wpmath0000003_4

Double_Mersenne_number.html

  1. M M p = 2 2 p - 1 - 1 M_{M_{p}}=2^{2^{p}-1}-1
  2. M M 2 = M 3 = 7 M_{M_{2}}=M_{3}=7
  3. M M 3 = M 7 = 127 M_{M_{3}}=M_{7}=127
  4. M M 5 = M 31 = 2147483647 M_{M_{5}}=M_{31}=2147483647
  5. M M 7 = M 127 = 170141183460469231731687303715884105727 M_{M_{7}}=M_{127}=170141183460469231731687303715884105727
  6. M M p M_{M_{p}}
  7. M M p M_{M_{p}}
  8. M M 61 M_{M_{61}}
  9. × 10 6 94127911065419641 \times 10^{6}94127911065419641
  10. M ( p ) M(p)
  11. M p M_{p}
  12. M M 7 M_{M_{7}}

Double_negation.html

  1. * 𝟒 13. . p ( p ) \mathbf{*4\cdot 13}.\ \ \vdash.\ p\ \equiv\ \thicksim(\thicksim p)
  2. \Leftrightarrow
  3. \Leftrightarrow
  4. \Leftrightarrow
  5. P ¬ ¬ P P\vdash\neg\neg P
  6. ¬ ¬ P P \neg\neg P\vdash P
  7. P ¬ ¬ P \frac{P}{\neg\neg P}
  8. ¬ ¬ P P \frac{\neg\neg P}{P}
  9. P ¬ ¬ P P\to\neg\neg P
  10. ¬ ¬ P P \neg\neg P\to P
  11. ¬ ¬ P P \neg\neg P\leftrightarrow P
  12. ¬ ¬ ¬ A ¬ A \neg\neg\neg A\vdash\neg A

Downforce.html

  1. D = 1 2 ( W S ) H α F ρ V 2 D=\frac{1}{2}(WS)H\alpha F\rho V^{2}
  2. α \alpha

Dragon_curve.html

  1. \mapsto
  2. \mapsto
  3. f 1 ( z ) = ( 1 + i ) z 2 f_{1}(z)=\frac{(1+i)z}{2}
  4. f 2 ( z ) = 1 - ( 1 - i ) z 2 f_{2}(z)=1-\frac{(1-i)z}{2}
  5. S 0 = { 0 , 1 } S_{0}=\{0,1\}
  6. f 1 ( x , y ) = 1 2 ( cos 45 - sin 45 sin 45 cos 45 ) ( x y ) f_{1}(x,y)=\frac{1}{\sqrt{2}}\begin{pmatrix}\cos 45^{\circ}&-\sin 45^{\circ}\\ \sin 45^{\circ}&\cos 45^{\circ}\end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix}
  7. f 2 ( x , y ) = 1 2 ( cos 135 - sin 135 sin 135 cos 135 ) ( x y ) + ( 1 0 ) f_{2}(x,y)=\frac{1}{\sqrt{2}}\begin{pmatrix}\cos 135^{\circ}&-\sin 135^{\circ}% \\ \sin 135^{\circ}&\cos 135^{\circ}\end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix}+\begin{pmatrix}1\\ 0\end{pmatrix}
  8. 1 2 \textstyle{\frac{1}{2}}
  9. 2 \textstyle{\sqrt{2}}
  10. ln 2 ln 2 = 2 \textstyle{\frac{\ln 2}{\ln\sqrt{2}}=2}
  11. log 2 ( 1 + 73 - 6 87 3 + 73 + 6 87 3 3 ) 1.523627086202492. \log_{2}\left(\frac{1+\sqrt[3]{73-6\sqrt{87}}+\sqrt[3]{73+6\sqrt{87}}}{3}% \right)\cong 1.523627086202492.
  12. 4 x ( 2 x - 1 ) = 4 ( 2 x + 1 ) . \textstyle{4^{x}(2^{x}-1)=4(2^{x}+1).}
  13. f 1 ( z ) = ( 1 + i ) z 2 f_{1}(z)=\frac{(1+i)z}{2}
  14. f 2 ( z ) = 1 - ( 1 + i ) z 2 f_{2}(z)=1-\frac{(1+i)z}{2}
  15. S 0 = { 0 , 1 , 1 - i } S_{0}=\{0,1,1-i\}
  16. \mapsto
  17. \mapsto
  18. \mapsto
  19. f 1 ( z ) = λ z f_{1}(z)=\lambda z
  20. f 2 ( z ) = i 3 z + λ f_{2}(z)=\frac{i}{\sqrt{3}}z+\lambda
  21. f 3 ( z ) = λ z + λ * f_{3}(z)=\lambda z+\lambda^{*}
  22. where λ = 1 2 - i 2 3 and λ * = 1 2 + i 2 3 . \mbox{where }~{}\lambda=\frac{1}{2}-\frac{i}{2\sqrt{3}}\,\text{ and }\lambda^{% *}=\frac{1}{2}+\frac{i}{2\sqrt{3}}.
  23. p ( x ) = i = 0 n a i x i p(x)=\sum_{i=0}^{n}a_{i}x^{i}\,
  24. a i = ± 1 a_{i}=\pm 1
  25. f - ( z ) = 1 - w z f_{-}(z)=1-wz
  26. f + ( f - ( f - ( 0 ) ) ) = 1 + ( 1 - w ) w = 1 + 1 w - 1 w 2 f_{+}(f_{-}(f_{-}(0)))=1+(1-w)w=1+1w-1w^{2}

Drawing_straws.html

  1. 1 N \tfrac{1}{N}
  2. N - 1 N \tfrac{N-1}{N}
  3. N {N}
  4. N - 1 N \tfrac{N-1}{N}
  5. 1 N - 1 \tfrac{1}{N-1}
  6. N - 1 N × 1 N - 1 = 1 N \tfrac{N-1}{N}\times\tfrac{1}{N-1}=\tfrac{1}{N}
  7. N - 1 N × N - 2 N - 1 × N - 3 N - 2 × 1 N - 3 = 1 N \tfrac{N-1}{N}\times\tfrac{N-2}{N-1}\times\tfrac{N-3}{N-2}\times\tfrac{1}{N-3}% =\tfrac{1}{N}

Drude_model.html

  1. d d t 𝐩 ( t ) = q ( 𝐄 + 𝐩 ( t ) × 𝐁 m ) - 𝐩 ( t ) τ , \frac{d}{dt}\mathbf{p}(t)=q\left(\mathbf{E}+\frac{\mathbf{p}(t)\times\mathbf{B% }}{m}\right)-\frac{\mathbf{p}(t)}{\tau},
  2. 𝐉 \mathbf{J}
  3. 𝐄 \mathbf{E}
  4. 𝐉 = ( n q 2 τ m ) 𝐄 . \mathbf{J}=\left(\frac{nq^{2}\tau}{m}\right)\mathbf{E}.
  5. t t
  6. p , q , n , m p,q,n,m
  7. τ τ
  8. τ τ
  9. 𝐄 \mathbf{E}
  10. d 𝐩 d\mathbf{p}
  11. τ τ
  12. t t
  13. τ τ
  14. Δ 𝐩 = q 𝐄 τ . \Delta\langle\mathbf{p}\rangle=q\mathbf{E}\tau.
  15. 𝐩 = q 𝐄 τ . \langle\mathbf{p}\rangle=q\mathbf{E}\tau.
  16. 𝐩 = m 𝐯 , \langle\mathbf{p}\rangle=m\langle\mathbf{v}\rangle,
  17. 𝐉 = n q 𝐯 , \mathbf{J}=nq\langle\mathbf{v}\rangle,
  18. 𝐉 = ( n q 2 τ m ) 𝐄 . \mathbf{J}=\left(\frac{nq^{2}\tau}{m}\right)\mathbf{E}.
  19. 𝐩 ( t 0 + d t ) = ( 1 - d t τ ) ( 𝐩 ( t 0 ) + q 𝐄 d t ) , \langle\mathbf{p}(t_{0}+dt)\rangle=\left(1-\frac{dt}{\tau}\right)\left(\langle% \mathbf{p}(t_{0})\rangle+q\mathbf{E}dt\right),
  20. 1 d t τ 1−\frac{dt}{τ}
  21. d t 2 dt^{2}
  22. d d t 𝐩 ( t ) = q 𝐄 - 𝐩 ( t ) τ , \frac{d}{dt}\langle\mathbf{p}(t)\rangle=q\mathbf{E}-\frac{\langle\mathbf{p}(t)% \rangle}{\tau},
  23. 𝐩 ⟨\mathbf{p}⟩
  24. q q
  25. 𝐩 ( t ) = q τ 𝐄 + 𝐂 e - t / τ \langle\mathbf{p}(t)\rangle=q\tau\mathbf{E}+\mathbf{C}e^{-t/\tau}
  26. p ( t ) p(t)
  27. d 𝐩 = 0 d\frac{⟨}{\mathbf{p}}=0
  28. 𝐩 = q τ 𝐄 . \langle\mathbf{p}\rangle=q\tau\mathbf{E}.
  29. 𝐩 = m 𝐯 , \langle\mathbf{p}\rangle=m\langle\mathbf{v}\rangle,
  30. 𝐉 = n q 𝐯 , \mathbf{J}=nq\langle\mathbf{v}\rangle,
  31. 𝐉 = ( n q 2 τ m ) 𝐄 . \mathbf{J}=\left(\frac{nq^{2}\tau}{m}\right)\mathbf{E}.
  32. ω ω
  33. σ ( ω ) = σ 0 1 + i ω τ = σ 0 1 + ω 2 τ 2 - i ω τ σ 0 1 + ω 2 τ 2 . \sigma(\omega)=\frac{\sigma_{0}}{1+i\omega\tau}=\frac{\sigma_{0}}{1+\omega^{2}% \tau^{2}}-i\omega\tau\frac{\sigma_{0}}{1+\omega^{2}\tau^{2}}.
  34. E ( t ) = ( E 0 e i ω t ) ; E(t)=\Re(E_{0}e^{i\omega t});
  35. J ( t ) = ( σ ( ω ) E 0 e i ω t ) . J(t)=\Re(\sigma(\omega)E_{0}e^{i\omega t}).
  36. i i
  37. i −i
  38. τ τ
  39. σ ( ω ) σ(ω)
  40. τ τ
  41. σ ( ω ) σ(ω)
  42. τ < s u p > 1 τ<sup>−1

Drug_design.html

  1. Δ G bind = Δ G 0 + Δ G hb Σ h - b o n d s + Δ G ionic Σ i o n i c - i n t + Δ G lipophilic | A | + Δ G rot 𝑁𝑅𝑂𝑇 \Delta G_{\,\text{bind}}=\Delta G_{\,\text{0}}+\Delta G_{\,\text{hb}}\Sigma_{h% -bonds}+\Delta G_{\,\text{ionic}}\Sigma_{ionic-int}+\Delta G_{\,\text{% lipophilic}}\left|A\right|+\Delta G_{\,\text{rot}}\mathit{NROT}
  2. Δ G bind = - R T ln K d K d = [ Ligand ] [ Receptor ] [ Complex ] Δ G bind = Δ G desolvation + Δ G motion + Δ G configuration + Δ G interaction \begin{array}[]{lll}\Delta G_{\,\text{bind}}=-RT\ln K_{\,\text{d}}\\ K_{\,\text{d}}=\dfrac{[\,\text{Ligand}][\,\text{Receptor}]}{[\,\text{Complex}]% }\\ \Delta G_{\,\text{bind}}=\Delta G_{\,\text{desolvation}}+\Delta G_{\,\text{% motion}}+\Delta G_{\,\text{configuration}}+\Delta G_{\,\text{interaction}}\end% {array}

