wpmath0000015_9

Track_algorithm.html

  1. Capture (3D) { | X ( new ) - X ( s ) | < Separation | Y ( new ) - Y ( s ) | < Separation | Z ( new ) - Z ( s ) | < Separation \,\text{Capture (3D)}\;\begin{cases}\left|X(\,\text{new})-X(s)\right|<\,\text{% Separation}\\ \left|Y(\,\text{new})-Y(s)\right|<\,\text{Separation}\\ \left|Z(\,\text{new})-Z(s)\right|<\,\text{Separation}\end{cases}
  2. Capture (2D) { | X ( new ) - X ( s ) | < Separation | Y ( new ) - Y ( s ) | < Separation \,\text{Capture (2D)}\;\begin{cases}\left|X(\,\text{new})-X(s)\right|<\,\text{% Separation}\\ \left|Y(\,\text{new})-Y(s)\right|<\,\text{Separation}\end{cases}
  3. Lock criteria { ( Δ R Δ T ) - ( C × Doppler Frequency 2 × Transmit Frequency ) < Threshold \,\text{Lock criteria}\;\begin{cases}\mathrm{\left(\frac{\Delta R}{\Delta T}% \right)-\left(\frac{C\times\,\text{Doppler Frequency}}{2\times\,\text{Transmit% Frequency}}\right)<\,\text{Threshold}}\end{cases}
  4. Append { Size > | X s - ( X o + V x Δ T ) | Size > | Y s - ( Y o + V y Δ T ) | Size > | Z s - ( Z o + V z Δ T ) | \,\text{Append}\;\begin{cases}\,\text{Size}>\left|X_{s}-(X_{o}+V_{x}\cdot% \Delta T)\right|\\ \,\text{Size}>\left|Y_{s}-(Y_{o}+V_{y}\cdot\Delta T)\right|\\ \,\text{Size}>\left|Z_{s}-(Z_{o}+V_{z}\cdot\Delta T)\right|\end{cases}
  5. Velocity { V x = ( X 1 - X 0 ) / Δ T V y = ( Y 1 - Y 0 ) / Δ T V z = ( Z 1 - Z 0 ) / Δ T \,\text{Velocity}\;\begin{cases}Vx=(X_{1}-X_{0})/\Delta T\\ Vy=(Y_{1}-Y_{0})/\Delta T\\ Vz=(Z_{1}-Z_{0})/\Delta T\end{cases}

Trade_globalization.html

  1. I m p o r t s + E x p o r t s G D P \frac{Imports+Exports}{GDP}
  2. E x p o r t s G D P \frac{\sum Exports}{\sum GDP}

Trans,polycis-decaprenyl_diphosphate_synthase.html

  1. \rightleftharpoons

Trans,polycis-polyprenyl_diphosphate_synthase_((2Z,6E)-farnesyl_diphosphate_specific).html

  1. \rightleftharpoons

Trans-o-hydroxybenzylidenepyruvate_hydratase-aldolase.html

  1. \rightleftharpoons

Trans-resveratrol_di-O-methyltransferase.html

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

Transcendental_law_of_homogeneity.html

  1. a a
  2. d x dx
  3. a + d x = a a+dx=a
  4. u d v + v d u + d u d v = u d v + v d u , u\,dv+v\,du+du\,dv=u\,dv+v\,du,

Transgression_map.html

  1. \in
  2. \in

Transition_rate_matrix.html

  1. q i i = - j i q j i . q_{ii}=-\sum_{j\neq i}q_{ji}.
  2. j q i j = 0 \sum_{j}q_{ij}=0
  3. Q = ( - λ λ μ - ( μ + λ ) λ μ - ( μ + λ ) λ μ - ( μ + λ ) λ ) T . Q=\begin{pmatrix}-\lambda&\lambda\\ \mu&-(\mu+\lambda)&\lambda\\ &\mu&-(\mu+\lambda)&\lambda\\ &&\mu&-(\mu+\lambda)&\lambda&\\ &&&&\ddots\end{pmatrix}^{T}.

Transshipment_problem.html

  1. i = 1 , , m i=1,\ldots,m
  2. j = 1 , , n j=1,\ldots,n
  3. t r , s t_{r,s}
  4. a i a_{i}
  5. b m + j b_{m+j}
  6. x r , s x_{r,s}
  7. i = 1 m j = 1 n t i , j x i , j \sum\limits_{i=1}^{m}\sum\limits_{j=1}^{n}t_{i,j}x_{i,j}
  8. x r , s 0 x_{r,s}\geq 0
  9. r = 1 m \forall r=1\ldots m
  10. s = 1 n s=1\ldots n
  11. s = 1 m + n x i , s - r = 1 m + n x r , i = a i \sum_{s=1}^{m+n}{x_{i,s}}-\sum_{r=1}^{m+n}{x_{r,i}}=a_{i}
  12. i = 1 m \forall i=1\ldots m
  13. r = 1 m + n x r , m + j - s = 1 m + n x m + j , s = b m + j \sum_{r=1}^{m+n}{x_{r,m+j}}-\sum_{s=1}^{m+n}{x_{m+j,s}}=b_{m+j}
  14. j = 1 n \forall j=1\ldots n
  15. i = 1 m a i = j = 1 n b m + j \sum_{i=1}^{m}{a_{i}}=\sum_{j=1}^{n}{b_{m+j}}
  16. n m n\cdot m
  17. n + m - 2 n+m-2
  18. x r , s = 1 x^{\prime}_{r,s}=1
  19. x r , s x^{\prime}_{r,s}
  20. T i , m + j = r = 1 m + n s = 1 m + n t r , s x r , s T_{i,m+j}=\sum_{r=1}^{m+n}\sum_{s=1}^{m+n}{t_{r,s}\cdot x^{\prime}_{r,s}}
  21. s = 1 m + n x r , s = 1 \sum_{s=1}^{m+n}{x^{\prime}_{r,s}}=1
  22. r = 1 m + n x r , s = 1 \sum_{r=1}^{m+n}{x^{\prime}_{r,s}}=1
  23. x m + j , i = 1 x^{\prime}_{m+j,i}=1
  24. x r , s = 0 , 1 x^{\prime}_{r,s}=0,1
  25. x r , r = 1 x^{\prime}_{r,r}=1
  26. x m + j , i = 1 x^{\prime}_{m+j,i}=1
  27. x m + j , i = 1 x^{\prime}_{m+j,i}=1
  28. x r , r x^{\prime}_{r,r}
  29. x m + j , i = 1 x^{\prime}_{m+j,i}=1
  30. t m + j , i = - M t_{m+j,i}=-M
  31. t i , m + j t^{\prime}_{i,m+j}
  32. x i , m + j 0 x_{i,m+j}\geq 0
  33. z = m a x { t i , m + j : x i , m + j > 0 ( i = 1 m , j = 1 n ) } z=max\left\{t^{\prime}_{i,m+j}:x_{i,m+j}>0\;\;(i=1\ldots m,\;j=1\ldots n)\right\}
  34. i = 1 m x i , m + j = a i \sum_{i=1}^{m}{x_{i,m+j}}=a_{i}
  35. j = 1 n x i , m + j = b m + j \sum_{j=1}^{n}{x_{i,m+j}}=b_{m+j}
  36. i = 1 m a i = j = 1 n b m + j \sum_{i=1}^{m}{a_{i}}=\sum_{j=1}^{n}{b_{m+j}}
  37. { t i , m + j , i = 1 m , j = 1 n } \left\{t^{\prime}_{i,m+j},i=1\ldots m,\;j=1\ldots n\right\}
  38. L k , k = 1 q L_{k},k=1\ldots q
  39. L k L_{k}
  40. t i , m + j t^{\prime}_{i,m+j}
  41. L k L_{k}
  42. L 1 L_{1}
  43. t i , m + j t^{\prime}_{i,m+j}
  44. L 2 L_{2}
  45. M k M_{k}
  46. L k x i , m + j \sum_{L_{k}}{x_{i,m+j}}
  47. α M k - β M k + 1 = { - v e , i f α < 0 v e , i f α > 0 \alpha M_{k}-\beta M_{k+1}=\left\{\begin{array}[]{cc}-ve,&if\;\alpha<0\\ ve,&if\;\alpha>0\end{array}\right.
  48. β \beta
  49. x i , m + j x_{i,m+j}
  50. z 1 = k = 1 q M k L k x i , m + j z_{1}=\sum_{k=1}^{q}{M_{k}}\sum_{L_{k}}{x_{i,m+j}}
  51. i = 1 m x i , m + j = a i \sum_{i=1}^{m}{x_{i,m+j}}=a_{i}
  52. j = 1 n x i , m + j = b m + j \sum_{j=1}^{n}{x_{i,m+j}}=b_{m+j}
  53. i = 1 m a i = j = 1 n b m + j \sum_{i=1}^{m}{a_{i}}=\sum_{j=1}^{n}{b_{m+j}}
  54. α M k - β M k + 1 = { - v e , i f α < 0 v e , i f α > 0 \alpha M_{k}-\beta M_{k+1}=\left\{\begin{array}[]{cc}-ve,&if\;\alpha<0\\ ve,&if\;\alpha>0\end{array}\right.

Tree_accumulation.html

  1. ( A , B , A ) \otimes(A,B,A)

Tree_volume_measurement.html

  1. V = ( h 4 π ) ( r b 2 + r b 4 3 r u 2 3 + r b 2 3 r u 4 3 + r u 2 ) V=\left(\frac{h}{4}\pi\right)\left(r_{b}^{2}+r_{b}^{\frac{4}{3}}r_{u}^{\frac{2% }{3}}+r_{b}^{\frac{2}{3}}r_{u}^{\frac{4}{3}}+r_{u}^{2}\right)

Triacylglycerol_lipase.html

  1. \rightleftharpoons

Triakis_truncated_tetrahedral_honeycomb.html

  1. A ~ 3 {\tilde{A}}_{3}
  2. 3 ¯ \overline{3}

Triangular_network_coding.html

  1. O ( n 3 ) O(n^{3})
  2. n n
  3. O ( n 2 ) O(n^{2})

Tricetin_3',4',5'-O-trimethyltransferase.html

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

Tricin_synthase.html

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

Trihydroxypterocarpan_dimethylallyltransferase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Trimethylamine-corrinoid_protein_Co-methyltransferase.html

  1. \rightleftharpoons

Trimethylamine_monooxygenase.html

  1. \rightleftharpoons

Tritrans,polycis-undecaprenyl-diphosphate_synthase_(geranylgeranyl-diphosphate_specific).html

  1. \rightleftharpoons

TRNA-dihydrouridine16::17_synthase_(NAD(P)+).html

  1. \rightleftharpoons
  2. \rightleftharpoons

TRNA-dihydrouridine20_synthase_(NAD(P)+).html

  1. \rightleftharpoons

TRNA-dihydrouridine20a::20b_synthase_(NAD(P)+).html

  1. \rightleftharpoons
  2. \rightleftharpoons

TRNA-dihydrouridine47_synthase_(NAD(P)+).html

  1. \rightleftharpoons

TRNA-guanine15_transglycosylase.html

  1. \rightleftharpoons

TRNA-intron_endonuclease.html

  1. \rightleftharpoons

TRNA:m4X_modification_enzyme.html

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

TRNA1Val_(adenine37-N6)-methyltransferase.html

  1. \rightleftharpoons

TRNA_(adenine22-N1)-methyltransferase.html

  1. \rightleftharpoons

TRNA_(adenine57-N1::adenine58-N1)-methyltransferase.html

  1. \rightleftharpoons

TRNA_(adenine58-N1)-methyltransferase.html

  1. \rightleftharpoons

TRNA_(adenine9-N1)-methyltransferase.html

  1. \rightleftharpoons

TRNA_(carboxymethyluridine34-5-O)-methyltransferase.html

  1. \rightleftharpoons

TRNA_(cytidine32::guanosine34-2'-O)-methyltransferase.html

  1. \rightleftharpoons

TRNA_(cytidine32::uridine32-2'-O)-methyltransferase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

TRNA_(cytidine34-2'-O)-methyltransferase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

TRNA_(cytidine56-2'-O)-methyltransferase.html

  1. \rightleftharpoons

TRNA_(cytosine34-C5)-methyltransferase.html

  1. \rightleftharpoons

TRNA_(cytosine38-C5)-methyltransferase.html

  1. \rightleftharpoons

TRNA_(guanine10-N2)-dimethyltransferase.html

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

TRNA_(guanine10-N2)-methyltransferase.html

  1. \rightleftharpoons

TRNA_(guanine26-N2)-dimethyltransferase.html

  1. \rightleftharpoons

TRNA_(guanine26-N2::guanine27-N2)-dimethyltransferase.html

  1. \rightleftharpoons

TRNA_(guanine37-N1)-methyltransferase.html

  1. \rightleftharpoons

TRNA_(guanine6-N2)-methyltransferase.html

  1. \rightleftharpoons

TRNA_(guanine9-N1)-methyltransferase.html

  1. \rightleftharpoons

TRNA_(pseudouridine54-N1)-methyltransferase.html

  1. \rightleftharpoons

TRNA_dimethylallyltransferase.html

  1. \rightleftharpoons

TRNAHis_guanylyltransferase.html

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

TRNAMet_cytidine_acetyltransferase.html

  1. \rightleftharpoons

TRNASer_(uridine44-2'-O)-methyltransferase.html

  1. \rightleftharpoons

Tropine_acyltransferase.html

  1. \rightleftharpoons

True_shooting_percentage.html

  1. T S % = P T S 2 ( F G A + 0.44 × F T A ) TS\%=\frac{PTS}{2(FGA+0.44\times FTA)}

Truncated_16-cell_honeycomb.html

  1. F ~ 4 {\tilde{F}}_{4}
  2. B ~ 4 {\tilde{B}}_{4}
  3. D ~ 4 {\tilde{D}}_{4}

Tryptophan_7-halogenase.html

  1. \rightleftharpoons

Tryptophan_N-monooxygenase.html

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

Tryptophan_synthase_(indole-salvaging).html

  1. \rightleftharpoons

Tsen_rank.html

  1. n > d 1 i + + d m i . n>d_{1}^{i}+\cdots+d_{m}^{i}.\,

Tuberculosinol_synthase.html

  1. \rightleftharpoons

Tune_shift_with_amplitude.html

  1. ν \nu
  2. J J
  3. d ν d J \frac{d\nu}{dJ}

Tunstall_coding.html

  1. H ( U ) H(U)
  2. 𝒰 \mathcal{U}
  3. C C
  4. D D
  5. | 𝒰 | |\mathcal{U}|
  6. 𝒰 \mathcal{U}
  7. | D | < C |D|<C
  8. | 𝒰 | |\mathcal{U}|
  9. 𝒰 \mathcal{U}
  10. 3 12 3\over 12
  11. | 𝒰 | = 9 |\mathcal{U}|=9
  12. log 2 ( 9 ) = 4 \lceil\log_{2}(9)\rceil=4
  13. w 1 w_{1}
  14. | 𝒰 | = 9 |\mathcal{U}|=9
  15. 1 3 3 12 = 1 12 {1\over 3}\cdot{3\over 12}={1\over 12}
  16. log 2 ( 12 ) = 4 \lceil\log_{2}(12)\rceil=4
  17. | 𝒰 | - 1 = 8 |\mathcal{U}|-1=8

