wpmath0000005_1

Birkhoff_interpolation.html

  1. p ( n i ) ( x i ) = y i for i = 1 , , d , p^{(n_{i})}(x_{i})=y_{i}\qquad\mbox{for }~{}i=1,\ldots,d,
  2. ( x i , y i ) (x_{i},y_{i})
  3. n i n_{i}
  4. p ( j ) ( x i ) = y i , j for all ( i , j ) with e i j = 1. p^{(j)}(x_{i})=y_{i,j}\qquad\,\text{for all }(i,j)\,\text{ with }e_{ij}=1.
  5. [ 1 0 0 0 1 0 1 0 0 ] and [ 0 1 0 1 0 0 0 1 0 ] . \begin{bmatrix}1&0&0\\ 0&1&0\\ 1&0&0\end{bmatrix}\quad\,\text{and}\quad\begin{bmatrix}0&1&0\\ 1&0&0\\ 0&1&0\end{bmatrix}.
  6. S m = i = 1 k j = 1 m e i j . S_{m}=\sum_{i=1}^{k}\sum_{j=1}^{m}e_{ij}.

Bitmap_index.html

  1. 𝐂 n [ n 2 ] n ! ( n - [ n 2 ] ) ! [ n 2 ] ! \mathbf{C}_{n}^{\left[}\frac{n}{2}\right]\equiv\frac{n!}{(n-\left[\frac{n}{2}% \right])!\left[\frac{n}{2}\right]!}

Bjerrum_length.html

  1. k B T k_{B}T
  2. k B k_{B}
  3. T T
  4. λ B = e 2 4 π ε 0 ε r k B T , \lambda_{B}=\frac{e^{2}}{4\pi\varepsilon_{0}\varepsilon_{r}\ k_{B}T},
  5. e e
  6. ε r \varepsilon_{r}
  7. ε 0 \varepsilon_{0}
  8. T = 300 K T=300\mbox{ K}~{}
  9. ε r 80 \varepsilon_{r}\approx 80
  10. λ B 0.7 nm \lambda_{B}\approx 0.7\mbox{nm}~{}
  11. 4 π ε 0 = 1 4\pi\varepsilon_{0}=1
  12. λ B = e 2 ε r k B T . \lambda_{B}=\frac{e^{2}}{\varepsilon_{r}k_{B}T}.

BK-space.html

  1. c c
  2. c 0 c_{0}
  3. l l^{\infty}
  4. \|\cdot\|_{\infty}
  5. l p l^{p}
  6. p 1 p\geq 1
  7. p \|\cdot\|_{p}

Black_hole_electron.html

  1. r s = 2 G m c 2 r_{s}=\frac{2Gm}{c^{2}}
  2. r q = q 2 G 4 π ϵ 0 c 4 r_{q}=\sqrt{\frac{q^{2}G}{4\pi\epsilon_{0}c^{4}}}
  3. r α c 10 6 G e V 10 - 24 m r\approx\frac{\alpha\hbar c}{10^{6}GeV}\approx 10^{-24}m

Block_LU_decomposition.html

  1. ( A B C D ) = ( I 0 C A - 1 I ) ( A 0 0 D - C A - 1 B ) ( I A - 1 B 0 I ) \begin{pmatrix}A&B\\ C&D\end{pmatrix}=\begin{pmatrix}I&0\\ CA^{-1}&I\end{pmatrix}\begin{pmatrix}A&0\\ 0&D-CA^{-1}B\end{pmatrix}\begin{pmatrix}I&A^{-1}B\\ 0&I\end{pmatrix}
  2. ( A B C D ) = ( I C A - 1 ) A ( I A - 1 B ) + ( 0 0 0 D - C A - 1 B ) , \begin{pmatrix}A&B\\ C&D\end{pmatrix}=\begin{pmatrix}I\\ CA^{-1}\end{pmatrix}\,A\,\begin{pmatrix}I&A^{-1}B\end{pmatrix}+\begin{pmatrix}% 0&0\\ 0&D-CA^{-1}B\end{pmatrix},
  3. A \begin{matrix}A\end{matrix}
  4. I \begin{matrix}I\end{matrix}
  5. 0 \begin{matrix}0\end{matrix}
  6. ( A B C D ) = ( A 1 2 C A - * 2 ) ( A * 2 A - 1 2 B ) + ( 0 0 0 Q 1 2 ) ( 0 0 0 Q * 2 ) , \begin{pmatrix}A&B\\ C&D\end{pmatrix}=\begin{pmatrix}A^{\frac{1}{2}}\\ CA^{-\frac{*}{2}}\end{pmatrix}\begin{pmatrix}A^{\frac{*}{2}}&A^{-\frac{1}{2}}B% \end{pmatrix}+\begin{pmatrix}0&0\\ 0&Q^{\frac{1}{2}}\end{pmatrix}\begin{pmatrix}0&0\\ 0&Q^{\frac{*}{2}}\end{pmatrix},
  7. A \begin{matrix}A\end{matrix}
  8. Q = D - C A - 1 B \begin{matrix}Q=D-CA^{-1}B\end{matrix}
  9. A 1 2 A * 2 = A ; A 1 2 A - 1 2 = I ; A - * 2 A * 2 = I ; Q 1 2 Q * 2 = Q . \begin{matrix}A^{\frac{1}{2}}\,A^{\frac{*}{2}}=A;\end{matrix}\qquad\begin{% matrix}A^{\frac{1}{2}}\,A^{-\frac{1}{2}}=I;\end{matrix}\qquad\begin{matrix}A^{% -\frac{*}{2}}\,A^{\frac{*}{2}}=I;\end{matrix}\qquad\begin{matrix}Q^{\frac{1}{2% }}\,Q^{\frac{*}{2}}=Q.\end{matrix}
  10. ( A B C D ) = L U , \begin{pmatrix}A&B\\ C&D\end{pmatrix}=LU,
  11. L U = ( A 1 2 0 C A - * 2 0 ) ( A * 2 A - 1 2 B 0 0 ) + ( 0 0 0 Q 1 2 ) ( 0 0 0 Q * 2 ) . LU=\begin{pmatrix}A^{\frac{1}{2}}&0\\ CA^{-\frac{*}{2}}&0\end{pmatrix}\begin{pmatrix}A^{\frac{*}{2}}&A^{-\frac{1}{2}% }B\\ 0&0\end{pmatrix}+\begin{pmatrix}0&0\\ 0&Q^{\frac{1}{2}}\end{pmatrix}\begin{pmatrix}0&0\\ 0&Q^{\frac{*}{2}}\end{pmatrix}.
  12. L U \begin{matrix}LU\end{matrix}
  13. L = ( A 1 2 0 C A - * 2 Q 1 2 ) and U = ( A * 2 A - 1 2 B 0 Q * 2 ) . L=\begin{pmatrix}A^{\frac{1}{2}}&0\\ CA^{-\frac{*}{2}}&Q^{\frac{1}{2}}\end{pmatrix}\mathrm{~{}~{}and~{}~{}}U=\begin% {pmatrix}A^{\frac{*}{2}}&A^{-\frac{1}{2}}B\\ 0&Q^{\frac{*}{2}}\end{pmatrix}.

Block_reflector.html

  1. Q Q
  2. n ( \R ) \mathcal{M}_{n}(\R)
  3. Q = I - a u u T Q=I-auu^{T}
  4. I I
  5. n ( \R ) \mathcal{M}_{n}(\R)
  6. a a
  7. u u
  8. \R 𝓃 \mathcal{\R^{n}}

Blocking_(statistics).html

  1. Var ( X - Y ) = Var ( X ) + Var ( Y ) - 2 Cov ( X , Y ) . \operatorname{Var}(X-Y)=\operatorname{Var}(X)+\operatorname{Var}(Y)-2% \operatorname{Cov}(X,Y).

Bloom_syndrome.html

  1. μ m 2 s \tfrac{\mathrm{\mu m}^{2}}{\mathrm{s}}
  2. μ m 2 s \textstyle\tfrac{\mathrm{\mu m}^{2}}{\mathrm{s}}

Blowing_up.html

  1. X = { ( Q , ) P , Q } 𝐏 2 × 𝐆 ( 1 , 2 ) . X=\{(Q,\ell)\mid P,\,Q\in\ell\}\subseteq\mathbf{P}^{2}\times\mathbf{G}(1,2).
  2. ( Q , ) (Q,\ell)
  3. ( Q , ) (Q,\ell)
  4. \ell
  5. \ell
  6. 0 = [ L 0 : L 1 : L 2 ] \ell_{0}=[L_{0}:L_{1}:L_{2}]
  7. X = { ( [ X 0 : X 1 : X 2 ] , [ L 0 : L 1 : L 2 ] ) P 0 L 0 + P 1 L 1 + P 2 L 2 = 0 , X 0 L 0 + X 1 L 1 + X 2 L 2 = 0 } 𝐏 2 × 𝐏 2 . X=\{([X_{0}:X_{1}:X_{2}],[L_{0}:L_{1}:L_{2}])\mid P_{0}L_{0}+P_{1}L_{1}+P_{2}L% _{2}=0,\,X_{0}L_{0}+X_{1}L_{1}+X_{2}L_{2}=0\}\subseteq\mathbf{P}^{2}\times% \mathbf{P}^{2}.
  8. \ell
  9. { ( ( x , y ) , [ z : w ] ) x z + y w = 0 } 𝐀 2 × 𝐏 1 . \{((x,y),[z:w])\mid xz+yw=0\}\subseteq\mathbf{A}^{2}\times\mathbf{P}^{1}.
  10. { ( ( x , y ) , [ z : w ] ) det | x y w z | = 0 } . \left\{((x,y),[z:w])\mid\det\begin{vmatrix}x&y\\ w&z\end{vmatrix}=0\right\}.
  11. X = Proj r = 0 Sym k [ x , y ] r 𝔪 / 𝔪 2 . X=\operatorname{Proj}\,\bigoplus_{r=0}^{\infty}\operatorname{Sym}^{r}_{k[x,y]}% \mathfrak{m}/\mathfrak{m}^{2}.
  12. X = Proj k [ x , y ] [ z , w ] / ( x z - y w ) , X=\operatorname{Proj}\ k[x,y][z,w]/(xz-yw),
  13. 𝐏 2 # 𝐏 2 \mathbf{P}^{2}\#\mathbf{P}^{2}
  14. 𝐂𝐏 2 # 𝐂𝐏 2 ¯ \mathbf{CP}^{2}\#\overline{\mathbf{CP}^{2}}
  15. 𝐂𝐏 2 ¯ \overline{\mathbf{CP}^{2}}
  16. x 1 , , x n x_{1},\ldots,x_{n}
  17. y 1 , , y n y_{1},\ldots,y_{n}
  18. 𝐂 n ~ \tilde{\mathbf{C}^{n}}
  19. x i y j = x j y i x_{i}y_{j}=x_{j}y_{i}
  20. π : 𝐂 n × 𝐏 n - 1 𝐂 n \pi:\mathbf{C}^{n}\times\mathbf{P}^{n-1}\to\mathbf{C}^{n}
  21. π : 𝐂 n ~ 𝐂 n . \pi:\tilde{\mathbf{C}^{n}}\to\mathbf{C}^{n}.
  22. 𝐂 n ~ \tilde{\mathbf{C}^{n}}
  23. E = Z × 𝐏 n - 1 𝐂 n × 𝐏 n - 1 E=Z\times\mathbf{P}^{n-1}\subseteq\mathbf{C}^{n}\times\mathbf{P}^{n-1}
  24. 𝐂 n ~ E \tilde{\mathbf{C}^{n}}\setminus E
  25. 𝐂 n ~ \tilde{\mathbf{C}^{n}}
  26. x 1 = = x k = 0 x_{1}=\cdots=x_{k}=0
  27. y 1 , , y k y_{1},\ldots,y_{k}
  28. 𝐂 ~ n \tilde{\mathbf{C}}^{n}
  29. x i y j = x j y i x_{i}y_{j}=x_{j}y_{i}
  30. π : X ~ X \pi:\tilde{X}\to X
  31. π | E : E Z \pi|_{E}:E\to Z
  32. X ~ \tilde{X}
  33. X ~ \tilde{X}
  34. π - 1 ( V Z ) \pi^{-1}(V\setminus Z)
  35. X ~ \tilde{X}
  36. \mathcal{I}
  37. \mathcal{I}
  38. X ~ \tilde{X}
  39. π : X ~ X \pi\colon\tilde{X}\rightarrow X
  40. π - 1 𝒪 X ~ \pi^{-1}\mathcal{I}\cdot\mathcal{O}_{\tilde{X}}
  41. f - 1 𝒪 Y f^{-1}\mathcal{I}\cdot\mathcal{O}_{Y}
  42. X ~ = 𝐏𝐫𝐨𝐣 n = 0 n \tilde{X}=\mathbf{Proj}\bigoplus_{n=0}^{\infty}\mathcal{I}^{n}
  43. π : Bl X X \pi:\operatorname{Bl}_{\mathcal{I}}X\to X
  44. \mathcal{I}
  45. π - 1 𝒪 Bl X \pi^{-1}\mathcal{I}\cdot\mathcal{O}_{\operatorname{Bl}_{\mathcal{I}}X}
  46. n = 0 n + 1 \textstyle\bigoplus_{n=0}^{\infty}\mathcal{I}^{n+1}
  47. 𝒪 ( 1 ) \mathcal{O}(1)
  48. \mathcal{I}
  49. V × { 0 } in Y = X × 𝐂 or X × 𝐏 1 V\times\{0\}\ \,\text{in}\ Y=X\times\mathbf{C}\ \,\text{or}\ X\times\mathbf{P}% ^{1}
  50. Y ~ 𝐂 \tilde{Y}\to\mathbf{C}

Boltzmann_relation.html

  1. n e ( ϕ 2 ) = n e ( ϕ 1 ) e e ( ϕ 2 - ϕ 1 ) / k B T e n_{e}(\phi_{2})=n_{e}(\phi_{1})e^{e(\phi_{2}-\phi_{1})/k_{B}T_{e}}
  2. F fluid = - k B T e n e , F_{\rm fluid}=-k_{B}T_{e}\nabla n_{e},
  3. F electric = e n e ϕ F_{\rm electric}=en_{e}\nabla\phi

Bond_order.html

  1. B.O. = number of bonding electrons - number of antibonding electrons 2 \,\text{B.O.}=\frac{\,\text{number of bonding electrons}-\,\text{number of % antibonding electrons}}{2}
  2. s i j = exp [ d 1 - d i j b ] s_{ij}=\exp{\left[\frac{d_{1}-d_{ij}}{b}\right]}
  3. d 1 d_{1}
  4. d i j d_{ij}

Bonne_projection.html

  1. x = ρ sin E x=\rho\sin E\,
  2. y = cot φ 1 - ρ cos E y=\cot\varphi_{1}-\rho\cos E\,
  3. ρ = cot φ 1 + φ 1 - φ \rho=\cot\varphi_{1}+\varphi_{1}-\varphi\,
  4. E = ( λ - λ 0 ) cos φ ρ E=\frac{(\lambda-\lambda_{0})\cos\varphi}{\rho}
  5. φ = cot φ 1 + φ 1 - ρ \varphi=\cot\varphi_{1}+\varphi_{1}-\rho\,
  6. λ = λ 0 + ρ { arctan [ x / ( cot φ 1 - y ) ] } / cos φ \lambda=\lambda_{0}+\rho\{\arctan[x/(\cot\varphi_{1}-y)]\}/\cos\varphi
  7. ρ = \rho=
  8. [ x 2 + ( cot φ 1 - y ) 2 ] 1 / 2 [x^{2}+(\cot\varphi_{1}-y)^{2}]^{1/2}
  9. φ 1 \varphi_{1}

Booth's_multiplication_algorithm.html

  1. M × ′′ 0 0 1 1 1 1 1 0 ′′ = M × ( 2 5 + 2 4 + 2 3 + 2 2 + 2 1 ) = M × 62 M\times\,^{\prime\prime}0\;0\;1\;1\;1\;1\;1\;0\,^{\prime\prime}=M\times(2^{5}+% 2^{4}+2^{3}+2^{2}+2^{1})=M\times 62
  2. M × ′′ 0 1 0 0 0 0 -1 0 ′′ = M × ( 2 6 - 2 1 ) = M × 62. M\times\,^{\prime\prime}0\;1\;0\;0\;0\;0\mbox{-1}~{}\;0\,^{\prime\prime}=M% \times(2^{6}-2^{1})=M\times 62.
  3. ( 0 1 1 n 0 ) 2 ( 1 0 0 n 0 ) 2 - ( 0 0 1 n 0 ) 2 . (\ldots 0\overbrace{1\ldots 1}^{n}0\ldots)_{2}\equiv(\ldots 1\overbrace{0% \ldots 0}^{n}0\ldots)_{2}-(\ldots 0\overbrace{0\ldots 1}^{n}0\ldots)_{2}.
  4. M × ′′ 0 0 1 1 1 0 1 0 ′′ = M × ( 2 5 + 2 4 + 2 3 + 2 1 ) = M × 58 M\times\,^{\prime\prime}0\;0\;1\;1\;1\;0\;1\;0\,^{\prime\prime}=M\times(2^{5}+% 2^{4}+2^{3}+2^{1})=M\times 58
  5. M × ′′ 0 1 0 0 -1 1 -1 0 ′′ = M × ( 2 6 - 2 3 + 2 2 - 2 1 ) = M × 58. M\times\,^{\prime\prime}0\;1\;0\;0\mbox{-1}~{}\;1\mbox{-1}~{}\;0\,^{\prime% \prime}=M\times(2^{6}-2^{3}+2^{2}-2^{1})=M\times 58.

Bootstrap_aggregating.html

  1. D i D_{i}
  2. D i D_{i}
  3. D i D_{i}
  4. n n^{\prime}
  5. n n^{\prime}\to\infty
  6. n / n 0 n^{\prime}/n\to 0
  7. n n\to\infty
  8. d d
  9. C n , n b n n C^{bnn}_{n,n^{\prime}}
  10. n n
  11. n n^{\prime}
  12. ( C n , n b n n ) - ( C B a y e s ) = ( B 1 n n + B 2 1 ( n ) 4 / d ) { 1 + o ( 1 ) } , \mathcal{R}_{\mathcal{R}}(C^{bnn}_{n,n^{\prime}})-\mathcal{R}_{\mathcal{R}}(C^% {Bayes})=\left(B_{1}\frac{n^{\prime}}{n}+B_{2}\frac{1}{(n^{\prime})^{4/d}}% \right)\{1+o(1)\},
  13. B 1 B_{1}
  14. B 2 B_{2}
  15. n n^{\prime}
  16. n = B n d / ( d + 4 ) n^{\prime}=Bn^{d/(d+4)}
  17. B B
  18. n ( 1 - e - n / n ) n(1-e^{-n^{\prime}/n})

Borel's_lemma.html

  1. ( k t k F ) ( 0 , x ) = f k ( x ) , \displaystyle{\left(\frac{\partial^{k}}{\partial t^{k}}F\right)(0,x)=f_{k}(x),}
  2. F m ( t , x ) = t m m ! ψ ( t ε m ) f m ( x ) , \displaystyle{F_{m}(t,x)={t^{m}\over m!}\cdot\psi\left({t\over\varepsilon_{m}}% \right)\cdot f_{m}(x),}
  3. α F m 2 - m \displaystyle{\|\partial^{\alpha}F_{m}\|_{\infty}\leq 2^{-m}}
  4. F = m 0 F m \displaystyle{F=\sum_{m\geq 0}F_{m}}
  5. α F = m 0 α F m . \displaystyle{\partial^{\alpha}F=\sum_{m\geq 0}\partial^{\alpha}F_{m}.}
  6. t m F ( t , x ) | t = 0 = f m ( x ) . \displaystyle{\partial_{t}^{m}F(t,x)|_{t=0}=f_{m}(x).}

Borel–Carathéodory_theorem.html

  1. f f
  2. f r = max | z | r | f ( z ) | = max | z | = r | f ( z ) | \|f\|_{r}=\max_{|z|\leq r}|f(z)|=\max_{|z|=r}|f(z)|
  3. A = sup | z | R Re f ( z ) . A=\sup_{|z|\leq R}\operatorname{Re}f(z).
  4. w w / A - 1 w\mapsto w/A-1
  5. w R ( w + 1 ) / ( w - 1 ) w\mapsto R(w+1)/(w-1)
  6. w R w w - 2 A . w\mapsto\frac{Rw}{w-2A}.
  7. | R f ( z ) | | f ( z ) - 2 A | | z | . \frac{|Rf(z)|}{|f(z)-2A|}\leq|z|.
  8. R | f ( z ) | r | f ( z ) - 2 A | r | f ( z ) | + 2 A r R|f(z)|\leq r|f(z)-2A|\leq r|f(z)|+2Ar
  9. | f ( z ) | 2 A r R - r |f(z)|\leq\frac{2Ar}{R-r}
  10. | f ( z ) | - | f ( 0 ) | \displaystyle|f(z)|-|f(0)|

Borel–Weil_theorem.html

  1. Γ ( G / B , L λ ) . \Gamma(G/B,L_{\lambda}).
  2. f : G λ : f ( g b ) = χ λ ( b ) f ( g ) f:G\to\mathbb{C}_{\lambda}:f(gb)=\chi_{\lambda}(b)f(g)
  3. g f ( h ) = f ( g - 1 h ) g\cdot f(h)=f(g^{-1}h)
  4. χ n ( a b 0 a - 1 ) = a n . \chi_{n}\begin{pmatrix}a&b\\ 0&a^{-1}\end{pmatrix}=a^{n}.
  5. X i Y n - i , 0 i n X^{i}Y^{n-i},\quad 0\leq i\leq n

Bornological_space.html

  1. X = B ; X=\bigcup B;
  2. i = 1 n B i B . \bigcup_{i=1}^{n}B_{i}\in B.
  3. B 0 B_{0}
  4. B 0 B_{0}
  5. P κ ( X ) P_{\kappa}(X)
  6. X X
  7. κ \kappa
  8. B 1 B_{1}
  9. B 2 B_{2}
  10. X X
  11. Y Y
  12. f : X Y f\colon X\rightarrow Y
  13. f f
  14. B 1 B_{1}
  15. X X
  16. B 2 B_{2}
  17. Y Y
  18. f f
  19. f - 1 f^{-1}
  20. f f
  21. X X
  22. Y Y
  23. f : X Y f\colon X\rightarrow Y
  24. f f
  25. X X
  26. Y Y
  27. u : X Y u:X\to Y
  28. u - 1 ( D ) u^{-1}(D)
  29. X X
  30. X X
  31. X X
  32. X X
  33. 0
  34. B X B\subseteq X
  35. U X U\subseteq X
  36. λ > 0 \lambda>0
  37. B λ U B\subseteq\lambda U
  38. B X B\subseteq X
  39. X X
  40. X X
  41. X X
  42. X X
  43. X X
  44. X X^{\prime}
  45. X X
  46. X X
  47. X X
  48. X X
  49. Y Y
  50. X X
  51. 0
  52. τ ( X , X ) \tau(X,X^{\prime})
  53. X X
  54. X X
  55. X X
  56. X X
  57. X X
  58. X X
  59. X X
  60. X X^{\prime}
  61. β ( X , X ) \beta(X^{\prime},X)
  62. β ( X , X ) \beta(X^{\prime},X)
  63. β ( X , X ) \beta(X^{\prime},X)
  64. X X
  65. X X
  66. Y Y
  67. u : X Y u:X\to Y
  68. u u
  69. B \sub X B\sub X
  70. X X
  71. u ( B ) u(B)
  72. ( x n ) \sub X (x_{n})\sub X
  73. X X
  74. ( u ( x n ) ) (u(x_{n}))
  75. Y Y
  76. μ D \mu_{D}
  77. p D p_{D}
  78. X D X_{D}
  79. X = 2 X=\mathbb{R}^{2}
  80. μ D \mu_{D}
  81. X D X_{D}
  82. X D X_{D}
  83. X D X_{D}
  84. B r T B\subseteq rT
  85. X U / ker ( μ U ) X_{U}/\ker(\mu_{U})
  86. X ^ U \hat{X}_{U}
  87. X ^ U \hat{X}_{U}
  88. μ U \mu_{U}
  89. X ^ U \hat{X}_{U}
  90. D D^{\prime}
  91. X * X^{*}
  92. X D X_{D}
  93. u : X Y u:X\to Y
  94. u : X Y u:X\to Y

Borosilicate_glass.html

  1. 10 7.6 10^{7.6}

Bottomley_projection.html

  1. x = ρ sin E sin ϕ 1 x=\frac{\rho\sin E}{\sin\phi_{1}}\,
  2. y = π 2 - ρ cos E y=\tfrac{\pi}{2}-\rho\cos E\,
  3. ρ = π 2 - ϕ \rho=\tfrac{\pi}{2}-\phi\,
  4. E = λ sin ϕ 1 sin ρ ρ E=\frac{\lambda\sin\phi_{1}\sin\rho}{\rho}
  5. cos ( ϕ 1 ) \cos(\phi_{1})

Bottomness.html

  1. B = - ( n b - n b ¯ ) B^{\prime}=-(n_{b}-n_{\bar{b}})
  2. 1 / 3 {1}/{3}
  3. Δ B = \plusmn 1 \Delta B^{\prime}=\plusmn 1

Boundary-value_analysis.html

  1. X 1 , , X n X_{1},\dots,X_{n}
  2. \leq
  3. C 1 , C 2 C_{1},C_{2}
  4. X 1 C 1 X_{1}\in C_{1}
  5. X 2 C 2 X_{2}\in C_{2}
  6. X 1 X 2 X_{1}\leq X_{2}
  7. X 2 X 1 X_{2}\leq X_{1}
  8. C 1 , C 2 C_{1},C_{2}
  9. X 1 , X 2 X_{1},X_{2}
  10. [ a , b ] [a,b]
  11. INT _ MIN x + y INT _ MAX \,\text{INT}\_\,\text{MIN}\leq x+y\leq\,\text{INT}\_\,\text{MAX}
  12. x + y = INT _ MAX x+y=\,\text{INT}\_\,\text{MAX}
  13. INT _ MIN = x + y \,\text{INT}\_\,\text{MIN}=x+y
  14. ( a , b ) (a,b)
  15. x + y = INT _ MAX + 1 x+y=\,\text{INT}\_\,\text{MAX}+1
  16. ( a , b ) (a,b)
  17. x + y = INT _ MIN - 1 x+y=\,\text{INT}\_\,\text{MIN}-1
  18. ( a , b ) (a,b)

Boundary_parallel.html

  1. I × S 1 I\times S^{1}
  2. π : I × S 1 S 1 , ( x , z ) z . \pi:I\times S^{1}\rightarrow S^{1},\qquad(x,z)\mapsto z.

Bounded_set_(topological_vector_space).html

  1. S α N S\subseteq\alpha N
  2. α N := { α x x N } \alpha N:=\{\alpha x\mid x\in N\}
  3. | α | t S α N |\alpha|\geq t\Rightarrow S\subseteq\alpha N
  4. L p L^{p}
  5. 0 < p < 1 0<p<1

Boxcar_function.html

  1. boxcar ( x ) = ( b - a ) A f ( a , b ; x ) = A ( H ( x - a ) - H ( x - b ) ) , \operatorname{boxcar}(x)=(b-a)A\,f(a,b;x)=A(H(x-a)-H(x-b)),
  2. H ( x ) H(x)

Braided_monoidal_category.html

  1. γ \gamma
  2. 𝒞 \mathcal{C}
  3. γ A , B : A B B A \gamma_{A,B}:A\otimes B\rightarrow B\otimes A
  4. A B B A A\otimes B\cong B\otimes A
  5. A , B 𝒞 A,B\in\mathcal{C}
  6. 𝒞 \mathcal{C}
  7. γ \gamma
  8. 𝒞 \mathcal{C}
  9. γ \gamma
  10. A , B , C 𝒞 A,B,C\in\mathcal{C}
  11. α \alpha
  12. 𝒞 \mathcal{C}
  13. γ \gamma
  14. α , λ , ρ \alpha,\lambda,\rho
  15. 𝒞 \mathcal{C}
  16. γ \gamma
  17. N N
  18. ( γ B , C Id ) ( Id γ A , C ) ( γ A , B Id ) = ( Id γ A , B ) ( γ A , C Id ) ( Id γ B , C ) (\gamma_{B,C}\otimes\,\text{Id})\circ(\,\text{Id}\otimes\gamma_{A,C})\circ(% \gamma_{A,B}\otimes\,\text{Id})=(\,\text{Id}\otimes\gamma_{A,B})\circ(\gamma_{% A,C}\otimes\,\text{Id})\circ(\,\text{Id}\otimes\gamma_{B,C})
  19. A B C C B A A\otimes B\otimes C\rightarrow C\otimes B\otimes A
  20. γ \gamma
  21. γ B , A γ A , B = I d \gamma_{B,A}\circ\gamma_{A,B}=Id
  22. A A
  23. B B
  24. γ \gamma
  25. N N
  26. γ B , A γ A , B = Id \gamma_{B,A}\circ\gamma_{A,B}=\,\text{Id}
  27. A A
  28. B B
  29. γ B A , C ( γ A , B Id ) = γ A , C B ( Id γ B , C ) \gamma_{B\otimes A,C}\circ(\gamma_{A,B}\otimes\,\text{Id})=\gamma_{A,C\otimes B% }\circ(\,\text{Id}\otimes\gamma_{B,C})
  30. γ ( v w ) = w v \gamma(v\otimes w)=w\otimes v
  31. U q ( 𝔤 ) U_{q}(\mathfrak{g})
  32. γ \gamma

Brake_fade.html

  1. e μ θ e^{\mu\theta}
  2. e e
  3. μ \mu
  4. θ \theta

Breakdown_voltage.html

  1. V b = B p d ln A p d - ln ( ln ( 1 + 1 γ se ) ) V_{\mathrm{b}}=\frac{Bpd}{\ln Apd-\ln(\ln(1+\frac{1}{\gamma_{\mathrm{se}}}))}
  2. V b V_{\mathrm{b}}
  3. A A
  4. B B
  5. p p
  6. d d
  7. γ se \gamma_{\mathrm{se}}

Breather.html

  1. 2 u t 2 - 2 u x 2 + sin u = 0 , \frac{\partial^{2}u}{\partial t^{2}}-\frac{\partial^{2}u}{\partial x^{2}}+\sin u% =0,
  2. u = 4 arctan ( 1 - ω 2 cos ( ω t ) ω cosh ( 1 - ω 2 x ) ) , u=4\arctan\left(\frac{\sqrt{1-\omega^{2}}\;\cos(\omega t)}{\omega\;\cosh(\sqrt% {1-\omega^{2}}\;x)}\right),
  3. i u t + 2 u x 2 + | u | 2 u = 0 , i\,\frac{\partial u}{\partial t}+\frac{\partial^{2}u}{\partial x^{2}}+|u|^{2}u% =0,
  4. u = ( 2 b 2 cosh ( θ ) + 2 i b 2 - b 2 sinh ( θ ) 2 cosh ( θ ) - 2 2 - b 2 cos ( a b x ) - 1 ) a exp ( i a 2 t ) with θ = a 2 b 2 - b 2 t , u=\left(\frac{2\,b^{2}\cosh(\theta)+2\,i\,b\,\sqrt{2-b^{2}}\;\sinh(\theta)}{2% \,\cosh(\theta)-\sqrt{2}\,\sqrt{2-b^{2}}\cos(a\,b\,x)}-1\right)\;a\;\exp(i\,a^% {2}\,t)\quad\,\text{with}\quad\theta=a^{2}\,b\,\sqrt{2-b^{2}}\;t,

Breit_equation.html

  1. H ^ D ( i ) = [ q i ϕ ( 𝐫 i ) + c s = x , y , z α s ( i ) π s ( I ) + α 0 ( I ) m 0 c 2 ] \hat{H}_{D}(i)=\left[q_{i}\phi(\mathbf{r}_{i})+c\sum_{s=x,y,z}\alpha_{s}(i)\pi% _{s}(I)+\alpha_{0}(I)m_{0}c^{2}\right]
  2. B ^ i j = - 1 2 r i j [ 𝐚 ( i ) 𝐚 ( j ) + ( 𝐚 ( i ) 𝐫 i j ) ( 𝐚 ( j ) 𝐫 i j ) r i j 2 ] \hat{B}_{ij}=-\frac{1}{2r_{ij}}\left[\mathbf{a}(i)\cdot\mathbf{a}(j)+\frac{% \left(\mathbf{a}(i)\cdot\mathbf{r}_{ij}\right)\left(\mathbf{a}(j)\cdot\mathbf{% r}_{ij}\right)}{r_{ij}^{2}}\right]
  3. B ^ i j = H ^ 0 + H ^ 1 + + H ^ 6 \hat{B}_{ij}=\hat{H}_{0}+\hat{H}_{1}+...+\hat{H}_{6}
  4. H ^ 0 = i p ^ i 2 2 m i + V \hat{H}_{0}=\sum_{i}\frac{\hat{p}_{i}^{2}}{2m_{i}}+V
  5. m i m_{i}
  6. H ^ 1 = - 1 8 c 2 i p ^ i 4 m i 3 \hat{H}_{1}=-\frac{1}{8c^{2}}\sum_{i}\frac{\hat{p}_{i}^{4}}{m_{i}^{3}}
  7. E k i n 2 - ( m 0 c 2 ) 2 = m 2 v 2 c 2 E_{kin}^{2}-\left(m_{0}c^{2}\right)^{2}=m^{2}v^{2}c^{2}
  8. H ^ 2 = - i > j q i q j 2 r i j m i m j c 2 [ 𝐩 ^ i 𝐩 ^ j + ( 𝐫 𝐢𝐣 𝐩 ^ i ) ( 𝐫 𝐢𝐣 𝐩 ^ j ) r i j 2 ] \hat{H}_{2}=-\sum_{i>j}\frac{q_{i}q_{j}}{2r_{ij}m_{i}m_{j}c^{2}}\left[\mathbf{% \hat{p}}_{i}\cdot\mathbf{\hat{p}}_{j}+\frac{(\mathbf{r_{ij}}\cdot\mathbf{\hat{% p}}_{i})(\mathbf{r_{ij}}\cdot\mathbf{\hat{p}}_{j})}{r_{ij}^{2}}\right]
  9. H ^ 3 = μ B c i 1 m i 𝐬 i [ 𝐅 ( 𝐫 i ) × 𝐩 ^ i + j > i 2 q i r i j 3 𝐫 i j × 𝐩 ^ j ] \hat{H}_{3}=\frac{\mu_{B}}{c}\sum_{i}\frac{1}{m_{i}}\mathbf{s}_{i}\cdot\left[% \mathbf{F}(\mathbf{r}_{i})\times\mathbf{\hat{p}}_{i}+\sum_{j>i}\frac{2q_{i}}{r% _{ij}^{3}}\mathbf{r}_{ij}\times\mathbf{\hat{p}}_{j}\right]
  10. H ^ 4 = i h 8 π c 2 i q i m i 2 𝐩 ^ i 𝐅 ( 𝐫 i ) \hat{H}_{4}=\frac{ih}{8\pi c^{2}}\sum_{i}\frac{q_{i}}{m_{i}^{2}}\mathbf{\hat{p% }}_{i}\cdot\mathbf{F}(\mathbf{r}_{i})
  11. H ^ 5 = 4 μ B 2 i > j { - 8 π 3 ( 𝐬 i 𝐬 j ) δ ( 𝐫 i j ) + 1 r i j 3 [ 𝐬 i 𝐬 j - 3 ( 𝐬 i 𝐫 i j ) ( 𝐬 j 𝐫 i j ) r i j 2 ] } \hat{H}_{5}=4\mu_{B}^{2}\sum_{i>j}\left\{-\frac{8\pi}{3}(\mathbf{s}_{i}\cdot% \mathbf{s}_{j})\delta(\mathbf{r}_{ij})+\frac{1}{r_{ij}^{3}}\left[\mathbf{s}_{i% }\cdot\mathbf{s}_{j}-\frac{3(\mathbf{s}_{i}\cdot\mathbf{r}_{ij})(\mathbf{s}_{j% }\cdot\mathbf{r}_{ij})}{r_{ij}^{2}}\right]\right\}
  12. H ^ 6 = 2 μ B i [ 𝐇 ( 𝐫 i ) 𝐬 i + q i m i c 𝐀 ( 𝐫 i ) 𝐩 ^ i ] \hat{H}_{6}=2\mu_{B}\sum_{i}\left[\mathbf{H}(\mathbf{r}_{i})\cdot\mathbf{s}_{i% }+\frac{q_{i}}{m_{i}c}\mathbf{A}(\mathbf{r}_{i})\cdot\mathbf{\hat{p}}_{i}\right]
  13. V = i > j q i q j r i j V=\sum_{i>j}\frac{q_{i}q_{j}}{r_{ij}}
  14. μ B = e 2 m c \mu_{B}=\frac{e\hbar}{2mc}