Duality_(mathematics).html

  1. A S A\subseteq S
  2. A c A^{c}
  3. ( A c ) c = A (A^{c})^{c}=A
  4. A B A\subseteq B
  5. B c A c B^{c}\subseteq A^{c}
  6. B c B^{c}
  7. A c A^{c}
  8. 2 \mathbb{R}^{2}
  9. n \mathbb{R}^{n}
  10. C * 2 C^{*}\subseteq\mathbb{R}^{2}
  11. ( x 1 , x 2 ) (x_{1},x_{2})
  12. x 1 c 1 + x 2 c 2 0 x_{1}c_{1}+x_{2}c_{2}\geq 0
  13. ( c 1 , c 2 ) (c_{1},c_{2})
  14. C * * C^{**}
  15. C * * C^{**}
  16. C D C\subseteq D
  17. D * C * D^{*}\subseteq C^{*}
  18. D * D^{*}
  19. C * C^{*}
  20. V * V^{*}
  21. φ : V k \varphi:V\to k
  22. 2 \mathbb{R}^{2}
  23. V * * V^{**}
  24. V V * * V\to V^{**}
  25. V W V\to W
  26. W * V * W^{*}\to V^{*}
  27. W * W^{*}
  28. V * V^{*}
  29. V * V^{*}
  30. F E K F\subseteq E\subseteq K
  31. H G H\subseteq G
  32. K H K^{H}
  33. F F F\subseteq F^{\prime}
  34. G a l ( K / F ) G a l ( K / F ) . Gal(K/F^{\prime})\subseteq Gal(K/F).
  35. G a l ( K / E ) Gal(K/E)
  36. K H K^{H}
  37. f : S S . f:S\to S.
  38. f 1 , f 2 f_{1},f_{2}
  39. ( ¬ x i ) = ¬ x i \bigwedge(\neg x_{i})=\neg\bigvee x_{i}
  40. \square
  41. p \Diamond p
  42. \square
  43. \Diamond
  44. A α C = ( A α ) C \bigcap A_{\alpha}^{C}=\left(\bigcup A_{\alpha}\right)^{C}
  45. A α C \bigcap A_{\alpha}^{C}
  46. ( A α ) C \left(\bigcup A_{\alpha}\right)^{C}
  47. X X * * := ( X * ) * = H o m ( H o m ( X , D ) , D ) . X\to X^{**}:=(X^{*})^{*}=Hom(Hom(X,D),D).
  48. x X x\in X
  49. f : X D f:X\to D
  50. H o m ( X , D ) Hom(X,D)
  51. φ : V × V K \varphi:V\times V\to K
  52. 2 \mathbb{RP}^{2}
  53. V 3 V\subset\mathbb{R}^{3}
  54. ( 3 ) * (\mathbb{R}^{3})^{*}
  55. f : 3 f:\mathbb{R}^{3}\to\mathbb{R}
  56. f ( V ) = 0 f(V)=0
  57. ( 3 ) * (\mathbb{R}^{3})^{*}
  58. - , - : 3 × 3 , x , y = i = 1 3 x i y i \langle-,-\rangle:\mathbb{R}^{3}\times\mathbb{R}^{3}\to\mathbb{R},\langle x,y% \rangle=\sum_{i=1}^{3}x_{i}y_{i}
  59. 2 \mathbb{RP}^{2}
  60. V 3 V\subset\mathbb{R}^{3}
  61. { w 3 , v , w = 0 for all v V } \{w\in\mathbb{R}^{3},\langle v,w\rangle=0\,\text{ for all }v\in V\}
  62. L q L^{q}
  63. 1 / p + 1 / q = 1 1/p+1/q=1
  64. 1 p < 1\leq p<\infty
  65. L L^{\infty}
  66. L 1 L^{1}
  67. L 1 L^{1}
  68. - , - , \langle-,-,\rangle
  69. H H * , v ( w v , w , ) . H\to H^{*},v\mapsto(w\mapsto\langle v,w,\rangle).
  70. colim : C I C : Δ \operatorname{colim}:C^{I}\leftrightarrow C:\Delta\,
  71. Δ : C I C : lim . \Delta:C^{I}\leftrightarrow C:\lim.\,
  72. f ^ ( ξ ) := - f ( x ) e - 2 π i x ξ d x , \hat{f}(\xi):=\int_{-\infty}^{\infty}f(x)\ e^{-2\pi ix\xi}\,dx,
  73. f ( x ) = - f ^ ( ξ ) e 2 π i x ξ d ξ . f(x)=\int_{-\infty}^{\infty}\hat{f}(\xi)\ e^{2\pi ix\xi}\,d\xi.
  74. f ^ \hat{f}
  75. f ( - x ) = f ^ ^ ( x ) f(-x)=\hat{\hat{f}}(x)
  76. ( γ , ω ) γ ω (\gamma,\omega)\mapsto\int_{\gamma}\omega
  77. 𝐙 ^ \hat{\mathbf{Z}}
  78. S \ A S\backslash A
  79. C * * C^{*}*

Duality_(projective_geometry).html

  1. C C
  2. P P
  3. L L
  4. I I
  5. C = ( P , L , I ) C=(P,L,I)
  6. I I
  7. C C
  8. C C
  9. C C
  10. P G ( 2 , K ) PG(2,K)
  11. K K
  12. C C
  13. C C
  14. C C
  15. C C
  16. P G ( 2 , 𝐑 ) PG(2,\mathbf{R})
  17. m m
  18. n n
  19. c c
  20. n n
  21. d d
  22. m m
  23. C = ( P , L , I ) C=(P,L,I)
  24. σ σ
  25. Q Q
  26. m m
  27. Q I m QIm
  28. C C
  29. C C
  30. C = P G ( 2 , K ) C=PG(2,K)
  31. K K
  32. C C
  33. δ δ
  34. P G ( n , K ) PG(n,K)
  35. K K
  36. S T S⊆T
  37. S , T S,T
  38. P G ( n , K ) PG(n,K)
  39. r r
  40. n 1 r n−1−r
  41. r + 1 r+1
  42. n n
  43. V V
  44. K K
  45. P G ( n , K ) PG(n,K)
  46. K K
  47. P G ( n , K ) PG(n,K)
  48. P G ( n , K ) PG(n,K)
  49. n > 1 n>1
  50. K K
  51. π \pi
  52. P G ( n , K ) PG(n,K)
  53. n > 1 n>1
  54. π \pi
  55. P G ( n , K ) PG(n,K)
  56. θ θ
  57. P G ( n , K ) PG(n,K)
  58. θ θ
  59. K K
  60. π \pi
  61. σ = i d σ=id
  62. K K
  63. θ θ
  64. w w
  65. V V
  66. φ : V × V K φ:V×V→K
  67. φ ( v , w ) = T w ( v ) . \varphi(v,w)=T_{w}(v).
  68. φ φ
  69. σ σ
  70. P G ( n , K ) PG(n,K)
  71. n > 1 n>1
  72. K K
  73. K K
  74. P G ( n , K ) PG(n,K)
  75. n + 1 n+1
  76. K K
  77. n n
  78. n n
  79. n 1 n−1
  80. n n
  81. K K
  82. P G ( n , K ) PG(n,K)
  83. ( n 1 ) (n−1)
  84. 𝐮 \mathbf{u}
  85. 𝐮 \mathbf{u}
  86. K K
  87. P G ( 2 , K ) PG(2,K)
  88. K K
  89. ( a , b , c ) (a,b,c)↔
  90. a x + b y + c z = 0 ax+by+cz=0
  91. P G ( 3 , K ) PG(3,K)
  92. ( a , b , c , d ) (a,b,c,d)↔
  93. a x + b y + c z + d w = 0 ax+by+cz+dw=0
  94. σ = i d σ=id
  95. K K
  96. G G
  97. ( n + 1 ) × ( n + 1 ) (n+1)×(n+1)
  98. π ( 𝐱 P ) = ( G 𝐱 P ) 𝖳 = ( 𝐱 P ) 𝖳 = 𝐱 H . \pi(\mathbf{x}_{P})=(G\mathbf{x}_{P})^{\mathsf{T}}=(\mathbf{x}_{P})^{\mathsf{T% }}=\mathbf{x}_{H}.
  99. P G ( 2 , 𝐑 ) PG(2,\mathbf{R})
  100. P G ( 2 , 𝐑 ) PG(2,\mathbf{R})
  101. π \pi
  102. P G ( n , K ) PG(n,K)
  103. n > 1 n>1
  104. φ φ
  105. σ σ
  106. n + 1 n+1
  107. V V
  108. V V
  109. φ ( 𝐮 , 𝐱 ) = 𝐮 𝖳 G ( 𝐱 σ ) , \varphi(\mathbf{u},\mathbf{x})=\mathbf{u}^{\mathsf{T}}G(\mathbf{x}^{\sigma}),
  110. G G
  111. ( n + 1 ) × ( n + 1 ) (n+1)×(n+1)
  112. K K
  113. σ σ
  114. 𝐱 \mathbf{x}
  115. π ( 𝐱 ) = ( G ( 𝐱 σ ) ) 𝖳 . \pi(\mathbf{x})=(G(\mathbf{x}^{\sigma}))^{\mathsf{T}}.
  116. π \pi
  117. P G ( n , K ) PG(n,K)
  118. K K
  119. S S
  120. P G ( n , K ) PG(n,K)
  121. S S
  122. P G ( n , K ) PG(n,K)
  123. S = { 𝐮 in V : φ ( 𝐮 , 𝐱 ) = 0 for all 𝐱 in S } . S^{\bot}=\{\mathbf{u}\,\text{ in }V\colon\varphi(\mathbf{u},\mathbf{x})=0\,% \text{ for all }\mathbf{x}\,\text{ in }S\}.
  124. φ φ
  125. φ ( 𝐮 , 𝐱 ) = 0 φ(\mathbf{u},\mathbf{x})=0
  126. φ ( 𝐱 , 𝐮 ) = 0 φ(\mathbf{x},\mathbf{u})=0
  127. V V
  128. K K
  129. φ φ
  130. V V
  131. σ σ
  132. φ φ
  133. σ = i d σ=id
  134. K K
  135. φ ( 𝐮 , 𝐱 ) = φ ( 𝐱 , 𝐮 ) φ(\mathbf{u},\mathbf{x})=φ(\mathbf{x},\mathbf{u})
  136. 𝐮 , 𝐱 \mathbf{u},\mathbf{x}
  137. V V
  138. φ φ
  139. K K
  140. 𝐳 \mathbf{z}
  141. φ ( 𝐳 , 𝐳 ) 0 φ(\mathbf{z},\mathbf{z})≠0
  142. σ = i d σ=id
  143. K K
  144. φ ( 𝐮 , 𝐮 ) = 0 φ(\mathbf{u},\mathbf{u})=0
  145. 𝐮 \mathbf{u}
  146. V V
  147. n n
  148. K K
  149. 𝐮 , 𝐱 \mathbf{u},\mathbf{x}
  150. V V
  151. P P
  152. P G ( n , K ) PG(n,K)
  153. H H
  154. 𝐱 \mathbf{x}
  155. π \pi
  156. φ φ
  157. φ ( 𝐱 , 𝐱 ) = 0 φ(\mathbf{x},\mathbf{x})=0
  158. φ φ
  159. G G
  160. K K
  161. K K
  162. K K
  163. P G ( 2 s + 1 , K ) PG(2s+1,K)
  164. K K
  165. K K
  166. K = 𝐑 K=\mathbf{R}
  167. K = 𝐂 K=\mathbf{C}
  168. K K
  169. P P
  170. P P
  171. π \pi
  172. P G ( 2 , q ) PG(2,q)
  173. p p
  174. π \pi
  175. q + 1 q+1
  176. π \pi
  177. π \pi
  178. p p
  179. p = 2 p=2
  180. q q
  181. 𝐏 \mathbf{P}
  182. n n
  183. π \pi
  184. 𝐏 \mathbf{P}
  185. π \pi
  186. n + 1 n+1
  187. n n
  188. n + 1 n+1
  189. π \pi
  190. n + 1 n+1
  191. n n
  192. n n
  193. n n
  194. π \pi
  195. 1 1
  196. s + 1 s+1
  197. C C
  198. O O
  199. r r
  200. P P
  201. O O
  202. Q Q
  203. P Q P→Q
  204. C C
  205. p p
  206. Q Q
  207. O P OP
  208. P P
  209. C C
  210. q q
  211. O O
  212. O O
  213. q q
  214. q q
  215. P P
  216. q q
  217. O O
  218. Q Q
  219. P P
  220. C C
  221. q q
  222. M M
  223. q q
  224. O O
  225. q q
  226. M M
  227. C C
  228. O O
  229. O O
  230. O O
  231. s ( 0 ) s(≠0)
  232. 1 / s −1/s
  233. x x
  234. y y
  235. C C
  236. r = 1 r=1
  237. P P
  238. ( a , b ) (a,b)
  239. Q Q
  240. ( a a 2 + b 2 , b a 2 + b 2 ) . \left(\frac{a}{a^{2}+b^{2}},\frac{b}{a^{2}+b^{2}}\right).
  241. Q Q
  242. a x + b y = 1 ax+by=1
  243. ( a , b ) ( a , b , 1 ) (a,b)↦(a,b,1)
  244. π : P 2 P 2 \pi:\mathbb{R}P^{2}\rightarrow\mathbb{R}P^{2}
  245. π ( ( x , y , z ) 𝖳 ) = ( x , y , - z ) . \pi\left((x,y,z)^{\mathsf{T}}\right)=(x,y,-z).
  246. G = ( 1 0 0 0 1 0 0 0 - 1 ) . G=\left(\begin{matrix}1&0&0\\ 0&1&0\\ 0&0&-1\end{matrix}\right).
  247. 0 = P 𝖳 G P = x 2 + y 2 - z 2 , 0=P^{\mathsf{T}}GP=x^{2}+y^{2}-z^{2},
  248. 𝐏 \mathbf{P}
  249. = ( x , y , z ) =(x,y,z)
  250. z = 1 z=1
  251. C C
  252. P G ( 2 , F ) PG(2,F)
  253. F F
  254. P P
  255. C C
  256. A B ¯ \overline{AB}
  257. J K ¯ \overline{JK}
  258. A , B , J , K A,B,J,K
  259. P P
  260. P P
  261. C C
  262. P P
  263. P P
  264. C C
  265. P P
  266. C C
  267. P P
  268. π \pi
  269. P P
  270. q q
  271. Q Q
  272. Q Q
  273. p p
  274. P P
  275. P P
  276. Q Q
  277. π \pi

Dynatron_oscillator.html

  1. I P = I C - I G 2 I_{P}=I_{C}-I_{G2}\,
  2. r P = Δ V P Δ I P < 0 r_{P}={\Delta V_{P}\over\Delta I_{P}}<0\,
  3. - r P < R -r_{P}<R\,
  4. f = 1 2 π 1 L C f={1\over 2\pi}\sqrt{1\over LC}\,
  5. I C \scriptstyle I\text{C}
  6. I P \scriptstyle I\text{P}
  7. I G2 \scriptstyle I\text{G2}
  8. I G2 = I C - I P I\text{G2}=I\text{C}-I\text{P}\,