Two-body_Dirac_equations.html

  1. A μ A_{\mu}
  2. S S
  3. [ ( γ 1 ) μ ( p 1 - A ~ 1 ) μ + m 1 + S ~ 1 ] Ψ = 0 , [(\gamma_{1})_{\mu}(p_{1}-\tilde{A}_{1})^{\mu}+m_{1}+\tilde{S}_{1}]\Psi=0,
  4. [ ( γ 2 ) μ ( p 2 - A ~ 2 ) μ + m 2 + S ~ 2 ] Ψ = 0. [(\gamma_{2})_{\mu}(p_{2}-\tilde{A}_{2})^{\mu}+m_{2}+\tilde{S}_{2}]\Psi=0.
  5. η μ ν = ( - 1 , 1 , 1 , 1 ) \eta_{\mu\nu}=(-1,1,1,1)
  6. p μ = - i x μ p^{\mu}=-i{\partial\over\partial x_{\mu}}
  7. m 2 m_{2}\rightarrow\infty
  8. [ ( γ 1 ) μ ( p 1 - A ~ 1 ) μ + m 1 c + S ~ 1 ] Ψ = 0 , [(\gamma_{1})_{\mu}(p_{1}-\tilde{A}_{1})^{\mu}+m_{1}c+\tilde{S}_{1}]\Psi=0,
  9. [ ( γ 2 ) μ ( p 2 - A ~ 2 ) μ + m 2 c + S ~ 2 ] Ψ = 0. [(\gamma_{2})_{\mu}(p_{2}-\tilde{A}_{2})^{\mu}+m_{2}c+\tilde{S}_{2}]\Psi=0.
  10. p μ = - i x μ p^{\mu}=-i\hbar{\partial\over\partial x_{\mu}}
  11. x , p x,p
  12. x Ψ = 0 x\Psi=0
  13. p Ψ = 0 p\Psi=0
  14. P = p 1 + p 2 = ( w , 0 ) P=p_{1}+p_{2}=(w,\vec{0})
  15. w w
  16. x = x 1 - x 2 x=x_{1}-x_{2}
  17. x x_{\perp}
  18. P P
  19. x μ = ( η μ ν - P μ P ν / P 2 ) x ν , x_{\perp}^{\mu}=(\eta^{\mu\nu}-P^{\mu}P^{\nu}/P^{2})x_{\nu},\,
  20. P μ x μ = 0. P_{\mu}x_{\perp}^{\mu}=0.\,
  21. x = ( 0 , x = x 1 - x 2 ) x_{\perp}=(0,\vec{x}=\vec{x}_{1}-\vec{x}_{2})
  22. P p Ψ = ( - P 0 p 0 + P p ) Ψ = 0. P\cdot p\Psi=(-P^{0}p^{0}+\vec{P}\cdot p)\Psi=0.\,
  23. p p
  24. ( p 1 - p 2 ) / 2 (p_{1}-p_{2})/2
  25. P 0 = w , P = 0 P^{0}=w,\vec{P}=\vec{0}
  26. p 0 p^{0}
  27. p 0 Ψ = 0 p^{0}\Psi=0
  28. S ~ i \tilde{S}_{i}
  29. A ~ i μ \tilde{A}_{i}^{\mu}
  30. γ 1 \gamma_{1}
  31. γ 2 \gamma_{2}
  32. S i S_{i}
  33. A i μ A_{i}^{\mu}
  34. ϕ i ( p , x ) 0 \phi_{i}(p,x)\approx 0
  35. 0 \approx 0
  36. \mathcal{H}
  37. \mathcal{L}
  38. ( p x ˙ - ) (p\dot{x}-\mathcal{L})
  39. ( λ i ) (\lambda_{i})
  40. = p x ˙ - + λ i ϕ i \mathcal{H}=p\dot{x}-\mathcal{L}+\lambda_{i}\phi_{i}
  41. I = d τ ( τ ) = d τ d τ d τ ( τ ) = d τ ( τ ) . I=\int d\tau\mathcal{L(\tau)=}\int d\tau^{\prime}\frac{d\tau}{d\tau^{\prime}}% \mathcal{L(\tau)=}\int d\tau^{\prime}\mathcal{L(\tau}^{\prime}\mathcal{)}.
  42. ( τ ) = - ( m + S ( x ) ) - x ˙ 2 + x ˙ A ( x ) \mathcal{L(\tau)}=-(m+S(x))\sqrt{-\dot{x}^{2}}+\dot{x}\cdot A(x)\,
  43. p = x ˙ = ( m + S ( x ) ) x ˙ - x ˙ 2 + A ( x ) p=\frac{\partial\mathcal{L}}{\partial\dot{x}}=\frac{\mathcal{(}m+S(x))\dot{x}}% {\sqrt{-\dot{x}^{2}}}+A(x)
  44. ( p - A ) 2 + ( m + S ) 2 = 0. (p-A)^{2}+(m+S)^{2}=0.\,
  45. p x ˙ - = 0 , p\cdot\dot{x}-\mathcal{L}=0,\,
  46. = λ [ ( p - A ) 2 + ( m + S ) 2 ] λ ( p 2 + m 2 + Φ ( x , p ) ) . \mathcal{H=\lambda}\left[\left(p-A\right)^{2}+(m+S)^{2}\right]\equiv\lambda(p^% {2}+m^{2}+\Phi(x,p)).
  47. i = p i 2 + m i 2 + Φ i ( x 1 , x 2 , p 1 , p 2 ) 0 , \mathcal{H}_{i}=p_{i}^{2}+m_{i}^{2}+\Phi_{i}(x_{1},x_{2},p_{1},p_{2})\approx 0,\,
  48. = λ 1 [ p 1 2 + m 1 2 + Φ 1 ( x 1 , x 2 , p 1 , p 2 ) ] + λ 2 [ p 2 2 + m 2 2 + Φ 2 ( x 1 , x 2 , p 1 , p 2 ) ] \mathcal{H}=\lambda_{1}[p_{1}^{2}+m_{1}^{2}+\Phi_{1}(x_{1},x_{2},p_{1},p_{2})]% +\lambda_{2}[p_{2}^{2}+m_{2}^{2}+\Phi_{2}(x_{1},x_{2},p_{1},p_{2})]
  49. = λ 1 1 + λ 2 2 , =\lambda_{1}\mathcal{H}_{1}+\lambda_{2}\mathcal{H}_{2},\,
  50. i \mathcal{H}_{i}
  51. \mathcal{H}
  52. ˙ i = { i , } 0 \mathcal{\dot{H}}_{i}=\{\mathcal{H}_{i},\mathcal{H\}\approx}0\,
  53. { O 1 , O 2 } = O 1 x 1 μ O 2 p 1 μ - O 1 p 1 μ O 2 x 1 μ + O 1 x 2 μ O 2 p 2 μ - O 1 p 2 μ O 2 x 2 μ . \{O_{1},O_{2}\}=\frac{\partial O_{1}}{\partial x_{1}^{\mu}}\frac{\partial O_{2% }}{\partial p_{1\mu}}-\frac{\partial O_{1}}{\partial p_{1}^{\mu}}\frac{% \partial O_{2}}{\partial x_{1\mu}}+\frac{\partial O_{1}}{\partial x_{2}^{\mu}}% \frac{\partial O_{2}}{\partial p_{2\mu}}-\frac{\partial O_{1}}{\partial p_{2}^% {\mu}}\frac{\partial O_{2}}{\partial x_{2\mu}}.
  54. ˙ 1 = { 1 , } = λ 1 { 1 , 1 } + { 1 , λ 1 } 2 + λ 2 { 2 , 1 } + { λ 2 , 1 } 2 . \mathcal{\dot{H}}_{1}=\{\mathcal{H}_{1},\mathcal{H\}=}\lambda_{1}\{\mathcal{H}% _{1},\mathcal{H}_{1}\mathcal{\}+}\{\mathcal{H}_{1},\lambda_{1}\mathcal{\}% \mathcal{H}}_{2}\mathcal{+\lambda}_{2}\{\mathcal{H}_{2},\mathcal{H}_{1}% \mathcal{\}+}\{\mathcal{\lambda}_{2},\mathcal{H}_{1}\mathcal{\}H}_{2}.
  55. { 1 , 1 } = 0 \{\mathcal{H}_{1},\mathcal{H}_{1}\}=0
  56. 1 0 \mathcal{H}_{1}\approx 0
  57. 2 0 \mathcal{H}_{2}\approx 0
  58. ˙ 1 λ 2 { 2 , 1 } 0. \mathcal{\dot{H}}_{1}\approx\mathcal{\lambda}_{2}\{\mathcal{H}_{2},\mathcal{H}% _{1}\}\approx 0.\,
  59. Φ 1 = Φ 2 Φ ( x ) \Phi_{1}=\Phi_{2}\equiv\Phi(x_{\perp})
  60. { 2 , 1 } = 0 \{\mathcal{H}_{2},\mathcal{H}_{1}\}=0\,
  61. x x_{\perp}
  62. i 0 i Ψ = 0 \mathcal{H}_{i}\approx 0\rightarrow\mathcal{H}_{i}\Psi=0
  63. 0 Ψ = 0 \mathcal{H}\approx 0\rightarrow\mathcal{H}\Psi=0
  64. { O 1 , O 2 } 1 i [ O 1 , O 2 ] . \{O_{1},O_{2}\}\rightarrow\frac{1}{i}[O_{1},O_{2}].\,
  65. [ 2 , 1 ] = 0 , [\mathcal{H}_{2},\mathcal{H}_{1}]=0,\,
  66. p p
  67. p 1 = p 1 P P 2 P + p , p_{1}=\frac{p_{1}\cdot P}{P^{2}}P+p\,,
  68. p 2 = p 2 P P 2 P - p , p_{2}=\frac{p_{2}\cdot P}{P^{2}}P-p\,,
  69. P p = 0 P\cdot p=0
  70. P P
  71. p 1 p_{1}
  72. p 2 p_{2}
  73. p = ε 2 - P 2 p 1 - ε 1 - P 2 p 2 p=\frac{\varepsilon_{2}}{\sqrt{-P^{2}}}p_{1}-\frac{\varepsilon_{1}}{\sqrt{-P^{% 2}}}p_{2}
  74. ε 1 = - p 1 P - P 2 = - P 2 + p 1 2 - p 2 2 2 - P 2 \varepsilon_{1}=-\frac{p_{1}\cdot P}{\sqrt{-P^{2}}}=-\frac{P^{2}+p_{1}^{2}-p_{% 2}^{2}}{2\sqrt{-P^{2}}}
  75. ε 2 = - p 2 P - P 2 = - P 2 + p 2 2 - p 1 2 2 - P 2 \varepsilon_{2}=-\frac{p_{2}\cdot P}{\sqrt{-P^{2}}}=-\frac{P^{2}+p_{2}^{2}-p_{% 1}^{2}}{2\sqrt{-P^{2}}}
  76. p 1 p_{1}
  77. p 2 p_{2}
  78. P P
  79. 1 Ψ = 0 \mathcal{H}_{1}\Psi=0
  80. 2 Ψ = 0 \mathcal{H}_{2}\Psi=0
  81. ( p 1 2 - p 2 2 ) Ψ = - ( m 1 2 - m 2 2 ) Ψ (p_{1}^{2}-p_{2}^{2})\Psi=-(m_{1}^{2}-m_{2}^{2})\Psi
  82. Ψ \Psi
  83. ε 1 Ψ = - P 2 + m 1 2 - m 2 2 2 - P 2 Ψ \varepsilon_{1}\Psi=\frac{-P^{2}+m_{1}^{2}-m_{2}^{2}}{2\sqrt{-P^{2}}}\Psi
  84. ε 2 Ψ = - P 2 + m 2 2 - m 1 2 2 - P 2 Ψ \varepsilon_{2}\Psi=\frac{-P^{2}+m_{2}^{2}-m_{1}^{2}}{2\sqrt{-P^{2}}}\Psi
  85. Ψ = 0 \mathcal{H}\Psi=0
  86. Φ \Phi
  87. [ P , ] Ψ = 0 [P,\mathcal{H}]\Psi=0
  88. [ P , λ i ] = 0 [P,\lambda_{i}]=0
  89. i Ψ = 0 \mathcal{H}_{i}\Psi=0
  90. P P
  91. Ψ \Psi
  92. P P^{\prime}
  93. P = ( w , 0 ) , P^{\prime}=(w,\vec{0}),
  94. w w
  95. ( P 2 + w 2 ) Ψ = 0 , (P^{2}+w^{2})\Psi=0\,,
  96. Ψ \Psi
  97. ε 1 Ψ = w 2 + m 1 2 - m 2 2 2 w Ψ \varepsilon_{1}\Psi=\frac{w^{2}+m_{1}^{2}-m_{2}^{2}}{2w}\Psi
  98. ε 2 Ψ = w 2 + m 2 2 - m 1 2 2 w Ψ \varepsilon_{2}\Psi=\frac{w^{2}+m_{2}^{2}-m_{1}^{2}}{2w}\Psi
  99. p Ψ = ε 2 p 1 - ε 1 p 2 w Ψ p\Psi=\frac{\varepsilon_{2}p_{1}-\varepsilon_{1}p_{2}}{w}\Psi
  100. p 1 Ψ = ( ε 1 w P + p ) Ψ p_{1}\Psi=\left(\frac{\varepsilon_{1}}{w}P+p\right)\Psi
  101. p 2 Ψ = ( ε 2 w P - p ) Ψ p_{2}\Psi=\left(\frac{\varepsilon_{2}}{w}P-p\right)\Psi
  102. i Ψ = 0 \mathcal{H}_{i}\Psi=0
  103. ε 1 = w 2 + m 1 2 - m 2 2 2 w , \varepsilon_{1}=\frac{w^{2}+m_{1}^{2}-m_{2}^{2}}{2w},
  104. ε 2 = w 2 + m 2 2 - m 1 2 2 w , \varepsilon_{2}=\frac{w^{2}+m_{2}^{2}-m_{1}^{2}}{2w},
  105. p = ε 2 p 1 - ε 1 p 2 w , p=\frac{\varepsilon_{2}p_{1}-\varepsilon_{1}p_{2}}{w},
  106. p 1 = ( ε 1 w P + p ) , p_{1}=\left(\frac{\varepsilon_{1}}{w}P+p\right),
  107. p 2 = ( ε 2 w P - p ) , p_{2}=\left(\frac{\varepsilon_{2}}{w}P-p\right),
  108. P p = 0 P\cdot p=0
  109. p = m 2 p 1 - m 1 p 2 M \vec{p}=\frac{m_{2}\vec{p}_{1}-m_{1}\vec{p}_{2}}{M}
  110. p 1 = m 1 M P + p \vec{p}_{1}=\frac{m_{1}}{M}\vec{P}+\vec{p}
  111. p 2 = m 2 M P + p \vec{p}_{2}=\frac{m_{2}}{M}\vec{P}+\vec{p}
  112. \mathcal{H}
  113. P P
  114. p p
  115. Ψ = { λ 1 [ - ε 1 2 + m 1 2 + p 2 + Φ ( x ) ] + λ 2 [ - ε 2 2 + m 2 2 + p 2 + Φ ( x ) ] } Ψ \mathcal{H}\Psi=\{\lambda_{1}[-\varepsilon_{1}^{2}+m_{1}^{2}+p^{2}+\Phi(x_{% \perp})]+\lambda_{2}[-\varepsilon_{2}^{2}+m_{2}^{2}+p^{2}+\Phi(x_{\perp})]\}\Psi
  116. = ( λ 1 + λ 2 ) [ - b 2 ( - P 2 ; m 1 2 , m 2 2 ) + p 2 + Φ ( x ) ] Ψ = 0 , =(\lambda_{1}+\lambda_{2})[-b^{2}(-P^{2};m_{1}^{2},m_{2}^{2})+p^{2}+\Phi(x_{% \perp})]\Psi=0\,,
  117. b 2 ( - P 2 , m 1 2 , m 2 2 ) = ε 1 2 - m 1 2 = ε 2 2 - m 2 2 = - 1 4 P 2 ( P 4 + 2 P 2 ( m 1 2 + m 2 2 ) + ( m 1 2 - m 2 2 ) 2 ) . b^{2}(-P^{2},m_{1}^{2},m_{2}^{2})=\varepsilon_{1}^{2}-m_{1}^{2}=\varepsilon_{2% }^{2}-m_{2}^{2}\ =-\frac{1}{4P^{2}}(P^{4}+2P^{2}(m_{1}^{2}+m_{2}^{2})+(m_{1}^{% 2}-m_{2}^{2})^{2})\,.
  118. P P
  119. b 2 ( - P 2 , m 1 2 , m 2 2 ) b^{2}(-P^{2},m_{1}^{2},m_{2}^{2})
  120. p p
  121. ( λ 1 + λ 2 ) { p 2 + Φ ( x ) - b 2 ( w 2 , m 1 2 , m 2 2 ) } Ψ = 0 , (\lambda_{1}+\lambda_{2})\left\{p^{2}+\Phi(x_{\perp})-b^{2}(w^{2},m_{1}^{2},m_% {2}^{2})\right\}\Psi=0\,,
  122. b 2 ( w 2 , m 1 2 , m 2 2 ) b^{2}(w^{2},m_{1}^{2},m_{2}^{2})
  123. b 2 ( w 2 , m 1 2 , m 2 2 ) = 1 4 w 2 { w 4 - 2 w 2 ( m 1 2 + m 2 2 ) + ( m 1 2 - m 2 2 ) 2 } . b^{2}(w^{2},m_{1}^{2},m_{2}^{2})=\frac{1}{4w^{2}}\left\{w^{4}-2w^{2}(m_{1}^{2}% +m_{2}^{2})+(m_{1}^{2}-m_{2}^{2})^{2}\right\}\,.
  124. Ψ \Psi
  125. p 2 Ψ = p 2 Ψ p^{2}\Psi=p_{\perp}^{2}\Psi
  126. p = p - p P P / P 2 p_{\perp}=p-p\cdot PP/P^{2}
  127. { p 2 + Φ ( x ) } Ψ = b 2 ( w 2 , m 1 2 , m 2 2 ) Ψ , \{p_{\perp}^{2}+\Phi(x_{\perp})\}\Psi=b^{2}(w^{2},m_{1}^{2},m_{2}^{2})\Psi\,,
  128. p μ p_{\perp}^{\mu}
  129. x μ x_{\perp}^{\mu}
  130. P p = P x = 0 . P\cdot p_{\perp}=P\cdot x_{\perp}=0\,.
  131. P = ( w , 0 ) P=(w,\vec{0})
  132. p = ( 0 , p ) p_{\perp}=(0,\vec{p})
  133. x = ( 0 , x ) x_{\perp}=(0,\vec{x})
  134. { p 2 + Φ ( x ) } Ψ = b 2 ( w 2 , m 1 2 , m 2 2 ) Ψ , \{\vec{p}^{2}+\Phi(\vec{x})\}\Psi=b^{2}(w^{2},m_{1}^{2},m_{2}^{2})\Psi\,,
  135. ( p 2 + 2 μ V ( x ) ) Ψ = 2 μ E Ψ , \left(\vec{p}^{2}+2\mu V(\vec{x})\right)\Psi=2\mu E\Psi\,,
  136. Φ \Phi
  137. ( p 2 + m 2 ) ψ = ( p 2 - ε 2 + m 2 ) ψ = 0 (p^{2}+m^{2})\psi=(\vec{p}^{2}-\varepsilon^{2}+m^{2})\psi=0
  138. ( p 2 - ε 2 + m 2 + 2 m S + S 2 + 2 ε A - A 2 ) ψ = 0 (\vec{p}^{2}-\varepsilon^{2}+m^{2}+2mS+S^{2}+2\varepsilon A-A^{2})\psi=0~{}
  139. m m + S m\rightarrow m+S~{}
  140. ε ε - A \varepsilon\rightarrow\varepsilon-A
  141. Φ \Phi
  142. S ( r ) S(r)
  143. A ( r ) A(r)
  144. Φ = 2 m w S + S 2 + 2 ε w A - A 2 . \Phi=2m_{w}S+S^{2}+2\varepsilon_{w}A-A^{2}.
  145. m w = m 1 m 2 / w , m_{w}=m_{1}m_{2}/w,
  146. ε w = ( w 2 - m 1 2 - m 2 2 ) / 2 w , \varepsilon_{w}=(w^{2}-m_{1}^{2}-m_{2}^{2})/2w,
  147. S S
  148. A A
  149. m w m_{w}
  150. ε w \varepsilon_{w}
  151. ε w 2 - m w 2 = b 2 ( w ) , \varepsilon_{w}^{2}-m_{w}^{2}=b^{2}(w),
  152. A μ A^{\mu}
  153. 𝔭 = ε w P ^ + p , \mathfrak{p}=\varepsilon_{w}\hat{P}+p,
  154. A μ = P ^ μ A ( r ) A^{\mu}=\hat{P}^{\mu}A(r)
  155. r = x 2 , r=\sqrt{x_{\perp}^{2}}\,,
  156. S ( r ) S(r)
  157. = ( 𝔭 - A ) 2 + ( m w + S ) 2 0 , \mathcal{H=}\left(\mathfrak{p-}A\right)^{2}+(m_{w}+S)^{2}\approx 0\,,
  158. = p 2 + Φ - b 2 0 . \mathcal{H=}p_{\perp}^{2}+\Phi-b^{2}\approx 0\,.
  159. A ( r ) A(r)
  160. S ( r ) S(r)
  161. 1 = ( p 1 - A 1 ) 2 + ( m 1 + S 1 ) 2 = p 1 2 + m 1 2 + Φ 1 0 \mathcal{H}_{1}=(p_{1}-A_{1})^{2}+(m_{1}+S_{1})^{2}=p_{1}^{2}+m_{1}^{2}+\Phi_{% 1}\approx 0
  162. 2 = ( p 1 - A 2 ) 2 + ( m 2 + S 2 ) 2 = p 2 2 + m 2 2 + Φ 2 0 , \mathcal{H}_{2}=(p_{1}-A_{2})^{2}+(m_{2}+S_{2})^{2}=p_{2}^{2}+m_{2}^{2}+\Phi_{% 2}\approx 0,
  163. p 1 = ε 1 P ^ + p ; p 2 = ε 2 P ^ - p . p_{1}=\varepsilon_{1}\hat{P}+p;~{}~{}p_{2}=\varepsilon_{2}\hat{P}-p~{}.
  164. π 1 = p 1 - A 1 = [ P ^ ( ε 1 - 𝒜 1 ) + p ] , \pi_{1}=p_{1}-A_{1}=[\hat{P}(\varepsilon_{1}-\mathcal{A}_{1})+p],
  165. π 2 = p 2 - A 2 = [ P ^ ( ε 2 - 𝒜 1 ) - p ] , \pi_{2}=p_{2}-A_{2}=[\hat{P}(\varepsilon_{2}-\mathcal{A}_{1})-p],
  166. M 1 = m 1 + S 1 , M_{1}=m_{1}+S_{1},
  167. M 2 = m 2 + S 2 , M_{2}=m_{2}+S_{2},
  168. 1 = π 1 2 + M 1 2 , \mathcal{H}_{1}=\pi_{1}^{2}+M_{1}^{2},
  169. 2 = π 2 2 + M 2 2 . \mathcal{H}_{2}=\pi_{2}^{2}+M_{2}^{2}.
  170. Φ 1 = Φ 2 Φ ( x ) = - 2 p 1 A 1 + A 1 2 + 2 m 1 S 1 + S 1 2 = - 2 p 2 A 2 + A 2 2 + 2 m 2 S 2 + S 2 2 = 2 ε w A - A 2 + 2 m w S + S 2 , \begin{aligned}\displaystyle\Phi_{1}&\displaystyle=\Phi_{2}\equiv\Phi(x_{\perp% })\\ &\displaystyle=-2p_{1}\cdot A_{1}+A_{1}^{2}+2m_{1}S_{1}+S_{1}^{2}\\ &\displaystyle=-2p_{2}\cdot A_{2}+A_{2}^{2}+2m_{2}S_{2}+S_{2}^{2}\\ &\displaystyle=2\varepsilon_{w}A-A^{2}+2m_{w}S+S^{2},\end{aligned}
  171. P p 0 P\cdot p\approx 0
  172. π 1 2 - p 2 = - ( ε 1 - 𝒜 1 ) 2 = - ε 1 2 + 2 ε w A - A 2 , \pi_{1}^{2}-p^{2}=-\left(\varepsilon_{1}-\mathcal{A}_{1}\right)^{2}=-% \varepsilon_{1}^{2}+2\varepsilon_{w}A-A^{2},
  173. π 2 2 - p 2 = - ( ε 2 - 𝒜 2 ) 2 = - ε 2 2 + 2 ε w A - A 2 , \pi_{2}^{2}-p^{2}=-\left(\varepsilon_{2}-\mathcal{A}_{2}\right)^{2}=-% \varepsilon_{2}^{2}+2\varepsilon_{w}A-A^{2},
  174. M 1 = 2 m 1 2 + 2 m w S + S 2 , M_{1}{}^{2}=m_{1}^{2}+2m_{w}S+S^{2},
  175. M 2 2 = m 2 2 + 2 m w S + S 2 . M_{2}^{2}=m_{2}^{2}+2m_{w}S+S^{2}.
  176. ( π 1 2 + M 1 2 ) ψ = 0 , \left(\pi_{1}^{2}+M_{1}^{2}\right)\psi=0,
  177. ( π 2 2 + M 2 2 ) ψ = 0 , \left(\pi_{2}^{2}+M_{2}^{2}\right)\psi=0,
  178. P p 0 , P\cdot p\approx 0,
  179. ψ = ( p 2 + Φ - b 2 ) ψ = 0. \mathcal{H\psi=}\left(p_{\perp}^{2}+\Phi-b^{2}\right)\mathcal{\psi}=0.
  180. 1 1 1 2 ; γ 51 γ 52 ; γ 1 μ γ 2 μ ; γ 51 γ 1 μ γ 52 γ 2 μ ; σ 1 μ ν σ 2 μ ν , 1_{1}1_{2};\gamma_{51}\gamma_{52};\gamma_{1}^{\mu}\gamma_{2\mu};\gamma_{51}% \gamma_{1}^{\mu}\gamma_{52}\gamma_{2\mu};\sigma_{1\mu\nu}\sigma_{2}^{\mu\nu},
  181. σ i μ ν = 1 2 i [ γ i μ , γ i ν ] ; i = 1 , 2. \sigma_{i\mu\nu}=\frac{1}{2i}[\gamma_{i\mu},\gamma_{i\nu}];i=1,2.
  182. 𝒮 1 ψ = ( cosh ( Δ ) 𝐒 1 + sinh ( Δ ) 𝐒 2 ) ψ = 0 , \mathcal{S}_{1}\psi=(\cosh(\Delta)\mathbf{S}_{1}+\sinh(\Delta)\mathbf{S}_{2})% \psi=0\mathrm{,}
  183. 𝒮 2 ψ = ( cosh ( Δ ) 𝐒 2 + sinh ( Δ ) 𝐒 1 ) ψ = 0 , \mathcal{S}_{2}\psi=(\cosh(\Delta)\mathbf{S}_{2}+\sinh(\Delta)\mathbf{S}_{1})% \psi=0,
  184. Δ \Delta
  185. 𝐒 1 \mathbf{S}_{1}
  186. 𝐒 2 \mathbf{S}_{2}
  187. 𝐒 1 ψ ( 𝒮 10 cosh ( Δ ) + 𝒮 20 sinh ( Δ ) ) ψ = 0 , \mathbf{S}_{1}\psi\equiv(\mathcal{S}_{10}\cosh(\Delta)+\mathcal{S}_{20}\sinh(% \Delta)~{})\psi=0,
  188. 𝐒 2 ψ ( 𝒮 20 cosh ( Δ ) + 𝒮 10 sinh ( Δ ) ) ψ = 0 , \mathbf{S}_{2}\psi\equiv(\mathcal{S}_{20}\cosh(\Delta)+\mathcal{S}_{10}\sinh(% \Delta)~{})\psi=0,
  189. 𝒮 i 0 \mathcal{S}_{i0}
  190. 𝒮 i 0 = i 2 γ 5 i ( γ i p i + m i ) = 0 , \mathcal{S}_{i0}=\frac{i}{\sqrt{2}}\gamma_{5i}(\gamma_{i}\cdot p_{i}+m_{i})=0,
  191. [ 𝒮 1 , 𝒮 2 ] ψ = 0 , [\mathcal{S}_{1},\mathcal{S}_{2}]\psi=0,
  192. [ 𝐒 1 , 𝐒 2 ] ψ = 0 , [\mathbf{S}_{1},\mathbf{S}_{2}]\psi=0,
  193. Δ = Δ ( x ) . \Delta=\Delta(x_{\perp}).
  194. Δ \Delta
  195. Δ \Delta
  196. Δ ( x ) = - 1 1 1 2 ( x ) 2 𝒪 1 , scalar , \Delta_{\mathcal{L}}(x_{\perp})=-1_{1}1_{2}\frac{\mathcal{L}(x_{\perp})}{2}% \mathcal{O}_{1},\,\text{scalar}\mathrm{,}
  197. Δ 𝒢 ( x ) = γ 1 γ 2 𝒢 ( x ) 2 𝒪 1 , vector , \Delta_{\mathcal{G}}(x_{\perp})=\gamma_{1}\cdot\gamma_{2}\frac{\mathcal{G}(x_{% \perp})}{2}\mathcal{O}_{1},\,\text{vector}\mathrm{,}
  198. 𝒪 1 = - γ 51 γ 52 . \mathcal{O}_{1}=-\gamma_{51}\gamma_{52}.
  199. Δ ( x ) = Δ + Δ 𝒢 . \Delta(x_{\perp})=\Delta_{\mathcal{L}}+\Delta_{\mathcal{G}}.
  200. cosh 2 Δ - sinh 2 Δ = 1 \cosh^{2}\Delta-\sinh^{2}\Delta=1
  201. ( μ = / x μ ) , \left(\partial_{\mu}=\partial/\partial x^{\mu}\right),
  202. ( G γ 1 𝒫 2 - E 1 β 1 + M 1 - G i 2 Σ 2 ( β 2 - 𝒢 β 1 ) γ 52 ) ψ = 0 , \big(G\gamma_{1}\cdot\mathcal{P}_{2}-E_{1}\beta_{1}+M_{1}-G\frac{i}{2}\Sigma_{% 2}\cdot\partial(\mathcal{L}\beta_{2}\mathcal{-G}\beta_{1})\gamma_{52}\big)\psi% =0,
  203. ( - G γ 2 𝒫 1 - E 2 β 2 + M 2 + G i 2 Σ 1 ( β 1 - 𝒢 β 2 ) γ 51 ) ψ = 0 , \big(-G\gamma_{2}\cdot\mathcal{P}_{1}-E_{2}\beta_{2}+M_{2}+G\frac{i}{2}\Sigma_% {1}\cdot\partial(\mathcal{L}\beta_{1}\mathcal{-G}\beta_{2})\gamma_{51}\big)% \psi=0,
  204. G = exp 𝒢 , G=\exp\mathcal{G},
  205. β i = - γ i P ^ , \beta_{i}=-\gamma_{i}\cdot\hat{P},
  206. γ i μ = ( η μ ν + P ^ μ P ^ ν ) γ ν i , \gamma_{i\perp}^{\mu}=(\eta^{\mu\nu}+\hat{P}^{\mu}\hat{P}^{\nu})\gamma_{\nu i},
  207. Σ i = γ 5 i β i γ i , \Sigma_{i}=\gamma_{5i}\beta_{i}\gamma_{\perp i},
  208. 𝒫 i p - i 2 Σ i 𝒢 Σ i , i = 1 , 2. \mathcal{P}_{i}\equiv p_{\perp}-\frac{i}{2}\Sigma_{i}\cdot\partial\mathcal{G}% \Sigma_{i},i=1,2.
  209. M i M_{i}
  210. E i E_{i}
  211. m + S m+S
  212. ε - A \varepsilon-A
  213. ( γ 𝐩 - β ( ε - A ) + m + S ) ψ = 0. (\mathbf{\gamma}\cdot\mathbf{p-}\beta(\varepsilon-A)+m+S)\psi=0.
  214. γ 1 p \gamma_{1}\cdot p_{\perp}
  215. Σ i 𝒢 Σ i \Sigma_{i}\cdot\partial\mathcal{G}\Sigma_{i}
  216. , 𝒢 \mathcal{L},\mathcal{G}
  217. M i , E i M_{i},E_{i}
  218. M 1 = m 1 cosh + m 2 sinh , M_{1}=m_{1}\cosh\mathcal{L}+m_{2}\sinh\mathcal{L},
  219. M 2 = m 2 cosh + m 1 sinh , M_{2}=m_{2}\cosh\mathcal{L}+m_{1}\sinh\mathcal{L},
  220. E 1 = ε 1 cosh 𝒢 - ε 2 sinh 𝒢 , E_{1}=\varepsilon_{1}\cosh\mathcal{G}-\varepsilon_{2}\sinh\mathcal{G},
  221. E 2 = ε 2 cosh 𝒢 - ε 1 sinh 𝒢 . E_{2}=\varepsilon_{2}\cosh\mathcal{G}-\varepsilon_{1}\sinh\mathcal{G}.
  222. A ~ 1 μ = ( ( ε 1 - E 1 ) ) P ^ μ + ( 1 - G ) p μ - i 2 G γ 2 γ 2 μ , \tilde{A}_{1}^{\mu}=\big((\varepsilon_{1}-E_{1})\big)\hat{P}^{\mu}+(1-G)p_{% \perp}^{\mu}-\frac{i}{2}\partial G\cdot\gamma_{2}\gamma_{2}^{\mu},
  223. A 2 μ = ( ( ε 2 - E 2 ) ) P ^ μ - ( 1 - G ) p μ + i 2 G γ 1 γ 1 μ , A_{2}^{\mu}=\big((\varepsilon_{2}-E_{2})\big)\hat{P}^{\mu}-(1-G)p_{\perp}^{\mu% }+\frac{i}{2}\partial G\cdot\gamma_{1}\gamma_{1}^{\mu},
  224. P ^ μ ) \hat{P}^{\mu})
  225. P ^ μ ) \hat{P}^{\mu})
  226. S ~ i \tilde{S}_{i}
  227. S ~ 1 = M 1 - m 1 - i 2 G γ 2 , \tilde{S}_{1}=M_{1}-m_{1}-\frac{i}{2}G\gamma_{2}\cdot\partial\mathcal{L},
  228. S ~ 2 = M 2 - m 2 + i 2 G γ 1 . \tilde{S}_{2}=M_{2}-m_{2}+\frac{i}{2}G\gamma_{1}\cdot{\partial}\mathcal{L}{.}
  229. \mathcal{L}
  230. 𝒢 \mathcal{G}
  231. M i 2 = m i 2 + exp ( 2 𝒢 ) ( 2 m w S + S 2 ) , M_{i}^{2}=m_{i}^{2}+\exp(2\mathcal{G)(}2m_{w}S\mathcal{+}S^{2}),
  232. E i 2 = exp ( 2 𝒢 ( 𝒜 ) ) ( ε i - A ) 2 , E_{i}^{2}=\exp(2\mathcal{G(A))(}\varepsilon_{i}-A)^{2},
  233. exp ( ) = exp ( ( S , A ) ) = M 1 + M 2 m 1 + m 2 , \exp(\mathcal{L})=\exp(\mathcal{L}(S,A))=\frac{M_{1}+M_{2}}{m_{1}+m_{2}},
  234. G = exp 𝒢 = exp ( 𝒢 ( A ) ) = 1 ( 1 - 2 A / w ) . G=\exp\mathcal{G=}\exp(\mathcal{G(}A\mathcal{))=}\sqrt{\frac{1}{(1-2A/w)}}.
  235. V V
  236. T T
  237. T + V + V G T = 0 T+V+VGT=0
  238. G G
  239. 𝒯 + Φ + Φ 𝒢 𝒯 = 0 \mathcal{T}+\Phi+\Phi\mathcal{GT}=0
  240. 𝒢 \mathcal{G}
  241. 𝒯 \mathcal{T}
  242. 𝒢 \mathcal{G}
  243. x x_{\perp}

Two-state_vector_formalism.html

  1. Φ | \langle\Phi|
  2. | Ψ |\Psi\rangle

Tychonoff_cube.html

  1. I I
  2. [ 0 , 1 ] [0,1]
  3. κ ω \kappa\geq\omega
  4. κ \kappa
  5. I κ I^{\kappa}
  6. s S I s \prod_{s\in S}I_{s}
  7. κ \kappa
  8. S S
  9. s S s\in S
  10. I s = I I_{s}=I
  11. I ω I^{\omega}
  12. λ κ \lambda\leq\kappa
  13. I λ I^{\lambda}
  14. I κ I^{\kappa}
  15. I κ I^{\kappa}
  16. κ ω \kappa\geq\omega
  17. I κ I^{\kappa}
  18. κ ω \kappa\geq\omega
  19. x I κ x\in I^{\kappa}
  20. κ \kappa
  21. [ 0 , ω 1 ] [0,\omega_{1}]
  22. [ 0 , ω ] [0,\omega]
  23. ω \omega
  24. ω 1 \omega_{1}

Types_of_mesh.html

  1. Skewness = optimal cell size - cell size optimal cell size \,\text{ Skewness }=\frac{\,\text{ optimal cell size - cell size }}{\,\text{% optimal cell size}}
  2. Skewness ( for a quad ) = max [ θ m a x - 90 90 , 90 - θ m i n 90 ] \,\text{ Skewness ( for a quad ) }=\max{\left[\frac{\theta_{max}-90}{90},\frac% {90-\theta_{min}}{90}\right]}
  3. Equiangle Skew = max [ θ m a x - θ e 180 - θ e , θ e - θ m i n θ e ] \,\text{ Equiangle Skew }=\max{\left[\frac{\theta_{max}-\theta_{e}}{180-\theta% _{e}},\frac{\theta_{e}-\theta_{min}}{\theta_{e}}\right]}
  4. θ m a x \theta_{max}\,
  5. θ m i n \theta_{min}\,
  6. θ e \theta_{e}\,

Typical_subspace.html

  1. ρ \rho
  2. ρ = x p X ( x ) | x x | . \rho=\sum_{x}p_{X}\left(x\right)\left|x\right\rangle\left\langle x\right|.
  3. H ¯ ( x n ) \overline{H}\left(x^{n}\right)
  4. H ( X ) H\left(X\right)
  5. p X ( x ) p_{X}\left(x\right)
  6. T δ X n span { | x n : | H ¯ ( x n ) - H ( X ) | δ } , T_{\delta}^{X^{n}}\equiv\,\text{span}\left\{\left|x^{n}\right\rangle:\left|% \overline{H}\left(x^{n}\right)-H\left(X\right)\right|\leq\delta\right\},
  7. H ¯ ( x n ) - 1 n log ( p X n ( x n ) ) , \overline{H}\left(x^{n}\right)\equiv-\frac{1}{n}\log\left(p_{X^{n}}\left(x^{n}% \right)\right),
  8. H ( X ) - x p X ( x ) log p X ( x ) . H\left(X\right)\equiv-\sum_{x}p_{X}\left(x\right)\log p_{X}\left(x\right).
  9. Π ρ , δ n \Pi_{\rho,\delta}^{n}
  10. ρ \rho
  11. Π ρ , δ n x n T δ X n | x n x n | , \Pi_{\rho,\delta}^{n}\equiv\sum_{x^{n}\in T_{\delta}^{X^{n}}}\left|x^{n}\right% \rangle\left\langle x^{n}\right|,
  12. T δ X n T_{\delta}^{X^{n}}
  13. δ \delta
  14. T δ X n { x n : | H ¯ ( x n ) - H ( X ) | δ } . T_{\delta}^{X^{n}}\equiv\left\{x^{n}:\left|\overline{H}\left(x^{n}\right)-H% \left(X\right)\right|\leq\delta\right\}.
  15. Tr { Π ρ , δ n ρ n } 1 - ϵ , \,\text{Tr}\left\{\Pi_{\rho,\delta}^{n}\rho^{\otimes n}\right\}\geq 1-\epsilon,
  16. Tr { Π ρ , δ n } 2 n [ H ( X ) + δ ] , \,\text{Tr}\left\{\Pi_{\rho,\delta}^{n}\right\}\leq 2^{n\left[H\left(X\right)+% \delta\right]},
  17. 2 - n [ H ( X ) + δ ] Π ρ , δ n Π ρ , δ n ρ n Π ρ , δ n 2 - n [ H ( X ) - δ ] Π ρ , δ n , 2^{-n\left[H\left(X\right)+\delta\right]}\Pi_{\rho,\delta}^{n}\leq\Pi_{\rho,% \delta}^{n}\rho^{\otimes n}\Pi_{\rho,\delta}^{n}\leq 2^{-n\left[H\left(X\right% )-\delta\right]}\Pi_{\rho,\delta}^{n},
  18. ϵ , δ > 0 \epsilon,\delta>0
  19. n n
  20. { p X ( x ) , ρ x } x 𝒳 \left\{p_{X}\left(x\right),\rho_{x}\right\}_{x\in\mathcal{X}}
  21. ρ x \rho_{x}
  22. ρ x = y p Y | X ( y | x ) | y x y x | . \rho_{x}=\sum_{y}p_{Y|X}\left(y|x\right)\left|y_{x}\right\rangle\left\langle y% _{x}\right|.
  23. ρ x n \rho_{x^{n}}
  24. x n x 1 x n x^{n}\equiv x_{1}\cdots x_{n}
  25. ρ x n ρ x 1 ρ x n . \rho_{x^{n}}\equiv\rho_{x_{1}}\otimes\cdots\otimes\rho_{x_{n}}.
  26. x n x^{n}
  27. H ¯ ( y n | x n ) \overline{H}\left(y^{n}|x^{n}\right)
  28. H ( Y | X ) H\left(Y|X\right)
  29. p Y | X ( y | x ) p X ( x ) p_{Y|X}\left(y|x\right)p_{X}\left(x\right)
  30. T δ Y n | x n span { | y x n n : | H ¯ ( y n | x n ) - H ( Y | X ) | δ } , T_{\delta}^{Y^{n}|x^{n}}\equiv\,\text{span}\left\{\left|y_{x^{n}}^{n}\right% \rangle:\left|\overline{H}\left(y^{n}|x^{n}\right)-H\left(Y|X\right)\right|% \leq\delta\right\},
  31. H ¯ ( y n | x n ) - 1 n log ( p Y n | X n ( y n | x n ) ) , \overline{H}\left(y^{n}|x^{n}\right)\equiv-\frac{1}{n}\log\left(p_{Y^{n}|X^{n}% }\left(y^{n}|x^{n}\right)\right),
  32. H ( Y | X ) - x p X ( x ) y p Y | X ( y | x ) log p Y | X ( y | x ) . H\left(Y|X\right)\equiv-\sum_{x}p_{X}\left(x\right)\sum_{y}p_{Y|X}\left(y|x% \right)\log p_{Y|X}\left(y|x\right).
  33. Π ρ x n , δ \Pi_{\rho_{x^{n}},\delta}
  34. ρ x n \rho_{x^{n}}
  35. Π ρ x n , δ y n T δ Y n | x n | y x n n y x n n | , \Pi_{\rho_{x^{n}},\delta}\equiv\sum_{y^{n}\in T_{\delta}^{Y^{n}|x^{n}}}\left|y% _{x^{n}}^{n}\right\rangle\left\langle y_{x^{n}}^{n}\right|,
  36. T δ Y n | x n T_{\delta}^{Y^{n}|x^{n}}
  37. T δ Y n | x n { y n : | H ¯ ( y n | x n ) - H ( Y | X ) | δ } . T_{\delta}^{Y^{n}|x^{n}}\equiv\left\{y^{n}:\left|\overline{H}\left(y^{n}|x^{n}% \right)-H\left(Y|X\right)\right|\leq\delta\right\}.
  38. 𝔼 X n { Tr { Π ρ X n , δ ρ X n } } 1 - ϵ , \mathbb{E}_{X^{n}}\left\{\,\text{Tr}\left\{\Pi_{\rho_{X^{n}},\delta}\rho_{X^{n% }}\right\}\right\}\geq 1-\epsilon,
  39. Tr { Π ρ x n , δ } 2 n [ H ( Y | X ) + δ ] , \,\text{Tr}\left\{\Pi_{\rho_{x^{n}},\delta}\right\}\leq 2^{n\left[H\left(Y|X% \right)+\delta\right]},
  40. 2 - n [ H ( Y | X ) + δ ] Π ρ x n , δ Π ρ x n , δ ρ x n Π ρ x n , δ 2 - n [ H ( Y | X ) - δ ] Π ρ x n , δ , 2^{-n\left[H\left(Y|X\right)+\delta\right]}\ \Pi_{\rho_{x^{n}},\delta}\leq\Pi_% {\rho_{x^{n}},\delta}\ \rho_{x^{n}}\ \Pi_{\rho_{x^{n}},\delta}\leq 2^{-n\left[% H\left(Y|X\right)-\delta\right]}\ \Pi_{\rho_{x^{n}},\delta},
  41. ϵ , δ > 0 \epsilon,\delta>0
  42. n n
  43. p X n ( x n ) p_{X^{n}}\left(x^{n}\right)

