wpmath0000015_7

Renkonen_similarity_index.html

  1. p i = n i / n i p_{i}=n_{i}/\sum{n_{i}}

Representer_theorem.html

  1. f * f^{*}
  2. 𝒳 \mathcal{X}
  3. k k
  4. 𝒳 × 𝒳 \mathcal{X}\times\mathcal{X}
  5. H k H_{k}
  6. ( x 1 , y 1 ) , , ( x n , y n ) 𝒳 × \R (x_{1},y_{1}),\ldots,(x_{n},y_{n})\in\mathcal{X}\times\R
  7. g : [ 0 , ) \R g\colon[0,\infty)\to\R
  8. E : ( 𝒳 × \R 2 ) n \R { } E\colon(\mathcal{X}\times\R^{2})^{n}\to\R\cup\{\infty\}
  9. f * H k f^{*}\in H_{k}
  10. f * = argmin f H k { E ( ( x 1 , y 1 , f ( x 1 ) ) , , ( x n , y n , f ( x n ) ) ) + g ( f ) } , ( * ) f^{*}=\operatorname{argmin}_{f\in H_{k}}\left\{E\left((x_{1},y_{1},f(x_{1})),.% ..,(x_{n},y_{n},f(x_{n}))\right)+g\left(\lVert f\rVert\right)\right\},\quad(*)
  11. f * f^{*}
  12. f * ( ) = i = 1 n α i k ( , x i ) , f^{*}(\cdot)=\sum_{i=1}^{n}\alpha_{i}k(\cdot,x_{i}),
  13. α i \R \alpha_{i}\in\R
  14. 1 i n 1\leq i\leq n
  15. φ : 𝒳 \R 𝒳 φ ( x ) = k ( , x ) \begin{aligned}\displaystyle\varphi\colon\mathcal{X}&\displaystyle\to\R^{% \mathcal{X}}\\ \displaystyle\varphi(x)&\displaystyle=k(\cdot,x)\end{aligned}
  16. φ ( x ) = k ( , x ) \varphi(x)=k(\cdot,x)
  17. 𝒳 \R \mathcal{X}\to\R
  18. k k
  19. φ ( x ) ( x ) = k ( x , x ) = φ ( x ) , φ ( x ) , \varphi(x)(x^{\prime})=k(x^{\prime},x)=\langle\varphi(x^{\prime}),\varphi(x)\rangle,
  20. , \langle\cdot,\cdot\rangle
  21. H k H_{k}
  22. x 1 , , x n x_{1},...,x_{n}
  23. f H k f\in H_{k}
  24. span { φ ( x 1 ) , , φ ( x n ) } \operatorname{span}\left\{\varphi(x_{1}),...,\varphi(x_{n})\right\}
  25. f = i = 1 n α i φ ( x i ) + v , f=\sum_{i=1}^{n}\alpha_{i}\varphi(x_{i})+v,
  26. v , φ ( x i ) = 0 \langle v,\varphi(x_{i})\rangle=0
  27. i i
  28. f f
  29. x j x_{j}
  30. f ( x j ) = i = 1 n α i φ ( x i ) + v , φ ( x j ) = i = 1 n α i φ ( x i ) , φ ( x j ) , f(x_{j})=\left\langle\sum_{i=1}^{n}\alpha_{i}\varphi(x_{i})+v,\varphi(x_{j})% \right\rangle=\sum_{i=1}^{n}\alpha_{i}\langle\varphi(x_{i}),\varphi(x_{j})\rangle,
  31. v v
  32. E E
  33. v v
  34. v v
  35. i = 1 n α i φ ( x i ) \sum_{i=1}^{n}\alpha_{i}\varphi(x_{i})
  36. g g
  37. g ( f ) = g ( i = 1 n α i φ ( x i ) + v ) = g ( i = 1 n α i φ ( x i ) 2 + v 2 ) g ( i = 1 n α i φ ( x i ) ) . \begin{aligned}\displaystyle g\left(\lVert f\rVert\right)&\displaystyle=g\left% (\lVert\sum_{i=1}^{n}\alpha_{i}\varphi(x_{i})+v\rVert\right)\\ &\displaystyle=g\left(\sqrt{\lVert\sum_{i=1}^{n}\alpha_{i}\varphi(x_{i})\rVert% ^{2}+\lVert v\rVert^{2}}\right)\\ &\displaystyle\geq g\left(\lVert\sum_{i=1}^{n}\alpha_{i}\varphi(x_{i})\rVert% \right).\end{aligned}
  38. v = 0 v=0
  39. f * f^{*}
  40. v = 0 v=0
  41. f * ( ) = i = 1 n α i φ ( x i ) = i = 1 n α i k ( , x i ) , f^{*}(\cdot)=\sum_{i=1}^{n}\alpha_{i}\varphi(x_{i})=\sum_{i=1}^{n}\alpha_{i}k(% \cdot,x_{i}),
  42. E ( ( x 1 , y 1 , f ( x 1 ) ) , , ( x n , y n , f ( x n ) ) ) = 1 n i = 1 n ( f ( x i ) - y i ) 2 , g ( f ) = λ f 2 \begin{aligned}\displaystyle E\left((x_{1},y_{1},f(x_{1})),...,(x_{n},y_{n},f(% x_{n}))\right)&\displaystyle=\frac{1}{n}\sum_{i=1}^{n}(f(x_{i})-y_{i})^{2},\\ \displaystyle g(\lVert f\rVert)&\displaystyle=\lambda\lVert f\rVert^{2}\end{aligned}
  43. λ > 0 \lambda>0
  44. g ( ) g(\cdot)
  45. f ~ * = argmin { E ( ( x 1 , y 1 , f ~ ( x 1 ) ) , , ( x n , y n , f ~ ( x n ) ) ) + g ( f ) f ~ = f + h H k span { ψ p 1 p M } } , ( ) \tilde{f}^{*}=\operatorname{argmin}\left\{E\left((x_{1},y_{1},\tilde{f}(x_{1})% ),...,(x_{n},y_{n},\tilde{f}(x_{n}))\right)+g\left(\lVert f\rVert\right)\mid% \tilde{f}=f+h\in H_{k}\oplus\operatorname{span}\{\psi_{p}\mid 1\leq p\leq M\}% \right\},\quad(\dagger)
  46. f ~ = f + h \tilde{f}=f+h
  47. f H k f\in H_{k}
  48. h h
  49. { ψ p : 𝒳 \R 1 p M } \{\psi_{p}\colon\mathcal{X}\to\R\mid 1\leq p\leq M\}
  50. m × M m\times M
  51. ( ψ p ( x i ) ) i p \left(\psi_{p}(x_{i})\right)_{ip}
  52. M M
  53. f ~ * \tilde{f}^{*}
  54. ( ) (\dagger)
  55. f ~ * ( ) = i = 1 n α i k ( , x i ) + p = 1 M β p ψ p ( ) \tilde{f}^{*}(\cdot)=\sum_{i=1}^{n}\alpha_{i}k(\cdot,x_{i})+\sum_{p=1}^{M}% \beta_{p}\psi_{p}(\cdot)
  56. α i , β p \R \alpha_{i},\beta_{p}\in\R
  57. β p \beta_{p}
  58. 𝒳 \mathcal{X}
  59. k k
  60. 𝒳 × 𝒳 \mathcal{X}\times\mathcal{X}
  61. H k H_{k}
  62. R : H k \R R\colon H_{k}\to\R
  63. ( x 1 , y 1 ) , , ( x n , y n ) 𝒳 × \R (x_{1},y_{1}),...,(x_{n},y_{n})\in\mathcal{X}\times\R
  64. E : ( 𝒳 × \R 2 ) m \R { } E\colon(\mathcal{X}\times\R^{2})^{m}\to\R\cup\{\infty\}
  65. f * = argmin f H k { E ( ( x 1 , y 1 , f ( x 1 ) ) , , ( x n , y n , f ( x n ) ) ) + R ( f ) } ( ) f^{*}=\operatorname{argmin}_{f\in H_{k}}\left\{E\left((x_{1},y_{1},f(x_{1})),.% ..,(x_{n},y_{n},f(x_{n}))\right)+R(f)\right\}\quad(\ddagger)
  66. f * ( ) = i = 1 n α i k ( , x i ) , f^{*}(\cdot)=\sum_{i=1}^{n}\alpha_{i}k(\cdot,x_{i}),
  67. α i \R \alpha_{i}\in\R
  68. 1 i n 1\leq i\leq n
  69. h : [ 0 , ) \R h\colon[0,\infty)\to\R
  70. R ( f ) = h ( f ) . R(f)=h(\lVert f\rVert).
  71. R ( ) R(\cdot)
  72. ( ) (\ddagger)
  73. ( ) (\ddagger)
  74. H k H_{k}
  75. L 2 ( 𝒳 ) L^{2}(\mathcal{X})
  76. f * ( ) f^{*}(\cdot)
  77. n n
  78. α = ( α 1 , , α n ) \R n \alpha=(\alpha_{1},...,\alpha_{n})\in\R^{n}
  79. α \alpha

Residual_income_valuation.html

  1. r r
  2. V 0 = B V 0 + t = 1 R I t ( 1 + r ) t V_{0}=BV_{0}+\sum_{t=1}^{\infty}{RI_{t}\over(1+r)^{t}}
  3. g g
  4. m m
  5. T m = R I m ( r - g ) T_{m}={RI_{m}\over(r-g)}
  6. V 0 = B V 0 + t = 1 m - 1 R I t ( 1 + r ) t + T m ( 1 + r ) m - 1 V_{0}=BV_{0}+\sum_{t=1}^{m-1}{RI_{t}\over(1+r)^{t}}+{T_{m}\over(1+r)^{m-1}}

Residual_time.html

  1. t t
  2. { N ( t ) , t 0 } \{N(t),t\geq 0\}
  3. S i S_{i}
  4. J i J_{i}
  5. i i\in\mathbb{N}
  6. S i S_{i}
  7. N ( t ) = sup { n : J n t } N(t)=\sup\{n:J_{n}\leq t\}
  8. t t
  9. N ( t ) N(t)
  10. J N ( t ) t < J N ( t ) + 1 . J_{N(t)}\leq t<J_{N(t)+1}.\,
  11. Y ( t ) Y(t)
  12. t t
  13. Y ( t ) = J N ( t ) + 1 - t . Y(t)=J_{N(t)+1}-t.\,
  14. S i S_{i}
  15. F ( t ) = P r [ S i t ] F(t)=Pr[S_{i}\leq t]
  16. m ( t ) = 𝔼 [ N ( t ) ] m(t)=\mathbb{E}[N(t)]
  17. t t
  18. Y ( t ) Y(t)
  19. Pr [ Y ( t ) x ] = F ( t + x ) - 0 t [ 1 - F ( t + x - y ) ] d m ( y ) \Pr[Y(t)\leq x]=F(t+x)-\int_{0}^{t}\left[1-F(t+x-y)\right]dm(y)
  20. S i S_{i}
  21. F ( t ) = 1 - e - λ t F(t)=1-e^{-\lambda t}
  22. m ( t ) = λ t m(t)=\lambda t
  23. Pr [ Y ( t ) x ] = [ 1 - e - λ ( t + x ) ] - 0 t [ 1 - 1 + e - λ ( t + x - y ) ] d ( a y ) = 1 - e - λ t . \Pr[Y(t)\leq x]=\left[1-e^{-\lambda(t+x)}\right]-\int_{0}^{t}\left[1-1+e^{-% \lambda(t+x-y)}\right]d(ay)=1-e^{-\lambda t}.
  24. Z ( t ) = t - J N ( t ) Z(t)=t-J_{N(t)}

Resolution_proof_compression_by_splitting.html

  1. π \pi
  2. x x
  3. x x
  4. ¬ x \neg x\!
  5. π \pi
  6. x x
  7. y y
  8. π 1 π 2 \pi_{1}\pi_{2}\ldots
  9. π i + 1 \pi_{i+1}
  10. π i \pi_{i}
  11. π j \pi_{j}
  12. π j + 1 \pi_{j+1}
  13. { π 1 , π 2 , , π j } \{\pi_{1},\pi_{2},\ldots,\pi_{j}\}
  14. p p
  15. n n
  16. r r
  17. add ( r ) := max ( | r | - max ( | p | , | n | ) , 0 ) \operatorname{add}(r):=\max(|r|-\max(|p|,|n|),0)\,
  18. v v
  19. π \pi
  20. v v
  21. a d d ( v , π ) add(v,\pi)
  22. p ( v ) = add ( v , π i ) x add ( x , π i ) p(v)=\frac{\operatorname{add}(v,\pi_{i})}{\sum_{x}{\operatorname{add}(x,\pi_{i% })}}
  23. π \pi
  24. π x \pi_{x}
  25. x x
  26. π ¬ x \pi_{\neg x}
  27. ¬ x \neg x
  28. l l
  29. p x n p\oplus_{x}n
  30. p p
  31. n n
  32. x p x\in p
  33. ¬ x n \neg x\in n
  34. π l \pi_{l}
  35. π \pi
  36. π l ( c ) := { c , if c is an input π l ( p ) , if c = p x n and ( l = x or x π l ( p ) ) π l ( n ) , if c = p x n and ( l = ¬ x or ¬ x π l ( n ) ) π l ( p ) x π l ( p ) , if x π l ( p ) and ¬ x π l ( n ) \pi_{l}(c):=\begin{cases}c,&\,\text{if }c\,\text{ is an input}\\ \pi_{l}(p),&\,\text{if }c=p\oplus_{x}n\,\text{ and }(l=x\,\text{ or }x\notin% \pi_{l}(p))\\ \pi_{l}(n),&\,\text{if }c=p\oplus_{x}n\,\text{ and }(l=\neg x\mbox{ or }~{}% \neg x\notin\pi_{l}(n))\\ \pi_{l}(p)\oplus_{x}\pi_{l}(p),&\,\text{if }x\in\pi_{l}(p)\,\text{ and }\neg x% \in\pi_{l}(n)\end{cases}
  37. o o
  38. π \pi
  39. π x \pi_{x}
  40. π ¬ x \pi_{\neg x}
  41. π x ( o ) \pi_{x}(o)
  42. π ¬ x ( o ) \pi_{\neg x}(o)

Resolution_proof_reduction_via_local_context_rewriting.html

  1. p p
  2. q q
  3. α \alpha
  4. β \beta
  5. γ \gamma
  6. δ \delta
  7. η \eta
  8. p p
  9. β \beta
  10. γ \gamma
  11. q q
  12. δ \delta
  13. α \alpha
  14. β γ δ p α η q \cfrac{\cfrac{\beta\qquad\gamma}{\delta}\,p\qquad\alpha}{\eta}\,q
  15. s α , t γ s\notin\alpha,t\in\gamma
  16. s t C s ¯ t D t C D var ( s ) t ¯ E C D E var ( t ) s t C t ¯ E s C E var ( t ) t ¯ E s ¯ t D s ¯ D E var ( t ) C D E var ( s ) \cfrac{\cfrac{stC\qquad\overline{s}tD}{tCD}\,\operatorname{var}(s)\qquad% \overline{t}E}{CDE}\,\operatorname{var}(t)\Rightarrow\cfrac{\cfrac{stC\qquad% \overline{t}E}{sCE}\,\operatorname{var}(t)\qquad\cfrac{\overline{t}E\qquad% \overline{s}tD}{\overline{s}DE}\,\operatorname{var}(t)}{CDE}\,\operatorname{% var}(s)
  17. s α , t γ s\notin\alpha,t\notin\gamma
  18. s t C s ¯ D t C D var ( s ) t ¯ E C D E var ( t ) s t C t ¯ E s C E var ( t ) s ¯ D C D E var ( s ) \cfrac{\cfrac{stC\qquad\overline{s}D}{tCD}\,\operatorname{var}(s)\qquad% \overline{t}E}{CDE}\,\operatorname{var}(t)\Rightarrow\cfrac{\cfrac{stC\qquad% \overline{t}E}{sCE}\,\operatorname{var}(t)\qquad\overline{s}D}{CDE}\,% \operatorname{var}(s)
  19. s α , t γ s\in\alpha,t\in\gamma
  20. s t C s ¯ t D t C D var ( s ) s t ¯ E s C D E var ( t ) s t C s t ¯ E s C E var ( t ) \cfrac{\cfrac{stC\qquad\overline{s}tD}{tCD}\,\operatorname{var}(s)\qquad s% \overline{t}E}{sCDE}\,\operatorname{var}(t)\Rightarrow\cfrac{stC\qquad s% \overline{t}E}{sCE}\,\operatorname{var}(t)
  21. s α , t γ s\in\alpha,t\notin\gamma
  22. s t C s ¯ D t D C var ( s ) s t ¯ E s C D E var ( t ) s t C s t ¯ E s C E var ( t ) s ¯ D C D E var ( s ) \cfrac{\cfrac{stC\qquad\overline{s}D}{tDC}\,\operatorname{var}(s)\qquad s% \overline{t}E}{sCDE}\,\operatorname{var}(t)\Rightarrow\cfrac{\cfrac{stC\qquad s% \overline{t}E}{sCE}\,\operatorname{var}(t)\qquad\overline{s}D}{CDE}\,% \operatorname{var}(s)
  23. s ¯ α , t γ \overline{s}\in\alpha,t\notin\gamma
  24. s t C s ¯ D t D C var ( s ) s ¯ t ¯ E s ¯ C D E var ( t ) s ¯ D \cfrac{\cfrac{stC\qquad\overline{s}D}{tDC}\,\operatorname{var}(s)\qquad% \overline{s}\overline{t}E}{\overline{s}CDE}\,\operatorname{var}(t)\Rightarrow% \overline{s}D
  25. s t C s ¯ t D t C D var ( s ) t ¯ E C D E var ( t ) s t C t ¯ E s C E var ( t ) t ¯ E s ¯ t D s ¯ D E var ( t ) C D E var ( s ) \cfrac{\cfrac{stC\qquad\overline{s}tD}{tCD}\,\operatorname{var}(s)\qquad% \overline{t}E}{CDE}\,\operatorname{var}(t)\Leftarrow\cfrac{\cfrac{stC\qquad% \overline{t}E}{sCE}\,\operatorname{var}(t)\qquad\cfrac{\overline{t}E\qquad% \overline{s}tD}{\overline{s}DE}\,\operatorname{var}(t)}{CDE}\,\operatorname{% var}(s)
  26. t γ t\notin\gamma
  27. s t C s ¯ D t C D var ( s ) s t ¯ E s C D E var ( t ) s t C s t ¯ E s C E var ( t ) \cfrac{\cfrac{stC\qquad\overline{s}D}{tCD}\,\operatorname{var}(s)\qquad s% \overline{t}E}{sCDE}\,\operatorname{var}(t)\Rightarrow\cfrac{stC\qquad s% \overline{t}E}{sCE}\,\operatorname{var}(t)
  28. 𝐩𝐪 𝐩 ¯ 𝐨 𝐪𝐨 p 𝐩 𝐪 ¯ 𝐩𝐨 q q r p ¯ q ¯ p ¯ r q o r p o ¯ s r s o \cfrac{\cfrac{\cfrac{\cfrac{\mathbf{pq}\qquad\mathbf{\overline{p}o}}{\mathbf{% qo}}\,p\qquad\mathbf{p\overline{q}}}{\mathbf{po}}\,q\qquad\cfrac{qr\qquad% \overline{p}\overline{q}}{\overline{p}r}\,q}{or}\,p\qquad\overline{o}s}{rs}\,o
  29. 𝐩𝐪 𝐩 𝐪 ¯ 𝐩 q q r p ¯ q ¯ p ¯ r q o r p o ¯ s r s o \cfrac{\cfrac{\cfrac{\mathbf{pq}\qquad\mathbf{p\overline{q}}}{\mathbf{p}}\,q% \qquad\cfrac{qr\qquad\overline{p}\overline{q}}{\overline{p}r}\,q}{or}\,p\qquad% \overline{o}s}{rs}\,o
  30. o o
  31. o o
  32. p q p q ¯ p q q r p ¯ q ¯ p ¯ r q r p \cfrac{\cfrac{pq\qquad p\overline{q}}{p}\,q\qquad\cfrac{qr\qquad\overline{p}% \overline{q}}{\overline{p}r}\,q}{r}\,p
  33. π \pi
  34. π \pi
  35. π \pi
  36. n n
  37. n n
  38. n piv n clauseleft n\text{piv}\in n\text{clause}\text{left}
  39. n piv ¯ n clauseright \overline{n\text{piv}}\in n\text{clause}\text{right}
  40. n clause n\text{clause}
  41. n clauseleft n\text{clause}\text{left}
  42. n clauseright n\text{clause}\text{right}
  43. n n
  44. n n
  45. n piv n clauseleft n\text{piv}\notin n\text{clause}\text{left}
  46. n piv ¯ n clauseright \overline{n\text{piv}}\in n\text{clause}\text{right}
  47. n n
  48. n left n\text{left}
  49. n piv n clauseleft n\text{piv}\in n\text{clause}\text{left}
  50. n piv ¯ n clauseright \overline{n\text{piv}}\notin n\text{clause}\text{right}
  51. n n
  52. n right n\text{right}
  53. n piv n clauseleft n\text{piv}\notin n\text{clause}\text{left}
  54. n piv ¯ n clauseright \overline{n\text{piv}}\notin n\text{clause}\text{right}
  55. n left n\text{left}
  56. n right n\text{right}
  57. n n
  58. n left n\text{left}
  59. n right n\text{right}
  60. δ \delta
  61. δ \delta

Resolvent_(Galois_theory).html

  1. X 2 - Δ X^{2}-\Delta
  2. Δ \Delta
  3. n n
  4. ( X 1 , , X n ) (X_{1},\ldots,X_{n})
  5. n n
  6. F ( X ) = X n + i = 1 n ( - 1 ) i E i X n - i = i = 1 n ( X - X i ) , F(X)=X^{n}+\sum_{i=1}^{n}(-1)^{i}E_{i}X^{n-i}=\prod_{i=1}^{n}(X-X_{i}),
  7. G G
  8. G G
  9. G G
  10. G G
  11. G G
  12. G G
  13. G G
  14. ( 12 ) ( 34 ) (12)(34)
  15. ( 13 ) ( 24 ) (13)(24)
  16. ( 14 ) ( 23 ) (14)(23)
  17. X 1 X 2 X_{1}X_{2}
  18. 2 ( X 1 X 2 + X 3 X 4 ) 2(X_{1}X_{2}+X_{3}X_{4})
  19. G G
  20. ( 12 ) (12)
  21. ( 12 ) , ( 1324 ) \langle(12),(1324)\rangle
  22. P P
  23. G G
  24. m m
  25. m m
  26. P 1 , , P m P_{1},\ldots,P_{m}
  27. R G = i = 1 m ( Y - P i ) R_{G}=\prod_{i=1}^{m}(Y-P_{i})
  28. Y Y
  29. F F
  30. f ( X ) = X n + i = 1 n a i X n - i = i = 1 n ( X - x i ) , f(X)=X^{n}+\sum_{i=1}^{n}a_{i}X^{n-i}=\prod_{i=1}^{n}(X-x_{i}),
  31. K K
  32. F F
  33. f f
  34. R G ( f ) ( Y ) R_{G}^{(f)}(Y)
  35. f f
  36. G G
  37. G G
  38. R G ( f ) ( Y ) R_{G}^{(f)}(Y)
  39. K K
  40. K K
  41. R G ( f ) ( Y ) R_{G}^{(f)}(Y)
  42. f f
  43. G G
  44. i = 0 n - 1 X i ω i \sum_{i=0}^{n-1}X_{i}\omega^{i}
  45. ω \omega
  46. H H
  47. G G
  48. H H
  49. f f
  50. H H
  51. f f
  52. G G
  53. n n
  54. S n S_{n}
  55. S n S_{n}
  56. D 5 D_{5}
  57. A 5 A_{5}
  58. M 20 M_{20}

Resonating_valence_bond_theory.html

  1. H = - t i j c i σ c j σ + h.c. + U i n i n i H=-t\sum_{\langle ij\rangle}c^{\dagger}_{i\sigma}c_{j\sigma}+\,\text{h.c.}+U% \sum_{i}n_{i\uparrow}n_{i\downarrow}
  2. | RVB = C | C |\,\text{RVB}\rangle=\sum_{C}|C\rangle
  3. C C

Returns-based_style_analysis.html

  1. R t m = α + i = 1 I β i R t i + ϵ t R_{t}^{m}=\alpha+\sum\limits_{i=1}^{I}\beta^{i}R_{t}^{i}+\epsilon_{t}
  2. R t m R_{t}^{m}
  3. R t i R_{t}^{i}
  4. I I
  5. α \alpha
  6. ϵ t \epsilon_{t}
  7. i = 1 I β i = 1 ; β i > 0 i . \sum\limits_{i=1}^{I}\beta_{i}=1;\;\;\;\;\beta_{i}>0\;\;\forall i.

Reversible-deactivation_radical_polymerization.html

  1. M n = M m × [ M ] 0 - [ M ] t [ R-X ] 0 M\text{n}=M\text{m}\times\frac{[\,\text{M}]_{0}-[\,\text{M}]_{t}}{[\,\text{R-X% }]_{0}}

Reynolds_equation.html

  1. x ( ρ h 3 12 μ p x ) + y ( ρ h 3 12 μ p y ) = x ( ρ h ( u a + u b ) 2 ) + y ( ρ h ( v a + v b ) 2 ) + ρ ( w a - w b ) - ρ u a h x - ρ v a h y + h ρ t \frac{\partial}{\partial x}\left(\frac{\rho h^{3}}{12\mu}\frac{\partial p}{% \partial x}\right)+\frac{\partial}{\partial y}\left(\frac{\rho h^{3}}{12\mu}% \frac{\partial p}{\partial y}\right)=\frac{\partial}{\partial x}\left(\frac{% \rho h\left(u_{a}+u_{b}\right)}{2}\right)+\frac{\partial}{\partial y}\left(% \frac{\rho h\left(v_{a}+v_{b}\right)}{2}\right)+\rho\left(w_{a}-w_{b}\right)-% \rho u_{a}\frac{\partial h}{\partial x}-\rho v_{a}\frac{\partial h}{\partial y% }+h\frac{\partial\rho}{\partial t}
  2. p p
  3. x x
  4. y y
  5. z z
  6. h h
  7. μ \mu
  8. ρ \rho
  9. u , v , w u,v,w
  10. x , y , z x,y,z
  11. a , b a,b
  12. p z = 0 \frac{\partial p}{\partial z}=0
  13. h l h<<l
  14. h w h<<w

Rhamnopyranosyl-N-acetylglucosaminyl-diphospho-decaprenol_beta-1,3::1,4-galactofuranosyltransferase.html

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

Rhenium(VII)_sulfide.html

  1. 2 R e + 7 S Δ Re 2 S 7 \mathrm{2Re+7S\ \xrightarrow{\Delta}\ Re_{2}S_{7}}
  2. Re 2 O 7 + 7 H 2 S Δ Re 2 S 7 + 7 H 2 O \mathrm{Re_{2}O_{7}+7H_{2}S\ \xrightarrow{\Delta}\ Re_{2}S_{7}+7H_{2}O}
  3. Re 2 S 7 600 o C 2 ReS 2 + 3 S \mathrm{Re_{2}S_{7}\ \xrightarrow{600^{o}C}\ 2ReS_{2}+3S}
  4. 2 R e 2 S 7 + 21 O 2 Δ 2 Re 2 O 7 + 14 S O 2 \mathrm{2Re_{2}S_{7}+21O_{2}\ \xrightarrow{\Delta}\ 2Re_{2}O_{7}+14SO_{2}}

Rhizobium_leguminosarum_exopolysaccharide_glucosyl_ketal-pyruvate-transferase.html

  1. \rightleftharpoons

Rho_Geminorum.html

  1. M = m - 5 ( ( log 10 D L ) - 1 ) M=m-5((\log_{10}{D_{L}})-1)\!\,
  2. m m\!\,
  3. D L D_{L}\!\,

Riboflavin_reductase_(NAD(P)H).html

  1. \rightleftharpoons

Ribostamycin:4-(gamma-L-glutamylamino)-(S)-2-hydroxybutanoyl-(BtrI_acyl-carrier_protein)_4-(gamma-L-glutamylamino)-(S)-2-hydroxybutanoate_transferase.html

  1. \rightleftharpoons

Ricci_calculus.html

  1. X ϕ ^ , Y λ ¯ , Z η ~ , T μ X_{\hat{\phi}}\,,Y_{\bar{\lambda}}\,,Z_{\tilde{\eta}}\,,T_{\mu^{\prime}}\cdots
  2. v μ = v ν L ν . μ v^{\mu^{\prime}}=v^{\nu}L_{\nu}{}^{\mu^{\prime}}.
  3. A α β γ A_{\alpha\beta\gamma\cdots}
  4. A α β γ A^{\alpha\beta\gamma\cdots}
  5. A α δ β γ A_{\alpha}{}^{\beta}{}_{\gamma}{}^{\delta\cdots}
  6. A α B α α A α B α A_{\alpha}B^{\alpha}\equiv\sum_{\alpha}A_{\alpha}B^{\alpha}
  7. A α B α α A α B α . A^{\alpha}B_{\alpha}\equiv\sum_{\alpha}A^{\alpha}B_{\alpha}\,.
  8. A α B β A α B α α A α B α . A_{\alpha}B^{\beta}\rightarrow A_{\alpha}B^{\alpha}\equiv\sum_{\alpha}A_{% \alpha}B^{\alpha}\,.
  9. A α B α γ C γ β α γ A α B α γ C γ . β A_{\alpha}{}^{\gamma}B^{\alpha}C_{\gamma}{}^{\beta}\equiv\sum_{\alpha}\sum_{% \gamma}A_{\alpha}{}^{\gamma}B^{\alpha}C_{\gamma}{}^{\beta}\,.
  10. A α γ B α γ C γ β α γ A α γ B α γ C γ β A_{\alpha\gamma}{}^{\gamma}B^{\alpha}C_{\gamma}{}^{\beta}\not\equiv\sum_{% \alpha}\sum_{\gamma}A_{\alpha\gamma}{}^{\gamma}B^{\alpha}C_{\gamma}{}^{\beta}\,
  11. A i 1 i n B i 1 i n j 1 j m C j 1 j m A I B I J C J A_{i_{1}\cdots i_{n}}B^{i_{1}\cdots i_{n}j_{1}\cdots j_{m}}C_{j_{1}\cdots j_{m% }}\equiv A_{I}B^{IJ}C_{J}
  12. A | α β γ | B α β γ = A α β γ B | α β γ | = α < β < γ A α β γ B α β γ A_{|\alpha\beta\gamma|\cdots}B^{\alpha\beta\gamma\cdots}=A_{\alpha\beta\gamma% \cdots}B^{|\alpha\beta\gamma|\cdots}=\sum_{\alpha<\beta<\gamma}A_{\alpha\beta% \gamma\cdots}B^{\alpha\beta\gamma\cdots}
  13. A | α β γ | B α β γ | δ ϵ λ | C μ ν ζ δ ϵ λ | μ ν ζ | = α < β < γ δ < ϵ < < λ μ < ν < < ζ A α β γ B α β γ δ ϵ λ C μ ν ζ δ ϵ λ μ ν ζ A_{|\alpha\beta\gamma|}{}^{|\delta\epsilon\cdots\lambda|}B^{\alpha\beta\gamma}% {}_{\delta\epsilon\cdots\lambda|\mu\nu\cdots\zeta|}C^{\mu\nu\cdots\zeta}=\sum_% {\alpha<\beta<\gamma}~{}\sum_{\delta<\epsilon<\cdots<\lambda}~{}\sum_{\mu<\nu<% \cdots<\zeta}A_{\alpha\beta\gamma}{}^{\delta\epsilon\cdots\lambda}B^{\alpha% \beta\gamma}{}_{\delta\epsilon\cdots\lambda\mu\nu\cdots\zeta}C^{\mu\nu\cdots\zeta}
  14. A P B P Q C R Q R = P Q R A P B P Q C R Q R A_{\underset{\rightharpoondown}{P}}{}^{\underset{\rightharpoondown}{Q}}B^{P}{}% _{Q\underset{\rightharpoondown}{R}}C^{R}=\sum_{\underset{\rightharpoondown}{P}% }\sum_{\underset{\rightharpoondown}{Q}}\sum_{\underset{\rightharpoondown}{R}}A% _{P}{}^{Q}B^{P}{}_{QR}C^{R}
  15. P = | α β γ | , Q = | δ ϵ λ | , R = | μ ν ζ | \underset{\rightharpoondown}{P}=|\alpha\beta\gamma|\,,\quad\underset{% \rightharpoondown}{Q}=|\delta\epsilon\cdots\lambda|\,,\quad\underset{% \rightharpoondown}{R}=|\mu\nu\cdots\zeta|
  16. B γ = β g γ α A α β B^{\gamma}{}_{\beta\cdots}=g^{\gamma\alpha}A_{\alpha\beta\cdots}
  17. A α β = g α γ B γ β A_{\alpha\beta\cdots}=g_{\alpha\gamma}B^{\gamma}{}_{\beta\cdots}
  18. e α ¯ = L α ¯ e β β e^{\bar{\alpha}}=L^{\bar{\alpha}}{}_{\beta}e^{\beta}
  19. a α ¯ = a γ L γ α ¯ a_{\bar{\alpha}}=a_{\gamma}L^{\gamma}{}_{\bar{\alpha}}
  20. a α ¯ e α ¯ = a γ L γ L α ¯ α ¯ e β β = a γ δ γ e β β = a β e β a_{\bar{\alpha}}e^{\bar{\alpha}}=a_{\gamma}L^{\gamma}{}_{\bar{\alpha}}L^{\bar{% \alpha}}{}_{\beta}e^{\beta}=a_{\gamma}\delta^{\gamma}{}_{\beta}e^{\beta}=a_{% \beta}e^{\beta}
  21. e α ¯ = L γ e γ α ¯ e_{\bar{\alpha}}=L^{\gamma}{}_{\bar{\alpha}}e_{\gamma}
  22. a α ¯ = a β L α ¯ β a^{\bar{\alpha}}=a^{\beta}L^{\bar{\alpha}}{}_{\beta}
  23. a α ¯ e α ¯ = a β L α ¯ L γ β e γ α ¯ = a β δ γ e γ β = a γ e γ a^{\bar{\alpha}}e_{\bar{\alpha}}=a^{\beta}L^{\bar{\alpha}}{}_{\beta}L^{\gamma}% {}_{\bar{\alpha}}e_{\gamma}=a^{\beta}\delta^{\gamma}{}_{\beta}e_{\gamma}=a^{% \gamma}e_{\gamma}
  24. A α = β γ B α β γ A^{\alpha}{}_{\beta\gamma}=B^{\alpha}{}_{\beta\gamma}
  25. A α A^{\alpha}
  26. B β γ B_{\beta}{}^{\gamma}
  27. A α B β C γ δ γ + D α E δ β = T α δ β A^{\alpha}B_{\beta}{}^{\gamma}C_{\gamma\delta}+D^{\alpha}{}_{\beta}{}E_{\delta% }=T^{\alpha}{}_{\beta}{}_{\delta}
  28. A 0 B 1 C 00 0 + A 0 B 1 C 10 1 + A 0 B 1 C 20 2 + A 0 B 1 C 30 3 + D 0 E 0 1 = T 0 0 1 A^{0}B_{1}{}^{0}C_{00}+A^{0}B_{1}{}^{1}C_{10}+A^{0}B_{1}{}^{2}C_{20}+A^{0}B_{1% }{}^{3}C_{30}+D^{0}{}_{1}{}E_{0}=T^{0}{}_{1}{}_{0}
  29. A 1 B 0 C 00 0 + A 1 B 0 C 10 1 + A 1 B 0 C 20 2 + A 1 B 0 C 30 3 + D 1 E 0 0 = T 1 0 0 A^{1}B_{0}{}^{0}C_{00}+A^{1}B_{0}{}^{1}C_{10}+A^{1}B_{0}{}^{2}C_{20}+A^{1}B_{0% }{}^{3}C_{30}+D^{1}{}_{0}{}E_{0}=T^{1}{}_{0}{}_{0}
  30. A 1 B 2 C 02 0 + A 1 B 2 C 12 1 + A 1 B 2 C 22 2 + A 1 B 2 C 32 3 + D 1 E 2 2 = T 1 . 2 2 A^{1}B_{2}{}^{0}C_{02}+A^{1}B_{2}{}^{1}C_{12}+A^{1}B_{2}{}^{2}C_{22}+A^{1}B_{2% }{}^{3}C_{32}+D^{1}{}_{2}{}E_{2}=T^{1}{}_{2}{}_{2}.
  31. A α B β C γ δ γ + D α E δ β A λ B β C μ δ μ + D λ E δ β A^{\alpha}B_{\beta}{}^{\gamma}C_{\gamma\delta}+D^{\alpha}{}_{\beta}{}E_{\delta% }\rightarrow A^{\lambda}B_{\beta}{}^{\mu}C_{\mu\delta}+D^{\lambda}{}_{\beta}{}% E_{\delta}
  32. A α B β C γ δ γ + D α E δ β A λ B β C μ δ γ + D α E δ β . A^{\alpha}B_{\beta}{}^{\gamma}C_{\gamma\delta}+D^{\alpha}{}_{\beta}{}E_{\delta% }\nrightarrow A^{\lambda}B_{\beta}{}^{\gamma}C_{\mu\delta}+D^{\alpha}{}_{\beta% }{}E_{\delta}\,.
  33. A α B β C γ δ γ + D α E δ β = T α δ β A^{\alpha}B_{\beta}{}^{\gamma}C_{\gamma\delta}+D^{\alpha}{}_{\beta}{}E_{\delta% }=T^{\alpha}{}_{\beta}{}_{\delta}
  34. A α B β C γ δ γ + D α E δ β γ . A^{\alpha}B_{\beta}{}^{\gamma}C_{\gamma\delta}+D_{\alpha}{}_{\beta}{}^{\gamma}% E^{\delta}.
  35. α σ ( i ) \alpha_{\sigma(i)}
  36. A ( α 1 α 2 α p ) α p + 1 α q = 1 p ! σ A α σ ( 1 ) α σ ( p ) α p + 1 α q . A_{(\alpha_{1}\alpha_{2}\cdots\alpha_{p})\alpha_{p+1}\cdots\alpha_{q}}=\dfrac{% 1}{p!}\sum_{\sigma}A_{\alpha_{\sigma(1)}\cdots\alpha_{\sigma(p)}\alpha_{p+1}% \cdots\alpha_{q}}\,.
  37. A ( α β ) γ = 1 2 ! ( A α β γ + A β α γ ) A_{(\alpha\beta)\gamma\cdots}=\dfrac{1}{2!}\left(A_{\alpha\beta\gamma\cdots}+A% _{\beta\alpha\gamma\cdots}\right)
  38. A ( α β γ ) δ = 1 3 ! ( A α β γ δ + A γ α β δ + A β γ α δ + A α γ β δ + A γ β α δ + A β α γ δ ) A_{(\alpha\beta\gamma)\delta\cdots}=\dfrac{1}{3!}\left(A_{\alpha\beta\gamma% \delta\cdots}+A_{\gamma\alpha\beta\delta\cdots}+A_{\beta\gamma\alpha\delta% \cdots}+A_{\alpha\gamma\beta\delta\cdots}+A_{\gamma\beta\alpha\delta\cdots}+A_% {\beta\alpha\gamma\delta\cdots}\right)
  39. A ( α ( B β ) γ + C β ) γ ) = A ( α B β ) γ + A ( α C β ) γ A_{(\alpha}\left(B_{\beta)\gamma\cdots}+C_{\beta)\gamma\cdots}\right)=A_{(% \alpha}B_{\beta)\gamma\cdots}+A_{(\alpha}C_{\beta)\gamma\cdots}
  40. A ( α B β = γ ) 1 2 ! ( A α B β + γ A γ B β ) α A_{(\alpha}B^{\beta}{}_{\gamma)}=\dfrac{1}{2!}\left(A_{\alpha}B^{\beta}{}_{% \gamma}+A_{\gamma}B^{\beta}{}_{\alpha}\right)
  41. A ( α B | β | = γ ) 1 2 ! ( A α B β γ + A γ B β α ) A_{(\alpha}B_{|\beta|}{}_{\gamma)}=\dfrac{1}{2!}\left(A_{\alpha}B_{\beta\gamma% }+A_{\gamma}B_{\beta\alpha}\right)
  42. α σ ( i ) \alpha_{\sigma(i)}
  43. sgn ( σ ) \operatorname{sgn}(\sigma)
  44. A [ α 1 α p ] α p + 1 α q \displaystyle A_{[\alpha_{1}\cdots\alpha_{p}]\alpha_{p+1}\cdots\alpha_{q}}
  45. ε α 1 α n \varepsilon_{\alpha_{1}\dots\alpha_{n}}\,
  46. A [ α β ] γ = 1 2 ! ( A α β γ - A β α γ ) A_{[\alpha\beta]\gamma\cdots}=\dfrac{1}{2!}\left(A_{\alpha\beta\gamma\cdots}-A% _{\beta\alpha\gamma\cdots}\right)
  47. A [ α β γ ] δ = 1 3 ! ( A α β γ δ + A γ α β δ + A β γ α δ - A α γ β δ - A γ β α δ - A β α γ δ ) A_{[\alpha\beta\gamma]\delta\cdots}=\dfrac{1}{3!}\left(A_{\alpha\beta\gamma% \delta\cdots}+A_{\gamma\alpha\beta\delta\cdots}+A_{\beta\gamma\alpha\delta% \cdots}-A_{\alpha\gamma\beta\delta\cdots}-A_{\gamma\beta\alpha\delta\cdots}-A_% {\beta\alpha\gamma\delta\cdots}\right)
  48. 0 = F [ α β , γ ] = 1 3 ! ( F α β , γ + F γ α , β + F β γ , α - F β α , γ - F α γ , β - F γ β , α ) 0=F_{[\alpha\beta,\gamma]}=\dfrac{1}{3!}\left(F_{\alpha\beta,\gamma}+F_{\gamma% \alpha,\beta}+F_{\beta\gamma,\alpha}-F_{\beta\alpha,\gamma}-F_{\alpha\gamma,% \beta}-F_{\gamma\beta,\alpha}\right)\,
  49. A [ α ( B β ] γ + C β ] γ ) = A [ α B β ] γ + A [ α C β ] γ A_{[\alpha}\left(B_{\beta]\gamma\cdots}+C_{\beta]\gamma\cdots}\right)=A_{[% \alpha}B_{\beta]\gamma\cdots}+A_{[\alpha}C_{\beta]\gamma\cdots}
  50. A [ α B β = γ ] 1 2 ! ( A α B β - γ A γ B β ) α A_{[\alpha}B^{\beta}{}_{\gamma]}=\dfrac{1}{2!}\left(A_{\alpha}B^{\beta}{}_{% \gamma}-A_{\gamma}B^{\beta}{}_{\alpha}\right)
  51. A [ α B | β | = γ ] 1 2 ! ( A α B β γ - A γ B β α ) A_{[\alpha}B_{|\beta|}{}_{\gamma]}=\dfrac{1}{2!}\left(A_{\alpha}B_{\beta\gamma% }-A_{\gamma}B_{\beta\alpha}\right)
  52. A α β γ = A ( α β ) γ + A [ α β ] γ A_{\alpha\beta\gamma\cdots}=A_{(\alpha\beta)\gamma\cdots}+A_{[\alpha\beta]% \gamma\cdots}
  53. A ( α β ) γ A_{(\alpha\beta)\gamma\cdots}
  54. A [ α β ] γ A_{[\alpha\beta]\gamma\cdots}
  55. x γ x^{\gamma}
  56. A α β , γ = x γ A α β A_{\alpha\beta\cdots,\gamma}=\dfrac{\partial}{\partial x^{\gamma}}A_{\alpha% \beta\cdots}
  57. A α 1 α 2 α p , α p + 1 α q = q - p x α q x α p + 2 x α p + 1 A α 1 α 2 α p . A_{\alpha_{1}\alpha_{2}\cdots\alpha_{p}\,,\,\alpha_{p+1}\cdots\alpha_{q}}=% \dfrac{\partial^{q-p}}{\partial x^{\alpha_{q}}\cdots\partial x^{\alpha_{p+2}}% \partial x^{\alpha_{p+1}}}A_{\alpha_{1}\alpha_{2}\cdots\alpha_{p}}.
  58. x α = , γ δ α γ x^{\alpha}{}_{,\gamma}=\delta^{\alpha}{}_{\gamma}
  59. A α = ; β A α + , β Γ α A γ γ β A^{\alpha}{}_{;\beta}=A^{\alpha}{}_{,\beta}+\Gamma^{\alpha}{}_{\gamma\beta}A^{\gamma}
  60. Γ α β γ \Gamma^{\alpha}{}_{\beta\gamma}\,
  61. A α ; β = A α , β - Γ γ A γ α β . A_{\alpha;\beta}=A_{\alpha,\beta}-\Gamma^{\gamma}{}_{\alpha\beta}A_{\gamma}\,.
  62. T α 1 α r = β 1 β s ; γ T α 1 α r β 1 β s , γ + Γ α 1 T δ α 2 α r δ γ + β 1 β s + Γ α r T α 1 α r - 1 δ δ γ β 1 β s - Γ δ T α 1 α r β 1 γ - δ β 2 β s - Γ δ T α 1 α r β s γ . β 1 β s - 1 δ \begin{aligned}\displaystyle T^{\alpha_{1}\cdots\alpha_{r}}{}_{\beta_{1}\cdots% \beta_{s};\gamma}=T^{\alpha_{1}\cdots\alpha_{r}}{}_{\beta_{1}\cdots\beta_{s},% \gamma}&\displaystyle+\,\Gamma^{\alpha_{1}}{}_{\delta\gamma}T^{\delta\alpha_{2% }\cdots\alpha_{r}}{}_{\beta_{1}\cdots\beta_{s}}+\cdots+\Gamma^{\alpha_{r}}{}_{% \delta\gamma}T^{\alpha_{1}\cdots\alpha_{r-1}\delta}{}_{\beta_{1}\cdots\beta_{s% }}\\ &\displaystyle-\,\Gamma^{\delta}{}_{\beta_{1}\gamma}T^{\alpha_{1}\cdots\alpha_% {r}}{}_{\delta\beta_{2}\cdots\beta_{s}}-\cdots-\Gamma^{\delta}{}_{\beta_{s}% \gamma}T^{\alpha_{1}\cdots\alpha_{r}}{}_{\beta_{1}\cdots\beta_{s-1}\delta}\,.% \end{aligned}
  63. g μ ν g_{\mu\nu}\,
  64. g μ ν ; γ = 0 . g_{\mu\nu;\gamma}=0\,.
  65. v γ v^{\gamma}
  66. v γ A α ; γ . v^{\gamma}A_{\alpha;\gamma}\,.
  67. β \nabla_{\beta}
  68. A α A^{\alpha}
  69. β A α = A α x β + Γ α A γ γ β . \nabla_{\beta}A^{\alpha}=\frac{\partial A^{\alpha}}{\partial x^{\beta}}+\Gamma% ^{\alpha}{}_{\gamma\beta}A^{\gamma}.
  70. T T
  71. X ρ X^{\rho}
  72. ( X T ) α 1 α r = β 1 β s X γ T α 1 α r β 1 β s , γ - X α 1 T γ α 2 α r , γ - β 1 β s - X α r T α 1 α r - 1 γ , γ β 1 β s + X γ T α 1 α r , β 1 + γ β 2 β s + X γ T α 1 α r , β s . β 1 β s - 1 γ \begin{aligned}\displaystyle(\mathcal{L}_{X}T)^{\alpha_{1}\cdots\alpha_{r}}{}_% {\beta_{1}\cdots\beta_{s}}=X^{\gamma}T^{\alpha_{1}\cdots\alpha_{r}}{}_{\beta_{% 1}\cdots\beta_{s},\gamma}&\displaystyle-\,X^{\alpha_{1}}{}_{,\gamma}T^{\gamma% \alpha_{2}\cdots\alpha_{r}}{}_{\beta_{1}\cdots\beta_{s}}-\cdots-X^{\alpha_{r}}% {}_{,\gamma}T^{\alpha_{1}\cdots\alpha_{r-1}\gamma}{}_{\beta_{1}\cdots\beta_{s}% }\\ &\displaystyle+\,X^{\gamma}{}_{,\beta_{1}}T^{\alpha_{1}\cdots\alpha_{r}}{}_{% \gamma\beta_{2}\cdots\beta_{s}}+\cdots+X^{\gamma}{}_{,\beta_{s}}T^{\alpha_{1}% \cdots\alpha_{r}}{}_{\beta_{1}\cdots\beta_{s-1}\gamma}\,.\end{aligned}
  73. X ρ X^{\rho}
  74. ( X X ) ρ = [ X , X ] ρ = 0 . (\mathcal{L}_{X}X)^{\rho}=[X,X]^{\rho}=0\,.
  75. Λ \Lambda
  76. w w\,
  77. X ρ X^{\rho}
  78. ( X Λ ) α 1 α r = β 1 β s X γ Λ α 1 α r β 1 β s , γ - X α 1 Λ γ α 2 α r , γ - β 1 β s - X α r Λ α 1 α r - 1 γ , γ β 1 β s + X γ Λ α 1 α r , β 1 + γ β 2 β s + X γ Λ α 1 α r , β s β 1 β s - 1 γ + w X γ Λ α 1 α r , γ . β 1 β s \begin{aligned}\displaystyle(\mathcal{L}_{X}\Lambda)^{\alpha_{1}\cdots\alpha_{% r}}{}_{\beta_{1}\cdots\beta_{s}}=X^{\gamma}\Lambda^{\alpha_{1}\cdots\alpha_{r}% }{}_{\beta_{1}\cdots\beta_{s},\gamma}&\displaystyle-\,X^{\alpha_{1}}{}_{,% \gamma}\Lambda^{\gamma\alpha_{2}\cdots\alpha_{r}}{}_{\beta_{1}\cdots\beta_{s}}% -\cdots-X^{\alpha_{r}}{}_{,\gamma}\Lambda^{\alpha_{1}\cdots\alpha_{r-1}\gamma}% {}_{\beta_{1}\cdots\beta_{s}}\\ &\displaystyle+\,X^{\gamma}{}_{,\beta_{1}}\Lambda^{\alpha_{1}\cdots\alpha_{r}}% {}_{\gamma\beta_{2}\cdots\beta_{s}}+\cdots+X^{\gamma}{}_{,\beta_{s}}\Lambda^{% \alpha_{1}\cdots\alpha_{r}}{}_{\beta_{1}\cdots\beta_{s-1}\gamma}\\ &\displaystyle+\,wX^{\gamma}{}_{,\gamma}\Lambda^{\alpha_{1}\cdots\alpha_{r}}{}% _{\beta_{1}\cdots\beta_{s}}\,.\end{aligned}
  79. δ β α A β = A α \delta^{\alpha}_{\beta}\,A^{\beta}=A^{\alpha}\,
  80. δ ν μ B μ = B ν \delta^{\mu}_{\nu}\,B_{\mu}=B_{\nu}\,
  81. δ β α \delta^{\alpha}_{\beta}\,
  82. δ ρ ρ = δ 0 0 + δ 1 1 + δ 2 2 + δ 3 3 = 4 \delta^{\rho}_{\rho}=\delta^{0}_{0}+\delta^{1}_{1}+\delta^{2}_{2}+\delta^{3}_{% 3}=4\,
  83. Length = y 1 y 2 g α β d x α d y d x β d y d y \,\text{Length}=\int^{y_{2}}_{y_{1}}\sqrt{g_{\alpha\beta}\frac{dx^{\alpha}}{dy% }\frac{dx^{\beta}}{dy}}\,dy\,
  84. Duration = t 1 t 2 - 1 c 2 g α β d x α d t d x β d t d t \,\text{Duration}=\int^{t_{2}}_{t_{1}}\sqrt{\frac{-1}{c^{2}}g_{\alpha\beta}% \frac{dx^{\alpha}}{dt}\frac{dx^{\beta}}{dt}}\,dt\,
  85. g α β g β γ = δ γ α . g^{\alpha\beta}g_{\beta\gamma}=\delta^{\alpha}_{\gamma}\,.
  86. R ρ = σ μ ν Γ ρ - ν σ , μ Γ μ σ , ν ρ + Γ ρ Γ λ μ λ - ν σ Γ ρ Γ λ ν λ , μ σ R^{\rho}{}_{\sigma\mu\nu}=\Gamma^{\rho}{}_{\nu\sigma,\mu}-\Gamma^{\rho}_{\mu% \sigma,\nu}+\Gamma^{\rho}{}_{\mu\lambda}\Gamma^{\lambda}{}_{\nu\sigma}-\Gamma^% {\rho}{}_{\nu\lambda}\Gamma^{\lambda}{}_{\mu\sigma}\,,
  87. A ν ; ρ σ - A ν ; σ ρ = A β R β , ν ρ σ A_{\nu;\rho\sigma}-A_{\nu;\sigma\rho}=A_{\beta}R^{\beta}{}_{\nu\rho\sigma}\,,
  88. Γ α β μ \Gamma^{\alpha}{}_{\beta\mu}\,
  89. Γ λ - μ ν Γ λ ν μ \Gamma^{\lambda}{}_{\mu\nu}-\Gamma^{\lambda}{}_{\nu\mu}\,
  90. T α 1 α r - β 1 β s ; γ δ T α 1 α r = β 1 β s ; δ γ - R α 1 T ρ α 2 α r ρ γ δ - β 1 β s - R α r T α 1 α r - 1 ρ ρ γ δ β 1 β s + R σ T α 1 α r β 1 γ δ + σ β 2 β s + R σ T α 1 α r β s γ δ β 1 β s - 1 σ \begin{aligned}\displaystyle T^{\alpha_{1}\cdots\alpha_{r}}{}_{\beta_{1}\cdots% \beta_{s};\gamma\delta}-T^{\alpha_{1}\cdots\alpha_{r}}{}_{\beta_{1}\cdots\beta% _{s};\delta\gamma}=&\displaystyle-R^{\alpha_{1}}{}_{\rho\gamma\delta}T^{\rho% \alpha_{2}\cdots\alpha_{r}}{}_{\beta_{1}\cdots\beta_{s}}-\cdots-R^{\alpha_{r}}% {}_{\rho\gamma\delta}T^{\alpha_{1}\cdots\alpha_{r-1}\rho}{}_{\beta_{1}\cdots% \beta_{s}}\\ &\displaystyle+\,R^{\sigma}{}_{\beta_{1}\gamma\delta}T^{\alpha_{1}\cdots\alpha% _{r}}{}_{\sigma\beta_{2}\cdots\beta_{s}}+\cdots+R^{\sigma}{}_{\beta_{s}\gamma% \delta}T^{\alpha_{1}\cdots\alpha_{r}}{}_{\beta_{1}\cdots\beta_{s-1}\sigma}\end% {aligned}

