \mathbf{C}^{\mathtt{KED}-1}\cdot\mathbf{c}_{\mathbf{0}}^{\mathtt{KED}}
  40. λ 𝟎 𝙺𝙴𝙳 {\mathbf{\lambda}}_{\mathbf{0}}^{\mathtt{KED}}
  41. φ p \varphi_{p}
  42. 𝐂 𝙺𝙴𝙳 {\mathbf{C}}^{\mathtt{KED}}
  43. 𝐜 𝟎 𝙺𝙴𝙳 {\mathbf{c}}_{\mathbf{0}}^{\mathtt{KED}}
  44. 𝐂 𝙺𝙴𝙳 = [ C ( 𝐬 1 , 𝐬 1 ) C ( 𝐬 1 , 𝐬 n ) 1 q 1 ( 𝐬 1 ) q p ( 𝐬 1 ) C ( 𝐬 n , 𝐬 1 ) C ( 𝐬 n , 𝐬 n ) 1 q 1 ( 𝐬 n ) q p ( 𝐬 n ) 1 1 0 0 0 q 1 ( 𝐬 1 ) q 1 ( 𝐬 n ) 0 0 0 0 q p ( 𝐬 1 ) q p ( 𝐬 n ) 0 0 0 ] \mathbf{C}^{\mathtt{KED}}=\left[\begin{array}[]{ccccccc}C(\mathbf{s}_{1},% \mathbf{s}_{1})&\cdots&C(\mathbf{s}_{1},\mathbf{s}_{n})&1&q_{1}(\mathbf{s}_{1}% )&\cdots&q_{p}(\mathbf{s}_{1})\\ \vdots&&\vdots&\vdots&\vdots&&\vdots\\ C(\mathbf{s}_{n},\mathbf{s}_{1})&\cdots&C(\mathbf{s}_{n},\mathbf{s}_{n})&1&q_{% 1}(\mathbf{s}_{n})&\cdots&q_{p}(\mathbf{s}_{n})\\ 1&\cdots&1&0&0&\cdots&0\\ q_{1}(\mathbf{s}_{1})&\cdots&q_{1}(\mathbf{s}_{n})&0&0&\cdots&0\\ \vdots&&\vdots&0&\vdots&&\vdots\\ q_{p}(\mathbf{s}_{1})&\cdots&q_{p}(\mathbf{s}_{n})&0&0&\cdots&0\end{array}\right]
  45. 𝐜 𝟎 𝙺𝙴𝙳 \mathbf{c}_{\mathbf{0}}^{\mathtt{KED}}
  46. 𝐜 𝟎 𝙺𝙴𝙳 = { C ( 𝐬 0 , 𝐬 1 ) , , C ( 𝐬 0 , 𝐬 n ) , q 0 ( 𝐬 0 ) , q 1 ( 𝐬 0 ) , , q p ( 𝐬 0 ) } 𝐓 ; q 0 ( 𝐬 0 ) = 1 \mathbf{c}_{\mathbf{0}}^{\mathtt{KED}}=\left\{C(\mathbf{s}_{0},\mathbf{s}_{1})% ,\ldots,C(\mathbf{s}_{0},\mathbf{s}_{n}),q_{0}(\mathbf{s}_{0}),q_{1}(\mathbf{s% }_{0}),\ldots,q_{p}(\mathbf{s}_{0})\right\}^{\mathbf{T}};q_{0}(\mathbf{s}_{0})=1
  47. 200 \ll 200
  48. z ( 𝐬 i ) z(\mathbf{s}_{i})
  49. i = 1 , , n i=1,\ldots,n
  50. Δ z \Delta z
  51. q ( 𝐬 ) q(\mathbf{s})
  52. z * ( 𝐬 j ) z*(\mathbf{s}_{j})
  53. j = 1 , , l j=1,\ldots,l
  54. C 0 C_{0}
  55. C 1 C_{1}
  56. R R

Regular_tuning.html

  1. n n
  2. 12 - n 12-n

Regularization_perspectives_on_support_vector_machines.html

  1. f : 𝐗 𝐘 f:\mathbf{X}\to\mathbf{Y}
  2. S = { ( x 1 , y 1 ) , , ( x n , y n ) } S=\{(x_{1},y_{1}),\ldots,(x_{n},y_{n})\}
  3. x i x_{i}
  4. y i y_{i}
  5. ± 1 \pm 1