Tyrosine_ammonia-lyase.html

  1. T A L \xrightarrow{TAL}

Tyrosine_N-monooxygenase.html

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

U-invariant.html

  1. q ( F ) = | F / F 2 | q(F)=\left|{F^{\star}/F^{\star 2}}\right|
  2. u ( E ) n + 1 2 u ( F ) . u(E)\leq\frac{n+1}{2}u(F)\ .
  3. u ( F ) - 2 u ( E ) 3 2 u ( F ) u(F)-2\leq u(E)\leq\frac{3}{2}u(F)

U-rank.html

  1. U n ( p ) U_{n}(p)
  2. U n ( T ) = sup { U n ( p ) : p S ( T ) } U_{n}(T)=\sup\{U_{n}(p):p\in S(T)\}
  3. U n ( T ) < U_{n}(T)<\infty
  4. U 1 ( T ) = 1 U_{1}(T)=1

Ubiquinol_oxidase_(H+-transporting).html

  1. \rightleftharpoons

UDP-2,3-diacylglucosamine_diphosphatase.html

  1. \rightleftharpoons

UDP-2,4-diacetamido-2,4,6-trideoxy-beta-L-altropyranose_hydrolase.html

  1. \rightleftharpoons

UDP-2-acetamido-2-deoxy-ribo-hexuluronate_aminotransferase.html

  1. \rightleftharpoons

UDP-2-acetamido-3-amino-2,3-dideoxy-glucuronate_N-acetyltransferase.html

  1. \rightleftharpoons

UDP-3-O-(3-hydroxymyristoyl)glucosamine_N-acyltransferase.html

  1. \rightleftharpoons

UDP-3-O-acyl-N-acetylglucosamine_deacetylase.html

  1. \rightleftharpoons

UDP-4-amino-4,6-dideoxy-N-acetyl-alpha-D-glucosamine_N-acetyltransferase.html

  1. \rightleftharpoons

UDP-4-amino-4,6-dideoxy-N-acetyl-alpha-D-glucosamine_transaminase.html

  1. \rightleftharpoons

UDP-4-amino-4,6-dideoxy-N-acetyl-beta-L-altrosamine_N-acetyltransferase.html

  1. \rightleftharpoons

UDP-4-amino-4,6-dideoxy-N-acetyl-beta-L-altrosamine_transaminase.html

  1. \rightleftharpoons

UDP-4-amino-4-deoxy-L-arabinose_aminotransferase.html

  1. \rightleftharpoons

UDP-4-amino-4-deoxy-L-arabinose_formyltransferase.html

  1. \rightleftharpoons

UDP-D-xylose:beta-D-glucoside_alpha-1,3-D-xylosyltransferase.html

  1. \rightleftharpoons

UDP-GlcNAc:ribostamycin_N-acetylglucosaminyltransferase.html

  1. \rightleftharpoons

UDP-glucuronic_acid_dehydrogenase.html

  1. \rightleftharpoons

UDP-N,N'-diacetylbacillosamine_2-epimerase_(hydrolysing).html

  1. \rightleftharpoons

UDP-N-acetyl-2-amino-2-deoxyglucuronate_dehydrogenase.html

  1. \rightleftharpoons

UDP-N-acetyl-D-mannosamine_dehydrogenase.html

  1. \rightleftharpoons

UDP-N-acetylgalactosamine_diphosphorylase.html

  1. \rightleftharpoons

UDP-N-acetylglucosamine_2-epimerase_(hydrolysing).html

  1. \rightleftharpoons

UDP-N-acetylglucosamine_4,6-dehydratase_(configuration-inverting).html

  1. \rightleftharpoons

UDP-N-acetylglucosamine_kinase.html

  1. \rightleftharpoons

UDP-N-acetylglucosamine—decaprenyl-phosphate_N-acetylglucosaminephosphotransferase.html

  1. \rightleftharpoons

UDP-N-acetylglucosamine—undecaprenyl-phosphate_N-acetylglucosaminephosphotransferase.html

  1. \rightleftharpoons

Umbellic_acid.html

  1. P A L \xrightarrow{PAL}
  2. C 4 H \xrightarrow{C4H}
  3. C 2 H \xrightarrow{C2H}
  4. \longrightarrow

Undecaprenyl-diphosphooligosaccharide-protein_glycotransferase.html

  1. \rightleftharpoons

Undecaprenyl-phosphate_4-deoxy-4-formamido-L-arabinose_transferase.html

  1. \rightleftharpoons

Undecaprenyl-phosphate_glucose_phosphotransferase.html

  1. \rightleftharpoons

Undecaprenyl_phosphate_N,N'-diacetylbacillosamine_1-phosphate_transferase.html

  1. \rightleftharpoons

Unfolding_(DSP_implementation).html

  1. y ( n ) = a y ( n - 9 ) + x ( n ) \scriptstyle y(n)=ay(n-9)+x(n)
  2. n \scriptstyle n
  3. 2 k \scriptstyle 2k
  4. y ( 2 k ) = a y ( 2 k - 9 ) + x ( 2 k ) \scriptstyle y(2k)=ay(2k-9)+x(2k)
  5. n \scriptstyle n
  6. 2 k + 1 \scriptstyle 2k+1
  7. y ( 2 k + 1 ) = a y ( 2 k - 8 ) + x ( 2 k + 1 ) \scriptstyle y(2k+1)=ay(2k-8)+x(2k+1)
  8. x \scriptstyle x
  9. y \scriptstyle y
  10. y ( 2 k ) = a y ( 2 k - 9 ) + x ( 2 k ) \scriptstyle y(2k)=ay(2k-9)+x(2k)
  11. y ( 2 k + 1 ) = a y ( 2 k - 8 ) + x ( 2 k + 1 ) \scriptstyle y(2k+1)=ay(2k-8)+x(2k+1)
  12. i + w J \scriptstyle\lfloor\frac{i+w}{J}\rfloor
  13. 37 + 1 4 = 9 \scriptstyle\lfloor\frac{37+1}{4}\rfloor=9
  14. 37 + 3 4 = 10 \scriptstyle\lfloor\frac{37+3}{4}\rfloor=10
  15. w J + w + 1 J + w + 2 J w + J - 1 J = w \lfloor\frac{w}{J}\rfloor+\lfloor\frac{w+1}{J}\rfloor+\lfloor\frac{w+2}{J}% \rfloor...\lfloor\frac{w+J-1}{J}\rfloor=w

Uniformly_bounded_representation.html

  1. f x , y ( g ) = ( T g - 1 x , T g - 1 y ) , \displaystyle{f_{x,y}(g)=(T_{g}^{-1}x,T_{g}^{-1}y),}
  2. ( x , y ) 0 = φ ( f x , y ) \displaystyle{(x,y)_{0}=\varphi(f_{x,y})}
  3. M - 1 x x 0 M x \displaystyle{M^{-1}\|x\|\leq\|x\|_{0}\leq M\|x\|}
  4. M = sup g T g < . \displaystyle{M=\sup_{g}\|T_{g}\|<\infty.}
  5. ( x , y ) 0 = ( P x , y ) . \displaystyle{(x,y)_{0}=(Px,y).}
  6. ( T g x , T g y ) 0 = ( x , y ) 0 . \displaystyle{(T_{g}x,T_{g}y)_{0}=(x,y)_{0}.}
  7. ( S T g x , S T g y ) = ( P T g x , T g y ) = ( P x , y ) = ( S x , S y ) . \displaystyle{(ST_{g}x,ST_{g}y)=(PT_{g}x,T_{g}y)=(Px,y)=(Sx,Sy).}
  8. ( S T g S - 1 x , S T g S - 1 y ) = ( x , y ) . \displaystyle{(ST_{g}S^{-1}x,ST_{g}S^{-1}y)=(x,y).}
  9. U g = S T g S - 1 \displaystyle{U_{g}=ST_{g}S^{-1}}
  10. k σ ( s ) = ( 1 - cos s ) σ - 1 / 2 . \displaystyle{k_{\sigma}(s)=(1-\cos s)^{\sigma-1/2}.}
  11. ( f , g ) σ f g , \displaystyle{(f,g)_{\sigma}\leq\|f\|\cdot\|g\|,}
  12. f m ( t ) = e i m t \displaystyle{f_{m}(t)=e^{imt}}
  13. ( f m , f m ) σ = i = 1 | m | i - 1 / 2 - σ i - 1 / 2 + σ = Γ ( 1 / 2 + σ ) Γ ( | m | + 1 / 2 - σ ) Γ ( 1 / 2 - σ ) Γ ( m + 1 / 2 + σ ) . \displaystyle{(f_{m},f_{m})_{\sigma}=\prod_{i=1}^{|m|}{i-1/2-\sigma\over i-1/2% +\sigma}={\Gamma(1/2+\sigma)\Gamma(|m|+1/2-\sigma)\over\Gamma(1/2-\sigma)% \Gamma(m+1/2+\sigma)}.}
  14. ( F , G ) σ = - - F ( x ) G ( y ) ¯ | x - y | 2 σ - 1 d x d y . \displaystyle{(F,G)_{\sigma}^{\prime}=\int_{-\infty}^{\infty}\int_{-\infty}^{% \infty}F(x)\overline{G(y)}|x-y|^{2\sigma-1}\,dx\,dy.}
  15. ( F , G ) σ = C σ - F ^ ( t ) G ^ ( t ) ¯ | t | - 2 σ d t . \displaystyle{(F,G)_{\sigma}^{\prime}=C_{\sigma}\int_{-\infty}^{\infty}% \widehat{F}(t)\overline{\widehat{G}(t)}|t|^{-2\sigma}\,dt.}
  16. U f ( x ) = 2 σ / 2 - 3 / 4 π - 1 | x + i | 1 - 2 σ f ( x - i x + i ) . \displaystyle{Uf(x)=2^{\sigma/2-3/4}\pi^{-1}|x+i|^{1-2\sigma}f\left({x-i\over x% +i}\right).}
  17. U * F ( e i t ) = 2 3 / 4 - σ / 2 π | 1 - e i t | 1 - 2 σ F ( 1 + e i t 1 - e i t ) . \displaystyle{U^{*}F(e^{it})=2^{3/4-\sigma/2}\pi|1-e^{it}|^{1-2\sigma}F\left({% 1+e^{it}\over 1-e^{it}}\right).}
  18. g = ( α β β ¯ α ¯ ) , \displaystyle{g=\begin{pmatrix}\alpha&\beta\\ \overline{\beta}&\overline{\alpha}\end{pmatrix},}
  19. | α | 2 - | β | 2 = 1 , \displaystyle{|\alpha|^{2}-|\beta|^{2}=1,}
  20. π σ ( g - 1 ) f ( z ) = | β ¯ z + α ¯ | 1 - 2 σ f ( α z + β β ¯ z + α ¯ ) . \displaystyle{\pi_{\sigma}(g^{-1})f(z)=|\overline{\beta}z+\overline{\alpha}|^{% 1-2\sigma}f\left({\alpha z+\beta\over\overline{\beta}z+\overline{\alpha}}% \right).}
  21. g = ( a b c d ) , \displaystyle{g^{\prime}=\begin{pmatrix}a&b\\ c&d\end{pmatrix},}
  22. π σ ( ( g ) - 1 ) F ( x ) = | c x + d | 1 - 2 σ F ( a x + b c x + d ) . \displaystyle{\pi^{\prime}_{\sigma}((g^{\prime})^{-1})F(x)=|cx+d|^{1-2\sigma}F% \left({ax+b\over cx+d}\right).}
  23. U π σ ( g ) U * = π σ ( g ) . \displaystyle{U\pi_{\sigma}(g)U^{*}=\pi^{\prime}_{\sigma}(g^{\prime}).}
  24. π s ( g - 1 ) f ( z ) = | β ¯ z + α ¯ | 1 - 2 s f ( α z + β β ¯ z + α ¯ ) . \displaystyle{\pi_{s}(g^{-1})f(z)=|\overline{\beta}z+\overline{\alpha}|^{1-2s}% f\left({\alpha z+\beta\over\overline{\beta}z+\overline{\alpha}}\right).}
  25. π s ( ( g ) - 1 ) F ( x ) = | c x + d | 1 - 2 s F ( a x + b c x + d ) . \displaystyle{\pi^{\prime}_{s}((g^{\prime})^{-1})F(x)=|cx+d|^{1-2s}F\left({ax+% b\over cx+d}\right).}
  26. π s ( L 0 ) f m = m f m , π s ( L - 1 ) f m = - ( m + 1 / 2 + s ) f m + 1 , π s ( L 1 ) f m = - ( m - 1 / 2 - s ) f m - 1 . \displaystyle{\pi_{s}(L_{0})f_{m}=mf_{m},\,\,\pi_{s}(L_{-1})f_{m}=-(m+1/2+s)f_% {m+1},\,\,\pi_{s}(L_{1})f_{m}=-(m-1/2-s)f_{m-1}.}
  27. L i = T L i T - 1 \displaystyle{L_{i}^{\prime}=TL_{i}T^{-1}}
  28. f m = T f m = λ m f m . \displaystyle{f_{m}^{\prime}=Tf_{m}=\lambda_{m}f_{m}.}
  29. [ L m , L n ] = ( m - n ) L m + n , ( L i ) * = L - i . \displaystyle{[L_{m}^{\prime},L_{n}^{\prime}]=(m-n)L_{m+n}^{\prime},\,\,(L_{i}% ^{\prime})^{*}=L_{-i}^{\prime}.}
  30. v 1 = L - 1 v 0 . \displaystyle{v_{1}=L^{\prime}_{-1}v_{0}.}
  31. L 1 v 1 = c v 0 \displaystyle{L^{\prime}_{1}v_{1}=cv_{0}}
  32. v 1 2 = ( L - 1 v 0 , v 1 ) = ( v 0 , L 1 v 1 ) = c ¯ v 0 2 . \displaystyle{\|v_{1}\|^{2}=(L_{-1}^{\prime}v_{0},v_{1})=(v_{0},L_{1}^{\prime}% v_{1})=\overline{c}\|v_{0}\|^{2}.}
  33. c = 1 4 - s 2 = 1 4 - σ 2 + τ 2 - 2 i σ τ , \displaystyle{c={1\over 4}-s^{2}={1\over 4}-\sigma^{2}+\tau^{2}-2i\sigma\tau,}
  34. ( x , y ) 1 = H \ G ( g x , g y ) d g . \displaystyle{(x,y)_{1}=\int_{H\backslash G}(gx,gy)\,dg.}
  35. \ell
  36. T e 1 = 0 , T e g = e g , \displaystyle{Te_{1}=0,\,\,Te_{g}=e_{g^{\prime}},}
  37. π z ( g ) 1 + | z | 1 - | z | . \displaystyle{\|\pi_{z}(g)\|\leq{1+|z|\over 1-|z|}.}
  38. π z ( g ) f = λ ( g ) f + n = 0 z n + 1 T n [ T , λ ( g ) ] f 1 + 2 n = 0 n | z | n + 1 = 1 + | z | 1 - | z | . \displaystyle{\|\pi_{z}(g)f\|=\|\lambda(g)f+\sum_{n=0}^{\infty}z^{n+1}T^{n}[T,% \lambda(g)]f\|\leq 1+2\sum_{n=0}^{n}|z|^{n+1}={1+|z|\over 1-|z|}.}
  39. D f = ( 3 z + z - 1 ) f \displaystyle{Df=(3z+z^{-1})f}
  40. f ( 1 ) = 1 , f ( g ) = 3 4 ( 3 z ) - L ( g ) ( g 1 ) . \displaystyle{f(1)=1,\,\,f(g)={3\over 4}(3z)^{-L(g)}\,\,(g\neq 1).}
  41. G G
  42. N N
  43. G / N G/N
  44. G G
  45. G G
  46. N N
  47. G / N G/N

United_Kingdom_Model_for_End-Stage_Liver_Disease.html

  1. ( 5.395 × ln I N R ) + ( 1.485 × ln c r e a t i n i n e ) + ( 3.13 × ln b i l i r u b i n ) - ( 81.565 × ln N a ) + 435 (5.395\times\ln INR)+(1.485\times\ln creatinine)+(3.13\times\ln bilirubin)-(81% .565\times\ln Na)+435

Universal_geometric_algebra.html

  1. 𝒢 ( n , n ) \mathcal{G}(n,n)
  2. 2 n 2n
  3. n = n=∞
  4. 𝒢 ( , ) \mathcal{G}(\infty,\infty)
  5. 𝒢 ( p , q ) \mathcal{G}(p,q)
  6. r r
  7. r = 0 r=0
  8. r = 1 r=1
  9. r = 2 r=2
  10. I I
  11. a I = 0 a\wedge I=0
  12. I I
  13. I I
  14. n n
  15. ( n + 1 ) (n+1)
  16. x M n : I n ( x ) = x A ( x ) I n ( x ) M n = x \forall x\in M^{n}:\exists I_{n}(x)=x\wedge A(x)\mid I_{n}(x)M_{n}=x
  17. M < s u p > n M<sup>n

Unsaturated_chondroitin_disaccharide_hydrolase.html

  1. \rightleftharpoons

Unsaturated_rhamnogalacturonyl_hydrolase.html

  1. \rightleftharpoons

Unspecific_peroxygenase.html

  1. \rightleftharpoons

Uracil::thymine_dehydrogenase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Uracil_dehydrogenase.html

  1. \rightleftharpoons

Uroporphyrinogen-III_C-methyltransferase.html

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

Vacancy_(Kylee_song).html

  1. \cdot
  2. \cdot
  3. \cdot
  4. \cdot
  5. \cdot
  6. \cdot
  7. \cdot
  8. \cdot
  9. \cdot
  10. \cdot
  11. \cdot
  12. \cdot

Vague_set.html

  1. V V
  2. t v ( x ) t_{v}(x)
  3. f v ( x ) f_{v}(x)
  4. 0 t v ( x ) + f v ( x ) 1 0\leq t_{v}(x)+f_{v}(x)\leq 1
  5. [ t v ( x ) , 1 - f v ( x ) ] [t_{v}(x),1-f_{v}(x)]
  6. 1 - f v ( x ) = t v ( x ) 1-f_{v}(x)=t_{v}(x)
  7. ( 1 - f v ( x ) ) - t v ( x ) (1-f_{v}(x))-t_{v}(x)

Valine_dehydrogenase_(NAD+).html

  1. \rightleftharpoons

Valine_N-monooxygenase.html

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

Variable_elimination.html

  1. p ( X | E = e ) p(X|E=e)
  2. X X
  3. E E
  4. U U
  5. E E
  6. e e
  7. v v
  8. ϕ \phi
  9. ϕ \phi
  10. v v
  11. v v
  12. ϕ \phi
  13. Φ \Phi
  14. v v
  15. ϕ \phi
  16. Ψ \Psi
  17. Φ \Phi
  18. τ = v Ψ \tau=\sum_{v}\Psi
  19. ( ϕ - Ψ ) { τ } (\phi-\Psi)\cup\{\tau\}
  20. p ( X | E = e ) p(X|E=e)
  21. ϕ \phi
  22. X X
  23. E E
  24. e e
  25. σ \sigma
  26. U - X E U-XE
  27. X E XE
  28. X E X\cup E
  29. ϕ , X , E , e , σ \phi,X,E,e,\sigma
  30. v v
  31. σ \sigma
  32. ϕ \phi
  33. ( v , ϕ ) (v,\phi)
  34. p ( X , E = e ) p(X,E=e)
  35. Ψ ϕ \Psi\in\phi
  36. p ( X , E = e ) / X p ( X , E = e ) p(X,E=e)/\sum_{X}p(X,E=e)

Variable_geometry_turbomachine.html

  1. N N
  2. Q Q
  3. D D
  4. ( a , b a n d c ) (a,b\,and\,c)
  5. ( a , b a n d c ) (a,b\,and\,c)
  6. β \beta
  7. ψ = f 4 ( ϕ , β ) , \psi\ =f_{4}(\phi,\beta),\,
  8. η = f 5 ( ϕ , β ) , \eta\ =f_{5}(\phi,\beta),\,
  9. ϕ = ( Q N D 3 ) . \phi\ =\!\left({Q\over{ND^{3}}}\right).\,
  10. β = f 6 ( ψ , ϕ ) , \beta\ =f_{6}(\psi,\phi),\,
  11. β = f 7 ( η , ϕ ) , \beta\ =f_{7}(\eta,\phi),\,
  12. β \beta
  13. η = f 8 ( ϕ , ψ ) = f 8 ( Q N D 3 , g H N 2 D 2 ) . \eta\ =f_{8}(\phi,\psi)=f_{8}\!\left({Q\over{ND^{3}}},{gH\over{N^{2}D^{2}}}% \right).\,

Variable_neighborhood_search.html

  1. min { f ( x ) | x X , X S } \min{\{f(x)|x\in X,X\subseteq S\}}
  2. S = R n {S=R^{n}}
  3. x * X {x^{*}\in X}
  4. f ( x * ) f ( x ) , x X {f(x^{*})\leq f(x),\qquad\forall{x}\,\in X}
  5. X = X=\varnothing
  6. x * x^{*}
  7. f ( x * ) f ( x ) + ϵ , x X {f(x^{*})\leq f(x)+\epsilon,\qquad\forall{x}\,\in X}
  8. ( f ( x * ) - f ( x ) ) / f ( x * ) < ϵ , x X {(f(x^{*})-f(x))/f(x^{*})<\epsilon,\qquad\forall{x}\,\in X}
  9. x h x_{h}
  10. ( f ( x h ) - f ( x ) ) / f ( x h ) ϵ , x X {(f(x_{h})-f(x))/f(x_{h})\leq\epsilon,\qquad\forall{x}\,\in X}
  11. ϵ \epsilon
  12. x L x_{L}
  13. f ( x L ) f ( x ) , x N ( x L ) X {f(x_{L})\leq f(x),\qquad\forall{x}\,\in N(x_{L})\cap X}
  14. N ( x L ) N(x_{L})
  15. x L x_{L}
  16. x i N ( x ) x^{i}\in N(x)
  17. 𝒩 k ( k = 1 , , k m a x ) \mathcal{N}_{k}(k=1,...,k_{max})
  18. 𝒩 k ( x ) \mathcal{N}_{k}(x)
  19. 𝒩 k ( x ) , k = 1 , , k m a x \mathcal{N^{\prime}}_{k}(x),k=1,...,k^{\prime}_{max}
  20. 𝒩 k ( x ) \mathcal{N}_{k}(x)
  21. 𝒩 k ( x ) \mathcal{N^{\prime}}_{k}(x)
  22. x o p t x_{opt}
  23. 𝒩 k ( x ) \mathcal{N}_{k}(x)
  24. x 𝒩 k ( x ) X x\in\mathcal{N^{\prime}}_{k}(x)\subseteq X
  25. f ( x ) < f ( x ) f(x)<f(x^{\prime})
  26. 𝒩 k \mathcal{N}_{k}
  27. k m a x k_{max}
  28. 𝒩 k ( x ) \mathcal{N}_{k}(x)
  29. t m a x t_{max}
  30. t m a x t_{max}
  31. t m a x t_{max}
  32. k m a x k_{max}

Variable_speed_wind_turbine.html

  1. λ = ω R v \lambda=\frac{\omega R}{v}
  2. ω \omega
  3. R R
  4. v v
  5. C p C_{p}
  6. λ \lambda
  7. λ \lambda
  8. C p - λ C_{p}-\lambda
  9. P = 1 2 ρ π R 2 v 3 × C P P=\frac{1}{2}\rho\pi R^{2}v^{3}\times C_{P}
  10. P P
  11. ρ \rho
  12. R R
  13. v v
  14. C p C_{p}
  15. T T
  16. ω \omega
  17. T = P ω T=\frac{P}{\omega}
  18. v v
  19. λ \lambda
  20. λ = ω R v \lambda=\frac{\omega R}{v}
  21. P = 1 2 λ 3 ρ π R 5 ω 3 × C P P=\frac{1}{2\lambda^{3}}\rho\pi R^{5}\omega^{3}\times C_{P}
  22. T = 1 2 λ 3 ρ π R 5 ω 2 × C P = 1 2 λ ρ π R 3 v 2 × C P T=\frac{1}{2\lambda^{3}}\rho\pi R^{5}\omega^{2}\times C_{P}=\frac{1}{2\lambda}% \rho\pi R^{3}v^{2}\times C_{P}
  23. C p C_{p}
  24. \rightarrow
  25. P = const . = T ω P=\mathrm{const.}=T\omega
  26. v v
  27. \rightarrow
  28. T ω 2 × C p T\propto\omega^{2}\times C_{p}
  29. C p C_{p}
  30. \rightarrow
  31. T ω 2 T\propto\omega^{2}
  32. θ \theta
  33. W W
  34. ω \omega
  35. ω \omega
  36. C p = C p m a x C_{p}=C_{p~{}max}
  37. C p m a x C_{p~{}max}
  38. β \beta
  39. N = 120 f P N=\frac{120f}{P}
  40. N N
  41. P P
  42. f f