Bremermann's_limit.html

  1. Δ E \Delta E
  2. Δ t = π / 2 Δ E \Delta t=\pi\hbar/2\Delta E
  3. Δ t = π / 2 E \Delta t=\pi\hbar/2E

Brent's_method.html

  1. s = { b k - b k - b k - 1 f ( b k ) - f ( b k - 1 ) f ( b k ) , if f ( b k ) f ( b k - 1 ) m otherwise s=\begin{cases}b_{k}-\frac{b_{k}-b_{k-1}}{f(b_{k})-f(b_{k-1})}f(b_{k}),&\mbox{% if }~{}f(b_{k})\neq f(b_{k-1})\\ m&\mbox{otherwise }\end{cases}
  2. m = a k + b k 2 . m=\frac{a_{k}+b_{k}}{2}.
  3. δ \delta
  4. | δ | < | b k - b k - 1 | |\delta|<|b_{k}-b_{k-1}|
  5. | δ | < | b k - 1 - b k - 2 | |\delta|<|b_{k-1}-b_{k-2}|
  6. | s - b k | < 1 2 | b k - b k - 1 | |s-b_{k}|<\begin{matrix}\frac{1}{2}\end{matrix}|b_{k}-b_{k-1}|
  7. | s - b k | < 1 2 | b k - 1 - b k - 2 | |s-b_{k}|<\begin{matrix}\frac{1}{2}\end{matrix}|b_{k-1}-b_{k-2}|
  8. 2 log 2 ( | b k - 1 - b k - 2 | / δ ) 2\log_{2}(|b_{k-1}-b_{k-2}|/\delta)
  9. 2 log 2 ( | b k - 1 - b k - 2 | / δ ) 2\log_{2}(|b_{k-1}-b_{k-2}|/\delta)
  10. δ \delta
  11. s := b - f ( b ) b - a f ( b ) - f ( a ) s:=b-f(b)\frac{b-a}{f(b)-f(a)}
  12. 3 a + b 4 \frac{3a+b}{4}
  13. b b
  14. s b s−b
  15. b c b−c
  16. s b s−b
  17. c d c−d
  18. b c b−c
  19. c d c−d
  20. s := a + b 2 s:=\frac{a+b}{2}

Brian_Goodwin.html

  1. d X d t = k 1 K 1 + Z n - k 2 X \frac{dX}{dt}={k_{1}\over K_{1}+Z^{n}}-k_{2}X
  2. d Y d t = k 3 X - k 4 Y \frac{dY}{dt}=k_{3}X-k_{4}Y
  3. d Z d t = k 5 Y - k 6 Z \frac{dZ}{dt}=k_{5}Y-k_{6}Z
  4. ρ 2 ξ t 2 = x ( P 1 ( χ ) ξ x ) + x ( P 2 ( χ ) 2 ξ x t ) - P 3 ( χ ) - F 0 χ x \rho{\partial^{2}\xi\over\partial t^{2}}={\partial\over\partial x}\left(P_{1}(% \chi){\partial\xi\over\partial x}\right)+{\partial\over\partial x}\left(P_{2}(% \chi){\partial^{2}\xi\over\partial x\partial t}\right)-P_{3}(\chi)-F_{0}{% \partial\chi\over\partial x}
  5. χ t = ( a 0 + a ξ x ) ( K - χ ) - k 1 ( β + χ ) χ n + D 2 ξ x 2 {\partial\chi\over\partial t}=\left(a_{0}+a{\partial\xi\over\partial x}\right)% (K-\chi)-k_{1}(\beta+\chi)\chi^{n}+D{\partial^{2}\xi\over\partial x^{2}}

Bridge_(graph_theory).html

  1. n n
  2. n - 1 n-1
  3. n - 1 n-1
  4. G G
  5. F F
  6. F F
  7. v v
  8. N D ( v ) ND(v)
  9. L ( v ) L(v)
  10. v v
  11. v v
  12. L ( w ) L(w)
  13. v v
  14. v v
  15. F F
  16. H ( v ) H(v)
  17. v v
  18. H ( w ) H(w)
  19. v v
  20. v v
  21. F F
  22. w w
  23. v v
  24. L ( w ) = w L(w)=w
  25. H ( w ) < w + N D ( w ) H(w)<w+ND(w)
  26. v v
  27. w w

Bridgman's_thermodynamic_equations.html

  1. C P = ( H T ) P C_{P}=\left(\frac{\partial H}{\partial T}\right)_{P}
  2. C P = ( H ) P ( T ) P C_{P}=\frac{(\partial H)_{P}}{(\partial T)_{P}}
  3. ( H ) P = C P (\partial H)_{P}=C_{P}
  4. ( T ) P = 1 (\partial T)_{P}=1
  5. ( V T ) P = α V \left(\frac{\partial V}{\partial T}\right)_{P}=\alpha V
  6. ( V P ) T = - β T V \left(\frac{\partial V}{\partial P}\right)_{T}=-\beta_{T}V
  7. ( H T ) P = C P = c P N \left(\frac{\partial H}{\partial T}\right)_{P}=C_{P}=c_{P}N
  8. ( T ) P = - ( P ) T = 1 (\partial T)_{P}=-(\partial P)_{T}=1
  9. ( V ) P = - ( P ) V = ( V T ) P (\partial V)_{P}=-(\partial P)_{V}=\left(\frac{\partial V}{\partial T}\right)_% {P}
  10. ( S ) P = - ( P ) S = C p T (\partial S)_{P}=-(\partial P)_{S}=\frac{C_{p}}{T}
  11. ( U ) P = - ( P ) U = C P - P ( V T ) P (\partial U)_{P}=-(\partial P)_{U}=C_{P}-P\left(\frac{\partial V}{\partial T}% \right)_{P}
  12. ( H ) P = - ( P ) H = C P (\partial H)_{P}=-(\partial P)_{H}=C_{P}
  13. ( G ) P = - ( P ) G = - S (\partial G)_{P}=-(\partial P)_{G}=-S
  14. ( A ) P = - ( P ) A = - S - P ( V T ) P (\partial A)_{P}=-(\partial P)_{A}=-S-P\left(\frac{\partial V}{\partial T}% \right)_{P}
  15. ( V ) T = - ( T ) V = - ( V P ) T (\partial V)_{T}=-(\partial T)_{V}=-\left(\frac{\partial V}{\partial P}\right)% _{T}
  16. ( S ) T = - ( T ) S = ( V T ) P (\partial S)_{T}=-(\partial T)_{S}=\left(\frac{\partial V}{\partial T}\right)_% {P}
  17. ( U ) T = - ( T ) U = T ( V T ) P + P ( V P ) T (\partial U)_{T}=-(\partial T)_{U}=T\left(\frac{\partial V}{\partial T}\right)% _{P}+P\left(\frac{\partial V}{\partial P}\right)_{T}
  18. ( H ) T = - ( T ) H = - V + T ( V T ) P (\partial H)_{T}=-(\partial T)_{H}=-V+T\left(\frac{\partial V}{\partial T}% \right)_{P}
  19. ( G ) T = - ( T ) G = - V (\partial G)_{T}=-(\partial T)_{G}=-V
  20. ( A ) T = - ( T ) A = P ( V P ) T (\partial A)_{T}=-(\partial T)_{A}=P\left(\frac{\partial V}{\partial P}\right)% _{T}
  21. ( S ) V = - ( V ) S = C P T ( V P ) T + ( V T ) P 2 (\partial S)_{V}=-(\partial V)_{S}=\frac{C_{P}}{T}\left(\frac{\partial V}{% \partial P}\right)_{T}+\left(\frac{\partial V}{\partial T}\right)_{P}^{2}
  22. ( U ) V = - ( V ) U = C P ( V P ) T + T ( V T ) P 2 (\partial U)_{V}=-(\partial V)_{U}=C_{P}\left(\frac{\partial V}{\partial P}% \right)_{T}+T\left(\frac{\partial V}{\partial T}\right)_{P}^{2}
  23. ( H ) V = - ( V ) H = C P ( V P ) T + T ( V T ) P 2 - V ( V T ) P (\partial H)_{V}=-(\partial V)_{H}=C_{P}\left(\frac{\partial V}{\partial P}% \right)_{T}+T\left(\frac{\partial V}{\partial T}\right)_{P}^{2}-V\left(\frac{% \partial V}{\partial T}\right)_{P}
  24. ( G ) V = - ( V ) G = - V ( V T ) P - S ( V P ) T (\partial G)_{V}=-(\partial V)_{G}=-V\left(\frac{\partial V}{\partial T}\right% )_{P}-S\left(\frac{\partial V}{\partial P}\right)_{T}
  25. ( A ) V = - ( V ) A = - S ( V P ) T (\partial A)_{V}=-(\partial V)_{A}=-S\left(\frac{\partial V}{\partial P}\right% )_{T}
  26. ( U ) S = - ( S ) U = P C P T ( V P ) T + P ( V T ) P 2 (\partial U)_{S}=-(\partial S)_{U}=\frac{PC_{P}}{T}\left(\frac{\partial V}{% \partial P}\right)_{T}+P\left(\frac{\partial V}{\partial T}\right)_{P}^{2}
  27. ( H ) S = - ( S ) H = - V C P T (\partial H)_{S}=-(\partial S)_{H}=-\frac{VC_{P}}{T}
  28. ( G ) S = - ( S ) G = - V C P T + S ( V T ) P (\partial G)_{S}=-(\partial S)_{G}=-\frac{VC_{P}}{T}+S\left(\frac{\partial V}{% \partial T}\right)_{P}
  29. ( A ) S = - ( S ) A = P C P T ( V P ) T + P ( V T ) P 2 + S ( V T ) P (\partial A)_{S}=-(\partial S)_{A}=\frac{PC_{P}}{T}\left(\frac{\partial V}{% \partial P}\right)_{T}+P\left(\frac{\partial V}{\partial T}\right)_{P}^{2}+S% \left(\frac{\partial V}{\partial T}\right)_{P}
  30. ( H ) U = - ( U ) H = - V C P + P V ( V T ) P - P C P ( V P ) T - P T ( V T ) P 2 (\partial H)_{U}=-(\partial U)_{H}=-VC_{P}+PV\left(\frac{\partial V}{\partial T% }\right)_{P}-PC_{P}\left(\frac{\partial V}{\partial P}\right)_{T}-PT\left(% \frac{\partial V}{\partial T}\right)_{P}^{2}
  31. ( G ) U = - ( U ) G = - V C P + P V ( V T ) P + S T ( V T ) P + S P ( V P ) T (\partial G)_{U}=-(\partial U)_{G}=-VC_{P}+PV\left(\frac{\partial V}{\partial T% }\right)_{P}+ST\left(\frac{\partial V}{\partial T}\right)_{P}+SP\left(\frac{% \partial V}{\partial P}\right)_{T}
  32. ( A ) U = - ( U ) A = P ( C P + S ) ( V P ) T + P T ( V T ) P 2 + S T ( V T ) P (\partial A)_{U}=-(\partial U)_{A}=P(C_{P}+S)\left(\frac{\partial V}{\partial P% }\right)_{T}+PT\left(\frac{\partial V}{\partial T}\right)_{P}^{2}+ST\left(% \frac{\partial V}{\partial T}\right)_{P}
  33. ( G ) H = - ( H ) G = - V ( C P + S ) + T S ( V T ) P (\partial G)_{H}=-(\partial H)_{G}=-V(C_{P}+S)+TS\left(\frac{\partial V}{% \partial T}\right)_{P}
  34. ( A ) H = - ( H ) A = - [ S + P ( V T ) P ] [ V - T ( V T ) P ] + P C P ( V P ) T (\partial A)_{H}=-(\partial H)_{A}=-\left[S+P\left(\frac{\partial V}{\partial T% }\right)_{P}\right]\left[V-T\left(\frac{\partial V}{\partial T}\right)_{P}% \right]+PC_{P}\left(\frac{\partial V}{\partial P}\right)_{T}
  35. ( A ) G = - ( G ) A = - S [ V + P ( V P ) T ] - P V ( V T ) P (\partial A)_{G}=-(\partial G)_{A}=-S\left[V+P\left(\frac{\partial V}{\partial P% }\right)_{T}\right]-PV\left(\frac{\partial V}{\partial T}\right)_{P}

Brinkmann_coordinates.html

  1. d s 2 = H ( u , x , y ) d u 2 + 2 d u d v + d x 2 + d y 2 ds^{2}\,=H(u,x,y)du^{2}+2dudv+dx^{2}+dy^{2}
  2. v \partial_{v}
  3. d v dv
  4. u \partial_{u}
  5. H ( u , x , y ) H(u,x,y)
  6. x , y \partial_{x},\partial_{y}
  7. u = u 0 , v = v 0 u=u_{0},v=v_{0}
  8. u , v , x , y u,v,x,y
  9. - < v , x , y < , u 0 < u < u 1 -\infty<v,x,y<\infty,u_{0}<u<u_{1}

Brouwer–Heyting–Kolmogorov_interpretation.html

  1. P Q P\wedge Q
  2. P Q P\vee Q
  3. P Q P\to Q
  4. x S : φ ( x ) \exists x\in S:\varphi(x)
  5. x S : φ ( x ) \forall x\in S:\varphi(x)
  6. ¬ P \neg P
  7. P P\to\bot
  8. \bot
  9. \bot
  10. P Q P\to Q
  11. P P P\to P
  12. ¬ ( P ¬ P ) \neg(P\wedge\neg P)
  13. ( P ( P ) ) (P\wedge(P\to\bot))\to\bot
  14. ( P ( P ) ) (P\wedge(P\to\bot))\to\bot
  15. ( P ( P ) ) (P\wedge(P\to\bot))
  16. \bot
  17. ( P ( P ) ) (P\wedge(P\to\bot))
  18. P P\to\bot
  19. P P\to\bot
  20. \bot
  21. ( P ( P ) ) (P\wedge(P\to\bot))\to\bot
  22. \bot
  23. \bot
  24. f ( a , b ) = b ( a ) f(\langle a,b\rangle)=b(a)
  25. P ( ¬ P ) P\vee(\neg P)
  26. P ( P ) P\vee(P\to\bot)
  27. P ( ¬ P ) P\vee(\neg P)
  28. P P\to\bot
  29. P P\to\bot
  30. P ( ¬ P ) P\vee(\neg P)
  31. \bot
  32. \bot

Brownian_bridge.html

  1. B t := ( W t | W 1 = 0 ) , t [ 0 , 1 ] B_{t}:=(W_{t}|W_{1}=0),\;t\in[0,1]
  2. W ( t ) = B ( t ) + t Z W(t)=B(t)+tZ\,
  3. W ( t ) = B ( t T ) + t T Z . W(t)=B\left(\frac{t}{T}\right)+\frac{t}{\sqrt{T}}Z.
  4. B ( t ) = ( 1 - t ) W ( t 1 - t ) . B(t)=(1-t)W\left(\frac{t}{1-t}\right).
  5. W ( t ) = ( 1 + t ) B ( t 1 + t ) . W(t)=(1+t)B\left(\frac{t}{1+t}\right).
  6. B t = k = 1 Z k 2 sin ( k π t ) k π B_{t}=\sum_{k=1}^{\infty}Z_{k}\frac{\sqrt{2}\sin(k\pi t)}{k\pi}
  7. Z 1 , Z 2 , Z_{1},Z_{2},\ldots
  8. a + t - t 1 t 2 - t 1 ( b - a ) a+\frac{t-t_{1}}{t_{2}-t_{1}}(b-a)
  9. ( t 2 - t ) ( s - t 1 ) t 2 - t 1 . \frac{(t_{2}-t)(s-t_{1})}{t_{2}-t_{1}}.

Brownian_noise.html

  1. d W ( t ) dW(t)
  2. W ( t ) = 0 t d W ( t ) W(t)=\int_{0}^{t}dW(t)
  3. d W ( t ) dW(t)
  4. S 0 = | [ d W ( t ) d t ] ( ω ) | 2 = const S_{0}=\left|\mathcal{F}\left[\frac{dW(t)}{dt}\right](\omega)\right|^{2}=\,% \text{const}
  5. \mathcal{F}
  6. S 0 S_{0}
  7. [ d W ( t ) d t ] ( ω ) = i ω [ W ( t ) ] ( ω ) \mathcal{F}\left[\frac{dW(t)}{dt}\right](\omega)=i\omega\mathcal{F}[W(t)](\omega)
  8. S ( ω ) = | [ W ( t ) ] ( ω ) | 2 = S 0 ω 2 S(\omega)=\left|\mathcal{F}[W(t)](\omega)\right|^{2}=\frac{S_{0}}{\omega^{2}}

Broyden–Fletcher–Goldfarb–Shanno_algorithm.html

  1. B k 𝐩 k = - f ( 𝐱 k ) B_{k}\mathbf{p}_{k}=-\nabla f(\mathbf{x}_{k})
  2. B k B_{k}
  3. f ( 𝐱 k ) \nabla f(\mathbf{x}_{k})
  4. B k + 1 = B k + U k + V k B_{k+1}=B_{k}+U_{k}+V_{k}\,\!
  5. C = 𝐚𝐛 T C=\mathbf{a}\mathbf{b}^{\mathrm{T}}
  6. B k + 1 ( 𝐱 k + 1 - 𝐱 k ) = f ( 𝐱 k + 1 ) - f ( 𝐱 k ) . B_{k+1}(\mathbf{x}_{k+1}-\mathbf{x}_{k})=\nabla f(\mathbf{x}_{k+1})-\nabla f(% \mathbf{x}_{k}).
  7. 𝐱 0 \mathbf{x}_{0}
  8. B 0 B_{0}
  9. 𝐱 k \mathbf{x}_{k}
  10. 𝐩 k \mathbf{p}_{k}
  11. B k 𝐩 k = - f ( 𝐱 k ) . B_{k}\mathbf{p}_{k}=-\nabla f(\mathbf{x}_{k}).
  12. α k \alpha_{k}
  13. 𝐱 k + 1 = 𝐱 k + α k 𝐩 k . \mathbf{x}_{k+1}=\mathbf{x}_{k}+\alpha_{k}\mathbf{p}_{k}.
  14. 𝐬 k = α k 𝐩 k . \mathbf{s}_{k}=\alpha_{k}\mathbf{p}_{k}.
  15. 𝐲 k = f ( 𝐱 k + 1 ) - f ( 𝐱 k ) . \mathbf{y}_{k}={\nabla f(\mathbf{x}_{k+1})-\nabla f(\mathbf{x}_{k})}.
  16. B k + 1 = B k + 𝐲 k 𝐲 k T 𝐲 k T 𝐬 k - B k 𝐬 k 𝐬 k T B k 𝐬 k T B k 𝐬 k . B_{k+1}=B_{k}+\frac{\mathbf{y}_{k}\mathbf{y}_{k}^{\mathrm{T}}}{\mathbf{y}_{k}^% {\mathrm{T}}\mathbf{s}_{k}}-\frac{B_{k}\mathbf{s}_{k}\mathbf{s}_{k}^{\mathrm{T% }}B_{k}}{\mathbf{s}_{k}^{\mathrm{T}}B_{k}\mathbf{s}_{k}}.
  17. f ( 𝐱 ) f(\mathbf{x})
  18. | f ( 𝐱 k ) | \left|\nabla f(\mathbf{x}_{k})\right|
  19. B 0 B_{0}
  20. B 0 = I B_{0}=I
  21. B k B_{k}
  22. B k B_{k}
  23. B k + 1 - 1 = ( I - s k y k T y k T s k ) B k - 1 ( I - y k s k T y k T s k ) + s k s k T y k T s k . B_{k+1}^{-1}=\left(I-\frac{s_{k}y_{k}^{T}}{y_{k}^{T}s_{k}}\right)B_{k}^{-1}% \left(I-\frac{y_{k}s_{k}^{T}}{y_{k}^{T}s_{k}}\right)+\frac{s_{k}s_{k}^{T}}{y_{% k}^{T}\,s_{k}}.
  24. B k - 1 B_{k}^{-1}
  25. 𝐲 k T B k - 1 𝐲 k \mathbf{y}_{k}^{\mathrm{T}}B_{k}^{-1}\mathbf{y}_{k}
  26. 𝐬 k T 𝐲 k \mathbf{s}_{k}^{\mathrm{T}}\mathbf{y}_{k}
  27. B k + 1 - 1 = B k - 1 + ( 𝐬 k T 𝐲 k + 𝐲 k T B k - 1 𝐲 k ) ( 𝐬 k 𝐬 k T ) ( 𝐬 k T 𝐲 k ) 2 - B k - 1 𝐲 k 𝐬 k T + 𝐬 k 𝐲 k T B k - 1 𝐬 k T 𝐲 k . B_{k+1}^{-1}=B_{k}^{-1}+\frac{(\mathbf{s}_{k}^{\mathrm{T}}\mathbf{y}_{k}+% \mathbf{y}_{k}^{\mathrm{T}}B_{k}^{-1}\mathbf{y}_{k})(\mathbf{s}_{k}\mathbf{s}_% {k}^{\mathrm{T}})}{(\mathbf{s}_{k}^{\mathrm{T}}\mathbf{y}_{k})^{2}}-\frac{B_{k% }^{-1}\mathbf{y}_{k}\mathbf{s}_{k}^{\mathrm{T}}+\mathbf{s}_{k}\mathbf{y}_{k}^{% \mathrm{T}}B_{k}^{-1}}{\mathbf{s}_{k}^{\mathrm{T}}\mathbf{y}_{k}}.

Bryant_Tuckerman.html

  1. 2 19937 - 1 2^{19937}-1

Bubble_ring.html

  1. ω 2 = ( k a K 1 k a ) [ ( 1 - k 2 a 2 ) T p a 3 - Γ 2 4 π 2 a 4 ] \omega^{2}=\left(\frac{kaK_{1}}{ka}\right)\left[(1-k^{2}a^{2})\frac{T}{pa^{3}}% -\frac{\Gamma^{2}}{4\pi^{2}a^{4}}\right]

Buffering_agent.html

  1. pH = pK a + log ( [ A - ] [ HA ] ) \textrm{pH}=\textrm{pK}_{a}+\log\left(\frac{[\textrm{A}^{-}]}{[\textrm{HA}]}\right)
  2. pH = pK a + log ( m o l A - m o l HA ) \textrm{pH}=\textrm{pK}_{a}+\log\left(\frac{mol\textrm{A}^{-}}{mol\textrm{HA}}\right)
  3. pH n e w = pK a + log ( m o l A - + 0.5 m o l A - m o l HA - 0.5 m o l HA ) \textrm{pH}_{new}=\textrm{pK}_{a}+\log\left(\frac{mol\textrm{A}^{-}+0.5mol% \textrm{A}^{-}}{mol\textrm{HA}-0.5mol\textrm{HA}}\right)
  4. pH n e w = pK a + log ( n + 0.5 n n - 0.5 n ) \textrm{pH}_{new}=\textrm{pK}_{a}+\log\left(\frac{\textrm{n}+0.5\textrm{n}}{% \textrm{n}-0.5\textrm{n}}\right)
  5. pH n e w = pK a + log ( 1.5 n 0.5 n ) \textrm{pH}_{new}=\textrm{pK}_{a}+\log\left(\frac{1.5\textrm{n}}{0.5\textrm{n}% }\right)
  6. pH n e w = pK a + log ( [ A - ] o r i g i n a l [ HA ] o r i g i n a l ) + log ( 3 ) \textrm{pH}_{new}=\textrm{pK}_{a}+\log\left(\frac{[\textrm{A}^{-}]_{original}}% {[\textrm{HA}]_{original}}\right)+\log\left(3\right)

Bulk_Richardson_number.html

  1. R i = g Δ θ v / θ v [ ( Δ U ) 2 + ( Δ V ) 2 ] / Δ Z Ri=\frac{g\Delta\theta_{v}/\theta_{v}}{[(\Delta U)^{2}+(\Delta V)^{2}]/\Delta Z}

Bundle_gerbe.html

  1. U ( 1 ) U(1)
  2. M M
  3. U ( 1 ) U(1)
  4. H 2 ( M , ) H^{2}(M,\mathbb{Z})
  5. M M
  6. H 3 ( M , ) H^{3}(M,\mathbb{Z})
  7. d + H d+H
  8. d d
  9. H H

Burnside_theorem.html

  1. p a q b p^{a}q^{b}
  2. p r p^{r}

Business_valuation.html

  1. k e = R f + β ( R m - R f ) + S C R P + C S R P k_{e}=R_{f}+\beta(R_{m}-R_{f})+SCRP+CSRP
  2. R f R_{f}
  3. β \beta
  4. k e k_{e}
  5. R m R_{m}

Butterfly_theorem.html

  1. X X XX^{\prime}\,
  2. X X ′′ XX^{\prime\prime}\,
  3. X X\,
  4. A M AM\,
  5. D M DM\,
  6. Y Y YY^{\prime}\,
  7. Y Y ′′ YY^{\prime\prime}\,
  8. Y Y\,
  9. B M BM\,
  10. C M CM\,
  11. M X X M Y Y , \triangle MXX^{\prime}\sim\triangle MYY^{\prime},\,
  12. M X M Y = X X Y Y , {MX\over MY}={XX^{\prime}\over YY^{\prime}},
  13. M X X ′′ M Y Y ′′ , \triangle MXX^{\prime\prime}\sim\triangle MYY^{\prime\prime},\,
  14. M X M Y = X X ′′ Y Y ′′ , {MX\over MY}={XX^{\prime\prime}\over YY^{\prime\prime}},
  15. A X X C Y Y ′′ , \triangle AXX^{\prime}\sim\triangle CYY^{\prime\prime},\,
  16. X X Y Y ′′ = A X C Y , {XX^{\prime}\over YY^{\prime\prime}}={AX\over CY},
  17. D X X ′′ B Y Y , \triangle DXX^{\prime\prime}\sim\triangle BYY^{\prime},\,
  18. X X ′′ Y Y = D X B Y , {XX^{\prime\prime}\over YY^{\prime}}={DX\over BY},
  19. ( M X M Y ) 2 = X X Y Y X X ′′ Y Y ′′ , \left({MX\over MY}\right)^{2}={XX^{\prime}\over YY^{\prime}}{XX^{\prime\prime}% \over YY^{\prime\prime}},
  20. = A X . D X C Y . B Y , {}={AX.DX\over CY.BY},
  21. = P X . Q X P Y . Q Y , {}={PX.QX\over PY.QY},
  22. = ( P M - X M ) . ( M Q + X M ) ( P M + M Y ) . ( Q M - M Y ) , {}={(PM-XM).(MQ+XM)\over(PM+MY).(QM-MY)},
  23. = ( P M ) 2 - ( M X ) 2 ( P M ) 2 - ( M Y ) 2 , {}={(PM)^{2}-(MX)^{2}\over(PM)^{2}-(MY)^{2}},
  24. P M PM\,
  25. M Q MQ\,
  26. ( M X ) 2 ( M Y ) 2 = ( P M ) 2 - ( M X ) 2 ( P M ) 2 - ( M Y ) 2 . {(MX)^{2}\over(MY)^{2}}={(PM)^{2}-(MX)^{2}\over(PM)^{2}-(MY)^{2}}.
  27. M X = M Y , MX=MY,\,
  28. M M\,
  29. X Y . XY.\,

C0-semigroup.html

  1. X X
  2. T : + L ( X ) T:\mathbb{R}_{+}\to L(X)
  3. T ( 0 ) = I T(0)=I
  4. X X
  5. t , s 0 : T ( t + s ) = T ( t ) T ( s ) \forall t,s\geq 0:\ T(t+s)=T(t)T(s)
  6. x 0 X : T ( t ) x 0 - x 0 0 \forall x_{0}\in X:\ \|T(t)x_{0}-x_{0}\|\to 0
  7. t 0 t\downarrow 0
  8. T T
  9. + , + \mathbb{R}_{+},+
  10. T T
  11. A x = lim t 0 1 t ( T ( t ) - I ) x A\,x=\lim_{t\downarrow 0}\frac{1}{t}\,(T(t)-I)\,x
  12. lim t 0 + T ( t ) - I = 0 \lim_{t\to 0^{+}}\|T(t)-I\|=0
  13. 𝒟 ( A ) = X \mathcal{D}(A)=X
  14. T ( t ) = e A t := k = 0 A k k ! t k . T(t)=e^{At}:=\sum_{k=0}^{\infty}\frac{A^{k}}{k!}t^{k}.
  15. A : X X A\colon X\to X
  16. T ( t ) := e A t T(t):=e^{At}
  17. e A t e^{At}
  18. u ( t ) = A u ( t ) , u ( 0 ) = x , u^{\prime}(t)=Au(t),~{}~{}~{}u(0)=x,
  19. 0 t u ( s ) d s D ( A ) and A 0 t u ( s ) d s = u ( t ) - x . \int_{0}^{t}u(s)\,ds\in D(A)\,\text{ and }A\int_{0}^{t}u(s)\,ds=u(t)-x.
  20. ω 0 = inf t > 0 1 t log T ( t ) . \omega_{0}=\inf_{t>0}\frac{1}{t}\log\|T(t)\|.
  21. T ( t ) M e ω t \|T(t)\|\leq Me^{\omega t}
  22. T ( t ) M e - ω t , \|T(t)\|\leq M{\rm e}^{-\omega t},
  23. lim t T ( t ) = 0 \lim_{t\to\infty}\|T(t)\|=0
  24. T ( t 0 ) < 1 \|T(t_{0})\|<1
  25. 0 T ( t ) x p d t < \int_{0}^{\infty}\|T(t)x\|^{p}\,dt<\infty
  26. 0 T ( t ) x p d t < . \int_{0}^{\infty}\|T(t)x\|^{p}\,dt<\infty.
  27. H ( + ; L ( X ) ) H^{\infty}(\mathbb{C}_{+};L(X))
  28. s ( A ) := sup { Re λ : λ σ ( A ) } s(A):=\sup\{{\rm Re}\lambda:\lambda\in\sigma(A)\}
  29. lim t T ( t ) x = 0 \lim_{t\to\infty}\|T(t)x\|=0
  30. T ( t ) M \|T(t)\|\leq M

C4.5_algorithm.html

  1. S = s 1 , s 2 , S={s_{1},s_{2},...}
  2. s i s_{i}
  3. ( x 1 , i , x 2 , i , , x p , i ) (x_{1,i},x_{2,i},...,x_{p,i})
  4. x j x_{j}
  5. s i s_{i}

C_parity.html

  1. 𝒞 \mathcal{C}
  2. 𝒞 | ψ = | ψ ¯ . \mathcal{C}\,|\psi\rangle=|\bar{\psi}\rangle.
  3. 1 = ψ | ψ = ψ ¯ | ψ ¯ = ψ | 𝒞 𝒞 | ψ , 1=\langle\psi|\psi\rangle=\langle\bar{\psi}|\bar{\psi}\rangle=\langle\psi|% \mathcal{C}^{\dagger}\mathcal{C}|\psi\rangle,
  4. 𝒞 \mathcal{C}
  5. 𝒞 𝒞 = 1. \mathcal{C}\mathcal{C}^{\dagger}=\mathbf{1}.
  6. 𝒞 \mathcal{C}
  7. 𝒞 2 | ψ = 𝒞 | ψ ¯ = | ψ , \mathcal{C}^{2}|\psi\rangle=\mathcal{C}|\bar{\psi}\rangle=|\psi\rangle,
  8. 𝒞 2 = 𝟏 \mathcal{C}^{2}=\mathbf{1}
  9. 𝒞 = 𝒞 - 1 \mathcal{C}=\mathcal{C}^{-1}
  10. 𝒞 = 𝒞 , \mathcal{C}=\mathcal{C}^{\dagger},
  11. 𝒞 | ψ = η C | ψ \mathcal{C}\,|\psi\rangle=\eta_{C}\,|{\psi}\rangle
  12. 𝒞 \mathcal{C}
  13. 𝒞 2 | ψ = η C 𝒞 | ψ = η C 2 | ψ = | ψ \mathcal{C}^{2}|\psi\rangle=\eta_{C}\mathcal{C}|{\psi}\rangle=\eta_{C}^{2}|% \psi\rangle=|\psi\rangle
  14. η C = ± 1 \eta_{C}=\pm 1
  15. 𝒞 | ψ \mathcal{C}|\psi\rangle
  16. | ψ |\psi\rangle
  17. 𝒞 | π + π - = ( - 1 ) L | π + π - \mathcal{C}\,|\pi^{+}\,\pi^{-}\rangle=(-1)^{L}\,|\pi^{+}\,\pi^{-}\rangle
  18. 𝒞 | f f ¯ = ( - 1 ) L ( - 1 ) S + 1 ( - 1 ) | f f ¯ = ( - 1 ) L + S | f f ¯ \mathcal{C}\,|f\,\bar{f}\rangle=(-1)^{L}(-1)^{S+1}(-1)\,|f\,\bar{f}\rangle=(-1% )^{L+S}\,|f\,\bar{f}\rangle
  19. π 0 3 γ \pi^{0}\rightarrow 3\gamma
  20. π 0 \pi^{0}
  21. η C = ( - 1 ) 2 = 1 \eta_{C}=(-1)^{2}=1
  22. π - + p π 0 + n \pi^{-}+p\rightarrow\pi^{0}+n
  23. η π + π - π 0 \eta\rightarrow\pi^{+}\pi^{-}\pi^{0}
  24. p p ¯ p\bar{p}