Ricci_scalars_(Newman–Penrose_formalism).html

  1. { Φ 00 , Φ 11 , Φ 22 } \{\Phi_{00},\Phi_{11},\Phi_{22}\}
  2. { Φ 01 = Φ 10 ¯ , Φ 02 = Φ 20 ¯ , Φ 12 = Φ 21 ¯ } \{\Phi_{01}=\overline{\Phi_{10}}\,,\Phi_{02}=\overline{\Phi_{20}}\,,\Phi_{12}=% \overline{\Phi_{21}}\}
  3. Λ \Lambda
  4. { l a , n a , m a , m ¯ a } \{l^{a},n^{a},m^{a},\bar{m}^{a}\}
  5. { ( - , + , + , + ) ; l a n a = - 1 , m a m ¯ a = 1 } \{(-,+,+,+);l^{a}n_{a}=-1\,,m^{a}\bar{m}_{a}=1\}
  6. Φ 00 := 1 2 R a b l a l b , Φ 11 := 1 4 R a b ( l a n b + m a m ¯ b ) , Φ 22 := 1 2 R a b n a n b , Λ := R 24 ; \Phi_{00}:=\frac{1}{2}R_{ab}l^{a}l^{b}\,,\quad\Phi_{11}:=\frac{1}{4}R_{ab}(\,l% ^{a}n^{b}+m^{a}\bar{m}^{b})\,,\quad\Phi_{22}:=\frac{1}{2}R_{ab}n^{a}n^{b}\,,% \quad\Lambda:=\frac{R}{24}\,;
  7. Φ 01 := 1 2 R a b l a m b , Φ 10 := 1 2 R a b l a m ¯ b = Φ 01 ¯ , \Phi_{01}:=\frac{1}{2}R_{ab}l^{a}m^{b}\,,\quad\;\Phi_{10}:=\frac{1}{2}R_{ab}l^% {a}\bar{m}^{b}=\overline{\Phi_{01}}\,,
  8. Φ 02 := 1 2 R a b m a m b , Φ 20 := 1 2 R a b m ¯ a m ¯ b = Φ 02 ¯ , \Phi_{02}:=\frac{1}{2}R_{ab}m^{a}m^{b}\,,\quad\Phi_{20}:=\frac{1}{2}R_{ab}\bar% {m}^{a}\bar{m}^{b}=\overline{\Phi_{02}}\,,
  9. Φ 12 := 1 2 R a b m ¯ a n b , Φ 21 := 1 2 R a b m a n b = Φ 12 ¯ . \Phi_{12}:=\frac{1}{2}R_{ab}\bar{m}^{a}n^{b}\,,\quad\;\Phi_{21}:=\frac{1}{2}R_% {ab}m^{a}n^{b}=\overline{\Phi_{12}}\,.
  10. R a b R_{ab}
  11. Q a b = R a b - 1 4 g a b R Q_{ab}=R_{ab}-\frac{1}{4}g_{ab}R
  12. G a b = R a b - 1 2 g a b R G_{ab}=R_{ab}-\frac{1}{2}g_{ab}R
  13. l a l a = n a n a = m a m a = m ¯ a m ¯ a = 0 , l_{a}l^{a}=n_{a}n^{a}=m_{a}m^{a}=\bar{m}_{a}\bar{m}^{a}=0\,,
  14. l a m a = l a m ¯ a = n a m a = n a m ¯ a = 0 . l_{a}m^{a}=l_{a}\bar{m}^{a}=n_{a}m^{a}=n_{a}\bar{m}^{a}=0\,.
  15. Λ = 0 \Lambda=0
  16. 24 Λ = 0 = R a b g a b = R a b ( - 2 l a n b + 2 m a m ¯ b ) R a b l a n b = R a b m a m ¯ b , 24\Lambda\,=0=\,R_{ab}g^{ab}\,=\,R_{ab}\Big(-2l^{a}n^{b}+2m^{a}\bar{m}^{b}\Big% )\;\Rightarrow\;R_{ab}l^{a}n^{b}\,=\,R_{ab}m^{a}\bar{m}^{b}\,,
  17. Φ 11 \Phi_{11}
  18. Φ 11 := 1 4 R a b ( l a n b + m a m ¯ b ) = 1 2 R a b l a n b = 1 2 R a b m a m ¯ a . \Phi_{11}:=\frac{1}{4}R_{ab}(\,l^{a}n^{b}+m^{a}\bar{m}^{b})=\frac{1}{2}R_{ab}l% ^{a}n^{b}=\frac{1}{2}R_{ab}m^{a}\bar{m}^{a}\,.
  19. { ( + , - , - , - ) ; l a n a = 1 , m a m ¯ a = - 1 } \{(+,-,-,-);l^{a}n_{a}=1\,,m^{a}\bar{m}_{a}=-1\}
  20. Φ i j \Phi_{ij}
  21. Φ i j - Φ i j \Phi_{ij}\mapsto-\Phi_{ij}
  22. Φ i j \Phi_{ij}
  23. Φ 00 = D ρ - δ ¯ κ - ( ρ 2 + σ σ ¯ ) - ( ε + ε ¯ ) ρ + κ ¯ τ + κ ( 3 α + β ¯ - π ) , \Phi_{00}=D\rho-\bar{\delta}\kappa-(\rho^{2}+\sigma\bar{\sigma})-(\varepsilon+% \bar{\varepsilon})\rho+\bar{\kappa}\tau+\kappa(3\alpha+\bar{\beta}-\pi)\,,
  24. Φ 10 = D α - δ ¯ ε - ( ρ + ε ¯ - 2 ε ) α - β σ ¯ + β ¯ ε + κ λ + κ ¯ γ - ( ε + ρ ) π , \Phi_{10}=D\alpha-\bar{\delta}\varepsilon-(\rho+\bar{\varepsilon}-2\varepsilon% )\alpha-\beta\bar{\sigma}+\bar{\beta}\varepsilon+\kappa\lambda+\bar{\kappa}% \gamma-(\varepsilon+\rho)\pi\,,
  25. Φ 02 = δ τ - Δ σ - ( μ σ + λ ¯ ρ ) - ( τ + β - α ¯ ) τ + ( 3 γ - γ ¯ ) σ + κ ν ¯ , \Phi_{02}=\delta\tau-\Delta\sigma-(\mu\sigma+\bar{\lambda}\rho)-(\tau+\beta-% \bar{\alpha})\tau+(3\gamma-\bar{\gamma})\sigma+\kappa\bar{\nu}\,,
  26. Φ 20 = D λ - δ ¯ π - ( ρ λ + σ ¯ μ ) - π 2 - ( α - β ¯ ) π + ν κ ¯ + ( 3 ε - ε ¯ ) λ , \Phi_{20}=D\lambda-\bar{\delta}\pi-(\rho\lambda+\bar{\sigma}\mu)-\pi^{2}-(% \alpha-\bar{\beta})\pi+\nu\bar{\kappa}+(3\varepsilon-\bar{\varepsilon})\lambda\,,
  27. Φ 12 = δ γ - Δ β - ( τ - α ¯ - β ) γ - μ τ + σ ν + ε ν ¯ + ( γ - γ ¯ - μ ) β - α λ ¯ , \Phi_{12}=\delta\gamma-\Delta\beta-(\tau-\bar{\alpha}-\beta)\gamma-\mu\tau+% \sigma\nu+\varepsilon\bar{\nu}+(\gamma-\bar{\gamma}-\mu)\beta-\alpha\bar{% \lambda}\,,
  28. Φ 22 = δ ν - Δ μ - ( μ 2 + λ λ ¯ ) - ( γ + γ ¯ ) μ + ν ¯ π - ( τ - 3 β - α ¯ ) ν , \Phi_{22}=\delta\nu-\Delta\mu-(\mu^{2}+\lambda\bar{\lambda})-(\gamma+\bar{% \gamma})\mu+\bar{\nu}\pi-(\tau-3\beta-\bar{\alpha})\nu\,,
  29. 2 Φ 11 = D γ - Δ ε + δ α - δ ¯ β - ( τ + π ¯ ) α - α α ¯ - ( τ ¯ + π ) β - β β ¯ + 2 α β + ( ε + ε ¯ ) γ - ( ρ - ρ ¯ ) γ + ( γ + γ ¯ ) ε - ( μ - μ ¯ ) ε - τ π + ν κ - ( μ ρ - λ σ ) , 2\Phi_{11}=D\gamma-\Delta\varepsilon+\delta\alpha-\bar{\delta}\beta-(\tau+\bar% {\pi})\alpha-\alpha\bar{\alpha}-(\bar{\tau}+\pi)\beta-\beta\bar{\beta}+2\alpha% \beta+(\varepsilon+\bar{\varepsilon})\gamma-(\rho-\bar{\rho})\gamma+(\gamma+% \bar{\gamma})\varepsilon-(\mu-\bar{\mu})\varepsilon-\tau\pi+\nu\kappa-(\mu\rho% -\lambda\sigma)\,,
  30. Λ \Lambda
  31. Λ = R 24 \Lambda=\frac{R}{24}
  32. R R
  33. g a b = - l a n b - n a l b + m a m ¯ b + m ¯ a m b g_{ab}=-l_{a}n_{b}-n_{a}l_{b}+m_{a}\bar{m}_{b}+\bar{m}_{a}m_{b}
  34. Φ i j \Phi_{ij}
  35. R a b R_{ab}
  36. G a b G_{ab}
  37. Φ i j \Phi_{ij}
  38. G a b = 8 π T a b G_{ab}=8\pi T_{ab}
  39. T a b = 0 T_{ab}=0
  40. Φ i j = 0 \Phi_{ij}=0
  41. Φ i j \Phi_{ij}
  42. Φ i j = 2 ϕ i ϕ j ¯ , ( i , j { 0 , 1 , 2 } ) , \Phi_{ij}=\,2\,\phi_{i}\,\overline{\phi_{j}}\,,\quad(i,j\in\{0,1,2\})\,,
  43. ϕ i \phi_{i}
  44. F a b F_{ab}
  45. ϕ 0 := - F a b l a m b , ϕ 1 := - 1 2 F a b ( l a n a - m a m ¯ b ) , ϕ 2 := F a b n a m ¯ b . \phi_{0}:=-F_{ab}l^{a}m^{b}\,,\quad\phi_{1}:=-\frac{1}{2}F_{ab}\big(l^{a}n^{a}% -m^{a}\bar{m}^{b}\big)\,,\quad\phi_{2}:=F_{ab}n^{a}\bar{m}^{b}\,.
  46. Φ i j = 2 ϕ i ϕ j ¯ \Phi_{ij}=2\,\phi_{i}\,\overline{\phi_{j}}
  47. Φ i j = Tr ( ϝ i ϝ ¯ j ) \Phi_{ij}=\,\,\text{Tr}\,(\digamma_{i}\,\bar{\digamma}_{j})
  48. ϝ i ( i { 0 , 1 , 2 } ) \digamma_{i}(i\in\{0,1,2\})

Rice's_formula.html

  1. 𝔼 ( D u ) = - | x | p ( u , x ) d x \mathbb{E}(D_{u})=\int_{-\infty}^{\infty}|x^{\prime}|p(u,x^{\prime})\,\mathrm{% d}x^{\prime}
  2. 𝔼 ( D 0 ) = 1 π - ρ ′′ ( 0 ) \mathbb{E}(D_{0})=\frac{1}{\pi}\sqrt{-\rho^{\prime\prime}(0)}
  3. { sup t [ 0 , 1 ] X ( t ) u } \mathbb{P}\left\{\sup_{t\in[0,1]}X(t)\geq u\right\}

Richmond_surface.html

  1. f ( z ) = 1 / z m , g ( z ) = z m f(z)=1/z^{m},g(z)=z^{m}
  2. X ( z ) = [ ( - 1 / 2 z ) - z 2 m + 1 / ( 4 m + 2 ) ] Y ( z ) = [ ( - i / 2 z ) + i z 2 m + 1 / ( 4 m + 2 ) ] Z ( z ) = [ z n / n ] \begin{aligned}\displaystyle X(z)&\displaystyle=\Re[(-1/2z)-z^{2m+1}/(4m+2)]\\ \displaystyle Y(z)&\displaystyle=\Re[(-i/2z)+iz^{2m+1}/(4m+2)]\\ \displaystyle Z(z)&\displaystyle=\Re[z^{n}/n]\end{aligned}
  3. X ( u , v ) = ( 1 / 3 ) u 3 - u v 2 + u u 2 + v 2 Y ( u , v ) = - u 2 v + ( 1 / 3 ) v 3 - u u 2 + v 2 Z ( u , v ) = 2 u \begin{aligned}\displaystyle X(u,v)&\displaystyle=(1/3)u^{3}-uv^{2}+\frac{u}{u% ^{2}+v^{2}}\\ \displaystyle Y(u,v)&\displaystyle=-u^{2}v+(1/3)v^{3}-\frac{u}{u^{2}+v^{2}}\\ \displaystyle Z(u,v)&\displaystyle=2u\end{aligned}

Riemann–Silberstein_vector.html

  1. 𝐅 = 𝐄 + i c 𝐁 \mathbf{F}=\mathbf{E}+ic\mathbf{B}
  2. ϵ 0 / 2 \sqrt{\epsilon_{0}/2}
  3. 𝔈 + i 𝔐 \mathfrak{E}+i\ \mathfrak{M}
  4. curl ( 𝔈 + i 𝔐 ) = i c ( 𝔈 + i 𝔐 ) t \operatorname{curl}(\mathfrak{E}+i\ \mathfrak{M})=\frac{i}{c}\ \frac{\partial(% \mathfrak{E}+i\ \mathfrak{M})}{\partial t}
  5. ( 1 c t + s y m b o l ) 𝐅 = 1 ϵ 0 ( ρ - 1 c 𝐉 ) . \left(\frac{1}{c}\dfrac{\partial}{\partial t}+symbol{\nabla}\right)\mathbf{F}=% \frac{1}{\epsilon_{0}}\left(\rho-\frac{1}{c}\mathbf{J}\right).
  6. 𝐅 2 = 𝐄 2 - c 2 𝐁 2 + 2 i c 𝐄 𝐁 \mathbf{F}^{2}=\mathbf{E}^{2}-c^{2}\mathbf{B}^{2}+2ic\mathbf{E}\cdot\mathbf{B}
  7. ϵ 0 2 𝐅 𝐅 = ϵ 0 2 ( 𝐄 2 + c 2 𝐁 2 ) + 1 c 𝐒 , \frac{\epsilon_{0}}{2}\mathbf{F}^{\dagger}\mathbf{F}=\frac{\epsilon_{0}}{2}% \left(\mathbf{E}^{2}+c^{2}\mathbf{B}^{2}\right)+\frac{1}{c}\mathbf{S},
  8. \hbar
  9. i t 𝐅 = c ( 𝐒 i ) 𝐅 = c ( 𝐒 𝐩 ) 𝐅 i\hbar\partial_{t}{\mathbf{F}}=c(\mathbf{S}\cdot{\hbar\over i}\nabla)\mathbf{F% }=c(\mathbf{S}\cdot\mathbf{p})\mathbf{F}
  10. 𝐒 {\mathbf{S}}
  11. 𝐅 = 1 c 𝐅 * ( x ) 𝐅 ( x ) | x - x | 2 d x 3 d x 3 = 1 \|\mathbf{F}\|={1\over\hbar c}\int{\mathbf{F}^{*}(x)\cdot\mathbf{F}(x^{\prime}% )\over|x-x^{\prime}|^{2}}dx^{3}dx^{\prime 3}=1
  12. 𝐅 = 0 \nabla\cdot\mathbf{F}=0
  13. t = 0 t=0
  14. 𝐅 ( 0 ) = × 𝐆 \mathbf{F}(0)=\nabla\times\mathbf{G}
  15. 𝐆 \mathbf{G}
  16. E / c E/c
  17. H / c H/c
  18. r r
  19. A , B A,B
  20. σ A σ B 1 2 | [ A ^ , B ^ ] | . \sigma_{A}\sigma_{B}\geq\frac{1}{2}\left|\langle[\hat{A},\hat{B}]\rangle\right|.
  21. σ r σ p \sigma_{r}\sigma_{p}
  22. r 2 = x 2 + y 2 + z 2 r^{2}=x^{2}+y^{2}+z^{2}
  23. p 2 = ( 𝐒 𝐩 ) 2 p^{2}=(\mathbf{S}\cdot\mathbf{p})^{2}
  24. r ~ \tilde{r}
  25. r r
  26. r ~ = α 1 x + α 2 y + α 3 z \tilde{r}=\alpha_{1}x+\alpha_{2}y+\alpha_{3}z
  27. α i \alpha_{i}
  28. α i 2 = 1 \alpha_{i}^{2}=1
  29. α i α k + α k α i = 2 δ i k \alpha_{i}\alpha_{k}+\alpha_{k}\alpha_{i}=2\delta_{ik}
  30. r ~ 2 = r 2 \tilde{r}^{2}=r^{2}
  31. 3 × 3 3\times 3
  32. 3 / 2 1 3/2\approx 1
  33. 1 / 2 1/2
  34. 1 / 2 1/2
  35. p ~ 2 = ( ~ L 𝐩 ) 2 \tilde{p}^{2}=(\mathbf{\tilde{}}L\cdot\mathbf{p})^{2}
  36. α i \alpha_{i}
  37. L ~ i \tilde{L}_{i}
  38. e - a r 2 e^{-ar^{2}}
  39. x p y xp_{y}
  40. L z α z = α z L z = 0 L_{z}\alpha_{z}=\alpha_{z}L_{z}=0
  41. 4 × 4 4\times 4
  42. 2 3 2\sqrt{3}
  43. L 2 , 1 L2,1
  44. 48 49 = 7 2 8 2 48\approx 49=7^{2}\approx 8^{2}
  45. || 𝐀𝐱 || || 𝐀 || || 𝐱 || || 𝐀 || || 𝐱 || ||\mathbf{A}\mathbf{x}||\leq||\mathbf{A}||||\mathbf{x}||\approx||\mathbf{A}|||% |\mathbf{x}||
  46. | [ r ~ , p ~ ] | 8 . \left|\langle[\tilde{r},\tilde{p}]\rangle\right|\geq 8\hbar.
  47. σ r σ p 4 \sigma_{r}\sigma_{p}\geq 4\hbar
  48. σ r σ p 3 2 \sigma_{r}\sigma_{p}\geq\frac{3}{2}\hbar
  49. 8 / 3 8/3
  50. r 2 r^{2}

Riesz_rearrangement_inequality.html

  1. I ( f , g , h ) = n × n f ( x ) g ( x - y ) h ( y ) d x d y I(f,g,h)=\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}f(x)g(x-y)h(y)\,dxdy
  2. I ( f , g , h ) I ( f * , g * , h * ) I(f,g,h)\leq I(f^{*},g^{*},h^{*})
  3. f * , g * , h * f^{*},g^{*},h^{*}

Riesz–Markov–Kakutani_representation_theorem.html

  1. μ ( E ) = inf { μ ( U ) : E U , U open } \mu(E)=\inf\{\mu(U):E\subseteq U,U\mbox{ open}~{}\}
  2. μ ( E ) = sup { μ ( K ) : K E , K compact } \mu(E)=\sup\{\mu(K):K\subseteq E,K\mbox{ compact}~{}\}
  3. ψ ( f ) = X f ( x ) d μ ( x ) \psi(f)=\int_{X}f(x)\,d\mu(x)\quad
  4. A [ f ] = 0 1 f ( x ) d α ( x ) , A[f]=\int_{0}^{1}f(x)\,d\alpha(x),
  5. ψ ( f ) = X f ( x ) d μ ( x ) \psi(f)=\int_{X}f(x)\,d\mu(x)\quad
  6. ψ = | μ | ( X ) . \|\psi\|=|\mu|(X).

Right_kite.html

  1. tan A 2 = b a , tan C 2 = a b \tan{\frac{A}{2}}=\frac{b}{a},\qquad\tan{\frac{C}{2}}=\frac{a}{b}
  2. K = a b . \displaystyle K=ab.
  3. p = a 2 + b 2 p=\sqrt{a^{2}+b^{2}}
  4. K = p q 2 K=\frac{pq}{2}
  5. q = 2 a b a 2 + b 2 . q=\frac{2ab}{\sqrt{a^{2}+b^{2}}}.
  6. R = 1 2 a 2 + b 2 R=\tfrac{1}{2}\sqrt{a^{2}+b^{2}}
  7. r = K s = a b a + b r=\frac{K}{s}=\frac{ab}{a+b}
  8. K = r ( r + 4 R 2 + r 2 ) . K=r(r+\sqrt{4R^{2}+r^{2}}).

Rigidity_matroid.html

  1. d n - ( d + 1 2 ) dn-{\left({{d+1}\atop{2}}\right)}
  2. d n - ( d + 1 2 ) dn-{\left({{d+1}\atop{2}}\right)}
  3. ( v , i ) (v,i)
  4. v v
  5. i i
  6. d d
  7. ( d + 1 2 ) {\left({{d+1}\atop{2}}\right)}
  8. d n - ( d + 1 2 ) dn-{\left({{d+1}\atop{2}}\right)}
  9. e e
  10. S S
  11. e e
  12. S S
  13. d n - ( d + 1 2 ) dn-{\left({{d+1}\atop{2}}\right)}
  14. ( d + 1 ) (d+1)
  15. ( k , l ) (k,l)
  16. n n
  17. k n - l kn-l
  18. ( k , l ) (k,l)
  19. ( k , l ) (k,l)
  20. k n - l kn-l
  21. ( d , ( d + 1 2 ) ) (d,{\left({{d+1}\atop{2}}\right)})
  22. ( d , ( d + 1 2 ) ) (d,{\left({{d+1}\atop{2}}\right)})
  23. n n
  24. G G
  25. G G
  26. G G
  27. ( d , ( d + 1 2 ) ) (d,{\left({{d+1}\atop{2}}\right)})

Ring_of_p-adic_periods.html

  1. 𝐁 d R \mathbf{B}_{dR}
  2. 𝐂 p \mathbf{C}_{p}
  3. 𝐐 p ¯ \overline{\mathbf{Q}_{p}}
  4. 𝐄 ~ + = lim x x p 𝒪 𝐂 p / ( p ) \tilde{\mathbf{E}}^{+}=\underleftarrow{\lim}_{x\mapsto x^{p}}\mathcal{O}_{% \mathbf{C}_{p}}/(p)
  5. 𝐄 ~ + \tilde{\mathbf{E}}^{+}
  6. ( x 1 , x 2 , ) (x_{1},x_{2},\cdots)
  7. x i 𝒪 𝐂 p / ( p ) x_{i}\in\mathcal{O}_{\mathbf{C}_{p}}/(p)
  8. x i + 1 p x i ( mod p ) x_{i+1}^{p}\equiv x_{i}\;\;(\mathop{{\rm mod}}p)
  9. f : 𝐄 ~ + 𝒪 𝐂 p / ( p ) f:\tilde{\mathbf{E}}^{+}\to\mathcal{O}_{\mathbf{C}_{p}}/(p)
  10. f ( x 1 , x 2 , ) = x 1 f(x_{1},x_{2},\ldots)=x_{1}
  11. t : 𝐄 ~ + 𝒪 𝐂 p t:\tilde{\mathbf{E}}^{+}\to\mathcal{O}_{\mathbf{C}_{p}}
  12. t ( x , x 2 , ) = lim i x ~ i p i t(x_{,}x_{2},\ldots)=\lim_{i\to\infty}\tilde{x}_{i}^{p^{i}}
  13. x ~ i \tilde{x}_{i}
  14. x i x_{i}
  15. 𝒪 𝐂 p \mathcal{O}_{\mathbf{C}_{p}}
  16. t t
  17. 𝒪 𝐂 p 𝒪 𝐂 p / ( p ) \mathcal{O}_{\mathbf{C}_{p}}\to\mathcal{O}_{\mathbf{C}_{p}}/(p)
  18. f f
  19. θ : W ( 𝐄 ~ + ) 𝒪 𝐂 p \theta:W(\tilde{\mathbf{E}}^{+})\to\mathcal{O}_{\mathbf{C}_{p}}
  20. θ ( [ x ] ) = t ( x ) \theta([x])=t(x)
  21. x 𝐄 ~ + x\in\tilde{\mathbf{E}}^{+}
  22. [ x ] [x]
  23. x x
  24. 𝐁 d R + \mathbf{B}_{dR}^{+}
  25. 𝐁 ~ + = W ( 𝐄 ~ + ) [ 1 / p ] \tilde{\mathbf{B}}^{+}=W(\tilde{\mathbf{E}}^{+})[1/p]
  26. ker ( θ : 𝐁 ~ + 𝐂 p ) \ker\left(\theta:\tilde{\mathbf{B}}^{+}\to\mathbf{C}_{p}\right)
  27. 𝐁 d R \mathbf{B}_{dR}
  28. 𝐁 d R + \mathbf{B}_{dR}^{+}

Ring_of_polynomial_functions.html

  1. k [ t 1 , , t n ] k[t_{1},\dots,t_{n}]
  2. t i t_{i}
  3. k n k^{n}
  4. t i ( x ) = x i t_{i}(x)=x_{i}
  5. x = ( x 1 , , x n ) . x=(x_{1},\dots,x_{n}).
  6. V * V^{*}
  7. V k V\to k
  8. t i t_{i}
  9. t i t_{i}
  10. S q ( V ) S^{q}(V)
  11. λ : 1 q V k \textstyle\lambda:\prod_{1}^{q}V\to k
  12. λ ( v 1 , , v q ) \lambda(v_{1},\dots,v_{q})
  13. v i v_{i}
  14. S q ( V ) S^{q}(V)
  15. f ( v ) = λ ( v , , v ) . f(v)=\lambda(v,\dots,v).
  16. e i , 1 i n e_{i},\,1\leq i\leq n
  17. t i t_{i}
  18. λ ( v 1 , , v q ) = i 1 , , i q = 1 n λ ( e i 1 , , e i q ) t i 1 ( v 1 ) t i q ( v q ) \lambda(v_{1},\dots,v_{q})=\sum_{i_{1},\dots,i_{q}=1}^{n}\lambda(e_{i_{1}},% \dots,e_{i_{q}})t_{i_{1}}(v_{1})\cdots t_{i_{q}}(v_{q})
  19. ϕ : S q ( V ) k [ V ] q , ϕ ( λ ) ( v ) = λ ( v , , v ) . \phi:S^{q}(V)\to k[V]_{q},\,\phi(\lambda)(v)=\lambda(v,\cdots,v).
  20. f = i 1 , , i q = 1 n a i 0 i q t i 1 t i q f=\sum_{i_{1},\dots,i_{q}=1}^{n}a_{i_{0}\cdots i_{q}}t_{i_{1}}\cdots t_{i_{q}}
  21. a i 0 i q a_{i_{0}\cdots i_{q}}
  22. i 0 , , i q i_{0},\dots,i_{q}
  23. ψ ( f ) ( v 1 , , v q ) = i 1 , , i q = 1 n a i 0 i q t i 1 ( v 1 ) t i q ( v q ) . \psi(f)(v_{1},\dots,v_{q})=\sum_{i_{1},\cdots,i_{q}=1}^{n}a_{i_{0}\cdots i_{q}% }t_{i_{1}}(v_{1})\cdots t_{i_{q}}(v_{q}).