Variance-based_sensitivity_analysis.html

  1. X i [ 0 , 1 ] X_{i}\in[0,1]
  2. i = 1 , 2 , , d i=1,2,...,d
  3. f ( 𝐗 ) = f 0 + i = 1 d f i ( X i ) + i < j d f i j ( X i , X j ) + + f 12 d f(\mathbf{X})=f_{0}+\sum_{i=1}^{d}f_{i}(X_{i})+\sum_{i<j}^{d}f_{ij}(X_{i},X_{j% })+\cdots+f_{12\dots d}
  4. 0 1 f i 1 i 2 i s ( X i 1 , X i 2 , , X i s ) d X i k = 0 , for k = i 1 , , i s \int_{0}^{1}f_{i_{1}i_{2}\dots i_{s}}(X_{i_{1}},X_{i_{2}},\dots,X_{i_{s}})dX_{% i_{k}}=0,\,\text{ for }k=i_{1},...,i_{s}
  5. f 0 = E ( Y ) f_{0}=E(Y)
  6. f i ( X i ) = E ( Y | X i ) - f 0 f_{i}(X_{i})=E(Y|X_{i})-f_{0}
  7. f i j ( X i , X j ) = E ( Y | X i , X j ) - f 0 - f i - f j f_{ij}(X_{i},X_{j})=E(Y|X_{i},X_{j})-f_{0}-f_{i}-f_{j}
  8. 0 1 f 2 ( 𝐗 ) d 𝐗 - f 0 2 = s = 1 d i 1 < < i s d f i 1 i s 2 d X i 1 d X i s \int_{0}^{1}f^{2}(\mathbf{X})d\mathbf{X}-f_{0}^{2}=\sum_{s=1}^{d}\sum_{i_{1}<% \dots<i_{s}}^{d}\int f_{i_{1}\dots i_{s}}^{2}dX_{i_{1}}\dots dX_{i_{s}}
  9. Var ( Y ) = i = 1 d V i + i < j d V i j + + V 12 d \operatorname{Var}(Y)=\sum_{i=1}^{d}V_{i}+\sum_{i<j}^{d}V_{ij}+\cdots+V_{12% \dots d}
  10. V i = Var X i ( E 𝐗 i ( Y X i ) ) V_{i}=\operatorname{Var}_{X_{i}}\left(E_{\,\textbf{X}_{\sim i}}(Y\mid X_{i})\right)
  11. V i j = Var X i j ( E 𝐗 i j ( Y X i j ) ) V_{ij}=\operatorname{Var}_{X_{ij}}\left(E_{\,\textbf{X}_{\sim ij}}\left(Y\mid X% _{ij}\right)\right)
  12. S i = V i Var ( Y ) S_{i}=\frac{V_{i}}{\operatorname{Var}(Y)}
  13. i = 1 d S i + i < j d S i j + + S 12 d = 1 \sum_{i=1}^{d}S_{i}+\sum_{i<j}^{d}S_{ij}+\cdots+S_{12\dots d}=1
  14. S T i = E 𝐗 i ( Var X i ( Y 𝐗 i ) ) Var ( Y ) = 1 - Var 𝐗 i ( E X i ( Y 𝐗 i ) ) Var ( Y ) S_{Ti}=\frac{E_{\,\textbf{X}_{\sim i}}\left(\operatorname{Var}_{X_{i}}(Y\mid% \mathbf{X}_{\sim i})\right)}{\operatorname{Var}(Y)}=1-\frac{\operatorname{Var}% _{\,\textbf{X}_{\sim i}}\left(E_{X_{i}}(Y\mid\mathbf{X}_{\sim i})\right)}{% \operatorname{Var}(Y)}
  15. i = 1 d S T i 1 \sum_{i=1}^{d}S_{Ti}\geq 1
  16. Var X i ( E 𝐗 i ( Y | X i ) ) 1 N j = 1 N f ( 𝐁 ) j ( f ( 𝐀 B i ) j - f ( 𝐀 ) j ) \operatorname{Var}_{X_{i}}(E_{\mathbf{X}_{\sim i}}(Y|X_{i}))\approx{\frac{1}{N% }\sum_{j=1}^{N}f\left(\mathbf{B}\right)_{j}\left(f\left(\mathbf{A}^{i}_{B}% \right)_{j}-f\left(\mathbf{A}\right)_{j}\right)}
  17. E 𝐗 i ( Var X i ( Y 𝐗 i ) ) 1 2 N j = 1 N ( f ( 𝐀 ) j - f ( 𝐀 B i ) j ) 2 E_{\mathbf{X}_{\sim i}}\left(\operatorname{Var}_{X_{i}}\left(Y\mid\mathbf{X}_{% \sim i}\right)\right)\approx{\frac{1}{2N}\sum_{j=1}^{N}\left(f\left(\mathbf{A}% \right)_{j}-f\left(\mathbf{A}^{i}_{B}\right)_{j}\right)^{2}}

Variation_diminishing_property.html

  1. lim 𝐫 𝐑 𝐫 = 𝐁 \mathbf{\lim_{r\to\infty}R_{r}}=\mathbf{B}

Variational_method_(quantum_mechanics).html

  1. λ 1 , λ 2 Spec ( H ) ψ λ 1 ψ λ 2 = δ λ 1 λ 2 \sum_{\lambda_{1},\,\lambda_{2}\in\mathrm{Spec}(H)}\langle\psi_{\lambda_{1}}% \mid\psi_{\lambda_{2}}\rangle=\delta_{\lambda_{1}\lambda_{2}}
  2. δ i , j \delta_{i,j}
  3. H ^ | ψ λ = λ | ψ λ . \hat{H}\left|\psi_{\lambda}\right\rangle=\lambda\left|\psi_{\lambda}\right\rangle.
  4. ψ H ψ = λ 1 , λ 2 Spec ( H ) ψ | ψ λ 1 ψ λ 1 | H | ψ λ 2 ψ λ 2 | ψ = λ Spec ( H ) λ | ψ λ ψ | 2 λ Spec ( H ) E 0 | ψ λ ψ | 2 = E 0 \begin{aligned}\displaystyle\left\langle\psi\mid H\mid\psi\right\rangle&% \displaystyle=\sum_{\lambda_{1},\lambda_{2}\in\mathrm{Spec}(H)}\left\langle% \psi|\psi_{\lambda_{1}}\right\rangle\left\langle\psi_{\lambda_{1}}|H|\psi_{% \lambda_{2}}\right\rangle\left\langle\psi_{\lambda_{2}}|\psi\right\rangle\\ &\displaystyle=\sum_{\lambda\in\mathrm{Spec}(H)}\lambda\left|\left\langle\psi_% {\lambda}\mid\psi\right\rangle\right|^{2}\geq\sum_{\lambda\in\mathrm{Spec}(H)}% E_{0}\left|\left\langle\psi_{\lambda}\mid\psi\right\rangle\right|^{2}=E_{0}% \end{aligned}
  5. ψ ( α i ) ψ ( α i ) = 1 \left\langle\psi(\alpha_{i})\mid\psi(\alpha_{i})\right\rangle=1
  6. ε ( α i ) = ψ ( α i ) | H | ψ ( α i ) . \varepsilon(\alpha_{i})=\left\langle\psi(\alpha_{i})|H|\psi(\alpha_{i})\right\rangle.
  7. | ψ = | ψ test - ψ gr ψ test | ψ gr \left|\psi\right\rangle=\left|\psi_{\,\text{test}}\right\rangle-\left\langle% \psi_{\mathrm{gr}}\mid\psi_{\,\text{test}}\right\rangle\left|\psi_{\,\text{gr}% }\right\rangle
  8. ψ gr \psi_{\,\text{gr}}
  9. E ground ϕ | H | ϕ . E\text{ground}\leq\left\langle\phi|H|\phi\right\rangle.
  10. ϕ = n c n ψ n . \phi=\sum_{n}c_{n}\psi_{n}.\,
  11. ϕ | H | ϕ \displaystyle\left\langle\phi|H|\phi\right\rangle
  12. E n E g E_{n}\geq E_{g}
  13. ϕ | H | ϕ E g n | c n | 2 = E g . \left\langle\phi|H|\phi\right\rangle\geq E_{g}\sum_{n}|c_{n}|^{2}=E_{g}.\,
  14. ε [ Ψ ] = Ψ | H ^ | Ψ Ψ Ψ . \varepsilon\left[\Psi\right]=\frac{\left\langle\Psi|\hat{H}|\Psi\right\rangle}% {\left\langle\Psi\mid\Psi\right\rangle}.
  15. ε E 0 \varepsilon\geq E_{0}
  16. E 0 E_{0}
  17. ε = E 0 \varepsilon=E_{0}
  18. Ψ \Psi
  19. Ψ \Psi
  20. Ψ \Psi^{\dagger}
  21. H = - 2 2 m ( 1 2 + 2 2 ) - e 2 4 π ϵ 0 ( 2 r 1 + 2 r 2 - 1 | 𝐫 1 - 𝐫 2 | ) H=-\frac{\hbar^{2}}{2m}(\nabla_{1}^{2}+\nabla_{2}^{2})-\frac{e^{2}}{4\pi% \epsilon_{0}}\left(\frac{2}{r_{1}}+\frac{2}{r_{2}}-\frac{1}{|\mathbf{r}_{1}-% \mathbf{r}_{2}|}\right)
  22. ψ ( 𝐫 1 , 𝐫 2 ) = Z 3 π a 0 3 e - Z ( r 1 + r 2 ) / a 0 . \psi(\mathbf{r}_{1},\mathbf{r}_{2})=\frac{Z^{3}}{\pi a_{0}^{3}}e^{-Z(r_{1}+r_{% 2})/a_{0}}.

Varignon's_theorem_(mechanics).html

  1. s y m b o l u 1 , s y m b o l u 2 , , s y m b o l u n symbol{u_{1}},symbol{u_{2}},...,symbol{u_{n}}
  2. s y m b o l R = i s y m b o l u i symbol{R}=\sum_{i}symbol{u_{i}}
  3. i s y m b o l M O 1 u i = ( s y m b o l O - s y m b o l O 1 ) × s y m b o l u i \sum_{i}symbol{M_{O_{1}}^{u_{i}}}=\sum(symbol{O}-symbol{O_{1}})\times symbol{u% _{i}}
  4. ( 𝐎 - 𝐎 𝟏 ) (\mathbf{O}-\mathbf{O_{1}})
  5. i s y m b o l M O 1 u i = ( s y m b o l O - s y m b o l O 1 ) × ( i s y m b o l u i ) = ( s y m b o l O - s y m b o l O 1 ) × s y m b o l R = s y m b o l M O i R \sum_{i}symbol{M_{O_{1}}^{u_{i}}}=(symbol{O}-symbol{O_{1}})\times\left(\sum_{i% }symbol{u_{i}}\right)=(symbol{O}-symbol{O_{1}})\times symbol{R}=symbol{M_{O_{i% }}^{R}}

Vámos_matroid.html

  1. F F
  2. F F
  3. M M
  4. M M
  5. L / K L/K
  6. L L
  7. x 4 + 4 x 3 + 10 x 2 + 15 x + 5 x y + 15 y + 10 y 2 + 4 y 3 + y 4 . x^{4}+4x^{3}+10x^{2}+15x+5xy+15y+10y^{2}+4y^{3}+y^{4}.

VD::VT.html

  1. V D V T = P a C O 2 - P E C O 2 P a C O 2 \frac{V_{D}}{V_{T}}=\frac{P_{a}CO_{2}-P_{E}CO_{2}}{P_{a}CO_{2}}

Vector_logic.html

  1. t s t\mapsto s
  2. f n f\mapsto n
  3. q 2 q\geq 2
  4. u T v = u , v u^{T}v=\langle u,v\rangle
  5. u , v = 1 \langle u,v\rangle=1
  6. u = v u=v
  7. u , v = 0 \langle u,v\rangle=0
  8. u v u\neq v
  9. M o n : V 2 V 2 Mon:V_{2}\to V_{2}
  10. I = s s T + n n T I=ss^{T}+nn^{T}
  11. I s = s s T s + n n T s = s s , s + n n , s = s Is=ss^{T}s+nn^{T}s=s\langle s,s\rangle+n\langle n,s\rangle=s
  12. I n = n In=n
  13. N = n s T + s n T N=ns^{T}+sn^{T}
  14. D y a d : V 2 V 2 V 2 Dyad:V_{2}\otimes V_{2}\to V_{2}
  15. C ( u v ) C(u\otimes v)
  16. C = s ( s s ) T + n ( s n ) T + n ( n s ) T + n ( n n ) T C=s(s\otimes s)^{T}+n(s\otimes n)^{T}+n(n\otimes s)^{T}+n(n\otimes n)^{T}
  17. C ( s s ) = s , C(s\otimes s)=s,
  18. C ( s n ) = C ( n s ) = C ( n n ) = n . C(s\otimes n)=C(n\otimes s)=C(n\otimes n)=n.
  19. D = s ( s s ) T + s ( s n ) T + s ( n s ) T + n ( n n ) T , D=s(s\otimes s)^{T}+s(s\otimes n)^{T}+s(n\otimes s)^{T}+n(n\otimes n)^{T},
  20. D ( s s ) = D ( s n ) = D ( n s ) = s D(s\otimes s)=D(s\otimes n)=D(n\otimes s)=s
  21. D ( n n ) = n . D(n\otimes n)=n.
  22. L = D ( N I ) L=D(N\otimes I)
  23. L = s ( s s ) T + n ( s n ) T + s ( n s ) T + n ( n n ) T , L=s(s\otimes s)^{T}+n(s\otimes n)^{T}+s(n\otimes s)^{T}+n(n\otimes n)^{T},
  24. L ( s s ) = L ( n s ) = L ( n n ) = s L(s\otimes s)=L(n\otimes s)=L(n\otimes n)=s
  25. L ( s n ) = n . L(s\otimes n)=n.
  26. E = s ( s s ) T + n ( s n ) T + n ( n s ) T + s ( n n ) T E=s(s\otimes s)^{T}+n(s\otimes n)^{T}+n(n\otimes s)^{T}+s(n\otimes n)^{T}
  27. E ( s s ) = E ( n n ) = s E(s\otimes s)=E(n\otimes n)=s
  28. E ( s n ) = E ( n s ) = n . E(s\otimes n)=E(n\otimes s)=n.
  29. X = N E X=NE
  30. X = n ( s s ) T + s ( s n ) T + s ( n s ) T + n ( n n ) T , X=n(s\otimes s)^{T}+s(s\otimes n)^{T}+s(n\otimes s)^{T}+n(n\otimes n)^{T},
  31. X ( s s ) = X ( n n ) = n X(s\otimes s)=X(n\otimes n)=n
  32. X ( s n ) = X ( n s ) = s . X(s\otimes n)=X(n\otimes s)=s.
  33. S = N C S=NC
  34. P = N D P=ND
  35. C ( u v ) = N D ( N u N v ) C(u\otimes v)=ND(Nu\otimes Nv)
  36. C ( u v ) = N D ( N N ) ( u v ) . C(u\otimes v)=ND(N\otimes N)(u\otimes v).
  37. C = N D ( N N ) C=ND(N\otimes N)
  38. L ( u v ) = D ( N I ) ( u v ) = D ( N u v ) = D ( N u N N v ) = L(u\otimes v)=D(N\otimes I)(u\otimes v)=D(Nu\otimes v)=D(Nu\otimes NNv)=
  39. D ( N N v N u ) = D ( N I ) ( N v N u ) = L ( N v N u ) D(NNv\otimes Nu)=D(N\otimes I)(Nv\otimes Nu)=L(Nv\otimes Nu)
  40. f = ϵ s + δ n f=\epsilon s+\delta n
  41. ϵ , δ [ 0 , 1 ] , ϵ + δ = 1 \epsilon,\delta\in[0,1],\epsilon+\delta=1
  42. u = α s + β n u=\alpha s+\beta n
  43. v = α s + β n v=\alpha^{\prime}s+\beta^{\prime}n
  44. G G
  45. V a l ( scalars ) = s T G ( vectors ) Val(\mathrm{scalars})=s^{T}G(\mathrm{vectors})
  46. N O T ( α ) = s T N u = 1 - α NOT(\alpha)=s^{T}Nu=1-\alpha
  47. O R ( α , α ) = s T D ( u v ) = α + α - α α OR(\alpha,\alpha^{\prime})=s^{T}D(u\otimes v)=\alpha+\alpha^{\prime}-\alpha% \alpha^{\prime}
  48. A N D ( α , α ) = s T C ( u v ) = α α AND(\alpha,\alpha^{\prime})=s^{T}C(u\otimes v)=\alpha\alpha^{\prime}
  49. I M P L ( α , α ) = s T L ( u v ) = 1 - α ( 1 - α ) IMPL(\alpha,\alpha^{\prime})=s^{T}L(u\otimes v)=1-\alpha(1-\alpha^{\prime})
  50. X O R ( α , α ) = s T X ( u v ) = α + α - 2 α α XOR(\alpha,\alpha^{\prime})=s^{T}X(u\otimes v)=\alpha+\alpha^{\prime}-2\alpha% \alpha^{\prime}
  51. N O R ( α , α ) = 1 - O R ( α , α ) NOR(\alpha,\alpha^{\prime})=1-OR(\alpha,\alpha^{\prime})
  52. N A N D ( α , α ) = 1 - A N D ( α , α ) NAND(\alpha,\alpha^{\prime})=1-AND(\alpha,\alpha^{\prime})
  53. E Q U I ( α , α ) = 1 - X O R ( α , α ) EQUI(\alpha,\alpha^{\prime})=1-XOR(\alpha,\alpha^{\prime})
  54. f ( x ) = f ( 1 ) x + f ( 0 ) ( 1 - x ) f(x)=f(1)x+f(0)(1-x)
  55. f ( x , y ) = f ( 1 , 1 ) x y + f ( 1 , 0 ) x ( 1 - y ) + f ( 0 , 1 ) ( 1 - x ) y + f ( 0 , 0 ) ( 1 - x ) ( 1 - y ) f(x,y)=f(1,1)xy+f(1,0)x(1-y)+f(0,1)(1-x)y+f(0,0)(1-x)(1-y)
  56. f ( 1 , 1 ) = 0 f(1,1)=0
  57. f ( 1 , 0 ) = f ( 0 , 1 ) = f ( 0 , 0 ) = 1 f(1,0)=f(0,1)=f(0,0)=1
  58. S = n ( s s ) T + s [ ( s n ) T + ( n s ) T + ( n n ) T ] S=n(s\otimes s)^{T}+s[(s\otimes n)^{T}+(n\otimes s)^{T}+(n\otimes n)^{T}]

Vector_measuring_current_meter.html

  1. θ \theta
  2. u u
  3. v v
  4. u ¯ = i = 1 N c o s ( θ i ) + j = 1 M s i n ( θ j ) \bar{u}=\sum_{i=1}^{N}cos(\theta_{i})+\sum_{j=1}^{M}sin(\theta_{j})
  5. v ¯ = - i = 1 N s i n ( θ i ) + j = 1 M c o s ( θ j ) \bar{v}=-\sum_{i=1}^{N}sin(\theta_{i})+\sum_{j=1}^{M}cos(\theta_{j})
  6. θ \theta
  7. θ i \theta i
  8. θ j \theta j
  9. u u
  10. v v

Vector_radiative_transfer.html

  1. ν \nu
  2. d I ( n ^ , ν ) d s = - 𝐊 I + a B ( ν , T ) + 4 π 𝐙 ( n ^ , n ^ , ν ) I d n ^ \frac{\mathrm{d}\vec{I}(\hat{n},\nu)}{\mathrm{d}s}=-\mathbf{K}\vec{I}+\vec{a}B% (\nu,T)+\int_{4\pi}\mathbf{Z}(\hat{n},\hat{n}^{\prime},\nu)\vec{I}\mathrm{d}% \hat{n}^{\prime}
  3. n ^ \hat{n}
  4. a \vec{a}
  5. a \vec{a}
  6. a \vec{a}
  7. n ^ \hat{n}
  8. 𝐊 ( n ^ , ν ) = a ( ν ) 𝐈 + 4 π 𝐙 ( n ^ , n ^ , ν ) d n ^ \mathbf{K}(\hat{n},\nu)=\vec{a}(\nu)\mathbf{I}+\int_{4\pi}\mathbf{Z}(\hat{n}^{% \prime},\hat{n},\nu)\mathrm{d}\hat{n}^{\prime}
  9. I = ( I , Q , U , V ) \vec{I}=(I,Q,U,V)

Venoyl-CoA_hydratase_2.html

  1. \rightleftharpoons

Verification_and_validation_of_computer_simulation_models.html

  1. t 0 = ( E ( Y ) - u 0 ) / ( S / n ) t_{0}={(E(Y)-u_{0})}/{(S/\sqrt{n})}
  2. t a / 2 , n - 1 t_{a/2,n-1}
  3. | t 0 | > t a / 2 , n - 1 \left|t_{0}\right|>t_{a/2,n-1}
  4. a = E ( Y ) - t a / 2 , n - 1 S / n a n d b = E ( Y ) + t a / 2 , n - 1 S / n a=E(Y)-t_{a/2,n-1}S/\sqrt{n}\qquad and\qquad b=E(Y)+t_{a/2,n-1}S/\sqrt{n}
  5. t a / 2 , n - 1 t_{a/2,n-1}

Versatile_peroxidase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Vertical_electrical_sounding.html

  1. ρ k = k U M N I A B \rho_{k}=k\frac{U_{MN}}{I_{AB}}
  2. U M N {U_{MN}}
  3. I A B {I_{AB}}
  4. k = 2 π 1 r A M - 1 r B M - 1 r A N + 1 r B N k=\frac{2\pi}{\frac{1}{r_{AM}}-\frac{1}{r_{BM}}-\frac{1}{r_{AN}}+\frac{1}{r_{% BN}}}

Very-long-chain_3-oxoacyl-CoA_reductase.html

  1. \rightleftharpoons

Very-long-chain_3-oxoacyl-CoA_synthase.html

  1. \rightleftharpoons

Very-long-chain_acyl-CoA_dehydrogenase.html

  1. \rightleftharpoons

Very-long-chain_enoyl-CoA_reductase.html

  1. \rightleftharpoons

Vibronic_spectroscopy.html

  1. v ′′ = 0 v^{\prime\prime}=0
  2. v = 0 , 1 , 2 , 3 , v^{\prime}=0,1,2,3,...
  3. v ′′ = 0 v^{\prime\prime}=0
  4. G ( v ) G(v)
  5. G ( v ) = ν ¯ e l e c t r o n i c + ω e ( v + 1 2 ) G(v)=\bar{\nu}_{electronic}+\omega_{e}(v+{1\over 2})\,
  6. G ( v ) = ν ¯ e l e c t r o n i c + ω e ( v + 1 2 ) - ω e χ e ( v + 1 2 ) 2 G(v)=\bar{\nu}_{electronic}+\omega_{e}(v+{1\over 2})-\omega_{e}\chi_{e}(v+{1% \over 2})^{2}\,
  7. G ( J , J ′′ ) = ν ¯ v - v ′′ + B J ( J + 1 ) - B ′′ J ′′ ( J ′′ + 1 ) G(J^{\prime},J^{\prime\prime})=\bar{\nu}_{v^{\prime}-v^{\prime\prime}}+B^{% \prime}J^{\prime}(J^{\prime}+1)-B^{\prime\prime}J^{\prime\prime}(J^{\prime% \prime}+1)
  8. J = J ′′ - 1 J^{\prime}=J^{\prime\prime}-1
  9. ν ¯ P = ν ¯ v - v ′′ + B ( J ′′ - 1 ) J ′′ - B ′′ J ′′ ( J ′′ + 1 ) \bar{\nu}_{P}=\bar{\nu}_{v^{\prime}-v^{\prime\prime}}+B^{\prime}(J^{\prime% \prime}-1)J^{\prime\prime}-B^{\prime\prime}J^{\prime\prime}(J^{\prime\prime}+1)
  10. = ν ¯ v - v ′′ - ( B + B ′′ ) J ′′ + ( B - B ′′ ) J ′′ 2 =\bar{\nu}_{v^{\prime}-v^{\prime\prime}}-(B^{\prime}+B^{\prime\prime})J^{% \prime\prime}+(B^{\prime}-B^{\prime\prime}){J^{\prime\prime}}^{2}
  11. J ′′ = J - 1 J^{\prime\prime}=J^{\prime}-1
  12. ν ¯ R = ν ¯ v - v ′′ + B J ( J + 1 ) - B ′′ J ( J - 1 ) \bar{\nu}_{R}=\bar{\nu}_{v^{\prime}-v^{\prime\prime}}+B^{\prime}J^{\prime}(J^{% \prime}+1)-B^{\prime\prime}J^{\prime}(J^{\prime}-1)
  13. = ν ¯ v - v ′′ + ( B + B ′′ ) J + ( B - B ′′ ) J 2 =\bar{\nu}_{v^{\prime}-v^{\prime\prime}}+(B^{\prime}+B^{\prime\prime})J^{% \prime}+(B^{\prime}-B^{\prime\prime}){J^{\prime}}^{2}
  14. ν ¯ P , R = ν ¯ v , v ′′ + ( B + B ′′ ) m + ( B - B ′′ ) m 2 , m = ± 1 , ± 2 e t c . \bar{\nu}_{P,R}=\bar{\nu}_{v^{\prime},v^{\prime\prime}}+(B^{\prime}+B^{\prime% \prime})m+(B^{\prime}-B^{\prime\prime})m^{2},\quad m=\pm 1,\pm 2\ etc.
  15. ν ¯ v , v ′′ \bar{\nu}_{v^{\prime},v^{\prime\prime}}
  16. x = - B + B ′′ 2 ( B - B ′′ ) x=-\frac{B^{\prime}+B^{\prime\prime}}{2(B^{\prime}-B^{\prime\prime})}
  17. ν ¯ Q = ν ¯ v , v ′′ + ( B - B ′′ ) J ( J + 1 ) J = 1 , 2 e t c . \bar{\nu}_{Q}=\bar{\nu}_{v^{\prime},v^{\prime\prime}}+(B^{\prime}-B^{\prime% \prime})J(J+1)\quad J=1,2\ etc.
  18. d 3 Π u a 3 Π g d^{3}\Pi_{u}\Leftrightarrow a^{3}\Pi_{g}
  19. E = h c ν ¯ E=hc\bar{\nu}