Cache-oblivious_algorithm.html

  1. O ( 1 ) O(1)
  2. L L
  3. w w
  4. b b
  5. w w
  6. b b
  7. Z Z
  8. Z = Ω ( L 2 ) Z=\Omega(L^{2})
  9. t t
  10. W ( n ) W(n)
  11. Q ( n , L , Z ) Q(n,L,Z)
  12. Q ( n , L , Z ) Q(n,L,Z)
  13. L L
  14. Z Z
  15. n n
  16. n / L n/L
  17. L L
  18. L L
  19. O ( n ) O(n)
  20. L L
  21. A A
  22. B B
  23. Θ ( m n ) \Theta(mn)
  24. O ( m n ) O(mn)
  25. O ( 1 + m n / L ) O(1+mn/L)
  26. m m
  27. n n
  28. m × n m\times n
  29. n × m n\times m
  30. O ( m n ) O(mn)
  31. O ( m n / L ) O(mn/L)

Calculator_input_methods.html

  1. 1 + 2 × 3 1+2\times 3
  2. sin 30 × cos 30 \sin 30\times\cos 30
  3. 5 - 3 5-3
  4. 15 + 10 + 10 + 10 15+10+10+10
  5. 3 , 8 6 ¯ 3,8\overline{6}
  6. 1 + 2 × 3 1+2\times 3
  7. sin 30 × cos 30 \sin 30\times\cos 30
  8. ( 1 + 2 ) × ( 3 + 4 ) (1+2)\times(3+4)
  9. 15 + 10 + 10 + 10 15+10+10+10
  10. 1 + 2 × 3 1+2\times 3
  11. sin 30 × cos 30 \sin 30\times\cos 30
  12. ( 1 + 2 ) × ( 3 + 4 ) (1+2)\times(3+4)
  13. 15 + 10 + 10 + 10 15+10+10+10
  14. 1 + 2 × 3 1+2\times 3
  15. sin 30 × cos 30 \sin 30\times\cos 30
  16. 1 + 2 × 3 1+2\times 3
  17. sin 30 × cos 30 \sin 30\times\cos 30
  18. 5 - 3 5-3
  19. 15 + 10 + 10 + 10 15+10+10+10
  20. 1 + 2 × 3 1+2\times 3
  21. sin 30 × cos 30 \sin 30\times\cos 30
  22. 5 - 3 5-3
  23. 15 + 10 + 10 + 10 15+10+10+10

Canonical_ring.html

  1. R ( V , K ) = R ( V , K V ) R(V,K)=R(V,K_{V})\,
  2. n 0 n\geq 0
  3. R n := H 0 ( V , K n ) , R_{n}:=H^{0}(V,K^{n}),
  4. R 0 R_{0}
  5. P n = h 0 ( V , K n ) = dim H 0 ( V , K n ) P_{n}=h^{0}(V,K^{n})=\operatorname{dim}\ H^{0}(V,K^{n})
  6. K n K^{n}
  7. 𝐏 ( H 0 ( V , K n ) ) = 𝐏 P n - 1 \mathbf{P}(H^{0}(V,K^{n}))=\mathbf{P}^{P_{n}-1}

Cant_(road::rail).html

  1. E a E_{a}
  2. v v
  3. r r
  4. w w
  5. v 2 = E a r g w 2 - E a 2 E a r g w v^{2}=\frac{E_{a}rg}{\sqrt{w^{2}-E_{a}^{2}}}\approx\frac{E_{a}rg}{w}
  6. g g
  7. E u E_{u}
  8. v m a x v_{max}
  9. v m a x ( E a + E u ) r g w = ( E a + E u ) g d w v_{max}\approx\sqrt{\frac{(E_{a}+E_{u})rg}{w}}=\sqrt{\frac{(E_{a}+E_{u})g}{dw}}
  10. d = 1 / r d=1/r
  11. g = 32.17 ft / s 2 g=32.17\,\mathrm{ft/s^{2}}
  12. w = 56.5 in w=56.5\,\mathrm{in}
  13. v m a x 3600 63360 32.17 12 ( E a + E u ) 56.5 d π 1200 180 E a + E u 0.00066 d v_{max}\approx\frac{3600}{63360}\sqrt{\frac{32.17\cdot 12(E_{a}+E_{u})}{56.5% \cdot d\frac{\pi}{1200\cdot 180}}}\approx\sqrt{\frac{E_{a}+E_{u}}{0.00066d}}
  14. E a E_{a}
  15. E u E_{u}
  16. d d
  17. v m a x v_{max}
  18. v m a x = E a + 3 0.00066 d v_{max}=\sqrt{\frac{E_{a}+3}{0.00066d}}

Capillary_electrophoresis.html

  1. u p u_{p}
  2. u p = μ p E u_{p}=\mu_{p}E\,
  3. μ p = ( L t r ) ( L t V ) \mu_{p}=\left(\frac{L}{t_{r}}\right)\left(\frac{L_{t}}{V}\right)
  4. L L
  5. t r t_{r}
  6. V V
  7. L t L_{t}
  8. u o u_{o}
  9. u o = μ o E u_{o}=\mu_{o}E
  10. μ o \mu_{o}
  11. μ o = ϵ ζ η \mu_{o}=\frac{\epsilon\zeta}{\eta}
  12. ζ \zeta
  13. ϵ \epsilon
  14. u u
  15. u p + u o = ( μ p + μ o ) E u_{p}+u_{o}=(\mu_{p}+\mu_{o})E
  16. N = μ V 2 D m N=\frac{\mu V}{2D_{m}}
  17. N N
  18. μ \mu
  19. D m D_{m}
  20. R s R_{s}
  21. R s = 1 4 ( μ p N μ p + μ o ) R_{s}=\frac{1}{4}\left(\frac{\triangle\mu_{p}\sqrt{N}}{\mu_{p}+\mu_{o}}\right)

Capital_account.html

  1. = Change in foreign ownership of domestic assets \displaystyle=\mbox{Change in foreign ownership of domestic assets}
  2. = Foreign direct investment \displaystyle=\mbox{Foreign direct investment}

Capital_Cost_Allowance.html

  1. C C A = t d U C C CCA=tdUCC
  2. C C A = t d U C C - 1 2 t d ( a - b - c ) CCA=tdUCC-\frac{1}{2}td\left(a-b-c\right)
  3. I d Id
  4. I d ( 1 - d ) Id(1-d)
  5. I d ( 1 - d ) 2 Id(1-d)^{2}
  6. I d ( 1 - d ) n - 1 Id(1-d)^{n-1}
  7. I t d ( 1 - d ) n - 1 Itd(1-d)^{n-1}
  8. I t d n = 1 ( 1 - d ) n - 1 ( 1 + i ) n Itd\sum_{n=1}^{\infty}\frac{(1-d)^{n-1}}{(1+i)^{n}}
  9. P V = I t d i + d PV=\frac{Itd}{i+d}
  10. I ( 1 - t d i + d ) I\left(1-\frac{td}{i+d}\right)
  11. P V \displaystyle PV
  12. I [ 1 - ( t d i + d ) ( 1 + 1 2 i 1 + i ) ] I\left[1-\left(\frac{td}{i+d}\right)\left(\frac{1+\frac{1}{2}i}{1+i}\right)\right]

Cardinal_utility.html

  1. u ( x i ) u(x_{i})
  2. x i x_{i}
  3. v ( x i ) v(x_{i})
  4. v ( x i ) = a u ( x i ) + b v(x_{i})=au(x_{i})+b\!
  5. v ( x ) = a u ( x ) + b . v(x)=au(x)+b.
  6. A r e l a t i v e t o B ArelativetoB
  7. B r e l a t i v e t o C BrelativetoC
  8. A t o B AtoB
  9. B t o C BtoC
  10. A A
  11. B B
  12. B B
  13. A A
  14. B B
  15. B B
  16. C C
  17. $\mathbf{ }$
  18. A A
  19. B B
  20. A , B A,B
  21. C C
  22. X X
  23. L 1 = ( 0.6 , 0 , 0.4 ) , L_{1}=(0.6,0,0.4),
  24. L 2 = ( 0 , 1 , 0 ) . L_{2}=(0,1,0)\ .
  25. L 1 L 2 , L_{1}\succ L_{2},
  26. L 1 = ( 0.5 , 0 , 0.5 ) . L_{1}^{\prime}=(0.5,0,0.5).
  27. E U ( L 1 ) = E U ( L 2 ) EU(L_{1}^{\prime})=EU(L_{2})\!
  28. ( 0.5 ) * u ( x 1 ) + ( 0.5 ) * u ( x 3 ) = 1 * u ( x 2 ) . (0.5)*u(x_{1})+(0.5)*u(x_{3})=1*u(x_{2}).
  29. ( 0.5 ) * u ( x 3 ) = 1. (0.5)*u(x_{3})=1.
  30. u ( x 3 ) = 2. u(x_{3})=2.

Cardinality_of_the_continuum.html

  1. \mathbb{R}
  2. | | |\mathbb{R}|
  3. 𝔠 \mathfrak{c}
  4. \mathbb{R}
  5. \mathbb{N}
  6. \mathbb{R}
  7. \mathbb{N}
  8. \mathbb{N}
  9. 0 \aleph_{0}
  10. 𝔠 = 2 0 > 0 . \mathfrak{c}=2^{\aleph_{0}}>\aleph_{0}\,.
  11. n \mathbb{R}^{n}
  12. | ( a , b ) | = | | = | n | . |(a,b)|=|\mathbb{R}|=|\mathbb{R}^{n}|.
  13. 0 \aleph_{0}
  14. 1 \aleph_{1}
  15. 0 \aleph_{0}
  16. 𝔠 \mathfrak{c}
  17. 𝔠 = 1 \mathfrak{c}=\aleph_{1}
  18. 𝔠 {\mathfrak{c}}
  19. 0 \aleph_{0}
  20. 0 < 𝔠 . \aleph_{0}<\mathfrak{c}.
  21. 𝔠 {\mathfrak{c}}
  22. { q q x } \{q\in\mathbb{Q}\mid q\leq x\}
  23. 𝔠 2 0 \mathfrak{c}\leq 2^{\aleph_{0}}
  24. 2 0 2^{\aleph_{0}}
  25. 2 0 𝔠 2^{\aleph_{0}}\leq\mathfrak{c}
  26. 𝔠 = | P ( ) | = 2 0 . \mathfrak{c}=|P(\mathbb{N})|=2^{\aleph_{0}}.
  27. 𝔠 = 2 0 \mathfrak{c}=2^{\aleph_{0}}
  28. 𝔠 2 = 𝔠 \mathfrak{c}^{2}=\mathfrak{c}
  29. 𝔠 2 = ( 2 0 ) 2 = 2 2 × 0 = 2 0 = 𝔠 . \mathfrak{c}^{2}=(2^{\aleph_{0}})^{2}=2^{2\times{\aleph_{0}}}=2^{\aleph_{0}}=% \mathfrak{c}.
  30. 𝔠 0 = 0 0 = n 0 = 𝔠 n = 0 𝔠 = n 𝔠 = 𝔠 , \mathfrak{c}^{\aleph_{0}}={\aleph_{0}}^{\aleph_{0}}=n^{\aleph_{0}}=\mathfrak{c% }^{n}=\aleph_{0}\mathfrak{c}=n\mathfrak{c}=\mathfrak{c},
  31. 𝔠 𝔠 = ( 2 0 ) 𝔠 = 2 𝔠 × 0 = 2 𝔠 , \mathfrak{c}^{\mathfrak{c}}=(2^{\aleph_{0}})^{\mathfrak{c}}=2^{\mathfrak{c}% \times\aleph_{0}}=2^{\mathfrak{c}},
  32. 2 𝔠 2^{\mathfrak{c}}
  33. 2 𝔠 > 𝔠 2^{\mathfrak{c}}>\mathfrak{c}
  34. 𝔠 = 2 0 {\mathfrak{c}}=2^{\aleph_{0}}
  35. π \pi
  36. \mathbb{N}
  37. π \pi
  38. 0 , \aleph_{0},
  39. 0 \aleph_{0}
  40. 𝔠 0 10 0 2 0 ( 2 4 ) 0 = 2 0 + 4 0 = 2 0 {\mathfrak{c}}\leq\aleph_{0}\cdot 10^{\aleph_{0}}\leq 2^{\aleph_{0}}\cdot{(2^{% 4})}^{\aleph_{0}}=2^{\aleph_{0}+4\cdot\aleph_{0}}=2^{\aleph_{0}}
  41. 0 + 4 0 = 0 . \aleph_{0}+4\cdot\aleph_{0}=\aleph_{0}\,.
  42. 2 = { 0 , 1 } 2=\{0,1\}
  43. { 3 , 7 } \{3,7\}
  44. 2 0 𝔠 . 2^{\aleph_{0}}\leq{\mathfrak{c}}\,.
  45. 𝔠 = 2 0 . {\mathfrak{c}}=2^{\aleph_{0}}\,.
  46. 0 = 0 \beth_{0}=\aleph_{0}
  47. k + 1 = 2 k \beth_{k+1}=2^{\beth_{k}}
  48. 𝔠 {\mathfrak{c}}
  49. 𝔠 = 1 . \mathfrak{c}=\beth_{1}.
  50. 2 𝔠 = 2 . 2^{\mathfrak{c}}=\beth_{2}.
  51. 𝔠 {\mathfrak{c}}
  52. 1 \aleph_{1}
  53. A A\,
  54. 0 \aleph_{0}
  55. 𝔠 {\mathfrak{c}}
  56. A : 0 < | A | < 𝔠 . \nexists A\quad:\quad\aleph_{0}<|A|<\mathfrak{c}.
  57. 𝔠 {\mathfrak{c}}
  58. n \aleph_{n}
  59. n = 1 n=1
  60. 𝔠 ω . \mathfrak{c}\neq\aleph_{\omega}.
  61. 𝔠 \mathfrak{c}
  62. 1 \aleph_{1}
  63. ω 1 \aleph_{\omega_{1}}
  64. ω 1 \omega_{1}
  65. 𝔠 {\mathfrak{c}}
  66. \mathbb{R}
  67. \mathbb{R}
  68. [ 0 , 1 ] [0,1]
  69. a , b a,b\in\mathbb{R}
  70. a < b a<b
  71. f : ( a , b ) x arctan x + π 2 π ( b - a ) + a \begin{aligned}\displaystyle f\colon\mathbb{R}&\displaystyle\to(a,b)\\ \displaystyle x&\displaystyle\mapsto\frac{\arctan x+\frac{\pi}{2}}{\pi}\cdot(b% -a)+a\end{aligned}
  72. a a\in\mathbb{R}
  73. f : ( a , ) x { arctan x + π 2 + a if x < 0 x + π 2 + a if x 0 \begin{aligned}\displaystyle f\colon\mathbb{R}&\displaystyle\to(a,\infty)\\ \displaystyle x&\displaystyle\mapsto\begin{cases}\arctan x+\frac{\pi}{2}+a&% \mbox{if }~{}x<0\\ x+\frac{\pi}{2}+a&\mbox{if }~{}x\geq 0\end{cases}\end{aligned}
  74. b b\in\mathbb{R}
  75. f : ( - , b ) x { x - π 2 + b if x < 0 arctan x - π 2 + b if x 0 \begin{aligned}\displaystyle f\colon\mathbb{R}&\displaystyle\to(-\infty,b)\\ \displaystyle x&\displaystyle\mapsto\begin{cases}x-\frac{\pi}{2}+b&\mbox{if }~% {}x<0\\ \arctan x-\frac{\pi}{2}+b&\mbox{if }~{}x\geq 0\end{cases}\end{aligned}
  76. 0 \aleph_{0}
  77. \mathbb{R}
  78. \mathbb{R}
  79. 𝔠 \mathfrak{c}
  80. 𝔠 - 0 = 𝔠 \mathfrak{c}-\aleph_{0}=\mathfrak{c}
  81. | | = 𝔠 \left|\mathbb{C}\right|=\mathfrak{c}
  82. n \mathbb{R}^{n}
  83. \mathbb{C}
  84. | 2 | = 𝔠 \left|\mathbb{R}^{2}\right|=\mathfrak{c}
  85. c c\in\mathbb{C}
  86. a + b i a+bi
  87. a , b a,b\in\mathbb{R}
  88. f : 2 ( a , b ) a + b i \begin{aligned}\displaystyle f\colon\mathbb{R}^{2}&\displaystyle\to\mathbb{C}% \\ \displaystyle(a,b)&\displaystyle\mapsto a+bi\end{aligned}
  89. 𝒫 ( ) \mathcal{P}(\mathbb{N})
  90. \mathbb{N}\rightarrow\mathbb{Z}
  91. \mathbb{Z}^{\mathbb{N}}
  92. \mathbb{R}^{\mathbb{N}}
  93. \mathbb{R}
  94. \mathbb{R}
  95. n \mathbb{R}^{n}
  96. n \mathbb{R}^{n}
  97. \mathbb{R}
  98. \mathbb{R}
  99. 𝔠 {\mathfrak{c}}
  100. \mathbb{R}
  101. 𝒫 ( ) \mathcal{P}(\mathbb{R})
  102. 2 2^{\mathbb{R}}
  103. 𝒫 ( ) \mathcal{P}(\mathbb{R})
  104. \mathbb{R}^{\mathbb{R}}
  105. \mathbb{R}
  106. \mathbb{R}
  107. \mathbb{R}
  108. \mathbb{R}
  109. \mathbb{N}
  110. \mathbb{Q}
  111. \mathbb{R}
  112. 2 𝔠 = 2 2^{\mathfrak{c}}=\beth_{2}

Carl_Størmer.html

  1. π = 16 arctan 1 5 - 4 arctan 1 239 , \textstyle\pi=16\arctan\frac{1}{5}-4\arctan\frac{1}{239},
  2. π = 176 arctan 1 57 + 28 arctan 1 239 - 48 arctan 1 682 + 96 arctan 1 12943 \textstyle\pi=176\arctan\frac{1}{57}+28\arctan\frac{1}{239}-48\arctan\frac{1}{% 682}+96\arctan\frac{1}{12943}

Carroll's_paradox.html

  1. r 1 r_{1}
  2. r 2 r_{2}
  3. l = | r 2 - r 1 | l=|r_{2}-r_{1}|

Cartan's_equivalence_method.html

  1. ϕ : M N \phi:M\rightarrow N
  2. ϕ * h = g \phi^{*}h=g
  3. ϕ * γ i ( y ) = g j i ( x ) θ j ( x ) , ( g j i ) G \phi^{*}\gamma^{i}(y)=g^{i}_{j}(x)\theta^{j}(x),\ (g^{i}_{j})\in G

Cartan_subalgebra.html

  1. 𝔥 \mathfrak{h}
  2. 𝔤 \mathfrak{g}
  3. [ X , Y ] 𝔥 [X,Y]\in\mathfrak{h}
  4. X 𝔥 X\in\mathfrak{h}
  5. Y 𝔥 Y\in\mathfrak{h}
  6. 𝔤 \mathfrak{g}
  7. 𝔥 \mathfrak{h}
  8. 𝔥 \mathfrak{h}
  9. 𝔥 \mathfrak{h}
  10. 𝔥 \mathfrak{h}
  11. 𝔤 \mathfrak{g}
  12. 𝔤 \mathfrak{g}
  13. 𝔤 \mathfrak{g}
  14. 𝔤 \mathfrak{g}
  15. 𝔤 \mathfrak{g}
  16. 𝔤 \mathfrak{g}
  17. 𝔤 \mathfrak{g}
  18. 𝔤 \mathfrak{g}
  19. 𝔤 \mathfrak{g}
  20. ( 0 A 0 0 ) {0\ A\choose 0\ 0}
  21. 𝔥 \mathfrak{h}
  22. ( 𝔤 , 𝔥 ) (\mathfrak{g},\mathfrak{h})