Riordan_array.html

  1. D D
  2. d ( t ) d(t)
  3. h ( t ) h(t)
  4. d n , k = [ t n ] d ( t ) ( t h ( t ) ) k d_{n,k}=[t^{n}]d(t)\left(th(t)\right)^{k}

Risk_of_ruin.html

  1. p ( ruin ) = ( 2 1 + μ r - 1 ) s r p(\mathrm{ruin})=\left(\frac{2}{1+\frac{\mu}{r}}-1\right)^{\frac{s}{r}}
  2. r = μ 2 + σ 2 r=\sqrt{\mu^{2}+\sigma^{2}}

Rock_mass_plasticity.html

  1. s y m b o l 𝐯 \displaystyle symbol{\nabla}\mathbf{v}
  2. 𝐯 \mathbf{v}
  3. s y m b o l S symbol{S}
  4. 𝐞 i \mathbf{e}_{i}
  5. s y m b o l A : s y m b o l B = i , j = 1 3 A i j B i j = trace ( s y m b o l A s y m b o l B T ) . symbol{A}:symbol{B}=\sum_{i,j=1}^{3}A_{ij}~{}B_{ij}=\operatorname{trace}(% symbol{A}symbol{B}^{T})~{}.
  6. ρ ˙ + ρ s y m b o l 𝐯 = 0 Balance of Mass ρ 𝐯 ˙ - s y m b o l s y m b o l σ - ρ 𝐛 = 0 Balance of Linear Momentum s y m b o l σ = s y m b o l σ T Balance of Angular Momentum ρ e ˙ - s y m b o l σ : ( s y m b o l 𝐯 ) + s y m b o l 𝐪 - ρ s = 0 Balance of Energy. {\begin{aligned}\displaystyle\dot{\rho}+\rho~{}symbol{\nabla}\cdot\mathbf{v}&% \displaystyle=0&&\displaystyle\qquad\,\text{Balance of Mass}\\ \displaystyle\rho~{}\dot{\mathbf{v}}-symbol{\nabla}\cdot symbol{\sigma}-\rho~{% }\mathbf{b}&\displaystyle=0&&\displaystyle\qquad\,\text{Balance of Linear % Momentum}\\ \displaystyle symbol{\sigma}&\displaystyle=symbol{\sigma}^{T}&&\displaystyle% \qquad\,\text{Balance of Angular Momentum}\\ \displaystyle\rho~{}\dot{e}-symbol{\sigma}:(symbol{\nabla}\mathbf{v})+symbol{% \nabla}\cdot\mathbf{q}-\rho~{}s&\displaystyle=0&&\displaystyle\qquad\,\text{% Balance of Energy.}\end{aligned}}
  7. ρ ( 𝐱 , t ) \rho(\mathbf{x},t)
  8. ρ ˙ \dot{\rho}
  9. ρ \rho
  10. 𝐯 ( 𝐱 , t ) = 𝐮 ˙ ( 𝐱 , t ) \mathbf{v}(\mathbf{x},t)=\dot{\mathbf{u}}(\mathbf{x},t)
  11. 𝐮 \mathbf{u}
  12. 𝐯 ˙ \dot{\mathbf{v}}
  13. 𝐯 \mathbf{v}
  14. s y m b o l σ ( 𝐱 , t ) symbol{\sigma}(\mathbf{x},t)
  15. 𝐛 ( 𝐱 , t ) \mathbf{b}(\mathbf{x},t)
  16. e ( 𝐱 , t ) e(\mathbf{x},t)
  17. e ˙ \dot{e}
  18. e e
  19. 𝐪 ( 𝐱 , t ) \mathbf{q}(\mathbf{x},t)
  20. s ( 𝐱 , t ) s(\mathbf{x},t)
  21. 𝐱 \mathbf{x}
  22. Ω [ s y m b o l σ 𝐰 - ρ 𝐛 𝐰 + ρ 𝐯 ˙ 𝐰 ] dV = Ω 𝐭 𝐰 dS \int_{\Omega}[symbol{\sigma}\cdot\nabla{\mathbf{w}}-\rho\,\mathbf{b}\cdot% \mathbf{w}+\rho\,\dot{\mathbf{v}}\cdot\mathbf{w}]\,\,\text{dV}=\int_{\partial% \Omega}\mathbf{t}\cdot\mathbf{w}\,\,\text{dS}
  23. Ω \Omega
  24. Ω \partial\Omega
  25. 𝐰 \mathbf{w}
  26. 𝐭 \mathbf{t}
  27. Ω \partial\Omega
  28. 𝐧 \cdotsymbol σ + + 𝐧 \cdotsymbol σ - 1 = 𝟎 or 𝐧 [ [ s y m b o l σ ] ] = 𝟎 \mathbf{n}\cdotsymbol{\sigma}^{+}+\mathbf{n}\cdotsymbol{\sigma}^{-1}=\mathbf{0% }\qquad\,\text{or}\qquad\mathbf{n}\cdot[[symbol{\sigma}]]=\mathbf{0}
  29. s y m b o l σ + , s y m b o l σ - symbol{\sigma}^{+},symbol{\sigma}^{-}
  30. Ω + , Ω - \Omega^{+},\Omega^{-}
  31. 𝐧 \mathbf{n}
  32. s y m b o l ε = 1 2 [ 𝐮 + ( 𝐮 ) T ] . symbol{\varepsilon}=\tfrac{1}{2}\left[\nabla\mathbf{u}+(\nabla\mathbf{u})^{T}% \right]\,.
  33. s y m b o l σ = 𝖧 : s y m b o l ε \,\,symbol{\sigma}=\mathsf{H}:symbol{\varepsilon}\,\,
  34. σ i j = H i j k l ε k l \,\,\sigma_{ij}=H_{ijkl}\,\varepsilon_{kl}\,\,
  35. s y m b o l σ = 2 μ s y m b o l ε + λ tr ( s y m b o l ε ) s y m b o l I \,\,symbol{\sigma}=2\mu\,symbol{\varepsilon}+\lambda\,\,\text{tr}(symbol{% \varepsilon})\,symbol{I}\,\,
  36. σ i j = 2 μ ε i j + λ ε k k δ i j \,\,\sigma_{ij}=2\mu\varepsilon_{ij}+\lambda\varepsilon_{kk}\delta_{ij}
  37. μ , λ \mu,\lambda
  38. s y m b o l σ = - p s y m b o l I + 2 μ s y m b o l ε ˙ + λ tr ( s y m b o l ε ˙ ) s y m b o l I \,\,symbol{\sigma}=-p\,symbol{I}+2\mu\,\dot{symbol{\varepsilon}}+\lambda\,\,% \text{tr}(\dot{symbol{\varepsilon}})\,symbol{I}\,\,
  39. σ i j = - P δ i j + 2 μ ε ˙ i j + λ ε ˙ k k δ i j \,\,\sigma_{ij}=-P\,\delta_{ij}+2\mu\dot{\varepsilon}_{ij}+\lambda\dot{% \varepsilon}_{kk}\delta_{ij}
  40. μ \mu
  41. λ \lambda
  42. s y m b o l σ = 2 μ s y m b o l ε + λ tr ( s y m b o l ε ) s y m b o l I + λ s y m b o l ε \cdotsymbol ε \,\,symbol{\sigma}=2\mu\,symbol{\varepsilon}+\lambda\,\,\text{tr}(symbol{% \varepsilon})\,symbol{I}+\lambda^{\prime}\,symbol{\varepsilon}\cdotsymbol{% \varepsilon}\,\,
  43. σ i j = 2 μ ε i j + λ ε k k δ i j + λ ε i k ε k j \,\,\sigma_{ij}=2\mu\varepsilon_{ij}+\lambda\varepsilon_{kk}\delta_{ij}+% \lambda^{\prime}\,\varepsilon_{ik}\,\varepsilon_{kj}
  44. s y m b o l σ ˙ = 𝖧 ( s y m b o l σ ) : s y m b o l ε ˙ \,\,\dot{symbol{\sigma}}=\mathsf{H}(symbol{\sigma}):\dot{symbol{\varepsilon}}\,\,
  45. σ ˙ i j = H i j k l ( σ m n ) ε ˙ k l \,\,\dot{\sigma}_{ij}=H_{ijkl}(\sigma_{mn})\,\dot{\varepsilon}_{kl}\,\,
  46. F ( s y m b o l σ , s y m b o l σ ˙ , s y m b o l ε , s y m b o l ε ˙ , 𝐱 , t ) = 0 . F(symbol{\sigma},\dot{symbol{\sigma}},symbol{\varepsilon},\dot{symbol{% \varepsilon}},\mathbf{x},t)=0\,.
  47. σ = σ 0 \sigma=\sigma_{0}
  48. σ = σ y \sigma=\sigma_{y}
  49. d σ > 0 d\sigma>0
  50. d ε p > 0 d\varepsilon_{p}>0
  51. d σ < 0 d\sigma<0
  52. d ε = d ε e + d ε p d\varepsilon=d\varepsilon_{e}+d\varepsilon_{p}
  53. d σ d ε = d σ ( d ε e + d ε p ) 0 d\sigma\,d\varepsilon=d\sigma\,(d\varepsilon_{e}+d\varepsilon_{p})\geq 0
  54. s y m b o l σ = 𝖢 : s y m b o l ε symbol{\sigma}=\mathsf{C}:symbol{\varepsilon}
  55. 𝖢 \mathsf{C}
  56. f ( s y m b o l σ ) = 0 . f(symbol{\sigma})=0\,.
  57. f ( s y m b o l σ , s y m b o l ε p ) = 0 . f(symbol{\sigma},symbol{\varepsilon}_{p})=0\,.
  58. d σ > 0 d\sigma>0
  59. f 0 f\geq 0
  60. f / \partialsymbol σ \partial f/\partialsymbol{\sigma}
  61. d s y m b o l ε p : d s y m b o l σ 0 dsymbol{\varepsilon}_{p}:dsymbol{\sigma}\geq 0
  62. d s y m b o l σ : f s y m b o l σ 0 . dsymbol{\sigma}:\frac{\partial f}{\partial symbol{\sigma}}\geq 0\,.
  63. f < 0 f<0
  64. d s y m b o l σ : f s y m b o l σ < 0 . dsymbol{\sigma}:\frac{\partial f}{\partial symbol{\sigma}}<0\,.
  65. d s y m b o l ε = d s y m b o l ε e + d s y m b o l ε p . dsymbol{\varepsilon}=dsymbol{\varepsilon}_{e}+dsymbol{\varepsilon}_{p}\,.
  66. d s y m b o l σ : d s y m b o l ε 0 . dsymbol{\sigma}:dsymbol{\varepsilon}\geq 0\,.
  67. d s y m b o l ε p = d λ f s y m b o l σ dsymbol{\varepsilon}_{p}=d\lambda\,\frac{\partial f}{\partial symbol{\sigma}}
  68. d λ > 0 d\lambda>0
  69. f f
  70. d s y m b o l σ = 0 dsymbol{\sigma}=0
  71. d s y m b o l ε p > 0 dsymbol{\varepsilon}_{p}>0
  72. d s y m b o l ε e = 0 dsymbol{\varepsilon}_{e}=0
  73. d s y m b o l σ : f s y m b o l σ = 0 and d s y m b o l σ : d s y m b o l ε p = 0 . dsymbol{\sigma}:\frac{\partial f}{\partial symbol{\sigma}}=0\quad\,\text{and}% \quad dsymbol{\sigma}:dsymbol{\varepsilon}_{p}=0\,.
  74. d s y m b o l σ : d s y m b o l ε p 0 . dsymbol{\sigma}:dsymbol{\varepsilon}_{p}\geq 0\,.
  75. d λ d\lambda
  76. d f = 0 df=0
  77. f ( s y m b o l σ , s y m b o l ε p ) = 0 f(symbol{\sigma},symbol{\varepsilon}_{p})=0
  78. d f = f s y m b o l σ : d s y m b o l σ + f s y m b o l ε p : d s y m b o l ε p = 0 . df=\frac{\partial f}{\partial symbol{\sigma}}:dsymbol{\sigma}+\frac{\partial f% }{\partial symbol{\varepsilon}_{p}}:dsymbol{\varepsilon}_{p}=0\,.

Rose–Vinet_equation_of_state.html

  1. B 0 B_{0}
  2. B 0 B_{0}^{\prime}
  3. V 0 V_{0}
  4. P = 0 P=0
  5. η = V V 0 3 \eta=\sqrt[3]{\frac{V}{V_{0}}}
  6. P = 3 B 0 ( 1 - η η 2 ) e 3 2 ( B 0 - 1 ) ( 1 - η ) P=3B_{0}\left(\frac{1-\eta}{\eta^{2}}\right)e^{\frac{3}{2}(B_{0}^{\prime}-1)(1% -\eta)}

Ross's_conjecture.html

  1. λ E ( S 2 ) 2 { 1 - λ E ( S ) } \frac{\lambda\operatorname{E}(S^{2})}{2\{1-\lambda\operatorname{E}(S)\}}

Rost_invariant.html

  1. 0 H 3 ( K , 𝐐 / 𝐙 ( 2 ) ) H e t 3 ( P K , 𝐐 / 𝐙 ( 2 ) ) 𝐐 / 𝐙 0\rightarrow H^{3}(K,\mathbf{Q}/\mathbf{Z}(2))\rightarrow H^{3}_{et}(P_{K},% \mathbf{Q}/\mathbf{Z}(2))\rightarrow\mathbf{Q}/\mathbf{Z}

Rota's_conjecture.html

  1. S S
  2. F F
  3. S S
  4. M M
  5. S S
  6. M M
  7. S S
  8. F F
  9. M M
  10. F F
  11. F F
  12. U 4 2 U{}^{2}_{4}
  13. U 5 2 U{}^{2}_{5}
  14. U 5 3 U{}^{3}_{5}
  15. U 6 2 U{}^{2}_{6}
  16. U 6 4 U{}^{4}_{6}
  17. n / 2 n/2

Rotational_sampling_in_wind_turbines.html

  1. T g r a v = r m g cos ( Ω 0 t ) T_{grav}=rmg\cos(\Omega_{0}t)
  2. Ω 0 \Omega_{0}
  3. n n
  4. 360 / n 360/n
  5. 𝐫 × 𝐅 \,\textbf{r}\times\mathbf{F}
  6. T g r a v = k r m g cos ( k Ω 0 t ) T_{grav}=\sum_{k}rmg\cos(k\Omega_{0}t)
  7. Ω 0 \Omega_{0}
  8. T g r a v = k ( r m g cos ( k Ω 0 t ) + r m g cos ( k Ω 0 t - 360 / n ) + r m g cos ( k Ω 0 t - 2 ( 360 / n ) ) + ) T_{grav}=\sum_{k}\left(rmg\cos(k\Omega_{0}t)+rmg\cos(k\Omega_{0}t-360/n)+rmg% \cos(k\Omega_{0}t-2(360/n))+\dots\right)
  9. T g r a v = n k = 1 cos ( n k Ω 0 t ) T_{grav}=n\sum_{k=1}\cos(nk\Omega_{0}t)
  10. k Ω 0 k^{\prime}\Omega_{0}

Rotational_viscosity.html

  1. J i j J_{ij}
  2. J i j t + ( v k J i j ) x k = ( x j P k i x k - x i P k j x k ) + ( P j i - P i j ) \frac{\partial J_{ij}}{\partial t}+\frac{\partial(v_{k}J_{ij})}{\partial x_{k}% }=\left(x_{j}\frac{\partial P_{ki}}{\partial x_{k}}-x_{i}\frac{\partial P_{kj}% }{\partial x_{k}}\right)+(P_{ji}-P_{ij})
  3. v i v_{i}
  4. P i j P_{ij}
  5. L i j L_{ij}
  6. S i j S_{ij}
  7. J i j = L i j + S i j J_{ij}=L_{ij}+S_{ij}
  8. L i j = ρ ( x i v j - x j v i ) L_{ij}=\rho(x_{i}v_{j}-x_{j}v_{i})
  9. ρ \rho
  10. ( ρ v i ) t + ( ρ v i v k ) x k = - P k i x k \frac{\partial(\rho v_{i})}{\partial t}+\frac{\partial(\rho v_{i}v_{k})}{% \partial x_{k}}=-\frac{\partial P_{ki}}{\partial x_{k}}
  11. L i j t + ( v k L i j ) x k = ( x j P k i x k - x i P k j x k ) \frac{\partial L_{ij}}{\partial t}+\frac{\partial(v_{k}L_{ij})}{\partial x_{k}% }=\left(x_{j}\frac{\partial P_{ki}}{\partial x_{k}}-x_{i}\frac{\partial P_{kj}% }{\partial x_{k}}\right)
  12. S i j t + ( v k S i j ) x k = P j i - P i j \frac{\partial S_{ij}}{\partial t}+\frac{\partial(v_{k}S_{ij})}{\partial x_{k}% }=P_{ji}-P_{ij}
  13. P i j - P j i = - η r ( v i x j - v j x i - 2 ω i j ) P_{ij}-P_{ji}=-\eta_{r}\left(\frac{\partial v_{i}}{\partial x_{j}}-\frac{% \partial v_{j}}{\partial x_{i}}-2\omega_{ij}\right)
  14. ω i j \omega_{ij}
  15. η r \eta_{r}

Rotatum.html

  1. P = d τ d t \vec{P}=\frac{d\vec{\tau}}{dt}
  2. d d t \frac{\mathrm{d}}{\mathrm{d}t}
  3. t t
  4. τ = d 𝐋 d t \mathbf{\tau}=\frac{\mathrm{d}\mathbf{L}}{\mathrm{d}t}
  5. \Rho = d τ d t = d d t ( d 𝐋 d t ) = d 2 𝐋 d t 2 = d 2 ( I ω ) d t 2 \mathbf{\Rho}=\frac{\mathrm{d}\mathbf{\tau}}{\mathrm{d}t}=\frac{\mathrm{d}}{% \mathrm{d}t}\left(\frac{\mathrm{d}\mathbf{L}}{\mathrm{d}t}\right)=\frac{% \mathrm{d}^{2}\mathbf{L}}{\mathrm{d}t^{2}}=\frac{\mathrm{d}^{2}(I\cdot\mathbf{% \omega})}{\mathrm{d}t^{2}}
  6. I I
  7. ω \omega
  8. \Rho = I d 2 ω d t 2 \mathbf{\Rho}=I\frac{\mathrm{d}^{2}\omega}{\mathrm{d}t^{2}}
  9. \Rho = I ζ \mathbf{\Rho}=I\zeta

Routing_(hydrology).html

  1. I = O + Δ S Δ t I=O+\frac{\Delta S}{\Delta t}
  2. Δ t \Delta t
  3. Δ t \Delta t

RRNA_small_subunit_pseudouridine_methyltransferase_Nep1.html

  1. \rightleftharpoons

Rubredoxin—NAD(P)+_reductase.html

  1. \rightleftharpoons

Ruelle_zeta_function.html

  1. ζ ( z ) = exp ( m 1 z m m x Fix ( f m ) Tr ( k = 0 m - 1 ϕ ( f k ( x ) ) ) ) \zeta(z)=\exp\left({\sum_{m\geq 1}\frac{z^{m}}{m}\sum_{x\in\mathrm{Fix}(f^{m})% }\mathrm{Tr}\left({\prod_{k=0}^{m-1}\phi(f^{k}(x))}\right)}\right)
  2. ζ ( z ) = exp ( m 1 z m m | Fix ( f m ) | ) \zeta(z)=\exp\left({\sum_{m\geq 1}\frac{z^{m}}{m}\left|{\mathrm{Fix}(f^{m})}% \right|}\right)

Rule_of_mixtures.html

  1. E E
  2. E c = f E f + ( 1 - f ) E m E_{c}=fE_{f}+\left(1-f\right)E_{m}
  3. f = V f V f + V m f=\frac{V_{f}}{V_{f}+V_{m}}
  4. E f E_{f}
  5. E m E_{m}
  6. E c = ( f E f + 1 - f E m ) - 1 . E_{c}=\left(\frac{f}{E_{f}}+\frac{1-f}{E_{m}}\right)^{-1}.
  7. σ \sigma_{\infty}
  8. ϵ f \epsilon_{f}
  9. ϵ m \epsilon_{m}
  10. σ f E f = ϵ f = ϵ m = σ m E m \frac{\sigma_{f}}{E_{f}}=\epsilon_{f}=\epsilon_{m}=\frac{\sigma_{m}}{E_{m}}
  11. σ f \sigma_{f}
  12. E f E_{f}
  13. σ m \sigma_{m}
  14. E m E_{m}
  15. f f
  16. 1 - f 1-f
  17. σ = E c ϵ c \sigma_{\infty}=E_{c}\epsilon_{c}
  18. E c E_{c}
  19. ϵ c \epsilon_{c}
  20. E c ϵ c = f E f ϵ f + ( 1 - f ) E m ϵ m . E_{c}\epsilon_{c}=fE_{f}\epsilon_{f}+\left(1-f\right)E_{m}\epsilon_{m}.
  21. ϵ c = ϵ f = ϵ m \epsilon_{c}=\epsilon_{f}=\epsilon_{m}
  22. E c = f E f + ( 1 - f ) E m . E_{c}=fE_{f}+\left(1-f\right)E_{m}.
  23. σ = σ f = σ m \sigma_{\infty}=\sigma_{f}=\sigma_{m}
  24. ϵ c = f ϵ f + ( 1 - f ) ϵ m . \epsilon_{c}=f\epsilon_{f}+\left(1-f\right)\epsilon_{m}.
  25. E c = σ ϵ c = σ f f ϵ f + ( 1 - f ) ϵ m = ( f E f + 1 - f E m ) - 1 E_{c}=\frac{\sigma_{\infty}}{\epsilon_{c}}=\frac{\sigma_{f}}{f\epsilon_{f}+% \left(1-f\right)\epsilon_{m}}=\left(\frac{f}{E_{f}}+\frac{1-f}{E_{m}}\right)^{% -1}
  26. σ f = E ϵ f \sigma_{f}=E\epsilon_{f}
  27. σ m = E ϵ m \sigma_{m}=E\epsilon_{m}
  28. ( f ρ f + 1 - f ρ m ) - 1 ρ c f ρ f + ( 1 - f ) ρ m \left(\frac{f}{\rho_{f}}+\frac{1-f}{\rho_{m}}\right)^{-1}\leq\rho_{c}\leq f% \rho_{f}+\left(1-f\right)\rho_{m}
  29. ( f σ U T S , f + 1 - f σ U T S , m ) - 1 σ U T S , c f σ U T S , f + ( 1 - f ) σ U T S , m \left(\frac{f}{\sigma_{UTS,f}}+\frac{1-f}{\sigma_{UTS,m}}\right)^{-1}\leq% \sigma_{UTS,c}\leq f\sigma_{UTS,f}+\left(1-f\right)\sigma_{UTS,m}
  30. ( f k f + 1 - f k m ) - 1 k c f k f + ( 1 - f ) k m \left(\frac{f}{k_{f}}+\frac{1-f}{k_{m}}\right)^{-1}\leq k_{c}\leq fk_{f}+\left% (1-f\right)k_{m}
  31. ( f σ f + 1 - f σ m ) - 1 σ c f σ f + ( 1 - f ) σ m \left(\frac{f}{\sigma_{f}}+\frac{1-f}{\sigma_{m}}\right)^{-1}\leq\sigma_{c}% \leq f\sigma_{f}+\left(1-f\right)\sigma_{m}

Rulkov_map.html

  1. n n
  2. x n + 1 = α 1 + x n 2 + y n x_{n+1}=\frac{\alpha}{1+x_{n}^{2}}+y_{n}
  3. y n + 1 = y n - μ ( x n - σ ) y_{n+1}=y_{n}-\mu(x_{n}-\sigma)
  4. x x
  5. y y
  6. μ \mu
  7. ( 0 < μ 1 ) (0<\mu<<1)
  8. x x
  9. y y
  10. σ \sigma
  11. α \alpha
  12. σ \sigma
  13. α \alpha
  14. α > 4 \alpha>4
  15. y y
  16. x x
  17. y y
  18. x x
  19. y y
  20. y y

Rumor_spread_in_social_network.html

  1. S + I 𝛼 2 I S+I\xrightarrow{\alpha}2I
  2. I + I 𝛽 I + R I+I\xrightarrow{\beta}I+R
  3. I + R 𝛽 2 R I+R\xrightarrow{\beta}2R
  4. N = I + S + R N=I+S+R
  5. Δ S - Δ t α I S / N \Delta S\approx-\Delta t\alpha IS/N
  6. Δ I Δ t [ α I S N - β I 2 N - β I R N ] \Delta I\approx\Delta t[{\alpha IS\over N}-{\beta I^{2}\over N}-{\beta IR\over N}]
  7. Δ R Δ t [ β I 2 N + β I R N ] \Delta R\approx\Delta t[{\beta I^{2}\over N}+{\beta IR\over N}]
  8. S S
  9. I I
  10. R R
  11. N N
  12. x = I / N x=I/N
  13. y = S / N y=S/N
  14. d x d t = x α y - β x 2 - β x ( 1 - x - y ) {dx\over dt}=x\alpha y-\beta x^{2}-\beta x(1-x-y)
  15. d y d t = - α x y {dy\over dt}=-\alpha xy
  16. d x d t = ( α + β ) x y - β x {dx\over dt}=(\alpha+\beta)xy-\beta x
  17. d y d t = - α x y {dy\over dt}=-\alpha xy
  18. α + β \alpha+\beta
  19. α \alpha
  20. x , y 0 x,y\geq 0
  21. d y d t 0 {dy\over dt}\leq 0
  22. R 0 = α + β β > 1 R_{0}={\alpha+\beta\over\beta}>1
  23. α β > 0 {\alpha\over\beta}>0
  24. X i ( t ) X_{i}(t)
  25. X ( t ) X(t)
  26. S = { S , I , R } N S=\{S,I,R\}^{N}
  27. f f
  28. x x
  29. S S
  30. f ( x , i , j ) f(x,i,j)
  31. x x
  32. y = f ( x , i , j ) y=f(x,i,j)
  33. P ( x , y ) P(x,y)
  34. P ( x , y ) = α A j i / k i P(x,y)=\alpha A_{ji}/k_{i}
  35. P ( x , y ) = β A j i / k i P(x,y)=\beta A_{ji}/k_{i}
  36. P ( x , y ) = β A j i / k i P(x,y)=\beta A_{ji}/k_{i}
  37. y y
  38. P ( x , y ) = 0 P(x,y)=0
  39. r = 1 - e - ( α + β β ) r r_{\infty}=1-e^{-({\alpha+\beta\over\beta})r_{\infty}}
  40. r r_{\infty}
  41. p p

Run-time_estimation_of_system_and_sub-system_level_power_consumption.html

  1. P o w e r c p u = 1 ( I F e t c h m i s s ) + 2 ( D a t a D e p ) + 3 ( D a t a T L B m i s s ) + 4 ( I n s T L B m i s s ) + 5 ( I n s t E x e c ) + K c p u Powe{{r}_{cpu}}={{\propto}_{1}}\left(IFetc{{h}_{miss}}\right)+{{\propto}_{2}}% \left(DataDep\right)+{{\propto}_{3}}\left(DataTL{{B}_{miss}}\right)+{{\propto}% _{4}}\left(InsTL{{B}_{miss}}\right)+{{\propto}_{5}}\left(InstExec\right)+{{K}_% {cpu}}
  2. 1 , 2 , 3 , 4 , 5 {{\propto}_{1}},{{\propto}_{2}},{{\propto}_{3}},{{\propto}_{4}},{{\propto}_{5}}
  3. K c p u {{K}_{cpu}}
  4. P o w e r m e m o r y = β 1 ( I F e t c h m i s s ) + β 2 ( D a t a D e p ) + K m e m o r y Powe{{r}_{memory}}={{\beta}_{1}}\left(IFetc{{h}_{miss}}\right)+{{\beta}_{2}}% \left(DataDep\right)+{{K}_{memory}}
  5. β 1 , β 2 {{\beta}_{1}},{{\beta}_{2}}
  6. K m e m o r y {{K}_{memory}}
  7. ε 1 {{\,\text{ }\!\!\varepsilon\!\!\,\text{ }}_{1}}
  8. ε 2 {{\,\text{ }\!\!\varepsilon\!\!\,\text{ }}_{2}}
  9. ε 3 {{\,\text{ }\!\!\varepsilon\!\!\,\text{ }}_{3}}
  10. ε 4 {{\,\text{ }\!\!\varepsilon\!\!\,\text{ }}_{4}}
  11. P c o r e = { F 1 ( g 1 ( r 1 ) , , g n ( r n ) ) , i f c o n d i t i o n F 2 ( g 1 ( r 1 ) , , g n ( R n ) ) , e l s e {{P}_{core}}=\left\{\begin{matrix}{{F}_{1}}\left({{g}_{1}}\left({{r}_{1}}% \right),\ldots\ldots,{{g}_{n}}\left({{r}_{n}}\right)\right),if\,\text{ }% condition\\ {{F}_{2}}\left({{g}_{1}}\left({{r}_{1}}\right),\ldots\ldots,{{g}_{n}}\left({{R% }_{n}}\right)\right),else\\ \end{matrix}\right.
  12. r i = ε i / ( c y c l e c o u n t ) {{r}_{i}}={{\varepsilon}_{i}}/(cycle\,\text{ }count)
  13. F n = P 0 + P 1 * g 1 ( r 1 ) + + P 2 * g n ( r n ) {{F}_{n}}={{P}_{0}}+{{P}_{1}}*{{g}_{1}}\left({{r}_{1}}\right)+\ldots\ldots+{{P% }_{2}}*{{g}_{n}}({{r}_{n}})
  14. P s y s t e m = P C P U + P R A M + P D i s k {{P}_{system}}={{P}_{CPU}}+{{P}_{RAM}}+{{P}_{Disk}}
  15. P C P U = [ , β , γ ] * V e c o r o f C P U p e r f o r m a n c e C o u n t e r s + λ c o n s t a n t C P U {{P}_{CPU}}=\left[\propto,\beta,\ldots\gamma\right]*Vecor\,\text{ }of\,\text{ % }CPU\,\text{ }performance\,\text{ }Counters+{{\lambda}_{constantCPU}}
  16. P R A M = [ Δ , Γ , Θ ] * V e c t o r o f P e r f o r m a n c e C o u n t e r s + λ c o n s t a n t R A M {{P}_{RAM}}=\left[\,\text{ }\!\!\Delta\!\!\,\text{ },\,\text{ }\!\!\Gamma\!\!% \,\text{ },\ldots\,\text{ }\!\!\Theta\!\!\,\text{ }\right]*\,\text{ }Vector\,% \text{ }of\,\text{ }Performance\,\text{ }Counters+{{\lambda}_{constantRAM}}
  17. P D i s k = [ φ , χ ] * V e c t o r o f D i s k p e r f o r m a n c e c o u n t e r + λ c o n s t a n t D i s k {{P}_{Disk}}=\left[\varphi,\chi\right]*\,\text{ }Vector\,\text{ }of\,\text{ }% Disk\,\text{ }performance\,\text{ }counter+{{\lambda}_{constantDisk}}
  18. G x + C x = B u Gx+Cx=Bu
  19. P * ( t 1 - t 2 ) = E * ( S O D ( V 1 ) - S O D ( V 2 ) ) P*\left({{t}_{1}}-{{t}_{2}}\right)=E*(SOD\left({{V}_{1}}\right)-SOD\left({{V}_% {2}}\right))
  20. E = P * T E=P*T
  21. P O S = K 1 * I P C O S + K 0 {{P}_{OS}}={{K}_{1}}*\,\text{ }IP{{C}_{OS}}+{{K}_{0}}

S-(hydroxymethyl)mycothiol_dehydrogenase.html

  1. \rightleftharpoons

S-adenozilmetionin:tRNA_ribosyltransferase-isomerase.html

  1. \rightleftharpoons

S-equivalence.html

  1. 0 = E 0 E 1 E n = E 0=E_{0}\subseteq E_{1}\subseteq\ldots\subseteq E_{n}=E
  2. E i E_{i}
  3. E i / E i - 1 E_{i}/E_{i-1}
  4. g r E = i E i / E i - 1 grE=\bigoplus_{i}E_{i}/E_{i-1}

S-methyl-5'-thioadenosine_deaminase.html

  1. \rightleftharpoons

S-methyl-5'-thioinosine_phosphorylase.html

  1. \rightleftharpoons

S-specific_spore_photoproduct_lyase.html

  1. \rightleftharpoons

Salicylate_decarboxylase.html

  1. \rightleftharpoons

Sarcosine::dimethylglycine_N-methyltransferase.html

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

Saturation_dome.html

  1. x = v - v f v g - v f x={v-v_{f}\over v_{g}-v_{f}}

Sayre_equation.html

  1. F h k l = h k l F h k l F h - h , k - k , l - l F_{hkl}=\sum_{h^{\prime}k^{\prime}l^{\prime}}F_{h^{\prime}k^{\prime}l^{\prime}% }F_{h-h^{\prime},k-k^{\prime},l-l^{\prime}}
  2. h , k , l h,k,l
  3. π \pi
  4. S h S h S h - h S_{h}\approx S_{h^{\prime}}S_{h-h^{\prime}}
  5. S S
  6. π \pi
  7. \approx

Scaled_correlation.html

  1. K K
  2. T T
  3. s s
  4. K = round ( T s ) . K=\operatorname{round}\left(\frac{T}{s}\right).
  5. r k r_{k}
  6. k k
  7. r ¯ s \bar{r}_{s}
  8. r ¯ s = 1 K k = 1 K r k . \bar{r}_{s}=\frac{1}{K}\sum\limits_{k=1}^{K}r_{k}.

Schizophrenic_number.html

  1. f ( 49 ) \sqrt{f(49)}
  2. f ( n ) = 10 f ( n - 1 ) + n f(n)=10f(n-1)+n
  3. π \pi

Schwarz_minimal_surface.html

  1. cos ( x ) + cos ( y ) + cos ( z ) = 0 \cos(x)+\cos(y)+\cos(z)=0
  2. sin ( x ) sin ( y ) sin ( z ) + sin ( x ) cos ( y ) cos ( z ) + cos ( x ) sin ( y ) cos ( z ) + cos ( x ) cos ( y ) sin ( z ) = 0. \sin(x)\sin(y)\sin(z)+\sin(x)\cos(y)\cos(z)+\cos(x)\sin(y)\cos(z)+\cos(x)\cos(% y)\sin(z)=0.

Schwinger_limit.html

  1. E S = m e 2 c 3 q e 1.3 × 10 18 V / m , E_{S}=\frac{m_{e}^{2}c^{3}}{q_{e}\hbar}\simeq 1.3\times 10^{18}\,\mathrm{V}/% \mathrm{m},

Sclareol_cyclase.html

  1. \rightleftharpoons

Secoisolariciresinol_dehydrogenase.html

  1. \rightleftharpoons

Segment_protection.html

  1. A l p h a Alpha
  2. B e t a Beta
  3. A l p h a Alpha
  4. A l p h a Alpha
  5. B e t a Beta
  6. A l p h a Alpha

Segmented_scan.html

  1. [ 1 2 3 4 5 6 i n p u t 1 0 0 1 0 1 f l a g b i t s 1 3 6 4 9 6 s e g m e n t e d s c a n + ] \begin{bmatrix}1&2&3&4&5&6&input\\ 1&0&0&1&0&1&flag\ bits\\ 1&3&6&4&9&6&segmented\ scan\ +\end{bmatrix}

Segré–Silberberg_effect.html

  1. d / D d/D
  2. d / D 0.07 d/D\geq 0.07

Seismic_inverse_Q_filtering.html

  1. d U ( r , w ) d r - i k ( w ) U ( r , w ) = 0 ( 1.1 ) \frac{dU(r,w)}{dr}-ik(w)U(r,w)=0\quad(1.1)
  2. U ( r + r , w ) = U ( r , w ) exp ( i k ( w ) r ) ( 1.2 ) U(r+\bigtriangleup r,w)=U(r,w)\exp(ik(w)\bigtriangleup r)\quad(1.2)
  3. α = | w | ( 2 c r Q r ) ( 1.3 ) \alpha=\frac{|w|}{(2c_{r}Q_{r})}\quad(1.3)
  4. 1 c ( w ) = 1 c r ( 1 - 1 π Q r l n | w w r | ) ( 1.4 ) \frac{1}{c(w)}=\frac{1}{c_{r}}(1-\frac{1}{\pi Q_{r}}ln|\frac{w}{w_{r}}|)\quad(% 1.4)
  5. 1 c ( w ) = 1 c r | w w r | - γ ( 1.5 ) \frac{1}{c(w)}=\frac{1}{c_{r}}|\frac{w}{w_{r}}|^{-\gamma}\quad(1.5)
  6. γ = ( π Q r ) - 1 \gamma=(\pi Q_{r})^{-1}
  7. k ( w ) = ( 1 - i 2 Q ( w ) ) | w | c r | w w r | - γ ) ( 1.6 ) k(w)=(1-\frac{i}{2Q(w)})\frac{|w|}{c_{r}}|\frac{w}{w_{r}}|^{-\gamma})\quad(1.6)
  8. U ( r + r , w ) = U ( r , w ) exp [ r 2 Q ( w ) + i | w | r c r | w w r | - γ ] ( 1.7 ) U(r+\bigtriangleup r,w)=U(r,w)\exp[\frac{\bigtriangleup r}{2Q(w)}+i\frac{|w|% \bigtriangleup r}{c_{r}}|\frac{w}{w_{r}}|^{-\gamma}]\quad(1.7)
  9. U ( t + t , w ) = U ( t , w ) exp ( | w w h | - γ | w | t 2 Q ( w ) ) exp ( i | w w h | - γ w t ) ( 1.8. a ) U(t+\bigtriangleup t,w)=U(t,w)\exp\bigg(|\frac{w}{w_{h}}|^{-\gamma}\frac{|w|% \bigtriangleup t}{2Q(w)}\bigg)\exp\bigg(i|\frac{w}{w_{h}}|^{-\gamma}w% \bigtriangleup t\bigg)\quad(1.8.a)
  10. U ( t + t , w ) = U ( t , w ) exp ( - | w w r | γ | w | t 2 Q ( w ) ) exp ( i | w w r | γ w t ) ( 1.8. b ) U(t+\bigtriangleup t,w)=U(t,w)\exp\bigg(-|\frac{w}{w_{r}}|^{\gamma}\frac{|w|% \bigtriangleup t}{2Q(w)}\bigg)\exp\bigg(i|\frac{w}{w_{r}}|^{\gamma}w% \bigtriangleup t\bigg)\quad(1.8.b)
  11. U ( t + t ) = 0 U ( t + t , w ) d w . ( 1.9 ) U(t+\bigtriangleup t)=\int_{0}^{\infty}U(t+\bigtriangleup t,w)dw.\quad(1.9)
  12. π \pi

Self-consistency_principle_in_high_energy_Physics.html

  1. σ ( E ) \sigma(E)
  2. ρ ( m ) \rho(m)
  3. l o g [ ρ ( m ) ] = l o g [ σ ( E ) ] log[\rho(m)]=log[\sigma(E)]
  4. ρ ( m ) \textstyle\rho(m)
  5. σ ( E ) \textstyle\sigma(E)
  6. ρ ( m ) = a m - 5 / 2 e x p ( β o m ) \rho(m)=am^{-5/2}exp(\beta_{o}m)
  7. σ ( E ) = b E α - 1 e x p ( β o E ) \sigma(E)=bE^{\alpha-1}exp(\beta_{o}E)
  8. a \textstyle a
  9. α \textstyle\alpha
  10. α = a V ( 2 π β ) 3 / 5 \alpha=\frac{aV}{(2\pi\beta)^{3/5}}
  11. Z q ( V o , T ) = ( 1 β - β o ) α - 1 Z_{q}(V_{o},T)=\bigg(\frac{1}{\beta-\beta_{o}}\bigg)^{\alpha}-1
  12. β \beta
  13. β o \beta_{o}
  14. T o = 1 / β o \textstyle T_{o}=1/\beta_{o}