Vincent's_theorem.html

  1. x = a 1 + 1 x , x = a 2 + 1 x ′′ , x ′′ = a 3 + 1 x ′′′ , x=a_{1}+\frac{1}{x^{\prime}},\quad x^{\prime}=a_{2}+\frac{1}{x^{\prime\prime}}% ,\quad x^{\prime\prime}=a_{3}+\frac{1}{x^{\prime\prime\prime}},\ldots
  2. a 1 , a 2 , a 3 , a_{1},a_{2},a_{3},\ldots
  3. a 1 + 1 a 2 + 1 a 3 + 1 a_{1}+\cfrac{1}{a_{2}+\cfrac{1}{a_{3}+\cfrac{1}{\ddots}}}
  4. a 1 , a 2 , a 3 , a_{1},a_{2},a_{3},\ldots
  5. f n ( x ) f_{n}(x)
  6. n N n\geq N
  7. f n ( x ) f_{n}(x)
  8. f n ( x ) f_{n}(x)
  9. n N n\geq N
  10. a 1 1 a_{1}\geq 1
  11. a 1 0 a_{1}\geq 0
  12. | b - a | < δ |b-a|<\delta
  13. f ( x ) = ( 1 + x ) deg ( p ) p ( a + b x 1 + x ) f(x)=(1+x)^{\deg(p)}p\left(\frac{a+bx}{1+x}\right)
  14. x a + b x 1 + x x\leftarrow\frac{a+bx}{1+x}
  15. v a r a b ( p ) = v a r ( ( 1 + x ) deg ( p ) p ( a + b x 1 + x ) ) , var_{ab}(p)=var\left((1+x)^{\deg(p)}p\left(\frac{a+bx}{1+x}\right)\right),
  16. v a r a b ( p ) = v a r b a ( p ) ρ a b ( p ) . var_{ab}(p)=var_{ba}(p)\geq\rho_{ab}(p).
  17. S 3 = { x = - α + i β : | β | 3 | α | , α > 0 } S_{\sqrt{3}}=\left\{x=-\alpha+i\beta\ :\ |\beta|\leq\sqrt{3}|\alpha|,\alpha>0\right\}
  18. S 3 S_{\sqrt{3}}
  19. M ( x ) = a x + b c x + d , a , b , c , d > 0 M(x)=\frac{ax+b}{cx+d},\qquad a,b,c,d\in\mathbb{Z}_{>0}
  20. | x - 1 2 ( a c + b d ) | = 1 2 ( b d - a c ) \left|x-\tfrac{1}{2}\left(\tfrac{a}{c}+\tfrac{b}{d}\right)\right|=\tfrac{1}{2}% \left(\tfrac{b}{d}-\tfrac{a}{c}\right)
  21. a c \frac{a}{c}
  22. b d , \frac{b}{d},
  23. M - 1 ( x ) = d x - b - c x + a M^{-1}(x)=\frac{dx-b}{-cx+a}
  24. 1 2 ( a c + b d ) + i 2 ( b d - a c ) \tfrac{1}{2}\left(\tfrac{a}{c}+\tfrac{b}{d}\right)+\tfrac{i}{2}\left(\tfrac{b}% {d}-\tfrac{a}{c}\right)
  25. i d c . −i\frac{d}{c}.
  26. R e ( x ) < 0 Re(x)<0
  27. 1 2 ( a c + b d ) ± i 2 3 ( b d - a c ) \tfrac{1}{2}\left(\tfrac{a}{c}+\tfrac{b}{d}\right)\pm\tfrac{i}{2\sqrt{3}}\left% (\tfrac{b}{d}-\tfrac{a}{c}\right)
  28. 1 3 ( b d - a c ) \tfrac{1}{\sqrt{3}}\left(\tfrac{b}{d}-\tfrac{a}{c}\right)
  29. M - 1 ( x ) = d x - b - c x + a M^{-1}(x)=\frac{dx-b}{-cx+a}
  30. I m ( x ) = ± 3 R e ( x ) Im(x)=±\sqrt{3}Re(x)
  31. 1 2 ( a c + b d ) - 3 i 2 3 ( b d - a c ) \tfrac{1}{2}\left(\tfrac{a}{c}+\tfrac{b}{d}\right)-\tfrac{3i}{2\sqrt{3}}\left(% \tfrac{b}{d}-\tfrac{a}{c}\right)
  32. - d 2 c ( 1 - i 3 ) . \tfrac{-d}{2c}\left(1-i\sqrt{3}\right).
  33. S 3 S_{\sqrt{3}}
  34. M ( x ) = a x + b c x + d , a , b , c , d M(x)=\frac{ax+b}{cx+d},\qquad a,b,c,d\in\mathbb{N}
  35. f ( x ) = ( c x + d ) deg ( p ) p ( a x + b c x + d ) f(x)=(cx+d)^{\deg(p)}p\left(\frac{ax+b}{cx+d}\right)
  36. b d \frac{b}{d}
  37. a c \frac{a}{c}
  38. M ( x ) = a x + b c x + d M(x)=\frac{ax+b}{cx+d}
  39. M ( x ) = a x + b c x + d M(x)=\frac{ax+b}{cx+d}
  40. x deg ( p ) p ( 1 x ) x^{\deg(p)}p\left(\frac{1}{x}\right)
  41. l b = 1 u b lb=\frac{1}{ub}
  42. x l b c o m p u t e d * x x\leftarrow lb_{computed}*x
  43. l b c o m p u t e d > 16 lb_{computed}>16
  44. l b > l b c o m p u t e d lb>lb_{computed}
  45. l b c o m p u t e d lb_{computed}
  46. l b c o m p u t e d > 1 lb_{computed}>1
  47. x x + l b c o m p u t e d x\leftarrow x+lb_{computed}
  48. l b c o m p u t e d 1 lb_{computed}\leq 1
  49. x 1 1 + x x\leftarrow\frac{1}{1+x}
  50. { ( 1 + x ) deg ( p ) p ( 1 1 + x ) , M ( 1 1 + x ) } , \left\{(1+x)^{\deg(p)}p\left(\tfrac{1}{1+x}\right),M(\tfrac{1}{1+x})\right\},
  51. p ( x ) [ x ] , p ( 0 ) 0 p(x)\in\mathbb{Z}[x],p(0)\neq 0
  52. M ( x ) = a x + b c x + d = x , a , b , c , d . M(x)=\frac{ax+b}{cx+d}=x,\qquad a,b,c,d\in\mathbb{N}.
  53. M ( x ) = x M(x)=x
  54. { { x 3 - x 2 - 2 x + 1 , x + 2 x + 1 } , { x 3 + 6 x 2 + 5 x + 1 , x + 2 } } . \left\{\left\{x^{3}-x^{2}-2x+1,\tfrac{x+2}{x+1}\right\},\{x^{3}+6x^{2}+5x+1,x+% 2\}\right\}.
  55. { { x 3 + x 2 - 2 x - 1 , 2 x + 3 x + 2 } , { x 3 + 2 x 2 - x - 1 , x + 3 x + 2 } , { x 3 + 6 x 2 + 5 x + 1 , x + 2 } } . \left\{\left\{x^{3}+x^{2}-2x-1,\tfrac{2x+3}{x+2}\right\},\left\{x^{3}+2x^{2}-x% -1,\tfrac{x+3}{x+2}\right\},\{x^{3}+6x^{2}+5x+1,x+2\}\right\}.
  56. { { x 3 + 2 x 2 - x - 1 , x + 3 x + 2 } , { x 3 + 6 x 2 + 5 x + 1 , x + 2 } } . \left\{\left\{x^{3}+2x^{2}-x-1,\tfrac{x+3}{x+2}\right\},\{x^{3}+6x^{2}+5x+1,x+% 2\}\right\}.
  57. { { x 3 + 6 x 2 + 5 x + 1 , x + 2 } } . \left\{\left\{x^{3}+6x^{2}+5x+1,x+2\right\}\right\}.
  58. ( 1 , 3 2 ) (1,\frac{3}{2})
  59. ( 3 2 , 2 ) (\frac{3}{2},2)
  60. 1 2 \frac{1}{2}
  61. 1 2 \frac{1}{2}
  62. 1 2 \frac{1}{2}
  63. 1 2 \frac{1}{2}
  64. { 2 deg ( p ) p ( x 2 ) , ( a , 1 2 ( a + b ) ) } , { 2 deg ( p ) p ( 1 2 ( x + 1 ) ) , ( 1 2 ( a + b ) , b ) } \left\{2^{\deg(p)}p(\tfrac{x}{2}),(a,\tfrac{1}{2}(a+b))\right\},\quad\left\{2^% {\deg(p)}p(\tfrac{1}{2}(x+1)),(\tfrac{1}{2}(a+b),b)\right\}
  65. 1 2 \frac{1}{2}
  66. 1 2 \frac{1}{2}
  67. x 2 \frac{x}{2}
  68. x + 1 2 x\frac{+1}{2}
  69. u b = 4 ub=4
  70. ( a , b ) = ( 0 , 4 ) (a,b)=(0,4)
  71. { { 64 x 3 - 112 x + 56 , ( 0 , 2 ) } , { 64 x 3 + 192 x 2 + 80 x + 8 , ( 2 , 4 ) } } . \left\{\left\{64x^{3}-112x+56,(0,2)\right\},\left\{64x^{3}+192x^{2}+80x+8,(2,4% )\right\}\right\}.
  72. { { 64 x 3 - 448 x + 448 , ( 0 , 1 ) } , { 64 x 3 + 192 x 2 - 256 x + 64 , ( 1 , 2 ) } , { 64 x 3 + 192 x 2 + 80 x + 8 , ( 2 , 4 ) } } . \left\{\left\{64x^{3}-448x+448,(0,1)\right\},\left\{64x^{3}+192x^{2}-256x+64,(% 1,2)\right\},\left\{64x^{3}+192x^{2}+80x+8,(2,4)\right\}\right\}.
  73. { { 64 x 3 + 192 x 2 - 256 x + 64 , ( 1 , 2 ) } , { 64 x 3 + 192 x 2 + 80 x + 8 , ( 2 , 4 ) } } . \left\{\left\{64x^{3}+192x^{2}-256x+64,(1,2)\right\},\left\{64x^{3}+192x^{2}+8% 0x+8,(2,4)\right\}\right\}.
  74. { { 64 x 3 + 384 x 2 - 1024 x + 512 , ( 1 , 3 2 ) } , { 64 x 3 + 576 x 2 - 64 x - 64 , ( 3 2 , 2 ) } , { 64 x 3 + 192 x 2 + 80 x + 8 , ( 2 , 4 ) } } . \left\{\left\{64x^{3}+384x^{2}-1024x+512,\left(1,\tfrac{3}{2}\right)\right\},% \left\{64x^{3}+576x^{2}-64x-64,\left(\tfrac{3}{2},2\right)\right\},\left\{64x^% {3}+192x^{2}+80x+8,(2,4)\right\}\right\}.
  75. { { 64 x 3 + 576 x 2 - 64 x - 64 , ( 3 2 , 2 ) } , { 64 x 3 + 192 x 2 + 80 x + 8 , ( 2 , 4 ) } } . \left\{\left\{64x^{3}+576x^{2}-64x-64,\left(\tfrac{3}{2},2\right)\right\},% \left\{64x^{3}+192x^{2}+80x+8,(2,4)\right\}\right\}.
  76. { { 64 x 3 + 192 x 2 + 80 x + 8 , ( 2 , 4 ) } } . \left\{\left\{64x^{3}+192x^{2}+80x+8,(2,4)\right\}\right\}.
  77. p ( x ) = x < s u p > 3 7 x + 7 p(x)=x<sup>3−7x+7

Violaxanthin_de-epoxidase.html

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

Viral_phylodynamics.html

  1. R 0 R_{0}
  2. R 0 R_{0}
  3. R 0 R_{0}
  4. R 0 R_{0}
  5. R 0 R_{0}
  6. N N
  7. N N
  8. n n
  9. λ n = ( n 2 ) 1 N \lambda_{n}={n\choose 2}\frac{1}{N}
  10. T n T_{n}
  11. n - 1 n-1
  12. λ n - 1 λ 2 \lambda_{n-1}\cdots\lambda_{2}
  13. T n - 1 T 2 T_{n-1}\cdots T_{2}
  14. { λ n - i } i = 0 , , n - 2 \{\lambda_{n-i}\}_{i=0,\cdots,n-2}
  15. T i T_{i}
  16. E [ TMRCA ] \displaystyle\mathrm{E}[\mathrm{TMRCA}]
  17. lim n inf E [ TMRCA ] = 2 N . \lim_{n\rightarrow\inf}\mathrm{E}[\mathrm{TMRCA}]=2N.
  18. O ( 1 / n ) O(1/n)
  19. 1 - 1 / 74 = 99 % 1-1/74=99\%
  20. N ( t ) N(t)
  21. λ n ( t ) \lambda_{n}(t)
  22. λ n ( t ) = ( n 2 ) 1 N ( t ) \lambda_{n}(t)={n\choose 2}\frac{1}{N(t)}
  23. t τ = 0 t d τ N ( τ ) t\rightarrow\int_{\tau=0}^{t}\frac{\mathrm{d}\tau}{N(\tau)}
  24. r r
  25. t t
  26. N ( t ) = N 0 e - r t N(t)=N_{0}e^{-rt}
  27. λ n ( t ) = ( n 2 ) 1 N 0 e - r t \lambda_{n}(t)={n\choose 2}\frac{1}{N_{0}e^{-rt}}
  28. t = 0 t=0
  29. D D
  30. R 0 R_{0}
  31. r = R 0 - 1 D r=\frac{R_{0}-1}{D}
  32. R 0 R_{0}
  33. S ( t ) S(t)
  34. I ( t ) I(t)
  35. R ( t ) R(t)
  36. d S d t = - β S I \frac{dS}{dt}=-\beta SI
  37. d I d t = β S I - γ I \frac{dI}{dt}=\beta SI-\gamma I
  38. d R d t = γ I \frac{dR}{dt}=\gamma I
  39. β \beta
  40. γ \gamma
  41. f ( t ) = β S I f(t)=\beta SI
  42. λ n ( t ) = ( n 2 ) 2 f ( t ) I ( t ) 2 \lambda_{n}(t)={n\choose 2}\frac{2f(t)}{I(t)^{2}}
  43. 2 ( n 2 ) / I ( t ) 2 2{n\choose 2}/{I(t)^{2}}
  44. ( n 2 ) / ( I ( t ) 2 ) 2 ( n 2 ) / I ( t ) 2 {n\choose 2}/{I(t)\choose 2}\approx 2{n\choose 2}/{I(t)^{2}}
  45. f ( t ) f(t)
  46. λ n ( t ) = ( n 2 ) 2 β S ( t ) I ( t ) \lambda_{n}(t)={n\choose 2}\frac{2\beta S(t)}{I(t)}
  47. S ( t ) S(t)
  48. S ( 0 ) 1 S(0)\approx 1
  49. λ n ( t ) ( n 2 ) 2 β I ( t ) \lambda_{n}(t)\approx{n\choose 2}\frac{2\beta}{I(t)}
  50. N e = I ( t ) / 2 β N_{e}=I(t)/2\beta
  51. τ \tau
  52. λ n = ( n 2 ) 1 N e τ \lambda_{n}={n\choose 2}\frac{1}{N_{e}\tau}
  53. N e N_{e}
  54. N N
  55. σ 2 \sigma^{2}
  56. τ \tau
  57. N N
  58. σ 2 \sigma^{2}
  59. ν \nu
  60. ν \nu
  61. E [ ν ] \mathrm{E}[\nu]
  62. Var [ ν ] \mathrm{Var}[\nu]
  63. σ 2 \sigma^{2}
  64. σ 2 = Var [ ν ] E [ ν ] 2 + 1 \sigma^{2}=\frac{\mathrm{Var}[\nu]}{\mathrm{E}[\nu]^{2}}+1
  65. N N
  66. I I
  67. β / γ \beta/\gamma
  68. γ \gamma
  69. ( β / γ ) 2 (\beta/\gamma)^{2}
  70. σ 2 \sigma^{2}
  71. N e N_{e}
  72. I 2 \frac{I}{2}
  73. λ n = ( n 2 ) 2 γ I \lambda_{n}={n\choose 2}\frac{2\gamma}{I}
  74. I 1 I n I_{1}\cdots I_{n}
  75. N e N_{e}
  76. N e N_{e}

Viscous_stress_tensor.html

  1. ϵ \epsilon
  2. E E
  3. μ \mu
  4. ϵ = μ E \epsilon=\mu E
  5. μ \mu
  6. ϵ \epsilon
  7. E E
  8. μ \mu
  9. μ \mu
  10. ϵ \epsilon
  11. E E
  12. ϵ \epsilon
  13. E E
  14. ϵ ( p , t ) = [ ϵ 11 ϵ 12 ϵ 13 ϵ 21 ϵ 22 ϵ 23 ϵ 31 ϵ 32 ϵ 33 ] \epsilon(p,t)=\begin{bmatrix}\epsilon_{11}&\epsilon_{12}&\epsilon_{13}\\ \epsilon_{21}&\epsilon_{22}&\epsilon_{23}\\ \epsilon_{31}&\epsilon_{32}&\epsilon_{33}\end{bmatrix}
  15. p p
  16. t t
  17. p p
  18. d A dA
  19. d F dF
  20. d A dA
  21. d F dF
  22. d F i = j ϵ i j d A j dF_{i}=\sum_{j}\epsilon_{ij}\,dA_{j}\,
  23. σ \sigma
  24. ϵ \epsilon
  25. τ \tau
  26. p p
  27. σ i j = - p δ i j + ϵ i j \sigma_{ij}=-p\delta_{ij}+\epsilon_{ij}
  28. δ i j \delta_{ij}
  29. δ i j \delta_{ij}
  30. i = j i=j
  31. i j i\neq j
  32. ϵ \epsilon
  33. τ i j = ϵ i j - ϵ j i \tau_{ij}=\epsilon_{ij}-\epsilon_{ji}
  34. E ( p , t ) E(p,t)
  35. p p
  36. t t
  37. E ( p , t ) E(p,t)
  38. e ( p , t ) e(p,t)
  39. v ( p , t ) v(p,t)
  40. E = e t = 1 2 ( ( v ) + ( v ) ) , E=\frac{\partial e}{\partial t}=\frac{1}{2}((\nabla v)+(\nabla v)^{\top}),
  41. v \nabla v
  42. v \nabla v
  43. ( v ) j i = v j x i (\nabla v)_{ji}=\frac{\partial v_{j}}{\partial x_{i}}
  44. E i j = e i j t = 1 2 ( v j x i + v i x j ) . E_{ij}=\frac{\partial e_{ij}}{\partial t}=\frac{1}{2}\left(\frac{\partial v_{j% }}{\partial x_{i}}+\frac{\partial v_{i}}{\partial x_{j}}\right).
  45. E ( p , t ) E(p,t)
  46. p p
  47. p p
  48. × v \nabla\times v
  49. v \nabla v
  50. p p
  51. E E
  52. p p
  53. E E
  54. ϵ \epsilon
  55. E ( p , t ) E(p,t)
  56. ϵ ( p , t ) \epsilon(p,t)
  57. E E
  58. ϵ \epsilon
  59. ϵ ( p , t ) \epsilon(p,t)
  60. E ( p , t ) E(p,t)
  61. p p
  62. μ \mathbf{\mu}
  63. ϵ i j = k μ i j k E k \epsilon_{ij}=\sum_{k\ell}\mu_{ijk\ell}E_{k\ell}
  64. μ \mathbf{\mu}
  65. v v
  66. σ \sigma
  67. E ( p , t ) E(p,t)
  68. μ \mathbf{\mu}
  69. ϵ \epsilon
  70. ϵ v \epsilon^{\,\text{v}}
  71. ϵ s \epsilon^{\,\text{s}}
  72. ϵ i j = ϵ i j v + ϵ i j s \epsilon_{ij}=\epsilon_{ij}^{\,\text{v}}+\epsilon_{ij}^{\,\text{s}}
  73. ϵ i j v = 1 3 δ i j k ϵ k k \epsilon_{ij}^{\,\text{v}}=\tfrac{1}{3}\delta_{ij}\sum_{k}\epsilon_{kk}
  74. ϵ i j s = ϵ i j - 1 3 δ i j k ϵ k k \epsilon_{ij}^{\,\text{s}}=\epsilon_{ij}-\tfrac{1}{3}\delta_{ij}\sum_{k}% \epsilon_{kk}
  75. ϵ v \epsilon^{\,\text{v}}
  76. ϵ v ( p , t ) = μ v E v ( p , t ) , and \epsilon^{\,\text{v}}(p,t)=\mu^{\,\text{v}}E^{\,\text{v}}(p,t),\,\text{and}
  77. ϵ s ( p , t ) = μ s E s ( p , t ) , \epsilon^{\,\text{s}}(p,t)=\mu^{\,\text{s}}E^{\,\text{s}}(p,t),
  78. E v E^{\,\text{v}}
  79. E s E^{\,\text{s}}
  80. E E
  81. μ v \mu^{\,\text{v}}
  82. μ s \mu^{\,\text{s}}
  83. μ \mu
  84. E s E^{\,\text{s}}
  85. E E
  86. ϵ s \epsilon^{\,\text{s}}
  87. ϵ \epsilon
  88. E v E^{\,\text{v}}
  89. E E
  90. ϵ v \epsilon^{\,\text{v}}
  91. v = k v k x k , \nabla\cdot v=\sum_{k}\frac{\partial v_{k}}{\partial x_{k}},
  92. ϵ v \epsilon^{\,\text{v}}
  93. ϵ \epsilon
  94. μ v \mu^{\,\text{v}}
  95. η \eta
  96. μ s \mu^{\,\text{s}}

Viscous_vortex_domains_method.html

  1. 𝐮 = 𝐕 + 𝐕 d ; 𝐕 d = - ν 𝛀 | 𝛀 | ; 𝛀 = [ × 𝐕 ] \mathbf{u}=\mathbf{V}+\mathbf{V}_{d};~{}~{}~{}\mathbf{V}_{d}=-\nu\dfrac{\nabla% \mathbf{\Omega}}{|\mathbf{\Omega}|};~{}~{}~{}\mathbf{\Omega}=[\nabla\times% \mathbf{V}]
  2. γ \gamma
  3. 𝐕 ( 𝐫 ) = 1 2 π i γ i [ 𝐞 z × 𝐫 - 𝐫 i ( 𝐫 - 𝐫 i ) 2 + δ 2 ] \mathbf{V}(\mathbf{r})=\dfrac{1}{2\pi}\sum_{i}\gamma_{i}\cdot\left[\mathbf{e}_% {z}\times\dfrac{\mathbf{r}-\mathbf{r}_{i}}{(\mathbf{r}-\mathbf{r}_{i})^{2}+% \delta^{2}}\right]
  4. 𝐕 d ( 𝐫 ) = ν ( 𝐈 2 ( 𝐫 ) I 1 ( 𝐫 ) + 𝐈 3 ( 𝐫 ) 2 π ε 2 - I 0 ( 𝐫 ) ) \mathbf{V}_{d}(\mathbf{r})=\nu\left(\dfrac{\mathbf{I}_{2}(\mathbf{r})}{I_{1}(% \mathbf{r})}+\dfrac{\mathbf{I}_{3}(\mathbf{r})}{2\pi\varepsilon^{2}-I_{0}(% \mathbf{r})}\right)
  5. 𝐈 2 ( 𝐫 ) = i 𝐫 - 𝐫 i ε | 𝐫 - 𝐫 i | γ i exp ( - | 𝐫 - 𝐫 i | / ε ) \mathbf{I}_{2}(\mathbf{r})=\sum\limits_{i}\dfrac{\mathbf{r}-\mathbf{r}_{i}}{% \varepsilon\left|\mathbf{r}-\mathbf{r}_{i}\right|}\cdot\gamma_{i}\cdot\exp(-% \left|\mathbf{r}-\mathbf{r}_{i}\right|/\varepsilon)
  6. I 1 ( 𝐫 ) = i γ i exp ( - | 𝐫 - 𝐫 i | / ε ) I_{1}(\mathbf{r})={\sum\limits_{i}\gamma_{i}\cdot\exp(-\left|\mathbf{r}-% \mathbf{r}_{i}\right|/\varepsilon)}
  7. 𝐈 3 ( 𝐫 ) = k d 𝐒 k exp ( - | 𝐫 - 𝐫 k | / ε ) \mathbf{I}_{3}(\mathbf{r})={\sum\limits_{k}d\mathbf{S}_{k}\cdot\exp(-\left|% \mathbf{r}-\mathbf{r}_{k}\right|/\varepsilon)}
  8. I 0 ( 𝐫 ) = ε 2 k | 𝐫 - 𝐫 k | / ε + 1 ( 𝐫 - 𝐫 k ) 2 ( ( 𝐫 - 𝐫 k ) d 𝐒 k ) exp ( - | 𝐫 - 𝐫 k | / ε ) I_{0}(\mathbf{r})={\varepsilon^{2}\sum\limits_{k}\dfrac{\left|\mathbf{r}-% \mathbf{r}_{k}\right|/\varepsilon+1}{(\mathbf{r}-\mathbf{r}_{k})^{2}}\cdot((% \mathbf{r}-\mathbf{r}_{k})\cdot d\mathbf{S}_{k})\cdot\exp(-\left|\mathbf{r}-% \mathbf{r}_{k}\right|/\varepsilon)}

Visual_impairment_due_to_intracranial_pressure.html

  1. I O P = F C + P V IOP=\frac{F}{C}+PV

Vitamin_D3_24-hydroxylase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Von_Kries_Coefficient_Law.html

  1. R c = R R_{c}=R
  2. G c = G G_{c}=G
  3. B c = B B_{c}=B
  4. R c , G c , R_{c},G_{c},
  5. B c B_{c}
  6. R , G , R,G,
  7. B B
  8. R c , G c , R_{c},G_{c},
  9. B c B_{c}
  10. R w r , G w r , R_{wr},G_{wr},
  11. B w r B_{wr}
  12. R w , G w , R_{w},G_{w},
  13. B w B_{w}
  14. R R w = R c R w r \frac{R}{R_{w}}=\frac{R_{c}}{R_{wr}}
  15. G G w = G c G w r \frac{G}{G_{w}}=\frac{G_{c}}{G_{wr}}
  16. B B w = B c B w r \frac{B}{B_{w}}=\frac{B_{c}}{B_{wr}}
  17. = R w r R w =\frac{R_{wr}}{R_{w}}
  18. = G w r G w =\frac{G_{wr}}{G_{w}}
  19. = B w r B w =\frac{B_{wr}}{B_{w}}
  20. c c^{\prime}
  21. c = D 1 S T f 1 = D 2 S T f 2 c^{\prime}=D_{1}\,S^{T}\,f_{1}=D_{2}\,S^{T}\,f_{2}
  22. S S
  23. f f
  24. D = D 1 - 1 D 2 = [ L 2 / L 1 0 0 0 M 2 / M 1 0 0 0 S 2 / S 1 ] D=D_{1}^{-1}D_{2}=\begin{bmatrix}L_{2}/L_{1}&0&0\\ 0&M_{2}/M_{1}&0\\ 0&0&S_{2}/S_{1}\end{bmatrix}