  6. f = arg min f { 1 n i = 1 n V ( y i , f ( x i ) ) + λ || f || 2 } f=\,\text{arg}\min_{f\in\mathcal{H}}\left\{\frac{1}{n}\sum_{i=1}^{n}V(y_{i},f(% x_{i}))+\lambda||f||^{2}_{\mathcal{H}}\right\}
  7. \mathcal{H}
  8. V : 𝐘 × 𝐘 V:\mathbf{Y}\times\mathbf{Y}\to\mathbb{R}
  9. | | | | ||\cdot||_{\mathcal{H}}
  10. λ \lambda\in\mathbb{R}
  11. \mathcal{H}
  12. K : 𝐗 × 𝐗 K:\mathbf{X}\times\mathbf{X}\to\mathbb{R}
  13. n × n n\times n
  14. 𝐊 \mathbf{K}
  15. f ( x i ) = f = 1 n c j 𝐊 i j f(x_{i})=\sum_{f=1}^{n}c_{j}\mathbf{K}_{ij}
  16. || f || 2 = f , f = i = 1 n j = 1 n c i c j K ( x i , x j ) = c T 𝐊 c ||f||^{2}_{\mathcal{H}}=\langle f,f\rangle_{\mathcal{H}}=\sum_{i=1}^{n}\sum_{j% =1}^{n}c_{i}c_{j}K(x_{i},x_{j})=c^{T}\mathbf{K}c
  17. f ( x i ) = y i f(x_{i})=y_{i}
  18. f ( x i ) y i f(x_{i})\neq y_{i}
  19. - y i f ( x i ) -y_{i}f(x_{i})
  20. V ( y i , f ( x i ) ) = ( 1 - y f ( x ) ) + V(y_{i},f(x_{i}))=(1-yf(x))_{+}
  21. ( s ) + = m a x ( s , 0 ) (s)_{+}=max(s,0)
  22. f b ( x ) = { 1 p ( 1 | x ) > p ( - 1 | x ) - 1 p ( 1 | x ) < p ( - 1 | x ) f_{b}(x)=\left\{\begin{matrix}1&p(1|x)>p(-1|x)\\ -1&p(1|x)<p(-1|x)\end{matrix}\right.
  23. V ( y i , f ( x i ) ) = ( 1 - y f ( x ) ) + V(y_{i},f(x_{i}))=(1-yf(x))_{+}
  24. ( s ) + = m a x ( s , 0 ) (s)_{+}=max(s,0)
  25. f = arg min f { 1 n i = 1 n ( 1 - y f ( x ) ) + + λ || f || 2 } f=\,\text{arg}\min_{f\in\mathcal{H}}\left\{\frac{1}{n}\sum_{i=1}^{n}(1-yf(x))_% {+}+\lambda||f||^{2}_{\mathcal{H}}\right\}
  26. 1 / ( 2 λ ) 1/(2\lambda)
  27. f = arg min f { C i = 1 n ( 1 - y f ( x ) ) + + 1 2 || f || 2 } f=\,\text{arg}\min_{f\in\mathcal{H}}\left\{C\sum_{i=1}^{n}(1-yf(x))_{+}+\frac{% 1}{2}||f||^{2}_{\mathcal{H}}\right\}
  28. C = 1 / ( 2 λ n ) C=1/(2\lambda n)

RekenTest.html

  1. 125 + 250 = 125+250=
  2. 1.25 + 2.50 = 1.25+2.50=
  3. 25 % * 100 = 25\%*100=
  4. $ 2.75 + $ 1.25 = \$2.75+\$1.25=
  5. 1 1 2 + 2 3 4 = 1\tfrac{1}{2}+2\tfrac{3}{4}=

Relativistic_angular_momentum.html