Self-dual_Palatini_action.html

  1. e I α e_{I}^{\alpha}
  2. ω α I J \omega_{\alpha}^{\;\;IJ}
  3. D α D_{\alpha}
  4. g α β = e α I e β J η I J . g_{\alpha\beta}=e^{I}_{\alpha}e^{J}_{\beta}\eta_{IJ}.
  5. Ω α β I J = α ω β I J - β ω α I J + ω α I K ω β K J - ω β I K ω α K J E q ( 1 ) . \Omega_{\alpha\beta}^{\;\;\;\;IJ}=\partial_{\alpha}\omega_{\beta}^{\;\;IJ}-% \partial_{\beta}\omega_{\alpha}^{\;\;IJ}+\omega_{\alpha}^{IK}\omega_{\beta K}^% {\;\;\;\;J}-\omega_{\beta}^{IK}\omega_{\alpha K}^{\;\;\;\;J}\;\;\;\;\;Eq(1).
  6. e I α e J β Ω α β I J e_{I}^{\alpha}e_{J}^{\beta}\Omega_{\alpha\beta}^{\;\;\;\;IJ}
  7. S = d 4 x e e I α e J β Ω α β I J [ ω ] S=\int d^{4}x\;e\;e_{I}^{\alpha}e_{J}^{\beta}\;\Omega_{\alpha\beta}^{\;\;\;\;% IJ}[\omega]
  8. e = - g e=\sqrt{-g}
  9. ω α β I J \omega_{\alpha\beta}^{\;\;\;IJ}
  10. D α e I β = 0 D_{\alpha}e_{I}^{\beta}=0
  11. α \nabla_{\alpha}
  12. Ω α β I J \Omega_{\alpha\beta}^{\;\;\;\;IJ}
  13. R α β I J R_{\alpha\beta}^{\;\;\;\;IJ}
  14. α \nabla_{\alpha}
  15. e I α e J β R α β I J e_{I}^{\alpha}e_{J}^{\beta}R_{\alpha\beta}^{\;\;\;\;IJ}
  16. R R
  17. R α β - 1 2 g α β R = 0 R_{\alpha\beta}-{1\over 2}g_{\alpha\beta}R=0
  18. ϵ I J K L \epsilon_{IJKL}
  19. I J K L IJKL
  20. 0123 0123
  21. ϵ I J K L \epsilon_{IJKL}
  22. η I J \eta^{IJ}
  23. T I J T^{IJ}
  24. * T I J = 1 2 ϵ K L I J T K L . *T^{IJ}={1\over 2}\epsilon_{KL}^{\;\;\;\;\;\;IJ}T^{KL}.
  25. T I J T^{IJ}
  26. T I J + := 1 2 ( T I J - i 2 ϵ K L I J T K L ) \;{}^{+}T^{IJ}:={1\over 2}\Big(T^{IJ}-{i\over 2}\epsilon_{KL}^{\;\;\;\;\;\;IJ}% T^{KL}\Big)
  27. T I J - := 1 2 ( T I J + i 2 ϵ K L I J T K L ) \;{}^{-}T^{IJ}:={1\over 2}\Big(T^{IJ}+{i\over 2}\epsilon_{KL}^{\;\;\;\;\;\;IJ}% T^{KL}\Big)
  28. i i
  29. T I J T^{IJ}
  30. T I J = 1 2 ( T I J - i 2 ϵ K L I J T K L ) + 1 2 ( T I J + i 2 ϵ K L I J T K L ) = + T I J + - T I J T^{IJ}={1\over 2}(T^{IJ}-{i\over 2}\epsilon_{KL}^{\;\;\;\;\;\;\;IJ}T^{KL})+{1% \over 2}(T^{IJ}+{i\over 2}\epsilon_{KL}^{\;\;\;\;\;\;\;IJ}T^{KL})=\;^{+}T^{IJ}% +\;^{-}T^{IJ}
  31. T I J + \;{}^{+}T^{IJ}
  32. T I J - \;{}^{-}T^{IJ}
  33. T I J T^{IJ}
  34. P ( ± ) = 1 2 ( 1 i * ) . P^{(\pm)}={1\over 2}(1\mp i*).
  35. P + P^{+}
  36. ( P + T ) I J = ( 1 2 ( 1 - i * ) T ) I J = 1 2 ( δ K I δ L J - i 1 2 ϵ K L I J ) T K L = 1 2 ( T I J - i 2 ϵ K L I J T K L ) = + T I J . (P^{+}T)^{IJ}=({1\over 2}(1-i*)T)^{IJ}={1\over 2}(\delta^{I}_{\;K}\delta^{J}_{% \;\;L}-i{1\over 2}\epsilon_{KL}^{\;\;\;\;\;\;\;IJ})T^{KL}={1\over 2}(T^{IJ}-{i% \over 2}\epsilon_{KL}^{\;\;\;\;\;\;\;IJ}T^{KL})=\;^{+}T^{IJ}.
  37. T I J ± = ( P ( ± ) T ) I J . \;{}^{\pm}T^{IJ}=(P^{(\pm)}T)^{IJ}.
  38. [ F , G ] I J := F I K G K J - G I K F K J , [F,G]^{IJ}:=F^{IK}G_{K}^{\;\;J}-G^{IK}F_{K}^{\;\;J},
  39. E q .1 Eq.1
  40. P ( ± ) [ F , G ] I J = [ P ( ± ) F , G ] I J = [ F , P ( ± ) G ] I J = [ P ( ± ) F , P ( ± ) G ] I J E q .2 P^{(\pm)}[F,G]^{IJ}=[P^{(\pm)}F,G]^{IJ}=[F,P^{(\pm)}G]^{IJ}=[P^{(\pm)}F,P^{(% \pm)}G]^{IJ}\;\;\;\;\;Eq.2
  41. [ F , G ] = [ P + F , P + G ] + [ P - F , P - G ] . [F,G]=[P^{+}F,P^{+}G]+[P^{-}F,P^{-}G].
  42. s o ( 1 , 3 ) = s o ( 1 , 3 ) + + s o ( 1 , 3 ) - so(1,3)_{\mathbb{C}}=so(1,3)_{\mathbb{C}}^{+}+so(1,3)_{\mathbb{C}}^{-}
  43. s o ( 1 , 3 ) ± so(1,3)_{\mathbb{C}}^{\pm}
  44. s o ( 1 , 3 ) so(1,3)_{\mathbb{C}}
  45. A α I J A_{\alpha}^{\;\;\;IJ}
  46. ω α I J \omega_{\alpha}^{\;\;IJ}
  47. A α I J = 1 2 ( ω α I J - i 2 ϵ K L I J ω K L ) . A_{\alpha}^{\;\;\;IJ}={1\over 2}\big(\omega_{\alpha}^{\;\;IJ}-{i\over 2}% \epsilon_{KL}^{\;\;\;\;\;IJ}\omega^{KL}\big).
  48. A α I J = ( P + ω ) I J . A_{\alpha}^{\;\;\;IJ}=(P^{+}\omega\big)^{IJ}.
  49. F α β I J F_{\alpha\beta}^{\;\;IJ}
  50. F α β I J = α A β I J - β A α I J + A α I K A β K J - A β I K A α K J . F_{\alpha\beta}^{\;\;IJ}=\partial_{\alpha}A_{\beta}^{\;\;IJ}-\partial_{\beta}A% _{\alpha}^{\;\;IJ}+A_{\alpha}^{\;\;IK}A_{\beta K}^{\;\;\;\;\;J}-A_{\beta}^{IK}% A_{\alpha K}^{\;\;\;\;\;J}.
  51. E q .2 Eq.2
  52. F α β I J = α ( P + ω β ) I J - β ( P + ω α ) I J + [ P + ω α , P + ω β ] I J F_{\alpha\beta}^{\;\;\;\;IJ}=\partial_{\alpha}(P^{+}\omega_{\beta})^{IJ}-% \partial_{\beta}(P^{+}\omega_{\alpha})^{IJ}+[P^{+}\omega_{\alpha},P^{+}\omega_% {\beta}]^{IJ}
  53. = ( P + 2 [ α ω β ] ) I J + ( P + [ ω α , ω β ] ) I J =(P^{+}2\partial_{[\alpha}\omega_{\beta]})^{IJ}+(P^{+}[\omega_{\alpha},\omega_% {\beta}])^{IJ}
  54. = ( P + Ω α β ) I J . =(P^{+}\Omega_{\alpha\beta})^{IJ}.
  55. S = d 4 x e e I α e J β F α β I J . S=\int d^{4}x\;e\;e_{I}^{\alpha}e_{J}^{\beta}\;F_{\alpha\beta}^{\;\;\;\;IJ}.
  56. A α I J A_{\alpha}^{\;\;IJ}
  57. R α β + - 1 2 g α β + R = 0. \;{}^{+}R_{\alpha\beta}-{1\over 2}g_{\alpha\beta}\;^{+}R=0.
  58. η I J \eta_{IJ}
  59. ( - , + , + , + ) (-,+,+,+)
  60. ϵ I J K L = - ϵ I J K L . \epsilon^{IJKL}=-\epsilon_{IJKL}.
  61. ϵ 0123 = η 0 I η 1 J η 2 K η 3 L ϵ I J K L \epsilon^{0123}=\eta^{0I}\eta^{1J}\eta^{2K}\eta^{3L}\epsilon_{IJKL}
  62. = ( - 1 ) ( + 1 ) ( + 1 ) ( + 1 ) ϵ 0123 = - ϵ 0123 . =(-1)(+1)(+1)(+1)\epsilon_{0123}=-\epsilon_{0123}.
  63. ϵ I J K O ϵ L M N O = - 6 δ [ L I δ M J δ N ] K E q .3 \epsilon^{IJKO}\epsilon_{LMNO}=-6\delta^{I}_{[L}\delta^{J}_{M}\delta^{K}_{N]}% \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;Eq.3
  64. ϵ I J M N ϵ K L M N = - 4 δ [ K I δ L ] J = - 2 ( δ K I δ L J - δ L I δ K J ) E q .4 \epsilon^{IJMN}\epsilon_{KLMN}=-4\delta^{I}_{[K}\delta^{J}_{L]}=-2(\delta^{I}_% {K}\delta^{J}_{L}-\delta^{I}_{L}\delta^{J}_{K})\;\;\;\;Eq.4
  65. E q .4 Eq.4
  66. * * T I J = 1 4 ϵ K L I J ϵ M N K L T M N = - T I J **T^{IJ}={1\over 4}\epsilon_{KL}^{\;\;\;\;\;\;IJ}\epsilon_{MN}^{\;\;\;\;\;\;\;% KL}T^{\;\;MN}=-T^{IJ}
  67. E q .4 Eq.4
  68. ( + , + , + , + ) (+,+,+,+)
  69. S I J S^{IJ}
  70. * S I J = i S I J . *S^{IJ}=iS^{IJ}.
  71. * S I J = S I J *S^{IJ}=S^{IJ}
  72. S I J S^{IJ}
  73. S I J = 1 2 T I J + i 1 2 U I J . S^{IJ}={1\over 2}T^{IJ}+i{1\over 2}U^{IJ}.
  74. U U
  75. V V
  76. * ( T I J + i U I J ) = 1 2 ϵ K L I J ( T K L + i U K L ) = i ( T I J + i U I J ) . *(T^{IJ}+iU^{IJ})={1\over 2}\epsilon_{KL}^{\;\;\;\;\;\;IJ}(T^{KL}+iU^{KL})=i(T% ^{IJ}+iU^{IJ}).
  77. U I J = - 1 2 ϵ K L I J T K L U^{IJ}=-{1\over 2}\epsilon_{KL}^{\;\;\;\;\;\;IJ}T^{KL}
  78. S I J = 1 2 ( T I J - i 2 ϵ K L I J T K L ) S^{IJ}={1\over 2}(T^{IJ}-{i\over 2}\epsilon_{KL}^{\;\;\;\;\;\;IJ}T^{KL})
  79. T I J T^{IJ}
  80. 2 S I J 2S^{IJ}
  81. E q .3 Eq.3
  82. * [ F , * G ] I J = 1 2 ϵ M N I J ( F M K ( * G ) K N - ( * G ) M K F K N ) *[F,*G]^{IJ}={1\over 2}\epsilon_{MN}^{\;\;\;\;\;\;IJ}(F^{MK}(*G)_{K}^{\;\;\;N}% -(*G)^{MK}F_{K}^{\;\;\;N})
  83. = 1 2 ϵ M N I J ( F M K 1 2 ϵ O P K N G O P - 1 2 ϵ O P M K G O P F K N ) ={1\over 2}\epsilon_{MN}^{\;\;\;\;\;\;\;IJ}(F^{MK}{1\over 2}\epsilon_{OPK}^{\;% \;\;\;\;\;\;\;\;N}G^{OP}-{1\over 2}\epsilon_{OP}^{\;\;\;\;\;\;MK}G^{OP}F_{K}^{% \;\;\;N})
  84. = 1 4 ( ϵ M N I J ϵ O P K N + ϵ N M I J ϵ O P N K ) F K M G O P ={1\over 4}(\epsilon_{MN}^{\;\;\;\;\;\;\;IJ}\epsilon_{OP}^{\;\;\;\;\;\;KN}+% \epsilon_{NM}^{\;\;\;\;\;\;\;IJ}\epsilon_{OP}^{\;\;\;\;\;\;NK})F^{M}_{\;\;\;K}% G^{OP}
  85. = 1 2 ϵ M N I J ϵ O P K N F K M G O P ={1\over 2}\epsilon_{MN}^{\;\;\;\;\;\;\;IJ}\epsilon_{OP}^{\;\;\;\;\;\;KN}F^{M}% _{\;\;\;K}G^{OP}
  86. = 1 2 ϵ M I J N ϵ O P K N F M K G O P ={1\over 2}\epsilon^{MIJN}\epsilon_{OPKN}F_{M}^{\;\;\;K}G^{OP}
  87. = - 1 2 ϵ K I J N ϵ O P M N F K M G O P =-{1\over 2}\epsilon^{KIJN}\epsilon_{OPMN}F^{M}_{\;\;\;K}G^{OP}
  88. = 1 2 ( δ O K δ P I δ M J + δ M K δ O I δ P J + δ P K δ M I δ O J - δ P K δ O I δ M J - δ M K δ P I δ O J - δ O K δ M I δ P J ) F K M G O P ={1\over 2}(\delta^{K}_{O}\delta^{I}_{P}\delta^{J}_{M}+\delta^{K}_{M}\delta^{I% }_{O}\delta^{J}_{P}+\delta^{K}_{P}\delta^{I}_{M}\delta^{J}_{O}-\delta^{K}_{P}% \delta^{I}_{O}\delta^{J}_{M}-\delta^{K}_{M}\delta^{I}_{P}\delta^{J}_{O}-\delta% ^{K}_{O}\delta^{I}_{M}\delta^{J}_{P})F^{M}_{\;\;\;K}G^{OP}
  89. = 1 2 ( F K J G K I + F K K G I J + F K I G J K - F K J G I K - F K K G J I - F K I G K J ) ={1\over 2}(F^{J}_{\;\;\;K}G^{KI}+F^{K}_{\;\;\;K}G^{IJ}+F^{I}_{\;\;K}G^{JK}-F^% {J}_{\;\;\;K}G^{IK}-F^{K}_{\;\;\;K}G^{JI}-F^{I}_{\;\;K}G^{KJ})
  90. = - F I K G K J + G I K F K J =-F^{IK}G_{K}^{\;\;\;J}+G^{IK}F_{K}^{\;\;J}
  91. = - [ F , G ] I J =-[F,G]^{IJ}
  92. * [ F , * G ] I J = - [ F , G ] I J E q .5. *[F,*G]^{IJ}=-[F,G]^{IJ}\;\;\;\;\;\;Eq.5.
  93. * [ * F , G ] I J = - * [ G , * F ] I J = + [ G , F ] I J = - [ F , G ] I J . *[*F,G]^{IJ}=-*[G,*F]^{IJ}=+[G,F]^{IJ}=-[F,G]^{IJ}.
  94. * F *F
  95. G G
  96. E q .5 Eq.5
  97. * ( - [ F , G ] I J ) = * ( * [ * F , G ] I J ) = * * [ * F , G ] I J = - [ * F , G ] I J . *(-[F,G]^{IJ})=*(*[*F,G]^{IJ})=**[*F,G]^{IJ}=-[*F,G]^{IJ}.
  98. * * = - 1 **=-1
  99. * [ F , G ] I J = [ * F , G ] I J *[F,G]^{IJ}=[*F,G]^{IJ}
  100. * [ F , G ] I J = [ F , * G ] I J . *[F,G]^{IJ}=[F,*G]^{IJ}.
  101. * [ F , G ] I J = [ * F , G ] I J *[F,G]^{IJ}=[*F,G]^{IJ}
  102. G G
  103. * G *G
  104. * [ F , * G ] I J = [ * F , * G ] I J *[F,*G]^{IJ}=[*F,*G]^{IJ}
  105. - [ F , G ] I J = * [ F , * G ] I J -[F,G]^{IJ}=*[F,*G]^{IJ}
  106. E q .5 Eq.5
  107. * [ F , * G ] I J = [ * F , * G ] I J *[F,*G]^{IJ}=[*F,*G]^{IJ}
  108. - [ F , G ] I J = [ * F , * G ] I J . -[F,G]^{IJ}=[*F,*G]^{IJ}.
  109. * [ F , * G ] I J = - [ F , G ] I J = * [ * F , G ] I J *[F,*G]^{IJ}=-[F,G]^{IJ}=*[*F,G]^{IJ}
  110. * [ F , G ] I J = [ * F , G ] I J = [ F , * G ] I J E q .6 *[F,G]^{IJ}=[*F,G]^{IJ}=[F,*G]^{IJ}\;\;\;\;\;Eq.6
  111. [ * F , * G ] I J = - [ F , G ] I J E q .7 [*F,*G]^{IJ}=-[F,G]^{IJ}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;Eq.7
  112. ( P ( ± ) [ F , G ] ) I J = 1 2 ( [ F , G ] I J i * [ F , G ] I J ) (P^{(\pm)}[F,G])^{IJ}={1\over 2}([F,G]^{IJ}\mp i*[F,G]^{IJ})
  113. = 1 2 ( [ F , G ] I J + [ i * F , G ] I J ) ={1\over 2}([F,G]^{IJ}+[\mp i*F,G]^{IJ})
  114. = [ P ( ± ) F , G ] I J E q .8 =[P^{(\pm)}F,G]^{IJ}\;\;\;\;\;\;\;\;\;\;\;\;Eq.8
  115. E q .6 Eq.6
  116. ( P ( ± ) [ F , G ] ) I J = [ F , P ( ± ) G ] I J (P^{(\pm)}[F,G])^{IJ}=[F,P^{(\pm)}G]^{IJ}
  117. [ P + F , P - G ] I J [P^{+}F,P^{-}G]^{IJ}
  118. [ P + F , P - G ] I J = 1 4 [ ( 1 - i * ) F , ( 1 + i * ) G ] I J [P^{+}F,P^{-}G]^{IJ}={1\over 4}[(1-i*)F,(1+i*)G]^{IJ}
  119. = 1 4 [ F , G ] I J - 1 4 i [ * F , G ] I J + 1 4 i [ F , * G ] I J + 1 4 [ * F , * G ] I J ={1\over 4}[F,G]^{IJ}-{1\over 4}i[*F,G]^{IJ}+{1\over 4}i[F,*G]^{IJ}+{1\over 4}% [*F,*G]^{IJ}
  120. = 1 4 [ F , G ] I J - 1 4 i [ * F , G ] I J + 1 4 i [ * F , G ] I J - 1 4 [ F , G ] I J ={1\over 4}[F,G]^{IJ}-{1\over 4}i[*F,G]^{IJ}+{1\over 4}i[*F,G]^{IJ}-{1\over 4}% [F,G]^{IJ}
  121. = 0 E q .9 =0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;% \;\;Eq.9
  122. E q .6 Eq.6
  123. E q .7 Eq.7
  124. [ P - F , P + G ] I J = 0 E q .10. [P^{-}F,P^{+}G]^{IJ}=0\;\;\;\;\;\;\;\;\;\;\;\;\;Eq.10.
  125. E q .8 Eq.8
  126. ( P ( ± ) [ F , G ] ) I J = [ P ( ± ) F , G ] I J = [ P ( ± ) F , P ( ± ) G + P ( ) G ] I J = [ P ( ± ) F , P ( ± ) G ] I J (P^{(\pm)}[F,G])^{IJ}=[P^{(\pm)}F,G]^{IJ}=[P^{(\pm)}F,P^{(\pm)}G+P^{(\mp)}G]^{% IJ}=[P^{(\pm)}F,P^{(\pm)}G]^{IJ}
  127. G G
  128. G = P ( ± ) G + P ( ) G G=P^{(\pm)}G+P^{(\mp)}G
  129. E q .9 / E q .10 Eq.9/Eq.10
  130. ( P ( ± ) [ F , G ] ) I J = [ P ( ± ) F , G ] I J = [ F , P ( ± ) G ] I J = [ P ( ± ) F , P ( ± ) G ] I J (P^{(\pm)}[F,G])^{IJ}=[P^{(\pm)}F,G]^{IJ}=[F,P^{(\pm)}G]^{IJ}=[P^{(\pm)}F,P^{(% \pm)}G]^{IJ}
  131. E q .2 Eq.2
  132. [ F , G ] I J = [ P + F + P - F , P + G + P - F ] I J [F,G]^{IJ}=[P^{+}F+P^{-}F,P^{+}G+P^{-}F]^{IJ}
  133. = [ P + F , P + G ] I J + [ P - F , P - G ] I J . =[P^{+}F,P^{+}G]^{IJ}+[P^{-}F,P^{-}G]^{IJ}.
  134. [ F , G ] I J [F,G]^{IJ}
  135. [ F , G ] I J [F,G]^{IJ}
  136. S ( e , ω ) = d 4 x e e I a e J b Ω a b I J [ ω ] E q 11 S(e,\omega)=\int d^{4}xee^{a}_{I}e^{b}_{J}\Omega_{ab}^{\;\;\;\;IJ}[\omega]\;\;% \;Eq\;11
  137. Ω a b I J \Omega_{ab}^{\;\;\;\;IJ}
  138. ω a I J \omega_{a}^{IJ}
  139. D b e a I = 0. D_{b}e_{a}^{I}=0.
  140. S ( e , A ) = d 4 x e e I a e J b F a b I J [ A ] S(e,A)=\int d^{4}xee^{a}_{I}e^{b}_{J}F_{ab}^{\;\;\;\;IJ}[A]
  141. F F
  142. A A
  143. ω \omega
  144. A a I J = 1 2 ( ω a I J - i 2 ϵ M N I J ω a M N ) . A_{a}^{IJ}={1\over 2}(\omega_{a}^{IJ}-{i\over 2}\epsilon^{IJ}_{\;\;\;\;MN}% \omega_{a}^{MN}).
  145. F [ A ] F[A]
  146. Ω [ ω ] \Omega[\omega]
  147. E I a = q b a e I b , E^{a}_{I}=q^{a}_{b}e^{b}_{I},
  148. q b a = δ b a + n a n b q^{a}_{b}=\delta^{a}_{b}+n^{a}n_{b}
  149. n a n^{a}
  150. E I a = ( δ b a + n b n a ) e I b E^{a}_{I}=(\delta_{b}^{a}+n_{b}n^{a})e^{b}_{I}
  151. d 4 x ( e E I a E J b F a b I J - 2 e E I a e J d n d n b F a b I J ) \int d^{4}x(eE^{a}_{I}E^{b}_{J}F_{ab}^{\;\;\;IJ}-2eE^{a}_{I}e^{d}_{J}n_{d}n^{b% }F_{ab}^{\;\;\;IJ})
  152. = d 4 x ( e ( δ c a + n c n a ) e I c ( δ d b + n d n b ) e J d F a b I J - 2 e ( δ c a + n c n a ) e I c e J d n d n b F a b I J ) =\int d^{4}x(e(\delta_{c}^{a}+n_{c}n^{a})e^{c}_{I}(\delta_{d}^{b}+n_{d}n^{b})e% ^{d}_{J}F_{ab}^{\;\;\;IJ}-2e(\delta_{c}^{a}+n_{c}n^{a})e^{c}_{I}e^{d}_{J}n_{d}% n^{b}F_{ab}^{\;\;\;IJ})
  153. = d 4 x ( e e I a e J b F a b I J + e n c n a e I c e J b F a b I J + e e I a n d n b e J d F a b I J + e n c n a n d n b E I c E J d F a b I J =\int d^{4}x(ee^{a}_{I}e^{b}_{J}F_{ab}^{\;\;\;IJ}+en_{c}n^{a}e^{c}_{I}e^{b}_{J% }F_{ab}^{\;\;\;IJ}+ee^{a}_{I}n_{d}n^{b}e^{d}_{J}F_{ab}^{\;\;\;IJ}+en_{c}n^{a}n% _{d}n^{b}E^{c}_{I}E^{d}_{J}F_{ab}^{\;\;\;IJ}
  154. - 2 e e I a e J d n d n b F a b I J - 2 n c n a e I c e J d n d n b F a b I J ) \qquad-\;2ee^{a}_{I}e^{d}_{J}n_{d}n^{b}F_{ab}^{\;\;\;IJ}-2n_{c}n^{a}e^{c}_{I}e% ^{d}_{J}n_{d}n^{b}F_{ab}^{\;\;\;IJ})
  155. = d 4 x e e I a e J b F a b I J =\int d^{4}xee^{a}_{I}e^{b}_{J}F_{ab}^{\;\;\;IJ}
  156. = S ( E , A ) =S(E,A)
  157. F a b I J = F b a J I F_{ab}^{\;\;\;IJ}=F_{ba}^{\;\;\;JI}
  158. n a n b F a b i = 0 n^{a}n^{b}F_{ab}^{i}=0
  159. S ( E , A ) = d 4 x ( e E I a E J b F a b I J - 2 e E I a e J d n d n b F a b I J ) E q 12 S(E,A)=\int d^{4}x(eE^{a}_{I}E^{b}_{J}F_{ab}^{\;\;\;IJ}-2eE^{a}_{I}e^{d}_{J}n_% {d}n^{b}F_{ab}^{\;\;\;IJ})\;\;\;Eq\;12
  160. e = N q e=N\sqrt{q}
  161. E ~ I a = q E I a \tilde{E}_{I}^{a}=\sqrt{q}E_{I}^{a}
  162. S I J S^{IJ}
  163. * S I J := 1 2 ϵ M N I J S M N = i S I J *S^{IJ}:={1\over 2}\epsilon^{IJ}_{\;\;\;\;MN}S^{MN}=iS^{IJ}
  164. F a b I J F_{ab}^{\;\;\;IJ}
  165. F a b I J = - i 1 2 ϵ M N I J F a b M N F_{ab}^{\;\;\;IJ}=-i{1\over 2}\epsilon^{IJ}_{\;\;\;\;MN}F_{ab}^{\;\;\;MN}
  166. \;
  167. S ( E , A ) = d 4 x ( - i 1 2 ( N q ) E ~ I a E ~ J b ϵ M N I J F a b M N - 2 N n b E ~ I a n J F a b I J ) S(E,A)=\int d^{4}x(-i{1\over 2}({N\over\sqrt{q}})\tilde{E}^{a}_{I}\tilde{E}^{b% }_{J}\epsilon^{IJ}_{\;\;\;\;MN}F_{ab}^{\;\;\;MN}-2Nn^{b}\tilde{E}^{a}_{I}n_{J}% F_{ab}^{\;\;\;IJ})
  168. n J = e J d n d n_{J}=e_{J}^{d}n_{d}
  169. E ~ 0 a = 0 \tilde{E}^{a}_{0}=0
  170. n I = δ 0 I n^{I}=\delta_{0}^{I}
  171. n I = η I J n J = η 00 δ 0 I = - δ 0 I n_{I}=\eta_{IJ}n^{J}=\eta_{00}\delta_{0}^{I}=-\delta_{0}^{I}
  172. ϵ I J K L n L = ϵ I J K \epsilon_{IJKL}n^{L}=\epsilon_{IJK}
  173. ϵ I J K 0 = ϵ I J K \epsilon_{IJK0}=\epsilon_{IJK}
  174. S ( E , A ) = d 4 x ( - i 1 2 ( N q ) E ~ I a E ~ J b ( ϵ M 0 I J F a b M 0 + ϵ 0 M I J F a b 0 M ) - 2 N n b E ~ I a n J F a b I J ) S(E,A)=\int d^{4}x(-i{1\over 2}({N\over\sqrt{q}})\tilde{E}^{a}_{I}\tilde{E}^{b% }_{J}(\epsilon^{IJ}_{\;\;\;\;M0}F_{ab}^{\;\;\;M0}+\epsilon^{IJ}_{\;\;\;\;0M}F_% {ab}^{\;\;\;0M})-2Nn^{b}\tilde{E}^{a}_{I}n_{J}F_{ab}^{\;\;\;IJ})
  175. = d 4 x ( - i ( N q ) E ~ I a E ~ J b ϵ M I J F a b M 0 + 2 N n b E ~ I a F a b I 0 ) =\int d^{4}x(-i({N\over\sqrt{q}})\tilde{E}^{a}_{I}\tilde{E}^{b}_{J}\epsilon^{% IJ}_{\;\;\;\;M}F_{ab}^{\;\;\;M0}+2Nn^{b}\tilde{E}^{a}_{I}F_{ab}^{\;\;\;I0})
  176. I , J , M I,J,M
  177. 1 , 2 , 3 1,2,3
  178. A a I J A_{a}^{IJ}
  179. A a i 0 = - i 1 2 ϵ j k i 0 A a j k = i 1 2 ϵ j k i A a j k = i A a i . A_{a}^{i0}=-i{1\over 2}\epsilon^{i0}_{\;\;\;jk}A_{a}^{jk}=i{1\over 2}\epsilon^% {i}_{\;\;jk}A_{a}^{jk}=iA_{a}^{i}.
  180. ϵ j k i 0 = - ϵ 0 j k i = - ϵ j k 0 i = - ϵ j k i \epsilon^{i0}_{\;\;\;jk}=-\epsilon^{i}_{\;\;0jk}=-\epsilon^{i}_{\;\;jk0}=-% \epsilon^{i}_{\;\;jk}
  181. F a b i 0 = a A b i 0 - b A a i 0 + A a i k A b k 0 - A b i k A a k 0 F_{ab}^{\;\;\;i0}=\partial_{a}A_{b}^{i0}-\partial_{b}A_{a}^{i0}+A_{a}^{ik}A_{% bk}^{\;\;\;0}-A_{b}^{ik}A_{ak}^{\;\;\;0}
  182. = i ( a A b i - b A a i + A a i k A b k - A b i k A a k ) =i(\partial_{a}A_{b}^{i}-\partial_{b}A_{a}^{i}+A_{a}^{ik}A_{bk}-A_{b}^{ik}A_{% ak})
  183. = i ( a A b i - b A a i + ϵ i j k A a j A b k ) =i(\partial_{a}A_{b}^{i}-\partial_{b}A_{a}^{i}+\epsilon_{ijk}A_{a}^{j}A_{b}^{k})
  184. = i F a b i . =iF_{ab}^{i}.
  185. N n b Nn^{b}
  186. t b - n b t^{b}-n^{b}
  187. t A b i = t a a A b i + A a i b t a \mathcal{L}_{t}A_{b}^{i}=t^{a}\partial_{a}A_{b}^{i}+A_{a}^{i}\partial_{b}t^{a}
  188. 𝒟 b ( t a A a i ) = b ( t a A a i ) + ϵ i j k A b j ( t a A a k ) \mathcal{D}_{b}(t^{a}A_{a}^{i})=\partial_{b}(t^{a}A_{a}^{i})+\epsilon_{ijk}A^{% j}_{b}(t^{a}A_{a}^{k})
  189. t A b i - 𝒟 b ( t a A a i ) = t a ( a A b i - b A a i + ϵ i j k A a j A b k ) = t a F a b i . \mathcal{L}_{t}A_{b}^{i}-\mathcal{D}_{b}(t^{a}A_{a}^{i})=t^{a}(\partial_{a}A_{% b}^{i}-\partial_{b}A_{a}^{i}+\epsilon_{ijk}A_{a}^{j}A^{k}_{b})=t^{a}F_{ab}^{i}.
  190. S = d 4 x ( - i ( N q ) E ~ I a E ~ J b ϵ M I J F a b M 0 - 2 ( t a - N a ) E ~ I b F a b I 0 ) S=\int d^{4}x(-i({N\over\sqrt{q}})\tilde{E}^{a}_{I}\tilde{E}^{b}_{J}\epsilon^{% IJ}_{\;\;\;\;M}F_{ab}^{\;\;\;M0}-2(t^{a}-N^{a})\tilde{E}^{b}_{I}F_{ab}^{\;\;\;% I0})
  191. = d 4 x ( - 2 i E ~ i b t A b i + 2 i E ~ i b 𝒟 b ( t a A a i ) + 2 i N a E ~ i b F a b i - ( N q ) ϵ i j k E ~ i a E ~ j b F a b k ) =\int d^{4}x(-2i\tilde{E}_{i}^{b}\mathcal{L}_{t}A_{b}^{i}+2i\tilde{E}_{i}^{b}% \mathcal{D}_{b}(t^{a}A_{a}^{i})+2iN^{a}\tilde{E}^{b}_{i}F_{ab}^{i}-({N\over% \sqrt{q}})\epsilon_{ijk}\tilde{E}^{a}_{i}\tilde{E}^{b}_{j}F_{ab}^{k})
  192. a a
  193. b b
  194. d 4 x E ~ i b 𝒟 b ( t a A a i ) = d t d 3 x E ~ i b ( b ( t a A a i ) + ϵ i j k A b j ( t a A a k ) ) \int d^{4}x\tilde{E}_{i}^{b}\mathcal{D}_{b}(t^{a}A_{a}^{i})=\int dtd^{3}x% \tilde{E}_{i}^{b}(\partial_{b}(t^{a}A_{a}^{i})+\epsilon_{ijk}A_{b}^{j}(t^{a}A_% {a}^{k}))
  195. = - d t d 3 x t a A a i ( b E ~ i b + ϵ i j k A b j E ~ k b ) =-\int dtd^{3}xt^{a}A_{a}^{i}(\partial_{b}\tilde{E}_{i}^{b}+\epsilon_{ijk}A_{b% }^{j}\tilde{E}_{k}^{b})
  196. = - d 4 x t a A a i 𝒟 b E ~ i b =-\int d^{4}xt^{a}A_{a}^{i}\mathcal{D}_{b}\tilde{E}_{i}^{b}
  197. V ~ i b \tilde{V}_{i}^{b}
  198. 𝒟 b V ~ i b = b V ~ i b + ϵ i j k A b j V ~ k b . \mathcal{D}_{b}\tilde{V}_{i}^{b}=\partial_{b}\tilde{V}_{i}^{b}+\epsilon_{ijk}A% _{b}^{j}\tilde{V}_{k}^{b}.
  199. S = d 4 x ( - 2 i E ~ i b t A b i - 2 i ( t a A a i ) 𝒟 b E ~ i b + 2 i N a E ~ i b F a b i + ( N q ) ϵ i j k E ~ i a E ~ j b F a b k ) S=\int d^{4}x(-2i\tilde{E}_{i}^{b}\mathcal{L}_{t}A_{b}^{i}-2i(t^{a}A_{a}^{i})% \mathcal{D}_{b}\tilde{E}_{i}^{b}+2iN^{a}\tilde{E}^{b}_{i}F_{ab}^{i}+({N\over% \sqrt{q}})\epsilon_{ijk}\tilde{E}^{a}_{i}\tilde{E}^{b}_{j}F_{ab}^{k})
  200. p q ˙ p\dot{q}
  201. E ~ i a \tilde{E}_{i}^{a}
  202. A a i A_{a}^{i}
  203. { A a i ( x ) , E ~ j b ( y ) } = i 2 δ a b δ j i δ 3 ( x , y ) . \{A_{a}^{i}(x),\tilde{E}_{j}^{b}(y)\}={i\over 2}\delta^{b}_{a}\delta^{i}_{j}% \delta^{3}(x,y).
  204. ( t a A a i ) (t^{a}A_{a}^{i})
  205. N b N^{b}
  206. N N
  207. 𝒟 a E ~ i a = 0 , \mathcal{D}_{a}\tilde{E}_{i}^{a}=0,
  208. F a b i E ~ i b = 0 , F_{ab}^{i}\tilde{E}^{b}_{i}=0,
  209. ϵ i j k E ~ i a E ~ j b F a b k = 0 E q 13. \epsilon_{ijk}\tilde{E}^{a}_{i}\tilde{E}^{b}_{j}F_{ab}^{k}=0\;\;Eq\;\;13.
  210. N N
  211. 13 \;13
  212. q \sqrt{q}
  213. A a i A_{a}^{i}
  214. A a i = 1 2 ϵ j k i A a j k = 1 2 ϵ j k i ( ω a j k - i 1 2 ( ϵ m 0 j k A^{i}_{a}={1\over 2}\epsilon^{i}_{\;\;jk}A^{jk}_{a}={1\over 2}\epsilon^{i}_{\;% \;jk}\big(\omega^{jk}_{a}-i{1\over 2}(\epsilon^{jk}_{\;\;\;m0}
  215. ω a m 0 + ϵ 0 m j k ω a 0 m ) ) = Γ a i - i ω 0 i a \omega^{m0}_{a}+\epsilon^{jk}_{\;\;\;0m}\omega^{0m}_{a})\big)=\Gamma_{a}^{i}-i% \omega^{0i}_{a}
  216. E c i ω a 0 i = - q a b E c i ω b i 0 E_{ci}\omega^{0i}_{a}=-q^{b}_{a}E_{ci}\omega_{b}^{i0}
  217. = - q a b E c i e d i b e d 0 = q a b q c d b n d = K a c =-q^{b}_{a}E_{ci}e^{di}\nabla_{b}e_{d}^{0}=q^{b}_{a}q^{d}_{c}\nabla_{b}n_{d}=K% _{ac}
  218. e d 0 = η 0 I g d c e I c = - g d c e 0 c = - n d e^{0}_{d}=\eta^{0I}g_{dc}e_{I}^{c}=-g_{dc}e_{0}^{c}=-n_{d}
  219. ω a 0 i = K a i \omega^{0i}_{a}=K_{a}^{i}
  220. A a i = Γ a i - i K a i . A_{a}^{i}=\Gamma_{a}^{i}-iK_{a}^{i}.

Semblance_analysis.html

  1. V rms = t i V i 2 t i V_{\mathrm{rms}}=\sqrt{\frac{\sum t_{i}\cdot{V_{i}}^{2}}{\sum t_{i}}}

Semibatch_reactor.html

  1. V d C U d t + C V d V d t V\frac{dC_{U}}{dt}+C_{V}\frac{dV}{dt}
  2. k 1 k_{1}
  3. C A α C_{A}^{\alpha}
  4. V d C W d t + C W d V d t V\frac{dC_{W}}{dt}+C_{W}\frac{dV}{dt}
  5. k 2 k_{2}
  6. C A β C_{A}^{\beta}
  7. d ( V C U ) d ( V C W ) \frac{d(VC_{U})}{d(VC_{W})}
  8. k 1 k 2 C A α - β \frac{k_{1}}{k_{2}}C_{A}^{\alpha-\beta}
  9. d C U d C W \frac{dC_{U}}{dC_{W}}
  10. k 1 k 2 C A α - β \frac{k_{1}}{k_{2}}C_{A}^{\alpha-\beta}
  11. β > α \beta>\alpha

Semiconductor_Bloch_equations.html

  1. 𝐏 \mathbf{P}
  2. 𝐄 \mathbf{E}
  3. 𝐏 \mathbf{P}
  4. P 𝐤 P_{\mathbf{k}}
  5. 𝐏 = 𝐝 𝐤 P 𝐤 + c . c . , \mathbf{P}=\mathbf{d}\,\sum_{\mathbf{k}}P_{\mathbf{k}}+\operatorname{c.c.}\;,
  6. 𝐤 \hbar{\mathbf{k}}
  7. 𝐝 \mathbf{d}
  8. P 𝐤 P_{\mathbf{k}}
  9. H ^ System \hat{H}_{\mathrm{System}}
  10. 𝒪 ^ \hat{\mathcal{O}}
  11. i d d t 𝒪 ^ = [ 𝒪 ^ , H ^ System ] - . \mathrm{i}\hbar\frac{\mathrm{d}}{\mathrm{d}t}\langle\hat{\mathcal{O}}\rangle=% \langle[\hat{\mathcal{O}},\hat{H}_{\mathrm{System}}]_{-}\rangle\;.
  12. H ^ System \hat{H}_{\mathrm{System}}
  13. 𝒪 ^ \hat{\mathcal{O}}
  14. a ^ c , 𝐤 \hat{a}^{\dagger}_{c,{\mathbf{k}}}
  15. a ^ c , 𝐤 \hat{a}_{c,{\mathbf{k}}}
  16. a ^ v , 𝐤 \hat{a}^{\dagger}_{v,{\mathbf{k}}}
  17. a ^ v , 𝐤 \hat{a}_{v,{\mathbf{k}}}
  18. P 𝐤 = a ^ c , 𝐤 a ^ v , 𝐤 , P 𝐤 = a ^ v , 𝐤 a ^ c , 𝐤 , P^{\star}_{\mathbf{k}}=\langle\hat{a}^{\dagger}_{c,\mathbf{k}}\hat{a}_{v,% \mathbf{k}}\rangle\,,\qquad P_{\mathbf{k}}=\langle\hat{a}^{\dagger}_{v,\mathbf% {k}}\hat{a}_{c,\mathbf{k}}\rangle\,,
  19. P 𝐤 P^{\star}_{\mathbf{k}}
  20. P 𝐤 P_{\mathbf{k}}
  21. f 𝐤 e = a ^ c , 𝐤 a ^ c , 𝐤 . f^{e}_{\mathbf{k}}=\langle\hat{a}^{\dagger}_{c,\mathbf{k}}\hat{a}_{c,\mathbf{k% }}\rangle\;.
  22. f 𝐤 h = 1 - a ^ v , 𝐤 a ^ v , 𝐤 = a ^ v , 𝐤 a ^ v , 𝐤 f^{h}_{\mathbf{k}}=1-\langle\hat{a}^{\dagger}_{v,\mathbf{k}}\hat{a}_{v,\mathbf% {k}}\rangle=\langle\hat{a}_{v,\mathbf{k}}\hat{a}^{\dagger}_{v,\mathbf{k}}\rangle
  23. t f 𝐤 e = 2 Im [ Ω 𝐤 P 𝐤 ] + t f 𝐤 e | scatter , \hbar\frac{\partial}{\partial t}f^{e}_{\mathbf{k}}=2\operatorname{Im}\left[% \Omega^{\star}_{\mathbf{k}}P_{\mathbf{k}}\right]+\hbar\left.\frac{\partial}{% \partial t}f^{e}_{\mathbf{k}}\right|_{\mathrm{scatter}}\,,
  24. t f 𝐤 h = 2 Im [ Ω 𝐤 P 𝐤 ] + t f 𝐤 h | scatter . \hbar\frac{\partial}{\partial t}f^{h}_{\mathbf{k}}=2\operatorname{Im}\left[% \Omega^{\star}_{\mathbf{k}}P_{\mathbf{k}}\right]+\hbar\left.\frac{\partial}{% \partial t}f^{h}_{\mathbf{k}}\right|_{\mathrm{scatter}}\;.
  25. Ω 𝐤 = 𝐝 𝐄 + 𝐤 𝐤 V 𝐤 - 𝐤 P 𝐤 \Omega_{\mathbf{k}}=\mathbf{d}\cdot\mathbf{E}+\sum_{\mathbf{k}^{\prime}\neq% \mathbf{k}}V_{\mathbf{k}-\mathbf{k}^{\prime}}P_{\mathbf{k}^{\prime}}
  26. ε ~ 𝐤 = ε 𝐤 - 𝐤 𝐤 V 𝐤 - 𝐤 [ f 𝐤 e + f 𝐤 h ] , \tilde{\varepsilon}_{\mathbf{k}}=\varepsilon_{\mathbf{k}}-\sum_{\mathbf{k}^{% \prime}\neq\mathbf{k}}V_{\mathbf{k}-\mathbf{k}^{\prime}}\left[f^{e}_{\mathbf{k% }^{\prime}}+f^{h}_{\mathbf{k}^{\prime}}\right]\,,
  27. ε 𝐤 \varepsilon_{\mathbf{k}}
  28. V 𝐤 V_{\mathbf{k}}
  29. 𝐤 \mathbf{k}
  30. | scatter \left.\cdots\right|_{\mathrm{scatter}}
  31. P 𝐤 P_{\mathbf{k}}
  32. f 𝐤 e f^{e}_{\mathbf{k}}
  33. f 𝐤 h f^{h}_{\mathbf{k}}
  34. f 𝐤 e f^{e}_{\mathbf{k}}
  35. f 𝐤 h f^{h}_{\mathbf{k}}
  36. 𝐤 \hbar\mathbf{k}
  37. α ( E ) \alpha(E)
  38. γ = 0.13 meV \hbar\gamma=0.13\,\mathrm{meV}
  39. α ( E ) \alpha(E)
  40. E E
  41. E gap = 1.490 meV E_{\mathrm{gap}}=1.490\,\mathrm{meV}
  42. P 𝐤 P_{\mathbf{k}}
  43. P 𝐤 P_{\mathbf{k}}
  44. V 𝐤 V_{\mathbf{k}}
  45. P 𝐤 P_{\mathbf{k}}
  46. V 𝐤 V_{\mathbf{k}}
  47. 𝐤 {\mathbf{k}}
  48. P 𝐤 P_{\mathbf{k}}
  49. P 𝐤 P_{\mathbf{k}}
  50. P 𝐤 P_{\mathbf{k}}
  51. P 𝐤 P_{\mathbf{k}}

Semiconductor_laser_theory.html

  1. E ( t ) E(t)
  2. α \alpha
  3. g g
  4. ω \hbar\omega
  5. n b n_{\mathrm{b}}
  6. c c
  7. ϵ 0 \epsilon_{0}
  8. ϵ \epsilon
  9. E ( ω ) E(\omega)
  10. Im \operatorname{Im}
  11. τ p \tau_{p}
  12. τ p \tau_{p}

Semiconductor_luminescence_equations.html

  1. ω \omega
  2. B ^ ω \hat{B}^{\dagger}_{\omega}
  3. B ^ ω \hat{B}_{\omega}
  4. B B
  5. B ^ ω B ^ ω \hat{B}^{\dagger}_{\omega}\,\hat{B}_{\omega}
  6. B ^ ω \langle\hat{B}_{\omega}\rangle
  7. L ( ω ) = t B ^ ω B ^ ω = 2 Re [ 𝐤 ω Π 𝐤 , ω ] . \mathrm{L}(\omega)=\frac{\partial}{\partial t}\langle\hat{B}^{\dagger}_{\omega% }\hat{B}_{\omega}\rangle=2\,\mathrm{Re}\left[\sum_{\mathbf{k}}\mathcal{F}_{% \omega}^{\star}\,\Pi_{\mathbf{k},\omega}\right]\,.
  8. Π 𝐤 , ω Δ B ^ ω P ^ 𝐤 \Pi_{\mathbf{k},\omega}\equiv\Delta\langle\hat{B}^{\dagger}_{\omega}\hat{P}_{% \mathbf{k}}\rangle
  9. ( B ^ ω ) (\hat{B}^{\dagger}_{\omega})
  10. 𝐤 \mathbf{k}
  11. P ^ 𝐤 \hat{P}_{\mathbf{k}}
  12. Π 𝐤 , ω \Pi_{\mathbf{k},\omega}
  13. ω \omega
  14. Δ \Delta
  15. Π 𝐤 , ω \Pi_{\mathbf{k},\omega}
  16. B ^ ω P 𝐤 \langle\hat{B}^{\dagger}_{\omega}P_{\mathbf{k}}\rangle
  17. ω \mathcal{F}_{\omega}
  18. ϵ ~ 𝐤 \tilde{\epsilon}_{\mathbf{k}}
  19. V 𝐤 V_{\mathbf{k}}
  20. T [ Π ] T[\Pi]
  21. Ω 𝐤 , ω spont \Omega^{\mathrm{spont}}_{\mathbf{k},\omega}
  22. Ω ω stim \Omega_{\omega}^{\mathrm{stim}}
  23. f 𝐤 e f^{e}_{\mathbf{k}}
  24. f 𝐤 h f^{h}_{\mathbf{k}}
  25. Π 𝐤 , ω \Pi_{\mathbf{k},\omega}
  26. ( 1 - f 𝐤 e - f 𝐤 h ) \left(1-f^{e}_{\mathbf{k}}-f^{h}_{\mathbf{k}}\right)
  27. t f 𝐤 e | L = t f 𝐤 h | L = - 2 Re [ ω ω Π 𝐤 , ω ] . \left.\frac{\partial}{\partial t}f^{e}_{\mathbf{k}}\right|_{\mathrm{L}}=\left.% \frac{\partial}{\partial t}f^{h}_{\mathbf{k}}\right|_{\mathrm{L}}=-2\,\mathrm{% Re}\left[\sum_{\omega}\mathcal{F}^{\star}_{\omega}\,\Pi_{\mathbf{k},\omega}% \right]\,.
  28. t ω B ^ ω B ^ ω = - t 𝐤 f 𝐤 e \frac{\partial}{\partial t}\sum_{\omega}\langle\hat{B}^{\dagger}_{\omega}\hat{% B}_{\omega}\rangle=-\frac{\partial}{\partial t}\sum_{\mathbf{k}}f^{e}_{\mathbf% {k}}
  29. Ω 𝐤 , ω spont = i ω ( f 𝐤 e f 𝐤 h + 𝐤 c X 𝐤 , 𝐤 ) . \Omega^{\mathrm{spont}}_{\mathbf{k},\omega}=\mathrm{i}\mathcal{F}_{\omega}% \Bigl(f^{e}_{\mathbf{k}}f^{h}_{\mathbf{k}}+\sum_{\mathbf{k^{\prime}}}c_{% \mathrm{X}}^{\mathbf{k},\mathbf{k^{\prime}}}\Bigr)\,.
  30. f 𝐤 e f 𝐤 h f^{e}_{\mathbf{k}}\,f^{h}_{\mathbf{k}}
  31. 𝐤 \mathbf{k}
  32. 𝐤 \mathbf{k}
  33. c X 𝐤 , 𝐤 c_{\mathrm{X}}^{\mathbf{k},\mathbf{k^{\prime}}}
  34. c X 𝐤 , 𝐤 c_{\mathrm{X}}^{\mathbf{k},\mathbf{k^{\prime}}}
  35. Δ Ω ω stim = i ω ω Δ B ^ ω B ^ ω \Delta\Omega_{\omega}^{\mathrm{stim}}=\mathrm{i}\sum_{\omega}\mathcal{F}_{% \omega^{\prime}}\,\Delta\langle\hat{B}^{\dagger}_{\omega}\hat{B}_{\omega^{% \prime}}\rangle
  36. i t c X 𝐤 , 𝐤 = ( ϵ ~ 𝐤 - ϵ ~ 𝐤 ) c X 𝐤 , 𝐤 + S X 𝐤 , 𝐤 + ( 1 - f 𝐤 e - f 𝐤 h ) 𝐥 V 𝐥 - 𝐤 c X 𝐤 , 𝐥 - ( 1 - f 𝐤 e - f 𝐤 h ) 𝐥 V 𝐥 - 𝐤 c X 𝐥 , 𝐤 + D X , rest 𝐤 , 𝐤 + T X 𝐤 , 𝐤 . \begin{aligned}\displaystyle\mathrm{i}\hbar\frac{\partial}{\partial t}c_{% \mathrm{X}}^{\mathbf{k},\mathbf{k^{\prime}}}=&\displaystyle\left(\tilde{% \epsilon}_{\mathbf{k}}-\tilde{\epsilon}_{\mathbf{k^{\prime}}}\right)\,c_{% \mathrm{X}}^{\mathbf{k},\mathbf{k^{\prime}}}+S_{\mathrm{X}}^{\mathbf{k},% \mathbf{k^{\prime}}}\\ &\displaystyle+\Bigl(1-f^{e}_{\mathbf{k^{\prime}}}-f^{h}_{\mathbf{k^{\prime}}}% \Bigr)\sum_{\mathbf{l}}V_{\mathbf{l}-\mathbf{k}^{\prime}}\,c_{\mathrm{X}}^{% \mathbf{k},\mathbf{l}}-\Bigl(1-f^{e}_{\mathbf{k}}-f^{h}_{\mathbf{k}}\Bigr)\sum% _{\mathbf{l}}V_{\mathbf{l}-\mathbf{k}^{\prime}}\,c_{\mathrm{X}}^{\mathbf{l},% \mathbf{k^{\prime}}}\\ &\displaystyle+D_{\mathrm{X,\,rest}}^{\mathbf{k},\mathbf{k^{\prime}}}+T_{% \mathrm{X}}^{\mathbf{k},\mathbf{k^{\prime}}}\,.\end{aligned}
  37. D X , rest 𝐤 , 𝐤 D_{\mathrm{X,\,rest}}^{\mathbf{k},\mathbf{k^{\prime}}}
  38. T X 𝐤 , 𝐤 T_{\mathrm{X}}^{\mathbf{k},\mathbf{k^{\prime}}}
  39. Ω 𝐤 , ω spont \Omega^{\mathrm{spont}}_{\mathbf{k},\omega}
  40. 𝐤 \mathbf{k}
  41. Π ω , 𝐤 \Pi_{\omega,\mathbf{k}}
  42. 𝐤 \mathbf{k}
  43. Π ω , 𝐤 \Pi_{\omega,\mathbf{k}}
  44. 𝐤 \mathbf{k}
  45. Π ω , 𝐤 \Pi_{\omega,\mathbf{k}}
  46. Π ω , 𝐤 \Pi_{\omega,\mathbf{k}}
  47. Π ω , 𝐤 \Pi_{\omega,\mathbf{k}}
  48. Π ω , 𝐤 \Pi_{\omega,\mathbf{k}}
  49. ω \omega
  50. ω \omega
  51. Δ B ^ ω B ^ ω \Delta\langle\hat{B}_{\omega}\hat{B}_{\omega^{\prime}}\rangle