Cartesian_tensor.html

  1. 𝐚 = a x 𝐞 x + a y 𝐞 y + a z 𝐞 z \mathbf{a}=a\text{x}\mathbf{e}\text{x}+a\text{y}\mathbf{e}\text{y}+a\text{z}% \mathbf{e}\text{z}
  2. 𝐞 x = ( 1 0 0 ) , 𝐞 y = ( 0 1 0 ) , 𝐞 z = ( 0 0 1 ) \mathbf{e}\text{x}=\begin{pmatrix}1\\ 0\\ 0\end{pmatrix}\,,\quad\mathbf{e}\text{y}=\begin{pmatrix}0\\ 1\\ 0\end{pmatrix}\,,\quad\mathbf{e}\text{z}=\begin{pmatrix}0\\ 0\\ 1\end{pmatrix}
  3. 𝐚 = ( a x a y a z ) \mathbf{a}=\begin{pmatrix}a\text{x}\\ a\text{y}\\ a\text{z}\end{pmatrix}
  4. 𝐚 = a 1 𝐞 1 + a 2 𝐞 2 + a 3 𝐞 3 = i = 1 3 a i 𝐞 i \mathbf{a}=a_{1}\mathbf{e}_{1}+a_{2}\mathbf{e}_{2}+a_{3}\mathbf{e}_{3}=\sum_{i% =1}^{3}a_{i}\mathbf{e}_{i}
  5. 𝐚 = i = 1 3 a i 𝐞 i a i 𝐞 i \mathbf{a}=\sum_{i=1}^{3}a_{i}\mathbf{e}_{i}\equiv a_{i}\mathbf{e}_{i}
  6. 𝐓 = ( a x 𝐞 x + a y 𝐞 y + a z 𝐞 z ) ( b x 𝐞 x + b y 𝐞 y + b z 𝐞 z ) = a x b x 𝐞 x 𝐞 x + a x b y 𝐞 x 𝐞 y + a x b z 𝐞 x 𝐞 z + a y b x 𝐞 y 𝐞 x + a y b y 𝐞 y 𝐞 y + a y b z 𝐞 y 𝐞 z + a z b x 𝐞 z 𝐞 x + a z b y 𝐞 z 𝐞 y + a z b z 𝐞 z 𝐞 z \begin{array}[]{ccl}\mathbf{T}&=&\left(a\text{x}\mathbf{e}\text{x}+a\text{y}% \mathbf{e}\text{y}+a\text{z}\mathbf{e}\text{z}\right)\otimes\left(b\text{x}% \mathbf{e}\text{x}+b\text{y}\mathbf{e}\text{y}+b\text{z}\mathbf{e}\text{z}% \right)\\ &&\\ &=&a\text{x}b\text{x}\mathbf{e}\text{x}\otimes\mathbf{e}\text{x}+a\text{x}b% \text{y}\mathbf{e}\text{x}\otimes\mathbf{e}\text{y}+a\text{x}b\text{z}\mathbf{% e}\text{x}\otimes\mathbf{e}\text{z}\\ &&{}+a\text{y}b\text{x}\mathbf{e}\text{y}\otimes\mathbf{e}\text{x}+a\text{y}b% \text{y}\mathbf{e}\text{y}\otimes\mathbf{e}\text{y}+a\text{y}b\text{z}\mathbf{% e}\text{y}\otimes\mathbf{e}\text{z}\\ &&{}+a\text{z}b\text{x}\mathbf{e}\text{z}\otimes\mathbf{e}\text{x}+a\text{z}b% \text{y}\mathbf{e}\text{z}\otimes\mathbf{e}\text{y}+a\text{z}b\text{z}\mathbf{% e}\text{z}\otimes\mathbf{e}\text{z}\\ \end{array}
  7. 𝐞 x 𝐞 x 𝐞 xx = ( 1 0 0 0 0 0 0 0 0 ) , 𝐞 x 𝐞 y 𝐞 xy = ( 0 1 0 0 0 0 0 0 0 ) , 𝐞 z 𝐞 z 𝐞 zz = ( 0 0 0 0 0 0 0 0 1 ) {\mathbf{e}\text{x}\otimes\mathbf{e}\text{x}}\equiv\mathbf{e}\text{xx}=\begin{% pmatrix}1&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}\,,\quad{\mathbf{e}\text{x}\otimes\mathbf{e}\text{y}}\equiv% \mathbf{e}\text{xy}=\begin{pmatrix}0&1&0\\ 0&0&0\\ 0&0&0\end{pmatrix}\,,\cdots\quad{\mathbf{e}\text{z}\otimes\mathbf{e}\text{z}}% \equiv\mathbf{e}\text{zz}=\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&1\end{pmatrix}
  8. 𝐓 = ( a x b x a x b y a x b z a y b x a y b y a y b z a z b x a z b y a z b z ) \mathbf{T}=\begin{pmatrix}a\text{x}b\text{x}&a\text{x}b\text{y}&a\text{x}b% \text{z}\\ a\text{y}b\text{x}&a\text{y}b\text{y}&a\text{y}b\text{z}\\ a\text{z}b\text{x}&a\text{z}b\text{y}&a\text{z}b\text{z}\end{pmatrix}
  9. 𝐓 = T xx 𝐞 xx + T xy 𝐞 xy + T xz 𝐞 xz + T yx 𝐞 yx + T yy 𝐞 yy + T yz 𝐞 yz + T zx 𝐞 zx + T zy 𝐞 zy + T zz 𝐞 zz \begin{array}[]{ccl}\mathbf{T}&=&T\text{xx}\mathbf{e}\text{xx}+T\text{xy}% \mathbf{e}\text{xy}+T\text{xz}\mathbf{e}\text{xz}\\ &&{}+T\text{yx}\mathbf{e}\text{yx}+T\text{yy}\mathbf{e}\text{yy}+T\text{yz}% \mathbf{e}\text{yz}\\ &&{}+T\text{zx}\mathbf{e}\text{zx}+T\text{zy}\mathbf{e}\text{zy}+T\text{zz}% \mathbf{e}\text{zz}\end{array}
  10. 𝐓 = T i j 𝐞 i j i j T i j 𝐞 i 𝐞 j , \mathbf{T}=T_{ij}\mathbf{e}_{ij}\equiv\sum_{ij}T_{ij}\mathbf{e}_{i}\otimes% \mathbf{e}_{j}\,,
  11. 𝐓 = ( T xx T xy T xz T yx T yy T yz T zx T zy T zz ) \mathbf{T}=\begin{pmatrix}T\text{xx}&T\text{xy}&T\text{xz}\\ T\text{yx}&T\text{yy}&T\text{yz}\\ T\text{zx}&T\text{zy}&T\text{zz}\end{pmatrix}
  12. ( v x v y v z ) = ( T xx T xy T xz T yx T yy T yz T zx T zy T zz ) ( u x u y u z ) , v i = T i j u j \begin{pmatrix}v\text{x}\\ v\text{y}\\ v\text{z}\end{pmatrix}=\begin{pmatrix}T\text{xx}&T\text{xy}&T\text{xz}\\ T\text{yx}&T\text{yy}&T\text{yz}\\ T\text{zx}&T\text{zy}&T\text{zz}\end{pmatrix}\begin{pmatrix}u\text{x}\\ u\text{y}\\ u\text{z}\end{pmatrix}\,,\quad v_{i}=T_{ij}u_{j}
  13. 𝐯 = 𝐓 ( ρ 𝐫 + σ 𝐬 ) = ρ 𝐓 ( 𝐫 ) + σ 𝐓 ( 𝐬 ) \mathbf{v}=\mathbf{T}(\rho\mathbf{r}+\sigma\mathbf{s})=\rho\mathbf{T}(\mathbf{% r})+\sigma\mathbf{T}(\mathbf{s})
  14. v i = T i j ( ρ r j + σ s j ) = ρ T i j r j + σ T i j s j v_{i}=T_{ij}(\rho r_{j}+\sigma s_{j})=\rho T_{ij}r_{j}+\sigma T_{ij}s_{j}
  15. r = ( p x p y p z ) ( T xx T xy T xz T yx T yy T yz T zx T zy T zz ) ( q x q y q z ) , r = p i T i j q j r=\begin{pmatrix}p\text{x}&p\text{y}&p\text{z}\end{pmatrix}\begin{pmatrix}T% \text{xx}&T\text{xy}&T\text{xz}\\ T\text{yx}&T\text{yy}&T\text{yz}\\ T\text{zx}&T\text{zy}&T\text{zz}\end{pmatrix}\begin{pmatrix}q\text{x}\\ q\text{y}\\ q\text{z}\end{pmatrix}\,,\quad r=p_{i}T_{ij}q_{j}
  16. 𝐯 = 𝐓 𝐮 \mathbf{v}=\mathbf{T}\cdot\mathbf{u}
  17. r = 𝐩 𝐓 𝐪 r=\mathbf{p}\cdot\mathbf{T}\cdot\mathbf{q}
  18. 𝐚 𝐛 a i b i \mathbf{a}\cdot\mathbf{b}\equiv a_{i}b_{i}
  19. 𝐭 = s y m b o l σ 𝐧 \mathbf{t}=symbol{\sigma}\cdot\mathbf{n}
  20. 𝐉 = 𝐈 \cdotsymbol ω \mathbf{J}=\mathbf{I}\cdotsymbol{\omega}
  21. T = 1 2 s y m b o l ω 𝐈 \cdotsymbol ω T=\frac{1}{2}symbol{\omega}\cdot\mathbf{I}\cdotsymbol{\omega}
  22. 𝐉 = s y m b o l σ 𝐄 \mathbf{J}=symbol{\sigma}\cdot\mathbf{E}
  23. 𝐁 = s y m b o l μ 𝐇 \mathbf{B}=symbol{\mu}\cdot\mathbf{H}
  24. J i = σ i j E j j σ i j E j J_{i}=\sigma_{ij}E_{j}\equiv\sum_{j}\sigma_{ij}E_{j}
  25. ( J x J y J z ) = ( σ xx σ xy σ xz σ yx σ yy σ yz σ zx σ zy σ zz ) ( E x E y E z ) \begin{pmatrix}J\text{x}\\ J\text{y}\\ J\text{z}\end{pmatrix}=\begin{pmatrix}\sigma\text{xx}&\sigma\text{xy}&\sigma% \text{xz}\\ \sigma\text{yx}&\sigma\text{yy}&\sigma\text{yz}\\ \sigma\text{zx}&\sigma\text{zy}&\sigma\text{zz}\end{pmatrix}\begin{pmatrix}E% \text{x}\\ E\text{y}\\ E\text{z}\end{pmatrix}
  26. T = 1 2 ω i I i j ω j 1 2 i j ω i I i j ω j , T=\frac{1}{2}\omega_{i}I_{ij}\omega_{j}\equiv\frac{1}{2}\sum_{ij}\omega_{i}I_{% ij}\omega_{j}\,,
  27. T = 1 2 ( ω x ω y ω z ) ( I xx I xy I xz I yx I yy I yz I zx I zy I zz ) ( ω x ω y ω z ) . T=\frac{1}{2}\begin{pmatrix}\omega\text{x}&\omega\text{y}&\omega\text{z}\end{% pmatrix}\begin{pmatrix}I\text{xx}&I\text{xy}&I\text{xz}\\ I\text{yx}&I\text{yy}&I\text{yz}\\ I\text{zx}&I\text{zy}&I\text{zz}\end{pmatrix}\begin{pmatrix}\omega\text{x}\\ \omega\text{y}\\ \omega\text{z}\end{pmatrix}\,.
  28. ( 𝐞 i ) j = δ i j (\mathbf{e}_{i})_{j}=\delta_{ij}
  29. 𝐚 = a i 𝐞 i i a i 𝐞 i . \mathbf{a}=a_{i}\mathbf{e}_{i}\equiv\sum_{i}a_{i}\mathbf{e}_{i}\,.
  30. 𝐓 = a i b j 𝐞 i j i j a i b j 𝐞 i 𝐞 j , \mathbf{T}=a_{i}b_{j}\mathbf{e}_{ij}\equiv\sum_{ij}a_{i}b_{j}\mathbf{e}_{i}% \otimes\mathbf{e}_{j}\,,
  31. 𝐓 = T i j 𝐞 i j i j T i j 𝐞 i 𝐞 j . \mathbf{T}=T_{ij}\mathbf{e}_{ij}\equiv\sum_{ij}T_{ij}\mathbf{e}_{i}\otimes% \mathbf{e}_{j}\,.
  32. 𝐱 = x i 𝐞 i , 𝐱 = x i 𝐞 i \mathbf{x}=x^{i}\mathbf{e}_{i}\,,\quad\mathbf{x}=x_{i}\mathbf{e}^{i}
  33. x ¯ \overline{x}
  34. 𝐞 ¯ \overline{\mathbf{e}}
  35. x ¯ \overline{x}
  36. 𝐞 ¯ \overline{\mathbf{e}}
  37. 𝐱 = x ¯ i 𝐞 ¯ i , 𝐱 = x ¯ i 𝐞 ¯ i \mathbf{x}=\bar{x}^{i}\bar{\mathbf{e}}_{i}\,,\quad\mathbf{x}=\bar{x}_{i}\bar{% \mathbf{e}}^{i}
  38. x ¯ = i x ¯ ( x 1 , x 2 , ) i x = i x ( x ¯ 1 , x ¯ 2 , ) i \bar{x}{}^{i}=\bar{x}{}^{i}\left(x^{1},x^{2},\cdots\right)\quad% \rightleftharpoons\quad x{}^{i}=x{}^{i}\left(\bar{x}^{1},\bar{x}^{2},\cdots\right)
  39. x ¯ = i x ¯ ( x 1 , x 2 , ) i x = i x ( x ¯ 1 , x ¯ 2 , ) i \bar{x}{}_{i}=\bar{x}{}_{i}\left(x_{1},x_{2},\cdots\right)\quad% \rightleftharpoons\quad x{}_{i}=x{}_{i}\left(\bar{x}_{1},\bar{x}_{2},\cdots\right)
  40. 𝐞 ¯ = j 𝐞 ¯ ( 𝐞 1 , 𝐞 2 ) j 𝐞 = j 𝐞 ( 𝐞 ¯ 1 , 𝐞 ¯ 2 ) j \bar{\mathbf{e}}{}_{j}=\bar{\mathbf{e}}{}_{j}\left(\mathbf{e}_{1},\mathbf{e}_{% 2}\cdots\right)\quad\rightleftharpoons\quad\mathbf{e}{}_{j}=\mathbf{e}{}_{j}% \left(\bar{\mathbf{e}}_{1},\bar{\mathbf{e}}_{2}\cdots\right)
  41. 𝐞 ¯ = j 𝐞 ¯ ( 𝐞 1 , 𝐞 2 ) j 𝐞 = j 𝐞 ( 𝐞 ¯ 1 , 𝐞 ¯ 2 ) j \bar{\mathbf{e}}{}^{j}=\bar{\mathbf{e}}{}^{j}\left(\mathbf{e}^{1},\mathbf{e}^{% 2}\cdots\right)\quad\rightleftharpoons\quad\mathbf{e}{}^{j}=\mathbf{e}{}^{j}% \left(\bar{\mathbf{e}}^{1},\bar{\mathbf{e}}^{2}\cdots\right)
  42. x ¯ j = x i ( s y m b o l 𝖫 ) i = j x i 𝖫 i j \bar{x}^{j}=x^{i}(symbol{\mathsf{L}})_{i}{}^{j}=x^{i}\mathsf{L}_{i}{}^{j}
  43. x ¯ j = x k ( s y m b o l 𝖫 - 1 ) j k \bar{x}_{j}=x_{k}(symbol{\mathsf{L}}^{-1})_{j}{}^{k}
  44. 𝐞 ¯ j = ( s y m b o l 𝖫 - 1 ) j 𝐞 k k \bar{\mathbf{e}}_{j}=(symbol{\mathsf{L}}^{-1})_{j}{}^{k}\mathbf{e}_{k}
  45. 𝐞 ¯ j = ( s y m b o l 𝖫 ) i 𝐞 i j = 𝖫 i 𝐞 i j \bar{\mathbf{e}}^{j}=(symbol{\mathsf{L}})_{i}{}^{j}\mathbf{e}^{i}=\mathsf{L}_{% i}{}^{j}\mathbf{e}^{i}
  46. x ¯ j 𝐞 ¯ j = x i 𝖫 i ( s y m b o l 𝖫 - 1 ) j j 𝐞 k k = x i δ i 𝐞 k k = x i 𝐞 i \bar{x}^{j}\bar{\mathbf{e}}_{j}=x^{i}\mathsf{L}_{i}{}^{j}(symbol{\mathsf{L}}^{% -1})_{j}{}^{k}\mathbf{e}_{k}=x^{i}\delta_{i}{}^{k}\mathbf{e}_{k}=x^{i}\mathbf{% e}_{i}
  47. x ¯ j 𝐞 ¯ j = x i ( s y m b o l 𝖫 - 1 ) j 𝖫 k i 𝐞 k j = x i δ i 𝐞 k k = x i 𝐞 i \bar{x}_{j}\bar{\mathbf{e}}^{j}=x_{i}(symbol{\mathsf{L}}^{-1})_{j}{}^{i}% \mathsf{L}_{k}{}^{j}\mathbf{e}^{k}=x_{i}\delta^{i}{}_{k}\mathbf{e}^{k}=x_{i}% \mathbf{e}^{i}
  48. s y m b o l 𝖫 T = s y m b o l 𝖫 - 1 ( s y m b o l 𝖫 - 1 ) i = j ( s y m b o l 𝖫 T ) i = j ( s y m b o l 𝖫 ) j = i 𝖫 j i symbol{\mathsf{L}}^{\mathrm{T}}=symbol{\mathsf{L}}^{-1}\Rightarrow(symbol{% \mathsf{L}}^{-1})_{i}{}^{j}=(symbol{\mathsf{L}}^{\mathrm{T}})_{i}{}^{j}=(% symbol{\mathsf{L}})^{j}{}_{i}=\mathsf{L}^{j}{}_{i}
  49. x ¯ \overline{x}
  50. x ¯ i x k = x k ( x j 𝖫 j i ) = 𝖫 j i x j x k = δ k j 𝖫 j i = 𝖫 k i \frac{\partial\bar{x}_{i}}{\partial x_{k}}=\frac{\partial}{\partial x_{k}}(x_{% j}\mathsf{L}_{ji})=\mathsf{L}_{ji}\frac{\partial x_{j}}{\partial x_{k}}=\delta% _{kj}\mathsf{L}_{ji}=\mathsf{L}_{ki}
  51. 𝖫 i j 𝖫 i j = x ¯ j x i \mathsf{L}_{i}{}^{j}\equiv\mathsf{L}_{ij}=\frac{\partial\bar{x}_{j}}{\partial x% _{i}}
  52. x ¯ \overline{x}
  53. x j x ¯ k = x ¯ k ( x ¯ i ( s y m b o l 𝖫 - 1 ) i j ) = x ¯ i x ¯ k ( s y m b o l 𝖫 - 1 ) i j = δ k i ( s y m b o l 𝖫 - 1 ) i j = ( s y m b o l 𝖫 - 1 ) k j \frac{\partial x_{j}}{\partial\bar{x}_{k}}=\frac{\partial}{\partial\bar{x}_{k}% }(\bar{x}_{i}(symbol{\mathsf{L}}^{-1})_{ij})=\frac{\partial\bar{x}_{i}}{% \partial\bar{x}_{k}}(symbol{\mathsf{L}}^{-1})_{ij}=\delta_{ki}(symbol{\mathsf{% L}}^{-1})_{ij}=(symbol{\mathsf{L}}^{-1})_{kj}
  54. ( s y m b o l 𝖫 - 1 ) i j ( s y m b o l 𝖫 - 1 ) i j = x j x ¯ i (symbol{\mathsf{L}}^{-1})_{i}{}^{j}\equiv(symbol{\mathsf{L}}^{-1})_{ij}=\frac{% \partial x_{j}}{\partial\bar{x}_{i}}
  55. 𝐱 ¯ = s y m b o l 𝖫 𝐱 \bar{\mathbf{x}}=symbol{\mathsf{L}}\mathbf{x}
  56. ( x ¯ 1 x ¯ 2 x ¯ 3 ) = ( x ¯ 1 x 1 x ¯ 1 x 2 x ¯ 1 x 3 x ¯ 2 x 1 x ¯ 2 x 2 x ¯ 2 x 3 x ¯ 3 x 1 x ¯ 3 x 2 x ¯ 3 x 3 ) ( x 1 x 2 x 3 ) \begin{pmatrix}\bar{x}_{1}\\ \bar{x}_{2}\\ \bar{x}_{3}\end{pmatrix}=\begin{pmatrix}\frac{\partial\bar{x}_{1}}{\partial x_% {1}}&\frac{\partial\bar{x}_{1}}{\partial x_{2}}&\frac{\partial\bar{x}_{1}}{% \partial x_{3}}\\ \frac{\partial\bar{x}_{2}}{\partial x_{1}}&\frac{\partial\bar{x}_{2}}{\partial x% _{2}}&\frac{\partial\bar{x}_{2}}{\partial x_{3}}\\ \frac{\partial\bar{x}_{3}}{\partial x_{1}}&\frac{\partial\bar{x}_{3}}{\partial x% _{2}}&\frac{\partial\bar{x}_{3}}{\partial x_{3}}\end{pmatrix}\begin{pmatrix}x_% {1}\\ x_{2}\\ x_{3}\end{pmatrix}
  57. 𝐱 = s y m b o l 𝖫 - 1 𝐱 ¯ = s y m b o l 𝖫 T 𝐱 ¯ \mathbf{x}=symbol{\mathsf{L}}^{-1}\bar{\mathbf{x}}=symbol{\mathsf{L}}^{\mathrm% {T}}\bar{\mathbf{x}}
  58. x ¯ \overline{x}
  59. 𝐞 ¯ i 𝐞 ¯ j = 𝐞 i 𝐞 j = δ i j , | 𝐞 i | = | 𝐞 ¯ i | = 1 , \bar{\mathbf{e}}_{i}\cdot\bar{\mathbf{e}}_{j}=\mathbf{e}_{i}\cdot\mathbf{e}_{j% }=\delta_{ij}\,,\quad\left|\mathbf{e}_{i}\right|=\left|\bar{\mathbf{e}}_{i}% \right|=1\,,
  60. x ¯ \overline{x}
  61. x ¯ i = 𝐞 ¯ i 𝐱 = 𝐞 ¯ i x j 𝐞 j = x i 𝖫 i j , \bar{x}_{i}=\bar{\mathbf{e}}_{i}\cdot\mathbf{x}=\bar{\mathbf{e}}_{i}\cdot x_{j% }\mathbf{e}_{j}=x_{i}\mathsf{L}_{ij}\,,
  62. x i = 𝐞 i 𝐱 = 𝐞 i x ¯ j 𝐞 ¯ j = x ¯ j ( s y m b o l 𝖫 - 1 ) j i . x_{i}=\mathbf{e}_{i}\cdot\mathbf{x}=\mathbf{e}_{i}\cdot\bar{x}_{j}\bar{\mathbf% {e}}_{j}=\bar{x}_{j}(symbol{\mathsf{L}}^{-1})_{ji}\,.
  63. x ¯ \overline{x}
  64. 𝖫 i j = 𝐞 ¯ i 𝐞 j = cos θ i j \mathsf{L}_{ij}=\bar{\mathbf{e}}_{i}\cdot\mathbf{e}_{j}=\cos\theta_{ij}
  65. ( s y m b o l 𝖫 - 1 ) i j = 𝐞 i 𝐞 ¯ j = cos θ j i (symbol{\mathsf{L}}^{-1})_{ij}=\mathbf{e}_{i}\cdot\bar{\mathbf{e}}_{j}=\cos% \theta_{ji}
  66. x ¯ \overline{x}
  67. 𝐱 ¯ = s y m b o l 𝖫 𝐱 \bar{\mathbf{x}}=symbol{\mathsf{L}}\mathbf{x}
  68. ( x ¯ 1 x ¯ 2 x ¯ 3 ) = ( 𝐞 ¯ 1 𝐞 1 𝐞 ¯ 1 𝐞 2 𝐞 ¯ 1 𝐞 3 𝐞 ¯ 2 𝐞 1 𝐞 ¯ 2 𝐞 2 𝐞 ¯ 2 𝐞 3 𝐞 ¯ 3 𝐞 1 𝐞 ¯ 3 𝐞 2 𝐞 ¯ 3 𝐞 3 ) ( x 1 x 2 x 3 ) = ( cos θ 11 cos θ 12 cos θ 13 cos θ 21 cos θ 22 cos θ 23 cos θ 31 cos θ 32 cos θ 33 ) ( x 1 x 2 x 3 ) \begin{pmatrix}\bar{x}_{1}\\ \bar{x}_{2}\\ \bar{x}_{3}\end{pmatrix}=\begin{pmatrix}\bar{\mathbf{e}}_{1}\cdot\mathbf{e}_{1% }&\bar{\mathbf{e}}_{1}\cdot\mathbf{e}_{2}&\bar{\mathbf{e}}_{1}\cdot\mathbf{e}_% {3}\\ \bar{\mathbf{e}}_{2}\cdot\mathbf{e}_{1}&\bar{\mathbf{e}}_{2}\cdot\mathbf{e}_{2% }&\bar{\mathbf{e}}_{2}\cdot\mathbf{e}_{3}\\ \bar{\mathbf{e}}_{3}\cdot\mathbf{e}_{1}&\bar{\mathbf{e}}_{3}\cdot\mathbf{e}_{2% }&\bar{\mathbf{e}}_{3}\cdot\mathbf{e}_{3}\end{pmatrix}\begin{pmatrix}x_{1}\\ x_{2}\\ x_{3}\end{pmatrix}=\begin{pmatrix}\cos\theta_{11}&\cos\theta_{12}&\cos\theta_{% 13}\\ \cos\theta_{21}&\cos\theta_{22}&\cos\theta_{23}\\ \cos\theta_{31}&\cos\theta_{32}&\cos\theta_{33}\end{pmatrix}\begin{pmatrix}x_{% 1}\\ x_{2}\\ x_{3}\end{pmatrix}
  69. 𝐱 = s y m b o l 𝖫 - 1 𝐱 ¯ = s y m b o l 𝖫 T 𝐱 ¯ \mathbf{x}=symbol{\mathsf{L}}^{-1}\bar{\mathbf{x}}=symbol{\mathsf{L}}^{\mathrm% {T}}\bar{\mathbf{x}}
  70. x ¯ \overline{x}
  71. x ¯ \overline{x}
  72. 𝐞 ¯ \overline{\mathbf{e}}
  73. 𝐞 ¯ \overline{\mathbf{e}}
  74. x ¯ \overline{x}
  75. P = 𝐯 𝐅 P=\mathbf{v}\cdot\mathbf{F}
  76. 𝐯 = s y m b o l ω × 𝐱 \mathbf{v}=symbol{\omega}\times\mathbf{x}
  77. U = - 𝐦 𝐁 U=-\mathbf{m}\cdot\mathbf{B}
  78. 𝐉 = 𝐫 × 𝐩 \mathbf{J}=\mathbf{r}\times\mathbf{p}
  79. s y m b o l τ = 𝐩 × 𝐄 symbol{\tau}=\mathbf{p}\times\mathbf{E}
  80. 𝐣 S = 𝐌 × 𝐧 \mathbf{j}_{\mathrm{S}}=\mathbf{M}\times\mathbf{n}
  81. 𝐞 x 𝐞 y = 𝐞 y 𝐞 z = 𝐞 z 𝐞 x 𝐞 y 𝐞 x = 𝐞 z 𝐞 y = 𝐞 x 𝐞 z = 0 \begin{array}[]{cccc}\mathbf{e}\text{x}\cdot\mathbf{e}\text{y}&=\mathbf{e}% \text{y}\cdot\mathbf{e}\text{z}&=\mathbf{e}\text{z}\cdot\mathbf{e}\text{x}\\ \mathbf{e}\text{y}\cdot\mathbf{e}\text{x}&=\mathbf{e}\text{z}\cdot\mathbf{e}% \text{y}&=\mathbf{e}\text{x}\cdot\mathbf{e}\text{z}&=0\end{array}
  82. 𝐞 x 𝐞 x = 𝐞 y 𝐞 y = 𝐞 z 𝐞 z = 1. \mathbf{e}\text{x}\cdot\mathbf{e}\text{x}=\mathbf{e}\text{y}\cdot\mathbf{e}% \text{y}=\mathbf{e}\text{z}\cdot\mathbf{e}\text{z}=1.
  83. 𝐞 i 𝐞 j = δ i j \mathbf{e}_{i}\cdot\mathbf{e}_{j}=\delta_{ij}
  84. g i j = 𝐞 i 𝐞 j . g_{ij}=\mathbf{e}_{i}\cdot\mathbf{e}_{j}.
  85. 𝐠 = ( g xx g xy g xz g yx g yy g zz g zx g zy g zz ) = ( 𝐞 x 𝐞 x 𝐞 x 𝐞 y 𝐞 x 𝐞 z 𝐞 y 𝐞 x 𝐞 y 𝐞 y 𝐞 y 𝐞 z 𝐞 z 𝐞 x 𝐞 z 𝐞 y 𝐞 z 𝐞 z ) = ( 1 0 0 0 1 0 0 0 1 ) \mathbf{g}=\begin{pmatrix}g\text{xx}&g\text{xy}&g\text{xz}\\ g\text{yx}&g\text{yy}&g\text{zz}\\ g\text{zx}&g\text{zy}&g\text{zz}\\ \end{pmatrix}=\begin{pmatrix}\mathbf{e}\text{x}\cdot\mathbf{e}\text{x}&\mathbf% {e}\text{x}\cdot\mathbf{e}\text{y}&\mathbf{e}\text{x}\cdot\mathbf{e}\text{z}\\ \mathbf{e}\text{y}\cdot\mathbf{e}\text{x}&\mathbf{e}\text{y}\cdot\mathbf{e}% \text{y}&\mathbf{e}\text{y}\cdot\mathbf{e}\text{z}\\ \mathbf{e}\text{z}\cdot\mathbf{e}\text{x}&\mathbf{e}\text{z}\cdot\mathbf{e}% \text{y}&\mathbf{e}\text{z}\cdot\mathbf{e}\text{z}\\ \end{pmatrix}=\begin{pmatrix}1&0&0\\ 0&1&0\\ 0&0&1\\ \end{pmatrix}
  86. g i j = δ i j g_{ij}=\delta_{ij}
  87. 𝐚 𝐛 = a ¯ j b ¯ j = a i 𝖫 i j b k ( s y m b o l 𝖫 - 1 ) j k = a i δ i b k k = a i b i \mathbf{a}\cdot\mathbf{b}=\bar{a}_{j}\bar{b}_{j}=a_{i}\mathsf{L}_{ij}b_{k}(% symbol{\mathsf{L}}^{-1})_{jk}=a_{i}\delta_{i}{}_{k}b_{k}=a_{i}b_{i}
  88. 𝐞 x × 𝐞 y = 𝐞 z 𝐞 y × 𝐞 z = 𝐞 x 𝐞 z × 𝐞 x = 𝐞 y \mathbf{e}\text{x}\times\mathbf{e}\text{y}=\mathbf{e}\text{z}\,\quad\mathbf{e}% \text{y}\times\mathbf{e}\text{z}=\mathbf{e}\text{x}\,\quad\mathbf{e}\text{z}% \times\mathbf{e}\text{x}=\mathbf{e}\text{y}
  89. 𝐞 y × 𝐞 x = - 𝐞 z 𝐞 z × 𝐞 y = - 𝐞 x 𝐞 x × 𝐞 z = - 𝐞 y \mathbf{e}\text{y}\times\mathbf{e}\text{x}=-\mathbf{e}\text{z}\,\quad\mathbf{e% }\text{z}\times\mathbf{e}\text{y}=-\mathbf{e}\text{x}\,\quad\mathbf{e}\text{x}% \times\mathbf{e}\text{z}=-\mathbf{e}\text{y}
  90. 𝐞 x × 𝐞 x = 𝐞 y × 𝐞 y = 𝐞 z × 𝐞 z = s y m b o l 0 \mathbf{e}\text{x}\times\mathbf{e}\text{x}=\mathbf{e}\text{y}\times\mathbf{e}% \text{y}=\mathbf{e}\text{z}\times\mathbf{e}\text{z}=symbol{0}
  91. 𝐞 i × 𝐞 j = [ + 𝐞 k cyclic permutations: ( i , j , k ) = ( 1 , 2 , 3 ) , ( 2 , 3 , 1 ) , ( 3 , 1 , 2 ) - 𝐞 k anticyclic permutations: ( i , j , k ) = ( 2 , 1 , 3 ) , ( 3 , 2 , 1 ) , ( 1 , 3 , 2 ) s y m b o l 0 i = j \mathbf{e}_{i}\times\mathbf{e}_{j}=\left[\begin{array}[]{cc}+\mathbf{e}_{k}&\,% \text{cyclic permutations: }(i,j,k)=(1,2,3),(2,3,1),(3,1,2)\\ -\mathbf{e}_{k}&\,\text{anticyclic permutations: }(i,j,k)=(2,1,3),(3,2,1),(1,3% ,2)\\ symbol{0}&i=j\end{array}\right.
  92. 𝐞 k 𝐞 i × 𝐞 j = [ + 1 cyclic permutations: ( i , j , k ) = ( 1 , 2 , 3 ) , ( 2 , 3 , 1 ) , ( 3 , 1 , 2 ) - 1 anticyclic permutations: ( i , j , k ) = ( 2 , 1 , 3 ) , ( 3 , 2 , 1 ) , ( 1 , 3 , 2 ) 0 i = j or j = k or k = i {\mathbf{e}_{k}\cdot\mathbf{e}_{i}\times\mathbf{e}_{j}}=\left[\begin{array}[]{% cc}+1&\,\text{cyclic permutations: }(i,j,k)=(1,2,3),(2,3,1),(3,1,2)\\ -1&\,\text{anticyclic permutations: }(i,j,k)=(2,1,3),(3,2,1),(1,3,2)\\ 0&i=j\,\text{ or }j=k\,\text{ or }k=i\end{array}\right.
  93. ε i j k = 𝐞 i 𝐞 j × 𝐞 k \varepsilon_{ijk}=\mathbf{e}_{i}\cdot\mathbf{e}_{j}\times\mathbf{e}_{k}
  94. 𝐜 𝐚 × 𝐛 = c i 𝐞 i a j 𝐞 j × b k 𝐞 k = ε i j k c i a j b k \mathbf{c}\cdot\mathbf{a}\times\mathbf{b}=c_{i}\mathbf{e}_{i}\cdot a_{j}% \mathbf{e}_{j}\times b_{k}\mathbf{e}_{k}=\varepsilon_{ijk}c_{i}a_{j}b_{k}
  95. 𝐜 𝐚 × 𝐛 = | c x a x b x c y a y b y c z a z b z | \mathbf{c}\cdot\mathbf{a}\times\mathbf{b}=\begin{vmatrix}c\text{x}&a\text{x}&b% \text{x}\\ c\text{y}&a\text{y}&b\text{y}\\ c\text{z}&a\text{z}&b\text{z}\end{vmatrix}
  96. ( 𝐚 × 𝐛 ) i = 𝐞 i 𝐚 × 𝐛 = ε j k ( 𝐞 i ) a j b k = ε j k δ i a j b k = ε i j k a j b k 𝐚 × 𝐛 = ( 𝐚 × 𝐛 ) i 𝐞 i = ε i j k a j b k 𝐞 i \begin{array}[]{ll}&(\mathbf{a}\times\mathbf{b})_{i}={\mathbf{e}_{i}\cdot% \mathbf{a}\times\mathbf{b}}=\varepsilon_{\ell jk}{(\mathbf{e}_{i})}_{\ell}a_{j% }b_{k}=\varepsilon_{\ell jk}\delta_{i\ell}a_{j}b_{k}=\varepsilon_{ijk}a_{j}b_{% k}\\ \Rightarrow&{\mathbf{a}\times\mathbf{b}}=(\mathbf{a}\times\mathbf{b})_{i}% \mathbf{e}_{i}=\varepsilon_{ijk}a_{j}b_{k}\mathbf{e}_{i}\end{array}
  97. ε ¯ p q r = det ( s y m b o l 𝖫 ) ε i j k 𝖫 i p 𝖫 j q 𝖫 k r . \bar{\varepsilon}_{pqr}=\det(symbol{\mathsf{L}})\varepsilon_{ijk}\mathsf{L}_{% ip}\mathsf{L}_{jq}\mathsf{L}_{kr}\,.
  98. ( 𝐚 ¯ × 𝐛 ¯ ) i \displaystyle(\bar{\mathbf{a}}\times\bar{\mathbf{b}})_{i}
  99. ε i j k ε p q k = δ i p δ j q - δ i q δ j p \varepsilon_{ijk}\varepsilon_{pqk}=\delta_{ip}\delta_{jq}-\delta_{iq}\delta_{jp}
  100. 𝐚 ( 𝐛 + 𝐜 ) = a i ( b i + c i ) = a i b i + a i c i = 𝐚 𝐛 + 𝐚 𝐜 \mathbf{a}\cdot(\mathbf{b}+\mathbf{c})=a_{i}(b_{i}+c_{i})=a_{i}b_{i}+a_{i}c_{i% }=\mathbf{a}\cdot\mathbf{b}+\mathbf{a}\cdot\mathbf{c}
  101. 𝐚 × ( 𝐛 + 𝐜 ) = 𝐞 i ε i j k a j ( b k + c k ) = 𝐞 i ε i j k a j b k + 𝐞 i ε i j k a j c k = 𝐚 × 𝐛 + 𝐚 × 𝐜 \mathbf{a}\times(\mathbf{b}+\mathbf{c})=\mathbf{e}_{i}\varepsilon_{ijk}a_{j}(b% _{k}+c_{k})=\mathbf{e}_{i}\varepsilon_{ijk}a_{j}b_{k}+\mathbf{e}_{i}% \varepsilon_{ijk}a_{j}c_{k}=\mathbf{a}\times\mathbf{b}+\mathbf{a}\times\mathbf% {c}
  102. [ 𝐚 × ( 𝐛 × 𝐜 ) ] i = ε i j k a j ( ε k m b c m ) = ( ε i j k ε k m ) a j b c m \left[\mathbf{a}\times(\mathbf{b}\times\mathbf{c})\right]_{i}=\varepsilon_{ijk% }a_{j}(\varepsilon_{k\ell m}b_{\ell}c_{m})=(\varepsilon_{ijk}\varepsilon_{k% \ell m})a_{j}b_{\ell}c_{m}
  103. [ 𝐚 × ( 𝐛 × 𝐜 ) ] i \displaystyle\left[\mathbf{a}\times(\mathbf{b}\times\mathbf{c})\right]_{i}
  104. 𝐚 × ( 𝐛 × 𝐜 ) = ( 𝐚 𝐜 ) 𝐛 - ( 𝐚 𝐛 ) 𝐜 \mathbf{a}\times(\mathbf{b}\times\mathbf{c})=(\mathbf{a}\cdot\mathbf{c})% \mathbf{b}-(\mathbf{a}\cdot\mathbf{b})\mathbf{c}
  105. ( 𝐚 × 𝐛 ) × 𝐜 = ( 𝐜 𝐚 ) 𝐛 - ( 𝐜 𝐛 ) 𝐚 (\mathbf{a}\times\mathbf{b})\times\mathbf{c}=(\mathbf{c}\cdot\mathbf{a})% \mathbf{b}-(\mathbf{c}\cdot\mathbf{b})\mathbf{a}
  106. ( 𝐚 × 𝐛 ) ( 𝐜 × 𝐝 ) , ( 𝐚 × 𝐛 ) × ( 𝐜 × 𝐝 ) , (\mathbf{a}\times\mathbf{b})\cdot(\mathbf{c}\times\mathbf{d}),\quad(\mathbf{a}% \times\mathbf{b})\times(\mathbf{c}\times\mathbf{d}),\ldots
  107. a ¯ \overline{a}
  108. b ¯ \overline{b}
  109. 𝐚 𝐛 = a i 𝐞 i b j 𝐞 j = a i b j 𝐞 i 𝐞 j \mathbf{a}\otimes\mathbf{b}=a_{i}\mathbf{e}_{i}\otimes b_{j}\mathbf{e}_{j}=a_{% i}b_{j}\mathbf{e}_{i}\otimes\mathbf{e}_{j}
  110. a ¯ p b ¯ q = a i 𝖫 i b j p 𝖫 j = q 𝖫 i 𝖫 j p a i q b j \bar{a}_{p}\bar{b}_{q}=a_{i}\mathsf{L}_{i}{}_{p}b_{j}\mathsf{L}_{j}{}_{q}=% \mathsf{L}_{i}{}_{p}\mathsf{L}_{j}{}_{q}a_{i}b_{j}
  111. 𝐞 ¯ p 𝐞 ¯ q = ( s y m b o l 𝖫 - 1 ) p i 𝐞 i ( s y m b o l 𝖫 - 1 ) q j 𝐞 ¯ j = ( s y m b o l 𝖫 - 1 ) p i ( s y m b o l 𝖫 - 1 ) q j 𝐞 i 𝐞 ¯ j = 𝖫 i p 𝖫 j q 𝐞 i 𝐞 ¯ j \bar{\mathbf{e}}_{p}\otimes\bar{\mathbf{e}}_{q}=(symbol{\mathsf{L}}^{-1})_{pi}% \mathbf{e}_{i}\otimes(symbol{\mathsf{L}}^{-1})_{qj}\bar{\mathbf{e}}_{j}=(% symbol{\mathsf{L}}^{-1})_{pi}(symbol{\mathsf{L}}^{-1})_{qj}\mathbf{e}_{i}% \otimes\bar{\mathbf{e}}_{j}=\mathsf{L}_{ip}\mathsf{L}_{jq}\mathbf{e}_{i}% \otimes\bar{\mathbf{e}}_{j}
  112. a ¯ p b ¯ q 𝐞 ¯ p 𝐞 ¯ q = 𝖫 k p 𝖫 q a k b ( s y m b o l 𝖫 - 1 ) p i ( s y m b o l 𝖫 - 1 ) q j 𝐞 i 𝐞 j = 𝖫 k p ( s y m b o l 𝖫 - 1 ) p i 𝖫 q ( s y m b o l 𝖫 - 1 ) q j a k b 𝐞 i 𝐞 j = δ k δ j i a k b 𝐞 i 𝐞 j = a i b j 𝐞 i 𝐞 j \begin{array}[]{cl}\bar{a}_{p}\bar{b}_{q}\bar{\mathbf{e}}_{p}\otimes\bar{% \mathbf{e}}_{q}&=\mathsf{L}_{kp}\mathsf{L}_{\ell q}a_{k}b_{\ell}\,(symbol{% \mathsf{L}}^{-1})_{pi}(symbol{\mathsf{L}}^{-1})_{qj}\mathbf{e}_{i}\otimes% \mathbf{e}_{j}\\ &=\mathsf{L}_{kp}(symbol{\mathsf{L}}^{-1})_{pi}\mathsf{L}_{\ell q}(symbol{% \mathsf{L}}^{-1})_{qj}\,a_{k}b_{\ell}\mathbf{e}_{i}\otimes\mathbf{e}_{j}\\ &=\delta_{k}{}_{i}\delta_{\ell j}\,a_{k}b_{\ell}\mathbf{e}_{i}\otimes\mathbf{e% }_{j}\\ &=a_{i}b_{j}\mathbf{e}_{i}\otimes\mathbf{e}_{j}\end{array}
  113. 𝐑 = R i j 𝐞 i 𝐞 j , \mathbf{R}=R_{ij}\mathbf{e}_{i}\otimes\mathbf{e}_{j}\,,
  114. R ¯ p q = 𝖫 i 𝖫 j p R i j q \bar{R}_{pq}=\mathsf{L}_{i}{}_{p}\mathsf{L}_{j}{}_{q}R_{ij}
  115. 𝐞 ¯ p 𝐞 ¯ q = ( s y m b o l 𝖫 - 1 ) i p 𝐞 i ( s y m b o l 𝖫 - 1 ) j q 𝐞 j \bar{\mathbf{e}}_{p}\otimes\bar{\mathbf{e}}_{q}=(symbol{\mathsf{L}}^{-1})_{ip}% \mathbf{e}_{i}\otimes(symbol{\mathsf{L}}^{-1})_{jq}\mathbf{e}_{j}
  116. 𝐓 = T j 1 j 2 j p 𝐞 j 1 𝐞 j 2 𝐞 j p \mathbf{T}=T_{j_{1}j_{2}\cdots j_{p}}\mathbf{e}_{j_{1}}\otimes\mathbf{e}_{j_{2% }}\otimes\cdots\mathbf{e}_{j_{p}}
  117. T ¯ j 1 j 2 j p = 𝖫 i 1 j 1 𝖫 i 2 j 2 𝖫 i p j p T i 1 i 2 i p \bar{T}_{j_{1}j_{2}\cdots j_{p}}=\mathsf{L}_{i_{1}j_{1}}\mathsf{L}_{i_{2}j_{2}% }\cdots\mathsf{L}_{i_{p}j_{p}}T_{i_{1}i_{2}\cdots i_{p}}
  118. 𝐞 ¯ j 1 𝐞 ¯ j 2 𝐞 ¯ j p = ( s y m b o l 𝖫 - 1 ) j 1 i 1 𝐞 i 1 ( s y m b o l 𝖫 - 1 ) j 2 i 2 𝐞 i 2 ( s y m b o l 𝖫 - 1 ) j p i p 𝐞 i p \bar{\mathbf{e}}_{j_{1}}\otimes\bar{\mathbf{e}}_{j_{2}}\cdots\otimes\bar{% \mathbf{e}}_{j_{p}}=(symbol{\mathsf{L}}^{-1})_{j_{1}i_{1}}\mathbf{e}_{i_{1}}% \otimes(symbol{\mathsf{L}}^{-1})_{j_{2}i_{2}}\mathbf{e}_{i_{2}}\cdots\otimes(% symbol{\mathsf{L}}^{-1})_{j_{p}i_{p}}\mathbf{e}_{i_{p}}
  119. S ¯ j 1 j 2 j p = det ( s y m b o l 𝖫 ) 𝖫 i 1 j 1 𝖫 i 2 j 2 𝖫 i p j p S i 1 i 2 i p . \bar{S}_{j_{1}j_{2}\cdots j_{p}}=\det(symbol{\mathsf{L}})\mathsf{L}_{i_{1}j_{1% }}\mathsf{L}_{i_{2}j_{2}}\cdots\mathsf{L}_{i_{p}j_{p}}S_{i_{1}i_{2}\cdots i_{p% }}\,.
  120. 𝐜 = 𝐚 × 𝐛 = 𝐓 𝐛 \mathbf{c}=\mathbf{a}\times\mathbf{b}=\mathbf{T}\cdot\mathbf{b}
  121. 𝐓 = ( 0 - a z a y a z 0 - a x - a y a x 0 ) \mathbf{T}=\begin{pmatrix}0&-a\text{z}&a\text{y}\\ a\text{z}&0&-a\text{x}\\ -a\text{y}&a\text{x}&0\\ \end{pmatrix}
  122. s y m b o l Ω = ( 0 - ω z ω y ω z 0 - ω x - ω y ω x 0 ) symbol{\Omega}=\begin{pmatrix}0&-\omega\text{z}&\omega\text{y}\\ \omega\text{z}&0&-\omega\text{x}\\ -\omega\text{y}&\omega\text{x}&0\\ \end{pmatrix}
  123. 𝐅 = q ( 𝐄 + 𝐯 × 𝐁 ) = q ( 𝐄 - 𝐁 × 𝐯 ) \mathbf{F}=q(\mathbf{E}+\mathbf{v}\times\mathbf{B})=q(\mathbf{E}-\mathbf{B}% \times\mathbf{v})
  124. ( F x F y F z ) = q ( E x E y E z ) - q ( 0 - B z B y B z 0 - B x - B y B x 0 ) ( v x v y v z ) \begin{pmatrix}F\text{x}\\ F\text{y}\\ F\text{z}\\ \end{pmatrix}=q\begin{pmatrix}E\text{x}\\ E\text{y}\\ E\text{z}\\ \end{pmatrix}-q\begin{pmatrix}0&-B\text{z}&B\text{y}\\ B\text{z}&0&-B\text{x}\\ -B\text{y}&B\text{x}&0\\ \end{pmatrix}\begin{pmatrix}v\text{x}\\ v\text{y}\\ v\text{z}\\ \end{pmatrix}
  125. 𝐉 = ( 0 - J z J y J z 0 - J x - J y J x 0 ) = ( 0 - ( x p y - y p x ) ( z p x - x p z ) ( x p y - y p x ) 0 - ( y p z - z p y ) - ( z p x - x p z ) ( y p z - z p y ) 0 ) \mathbf{J}=\begin{pmatrix}0&-J\text{z}&J\text{y}\\ J\text{z}&0&-J\text{x}\\ -J\text{y}&J\text{x}&0\\ \end{pmatrix}=\begin{pmatrix}0&-(xp\text{y}-yp\text{x})&(zp\text{x}-xp\text{z}% )\\ (xp\text{y}-yp\text{x})&0&-(yp\text{z}-zp\text{y})\\ -(zp\text{x}-xp\text{z})&(yp\text{z}-zp\text{y})&0\\ \end{pmatrix}
  126. 𝐀 ( 𝐫 , t ) = A x ( 𝐫 , t ) 𝐞 x + A y ( 𝐫 , t ) 𝐞 y + A z ( 𝐫 , t ) 𝐞 z \mathbf{A}(\mathbf{r},t)=A\text{x}(\mathbf{r},t)\mathbf{e}\text{x}+A\text{y}(% \mathbf{r},t)\mathbf{e}\text{y}+A\text{z}(\mathbf{r},t)\mathbf{e}\text{z}
  127. 𝐁 ( 𝐫 , t ) = B x ( 𝐫 , t ) 𝐞 x + B y ( 𝐫 , t ) 𝐞 y + B z ( 𝐫 , t ) 𝐞 z \mathbf{B}(\mathbf{r},t)=B\text{x}(\mathbf{r},t)\mathbf{e}\text{x}+B\text{y}(% \mathbf{r},t)\mathbf{e}\text{y}+B\text{z}(\mathbf{r},t)\mathbf{e}\text{z}
  128. = 𝐞 x x + 𝐞 y y + 𝐞 z z \nabla=\mathbf{e}\text{x}\frac{\partial}{\partial x}+\mathbf{e}\text{y}\frac{% \partial}{\partial y}+\mathbf{e}\text{z}\frac{\partial}{\partial z}
  129. i i x i \nabla_{i}\equiv\partial_{i}\equiv\frac{\partial}{\partial x_{i}}
  130. ( Φ ) i = i Φ \left(\nabla\Phi\right)_{i}=\nabla_{i}\Phi
  131. 𝐚 ( Φ ) = a j ( Φ ) j \mathbf{a}\cdot(\nabla\Phi)=a_{j}(\nabla\Phi)_{j}
  132. 𝐀 = i A i \nabla\cdot\mathbf{A}=\nabla_{i}A_{i}
  133. 𝐀 = A i i \mathbf{A}\cdot\nabla=A_{i}\nabla_{i}
  134. D D t = t + 𝐮 \frac{D}{Dt}=\frac{\partial}{\partial t}+\mathbf{u}\cdot\nabla
  135. ( × 𝐀 ) i = ε i j k j A k \left(\nabla\times\mathbf{A}\right)_{i}=\varepsilon_{ijk}\nabla_{j}A_{k}
  136. ( × 𝐀 ) i j = i A j - j A i = 2 [ i A j ] \left(\nabla\times\mathbf{A}\right)_{ij}=\nabla_{i}A_{j}-\nabla_{j}A_{i}=2% \nabla_{[i}A_{j]}
  137. ε i j k A j k \varepsilon_{ijk}A_{j}\nabla_{k}
  138. A i j - A j i = 2 A [ i j ] A_{i}\nabla_{j}-A_{j}\nabla_{i}=2A_{[i}\nabla_{j]}
  139. ( Φ ) = i ( i Φ ) \nabla\cdot(\nabla\Phi)=\nabla_{i}(\nabla_{i}\Phi)
  140. ( ) Φ = ( i i ) Φ (\nabla\cdot\nabla)\Phi=(\nabla_{i}\nabla_{i})\Phi
  141. ( ) 𝐀 = ( i i ) 𝐀 (\nabla\cdot\nabla)\mathbf{A}=(\nabla_{i}\nabla_{i})\mathbf{A}
  142. [ × ( 𝐀 × 𝐁 ) ] i \displaystyle\left[\nabla\times(\mathbf{A}\times\mathbf{B})\right]_{i}
  143. × ( 𝐀 × 𝐁 ) = ( 𝐁 ) 𝐀 + 𝐀 ( 𝐁 ) - 𝐁 ( 𝐀 ) - ( 𝐀 ) 𝐁 \nabla\times(\mathbf{A}\times\mathbf{B})=(\mathbf{B}\cdot\nabla)\mathbf{A}+% \mathbf{A}(\nabla\cdot\mathbf{B})-\mathbf{B}(\nabla\cdot\mathbf{A})-(\mathbf{A% }\cdot\nabla)\mathbf{B}
  144. ( 𝐀 ) i j ( 𝐀 ) i j = i A j (\nabla\mathbf{A})_{ij}\equiv(\nabla\otimes\mathbf{A})_{ij}=\nabla_{i}A_{j}
  145. ( 𝐓 ) j = i T i j (\nabla\cdot\mathbf{T})_{j}=\nabla_{i}T_{ij}
  146. 𝐓 = T j 1 j 2 j q i 1 i 2 i p 𝐞 i 1 i 2 i p j 1 j 2 j q \mathbf{T}=T_{j_{1}j_{2}\cdots j_{q}}^{i_{1}i_{2}\cdots i_{p}}\mathbf{e}_{i_{1% }i_{2}\cdots i_{p}}^{j_{1}j_{2}\cdots j_{q}}
  147. 𝐞 i 1 i 2 i p j 1 j 2 j q = 𝐞 i 1 𝐞 i 2 𝐞 i p 𝐞 j 1 𝐞 j 2 𝐞 j q \mathbf{e}_{i_{1}i_{2}\cdots i_{p}}^{j_{1}j_{2}\cdots j_{q}}=\mathbf{e}_{i_{1}% }\otimes\mathbf{e}_{i_{2}}\otimes\cdots\mathbf{e}_{i_{p}}\otimes\mathbf{e}^{j_% {1}}\otimes\mathbf{e}^{j_{2}}\otimes\cdots\mathbf{e}^{j_{q}}
  148. T ¯ 1 2 q k 1 k 2 k p = 𝖫 i 1 𝖫 i 2 k 1 k 2 𝖫 i p ( s y m b o l 𝖫 - 1 ) 1 k p ( s y m b o l 𝖫 - 1 ) 2 j 1 j 2 ( s y m b o l 𝖫 - 1 ) q T j 1 j 2 j q i 1 i 2 i p j q \bar{T}_{\ell_{1}\ell_{2}\cdots\ell_{q}}^{k_{1}k_{2}\cdots k_{p}}=\mathsf{L}_{% i_{1}}{}^{k_{1}}\mathsf{L}_{i_{2}}{}^{k_{2}}\cdots\mathsf{L}_{i_{p}}{}^{k_{p}}% (symbol{\mathsf{L}}^{-1})_{\ell_{1}}{}^{j_{1}}(symbol{\mathsf{L}}^{-1})_{\ell_% {2}}{}^{j_{2}}\cdots(symbol{\mathsf{L}}^{-1})_{\ell_{q}}{}^{j_{q}}T_{j_{1}j_{2% }\cdots j_{q}}^{i_{1}i_{2}\cdots i_{p}}
  149. 𝐞 ¯ k 1 k 2 k p 1 2 q = ( s y m b o l 𝖫 - 1 ) k 1 ( s y m b o l 𝖫 - 1 ) k 2 i 1 i 2 ( s y m b o l 𝖫 - 1 ) k p 𝖫 j 1 i p 𝖫 j 2 1 2 𝖫 j q 𝐞 i 1 i 2 i p j 1 j 2 j q q \bar{\mathbf{e}}_{k_{1}k_{2}\cdots k_{p}}^{\ell_{1}\ell_{2}\cdots\ell_{q}}=(% symbol{\mathsf{L}}^{-1})_{k_{1}}{}^{i_{1}}(symbol{\mathsf{L}}^{-1})_{k_{2}}{}^% {i_{2}}\cdots(symbol{\mathsf{L}}^{-1})_{k_{p}}{}^{i_{p}}\mathsf{L}_{j_{1}}{}^{% \ell_{1}}\mathsf{L}_{j_{2}}{}^{\ell_{2}}\cdots\mathsf{L}_{j_{q}}{}^{\ell_{q}}% \mathbf{e}_{i_{1}i_{2}\cdots i_{p}}^{j_{1}j_{2}\cdots j_{q}}