  1. 𝐋 = 𝐱 × 𝐩 \mathbf{L}=\mathbf{x}\times\mathbf{p}
  2. L 3 = x 1 p 2 - x 2 p 1 L_{3}=x_{1}p_{2}-x_{2}p_{1}
  3. L 1 = x 2 p 3 - x 3 p 2 L_{1}=x_{2}p_{3}-x_{3}p_{2}
  4. L 2 = x 3 p 1 - x 1 p 3 . L_{2}=x_{3}p_{1}-x_{1}p_{3}\,.
  5. 𝐋 = 𝐱 𝐩 \mathbf{L}=\mathbf{x}\wedge\mathbf{p}
  6. L i j = x i p j - x j p i = 2 x [ i p j ] L^{ij}=x^{i}p^{j}-x^{j}p^{i}=2x^{[i}p^{j]}
  7. 𝐋 \displaystyle\mathbf{L}
  8. 𝐋 = 𝐱 𝐩 \mathbf{L}=\mathbf{x}\wedge\mathbf{p}
  9. 𝐍 = m ( 𝐱 - t 𝐮 ) = m 𝐱 - t 𝐩 \mathbf{N}=m\left(\mathbf{x}-t\mathbf{u}\right)=m\mathbf{x}-t\mathbf{p}
  10. n 𝐍 n = n m n ( 𝐱 n - t 𝐮 n ) = ( 𝐱 com n m n - t n m n 𝐮 n ) \sum_{n}\mathbf{N}_{n}=\sum_{n}m_{n}\left(\mathbf{x}_{n}-t\mathbf{u}_{n}\right% )=\left(\mathbf{x}_{\mathrm{com}}\sum_{n}m_{n}-t\sum_{n}m_{n}\mathbf{u}_{n}\right)
  11. 𝐱 com = n m n 𝐱 n n m n \mathbf{x}_{\mathrm{com}}=\frac{\sum_{n}m_{n}\mathbf{x}_{n}}{\sum_{n}m_{n}}
  12. E = γ ( 𝐯 ) m 0 c 2 , 𝐩 = γ ( 𝐯 ) m 0 𝐯 E=\gamma(\mathbf{v})m_{0}c^{2},\quad\mathbf{p}=\gamma(\mathbf{v})m_{0}\mathbf{v}
  13. m = γ ( 𝐯 ) m 0 , 𝐮 = γ ( 𝐯 ) 𝐯 m=\gamma(\mathbf{v})m_{0},\quad\mathbf{u}=\gamma(\mathbf{v})\mathbf{v}
  14. 𝐍 = m 𝐱 - 𝐩 t = E c 2 𝐱 - 𝐩 t = γ ( 𝐯 ) m 0 ( 𝐱 - 𝐯 t ) \mathbf{N}=m\mathbf{x}-\mathbf{p}t=\frac{E}{c^{2}}\mathbf{x}-\mathbf{p}t=% \gamma(\mathbf{v})m_{0}(\mathbf{x}-\mathbf{v}t)
  15. t = γ ( V ) ( t - V x c 2 ) , E = γ ( V ) ( E - V p x ) t^{\prime}=\gamma(V)\left(t-\frac{Vx}{c^{2}}\right)\,,\quad E^{\prime}=\gamma(% V)\left(E-Vp_{x}\right)
  16. x = γ ( V ) ( x - V t ) , p x = γ ( V ) ( p x - V E c 2 ) x^{\prime}=\gamma(V)(x-Vt)\,,\quad p_{x}^{\prime}=\gamma(V)\left(p_{x}-\frac{% VE}{c^{2}}\right)
  17. y = y , p y = p y y^{\prime}=y\,,\quad p_{y}^{\prime}=p_{y}
  18. z = z , p z = p z z^{\prime}=z\,,\quad p_{z}^{\prime}=p_{z}
  19. L x = y p z - z p y = y p z - z p y = L x L_{x}^{\prime}=y^{\prime}p_{z}^{\prime}-z^{\prime}p_{y}^{\prime}=yp_{z}-zp_{y}% =L_{x}
  20. L y = z p x - x p z = γ ( V ) [ ( z p x - x p z ) + V ( p z t - z E / c 2 ) ] = γ ( V ) [ L y - V ( m z - p z t ) ] L_{y}^{\prime}=z^{\prime}p_{x}^{\prime}-x^{\prime}p_{z}^{\prime}=\gamma(V)[(zp% _{x}-xp_{z})+V(p_{z}t-zE/c^{2})]=\gamma(V)[L_{y}-V(mz-p_{z}t)]
  21. L z = x p y - y p x = γ ( V ) [ ( x p y - y p x ) + V ( y E / c 2 - p y t ) ] = γ ( V ) [ L z + V ( m y - p y t ) ] L_{z}^{\prime}=x^{\prime}p_{y}^{\prime}-y^{\prime}p_{x}^{\prime}=\gamma(V)[(xp% _{y}-yp_{x})+V(yE/c^{2}-p_{y}t)]=\gamma(V)[L_{z}+V(my-p_{y}t)]
  22. N x = m x - p x t = γ ( u ) m 0 ( x - u x t ) N_{x}=mx-p_{x}t=\gamma(u)m_{0}(x-u_{x}t)
  23. N y = m y - p y t = γ ( u ) m 0 ( y - u y t ) N_{y}=my-p_{y}t=\gamma(u)m_{0}(y-u_{y}t)
  24. N z = m z - p z t = γ ( u ) m 0 ( z - u z t ) N_{z}=mz-p_{z}t=\gamma(u)m_{0}(z-u_{z}t)
  25. - V ( m z - p z t ) = V z N x - V x N z = ( 𝐕 × 𝐍 ) y -V(mz-p_{z}t)=V_{z}N_{x}-V_{x}N_{z}=\left(\mathbf{V}\times\mathbf{N}\right)_{y}
  26. V ( m y - p y t ) = V x N y - V y N x = ( 𝐕 × 𝐍 ) z V(my-p_{y}t)=V_{x}N_{y}-V_{y}N_{x}=\left(\mathbf{V}\times\mathbf{N}\right)_{z}
  27. 𝐋 = 𝐋 \mathbf{L}_{\parallel}^{\prime}=\mathbf{L}_{\parallel}
  28. 𝐋 = γ ( 𝐕 ) ( 𝐋 + 𝐕 × 𝐍 ) \mathbf{L}_{\perp}^{\prime}=\gamma(\mathbf{V})\left(\mathbf{L}_{\perp}+\mathbf% {V}\times\mathbf{N}\right)
  29. 𝐋 = 𝐋 + 𝐋 , 𝐋 = 𝐋 + 𝐋 . \mathbf{L}=\mathbf{L}_{\parallel}+\mathbf{L}_{\perp}\,,\quad\mathbf{L}^{\prime% }=\mathbf{L}_{\parallel}^{\prime}+\mathbf{L}_{\perp}^{\prime}\,.