Eadie–Hofstee_diagram.html

  1. v = - K m v [ S ] + V max v=-K_{m}{v\over[S]}+V_{\max}
  2. v = V max [ S ] K m + [ S ] v={{V_{\max}{}[S]}\over{K_{m}+[S]}}
  3. V max V_{\max}
  4. V max v = V max ( K m + [ S ] ) V max [ S ] = K m + [ S ] [ S ] {V_{\max}\over v}={{V_{\max}{}(K_{m}+[S])}\over{V_{\max}{}[S]}}={{K_{m}+[S]}% \over{[S]}}
  5. V max = v K m [ S ] + v [ S ] [ S ] = v K m [ S ] + v V_{\max}={{{vK_{m}}\over{[S]}}+{{v[S]}\over{[S]}}}={{vK_{m}}\over{[S]}}+v
  6. v = - K m v [ S ] + V max v=-K_{m}{v\over{[S]}}+V_{\max}

Earnshaw's_theorem.html

  1. 𝐅 = ( - U ) = - 2 U = 0. \nabla\cdot\mathbf{F}=\nabla\cdot(-\nabla U)=-\nabla^{2}U=0.
  2. U = - 𝐌 𝐁 = - ( M x B x + M y B y + M z B z ) . U=-\mathbf{M}\cdot\mathbf{B}=-(M_{x}B_{x}+M_{y}B_{y}+M_{z}B_{z}).
  3. 2 U = 2 U x 2 + 2 U y 2 + 2 U z 2 > 0. \nabla^{2}U=\frac{\partial^{2}U}{\partial x^{2}}+\frac{\partial^{2}U}{\partial y% ^{2}}+\frac{\partial^{2}U}{\partial z^{2}}>0.
  4. 2 B x = 2 B y = 2 B z = 0. \nabla^{2}B_{x}=\nabla^{2}B_{y}=\nabla^{2}B_{z}=0.
  5. U = - 𝐌 𝐁 = - ( M x B x + M y B y + M z B z ) , U=-\mathbf{M}\cdot\mathbf{B}=-(M_{x}B_{x}+M_{y}B_{y}+M_{z}B_{z}),
  6. 2 U = 0 , \nabla^{2}U=0,
  7. U = - 𝐌 𝐁 = - k 𝐁 𝐁 = - k ( B x 2 + B y 2 + B z 2 ) , U=-\mathbf{M}\cdot\mathbf{B}=-k\mathbf{B}\cdot\mathbf{B}=-k\left(B_{x}^{2}+B_{% y}^{2}+B_{z}^{2}\right),
  8. 2 ( B x 2 + B y 2 + B z 2 ) 0 , \nabla^{2}\left(B_{x}^{2}+B_{y}^{2}+B_{z}^{2}\right)\geq 0,
  9. 𝐌 = k 𝐁 | 𝐁 | , \mathbf{M}=k{\mathbf{B}\over|\mathbf{B}|},
  10. U = - 𝐌 𝐁 = - k 𝐁 𝐁 | 𝐁 | = - k | 𝐁 | 2 | 𝐁 | = - k ( B x 2 + B y 2 + B z 2 ) 1 2 ; U=-\mathbf{M}\cdot\mathbf{B}=-k{{\mathbf{B}\cdot\mathbf{B}}\over|\mathbf{B}|}=% -k{|\mathbf{B}|^{2}\over|\mathbf{B}|}=-k\left(B_{x}^{2}+B_{y}^{2}+B_{z}^{2}% \right)^{\frac{1}{2}};
  11. ( U ) = 2 U = 2 U x 2 + 2 U y 2 + 2 U z 2 = 0. \nabla\cdot(\nabla U)=\nabla^{2}U={\partial^{2}U\over{\partial x}^{2}}+{% \partial^{2}U\over{\partial y}^{2}}+{\partial^{2}U\over{\partial z}^{2}}=0.
  12. U = - 𝐌 𝐁 = - M x B x - M y B y - M z B z . U=-\mathbf{M}\cdot\mathbf{B}=-M_{x}B_{x}-M_{y}B_{y}-M_{z}B_{z}.
  13. 2 U = - 2 ( M x B x + M y B y + M z B z ) x 2 - 2 ( M x B x + M y B y + M z B z ) y 2 - 2 ( M x B x + M y B y + M z B z ) z 2 \nabla^{2}U=-{\partial^{2}(M_{x}B_{x}+M_{y}B_{y}+M_{z}B_{z})\over{\partial x}^% {2}}-{\partial^{2}(M_{x}B_{x}+M_{y}B_{y}+M_{z}B_{z})\over{\partial y}^{2}}-{% \partial^{2}(M_{x}B_{x}+M_{y}B_{y}+M_{z}B_{z})\over{\partial z}^{2}}
  14. 2 U = - M x ( 2 B x x 2 + 2 B x y 2 + 2 B x z 2 ) - M y ( 2 B y x 2 + 2 B y y 2 + 2 B y z 2 ) - M z ( 2 B z x 2 + 2 B z y 2 + 2 B z z 2 ) = - M x 2 B x - M y 2 B y - M z 2 B z \begin{aligned}\displaystyle\nabla^{2}U&\displaystyle=-M_{x}\left({\partial^{2% }B_{x}\over{\partial x}^{2}}+{\partial^{2}B_{x}\over{\partial y}^{2}}+{% \partial^{2}B_{x}\over{\partial z}^{2}}\right)-M_{y}\left({\partial^{2}B_{y}% \over{\partial x}^{2}}+{\partial^{2}B_{y}\over{\partial y}^{2}}+{\partial^{2}B% _{y}\over{\partial z}^{2}}\right)-M_{z}\left({\partial^{2}B_{z}\over{\partial x% }^{2}}+{\partial^{2}B_{z}\over{\partial y}^{2}}+{\partial^{2}B_{z}\over{% \partial z}^{2}}\right)\\ &\displaystyle=-M_{x}\nabla^{2}B_{x}-M_{y}\nabla^{2}B_{y}-M_{z}\nabla^{2}B_{z}% \end{aligned}
  15. 2 U = - M x 0 - M y 0 - M z 0 = 0 , \nabla^{2}U=-M_{x}0-M_{y}0-M_{z}0=0,
  16. U = - k | 𝐁 | 2 = - k ( B x 2 + B y 2 + B z 2 ) . U=-k|\mathbf{B}|^{2}=-k\left(B_{x}^{2}+B_{y}^{2}+B_{z}^{2}\right).
  17. 2 | 𝐁 | 2 = 2 ( B x 2 + B y 2 + B z 2 ) = 2 ( | B x | 2 + | B y | 2 + | B z | 2 + B x 2 B x + B y 2 B y + B z 2 B z ) \begin{aligned}\displaystyle\nabla^{2}|\mathbf{B}|^{2}&\displaystyle=\nabla^{2% }\left(B_{x}^{2}+B_{y}^{2}+B_{z}^{2}\right)\\ &\displaystyle=2\left(|\nabla B_{x}|^{2}+|\nabla B_{y}|^{2}+|\nabla B_{z}|^{2}% +B_{x}\nabla^{2}B_{x}+B_{y}\nabla^{2}B_{y}+B_{z}\nabla^{2}B_{z}\right)\end{aligned}
  18. 2 | 𝐁 | 2 = 2 ( | B x | 2 + | B y | 2 + | B z | 2 ) ; \nabla^{2}|\mathbf{B}|^{2}=2\left(|\nabla B_{x}|^{2}+|\nabla B_{y}|^{2}+|% \nabla B_{z}|^{2}\right);
  19. 2 | 𝐁 | 2 0. \nabla^{2}|\mathbf{B}|^{2}\geq 0.
  20. 2 B x = 2 B x x 2 + 2 B x y 2 + 2 B x z 2 = x B x x + y B x y + z B x z \begin{aligned}\displaystyle\nabla^{2}B_{x}&\displaystyle={\partial^{2}B_{x}% \over\partial x^{2}}+{\partial^{2}B_{x}\over\partial y^{2}}+{\partial^{2}B_{x}% \over\partial z^{2}}\\ &\displaystyle={\partial\over\partial x}{\partial B_{x}\over\partial x}+{% \partial\over\partial y}{\partial B_{x}\over\partial y}+{\partial\over\partial z% }{\partial B_{x}\over\partial z}\end{aligned}
  21. B x y = B y x , \frac{\partial B_{x}}{\partial y}=\frac{\partial B_{y}}{\partial x},
  22. B x z = B z x , \frac{\partial B_{x}}{\partial z}=\frac{\partial B_{z}}{\partial x},
  23. 2 B x = x B x x + y B y x + z B z x . \nabla^{2}B_{x}={\partial\over\partial x}{\partial B_{x}\over\partial x}+{% \partial\over\partial y}{\partial B_{y}\over\partial x}+{\partial\over\partial z% }{\partial B_{z}\over\partial x}.
  24. 2 B x = x ( B x x + B y y + B z z ) = x ( 𝐁 ) . \nabla^{2}B_{x}={\partial\over\partial x}\left({\partial B_{x}\over\partial x}% +{\partial B_{y}\over\partial y}+{\partial B_{z}\over\partial z}\right)={% \partial\over\partial x}(\nabla\cdot\mathbf{B}).
  25. 𝐁 = 0 , \nabla\cdot\mathbf{B}=0,
  26. 2 B x = x ( 𝐁 ) = 0. \nabla^{2}B_{x}={\partial\over\partial x}(\nabla\cdot\mathbf{B})=0.
  27. 2 𝐁 = ( 𝐁 ) - × ( × 𝐁 ) , \nabla^{2}\mathbf{B}=\nabla(\nabla\cdot\mathbf{B})-\nabla\times\left(\nabla% \times\mathbf{B}\right),

Eckmann–Hilton_argument.html

  1. X X
  2. a , b , c , d X , ( a * b ) . ( c * d ) = ( a . c ) * ( b . d ) \forall a,b,c,d\in X,\ (a*b).(c*d)=(a.c)*(b.d)
  3. X X
  4. ( X , . ) × ( X , . ) ( X , . ) (X,.)\times(X,.)\to(X,.)
  5. X X
  6. X X
  7. , \circ,\bullet
  8. ( a b ) ( c d ) = ( a c ) ( b d ) (a\circ b)\bullet(c\circ d)=(a\bullet c)\circ(b\bullet d)
  9. n > 1 n>1
  10. 1 . = 1 . .1 . = ( 1 * * 1 . ) . ( 1 . * 1 * ) = ( 1 * .1 . ) * ( 1 . .1 * ) = 1 * * 1 * = 1 * . 1_{.}=1_{.}.1_{.}=(1_{*}*1_{.}).(1_{.}*1_{*})=(1_{*}.1_{.})*(1_{.}.1_{*})=1_{*% }*1_{*}=1_{*}.
  11. a , b X a,b\in X
  12. a . b = ( 1 * a ) . ( b * 1 ) = ( 1. b ) * ( a .1 ) = b * a = ( b .1 ) * ( 1. a ) = ( b * 1 ) . ( 1 * a ) = b . a . a.b=(1*a).(b*1)=(1.b)*(a.1)=b*a=(b.1)*(1.a)=(b*1).(1*a)=b.a.
  13. ( a * b ) * c = ( a * b ) * ( 1 * c ) = ( a * 1 ) * ( b * c ) = a * ( b * c ) . (a*b)*c=(a*b)*(1*c)=(a*1)*(b*c)=a*(b*c).