Cassini_and_Catalan_identities.html

  1. F n - 1 F n + 1 - F n 2 = ( - 1 ) n . F_{n-1}F_{n+1}-F_{n}^{2}=(-1)^{n}.\,
  2. F n 2 - F n - r F n + r = ( - 1 ) n - r F r 2 . F_{n}^{2}-F_{n-r}F_{n+r}=(-1)^{n-r}F_{r}^{2}.\,
  3. F n + i F n + j - F n F n + i + j = ( - 1 ) n F i F j . F_{n+i}F_{n+j}-F_{n}F_{n+i+j}=(-1)^{n}F_{i}F_{j}.\,
  4. F n - 1 F n + 1 - F n 2 = det [ F n + 1 F n F n F n - 1 ] = det [ 1 1 1 0 ] n = ( det [ 1 1 1 0 ] ) n = ( - 1 ) n . F_{n-1}F_{n+1}-F_{n}^{2}=\det\left[\begin{matrix}F_{n+1}&F_{n}\\ F_{n}&F_{n-1}\end{matrix}\right]=\det\left[\begin{matrix}1&1\\ 1&0\end{matrix}\right]^{n}=\left(\det\left[\begin{matrix}1&1\\ 1&0\end{matrix}\right]\right)^{n}=(-1)^{n}.

Catalytic_reforming.html

  1. \Rightarrow
  2. \Rightarrow

Categorification.html

  1. λ \lambda
  2. S λ φ s λ , S^{\lambda}\stackrel{\varphi}{\to}s_{\lambda},
  3. [ Ind S m S n S n + m ( S μ S ν ) ] and s μ s ν [\mathrm{Ind}_{S_{m}\otimes S_{n}}^{S_{n+m}}(S^{\mu}\otimes S^{\nu})]\qquad\,% \text{ and }\qquad s_{\mu}s_{\nu}
  4. \mathcal{B}
  5. K ( ) K(\mathcal{B})
  6. \mathcal{B}
  7. A A
  8. 𝐚 = { a i } i I \mathbf{a}=\{a_{i}\}_{i\in I}
  9. A A
  10. 𝐚 \mathbf{a}
  11. a i a j = k c i j k a k , a_{i}a_{j}=\sum_{k}c_{ij}^{k}a_{k},
  12. c i j k 0 . c_{ij}^{k}\in\mathbb{Z}_{\geq 0}.
  13. B B
  14. A A
  15. ( A , 𝐚 , B ) (A,\mathbf{a},B)
  16. \mathcal{B}
  17. ϕ : K ( ) B \phi:K(\mathcal{B})\to B
  18. F i : F_{i}:\mathcal{B}\to\mathcal{B}
  19. F i F_{i}
  20. a i a_{i}
  21. B B
  22. φ [ F i ] = a i φ \varphi[F_{i}]=a_{i}\varphi
  23. F i F j k F k c i j k , F_{i}F_{j}\cong\bigoplus_{k}F_{k}^{c_{ij}^{k}},
  24. F i F j F_{i}F_{j}
  25. F k F_{k}
  26. a i a j a_{i}a_{j}
  27. a k a_{k}

Category:Rational_numbers.html

  1. a b \frac{a}{b}
  2. a a
  3. b b
  4. b b

Category:Wikipedian_mathematicians.html

  1. e i π e^{i\pi}\,\;

Catmull–Clark_subdivision_surface.html

  1. F + 2 R + ( n - 3 ) P n . \frac{F+2R+(n-3)P}{n}.
  2. 𝒞 1 \mathcal{C}^{1}
  3. 𝒞 2 \mathcal{C}^{2}
  4. 𝒞 n \mathcal{C}^{n}

Cauchy_boundary_condition.html

  1. y ′′ ( s ) = f ( y ( s ) , y ( s ) , s ) y^{\prime\prime}(s)=f(y(s),y^{\prime}(s),s)
  2. y ( s ) y(s)
  3. y y
  4. y y^{\prime}
  5. s = a s=a
  6. y ( a ) = α , y(a)=\alpha\ ,
  7. y ( a ) = β . y^{\prime}(a)=\beta\ .
  8. a a
  9. s s
  10. y ′′ y^{\prime\prime}
  11. y y
  12. y y^{\prime}
  13. s s
  14. A ( x , y ) ψ x x + B ( x , y ) ψ x y + C ( x , y ) ψ y y = F ( x , y , ψ , ψ x , ψ y ) A(x,y)\psi_{xx}+B(x,y)\psi_{xy}+C(x,y)\psi_{yy}=F(x,y,\psi,\psi_{x},\psi_{y})
  15. ψ ( x , y ) \psi(x,y)
  16. ψ x \psi_{x}
  17. ψ \psi
  18. x x
  19. A , B , C , F A,B,C,F
  20. ψ \psi
  21. Ω \Omega
  22. x - y x-y
  23. ψ ( x , y ) = α ( x , y ) , 𝐧 ψ = β ( x , y ) \psi(x,y)=\alpha(x,y),\mathbf{n}\cdot\nabla\psi=\beta(x,y)
  24. ( x , y ) Ω (x,y)\in\partial\Omega
  25. 𝐧 ψ \mathbf{n}\cdot\nabla\psi
  26. α \alpha
  27. β \beta

Cauchy_problem.html

  1. u ( x ) = f 0 ( x ) for all x S ; k u ( x ) n k = f k ( x ) for k = 1 , , κ - 1 and all x S , \begin{aligned}\displaystyle u(x)&\displaystyle=f_{0}(x)&&\displaystyle\,\text% {for all }x\in S;\\ \displaystyle\frac{\partial^{k}u(x)}{\partial n^{k}}&\displaystyle=f_{k}(x)&&% \displaystyle\,\text{for }k=1,\ldots,\kappa-1\,\text{ and all }x\in S,\end{aligned}
  2. f k f_{k}
  3. S S

Cayley's_formula.html

  1. 2 2 - 2 = 1 2^{2-2}=1
  2. 3 3 - 2 = 3 3^{3-2}=3
  3. 4 4 - 2 = 16 4^{4-2}=16
  4. n n - 2 n^{n-2}

Cayley_table.html

  1. ( a b ) c = a ( b c ) (ab)c=a(bc)

CCFL_inverter.html

  1. f o = 1 2 π L p C o 1 2 π L sc ( C w + C a + C s ) f_{o}=\frac{1}{2\pi\sqrt{L_{p}\cdot C_{o}}}\approx\frac{1}{2\pi\sqrt{L_{% \mathrm{sc}}\cdot(C_{w}+C_{a}+C_{s})}}

CCR_and_CAR_algebras.html

  1. V V
  2. ( , ) (\cdot,\cdot)
  3. V V
  4. f g - g f = i ( f , g ) fg-gf=i(f,g)\,
  5. f * = f , f^{*}=f,\,
  6. f , g f,~{}g
  7. V V
  8. V V
  9. V V
  10. ( , ) (\cdot,\cdot)
  11. V V
  12. f g + g f = ( f , g ) , fg+gf=(f,g),\,
  13. f * = f , f^{*}=f,\,
  14. f , g f,~{}g
  15. V V
  16. H H
  17. ( , ) (\cdot,\cdot)
  18. H H
  19. { W ( f ) : f H } \{W(f):~{}f\in H\}
  20. W ( f ) W ( g ) = e - i ( f , g ) W ( f + g ) , W(f)W(g)=e^{-i(f,g)}W(f+g),\,
  21. W ( f ) * = W ( - f ) . W(f)^{*}=W(-f).\,
  22. W ( f ) W(f)
  23. W ( 0 ) = 1 W(0)=1
  24. H H
  25. ( , ) (\cdot,\cdot)
  26. H H
  27. W ( f ) ( 1 , g , g 2 2 ! , g 3 3 ! , ) = e - 1 2 || f || 2 - f , g ( 1 , f + g , ( f + g ) 2 2 ! , ( f + g ) 3 3 ! , ) , W(f)\left(1,g,\frac{g^{\otimes 2}}{2!},\frac{g^{\otimes 3}}{3!},\ldots\right)=% e^{-\frac{1}{2}||f||^{2}-\langle f,g\rangle}\left(1,f+g,\frac{(f+g)^{\otimes 2% }}{2!},\frac{(f+g)^{\otimes 3}}{3!},\ldots\right),\,
  28. f , g H f,g\in H
  29. B ( f ) B(f)
  30. f H f\in H
  31. ( W ( t f ) ) t (W(tf))_{t\in\mathbb{R}}
  32. B ( f ) B ( g ) - B ( g ) B ( f ) = 2 i Im f , g . B(f)B(g)-B(g)B(f)=2i\mathrm{Im}\langle f,g\rangle.\,
  33. f B ( f ) f\mapsto B(f)
  34. B ( f ) B(f)
  35. ( H , 2 I m , ) (H,2\mathrm{Im}\langle\cdot,\cdot\rangle)
  36. H H
  37. { b ( f ) , b * ( f ) : f H } \{b(f),b^{*}(f):~{}f\in H\}
  38. b ( f ) b * ( g ) + b * ( g ) b ( f ) = f , g , b(f)b^{*}(g)+b^{*}(g)b(f)=\langle f,g\rangle,\,
  39. b ( f ) b ( g ) + b ( g ) b ( f ) = 0 , b(f)b(g)+b(g)b(f)=0,\,
  40. λ b * ( f ) = b * ( λ f ) , \lambda b^{*}(f)=b^{*}(\lambda f),\,
  41. b ( f ) * = b * ( f ) , b(f)^{*}=b^{*}(f),\,
  42. f , g H f,g\in H
  43. λ \lambda\in\mathbb{C}
  44. H H
  45. H H
  46. M 2 ( ) {M_{2^{\infty}}(\mathbb{C})}
  47. F a ( H ) F_{a}(H)
  48. H H
  49. P a P_{a}
  50. P a : n = 0 H n F a ( H ) . P_{a}:\bigoplus_{n=0}^{\infty}H^{\otimes n}\to F_{a}(H).\,
  51. F a ( H ) F_{a}(H)
  52. b * ( f ) P a ( g 1 g 2 g n ) = P a ( f g 1 g 2 g n ) b^{*}(f)P_{a}(g_{1}\otimes g_{2}\otimes\cdots\otimes g_{n})=P_{a}(f\otimes g_{% 1}\otimes g_{2}\otimes\cdots g_{n})\,
  53. f , g 1 , , g n H f,g_{1},\ldots,g_{n}\in H
  54. n n\in\mathbb{N}
  55. B ( f ) := b * ( f ) + b ( f ) B(f):=b^{*}(f)+b(f)
  56. B ( f ) B ( g ) + B ( g ) B ( f ) = 2 R e f , g , B(f)B(g)+B(g)B(f)=2\mathrm{Re}\langle f,g\rangle,\,
  57. V V
  58. 2 \mathbb{Z}_{2}
  59. ( , ) (\cdot,\cdot)
  60. ( g , f ) = - ( - 1 ) | f | | g | ( f , g ) (g,f)=-(-1)^{|f||g|}(f,g)
  61. ( f , g ) (f,g)
  62. f f
  63. g g
  64. V V
  65. f g - ( - 1 ) | f | | g | g f = i ( f , g ) fg-(-1)^{|f||g|}gf=i(f,g)\,
  66. f * = f , g * = g f^{*}=f,~{}g^{*}=g\,
  67. f , g f,~{}g
  68. V V

Cellular_traffic.html

  1. efficiency = N c BW A c , \mathrm{efficiency}=\frac{N_{\mathrm{c}}}{\mathrm{BW}\cdot A_{\mathrm{c}}},
  2. BW A c B \frac{\mathrm{BW}}{A_{\mathrm{c}}\cdot\mathrm{B}}

Center_embedding.html

  1. \Rightarrow
  2. \Rightarrow
  3. \Rightarrow

Centered_decagonal_number.html

  1. 5 n 2 + 5 n + 1 5n^{2}+5n+1\,
  2. C D n + 1 = C D n + 10 n , CD_{n+1}=CD_{n}+10n,
  3. C D 0 = 0. CD_{0}=0.

Centered_heptagonal_number.html

  1. 7 n 2 - 7 n + 2 2 {7n^{2}-7n+2}\over 2

Centered_nonagonal_number.html

  1. N c ( n ) = ( 3 n - 2 ) ( 3 n - 1 ) 2 . Nc(n)=\frac{(3n-2)(3n-1)}{2}.

Centered_octagonal_number.html

  1. ( 2 n - 1 ) 2 = 4 n 2 - 4 n + 1. (2n-1)^{2}=4n^{2}-4n+1.

Central_force.html

  1. F = 𝐅 ( 𝐫 ) = F ( || 𝐫 || ) 𝐫 ^ \vec{F}=\mathbf{F}(\mathbf{r})=F(||\mathbf{r}||)\hat{\mathbf{r}}
  2. F \scriptstyle\vec{\,\text{ F }}
  3. 𝐫 ^ \scriptstyle\hat{\mathbf{r}}
  4. 𝐅 ( 𝐫 ) = - V ( 𝐫 ) , where V ( 𝐫 ) = | 𝐫 | + F ( r ) d r \mathbf{F}(\mathbf{r})=-\mathbf{\nabla}V(\mathbf{r})\,\text{, where }V(\mathbf% {r})=\int_{|\mathbf{r}|}^{+\infty}F(r)\,\mathrm{d}r
  5. E = 1 2 m | 𝐫 ˙ | 2 + V ( 𝐫 ) = constant E=\frac{1}{2}m|\mathbf{\dot{r}}|^{2}+V(\mathbf{r})=\,\text{constant}
  6. 𝐋 = 𝐫 × m 𝐫 ˙ = constant \mathbf{L}=\mathbf{r}\times m\mathbf{\dot{r}}=\,\text{constant}
  7. × 𝐅 ( 𝐫 ) = 𝟎 . \nabla\times\mathbf{F}(\mathbf{r})=\mathbf{0}\,\text{.}

Centrality.html

  1. k = 0 β k A R k \sum_{k=0}^{\infty}\beta^{k}A_{R}^{k}
  2. k = 0 ( β A R ) k k ! \sum_{k=0}^{\infty}\frac{(\beta A_{R})^{k}}{k!}
  3. k k
  4. A R A_{R}
  5. β \beta
  6. β \beta
  7. β \beta
  8. v v
  9. G := ( V , E ) G:=(V,E)
  10. | V | |V|
  11. | E | |E|
  12. C D ( v ) = deg ( v ) C_{D}(v)=\deg(v)
  13. Θ ( V 2 ) \Theta(V^{2})
  14. Θ ( E ) \Theta(E)
  15. v * v*
  16. G G
  17. X := ( Y , Z ) X:=(Y,Z)
  18. | Y | |Y|
  19. y * y*
  20. X X
  21. H = j = 1 | Y | [ C D ( y * ) - C D ( y j ) ] H=\sum^{|Y|}_{j=1}[C_{D}(y*)-C_{D}(y_{j})]
  22. G G
  23. C D ( G ) = i = 1 | V | [ C D ( v * ) - C D ( v i ) ] H C_{D}(G)=\frac{\displaystyle{\sum^{|V|}_{i=1}{[C_{D}(v*)-C_{D}(v_{i})]}}}{H}
  24. H H
  25. X X
  26. H = ( n - 1 ) ( n - 2 ) H=(n-1)(n-2)
  27. C ( x ) = 1 y d ( y , x ) . C(x)=\frac{1}{\sum_{y}d(y,x)}.
  28. 1 / = 0 1/\infty=0
  29. H ( x ) = y x 1 d ( y , x ) . H(x)=\sum_{y\neq x}\frac{1}{d(y,x)}.
  30. D ( x ) = y x 1 2 d ( y , x ) . D(x)=\sum_{y\neq x}\frac{1}{2^{d(y,x)}}.
  31. d ( x , y ) d(x,y)
  32. v v
  33. G := ( V , E ) G:=(V,E)
  34. V V
  35. C B ( v ) = s v t V σ s t ( v ) σ s t C_{B}(v)=\sum_{s\neq v\neq t\in V}\frac{\sigma_{st}(v)}{\sigma_{st}}
  36. σ s t \sigma_{st}
  37. s s
  38. t t
  39. σ s t ( v ) \sigma_{st}(v)
  40. v v
  41. ( n - 1 ) ( n - 2 ) (n-1)(n-2)
  42. ( n - 1 ) ( n - 2 ) / 2 (n-1)(n-2)/2
  43. ( n - 1 ) ( n - 2 ) / 2 (n-1)(n-2)/2
  44. Θ ( V 3 ) \Theta(V^{3})
  45. O ( V 2 log V + V E ) O(V^{2}\log V+VE)
  46. O ( V E ) O(VE)
  47. G := ( V , E ) G:=(V,E)
  48. | V | |V|
  49. A = ( a v , t ) A=(a_{v,t})
  50. a v , t = 1 a_{v,t}=1
  51. v v
  52. t t
  53. a v , t = 0 a_{v,t}=0
  54. v v
  55. x v = 1 λ t M ( v ) x t = 1 λ t G a v , t x t x_{v}=\frac{1}{\lambda}\sum_{t\in M(v)}x_{t}=\frac{1}{\lambda}\sum_{t\in G}a_{% v,t}x_{t}
  56. M ( v ) M(v)
  57. v v
  58. λ \lambda
  59. 𝐀𝐱 = λ 𝐱 \mathbf{Ax}={\lambda}\mathbf{x}
  60. λ \lambda
  61. v t h v^{th}
  62. v v
  63. x i = k = 1 j = 1 N α k ( A k ) j i x_{i}=\sum_{k=1}^{\infty}\sum_{j=1}^{N}\alpha^{k}(A^{k})_{ji}
  64. α \alpha
  65. ( 0 , 1 ) (0,1)
  66. x i = α j = 1 N a i j ( x j + 1 ) . x_{i}=\alpha\sum_{j=1}^{N}a_{ij}(x_{j}+1).
  67. x j x_{j}
  68. x j + 1 x_{j}+1
  69. A A
  70. α \alpha
  71. 1 / λ 1/\lambda
  72. x i = α j a j i x j L ( j ) + 1 - α N , x_{i}=\alpha\sum_{j}a_{ji}\frac{x_{j}}{L(j)}+\frac{1-\alpha}{N},
  73. L ( j ) = j a i j L(j)=\sum_{j}a_{ij}
  74. j j
  75. L ( j ) L(j)
  76. a j i a_{ji}
  77. P C t ( v ) = 1 N - 2 s v r σ s r ( v ) σ s r x t s [ x t i ] - x t v PC^{t}(v)=\frac{1}{N-2}\sum_{s\neq v\neq r}\frac{\sigma_{sr}(v)}{\sigma_{sr}}% \frac{{x^{t}}_{s}}{{\sum{[{x^{t}}_{i}}]}-{x^{t}}_{v}}
  78. σ s r \sigma_{sr}
  79. s s
  80. r r
  81. σ s r ( v ) \sigma_{sr}(v)
  82. v v
  83. i i
  84. t t
  85. x t i {x^{t}}_{i}
  86. x t i = 0 {x^{t}}_{i}=0
  87. t t
  88. x t i = 1 {x^{t}}_{i}=1
  89. t t
  90. O ( N M ) O(NM)
  91. O ( N 3 ) O(N^{3})
  92. v v
  93. G := ( V , E ) G:=(V,E)
  94. | V | |V|
  95. | E | |E|
  96. X ( v ) X(v)
  97. X ( v ) X(v)
  98. v v
  99. C x ( p i ) C_{x}(p_{i})
  100. i i
  101. C x ( p * ) C_{x}(p_{*})
  102. max i = 1 N C x ( p * ) - C x ( p i ) \max\sum_{i=1}^{N}C_{x}(p_{*})-C_{x}(p_{i})
  103. C x C_{x}
  104. C x = i = 1 N C x ( p * ) - C x ( p i ) max i = 1 N C x ( p * ) - C x ( p i ) C_{x}=\frac{\sum_{i=1}^{N}C_{x}(p_{*})-C_{x}(p_{i})}{\max\sum_{i=1}^{N}C_{x}(p% _{*})-C_{x}(p_{i})}

Cerium(IV)_oxide.html

  1. x x
  2. ( x 0.35 - x ) = 106000 [ Pa 0.217 ] × P O 2 - 0.217 exp ( - 195.6 [ k J mol - 1 ] R T ) \Bigg(\frac{x}{0.35-x}\Bigg)=106000\,\,[\mathrm{Pa}^{0.217}]\times P_{O_{2}}^{% \,\,-0.217}\exp\Bigg(\frac{-195.6\,\,[\mathrm{k\,J\,mol}^{-1}]}{RT}\Bigg)
  3. x x

Certificate_of_Entitlement.html

  1. ( Total COE Quota ) q y = \displaystyle(\,\text{Total COE Quota})_{qy}=
  2. y y
  3. q y qy
  4. y y
  5. y - 1 y-1
  6. g g

Chain-growth_polymerization.html

  1. ( - M - ) n ( p o l y m e r ) + M ( m o n o m e r ) ( - M - ) n + 1 (-M-)_{n}(polymer)+M(monomer)\rightarrow(-M-)_{n+1}

Channel_catfish.html

  1. L 1 = 45.23 L_{1}=45.23
  2. W = ( L / L 1 ) b W=(L/L_{1})^{b}\!\,
  3. L 1 L_{1}
  4. L 1 = 45.23 L_{1}=45.23

Character_group.html

  1. f : G \ { 0 } f:G\rightarrow\mathbb{C}\backslash\{0\}
  2. g 1 , g 2 G f ( g 1 g 2 ) = f ( g 1 ) f ( g 2 ) \forall g_{1},g_{2}\in G\;\;f(g_{1}g_{2})=f(g_{1})f(g_{2})
  3. g G k \forall g\in G\;\;\exists k\in\mathbb{N}
  4. g k = e g^{k}=e
  5. f ( g ) k = f ( g k ) = f ( e ) = 1 f(g)^{k}=f(g^{k})=f(e)=1
  6. g G f 1 ( g ) = 1 \forall g\in G\;\;f_{1}(g)=1
  7. f i ( g ) 1 f_{i}(g)\neq 1
  8. g G g\in G
  9. ( f j f k ) ( g ) = f j ( g ) f k ( g ) (f_{j}f_{k})(g)=f_{j}(g)f_{k}(g)
  10. g G g\in G
  11. G ^ \hat{G}
  12. G ^ \hat{G}
  13. g G | f k ( g ) | = 1 \forall g\in G\;\;|f_{k}(g)|=1
  14. n × n n\times n
  15. A j k = f j ( g k ) A_{jk}=f_{j}(g_{k})
  16. g k g_{k}
  17. k = 1 n A j k = k = 1 n f j ( g k ) = 0 \sum_{k=1}^{n}A_{jk}=\sum_{k=1}^{n}f_{j}(g_{k})=0
  18. j 1 j\neq 1
  19. k = 1 n A 1 k = n \sum_{k=1}^{n}A_{1k}=n
  20. j = 1 n A j k = j = 1 n f j ( g k ) = 0 \sum_{j=1}^{n}A_{jk}=\sum_{j=1}^{n}f_{j}(g_{k})=0
  21. k 1 k\neq 1
  22. j = 1 n A j 1 = j = 1 n f j ( e ) = n \sum_{j=1}^{n}A_{j1}=\sum_{j=1}^{n}f_{j}(e)=n
  23. A A^{\ast}
  24. A A = A A = n I AA^{\ast}=A^{\ast}A=nI
  25. k = 1 n f k * ( g i ) f k ( g j ) = n δ i j \sum_{k=1}^{n}{f_{k}}^{*}(g_{i})f_{k}(g_{j})=n\delta_{ij}
  26. δ i j \delta_{ij}
  27. f k * ( g i ) f^{*}_{k}(g_{i})
  28. f k ( g i ) f_{k}(g_{i})

Charge_invariance.html

  1. γ = 1 1 - ( v c ) 2 \gamma=\frac{1}{\sqrt{1-(\frac{v}{c})^{2}}}
  2. v v
  3. c c
  4. E = γ E 0 E=\gamma E_{0}\,\!
  5. m = γ m 0 m=\gamma m_{0}\,\!
  6. e = e 0 e=e_{0}\,\!
  7. 2 = 0 2 \frac{\hbar}{2}=\frac{\hbar_{0}}{2}\,\!
  8. γ \gamma
  9. γ \gamma
  10. E 0 E_{0}
  11. m 0 m_{0}
  12. v v
  13. γ \gamma
  14. E E
  15. m m
  16. j μ = ( c ρ , j ) j^{\mu}=(c\rho,{\vec{j}})
  17. μ j μ = 0 \partial_{\mu}j^{\mu}=0

Charles_Anderson-Pelham,_2nd_Earl_of_Yarborough.html

  1. ( 32 13 ) ( 52 13 ) \frac{{\left({{32}\atop{13}}\right)}}{{\left({{52}\atop{13}}\right)}}
  2. 347 , 373 , 600 635 , 013 , 559 , 600 \frac{347,373,600}{635,013,559,600}
  3. 1 1828 \frac{1}{1828}

Charm_(quantum_number).html

  1. C = n c - n c ¯ . C=n\text{c}-n_{\mathrm{\bar{c}}}.

Chebyshev_distance.html

  1. p i p_{i}
  2. q i q_{i}
  3. D Chebyshev ( p , q ) := max i ( | p i - q i | ) . D_{\rm Chebyshev}(p,q):=\max_{i}(|p_{i}-q_{i}|).
  4. lim k ( i = 1 n | p i - q i | k ) 1 / k , \lim_{k\to\infty}\bigg(\sum_{i=1}^{n}\left|p_{i}-q_{i}\right|^{k}\bigg)^{1/k},
  5. ( x 1 , y 1 ) (x_{1},y_{1})
  6. ( x 2 , y 2 ) (x_{2},y_{2})
  7. D Chess = max ( | x 2 - x 1 | , | y 2 - y 1 | ) . D_{\rm Chess}=\max\left(\left|x_{2}-x_{1}\right|,\left|y_{2}-y_{1}\right|% \right).
  8. 2 \sqrt{2}

Check_mark.html

  1. / \cdot\!/\!\cdot

Checking_whether_a_coin_is_fair.html

  1. f ( r | H = h , T = t ) = Pr ( H = h | r , N = h + t ) g ( r ) 0 1 Pr ( H = h | r , N = h + t ) g ( r ) d r . f(r|H=h,T=t)=\frac{\Pr(H=h|r,N=h+t)\,g(r)}{\int_{0}^{1}\Pr(H=h|r^{\prime},N=h+% t)\,g(r^{\prime})\,dr^{\prime}}.\!
  2. Pr ( H = h | r , N = h + t ) = ( N h ) r h ( 1 - r ) t . \Pr(H=h|r,N=h+t)={N\choose h}\,r^{h}\,(1-r)^{t}.\!
  3. f ( r | H = h , T = t ) = ( N h ) r h ( 1 - r ) t 0 1 ( N h ) r h ( 1 - r ) t d r = r h ( 1 - r ) t 0 1 r h ( 1 - r ) t d r . f(r|H=h,T=t)=\frac{{N\choose h}\,r^{h}\,(1-r)^{t}}{\int_{0}^{1}{N\choose h}\,r% ^{h}\,(1-r)^{t}\,dr}=\frac{r^{h}\,(1-r)^{t}}{\int_{0}^{1}r^{h}\,(1-r)^{t}\,dr}.
  4. f ( r | H = h , T = t ) = 1 B ( h + 1 , t + 1 ) r h ( 1 - r ) t . f(r|H=h,T=t)=\frac{1}{\mathrm{B}(h+1,t+1)}\;r^{h}\,(1-r)^{t}.\!
  5. f ( r | H = h , T = t ) = ( h + t + 1 ) ! h ! t ! r h ( 1 - r ) t . f(r|H=h,T=t)=\frac{(h+t+1)!}{h!\,\,t!}\;r^{h}\,(1-r)^{t}.\!
  6. f ( r | H = 7 , T = 3 ) = ( 10 + 1 ) ! 7 ! 3 ! r 7 ( 1 - r ) 3 = 1320 r 7 ( 1 - r ) 3 f(r|H=7,T=3)=\frac{(10+1)!}{7!\,\,3!}\;r^{7}\,(1-r)^{3}=1320\,r^{7}\,(1-r)^{3}\!
  7. Pr ( 0.45 < r < 0.55 ) = 0.45 0.55 f ( r | H = 7 , T = 3 ) d r 13 % \Pr(0.45<r<0.55)=\int_{0.45}^{0.55}f(r|H=7,T=3)\,dr\approx 13\%\!
  8. E [ r ] = 0 1 r f ( r | H = 7 , T = 3 ) d r = h + 1 h + t + 2 = 2 3 . \operatorname{E}[r]=\int_{0}^{1}r\cdot f(r|H=7,T=3)\,\mathrm{d}r=\frac{h+1}{h+% t+2}=\frac{2}{3}\,.
  9. r r\,\!
  10. p = h h + t p\,\!=\frac{h}{h+t}
  11. | p - r | < E |p-r|<E
  12. | p - r | < E |p-r|<E
  13. p p\,\!
  14. r r
  15. r r\,\!
  16. s p = p ( 1 - p ) n s_{p}=\sqrt{\frac{p\,(1-p)}{n}}
  17. s p s_{p}
  18. p = ( 1 - p ) = 0.5 p=(1-p)=0.5
  19. s p s_{p}\,\!
  20. = p ( 1 - p ) n = 0.5 × 0.5 n = 1 2 n =\sqrt{\frac{p\,(1-p)}{n}}=\sqrt{\frac{0.5\times 0.5}{n}}=\frac{1}{2\,\sqrt{n}}
  21. E = Z s p = Z 2 n E=Z\,s_{p}=\frac{Z}{2\,\sqrt{n}}
  22. n = Z 2 4 E 2 n=\frac{Z^{2}}{4\,E^{2}}\!
  23. n = Z 2 4 E 2 = Z 2 4 × 0.01 2 = 2500 Z 2 n=\frac{Z^{2}}{4\,E^{2}}=\frac{Z^{2}}{4\times 0.01^{2}}=2500\ Z^{2}
  24. n = 2500 n=2500\,
  25. n = 10000 n=10000\,
  26. n = 27225 n=27225\,
  27. p p\,\!
  28. r r\,\!
  29. E = Z 2 n E=\frac{Z}{2\,\sqrt{n}}
  30. E = Z 2 10000 = Z 200 E=\frac{Z}{2\,\sqrt{10000}}=\frac{Z}{200}
  31. E = 0.0050 E=0.0050\,
  32. E = 0.0100 E=0.0100\,
  33. E = 0.0165 E=0.0165\,
  34. r r\,\!
  35. p = h h + t = 5961 12000 = 0.4968 p=\frac{h}{h+t}\,=\frac{5961}{12000}\,=0.4968
  36. Z = 4.4172 Z=4.4172\,\!
  37. E = Z 2 n = 4.4172 2 12000 = 0.0202 E=\frac{Z}{2\,\sqrt{n}}\,=\frac{4.4172}{2\,\sqrt{12000}}\,=0.0202
  38. p - E < r < p + E p-E<r<p+E\,\!
  39. 0.4766 < r < 0.5170 0.4766<r<0.5170\,\!
  40. r r\,\!