  30. 𝐋 = γ ( 𝐕 ) ( 𝐋 + 𝐕 𝐍 ) \mathbf{L}_{\perp}^{\prime}=\gamma(\mathbf{V})\left(\mathbf{L}_{\perp}+\mathbf% {V}\wedge\mathbf{N}\right)
  31. - V ( m z - p z t ) = V z N x - V x N z = ( 𝐕 𝐍 ) z x -V(mz-p_{z}t)=V_{z}N_{x}-V_{x}N_{z}=\left(\mathbf{V}\wedge\mathbf{N}\right)_{zx}
  32. V ( m y - p y t ) = V x N y - V y N x = ( 𝐕 𝐍 ) x y V(my-p_{y}t)=V_{x}N_{y}-V_{y}N_{x}=\left(\mathbf{V}\wedge\mathbf{N}\right)_{xy}
  33. N x = m x - p x t = γ ( V ) ( m - V p x c 2 ) γ ( V ) ( x - v t ) - γ ( V ) ( p x - V E c 2 ) γ ( V ) ( t - V x c 2 ) = N x N_{x}^{\prime}=m^{\prime}x^{\prime}-p_{x}^{\prime}t^{\prime}=\gamma(V)\left(m-% \frac{Vp_{x}}{c^{2}}\right)\gamma(V)(x-vt)-\gamma(V)\left(p_{x}-\frac{VE}{c^{2% }}\right)\gamma(V)\left(t-\frac{Vx}{c^{2}}\right)=N_{x}
  34. N y = m y - p y t = γ ( V ) ( m - V p x c 2 ) y - p y γ ( V ) ( t - V x c 2 ) = γ ( V ) ( N y + V L z c 2 ) N_{y}^{\prime}=m^{\prime}y^{\prime}-p_{y}^{\prime}t^{\prime}=\gamma(V)\left(m-% \frac{Vp_{x}}{c^{2}}\right)y-p_{y}\gamma(V)\left(t-\frac{Vx}{c^{2}}\right)=% \gamma(V)\left(N_{y}+\frac{VL_{z}}{c^{2}}\right)
  35. N z = m z - p z t = γ ( V ) ( m - V p x c 2 ) z - p z γ ( V ) ( t - V x c 2 ) = γ ( V ) ( N z - V L y c 2 ) N_{z}^{\prime}=m^{\prime}z^{\prime}-p_{z}^{\prime}t^{\prime}=\gamma(V)\left(m-% \frac{Vp_{x}}{c^{2}}\right)z-p_{z}\gamma(V)\left(t-\frac{Vx}{c^{2}}\right)=% \gamma(V)\left(N_{z}-\frac{VL_{y}}{c^{2}}\right)
  36. 𝐍 = 𝐍 \mathbf{N}_{\parallel}^{\prime}=\mathbf{N}_{\parallel}
  37. 𝐍 = γ ( 𝐕 ) ( 𝐍 - 1 c 2 𝐕 × 𝐋 ) \mathbf{N}_{\perp}^{\prime}=\gamma(\mathbf{V})\left(\mathbf{N}_{\perp}-\frac{1% }{c^{2}}\mathbf{V}\times\mathbf{L}\right)
  38. 𝐌 = 𝐗 𝐏 \mathbf{M}=\mathbf{X}\wedge\mathbf{P}
  39. M α β = X α P β - X β P α = 2 X [ α P β ] M^{\alpha\beta}=X^{\alpha}P^{\beta}-X^{\beta}P^{\alpha}=2X^{[\alpha}P^{\beta]}
  40. M i j = x i p j - x j p i = L i j M^{ij}=x^{i}p^{j}-x^{j}p^{i}=L^{ij}
  41. M 0 i = x 0 p i - x i p 0 = c ( t p i - x i E c 2 ) = c ( t p i - m x i ) = γ ( u ) m 0 c ( t u i - x i ) = - c N i M^{0i}=x^{0}p^{i}-x^{i}p^{0}=c\,\left(tp^{i}-x^{i}\frac{E}{c^{2}}\right)=c\,% \left(tp^{i}-mx^{i}\right)=\gamma(u)m_{0}c\,\left(tu^{i}-x^{i}\right)=-cN^{i}
  42. 𝐌 \displaystyle\mathbf{M}
  43. 𝐌 total = n 𝐌 n = n 𝐗 n 𝐏 n . \mathbf{M}_{\mathrm{total}}=\sum_{n}\mathbf{M}_{n}=\sum_{n}\mathbf{X}_{n}% \wedge\mathbf{P}_{n}\,.