Ecological_fallacy.html

  1. cov ( i = 1 N Y i , i = 1 N X i ) = i = 1 N cov ( Y i , X i ) + i = 1 N l i cov ( Y l , X i ) \operatorname{cov}\left(\sum_{i=1}^{N}Y_{i},\sum_{i=1}^{N}X_{i}\right)=\sum_{i% =1}^{N}\operatorname{cov}(Y_{i},X_{i})+\sum_{i=1}^{N}\sum_{l\neq i}% \operatorname{cov}(Y_{l},X_{i})
  2. Y i Y_{i}
  3. X i X_{i}
  4. Y i = α + β X i + u i , Y_{i}=\alpha+\beta X_{i}+u_{i},
  5. cov [ u i , X i ] = 0. \operatorname{cov}[u_{i},X_{i}]=0.
  6. i = 1 N Y i = α + β i = 1 N X i + i = 1 N u i , \sum_{i=1}^{N}Y_{i}=\alpha+\beta\sum_{i=1}^{N}X_{i}+\sum_{i=1}^{N}u_{i},
  7. cov [ i = 1 N u i , i = 1 N X i ] 0. \operatorname{cov}\left[\sum_{i=1}^{N}u_{i},\sum_{i=1}^{N}X_{i}\right]\neq 0.
  8. cov [ u i , k = 1 N X k ] = 0 for all i . \operatorname{cov}\left[u_{i},\sum_{k=1}^{N}X_{k}\right]=0\quad\,\text{ for % all }i.
  9. X i X_{i}
  10. k = 1 N X k \sum_{k=1}^{N}X_{k}
  11. Y i Y_{i}
  12. P [ Suicide Protestant ] P[\,\text{Suicide}\mid\,\text{Protestant}]
  13. P [ Suicide Protestant ] P [ Suicide ] P ( Protestant ) P[\,\text{Suicide}\mid\,\text{Protestant}]\neq\frac{P[\,\text{Suicide}]}{P(\,% \text{Protestant})}
  14. P [ Suicide ] = \color B l u e P [ Suicide Protestant ] P ( Protestant ) + \color B l u e P [ Suicide not Protestant ] ( 1 - P ( Protestant ) ) \begin{aligned}\displaystyle P[\,\text{Suicide}]={\color{Blue}P[\,\text{% Suicide}\mid\,\text{Protestant}]}P(\,\text{Protestant})+{\color{Blue}P[\,\text% {Suicide}\mid\,\text{not Protestant}]}(1-P(\,\text{Protestant}))\end{aligned}
  15. P [ Suicide not Protestant ] P[\,\text{Suicide}\mid\,\text{not Protestant}]
  16. P [ Suicide Protestant ] P[\,\text{Suicide}\mid\,\text{Protestant}]
  17. E [ Y Z = z , X = 1 ] > E [ Y Z = z , X = 0 ] for all z , while E [ Y X = 1 ] < E [ Y X = 0 ] E[Y\mid Z=z,X=1]>E[Y\mid Z=z,X=0]\,\text{for all }z,\,\text{ while }E[Y\mid X=% 1]<E[Y\mid X=0]
  18. E [ Y Z = z , X = 1 ] - E [ Y Z = z , X = 0 ] E[Y\mid Z=z,X=1]-E[Y\mid Z=z,X=0]
  19. Z Z
  20. X X
  21. Z Z

EconMult.html

  1. V = ( v 1 , 1 v 1 , n v j , 1 v j , n ) V=\begin{pmatrix}v_{1,1}&\cdots&v_{1,n}\\ \vdots&\ddots&\vdots\\ v_{j,1}&\cdots&v_{j,n}\end{pmatrix}
  2. D = ( d 1 , 1 d 1 , n d j , 1 d j , n ) D=\begin{pmatrix}d_{1,1}&\cdots&d_{1,n}\\ \vdots&\ddots&\vdots\\ d_{j,1}&\cdots&d_{j,n}\end{pmatrix}
  3. X = ( x 1 x i ) X=\begin{pmatrix}x_{1}\\ \vdots\\ x_{i}\end{pmatrix}
  4. i n > 0 i\geq n>0
  5. j > 0 j>0\,\!
  6. Y V = ( ( q 1 , 1 , 1 d 1 , 1 v 1 , 1 α 1 , 1 , 1 - 1 x 1 β 1 , 1 , 1 q 1 , 1 , i d 1 , 1 v 1 , 1 α 1 , 1 , i - 1 x i β 1 , 1 , i ) ( q 1 , n , 1 d 1 , n v 1 , n α 1 , n , 1 - 1 x 1 β 1 , n , 1 q 1 , n , i d 1 , n v 1 , n α 1 , n , i - 1 x i β 1 , n , i ) ( q j , 1 , 1 d j , 1 v j , 1 α j , 1 , 1 - 1 x 1 β j , 1 , 1 q j , 1 , i d j , 1 v j , 1 α j , 1 , i - 1 x i β j , 1 , i ) ( q j , n , 1 d j , n v j , n α j , n , 1 - 1 x 1 β j , n , 1 q j , n , i d j , n v j , n α j , n , i - 1 x i β j , n , i ) ) Y_{V}=\begin{pmatrix}\begin{pmatrix}q_{1,1,1}d_{1,1}v_{1,1}^{\alpha_{1,1,1}-1}% x_{1}^{\beta_{1,1,1}}\\ \vdots\\ q_{1,1,i}d_{1,1}v_{1,1}^{\alpha_{1,1,i}-1}x_{i}^{\beta_{1,1,i}}\end{pmatrix}&% \cdots&\begin{pmatrix}q_{1,n,1}d_{1,n}v_{1,n}^{\alpha_{1,n,1}-1}x_{1}^{\beta_{% 1,n,1}}\\ \vdots\\ q_{1,n,i}d_{1,n}v_{1,n}^{\alpha_{1,n,i}-1}x_{i}^{\beta_{1,n,i}}\end{pmatrix}\\ \vdots&\ddots&\vdots\\ \begin{pmatrix}q_{j,1,1}d_{j,1}v_{j,1}^{\alpha_{j,1,1}-1}x_{1}^{\beta_{j,1,1}}% \\ \vdots\\ q_{j,1,i}d_{j,1}v_{j,1}^{\alpha_{j,1,i}-1}x_{i}^{\beta_{j,1,i}}\end{pmatrix}&% \cdots&\begin{pmatrix}q_{j,n,1}d_{j,n}v_{j,n}^{\alpha_{j,n,1}-1}x_{1}^{\beta_{% j,n,1}}\\ \vdots\\ q_{j,n,i}d_{j,n}v_{j,n}^{\alpha_{j,n,i}-1}x_{i}^{\beta_{j,n,i}}\end{pmatrix}% \end{pmatrix}
  7. Y = ( ( q 1 , 1 , 1 d 1 , 1 v 1 , 1 α 1 , 1 , 1 x 1 β 1 , 1 , 1 q 1 , 1 , i d 1 , 1 v 1 , 1 α 1 , 1 , i x i β 1 , 1 , i ) ( q 1 , n , 1 d 1 , n v 1 , n α 1 , n , 1 x 1 β 1 , n , 1 q 1 , n , i d 1 , n v 1 , n α 1 , n , i x i β 1 , n , i ) ( q j , 1 , 1 d j , 1 v j , 1 α j , 1 , 1 x 1 β j , 1 , 1 q j , 1 , i d j , 1 v j , 1 α j , 1 , i x i β j , 1 , i ) ( q j , n , 1 d j , n v j , n α j , n , 1 x 1 β j , n , 1 q j , n , i d j , n v j , n α j , n , i x i β j , n , i ) ) Y=\begin{pmatrix}\begin{pmatrix}q_{1,1,1}d_{1,1}v_{1,1}^{\alpha_{1,1,1}}x_{1}^% {\beta_{1,1,1}}\\ \vdots\\ q_{1,1,i}d_{1,1}v_{1,1}^{\alpha_{1,1,i}}x_{i}^{\beta_{1,1,i}}\end{pmatrix}&% \cdots&\begin{pmatrix}q_{1,n,1}d_{1,n}v_{1,n}^{\alpha_{1,n,1}}x_{1}^{\beta_{1,% n,1}}\\ \vdots\\ q_{1,n,i}d_{1,n}v_{1,n}^{\alpha_{1,n,i}}x_{i}^{\beta_{1,n,i}}\end{pmatrix}\\ \vdots&\ddots&\vdots\\ \begin{pmatrix}q_{j,1,1}d_{j,1}v_{j,1}^{\alpha_{j,1,1}}x_{1}^{\beta_{j,1,1}}\\ \vdots\\ q_{j,1,i}d_{j,1}v_{j,1}^{\alpha_{j,1,i}}x_{i}^{\beta_{j,1,i}}\end{pmatrix}&% \cdots&\begin{pmatrix}q_{j,n,1}d_{j,n}v_{j,n}^{\alpha_{j,n,1}}x_{1}^{\beta_{j,% n,1}}\\ \vdots\\ q_{j,n,i}d_{j,n}v_{j,n}^{\alpha_{j,n,i}}x_{i}^{\beta_{j,n,i}}\end{pmatrix}\end% {pmatrix}
  8. Q = ( ( q 1 , 1 , 1 q 1 , 1 , i ) ( q 1 , n , 1 q 1 , n , i ) ( q j , 1 , 1 q j , 1 , i ) ( q j , n , 1 q j , n , i ) ) Q=\begin{pmatrix}\begin{pmatrix}q_{1,1,1}\\ \vdots\\ q_{1,1,i}\end{pmatrix}&\cdots&\begin{pmatrix}q_{1,n,1}\\ \vdots\\ q_{1,n,i}\end{pmatrix}\\ \vdots&\ddots&\vdots\\ \begin{pmatrix}q_{j,1,1}\\ \vdots\\ q_{j,1,i}\end{pmatrix}&\cdots&\begin{pmatrix}q_{j,n,1}\\ \vdots\\ q_{j,n,i}\end{pmatrix}\end{pmatrix}

Eddy_current.html

  1. P = π 2 B p 2 d 2 f 2 6 k ρ D , P=\frac{\pi^{2}B\text{p}^{\,2}d^{2}f^{2}}{6k\rho D},
  2. δ = 1 π f μ σ , \delta=\frac{1}{\sqrt{\pi f\mu\sigma}},
  3. × 𝐇 = 𝐉 . \nabla\times\mathbf{H}=\mathbf{J}.
  4. \color w h i t e - ( 𝐇 ) - 2 𝐇 = × 𝐉 . {\color{white}-}\nabla\left(\nabla\cdot\mathbf{H}\right)-\nabla^{2}\mathbf{H}=% \nabla\times\mathbf{J}.
  5. - 2 𝐇 = × 𝐉 . -\nabla^{2}\mathbf{H}=\nabla\times\mathbf{J}.
  6. = =
  7. - 2 𝐇 = σ \timessymbol E . -\nabla^{2}\mathbf{H}=\sigma\nabla\timessymbol{E}.
  8. \color w h i t e - 2 𝐇 = σ 𝐁 t . {\color{white}-}\nabla^{2}\mathbf{H}=\sigma\frac{\partial\mathbf{B}}{\partial t}.
  9. \color w h i t e - 2 𝐇 = μ 0 σ ( 𝐌 t + 𝐇 t ) . {\color{white}-}\nabla^{2}\mathbf{H}=\mu_{0}\sigma\left(\frac{\partial\mathbf{% M}}{\partial t}+\frac{\partial\mathbf{H}}{\partial t}\right).

Edge-transitive_graph.html

  1. K m , n K_{m,n}

Edge_detection.html

  1. f f
  2. x = 0 x=0
  3. f ( x ) = I r - I l 2 ( erf ( x 2 σ ) + 1 ) + I l . f(x)=\frac{I_{r}-I_{l}}{2}\left(\operatorname{erf}\left(\frac{x}{\sqrt{2}% \sigma}\right)+1\right)+I_{l}.
  4. I l = lim x - f ( x ) I_{l}=\lim_{x\rightarrow-\infty}f(x)
  5. I r = lim x f ( x ) I_{r}=\lim_{x\rightarrow\infty}f(x)
  6. σ \sigma
  7. L x ( x , y ) = - 1 / 2 L ( x - 1 , y ) + 0 L ( x , y ) + 1 / 2 L ( x + 1 , y ) L_{x}(x,y)=-1/2\cdot L(x-1,y)+0\cdot L(x,y)+1/2\cdot L(x+1,y)\,
  8. L y ( x , y ) = - 1 / 2 L ( x , y - 1 ) + 0 L ( x , y ) + 1 / 2 L ( x , y + 1 ) , L_{y}(x,y)=-1/2\cdot L(x,y-1)+0\cdot L(x,y)+1/2\cdot L(x,y+1),\,
  9. L x = [ - 1 / 2 0 1 / 2 ] * L and L y = [ + 1 / 2 0 - 1 / 2 ] * L . L_{x}=\begin{bmatrix}-1/2&0&1/2\end{bmatrix}*L\quad\mbox{and}~{}\quad L_{y}=% \begin{bmatrix}+1/2\\ 0\\ -1/2\end{bmatrix}*L.
  10. L x = [ - 1 0 + 1 - 2 0 + 2 - 1 0 + 1 ] * L and L y = [ + 1 + 2 + 1 0 0 0 - 1 - 2 - 1 ] * L . L_{x}=\begin{bmatrix}-1&0&+1\\ -2&0&+2\\ -1&0&+1\end{bmatrix}*L\quad\mbox{and}~{}\quad L_{y}=\begin{bmatrix}+1&+2&+1\\ 0&0&0\\ -1&-2&-1\end{bmatrix}*L.
  11. | L | = L x 2 + L y 2 |\nabla L|=\sqrt{L_{x}^{2}+L_{y}^{2}}
  12. θ = atan2 ( L y , L x ) . \theta=\operatorname{atan2}(L_{y},L_{x}).
  13. ( u , v ) (u,v)
  14. v v
  15. L ( x , y ; t ) L(x,y;t)
  16. t t
  17. v v
  18. L v L_{v}
  19. v v
  20. v ( L v ) = 0 \partial_{v}(L_{v})=0
  21. v v
  22. L v L_{v}
  23. v v ( L v ) 0. \partial_{vv}(L_{v})\leq 0.
  24. L x L_{x}
  25. L y L_{y}
  26. L y y y L_{yyy}
  27. L v 2 L v v = L x 2 L x x + 2 L x L y L x y + L y 2 L y y = 0 , L_{v}^{2}L_{vv}=L_{x}^{2}\,L_{xx}+2\,L_{x}\,L_{y}\,L_{xy}+L_{y}^{2}\,L_{yy}=0,
  28. L v 3 L v v v = L x 3 L x x x + 3 L x 2 L y L x x y + 3 L x L y 2 L x y y + L y 3 L y y y 0 L_{v}^{3}L_{vvv}=L_{x}^{3}\,L_{xxx}+3\,L_{x}^{2}\,L_{y}\,L_{xxy}+3\,L_{x}\,L_{% y}^{2}\,L_{xyy}+L_{y}^{3}\,L_{yyy}\leq 0
  29. L x L_{x}
  30. L y L_{y}
  31. L y y y L_{yyy}
  32. L L
  33. L L
  34. L x x ( x , y ) = L ( x - 1 , y ) - 2 L ( x , y ) + L ( x + 1 , y ) . L_{xx}(x,y)=L(x-1,y)-2L(x,y)+L(x+1,y).\,
  35. L x y ( x , y ) = ( L ( x - 1 , y - 1 ) - L ( x - 1 , y + 1 ) - L ( x + 1 , y - 1 ) + L ( x + 1 , y + 1 ) ) / 4 , L_{xy}(x,y)=(L(x-1,y-1)-L(x-1,y+1)-L(x+1,y-1)+L(x+1,y+1))/4,\,
  36. L y y ( x , y ) = L ( x , y - 1 ) - 2 L ( x , y ) + L ( x , y + 1 ) . L_{yy}(x,y)=L(x,y-1)-2L(x,y)+L(x,y+1).\,
  37. L x x = [ 1 - 2 1 ] * L and L x y = [ - 1 / 4 0 1 / 4 0 0 0 1 / 4 0 - 1 / 4 ] * L and L y y = [ 1 - 2 1 ] * L . L_{xx}=\begin{bmatrix}1&-2&1\end{bmatrix}*L\quad\mbox{and}~{}\quad L_{xy}=% \begin{bmatrix}-1/4&0&1/4\\ 0&0&0\\ 1/4&0&-1/4\end{bmatrix}*L\quad\mbox{and}~{}\quad L_{yy}=\begin{bmatrix}1\\ -2\\ 1\end{bmatrix}*L.