Voter_model.html

  1. η t \scriptstyle\eta_{t}
  2. S = { 0 , 1 } Z d \scriptstyle S=\{0,1\}^{Z^{d}}
  3. c ( x , η ) \scriptstyle c(x,\eta)
  4. Z d \scriptstyle Z^{d}
  5. c ( \scriptstyle c(
  6. ) \scriptstyle)
  7. η \scriptstyle\eta
  8. S \scriptstyle S
  9. η S \scriptstyle\eta\in S
  10. η ( x ) \scriptstyle\eta(x)
  11. η ( . ) \scriptstyle\eta(.)
  12. η t ( x ) \scriptstyle\eta_{t}(x)
  13. η ( . ) \scriptstyle\eta(.)
  14. t \scriptstyle t
  15. x \scriptstyle x
  16. c ( x , η ) \scriptstyle c(x,\eta)
  17. x \scriptstyle x
  18. c ( x , η ) = 0 \scriptstyle c(x,\eta)=0
  19. x Z d \scriptstyle x\in Z^{d}
  20. η 0 \scriptstyle\eta\equiv 0
  21. η 1 \scriptstyle\eta\equiv 1
  22. c ( x , η ) = c ( x , ζ ) \scriptstyle c(x,\eta)=c(x,\zeta)
  23. x Z d \scriptstyle x\in Z^{d}
  24. η ( y ) + ζ ( y ) = 1 \scriptstyle\eta(y)+\zeta(y)=1
  25. y Z d \scriptstyle y\in Z^{d}
  26. c ( x , η ) c ( x , ζ ) \scriptstyle c(x,\eta)\leq c(x,\zeta)
  27. η ζ \scriptstyle\eta\leq\zeta
  28. η ( x ) = ζ ( x ) = 0 \scriptstyle\eta(x)=\zeta(x)=0
  29. c ( x , η ) \scriptstyle c(x,\eta)
  30. Z d \scriptstyle Z^{d}
  31. η 0 \scriptstyle\eta\equiv 0
  32. η 1 \scriptstyle\eta\equiv 1
  33. η ζ \scriptstyle\eta\leq\zeta
  34. x , η ( x ) ζ ( x ) \scriptstyle\forall x,\eta(x)\leq\zeta(x)
  35. η ζ \scriptstyle\eta\leq\zeta
  36. c ( x , η ) c ( x , ζ ) \scriptstyle c(x,\eta)\leq c(x,\zeta)
  37. η ( x ) = ζ ( x ) = 0 \scriptstyle\eta(x)=\zeta(x)=0
  38. c ( x , η ) c ( x , ζ ) \scriptstyle c(x,\eta)\geq c(x,\zeta)
  39. η ( x ) = ζ ( x ) = 1 \scriptstyle\eta(x)=\zeta(x)=1
  40. δ 0 \scriptstyle\delta_{0}
  41. δ 1 \scriptstyle\delta_{1}
  42. η 0 \scriptstyle\eta\equiv 0
  43. η 1 \scriptstyle\eta\equiv 1
  44. x , y Z d \scriptstyle x,y\in Z^{d}
  45. lim t P [ η t ( x ) η t ( y ) ] = 0 \lim_{t\rightarrow\infty}P[\eta_{t}(x)\neq\eta_{t}(y)]=0
  46. { x : η ( x ) = 0 } \scriptstyle\{x:\eta(x)=0\}
  47. { x : η ( x ) = 1 } \scriptstyle\{x:\eta(x)=1\}
  48. p ( \scriptstyle p(
  49. ) \scriptstyle)
  50. Z d \scriptstyle Z^{d}
  51. p ( x , y ) 0 and y p ( x , y ) = 1 p(x,y)\geq 0\quad\,\text{and}\sum_{y}p(x,y)=1
  52. η \scriptstyle\eta
  53. c ( x , η ) = { y p ( x , y ) η ( y ) for all η ( x ) = 0 y p ( x , y ) ( 1 - η ( y ) ) for all η ( x ) = 1 c(x,\eta)=\left\{\begin{array}[]{l}\sum_{y}p(x,y)\eta(y)\quad\,\text{for all}% \quad\eta(x)=0\\ \sum_{y}p(x,y)(1-\eta(y))\quad\,\text{for all}\quad\eta(x)=1\\ \end{array}\right.
  54. η x \scriptstyle\eta_{x}
  55. x \scriptstyle x
  56. η η x at rate y : η ( y ) η ( x ) p ( x , y ) . \eta\rightarrow\eta_{x}\quad\,\text{at rate}\sum_{y:\eta(y)\neq\eta(x)}p(x,y).
  57. A t Z d \scriptstyle A_{t}\subset Z^{d}
  58. A t \scriptstyle A_{t}
  59. t \scriptstyle t
  60. A t \scriptstyle A_{t}
  61. Z d \scriptstyle Z^{d}
  62. p ( \scriptstyle p(
  63. ) \scriptstyle)
  64. p ( \scriptstyle p(
  65. ) \scriptstyle)
  66. P η ( η t 1 on A ) = P A ( η ( A t ) 1 ) , P^{\eta}(\eta_{t}\equiv 1\quad\,\text{on }A)=P^{A}(\eta(A_{t})\equiv 1),
  67. η { 0 , 1 } Z d \scriptstyle\eta\in\{0,1\}^{Z^{d}}
  68. η t \scriptstyle\eta_{t}
  69. A = { x Z d , η ( x ) = 1 } Z d \scriptstyle A=\{x\in Z^{d},\eta(x)=1\}\subset Z^{d}
  70. A t \scriptstyle A_{t}
  71. p ( x , y ) \scriptstyle p(x,y)
  72. Z d \scriptstyle Z^{d}
  73. p ( x , y ) = p ( 0 , x - y ) \scriptstyle p(x,y)=p(0,x-y)
  74. η S = { 0 , 1 } Z d \scriptstyle\forall\eta\in S=\{0,1\}^{Z^{d}}
  75. P η [ η t ( x ) η t ( y ) ] = P [ η ( X t ) η ( Y t ) ] P^{\eta}[\eta_{t}(x)\neq\eta_{t}(y)]=P[\eta(X_{t})\neq\eta(Y_{t})]
  76. X t \scriptstyle X_{t}
  77. Y t \scriptstyle Y_{t}
  78. Z d \scriptstyle Z^{d}
  79. X 0 = x \scriptstyle X_{0}=x
  80. Y 0 = y \scriptstyle Y_{0}=y
  81. η ( X t ) \scriptstyle\eta(X_{t})
  82. t \scriptstyle t
  83. X t \scriptstyle X_{t}
  84. Y t \scriptstyle Y_{t}
  85. X ( t ) - Y ( t ) \scriptstyle X(t)-Y(t)
  86. X ( t ) - Y ( t ) \scriptstyle X(t)-Y(t)
  87. d 2 \scriptstyle d\leq 2
  88. X t \scriptstyle X_{t}
  89. Y t \scriptstyle Y_{t}
  90. P η [ η t ( x ) η t ( y ) ] = P [ η ( X t ) η ( Y t ) ] P [ X t Y t ] 0 as t 0 P^{\eta}[\eta_{t}(x)\neq\eta_{t}(y)]=P[\eta(X_{t})\neq\eta(Y_{t})]\leq P[X_{t}% \neq Y_{t}]\rightarrow 0\quad\,\text{as}\quad t\to 0
  91. d 3 d\geq 3
  92. d 3 \scriptstyle d\geq 3
  93. X ( t ) - Y ( t ) \scriptstyle X(t)-Y(t)
  94. x y \scriptstyle x\neq y
  95. lim t P [ η t ( x ) η t ( y ) ] = C lim t P [ X t Y t ] > 0 \lim_{t\rightarrow\infty}P[\eta_{t}(x)\neq\eta_{t}(y)]=C\lim_{t\rightarrow% \infty}P[X_{t}\neq Y_{t}]>0
  96. C C
  97. X ~ ( t ) = X ( t ) - Y ( t ) \scriptstyle\tilde{X}(t)=X(t)-Y(t)
  98. η t \scriptstyle\eta_{t}
  99. X ~ t \scriptstyle\tilde{X}_{t}
  100. X ~ t \scriptstyle\tilde{X}_{t}
  101. d = 1 \scriptstyle d=1
  102. x | x | p ( 0 , x ) \scriptstyle\sum_{x}|x|p(0,x)\leq\infty
  103. d = 2 \scriptstyle d=2
  104. x | x | 2 p ( 0 , x ) \scriptstyle\sum_{x}|x|^{2}p(0,x)\leq\infty
  105. d 3 \scriptstyle d\geq 3
  106. μ \scriptstyle\mu
  107. S = { 0 , 1 } Z d \scriptstyle S=\{0,1\}^{Z^{d}}
  108. X ~ t \scriptstyle\tilde{X}_{t}
  109. μ S ( t ) ρ δ 1 + ( 1 - ρ ) δ 0 as t \scriptstyle\mu S(t)\Rightarrow\rho\delta_{1}+(1-\rho)\delta_{0}\quad\,\text{% as}\quad t\to\infty
  110. X ~ t \scriptstyle\tilde{X}_{t}
  111. μ S ( t ) μ ρ \scriptstyle\mu S(t)\Rightarrow\mu_{\rho}
  112. μ S ( t ) \scriptstyle\mu S(t)
  113. η t \scriptstyle\eta_{t}
  114. \scriptstyle\Rightarrow
  115. μ ρ \scriptstyle\mu_{\rho}
  116. ρ = μ ( { η : η ( x ) = 1 } ) \scriptstyle\rho=\mu(\{\eta:\eta(x)=1\})
  117. { 0 , 1 } Z d \scriptstyle\{0,1\}^{Z^{d}}
  118. p ( x , y ) = { 1 / 2 d if | x - y | = 1 and η ( x ) η ( y ) 0 otherwise p(x,y)=\begin{cases}1/2d&\,\text{if }|x-y|=1\,\text{ and }\eta(x)\neq\eta(y)\\ 0&\,\text{otherwise}\end{cases}
  119. η t ( x ) 1 - η t ( x ) at rate ( 2 d ) - 1 | { y : | y - x | = 1 , η t ( y ) η t ( x ) } | \eta_{t}(x)\to 1-\eta_{t}(x)\quad\,\text{at rate}\quad(2d)^{-1}|\{y:|y-x|=1,% \eta_{t}(y)\neq\eta_{t}(x)\}|
  120. d 2 \scriptstyle d\leq 2
  121. d 3 \scriptstyle d\geq 3
  122. Z d \scriptstyle Z^{d}
  123. d 2 \scriptstyle d\leq 2
  124. d 3 \scriptstyle d\geq 3
  125. d = 1 \scriptstyle d=1
  126. S = Z 1 \scriptstyle S=Z^{1}
  127. p ( x , x + 1 ) = p ( x , x - 1 ) = 1 2 \scriptstyle p(x,x+1)=p(x,x-1)=\frac{1}{2}
  128. x \scriptstyle x
  129. μ S ( t ) ρ δ 1 + ( 1 - ρ ) δ 0 \scriptstyle\mu S(t)\Rightarrow\rho\delta_{1}+(1-\rho)\delta_{0}
  130. η \scriptstyle\eta
  131. { x : η ( x ) = 0 } \scriptstyle\{x:\eta(x)=0\}
  132. { x : η ( x ) = 1 } \scriptstyle\{x:\eta(x)=1\}
  133. η \scriptstyle\eta
  134. C ( η ) = lim n 2 n number of clusters in [ - n , n ] C(\eta)=\lim_{n\rightarrow\infty}\frac{2n}{\,\text{number of clusters in}[-n,n]}
  135. μ \scriptstyle\mu
  136. μ \scriptstyle\mu
  137. P ( C ( η ) = 1 P [ η t ( 0 ) η t ( 1 ) ] ) = 1. P\left(C(\eta)=\frac{1}{P[\eta_{t}(0)\neq\eta_{t}(1)]}\right)=1.
  138. T t x = 0 t η s ρ ( x ) d s . T_{t}^{x}=\int_{0}^{t}\eta^{\rho}_{s}(x)\mathrm{d}s.
  139. P ( η t ( x ) = 1 ) = ρ \scriptstyle P(\eta_{t}(x)=1)=\rho
  140. t \scriptstyle t\rightarrow\infty
  141. T t x / t ρ \scriptstyle T_{t}^{x}/t\rightarrow\rho
  142. d 2 \scriptstyle d\geq 2
  143. P ( ρ r lim inf t T t t lim sup t T t t ρ r ) = 1 ; r > 1 P\left(\frac{\rho}{r}\leq\lim\inf_{t\rightarrow\infty}\frac{T_{t}}{t}\leq\lim% \sup_{t\rightarrow\infty}\frac{T_{t}}{t}\leq\rho r\right)=1;\quad\forall r>1
  144. r 1 \scriptstyle r\searrow 1
  145. 𝒩 \scriptstyle\mathcal{N}
  146. 0 Z d \scriptstyle 0\in Z^{d}
  147. Z d \scriptstyle Z^{d}
  148. R d \scriptstyle R^{d}
  149. 𝒩 \scriptstyle\mathcal{N}
  150. Z d \scriptstyle Z^{d}
  151. 𝒩 \scriptstyle\mathcal{N}
  152. ( 1 , 0 , 0 , , 0 ) , , ( 0 , , 0 , 1 ) \scriptstyle(1,0,0,\dots,0),\dots,(0,\dots,0,1)
  153. T \scriptstyle T
  154. 𝒩 \scriptstyle\mathcal{N}
  155. T \scriptstyle T
  156. c ( x , η ) = { 1 if | { y x + 𝒩 : η ( y ) η ( x ) } | T 0 otherwise c(x,\eta)=\left\{\begin{array}[]{l}1\quad\,\text{if}\quad|\{y\in x+\mathcal{N}% :\eta(y)\neq\eta(x)\}|\geq T\\ 0\quad\,\text{otherwise}\\ \end{array}\right.
  157. x \scriptstyle x
  158. x \scriptstyle x
  159. d = 1 \scriptstyle d=1
  160. 𝒩 = { - 1 , 0 , 1 } \scriptstyle\mathcal{N}=\{-1,0,1\}
  161. T = 2 \scriptstyle T=2
  162. 1 1 0 0 1 1 0 0 \scriptstyle\dots 1\quad 1\quad 0\quad 0\quad 1\quad 1\quad 0\quad 0\dots
  163. | 𝒩 | \scriptstyle|\mathcal{N}|
  164. T > | 𝒩 | - 1 2 \scriptstyle T>\frac{|\mathcal{N}|-1}{2}
  165. d = 1 \scriptstyle d=1
  166. T = | 𝒩 | - 1 2 \scriptstyle T=\frac{|\mathcal{N}|-1}{2}
  167. T = θ | 𝒩 | \scriptstyle T=\theta|\mathcal{N}|
  168. θ \scriptstyle\theta
  169. θ < 1 4 \scriptstyle\theta<\frac{1}{4}
  170. | 𝒩 | \scriptstyle|\mathcal{N}|
  171. T > | 𝒩 | - 1 2 \scriptstyle T>\frac{|\mathcal{N}|-1}{2}
  172. d = 1 \scriptstyle d=1
  173. 𝒩 = { - T , , T } , T 1 \scriptstyle\mathcal{N}=\{-T,\dots,T\},T\geq 1
  174. U n \scriptstyle U_{n}
  175. V n \scriptstyle V_{n}
  176. n 1 \scriptstyle n\geq 1
  177. 0 = V 0 < U 1 < V 1 < U 2 < V 2 < \scriptstyle 0=V_{0}<U_{1}<V_{1}<U_{2}<V_{2}<\dots
  178. { U k + 1 - V k , k 0 } \scriptstyle\{U_{k+1}-V_{k},k\geq 0\}
  179. E ( U k + 1 - V k ) < \scriptstyle\mathrm{E}(U_{k+1}-V_{k})<\infty
  180. { V k - U k , k 1 } \scriptstyle\{V_{k}-U_{k},k\geq 1\}
  181. E ( V k - U k ) = \scriptstyle\mathrm{E}(V_{k}-U_{k})=\infty
  182. { η t ( . ) \scriptstyle\{\eta_{t}(.)
  183. { - T , , T } } \scriptstyle\{-T,\dots,T\}\}
  184. t k = 1 [ U k , V k ] \scriptstyle t\in\cup_{k=1}^{\infty}[U_{k},V_{k}]
  185. P ( A ) P ( t k = 1 [ U k , V k ] ) 1 as t P(A)\geq P(t\in\cup_{k=1}^{\infty}[U_{k},V_{k}])\to 1\quad\,\text{as}\quad t\to\infty
  186. lim t P ( η t ( 1 ) η t ( 0 ) ) = 0 \scriptstyle\lim_{t\rightarrow\infty}P(\eta_{t}(1)\neq\eta_{t}(0))=0
  187. T = | 𝒩 | - 1 2 \scriptstyle T=\frac{|\mathcal{N}|-1}{2}
  188. d = 2 , T = 2 \scriptstyle d=2,T=2
  189. 𝒩 = { ( 0 , 0 ) , ( 0 , 1 ) , ( 1 , 0 ) , ( 0 , - 1 ) , ( - 1 , 0 ) } \scriptstyle\mathcal{N}=\{(0,0),(0,1),(1,0),(0,-1),(-1,0)\}
  190. η \scriptstyle\eta
  191. i , j \scriptstyle i,j
  192. η ( 4 i , j ) = η ( 4 i + 1 , j ) = 1 , η ( 4 i + 2 , j ) = η ( 4 i + 3 , j ) = 0 \eta(4i,j)=\eta(4i+1,j)=1,\quad\eta(4i+2,j)=\eta(4i+3,j)=0
  193. 000111 \scriptstyle\dots 000111\dots
  194. λ > 0 \scriptstyle\lambda>0
  195. [ 0 , 1 ] Z d \scriptstyle[0,1]^{Z^{d}}
  196. c ( x , η ) = { λ if η ( x ) = 0 and | { y x + 𝒩 : η ( y ) = 1 } | T ; 1 if η ( x ) = 1 ; 0 otherwise c(x,\eta)=\left\{\begin{array}[]{l}\lambda\quad\,\text{if}\quad\eta(x)=0\quad% \,\text{and}|\{y\in x+\mathcal{N}:\eta(y)=1\}|\geq T;\\ 1\quad\,\text{if}\quad\eta(x)=1;\\ 0\quad\,\text{otherwise}\end{array}\right.
  197. d , 𝒩 \scriptstyle d,\mathcal{N}
  198. T \scriptstyle T
  199. λ = 1 \scriptstyle\lambda=1
  200. T = 1 \scriptstyle T=1
  201. c ( x , η ) \scriptstyle c(x,\eta)
  202. c ( x , η ) = { 1 if exists one y with | x - y | N and η ( x ) η ( y ) 0 otherwise c(x,\eta)=\left\{\begin{array}[]{l}1\quad\,\text{if exists one}\quad y\quad\,% \text{with}\quad|x-y|\leq N\quad\,\text{and}\quad\eta(x)\neq\eta(y)\\ 0\quad\,\text{otherwise}\\ \end{array}\right.
  203. N \scriptstyle N
  204. 𝒩 \scriptstyle\mathcal{N}
  205. N \scriptstyle N
  206. 𝒩 1 = { - 2 , - 1 , 0 , 1 , 2 } \scriptstyle\mathcal{N}_{1}=\{-2,-1,0,1,2\}
  207. N 1 = 2 \scriptstyle N_{1}=2
  208. 𝒩 2 = { ( 0 , 0 ) , ( 0 , 1 ) , ( 1 , 0 ) , ( 0 , - 1 ) , ( - 1 , 0 ) } \scriptstyle\mathcal{N}_{2}=\{(0,0),(0,1),(1,0),(0,-1),(-1,0)\}
  209. N 2 = 1 \scriptstyle N_{2}=1
  210. d = 1 \scriptstyle d=1
  211. 𝒩 = { - 1 , 0 , 1 } \scriptstyle\mathcal{N}=\{-1,0,1\}
  212. d \scriptstyle d
  213. 𝒩 \scriptstyle\mathcal{N}
  214. N 1 \scriptstyle N\geq 1
  215. ( N , d ) ( 1 , 1 ) \scriptstyle(N,d)\neq(1,1)
  216. Z d \scriptstyle Z^{d}
  217. N \scriptstyle N

VTPR.html

  1. P = R T v - b - a α ( T ) v 2 + 2 b v - b 2 P=\frac{R\;T}{v-b}-\frac{a\;\alpha(T)}{v^{2}+2bv-b^{2}}
  2. α ( T r ) = T r N ( M - 1 ) e x p ( L ( 1 - T r M N ) ) \alpha(T_{r})=T_{r}^{N\left(M-1\right)}exp\left(L\left(1-T_{r}^{MN}\right)\right)
  3. a ( T ) = b ( i x i a i i ( T ) b i i + g r e s E - 0.53087 ) a(T)=b\cdot\left(\sum_{i}{x_{i}}\frac{a_{ii}(T)}{b_{ii}}+\frac{g^{E}_{res}}{-0% .53087}\right)
  4. P r e f = 1 a t m P_{ref}=1\,atm
  5. b i j 3 / 4 = b i i 3 / 4 + b j j 3 / 4 2 b_{ij}^{3/4}=\frac{b_{ii}^{3/4}+b_{jj}^{3/4}}{2}
  6. b i i = 0.0778 R T c , i P c , i b_{ii}=0.0778\cdot\frac{R\cdot T_{c,i}}{P_{c,i}}
  7. b = i j x i x j b i j b=\sum_{i}\sum_{j}x_{i}\;x_{j}\;b_{ij}
  8. c i = v P R , i - v e x p , i c_{i}=v_{PR,i}-v_{exp,i}
  9. P = R T v + c - b - α a ( T ) ( v + c ) ( v + c + b ) + b ( v + c - b ) P=\frac{R\cdot T}{v+c-b}-\frac{\alpha\cdot a(T)}{(v+c)\cdot(v+c+b)+b\cdot(v+c-% b)}
  10. g r e s E = R T x i ln γ r e s , i g_{res}^{E}=R\cdot T\cdot\sum{x_{i}\cdot\ln\;\gamma_{res,i}}

Wald's_martingale.html

  1. { exp ( λ W t - 1 2 λ 2 t ) , t 0 } \left\{\exp\left(\lambda W_{t}-\frac{1}{2}\lambda^{2}t\right),t\geq 0\right\}

Wall's_finiteness_obstruction.html

  1. K ~ 0 ( [ π 1 ( X ) ] ) \widetilde{K}_{0}(\mathbb{Z}[\pi_{1}(X)])
  2. r : K X r:K\to X
  3. i : X K i:X\to K
  4. r i 1 X r\circ i\simeq 1_{X}
  5. r ¯ : K ¯ X \bar{r}:\bar{K}\to X
  6. K ¯ \bar{K}
  7. π n ( r ) \pi_{n}(r)
  8. K ¯ \bar{K}
  9. π n ( r ) \pi_{n}(r)
  10. H n ( X ~ , K ~ ) H_{n}(\widetilde{X},\widetilde{K})
  11. C * ( X ~ ) C_{*}(\widetilde{X})
  12. A * A_{*}
  13. [ π 1 ( X ) ] \mathbb{Z}[\pi_{1}(X)]
  14. H n ( X ~ , K ~ ) H n ( A * ) H_{n}(\widetilde{X},\widetilde{K})\cong H_{n}(A_{*})
  15. w ( X ) = i ( - 1 ) i [ A i ] K ~ 0 ( [ π 1 ( X ) ] ) w(X)=\sum_{i}(-1)^{i}[A_{i}]\in\widetilde{K}_{0}(\mathbb{Z}[\pi_{1}(X)])

Wannier_equation.html

  1. 𝐫 \mathbf{r}
  2. - [ 2 2 2 μ + V ( 𝐫 ) ] ϕ λ ( 𝐫 ) = E λ ϕ λ ( 𝐫 ) , -\left[\frac{\hbar^{2}\nabla^{2}}{2\mu}+V(\mathbf{r})\right]\phi_{\lambda}(% \mathbf{r})=E_{\lambda}\phi_{\lambda}(\mathbf{r})\,,
  3. V ( 𝐫 ) = e 2 4 π ε r ε 0 | 𝐫 | . V(\mathbf{r})=\frac{e^{2}}{4\pi\varepsilon_{r}\varepsilon_{0}|\mathbf{r}|}\,.
  4. \hbar
  5. \nabla
  6. μ \mu
  7. - | e | -|e|
  8. + | e | +|e|
  9. ε r \varepsilon_{r}
  10. ε 0 \varepsilon_{0}
  11. ϕ λ ( 𝐫 ) \phi_{\lambda}(\mathbf{r})
  12. E λ E_{\lambda}
  13. λ \lambda
  14. E λ E_{\lambda}
  15. ε r \varepsilon_{r}
  16. m e m_{e}
  17. μ m e \mu\ll m_{e}
  18. E 𝐤 ϕ λ ( 𝐤 ) - 𝐤 V 𝐤 - 𝐤 ϕ λ ( 𝐤 ) = E λ ϕ λ ( 𝐤 ) , E_{\mathbf{k}}\phi_{\lambda}(\mathbf{k})-\sum_{\mathbf{k^{\prime}}}V_{\mathbf{% k}-\mathbf{k^{\prime}}}\phi_{\lambda}(\mathbf{k^{\prime}})=E_{\lambda}\phi_{% \lambda}(\mathbf{k})\,,
  19. 𝐤 \mathbf{k}
  20. E 𝐤 E_{\mathbf{k}}
  21. V 𝐤 V_{\mathbf{k}}
  22. ϕ λ ( 𝐤 ) \phi_{\lambda}(\mathbf{k})
  23. V ( 𝐫 ) V(\mathbf{r})
  24. ϕ λ ( 𝐫 ) \phi_{\lambda}(\mathbf{r})
  25. 𝐤 \mathbf{k}
  26. ϵ ~ 𝐤 ϕ λ R ( 𝐤 ) - 𝐤 V 𝐤 - 𝐤 eff ϕ λ R ( 𝐤 ) = ϵ λ ϕ λ R ( 𝐤 ) , \tilde{\epsilon}_{\mathbf{k}}\phi_{\lambda}^{\mathrm{R}}(\mathbf{k})-\sum_{% \mathbf{k^{\prime}}}V_{\mathbf{k}-\mathbf{k^{\prime}}}^{\mathrm{eff}}\phi_{% \lambda}^{\mathrm{R}}(\mathbf{k^{\prime}})=\epsilon_{\lambda}\phi_{\lambda}^{% \mathrm{R}}(\mathbf{k})\,,
  27. ϵ ~ 𝐤 = E 𝐤 - 𝐤 V 𝐤 - 𝐤 ( f 𝐤 e + f 𝐤 h ) , \tilde{\epsilon}_{\mathbf{k}}=E_{\mathbf{k}}-\sum_{\mathbf{k}^{\prime}}V_{{% \mathbf{k}}^{\prime}-{\mathbf{k}}}\left(f^{e}_{\mathbf{k}^{\prime}}+f^{h}_{% \mathbf{k}^{\prime}}\right)\,,
  28. f 𝐤 e f^{e}_{\mathbf{k}}
  29. f 𝐤 h f^{h}_{\mathbf{k}}
  30. V 𝐤 - 𝐤 eff ( 1 - f 𝐤 e - f 𝐤 h ) V 𝐤 - 𝐤 , V_{\mathbf{k}-\mathbf{k^{\prime}}}^{\mathrm{eff}}\equiv(1-f^{\mathrm{e}}_{% \mathbf{k}}-f^{\mathrm{h}}_{\mathbf{k}})V_{\mathbf{k}-\mathbf{k^{\prime}}}\,,
  31. ( 1 - f 𝐤 e - f 𝐤 h ) (1-f^{\mathrm{e}}_{\mathbf{k}}-f^{\mathrm{h}}_{\mathbf{k}})
  32. f 𝐤 e + f 𝐤 h > 1 f^{\mathrm{e}}_{\mathbf{k}}+f^{\mathrm{h}}_{\mathbf{k}}>1
  33. ϕ λ L ( 𝐤 ) \phi_{\lambda}^{\mathrm{L}}(\mathbf{k})
  34. ϕ λ R ( 𝐤 ) \phi_{\lambda}^{\mathrm{R}}(\mathbf{k})
  35. ϕ λ L ( 𝐤 ) = ϕ λ R ( 𝐤 ) / ( 1 - f 𝐤 e - f 𝐤 h ) \phi_{\lambda}^{\mathrm{L}}(\mathbf{k})=\phi_{\lambda}^{\mathrm{R}}(\mathbf{k}% )/(1-f^{\mathrm{e}}_{\mathbf{k}}-f^{\mathrm{h}}_{\mathbf{k}})
  36. E λ E_{\lambda}
  37. 𝐤 [ ϕ λ L ( 𝐤 ) ] ϕ ν R ( 𝐤 ) = 𝐤 [ ϕ λ R ( 𝐤 ) ] ϕ ν L ( 𝐤 ) = δ λ , ν \sum_{\mathbf{k}}\left[\phi^{L}_{\lambda}(\mathbf{k})\right]^{\star}\,\phi^{R}% _{\nu}(\mathbf{k})=\sum_{\mathbf{k}}\left[\phi^{R}_{\lambda}(\mathbf{k})\right% ]^{\star}\,\phi^{L}_{\nu}(\mathbf{k})=\delta_{\lambda,\nu}
  38. E λ E_{\lambda}
  39. E λ E_{\lambda}
  40. λ \lambda
  41. V 𝐤 V_{\mathbf{k}}

Ward's_conjecture.html

  1. F = - F F=-\star F

Water_chlorination.html

  1. \overrightarrow{\leftarrow}

Wave_maps_equation.html

  1. D α α u = 0 D^{\alpha}\partial_{\alpha}u=0\,
  2. D D

Wave_surface.html

  1. a 2 x 2 x 2 + y 2 + z 2 - a 2 w 2 + b 2 y 2 x 2 + y 2 + z 2 - b 2 w 2 + c 2 z 2 x 2 + y 2 + z 2 - c 2 w 2 = 0 \frac{a^{2}x^{2}}{x^{2}+y^{2}+z^{2}-a^{2}w^{2}}+\frac{b^{2}y^{2}}{x^{2}+y^{2}+% z^{2}-b^{2}w^{2}}+\frac{c^{2}z^{2}}{x^{2}+y^{2}+z^{2}-c^{2}w^{2}}=0

Wavelet_Tree.html

  1. 𝐫𝐚𝐧𝐤 q \mathbf{rank}_{q}
  2. 𝐬𝐞𝐥𝐞𝐜𝐭 q \mathbf{select}_{q}
  3. Σ \Sigma
  4. σ = | Σ | \sigma=|\Sigma|
  5. s Σ * s\in\Sigma^{*}
  6. n H 0 ( s ) + o ( | s | log σ ) nH_{0}(s)+o(|s|\log\sigma)
  7. H 0 ( s ) H_{0}(s)
  8. s s
  9. 𝐚𝐜𝐜𝐞𝐬𝐬 \mathbf{access}
  10. 𝐫𝐚𝐧𝐤 q \mathbf{rank}_{q}
  11. 𝐬𝐞𝐥𝐞𝐜𝐭 q \mathbf{select}_{q}
  12. O ( log σ ) O(\log\sigma)

Weber_electrodynamics.html

  1. q 1 q_{1}
  2. q 2 q_{2}
  3. 𝐅 = q 1 q 2 𝐫 ^ 4 π ϵ 0 r 2 ( 1 - r ˙ 2 2 c 2 + r r ¨ c 2 ) \mathbf{F}=\frac{q_{1}q_{2}\mathbf{\hat{r}}}{4\pi\epsilon_{0}r^{2}}\left(1-% \frac{\dot{r}^{2}}{2c^{2}}+\frac{r\ddot{r}}{c^{2}}\right)
  4. 𝐫 \mathbf{r}
  5. q 1 q_{1}
  6. q 2 q_{2}
  7. r r
  8. c c
  9. r ˙ c \dot{r}\ll c
  10. U W e b = q 1 q 2 4 π ϵ 0 r ( 1 - r ˙ 2 2 c 2 ) U_{Web}=\frac{q_{1}q_{2}}{4\pi\epsilon_{0}r}\left(1-\frac{\dot{r}^{2}}{2c^{2}}\right)
  11. v 1 v_{1}
  12. v 2 v_{2}
  13. q 1 q_{1}
  14. q 2 q_{2}
  15. U M a x = q 1 q 2 4 π ϵ 0 r ( 1 - 𝐯 𝟏 𝐯 𝟐 + ( 𝐯 𝟏 𝐫 ^ ) ( 𝐯 𝟐 𝐫 ^ ) 2 c 2 ) U_{Max}=\frac{q_{1}q_{2}}{4\pi\epsilon_{0}r}\left(1-\frac{\mathbf{v_{1}}\cdot% \mathbf{v_{2}}+(\mathbf{v_{1}}\cdot\mathbf{\hat{r}})(\mathbf{v_{2}}\cdot% \mathbf{\hat{r}})}{2c^{2}}\right)
  16. ( v / c ) 2 (v/c)^{2}

Wehrl_entropy.html

  1. Q Q
  2. S Q = - Q ( x , p ) log Q ( x , p ) d x d p . S_{Q}=-\int Q(x,p)\log Q(x,p)\,dx\,dp~{}.
  3. exp ( 2 ( x - i p ) ( x + i p ) ) , \star\equiv\exp\left(\frac{\hbar}{2}({\stackrel{\leftarrow}{\partial}}_{x}-i{% \stackrel{\leftarrow}{\partial}}_{p})({\stackrel{\rightarrow}{\partial}}_{x}+i% {\stackrel{\rightarrow}{\partial}}_{p})\right)~{},
  4. ħ ħ
  5. Q Q

Weighted_constraint_satisfaction_problem.html

  1. X , C , k \langle X,C,k\rangle
  2. X X
  3. C C
  4. k > 0 k>0
  5. \infty
  6. c S C c_{S}\in C
  7. S S
  8. l ( S ) l(S)
  9. 0 , , k \langle 0,...,k\rangle
  10. l ( S ) l(S)
  11. S S
  12. I l ( S ) I\in l(S)
  13. k k
  14. c S ( I ) = k c_{S}(I)=k
  15. \oplus
  16. α , β 0 , , k , α β = m i n ( k , α + β ) \forall\alpha,\beta\in\langle 0,...,k\rangle,\alpha\oplus\beta=min(k,\alpha+\beta)
  17. \oplus
  18. \ominus
  19. 0 β α < k 0\leq\beta\leq\alpha<k
  20. α β = α - β \alpha\ominus\beta=\alpha-\beta
  21. 0 β < k 0\leq\beta<k
  22. k β = k k\ominus\beta=k
  23. c c
  24. c x c_{x}
  25. x x
  26. I l ( S ) I\in l(S)
  27. c S c_{S}
  28. I I
  29. c S c_{S}
  30. c c_{\emptyset}
  31. I I
  32. S S
  33. c c
  34. c c
  35. l b lb
  36. u b ub
  37. l b u b lb\geq ub
  38. G A C w - W S T R GAC^{w}-WSTR