Semiconductor_optical_gain.html

  1. g ( ε ) g(\varepsilon)
  2. g ( ε ) = g 0 ε [ f e ( ε ) + f h ( ε ) - 1 ] , g(\varepsilon)=g_{0}\sqrt{\varepsilon}\,[f^{\mathrm{e}}(\varepsilon)+f^{% \mathrm{h}}(\varepsilon)-1]~{},
  3. ε \varepsilon
  4. f e f^{\mathrm{e}}
  5. f h f^{\mathrm{h}}
  6. g 0 g_{0}
  7. g 0 ( ε ) = ν | μ ( ε ) | 2 4 ε 0 π n ( 2 m r 2 ) 3 / 2 , g_{0}(\varepsilon)=\frac{\nu|\mu(\varepsilon)|^{2}}{4\varepsilon_{0}\pi n}% \left(\frac{2m_{\mathrm{r}}}{\hbar^{2}}\right)^{3/2}~{},
  8. ν \nu
  9. | μ ( ε ) | 2 |\mu(\varepsilon)|^{2}
  10. m r m_{\mathrm{r}}
  11. ε 0 \varepsilon_{0}
  12. n n
  13. g ( ε ) g(\varepsilon)
  14. ε \sqrt{\varepsilon}
  15. p 𝐤 p_{\mathbf{k}}
  16. n 𝐤 n_{\mathbf{k}}
  17. f 𝐤 e f^{e}_{\mathbf{k}}
  18. f 𝐤 h f^{h}_{\mathbf{k}}
  19. t p 𝐤 = - i δ k p 𝐤 - i [ 1 - f 𝐤 e - f 𝐤 h ] Ω 𝐤 - t p 𝐤 | coll \frac{\mathrm{\partial}}{\mathrm{\partial}t}p_{\mathbf{k}}=-\mathrm{i}\,\delta% _{k}p_{\mathbf{k}}-\mathrm{i}\,[1-f^{e}_{\mathbf{k}}-f^{h}_{\mathbf{k}}]\Omega% _{\mathbf{k}}-\left.\frac{\mathrm{\partial}}{\mathrm{\partial}t}p_{\mathbf{k}}% \right|_{\mathrm{coll}}
  20. δ 𝐤 \delta_{\mathbf{k}}
  21. Ω 𝐤 \Omega_{\mathbf{k}}
  22. δ 𝐤 \delta_{\mathbf{k}}
  23. Ω 𝐤 \Omega_{\mathbf{k}}
  24. t p 𝐤 | coll \left.\frac{\mathrm{\partial}}{\mathrm{\partial}t}p_{\mathbf{k}}\right|_{% \mathrm{coll}}
  25. T 2 T_{2}
  26. 𝐤 \mathbf{k}
  27. I ASE I_{\mathrm{ASE}}
  28. l l
  29. I ASE ( l ) I_{\mathrm{ASE}}(l)

Seminormal_ring.html

  1. x 3 = y 2 x^{3}=y^{2}
  2. s 2 = x s^{2}=x
  3. s 3 = y s^{3}=y

Sepiapterin_reductase_(L-threo-7,8-dihydrobiopterin_forming).html

  1. \rightleftharpoons
  2. \rightleftharpoons

SequenceL.html

  1. k = 1 p A ( i , k ) B ( k , j ) \sum_{k=1}^{p}A(i,k)B(k,j)
  2. [ 1 , 2 , 3 ] + 10 = = [ 11 , 12 , 13 ] [1,2,3]+10==[11,12,13]
  3. f ( x , [ 1 , 2 , 3 ] , z ) = = [ f ( x , 1 , z ) , f ( x , 2 , z ) , f ( x , 3 , z ) ] f(x,[1,2,3],z)==[f(x,1,z),f(x,2,z),f(x,3,z)]
  4. [ 1 , 2 , 3 ] + [ 10 , 20 , 30 ] = = [ 11 , 22 , 33 ] [1,2,3]+[10,20,30]==[11,22,33]

Series_and_parallel_springs.html

  1. k 1 k_{1}
  2. k 2 k_{2}
  3. c c
  4. 1 / k 1/k
  5. k e q = k 1 + k 2 k_{eq}=k_{1}+k_{2}
  6. 1 k e q = 1 k 1 + 1 k 2 \frac{1}{k_{eq}}=\frac{1}{k_{1}}+\frac{1}{k_{2}}
  7. 1 c e q = 1 c 1 + 1 c 2 \frac{1}{c_{eq}}=\frac{1}{c_{1}}+\frac{1}{c_{2}}
  8. c e q = c 1 + c 2 c_{eq}=c_{1}+c_{2}
  9. x e q = x 1 = x 2 x_{eq}=x_{1}=x_{2}
  10. x e q = x 1 + x 2 x_{eq}=x_{1}+x_{2}
  11. F e q = F 1 + F 2 F_{eq}=F_{1}+F_{2}
  12. F e q = F 1 = F 2 F_{eq}=F_{1}=F_{2}
  13. E e q = E 1 + E 2 E_{eq}=E_{1}+E_{2}
  14. E e q = E 1 + E 2 E_{eq}=E_{1}+E_{2}
  15. x 1 = x 2 x_{1}=x_{2}\,
  16. x 1 x 2 = k 2 k 1 = c 1 c 2 \frac{x_{1}}{x_{2}}=\frac{k_{2}}{k_{1}}=\frac{c_{1}}{c_{2}}
  17. F 1 F 2 = k 1 k 2 = c 2 c 1 \frac{F_{1}}{F_{2}}=\frac{k_{1}}{k_{2}}=\frac{c_{2}}{c_{1}}
  18. F 1 = F 2 F_{1}=F_{2}\,
  19. E 1 E 2 = k 1 k 2 = c 2 c 1 \frac{E_{1}}{E_{2}}=\frac{k_{1}}{k_{2}}=\frac{c_{2}}{c_{1}}
  20. E 1 E 2 = k 2 k 1 = c 1 c 2 \frac{E_{1}}{E_{2}}=\frac{k_{2}}{k_{1}}=\frac{c_{1}}{c_{2}}
  21. F b = - k e q ( x 1 + x 2 ) . F_{b}=-k_{eq}(x_{1}+x_{2}).\,
  22. F 1 = - k 1 x 1 = F 2 = - k 2 x 2 . F_{1}=-k_{1}x_{1}=F_{2}=-k_{2}x_{2}.\,
  23. x 2 x_{2}\,
  24. x 2 = k 1 k 2 x 1 . x_{2}=\frac{k_{1}}{k_{2}}x_{1}.\,
  25. F b F_{b}\,
  26. = - k e q ( x 1 + k 1 k 2 x 1 ) =-k_{eq}\left(x_{1}+\frac{k_{1}}{k_{2}}x_{1}\right)\,
  27. F b = - k e q ( k 2 + k 1 k 2 ) x 1 F_{b}=-k_{eq}\left(\frac{k_{2}+k_{1}}{k_{2}}\right)x_{1}\,
  28. - k e q ( k 2 + k 1 k 2 ) x 1 = - k 1 x 1 . -k_{eq}\left(\frac{k_{2}+k_{1}}{k_{2}}\right)x_{1}=-k_{1}x_{1}.\,
  29. k e q = k 1 k 2 k 2 + k 1 . k_{eq}=\frac{k_{1}k_{2}}{k_{2}+k_{1}}.\,
  30. 1 k e q = 1 k 1 + 1 k 2 . \frac{1}{k_{eq}}=\frac{1}{k_{1}}+\frac{1}{k_{2}}.\,
  31. F b F_{b}\,
  32. = F 1 + F 2 =F_{1}+F_{2}\,
  33. = - k 1 x - k 2 x =-k_{1}x-k_{2}x\,
  34. F b = - ( k 1 + k 2 ) x . F_{b}=-(k_{1}+k_{2})x.\,
  35. k e q = k 1 + k 2 . k_{eq}=k_{1}+k_{2}.\,
  36. F 1 = F 2 F_{1}=F_{2}\,
  37. - k 1 x 1 = - k 2 x 2 . -k_{1}x_{1}=-k_{2}x_{2}.\,
  38. x 1 x 2 = k 2 k 1 . \frac{x_{1}}{x_{2}}=\frac{k_{2}}{k_{1}}.\,
  39. E 1 E 2 = 1 2 k 1 x 1 2 1 2 k 2 x 2 2 , \frac{E_{1}}{E_{2}}=\frac{\frac{1}{2}k_{1}x_{1}^{2}}{\frac{1}{2}k_{2}x_{2}^{2}% },\,
  40. E 1 E 2 = k 1 k 2 ( k 2 k 1 ) 2 = k 2 k 1 . \frac{E_{1}}{E_{2}}=\frac{k_{1}}{k_{2}}\left(\frac{k_{2}}{k_{1}}\right)^{2}=% \frac{k_{2}}{k_{1}}.\,
  41. E 1 E 2 = 1 2 k 1 x 2 1 2 k 2 x 2 \frac{E_{1}}{E_{2}}=\frac{\frac{1}{2}k_{1}x^{2}}{\frac{1}{2}k_{2}x^{2}}\,
  42. E 1 E 2 = k 1 k 2 . \frac{E_{1}}{E_{2}}=\frac{k_{1}}{k_{2}}.\,

Sesquipower.html

  1. f i + 1 = f i g i f i f_{i+1}=f_{i}g_{i}f_{i}
  2. f = f 1 g 1 f 1 g 2 f 1 g 1 f 1 g 3 f 1 f=f_{1}g_{1}f_{1}g_{2}f_{1}g_{1}f_{1}g_{3}f_{1}\cdots

Set_estimation.html

  1. P = P 0 f - 1 ( Y ) P=P_{0}\cap f^{-1}(Y)
  2. ϕ ( p 1 , p 2 , t ) = ( t p 1 ) 2 + t p 2 2 + s i n ( p 1 + t p 2 ) , \phi(p_{1},p_{2},t)=(tp_{1})^{2}+tp_{2}^{2}+sin(p_{1}+tp_{2}),
  3. Y = [ y 1 ] × [ y 2 ] × [ y 3 ] Y=[y_{1}]\times[y_{2}]\times[y_{3}]
  4. f ( p 1 , p 2 ) = [ p 1 2 - p 2 2 + s i n ( p 1 - p 2 ) p 1 2 + p 2 2 + s i n ( p 1 + p 2 ) ( 2 p 1 ) 2 + 2 p 2 2 + s i n ( p 1 + 2 p 2 ) ] f(p_{1},p_{2})=\begin{bmatrix}p_{1}^{2}-p_{2}^{2}+sin(p_{1}-p_{2})\\ p_{1}^{2}+p_{2}^{2}+sin(p_{1}+p_{2})\\ (2p_{1})^{2}+2p_{2}^{2}+sin(p_{1}+2p_{2})\end{bmatrix}

Seven_Steps_to_Heaven_(composition).html

  1. E b 9 6 Eb^{6}_{9}
  2. E 6 b 9 E^{b9}_{6}
  3. E b 9 6 Eb^{6}_{9}
  4. E 6 b 9 E^{b9}_{6}
  5. E b 9 6 Eb^{6}_{9}
  6. E 6 b 9 E^{b9}_{6}
  7. E b 9 6 Eb^{6}_{9}
  8. E 6 b 9 E^{b9}_{6}

Sh2-106.html

  1. M \begin{smallmatrix}M_{\odot}\end{smallmatrix}

Short-chain_acyl-CoA_dehydrogenase.html

  1. \rightleftharpoons

Short-circuit_test.html

  1. 𝐖 \mathbf{W}
  2. 𝐕 𝟏 \mathbf{V_{1}}
  3. 𝐈 𝟏 \mathbf{I_{1}}
  4. 𝐑 𝟎𝟏 \mathbf{R_{01}}
  5. 𝐙 𝟎𝟏 \mathbf{Z_{01}}
  6. 𝐗 𝟎𝟏 \mathbf{X_{01}}
  7. 𝐖 = 𝐈 𝟏 2 𝐑 𝟎𝟏 \mathbf{W}={\mathbf{I_{1}}}^{2}\mathbf{R_{01}}
  8. 𝐑 𝟎𝟏 = 𝐖 𝐈 𝟏 2 \mathbf{R_{01}}=\frac{\mathbf{W}}{\mathbf{I_{1}}^{2}}
  9. 𝐙 𝟎𝟏 = 𝐕 𝟏 𝐈 𝟏 \mathbf{Z_{01}}=\frac{\mathbf{V_{1}}}{\mathbf{I_{1}}}
  10. 𝐗 𝟎𝟏 = 𝐙 𝟎𝟏 2 - 𝐑 𝟎𝟏 2 {\mathbf{X_{01}}}=\sqrt{\mathbf{Z_{01}}^{2}-\mathbf{R_{01}}^{2}}

Sidi's_generalized_secant_method.html

  1. f ( x ) = 0 f(x)=0
  2. f f
  3. f f
  4. f f
  5. α \alpha
  6. f f
  7. f ( α ) = 0 f(\alpha)=0
  8. { x i } \{x_{i}\}
  9. α \alpha
  10. x 1 , , x k + 1 x_{1},\dots,x_{k+1}
  11. x k + 2 x_{k+2}
  12. x k + 3 x_{k+3}
  13. f f
  14. x n , , x n + k x_{n},\dots,x_{n+k}
  15. f ( x n ) , , f ( x n + k ) f(x_{n}),\dots,f(x_{n+k})
  16. x 1 , , x k + 1 x_{1},\dots,x_{k+1}
  17. x n + k + 1 x_{n+k+1}
  18. p n , k ( x ) p_{n,k}(x)
  19. ( x n , f ( x n ) ) , ( x n + k , f ( x n + k ) ) (x_{n},f(x_{n})),\dots(x_{n+k},f(x_{n+k}))
  20. x n + k + 1 x_{n+k+1}
  21. α \alpha
  22. x n + k + 1 = x n + k - f ( x n + k ) p n , k ( x n + k ) x_{n+k+1}=x_{n+k}-\frac{f(x_{n+k})}{p_{n,k}^{\prime}(x_{n+k})}
  23. p n , k ( x n + k ) p_{n,k}^{\prime}(x_{n+k})
  24. p n , k p_{n,k}
  25. x n + k x_{n+k}
  26. x n + k + 1 x_{n+k+1}
  27. f ( x n + k + 1 ) f(x_{n+k+1})
  28. f f
  29. f f
  30. α \alpha
  31. p n , k ( x ) p_{n,k}(x)
  32. f f
  33. I I
  34. α \alpha
  35. f C k + 1 ( I ) f\in C^{k+1}(I)
  36. α \alpha
  37. f f
  38. f ( α ) 0 f^{\prime}(\alpha)\neq 0
  39. x 1 , , x k + 1 x_{1},\dots,x_{k+1}
  40. α \alpha
  41. { x i } \{x_{i}\}
  42. α \alpha
  43. lim n x n = α \lim\limits_{n\to\infty}x_{n}=\alpha
  44. lim n x n + 1 - α i = 0 k ( x n - i - α ) = L = ( - 1 ) k + 1 ( k + 1 ) ! f ( k + 1 ) ( α ) f ( α ) , \lim_{n\to\infty}\frac{x_{n+1}-\alpha}{\prod^{k}_{i=0}(x_{n-i}-\alpha)}=L=% \frac{(-1)^{k+1}}{(k+1)!}\frac{f^{(k+1)}(\alpha)}{f^{\prime}(\alpha)},
  45. α \alpha
  46. ψ k \psi_{k}
  47. lim n | x n + 1 - α | | x n - α | ψ k = | L | ( ψ k - 1 ) / k \lim\limits_{n\to\infty}\frac{|x_{n+1}-\alpha|}{|x_{n}-\alpha|^{\psi_{k}}}=|L|% ^{(\psi_{k}-1)/k}
  48. ψ k \psi_{k}
  49. s k + 1 - s k - s k - 1 - - s - 1 s^{k+1}-s^{k}-s^{k-1}-\dots-s-1
  50. ψ 1 = ( 1 + 5 ) / 2 \psi_{1}=(1+\sqrt{5})/2
  51. ψ 2 \psi_{2}
  52. ψ 3 \psi_{3}
  53. lim k ψ k = 2 \lim\limits_{k\to\infty}\psi_{k}=2
  54. p n , 1 ( x ) p_{n,1}(x)
  55. f f
  56. α \alpha
  57. p n , k ( x ) p_{n,k}(x)
  58. f ( x ) f(x)
  59. x = α x=\alpha
  60. p n , k ( x ) p_{n,k}^{\prime}(x)
  61. f ( x ) f^{\prime}(x)
  62. x = α x=\alpha
  63. p n , k p_{n,k}^{\prime}
  64. f f^{\prime}
  65. x n + k + 1 = x n + k - f ( x n + k ) f ( x n + k ) x_{n+k+1}=x_{n+k}-\frac{f(x_{n+k})}{f^{\prime}(x_{n+k})}
  66. x 1 x_{1}
  67. f f^{\prime}
  68. f f
  69. p n , k ( x ) p_{n,k}(x)
  70. x n + k + 1 x_{n+k+1}
  71. p n , k ( x ) = 0 p_{n,k}(x)=0
  72. p n , k ( x ) = 0 p_{n,k}(x)=0
  73. x n + k + 1 x_{n+k+1}

Siegel_identity.html

  1. x 3 - x 1 x 2 - x 1 + x 2 - x 3 x 2 - x 1 = 1. \frac{x_{3}-x_{1}}{x_{2}-x_{1}}+\frac{x_{2}-x_{3}}{x_{2}-x_{1}}=1.
  2. x 3 - x 1 x 2 - x 1 t - x 2 t - x 3 + x 2 - x 3 x 2 - x 1 t - x 1 t - x 3 = 1. \frac{x_{3}-x_{1}}{x_{2}-x_{1}}\cdot\frac{t-x_{2}}{t-x_{3}}+\frac{x_{2}-x_{3}}% {x_{2}-x_{1}}\cdot\frac{t-x_{1}}{t-x_{3}}=1.

Silver_cyanate.html

  1. AgNO 3 + KOCN AgOCN + KNO 3 \mathrm{AgNO_{3}+KOCN\longrightarrow AgOCN\downarrow+\ KNO_{3}}
  2. AgNO 3 + H 2 N - C ( O ) - NH 2 \mathrm{AgNO_{3}+H_{2}N\,\text{-}C(O)\,\text{-}NH_{2}\longrightarrow}
  3. AgOCN + NH 4 NO 3 \mathrm{AgOCN\downarrow+\ NH_{4}NO_{3}}
  4. AgOCN + 2 H N O 3 + H 2 O \mathrm{AgOCN+2HNO_{3}+H_{2}O\longrightarrow}
  5. AgNO 3 + CO 2 + NH 4 NO 3 \mathrm{AgNO_{3}+CO_{2}\uparrow+\ NH_{4}NO_{3}}

Similarity_learning.html

  1. ( x i 1 , x i 2 ) (x_{i}^{1},x_{i}^{2})
  2. y i R y_{i}\in R
  3. f ( x i 1 , x i 2 ) y i f(x_{i}^{1},x_{i}^{2})\sim y_{i}
  4. ( x i 1 , x i 2 , y i ) (x_{i}^{1},x_{i}^{2},y_{i})
  5. m i n W i l o s s ( w ; x i 1 , x i 2 , y i ) + r e g ( w ) min_{W}\sum_{i}loss(w;x_{i}^{1},x_{i}^{2},y_{i})+reg(w)
  6. ( x i , x i + ) (x_{i},x_{i}^{+})
  7. ( x i , x i - ) (x_{i},x_{i}^{-})
  8. ( x i 1 , x i 2 ) (x_{i}^{1},x_{i}^{2})
  9. y i { 0 , 1 } y_{i}\in\{0,1\}
  10. ( x i , x i + , x i - ) (x_{i},x_{i}^{+},x_{i}^{-})
  11. x i x_{i}
  12. x i + x_{i}^{+}
  13. x i - x_{i}^{-}
  14. f f
  15. ( x , x + , x - ) (x,x^{+},x^{-})
  16. f ( x , x + ) > f ( x , x - ) f(x,x^{+})>f(x,x^{-})
  17. f W ( x , z ) = x T W z f_{W}(x,z)=x^{T}Wz
  18. x i x_{i}
  19. R d R^{d}
  20. W W
  21. S + d S_{+}^{d}
  22. D W ( x 1 , x 2 ) 2 = ( x 1 - x 2 ) W ( x 1 - x 2 ) D_{W}(x_{1},x_{2})^{2}=(x_{1}-x_{2})^{\top}W(x_{1}-x_{2})
  23. W W
  24. D W D_{W}
  25. W S + d W\in S_{+}^{d}
  26. W = L L W=L^{\top}L
  27. L R e × d L\in R^{e\times d}
  28. e r a n k ( W ) e\geq rank(W)
  29. D W D_{W}
  30. D W ( x 1 , x 2 ) 2 = ( x 1 - x 2 ) L L ( x 1 - x 2 ) = L ( x 1 - x 2 ) 2 2 D_{W}(x_{1},x_{2})^{2}=(x_{1}-x_{2})^{\top}L^{\top}L(x_{1}-x_{2})=\|L(x_{1}-x_% {2})\|_{2}^{2}
  31. D W ( x 1 , x 2 ) 2 = x 1 - x 2 2 2 D_{W}(x_{1},x_{2})^{2}=\|x_{1}^{\prime}-x_{2}^{\prime}\|_{2}^{2}
  32. x 1 = L x 1 x_{1}^{\prime}=Lx_{1}
  33. x 2 = L x 2 x_{2}^{\prime}=Lx_{2}

Simon_Brendle.html

  1. S 3 S^{3}

Simple_chemical_reacting_system.html

  1. d ( ρ m f u ) d t + d i v ( ρ m f u u ) = d i v ( R f u . g r a d m f u ) + S f u {d(\rho m_{fu})\over dt}+div(\rho m_{fu}u)=div(R_{fu}.gradm_{fu})+S_{fu}
  2. d ( ρ m o x ) d t + d i v ( ρ m o x u ) = d i v ( R o x . g r a d m o x ) + S o x {d(\rho m_{ox})\over dt}+div(\rho m_{ox}u)=div(R_{ox}.gradm_{ox})+S_{ox}
  3. d ( ρ ϕ ) d t + d i v ( ρ ϕ u ) = d i v ( R ϕ . g r a d ϕ ) + ( s . S f u - S o x ) {d(\rho\phi)\over dt}+div(\rho\phi u)=div(R_{\phi}.grad\phi)+(s.S_{fu}-S_{ox})
  4. d ( ρ ϕ ) d t + d i v ( ρ ϕ u ) = d i v ( R ϕ . g r a d ϕ ) {d(\rho\phi)\over dt}+div(\rho\phi u)=div(R_{\phi}.grad\phi)
  5. f = ϕ - ϕ 0 ϕ 1 - ϕ 0 f=\frac{\phi-\phi_{0}}{\phi_{1}-\phi_{0}}
  6. f = [ s m f u - m o x ] - [ s m f u - m o x ] 0 [ s m f u - m o x ] 1 - [ s m f u - m o x ] 0 f=\frac{[sm_{fu}-m_{ox}]-[sm_{fu}-m_{ox}]_{0}}{[sm_{fu}-m_{ox}]_{1}-[sm_{fu}-m% _{ox}]_{0}}
  7. f = s m f u - m o x + m o x , 0 s m f u , 1 + m o x , 0 f=\frac{sm_{fu}-m_{ox}+m_{ox,0}}{sm_{fu,1}+m_{ox,0}}
  8. f s t = m o x , 0 s m f u , 1 + m o x , 0 f_{st}=\frac{m_{ox,0}}{sm_{fu,1}+m_{ox,0}}
  9. f = - m o x + m o x , 0 s m f u , 1 + m o x , 0 f=\frac{-m_{ox}+m_{ox,0}}{sm_{fu,1}+m_{ox,0}}
  10. f = s m f u + m o x , 0 s m f u , 1 + m o x , 0 f=\frac{sm_{fu}+m_{ox,0}}{sm_{fu,1}+m_{ox,0}}
  11. d ( ρ u f ) d t + d i v ( ρ f u ) = d i v ( R f . g r a d f ) {d(\rho uf)\over dt}+div(\rho fu)=div(R_{f}.gradf)
  12. m f u = f - f s t 1 - f s t ( m f u , 1 ) , f s t < f < 1 , m o x = 0 m_{fu}=\frac{f-f_{st}}{1-f_{st}}\left(m_{fu,1}\right),f_{st}<f<1,m_{ox}=0
  13. m o x = f s t - f f s t ( m o x , 0 ) , 0 < f s t < f , m f u = 0 m_{ox}=\frac{f_{st}-f}{f_{st}}\left(m_{ox,0}\right),0<f_{st}<f,m_{fu}=0