Edit_distance.html

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  35. w w
  36. x x
  37. w w
  38. x x
  39. y y
  40. u u
  41. x x
  42. y y
  43. v v
  44. u u
  45. y y
  46. x x
  47. v v
  48. d d
  49. a a
  50. a a
  51. d d
  52. a a
  53. b b
  54. a a
  55. b b
  56. d d
  57. a a
  58. b b
  59. d d
  60. b b
  61. a a
  62. d d
  63. a a
  64. c c
  65. d d
  66. a a
  67. b b
  68. d d
  69. b b
  70. c c
  71. a a
  72. b b
  73. a a
  74. u v uv
  75. b b
  76. u w uw
  77. d d
  78. a a
  79. b b
  80. d d
  81. v v
  82. w w
  83. a = a 1 a n a=a_{1}\ldots a_{n}
  84. b = b 1 b m b=b_{1}\ldots b_{m}
  85. d m n d_{mn}
  86. d i 0 = k = 1 i w del ( b k ) , for 1 i m d 0 j = k = 1 j w ins ( a k ) , for 1 j n d i j = { d i - 1 , j - 1 for a j = b i min { d i - 1 , j + w del ( b i ) d i , j - 1 + w ins ( a j ) d i - 1 , j - 1 + w sub ( a j , b i ) for a j b i for 1 i m , 1 j n . \begin{aligned}\displaystyle d_{i0}&\displaystyle=\sum_{k=1}^{i}w_{\mathrm{del% }}(b_{k}),&&\displaystyle\quad\,\text{for}\;1\leq i\leq m\\ \displaystyle d_{0j}&\displaystyle=\sum_{k=1}^{j}w_{\mathrm{ins}}(a_{k}),&&% \displaystyle\quad\,\text{for}\;1\leq j\leq n\\ \displaystyle d_{ij}&\displaystyle=\begin{cases}d_{i-1,j-1}&\,\text{for}\;a_{j% }=b_{i}\\ \min\begin{cases}d_{i-1,j}+w_{\mathrm{del}}(b_{i})\\ d_{i,j-1}+w_{\mathrm{ins}}(a_{j})\\ d_{i-1,j-1}+w_{\mathrm{sub}}(a_{j},b_{i})\end{cases}&\,\text{for}\;a_{j}\neq b% _{i}\end{cases}&&\displaystyle\quad\,\text{for}\;1\leq i\leq m,1\leq j\leq n.% \end{aligned}
  87. d m n d_{mn}
  88. m m
  89. n n
  90. s s
  91. m m
  92. n n
  93. p p
  94. k k
  95. s s
  96. p p
  97. k k

Educational_assessment.html

  1. R x = V t / V x R\text{x}=V\text{t}/V\text{x}
  2. R x R\text{x}
  3. V t V\text{t}
  4. V x V\text{x}
  5. R x R\text{x}

Edward_Nelson.html

  1. χ \chi

Edward_Victor_Appleton.html

  1. h - h = D h^{\prime}-h=D
  2. h - h λ = D λ = N \frac{h-h^{\prime}}{\lambda}=\frac{D}{\lambda}=N
  3. N - N = D λ - D λ = 1 N-N^{\prime}=\frac{D}{\lambda}-\frac{D}{\lambda^{\prime}}=1
  4. D = h - h = 1 1 λ - 1 λ D=h-h^{\prime}=\frac{1}{\frac{1}{\lambda}-\frac{1}{\lambda^{\prime}}}

Effect_size.html

  1. ρ ^ \hat{\rho}
  2. ρ \rho
  3. η 2 = S S Treatment S S Total . \eta^{2}=\frac{SS\text{Treatment}}{SS\text{Total}}.
  4. ω 2 = S S treatment - d f treatment * M S error S S total + M S error . \omega^{2}=\frac{SS\text{treatment}-df\text{treatment}*MS\text{error}}{SS\text% {total}+MS\text{error}}.
  5. f 2 = R 2 1 - R 2 f^{2}={R^{2}\over 1-R^{2}}
  6. f 2 = η 2 1 - η 2 f^{2}={\eta^{2}\over 1-\eta^{2}}
  7. f 2 = ω 2 1 - ω 2 f^{2}={\omega^{2}\over 1-\omega^{2}}
  8. f 2 f^{2}
  9. f 2 = R A B 2 - R A 2 1 - R A B 2 f^{2}={R^{2}_{AB}-R^{2}_{A}\over 1-R^{2}_{AB}}
  10. f ^ \hat{f}
  11. f ^ effect = ( d f effect / N ) ( F effect - 1 ) . \hat{f}\text{effect}={\sqrt{(df\text{effect}/N)(F\text{effect}-1)}}.
  12. f 2 f^{2}
  13. S S ( μ 1 , μ 2 , , μ K ) K × σ 2 , {SS(\mu_{1},\mu_{2},\dots,\mu_{K})}\over{K\times\sigma^{2}},
  14. q = 1 2 l o g 1 + r 1 1 - r 1 - 1 2 l o g 1 + r 2 1 - r 2 q=\frac{1}{2}log\frac{1+r_{1}}{1-r_{1}}-\frac{1}{2}log\frac{1+r_{2}}{1-r_{2}}
  15. v a r ( q ) = 1 N 1 - 3 + 1 N 2 - 3 var(q)=\frac{1}{N_{1}-3}+\frac{1}{N_{2}-3}
  16. θ = μ 1 - μ 2 σ , \theta=\frac{\mu_{1}-\mu_{2}}{\sigma},
  17. n \sqrt{n}
  18. d = x ¯ 1 - x ¯ 2 s . d=\frac{\bar{x}_{1}-\bar{x}_{2}}{s}.
  19. s = ( n 1 - 1 ) s 1 2 + ( n 2 - 1 ) s 2 2 n 1 + n 2 - 2 s=\sqrt{\frac{(n_{1}-1)s^{2}_{1}+(n_{2}-1)s^{2}_{2}}{n_{1}+n_{2}-2}}
  20. s 1 2 = 1 n 1 - 1 i = 1 n 1 ( x 1 , i - x ¯ 1 ) 2 , s_{1}^{2}=\frac{1}{n_{1}-1}\sum_{i=1}^{n_{1}}(x_{1,i}-\bar{x}_{1})^{2},
  21. s = ( n 1 - 1 ) s 1 2 + ( n 2 - 1 ) s 2 2 n 1 + n 2 s=\sqrt{\frac{(n_{1}-1)s^{2}_{1}+(n_{2}-1)s^{2}_{2}}{n_{1}+n_{2}}}
  22. t = X ¯ 1 - X ¯ 2 S E = X ¯ 1 - X ¯ 2 S D N = N * ( X ¯ 1 - X ¯ 2 ) S D t=\frac{\bar{X}_{1}-\bar{X}_{2}}{SE}=\frac{\bar{X}_{1}-\bar{X}_{2}}{\frac{SD}{% \sqrt{N}}}=\frac{\sqrt{N}*(\bar{X}_{1}-\bar{X}_{2})}{SD}
  23. d = X ¯ 1 - X ¯ 2 S D = t N d=\frac{\bar{X}_{1}-\bar{X}_{2}}{SD}=\frac{t}{\sqrt{N}}
  24. Δ = x ¯ 1 - x ¯ 2 s 2 \Delta=\frac{\bar{x}_{1}-\bar{x}_{2}}{s_{2}}
  25. g = x ¯ 1 - x ¯ 2 s * g=\frac{\bar{x}_{1}-\bar{x}_{2}}{s^{*}}
  26. s * s^{*}
  27. s * = ( n 1 - 1 ) s 1 2 + ( n 2 - 1 ) s 2 2 n 1 + n 2 - 2 . s^{*}=\sqrt{\frac{(n_{1}-1)s_{1}^{2}+(n_{2}-1)s_{2}^{2}}{n_{1}+n_{2}-2}}.
  28. g * = J ( n 1 + n 2 - 2 ) g ( 1 - 3 4 ( n 1 + n 2 ) - 9 ) g g^{*}=J(n_{1}+n_{2}-2)\,\,g\,\approx\,\left(1-\frac{3}{4(n_{1}+n_{2})-9}\right% )\,\,g
  29. g * g^{*}
  30. J ( a ) = Γ ( a / 2 ) a / 2 Γ ( ( a - 1 ) / 2 ) . J(a)=\frac{\Gamma(a/2)}{\sqrt{a/2\,}\,\Gamma((a-1)/2)}.
  31. Ψ = ( 1 k - 1 ) Σ ( x ¯ j - X ¯ ) 2 M S e r r o r \Psi=\sqrt{\left(\frac{1}{k-1}\right)\frac{\Sigma(\bar{x}_{j}-\bar{X})^{2}}{MS% _{error}}}
  32. n 1 n 2 / ( n 1 + n 2 ) g \sqrt{n_{1}n_{2}/(n_{1}+n_{2})}\,g
  33. n 1 n 2 / ( n 1 + n 2 ) θ \sqrt{n_{1}n_{2}/(n_{1}+n_{2})}\theta
  34. σ ^ 2 ( g * ) = n 1 + n 2 n 1 n 2 + ( g * ) 2 2 ( n 1 + n 2 ) . \hat{\sigma}^{2}(g^{*})=\frac{n_{1}+n_{2}}{n_{1}n_{2}}+\frac{(g^{*})^{2}}{2(n_% {1}+n_{2})}.
  35. ϕ = χ 2 N \phi=\sqrt{\frac{\chi^{2}}{N}}
  36. ϕ c = χ 2 N ( k - 1 ) \phi_{c}=\sqrt{\frac{\chi^{2}}{N(k-1)}}
  37. w = i = 1 N ( p 0 i - p 1 i ) 2 p 0 i w=\sqrt{\sum_{i=1}^{N}{\frac{(p_{0i}-p_{1i})^{2}}{p_{0i}}}}
  38. h = 2 ( a r c s i n p 1 - a r c s i n p 2 ) h=2(arcsin\sqrt{p_{1}}-arcsin\sqrt{p_{2}})
  39. d d
  40. d d
  41. d = # ( x i > x j ) - # ( x i < x j ) m n d=\frac{\#(x_{i}>x_{j})-\#(x_{i}<x_{j})}{mn}
  42. n n
  43. m m
  44. x i x_{i}
  45. x j x_{j}
  46. # \#
  47. d d
  48. U U
  49. d d
  50. d = 2 U m n - 1 d=\frac{2U}{mn}-1
  51. d d
  52. d {d}
  53. f 2 {f}^{2}
  54. t := M S E = M S D / n = n M - μ σ + n μ - μ baseline σ S D σ t:=\frac{M}{SE}=\frac{M}{SD/\sqrt{n}}=\frac{\sqrt{n}\frac{M-\mu}{\sigma}+\sqrt% {n}\frac{\mu-\mu\text{baseline}}{\sigma}}{\frac{SD}{\sigma}}
  55. n c p = n μ - μ baseline σ ncp=\sqrt{n}\frac{\mu-\mu\text{baseline}}{\sigma}
  56. d := M - μ baseline S D d:=\frac{M-\mu\text{baseline}}{SD}
  57. μ - μ baseline σ . \frac{\mu-\mu\text{baseline}}{\sigma}.
  58. d ~ = n c p n . \tilde{d}=\frac{ncp}{\sqrt{n}}.
  59. t := M 1 - M 2 S D within / n 1 n 2 n 1 + n 2 , t:=\frac{M_{1}-M_{2}}{SD\text{within}/\sqrt{\frac{n_{1}n_{2}}{n_{1}+n_{2}}}},
  60. S D within := S S within d f within = ( n 1 - 1 ) S D 1 2 + ( n 2 - 1 ) S D 2 2 n 1 + n 2 - 2 . SD\text{within}:=\sqrt{\frac{SS\text{within}}{df\text{within}}}=\sqrt{\frac{(n% _{1}-1)SD_{1}^{2}+(n_{2}-1)SD_{2}^{2}}{n_{1}+n_{2}-2}}.
  61. n c p = n 1 n 2 n 1 + n 2 μ 1 - μ 2 σ ncp=\sqrt{\frac{n_{1}n_{2}}{n_{1}+n_{2}}}\frac{\mu_{1}-\mu_{2}}{\sigma}
  62. d := M 1 - M 2 S D within d:=\frac{M_{1}-M_{2}}{SD\text{within}}
  63. μ 1 - μ 2 σ . \frac{\mu_{1}-\mu_{2}}{\sigma}.
  64. d ~ = n c p n 1 n 2 n 1 + n 2 . \tilde{d}=\frac{ncp}{\sqrt{\frac{n_{1}n_{2}}{n_{1}+n_{2}}}}.
  65. σ \sigma
  66. F := S S between σ 2 / d f between S S within σ 2 / d f within F:=\frac{\frac{SS\text{between}}{\sigma^{2}}/df\text{between}}{\frac{SS\text{% within}}{\sigma^{2}}/df\text{within}}
  67. M i ( X i , j ) := w = 1 n i X i , w n i ; μ i ( X i , j ) := μ i . M_{i}\left(X_{i,j}\right):=\frac{\sum_{w=1}^{n_{i}}X_{i,w}}{n_{i}};\;\mu_{i}% \left(X_{i,j}\right):=\mu_{i}.
  68. S S between / σ 2 = S S ( M i ( X i , j ) ; i = 1 , 2 , , K , j = 1 , 2 , , n i ) σ 2 = S S ( M i ( X i , j - μ i ) σ + μ i σ ; i = 1 , 2 , , K , j = 1 , 2 , , n i ) χ 2 ( d f = K - 1 , n c p = S S ( μ i ( X i , j ) σ ; i = 1 , 2 , , K , j = 1 , 2 , , n i ) ) \begin{array}[]{ll}SS\text{between}/\sigma^{2}&=\frac{SS\left(M_{i}\left(X_{i,% j}\right);i=1,2,\dots,K,\;j=1,2,\dots,n_{i}\right)}{\sigma^{2}}\\ &=SS\left(\frac{M_{i}\left(X_{i,j}-\mu_{i}\right)}{\sigma}+\frac{\mu_{i}}{% \sigma};i=1,2,\dots,K,\;j=1,2,\dots,n_{i}\right)\\ &\sim\chi^{2}\left(df=K-1,\;ncp=SS\left(\frac{\mu_{i}\left(X_{i,j}\right)}{% \sigma};i=1,2,\dots,K,\;j=1,2,\dots,n_{i}\right)\right)\end{array}
  69. χ 2 \chi^{2}
  70. S S ( μ i ( X i , j ) / σ ; i = 1 , 2 , , K , j = 1 , 2 , , n i ) . SS\left(\mu_{i}(X_{i,j})/\sigma;i=1,2,\dots,K,\;j=1,2,\dots,n_{i}\right).
  71. n := n 1 = n 2 = = n K n:=n_{1}=n_{2}=\cdots=n_{K}
  72. Cohens f ~ 2 := S S ( μ 1 , μ 2 , , μ K ) K σ 2 = S S ( μ i ( X i , j ) / σ ; i = 1 , 2 , , K , j = 1 , 2 , , n i ) n K = n c p n K = n c p N . \,\text{Cohens }\tilde{f}^{2}:=\frac{SS(\mu_{1},\mu_{2},\dots,\mu_{K})}{K\cdot% \sigma^{2}}=\frac{SS\left(\mu_{i}\left(X_{i,j}\right)/\sigma;i=1,2,\dots,K,\;j% =1,2,\dots,n_{i}\right)}{n\cdot K}=\frac{ncp}{n\cdot K}=\frac{ncp}{N}.
  73. n c p F ncp_{F}
  74. n c p t ncp_{t}
  75. n c p F = n c p t 2 ncp_{F}=ncp_{t}^{2}
  76. f ~ = | d ~ 2 | \tilde{f}=\left|\frac{\tilde{d}}{2}\right|