Chemical_ionization.html

  1. C H 4 + e - C H 4 + + 2 e - CH_{4}+e^{-}\to CH_{4}^{+}+2e^{-}
  2. C H 4 + C H 4 + C H 5 + + C H 3 CH_{4}+CH_{4}^{+}\to CH_{5}^{+}+CH_{3}
  3. C H 4 + C H 3 + C 2 H 5 + + H 2 CH_{4}+CH_{3}^{+}\to C_{2}H_{5}^{+}+H_{2}
  4. M + C H 5 + C H 4 + [ M + H ] + M+CH_{5}^{+}\to CH_{4}+[M+H]^{+}
  5. A H + C H 3 + C H 4 + A + AH+CH_{3}^{+}\to CH_{4}+A^{+}
  6. H - H^{-}
  7. M + C 2 H 5 + [ M + C 2 H 5 ] + M+C_{2}H_{5}^{+}\to[M+C_{2}H_{5}]^{+}
  8. A + C H 4 + C H 4 + A + A+CH_{4}^{+}\to CH_{4}+A^{+}

Chemical_shift.html

  1. ω 0 \omega_{0}
  2. ω 0 = γ B 0 \omega_{0}=\gamma B_{0}\,
  3. B 0 B_{0}
  4. γ \gamma
  5. μ \mu
  6. I I
  7. μ N \mu_{N}
  8. γ = μ μ N h I \gamma=\frac{\mu\,\mu_{N}}{hI}\,
  9. ω 0 = γ B 0 = 2.79 × 5.05 × 10 - 27 J / T 6.62 × 10 - 34 Js × ( 1 / 2 ) × 1 T = 42.5 MHz \omega_{0}=\gamma B_{0}=\frac{{2.79\times 5.05\times 10^{-27}\,{\rm{J/T}}}}{{6% .62\times 10^{-34}\,{\rm{Js}}\times\left({1/2}\right)}}\times 1\,{\rm{T}}=42.5% \,{\rm{MHz}}\,
  10. δ = ( ν s a m p l e - ν r e f e r e n c e ) ν r e f e r e n c e \delta={(\nu_{sample}-\nu_{reference})\over\nu_{reference}}
  11. ν s a m p l e \nu_{sample}
  12. ν r e f e r e n c e \nu_{reference}
  13. 300 Hz 300 × 10 6 Hz = 1 × 10 - 6 = 1 ppm \frac{300\,\rm Hz}{300\times 10^{6}\,\rm Hz}=1\times 10^{-6}=1\,\rm ppm\,

Chi-squared.html

  1. χ 2 \chi^{2}

Chi_distribution.html

  1. P ( k / 2 , x 2 / 2 ) P(k/2,x^{2}/2)\,
  2. μ = 2 Γ ( ( k + 1 ) / 2 ) Γ ( k / 2 ) \mu=\sqrt{2}\,\frac{\Gamma((k+1)/2)}{\Gamma(k/2)}
  3. k - 1 \sqrt{k-1}\,
  4. k 1 k\geq 1
  5. σ 2 = k - μ 2 \sigma^{2}=k-\mu^{2}\,
  6. γ 1 = μ σ 3 ( 1 - 2 σ 2 ) \gamma_{1}=\frac{\mu}{\sigma^{3}}\,(1-2\sigma^{2})
  7. 2 σ 2 ( 1 - μ σ γ 1 - σ 2 ) \frac{2}{\sigma^{2}}(1-\mu\sigma\gamma_{1}-\sigma^{2})
  8. ln ( Γ ( k / 2 ) ) + \ln(\Gamma(k/2))+\,
  9. 1 2 ( k - ln ( 2 ) - ( k - 1 ) ψ 0 ( k / 2 ) ) \frac{1}{2}(k\!-\!\ln(2)\!-\!(k\!-\!1)\psi_{0}(k/2))
  10. X i X_{i}
  11. μ i \mu_{i}
  12. σ i \sigma_{i}
  13. Y = i = 1 k ( X i - μ i σ i ) 2 Y=\sqrt{\sum_{i=1}^{k}\left(\frac{X_{i}-\mu_{i}}{\sigma_{i}}\right)^{2}}
  14. k k
  15. X i X_{i}
  16. f ( x ; k ) = 2 1 - k 2 x k - 1 e - x 2 2 Γ ( k 2 ) f(x;k)=\frac{2^{1-\frac{k}{2}}x^{k-1}e^{-\frac{x^{2}}{2}}}{\Gamma(\frac{k}{2})}
  17. Γ ( z ) \Gamma(z)
  18. F ( x ; k ) = P ( k / 2 , x 2 / 2 ) F(x;k)=P(k/2,x^{2}/2)\,
  19. P ( k , x ) P(k,x)
  20. M ( t ) = M ( k 2 , 1 2 , t 2 2 ) + M(t)=M\left(\frac{k}{2},\frac{1}{2},\frac{t^{2}}{2}\right)+
  21. t 2 Γ ( ( k + 1 ) / 2 ) Γ ( k / 2 ) M ( k + 1 2 , 3 2 , t 2 2 ) t\sqrt{2}\,\frac{\Gamma((k+1)/2)}{\Gamma(k/2)}M\left(\frac{k+1}{2},\frac{3}{2}% ,\frac{t^{2}}{2}\right)
  22. φ ( t ; k ) = M ( k 2 , 1 2 , - t 2 2 ) + \varphi(t;k)=M\left(\frac{k}{2},\frac{1}{2},\frac{-t^{2}}{2}\right)+
  23. i t 2 Γ ( ( k + 1 ) / 2 ) Γ ( k / 2 ) M ( k + 1 2 , 3 2 , - t 2 2 ) it\sqrt{2}\,\frac{\Gamma((k+1)/2)}{\Gamma(k/2)}M\left(\frac{k+1}{2},\frac{3}{2% },\frac{-t^{2}}{2}\right)
  24. M ( a , b , z ) M(a,b,z)
  25. { x f ( x ) + f ( x ) ( - ν + x 2 + 1 ) = 0 , f ( 1 ) = 2 1 - ν 2 e Γ ( ν 2 ) } \left\{xf^{\prime}(x)+f(x)\left(-\nu+x^{2}+1\right)=0,f(1)=\frac{2^{1-\frac{% \nu}{2}}}{\sqrt{e}\Gamma\left(\frac{\nu}{2}\right)}\right\}
  26. μ j = 2 j / 2 Γ ( ( k + j ) / 2 ) Γ ( k / 2 ) \mu_{j}=2^{j/2}\frac{\Gamma((k+j)/2)}{\Gamma(k/2)}
  27. Γ ( z ) \Gamma(z)
  28. μ 1 = 2 Γ ( ( k + 1 ) / 2 ) Γ ( k / 2 ) \mu_{1}=\sqrt{2}\,\,\frac{\Gamma((k\!+\!1)/2)}{\Gamma(k/2)}
  29. μ 2 = k \mu_{2}=k\,
  30. μ 3 = 2 2 Γ ( ( k + 3 ) / 2 ) Γ ( k / 2 ) = ( k + 1 ) μ 1 \mu_{3}=2\sqrt{2}\,\,\frac{\Gamma((k\!+\!3)/2)}{\Gamma(k/2)}=(k+1)\mu_{1}
  31. μ 4 = ( k ) ( k + 2 ) \mu_{4}=(k)(k+2)\,
  32. μ 5 = 4 2 Γ ( ( k + 5 ) / 2 ) Γ ( k / 2 ) = ( k + 1 ) ( k + 3 ) μ 1 \mu_{5}=4\sqrt{2}\,\,\frac{\Gamma((k\!+\!5)/2)}{\Gamma(k/2)}=(k+1)(k+3)\mu_{1}
  33. μ 6 = ( k ) ( k + 2 ) ( k + 4 ) \mu_{6}=(k)(k+2)(k+4)\,
  34. Γ ( x + 1 ) = x Γ ( x ) \Gamma(x+1)=x\Gamma(x)\,
  35. μ = 2 Γ ( ( k + 1 ) / 2 ) Γ ( k / 2 ) \mu=\sqrt{2}\,\,\frac{\Gamma((k+1)/2)}{\Gamma(k/2)}
  36. σ 2 = k - μ 2 \sigma^{2}=k-\mu^{2}\,
  37. γ 1 = μ σ 3 ( 1 - 2 σ 2 ) \gamma_{1}=\frac{\mu}{\sigma^{3}}\,(1-2\sigma^{2})
  38. γ 2 = 2 σ 2 ( 1 - μ σ γ 1 - σ 2 ) \gamma_{2}=\frac{2}{\sigma^{2}}(1-\mu\sigma\gamma_{1}-\sigma^{2})
  39. S = ln ( Γ ( k / 2 ) ) + 1 2 ( k - ln ( 2 ) - ( k - 1 ) ψ 0 ( k / 2 ) ) S=\ln(\Gamma(k/2))+\frac{1}{2}(k\!-\!\ln(2)\!-\!(k\!-\!1)\psi_{0}(k/2))
  40. ψ 0 ( z ) \psi_{0}(z)
  41. X χ k ( x ) X\sim\chi_{k}(x)
  42. X 2 χ k 2 X^{2}\sim\chi^{2}_{k}
  43. lim k χ k ( x ) - μ k σ k 𝑑 N ( 0 , 1 ) \lim_{k\to\infty}\tfrac{\chi_{k}(x)-\mu_{k}}{\sigma_{k}}\xrightarrow{d}\ N(0,1)\,
  44. X N ( 0 , 1 ) X\sim N(0,1)\,
  45. | X | χ 1 ( x ) |X|\sim\chi_{1}(x)\,
  46. X χ 1 ( x ) X\sim\chi_{1}(x)\,
  47. σ X H N ( σ ) \sigma X\sim HN(\sigma)\,
  48. σ > 0 \sigma>0\,
  49. χ 2 ( x ) Rayleigh ( 1 ) \chi_{2}(x)\sim\mathrm{Rayleigh}(1)\,
  50. χ 3 ( x ) Maxwell ( 1 ) \chi_{3}(x)\sim\mathrm{Maxwell}(1)\,
  51. s y m b o l N i = 1 , , k ( 0 , 1 ) 2 χ k ( x ) \|symbol{N}_{i=1,\ldots,k}{(0,1)}\|_{2}\sim\chi_{k}(x)
  52. k k
  53. k k
  54. i = 1 k ( X i - μ i σ i ) 2 \sum_{i=1}^{k}\left(\frac{X_{i}-\mu_{i}}{\sigma_{i}}\right)^{2}
  55. i = 1 k ( X i σ i ) 2 \sum_{i=1}^{k}\left(\frac{X_{i}}{\sigma_{i}}\right)^{2}
  56. i = 1 k ( X i - μ i σ i ) 2 \sqrt{\sum_{i=1}^{k}\left(\frac{X_{i}-\mu_{i}}{\sigma_{i}}\right)^{2}}
  57. i = 1 k ( X i σ i ) 2 \sqrt{\sum_{i=1}^{k}\left(\frac{X_{i}}{\sigma_{i}}\right)^{2}}

Chinese_mathematics.html

  1. 355 113 \tfrac{355}{113}
  2. 1 2 + 2 2 + 3 2 + + n 2 = n ( n + 1 ) ( 2 n + 1 ) 3 ! 1^{2}+2^{2}+3^{2}+\cdots+n^{2}={n(n+1)(2n+1)\over 3!}
  3. 1 + 8 + 30 + 80 + + n 2 ( n + 1 ) ( n + 2 ) 3 ! = n ( n + 1 ) ( n + 2 ) ( n + 3 ) ( 4 n + 1 ) 5 ! 1+8+30+80+\cdots+{n^{2}(n+1)(n+2)\over 3!}={n(n+1)(n+2)(n+3)(4n+1)\over 5!}

Chip_log.html

  1. distance = 1 mile × 30 3600 \,\text{distance}=1\,\text{mile}\times\frac{30}{3600}

Chiral_symmetry.html

  1. = u ¯ i u + d ¯ i d + gluons . \mathcal{L}=\overline{u}\,i\displaystyle{\not}D\,u+\overline{d}\,i% \displaystyle{\not}D\,d+\mathcal{L}_{\mathrm{gluons}}~{}.
  2. = u ¯ L i u L + u ¯ R i u R + d ¯ L i d L + d ¯ R i d R + gluons . \mathcal{L}=\overline{u}_{L}\,i\displaystyle{\not}D\,u_{L}+\overline{u}_{R}\,i% \displaystyle{\not}D\,u_{R}+\overline{d}_{L}\,i\displaystyle{\not}D\,d_{L}+% \overline{d}_{R}\,i\displaystyle{\not}D\,d_{R}+\mathcal{L}_{\mathrm{gluons}}~{}.
  3. \displaystyle{\not}D
  4. q = [ u d ] , q=\begin{bmatrix}u\\ d\end{bmatrix},
  5. = q ¯ L i q L + q ¯ R i q R + gluons . \mathcal{L}=\overline{q}_{L}\,i\displaystyle{\not}D\,q_{L}+\overline{q}_{R}\,i% \displaystyle{\not}D\,q_{R}+\mathcal{L}_{\mathrm{gluons}}~{}.
  6. S U ( 2 ) L × S U ( 2 ) R × U ( 1 ) V × U ( 1 ) A . SU(2)_{L}\times SU(2)_{R}\times U(1)_{V}\times U(1)_{A}~{}.
  7. q L e i θ q L q R e i θ q R , q_{L}\rightarrow e^{i\theta}q_{L}\qquad q_{R}\rightarrow e^{i\theta}q_{R}~{},
  8. q L e i θ q L q R e - i θ q R , q_{L}\rightarrow e^{i\theta}q_{L}\qquad q_{R}\rightarrow e^{-i\theta}q_{R}~{},
  9. q ¯ R a q L b = v δ a b \langle\bar{q}^{a}_{R}q^{b}_{L}\rangle=v\delta^{ab}
  10. N N
  11. S U ( N ) L × S U ( N ) R × U ( 1 ) V × U ( 1 ) A , SU(N)_{L}\times SU(N)_{R}\times U(1)_{V}\times U(1)_{A}~{},
  12. N N

Chladni's_law.html

  1. f = C ( m + 2 n ) p f=C(m+2n)^{p}

Choice_function.html

  1. X X
  2. X X
  3. X . X.
  4. X X
  5. X \bigcup X
  6. X X
  7. X X
  8. f : A B f:A\rightarrow B
  9. φ : A 𝒫 ( B ) \varphi:A\rightarrow\mathcal{P}(B)
  10. 𝒫 ( B ) \mathcal{P}(B)
  11. a A ( f ( a ) φ ( a ) ) . \forall a\in A\,(f(a)\in\varphi(a))\,.
  12. τ \tau
  13. P ( x ) P(x)
  14. τ x ( P ) \tau_{x}(P)
  15. P P
  16. P ( τ x ( P ) ) P(\tau_{x}(P))
  17. ( x ) ( P ( x ) ) (\exists x)(P(x))
  18. A ( a ) A ( ε ( A ) ) A(a)\to A(\varepsilon(A))
  19. ε \varepsilon

Chopper_(electronics).html

  1. f f

Christoffel_symbols.html

  1. g g
  2. e i = x i = i , i = 1 , 2 , , n \mathrm{e}_{i}=\frac{\partial}{\partial x^{i}}=\partial_{i},\quad i=1,2,\dots,n
  3. Γ c a b = g c d Γ d , a b \Gamma_{cab}=g_{cd}\Gamma^{d}{}_{ab}\,,
  4. Γ c a b = 1 2 ( g c a x b + g c b x a - g a b x c ) = 1 2 ( g c a , b + g c b , a - g a b , c ) = 1 2 ( b g c a + a g c b - c g a b ) . \Gamma_{cab}=\frac{1}{2}\left(\frac{\partial g_{ca}}{\partial x^{b}}+\frac{% \partial g_{cb}}{\partial x^{a}}-\frac{\partial g_{ab}}{\partial x^{c}}\right)% =\frac{1}{2}\,(g_{ca,b}+g_{cb,a}-g_{ab,c})=\frac{1}{2}\,\left(\partial_{b}g_{% ca}+\partial_{a}g_{cb}-\partial_{c}g_{ab}\right)\,.
  5. Γ c a b = [ a b , c ] . \Gamma_{cab}=[ab,c].
  6. [ a b , c ] = [ b a , c ] [ab,c]=[ba,c]
  7. Γ k = i j Γ k j i \Gamma^{k}{}_{ij}=\Gamma^{k}{}_{ji}\,
  8. Γ k i j \Gamma^{k}{}_{ij}
  9. Γ i j k \Gamma^{k}_{ij}
  10. { k i j } \{\begin{smallmatrix}k\\ ij\end{smallmatrix}\}
  11. i e j = Γ k e k i j \nabla_{i}\mathrm{e}_{j}=\Gamma^{k}{}_{ij}\mathrm{e}_{k}
  12. i \nabla_{i}
  13. e i \mathrm{e}_{i}
  14. i e i \nabla_{i}\equiv\nabla_{\mathrm{e}_{i}}
  15. e i = i \mathrm{e}_{i}=\partial_{i}
  16. g i k g_{ik}
  17. 0 = g i k = g i k x - g m k Γ m - i g i m Γ m = k g i k x - 2 g m ( k Γ m . i ) 0=\nabla_{\ell}g_{ik}=\frac{\partial g_{ik}}{\partial x^{\ell}}-g_{mk}\Gamma^{% m}{}_{i\ell}-g_{im}\Gamma^{m}{}_{k\ell}=\frac{\partial g_{ik}}{\partial x^{% \ell}}-2g_{m(k}\Gamma^{m}{}_{i)\ell}.
  18. 0 = g i k ; = g i k , - g m k Γ m - i g i m Γ m . k 0=\,g_{ik;\ell}=g_{ik,\ell}-g_{mk}\Gamma^{m}{}_{i\ell}-g_{im}\Gamma^{m}{}_{k% \ell}.
  19. Γ i = k 1 2 g i m ( g m k x + g m x k - g k x m ) = 1 2 g i m ( g m k , + g m , k - g k , m ) , \Gamma^{i}{}_{k\ell}=\frac{1}{2}g^{im}\left(\frac{\partial g_{mk}}{\partial x^% {\ell}}+\frac{\partial g_{m\ell}}{\partial x^{k}}-\frac{\partial g_{k\ell}}{% \partial x^{m}}\right)={1\over 2}g^{im}(g_{mk,\ell}+g_{m\ell,k}-g_{k\ell,m}),
  20. ( g j k ) (g^{jk})
  21. ( g j k ) (g_{jk})\,
  22. g j i g i k = δ j k g^{ji}g_{ik}=\delta^{j}{}_{k}
  23. 𝐮 i \mathbf{u}_{i}
  24. 𝐮 i 𝐮 j = ω k 𝐮 k i j . \nabla_{\mathbf{u}_{i}}\mathbf{u}_{j}=\omega^{k}{}_{ij}\mathbf{u}_{k}.
  25. ω i = k 1 2 g i m ( g m k , + g m , k - g k , m + c m k + c m k - c k m ) , \omega^{i}{}_{k\ell}=\frac{1}{2}g^{im}\left(g_{mk,\ell}+g_{m\ell,k}-g_{k\ell,m% }+c_{mk\ell}+c_{m\ell k}-c_{k\ell m}\right),
  26. c k m = g m p c k p c_{k\ell m}=g_{mp}{c_{k\ell}}^{p}
  27. [ 𝐮 k , 𝐮 ] = c k 𝐮 m m [\mathbf{u}_{k},\mathbf{u}_{\ell}]=c_{k\ell}{}^{m}\mathbf{u}_{m}\,
  28. 𝐮 k \mathbf{u}_{k}
  29. [ , ] [{\,},{\,}]
  30. X i 𝐮 i \mathrm{X}_{i}\equiv\mathbf{u}_{i}
  31. g a b η a b = X a , X b g_{ab}\equiv\eta_{ab}=\langle X_{a},X_{b}\rangle
  32. g m k , η m k , = 0 g_{mk,\ell}\equiv\eta_{mk,\ell}=0
  33. ω i = k 1 2 η i m ( c m k + c m k - c k m ) \omega^{i}{}_{k\ell}=\frac{1}{2}\eta^{im}\left(c_{mk\ell}+c_{m\ell k}-c_{k\ell m% }\right)
  34. ω a b c = - ω b a c , \omega_{abc}=-\omega_{bac}\,,
  35. ω a b c = η a d ω d b c \omega_{abc}=\eta_{ad}\omega^{d}{}_{bc}
  36. ω a b c \omega^{a}{}_{bc}
  37. ω k := i j 𝐮 k ( j 𝐮 i ) , \omega^{k}{}_{ij}:={{\mathbf{u}}}^{k}\cdot\left(\nabla_{j}{{\mathbf{u}}}_{i}% \right)\,,
  38. 𝐮 i \mathbf{u}_{i}
  39. 𝐮 k = η k 𝐮 \mathbf{u}^{k}=\eta^{k\ell}\mathbf{u}_{\ell}
  40. X i X^{i}
  41. Y k Y^{k}
  42. ( X Y ) k = X i ( i Y ) k = X i ( Y k x i + Γ k Y m i m ) . \left(\nabla_{X}Y\right)^{k}=X^{i}(\nabla_{i}Y)^{k}=X^{i}\left(\frac{\partial Y% ^{k}}{\partial x^{i}}+\Gamma^{k}{}_{im}Y^{m}\right).
  43. g ( X , Y ) = X i Y i = g i k X i Y k = g i k X i Y k . g(X,Y)=X^{i}Y_{i}=g_{ik}X^{i}Y^{k}=g^{ik}X_{i}Y_{k}.
  44. g i k g i k g_{ik}\neq g^{ik}
  45. g i = k δ i k g^{i}{}_{k}=\delta^{i}{}_{k}
  46. g i k g^{ik}
  47. g i k g_{ik}
  48. g i j g j k = δ i k g^{ij}g_{jk}=\delta^{i}{}_{k}
  49. X Y - Y X = [ X , Y ] \nabla_{X}Y-\nabla_{Y}X=[X,Y]
  50. Γ i = j k Γ i . k j \Gamma^{i}{}_{jk}=\Gamma^{i}{}_{kj}.
  51. V m = V m x + Γ m V k k . \nabla_{\ell}V^{m}=\frac{\partial V^{m}}{\partial x^{\ell}}+\Gamma^{m}{}_{k% \ell}V^{k}.
  52. φ \varphi
  53. i φ = φ x i \nabla_{i}\varphi=\frac{\partial\varphi}{\partial x^{i}}
  54. ω m \omega_{m}
  55. ω m = ω m x - Γ k ω k m . \nabla_{\ell}\omega_{m}=\frac{\partial\omega_{m}}{\partial x^{\ell}}-\Gamma^{k% }{}_{m\ell}\omega_{k}.
  56. i j φ = j i φ \nabla_{i}\nabla_{j}\varphi=\nabla_{j}\nabla_{i}\varphi
  57. A i k A^{ik}
  58. A i k = A i k x + Γ i A m k m + Γ k A i m m , \nabla_{\ell}A^{ik}=\frac{\partial A^{ik}}{\partial x^{\ell}}+\Gamma^{i}{}_{m% \ell}A^{mk}+\Gamma^{k}{}_{m\ell}A^{im},
  59. A i k = ; A i k + , A m k Γ i + m A i m Γ k . m A^{ik}{}_{;\ell}=A^{ik}{}_{,\ell}+A^{mk}\Gamma^{i}{}_{m\ell}+A^{im}\Gamma^{k}{% }_{m\ell}.
  60. A i = k ; A i + k , A m Γ i k - m A i Γ m m , k A^{i}{}_{k;\ell}=A^{i}{}_{k,\ell}+A^{m}{}_{k}\Gamma^{i}{}_{m\ell}-A^{i}{}_{m}% \Gamma^{m}{}_{k\ell},
  61. A i k ; = A i k , - A m k Γ m - i A i m Γ m . k A_{ik;\ell}=A_{ik,\ell}-A_{mk}\Gamma^{m}{}_{i\ell}-A_{im}\Gamma^{m}{}_{k\ell}.
  62. ( y 1 , , y n ) (y^{1},\dots,y^{n})
  63. ( x 1 , , x n ) (x^{1},\dots,x^{n})
  64. y i = x k y i x k \frac{\partial}{\partial y^{i}}=\frac{\partial x^{k}}{\partial y^{i}}\frac{% \partial}{\partial x^{k}}
  65. Γ ¯ k = i j x p y i x q y j Γ r y k x r p q + y k x m 2 x m y i y j {\bar{\Gamma}}^{k}{}_{ij}=\frac{\partial x^{p}}{\partial y^{i}}\,\frac{% \partial x^{q}}{\partial y^{j}}\,\Gamma^{r}{}_{pq}\,\frac{\partial y^{k}}{% \partial x^{r}}+\frac{\partial y^{k}}{\partial x^{m}}\,\frac{\partial^{2}x^{m}% }{\partial y^{i}\partial y^{j}}

Chromatic_circle.html

  1. Z 12 Z_{12}

Chromatic_polynomial.html

  1. k 3 k^{3}
  2. k 2 ( k - 1 ) k^{2}(k-1)
  3. k ( k - 1 ) 2 k(k-1)^{2}
  4. k ( k - 1 ) ( k - 2 ) k(k-1)(k-2)
  5. P ( G , k ) P(G,k)
  6. P ( G , 4 ) > 0 P(G,4)>0
  7. k = 0 , 1 , 2 , 3 k=0,1,2,3
  8. P G ( k ) P_{G}(k)
  9. χ G ( k ) \chi_{G}(k)
  10. π G ( k ) \pi_{G}(k)
  11. P ( G , k ) P(G,k)
  12. P 3 P_{3}
  13. k k
  14. P ( P 3 , k ) P(P_{3},k)
  15. { ( 0 , P ( G , 0 ) ) , ( 1 , P ( G , 1 ) ) , , ( n , P ( G , n ) ) } . \left\{(0,P(G,0)),(1,P(G,1)),\cdots,(n,P(G,n))\right\}.
  16. P ( P 3 , t ) = t ( t - 1 ) 2 , P ( P 3 , 3 ) = 12. P(P_{3},t)=t(t-1)^{2},\qquad P(P_{3},3)=12.
  17. χ ( G ) = min { k : P ( G , k ) > 0 } . \chi(G)=\min\{k:P(G,k)>0\}.
  18. K 3 K_{3}
  19. t ( t - 1 ) ( t - 2 ) t(t-1)(t-2)
  20. K n K_{n}
  21. t ( t - 1 ) ( t - 2 ) ( t - ( n - 1 ) ) t(t-1)(t-2)...(t-(n-1))
  22. P n P_{n}
  23. t ( t - 1 ) n - 1 t(t-1)^{n-1}
  24. t ( t - 1 ) n - 1 t(t-1)^{n-1}
  25. C n C_{n}
  26. ( t - 1 ) n + ( - 1 ) n ( t - 1 ) (t-1)^{n}+(-1)^{n}(t-1)
  27. t ( t - 1 ) ( t - 2 ) ( t 7 - 12 t 6 + 67 t 5 - 230 t 4 + 529 t 3 - 814 t 2 + 775 t - 352 ) t(t-1)(t-2)\left(t^{7}-12t^{6}+67t^{5}-230t^{4}+529t^{3}-814t^{2}+775t-352\right)
  28. P ( G , t ) P(G,t)
  29. P ( G , k ) P(G,k)
  30. k = 0 , 1 , , n k=0,1,\cdots,n
  31. ( - 1 ) | V ( G ) | P ( G , - 1 ) (-1)^{|V(G)|}P(G,-1)
  32. P ( G , 1 ) P^{\prime}(G,1)
  33. θ ( G ) \theta(G)
  34. G 1 , , G c G_{1},\cdots,G_{c}
  35. t 0 , , t c - 1 t^{0},\cdots,t^{c-1}
  36. t c , , t n t^{c},\cdots,t^{n}
  37. t n t^{n}
  38. P ( G , t ) P(G,t)
  39. t n - 1 t^{n-1}
  40. P ( G , t ) P(G,t)
  41. - m -m
  42. P ( G , t ) = P ( G 1 , t ) P ( G 2 , t ) P ( G c , t ) \scriptstyle P(G,t)=P(G_{1},t)P(G_{2},t)\cdots P(G_{c},t)
  43. P ( G , t ) = t ( t - 1 ) n - 1 . P(G,t)=t(t-1)^{n-1}.
  44. ( x - 2 ) ( x - 1 ) 3 x (x-2)(x-1)^{3}x
  45. ( x - 1 ) n - 1 x , (x-1)^{n-1}x,
  46. ( x - 1 ) 3 x (x-1)^{3}x
  47. P ( G , t ) = P ( H , t ) P(G,t)=P(H,t)
  48. P ( G , x ) = 0 P(G,x)=0
  49. P ( G , x ) > 0 P(G,x)>0
  50. ϕ \phi
  51. ϕ 2 \phi^{2}
  52. G n G_{n}
  53. P ( G n , ϕ 2 ) ϕ 5 - n . P(G_{n},\phi^{2})\leq\phi^{5-n}.
  54. P ( G , t ) P(G,t)
  55. P ( G , k ) P(G,k)
  56. P ( G , t ) P(G,t)
  57. u u
  58. v v
  59. G / u v G/uv
  60. P ( G , k ) = P ( G - u v , k ) - P ( G / u v , k ) P(G,k)=P(G-uv,k)-P(G/uv,k)
  61. u u
  62. v v
  63. G - u v G-uv
  64. u v uv
  65. P ( G , k ) = P ( G + u v , k ) + P ( G / u v , k ) P(G,k)=P(G+uv,k)+P(G/uv,k)
  66. u u
  67. v v
  68. G + u v G+uv
  69. u v uv
  70. ϕ n + m = ( 1 + 5 2 ) n + m O ( 1.62 n + m ) , \phi^{n+m}=\left(\frac{1+\sqrt{5}}{2}\right)^{n+m}\in O\left(1.62^{n+m}\right),
  71. t ( G ) t(G)
  72. i i
  73. j j
  74. i i
  75. j j
  76. { x R d : x i = x j } \{x\in R^{d}:x_{i}=x_{j}\}
  77. k k
  78. [ 0 , k ] n [0,k]^{n}
  79. [ 0 , k ] [0,k]
  80. P ( G , 3 ) P(G,3)
  81. k = 0 , 1 , 2 k=0,1,2
  82. P ( G , k ) P(G,k)
  83. k > 3 k>3
  84. k = 3 k=3
  85. P ( G , x ) P(G,x)
  86. P ( G , t ) = a 1 t + a 2 t 2 + + a n t n , P(G,t)=a_{1}t+a_{2}t^{2}+\dots+a_{n}t^{n},
  87. a n a_{n}
  88. P ( G , x ) P(G,x)
  89. k = 3 , 4 , k=3,4,\dots
  90. P ( G , x ) P(G,x)

Circle_graph.html

  1. 21 2 k - 24 k - 24 21\cdot 2^{k}-24k-24

Circuit_rank.html

  1. r = 2 r=2
  2. r r
  3. r = m - n + c r=m-n+c
  4. m m
  5. n n
  6. c c
  7. G G
  8. G G
  9. G G
  10. G G
  11. G G
  12. r = rank [ H 1 ( G , \Z ) ] . r=\operatorname{rank}\left[H_{1}(G,\Z)\right].
  13. G G
  14. G G
  15. m m
  16. n n
  17. m - n + 1 2 n - 5 . \frac{m-n+1}{2n-5}.
  18. m - n + 1 m-n+1
  19. 2 n - 5 2n-5
  20. n n
  21. r r
  22. r r
  23. r r
  24. r r