Singular_integral_operators_of_convolution_type.html

  1. f ( θ ) = n 𝐙 a n e i n θ . f(\theta)=\sum_{n\in\mathbf{Z}}a_{n}e^{in\theta}.
  2. f r ( θ ) = F ( r e i θ ) , f_{r}(\theta)=F(re^{i\theta}),
  3. f r - f 2 0. \|f_{r}-f\|_{2}\rightarrow 0.
  4. F ( z ) = 1 2 π i | ζ | = 1 f ( ζ ) ζ - z d ζ = 1 2 π - π π f ( θ ) 1 - e - i θ z d θ . F(z)={1\over 2\pi i}\int_{|\zeta|=1}{f(\zeta)\over\zeta-z}\,d\zeta={1\over 2% \pi}\int_{-\pi}^{\pi}{f(\theta)\over 1-e^{-i\theta}z}\,d\theta.
  5. F ( r e i φ ) = 1 2 π - π π f ( φ - θ ) 1 - r e i θ d θ . \displaystyle{F(re^{i\varphi})={1\over 2\pi}\int_{-\pi}^{\pi}{f(\varphi-\theta% )\over 1-re^{i\theta}}\,d\theta.}
  6. H ε f ( φ ) = i π ε | θ | π f ( φ - θ ) 1 - e i θ d θ = 1 π | ζ - e i φ | δ f ( ζ ) ζ - e i φ d ζ , \displaystyle{H_{\varepsilon}f(\varphi)={i\over\pi}\int_{\varepsilon\leq|% \theta|\leq\pi}{f(\varphi-\theta)\over 1-e^{i\theta}}\,d\theta={1\over\pi}\int% _{|\zeta-e^{i\varphi}|\geq\delta}{f(\zeta)\over\zeta-e^{i\varphi}}\,d\zeta,}
  7. H ε 1 = i π ε π 2 ( 1 - e i θ ) - 1 d θ = i π ε π 1 d θ = i - i ε π . \displaystyle{H_{\varepsilon}{1}={i\over\pi}\int_{\varepsilon}^{\pi}2\Re(1-e^{% i\theta})^{-1}\,d\theta={i\over\pi}\int_{\varepsilon}^{\pi}1\,d\theta=i-{i% \varepsilon\over\pi}.}
  8. H ε f ( z ) - i ( 1 - ε ) π f ( z ) = 1 π i | ζ - z | δ f ( ζ ) - f ( z ) ζ - z d ζ . \displaystyle{H_{\varepsilon}f(z)-{i(1-\varepsilon)\over\pi}f(z)={1\over\pi i}% \int_{|\zeta-z|\geq\delta}{f(\zeta)-f(z)\over\zeta-z}\,d\zeta.}
  9. H ε f i f \displaystyle{H_{\varepsilon}f\rightarrow if}
  10. H ε f ¯ = - u - 1 H ε ( u f ¯ ) . \displaystyle{\overline{H_{\varepsilon}f}=-u^{-1}H_{\varepsilon}(u\overline{f}% ).}
  11. H ε f - i f \displaystyle{H_{\varepsilon}f\rightarrow-if}
  12. H = i ( 2 P - I ) . \displaystyle{H=i(2P-I).}
  13. H ε f H f \displaystyle{H_{\varepsilon}f\rightarrow Hf}
  14. H ε f H f \displaystyle{H_{\varepsilon}f\rightarrow Hf}
  15. ( 1 - e i θ ) - 1 = [ ( 1 - e i θ ) - 1 - i θ - 1 ] + i θ - 1 . \displaystyle{(1-e^{i\theta})^{-1}=[(1-e^{i\theta})^{-1}-i\theta^{-1}]+i\theta% ^{-1}.}
  16. S ε f ( φ ) = ε | θ | π f ( φ - θ ) θ - 1 d θ \displaystyle{S_{\varepsilon}f(\varphi)=\int_{\varepsilon\leq|\theta|\leq\pi}f% (\varphi-\theta)\theta^{-1}\,d\theta}
  17. 1 π | a b sin t t d t | \displaystyle{{1\over\pi}\left|\int_{a}^{b}{\sin t\over t}\,dt\right|}
  18. H f = P . V . 1 π f ( ζ ) ζ - e i φ d ζ . \displaystyle{Hf=\mathrm{P.V.}\,{1\over\pi}\int{f(\zeta)\over\zeta-e^{i\varphi% }}\,d\zeta.}
  19. T r ( a n e i n θ ) = r | n | a n e i n θ , T_{r}\left(\sum a_{n}e^{in\theta}\right)=\sum r^{|n|}a_{n}e^{in\theta},
  20. H ( e i θ ) = e i h ( θ ) , h ( θ + 2 π ) = h ( θ ) + 2 π , H(e^{i\theta})=e^{ih(\theta)},\,\,\,h(\theta+2\pi)=h(\theta)+2\pi,
  21. H ε h f ( e i φ ) = 1 π | e i h ( θ ) - e i h ( φ ) | ε f ( e i θ ) e i θ - e i φ e i θ d θ , H_{\varepsilon}^{h}f(e^{i\varphi})=\frac{1}{\pi}\int_{|e^{ih(\theta)}-e^{ih(% \varphi)}|\geq\varepsilon}\frac{f(e^{i\theta})}{e^{i\theta}-e^{i\varphi}}e^{i% \theta}\,d\theta,
  22. ( V H ε h V - 1 - H ε ) f ( e i φ ) = 1 π | e i θ - e i φ | ε [ g ( θ ) e i g ( θ ) e i g ( θ ) - e i g ( φ ) - e i θ e i θ - e i φ ] f ( e i θ ) d θ . (VH^{h}_{\varepsilon}V^{-1}-H_{\varepsilon})f(e^{i\varphi})={1\over\pi}\int_{|% e^{i\theta}-e^{i\varphi}|\geq\varepsilon}\left[{g^{\prime}(\theta)e^{ig(\theta% )}\over e^{ig(\theta)}-e^{ig(\varphi)}}-{e^{i\theta}\over e^{i\theta}-e^{i% \varphi}}\right]\,f(e^{i\theta})\,d\theta.
  23. H ε h f ( ζ ) = 1 π i | H ( z ) - H ( ζ ) | ε f ( z ) z - ζ d z = 1 π i | H ( z ) - H ( ζ ) | ε f ( z ) - f ( ζ ) z - ζ d z + f ( ζ ) π i | H ( z ) - H ( ζ ) | ε d z z - ζ . H^{h}_{\varepsilon}f(\zeta)={1\over\pi i}\int_{|H(z)-H(\zeta)|\geq\varepsilon}% \frac{f(z)}{z-\zeta}dz={1\over\pi i}\int_{|H(z)-H(\zeta)|\geq\varepsilon}{f(z)% -f(\zeta)\over z-\zeta}\,dz+\frac{f(\zeta)}{\pi i}\int_{|H(z)-H(\zeta)|\geq% \varepsilon}{dz\over z-\zeta}.
  24. 1 π i f ( z ) - f ( ζ ) z - ζ d z . {1\over\pi i}\int{f(z)-f(\zeta)\over z-\zeta}\,dz.
  25. lim ε 0 H ε h f ( ζ ) = f ( ζ ) + 1 π i f ( z ) - f ( ζ ) z - ζ d z , \lim_{\varepsilon\to 0}H_{\varepsilon}^{h}f(\zeta)=f(\zeta)+{1\over\pi i}\int{% f(z)-f(\zeta)\over z-\zeta}\,dz,
  26. ( V H V - 1 - H ) f ( e i φ ) = 1 π [ g ( θ ) e i g ( θ ) e i g ( θ ) - e i g ( φ ) - e i θ e i θ - e i φ ] f ( e i θ ) d θ . (VHV^{-1}-H)f(e^{i\varphi})=\frac{1}{\pi}\int\left[{g^{\prime}(\theta)e^{ig(% \theta)}\over e^{ig(\theta)}-e^{ig(\varphi)}}-{e^{i\theta}\over e^{i\theta}-e^% {i\varphi}}\right]\,f(e^{i\theta})\,d\theta.
  27. f ( e i θ ) = n 𝐙 a n e i n θ , f(e^{i\theta})=\sum_{n\in\mathbf{Z}}a_{n}e^{in\theta},
  28. P r f ( e i θ ) = n 𝐙 a n r | n | e i n θ = 1 2 π 0 2 π ( 1 - r 2 ) f ( e i θ ) 1 - 2 r cos θ + r 2 d θ = K r f ( e i θ ) , \displaystyle{P_{r}f(e^{i\theta})=\sum_{n\in\mathbf{Z}}a_{n}r^{|n|}e^{in\theta% }={1\over 2\pi}\int_{0}^{2\pi}{(1-r^{2})f(e^{i\theta})\over 1-2r\cos\theta+r^{% 2}}\,d\theta=K_{r}\star f(e^{i\theta}),}
  29. K r ( e i θ ) = n 𝐙 r | n | e i n θ = 1 - r 2 1 - 2 r cos θ + r 2 . \displaystyle{K_{r}(e^{i\theta})=\sum_{n\in\mathbf{Z}}r^{|n|}e^{in\theta}={1-r% ^{2}\over 1-2r\cos\theta+r^{2}}.}
  30. P r f - f p 0. \displaystyle{\|P_{r}f-f\|_{p}\rightarrow 0.}
  31. K r 1 = 1 2 π 0 2 π K r ( e i θ ) d θ = 1. \displaystyle{\|K_{r}\|_{1}={1\over 2\pi}\int_{0}^{2\pi}K_{r}(e^{i\theta})\,d% \theta=1.}
  32. ψ r ( e i θ ) = 1 + 1 - r 1 + r cot ( θ 2 ) K r ( e i θ ) 1 + 1 - r 1 + r cot ( 1 - r 2 ) K r ( e i θ ) \begin{aligned}\displaystyle\psi_{r}(e^{i\theta})&\displaystyle=1+\frac{1-r}{1% +r}\cot\left(\tfrac{\theta}{2}\right)K_{r}(e^{i\theta})\\ &\displaystyle\leq 1+\frac{1-r}{1+r}\cot\left(\tfrac{1-r}{2}\right)K_{r}(e^{i% \theta})\end{aligned}
  33. H 𝐑 f ^ = ( i χ [ 0 , ) - i χ ( - , 0 ] ) f ^ , \widehat{H_{\mathbf{R}}f}=\left(i\chi_{[0,\infty)}-i\chi_{(-\infty,0]}\right)% \widehat{f},
  34. f ^ ( t ) = 1 2 π - f ( x ) e - i t x d x . \displaystyle{\widehat{f}(t)={1\over\sqrt{2\pi}}\int_{-\infty}^{\infty}f(x)e^{% -itx}\,dx.}
  35. H 𝐑 = i ( 2 P 𝐑 - I ) . \displaystyle{H_{\mathbf{R}}=i(2P_{\mathbf{R}}-I).}
  36. C ( x ) = x - i x + i \displaystyle{C(x)={x-i\over x+i}}
  37. U f ( x ) = π - 1 / 2 ( x + i ) - 1 f ( C ( x ) ) . \displaystyle{Uf(x)=\pi^{-1/2}(x+i)^{-1}f(C(x)).}
  38. U f w ( x ) = 1 π 1 ( 1 - w ) ( x - z ¯ ) Uf_{w}(x)=\frac{1}{\sqrt{\pi}}\frac{1}{(1-w)(x-\overline{z})}
  39. z = C - 1 ( w ¯ ) . \displaystyle{z=C^{-1}(\overline{w}).}
  40. g z ( t ) = e i t z χ [ 0 , ) ( t ) \displaystyle{g_{z}(t)=e^{itz}\chi_{[0,\infty)}(t)}
  41. h z ( x ) = g z ^ ( - x ) = i 2 π ( x + z ) - 1 , \displaystyle{h_{z}(x)=\widehat{g_{z}}(-x)={i\over\sqrt{2\pi}}(x+z)^{-1},}
  42. U H 𝐓 U * = H 𝐑 . \displaystyle{UH_{\mathbf{T}}U^{*}=H_{\mathbf{R}}.}
  43. F ( z ) = 1 2 π i - f ( s ) s - z d s . \displaystyle{F(z)={1\over 2\pi i}\int_{-\infty}^{\infty}{f(s)\over s-z}\,ds.}
  44. 1 2 π i - f ( s ) s - z d s = 1 2 π - f ( s ) g z ^ ( s ) d s = 1 2 π - f ^ ( s ) g z ( s ) d s = V y P f ( x ) . \displaystyle{{1\over 2\pi i}\int_{-\infty}^{\infty}{f(s)\over s-z}\,ds={1% \over\sqrt{2\pi}}\int_{-\infty}^{\infty}f(s)\widehat{g_{z}}(s)\,ds={1\over% \sqrt{2\pi}}\int_{-\infty}^{\infty}\widehat{f}(s)g_{z}(s)\,ds=V_{y}Pf(x).}
  45. f y + t = V t P f y \displaystyle{f_{y+t}=V_{t}Pf_{y}}
  46. V t f y = f y + t = V y f t , \displaystyle{V_{t}f_{y}=f_{y+t}=V_{y}f_{t},}
  47. f t = lim y 0 f y + t = lim y 0 V t f y = V t f . \displaystyle{f_{t}=\lim_{y\rightarrow 0}f_{y+t}=\lim_{y\rightarrow 0}V_{t}f_{% y}=V_{t}f.}
  48. H ε , R f ( x ) = 1 π ε | y - x | R f ( y ) x - y d y = 1 π ε | y | R f ( x - y ) y d y H ε f ( x ) = 1 π | y - x | ε f ( y ) x - y d y = 1 π | y | ε f ( x - y ) y d y . \begin{aligned}\displaystyle H_{\varepsilon,R}f(x)&\displaystyle={1\over\pi}% \int_{\varepsilon\leq|y-x|\leq R}{f(y)\over x-y}\,dy={1\over\pi}\int_{% \varepsilon\leq|y|\leq R}{f(x-y)\over y}\,dy\\ \displaystyle H_{\varepsilon}f(x)&\displaystyle={1\over\pi}\int_{|y-x|\geq% \varepsilon}{f(y)\over x-y}\,dy={1\over\pi}\int_{|y|\geq\varepsilon}{f(x-y)% \over y}\,dy.\end{aligned}
  49. 1 2 π | a b 2 sin t t d t | . \displaystyle{{1\over\sqrt{2\pi}}\left|\int_{a}^{b}{2\sin t\over t}\,dt\right|.}
  50. H ε f ¯ = - H ε ( f ¯ ) , \displaystyle{\overline{H_{\varepsilon}f}=-H_{\varepsilon}(\overline{f}),}
  51. | F ( m ) ( z ) | K N , m ( 1 + | z | ) - N \displaystyle{|F^{(m)}(z)|\leq K_{N,m}(1+|z|)^{-N}}
  52. 1 π Γ F ( z ) z - x d z . \displaystyle{{1\over\pi}\int_{\Gamma}{F(z)\over z-x}\,dz.}
  53. H ε f ( x ) = 1 π | y - x | ε f ( y ) - f ( x ) y - x d y = 1 π | y - x | ε 0 1 f ( x + t ( y - x ) ) d t d y H_{\varepsilon}f(x)=\frac{1}{\pi}\int_{|y-x|\geq\varepsilon}\frac{f(y)-f(x)}{y% -x}\,dy=\frac{1}{\pi}\int_{|y-x|\geq\varepsilon}\int_{0}^{1}f^{\prime}(x+t(y-x% ))\,dt\,dy
  54. G ( x ) = 1 2 π 0 1 - | f ( x + t y ) | d y G(x)=\frac{1}{2\pi}\int_{0}^{1}\int_{-\infty}^{\infty}|f^{\prime}(x+ty)|\,dy
  55. H ε f H f . \displaystyle{H_{\varepsilon}f\rightarrow Hf.}
  56. T y f ( x ) = - P y ( x - t ) f ( t ) d t , T_{y}f(x)=\int_{-\infty}^{\infty}P_{y}(x-t)f(t)\,dt,
  57. P y ( x ) = y π ( x 2 + y 2 ) . P_{y}(x)=\frac{y}{\pi(x^{2}+y^{2})}.
  58. P y ^ ( t ) = e - y | t | , \displaystyle{\widehat{P_{y}}(t)=e^{-y|t|},}
  59. g ε ( x ) = { x π ( x 2 + ε 2 ) | x | ε x π ( x 2 + ε 2 ) - 1 π x | x | > ε g_{\varepsilon}(x)=\begin{cases}\frac{x}{\pi(x^{2}+\varepsilon^{2})}&|x|\leq% \varepsilon\\ \frac{x}{\pi(x^{2}+\varepsilon^{2})}-\frac{1}{\pi x}&|x|>\varepsilon\end{cases}
  60. R f ^ ( z ) = z ¯ | z | f ^ ( z ) , R * f ^ ( z ) = z | z | f ^ ( z ) . \displaystyle{\widehat{Rf}(z)={\overline{z}\over|z|}\widehat{f}(z),\,\,\,% \widehat{R^{*}f}(z)={z\over|z|}\widehat{f}(z).}
  61. R = - i R 1 + R 2 , R * = - i R 1 - R 2 , \displaystyle{R=-iR_{1}+R_{2},\,\,\,R^{*}=-iR_{1}-R_{2},}
  62. R k f ( w ) = lim ε 0 | z - w | ε M k ( w - z ) f ( z ) d x d y , \displaystyle{R^{k}f(w)=\lim_{\varepsilon\rightarrow 0}\int_{|z-w|\geq% \varepsilon}M_{k}(w-z)f(z)\,dx\,dy,}
  63. M k ( z ) = k 2 π i k z k | z | k + 2 ( k 1 ) , M - k ( z ) = M k ( z ) ¯ . \displaystyle{M_{k}(z)={k\over 2\pi i^{k}}{z^{k}\over|z|^{k+2}}\,\,\,\,(k\geq 1% ),\,\,\,\,M_{-k}(z)=\overline{M_{k}(z)}.}
  64. R ε ( k ) f ( w ) = | z - w | ε M k ( w - z ) f ( z ) d x d y , \displaystyle{R^{(k)}_{\varepsilon}f(w)=\int_{|z-w|\geq\varepsilon}M_{k}(w-z)f% (z)\,dx\,dy,}
  65. T s f ( x ) = 1 2 π 𝐑 2 s f ( x ) ( | x - t | 2 + s 2 ) 3 / 2 d t . \displaystyle{T_{s}f(x)={1\over 2\pi}\int_{\mathbf{R}^{2}}{sf(x)\over(|x-t|^{2% }+s^{2})^{3/2}}\,dt.}
  66. P s ( x ) = s 2 π ( | x | 2 + s 2 ) 3 / 2 . \displaystyle{P_{s}(x)={s\over 2\pi(|x|^{2}+s^{2})^{3/2}}.}
  67. T s 1 F ( x , y , s 2 ) = F ( x , y , s 1 + s 2 ) . \displaystyle{T_{s_{1}}F(x,y,s_{2})=F(x,y,s_{1}+s_{2}).}
  68. T ε R k - R ε ( k ) \displaystyle{T_{\varepsilon}R^{k}-R^{(k)}_{\varepsilon}}
  69. U θ f ( z ) = f ( e i θ z ) . \displaystyle{U_{\theta}f(z)=f(e^{i\theta}z).}
  70. B = 1 2 π 0 2 π φ ( θ ) U θ A ( 1 ) U θ * d θ . \displaystyle{B={1\over 2\pi}\int_{0}^{2\pi}\varphi(\theta)U_{\theta}A^{(1)}U_% {\theta}^{*}\,d\theta.}
  71. ( B f , g ) = 1 2 π 0 2 π φ ( θ ) ( U θ A ( 1 ) U θ * f , g ) d θ \displaystyle{(Bf,g)={1\over 2\pi}\int_{0}^{2\pi}\varphi(\theta)(U_{\theta}A^{% (1)}U_{\theta}^{*}f,g)\,d\theta}
  72. B 1 2 π 0 2 π | φ ( θ ) | A d θ . \displaystyle{\|B\|\leq{1\over 2\pi}\int_{0}^{2\pi}|\varphi(\theta)|\cdot\|A\|% \,d\theta.}
  73. R = 1 2 π 0 2 π e - i θ U θ H ( 1 ) U θ * d θ , R ε = 1 2 π 0 2 π e - i θ U θ H ε ( 1 ) U θ * d θ . \begin{aligned}\displaystyle R&\displaystyle={1\over 2\pi}\int_{0}^{2\pi}e^{-i% \theta}U_{\theta}H^{(1)}U_{\theta}^{*}\,d\theta,\\ \displaystyle R_{\varepsilon}&\displaystyle={1\over 2\pi}\int_{0}^{2\pi}e^{-i% \theta}U_{\theta}H^{(1)}_{\varepsilon}U_{\theta}^{*}\,d\theta.\end{aligned}
  74. z ¯ z = ( z ¯ | z | ) 2 , {\overline{z}\over z}=\left({\overline{z}\over|z|}\right)^{2},
  75. T ε f ( w ) = - 1 π | z - w | ε f ( z ) ( w - z ) 2 d x d y . T_{\varepsilon}f(w)=-\frac{1}{\pi}\iint_{|z-w|\geq\varepsilon}\frac{f(z)}{(w-z% )^{2}}dxdy.
  76. T f ( w ) = - 1 π P . V . f ( z ) ( w - z ) 2 d x d y = - 1 π lim ε 0 | z - w | ε f ( z ) ( w - z ) 2 d x d y . Tf(w)=-\frac{1}{\pi}P.V.\iint\frac{f(z)}{(w-z)^{2}}dxdy=-\frac{1}{\pi}\lim_{% \varepsilon\to 0}\iint_{|z-w|\geq\varepsilon}\frac{f(z)}{(w-z)^{2}}dxdy.
  77. T ( z ¯ f ) = z T f , T * ( z f ) = z ¯ T * f . \begin{aligned}\displaystyle T(\partial_{\overline{z}}f)&\displaystyle=% \partial_{z}Tf,\\ \displaystyle T^{*}(\partial_{z}f)=\partial_{\overline{z}}T^{*}f.\end{aligned}
  78. T ε ( Ω ) f ( w ) = - 1 π D \ V ε [ φ ( w ) φ ( z ) ( φ ( z ) - φ ( w ) ) 2 f ( z ) ] d x d y , T ε ( D ) f ( w ) = - 1 π D \ U ε f ( z ) ( z - w ) 2 d x d y , \begin{aligned}\displaystyle T_{\varepsilon}(\Omega)f(w)&\displaystyle=-\frac{% 1}{\pi}\iint_{D\backslash V_{\varepsilon}}\left[{\varphi^{\prime}(w)\varphi^{% \prime}(z)\over(\varphi(z)-\varphi(w))^{2}}f(z)\right]dxdy,\\ \displaystyle T_{\varepsilon}(D)f(w)&\displaystyle=-{1\over\pi}\iint_{D% \backslash U_{\varepsilon}}{f(z)\over(z-w)^{2}}dxdy,\end{aligned}
  79. T ε ( D ) f ( w ) = - 1 π D \ V ε f ( z ) ( z - w ) 2 d x d y , T^{\prime}_{\varepsilon}(D)f(w)=-{1\over\pi}\iint_{D\backslash V_{\varepsilon}% }\frac{f(z)}{(z-w)^{2}}dxdy,
  80. K ( w , z ) = - 1 π [ φ ( w ) φ ( z ) ( φ ( z ) - φ ( w ) ) 2 - 1 ( z - w ) 2 ] . K(w,z)=-{1\over\pi}\left[{\varphi^{\prime}(w)\varphi^{\prime}(z)\over(\varphi(% z)-\varphi(w))^{2}}-{1\over(z-w)^{2}}\right].
  81. ( T ε ( D ) - T ε ( D ) ) f ( w ) = 1 π U ε z f ( z ) z - w d x d y - 1 π V ε z f ( z ) z - w d x d y + 1 2 π i U ε f ( z ) z - w d z ¯ - 1 2 π i V ε f ( z ) z - w d z ¯ . \left(T_{\varepsilon}(D)-T^{\prime}_{\varepsilon}(D)\right)f(w)=\frac{1}{\pi}% \iint_{U_{\varepsilon}}{\partial_{z}f(z)\over z-w}dxdy-{1\over\pi}\iint_{V_{% \varepsilon}}{\partial_{z}f(z)\over z-w}dxdy+{1\over 2\pi i}\int_{\partial U_{% \varepsilon}}\frac{f(z)}{z-w}d\overline{z}-\frac{1}{2\pi i}\int_{\partial V_{% \varepsilon}}{f(z)\over z-w}\,d\overline{z}.
  82. ( T f ) g = - 1 π lim | z - w | ε f ( w ) g ( z ) ( w - z ) 2 = f ( T g ) . \iint(Tf)g=-{1\over\pi}\lim\int_{|z-w|\geq\varepsilon}\frac{f(w)g(z)}{(w-z)^{2% }}=\iint f(Tg).
  83. T χ ( w ) = - ε 2 1 - χ ( w ) ( w - z ) 2 . T\chi(w)=-\varepsilon^{2}\frac{1-\chi(w)}{(w-z)^{2}}.
  84. T ε ( f ) ( z ) = 1 π ε 2 f ( T χ ) = 1 π ε 2 ( T f ) χ = 𝐀𝐯 D ( z , ε ) T f . T_{\varepsilon}(f)(z)={1\over\pi\varepsilon^{2}}\iint f(T\chi)={1\over\pi% \varepsilon^{2}}\iint(Tf)\chi=\mathbf{Av}_{D(z,\varepsilon)}\,Tf.
  85. R j f ( x ) = c n lim ε 0 | y | ε f ( x - y ) y j | y | n + 1 d y = c n n - 1 j f ( x - y ) 1 | y | n - 1 d y , R_{j}f(x)=c_{n}\lim_{\varepsilon\to 0}\int_{|y|\geq\varepsilon}f(x-y){y_{j}% \over|y|^{n+1}}dy=\frac{c_{n}}{n-1}\int\partial_{j}f(x-y){1\over|y|^{n-1}}dy,
  86. c n = Γ ( n + 1 2 ) π - n + 1 2 . c_{n}=\Gamma\left(\tfrac{n+1}{2}\right)\pi^{-\frac{n+1}{2}}.
  87. R j f ^ ( t ) = i t j | t | f ^ ( t ) . \widehat{R_{j}f}(t)={it_{j}\over|t|}\widehat{f}(t).
  88. R 1 2 + + R n 2 = - I . R_{1}^{2}+\cdots+R_{n}^{2}=-I.
  89. R j , ε f ( x ) = c n | y | ε f ( x - y ) y j | y | n + 1 d y R_{j,\varepsilon}f(x)=c_{n}\int_{|y|\geq\varepsilon}f(x-y){y_{j}\over|y|^{n+1}% }dy
  90. R j = G φ ( g ) g H ( 1 ) g - 1 d g , R j , ε = G φ ( g ) g H ε ( 1 ) g - 1 d g , R j , ε , R = G φ ( g ) g H ε , R ( 1 ) g - 1 d g . \begin{aligned}\displaystyle R_{j}&\displaystyle=\int_{G}\varphi(g)gH^{(1)}g^{% -1}\,dg,\\ \displaystyle R_{j,\varepsilon}&\displaystyle=\int_{G}\varphi(g)gH_{% \varepsilon}^{(1)}g^{-1}\,dg,\\ \displaystyle R_{j,\varepsilon,R}&\displaystyle=\int_{G}\varphi(g)gH_{% \varepsilon,R}^{(1)}g^{-1}\,dg.\end{aligned}
  91. T y f ( x ) = c n 𝐑 n y f ( x ) ( | x - t | 2 + y 2 ) n + 1 2 d t . T_{y}f(x)=c_{n}\int_{\mathbf{R}^{n}}\frac{yf(x)}{\left(|x-t|^{2}+y^{2}\right)^% {\frac{n+1}{2}}}dt.
  92. P y ( x ) = c n y ( | x | 2 + y 2 ) n + 1 2 . P_{y}(x)=c_{n}\frac{y}{\left(|x|^{2}+y^{2}\right)^{\frac{n+1}{2}}}.
  93. p p
  94. H H
  95. f ( e i θ ) = m = 1 N a m e i m θ + a - m e - i m θ , a - m = a m ¯ . f\left(e^{i\theta}\right)=\sum_{m=1}^{N}a_{m}e^{im\theta}+a_{-m}e^{-im\theta},% \qquad a_{-m}=\overline{a_{m}}.
  96. 1 2 π 0 2 π ( f + i H f ) 2 n d θ = 0. \frac{1}{2\pi}\int_{0}^{2\pi}(f+iHf)^{2n}\,d\theta=0.
  97. H f 2 n 2 n k = 0 n - 1 ( 2 n 2 k ) | ( ( H f ) 2 k , f 2 n - 2 k ) | k = 0 n - 1 ( 2 n 2 k ) H f 2 n 2 k f 2 n 2 n - 2 k . \|Hf\|_{2n}^{2n}\leq\sum_{k=0}^{n-1}{2n\choose 2k}\left|\left((Hf)^{2k},f^{2n-% 2k}\right)\right|\leq\sum_{k=0}^{n-1}{2n\choose 2k}\|Hf\|_{2n}^{2k}\cdot\|f\|_% {2n}^{2n-2k}.
  98. p p
  99. p p
  100. H f 2 n + 1 2 = ( H f ) 2 2 n f 2 2 n + 2 H ( f H ( f ) ) 2 n f 2 n + 1 2 + 2 H 2 n f 2 n + 1 H f 2 n + 1 . \|Hf\|^{2}_{2^{n+1}}=\left\|(Hf)^{2}\right\|_{2^{n}}\leq\left\|f^{2}\right\|_{% 2^{n}}+2\|H(fH(f))\|_{2^{n}}\leq\|f\|_{2^{n+1}}^{2}+2\|H\|_{2^{n}}\|f\|_{2^{n+% 1}}\|Hf\|_{2^{n+1}}.
  101. R 2 > 1 + 2 H 2 n R R^{2}>1+2\|H\|_{2^{n}}R
  102. R R
  103. ± i ±i
  104. H H
  105. p p
  106. Φ Φ
  107. 𝐓 \mathbf{T}
  108. S O ( n ) SO(n)
  109. ( G Φ ( x ) d x f , g ) = G ( Φ ( x ) f , g ) d x \left(\int_{G}\Phi(x)\,dx\,f,g\right)=\int_{G}(\Phi(x)f,g)\,dx
  110. g g
  111. 1 p + 1 q . \frac{1}{p}+\frac{1}{q}.
  112. A ( ε ) = 1 2 ε x - ε x + ε | f ( t ) - f ( x ) | d t 0 \displaystyle{A(\varepsilon)={1\over 2\varepsilon}\int_{x-\varepsilon}^{x+% \varepsilon}|f(t)-f(x)|\,dt\to 0}
  113. 2 π | T r f ( x ) - f ( x ) | = 0 2 π | ( f ( x - y ) - f ( x ) ) P r ( y ) | d y | y | ε + | y | ε . 2\pi|T_{r}f(x)-f(x)|=\int_{0}^{2\pi}|(f(x-y)-f(x))P_{r}(y)|\,dy\leq\int_{|y|% \leq\varepsilon}+\int_{|y|\geq\varepsilon}.
  114. sup y [ - ε , ε ] | P 1 - ε ( y ) | ε - 1 . sup y ( - ε , ε ) | P 1 - ε ( y ) | 0. \begin{aligned}\displaystyle\sup_{y\in[-\varepsilon,\varepsilon]}|P_{1-% \varepsilon}(y)|&\displaystyle\leq\varepsilon^{-1}.\\ \displaystyle\sup_{y\notin(-\varepsilon,\varepsilon)}|P_{1-\varepsilon}(y)|&% \displaystyle\to 0.\end{aligned}
  115. Q r ( θ ) = 2 r sin θ 1 - 2 r cos θ + r 2 . \displaystyle{Q_{r}(\theta)={2r\sin\theta\over 1-2r\cos\theta+r^{2}}.}
  116. 2 π | T 1 - ε H f ( x ) - H ε f ( x ) | | y | ε | f ( x - y ) - f ( x ) | | Q r ( y ) | d y + | y | ε | f ( x - y ) - f ( x ) | | Q 1 ( y ) - Q r ( y ) | d y . \displaystyle{2\pi|T_{1-\varepsilon}Hf(x)-H_{\varepsilon}f(x)|\leq\int_{|y|% \leq\varepsilon}|f(x-y)-f(x)|\cdot|Q_{r}(y)|\,dy+\int_{|y|\geq\varepsilon}|f(x% -y)-f(x)|\cdot|Q_{1}(y)-Q_{r}(y)|\,dy.}
  117. sup y [ - ε , ε ] | Q 1 - ε ( y ) | ε - 1 . sup y ( - ε , ε ) | Q 1 ( y ) - Q 1 - ε ( y ) | 0. \begin{aligned}\displaystyle\sup_{y\in[-\varepsilon,\varepsilon]}|Q_{1-% \varepsilon}(y)|&\displaystyle\leq\varepsilon^{-1}.\\ \displaystyle\sup_{y\notin(-\varepsilon,\varepsilon)}|Q_{1}(y)-Q_{1-% \varepsilon}(y)|&\displaystyle\to 0.\end{aligned}
  118. f * ( t ) = sup 0 < h π 1 2 h t - h t + h | f ( s ) | d s . \displaystyle{f^{*}(t)=\sup_{0<h\leq\pi}{1\over 2h}\int_{t-h}^{t+h}|f(s)|\,ds.}
  119. E f ( λ ) = { x : | f ( x ) | > λ } , f λ = χ E ( λ ) f , \displaystyle{E_{f}(\lambda)=\{x:\,|f(x)|>\lambda\},\,\,f_{\lambda}=\chi_{E(% \lambda)}f,}
  120. m ( E f * ( λ ) ) 8 λ E f ( λ ) | f | 8 f 1 λ , m(E_{f^{*}}(\lambda))\leq{8\over\lambda}\int_{E_{f}(\lambda)}|f|\leq{8\|f\|_{1% }\over\lambda},
  121. lim h 0 x - h x + h | f ( t ) - f ( x ) | d t 2 h 0. \displaystyle{\lim_{h\to 0}\frac{\int^{x+h}_{x-h}|f(t)-f(x)|\,dt}{2h}\to 0.}
  122. ω ( f ) ( x ) = lim sup h 0 x - h x + h | f ( t ) - f ( x ) | d t 2 h f * ( x ) + | f ( x ) | . \omega(f)(x)=\limsup_{h\to 0}\frac{\int^{x+h}_{x-h}|f(t)-f(x)|\,dt}{2h}\leq f^% {*}(x)+|f(x)|.
  123. m { x : ω ( f ) ( x ) > λ } = m { x : ω ( f - g ) ( x ) > λ } m { x : ( f - g ) * ( x ) > λ } + m { x : | f ( x ) - g ( x ) | > λ } C λ - 1 f - g 1 . m\{x:\,\omega(f)(x)>\lambda\}=m\{x:\,\omega(f-g)(x)>\lambda\}\leq m\{x:\,(f-g)% ^{*}(x)>\lambda\}+m\{x:\,|f(x)-g(x)|>\lambda\}\leq C\lambda^{-1}\|f-g\|_{1}.
  124. | T r f | f * . \displaystyle{|T_{r}f|\leq f^{*}.}
  125. Ω ( f ) = lim sup r 1 | T r f - f | . \displaystyle{\Omega(f)=\limsup_{r\rightarrow 1}|T_{r}f-f|.}
  126. m { x : Ω ( f ) ( x ) > λ } = m { x : Ω ( f - g ) ( x ) > λ } m { x : ( f - g ) * ( x ) > λ } + m { x : | f ( x ) - g ( x ) | > λ } C λ - 1 f - g 1 . m\{x:\,\Omega(f)(x)>\lambda\}=m\{x:\,\Omega(f-g)(x)>\lambda\}\leq m\{x:\,(f-g)% ^{*}(x)>\lambda\}+m\{x:\,|f(x)-g(x)|>\lambda\}\leq C\lambda^{-1}\|f-g\|_{1}.
  127. F ( z ) = exp ( - f ( z ) - i H f ( z ) ) , \displaystyle{F(z)=\exp(-f(z)-iHf(z)),}
  128. | H ε f - T 1 - ε H f | 4 f * . \displaystyle{|H_{\varepsilon}f-T_{1-\varepsilon}Hf|\leq 4f^{*}.}
  129. ω ( f ) = lim sup ε 0 | H ε f - T 1 - ε H f | . \displaystyle{\omega(f)=\limsup_{\varepsilon\rightarrow 0}|H_{\varepsilon}f-T_% {1-\varepsilon}Hf|.}
  130. m { x : ω ( f ) ( x ) > λ } = m { x : ω ( f - g ) ( x ) > λ } m { x : 4 ( f - g ) * ( x ) > λ } C λ - 1 f - g 1 . m\{x:\,\omega(f)(x)>\lambda\}=m\{x:\,\omega(f-g)(x)>\lambda\}\leq m\{x:\,4(f-g% )^{*}(x)>\lambda\}\leq C\lambda^{-1}\|f-g\|_{1}.
  131. g ( x ) = χ J n ( f ) ( x J n ) , g ( x ) = f ( x ) ( x Ω c ) . \displaystyle{g(x)=\chi_{J_{n}}(f)\,\,\,(x\in J_{n}),\,\,\,\,\,g(x)=f(x)\,\,\,% (x\in\Omega^{c}).}
  132. g 1 f 1 . \displaystyle{\|g\|_{1}\leq\|f\|_{1}.}
  133. g p p ( 2 α ) p - 1 f 1 . \displaystyle{\|g\|_{p}^{p}\leq(2\alpha)^{p-1}\|f\|_{1}.}
  134. b 1 2 f 1 . \displaystyle{\|b\|_{1}\leq 2\|f\|_{1}.}
  135. m ( Ω ) α - 1 f 1 . \displaystyle{m(\Omega)\leq\alpha^{-1}\|f\|_{1}.}
  136. f ( x ) = g ( x ) + b ( x ) \displaystyle{f(x)=g(x)+b(x)}
  137. W ( f ) = lim ε 0 | x | ε K ( x ) f ( x ) d x W(f)=\lim_{\varepsilon\to 0}\int_{|x|\geq\varepsilon}K(x)f(x)\,dx
  138. A = sup y 0 | x | 2 | y | | K ( x - y ) - K ( x ) | d x < , A=\sup_{y\neq 0}\int_{|x|\geq 2|y|}|K(x-y)-K(x)|\,dx<\infty,
  139. m { x : | T f ( x ) | 2 λ } ( 2 A + 4 T ) λ - 1 f 1 . m\{x:\,|Tf(x)|\geq 2\lambda\}\leq(2A+4\|T\|)\cdot\lambda^{-1}\|f\|_{1}.
  140. f ( x ) = g ( x ) + b ( x ) f(x)=g(x)+b(x)
  141. m { x : | T f ( x ) | 2 λ } m { x : | T g ( x ) | λ } + m { x : | T b ( x ) | λ } . m\{x:\,|Tf(x)|\geq 2\lambda\}\leq m\{x:\,|Tg(x)|\geq\lambda\}+m\{x:\,|Tb(x)|% \geq\lambda\}.
  142. m { x : | T g ( x ) | 2 λ } λ - 2 T g 2 2 λ - 2 T 2 g 2 2 2 λ - 1 μ T 2 f 1 . m\{x:\,|Tg(x)|\geq 2\lambda\}\leq\lambda^{-2}\|Tg\|_{2}^{2}\leq\lambda^{-2}\|T% \|^{2}\|g\|_{2}^{2}\leq 2\lambda^{-1}\mu\|T\|^{2}\|f\|_{1}.
  143. m { x : | T b ( x ) | λ } m { x : x J n * , | T b ( x ) | λ } + m ( J n * ) . m\{x:\,|Tb(x)|\geq\lambda\}\leq m\{x:\,x\notin\cup J_{n}^{*},\,\,\,|Tb(x)|\geq% \lambda\}+m(\cup J_{n}^{*}).
  144. m ( J n * ) m ( J n * ) = 2 m ( J n ) 2 λ - 1 μ - 1 f 1 . m(\cup J_{n}^{*})\leq\sum m(J_{n}^{*})=2\sum m(J_{n})\leq 2\lambda^{-1}\mu^{-1% }\|f\|_{1}.
  145. b = b n , b n = ( f - 𝐀𝐯 J n ( f ) ) χ J n . b=\sum b_{n},\qquad b_{n}=(f-\mathbf{Av}_{J_{n}}(f))\chi_{J_{n}}.
  146. m { x : x J m * , | T b ( x ) | λ } λ - 1 ( J m * ) c | T b ( x ) | d x λ - 1 n ( J n * ) c | T b n ( x ) | d x . m\{x:\,x\notin\cup J_{m}^{*},\,\,\,|Tb(x)|\geq\lambda\}\leq\lambda^{-1}\int_{(% \cup J_{m}^{*})^{c}}|Tb(x)|\,dx\leq\lambda^{-1}\sum_{n}\int_{(J_{n}^{*})^{c}}|% Tb_{n}(x)|\,dx.
  147. ( J n * ) c | T b n ( x ) | d x = ( J n * ) c | J n ( K ( x - y ) - K ( x - y n ) ) b n ( y ) d y | d x J n | b n ( y ) | ( J n * ) c | K ( x - y ) - K ( x - y n ) | d x d y A b n 1 . \int_{(J_{n}^{*})^{c}}|Tb_{n}(x)|\,dx=\int_{(J_{n}^{*})^{c}}\left|\int_{J_{n}}% (K(x-y)-K(x-y_{n}))b_{n}(y)\,dy\right|\,dx\leq\int_{J_{n}}|b_{n}(y)|\int_{(J_{% n}^{*})^{c}}|K(x-y)-K(x-y_{n})|\,dxdy\leq A\|b_{n}\|_{1}.
  148. m { x : x J m * , | T b ( x ) | λ } λ - 1 A b 1 2 A λ - 1 f 1 . m\left\{x:\,x\notin\cup J_{m}^{*},|Tb(x)|\geq\lambda\right\}\leq\lambda^{-1}A% \|b\|_{1}\leq 2A\lambda^{-1}\|f\|_{1}.
  149. m { x : | T f ( x ) | λ } ( 2 μ T 2 + 2 μ - 1 + 2 A ) λ - 1 f 1 . m\{x:\,|Tf(x)|\geq\lambda\}\leq\left(2\mu\|T\|^{2}+2\mu^{-1}+2A\right)\lambda^% {-1}\|f\|_{1}.
  150. μ = T - 1 . \mu=\|T\|^{-1}.
  151. f = f a + f a , f=f_{a}+f^{a},
  152. m { x : | T f ( x ) | > a } m { x : | T f a ( x ) | > a 2 } + m { x : | T f a ( x ) | > a 2 } 4 a - 2 T 2 f a 2 2 + C a - 1 f a 1 . m\{x:\,|Tf(x)|>a\}\leq m\left\{x:\,|Tf_{a}(x)|>\tfrac{a}{2}\right\}+m\left\{x:% \,|Tf^{a}(x)|>\tfrac{a}{2}\right\}\leq 4a^{-2}\|T\|^{2}\|f_{a}\|_{2}^{2}+Ca^{-% 1}\|f^{a}\|_{1}.
  153. T f p p = p 0 a p - 1 m { x : | T f ( x ) | > a } d a p 0 a p - 1 ( 4 a - 2 T 2 f a 2 2 + C a - 1 f a 1 ) d a = 4 T 2 | f ( x ) | < a | f ( x ) | 2 a p - 3 d x d a + 2 C | f ( x ) | a | f ( x ) | a p - 2 d x d a ( 4 T 2 ( 2 - p ) - 1 + C ( p - 1 ) - 1 ) | f | p = C p f p p . \begin{aligned}\displaystyle\|Tf\|_{p}^{p}&\displaystyle=p\int_{0}^{\infty}a^{% p-1}m\{x:\,|Tf(x)|>a\}\,da\\ &\displaystyle\leq p\int_{0}^{\infty}a^{p-1}\left(4a^{-2}\|T\|^{2}\|f_{a}\|_{2% }^{2}+Ca^{-1}\|f^{a}\|_{1}\right)da\\ &\displaystyle=4\|T\|^{2}\iint_{|f(x)|<a}|f(x)|^{2}a^{p-3}\,dx\,da+2C\iint_{|f% (x)|\geq a}|f(x)|a^{p-2}\,dx\,da\\ &\displaystyle\leq\left(4\|T\|^{2}(2-p)^{-1}+C(p-1)^{-1}\right)\int|f|^{p}\\ &\displaystyle=C_{p}\|f\|_{p}^{p}.\end{aligned}
  154. T f q C p f q . \|Tf\|_{q}\leq C_{p}\|f\|_{q}.

Singular_integral_operators_on_closed_curves.html

  1. f ( θ ) = n 𝐙 a n e i n θ . \displaystyle{f(\theta)=\sum_{n\in{\mathbf{Z}}}a_{n}e^{in\theta}.}
  2. f r ( θ ) = F ( r e i θ ) , \displaystyle{f_{r}(\theta)=F(re^{i\theta})},
  3. f r - f 2 0. \displaystyle{\|f_{r}-f\|_{2}\rightarrow 0.}
  4. F ( z ) = 1 2 π i | ζ | = 1 f ( ζ ) ζ - z d ζ = 1 2 π - π π f ( θ ) 1 - e - i θ z d θ . \displaystyle{F(z)={1\over 2\pi i}\int_{|\zeta|=1}{f(\zeta)\over\zeta-z}\,d% \zeta={1\over 2\pi}\int_{-\pi}^{\pi}{f(\theta)\over 1-e^{-i\theta}z}\,d\theta.}
  5. F ( r e i φ ) = 1 2 π - π π f ( φ - θ ) 1 - r e i θ d θ . \displaystyle{F(re^{i\varphi})={1\over 2\pi}\int_{-\pi}^{\pi}{f(\varphi-\theta% )\over 1-re^{i\theta}}\,d\theta.}
  6. H ε f ( φ ) = i π ε | θ | π f ( φ - θ ) 1 - e i θ d θ = 1 π | ζ - e i φ | δ f ( ζ ) ζ - e i φ d ζ , \displaystyle{H^{\varepsilon}f(\varphi)={i\over\pi}\int_{\varepsilon\leq|% \theta|\leq\pi}{f(\varphi-\theta)\over 1-e^{i\theta}}\,d\theta={1\over\pi}\int% _{|\zeta-e^{i\varphi}|\geq\delta}{f(\zeta)\over\zeta-e^{i\varphi}}\,d\zeta,}
  7. H ε 1 = i π ε π 2 ( 1 - e i θ ) - 1 d θ = i π ε π 1 d θ = i - i ε π . \displaystyle{H^{\varepsilon}{1}={i\over\pi}\int_{\varepsilon}^{\pi}2\Re(1-e^{% i\theta})^{-1}\,d\theta={i\over\pi}\int_{\varepsilon}^{\pi}1\,d\theta=i-{i% \varepsilon\over\pi}.}
  8. H ε f ( z ) - i ( 1 - ε ) π f ( z ) = 1 π i | ζ - z | δ f ( ζ ) - f ( z ) ζ - z d ζ . \displaystyle{H^{\varepsilon}f(z)-{i(1-\varepsilon)\over\pi}f(z)={1\over\pi i}% \int_{|\zeta-z|\geq\delta}{f(\zeta)-f(z)\over\zeta-z}\,d\zeta.}
  9. H ε f i f \displaystyle{H^{\varepsilon}f\rightarrow if}
  10. H ε f ¯ = - u - 1 H ε ( u f ¯ ) . \displaystyle{\overline{H^{\varepsilon}f}=-u^{-1}H^{\varepsilon}(u\overline{f}% ).}
  11. H ε f - i f \displaystyle{H^{\varepsilon}f\rightarrow-if}
  12. H = i ( 2 P - I ) . \displaystyle{H=i(2P-I).}
  13. H ε f H f \displaystyle{H^{\varepsilon}f\rightarrow Hf}
  14. H ε f H f \displaystyle{H^{\varepsilon}f\rightarrow Hf}
  15. H f = P . V . 1 π f ( ζ ) ζ - e i φ d ζ . \displaystyle{Hf=\mathrm{P.V.}\,{1\over\pi}\int{f(\zeta)\over\zeta-e^{i\varphi% }}\,d\zeta.}
  16. H ( e i θ ) = e i h ( θ ) , h ( θ + 2 π ) = h ( θ ) + 2 π , \displaystyle{H(e^{i\theta})=e^{ih(\theta)},\,\,\,h(\theta+2\pi)=h(\theta)+2% \pi,}
  17. H h ε f ( e i φ ) = 1 π | e i h ( θ ) - e i h ( φ ) | ε f ( e i θ ) e i θ - e i φ e i θ d θ , \displaystyle{H^{\varepsilon}_{h}f(e^{i\varphi})={1\over\pi}\int_{|e^{ih(% \theta)}-e^{ih(\varphi)}|\geq\varepsilon}{f(e^{i\theta})\over e^{i\theta}-e^{i% \varphi}}\,e^{i\theta}\,d\theta,}
  18. f ( e i θ ) = a n e i n θ \displaystyle{f(e^{i\theta})=\sum a_{n}e^{in\theta}}
  19. f ( r e i θ ) = P r f ( e i θ ) = a n r | n | e i n θ . \displaystyle{f(re^{i\theta})=P_{r}f(e^{i\theta})=\sum a_{n}r^{|n|}e^{in\theta% }.}
  20. P r f 2 = | a n | 2 r 2 | n | , \displaystyle{\|P_{r}f\|^{2}=\sum|a_{n}|^{2}r^{2|n|},}
  21. F R ( e i θ ) = F ( R e i θ ) = a n R - | n | a n e i n θ . \displaystyle{F_{R}(e^{i\theta})=F(Re^{i\theta})=\sum a_{n}R^{-|n|}a_{n}e^{in% \theta}.}
  22. 𝐯 ˙ 𝐯 ˙ = 1 , 𝐯 ¨ 𝐯 ˙ = 0. \displaystyle{\dot{\mathbf{v}}\cdot\dot{\mathbf{v}}=1,\,\,\,\ddot{\mathbf{v}}% \cdot\dot{\mathbf{v}}=0.}
  23. 𝐭 = 𝐯 ˙ , 𝐧 = ( - y ˙ , x ˙ ) . \displaystyle{\mathbf{t}=\dot{\mathbf{v}},\,\,\,\,\,\,\mathbf{n}=(-\dot{y},% \dot{x}).}
  24. 𝐯 ¨ = κ ( t ) 𝐧 ( t ) , κ ( t ) = 𝐯 ¨ 𝐧 = y ¨ x ˙ - x ¨ y ˙ . \displaystyle{\ddot{\mathbf{v}}=\kappa(t)\,\mathbf{n}(t),\,\,\,\,\,\kappa(t)=% \ddot{\mathbf{v}}\cdot\mathbf{n}=\ddot{y}\dot{x}-\ddot{x}\dot{y}.}
  25. 𝐧 ˙ = - κ 𝐭 , 𝐧 ¨ = κ ˙ 𝐧 - κ 2 𝐭 . \displaystyle{\dot{\mathbf{n}}=-\kappa\mathbf{t},\,\,\,\ddot{\mathbf{n}}=\dot{% \kappa}\mathbf{n}-\kappa^{2}\mathbf{t}.}
  26. 𝐯 s ( t ) = 𝐯 ( t ) + s 𝐧 ( t ) , \displaystyle{\mathbf{v}_{s}(t)=\mathbf{v}(t)+s\mathbf{n}(t),}
  27. 𝐯 ˙ s ( t ) = ( 1 - s κ ) 𝐭 , s | 𝐯 ˙ s | = - κ . \displaystyle{\dot{\mathbf{v}}_{s}(t)=(1-s\kappa)\mathbf{t},\,\,\,\,\partial_{% s}|\dot{\mathbf{v}}_{s}|=-\kappa.}
  28. s Ω s | f | 2 = - Ω s ( n f f ¯ + f n f ¯ ) - Ω s κ ( 1 - κ s ) - 1 | f | 2 = - 2 Ω s | f | 2 - Ω s κ ( 1 - κ s ) - 1 | f | 2 , \displaystyle{\partial_{s}\int_{\partial\Omega_{s}}|f|^{2}=-\int_{\partial% \Omega_{s}}(\partial_{n}f\overline{f}+f\overline{\partial_{n}f})-\int_{% \partial\Omega_{s}}\kappa(1-\kappa s)^{-1}|f|^{2}=-2\iint_{\Omega_{s}}|\nabla f% |^{2}-\int_{\partial\Omega_{s}}\kappa(1-\kappa s)^{-1}|f|^{2},}
  29. s f | Ω s 2 M f | Ω s 2 , \displaystyle{\partial_{s}\|f|_{\partial\Omega_{s}}\|^{2}\leq M\|f|_{\partial% \Omega_{s}}\|^{2},}
  30. s e - M s f | Ω s 2 0 , \displaystyle{\partial_{s}e^{-Ms}\|f|_{\partial\Omega_{s}}\|^{2}\leq 0,}
  31. f | Ω s e M s / 2 f | Ω . \displaystyle{\|f|_{\partial\Omega_{s}}\|\leq e^{Ms/2}\|f|_{\partial\Omega}\|.}
  32. Ω s | f | 2 \displaystyle{\int_{\partial\Omega_{s}}|f|^{2}}
  33. C g ( a ) = 1 2 π i Ω g ( z ) z - a d z . \displaystyle{Cg(a)={1\over 2\pi i}\int_{\partial\Omega}{g(z)\over z-a}\,dz.}
  34. H Ω ε f ( s ) = 1 π i | s - t | ε f ( t ) z ( t ) - z ( s ) z ˙ ( t ) d t . \displaystyle{H_{\partial\Omega}^{\varepsilon}f(s)={1\over\pi i}\int_{|s-t|% \geq\varepsilon}\,\,\,\,{f(t)\over z(t)-z(s)}\,\dot{z}(t)\,dt.}
  35. H Ω ε g ( s ) - H ε g ( s ) = 1 π i | t - s | ε K ( s , t ) g ( t ) d t , \displaystyle{H_{\partial\Omega}^{\varepsilon}g(s)-H^{\varepsilon}g(s)={1\over% \pi i}\int_{|t-s|\geq\varepsilon}K(s,t)\cdot g(t)\,dt,}
  36. K ( u , v ) = z ˙ ( t ) z ( t ) - z ( s ) - i e i t e i t - e i s = t log ( z ( t ) - z ( s ) e i t - e i s ) . \displaystyle{K(u,v)={\dot{z}(t)\over z(t)-z(s)}-{ie^{it}\over e^{it}-e^{is}}=% \partial_{t}\log\left({z(t)-z(s)\over e^{it}-e^{is}}\right).}
  37. H Ω g = lim ε 0 H Ω ε g . \displaystyle{H_{\partial\Omega}g=\lim_{\varepsilon\rightarrow 0}H^{% \varepsilon}_{\partial\Omega}g.}
  38. H Ω g ( s ) - H g ( s ) = 1 π i 0 2 π K ( s , t ) g ( t ) d t . \displaystyle{H_{\partial\Omega}g(s)-Hg(s)={1\over\pi i}\int_{0}^{2\pi}K(s,t)% \cdot g(t)\,dt.}
  39. C ( s , t ) = 1 π i ( z ˙ ( t ) z ( t ) - z ( s ) - z ˙ ( s ) ¯ z ( t ) ¯ - z ( s ) ¯ ) , \displaystyle{C(s,t)={1\over\pi i}\left({\dot{z}(t)\over z(t)-z(s)}-{\overline% {\dot{z}(s)}\over\overline{z(t)}-\overline{z(s)}}\right),}
  40. C ± H = ± i C ± . \displaystyle{C_{\pm}\circ H=\pm iC_{\pm}.}
  41. C ± f ± = F ± , C ± f = 0. \displaystyle{C_{\pm}f_{\pm}=F_{\pm},\,\,\,C_{\pm}f_{\mp}=0.}
  42. H f ± = ± i f ± . \displaystyle{Hf_{\pm}=\pm if_{\pm}.}
  43. Ω , | z - w | ε f - ( z ) z - w d z = - | z - w | = ε , z Ω F - ( z ) z - w d z . \displaystyle{\int_{\partial\Omega,\,\,|z-w|\geq\varepsilon}{f_{-}(z)\over z-w% }\,dz=-\int_{|z-w|=\varepsilon,\,\,z\in\Omega}{F_{-}(z)\over z-w}\,dz.}
  44. H 2 = - I . \displaystyle{H^{2}=-I.}
  45. - i H f = C - f | Ω - C + f | Ω . \displaystyle{-iHf=C_{-}f|_{\partial\Omega}-C_{+}f|_{\partial\Omega}.}
  46. H = i ( 2 E - I ) . \displaystyle{H=i(2E-I).}
  47. E f = C - f | Ω , ( I - E ) f = C + f | Ω . \displaystyle{Ef=C_{-}f|_{\partial\Omega},\,\,\,\,(I-E)f=C_{+}f|_{\partial% \Omega}.}
  48. P = E ( I + E - E * ) - 1 . \displaystyle{P=E(I+E-E^{*})^{-1}.}
  49. E P = P , P E = E . \displaystyle{EP=P,\,\,\,PE=E.}
  50. P ( I + E - E * ) = P + E - P = E . \displaystyle{P(I+E-E^{*})=P+E-P=E.}
  51. J f ( t ) = f ( t ) ¯ , U f ( t ) = z ˙ ( t ) f ( t ) . \displaystyle{Jf(t)=\overline{f(t)},\,\,\,Uf(t)=\dot{z}(t)\cdot f(t).}
  52. ( H ε ) * = J U H ε U * J . \displaystyle{(H^{\varepsilon})^{*}=JUH^{\varepsilon}U^{*}J.}
  53. H * = J U H U * J \displaystyle{H^{*}=JUHU^{*}J}
  54. I - E * = J U E U * J . \displaystyle{I-E^{*}=JUEU^{*}J.}
  55. A ( s , t ) = f ( s ) - f ( t ) z ( s ) - z ( t ) . \displaystyle{A(s,t)={f(s)-f(t)\over z(s)-z(t)}.}
  56. T f ( w ) = 1 2 π Ω n ( log | z - w | ) f ( z ) = 1 2 ( H f ) ( w ) . \displaystyle{Tf(w)={1\over 2\pi}\int_{\partial\Omega}\partial_{n}(\log|z-w|)f% (z)={1\over 2}\Re(Hf)(w).}
  57. 2 T h = ( H f ) + i ( H g ) = 1 2 ( H f + J H f + i H g + i J H g ) = 1 2 ( H + J H J ) h \displaystyle{2Th=\Re(Hf)+i\Re(Hg)={1\over 2}(Hf+JHf+iHg+iJHg)={1\over 2}(H+% JHJ)h}
  58. T = 1 4 ( H + J H J ) = 1 4 ( H + U H * U * ) , \displaystyle{T={1\over 4}(H+JHJ)={1\over 4}(H+UH^{*}U^{*}),}
  59. C h ( a ) - F ( a ) = C h ( a ) - C n h n ( a ) = C ( h - h n ) ( a ) + [ ( C - C n ) h n ] ( a ) . \displaystyle{Ch(a)-F(a)=Ch(a)-C_{n}h_{n}(a)=C(h-h_{n})(a)+[(C-C_{n})h_{n}](a).}
  60. [ ( C - C n ) h n ] ( a ) = 1 2 π i h n ( t ) ( z ˙ ( t ) z ( t ) - a - z ˙ n ( t ) z n ( t ) - a ) d t . \displaystyle{[(C-C_{n})h_{n}](a)={1\over 2\pi i}\int h_{n}(t)\left({\dot{z}(t% )\over z(t)-a}-{\dot{z}_{n}(t)\over z_{n}(t)-a}\right)\,dt.}