Effective_interest_rate.html

  1. r = ( 1 + i n ) n - 1 r\ =\ \left(1+\frac{i}{n}\right)^{n}-1
  2. r = e i - 1 r\ =\ e^{i}-1
  3. r = ( 1 + j ) 12 - 1 r\ =\ (1+j)^{12}-1

Effective_nuclear_charge.html

  1. Z eff Z_{\mathrm{eff}}
  2. Z Z^{\ast}
  3. Z eff = Z - S Z_{\mathrm{eff}}=Z-S
  4. Z eff = r H r Z Z_{\mathrm{eff}}=\frac{\langle r\rangle_{\rm H}}{\langle r\rangle_{Z}}
  5. r H \langle r\rangle_{\rm H}
  6. r Z \langle r\rangle_{Z}
  7. Z eff ( F - ) = 9 - 2 = 7 + Z_{\mathrm{eff}}(\mathrm{F}^{-})=9-2=7+
  8. Z eff ( Ne ) = 10 - 2 = 8 + Z_{\mathrm{eff}}(\mathrm{Ne})=10-2=8+
  9. Z eff ( Na + ) = 11 - 2 = 9 + Z_{\mathrm{eff}}(\mathrm{Na}^{+})=11-2=9+

Efficient_estimator.html

  1. Var [ T ] θ - 1 , \operatorname{Var}[\,T\,]\ \geq\ \mathcal{I}_{\theta}^{-1},
  2. θ \scriptstyle\mathcal{I}_{\theta}
  3. T ( X ) = 1 n i = 1 n x i . T(X)=\frac{1}{n}\sum_{i=1}^{n}x_{i}\ .
  4. T 1 T_{1}
  5. T 2 T_{2}
  6. θ \theta
  7. T 1 T_{1}
  8. T 2 T_{2}
  9. θ \theta
  10. T 2 T_{2}
  11. T 1 T_{1}
  12. T 2 T_{2}
  13. E [ ( T 1 - θ ) 2 ] E [ ( T 2 - θ ) 2 ] \mathrm{E}\left[(T_{1}-\theta)^{2}\right]\leq\mathrm{E}\left[(T_{2}-\theta)^{2% }\right]
  14. θ \theta
  15. e ( T 1 , T 2 ) = E [ ( T 2 - θ ) 2 ] E [ ( T 1 - θ ) 2 ] e(T_{1},T_{2})=\frac{\mathrm{E}\left[(T_{2}-\theta)^{2}\right]}{\mathrm{E}% \left[(T_{1}-\theta)^{2}\right]}
  16. e e
  17. θ \theta
  18. e e
  19. T 1 T_{1}
  20. θ \theta

Egyptian_Air_Force.html

  1. } \Big\}
  2. } \Big\}

Egyptian_fraction.html

  1. 1 2 + 1 3 + 1 16 \tfrac{1}{2}+\tfrac{1}{3}+\tfrac{1}{16}
  2. 4 5 \frac{4}{5}
  3. 3 4 \frac{3}{4}
  4. 4 5 \frac{4}{5}
  5. 1 2 \frac{1}{2}
  6. 1 4 \frac{1}{4}
  7. 1 20 \frac{1}{20}
  8. 3 4 \frac{3}{4}
  9. 1 2 \frac{1}{2}
  10. 1 4 \frac{1}{4}
  11. 4 5 \frac{4}{5}
  12. 1 20 \frac{1}{20}
  13. 3 11 \frac{3}{11}
  14. 2 7 \frac{2}{7}
  15. 3 11 \frac{3}{11}
  16. 1 4 \frac{1}{4}
  17. 1 44 \frac{1}{44}
  18. 2 7 \frac{2}{7}
  19. 1 4 \frac{1}{4}
  20. 1 28 \frac{1}{28}
  21. 1 44 \frac{1}{44}
  22. 1 28 \frac{1}{28}
  23. 3 11 \frac{3}{11}
  24. 2 7 \frac{2}{7}
  25. 5 8 \frac{5}{8}
  26. 1 2 \frac{1}{2}
  27. 1 8 \frac{1}{8}
  28. 13 12 \frac{13}{12}
  29. 1 2 \frac{1}{2}
  30. 1 3 \frac{1}{3}
  31. 1 4 \frac{1}{4}
  32. = 1 3 =\frac{1}{3}
  33. = 1 10 =\frac{1}{10}
  34. = 1 2 =\frac{1}{2}
  35. = 2 3 =\frac{2}{3}
  36. = 3 4 =\frac{3}{4}
  37. a a b - 1 = 1 b + 1 b ( a b - 1 ) . \tfrac{a}{ab-1}=\tfrac{1}{b}+\tfrac{1}{b(ab-1)}.
  38. 8 11 \tfrac{8}{11}
  39. 8 11 = 6 11 + 2 11 . \tfrac{8}{11}=\tfrac{6}{11}+\tfrac{2}{11}.
  40. 8 11 = 1 2 + 1 22 + 1 6 + 1 66 . \tfrac{8}{11}=\tfrac{1}{2}+\tfrac{1}{22}+\tfrac{1}{6}+\tfrac{1}{66}.
  41. x y = 1 y / x + ( - y ) mod x y y / x , \frac{x}{y}=\frac{1}{\lceil y/x\rceil}+\frac{(-y)\,\bmod\,x}{y\lceil y/x\rceil},
  42. \lceil\ldots\rceil
  43. 4 13 = 1 4 + 1 18 + 1 468 \tfrac{4}{13}=\tfrac{1}{4}+\tfrac{1}{18}+\tfrac{1}{468}
  44. 17 29 = 1 2 + 1 12 + 1 348 . \tfrac{17}{29}=\tfrac{1}{2}+\tfrac{1}{12}+\tfrac{1}{348}.
  45. 5 121 = 1 25 + 1 757 + 1 763309 + 1 873960180913 + 1 1527612795642093418846225 , \frac{5}{121}=\frac{1}{25}+\frac{1}{757}+\frac{1}{763309}+\frac{1}{87396018091% 3}+\frac{1}{1527612795642093418846225},
  46. 5 121 = 1 33 + 1 121 + 1 363 . \frac{5}{121}=\frac{1}{33}+\frac{1}{121}+\frac{1}{363}.
  47. y / x + 1 \lfloor y/x\rfloor+1
  48. y / x \lceil y/x\rceil
  49. a / b a/b
  50. b / 2 < c < b b/2<c<b
  51. a / b a/b
  52. a c / b c ac/bc
  53. a c ac
  54. b c bc
  55. n S 1 / n = 1. \sum_{n\in S}1/n=1.
  56. 1 x i + 1 x i = 1. \sum\frac{1}{x_{i}}+\prod\frac{1}{x_{i}}=1.
  57. 1 k + 1 k = 2 k + 1 + 2 k ( k + 1 ) \frac{1}{k}+\frac{1}{k}=\frac{2}{k+1}+\frac{2}{k(k+1)}
  58. 1 k + 1 k = 1 k + 1 k + 1 + 1 k ( k + 1 ) . \frac{1}{k}+\frac{1}{k}=\frac{1}{k}+\frac{1}{k+1}+\frac{1}{k(k+1)}.
  59. 4 5 = 1 5 + 1 6 + 1 7 + 1 8 + 1 30 + 1 31 + 1 32 + 1 42 + 1 43 + 1 56 + 1 930 + 1 931 + 1 992 + 1 1806 + 1 865830 . \frac{4}{5}=\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{30}+\frac% {1}{31}+\frac{1}{32}+\frac{1}{42}+\frac{1}{43}+\frac{1}{56}+\frac{1}{930}+% \frac{1}{931}+\frac{1}{992}+\frac{1}{1806}+\frac{1}{865830}.
  60. O ( y log y ( log log y ) 4 ( log log log y ) 2 ) O\left(y\log y(\log\log y)^{4}(\log\log\log y)^{2}\right)
  61. O ( log y ) O\left(\sqrt{\log y}\right)
  62. Ω ( log log y ) \Omega(\log\log y)
  63. O ( log log y ) O(\log\log y)
  64. [ 0 , π 2 6 - 1 ) [ 1 , π 2 6 ) . \left[0,\frac{\pi^{2}}{6}-1\right)\cup\left[1,\frac{\pi^{2}}{6}\right).
  65. x = 1 a 1 + 1 a 1 a 2 + 1 a 1 a 2 a 3 + . x=\frac{1}{a_{1}}+\frac{1}{a_{1}a_{2}}+\frac{1}{a_{1}a_{2}a_{3}}+\cdots.
  66. 5 12 = 1 4 + 1 10 + 1 15 = 1 5 + 1 6 + 1 20 . \frac{5}{12}=\frac{1}{4}+\frac{1}{10}+\frac{1}{15}=\frac{1}{5}+\frac{1}{6}+% \frac{1}{20}.
  67. 4 n = 1 x + 1 y + 1 z \frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}