Weyl's_theorem_on_complete_reducibility.html

  1. 𝔤 \mathfrak{g}
  2. 𝔤 \mathfrak{g}

Weyl_metrics.html

  1. ( 1 ) d s 2 = - e 2 ψ ( ρ , z ) d t 2 + e 2 γ ( ρ , z ) - 2 ψ ( ρ , z ) ( d ρ 2 + d z 2 ) + e - 2 ψ ( ρ , z ) ρ 2 d ϕ 2 , (1)\quad ds^{2}=-e^{2\psi(\rho,z)}dt^{2}+e^{2\gamma(\rho,z)-2\psi(\rho,z)}(d% \rho^{2}+dz^{2})+e^{-2\psi(\rho,z)}\rho^{2}d\phi^{2}\,,
  2. ψ ( ρ , z ) \psi(\rho,z)
  3. γ ( ρ , z ) \gamma(\rho,z)
  4. { ρ , z } \{\rho\,,z\}
  5. { t , ρ , z , ϕ } \{t,\rho,z,\phi\}
  6. ξ t = t \xi^{t}=\partial_{t}
  7. ξ ϕ = ϕ \xi^{\phi}=\partial_{\phi}
  8. { ρ , z } \{\rho\,,z\}
  9. T a b T_{ab}
  10. ( 2 ) R a b - 1 2 R g a b = 8 π T a b , (2)\quad R_{ab}-\frac{1}{2}Rg_{ab}=8\pi T_{ab}\,,
  11. ψ ( ρ , z ) \psi(\rho,z)
  12. γ ( ρ , z ) \gamma(\rho,z)
  13. T a b T_{ab}
  14. A a A_{a}
  15. F a b F_{ab}
  16. T a b T_{ab}
  17. ( T = g a b T a b = 0 ) (T=g^{ab}T_{ab}=0)
  18. ( 3 ) F a b = A b ; a - A a ; b = A b , a - A a , b (3)\quad F_{ab}=A_{b\,;\,a}-A_{a\,;\,b}=A_{b\,,\,a}-A_{a\,,\,b}
  19. ( 4 ) T a b = 1 4 π ( F a c F b c - 1 4 g a b F c d F c d ) , (4)\quad T_{ab}=\frac{1}{4\pi}\,\Big(\,F_{ac}F_{b}^{\;c}-\frac{1}{4}g_{ab}F_{% cd}F^{cd}\Big)\,,
  20. ( 5. a ) ( F a b ) ; b = 0 , F [ a b ; c ] = 0 . (5.a)\quad\big(F^{ab}\big)_{;\,b}=0\,,\quad F_{[ab\,;\,c]}=0\,.
  21. ( 5. b ) ( - g F a b ) , b = 0 , F [ a b , c ] = 0 (5.b)\quad\big(\sqrt{-g}\,F^{ab}\big)_{,\,b}=0\,,\quad F_{[ab\,,\,c]}=0
  22. Γ b c a = Γ c b a \Gamma^{a}_{bc}=\Gamma^{a}_{cb}
  23. R = - 8 π T = 0 R=-8\pi T=0
  24. ( 6 ) R a b = 8 π T a b . (6)\quad R_{ab}=8\pi T_{ab}\,.
  25. A a = Φ ( ρ , z ) [ d t ] a A_{a}=\Phi(\rho,z)[dt]_{a}
  26. Φ \Phi
  27. ( 7. a ) 2 ψ = ( ψ ) 2 + γ , ρ ρ + γ , z z (7.a)\quad\nabla^{2}\psi=\,(\nabla\psi)^{2}+\gamma_{,\,\rho\rho}+\gamma_{,\,zz}
  28. ( 7. b ) 2 ψ = e - 2 ψ ( Φ ) 2 (7.b)\quad\nabla^{2}\psi=\,e^{-2\psi}(\nabla\Phi)^{2}
  29. ( 7. c ) 1 ρ γ , ρ = ψ , ρ 2 - ψ , z 2 - e - 2 ψ ( Φ , ρ 2 - Φ , z 2 ) (7.c)\quad\frac{1}{\rho}\,\gamma_{,\,\rho}=\,\psi^{2}_{,\,\rho}-\psi^{2}_{,\,z% }-e^{-2\psi}\big(\Phi^{2}_{,\,\rho}-\Phi^{2}_{,\,z}\big)
  30. ( 7. d ) 1 ρ γ , z = 2 ψ , ρ ψ , z - 2 e - 2 ψ Φ , ρ Φ , z (7.d)\quad\frac{1}{\rho}\,\gamma_{,\,z}=\,2\psi_{,\,\rho}\psi_{,\,z}-2e^{-2% \psi}\Phi_{,\,\rho}\Phi_{,\,z}
  31. ( 7. e ) 2 Φ = 2 ψ Φ , (7.e)\quad\nabla^{2}\Phi=\,2\nabla\psi\nabla\Phi\,,
  32. R = 0 R=0
  33. R t t = 8 π T t t R_{tt}=8\pi T_{tt}
  34. R φ φ = 8 π T φ φ R_{\varphi\varphi}=8\pi T_{\varphi\varphi}
  35. R ρ ρ = 8 π T ρ ρ R_{\rho\rho}=8\pi T_{\rho\rho}
  36. R z z = 8 π T z z R_{zz}=8\pi T_{zz}
  37. R ρ z = 8 π T ρ z R_{\rho z}=8\pi T_{\rho z}
  38. 2 = ρ ρ + 1 ρ ρ + z z \nabla^{2}=\partial_{\rho\rho}+\frac{1}{\rho}\,\partial_{\rho}+\partial_{zz}
  39. = ρ e ^ ρ + z e ^ z \nabla=\partial_{\rho}\,\hat{e}_{\rho}+\partial_{z}\,\hat{e}_{z}
  40. ψ = ψ ( Φ ) \psi=\psi(\Phi)
  41. ( 7. f ) e ψ = Φ 2 - 2 C Φ + 1 . (7.f)\quad e^{\psi}=\,\Phi^{2}-2C\Phi+1\,.
  42. Φ = 0 \Phi=0
  43. T a b = 0 T_{ab}=0
  44. ( 8. a ) γ , ρ ρ + γ , z z = - ( ψ ) 2 (8.a)\quad\gamma_{,\,\rho\rho}+\gamma_{,\,zz}=-(\nabla\psi)^{2}
  45. ( 8. b ) 2 ψ = 0 (8.b)\quad\nabla^{2}\psi=0
  46. ( 8. c ) γ , ρ = ρ ( ψ , ρ 2 - ψ , z 2 ) (8.c)\quad\gamma_{,\,\rho}=\rho\,\Big(\psi^{2}_{,\,\rho}-\psi^{2}_{,\,z}\Big)
  47. ( 8. d ) γ , z = 2 ρ ψ , ρ ψ , z . (8.d)\quad\gamma_{,\,z}=2\,\rho\,\psi_{,\,\rho}\psi_{,\,z}\,.
  48. ψ ( ρ , z ) \psi(\rho,z)
  49. γ ( ρ , z ) \gamma(\rho,z)
  50. R = 0 R=0
  51. ( A .1. a ) ψ , ρ ρ + 1 ρ ψ , ρ + ψ , z z = ( ψ , ρ ) 2 + ( ψ , z ) 2 + γ , ρ ρ + γ , z z (A.1.a)\quad\psi_{,\,\rho\rho}+\frac{1}{\rho}\psi_{,\,\rho}+\psi_{,\,zz}=\,(% \psi_{,\,\rho})^{2}+(\psi_{,\,z})^{2}+\gamma_{,\,\rho\rho}+\gamma_{,\,zz}
  52. ( A .1. b ) ψ , ρ ρ + 1 ρ ψ , ρ + ψ , z z = e - 2 ψ ( Φ , ρ 2 + Φ , z 2 ) (A.1.b)\quad\psi_{,\,\rho\rho}+\frac{1}{\rho}\psi_{,\,\rho}+\psi_{,\,zz}=e^{-2% \psi}\big(\Phi^{2}_{,\,\rho}+\Phi^{2}_{,\,z}\big)
  53. ( A .1. c ) 1 ρ γ , ρ = ψ , ρ 2 - ψ , z 2 - e - 2 ψ ( Φ , ρ 2 - Φ , z 2 ) (A.1.c)\quad\frac{1}{\rho}\,\gamma_{,\,\rho}=\,\psi^{2}_{,\,\rho}-\psi^{2}_{,% \,z}-e^{-2\psi}\big(\Phi^{2}_{,\,\rho}-\Phi^{2}_{,\,z}\big)
  54. ( A .1. d ) 1 ρ γ , z = 2 ψ , ρ ψ , z - 2 e - 2 ψ Φ , ρ Φ , z (A.1.d)\quad\frac{1}{\rho}\,\gamma_{,\,z}=\,2\psi_{,\,\rho}\psi_{,\,z}-2e^{-2% \psi}\Phi_{,\,\rho}\Phi_{,\,z}
  55. ( A .1. e ) Φ , ρ ρ + 1 ρ Φ , ρ + Φ , z z = 2 ψ , ρ Φ , ρ + 2 ψ , z Φ , z (A.1.e)\quad\Phi_{,\,\rho\rho}+\frac{1}{\rho}\Phi_{,\,\rho}+\Phi_{,\,zz}=\,2% \psi_{,\,\rho}\Phi_{,\,\rho}+2\psi_{,\,z}\Phi_{,\,z}
  56. ( A .2. a ) ( ψ , ρ ) 2 + ( ψ , z ) 2 = - γ , ρ ρ - γ , z z (A.2.a)\quad(\psi_{,\,\rho})^{2}+(\psi_{,\,z})^{2}=-\gamma_{,\,\rho\rho}-% \gamma_{,\,zz}
  57. ( A .2. b ) ψ , ρ ρ + 1 ρ ψ , ρ + ψ , z z = 0 (A.2.b)\quad\psi_{,\,\rho\rho}+\frac{1}{\rho}\psi_{,\,\rho}+\psi_{,\,zz}=0
  58. ( A .2. c ) γ , ρ = ρ ( ψ , ρ 2 - ψ , z 2 ) (A.2.c)\quad\gamma_{,\,\rho}=\rho\,\Big(\psi^{2}_{,\,\rho}-\psi^{2}_{,\,z}\Big)
  59. ( A .2. d ) γ , z = 2 ρ ψ , ρ ψ , z . (A.2.d)\quad\gamma_{,\,z}=2\,\rho\,\psi_{,\,\rho}\psi_{,\,z}\,.
  60. ψ Φ \psi\sim\Phi
  61. ψ ( ρ , z ) \psi(\rho,z)
  62. Φ ( ρ , z ) \Phi(\rho,z)
  63. ψ = ψ ( Φ ) \psi=\psi(\Phi)
  64. ( B .1 ) ψ , i = ψ , Φ Φ , i , ψ = ψ , Φ Φ , 2 ψ = ψ , Φ 2 Φ + ψ , Φ Φ ( Φ ) 2 , (B.1)\quad\psi_{,\,i}=\psi_{,\,\Phi}\cdot\Phi_{,\,i}\quad,\quad\nabla\psi=\psi% _{,\,\Phi}\cdot\nabla\Phi\quad,\quad\nabla^{2}\psi=\psi_{,\,\Phi}\cdot\nabla^{% 2}\Phi+\psi_{,\,\Phi\Phi}\cdot(\nabla\Phi)^{2},
  65. ( B .2 ) Ψ , Φ 2 Φ = ( e - 2 ψ - ψ , Φ Φ ) ( Φ ) 2 , (B.2)\quad\Psi_{,\,\Phi}\cdot\nabla^{2}\Phi\,=\,\big(e^{-2\psi}-\psi_{,\,\Phi% \Phi}\big)\cdot(\nabla\Phi)^{2},
  66. ( B .3 ) 2 Φ = 2 ψ , Φ ( Φ ) 2 , (B.3)\quad\nabla^{2}\Phi\,=\,2\psi_{,\,\Phi}\cdot(\nabla\Phi)^{2},
  67. ( B .4 ) ψ , Φ Φ + 2 ( ψ , Φ ) 2 - e - 2 ψ = 0. (B.4)\quad\psi_{,\,\Phi\Phi}+2\,\big(\psi_{,\,\Phi}\big)^{2}-e^{-2\psi}=0.
  68. ψ \psi
  69. ζ := e 2 ψ \zeta:=e^{2\psi}
  70. ( B .5 ) ζ , Φ Φ - 2 = 0. (B.5)\quad\zeta_{,\,\Phi\Phi}-2=0.
  71. ζ = e 2 ψ = Φ 2 + C ~ Φ + B \zeta=e^{2\psi}=\Phi^{2}+\tilde{C}\Phi+B
  72. { B , C ~ } \{B,\tilde{C}\}
  73. lim ρ , z Φ = 0 \lim_{\rho,z\to\infty}\Phi=0
  74. lim ρ , z e 2 ψ = 1 \lim_{\rho,z\to\infty}e^{2\psi}=1
  75. B = 1 B=1
  76. C ~ \tilde{C}
  77. - 2 C -2C
  78. ( 7. f ) e 2 ψ = Φ 2 - 2 C Φ + 1 . (7.f)\quad e^{2\psi}=\Phi^{2}-2C\Phi+1\,.
  79. e ± 2 ψ = n = 0 ( ± 2 ψ ) n n ! e^{\pm 2\psi}=\sum_{n=0}^{\infty}\frac{(\pm 2\psi)^{n}}{n!}
  80. ψ 0 \psi\to 0
  81. ( 9 ) g t t = - ( 1 + 2 ψ ) - 𝒪 ( ψ 2 ) , g ϕ ϕ = 1 - 2 ψ + 𝒪 ( ψ 2 ) , (9)\quad g_{tt}=-(1+2\psi)-\mathcal{O}(\psi^{2})\,,\quad g_{\phi\phi}=1-2\psi+% \mathcal{O}(\psi^{2})\,,
  82. ( 10 ) d s 2 - ( 1 + 2 ψ ( ρ , z ) ) d t 2 + ( 1 - 2 ψ ( ρ , z ) ) [ e 2 γ ( d ρ 2 + d z 2 ) + ρ 2 d ϕ 2 ] . (10)\quad ds^{2}\approx-\Big(1+2\psi(\rho,z)\Big)\,dt^{2}+\Big(1-2\psi(\rho,z)% \Big)\Big[e^{2\gamma}(d\rho^{2}+dz^{2})+\rho^{2}d\phi^{2}\Big]\,.
  83. ( 11 ) d s 2 = - ( 1 + 2 Φ N ( ρ , z ) ) d t 2 + ( 1 - 2 Φ N ( ρ , z ) ) [ d ρ 2 + d z 2 + ρ 2 d ϕ 2 ] . (11)\quad ds^{2}=-\Big(1+2\Phi_{N}(\rho,z)\Big)\,dt^{2}+\Big(1-2\Phi_{N}(\rho,% z)\Big)\,\Big[d\rho^{2}+dz^{2}+\rho^{2}d\phi^{2}\Big]\,.
  84. Φ N ( ρ , z ) \Phi_{N}(\rho,z)
  85. L 2 Φ N = 4 π ϱ N \nabla^{2}_{L}\Phi_{N}=4\pi\varrho_{N}
  86. ψ ( ρ , z ) \psi(\rho,z)
  87. ψ ( ρ , z ) \psi(\rho,z)
  88. Φ N ( ρ , z ) \Phi_{N}(\rho,z)
  89. ψ ( ρ , z ) \psi(\rho,z)
  90. ψ ( ρ , z ) \psi(\rho,z)
  91. ψ ( ρ , z ) \psi(\rho,z)
  92. ( 12 ) ψ S S = 1 2 ln L - M L + M , γ S S = 1 2 ln L 2 - M 2 l + l - , (12)\quad\psi_{SS}=\frac{1}{2}\ln\frac{L-M}{L+M}\,,\quad\gamma_{SS}=\frac{1}{2% }\ln\frac{L^{2}-M^{2}}{l_{+}l_{-}}\,,
  93. ( 13 ) L = 1 2 ( l + + l - ) , l + = ρ 2 + ( z + M ) 2 , l - = ρ 2 + ( z - M ) 2 . (13)\quad L=\frac{1}{2}\big(l_{+}+l_{-}\big)\,,\quad l_{+}=\sqrt{\rho^{2}+(z+M% )^{2}}\,,\quad l_{-}=\sqrt{\rho^{2}+(z-M)^{2}}\,.
  94. ψ S S \psi_{SS}
  95. M M
  96. 2 M 2M
  97. z z
  98. σ = 1 / 2 \sigma=1/2
  99. z [ - M , M ] z\in[-M,M]
  100. ψ S S \psi_{SS}
  101. γ S S \gamma_{SS}
  102. ( 14 ) d s 2 = - L - M L + M d t 2 + ( L + M ) 2 l + l - ( d ρ 2 + d z 2 ) + L + M L - M ρ 2 d ϕ 2 , (14)\quad ds^{2}=-\frac{L-M}{L+M}dt^{2}+\frac{(L+M)^{2}}{l_{+}l_{-}}(d\rho^{2}% +dz^{2})+\frac{L+M}{L-M}\,\rho^{2}d\phi^{2}\,,
  103. ( 15 ) L + M = r , l + - l - = 2 M cos θ , z = ( r - M ) cos θ , (15)\quad L+M=r\,,\quad l_{+}-l_{-}=2M\cos\theta\,,\quad z=(r-M)\cos\theta\,,
  104. ρ = r 2 - 2 M r sin θ , l + l - = ( r - M ) 2 - M 2 cos 2 θ , \;\;\quad\rho=\sqrt{r^{2}-2Mr}\,\sin\theta\,,\quad l_{+}l_{-}=(r-M)^{2}-M^{2}% \cos^{2}\theta\,,
  105. { t , r , θ , ϕ } \{t,r,\theta,\phi\}
  106. ( 16 ) d s 2 = - ( 1 - 2 M r ) d t 2 + ( 1 - 2 M r ) - 1 d r 2 + r 2 d θ 2 + r 2 sin 2 θ d ϕ 2 . (16)\quad ds^{2}=-\Big(1-\frac{2M}{r}\Big)\,dt^{2}+\Big(1-\frac{2M}{r}\Big)^{-% 1}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta\,d\phi^{2}\,.
  107. ( t , ρ , z , ϕ ) = ( t , r sin θ , r cos θ , ϕ ) (t,\rho,z,\phi)=(t,r\sin\theta,r\cos\theta,\phi)
  108. { t , r , θ , ϕ } \{t,r,\theta,\phi\}
  109. ( t , ρ , z , ϕ ) (t,\rho,z,\phi)
  110. { t , ρ , z , ϕ } \{t,\rho,z,\phi\}
  111. 2 := ρ ρ + 1 ρ ρ + z z \nabla^{2}:=\partial_{\rho\rho}+\frac{1}{\rho}\partial_{\rho}+\partial_{zz}
  112. M > | Q | M>|Q|
  113. ( 17 ) ψ R N = 1 2 ln L 2 - ( M 2 - Q 2 ) ( L + M ) 2 , γ R N = 1 2 ln L 2 - ( M 2 - Q 2 ) l + l - , (17)\quad\psi_{RN}=\frac{1}{2}\ln\frac{L^{2}-(M^{2}-Q^{2})}{(L+M)^{2}}\,,\quad% \gamma_{RN}=\frac{1}{2}\ln\frac{L^{2}-(M^{2}-Q^{2})}{l_{+}l_{-}}\,,
  114. ( 18 ) L = 1 2 ( l + + l - ) , l + = ρ 2 + ( z + M 2 - Q 2 ) 2 , l - = ρ 2 + ( z - M 2 - Q 2 ) 2 . (18)\quad L=\frac{1}{2}\big(l_{+}+l_{-}\big)\,,\quad l_{+}=\sqrt{\rho^{2}+(z+% \sqrt{M^{2}-Q^{2}})^{2}}\,,\quad l_{-}=\sqrt{\rho^{2}+(z-\sqrt{M^{2}-Q^{2}})^{% 2}}\,.
  115. ψ R N \psi_{RN}
  116. γ R N \gamma_{RN}
  117. ( 19 ) d s 2 = - L 2 - ( M 2 - Q 2 ) ( L + M ) 2 d t 2 + ( L + M ) 2 l + l - ( d ρ 2 + d z 2 ) + ( L + M ) 2 L 2 - ( M 2 - Q 2 ) ρ 2 d ϕ 2 , (19)\quad ds^{2}=-\frac{L^{2}-(M^{2}-Q^{2})}{(L+M)^{2}}dt^{2}+\frac{(L+M)^{2}}% {l_{+}l_{-}}(d\rho^{2}+dz^{2})+\frac{(L+M)^{2}}{L^{2}-(M^{2}-Q^{2})}\rho^{2}d% \phi^{2}\,,
  118. ( 20 ) L + M = r , l + + l - = 2 M 2 - Q 2 cos θ , z = ( r - M ) cos θ , (20)\quad L+M=r\,,\quad l_{+}+l_{-}=2\sqrt{M^{2}-Q^{2}}\,\cos\theta\,,\quad z=% (r-M)\cos\theta\,,
  119. ρ = r 2 - 2 M r + Q 2 sin θ , l + l - = ( r - M ) 2 - ( M 2 - Q 2 ) cos 2 θ , \;\;\quad\rho=\sqrt{r^{2}-2Mr+Q^{2}}\,\sin\theta\,,\quad l_{+}l_{-}=(r-M)^{2}-% (M^{2}-Q^{2})\cos^{2}\theta\,,
  120. { t , r , θ , ϕ } \{t,r,\theta,\phi\}
  121. ( 21 ) d s 2 = - ( 1 - 2 M r + Q 2 r 2 ) d t 2 + ( 1 - 2 M r + Q 2 r 2 ) - 1 d r 2 + r 2 d θ 2 + r 2 sin 2 θ d ϕ 2 . (21)\quad ds^{2}=-\Big(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\Big)\,dt^{2}+\Big(1-% \frac{2M}{r}+\frac{Q^{2}}{r^{2}}\Big)^{-1}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2% }\theta\,d\phi^{2}\,.
  122. M = | Q | M=|Q|
  123. ( 22 ) ψ E R N = 1 2 ln L 2 ( L + M ) 2 , γ E R N = 0 , with L = ρ 2 + z 2 . (22)\quad\psi_{ERN}=\frac{1}{2}\ln\frac{L^{2}}{(L+M)^{2}}\,,\quad\gamma_{ERN}=% 0\,,\quad\,\text{with}\quad L=\sqrt{\rho^{2}+z^{2}}\,.
  124. ( 23 ) d s 2 = - L 2 ( L + M ) 2 d t 2 + ( L + M ) 2 L 2 ( d ρ 2 + d z 2 + ρ 2 d ϕ 2 ) , (23)\quad ds^{2}=-\frac{L^{2}}{(L+M)^{2}}dt^{2}+\frac{(L+M)^{2}}{L^{2}}(d\rho^% {2}+dz^{2}+\rho^{2}d\phi^{2})\,,
  125. ( 24 ) L + M = r , z = L cos θ , ρ = L sin θ , (24)\quad L+M=r\,,\quad z=L\cos\theta\,,\quad\rho=L\sin\theta\,,
  126. { t , r , θ , ϕ } \{t,r,\theta,\phi\}
  127. ( 25 ) d s 2 = - ( 1 - M r ) 2 d t 2 + ( 1 - M r ) - 2 d r 2 + r 2 d θ 2 + r 2 sin 2 θ d ϕ 2 . (25)\quad ds^{2}=-\Big(1-\frac{M}{r}\Big)^{2}dt^{2}+\Big(1-\frac{M}{r}\Big)^{-% 2}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta\,d\phi^{2}\,.
  128. Q M Q\to M
  129. γ ( ρ , z ) \gamma(\rho,z)
  130. ψ ( ρ , z ) \psi(\rho,z)
  131. ( 26 ) d s 2 = - e 2 λ ( ρ , z , ϕ ) d t 2 + e - 2 λ ( ρ , z , ϕ ) ( d ρ 2 + d z 2 + ρ 2 d ϕ 2 ) , (26)\quad ds^{2}\,=-e^{2\lambda(\rho,z,\phi)}dt^{2}+e^{-2\lambda(\rho,z,\phi)}% \Big(d\rho^{2}+dz^{2}+\rho^{2}d\phi^{2}\Big)\,,
  132. λ \lambda
  133. ψ \psi
  134. ϕ \phi
  135. ( 27 ) d s 2 = - e 2 ψ ( r , θ ) d t 2 + e 2 γ ( r , θ ) - 2 ψ ( r , θ ) ( d r 2 + r 2 d θ 2 ) + e - 2 ψ ( r , θ ) ρ 2 d ϕ 2 , (27)\quad ds^{2}\,=-e^{2\psi(r,\theta)}dt^{2}+e^{2\gamma(r,\theta)-2\psi(r,% \theta)}(dr^{2}+r^{2}d\theta^{2})+e^{-2\psi(r,\theta)}\rho^{2}d\phi^{2}\,,
  136. ( t , ρ , z , ϕ ) ( t , r sin θ , r cos θ , ϕ ) (t,\rho,z,\phi)\mapsto(t,r\sin\theta,r\cos\theta,\phi)
  137. ψ ( r , θ ) \psi(r,\theta)
  138. ( 28 ) r 2 ψ , r r + 2 r ψ , r + ψ , θ θ + cot θ ψ , θ = 0 . (28)\quad r^{2}\psi_{,\,rr}+2r\,\psi_{,\,r}+\psi_{,\,\theta\theta}+\cot\theta% \cdot\psi_{,\,\theta}\,=\,0\,.
  139. ( 29 ) ψ ( r , θ ) = - n = 0 a n P n ( cos θ ) r n + 1 , (29)\quad\psi(r,\theta)\,=-\sum_{n=0}^{\infty}a_{n}\frac{P_{n}(\cos\theta)}{r^% {n+1}}\,,
  140. P n ( cos θ ) P_{n}(\cos\theta)
  141. a n a_{n}
  142. γ ( r , θ ) \gamma(r,\theta)
  143. ( 30 ) γ ( r , θ ) = - l = 0 m = 0 a l a m (30)\quad\gamma(r,\theta)\,=-\sum_{l=0}^{\infty}\sum_{m=0}^{\infty}a_{l}a_{m}
  144. ( l + 1 ) ( m + 1 ) l + m + 2 \frac{(l+1)(m+1)}{l+m+2}
  145. P l P m - P l + 1 P m + 1 r l + m + 2 . \frac{P_{l}P_{m}-P_{l+1}P_{m+1}}{r^{l+m+2}}\,.