  44. M ¯ α β \displaystyle{\bar{M}}^{\alpha\beta}
  45. 𝐯 = s y m b o l ω × 𝐱 \mathbf{v}=symbol{\omega}\times\mathbf{x}
  46. 0 | 𝐯 | = | s y m b o l ω × 𝐱 | < c 0\leq|\mathbf{v}|=|symbol{\omega}\times\mathbf{x}|<c
  47. 𝐋 = 𝐈 \cdotsymbol ω L i = I i j ω j \mathbf{L}=\mathbf{I}\cdotsymbol{\omega}\quad\rightleftharpoons\quad L_{i}=I_{% ij}\omega_{j}
  48. 𝒥 α β γ = ( X α - Y α ) T β γ - ( X β - Y β ) T α γ \mathcal{J}^{\alpha\beta\gamma}=(X^{\alpha}-Y^{\alpha})T^{\beta\gamma}-(X^{% \beta}-Y^{\beta})T^{\alpha\gamma}
  49. 𝒱 \partial\mathcal{V}
  50. 𝒱 \mathcal{V}
  51. J α β = 𝒱 𝒥 α β γ d 3 Σ γ J^{\alpha\beta}=\oint_{\partial\mathcal{V}}\mathcal{J}^{\alpha\beta\gamma}d^{3% }\Sigma_{\gamma}
  52. J α β = 𝒱 𝒥 α β 0 d 3 Σ 0 = 𝒱 [ ( X α - Y α ) T β 0 - ( X β - Y β ) T α 0 ] d x d y d z J^{\alpha\beta}=\oint_{\partial\mathcal{V}}\mathcal{J}^{\alpha\beta 0}d^{3}% \Sigma_{0}=\oint_{\partial\mathcal{V}}[(X^{\alpha}-Y^{\alpha})T^{\beta 0}-(X^{% \beta}-Y^{\beta})T^{\alpha 0}]dxdydz
  53. γ T β γ = 0 \partial_{\gamma}T^{\beta\gamma}=0
  54. γ 𝒥 α β γ = 0 \partial_{\gamma}\mathcal{J}^{\alpha\beta\gamma}=0
  55. 𝒱 γ T β γ c d t d x d y d z = 𝒱 T β γ d 3 Σ γ = 0 \int_{\mathcal{V}}\partial_{\gamma}T^{\beta\gamma}\,cdt\,dx\,dy\,dz=\oint_{% \partial\mathcal{V}}T^{\beta\gamma}d^{3}\Sigma_{\gamma}=0
  56. 𝒱 γ 𝒥 α β γ c d t d x d y d z = 𝒱 𝒥 α β γ d 3 Σ γ = 0 \int_{\mathcal{V}}\partial_{\gamma}\mathcal{J}^{\alpha\beta\gamma}\,cdt\,dx\,% dy\,dz=\oint_{\partial\mathcal{V}}\mathcal{J}^{\alpha\beta\gamma}d^{3}\Sigma_{% \gamma}=0
  57. 𝐏 = ( m 0 c , 0 , 0 , 0 ) \mathbf{P}=(m_{0}c,0,0,0)
  58. X i com = 1 m 0 𝒱 X i T 00 d x d y d z X^{i}\text{com}=\frac{1}{m_{0}}\int_{\partial\mathcal{V}}X^{i}T^{00}dxdydz
  59. 𝐗 com = ( Y 0 , X 1 com , X 2 com , X 3 com ) \mathbf{X}\text{com}=(Y^{0},X^{1}\text{com},X^{2}\text{com},X^{3}\text{com})
  60. S 0 j = 0 , S j k = ϵ j k S , S = 𝒱 ϵ m n ( x m - x m com ) T 0 n d x d y d z S^{0j}=0\,,\quad S^{jk}=\epsilon^{jk\ell}S^{\ell}\,,\quad S^{\ell}=\int_{% \partial\mathcal{V}}\epsilon^{\ell mn}(x^{m}-x^{m}\text{com})T^{0n}dxdydz