Circular_distribution.html

  1. p ( ϕ ) ( 0 ϕ < 2 π ) , p(\phi)\qquad\qquad(0\leq\phi<2\pi),\,
  2. [ x ( ϕ ) , y ( ϕ ) ] = [ r ( ϕ ) cos ϕ , r ( ϕ ) sin ϕ ] , [x(\phi),y(\phi)]=[r(\phi)\cos\phi,\,r(\phi)\sin\phi],\,
  3. r ( ϕ ) r(\phi)\,
  4. r ( ϕ ) = a + b p ( ϕ ) , r(\phi)=a+bp(\phi),\,

Circular_sector.html

  1. r r
  2. L L
  3. π r 2 \pi r^{2}
  4. 2 π 2\pi
  5. 2 π 2\pi
  6. A = π r 2 θ 2 π = r 2 θ 2 A=\pi r^{2}\cdot\frac{\theta}{2\pi}=\frac{r^{2}\theta}{2}
  7. L L
  8. π r 2 \pi r^{2}
  9. L L
  10. 2 π r 2\pi r
  11. A = π r 2 L 2 π r = r L 2 A=\pi r^{2}\cdot\frac{L}{2\pi r}=\frac{r\cdot L}{2}
  12. A = 0 θ 0 r d S = 0 θ 0 r r ~ d r ~ d θ ~ = 0 θ 1 2 r 2 d θ ~ = r 2 θ 2 A=\int_{0}^{\theta}\int_{0}^{r}dS=\int_{0}^{\theta}\int_{0}^{r}\tilde{r}d% \tilde{r}d\tilde{\theta}=\int_{0}^{\theta}\frac{1}{2}r^{2}d\tilde{\theta}=% \frac{r^{2}\theta}{2}
  13. A = π r 2 θ 360 A=\pi r^{2}\cdot\frac{\theta^{\circ}}{360}
  14. L = θ r , L=\theta\cdot r,
  15. L = θ r π 180 L=\theta^{\circ}\cdot r\cdot\frac{\pi}{180}
  16. P = L + 2 r = θ r + 2 r = r ( θ + 2 ) P=L+2r=\theta r+2r=r\left(\theta+2\right)

Circular_shift.html

  1. σ ( i ) ( i + 1 ) \sigma(i)\equiv(i+1)
  2. σ ( i ) ( i - 1 ) \sigma(i)\equiv(i-1)

Circumscribed_circle.html

  1. ( x - a ) 2 + ( y - b ) 2 = r 2 . \left(x-a\right)^{2}+\left(y-b\right)^{2}=r^{2}.
  2. 𝐀 = ( A x , A y ) \mathbf{A}=(A_{x},A_{y})
  3. 𝐁 = ( B x , B y ) \mathbf{B}=(B_{x},B_{y})
  4. 𝐂 = ( C x , C y ) \mathbf{C}=(C_{x},C_{y})
  5. | 𝐯 - 𝐮 | 2 = r 2 |\mathbf{v}-\mathbf{u}|^{2}=r^{2}
  6. | 𝐀 - 𝐮 | 2 = r 2 |\mathbf{A}-\mathbf{u}|^{2}=r^{2}
  7. | 𝐁 - 𝐮 | 2 = r 2 |\mathbf{B}-\mathbf{u}|^{2}=r^{2}
  8. | 𝐂 - 𝐮 | 2 = r 2 |\mathbf{C}-\mathbf{u}|^{2}=r^{2}
  9. [ | 𝐯 | 2 - 2 v x - 2 v y - 1 | 𝐀 | 2 - 2 A x - 2 A y - 1 | 𝐁 | 2 - 2 B x - 2 B y - 1 | 𝐂 | 2 - 2 C x - 2 C y - 1 ] \begin{bmatrix}|\mathbf{v}|^{2}&-2v_{x}&-2v_{y}&-1\\ |\mathbf{A}|^{2}&-2A_{x}&-2A_{y}&-1\\ |\mathbf{B}|^{2}&-2B_{x}&-2B_{y}&-1\\ |\mathbf{C}|^{2}&-2C_{x}&-2C_{y}&-1\end{bmatrix}
  10. det [ | 𝐯 | 2 v x v y 1 | 𝐀 | 2 A x A y 1 | 𝐁 | 2 B x B y 1 | 𝐂 | 2 C x C y 1 ] = 0. \det\begin{bmatrix}|\mathbf{v}|^{2}&v_{x}&v_{y}&1\\ |\mathbf{A}|^{2}&A_{x}&A_{y}&1\\ |\mathbf{B}|^{2}&B_{x}&B_{y}&1\\ |\mathbf{C}|^{2}&C_{x}&C_{y}&1\end{bmatrix}=0.
  11. S x = 1 2 det [ | 𝐀 | 2 A y 1 | 𝐁 | 2 B y 1 | 𝐂 | 2 C y 1 ] , S y = 1 2 det [ A x | 𝐀 | 2 1 B x | 𝐁 | 2 1 C x | 𝐂 | 2 1 ] , \quad S_{x}=\frac{1}{2}\det\begin{bmatrix}|\mathbf{A}|^{2}&A_{y}&1\\ |\mathbf{B}|^{2}&B_{y}&1\\ |\mathbf{C}|^{2}&C_{y}&1\end{bmatrix},\quad S_{y}=\frac{1}{2}\det\begin{% bmatrix}A_{x}&|\mathbf{A}|^{2}&1\\ B_{x}&|\mathbf{B}|^{2}&1\\ C_{x}&|\mathbf{C}|^{2}&1\end{bmatrix},
  12. a = det [ A x A y 1 B x B y 1 C x C y 1 ] , b = det [ A x A y | 𝐀 | 2 B x B y | 𝐁 | 2 C x C y | 𝐂 | 2 ] a=\det\begin{bmatrix}A_{x}&A_{y}&1\\ B_{x}&B_{y}&1\\ C_{x}&C_{y}&1\end{bmatrix},\quad b=\det\begin{bmatrix}A_{x}&A_{y}&|\mathbf{A}|% ^{2}\\ B_{x}&B_{y}&|\mathbf{B}|^{2}\\ C_{x}&C_{y}&|\mathbf{C}|^{2}\end{bmatrix}
  13. n ^ = ( P 2 - P 1 ) × ( P 3 - P 1 ) | ( P 2 - P 1 ) × ( P 3 - P 1 ) | . \hat{n}=\frac{\left(P_{2}-P_{1}\right)\times\left(P_{3}-P_{1}\right)}{\left|% \left(P_{2}-P_{1}\right)\times\left(P_{3}-P_{1}\right)\right|}.
  14. n ^ \scriptstyle{\hat{n}}
  15. n ^ \scriptstyle{\hat{n}}
  16. R ( s ) = P c + cos ( s r ) ( P 0 - P c ) + sin ( s r ) [ n ^ × ( P 0 - P c ) ] . \mathrm{R}\left(s\right)=\mathrm{P_{c}}+\cos\left(\frac{\mathrm{s}}{\mathrm{r}% }\right)\left(P_{0}-P_{c}\right)+\sin\left(\frac{\mathrm{s}}{\mathrm{r}}\right% )\left[\hat{n}\times\left(P_{0}-P_{c}\right)\right].
  17. 𝐚 = 𝐀 - 𝐂 , \mathbf{a}=\mathbf{A}-\mathbf{C},
  18. 𝐛 = 𝐁 - 𝐂 . \mathbf{b}=\mathbf{B}-\mathbf{C}.
  19. r = 𝐚 𝐛 𝐚 - 𝐛 2 𝐚 × 𝐛 = 𝐚 - 𝐛 2 sin θ = 𝐀 - 𝐁 2 sin θ , r=\frac{\left\|\mathbf{a}\right\|\left\|\mathbf{b}\right\|\left\|\mathbf{a}-% \mathbf{b}\right\|}{2\left\|\mathbf{a}\times\mathbf{b}\right\|}=\frac{\left\|% \mathbf{a}-\mathbf{b}\right\|}{2\sin\theta}=\frac{\left\|\mathbf{A}-\mathbf{B}% \right\|}{2\sin\theta},
  20. p 0 = ( 𝐚 2 𝐛 - 𝐛 2 𝐚 ) × ( 𝐚 × 𝐛 ) 2 𝐚 × 𝐛 2 + 𝐂 . p_{0}=\frac{(\left\|\mathbf{a}\right\|^{2}\mathbf{b}-\left\|\mathbf{b}\right\|% ^{2}\mathbf{a})\times(\mathbf{a}\times\mathbf{b})}{2\left\|\mathbf{a}\times% \mathbf{b}\right\|^{2}}+\mathbf{C}.
  21. ( 𝐚 × 𝐛 ) × 𝐜 = ( 𝐚 𝐜 ) 𝐛 - ( 𝐛 𝐜 ) 𝐚 , (\mathbf{a}\times\mathbf{b})\times\mathbf{c}=(\mathbf{a}\cdot\mathbf{c})% \mathbf{b}-(\mathbf{b}\cdot\mathbf{c})\mathbf{a},
  22. 𝐚 × ( 𝐛 × 𝐜 ) = ( 𝐚 𝐜 ) 𝐛 - ( 𝐚 𝐛 ) 𝐜 , \mathbf{a}\times(\mathbf{b}\times\mathbf{c})=(\mathbf{a}\cdot\mathbf{c})% \mathbf{b}-(\mathbf{a}\cdot\mathbf{b})\mathbf{c},
  23. 𝐚 × 𝐛 = 𝐚 2 𝐛 2 - ( 𝐚 𝐛 ) 2 . \left\|\mathbf{a}\times\mathbf{b}\right\|=\sqrt{\mathbf{a}^{2}\mathbf{b}^{2}-(% \mathbf{a}\cdot\mathbf{b})^{2}}.
  24. U x = [ ( A x 2 + A y 2 ) ( B y - C y ) + ( B x 2 + B y 2 ) ( C y - A y ) + ( C x 2 + C y 2 ) ( A y - B y ) ] / D , U_{x}=\left[(A_{x}^{2}+A_{y}^{2})(B_{y}-C_{y})+(B_{x}^{2}+B_{y}^{2})(C_{y}-A_{% y})+(C_{x}^{2}+C_{y}^{2})(A_{y}-B_{y})\right]/D,
  25. U y = [ ( A x 2 + A y 2 ) ( C x - B x ) + ( B x 2 + B y 2 ) ( A x - C x ) + ( C x 2 + C y 2 ) ( B x - A x ) ] / D U_{y}=\left[(A_{x}^{2}+A_{y}^{2})(C_{x}-B_{x})+(B_{x}^{2}+B_{y}^{2})(A_{x}-C_{% x})+(C_{x}^{2}+C_{y}^{2})(B_{x}-A_{x})\right]/D
  26. D = 2 [ A x ( B y - C y ) + B x ( C y - A y ) + C x ( A y - B y ) ] . D=2\left[A_{x}(B_{y}-C_{y})+B_{x}(C_{y}-A_{y})+C_{x}(A_{y}-B_{y})\right].\,
  27. [ C y ( B x 2 + B y 2 ) - B y ( C x 2 + C y 2 ) ] / D , \left[C^{\prime}_{y}(B^{{}^{\prime}2}_{x}+B^{{}^{\prime}2}_{y})-B^{\prime}_{y}% (C^{{}^{\prime}2}_{x}+C^{{}^{\prime}2}_{y})\right]/D^{\prime},\,
  28. [ B x ( C x 2 + C y 2 ) - C x ( B x 2 + B y 2 ) ] / D \left[B^{\prime}_{x}(C^{{}^{\prime}2}_{x}+C^{{}^{\prime}2}_{y})-C^{\prime}_{x}% (B^{{}^{\prime}2}_{x}+B^{{}^{\prime}2}_{y})\right]/D^{\prime}\,
  29. D = 2 ( B x C y - B y C x ) . D^{\prime}=2(B^{\prime}_{x}C^{\prime}_{y}-B^{\prime}_{y}C^{\prime}_{x}).\,
  30. a ( b 2 + c 2 - a 2 ) : b ( c 2 + a 2 - b 2 ) : c ( a 2 + b 2 - c 2 ) . a(b^{2}+c^{2}-a^{2}):b(c^{2}+a^{2}-b^{2}):c(a^{2}+b^{2}-c^{2}).
  31. a 2 ( b 2 + c 2 - a 2 ) : b 2 ( c 2 + a 2 - b 2 ) : c 2 ( a 2 + b 2 - c 2 ) , a^{2}(b^{2}+c^{2}-a^{2}):\;b^{2}(c^{2}+a^{2}-b^{2}):\;c^{2}(a^{2}+b^{2}-c^{2}),\,
  32. α , β , γ , \alpha,\beta,\gamma,
  33. sin 2 α : sin 2 β : sin 2 γ . \sin 2\alpha:\sin 2\beta:\sin 2\gamma.
  34. U = a 2 ( b 2 + c 2 - a 2 ) A + b 2 ( c 2 + a 2 - b 2 ) B + c 2 ( a 2 + b 2 - c 2 ) C a 2 ( b 2 + c 2 - a 2 ) + b 2 ( c 2 + a 2 - b 2 ) + c 2 ( a 2 + b 2 - c 2 ) . U=\frac{a^{2}(b^{2}+c^{2}-a^{2})A+b^{2}(c^{2}+a^{2}-b^{2})B+c^{2}(a^{2}+b^{2}-% c^{2})C}{a^{2}(b^{2}+c^{2}-a^{2})+b^{2}(c^{2}+a^{2}-b^{2})+c^{2}(a^{2}+b^{2}-c% ^{2})}.
  35. P 1 = [ x 1 y 1 z 1 ] , P 2 = [ x 2 y 2 z 2 ] , P 3 = [ x 3 y 3 z 3 ] \mathrm{P_{1}}=\begin{bmatrix}x_{1}\\ y_{1}\\ z_{1}\end{bmatrix},\mathrm{P_{2}}=\begin{bmatrix}x_{2}\\ y_{2}\\ z_{2}\end{bmatrix},\mathrm{P_{3}}=\begin{bmatrix}x_{3}\\ y_{3}\\ z_{3}\end{bmatrix}
  36. r = | P 1 - P 2 | | P 2 - P 3 | | P 3 - P 1 | 2 | ( P 1 - P 2 ) × ( P 2 - P 3 ) | \mathrm{r}=\frac{\left|P_{1}-P_{2}\right|\left|P_{2}-P_{3}\right|\left|P_{3}-P% _{1}\right|}{2\left|\left(P_{1}-P_{2}\right)\times\left(P_{2}-P_{3}\right)% \right|}
  37. P c = α P 1 + β P 2 + γ P 3 \mathrm{P_{c}}=\alpha\,P_{1}+\beta\,P_{2}+\gamma\,P_{3}
  38. α = | P 2 - P 3 | 2 ( P 1 - P 2 ) ( P 1 - P 3 ) 2 | ( P 1 - P 2 ) × ( P 2 - P 3 ) | 2 \alpha=\frac{\left|P_{2}-P_{3}\right|^{2}\left(P_{1}-P_{2}\right)\cdot\left(P_% {1}-P_{3}\right)}{2\left|\left(P_{1}-P_{2}\right)\times\left(P_{2}-P_{3}\right% )\right|^{2}}
  39. β = | P 1 - P 3 | 2 ( P 2 - P 1 ) ( P 2 - P 3 ) 2 | ( P 1 - P 2 ) × ( P 2 - P 3 ) | 2 \beta=\frac{\left|P_{1}-P_{3}\right|^{2}\left(P_{2}-P_{1}\right)\cdot\left(P_{% 2}-P_{3}\right)}{2\left|\left(P_{1}-P_{2}\right)\times\left(P_{2}-P_{3}\right)% \right|^{2}}
  40. γ = | P 1 - P 2 | 2 ( P 3 - P 1 ) ( P 3 - P 2 ) 2 | ( P 1 - P 2 ) × ( P 2 - P 3 ) | 2 \gamma=\frac{\left|P_{1}-P_{2}\right|^{2}\left(P_{3}-P_{1}\right)\cdot\left(P_% {3}-P_{2}\right)}{2\left|\left(P_{1}-P_{2}\right)\times\left(P_{2}-P_{3}\right% )\right|^{2}}
  41. diameter = a b c 2 area = | A B | | B C | | C A | 2 | Δ A B C | = a b c 2 s ( s - a ) ( s - b ) ( s - c ) = 2 a b c ( a + b + c ) ( - a + b + c ) ( a - b + c ) ( a + b - c ) \begin{aligned}\displaystyle\,\text{diameter}&\displaystyle{}=\frac{abc}{2% \cdot\,\text{area}}=\frac{|AB||BC||CA|}{2|\Delta ABC|}\\ &\displaystyle{}=\frac{abc}{2\sqrt{s(s-a)(s-b)(s-c)}}\\ &\displaystyle{}=\frac{2abc}{\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}}\end{aligned}
  42. s ( s - a ) ( s - b ) ( s - c ) \sqrt{\scriptstyle{s(s-a)(s-b)(s-c)}}
  43. diameter = 2 area sin A sin B sin C . \,\text{diameter}=\sqrt{\frac{2\cdot\,\text{area}}{\sin A\sin B\sin C}}.
  44. O I = R ( R - 2 r ) , OI=\sqrt{R(R-2r)},
  45. O H = R 2 - 8 R 2 cos A cos B cos C = 9 R 2 - ( a 2 + b 2 + c 2 ) . OH=\sqrt{R^{2}-8R^{2}\cos A\cos B\cos C}=\sqrt{9R^{2}-(a^{2}+b^{2}+c^{2})}.
  46. I G < I O , IG<IO,
  47. 2 I N < I O , 2IN<IO,
  48. O I 2 = 2 R I N . OI^{2}=2R\cdot IN.
  49. r R = a b c 2 ( a + b + c ) . rR=\frac{abc}{2(a+b+c)}.
  50. 3 3 R a + b + c 3\sqrt{3}R\geq a+b+c
  51. 9 R 2 a 2 + b 2 + c 2 9R^{2}\geq a^{2}+b^{2}+c^{2}
  52. 27 4 R 2 m a 2 + m b 2 + m c 2 . \frac{27}{4}R^{2}\geq m_{a}^{2}+m_{b}^{2}+m_{c}^{2}.
  53. 4 R 2 h 2 ( t 2 - h 2 ) = t 4 ( m 2 - h 2 ) . 4R^{2}h^{2}(t^{2}-h^{2})=t^{4}(m^{2}-h^{2}).
  54. O I = R ( R - 2 r ) . OI=\sqrt{R(R-2r)}.
  55. M A 1 + M A 3 + + M A n - 2 + M A n < n 2 if n is odd ; MA_{1}+MA_{3}+\dots+MA_{n-2}+MA_{n}<\frac{n}{\sqrt{2}}\quad\,\text{if}\,n\,\,% \text{is odd};
  56. M A 1 + M A 3 + + M A n - 3 + M A n - 1 n 2 if n is even . MA_{1}+MA_{3}+\dots+MA_{n-3}+MA_{n-1}\leq\frac{n}{\sqrt{2}}\quad\,\text{if}\,n% \,\,\text{is even}.
  57. n 3 1 / cos ( π / n ) = 8.7000366.. \prod_{n\geq 3}1/\cos(\pi/n)=8.7000366..

Clairaut's_theorem.html

  1. g = G [ 1 + ( 5 2 m - f ) sin 2 φ ] , g=G\left[1+\left(\frac{5}{2}m-f\right)\sin^{2}\varphi\right]\ ,
  2. f = a - b a , f=\frac{a-b}{a}\ ,
  3. g = G [ 1 + k sin 2 ϕ 1 - e 2 sin 2 ϕ ] , g=G\left[\frac{1+k\sin^{2}\phi}{\sqrt{1-e^{2}\sin^{2}\phi}}\right]\ ,

Clapp_oscillator.html

  1. f 0 = 1 2 π 1 L ( 1 C 0 + 1 C 1 + 1 C 2 ) . f_{0}={1\over 2\pi}\sqrt{{1\over L}\left({1\over C_{0}}+{1\over C_{1}}+{1\over C% _{2}}\right)}\ .

Classical_electron_radius.html

  1. r e = 1 4 π ε 0 e 2 m e c 2 = 2.8179403267 ( 27 ) × 10 - 15 m , r\text{e}=\frac{1}{4\pi\varepsilon_{0}}\frac{e^{2}}{m_{\,\text{e}}c^{2}}=2.817% 9403267(27)\times 10^{-15}\,\text{ m},
  2. e e
  3. m e m_{\,\text{e}}
  4. c c
  5. ε 0 \varepsilon_{0}
  6. r e = e 2 m e c 2 = 2.8179403267 ( 27 ) × 10 - 13 cm r\text{e}=\frac{e^{2}}{m\text{e}c^{2}}=2.8179403267(27)\times 10^{-13}\,\text{% cm}
  7. e = 4.80 × 10 - 10 esu , m e = 9.11 × 10 - 28 g , c = 3.00 × 10 10 cm/s . e=4.80\times 10^{-10}\,\text{ esu},\quad m\text{e}=9.11\times 10^{-28}\,\text{% g},\quad c=3.00\times 10^{10}\,\text{ cm/s}.
  8. r e r\text{e}
  9. e e
  10. E = 3 5 1 4 π ε 0 e 2 r e . E=\frac{3}{5}\,\,\frac{1}{4\pi\varepsilon_{0}}\frac{e^{2}}{r\text{e}}.
  11. E = 1 2 1 4 π ε 0 e 2 r e . E=\frac{1}{2}\,\,\frac{1}{4\pi\varepsilon_{0}}\frac{e^{2}}{r\text{e}}.
  12. ρ ( r ) = q 4 π * R * r 2 . \rho(r)=\frac{q}{4\pi*R*r^{2}}.
  13. E = 1 4 π ε 0 e 2 r e . E=\frac{1}{4\pi\varepsilon_{0}}\frac{e^{2}}{r\text{e}}.
  14. E = m c 2 E=mc^{2}
  15. r e r\text{e}
  16. a 0 a_{0}
  17. λ e \lambda\text{e}
  18. m e m\text{e}
  19. c c
  20. e e
  21. m e m\text{e}
  22. e e
  23. h h
  24. m e m\text{e}
  25. h h
  26. c c
  27. α \alpha
  28. r e = α λ e 2 π = α 2 a 0 . r\text{e}={\alpha\lambda\text{e}\over 2\pi}=\alpha^{2}a_{0}.
  29. r = k e q 2 m c 2 , r=\frac{k\text{e}q^{2}}{mc^{2}},
  30. k e k\text{e}
  31. q q
  32. m m
  33. α \alpha
  34. \hbar

Classical_field_theory.html

  1. 𝐠 ( 𝐫 ) = 𝐅 ( 𝐫 ) m . \mathbf{g}(\mathbf{r})=\frac{\mathbf{F}(\mathbf{r})}{m}.
  2. 𝐅 ( 𝐫 ) = - G M m r 2 𝐫 ^ , \mathbf{F}(\mathbf{r})=-\frac{GMm}{r^{2}}\hat{\mathbf{r}},
  3. 𝐫 ^ \hat{\mathbf{r}}
  4. 𝐠 ( 𝐫 ) = 𝐅 ( 𝐫 ) m = - G M r 2 𝐫 ^ . \mathbf{g}(\mathbf{r})=\frac{\mathbf{F}(\mathbf{r})}{m}=-\frac{GM}{r^{2}}\hat{% \mathbf{r}}.
  5. 𝐠 ( 𝐫 ) = - Φ ( 𝐫 ) . \mathbf{g}(\mathbf{r})=-\nabla\Phi(\mathbf{r}).
  6. 𝐄 = 1 4 π ϵ 0 q r 2 𝐫 ^ . \mathbf{E}=\frac{1}{4\pi\epsilon_{0}}\frac{q}{r^{2}}\hat{\mathbf{r}}.
  7. 𝐄 ( 𝐫 ) = - V ( 𝐫 ) . \mathbf{E}(\mathbf{r})=-\nabla V(\mathbf{r}).
  8. 𝐅 ( 𝐫 ) = q 𝐯 × 𝐁 ( 𝐫 ) , \mathbf{F}(\mathbf{r})=q\mathbf{v}\times\mathbf{B}(\mathbf{r}),
  9. 𝐁 ( 𝐫 ) = μ 0 I 4 π d s y m b o l × d 𝐫 ^ r 2 . \mathbf{B}(\mathbf{r})=\frac{\mu_{0}I}{4\pi}\int\frac{dsymbol{\ell}\times d% \hat{\mathbf{r}}}{r^{2}}.
  10. 𝐁 ( 𝐫 ) = s y m b o l × 𝐀 ( 𝐫 ) \mathbf{B}(\mathbf{r})=symbol{\nabla}\times\mathbf{A}(\mathbf{r})
  11. 𝐄 = - s y m b o l V - 𝐀 t \mathbf{E}=-symbol{\nabla}V-\frac{\partial\mathbf{A}}{\partial t}
  12. 𝐁 = s y m b o l × 𝐀 . \mathbf{B}=symbol{\nabla}\times\mathbf{A}.
  13. u ˙ = 𝐅 - p ρ \dot{u}=\mathbf{F}-{\nabla p\over\rho}
  14. ρ ˙ + ( ρ u ) = 0 \dot{\rho}+\nabla\cdot(\rho u)=0
  15. c c
  16. ϕ \phi
  17. ( ϕ , ϕ , ϕ , , x ) \mathcal{L}(\phi,\partial\phi,\partial\partial\phi,...,x)
  18. ϕ \phi
  19. 𝒮 = d 4 x . \mathcal{S}=\int{\mathcal{L}\mathrm{d}^{4}x}.
  20. δ 𝒮 δ ϕ = ϕ - μ ( ( μ ϕ ) ) + . . . + ( - 1 ) m μ 1 μ 2 . . . μ m - 1 μ m ( ( μ 1 μ 2 μ m - 1 μ m ϕ ) ) = 0. \frac{\delta\mathcal{S}}{\delta\phi}=\frac{\partial\mathcal{L}}{\partial\phi}-% \partial_{\mu}\left(\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\phi)}% \right)+.~{}.~{}.+(-1)^{m}\partial_{\mu_{1}}\partial_{\mu_{2}}.~{}.~{}.% \partial_{\mu_{m-1}}\partial_{\mu_{m}}\left(\frac{\partial\mathcal{L}}{% \partial(\partial_{\mu_{1}}\partial_{\mu_{2}}...\partial_{\mu_{m-1}}\partial_{% \mu_{m}}\phi)}\right)=0.
  21. A a = ( - ϕ , A ) A_{a}=\left(-\phi,\vec{A}\right)
  22. j a = ( - ρ , j ) j_{a}=\left(-\rho,\vec{j}\right)
  23. F a b = a A b - b A a . F_{ab}=\partial_{a}A_{b}-\partial_{b}A_{a}.
  24. = - 1 4 μ 0 F a b F a b . \mathcal{L}=\frac{-1}{4\mu_{0}}F^{ab}F_{ab}.
  25. = - 1 4 μ 0 F a b F a b + j a A a . \mathcal{L}=\frac{-1}{4\mu_{0}}F^{ab}F_{ab}+j^{a}A_{a}.
  26. b ( ( b A a ) ) = A a . \partial_{b}\left(\frac{\partial\mathcal{L}}{\partial\left(\partial_{b}A_{a}% \right)}\right)=\frac{\partial\mathcal{L}}{\partial A_{a}}.
  27. / A a = μ 0 j a \partial\mathcal{L}/\partial A_{a}=\mu_{0}j^{a}
  28. F a b F^{ab}
  29. / ( b A a ) = F a b \partial\mathcal{L}/\partial(\partial_{b}A_{a})=F^{ab}
  30. b F a b = μ 0 j a . \partial_{b}F^{ab}=\mu_{0}j^{a}.
  31. F F
  32. A : A:
  33. 6 F [ a b , c ] = F a b , c + F c a , b + F b c , a = 0. 6F_{[ab,c]}\,=F_{ab,c}+F_{ca,b}+F_{bc,a}=0.
  34. = R - g \mathcal{L}=\,R\sqrt{-g}
  35. R = R a b g a b R\,=R_{ab}g^{ab}
  36. R a b \,R_{ab}
  37. g a b \,g_{ab}
  38. G a b = 0 G_{ab}\,=0
  39. G a b = R a b - R 2 g a b G_{ab}\,=R_{ab}-\frac{R}{2}g_{ab}
  40. c = 1 c=1
  41. E = m c 2 E=mc^{2}
  42. E = m E=m
  43. c 2 = 1 , c^{2}=1,

Classical_Heisenberg_model.html

  1. n = 3 n=3
  2. s i 3 , | s i | = 1 ( 1 ) \vec{s}_{i}\in\mathbb{R}^{3},|\vec{s}_{i}|=1\quad(1)
  3. = - i , j 𝒥 i j s i s j ( 2 ) \mathcal{H}=-\sum_{i,j}\mathcal{J}_{ij}\vec{s}_{i}\cdot\vec{s}_{j}\quad(2)
  4. 𝒥 i j = { J if i , j are neighbors 0 else. \mathcal{J}_{ij}=\begin{cases}J&\mbox{if }~{}i,j\mbox{ are neighbors}\\ 0&\mbox{else.}\end{cases}
  5. S t = S S x x . \vec{S}_{t}=\vec{S}\wedge\vec{S}_{xx}.
  6. J x , y | x - y | - α J_{x,y}\sim|x-y|^{-\alpha}
  7. α > 1 \alpha>1
  8. α 2 \alpha\geq 2
  9. 1 < α < 2 1<\alpha<2
  10. J x , y | x - y | - α J_{x,y}\sim|x-y|^{-\alpha}
  11. α > 2 \alpha>2
  12. α 4 \alpha\geq 4
  13. 2 < α < 4 2<\alpha<4
  14. T > 0 T>0

Clausius–Clapeyron_relation.html

  1. d P d T = L T Δ v = Δ s Δ v , \frac{\mathrm{d}P}{\mathrm{d}T}=\frac{L}{T\,\Delta v}=\frac{\Delta s}{\Delta v},
  2. d P / d T \mathrm{d}P/\mathrm{d}T
  3. L L
  4. T T
  5. Δ v \Delta v
  6. Δ s \Delta s
  7. s s
  8. v v
  9. T T
  10. d s = ( s v ) T d v + ( s T ) v d T . \mathrm{d}s=\left(\frac{\partial s}{\partial v}\right)_{T}\mathrm{d}v+\left(% \frac{\partial s}{\partial T}\right)_{v}\mathrm{d}T.
  11. d s = ( s v ) T d v . \mathrm{d}s=\left(\frac{\partial s}{\partial v}\right)_{T}\mathrm{d}v.
  12. d s = ( P T ) v d v . \mathrm{d}s=\left(\frac{\partial P}{\partial T}\right)_{v}\mathrm{d}v.
  13. P P
  14. Δ s = d P d T Δ v {\Delta s}=\frac{\mathrm{d}P}{\mathrm{d}T}{\Delta v}
  15. α \alpha
  16. β \beta
  17. d P d T = Δ s Δ v \frac{dP}{dT}=\frac{\Delta s}{\Delta v}
  18. Δ s s β - s α \Delta s\equiv s_{\beta}-s_{\alpha}
  19. Δ v v β - v α \Delta v\equiv v_{\beta}-v_{\alpha}
  20. d u = δ q - δ w = T d s - P d v \mathrm{d}u=\delta q-\delta w=T\;\mathrm{d}s-P\;\mathrm{d}v
  21. u u
  22. h h
  23. d h = d u + P d v dh=\mathrm{d}u+P\;\mathrm{d}v
  24. d h = T d s dh=T\;\mathrm{d}s
  25. d s = d h T \mathrm{d}s=\frac{\mathrm{d}h}{T}
  26. Δ s = Δ h T \Delta s=\frac{\Delta h}{T}
  27. L = Δ h L=\Delta h
  28. Δ s = L T \Delta s=\frac{L}{T}
  29. d P / d T = Δ s / Δ v \mathrm{d}P/\mathrm{d}T=\mathrm{\Delta s}/\mathrm{\Delta v}
  30. d P d T = L T Δ v . \frac{\mathrm{d}P}{\mathrm{d}T}=\frac{L}{T\Delta v}.
  31. d P / d T \mathrm{d}P/\mathrm{d}T
  32. L / T Δ v {L}/{T{\Delta v}}
  33. L L
  34. T T
  35. Δ v \Delta v
  36. α \alpha
  37. β \beta
  38. μ α = μ β . \mu_{\alpha}=\mu_{\beta}.
  39. d μ α = d μ β . \mathrm{d}\mu_{\alpha}=\mathrm{d}\mu_{\beta}.
  40. d μ = M ( - s d T + v d P ) \mathrm{d}\mu=M(-s\mathrm{d}T+v\mathrm{d}P)
  41. s s
  42. v v
  43. M M
  44. - ( s β - s α ) d T + ( v β - v α ) d P = 0 -(s_{\beta}-s_{\alpha})\mathrm{d}T+(v_{\beta}-v_{\alpha})\mathrm{d}P=0
  45. d P d T = s β - s α v β - v α = Δ s Δ v \frac{\mathrm{d}P}{\mathrm{d}T}=\frac{s_{\beta}-s_{\alpha}}{v_{\beta}-v_{% \alpha}}=\frac{\Delta s}{\Delta v}
  46. v g v_{\mathrm{g}}
  47. v c v_{\mathrm{c}}
  48. Δ v = v g ( 1 - v c v g ) v g \Delta v=v_{\mathrm{g}}\left(1-\tfrac{v_{\mathrm{c}}}{v_{\mathrm{g}}}\right)% \approx v_{\mathrm{g}}
  49. v g = R T / P v_{\mathrm{g}}=RT/P
  50. P P
  51. R R
  52. T T
  53. d P d T = Δ s Δ v \frac{\mathrm{d}P}{\mathrm{d}T}=\frac{\Delta s}{\Delta v}
  54. d P d T = P L T 2 R . \frac{\mathrm{d}P}{\mathrm{d}T}=\frac{PL}{T^{2}R}.
  55. L L
  56. ( P 1 , T 1 ) (P_{1},T_{1})
  57. ( P 2 , T 2 ) (P_{2},T_{2})
  58. α \alpha
  59. β \beta
  60. L L
  61. L L
  62. d P P = L R d T T 2 , \frac{\mathrm{d}P}{P}=\frac{L}{R}\frac{\mathrm{d}T}{T^{2}},
  63. P 1 P 2 d P P = L R d T T 2 \int_{P_{1}}^{P_{2}}\frac{\mathrm{d}P}{P}=\frac{L}{R}\int\frac{\mathrm{d}T}{T^% {2}}
  64. ln P | P = P 1 P 2 = - L R 1 T | T = T 1 T 2 \left.\ln P\right|_{P=P_{1}}^{P_{2}}=-\frac{L}{R}\cdot\left.\frac{1}{T}\right|% _{T=T_{1}}^{T_{2}}
  65. ln P 1 P 2 = - L R ( 1 T 1 - 1 T 2 ) \ln\frac{P_{1}}{P_{2}}=-\frac{L}{R}\left(\frac{1}{T_{1}}-\frac{1}{T_{2}}\right)
  66. ln P = - L R ( 1 T ) + c \ln P=-\frac{L}{R}\left(\frac{1}{T}\right)+c
  67. c c
  68. L L
  69. L L
  70. ln P \ln P
  71. 1 / T 1/T
  72. d e s d T = L v ( T ) e s R v T 2 \frac{\mathrm{d}e_{s}}{\mathrm{d}T}=\frac{L_{v}(T)e_{s}}{R_{v}T^{2}}
  73. e s e_{s}
  74. T T
  75. L v L_{v}
  76. R v R_{v}
  77. L v ( T ) L_{v}(T)
  78. e s ( T ) e_{s}(T)
  79. e s ( T ) = 6.1094 exp ( 17.625 T T + 243.04 ) e_{s}(T)=6.1094\exp\left(\frac{17.625T}{T+243.04}\right)
  80. T T
  81. Δ T {\Delta T}
  82. Δ P = L T Δ v Δ T {\Delta P}=\frac{L}{T\,\Delta v}{\Delta T}
  83. L L
  84. × 10 5 \times 10^{5}
  85. T T
  86. Δ v \Delta v
  87. × 10 5 \times 10^{−}5
  88. Δ P Δ T \frac{\Delta P}{\Delta T}
  89. d 2 P d T 2 = 1 v 2 - v 1 [ c p 2 - c p 1 T - 2 ( v 2 α 2 - v 1 α 1 ) d P d T + ( v 2 κ T 2 - v 1 κ T 1 ) ( d P d T ) 2 ] , \frac{\mathrm{d}^{2}P}{\mathrm{d}T^{2}}=\frac{1}{v_{2}-v_{1}}\left[\frac{c_{p2% }-c_{p1}}{T}-2(v_{2}\alpha_{2}-v_{1}\alpha_{1})\frac{\mathrm{d}P}{\mathrm{d}T}% +(v_{2}\kappa_{T2}-v_{1}\kappa_{T1})\left(\frac{\mathrm{d}P}{\mathrm{d}T}% \right)^{2}\right],
  90. c p c_{p}
  91. α = ( 1 / v ) ( d v / d T ) P \alpha=(1/v)(\mathrm{d}v/\mathrm{d}T)_{P}
  92. κ T = - ( 1 / v ) ( d v / d P ) T \kappa_{T}=-(1/v)(\mathrm{d}v/\mathrm{d}P)_{T}