Skew_and_direct_sums_of_permutations.html

  1. ( π σ ) ( i ) = { π ( i ) + n for 1 i m , σ ( i - m ) for m + 1 i m + n , (\pi\ominus\sigma)(i)=\begin{cases}\pi(i)+n&\,\text{for }1\leq i\leq m,\\ \sigma(i-m)&\,\text{for }m+1\leq i\leq m+n,\end{cases}
  2. ( π σ ) ( i ) = { π ( i ) for 1 i m , σ ( i - m ) + m for m + 1 i m + n . (\pi\oplus\sigma)(i)=\begin{cases}\pi(i)&\,\text{for }1\leq i\leq m,\\ \sigma(i-m)+m&\,\text{for }m+1\leq i\leq m+n.\end{cases}
  3. M π σ M_{\pi\ominus\sigma}
  4. π σ \pi\ominus\sigma
  5. M π σ = [ 0 M π M σ 0 ] M_{\pi\ominus\sigma}=\begin{bmatrix}0&M_{\pi}\\ M_{\sigma}&0\end{bmatrix}
  6. M π σ M_{\pi\oplus\sigma}
  7. π σ \pi\oplus\sigma
  8. M π σ = [ M π 0 0 M σ ] M_{\pi\oplus\sigma}=\begin{bmatrix}M_{\pi}&0\\ 0&M_{\sigma}\end{bmatrix}
  9. M 2413 = [ 1 1 1 1 ] M_{2413}=\begin{bmatrix}&1&&\\ &&&1\\ 1&&&\\ &&1&\end{bmatrix}
  10. M 35142 = [ 1 1 1 1 1 ] M_{35142}=\begin{bmatrix}&&1&&\\ &&&&1\\ 1&&&&\\ &&&1&\\ &1&&&\end{bmatrix}
  11. M 2413 35142 = M 796835142 = [ 1 1 1 1 1 1 1 1 1 ] M_{2413\ominus 35142}=M_{796835142}=\begin{bmatrix}&&&&&&1&&&\\ &&&&&&&&1\\ &&&&&1&&&\\ &&&&&&&1&\\ &&1&&&&&&\\ &&&&1&&&&\\ 1&&&&&&&&\\ &&&1&&&&&\\ &1&&&&&&&\end{bmatrix}
  12. M 2413 35142 = M 241379586 = [ 1 1 1 1 1 1 1 1 1 ] M_{2413\oplus 35142}=M_{241379586}=\begin{bmatrix}&1&&&&&&&\\ &&&1&&&&&\\ 1&&&&&&&&\\ &&1&&&&&&\\ &&&&&&1&&\\ &&&&&&&&1\\ &&&&1&&&&\\ &&&&&&&1&\\ &&&&&1&&&\end{bmatrix}
  13. π ( σ τ ) ( π σ ) τ \pi\oplus(\sigma\ominus\tau)\neq(\pi\oplus\sigma)\ominus\tau
  14. ( π σ ) - 1 = π - 1 σ - 1 (\pi\oplus\sigma)^{-1}=\pi^{-1}\oplus\sigma^{-1}
  15. ( π σ ) - 1 = σ - 1 π - 1 (\pi\ominus\sigma)^{-1}=\sigma^{-1}\ominus\pi^{-1}
  16. π σ = rev ( rev ( σ ) rev ( π ) ) \pi\oplus\sigma=\operatorname{rev}(\operatorname{rev}(\sigma)\ominus% \operatorname{rev}(\pi))

Skew_partition.html

  1. G G
  2. X X
  3. Y Y
  4. G [ X ] G[X]
  5. G [ Y ] G[Y]
  6. G G
  7. G G
  8. A A
  9. B B
  10. C C
  11. D D
  12. A A
  13. B B
  14. C C
  15. D D
  16. G [ A B ] G[A\cup B]
  17. G [ C D ] G[C\cup D]
  18. X = A B X=A\cup B
  19. Y = C D Y=C\cup D
  20. Y Y
  21. X X
  22. X X
  23. Y Y
  24. X X
  25. Y Y
  26. G G
  27. H H
  28. G G
  29. v v
  30. H H
  31. v v
  32. H H
  33. G G
  34. H H
  35. H H
  36. G G
  37. x x
  38. x x
  39. x x
  40. O ( n 101 ) O(n^{101})
  41. n n
  42. O ( n 4 m ) O(n^{4}m)
  43. m m

Skorokhod_problem.html

  1. 0 t W i ( s ) d Z i ( s ) = 0 \int_{0}^{t}W_{i}(s)\,\text{d}Z_{i}(s)=0

Slip_factor.html

  1. σ = V w 2 V w 2 \sigma=\frac{V^{\prime}_{w2}}{V_{w2}}
  2. σ = 1 - π z sin β 2 ( 1 - ϕ 2 cot β 2 ) \sigma=1-\frac{\pi}{z}\frac{\sin\beta_{2}}{(1-\phi_{2}\cot\beta_{2})}
  3. σ = 1 - π z \sigma=1-\frac{\pi}{z}
  4. σ = 1 - 1.98 z ( 1 - ϕ 2 cot β 2 ) \sigma=1-\frac{1.98}{z(1-\phi_{2}\cot\beta_{2})}
  5. σ = 1 - 1.98 z \sigma=1-\frac{1.98}{z}
  6. σ = [ 1 + 6.2 z . n 2 / 3 ] - 1 \sigma=[1+\frac{6.2}{z.n^{2/3}}]^{-1}
  7. I m p e l l e r T i p D i a m e t e r E y e T i p D i a m e t e r \frac{Impeller\ Tip\ Diameter}{Eye\ Tip\ Diameter}

Slope_number.html

  1. d / 2 \lceil d/2\rceil
  2. d / 2 \lceil d/2\rceil

Slotted_line.html

  1. | ρ | = VSWR - 1 VSWR + 1 |\rho|=\frac{\mathrm{VSWR}-1}{\mathrm{VSWR}+1}
  2. ρ = 4 π λ x - π \angle\rho=\frac{4\pi}{\lambda}x-\pi
  3. Z = Z 0 1 + ρ 1 - ρ Z=Z_{0}\frac{1+\rho}{1-\rho}

Slow_strain_rate_testing.html

  1. [ result from specimen in test environment result from specimen in inert environment ] \left[\frac{\,\text{result from specimen in test environment}}{\,\text{result % from specimen in inert environment}}\right]

Smooth_algebra.html

  1. u : A C / N u:A\to C/N
  2. v : A C v:A\to C
  3. k [ [ t 1 , , t n ] ] k[\![t_{1},\ldots,t_{n}]\!]
  4. char k = p > 0 \operatorname{char}k=p>0
  5. [ k : k p ] < [k:k^{p}]<\infty
  6. u : B C / N u:B\to C/N
  7. C / N C/N
  8. v : B C v:B\to C
  9. B = A [ [ t 1 , , t n ] ] B=A[\![t_{1},\ldots,t_{n}]\!]
  10. I = ( t 1 , , t n ) . I=(t_{1},\ldots,t_{n}).
  11. 𝔪 \mathfrak{m}
  12. 𝔪 \mathfrak{m}
  13. A k k A\otimes_{k}k^{\prime}
  14. k k^{\prime}

Snell_envelope.html

  1. ( Ω , , ( t ) t [ 0 , T ] , ) (\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\in[0,T]},\mathbb{P})
  2. \mathbb{Q}\ll\mathbb{P}
  3. U = ( U t ) t [ 0 , T ] U=(U_{t})_{t\in[0,T]}
  4. \mathbb{Q}
  5. X = ( X t ) t [ 0 , T ] X=(X_{t})_{t\in[0,T]}
  6. U U
  7. \mathbb{Q}
  8. U U
  9. X X
  10. U t X t U_{t}\geq X_{t}
  11. \mathbb{Q}
  12. t [ 0 , T ] t\in[0,T]
  13. V = ( V t ) t [ 0 , T ] V=(V_{t})_{t\in[0,T]}
  14. \mathbb{Q}
  15. X X
  16. V V
  17. U U
  18. ( Ω , , ( n ) n = 0 N , ) (\Omega,\mathcal{F},(\mathcal{F}_{n})_{n=0}^{N},\mathbb{P})
  19. \mathbb{Q}\ll\mathbb{P}
  20. ( U n ) n = 0 N (U_{n})_{n=0}^{N}
  21. \mathbb{Q}
  22. ( X n ) n = 0 N (X_{n})_{n=0}^{N}
  23. U N := X N , U_{N}:=X_{N},
  24. U n := X n 𝔼 [ U n + 1 n ] U_{n}:=X_{n}\lor\mathbb{E}^{\mathbb{Q}}[U_{n+1}\mid\mathcal{F}_{n}]
  25. n = N - 1 , , 0 n=N-1,...,0
  26. \lor
  27. X X
  28. U U
  29. U t U_{t}
  30. X X
  31. t t

Snowden_(physics).html

  1. 10 - 6 10^{-6}
  2. s - 1 a r c m i n - 2 s^{-1}arcmin^{-2}

Sobolev_spaces_for_planar_domains.html

  1. Ω Ω
  2. k k
  3. H [ u s u , u p = , u k , u b = 0 ] ( Ω ) H[u^{\prime}su^{\prime},u^{\prime}p=^{\prime},u^{\prime}k^{\prime},u^{\prime}b% =0^{\prime}](Ω)
  4. C [ u s u , u p = 21 e , u b = , u c ] ( Ω ) C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime},u^{\prime}b=^{\prime},u^% {\prime}c^{\prime}](Ω)
  5. k > 0 k>0
  6. H [ u s u , u p = , u k , u b = 0 ] ( Ω ) H[u^{\prime}su^{\prime},u^{\prime}p=^{\prime},u^{\prime}k^{\prime},u^{\prime}b% =0^{\prime}](Ω)
  7. Ω Ω
  8. k 1 k−1
  9. Ω ∂Ω
  10. H [ u s u , u p = , u k , u b = 0 ] ( Ω ) H[u^{\prime}su^{\prime},u^{\prime}p=^{\prime},u^{\prime}k^{\prime},u^{\prime}b% =0^{\prime}](Ω)
  11. H [ u s u , u p = 1 , u b = 0 ] ( Ω ) H[u^{\prime}su^{\prime},u^{\prime}p=1^{\prime},u^{\prime}b=0^{\prime}](Ω)
  12. Ω ( g ( D h ) + ( D g ) h ) d x d y = lim n 0 Ω ( g ( D h n ) + ( D g ) h n ) d x d y = 0. \iint_{\Omega}\left(g(Dh)+(Dg)h\right)\,dx\,dy=\lim_{n\to 0}\iint_{\Omega}% \left(g(Dh_{n})+(Dg)h_{n}\right)\,dx\,dy=0.
  13. Ω g k = 0 , k = h cos ( 𝐧 ( a , b ) ) , \int_{\partial\Omega}gk=0,\qquad k=h\cos\left(\mathbf{n}\cdot(a,b)\right),
  14. 𝐧 \mathbf{n}
  15. k k
  16. g = 0 g=0
  17. Ω ∂Ω
  18. Ω ¯ \overline{Ω}
  19. Ω Ω
  20. C [ u s u , u p = 21 e , u b = , u c ] ( Ω ) C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime},u^{\prime}b=^{\prime},u^% {\prime}c^{\prime}](Ω)
  21. g g
  22. Ω ¯ \overline{Ω}
  23. Ω ∂Ω
  24. Ω ¯ \overline{Ω}
  25. Ω Ω
  26. n n
  27. { g t = ψ 0 g + i = 1 N ψ n λ t n i g } g , \left\{g_{t}=\psi_{0}g+\sum_{i=1}^{N}\psi_{n}\lambda_{tn_{i}}g\right\}% \longrightarrow g,
  28. t t
  29. 0
  30. Ω Ω
  31. Ω Ω
  32. Ω Ω
  33. H [ u s u , u p = , u k , u b = 0 ] ( Ω ) H[u^{\prime}su^{\prime},u^{\prime}p=^{\prime},u^{\prime}k^{\prime},u^{\prime}b% =0^{\prime}](Ω)
  34. k k
  35. Ω ∂Ω
  36. f f
  37. 0
  38. F F
  39. h ( k ) 2 = j = 0 k ( k j ) x j y k - j h 2 , \|h\|_{(k)}^{2}=\sum_{j=0}^{k}{k\choose j}\left\|\partial_{x}^{j}\partial_{y}^% {k-j}h\right\|^{2},
  40. ( F , g ) = 0 (F,g)=0
  41. k 0 k≥0
  42. k = 0 k=0
  43. ( f , g ) = ( f , P k g ) . (f,g)=(f,P_{k}g).
  44. C [ u s u , u p = 21 e , u b = , u c ] ( Ω ) C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime},u^{\prime}b=^{\prime},u^% {\prime}c^{\prime}](Ω)
  45. k 0 k≤0
  46. k = 0 k=0
  47. C [ u s u , u p = 21 e , u b = , u c ] ( Ω ) C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime},u^{\prime}b=^{\prime},u^% {\prime}c^{\prime}](Ω)
  48. H [ u s u , u p = 0092 , u k , u b = 0 ] ( Ω ) H[u^{\prime}su^{\prime},u^{\prime}p=\u{2}0092^{\prime},u^{\prime}k^{\prime},u^% {\prime}b=0^{\prime}](Ω)
  49. H [ u s u , u p = , u k , u b = 0 ] ( Ω ) H[u^{\prime}su^{\prime},u^{\prime}p=^{\prime},u^{\prime}k^{\prime},u^{\prime}b% =0^{\prime}](Ω)
  50. H [ u s u , u p = , u k , u b = 0 ] ( Ω ) H[u^{\prime}su^{\prime},u^{\prime}p=^{\prime},u^{\prime}k^{\prime},u^{\prime}b% =0^{\prime}](Ω)
  51. C [ u s u , u p = 21 e , u b = , u c ] ( Ω ) C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime},u^{\prime}b=^{\prime},u^% {\prime}c^{\prime}](Ω)
  52. f , g C [ u s u , u p = 21 e , u b = , u c ] ( Ω ) f,g∈C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime},u^{\prime}b=^{\prime% },u^{\prime}c^{\prime}](Ω)
  53. P k ( I + Δ ) k f ( - k ) \displaystyle\left\|P_{k}(I+\Delta)^{k}f\right\|_{(-k)}
  54. H [ u s u , u p = 1 , u b = 0 ] ( Ω ) H[u^{\prime}su^{\prime},u^{\prime}p=1^{\prime},u^{\prime}b=0^{\prime}](Ω)
  55. 0
  56. Ω Ω
  57. H [ u s u , u p = 1 , u b = 0 ] ( Ω ) H[u^{\prime}su^{\prime},u^{\prime}p=1^{\prime},u^{\prime}b=0^{\prime}](Ω)
  58. ( 0 ) (0)
  59. f = g ∆f=g
  60. H [ u s u , u p = 1 , u b = 0 ] ( Ω ) H[u^{\prime}su^{\prime},u^{\prime}p=1^{\prime},u^{\prime}b=0^{\prime}](Ω)
  61. g g
  62. T T
  63. T = R 1 Δ - 1 R 0 , T=R_{1}\Delta^{-1}R_{0},
  64. H [ u s u , u p = 1 , u b = 0 ] H[u^{\prime}su^{\prime},u^{\prime}p=1^{\prime},u^{\prime}b=0^{\prime}]
  65. T T
  66. ( T f , f ) > 0 (Tf,f)>0
  67. f f
  68. T f n = μ n f n , μ n > 0 , μ n 0. Tf_{n}=\mu_{n}f_{n},\qquad\mu_{n}>0,\mu_{n}\to 0.
  69. H [ u s u , u p = 1 , u b = 0 ] ( Ω ) H[u^{\prime}su^{\prime},u^{\prime}p=1^{\prime},u^{\prime}b=0^{\prime}](Ω)
  70. Δ f n = λ n f n , λ n > 0 , λ n . \Delta f_{n}=\lambda_{n}f_{n},\qquad\lambda_{n}>0,\lambda_{n}\to\infty.
  71. Δ f = u , u H - 1 ( Ω ) , f H 0 1 ( Ω ) , \Delta f=u,\qquad u\in H^{-1}(\Omega),f\in H^{1}_{0}(\Omega),
  72. H [ u s u , u p = , u k , u b = 0 ] ( Ω ) H[u^{\prime}su^{\prime},u^{\prime}p=^{\prime},u^{\prime}k^{\prime},u^{\prime}b% =0^{\prime}](Ω)
  73. Ω Ω
  74. Ω ¯ \overline{Ω}
  75. f ( k ) 2 = j = 0 k ( k j ) x j y k - j f 2 . \|f\|_{(k)}^{2}=\sum_{j=0}^{k}{k\choose j}\left\|\partial_{x}^{j}\partial_{y}^% {k-j}f\right\|^{2}.
  76. C [ u s u , u p = 21 e , u b = , u c ] ( Ω ) C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime},u^{\prime}b=^{\prime},u^% {\prime}c^{\prime}](Ω)
  77. H [ u s u , u p = , u k , u b = 0 ] ( Ω ) H[u^{\prime}su^{\prime},u^{\prime}p=^{\prime},u^{\prime}k^{\prime},u^{\prime}b% =0^{\prime}](Ω)
  78. H [ u s u , u p = , u k , u b = 0 ] ( Ω ) H[u^{\prime}su^{\prime},u^{\prime}p=^{\prime},u^{\prime}k^{\prime},u^{\prime}b% =0^{\prime}](Ω)
  79. Ω ¯ \overline{Ω}
  80. C [ u s u , u p = 21 e , u b = , u c ] ( Ω ) C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime},u^{\prime}b=^{\prime},u^% {\prime}c^{\prime}](Ω)
  81. E E
  82. I × 𝐓 I×\mathbf{T}
  83. I I
  84. 𝐓 \mathbf{T}
  85. ψ ψ
  86. 0 ψ 1 0≤ψ≤1
  87. E ( ψ f ) + ( 1 ψ ) f E(ψf)+(1−ψ)f
  88. Ω Ω
  89. I I
  90. E ( f ) = m = 0 k a m f ( - x m + 1 ) . E(f)=\sum_{m=0}^{k}a_{m}f\left(-\frac{x}{m+1}\right).
  91. m = 0 k ( - m - 1 ) - k a m = 1. \sum_{m=0}^{k}(-m-1)^{-k}a_{m}=1.
  92. E ( f ) E(f)
  93. H 2 ( Ω ) H 1 ( Ω ) H 0 ( Ω ) H 0 - 1 ( Ω ) H 0 - 2 ( Ω ) \cdots\subset H^{2}(\Omega)\subset H^{1}(\Omega)\subset H^{0}(\Omega)\subset H% ^{-1}_{0}(\Omega)\subset H^{-2}_{0}(\Omega)\subset\cdots
  94. f f
  95. C [ u s u , u p = 21 e , u b = , u c ] ( Ω ) C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime},u^{\prime}b=^{\prime},u^% {\prime}c^{\prime}](Ω)
  96. ( f α , φ ) = ( - 1 ) | α | ( f , α φ ) , | α | k , φ C c ( Ω ) . \left(f_{\alpha},\varphi\right)=(-1)^{|\alpha|}\left(f,\partial^{\alpha}% \varphi\right),\qquad|\alpha|\leq k,\varphi\in C^{\infty}_{c}(\Omega).
  97. f f
  98. f f
  99. f f
  100. δ , δ × 𝐓 −−δ,δ×\mathbf{T}
  101. f f
  102. R > 1 R>1
  103. R 1 R→1
  104. f f
  105. Ω Ω
  106. f f
  107. f = ψ f + ( 1 ψ ) f f=ψf+(1−ψ)f
  108. Ω Ω
  109. 1 ψ 1−ψ
  110. f f
  111. f f
  112. C [ u s u , u p = 21 e , u b = , u c ] ( Ω ) C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime},u^{\prime}b=^{\prime},u^% {\prime}c^{\prime}](Ω)
  113. f f
  114. I × 𝐓 I×\mathbf{T}
  115. R t f ( x , y ) = f ( x , y + t ) . R_{t}f(x,y)=f(x,y+t).
  116. f f
  117. δ h f ^ ( m , n ) = h - 1 ( e - i h n - 1 ) f ^ ( m , n ) = - 0 1 i n e - i n h t d t f ^ ( m , n ) . \widehat{\delta_{h}f}(m,n)=h^{-1}(e^{-ihn}-1)\widehat{f}(m,n)=-\int_{0}^{1}ine% ^{-inht}\,dt\,\,\widehat{f}(m,n).
  118. u = f ∆u=f
  119. u u
  120. H [ u s u , u p = 1 , u b = 0 ] ( Ω ) H[u^{\prime}su^{\prime},u^{\prime}p=1^{\prime},u^{\prime}b=0^{\prime}](Ω)
  121. f f
  122. k 0 k≥0
  123. u u
  124. u = ψ u + ( 1 ψ ) u u=ψu+(1−ψ)u
  125. ψ ψ
  126. Ω Ω
  127. 1 ψ 1−ψ
  128. ψ u ψu
  129. v = ( 1 ψ ) u v=(1−ψ)u
  130. H [ u s u , u p = 1 , u b = 0 ] H[u^{\prime}su^{\prime},u^{\prime}p=1^{\prime},u^{\prime}b=0^{\prime}]
  131. Ω Ω
  132. Δ 1 = Δ - [ Δ , ψ ] = Δ + ( p x + q y - Δ ψ ) = Δ + X . \begin{aligned}\displaystyle\Delta_{1}&\displaystyle=\Delta-[\Delta,\psi]\\ &\displaystyle=\Delta+\left(p\partial_{x}+q\partial_{y}-\Delta\psi\right)\\ &\displaystyle=\Delta+X.\end{aligned}
  133. k k
  134. u ( k + 1 ) C Δ 1 u ( k - 1 ) + C u ( k ) , \|u\|_{(k+1)}\leq C\|\Delta_{1}u\|_{(k-1)}+C\|u\|_{(k)},
  135. C C
  136. k k
  137. k = 0 k=0
  138. u u
  139. Ω Ω
  140. u ( 1 ) 2 \displaystyle\|u\|_{(1)}^{2}
  141. Y Y
  142. Y Y
  143. Z Z
  144. h h
  145. 0
  146. H [ u s u , u p = 1 , u b = 0 ] ( Ω ) H[u^{\prime}su^{\prime},u^{\prime}p=1^{\prime},u^{\prime}b=0^{\prime}](Ω)
  147. u u
  148. v v
  149. δ h u ( k + 1 ) \displaystyle\|\delta_{h}u\|_{(k+1)}
  150. Y u Yu
  151. Y u ( k + 1 ) C Δ 1 u ( k ) + C u ( k + 1 ) . \|Yu\|_{(k+1)}\leq C\|\Delta_{1}u\|_{(k)}+C^{\prime}\|u\|_{(k+1)}.
  152. V u Vu
  153. V V
  154. V = Y r 2 + s 2 = a x + b y , a 2 + b 2 = 1. V=\frac{Y}{\sqrt{r^{2}+s^{2}}}=a\partial_{x}+b\partial_{y},\qquad a^{2}+b^{2}=1.
  155. V u Vu
  156. Y u Yu
  157. V u ( k + 1 ) C ′′ ( Δ 1 u ( k ) + u ( k + 1 ) ) . \|Vu\|_{(k+1)}\leq C^{\prime\prime}\left(\|\Delta_{1}u\|_{(k)}+\|u\|_{(k+1)}% \right).
  158. W W
  159. W = - b x + a y . W=-b\partial_{x}+a\partial_{y}.
  160. ξ Z ξZ
  161. ξ ξ
  162. W u Wu
  163. ( V ± i W ) u = ( a i b ) ( x ± i y ) u , (V\pm iW)u=(a\mp ib)\left(\partial_{x}\pm i\partial_{y}\right)u,
  164. u u
  165. W u Wu
  166. V W u VWu
  167. A = Δ - V 2 - W 2 , B = [ V , W ] , \begin{aligned}\displaystyle A&\displaystyle=\Delta-V^{2}-W^{2},\\ \displaystyle B&\displaystyle=[V,W],\end{aligned}
  168. W 2 u = Δ u - V 2 u - A u , V W u = W V u + B u , \begin{aligned}\displaystyle W^{2}u&\displaystyle=\Delta u-V^{2}u-Au,\\ \displaystyle VWu&\displaystyle=WVu+Bu,\end{aligned}
  169. V u Vu
  170. W u ( k + 1 ) C ( V W u ( k ) + W 2 u ( k ) ) C ( Δ - V 2 - A ) u ( k ) + C ( W V + B ) u ( k ) C 1 Δ 1 u ( k ) + C 1 u ( k + 1 ) . \begin{aligned}\displaystyle\|Wu\|_{(k+1)}&\displaystyle\leq C\left(\|VWu\|_{(% k)}+\left\|W^{2}u\right\|_{(k)}\right)\\ &\displaystyle\leq C\left\|\left(\Delta-V^{2}-A\right)u\right\|_{(k)}+C\|(WV+B% )u\|_{(k)}\\ &\displaystyle\leq C_{1}\|\Delta_{1}u\|_{(k)}+C_{1}\|u\|_{(k+1)}.\end{aligned}
  171. u ( k + 2 ) C ( V u ( k + 1 ) + W u ( k + 1 ) ) C Δ 1 u ( k ) + C u ( k + 1 ) . \begin{aligned}\displaystyle\|u\|_{(k+2)}&\displaystyle\leq C\left(\|Vu\|_{(k+% 1)}+\|Wu\|_{(k+1)}\right)\\ &\displaystyle\leq C^{\prime}\|\Delta_{1}u\|_{(k)}+C^{\prime}\|u\|_{(k+1)}.% \end{aligned}
  172. H [ u s u , u p = 1 , u b = 0 ] ( Ω ) H[u^{\prime}su^{\prime},u^{\prime}p=1^{\prime},u^{\prime}b=0^{\prime}](Ω)
  173. u = f ∆u=f
  174. f f
  175. u u
  176. H [ u s u , u p = 1 , u b = 0 ] ( Ω ) H[u^{\prime}su^{\prime},u^{\prime}p=1^{\prime},u^{\prime}b=0^{\prime}](Ω)
  177. u u
  178. u u
  179. Ω Ω
  180. { Δ f | Ω = 0 f | Ω = g g C ( Ω ) \begin{cases}\Delta f|_{\Omega}=0\\ f|_{\partial\Omega}=g&g\in C^{\infty}(\partial\Omega)\end{cases}
  181. g g
  182. G G
  183. F F
  184. F = G ∆F=∆G
  185. F = 0 F=0
  186. Ω ∂Ω
  187. f = G F f=G−F
  188. g g
  189. g g
  190. Ω Ω
  191. Ω ∂Ω
  192. κ κ
  193. Ω ∂Ω
  194. K K
  195. Ω Ω
  196. 0 Ω 0∈Ω
  197. U ( z ) U(z)
  198. Ω ¯ \overline{Ω}
  199. Ω Ω
  200. l o g | z | −log|z|
  201. Ω ∂Ω
  202. G ( z ) = l o g | z | + U ( z ) G(z)=log|z|+U(z)
  203. Ω ∂Ω
  204. Ω Ω
  205. 0
  206. V V
  207. U U
  208. Ω Ω
  209. U + i V U+iV
  210. U x \displaystyle U_{x}
  211. V ( z ) = 0 z - U y d x + V x d y , V(z)=\int_{0}^{z}-U_{y}dx+V_{x}dy,
  212. Ω ¯ \overline{Ω}
  213. U U
  214. V V
  215. Ω ¯ \overline{Ω}
  216. 0
  217. f = U + i V f=U+iV
  218. Ω ¯ \overline{Ω}
  219. Ω Ω
  220. f ( 0 ) = 0 f(0)=0
  221. H = a r g z + V ( z ) H=argz+V(z)
  222. 2 π
  223. F ( z ) = e G ( z ) + i H ( z ) = z e f ( z ) F(z)=e^{G(z)+iH(z)}=ze^{f(z)}
  224. Ω Ω
  225. Ω ¯ \overline{Ω}
  226. F ( 0 ) = 0 F(0)=0
  227. | F ( z ) | = 1 |F(z)|=1
  228. z Ω z∈∂Ω
  229. z z
  230. 1 1
  231. F ( z ) F(z)
  232. F ( z ) = 0 F(z)=0
  233. z = 0 z=0
  234. F F
  235. 𝐃 \mathbf{D}
  236. F F′
  237. Ω Ω
  238. H H
  239. G G
  240. Ω Ω
  241. G G
  242. F F
  243. F F
  244. F : Ω ¯ 𝐃 ¯ F:\overline{Ω}→\overline{\mathbf{D}}
  245. Ω 𝐃 Ω→\mathbf{D}
  246. Ω Ω
  247. u u
  248. U = 0 U=0
  249. 1 −1
  250. Ω Ω
  251. u u
  252. c > 0 c>0
  253. U = c u U=cu
  254. C ( - U y d x + U x d y ) = 2 π \int_{C}\left(-U_{y}\,dx+U_{x}\,dy\right)=2\pi
  255. V V
  256. U U
  257. V ( z ) = 1 z - u y d x + u x d y V(z)=\int_{1}^{z}-u_{y}\,dx+u_{x}\,dy
  258. 2 π
  259. F ( z ) = e U ( z ) + i V ( z ) \displaystyle{F(z)=e^{U(z)+iV(z)}}
  260. Ω ¯ \overline{Ω}
  261. Ω Ω
  262. | F | = 1 |F|=1
  263. F F
  264. 1 1
  265. 0
  266. F F
  267. Ω Ω
  268. F F
  269. Ω ¯ \overline{Ω}
  270. r | z | 1 r≤|z|≤1
  271. k 1 k≥1
  272. τ f ^ ( n ) = m f ^ ( m , n ) , \displaystyle{\widehat{\tau f}(n)=\sum_{m}\widehat{f}(m,n),}
  273. | τ f ^ ( n ) | 2 ( 1 + n 2 ) k - 1 2 ( m ( 1 + n 2 ) k - 1 2 ( 1 + m 2 + n 2 ) k ) ( m | f ^ ( m , n ) | 2 ( 1 + m 2 + n 2 ) k ) C k m | f ^ ( m , n ) | 2 ( 1 + m 2 + n 2 ) k , \begin{aligned}\displaystyle\left|\widehat{\tau f}(n)\right|^{2}\left(1+n^{2}% \right)^{k-\frac{1}{2}}&\displaystyle\leq\left(\sum_{m}\frac{\left(1+n^{2}% \right)^{k-\frac{1}{2}}}{\left(1+m^{2}+n^{2}\right)^{k}}\right)\left(\sum_{m}% \left|\widehat{f}(m,n)\right|^{2}\left(1+m^{2}+n^{2}\right)^{k}\right)\\ &\displaystyle\leq C_{k}\sum_{m}\left|\widehat{f}(m,n)\right|^{2}\left(1+m^{2}% +n^{2}\right)^{k},\end{aligned}
  274. C k = sup n m ( 1 + n 2 ) k - 1 2 ( 1 + m 2 + n 2 ) k < , c k = inf n m ( 1 + n 2 ) k - 1 2 ( 1 + m 2 + n 2 ) k > 0. \begin{aligned}\displaystyle C_{k}&\displaystyle=\sup_{n}\sum_{m}\frac{\left(1% +n^{2}\right)^{k-\frac{1}{2}}}{\left(1+m^{2}+n^{2}\right)^{k}}<\infty,\\ \displaystyle c_{k}&\displaystyle=\inf_{n}\sum_{m}\frac{\left(1+n^{2}\right)^{% k-\frac{1}{2}}}{\left(1+m^{2}+n^{2}\right)^{k}}>0.\end{aligned}
  275. τ τ
  276. E E
  277. E g ^ ( m , n ) = λ n - 1 g ^ ( n ) ( 1 + n 2 ) k - 1 2 ( 1 + n 2 + m 2 ) k , \widehat{Eg}(m,n)=\lambda_{n}^{-1}\widehat{g}(n)\frac{\left(1+n^{2}\right)^{k-% \frac{1}{2}}}{\left(1+n^{2}+m^{2}\right)^{k}},
  278. λ n = m ( 1 + n 2 ) k - 1 2 ( 1 + m 2 + n 2 ) k . \lambda_{n}=\sum_{m}\frac{\left(1+n^{2}\right)^{k-\frac{1}{2}}}{\left(1+m^{2}+% n^{2}\right)^{k}}.
  279. E g ( k ) 2 = m , n | E g ^ ( m , n ) | 2 ( 1 + m 2 + n 2 ) c k - 2 m , n | g ^ ( n ) | 2 ( 1 + n 2 ) 2 k - 1 ( 1 + m 2 + n 2 ) k c k - 2 C k g k - 1 2 2 . \begin{aligned}\displaystyle\|Eg\|_{(k)}^{2}&\displaystyle=\sum_{m,n}\left|% \widehat{Eg}(m,n)\right|^{2}\left(1+m^{2}+n^{2}\right)\\ &\displaystyle\leq c_{k}^{-2}\sum_{m,n}\left|\widehat{g}(n)\right|^{2}\frac{% \left(1+n^{2}\right)^{2k-1}}{\left(1+m^{2}+n^{2}\right)^{k}}\\ &\displaystyle\leq c_{k}^{-2}C_{k}\|g\|_{k-\frac{1}{2}}^{2}.\end{aligned}
  280. f [ k + 1 2 ] 2 = f ( k ) 2 + 0 2 π 0 2 π | f ( k ) ( s ) - f ( k ) ( t ) | 2 | e i s - e i t | 2 d s d t . \|f\|_{[k+\frac{1}{2}]}^{2}=\|f\|_{(k)}^{2}+\int_{0}^{2\pi}\int_{0}^{2\pi}% \frac{\left|f^{(k)}(s)-f^{(k)}(t)\right|^{2}}{\left|e^{is}-e^{it}\right|^{2}}% \,ds\,dt.
  281. ( 0 ) H 0 1 ( Ω ) H 1 ( Ω ) H 1 2 ( Ω ) ( 0 ) (0)\to H^{1}_{0}(\Omega)\to H^{1}(\Omega)\to H^{\frac{1}{2}}(\partial\Omega)% \to(0)
  282. ( 0 ) H 0 2 ( Ω ) H 2 ( Ω ) H 3 2 ( Ω ) H 1 2 ( Ω ) ( 0 ) , (0)\to H^{2}_{0}(\Omega)\to H^{2}(\Omega)\to H^{\frac{3}{2}}(\partial\Omega)% \oplus H^{\frac{1}{2}}(\partial\Omega)\to(0),
  283. ( 0 ) H 0 k ( Ω ) H k ( Ω ) j = 1 k H j - 1 2 ( Ω ) ( 0 ) . (0)\to H^{k}_{0}(\Omega)\to H^{k}(\Omega)\to\bigoplus_{j=1}^{k}H^{j-\frac{1}{2% }}(\partial\Omega)\to(0).
  284. f f
  285. Δ Δ
  286. Ω Ω
  287. H [ u s u , u p = 1 , u b = 0 ] ( Ω ) H[u^{\prime}su^{\prime},u^{\prime}p=1^{\prime},u^{\prime}b=0^{\prime}](Ω)
  288. Δ Δ
  289. Ω Ω
  290. Δ Δ
  291. D ( f , g ) D(f,g)
  292. D ( f , g ) = ( f , g ) D(f,g)=(∆f,g)
  293. f , g H [ u s u , u p = 1 , u b = 0 ] ( Ω ) f,g∈H[u^{\prime}su^{\prime},u^{\prime}p=1^{\prime},u^{\prime}b=0^{\prime}](Ω)
  294. D D
  295. C C
  296. λ λ
  297. f f
  298. u H : g H D ( f , g ) = ( u , g ) . \exists u\in H:\forall g\in H\quad D(f,g)=(u,g).
  299. D ( f , g ) = ( f x , g x ) + ( f y , g y ) . D(f,g)=\left(f_{x},g_{x}\right)+\left(f_{y},g_{y}\right).
  300. H = H [ u s u , u p = 1 , u b = 0 ] ( Ω ) H=H[u^{\prime}su^{\prime},u^{\prime}p=1^{\prime},u^{\prime}b=0^{\prime}](Ω)
  301. D ( f , g ) = ( Δ f , g ) , f , g H . D(f,g)=(\Delta f,g),\qquad f,g\in H.
  302. H H
  303. Ω Ω
  304. { Δ u = f , f , u C ( Ω - ) , n u = 0 on Ω \begin{cases}\Delta u=f,&f,u\in C^{\infty}(\Omega^{-}),\\ \partial_{n}u=0&\,\text{on }\partial\Omega\end{cases}
  305. ( Δ u , v ) = ( u x , v x ) + ( u y , v y ) - ( n u , v ) Ω . (\Delta u,v)=(u_{x},v_{x})+(u_{y},v_{y})-(\partial_{n}u,v)_{\partial\Omega}.
  306. Δ u = 0 Δu=0
  307. Ω Ω
  308. u u
  309. Ω Ω
  310. D ( f , g ) = ( u x , v x ) + ( u y , v y ) . \displaystyle{D(f,g)=(u_{x},v_{x})+(u_{y},v_{y}).}
  311. H [ u s u , u p = 2121 , u b = 0 ] ( Ω ) H[u^{\prime}su^{\prime},u^{\prime}p=\u{2}2121^{\prime},u^{\prime}b=0^{\prime}]% (Ω)
  312. L u Lu
  313. H [ u s u , u p = 2121 , u b = 0 ] ( Ω ) H[u^{\prime}su^{\prime},u^{\prime}p=\u{2}2121^{\prime},u^{\prime}b=0^{\prime}]% (Ω)
  314. D ( u , v ) = ( L u , v ) . \displaystyle{D(u,v)=(Lu,v).}
  315. I + L I+L
  316. H [ u s u , u p = 2121 , u b = 0 ] ( Ω ) H[u^{\prime}su^{\prime},u^{\prime}p=\u{2}2121^{\prime},u^{\prime}b=0^{\prime}]% (Ω)
  317. L L
  318. ( ( L + I ) u , v ) = ( u , v ) ( 1 ) . ((L+I)u,v)=(u,v)_{(1)}.
  319. ( L + I ) u ( - 1 ) = sup v ( 1 ) = 1 | ( ( L + I ) u , v ) | = sup v ( 1 ) = 1 | ( u , v ) ( 1 ) | = u ( 1 ) . \|(L+I)u\|_{(-1)}=\sup_{\|v\|_{(1)}=1}|((L+I)u,v)|=\sup_{\|v\|_{(1)}=1}|(u,v)_% {(1)}|=\|u\|_{(1)}.
  320. f f
  321. H [ u s u , u p = 2121 , u b = 0 ] ( Ω ) H[u^{\prime}su^{\prime},u^{\prime}p=\u{2}2121^{\prime},u^{\prime}b=0^{\prime}]% (Ω)
  322. v v
  323. ( f , v ) (f,v)
  324. ( f , v ) = ( u , v ) ( 1 ) . \displaystyle{(f,v)=(u,v)_{(1)}.}
  325. ( L + I ) u = f (L+I)u=f
  326. L + I L+I
  327. T T
  328. T = R 1 ( I + L ) - 1 R 0 , T=R_{1}(I+L)^{-1}R_{0},
  329. T T
  330. T T
  331. T T
  332. T T
  333. f = ( L + I ) u , g = ( L + I ) v f=(L+I)u,g=(L+I)v
  334. ( T f , g ) = ( u , ( I + L ) v ) = ( u , v ) ( 1 ) = ( ( I + L ) u , v ) = ( f , T g ) . (Tf,g)=(u,(I+L)v)=(u,v)_{(1)}=((I+L)u,v)=(f,Tg).
  335. T T
  336. ( 0 ) (0)
  337. ( T f , f ) = ( u , u ) ( 1 ) 0 , (Tf,f)=(u,u)_{(1)}\geq 0,
  338. T f = 0 Tf=0
  339. u = 0 u=0
  340. f = 0 f=0
  341. T T
  342. T f n = μ n f n Tf_{n}=\mu_{n}f_{n}
  343. T T
  344. L L
  345. L f n = λ n f n , λ n = μ n - 1 - 1. \displaystyle{Lf_{n}=\lambda_{n}f_{n},\qquad\lambda_{n}=\mu_{n}^{-1}-1.}
  346. 0
  347. L u = 0 Lu=0
  348. ( u x , u x ) + ( u y , u y ) = ( L u , u ) = 0 , (u_{x},u_{x})+(u_{y},u_{y})=(Lu,u)=0,
  349. u u
  350. L L
  351. D D
  352. Δ Δ
  353. u = T f u=Tf
  354. Δ u + u = f Δu+u=f
  355. C C
  356. u u
  357. u ( 2 ) C Δ u ( 0 ) + C u ( 1 ) . \|u\|_{(2)}\leq C\|\Delta u\|_{(0)}+C\|u\|_{(1)}.
  358. u ( 1 ) L u ( - 1 ) + u ( 0 ) , \|u\|_{(1)}\leq\|Lu\|_{(-1)}+\|u\|_{(0)},
  359. u ( 1 ) 2 = | ( L u , u ) | + u ( 0 ) 2 L u ( - 1 ) u ( 1 ) + u ( 0 ) u ( 1 ) . \begin{aligned}\displaystyle\|u\|_{(1)}^{2}&\displaystyle=|(Lu,u)|+\|u\|^{2}_{% (0)}\\ &\displaystyle\leq\|Lu\|_{(-1)}\|u\|_{(1)}+\|u\|_{(0)}\|u\|_{(1)}.\end{aligned}
  360. u = ψ u + ( 1 ψ ) u u=ψu+(1−ψ)u
  361. ψ ψ
  362. Ω Ω
  363. 1 ψ 1−ψ
  364. L L
  365. ( L f , g ) = ( f x , g x ) + ( f y , g y ) = ( Δ f , g ) Ω - ( n f , g ) Ω , f , g C ( Ω - ) . (Lf,g)=\left(f_{x},g_{x}\right)+\left(f_{y},g_{y}\right)=(\Delta f,g)_{\Omega}% -\left(\partial_{n}f,g\right)_{\partial\Omega},\qquad f,g\in C^{\infty}(\Omega% ^{-}).
  366. ( [ L , ψ ] f , g ) = ( [ Δ , ψ ] f , g ) , ([L,\psi]f,g)=([\Delta,\psi]f,g),
  367. [ L , ψ ] = - [ L , 1 - ψ ] = Δ ψ + 2 ψ x x + 2 ψ y y . \displaystyle{[L,\psi]=-[L,1-\psi]=\Delta\psi+2\psi_{x}\partial_{x}+2\psi_{y}% \partial_{y}.}
  368. v = ψ u v=ψu
  369. w = ( 1 ψ ) u w=(1−ψ)u
  370. v v
  371. w w
  372. v = ψ u v=ψu
  373. Ω Ω
  374. f C [ u s u , u p = 21 e , u b = , u c ] ( Ω ) f∈C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime},u^{\prime}b=^{\prime},% u^{\prime}c^{\prime}](Ω)
  375. ( L f , g ) = ( Δ f , g ) Ω . (Lf,g)=(\Delta f,g)_{\Omega}.
  376. f f
  377. v v
  378. L v = v Lv=∆v
  379. Δ v = L v = L ( ψ u ) = ψ L u + [ L , ψ ] u = ψ ( f - u ) + [ Δ , ψ ] u . \Delta v=Lv=L(\psi u)=\psi Lu+[L,\psi]u=\psi(f-u)+[\Delta,\psi]u.
  380. v v
  381. v = φ v v=φv
  382. φ C [ u s u , u p = 21 e , u b = , u c ] ( Ω ) φ∈C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime},u^{\prime}b=^{\prime},% u^{\prime}c^{\prime}](Ω)
  383. v H [ u s u , u p = 2 , u b = 0 ] ( Ω ) v∈H[u^{\prime}su^{\prime},u^{\prime}p=2^{\prime},u^{\prime}b=0^{\prime}](Ω)
  384. v ( 2 ) 2 = Δ v 2 + 2 v ( 1 ) 2 , \|v\|_{(2)}^{2}=\|\Delta v\|^{2}+2\|v\|_{(1)}^{2},
  385. v ( 2 ) C ( Δ ( v ) + v ( 1 ) ) . \|v\|_{(2)}\leq C\left(\|\Delta(v)\|+\|v\|_{(1)}\right).
  386. w = ( 1 ψ ) u w=(1−ψ)u
  387. δ h w ( 1 ) \displaystyle\|\delta_{h}w\|_{(1)}
  388. H [ u s u , u p = 2121 , u b = 0 ] ( Ω ) H[u^{\prime}su^{\prime},u^{\prime}p=\u{2}2121^{\prime},u^{\prime}b=0^{\prime}]% (Ω)
  389. | ( [ L , δ h ] g , h ) | A g ( 1 ) h ( 1 ) , \left|\left(\left[L,\delta_{h}\right]g,h\right)\right|\leq A\|g\|_{(1)}\|h\|_{% (1)},
  390. g g
  391. h h
  392. [ δ h , a α α ] = ( δ h ( a α ) R h ) α . \left[\delta^{h},\sum a_{\alpha}\partial^{\alpha}\right]=\left(\delta^{h}(a_{% \alpha})\circ R_{h}\right)\partial^{\alpha}.
  393. Y w ( 1 ) C L w ( 0 ) + C w ( 1 ) . \|Yw\|_{(1)}\leq C\|Lw\|_{(0)}+C^{\prime}\|w\|_{(1)}.
  394. V w Vw
  395. Y w Yw
  396. V w ( 1 ) C ′′ ( L w ( 0 ) + w ( 1 ) ) . \|Vw\|_{(1)}\leq C^{\prime\prime}\left(\|Lw\|_{(0)}+\|w\|_{(1)}\right).
  397. W W
  398. A = Δ - V 2 - W 2 B = [ V , W ] \begin{aligned}\displaystyle A&\displaystyle=\Delta-V^{2}-W^{2}\\ \displaystyle B&\displaystyle=[V,W]\end{aligned}
  399. ( L w , φ ) = ( w , φ ) (Lw,φ)=(∆w,φ)
  400. φ C [ u s u , u p = 21 e , u b = , u c ] ( Ω ) φ∈C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime},u^{\prime}b=^{\prime},% u^{\prime}c^{\prime}](Ω)
  401. L w Lw
  402. w ∆w
  403. Ω Ω
  404. ( W 2 w , φ ) = ( L w - V 2 w - A u , φ ) , ( V W w , φ ) = ( W V w + B w , φ ) . \begin{aligned}\displaystyle\left(W^{2}w,\varphi\right)&\displaystyle=\left(Lw% -V^{2}w-Au,\varphi\right),\\ \displaystyle(VWw,\varphi)&\displaystyle=(WVw+Bw,\varphi).\end{aligned}
  405. L w = w Lw=∆w
  406. φ φ
  407. V w Vw
  408. W w ( 1 ) C ( V W w ( 0 ) + W 2 w ( 0 ) ) C ( Δ - V 2 - A ) w ( 0 ) + C ( W V + B ) w ( 0 ) C 1 L w ( 0 ) + C 1 w ( 1 ) . \begin{aligned}\displaystyle\|Ww\|_{(1)}&\displaystyle\leq C\left(\|VWw\|_{(0)% }+\left\|W^{2}w\right\|_{(0)}\right)\\ &\displaystyle\leq C\left\|\left(\Delta-V^{2}-A\right)w\right\|_{(0)}+C\|(WV+B% )w\|_{(0)}\\ &\displaystyle\leq C_{1}\|Lw\|_{(0)}+C_{1}\|w\|_{(1)}.\end{aligned}
  409. w ( 2 ) C ( V w ( 1 ) + W w ( 1 ) ) C Δ w ( 0 ) + C w ( 1 ) . \begin{aligned}\displaystyle\|w\|_{(2)}&\displaystyle\leq C\left(\|Vw\|_{(1)}+% \|Ww\|_{(1)}\right)\\ &\displaystyle\leq C^{\prime}\|\Delta w\|_{(0)}+C^{\prime}\|w\|_{(1)}.\end{aligned}
  410. u ( 2 ) C ( Δ v + Δ w + v ( 1 ) + w ( 1 ) ) C ( ψ Δ u + ( 1 - ψ ) Δ u + 2 [ Δ , ψ ] u + u ( 1 ) ) C ′′ ( Δ u + u ( 1 ) ) . \begin{aligned}\displaystyle\|u\|_{(2)}&\displaystyle\leq C\left(\|\Delta v\|+% \|\Delta w\|+\|v\|_{(1)}+\|w\|_{(1)}\right)\\ &\displaystyle\leq C^{\prime}\left(\|\psi\Delta u\|+\|(1-\psi)\Delta u\|+2\|[% \Delta,\psi]u\|+\|u\|_{(1)}\right)\\ &\displaystyle\leq C^{\prime\prime}\left(\|\Delta u\|+\|u\|_{(1)}\right).\end{aligned}
  411. ( f , v ) = ( L u , v ) + ( u , v ) = ( u x , v x ) + ( u y , v y ) + ( u , v ) = ( ( Δ + I ) u , v ) + ( n u , v ) Ω = ( f , v ) + ( n u , v ) Ω . \begin{aligned}\displaystyle(f,v)&\displaystyle=(Lu,v)+(u,v)\\ &\displaystyle=(u_{x},v_{x})+(u_{y},v_{y})+(u,v)\\ &\displaystyle=((\Delta+I)u,v)+(\partial_{n}u,v)_{\partial\Omega}\\ &\displaystyle=(f,v)+(\partial_{n}u,v)_{\partial\Omega}.\end{aligned}
  412. n u | Ω = 0 , \partial_{n}u|_{\partial\Omega}=0,
  413. u ( k + 2 ) C Δ u ( k ) + C u ( k + 1 ) , \|u\|_{(k+2)}\leq C\|\Delta u\|_{(k)}+C\|u\|_{(k+1)},
  414. u u
  415. k 1 k≥1
  416. k = 1 k=1
  417. u u
  418. k = 0 k=0
  419. Z u | Ω = 0. Zu|_{\partial\Omega}=0.
  420. Z Z
  421. Y Y
  422. Z Z
  423. D D
  424. D u = f Du=f
  425. f f
  426. u u
  427. u u
  428. u u
  429. Ω ∂Ω
  430. { Δ f | Ω = 0 n f | Ω = g g C ( Ω ) \begin{cases}\Delta f|_{\Omega}=0\\ \partial_{n}f|_{\partial\Omega}=g&g\in C^{\infty}(\partial\Omega)\end{cases}
  431. g g
  432. G C < s u p > ( Ω ) G∈C<sup>∞(Ω^{−})