  61. 𝐒 = ( S 11 S 12 S 13 S 21 S 22 S 23 S 31 S 32 S 33 ) = ( 0 S x y S x z S y x 0 S y z S z x S z y 0 ) = ( 0 S x y - S z x - S x y 0 S y z S z x - S y z 0 ) \mathbf{S}=\begin{pmatrix}S^{11}&S^{12}&S^{13}\\ S^{21}&S^{22}&S^{23}\\ S^{31}&S^{32}&S^{33}\\ \end{pmatrix}=\begin{pmatrix}0&S_{xy}&S_{xz}\\ S_{yx}&0&S_{yz}\\ S_{zx}&S_{zy}&0\end{pmatrix}=\begin{pmatrix}0&S_{xy}&-S_{zx}\\ -S_{xy}&0&S_{yz}\\ S_{zx}&-S_{yz}&0\end{pmatrix}
  62. S z = S x y = 𝒱 [ ( x - x com ) T 0 y - ( y - y com ) T 0 x ] d x d y d z S_{z}=S_{xy}=\int_{\partial\mathcal{V}}[(x-x\text{com})T^{0y}-(y-y\text{com})T% ^{0x}]dxdydz
  63. S x = S y z = 𝒱 [ ( y - y com ) T 0 z - ( z - z com ) T 0 y ] d x d y d z S_{x}=S_{yz}=\int_{\partial\mathcal{V}}[(y-y\text{com})T^{0z}-(z-z\text{com})T% ^{0y}]dxdydz
  64. S y = S z x = 𝒱 [ ( z - z com ) T 0 x - ( x - x com ) T 0 z ] d x d y d z S_{y}=S_{zx}=\int_{\partial\mathcal{V}}[(z-z\text{com})T^{0x}-(x-x\text{com})T% ^{0z}]dxdydz
  65. S ρ σ = ε μ ν ρ σ U μ S ν S^{\rho\sigma}=\varepsilon^{\mu\nu\rho\sigma}U_{\mu}S_{\nu}
  66. P ν = m 0 U μ P_{\nu}=m_{0}U_{\mu}
  67. U α S α = P α S α = 0 U_{\alpha}S^{\alpha}=P_{\alpha}S^{\alpha}=0
  68. ( X α - Y α ) P α = 0 (X_{\alpha}-Y_{\alpha})P^{\alpha}=0
  69. J μ ν = M μ ν + S μ ν J^{\mu\nu}=M^{\mu\nu}+S^{\mu\nu}
  70. M μ ν = ( X μ - Y μ ) P ν - ( X ν - Y ν ) P μ M^{\mu\nu}=(X^{\mu}-Y^{\mu})P^{\nu}-(X^{\nu}-Y^{\nu})P^{\mu}
  71. X μ - Y μ = 1 m 0 2 P ν J μ ν X^{\mu}-Y^{\mu}=\frac{1}{m_{0}^{2}}P_{\nu}J^{\mu\nu}
  72. S ρ = 1 2 ε λ μ ν ρ U λ J μ ν , S_{\rho}=\frac{1}{2}\varepsilon_{\lambda\mu\nu\rho}U^{\lambda}J^{\mu\nu},
  73. s y m b o l Γ = d 𝐌 d τ = 𝐗 𝐅 symbol{\Gamma}=\frac{d\mathbf{M}}{d\tau}=\mathbf{X}\wedge\mathbf{F}
  74. Γ α β = X α F β - X β F α \Gamma_{\alpha\beta}=X_{\alpha}F_{\beta}-X_{\beta}F_{\alpha}
  75. 𝐚 ^ \hat{\mathbf{a}}
  76. 𝐧 ^ \hat{\mathbf{n}}
  77. φ 𝐚 ^ = φ ( a 1 , a 2 , a 3 ) \varphi\hat{\mathbf{a}}=\varphi(a_{1},a_{2},a_{3})
  78. θ 𝐧 ^ = θ ( n 1 , n 2 , n 3 ) \theta\hat{\mathbf{n}}=\theta(n_{1},n_{2},n_{3})