Ehrhart_polynomial.html

  1. L ( P , t ) = # ( t P L ) L(P,t)=\#(tP\cap L)\,
  2. L ( P , t ) = a d t d + a d - 1 t d - 1 + + a 0 L(P,t)=a_{d}t^{d}+a_{d-1}t^{d-1}+...+a_{0}
  3. L ( int ( P ) , t ) = ( - 1 ) d L ( P , - t ) , L(\,\text{int}(P),t)=(-1)^{d}L(P,-t),
  4. P = { x d : 0 x i 1 ; 1 i d } P=\{x\in\mathbb{Q}^{d}:0\leq x_{i}\leq 1;1\leq i\leq d\}
  5. L ( P , - t ) = ( - 1 ) d ( t - 1 ) d = ( - 1 ) d L ( int ( P ) , t ) , L(P,-t)=(-1)^{d}(t-1)^{d}=(-1)^{d}L(\,\text{int}(P),t),
  6. P P
  7. P = { x d : A x b } P=\{x\in\mathbb{Q}^{d}:Ax\leq b\}
  8. A k × d A\in\mathbb{R}^{k\times d}
  9. b k b\in\mathbb{Z}^{k}
  10. P P
  11. d \mathbb{Q}^{d}
  12. L ( P , t ) = # ( { x n : A x t b } ) . L(P,t)=\#(\{x\in\mathbb{Z}^{n}:Ax\leq tb\}).
  13. L ( int ( P ) , t ) = ( - 1 ) n L ( P , - t ) . L(\,\text{int}(P),t)=(-1)^{n}L(P,-t).
  14. L ( P , t ) = 7 4 t 2 + 5 2 t + 7 + ( - 1 ) t 8 . L(P,t)=\frac{7}{4}t^{2}+\frac{5}{2}t+\frac{7+(-1)^{t}}{8}.
  15. E h r P ( z ) = t 0 L ( P , t ) z t Ehr_{P}(z)=\sum_{t\geq 0}L(P,t)z^{t}
  16. h i h_{i}^{\ast}
  17. 0 j n 0\leq j\leq n
  18. E h r P ( z ) = j = 0 d h j z j ( 1 - z ) n + 1 , Ehr_{P}(z)=\frac{\sum_{j=0}^{d}h_{j}^{\ast}z^{j}}{(1-z)^{n+1}},
  19. j = 0 d h j 0 \sum_{j=0}^{d}h_{j}^{\ast}\neq 0
  20. h i h_{i}^{\ast}
  21. 0 j n . 0\leq j\leq n.
  22. E h r P ( z ) = t 0 L ( P , t ) z t = j = 0 d ( n + 1 ) h j z j ( 1 - z d ) n + 1 , Ehr_{P}(z)=\sum_{t\geq 0}L(P,t)z^{t}=\frac{\sum_{j=0}^{d(n+1)}h_{j}^{\ast}z^{j% }}{(1-z^{d})^{n+1}},

Eigenface.html

  1. n ( λ 1 + λ 2 + + λ k ) v > ϵ \frac{n(\lambda_{1}+\lambda_{2}+...+\lambda_{k})}{v}>\epsilon
  2. 𝐒𝐯 i = 𝐓𝐓 T 𝐯 i = λ i 𝐯 i \mathbf{Sv}_{i}=\mathbf{T}\mathbf{T}^{T}\mathbf{v}_{i}=\lambda_{i}\mathbf{v}_{i}
  3. 𝐓 T 𝐓𝐮 i = λ i 𝐮 i \mathbf{T}^{T}\mathbf{T}\mathbf{u}_{i}=\lambda_{i}\mathbf{u}_{i}
  4. 𝐓𝐓 T 𝐓𝐮 i = λ i 𝐓𝐮 i \mathbf{T}\mathbf{T}^{T}\mathbf{T}\mathbf{u}_{i}=\lambda_{i}\mathbf{T}\mathbf{% u}_{i}
  5. c o v a r i a n c e ( X ) = X X T n covariance(X)=\frac{XX^{T}}{n}
  6. X = U Σ V T X=U{\Sigma}V^{T}
  7. X X T XX^{T}
  8. X X T = U Σ Σ T U T = U Λ U T XX^{T}=U{\Sigma}{{\Sigma}^{T}}U^{T}=U{\Lambda}U^{T}
  9. X X T XX^{T}
  10. U U
  11. X X
  12. X X T = 1 n ( XX^{T}=\frac{1}{n}(
  13. X ) 2 X)^{2}
  14. U n U\in\Re^{n}
  15. M M
  16. w k = V k T ( U - M ) w_{k}=V_{k}^{T}(U-M)
  17. W = [ w 1 , w 2 , , w k , , w n ] W=[w_{1},w_{2},...,w_{k},...,w_{n}]
  18. W m W_{m}
  19. d = || W - W m || 2 d=||W-W_{m}||^{2}
  20. d < ϵ 1 d<\epsilon_{1}
  21. ϵ 1 < d < ϵ 2 \epsilon_{1}<d<\epsilon_{2}
  22. U U
  23. d > ϵ 2 , U d>\epsilon_{2},U

Eigenvalue_algorithm.html

  1. n × n n×n
  2. A A
  3. λ λ
  4. 𝐯 \mathbf{v}
  5. ( A - λ I ) k v = 0 , \left(A-\lambda I\right)^{k}{v}=0,
  6. 𝐯 \mathbf{v}
  7. n × 1 n×1
  8. I I
  9. n × n n×n
  10. k k
  11. λ λ
  12. 𝐯 \mathbf{v}
  13. A A
  14. k = 1 k=1
  15. A 𝐯 = λ 𝐯 A\mathbf{v}=λ\mathbf{v}
  16. λ λ
  17. A A
  18. k k
  19. 𝐯 \mathbf{v}
  20. k k
  21. n n
  22. 𝐯 \mathbf{v}
  23. λ . λ.
  24. λ λ
  25. A A
  26. k e r ( A - λ I ) ker(A-λI)
  27. λ λ
  28. λ λ
  29. λ λ
  30. λ λ
  31. p A ( z ) = det ( z I - A ) = i = 1 k ( z - λ i ) α i , p_{A}\left(z\right)={\rm det}\left(zI-A\right)=\prod_{i=1}^{k}(z-\lambda_{i})^% {\alpha_{i}},
  32. d e t det
  33. A A
  34. A A
  35. n n
  36. A A
  37. A A
  38. i j ( A - λ i I ) α i \textstyle\prod_{i\neq j}(A-\lambda_{i}I)^{\alpha_{i}}
  39. ( A - λ j I ) α j . \textstyle(A-\lambda_{j}I)^{\alpha_{j}}.
  40. k k
  41. i i
  42. j j
  43. V V
  44. A A
  45. V - 1 A V = [ λ 1 β 1 0 0 0 λ 2 β 2 0 0 0 λ 3 0 0 0 0 λ n ] , V^{-1}AV=\begin{bmatrix}\lambda_{1}&\beta_{1}&0&\ldots&0\\ 0&\lambda_{2}&\beta_{2}&\ldots&0\\ 0&0&\lambda_{3}&\ldots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\ldots&\lambda_{n}\end{bmatrix},
  46. W W
  47. λ λ
  48. A A
  49. 𝐯 \mathbf{v}
  50. λ λ
  51. M M
  52. M M
  53. A A
  54. A A
  55. A A
  56. A A
  57. A A
  58. 𝐯 \mathbf{v}
  59. A A
  60. A A
  61. A A
  62. x x
  63. κ ( ƒ , x ) κ(ƒ,x)
  64. A 𝐯 = 𝐛 A\mathbf{v}=\mathbf{b}
  65. A A
  66. 𝐛 \mathbf{b}
  67. A A
  68. κ ( A ) κ(A)
  69. A A
  70. κ ( A ) κ(A)
  71. A A
  72. A A
  73. κ ( A ) = 1 κ(A)=1
  74. λ λ
  75. n × n n×n
  76. A A
  77. V V
  78. λ λ
  79. κ ( V ) κ(V)
  80. A A
  81. λ λ
  82. A A
  83. V V
  84. κ ( λ , A ) = 1 κ(λ,A)=1
  85. A A
  86. λ λ
  87. λ λ
  88. A A
  89. λ λ
  90. A A
  91. λ λ
  92. A A
  93. A - μ I A-μI
  94. μ μ
  95. A - μ I A-μI
  96. μ μ
  97. A A
  98. μ = λ μ=λ
  99. λ λ
  100. μ μ
  101. λ λ
  102. A A
  103. A - λ I A-λI
  104. A A
  105. A - λ I A-λI
  106. μ μ
  107. μ μ
  108. A - λ [ u I ] I A-λ[u^{\prime}I^{\prime}]I
  109. λ [ u . ] . λ[u^{\prime}.^{\prime}].
  110. 2 n / < s u p > 3 + O ( n 2 ) 2{n}/{<sup>3}+O(n^{2})
  111. O ( n < s u p > 2 ) O(n<sup>2)
  112. p p
  113. p ( A ) = 0 , p(A)=0,
  114. A A
  115. p p
  116. A A
  117. P P
  118. P P
  119. P P
  120. A A
  121. α α
  122. ± α ±α
  123. P + = 1 2 ( I + A α ) P_{+}=\frac{1}{2}\left(I+\frac{A}{\alpha}\right)
  124. P - = 1 2 ( I - A α ) P_{-}=\frac{1}{2}\left(I-\frac{A}{\alpha}\right)
  125. A P + = α P + A P - = - α P - AP_{+}=\alpha P_{+}\quad AP_{-}=-\alpha P_{-}
  126. P + P + = P + P - P - = P - P + P - = P - P + = 0. P_{+}P_{+}=P_{+}\quad P_{-}P_{-}=P_{-}\quad P_{+}P_{-}=P_{-}P_{+}=0.
  127. A A
  128. + α
  129. - α
  130. A = [ a b c d ] , A=\begin{bmatrix}a&b\\ c&d\end{bmatrix},
  131. det [ λ - a - b - c λ - d ] = λ 2 - ( a + d ) λ + ( a d - b c ) = λ 2 - λ tr ( A ) + det ( A ) . {\rm det}\begin{bmatrix}\lambda-a&-b\\ -c&\lambda-d\end{bmatrix}=\lambda^{2}\,-\,\left(a+d\right)\lambda\,+\,\left(ad% -bc\right)=\lambda^{2}\,-\,\lambda\,{\rm tr}(A)\,+\,{\rm det}(A).
  132. λ = tr ( A ) ± tr 2 ( A ) - 4 d e t ( A ) 2 . \lambda=\frac{{\rm tr}(A)\pm\sqrt{{\rm tr}^{2}(A)-4{\rm det}(A)}}{2}.
  133. gap ( A ) = tr 2 ( A ) - 4 d e t ( A ) \textstyle{\rm gap}\left(A\right)=\sqrt{{\rm tr}^{2}(A)-4{\rm det}(A)}
  134. λ a = 1 2 ( 1 ± a - d gap ( A ) ) , λ b = ± c gap ( A ) \frac{\partial\lambda}{\partial a}=\frac{1}{2}\left(1\pm\frac{a-d}{{\rm gap}(A% )}\right),\qquad\frac{\partial\lambda}{\partial b}=\frac{\pm c}{{\rm gap}(A)}
  135. c c
  136. d d
  137. A A
  138. A = [ 4 3 - 2 - 3 ] , A=\begin{bmatrix}4&3\\ -2&-3\end{bmatrix},
  139. t r ( A ) = 4 - 3 = 1 tr(A)=4-3=1
  140. d e t ( A ) = 4 ( - 3 ) - 3 ( - 2 ) = - 6 det(A)=4(-3)-3(-2)=-6
  141. 0 = λ 2 - λ - 6 = ( λ - 3 ) ( λ + 2 ) , 0=\lambda^{2}-\lambda-6=(\lambda-3)(\lambda+2),
  142. A - 3 I = [ 1 3 - 2 - 6 ] , A + 2 I = [ 6 3 - 2 - 1 ] . A-3I=\begin{bmatrix}1&3\\ -2&-6\end{bmatrix},\qquad A+2I=\begin{bmatrix}6&3\\ -2&-1\end{bmatrix}.
  143. ( 1 , - 2 ) (1,-2)
  144. ( 3 , - 1 ) (3,-1)
  145. A A
  146. A A
  147. det ( α I - A ) = α 3 - α 2 tr ( A ) - α 1 2 ( tr ( A 2 ) - tr 2 ( A ) ) - det ( A ) = 0. {\rm det}\left(\alpha I-A\right)=\alpha^{3}-\alpha^{2}{\rm tr}(A)-\alpha\frac{% 1}{2}\left({\rm tr}(A^{2})-{\rm tr}^{2}(A)\right)-{\rm det}(A)=0.
  148. A A
  149. A = p B + q I A=pB+qI
  150. A A
  151. B B
  152. β β
  153. B B
  154. α = p β + q α=pβ+q
  155. A A
  156. q = tr ( A ) / 3 \textstyle q={\rm tr}(A)/3
  157. p = tr ( ( A - q I ) 2 / 6 ) 1 / 2 \textstyle p={\rm tr}\left((A-qI)^{2}/6\right)^{1/2}
  158. det ( β I - B ) = β 3 - 3 β - det ( B ) = 0. {\rm det}\left(\beta I-B\right)=\beta^{3}-3\beta-{\rm det}(B)=0.
  159. β = 2 c o s θ β=2cosθ
  160. c o s 3 θ = d e t ( B ) / 2 cos3θ=det(B)/2
  161. β = 2 c o s ( 1 3 arccos ( det ( B ) / 2 ) + 2 k π 3 ) , k = 0 , 1 , 2. \beta=2{\rm cos}\left(\frac{1}{3}{\rm arccos}\left({\rm det}(B)/2\right)+\frac% {2k\pi}{3}\right),\quad k=0,1,2.
  162. d e t ( B ) det(B)
  163. k k
  164. A A
  165. I I
  166. 𝐰 𝐯 = 𝐯 < s u p > * 𝐰 \mathbf{w•v}=\mathbf{v}<sup>*\mathbf{w}