Solar_Turbine_Plants.html

  1. C R = a p e r t u r e a r e a o f c o n c e n t r a t o r r e c e i v e r s u r f a c e a r e a = A c A r {CR}={{aperture\,area\,of\,concentrator}\over{receiver\,surface\,area}}={Ac% \over Ar}
  2. η 0 = h e a t e n e r g y r e c e i v e d b y t h e r e c e i v e r i n c i d e n t r a d i a t i o n o n t h e c o l l e c t o r = Q r Q c {\eta_{0}}={{heat\,energy\,received\,by\,the\,receiver}\over{incident\,% radiation\,on\,the\,collector}}={Qr\over Qc}
  3. Q c = I c . A c {Q_{c}}={I_{c}}.{A_{c}}
  4. I c = {I_{c}}=
  5. Q r = η o . . I c . A c {Q_{r}}={\eta_{o}}..{I_{c}}.{A_{c}}
  6. η c = u s e f u l h e a t r e c e i v e d b y t h e c o o l a n t i n c i d e n t r a d i a t i o n o n t h e c o l l e c t o r = Q u Q c {\eta_{c}}={{useful\,heat\,received\,by\,the\,coolant}\over{incident\,% radiation\,on\,the\,collector}}={Qu\over Qc}
  7. Q u = Q r - L = Q r - L o s s e s {Q_{u}}={Q_{r}}-{L}={Q_{r}}-{Losses}
  8. U {U}
  9. L = U . A r . ( T r - T a ) {L}={U}.{Ar}.({Tr}-{Ta})
  10. Q u = η 0 . I c . A c - U . A r . ( T r - T a ) {Q_{u}}={\eta_{0}}.{Ic}.{Ac}-{U}.{Ar}.({Tr}-{Ta})
  11. η c = η 0 - ( 1 c r ) . U T a I c . ( T r T a - 1 ) {\eta_{c}}={\eta_{0}}-({1\over{cr}}).{{U\,Ta\,}\over{Ic}}.({{Tr}\over{Ta}}-{1})
  12. η c = {\eta_{c}}=

Soler_model.html

  1. = ψ ¯ ( i / - m ) ψ + g 2 ( ψ ¯ ψ ) 2 \mathcal{L}=\overline{\psi}\left(i\partial\!\!\!/-m\right)\psi+\frac{g}{2}% \left(\overline{\psi}\psi\right)^{2}
  2. g g
  3. / = μ = 0 3 γ μ x μ \partial\!\!\!/=\sum_{\mu=0}^{3}\gamma^{\mu}\frac{\partial}{\partial x^{\mu}}
  4. ψ ¯ = ψ * γ 0 \overline{\psi}=\psi^{*}\gamma^{0}
  5. γ μ \gamma^{\mu}
  6. 0 μ 3 0\leq\mu\leq 3
  7. i t ψ = - i j = 1 3 α j x j ψ + m β ψ - g ( ψ ¯ ψ ) β ψ i\frac{\partial}{\partial t}\psi=-i\sum_{j=1}^{3}\alpha^{j}\frac{\partial}{% \partial x^{j}}\psi+m\beta\psi-g(\overline{\psi}\psi)\beta\psi
  8. α j \alpha^{j}
  9. 1 j 3 1\leq j\leq 3
  10. β \beta
  11. = ψ ¯ ( i / - m ) ψ + g ( ψ ¯ ψ ) k + 1 k + 1 \mathcal{L}=\overline{\psi}\left(i\partial\!\!\!/-m\right)\psi+g\frac{\left(% \overline{\psi}\psi\right)^{k+1}}{k+1}
  12. k > 0 k>0
  13. = ψ ¯ ( i / - m ) ψ + F ( ψ ¯ ψ ) \mathcal{L}=\overline{\psi}\left(i\partial\!\!\!/-m\right)\psi+F\left(% \overline{\psi}\psi\right)
  14. F F
  15. k = 1 k=1
  16. k k
  17. ϕ ( x ) e - i ω t , \phi(x)e^{-i\omega t},
  18. ϕ \phi
  19. x x
  20. ω \omega

Solid_Modeling_Solutions.html

  1. P ( t ) = i w i P i b i ( t ) i w i b i ( t ) P(t)=\frac{\sum_{i}w_{i}P_{i}b_{i}(t)}{\sum_{i}w_{i}b_{i}(t)}

Solomon_equations.html

  1. d I 1 z d t = - R z 1 ( I 1 z - I 1 z 0 ) - σ 12 ( I 2 z - I 2 z 0 ) {d{I_{1z}}\over dt}=-R_{z}^{1}(I_{1z}-I_{1z}^{0})-\sigma_{12}(I_{2z}-I_{2z}^{0})
  2. d I 2 z d t = - R z 2 ( I 2 z - I 2 z 0 ) - σ 12 ( I 1 z - I 1 z 0 ) {d{I_{2z}}\over dt}=-R_{z}^{2}(I_{2z}-I_{2z}^{0})-\sigma_{12}(I_{1z}-I_{1z}^{0})
  3. d I 1 z I 2 z d t = - R z 12 2 I 1 z I 2 z {d{I_{1z}I_{2z}}\over dt}=-R_{z}^{12}2I_{1z}I_{2z}
  4. σ 12 \sigma_{12}
  5. d I 1 z d t = - R z 1 ( I 1 z 0 - I 1 z 0 ) - σ 12 ( - I 2 z 0 - I 2 z 0 ) = 2 σ 12 I 2 z 0 {d{I_{1z}}\over dt}=-R_{z}^{1}(I_{1z}^{0}-I_{1z}^{0})-\sigma_{12}(-I_{2z}^{0}-% I_{2z}^{0})=2\sigma_{12}I_{2z}^{0}
  6. I 1 z ( t ) = 2 σ 12 t I 2 z 0 + I 1 z 0 I_{1z}(t)=2\sigma_{12}tI_{2z}^{0}+I_{1z}^{0}
  7. σ 12 \sigma_{12}

Soluble_quinoprotein_glucose_dehydrogenase.html

  1. \rightleftharpoons

Sorting_identity.html

  1. x 1 < x 2 < < x n x_{1}<x_{2}<\dots<x_{n}
  2. l = 1 n λ l x l = k = 1 n ( 1 - λ ) k - 1 λ n - k + 1 samp min ( x α 1 , x α 2 , , x α k ) \sum_{l=1}^{n}\lambda^{l}x_{l}=\sum_{k=1}^{n}(1-\lambda)^{k-1}\lambda^{n-k+1}% \sum_{\mathrm{samp}}\min(x_{\alpha_{1}},x_{\alpha_{2}},\dots,x_{\alpha_{k}})
  3. 0 < λ < 0<\lambda<\infty
  4. k k
  5. n n
  6. l = 1 n λ l x l = k = 1 n ( 1 - λ ) k - 1 λ n - k + 1 samp max ( x α 1 , x α 2 , , x α k ) \sum_{l=1}^{n}\lambda^{l}x_{l}=\sum_{k=1}^{n}(1-\lambda)^{k-1}\lambda^{n-k+1}% \sum_{\mathrm{samp}}\max(x_{\alpha_{1}},x_{\alpha_{2}},\dots,x_{\alpha_{k}})
  7. x 1 > x 2 > > x n x_{1}>x_{2}>\dots>x_{n}
  8. x i x_{i}
  9. λ \lambda\rightarrow\infty

Soyasapogenol_B_glucuronide_galactosyltransferase.html

  1. \rightleftharpoons

Soyasapogenol_glucuronosyltransferase.html

  1. \rightleftharpoons

Soyasaponin_III_rhamnosyltransferase.html

  1. \rightleftharpoons

Spaceflight_radiation_carcinogenesis.html

  1. H γ ( L ) = Q ( L ) D γ ( L ) H_{\gamma}(L)=Q(L)D_{\gamma}(L)
  2. H γ = D γ ( L ) Q ( L ) d L H_{\gamma}=\int D_{\gamma}(L)Q(L)dL
  3. D γ ( L ) = L F γ ( L ) D_{\gamma}(L)=LF_{\gamma}(L)
  4. H γ = d L Q ( L ) F γ ( L ) L H_{\gamma}=\int dLQ(L)F_{\gamma}(L)L
  5. E = γ w γ H γ E=\sum_{\gamma}w_{\gamma}H_{\gamma}
  6. E i = E ( t ) d t E_{i}=\int E(t)dt
  7. R i s k = i = 1 N E i R 0 ( a g e i , g e n d e r ) Risk=\sum_{i=1}^{N}E_{i}R_{0}(age_{i},gender)
  8. q ( E , a E , a ) = M ( a ) + m ( E , a E , a ) 1 + 1 2 [ M ( a ) + m ( E , a E , a ) ] q(E,a_{E},a)=\frac{M(a)+m(E,a_{E},a)}{1+\frac{1}{2}\left[M(a)+m(E,a_{E},a)% \right]}
  9. S ( E , a E , a ) = u = a E a - 1 [ 1 - q ( E , a E , u ) ] S(E,a_{E},a)=\prod_{u=a_{E}}^{a-1}\left[1-q(E,a_{E},u)\right]
  10. E L R = a = a E [ M ( a ) + m ( E , a E , a ) ] S ( E , a E , a ) - a = a E M ( a ) S ( 0 , a E , a ) ELR=\sum_{a=a_{E}}^{\infty}\left[M(a)+m(E,a_{E},a)\right]S(E,a_{E},a)-\sum_{a=% a_{E}}^{\infty}M(a)S(0,a_{E},a)
  11. R E I D = a = a E m ( E , a E , a ) S ( E , a E , a ) REID=\sum_{a=a_{E}}^{\infty}m(E,a_{E},a)S(E,a_{E},a)
  12. L L E = a = a E S ( 0 , a E , a ) - a = a E S ( E , a E , a ) LLE=\sum_{a=a_{E}}^{\infty}S(0,a_{E},a)-\sum_{a=a_{E}}^{\infty}S(E,a_{E},a)
  13. L L E - R E I D = L L E R E I D LLE-REID=\frac{LLE}{REID}
  14. m ( E , a x , a ) = m 0 ( E , a x , a ) D D R E F x D x s x T x B x D r m(E,a_{x},a)=\frac{m_{0}(E,a_{x},a)}{DDREF}\frac{x_{D}x_{s}x_{T}x_{B}}{x_{Dr}}
  15. m J ( E , a E , a ) l J ( E , a E , a ) d L d F d L L Q t r i a l - J ( L ) X L - J m_{J}(E,a_{E},a)_{lJ}(E,a_{E},a)\int dL\frac{dF}{dL}LQ_{trial-J}(L)X_{L-J}
  16. d F d L \frac{dF}{dL}
  17. m J ( E , a E , a ) = m l J ( E , a E , a ) j ( E ) L ( E ) Q t r i a l - J ( L ( E ) ) x L - J m_{J}(E,a_{E},a)=m_{lJ}(E,a_{E},a)\sum_{j}(E)L(E)Q_{trial-J}(L(E))x_{L-J}
  18. p ( R i ) \sqrt{p(R_{i})}
  19. χ 2 = n [ p 1 ( R n ) - p 2 ( R n ) ] 2 p 1 2 ( R n ) + p 2 2 ( R n ) \chi^{2}=\sum_{n}\frac{\left[p_{1}(R_{n})-p_{2}(R_{n})\right]^{2}}{\sqrt{p_{1}% ^{2}(R_{n})+p_{2}^{2}(R_{n})}}
  20. m ( E , a E , a ) = [ E R R ( a E , a ) M c ( a ) + ( 1 - v ) E A R ( a E , a ) ] L Q ( L ) F ( L ) L D D R E F m(E,a_{E},a)=\left[ERR(a_{E},a)M_{c}(a)+(1-v)EAR(a_{E},a)\right]{\frac{\sum_{L% }Q(L)F(L)L}{DDREF}}

Spectral_line_shape.html

  1. Δ E Δ t 2 . \Delta E\Delta t\gtrapprox\frac{\hbar}{2}.
  2. L = 1 1 + x 2 , L=\frac{1}{1+x^{2}},
  3. x = p 0 - p w / 2 , x=\frac{p^{0}-p}{w/2},
  4. G = e - ( l o g e 2 ) x 2 . G=e^{-(log_{e}2)x^{2}}.
  5. V ( x ; σ , γ ) = - G ( x ; σ ) L ( x - x ; γ ) d x , V(x;\sigma,\gamma)=\int_{-\infty}^{\infty}G(x^{\prime};\sigma)L(x-x^{\prime};% \gamma)\,dx^{\prime},
  6. I λ = k ϵ k , λ c k I_{\lambda}=\sum_{k}\epsilon_{k,\lambda}c_{k}
  7. d y d x \frac{dy}{dx}
  8. d 2 y d x 2 \frac{d^{2}y}{dx^{2}}
  9. d 4 y d x 4 \frac{d^{4}y}{dx^{4}}

Spermine_oxidase.html

  1. \rightleftharpoons

Spheroidene_monooxygenase.html

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

Sphinganine_C4-monooxygenase.html

  1. \rightleftharpoons

Sphingomyelin_deacylase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Spitzer's_formula.html

  1. n = 0 ϕ n ( α , β ) t n = exp [ n = 1 t n n ( u n ( α ) + v n ( β ) - 1 ) ] \sum_{n=0}^{\infty}\phi_{n}(\alpha,\beta)t^{n}=\exp\left[\sum_{n=1}^{\infty}% \frac{t^{n}}{n}\left(u_{n}(\alpha)+v_{n}(\beta)-1\right)\right]
  2. ϕ n ( α , β ) \displaystyle\phi_{n}(\alpha,\beta)

Splicing_rule.html

  1. r ( L ) = { x a d y : x a b q , p c d y L } . r(L)=\{xady:xabq,pcdy\in L\}\ .

Squalene_methyltransferase.html

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

Square_class.html

  1. F F
  2. F × / F × 2 F^{\times}/F^{\times 2}
  3. F = F=\mathbb{R}
  4. F × F^{\times}
  5. F × 2 F^{\times 2}
  6. V V
  7. F F
  8. q : V F q:V\to F
  9. v v
  10. V V
  11. q ( v ) = a F × q(v)=a\in F^{\times}
  12. u F × u\in F^{\times}
  13. q ( u v ) = a u 2 q(uv)=au^{2}

Square_sign.html

  1. \sqrt{\,\,}

ST_type_theory.html

  1. \in
  2. x x^{\prime}
  3. x = y x=y
  4. y x y\in x^{\prime}
  5. \in
  6. \in
  7. x = y z [ x z y z ] x=y\leftrightarrow\forall z^{\prime}[x\in z^{\prime}\leftrightarrow y\in z^{% \prime}]
  8. x [ x y x z ] [ y = z ] \forall x[x\in y^{\prime}\leftrightarrow x\in z^{\prime}]\rightarrow[y^{\prime% }=z^{\prime}]
  9. Φ ( x ) \Phi(x)
  10. x x
  11. z x [ x z Φ ( x ) ] \exists z^{\prime}\forall x[x\in z^{\prime}\leftrightarrow\Phi(x)]
  12. Φ ( x ) \Phi(x)
  13. R R
  14. x , y [ x y [ x R y y R x ] ] \forall x,y[x\neq y\rightarrow[xRy\vee yRx]]
  15. R R
  16. R R
  17. R R

Stabilized_inverse_Q_filtering.html

  1. U ( t + t , w ) = U ( t , w ) exp ( | w w r | - γ | w | t 2 Q ( w ) ) exp ( i | w w r | - γ w t ) ( 1 ) U(t+\bigtriangleup t,w)=U(t,w)\exp\bigg(|\frac{w}{w_{r}}|^{-\gamma}\frac{|w|% \bigtriangleup t}{2Q(w)}\bigg)\exp\bigg(i|\frac{w}{w_{r}}|^{-\gamma}w% \bigtriangleup t\bigg)\quad(1)
  2. γ = ( π Q r ) - 1 \gamma=(\pi Q_{r})^{-1}
  3. U ( t + t , w ) = U ( t , w ) exp ( | w | t 2 Q ( w ) ) exp ( i w t ) ( 2 ) U(t+\bigtriangleup t,w)=U(t,w)\exp\bigg(\frac{|w|\bigtriangleup t}{2Q(w)}\bigg% )\exp\bigg(iw\bigtriangleup t\bigg)\quad(2)
  4. U ( t + t ) = 0 U ( t + t , w ) d w . ( 2. b ) U(t+\bigtriangleup t)=\int_{0}^{\infty}U(t+\bigtriangleup t,w)dw.\quad(2.b)

Stable_process.html

  1. X ( t ) = m t X(t)=mt
  2. m m

Stack-sortable_permutation.html

  1. C n = 1 n + 1 ( 2 n n ) . C_{n}=\frac{1}{n+1}{\left({{2n}\atop{n}}\right)}.
  2. π n - O ( 1 ) \sqrt{\pi n}-O(1)
  3. 2 n 2\sqrt{n}
  4. ( n + 1 ) / 2 (n+1)/2
  5. 3 - 6 / ( n + 2 ) 3-6/(n+2)
  6. Θ ( n 3 / 2 ) \Theta(n^{3/2})
  7. Θ ( n 2 ) \Theta(n^{2})

Stadium_(geometry).html

  1. P = 2 ( π r + a ) P=2(\pi r+a)
  2. A = π r 2 + 2 r a = r ( π r + 2 a ) , A=\pi r^{2}+2ra=r(\pi r+2a),

Stage_loading.html

  1. L = g J Δ H U 2 L=\frac{gJ{\Delta}H}{U^{2}}
  2. g g
  3. J J
  4. Δ H {\Delta}H
  5. U U
  6. L = g J Δ H n U 2 L=\frac{gJ{\Delta}H}{nU^{2}}

Standard_dimension_ratio.html

  1. S D R = d o s SDR=\frac{d_{o}}{s}
  2. d o d_{o}
  3. s s

Standard_linear_solid_Q_model_for_attenuation_and_dispersion.html

  1. α = | w | ( 2 c r Q r ) ( 1 ) \alpha=\frac{|w|}{(2c_{r}Q_{r})}\quad(1)
  2. 1 c ( w ) = 1 c r ( 1 - 1 π Q r l n | w w r | ) ( 2 ) \frac{1}{c(w)}=\frac{1}{c_{r}}(1-\frac{1}{\pi Q_{r}}ln|\frac{w}{w_{r}}|)\quad(2)
  3. α = ( w τ r ) 2 c 0 Q c τ r [ 1 + ( w τ r ) 2 ] ( 3 ) \alpha=\frac{(w\tau_{r})^{2}}{c_{0}Q_{c}\tau_{r}[1+(w\tau_{r})^{2}]}\quad(3)
  4. 1 c ( w ) = 1 c 0 [ 1 - ( w τ r ) 2 Q c [ 1 + ( w τ r ) 2 ] ] ( 4 ) \frac{1}{c(w)}=\frac{1}{c_{0}}[1-\frac{(w\tau_{r})^{2}}{Q_{c}[1+(w\tau_{r})^{2% }]}]\quad(4)

Standard_Step_Method.html

  1. ( S f ) (S_{f})
  2. ( S 0 ) (S_{0})
  3. H = z + y + v 2 2 g H=z+y+\frac{v^{2}}{2g}
  4. ( F n ) (F_{n})
  5. F n = v ( g A B ) 0.5 F_{n}=\frac{v}{(g\frac{A}{B})^{0.5}}
  6. d y d x = S 0 - S f 1 - F n 2 \frac{dy}{dx}=\frac{S_{0}-S_{f}}{1-F_{n}^{2}}
  7. H 2 = H 1 - h f H_{2}=H_{1}-h_{f}
  8. H 2 = h v e l + h e l e H_{2}=h_{vel}+h_{ele}

Starch_synthase_(maltosyl-transferring).html

  1. \rightleftharpoons

Static_hashing.html

  1. ( n 2 ) n\choose 2

Statistical_manifold.html

  1. ( X , Σ , μ ) (X,\Sigma,\mu)
  2. ( Ω , , P ) (\Omega,\mathcal{F},P)
  3. Ω = X \Omega=X
  4. = Σ \mathcal{F}=\Sigma
  5. P = μ P=\mu
  6. μ \mu
  7. Σ \Sigma
  8. θ \theta
  9. θ \theta

Steinberg_formula.html

  1. w , w W ϵ ( w w ) P ( w ( λ + ρ ) + w ( μ + ρ ) - ( ν + 2 ρ ) ) \sum_{w,w^{\prime}\in W}\epsilon(ww^{\prime})P(w(\lambda+\rho)+w^{\prime}(\mu+% \rho)-(\nu+2\rho))

Steinberg_symbol.html

  1. ( , ) : F * × F * G (\cdot,\cdot):F^{*}\times F^{*}\rightarrow G
  2. ( , ) (\cdot,\cdot)
  3. a + b = 1 a+b=1
  4. ( a , b ) = 1 (a,b)=1
  5. F * F * / a 1 - a F^{*}\otimes F^{*}/\langle a\otimes 1-a\rangle
  6. K 2 F K_{2}F
  7. ( a , - a ) = 1 (a,-a)=1
  8. ( b , a ) = ( a , b ) - 1 (b,a)=(a,b)^{-1}
  9. ( a , a ) = ( a , - 1 ) (a,a)=(a,-1)
  10. ( a , b ) = ( a + b , - b / a ) (a,b)=(a+b,-b/a)
  11. ( a , b ) = { 1 , if z 2 = a x 2 + b y 2 has a non-zero solution ( x , y , z ) F 3 ; - 1 , if not. (a,b)=\begin{cases}1,&\mbox{ if }~{}z^{2}=ax^{2}+by^{2}\mbox{ has a non-zero % solution }~{}(x,y,z)\in F^{3};\\ -1,&\mbox{ if not.}\end{cases}

Stellar_aberration_(derivation_from_Lorentz_transformation).html

  1. β = v / c \scriptstyle\beta=v/c
  2. γ = 1 / 1 - β 2 \scriptstyle\gamma=1/\sqrt{1-\beta^{2}}
  3. x = γ ( x - β c t ) \scriptstyle x^{\prime}=\gamma\cdot(x-\beta\cdot ct)
  4. c t 1 = - 5 L y \scriptstyle c\,t_{1}=-5\,Ly
  5. x 1 = 4 L y , y 1 = 3 L y , z 1 = 0 \scriptstyle x_{1}=4\,Ly\;,\;y_{1}=3\,Ly\;,\;z_{1}=0
  6. c t 2 = 0 \scriptstyle c\,t_{2}=0
  7. x 2 = 0 , y 2 = 0 , z 2 = 0 \scriptstyle x_{2}=0\;,\;y_{2}=0\;,\;z_{2}=0
  8. δ \delta
  9. tan δ = 3 / 4 δ = 36.87 \scriptstyle\tan\delta=3/4\;\rightarrow\;\delta=36.87^{\circ}
  10. x 1 = γ ( x 1 - β c t 1 ) = 1 , 1547 ( 4 L j - 0 , 5 ( - 5 L j ) ) = 7 , 50555 L j ; y 1 = y 1 = 3 L j \scriptstyle x_{1}^{\prime}=\gamma\cdot(x_{1}-\beta\cdot c\,t_{1})=1,1547\cdot% (4Lj-0,5\cdot(-5Lj))=7,50555Lj;\quad y_{1}^{\prime}=y_{1}=3Lj
  11. δ \delta^{\prime}
  12. tan δ = y 1 / x 1 = 3 / 7 , 50555 δ = 21 , 79 \scriptstyle\tan\delta^{\prime}=y_{1}^{\prime}/x_{1}^{\prime}=3/7,50555\;% \rightarrow\;\delta^{\prime}=21,79^{\circ}
  13. tan ( δ / 2 ) = tan ( δ / 2 ) ( 1 - 0 , 5 ) / ( 1 + 0 , 5 ) = tan ( 36 , 87 / 2 ) 1 / 3 = 0 , 19245 \scriptstyle\tan(\delta^{\prime}/2)=\tan(\delta/2)\cdot\sqrt{(1-0,5)/(1+0,5)}=% \tan(36,87^{\circ}/2)\cdot\sqrt{1/3}=0,19245\;
  14. δ / 2 = 10 , 89 δ = 21 , 79 \scriptstyle\;\rightarrow\;\delta^{\prime}/2=10,89^{\circ}\;\rightarrow\;% \delta^{\prime}=21,79^{\circ}
  15. ( x 1 | y 1 | 0 | c t 1 ) (x_{1}|y_{1}|0|c\,t_{1})
  16. c t 1 < 0 c\,t_{1}<0
  17. ( x 1 | y 1 ) ( 0 | 0 ) (x_{1}|y_{1})\neq(0|0)
  18. ( 0 | 0 | 0 | 0 ) (0|0|0|0)
  19. t 1 < 0 t_{1}<0
  20. t 2 = 0 t_{2}=0
  21. d = c ( 0 - t 1 ) = - c t 1 d=c\cdot(0-t_{1})=-c\,t_{1}
  22. δ \delta
  23. sin δ = y 1 d = y 1 - c t 1 \sin\delta=\frac{y_{1}}{d}=\frac{y_{1}}{-c\,t_{1}}
  24. cos δ = x 1 - c t 1 \cos\delta=\frac{x_{1}}{-c\,t_{1}}
  25. ( v | 0 | 0 ) (v|0|0)
  26. t = t = 0 \scriptstyle t=t^{\prime}=0
  27. β = v c , γ = 1 1 - β 2 \beta=\frac{v}{c},\gamma=\frac{1}{\sqrt{1-\beta^{2}}}
  28. x 1 = γ ( x 1 - β c t 1 ) , y 1 = y 1 , z 1 = 0 , c t 1 = γ ( c t 1 - β x 1 ) x_{1}^{\prime}=\gamma\cdot(x_{1}-\beta\cdot c\,t_{1}),y_{1}^{\prime}=y_{1},z_{% 1}^{\prime}=0,c\,t_{1}^{\prime}=\gamma\cdot(c\,t_{1}-\beta\cdot x_{1})
  29. t 1 < 0 t_{1}^{\prime}<0
  30. t 2 = 0 t_{2}^{\prime}=0
  31. d = c ( 0 - t 1 ) = - c t 1 d\,^{\prime}=c\cdot(0-t_{1}^{\prime})=-c\,t_{1}^{\prime}
  32. δ \delta^{\prime}
  33. sin δ = y 1 d = y 1 - c t 1 = y 1 - γ ( c t 1 - β x 1 ) = y 1 - c t 1 γ ( 1 - β x 1 c t 1 ) = y 1 - c t 1 γ ( 1 + β x 1 - c t 1 ) = sin δ γ ( 1 + β cos δ ) \sin\delta^{\prime}=\frac{y_{1}^{\prime}}{d\,^{\prime}}=\frac{y_{1}}{-c\,t_{1}% ^{\prime}}=\frac{y_{1}}{-\gamma\cdot(c\,t_{1}-\beta\cdot x_{1})}=\frac{y_{1}}{% -c\,t_{1}\cdot\gamma\cdot\left(1-\beta\cdot\frac{x_{1}}{c\,t_{1}}\right)}=% \frac{\frac{y_{1}}{-c\,t_{1}}}{\gamma\cdot\left(1+\beta\cdot\frac{x_{1}}{-c\,t% _{1}}\right)}=\frac{\sin\delta}{\gamma\cdot(1+\beta\cdot\cos\delta)}
  34. cos δ = x 1 - c t 1 = γ ( x 1 - β c t 1 ) - γ ( c t 1 - β x 1 ) = - c t 1 γ ( x 1 - c t 1 + β ) - c t 1 γ ( 1 + β x 1 - c t 1 ) = x 1 - c t 1 + β 1 + β x 1 - c t 1 = cos δ + β 1 + β cos δ \cos\delta^{\prime}=\frac{x_{1}^{\prime}}{-c\,t_{1}^{\prime}}=\frac{\gamma% \cdot(x_{1}-\beta\cdot c\,t_{1})}{-\gamma\cdot(c\,t_{1}-\beta\cdot x_{1})}=% \frac{-c\,t_{1}\cdot\gamma\cdot\left(\frac{x_{1}}{-c\,t_{1}}+\beta\right)}{-c% \,t_{1}\cdot\gamma\cdot\left(1+\beta\cdot\frac{x_{1}}{-c\,t_{1}}\right)}=\frac% {\frac{x_{1}}{-c\,t_{1}}+\beta}{1+\beta\cdot\frac{x_{1}}{-c\,t_{1}}}=\frac{% \cos\delta+\beta}{1+\beta\cdot\cos\delta}
  35. tan δ = sin δ cos δ = sin δ γ ( cos δ + β ) \tan\delta^{\prime}=\frac{\sin\delta^{\prime}}{\cos\delta^{\prime}}=\frac{\sin% \delta}{\gamma\cdot(\cos\delta+\beta)}
  36. δ = ( cos δ ) 2 ( tan δ - tan δ ) = ( cos δ ) 2 tan δ ( 1 - β cos δ - 1 ) = - cos δ tan δ β = - β sin δ \triangle\delta\;=\;(\cos\delta)^{2}\cdot(\tan\delta^{\prime}-\tan\delta)=(% \cos\delta)^{2}\cdot\tan\delta\left(1-\frac{\beta}{\cos\delta}\;-1\right)=-% \cos\delta\cdot\tan\delta\cdot\beta=-\beta\cdot\sin\delta
  37. δ = 90 \;\delta\;=90^{\circ}\;
  38. tan δ = sin 90 γ ( cos 90 + β ) = 1 γ β 1 β cot δ = β δ = \arccot β π 2 - β \tan\delta^{\prime}=\frac{\sin 90^{\circ}}{\gamma\cdot(\cos 90^{\circ}+\beta)}% =\frac{1}{\gamma\cdot\beta}\;\approx\frac{1}{\beta}\quad\rightarrow\quad\cot% \delta^{\prime}=\beta\quad\rightarrow\quad\delta^{\prime}=\arccot\beta\approx% \frac{\pi}{2}-\beta
  39. δ = δ - δ = - β ( = - β sin ( 90 ) ) \triangle\delta=\delta^{\prime}-\delta=-\beta\quad\left(=-\beta\cdot\sin(90^{% \circ})\right)
  40. δ = - 90 \;\delta\;=-90^{\circ}\;
  41. tan δ = - 1 γ β - 1 β cot δ = - β δ = \arccot β - π 2 + β \tan\delta^{\prime}=\frac{-1}{\gamma\cdot\beta}\;\approx-\frac{1}{\beta}\quad% \rightarrow\quad\cot\delta^{\prime}=-\beta\quad\rightarrow\quad\delta^{\prime}% =\arccot\beta\approx-\frac{\pi}{2}+\beta\quad
  42. δ = δ - δ = + β ( = - β sin ( - 90 ) ) \triangle\delta=\delta^{\prime}-\delta=+\beta\quad\left(=-\beta\cdot\sin(-90^{% \circ})\right)
  43. δ = - v c sin δ 180 π \triangle\delta=-\frac{v}{c}\cdot\sin\delta\cdot\frac{180^{\circ}}{\pi}
  44. tan ( α / 2 ) = sin α / ( 1 + cos α ) \scriptstyle\tan(\alpha/2)=\sin\alpha/(1+\cos\alpha)
  45. tan δ 2 = 1 - β 1 + β tan δ 2 \tan\frac{\delta^{\prime}}{2}=\sqrt{\frac{1-\beta}{1+\beta}}\cdot\tan\frac{% \delta}{2}
  46. tan δ 2 = sin δ 1 + cos δ = sin δ γ ( 1 + β cos δ ) 1 + β cos δ 1 + β cos δ + cos δ + β 1 + β cos δ = sin δ γ ( 1 + β cos δ + cos δ + β ) = sin δ γ ( 1 + β ) ( 1 + cos δ ) = tan δ 2 γ ( 1 + β ) \tan\frac{\delta^{\prime}}{2}=\frac{\sin\delta^{\prime}}{1+\cos\delta^{\prime}% }=\frac{\frac{\sin\delta}{\gamma\cdot(1+\beta\cdot\cos\delta)}}{\frac{1+\beta% \cdot\cos\delta}{1+\beta\cdot\cos\delta}+\frac{\cos\delta+\beta}{1+\beta\cdot% \cos\delta}}=\frac{\sin\delta}{\gamma\cdot(1+\beta\cdot\cos\delta+\cos\delta+% \beta)}=\frac{\sin\delta}{\gamma\cdot(1+\beta)\cdot(1+\cos\delta)}=\frac{\tan% \frac{\delta}{2}}{\gamma\cdot(1+\beta)}
  47. 1 γ ( 1 + β ) = 1 - β 2 1 + β = ( 1 + β ) ( 1 - β ) 1 + β = 1 - β 1 + β \frac{1}{\gamma\cdot(1+\beta)}=\frac{\sqrt{1-\beta^{2}}}{1+\beta}=\frac{\sqrt{% (1+\beta)(1-\beta)}}{1+\beta}=\sqrt{\frac{1-\beta}{1+\beta}}
  48. θ \theta
  49. tan θ = sin θ γ ( cos θ + β ) \tan\theta\,^{\prime}=\frac{\sin\theta}{\gamma\cdot(\cos\theta+\beta)}
  50. tan θ 2 = 1 - β 1 + β tan θ 2 \tan\frac{\theta\,^{\prime}}{2}=\sqrt{\frac{1-\beta}{1+\beta}}\cdot\tan\frac{% \theta}{2}
  51. θ = - β sin θ \triangle\theta=-\beta\cdot\sin\theta
  52. θ = δ - 90 \theta=\delta-90^{\circ}
  53. sin ( δ - 90 ) = - sin ( 90 - δ ) = - cos δ , cos ( δ - 90 ) = cos ( 90 - δ ) = sin δ \scriptstyle\sin(\delta-90^{\circ})=-\sin(90^{\circ}-\delta)=-\cos\delta\;,\;% \cos(\delta-90^{\circ})=\cos(90^{\circ}-\delta)=\sin\delta
  54. tan ( δ - 90 ) = - tan ( 90 - δ ) = - cot δ \scriptstyle\tan(\delta-90^{\circ})=-\tan(90^{\circ}-\delta)=-\cot\delta
  55. tan ( δ - 90 ) = sin ( δ - 90 ) γ ( cos ( δ - 90 ) + β ) \tan(\delta^{\prime}-90^{\circ})=\frac{\sin(\delta-90^{\circ})}{\gamma\cdot% \left(\cos(\delta-90^{\circ})+\beta\right)}
  56. - cot δ = - cos δ γ ( sin δ + β ) -\cot\delta^{\prime}=\frac{-\cos\delta}{\gamma\cdot(\sin\delta+\beta)}
  57. cot δ = cos δ γ ( sin δ + β ) \cot\delta^{\prime}=\frac{\cos\delta}{\gamma\cdot(\sin\delta+\beta)}
  58. cot δ = 1 tan δ \cot\delta^{\prime}=\frac{1}{\tan\delta^{\prime}}
  59. tan δ = γ ( sin δ + β ) cos δ \tan\delta^{\prime}=\frac{\gamma\cdot(\sin\delta+\beta)}{\cos\delta}
  60. tan δ - 90 2 = 1 - β 1 + β tan δ - 90 2 \tan\frac{\delta^{\prime}-90^{\circ}}{2}=\sqrt{\frac{1-\beta}{1+\beta}}\cdot% \tan\frac{\delta-90^{\circ}}{2}
  61. θ = ( δ - 90 ) - ( δ - 90 ) = δ \scriptstyle\triangle\theta=(\delta^{\prime}-90^{\circ})-(\delta-90^{\circ})=\triangle\delta
  62. δ = θ = - β sin ( δ - 90 ) = + β cos δ \triangle\delta=\triangle\theta=-\beta\cdot\sin(\delta-90^{\circ})=+\beta\cdot\cos\delta
  63. tan ( δ - α ) = sin ( δ - α ) γ ( cos ( δ - α ) + β ) \tan(\delta^{\prime}-\alpha)=\frac{\sin(\delta-\alpha)}{\gamma\cdot\left(\cos(% \delta-\alpha)+\beta\right)}
  64. tan δ - α 2 = 1 - β 1 + β tan δ - α 2 \tan\frac{\delta^{\prime}-\alpha}{2}=\sqrt{\frac{1-\beta}{1+\beta}}\cdot\tan% \frac{\delta-\alpha}{2}
  65. δ = - β sin ( δ - α ) \triangle\delta=-\beta\cdot\sin(\delta-\alpha)
  66. δ = - v c sin ( δ - α ) 180 π \triangle\delta=-\frac{v}{c}\cdot\sin(\delta-\alpha)\cdot\frac{180^{\circ}}{\pi}
  67. v e = 2 π 1 A E 365 , 25 d = 29 , 78 k m / s v_{e}=\frac{2\cdot\pi\cdot 1\,AE}{365,25\,d}=29,78\,km/s
  68. v e c = 0 , 00009935 \frac{v_{e}}{c}=0,00009935
  69. v c 180 π = 20 , 5 ′′ \frac{v}{c}\cdot\frac{180^{\circ}}{\pi}=20,5^{\prime\prime}
  70. k A = 20 , 5 ′′ k_{A}=20,5^{\prime\prime}
  71. φ \varphi
  72. v r = cos φ 40000 k m 1 d = cos φ 463 m / s v_{r}=\frac{\cos\varphi\cdot 40000\,km}{1\,d}=\cos\varphi\cdot 463\,m/s
  73. v r c = cos φ 1 , 54 10 - 6 \frac{v_{r}}{c}=\cos\varphi\cdot 1,54\cdot 10^{-6}
  74. cos φ 0 , 32 ′′ \cos\varphi\cdot 0,32^{\prime\prime}
  75. v = 2 π 280000 9 , 461 10 15 m 230 10 6 365 , 25 24 3600 230 k m / s \scriptstyle v=\frac{2\pi\cdot 280000\cdot 9,461\cdot 10^{15}\,m}{230\cdot 10^% {6}\cdot 365,25\cdot 24\cdot 3600}\approx\,230\,km/s
  76. 2 , 6 2 π 1 a 230000000 a \scriptstyle 2,6^{\prime}\cdot\frac{2\pi\cdot 1\,a}{230000000\,a}