White_light_interferometry.html

  1. S ( ν ) = 1 π Δ ν exp [ - ( ( ν - ν 0 ) Δ ν ) 2 ] S(\nu)=\frac{1}{\sqrt{\pi}\Delta\nu}\exp\left[-\left(\frac{\left(\nu-\nu_{0}% \right)}{\Delta\nu}\right)^{2}\right]
  2. 2 Δ ν 2\Delta\nu
  3. k ( τ ) = - S ( ν ) exp ( - i 2 π ν τ ) d ν = exp ( - π 2 τ 2 Δ ν 2 ) exp ( - i 2 π ν 0 τ ) k(\tau)=\int\limits_{-\infty}^{\infty}S(\nu)\exp\left(-i2\pi\nu\tau\right)d\nu% =\exp\left(-\pi^{2}\tau^{2}\Delta\nu^{2}\right)\exp\left(-i2\pi\nu_{0}\tau\right)
  4. I ( z ) = I 0 R e { 1 + k ( τ ) } I(z)=I_{0}\cdot Re\{1+k(\tau)\}
  5. I 0 = I o b j + I r e f I_{0}=I_{obj}+I_{ref}
  6. I o b j I_{obj}
  7. I r e f I_{ref}
  8. ν 0 = c / λ 0 \nu_{0}=c/\lambda_{0}
  9. L c = c / π Δ ν L_{c}=c/\pi\Delta\nu
  10. I ( z ) = I 0 ( 1 + exp [ - 4 ( ( z - z 0 ) L c ) 2 ] cos ( 4 π z - z 0 λ 0 - φ 0 ) ) I(z)=I_{0}\left(1+\exp\left[-4\left(\frac{\left(z-z_{0}\right)}{L_{c}}\right)^% {2}\right]\cos\left(4\pi\frac{z-z_{0}}{\lambda_{0}}-\varphi_{0}\right)\right)
  11. τ = 2 ( z - z 0 ) / c \tau=2\cdot(z-z_{0})/c
  12. λ 0 / 2 \lambda_{0}/2
  13. E ( z ) = exp [ - 4 ( ( z - z 0 ) I c ) 2 ] E(z)=\exp\left[-4\left(\frac{\left(z-z_{0}\right)}{I_{c}}\right)^{2}\right]
  14. E ( z ) = ( exp [ - 4 ( ( z - z 0 ) I c ) 2 ] cos ( 4 π z - z 0 λ 0 ) ) 2 + ( exp [ - 4 ( ( z - z 0 ) I c ) 2 ] sin ( 4 π z - z 0 λ 0 ) ) 2 E(z)=\sqrt{\left(\exp\left[-4\left(\frac{\left(z-z_{0}\right)}{I_{c}}\right)^{% 2}\right]\cos\left(4\pi\frac{z-z_{0}}{\lambda_{0}}\right)\right)^{2}+\left(% \exp\left[-4\left(\frac{\left(z-z_{0}\right)}{I_{c}}\right)^{2}\right]\sin% \left(4\pi\frac{z-z_{0}}{\lambda_{0}}\right)\right)^{2}}

Whitehead's_lemma_(Lie_algebras).html

  1. 𝔤 \mathfrak{g}
  2. f : 𝔤 V f:\mathfrak{g}\to V
  3. f ( [ x , y ] ) = x f ( y ) - y f ( x ) f([x,y])=xf(y)-yf(x)
  4. f ( x ) = x v f(x)=xv

Whitham_equation.html

  1. η t + α η η x + - + K ( x - ξ ) η ( ξ , t ) ξ d ξ = 0. \frac{\partial\eta}{\partial t}+\alpha\eta\frac{\partial\eta}{\partial x}+\int% _{-\infty}^{+\infty}K(x-\xi)\,\frac{\partial\eta(\xi,t)}{\partial\xi}\,\,\text% {d}\xi=0.
  2. c ww ( k ) = g k tanh ( k h ) , c\text{ww}(k)=\sqrt{\frac{g}{k}\,\tanh(kh)},
  3. α ww = 3 2 g h , \alpha\text{ww}=\frac{3}{2}\sqrt{\frac{g}{h}},
  4. K ww ( s ) = 1 2 π - + c ww ( k ) e i k s d k . K\text{ww}(s)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}c\text{ww}(k)\,\,\text{e}^% {iks}\,\,\text{d}k.
  5. c kdv ( k ) = g h ( 1 - 1 6 k 2 h 2 ) , c\text{kdv}(k)=\sqrt{gh}\left(1-\frac{1}{6}k^{2}h^{2}\right),
  6. α kdv = 3 2 g h , \alpha\text{kdv}=\frac{3}{2}\sqrt{\frac{g}{h}},
  7. K fw ( s ) = 1 2 ν e - ν s K\text{fw}(s)=\frac{1}{2}\nu\,\text{e}^{-\nu s}
  8. c fw = ν 2 ν 2 + k 2 , c\text{fw}=\frac{\nu^{2}}{\nu^{2}+k^{2}},
  9. α fw = 3 2 . \alpha\text{fw}=\frac{3}{2}.
  10. ( 2 x 2 - ν 2 ) ( η t + 3 2 η η x ) + η x = 0. \left(\frac{\partial^{2}}{\partial x^{2}}-\nu^{2}\right)\left(\frac{\partial% \eta}{\partial t}+\frac{3}{2}\,\eta\,\frac{\partial\eta}{\partial x}\right)+% \frac{\partial\eta}{\partial x}=0.

William_Houlder_Zachariasen.html

  1. A A
  2. A m O n A_{m}O_{n}

Wiman's_sextic.html

  1. x 6 + y 6 + z 6 + ( x 2 + y 2 + z 2 ) ( x 4 + y 4 + z 4 ) = 12 x 2 y 2 z 2 x^{6}+y^{6}+z^{6}+(x^{2}+y^{2}+z^{2})(x^{4}+y^{4}+z^{4})=12x^{2}y^{2}z^{2}

Wirtinger_presentation.html

  1. w g i w - 1 = g j wg_{i}w^{-1}=g_{j}
  2. w w
  3. { g 1 , g 2 , , g k } \{g_{1},g_{2},\cdots,g_{k}\}
  4. π 1 ( 3 \ trefoil ) = x , y | y - 1 x y = x - 1 y x . \pi_{1}(\mathbb{R}^{3}\backslash\,\text{trefoil})=\langle x,y|y^{-1}xy=x^{-1}% yx\rangle.

Wolfe_duality.html

  1. minimize 𝑥 \displaystyle\underset{x}{\operatorname{minimize}}
  2. maximize 𝑢 \displaystyle\underset{u}{\operatorname{maximize}}
  3. f f
  4. g 1 , , g m g_{1},\ldots,g_{m}
  5. maximize x , u \displaystyle\underset{x,u}{\operatorname{maximize}}
  6. ( u , x ) (u,x)
  7. f ( x ) + j = 1 m u j g j ( x ) \nabla f(x)+\sum_{j=1}^{m}u_{j}\nabla g_{j}(x)

Work_stealing.html

  1. P P
  2. T 1 / P + O ( T ) T_{1}/P+O(T_{\infty})
  3. P P
  4. T 1 T_{1}
  5. T T_{\infty}
  6. O ( S 1 P ) O(S_{1}P)
  7. S 1 S_{1}
  8. O ( T 1 P avg + T P P avg ) O(\frac{T_{1}}{P_{\mathrm{avg}}}+\frac{T_{\infty}P}{P_{\mathrm{avg}}})
  9. P avg P_{\mathrm{avg}}

World_manifold.html

  1. T X TX
  2. X X
  3. F X FX
  4. T X TX
  5. G L + ( 4 , ) GL^{+}(4,\mathbb{R})
  6. X X
  7. T X TX
  8. F X FX
  9. F X FX
  10. G L + ( 4 , ) GL^{+}(4,\mathbb{R})
  11. F X FX
  12. X X
  13. S O ( 4 ) SO(4)
  14. F X / S O ( 4 ) FX/SO(4)
  15. g R g^{R}
  16. X X
  17. X X
  18. F X FX
  19. S O ( 1 , 3 ) SO(1,3)
  20. F X / S O ( 1 , 3 ) FX/SO(1,3)
  21. g g
  22. ( + , - - - ) (+,---)
  23. X X
  24. F X FX
  25. S O ( 3 ) SO(3)
  26. G L ( 4 , ) S O ( 4 ) GL(4,\mathbb{R})\to SO(4)
  27. \downarrow\qquad\qquad\qquad\quad\downarrow
  28. S O ( 1 , 3 ) S O ( 3 ) SO(1,3)\to SO(3)
  29. F X FX
  30. X X
  31. F X FX
  32. { h λ } \{h^{\lambda}\}
  33. S O ( 3 ) SO(3)
  34. h 0 = h 0 μ μ h_{0}=h^{\mu}_{0}\partial_{\mu}
  35. X X
  36. h 0 = h λ 0 d x λ h^{0}=h^{0}_{\lambda}dx^{\lambda}
  37. 𝔉 T X \mathfrak{F}\subset TX
  38. X X
  39. h 0 𝔉 = 0 h^{0}\rfloor\mathfrak{F}=0
  40. T X TX
  41. X X
  42. T X = 𝔉 T 0 X TX=\mathfrak{F}\oplus T^{0}X
  43. T 0 X T^{0}X
  44. h 0 h_{0}
  45. g g
  46. g g
  47. g R g^{R}
  48. X X
  49. ( g , g R , h 0 ) (g,g^{R},h^{0})
  50. g = 2 h 0 h 0 - g R g=2h^{0}\otimes h^{0}-g^{R}
  51. X X
  52. σ \sigma
  53. g R g^{R}
  54. X X
  55. g = 2 g R ( σ , σ ) σ σ - g R g=\frac{2}{g^{R}(\sigma,\sigma)}\sigma\otimes\sigma-g^{R}
  56. X X
  57. X X
  58. g g
  59. g R g^{R}
  60. ( g , g R , h 0 ) (g,g^{R},h^{0})
  61. g g
  62. X X
  63. X X
  64. X X
  65. g g
  66. g g
  67. 𝔉 \mathfrak{F}
  68. 𝔉 \mathfrak{F}^{\prime}
  69. X X
  70. 𝔉 \mathfrak{F}
  71. X X
  72. X X\to\mathbb{R}
  73. \mathbb{R}
  74. X X\to\mathbb{R}
  75. X X
  76. X = × M X=\mathbb{R}\times M

Wu_experiment.html

  1. ν ¯ \overline{ν}
  2. Co 27 60 Ni 28 60 + e - + ν ¯ e + 2 γ {}^{60}_{27}\,\text{Co}\rightarrow{}^{60}_{28}\,\text{Ni}+e^{-}+\bar{\nu}_{e}+% 2{\gamma}

Wu–Sprung_potential.html

  1. H = p 2 + f ( x ) H=p^{2}+f(x)
  2. E n E_{n}
  3. f ( x ) f(x)
  4. p d q = 2 π n ( E ) = 4 0 a d x E n - f ( x ) \oint p\,dq=2\pi n(E)=4\int_{0}^{a}dx\sqrt{E_{n}-f(x)}
  5. E = f ( a ) = f ( - a ) E=f(a)=f(-a)
  6. ξ ( 1 2 + i E n ) = 0 \xi(\frac{1}{2}+i\sqrt{E_{n}})=0
  7. f ( x ) = f ( - x ) f(x)=f(-x)
  8. f - 1 ( x ) = π d 1 / 2 d x 1 / 2 N ( x ) f^{-1}(x)=\sqrt{\pi}\frac{d^{1/2}}{dx^{1/2}}N(x)
  9. N ( x ) = n = 0 H ( x - E n ) N(x)=\sum_{n=0}^{\infty}H(x-E_{n})
  10. f - 1 ( x ) = 2 4 x + 1 + 1 4 π - x x d r x - r 2 ( Γ Γ ( 1 4 + i r 2 ) - ln π ) - n = 1 Λ ( n ) 2 n J 0 ( x ln n ) f^{-1}(x)=\frac{2}{\sqrt{4x+1}}+\frac{1}{4\pi}\int\nolimits_{-\sqrt{x}}^{\sqrt% {x}}\frac{dr}{\sqrt{x-r^{2}}}\left(\frac{\Gamma^{\prime}}{\Gamma}\left(\frac{1% }{4}+\frac{ir}{2}\right)-\ln\pi\right)-\sum\limits_{n=1}^{\infty}\frac{\Lambda% (n)}{2\sqrt{n}}J_{0}\left(\sqrt{x}\ln n\right)
  11. 1 π d 1 / 2 d x 1 / 2 f - 1 ( x ) = n = 0 δ ( x - E n ) \frac{1}{\sqrt{\pi}}\frac{d^{1/2}}{dx^{1/2}}f^{-1}(x)=\sum_{n=0}^{\infty}% \delta(x-E_{n})
  12. n = 0 δ ( x - γ n ) + n = 0 δ ( x + γ n ) = 1 2 π ζ ζ ( 1 2 + i x ) + 1 2 π ζ ζ ( 1 2 - i x ) - ln π 2 π + Γ Γ ( 1 4 + i x 2 ) 1 4 π + Γ Γ ( 1 4 - i x 2 ) 1 4 π + 1 π δ ( x - i 2 ) + 1 π δ ( x + i 2 ) \begin{array}[]{l}\sum\limits_{n=0}^{\infty}\delta\left(x-\gamma_{n}\right)+% \sum\limits_{n=0}^{\infty}\delta\left(x+\gamma_{n}\right)=\frac{1}{2\pi}\frac{% \zeta}{\zeta}\left(\frac{1}{2}+ix\right)+\frac{1}{2\pi}\frac{\zeta^{\prime}}{% \zeta}\left(\frac{1}{2}-ix\right)-\frac{\ln\pi}{2\pi}\\ {}+\frac{\Gamma^{\prime}}{\Gamma}\left(\frac{1}{4}+i\frac{x}{2}\right)\frac{1}% {4\pi}+\frac{\Gamma^{\prime}}{\Gamma}\left(\frac{1}{4}-i\frac{x}{2}\right)% \frac{1}{4\pi}+\frac{1}{\pi}\delta\left(x-\frac{i}{2}\right)+\frac{1}{\pi}% \delta\left(x+\frac{i}{2}\right)\end{array}
  13. 1 2 + i s \frac{1}{2}+is
  14. ζ ( s ) ζ ( s ) = - n = 1 Λ ( n ) e - s l n n \frac{\zeta^{\prime}(s)}{\zeta(s)}=-\sum_{n=1}^{\infty}\Lambda(n)e^{-slnn}
  15. Λ ( n ) \Lambda(n)
  16. ξ ( s ) ξ ( 0 ) = det ( H - s ( 1 - s ) + 1 4 ) det ( H + 1 4 ) \frac{\xi(s)}{\xi(0)}=\frac{\det(H-s(1-s)+\frac{1}{4})}{\det(H+\frac{1}{4})}
  17. N ( E ) E 2 π log ( E 2 π e ) N(E)\sim\frac{\sqrt{E}}{2\pi}\log\left(\frac{\sqrt{E}}{2\pi e}\right)
  18. | x | |x|\to\infty
  19. f ( - x ) = f ( x ) 4 π 2 e 2 ( 2 ϵ π x + B A ( ϵ ) ) 2 ϵ f(-x)=f(x)\sim 4\pi^{2}e^{2}\left(\frac{2\epsilon\sqrt{\pi}x+B}{A(\epsilon)}% \right)^{\frac{2}{\epsilon}}
  20. A ( ϵ ) = Γ ( 3 + ϵ 2 ) Γ ( 1 + ϵ 2 ) A(\epsilon)=\frac{\Gamma\left(\frac{3+\epsilon}{2}\right)}{\Gamma\left(1+\frac% {\epsilon}{2}\right)}
  21. B = A ( 0 ) B=A(0)
  22. ϵ 0 \epsilon\to 0
  23. 16 π 2 e 8 | x | 16\pi^{2}e^{8|x|}
  24. E n = 4 π 2 n 2 W 2 ( n e - 1 ) E_{n}=\frac{4\pi^{2}n^{2}}{W^{2}(ne^{-1})}

Wythoff_array.html

  1. 1 2 3 5 8 13 21 4 7 11 18 29 47 76 6 10 16 26 42 68 110 9 15 24 39 63 102 165 12 20 32 52 84 136 220 14 23 37 60 97 157 254 17 28 45 73 118 191 309 \begin{matrix}1&2&3&5&8&13&21&\cdots\\ 4&7&11&18&29&47&76&\cdots\\ 6&10&16&26&42&68&110&\cdots\\ 9&15&24&39&63&102&165&\cdots\\ 12&20&32&52&84&136&220&\cdots\\ 14&23&37&60&97&157&254&\cdots\\ 17&28&45&73&118&191&309&\cdots\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\\ \end{matrix}
  2. φ = 1 + 5 2 \varphi=\frac{1+\sqrt{5}}{2}
  3. i i
  4. ( i φ , i φ 2 ) (\lfloor i\varphi\rfloor,\lfloor i\varphi^{2}\rfloor)
  5. m m
  6. m = i φ m=\lfloor i\varphi\rfloor
  7. A m , n A_{m,n}
  8. m m
  9. n n
  10. A m , 1 = m φ φ A_{m,1}=\left\lfloor\lfloor m\varphi\rfloor\varphi\right\rfloor
  11. A m , 2 = m φ φ 2 A_{m,2}=\left\lfloor\lfloor m\varphi\rfloor\varphi^{2}\right\rfloor
  12. A m , n = A m , n - 2 + A m , n - 1 A_{m,n}=A_{m,n-2}+A_{m,n-1}
  13. n > 2 n>2
  14. A m , n A_{m,n}
  15. m m
  16. n n

Xanthan_ketal_pyruvate_transferase.html

  1. \rightleftharpoons

YDS_algorithm.html

  1. J := J 1 , , J n J:=J_{1},...,J_{n}
  2. J i J_{i}
  3. r i r_{i}
  4. d i d_{i}
  5. w i w_{i}
  6. I I
  7. Δ I = 1 | I | J i S I w i \Delta_{I}=\frac{1}{|I|}\sum_{J_{i}\in S_{I}}w_{i}
  8. I I
  9. S I J S_{I}\in J
  10. I I
  11. [ r i , d i ] I [r_{i},d_{i}]\in I
  12. A A
  13. a A = 0 a_{A}=0
  14. d A = 10 d_{A}=10
  15. w A = 5 w_{A}=5
  16. B B
  17. a B = 5 a_{B}=5
  18. d B = 10 d_{B}=10
  19. w B = 4 w_{B}=4
  20. 3 3
  21. 8 8
  22. A A
  23. B B
  24. A A
  25. B B
  26. a A a_{A}
  27. d A d_{A}
  28. 5 5
  29. 10 - ( 8 - 3 ) = 5 10-(8-3)=5
  30. A A
  31. a B a_{B}
  32. 3 3
  33. d B d_{B}
  34. 5 5
  35. 2 2

Yen's_algorithm.html

  1. N N
  2. ( i ) (i)
  3. i t h i^{th}
  4. i i
  5. 1 1
  6. N N
  7. ( 1 ) (1)
  8. ( N ) (N)
  9. d i j d_{ij}
  10. ( i ) (i)
  11. ( j ) (j)
  12. ( i ) ( j ) (i)\neq(j)
  13. d i j 0 d_{ij}\geq 0
  14. A k A^{k}
  15. k t h k^{th}
  16. ( 1 ) (1)
  17. ( N ) (N)
  18. k k
  19. 1 1
  20. K K
  21. A k = ( 1 ) - ( 2 k ) - ( 3 k ) - - ( Q k k ) - ( N ) A^{k}=(1)-(2^{k})-(3^{k})-\cdots-({Q_{k}}^{k})-(N)
  22. ( 2 k ) (2^{k})
  23. k t h k^{th}
  24. ( 3 k ) (3^{k})
  25. k t h k^{th}
  26. A k i {A^{k}}_{i}
  27. A k - 1 A^{k-1}
  28. ( i ) (i)
  29. i i
  30. 1 1
  31. Q k Q_{k}
  32. i i
  33. Q k Q_{k}
  34. k k
  35. k - 1 k-1
  36. A k A^{k}
  37. A k - 1 A^{k-1}
  38. i t h i_{th}
  39. ( i ) k - ( i + 1 ) k (i)^{k}-(i+1)^{k}
  40. A j A^{j}
  41. j j
  42. 1 1
  43. k - 1 k-1
  44. R k i {R^{k}}_{i}
  45. A k i {A^{k}}_{i}
  46. A k - 1 A^{k-1}
  47. i t h i_{th}
  48. A k - 1 A^{k-1}
  49. S k i {S^{k}}_{i}
  50. A k i {A^{k}}_{i}
  51. i t h i_{th}
  52. A k i {A^{k}}_{i}
  53. A 1 A^{1}
  54. A A
  55. B B
  56. A 1 A^{1}
  57. A k A^{k}
  58. k k
  59. 2 2
  60. K K
  61. A 1 A^{1}
  62. A k - 1 A^{k-1}
  63. k k
  64. A k i {A^{k}}_{i}
  65. A k A^{k}
  66. i i
  67. 1 1
  68. Q k k {Q^{k}}_{k}
  69. R k i {R^{k}}_{i}
  70. S k i {S^{k}}_{i}
  71. A k i {A^{k}}_{i}
  72. B B
  73. R k i {R^{k}}_{i}
  74. A k - 1 A^{k-1}
  75. i i
  76. A j A^{j}
  77. j j
  78. 1 1
  79. k - 1 k-1
  80. d i ( i + 1 ) d_{i(i+1)}
  81. A j A^{j}
  82. S k i {S^{k}}_{i}
  83. i i
  84. ( i ) (i)
  85. ( i + 1 ) (i+1)
  86. A k i = R k i + S k i {A^{k}}_{i}={R^{k}}_{i}+{S^{k}}_{i}
  87. B B
  88. A k A^{k}
  89. B B
  90. B B
  91. A A
  92. B B
  93. B B
  94. A A
  95. ( C ) (C)
  96. ( H ) (H)
  97. ( C ) (C)
  98. ( H ) (H)
  99. ( C ) - ( E ) - ( F ) - ( H ) (C)-(E)-(F)-(H)
  100. A A
  101. A 1 A^{1}
  102. ( C ) (C)
  103. A 1 A^{1}
  104. R 2 1 = ( C ) {R^{2}}_{1}=(C)
  105. ( C ) - ( E ) (C)-(E)
  106. A A
  107. S 2 1 {S^{2}}_{1}
  108. ( C ) - ( D ) - ( F ) - ( H ) (C)-(D)-(F)-(H)
  109. A 2 1 = R 2 1 + S 2 1 = ( C ) - ( D ) - ( F ) - ( H ) {A^{2}}_{1}={R^{2}}_{1}+{S^{2}}_{1}=(C)-(D)-(F)-(H)
  110. B B
  111. ( E ) (E)
  112. A 1 A^{1}
  113. R 2 2 = ( C ) - ( E ) {R^{2}}_{2}=(C)-(E)
  114. ( E ) - ( F ) (E)-(F)
  115. A A
  116. S 2 2 {S^{2}}_{2}
  117. ( E ) - ( G ) - ( H ) (E)-(G)-(H)
  118. A 2 2 = R 2 2 + S 2 2 = ( C ) - ( E ) - ( G ) - ( H ) {A^{2}}_{2}={R^{2}}_{2}+{S^{2}}_{2}=(C)-(E)-(G)-(H)
  119. B B
  120. ( F ) (F)
  121. A 1 A^{1}
  122. R 2 3 = ( C ) - ( E ) - ( F ) {R^{2}}_{3}=(C)-(E)-(F)
  123. ( F ) - ( H ) (F)-(H)
  124. A A
  125. S 2 3 {S^{2}}_{3}
  126. ( F ) - ( G ) - ( H ) (F)-(G)-(H)
  127. A 2 3 = R 2 3 + S 2 3 = ( C ) - ( E ) - ( F ) - ( G ) - ( H ) {A^{2}}_{3}={R^{2}}_{3}+{S^{2}}_{3}=(C)-(E)-(F)-(G)-(H)
  128. B B
  129. A 2 2 {A^{2}}_{2}
  130. A 2 A^{2}
  131. A 3 A^{3}
  132. ( C ) - ( D ) - ( F ) - ( H ) (C)-(D)-(F)-(H)
  133. A A
  134. B B
  135. N 2 + K N N^{2}+KN
  136. N 2 N^{2}
  137. K N KN
  138. A A
  139. B B
  140. K K
  141. N N
  142. O ( N 2 ) O(N^{2})
  143. O ( M + N log N ) O(M+N\log N)
  144. M M
  145. K l Kl
  146. l l
  147. l l
  148. O ( log N ) O(\log N)
  149. N N
  150. O ( K N ( M + N log N ) ) O(KN(M+N\log N))
  151. B B
  152. A k A^{k}
  153. B B
  154. K - k K-k
  155. A A
  156. A A
  157. A k A^{k}
  158. A k A^{k}
  159. A k - 1 A^{k-1}
  160. ( i ) (i)
  161. S k j {S^{k}}_{j}
  162. j = 0 , , i j=0,\ldots,i
  163. k - 1 k-1
  164. A k - 1 A^{k-1}
  165. S k h {S^{k}}_{h}
  166. h h
  167. ( i + 1 ) k - 1 (i+1)^{k-1}
  168. ( Q k ) k - 1 (Q_{k})^{k-1}
  169. A k A^{k}
  170. A k - 1 A^{k-1}
  171. A k - 2 A^{k-2}

Yoshio_Koide.html

  1. m m
  2. m 1 , m 2 m_{1},m_{2}
  3. 1 2 ( m 1 + m 2 ) \frac{1}{2}(m_{1}+m_{2})
  4. m 1 = m 2 m_{1}=m_{2}

Zariski's_lemma.html

  1. k [ t 1 , , t n ] k[t_{1},...,t_{n}]
  2. k n k^{n}
  3. f ( x ) = 0 f(x)=0
  4. 𝔪 \mathfrak{m}
  5. A = k [ t 1 , , t n ] A=k[t_{1},...,t_{n}]
  6. ϕ : A A / 𝔪 \phi:A\to A/\mathfrak{m}
  7. A / 𝔪 = k A/\mathfrak{m}=k
  8. f 𝔪 f\in\mathfrak{m}
  9. f ( ϕ ( t 1 ) , , ϕ ( t n ) ) = ϕ ( f ( t 1 , , t n ) ) = 0 f(\phi(t_{1}),\cdots,\phi(t_{n}))=\phi(f(t_{1},\cdots,t_{n}))=0
  10. x = ( ϕ ( t 1 ) , , ϕ ( t n ) ) x=(\phi(t_{1}),\cdots,\phi(t_{n}))
  11. 𝔪 \mathfrak{m}
  12. k [ x 1 , , x d ] k[x_{1},\ldots,x_{d}]
  13. x 1 , , x d x_{1},\ldots,x_{d}
  14. d = 0 d=0
  15. dim A = tr . deg k A \dim A=\operatorname{tr.deg}_{k}A

Zdeněk_Frolík.html

  1. X X
  2. X × Y X\times Y
  3. Y Y
  4. X X
  5. X × Y X\times Y
  6. Y Y

Zeaxanthin_7,8-dioxygenase.html

  1. \rightleftharpoons

Zeaxanthin_epoxidase.html

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

Zeaxanthin_glucosyltransferase.html

  1. \rightleftharpoons

Zeeman_conjecture.html

  1. K K
  2. K × [ 0 , 1 ] K\times[0,1]

Zero-inflated_model.html

  1. Pr ( y j = 0 ) = π + ( 1 - π ) e - λ \Pr(y_{j}=0)=\pi+(1-\pi)e^{-\lambda}
  2. Pr ( y j = h i ) = ( 1 - π ) λ h i e - λ h i ! , h i 1 \Pr(y_{j}=h_{i})=(1-\pi)\frac{\lambda^{h_{i}}e^{-\lambda}}{h_{i}!},\qquad h_{i% }\geq 1
  3. y j y_{j}
  4. λ i \lambda_{i}
  5. i i
  6. π \pi
  7. ( 1 - π ) λ (1-\pi)\lambda
  8. λ ( 1 - π ) ( 1 + λ π ) \lambda(1-\pi)(1+\lambda\pi)
  9. λ ^ m o = s 2 + m 2 - m m , \hat{\lambda}_{mo}=\frac{s^{2}+m^{2}-m}{m},
  10. π ^ m o = s 2 - m s 2 + m 2 - m , \hat{\pi}_{mo}=\frac{s^{2}-m}{s^{2}+m^{2}-m},
  11. m m
  12. s 2 s^{2}
  13. x ¯ ( 1 - e - λ ^ m l ) = λ ^ m l ( 1 - n 0 n ) . \bar{x}(1-e^{-\hat{\lambda}_{ml}})=\hat{\lambda}_{ml}\left(1-\frac{n_{0}}{n}% \right).
  14. x ¯ \bar{x}
  15. n 0 n \frac{n_{0}}{n}
  16. π \pi
  17. π ^ m l = 1 - x ¯ λ ^ m l . \hat{\pi}_{ml}=1-\frac{\bar{x}}{\hat{\lambda}_{ml}}.
  18. Pr ( y i = 0 ) > 0.5 \Pr(y_{i}=0)>0.5
  19. y i y_{i}
  20. G i ( z ) = n = 0 P ( y i = n ) z n {G_{i}(z)}=\sum\limits_{n=0}^{\infty}P(y_{i}=n)z^{n}
  21. y i y_{i}
  22. p 0 = Pr ( y i = 0 ) > 0.5 p_{0}=\Pr(y_{i}=0)>0.5
  23. | G i ( z ) | p 0 - i = 1 < m t p l > p i = 2 p 0 - 1 > 0 \left|{G_{i}(z)}\right|\geqslant{p_{0}}-\sum\limits_{i=1}^{\infty}<mtpl>{{p_{i% }}}=2{p_{0}}-1>0
  24. G ( z ) {G(z)}
  25. Y Y
  26. G Y ( z ) = n = 0 P ( Y = n ) z n = exp ( k = 1 α k λ ( z k - 1 ) ) , ( | z | 1 ) G_{Y}(z)=\sum\limits_{n=0}^{\infty}P(Y=n)z^{n}=\exp\left(\sum\limits_{k=1}^{% \infty}\alpha_{k}\lambda(z^{k}-1)\right),\quad(|z|\leq 1)
  27. ( α 1 λ , α 2 λ , ) ( k = 1 < m t p l > α k = 1 , k = 1 | α k | 0 ) (\alpha_{1}\lambda,\alpha_{2}\lambda,\ldots)\in\mathbb{R}^{\infty}\left({\sum% \limits_{k=1}^{\infty}<mtpl>{{\alpha_{k}}}=1,\sum\limits_{k=1}^{\infty}{\left|% {{\alpha_{k}}}\right|}0}\right)
  28. α k \alpha_{k}

Zerumbone_synthase.html

  1. \rightleftharpoons

Zinbiel_algebra.html

  1. ( a b ) c = a ( b c ) + a ( c b ) . (a\circ b)\circ c=a\circ(b\circ c)+a\circ(c\circ b).
  2. a b = a b + b a a\star b=a\circ b+b\circ a
  3. ( x 0 x p ) ( x p + 1 x p + q ) = x 0 ( p , q ) ( x 1 , , x p + q ) , (x_{0}\otimes\cdots\otimes x_{p})\circ(x_{p+1}\otimes\cdots\otimes x_{p+q})=x_% {0}\sum_{(p,q)}(x_{1},\ldots,x_{p+q}),

Zuc_stream_cipher.html

  1. 2 31 - 1 2^{31}-1