  79. M 0 a = - M a 0 = K a , M a b = ε a b c J c . M^{0a}=-M^{a0}=K_{a}\,,\quad M^{ab}=\varepsilon_{abc}J_{c}\,.
  80. ω 0 a = - ω a 0 = φ a a , ω a b = θ ε a b c n c , \omega_{0a}=-\omega_{a0}=\varphi a_{a}\,,\quad\omega_{ab}=\theta\varepsilon_{% abc}n_{c}\,,
  81. Λ ( φ , 𝐚 ^ , θ , 𝐧 ^ ) = exp ( - i 4 ω α β M α β ) = exp [ - i 2 ( φ 𝐚 ^ 𝐊 + θ 𝐧 ^ 𝐉 ) ] \Lambda(\varphi,\hat{\mathbf{a}},\theta,\hat{\mathbf{n}})=\exp\left(-\frac{i}{% 4}\omega_{\alpha\beta}M^{\alpha\beta}\right)=\exp\left[-\frac{i}{2}\left(% \varphi\hat{\mathbf{a}}\cdot\mathbf{K}+\theta\hat{\mathbf{n}}\cdot\mathbf{J}% \right)\right]
  82. 𝐀 = Λ ( φ , 𝐚 ^ , θ , 𝐧 ^ ) 𝐀 \mathbf{A}^{\prime}=\Lambda(\varphi,\hat{\mathbf{a}},\theta,\hat{\mathbf{n}})% \mathbf{A}

Relativistic_system_(mathematics).html

  1. Q Q\to\mathbb{R}
  2. \mathbb{R}
  3. Q Q
  4. \mathbb{R}
  5. t t
  6. \mathbb{R}
  7. Q Q
  8. Q = 4 Q=\mathbb{R}^{4}
  9. Q Q
  10. \mathbb{R}
  11. J 1 1 Q J^{1}_{1}Q
  12. Q Q
  13. J 1 1 Q Q J^{1}_{1}Q\to Q
  14. ( q 0 , q i ) (q^{0},q^{i})
  15. Q Q
  16. J 1 1 Q J^{1}_{1}Q
  17. ( q 0 , q i , q 0 i ) (q^{0},q^{i},q^{i}_{0})
  18. q 0 = q 0 ( q 0 , q k ) , q i = q i ( q 0 , q k ) , q 0 i = ( q i q j q 0 j + q i q 0 ) ( q 0 q j q 0 j + q 0 q 0 ) - 1 . q^{\prime 0}=q^{\prime 0}(q^{0},q^{k}),\quad q^{\prime i}=q^{\prime i}(q^{0},q% ^{k}),\quad{q^{\prime}}^{i}_{0}=\left(\frac{\partial q^{\prime i}}{\partial q^% {j}}q^{j}_{0}+\frac{\partial q^{\prime i}}{\partial q^{0}}\right)\left(\frac{% \partial q^{\prime 0}}{\partial q^{j}}q^{j}_{0}+\frac{\partial q^{\prime 0}}{% \partial q^{0}}\right)^{-1}.
  19. × T Q \mathbb{R}\times TQ
  20. ( τ , q λ , a τ λ ) (\tau,q^{\lambda},a^{\lambda}_{\tau})
  21. T Q TQ
  22. Q Q
  23. ( λ G μ α 2 α 2 N 2 N - μ G λ α 2 α 2 N ) q τ μ q τ α 2 q τ α 2 N - ( 2 N - 1 ) G λ μ α 3 α 2 N q τ τ μ q τ α 3 q τ α 2 N + F λ μ q τ μ = 0 , \left(\frac{\partial_{\lambda}G_{\mu\alpha_{2}\ldots\alpha_{2N}}}{2N}-\partial% _{\mu}G_{\lambda\alpha_{2}\ldots\alpha_{2N}}\right)q^{\mu}_{\tau}q^{\alpha_{2}% }_{\tau}\cdots q^{\alpha_{2N}}_{\tau}-(2N-1)G_{\lambda\mu\alpha_{3}\ldots% \alpha_{2N}}q^{\mu}_{\tau\tau}q^{\alpha_{3}}_{\tau}\cdots q^{\alpha_{2N}}_{% \tau}+F_{\lambda\mu}q^{\mu}_{\tau}=0,
  24. G α 1 α 2 N q τ α 1 q τ α 2 N = 1. G_{\alpha_{1}\ldots\alpha_{2N}}q^{\alpha_{1}}_{\tau}\cdots q^{\alpha_{2N}}_{% \tau}=1.
  25. Q Q
  26. G μ ν G_{\mu\nu}

Relaxed_intersection.html

  1. X 1 , , X m X_{1},\dots,X_{m}
  2. R n R^{n}
  3. X { q } = { q } X i X^{\{q\}}=\bigcap^{\{q\}}X_{i}
  4. x R n x\in R^{n}
  5. X i X_{i}
  6. q q
  7. λ ( x ) = card { i | x X i } . \lambda(x)=\,\text{card}\left\{i\ |\ x\in X_{i}\right\}.