Climate_sensitivity.html

  1. Δ T s = λ R F \Delta T_{s}=\lambda\cdot RF

Clock_skew.html

  1. T r e g + p a t h m a x + J + S - ( s d - s s ) T\geq reg+path_{max}+J+S-(s_{d}-s_{s})
  2. ( s d - s s ) r e g + p a t h m i n - J - H (s_{d}-s_{s})\leq reg+path_{min}-J-H
  3. p a t h m a x path_{max}
  4. ( s d - s s ) (s_{d}-s_{s})
  5. p a t h m i n path_{min}
  6. s d s_{d}
  7. s s s_{s}

Closed_category.html

  1. [ - - ] : V o p × V V \left[-\ -\right]:V^{op}\times V\to V
  2. B B
  3. C C
  4. A A
  5. L : [ B C ] [ [ A B ] [ A C ] ] L:\left[B\ C\right]\to\left[\left[A\ B\right]\left[A\ C\right]\right]
  6. i A : A [ I A ] i_{A}:A\cong\left[I\ A\right]
  7. j A : I [ A A ] . j_{A}:I\to\left[A\ A\right].\,
  8. I I

Closed_immersion.html

  1. f : Z X f:Z\to X
  2. f # : 𝒪 X f 𝒪 Z f^{\#}:\mathcal{O}_{X}\rightarrow f_{\ast}\mathcal{O}_{Z}
  3. Spec ( R / I ) Spec ( R ) \operatorname{Spec}(R/I)\to\operatorname{Spec}(R)
  4. R R / I R\to R/I
  5. f : Z X f:Z\to X
  6. U = Spec ( R ) X U=\operatorname{Spec}(R)\subset X
  7. I R I\subset R
  8. f - 1 ( U ) = Spec ( R / I ) f^{-1}(U)=\operatorname{Spec}(R/I)
  9. X = U j , U j = Spec R j X=\bigcup U_{j},U_{j}=\operatorname{Spec}R_{j}
  10. I j R j I_{j}\subset R_{j}
  11. f - 1 ( U j ) = Spec ( R j / I j ) f^{-1}(U_{j})=\operatorname{Spec}(R_{j}/I_{j})
  12. U j U_{j}
  13. \mathcal{I}
  14. f 𝒪 Z 𝒪 X / f_{\ast}\mathcal{O}_{Z}\cong\mathcal{O}_{X}/\mathcal{I}
  15. 𝒪 X / \mathcal{O}_{X}/\mathcal{I}
  16. X = U j X=\bigcup U_{j}
  17. f : f - 1 ( U j ) U j f:f^{-1}(U_{j})\rightarrow U_{j}
  18. Z Y X Z\to Y\to X
  19. Y X Y\to X
  20. Z Y Z\to Y
  21. i : Z X i:Z\to X
  22. 𝒪 X \mathcal{I}\subset\mathcal{O}_{X}
  23. i * i_{*}
  24. 𝒢 \mathcal{G}
  25. 𝒢 = 0 \mathcal{I}\mathcal{G}=0

Closed_monoidal_category.html

  1. A A
  2. B B
  3. A × B A\times B
  4. B A B^{A}
  5. A A
  6. B B
  7. 𝒞 \mathcal{C}
  8. B B
  9. B B
  10. A A B A\mapsto A\otimes B
  11. A ( B A ) . A\mapsto(B\Rightarrow A).
  12. Hom 𝒞 ( A B , C ) Hom 𝒞 ( A , B C ) \,\text{Hom}_{\mathcal{C}}(A\otimes B,C)\cong\,\text{Hom}_{\mathcal{C}}(A,B% \Rightarrow C)
  13. - B : 𝒞 𝒞 -\otimes B:\mathcal{C}\to\mathcal{C}
  14. [ B , - ] : 𝒞 𝒞 [B,-]:\mathcal{C}\to\mathcal{C}
  15. 𝒞 \mathcal{C}
  16. A B A\Rightarrow B
  17. eval A , B : ( A B ) A B \mathrm{eval}_{A,B}:(A\Rightarrow B)\otimes A\to B
  18. f : X A B f:X\otimes A\to B
  19. h : X A B h:X\to A\Rightarrow B
  20. f = eval A , B ( h id A ) . f=\mathrm{eval}_{A,B}\circ(h\otimes\mathrm{id}_{A}).
  21. : 𝒞 o p 𝒞 𝒞 \Rightarrow:\mathcal{C}^{op}\otimes\mathcal{C}\to\mathcal{C}
  22. A B A\Rightarrow B
  23. A A
  24. B B
  25. 𝒞 \mathcal{C}
  26. B A B^{A}
  27. A A
  28. A A
  29. B A B B\mapsto A\otimes B
  30. B ( B A ) B\mapsto(B\Leftarrow A)
  31. A B A\otimes B
  32. B A B\otimes A
  33. A B A\Rightarrow B
  34. A A
  35. B B
  36. A B A\Rightarrow B
  37. B A B^{A}
  38. A B A\Rightarrow B
  39. A A
  40. B B
  41. A B A\Rightarrow B
  42. B A * B\otimes A^{*}

Closeness_(mathematics).html

  1. ( X , d ) (X,d)
  2. p p
  3. A A
  4. d ( p , A ) = 0 d(p,A)=0
  5. d ( p , A ) := inf a A d ( p , a ) d(p,A):=\inf_{a\in A}d(p,a)
  6. B B
  7. A A
  8. d ( B , A ) = 0 d(B,A)=0
  9. d ( B , A ) := inf b B d ( b , A ) d(B,A):=\inf_{b\in B}d(b,A)
  10. p p
  11. A A
  12. B B
  13. A A
  14. B B
  15. A A
  16. B B
  17. p p
  18. p p
  19. A A
  20. A A\neq\emptyset
  21. p p
  22. A A
  23. B A B\supset A
  24. p p
  25. B B
  26. p p
  27. A B A\cup B
  28. p p
  29. A A
  30. p p
  31. B B
  32. A A
  33. B B
  34. C C
  35. A A
  36. B B
  37. A A\neq\emptyset
  38. B B\neq\emptyset
  39. A A
  40. B B
  41. B B
  42. A A
  43. A A
  44. B B
  45. B C B\subset C
  46. A A
  47. C C
  48. A A
  49. B C B\cup C
  50. A A
  51. B B
  52. A A
  53. C C
  54. A B A\cap B\neq\emptyset
  55. A A
  56. B B
  57. p p
  58. p p
  59. A A
  60. p cl ( A ) = A ¯ p\in\operatorname{cl}(A)=\overline{A}

Clover_(telescope).html

  1. r = 0.01 r=0.01

Club_filter.html

  1. κ \kappa
  2. club ( κ ) \operatorname{club}(\kappa)
  3. κ \kappa
  4. κ \kappa
  5. κ club ( κ ) \kappa\in\operatorname{club}(\kappa)
  6. x club ( κ ) x\in\operatorname{club}(\kappa)
  7. κ \kappa
  8. x x
  9. club ( κ ) \operatorname{club}(\kappa)
  10. x x
  11. κ \kappa
  12. κ \kappa
  13. C i i < α \langle C_{i}\rangle_{i<\alpha}
  14. α < κ \alpha<\kappa
  15. C = C i C=\bigcap C_{i}
  16. C C
  17. C i C_{i}
  18. C i C_{i}
  19. β < κ \beta<\kappa
  20. β 1 , i \langle\beta_{1,i}\rangle
  21. β 1 , 1 > β \beta_{1,1}>\beta
  22. β 1 , i C i \beta_{1,i}\in C_{i}
  23. i < α i<\alpha
  24. C i C_{i}
  25. α < κ \alpha<\kappa
  26. κ \kappa
  27. κ \kappa
  28. β 2 \beta_{2}
  29. β 2 , i \langle\beta_{2,i}\rangle
  30. β j , i \langle\beta_{j,i}\rangle
  31. i < α i<\alpha
  32. β j , i \langle\beta_{j,i}\rangle
  33. C i C_{i}
  34. β j , i \langle\beta_{j,i}\rangle
  35. C i C_{i}
  36. C C
  37. β \beta
  38. club ( κ ) \operatorname{club}(\kappa)
  39. C i \langle C_{i}\rangle
  40. i < κ i<\kappa
  41. C = Δ i < κ C i C=\Delta_{i<\kappa}C_{i}
  42. C C
  43. S α < κ S\subseteq\alpha<\kappa
  44. S = α \bigcup S=\alpha
  45. γ S \gamma\in S
  46. γ C β \gamma\in C_{\beta}
  47. β < γ \beta<\gamma
  48. C β C_{\beta}
  49. α C β \alpha\in C_{\beta}
  50. β < α \beta<\alpha
  51. α C \alpha\in C
  52. C C
  53. α < κ \alpha<\kappa
  54. ξ i \xi_{i}
  55. i < ω i<\omega
  56. ξ 0 = α \xi_{0}=\alpha
  57. ξ i + 1 \xi_{i+1}
  58. γ < ξ i C γ \bigcap_{\gamma<\xi_{i}}C_{\gamma}
  59. ξ i + 1 > ξ i \xi_{i+1}>\xi_{i}
  60. ξ i \xi_{i}
  61. ξ = i < ω ξ i > α \xi=\bigcup_{i<\omega}\xi_{i}>\alpha
  62. ξ C \xi\in C
  63. C i C_{i}
  64. i < ξ i<\xi

Club_set.html

  1. κ \kappa
  2. C κ C\subseteq\kappa
  3. κ \kappa
  4. α < κ \alpha<\kappa
  5. sup ( C α ) = α 0 \sup(C\cap\alpha)=\alpha\neq 0
  6. α C \alpha\in C
  7. C C
  8. κ \kappa
  9. C C
  10. κ \kappa
  11. C κ C\subseteq\kappa
  12. C C
  13. κ \kappa
  14. α < κ \alpha<\kappa
  15. β C \beta\in C
  16. α < β \alpha<\beta
  17. α < κ \alpha<\kappa
  18. κ \kappa
  19. κ \kappa
  20. X X
  21. λ \lambda
  22. C [ X ] λ C\subseteq[X]^{\lambda}
  23. C C
  24. C C
  25. X X
  26. λ \lambda
  27. C C
  28. κ \kappa\,
  29. λ . \lambda\,.
  30. α < λ \alpha<\lambda\,
  31. C ξ : ξ < α \langle C_{\xi}:\xi<\alpha\rangle\,
  32. κ . \kappa\,.
  33. ξ < α C ξ \bigcap_{\xi<\alpha}C_{\xi}\,
  34. β 0 < κ , \beta_{0}<\kappa\,,
  35. β n + 1 ξ > β n , \beta_{n+1}^{\xi}>\beta_{n}\,,
  36. λ \lambda\,
  37. κ , \kappa\,,
  38. κ , \kappa\,,
  39. β n + 1 . \beta_{n+1}\,.
  40. β 0 , β 1 , β 2 , . \beta_{0},\beta_{1},\beta_{2},\dots\,.
  41. β 0 ξ , β 1 ξ , β 2 ξ , , \beta_{0}^{\xi},\beta_{1}^{\xi},\beta_{2}^{\xi},\dots\,,
  42. C ξ C_{\xi}\,
  43. λ \lambda\,
  44. C ξ , C_{\xi}\,,
  45. β 0 , \beta_{0}\,,
  46. κ \kappa\,
  47. { S κ : C S such that C is closed unbounded in κ } \{S\subset\kappa:\exists C\subset S\,\text{ such that }C\,\text{ is closed % unbounded in }\kappa\}\,
  48. κ \kappa\,
  49. κ . \kappa\,.
  50. κ \kappa\,
  51. κ \kappa\,
  52. \mathcal{F}\,
  53. κ , \kappa\,,
  54. { ξ < κ : ξ α } \{\xi<\kappa:\xi\geq\alpha\}\,
  55. α < κ , \alpha<\kappa\,,
  56. \mathcal{F}\,

Cluster_chemistry.html

  1. E = Σ i Σ j ( A e - B r i j - C r i j - q i q j r i j - 1 ) E=\Sigma_{i}\Sigma_{j}(Ae^{-Br_{ij}}-Cr_{ij}-\frac{q_{i}q_{j}}{r_{ij}^{-1}})

Clustering_coefficient.html

  1. C = 3 × number of triangles number of connected triplets of vertices = number of closed triplets number of connected triplets of vertices . C=\frac{3\times\mbox{number of triangles}~{}}{\mbox{number of connected % triplets of vertices}~{}}=\frac{\mbox{number of closed triplets}~{}}{\mbox{% number of connected triplets of vertices}~{}}.
  2. G = ( V , E ) G=(V,E)
  3. V V
  4. E E
  5. e i j e_{ij}
  6. v i v_{i}
  7. v j v_{j}
  8. N i N_{i}
  9. v i v_{i}
  10. N i = { v j : e i j E and e j i E } . N_{i}=\{v_{j}:e_{ij}\in E\and e_{ji}\in E\}.
  11. k i k_{i}
  12. | N i | |N_{i}|
  13. N i N_{i}
  14. C i C_{i}
  15. v i v_{i}
  16. e i j e_{ij}
  17. e j i e_{ji}
  18. N i N_{i}
  19. k i ( k i - 1 ) k_{i}(k_{i}-1)
  20. k i k_{i}
  21. C i = | { e j k : v j , v k N i , e j k E } | k i ( k i - 1 ) . C_{i}=\frac{|\{e_{jk}:v_{j},v_{k}\in N_{i},e_{jk}\in E\}|}{k_{i}(k_{i}-1)}.
  22. e i j e_{ij}
  23. e j i e_{ji}
  24. v i v_{i}
  25. k i k_{i}
  26. k i ( k i - 1 ) 2 \frac{k_{i}(k_{i}-1)}{2}
  27. C i = 2 | { e j k : v j , v k N i , e j k E } | k i ( k i - 1 ) . C_{i}=\frac{2|\{e_{jk}:v_{j},v_{k}\in N_{i},e_{jk}\in E\}|}{k_{i}(k_{i}-1)}.
  28. λ G ( v ) \lambda_{G}(v)
  29. v V ( G ) v\in V(G)
  30. G G
  31. λ G ( v ) \lambda_{G}(v)
  32. G G
  33. v v
  34. τ G ( v ) \tau_{G}(v)
  35. v G v\in G
  36. τ G ( v ) \tau_{G}(v)
  37. v v
  38. v v
  39. C i = λ G ( v ) τ G ( v ) . C_{i}=\frac{\lambda_{G}(v)}{\tau_{G}(v)}.
  40. τ G ( v ) = C ( k i , 2 ) = 1 2 k i ( k i - 1 ) . \tau_{G}(v)=C({k_{i}},2)=\frac{1}{2}k_{i}(k_{i}-1).
  41. v i v_{i}
  42. v i v_{i}
  43. v i v_{i}
  44. n n
  45. C ¯ = 1 n i = 1 n C i . \bar{C}=\frac{1}{n}\sum_{i=1}^{n}C_{i}.
  46. k i ( k i - 1 ) k_{i}(k_{i}-1)
  47. C ¯ \bar{C}

Cobweb_model.html

  1. d P S d Q S > | d P D d Q D | . \frac{dP^{S}}{dQ^{S}}>\left|\frac{dP^{D}}{dQ^{D}}\right|.
  2. d Q S / Q S d P S / P S \frac{dQ^{S}/Q^{S}}{dP^{S}/P^{S}}
  3. d Q D / Q D d P D / P D \frac{dQ^{D}/Q^{D}}{dP^{D}/P^{D}}
  4. P S = P D = P > 0 P^{S}=P^{D}=P>0
  5. Q S = Q D = Q > 0 Q^{S}=Q^{D}=Q>0
  6. d Q S / Q d P S / P < | d Q D / Q d P D / P | , \frac{dQ^{S}/Q}{dP^{S}/P}<\left|\frac{dQ^{D}/Q}{dP^{D}/P}\right|,
  7. d Q S / Q d P S / P > | d Q D / Q d P D / P | . \frac{dQ^{S}/Q}{dP^{S}/P}>\left|\frac{dQ^{D}/Q}{dP^{D}/P}\right|.

Cobweb_plot.html

  1. x 0 x_{0}
  2. x 0 x_{0}
  3. x 0 , f ( x 0 ) x_{0},f(x_{0})
  4. f ( x 0 ) , f ( x 0 ) f(x_{0}),f(x_{0})
  5. f ( x 0 ) , f ( f ( x 0 ) ) f(x_{0}),f(f(x_{0}))

Code-excited_linear_prediction.html

  1. e [ n ] = e a [ n ] + e f [ n ] e[n]=e_{a}[n]+e_{f}[n]\,
  2. e a [ n ] e_{a}[n]
  3. e f [ n ] e_{f}[n]
  4. 1 / A ( z ) 1/A(z)
  5. A ( z ) A(z)
  6. W ( z ) = A ( z / γ 1 ) A ( z / γ 2 ) W(z)=\frac{A(z/\gamma_{1})}{A(z/\gamma_{2})}
  7. γ 1 > γ 2 \gamma_{1}>\gamma_{2}

Codex_Alexandrinus.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}
  4. 𝔓 \mathfrak{P}
  5. 𝔓 \mathfrak{P}
  6. 𝔓 \mathfrak{P}
  7. 𝔓 \mathfrak{P}
  8. 𝔓 \mathfrak{P}

Codex_Claromontanus.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}

Codex_Ephraemi_Rescriptus.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}
  4. 𝔓 \mathfrak{P}

Coefficient_of_determination.html

  1. R 2 = 1 - \color b l u e S S res \color r e d S S tot R^{2}=1-\frac{\color{blue}{SS\text{res}}}{\color{red}{SS\text{tot}}}
  2. R 2 R^{2}
  3. y ¯ \bar{y}
  4. y ¯ = 1 n i = 1 n y i \bar{y}=\frac{1}{n}\sum_{i=1}^{n}y_{i}
  5. S S tot = i ( y i - y ¯ ) 2 , SS\text{tot}=\sum_{i}(y_{i}-\bar{y})^{2},
  6. S S reg = i ( f i - y ¯ ) 2 , SS\text{reg}=\sum_{i}(f_{i}-\bar{y})^{2},
  7. S S res = i ( y i - f i ) 2 SS\text{res}=\sum_{i}(y_{i}-f_{i})^{2}\,
  8. S S R SS_{R}
  9. S S E SS_{E}
  10. R 2 1 - S S res S S tot . R^{2}\equiv 1-{SS_{\rm res}\over SS_{\rm tot}}.\,
  11. S S res + S S reg = S S tot . SS_{\rm res}+SS_{\rm reg}=SS_{\rm tot}.\,
  12. R 2 = S S reg S S tot = S S reg / n S S tot / n . R^{2}={SS_{\rm reg}\over SS_{\rm tot}}={SS_{\rm reg}/n\over SS_{\rm tot}/n}.
  13. f i = α + β q i f_{i}=\alpha+\beta q_{i}\,
  14. f ¯ = y ¯ . \bar{f}=\bar{y}.\,
  15. y 1 = = y n = y ¯ y_{1}=\ldots=y_{n}=\bar{y}
  16. Y i = β 0 + j = 1 p β j X i , j + ε i , {Y_{i}=\beta_{0}+\sum_{j=1}^{p}{\beta_{j}X_{i,j}}+\varepsilon_{i}},
  17. Y i {Y_{i}}
  18. X i , 1 , , X i , p X_{i,1},\dots,X_{i,p}
  19. ε i \varepsilon_{i}
  20. β 0 , , β p \beta_{0},\dots,\beta_{p}
  21. y y
  22. y ¯ \bar{y}
  23. min b S S res ( b ) min b i ( y i - X i b ) 2 \min_{b}SS\text{res}(b)\Rightarrow\min_{b}\sum_{i}(y_{i}-X_{i}b)^{2}\,
  24. X X
  25. S S t o t SS_{tot}
  26. S S res SS\text{res}
  27. R ¯ 2 \bar{R}^{2}
  28. R ¯ 2 = 1 - ( 1 - R 2 ) n - 1 n - p - 1 = R 2 - ( 1 - R 2 ) p n - p - 1 \bar{R}^{2}={1-(1-R^{2}){n-1\over n-p-1}}={R^{2}-(1-R^{2}){p\over n-p-1}}
  29. R ¯ 2 = 1 - S S res / d f e S S tot / d f t \bar{R}^{2}={1-{SS\text{res}/df_{e}\over SS\text{tot}/df_{t}}}
  30. R 2 = 1 - 𝑉𝐴𝑅 res 𝑉𝐴𝑅 tot R^{2}={1-{\,\textit{VAR}\text{res}\over\,\textit{VAR}\text{tot}}}
  31. 𝑉𝐴𝑅 res = S S res / n {\,\textit{VAR}\text{res}=SS\text{res}/n}
  32. 𝑉𝐴𝑅 tot = S S tot / n {\,\textit{VAR}\text{tot}=SS\text{tot}/n}
  33. 𝑉𝐴𝑅 res = S S res / ( n - p - 1 ) {\,\textit{VAR}\text{res}=SS\text{res}/(n-p-1)}
  34. 𝑉𝐴𝑅 tot = S S tot / ( n - 1 ) {\,\textit{VAR}\text{tot}=SS\text{tot}/(n-1)}
  35. R 2 = 1 - ( L ( 0 ) L ( θ ^ ) ) 2 / n R^{2}=1-\left({L(0)\over L(\hat{\theta})}\right)^{2/n}
  36. L ( θ ^ ) {L(\hat{\theta})}
  37. L ( θ ^ ) L(\hat{\theta})
  38. R max 2 = 1 - ( L ( 0 ) ) 2 / n R^{2}_{\max}=1-(L(0))^{2/n}
  39. norm of residuals = S S res \,\text{norm of residuals}=\sqrt{SS\text{res}}
  40. S S tot SS\text{tot}
  41. x = 1 , 2 , 3 , 4 , 5 x=1,\ 2,\ 3,\ 4,\ 5
  42. y = 1.9 , 3.7 , 5.8 , 8.0 , 9.6 y=1.9,\ 3.7,\ 5.8,\ 8.0,\ 9.6
  43. H 0 : R 2 = 0. H_{0}\colon R^{2}=0.

Coefficient_of_haze.html

  1. C O H = - 100 log 10 ( I 1 I 0 ) COH=-100\log_{10}\left(\frac{I_{1}}{I_{0}}\right)
  2. I 1 I_{1}
  3. I 0 I_{0}

Cofibration.html

  1. i : A X i\colon A\to X
  2. X × I X\times I
  3. ( A × I ) ( X × { 0 } ) (A\times I)\cup(X\times\{0\})
  4. ( X , A ) (X,A)
  5. A X A\to X
  6. S n - 1 D n S^{n-1}\to D^{n}
  7. n n

Cognitive_architecture.html

  1. 2 1000 2^{1000}

Cohen's_kappa.html

  1. κ = p o - p e 1 - p e , \kappa=\frac{p_{o}-p_{e}}{1-p_{e}},\!
  2. κ = 1 κ=1
  3. κ 0 κ≤0
  4. κ = p o - p e 1 - p e = 0.70 - 0.50 1 - 0.50 = 0.40 \kappa=\frac{p_{o}-p_{e}}{1-p_{e}}=\frac{0.70-0.50}{1-0.50}=0.40\!
  5. κ = 0.60 - 0.54 1 - 0.54 = 0.1304 \kappa=\frac{0.60-0.54}{1-0.54}=0.1304
  6. κ = 0.60 - 0.46 1 - 0.46 = 0.2593 \kappa=\frac{0.60-0.46}{1-0.46}=0.2593
  7. κ = 1 - ( 1 - i = 1 k j = 1 k w i j x i j ) ( 1 - i = 1 k j = 1 k w i j m i j ) \kappa=1-\frac{(1-\sum_{i=1}^{k}\sum_{j=1}^{k}w_{ij}x_{ij})}{(1-\sum_{i=1}^{k}% \sum_{j=1}^{k}w_{ij}m_{ij})}
  8. w i j w_{ij}
  9. x i j x_{ij}
  10. m i j m_{ij}
  11. κ max = P max - P exp 1 - P exp \kappa_{\max}=\frac{P_{\max}-P_{\exp}}{1-P_{\exp}}
  12. P exp = i = 1 k P i + P + i P_{\exp}=\sum_{i=1}^{k}P_{i+}P_{+i}
  13. P max = i = 1 k min ( P i + , P + i ) P_{\max}=\sum_{i=1}^{k}\min(P_{i+},P_{+i})
  14. P i + P_{i+}
  15. P + i P_{+i}

Coherent_duality.html

  1. f : X Y f:X\rightarrow Y
  2. f ! f^{!}
  3. f ! f^{!}
  4. R H o m A k L A ( A , M k L M ) RHom_{A\otimes^{L}_{k}A}(A,M\otimes^{L}_{k}M)
  5. R H o m A k L A ( A , M k L M ) RHom_{A\otimes^{L}_{k}A}(A,M\otimes^{L}_{k}M)
  6. ( R , ρ ) (R,\rho)
  7. ρ : R R H o m A k L A ( A , R k L R ) \rho:R\to RHom_{A\otimes^{L}_{k}A}(A,R\otimes^{L}_{k}R)
  8. R X R_{X}
  9. f : X Y f:X\to Y
  10. f ! := D X L f * D Y f^{!}:=D_{X}\circ Lf^{*}D_{Y}
  11. D X := R H o m 𝒪 X ( - , R X ) D_{X}:=RHom_{\mathcal{O}_{X}}(-,R_{X})

Cohomology_operation.html

  1. θ \theta
  2. ( n , q , π , G ) (n,q,\pi,G)\,
  3. θ : H n ( - , π ) H q ( - , G ) \theta:H^{n}(-,\pi)\to H^{q}(-,G)\,
  4. ( n , q , π , G ) (n,q,\pi,G)
  5. K ( π , n ) K ( G , q ) K(\pi,n)\to K(G,q)
  6. H q ( K ( π , n ) , G ) H^{q}(K(\pi,n),G)
  7. [ A , B ] [A,B]
  8. A A
  9. B B
  10. Nat ( H n ( - , π ) , H q ( - , G ) ) \displaystyle\mathrm{Nat}(H^{n}(-,\pi),H^{q}(-,G))

Coiflet.html

  1. N / 3 N/3
  2. N / 3 - 1 N/3-1
  3. ϕ , ϕ ~ , ψ , ψ ~ , h , h ~ , g , g ~ \phi,\tilde{\phi},\psi,\tilde{\psi},h,\tilde{h},g,\tilde{g}
  4. ψ ~ ( 0 , l ] = 0 for l =0,1,…,L-1 n ( - 1 ) n n l h [ n ] = 0 for l =0,1,…,L-1 H ( l ) ( π ) = 0 for l =0,1,…,L-1 \begin{array}[]{lcl}\\ \mathcal{M_{\tilde{\psi}}}(0,l]=0&\mbox{for }~{}l\mbox{ =0,1,...,L-1}\\ \sum_{n}(-1)^{n}n^{l}h[n]=0&\mbox{for }~{}l\mbox{ =0,1,...,L-1}\\ H^{(l)}(\pi)=0&\mbox{for }~{}l\mbox{ =0,1,...,L-1}\\ \end{array}
  5. ψ \psi
  6. h ~ \tilde{h}
  7. ϕ , ϕ ~ , ψ , ψ ~ , h , h ~ , g , g ~ \phi,\tilde{\phi},\psi,\tilde{\psi},h,\tilde{h},g,\tilde{g}
  8. ϕ ~ ( t 0 , l ] = δ [ l ] for l =0,1,…,L-1 ϕ ~ ( 0 , l ] = t 0 l for l =0,1,…,L-1 ϕ ^ ( l ) ( 0 ) = ( - j t 0 ) t for l =0,1,…,L-1 n ( n - t 0 ) l h [ n ] = δ [ l ] for l =0,1,…,L-1 n n l h [ n ] = t 0 l for l =0,1,…,L-1 H ( l ) ( 0 ) = ( - j t 0 ) t for l =0,1,…,L-1 \begin{array}[]{lcl}\\ \mathcal{M_{\tilde{\phi}}}(t_{0},l]=\delta[l]&\mbox{for }~{}l\mbox{ =0,1,...,L% -1}\\ \mathcal{M_{\tilde{\phi}}}(0,l]=t_{0}^{l}&\mbox{for }~{}l\mbox{ =0,1,...,L-1}% \\ \hat{\phi}^{(}l)(0)=(-jt_{0})^{t}&\mbox{for }~{}l\mbox{ =0,1,...,L-1}\\ \sum_{n}(n-t_{0})^{l}h[n]=\delta[l]&\mbox{for }~{}l\mbox{ =0,1,...,L-1}\\ \sum_{n}n^{l}h[n]=t_{0}^{l}&\mbox{for }~{}l\mbox{ =0,1,...,L-1}\\ H^{(l)}(0)=(-jt_{0})^{t}&\mbox{for }~{}l\mbox{ =0,1,...,L-1}\\ \end{array}
  9. ψ ~ \tilde{\psi}
  10. h ~ \tilde{h}
  11. ϕ , ψ , ϕ ~ , ψ ~ \phi,\psi,\tilde{\phi},\tilde{\psi}
  12. ψ ~ \tilde{\psi}
  13. ψ \psi
  14. ψ ~ ( t 0 , l ] = δ [ l ] for l =0,1,…, L ¯ - 1 ψ ( t 0 , l ] = δ [ l ] for l =0,1,…, L ¯ - 1 \begin{array}[]{lcl}\\ \mathcal{M_{\tilde{\psi}}}(t_{0},l]=\delta[l]&\mbox{for }~{}l\mbox{ =0,1,...,}% ~{}\bar{L}-1\\ \mathcal{M_{\psi}}(t_{0},l]=\delta[l]&\mbox{for }~{}l\mbox{ =0,1,..., }~{}\bar% {L}-1\\ \end{array}
  15. L ¯ \bar{L}
  16. L ¯ L \bar{L}\ll L
  17. 1 / 2 1/\sqrt{2}
  18. B k = ( - 1 ) k C N - 1 - k B_{k}=(-1)^{k}C_{N-1-k}

Cointegration.html

  1. x t x_{t}
  2. y t y_{t}
  3. y t - β x t = u t y_{t}-\beta x_{t}=u_{t}\,
  4. u t u_{t}
  5. u t u_{t}
  6. u t u_{t}
  7. u t u_{t}
  8. u ^ t \hat{u}_{t}
  9. u ^ t - 1 \hat{u}_{t-1}

Collectively_exhaustive_events.html

  1. A B = S A\cup B=S

Collision_theory.html

  1. k ( T ) = Z ρ exp ( - E a R T ) k(T)=Z\rho\exp\left(\frac{-E_{a}}{RT}\right)
  2. ρ \rho
  3. Z = N A σ A B 8 k B T π μ A B Z=N_{A}\sigma_{AB}\sqrt{\frac{8k_{B}T}{\pi\mu_{AB}}}
  4. r A B r_{AB}
  5. π r A B 2 c A \scriptstyle\pi r^{2}_{AB}c_{A}
  6. c A \scriptstyle c_{A}
  7. c A = 8 k B T π m A c_{A}=\sqrt{\frac{8k_{B}T}{\pi m_{A}}}
  8. k B \scriptstyle k_{B}
  9. m A \scriptstyle m_{A}
  10. μ A B \mu_{AB}
  11. m A m_{A}
  12. N A σ A B 8 k B T π μ A B [ A ] [ B ] = N A r A B 2 8 π k B T μ A B [ A ] [ B ] = Z [ A ] [ B ] N_{A}\sigma_{AB}\sqrt{\frac{8k_{B}T}{\pi\mu_{AB}}}[A][B]=N_{A}\ r^{2}_{AB}% \sqrt{\frac{8\pi k_{B}T}{\mu_{AB}}}[A][B]=Z[A][B]
  13. e - E a k B T e^{\frac{-E_{a}}{k_{B}T}}
  14. r = Z ρ [ A ] [ B ] exp ( - E a R T ) r=Z\rho[A][B]\exp\left(\frac{-E_{a}}{RT}\right)
  15. ρ \scriptstyle\rho
  16. r = k ( T ) [ A ] [ B ] \scriptstyle r=k(T)[A][B]
  17. k ( T ) = N A 2 σ A B 8 k B T π m A exp ( - E a R T ) k(T)=N_{A}^{2}\sigma_{AB}\sqrt{\frac{8k_{B}T}{\pi m_{A}}}\exp\left(\frac{-E_{a% }}{RT}\right)
  18. ρ = A o b s e r v e d Z c a l c u l a t e d \rho=\frac{A_{observed}}{Z_{calculated}}