  8. X { q } = λ - 1 ( [ m - q , m ] ) . X^{\{q\}}=\lambda^{-1}([m-q,m]).
  9. X 1 = [ 1 , 4 ] , X_{1}=[1,4],
  10. X 2 = [ 2 , 4 ] , X_{2}=\ [2,4],
  11. X 3 = [ 2 , 7 ] , X_{3}=[2,7],
  12. X 4 = [ 6 , 9 ] , X_{4}=[6,9],
  13. X 5 = [ 3 , 4 ] , X_{5}=[3,4],
  14. X 6 = [ 3 , 7 ] . X_{6}=[3,7].
  15. X { 0 } = , X^{\{0\}}=\emptyset,
  16. X { 1 } = [ 3 , 4 ] , X^{\{1\}}=[3,4],
  17. X { 2 } = [ 3 , 4 ] , X^{\{2\}}=[3,4],
  18. X { 3 } = [ 2 , 4 ] [ 6 , 7 ] , X^{\{3\}}=[2,4]\cup[6,7],
  19. X { 4 } = [ 2 , 7 ] , X^{\{4\}}=[2,7],
  20. X { 5 } = [ 1 , 9 ] , X^{\{5\}}=[1,9],
  21. X { 6 } = ] - , [ . X^{\{6\}}=]-\infty,\infty[.
  22. X i X_{i}
  23. λ \lambda
  24. X { q } = λ - 1 ( [ m - q , m ] ) X^{\{q\}}=\lambda^{-1}([m-q,m])
  25. λ ( x ) \lambda(x)
  26. R n R^{n}
  27. X 1 , , X m X_{1},\dots,X_{m}
  28. { q } X i = { m - 1 - q } X i \overset{\{q\}}{\bigcup}X_{i}=\bigcap^{\{m-1-q\}}X_{i}
  29. { 0 } X i = X i \bigcap^{\{0\}}X_{i}=\bigcap X_{i}
  30. { 0 } X i = X i \overset{\{0\}}{\bigcup}X_{i}=\bigcup X_{i}
  31. X ¯ \overline{X}
  32. X i X_{i}
  33. { q } X i ¯ = { q } X i ¯ \overline{\bigcap^{\{q\}}X_{i}}=\overset{\{q\}}{\bigcup}\overline{X_{i}}
  34. { q } X i ¯ = { q } X i ¯ . \overline{\overset{\{q\}}{\bigcup}X_{i}}=\bigcap^{\{q\}}\overline{X_{i}}.
  35. { q } X i ¯ = { m - q - 1 } X i ¯ = { m - q - 1 } X i ¯ \overline{\bigcap\limits^{\{q\}}X_{i}}=\overline{\overset{\{m-q-1\}}{\bigcup}X% _{i}}=\bigcap^{\{m-q-1\}}\overline{X_{i}}
  36. C 1 , , C m C_{1},\dots,C_{m}
  37. X 1 , , X m X_{1},\dots,X_{m}
  38. C ( [ x ] ) = { q } C i ( [ x ] ) . C([x])=\bigcap^{\{q\}}C_{i}([x]).
  39. X { q } X^{\{q\}}
  40. C ¯ ( [ x ] ) = { m - q - 1 } C ¯ i ( [ x ] ) \overline{C}([x])=\bigcap^{\{m-q-1\}}\overline{C}_{i}([x])
  41. X ¯ { q } \overline{X}^{\{q\}}
  42. C ¯ 1 , , C ¯ m \overline{C}_{1},\dots,\overline{C}_{m}
  43. X ¯ 1 , , X ¯ m . \overline{X}_{1},\dots,\overline{X}_{m}.
  44. R n R^{n}
  45. f i ( p ) = 1 2 π p 2 exp ( - ( t i - p 1 ) 2 2 p 2 ) f_{i}(p)=\frac{1}{\sqrt{2\pi p_{2}}}\exp(-\frac{(t_{i}-p_{1})^{2}}{2p_{2}})
  46. p R 2 p\in R^{2}
  47. f i ( p ) [ y i ] f_{i}(p)\in[y_{i}]
  48. t i t_{i}
  49. [ y i ] [y_{i}]
  50. { ( 1 , [ 0 ; 0.2 ] ) , ( 2 , [ 0.3 ; 2 ] ) , ( 3 , [ 0.3 ; 2 ] ) , ( 4 , [ 0.1 ; 0.2 ] ) , ( 5 , [ 0.4 ; 2 ] ) , ( 6 , [ - 1 ; 0.1 ] ) } \{(1,[0;0.2]),(2,[0.3;2]),(3,[0.3;2]),(4,[0.1;0.2]),(5,[0.4;2]),(6,[-1;0.1])\}
  51. λ - 1 ( q ) \lambda^{-1}(q)
  52. q q