Einstein_field_equations.html

  1. R μ ν R_{\mu\nu}\,
  2. g μ ν g_{\mu\nu}\,
  3. Λ \Lambda\,
  4. G G\,
  5. c c\,
  6. R R\,
  7. T μ ν T_{\mu\nu}\,
  8. g μ ν g_{\mu\nu}
  9. G μ ν = R μ ν - 1 2 R g μ ν , G_{\mu\nu}=R_{\mu\nu}-{1\over 2}Rg_{\mu\nu},
  10. G μ ν + g μ ν Λ = 8 π G c 4 T μ ν . G_{\mu\nu}+g_{\mu\nu}\Lambda={8\pi G\over c^{4}}T_{\mu\nu}.
  11. G μ ν + g μ ν Λ = 8 π T μ ν . G_{\mu\nu}+g_{\mu\nu}\Lambda=8\pi T_{\mu\nu}\,.
  12. g μ ν \displaystyle g_{\mu\nu}
  13. R μ ν = [ S 2 ] × [ S 3 ] × R α μ α ν R_{\mu\nu}=[S2]\times[S3]\times{R^{\alpha}}_{\mu\alpha\nu}
  14. ( + + + ) (+++)\,
  15. ( + - - ) (+--)\,
  16. ( - + + ) (-++)\,
  17. ( - + - ) (-+-)\,
  18. R μ ν - 1 2 g μ ν R - g μ ν Λ = - 8 π G c 4 T μ ν . R_{\mu\nu}-{1\over 2}g_{\mu\nu}\,R-g_{\mu\nu}\Lambda=-{8\pi G\over c^{4}}T_{% \mu\nu}.
  19. R - D 2 R + D Λ = 8 π G c 4 T R-\frac{D}{2}R+D\Lambda={8\pi G\over c^{4}}T\,
  20. D D
  21. - R + D Λ ( D / 2 - 1 ) = 8 π G c 4 T D / 2 - 1 . -R+\frac{D\Lambda}{(D/2-1)}={8\pi G\over c^{4}}\frac{T}{D/2-1}\,.
  22. - 1 2 g μ ν -{1\over{2}}g_{\mu\nu}\,
  23. R μ ν - g μ ν Λ D / 2 - 1 = 8 π G c 4 ( T μ ν - 1 D - 2 T g μ ν ) . R_{\mu\nu}-\frac{g_{\mu\nu}\Lambda}{D/2-1}={8\pi G\over c^{4}}\left(T_{\mu\nu}% -{1\over{D-2}}T\,g_{\mu\nu}\right).\,
  24. D = 4 D=4
  25. R μ ν - g μ ν Λ = 8 π G c 4 ( T μ ν - 1 2 T g μ ν ) . R_{\mu\nu}-g_{\mu\nu}\Lambda={8\pi G\over c^{4}}\left(T_{\mu\nu}-{1\over 2}T\,% g_{\mu\nu}\right).\,
  26. g μ ν g_{\mu\nu}\,
  27. Λ \Lambda
  28. R μ ν - 1 2 g μ ν R + g μ ν Λ = 8 π G c 4 T μ ν . R_{\mu\nu}-{1\over 2}g_{\mu\nu}\,R+g_{\mu\nu}\Lambda={8\pi G\over c^{4}}T_{\mu% \nu}\,.
  29. Λ \Lambda
  30. Λ \Lambda
  31. Λ \Lambda
  32. T μ ν ( vac ) = - Λ c 4 8 π G g μ ν . T_{\mu\nu}^{\mathrm{(vac)}}=-\frac{\Lambda c^{4}}{8\pi G}g_{\mu\nu}\,.
  33. ρ vac = Λ c 2 8 π G \rho_{\mathrm{vac}}=\frac{\Lambda c^{2}}{8\pi G}
  34. β T α β = T α β = ; β 0 \nabla_{\beta}T^{\alpha\beta}\,=T^{\alpha\beta}{}_{;\beta}\,=0
  35. R α β [ γ δ ; ε ] = 0 R_{\alpha\beta[\gamma\delta;\varepsilon]}=\,0
  36. g α γ g^{\alpha\gamma}
  37. g α β = ; γ 0 g^{\alpha\beta}{}_{;\gamma}=0
  38. R γ + β γ δ ; ε R γ + β ε γ ; δ R γ = β δ ε ; γ 0 R^{\gamma}{}_{\beta\gamma\delta;\varepsilon}+\,R^{\gamma}{}_{\beta\varepsilon% \gamma;\delta}+\,R^{\gamma}{}_{\beta\delta\varepsilon;\gamma}=\,0
  39. R γ - β γ δ ; ε R γ + β γ ε ; δ R γ = β δ ε ; γ 0 R^{\gamma}{}_{\beta\gamma\delta;\varepsilon}\,-R^{\gamma}{}_{\beta\gamma% \varepsilon;\delta}\,+R^{\gamma}{}_{\beta\delta\varepsilon;\gamma}\,=0
  40. R β δ ; ε - R β ε ; δ + R γ = β δ ε ; γ 0 R_{\beta\delta;\varepsilon}\,-R_{\beta\varepsilon;\delta}\,+R^{\gamma}{}_{% \beta\delta\varepsilon;\gamma}\,=0
  41. g β δ ( R β δ ; ε - R β ε ; δ + R γ ) β δ ε ; γ = 0 g^{\beta\delta}(R_{\beta\delta;\varepsilon}\,-R_{\beta\varepsilon;\delta}\,+R^% {\gamma}{}_{\beta\delta\varepsilon;\gamma})\,=0
  42. R δ - δ ; ε R δ + ε ; δ R γ δ = δ ε ; γ 0 R^{\delta}{}_{\delta;\varepsilon}\,-R^{\delta}{}_{\varepsilon;\delta}\,+R^{% \gamma\delta}{}_{\delta\varepsilon;\gamma}\,=0
  43. R ; ε - 2 R γ = ε ; γ 0 R_{;\varepsilon}\,-2R^{\gamma}{}_{\varepsilon;\gamma}\,=0
  44. ( R γ - ε 1 2 g γ R ε ) ; γ = 0 (R^{\gamma}{}_{\varepsilon}\,-\frac{1}{2}g^{\gamma}{}_{\varepsilon}R)_{;\gamma% }\,=0
  45. g ε δ g^{\varepsilon\delta}
  46. ( R γ δ - 1 2 g γ δ R ) ; γ = 0 (R^{\gamma\delta}\,-\frac{1}{2}g^{\gamma\delta}R)_{;\gamma}\,=0
  47. G α β = ; β 0 G^{\alpha\beta}{}_{;\beta}\,=0
  48. β T α β = T α β = ; β 0 \nabla_{\beta}T^{\alpha\beta}\,=T^{\alpha\beta}{}_{;\beta}\,=0
  49. Φ \Phi\!
  50. 2 Φ [ x , t ] = 4 π G ρ [ x , t ] \nabla^{2}\Phi[\vec{x},t]=4\pi G\rho[\vec{x},t]
  51. ρ \rho\!
  52. x ¨ [ t ] = - Φ [ x [ t ] , t ] . \ddot{\vec{x}}[t]=-\nabla\Phi[\vec{x}[t],t]\,.
  53. Φ , i i = 4 π G ρ \Phi_{,ii}=4\pi G\rho\,
  54. d 2 x i d t 2 = - Φ , i . \frac{d^{2}x^{i}}{{dt}^{2}}=-\Phi_{,i}\,.
  55. R μ ν = K ( T μ ν - 1 2 T g μ ν ) R_{\mu\nu}=K\left(T_{\mu\nu}-{1\over 2}Tg_{\mu\nu}\right)
  56. d 2 x α d τ 2 = - Γ β γ α d x β d τ d x γ d τ . \frac{d^{2}x^{\alpha}}{{d\tau}^{2}}=-\Gamma^{\alpha}_{\beta\gamma}\frac{dx^{% \beta}}{d\tau}\frac{dx^{\gamma}}{d\tau}\,.
  57. d x β d τ ( d t d τ , 0 , 0 , 0 ) \frac{dx^{\beta}}{d\tau}\approx\left(\frac{dt}{d\tau},0,0,0\right)
  58. d d t ( d t d τ ) 0 \frac{d}{dt}\left(\frac{dt}{d\tau}\right)\approx 0
  59. d 2 x i d t 2 - Γ 00 i \frac{d^{2}x^{i}}{{dt}^{2}}\approx-\Gamma^{i}_{00}
  60. d t d τ \frac{dt}{d\tau}
  61. Φ , i Γ 00 i = 1 2 g i α ( g α 0 , 0 + g 0 α , 0 - g 00 , α ) . \Phi_{,i}\approx\Gamma^{i}_{00}={1\over 2}g^{i\alpha}(g_{\alpha 0,0}+g_{0% \alpha,0}-g_{00,\alpha})\,.
  62. 2 Φ , i g i j ( - g 00 , j ) - g 00 , i 2\Phi_{,i}\approx g^{ij}(-g_{00,j})\approx-g_{00,i}\,
  63. g 00 - c 2 - 2 Φ . g_{00}\approx-c^{2}-2\Phi\,.
  64. R 00 = K ( T 00 - 1 2 T g 00 ) R_{00}=K\left(T_{00}-{1\over 2}Tg_{00}\right)
  65. T μ ν diag ( T 00 , 0 , 0 , 0 ) diag ( ρ c 4 , 0 , 0 , 0 ) . T_{\mu\nu}\approx\mathrm{diag}(T_{00},0,0,0)\approx\mathrm{diag}(\rho c^{4},0,% 0,0)\,.
  66. T = g α β T α β g 00 T 00 - 1 c 2 ρ c 4 = - ρ c 2 T=g^{\alpha\beta}T_{\alpha\beta}\approx g^{00}T_{00}\approx{-1\over c^{2}}\rho c% ^{4}=-\rho c^{2}\,
  67. K ( T 00 - 1 2 T g 00 ) K ( ρ c 4 - 1 2 ( - ρ c 2 ) ( - c 2 ) ) = 1 2 K ρ c 4 . K\left(T_{00}-{1\over 2}Tg_{00}\right)\approx K\left(\rho c^{4}-{1\over 2}(-% \rho c^{2})(-c^{2})\right)={1\over 2}K\rho c^{4}\,.
  68. R 00 = Γ 00 , ρ ρ - Γ ρ 0 , 0 ρ + Γ ρ λ ρ Γ 00 λ - Γ 0 λ ρ Γ ρ 0 λ . R_{00}=\Gamma^{\rho}_{00,\rho}-\Gamma^{\rho}_{\rho 0,0}+\Gamma^{\rho}_{\rho% \lambda}\Gamma^{\lambda}_{00}-\Gamma^{\rho}_{0\lambda}\Gamma^{\lambda}_{\rho 0}.
  69. R 00 Γ 00 , i i . R_{00}\approx\Gamma^{i}_{00,i}\,.
  70. Φ , i i Γ 00 , i i R 00 = K ( T 00 - 1 2 T g 00 ) 1 2 K ρ c 4 \Phi_{,ii}\approx\Gamma^{i}_{00,i}\approx R_{00}=K\left(T_{00}-{1\over 2}Tg_{0% 0}\right)\approx{1\over 2}K\rho c^{4}\,
  71. 1 2 K ρ c 4 = 4 π G ρ {1\over 2}K\rho c^{4}=4\pi G\rho\,
  72. K = 8 π G c 4 . K=\frac{8\pi G}{c^{4}}\,.
  73. T μ ν T_{\mu\nu}
  74. T μ ν = 0 T_{\mu\nu}=0
  75. R μ ν = 0 . R_{\mu\nu}=0\,.
  76. R μ ν = Λ D / 2 - 1 g μ ν . R_{\mu\nu}=\frac{\Lambda}{D/2-1}g_{\mu\nu}\,.
  77. R μ ν = 0 R_{\mu\nu}=0
  78. T μ ν T_{\mu\nu}
  79. T α β = - 1 μ 0 ( F α F ψ ψ + β 1 4 g α β F ψ τ F ψ τ ) T^{\alpha\beta}=\,-\frac{1}{\mu_{0}}\left(F^{\alpha}{}^{\psi}F_{\psi}{}^{\beta% }+{1\over 4}g^{\alpha\beta}F_{\psi\tau}F^{\psi\tau}\right)
  80. R α β - 1 2 R g α β + g α β Λ = 8 π G c 4 μ 0 ( F α F ψ ψ + β 1 4 g α β F ψ τ F ψ τ ) . R^{\alpha\beta}-{1\over 2}Rg^{\alpha\beta}+g^{\alpha\beta}\Lambda=\frac{8\pi G% }{c^{4}\mu_{0}}\left(F^{\alpha}{}^{\psi}F_{\psi}{}^{\beta}+{1\over 4}g^{\alpha% \beta}F_{\psi\tau}F^{\psi\tau}\right).
  81. F α β = ; β 0 F^{\alpha\beta}{}_{;\beta}\,=0
  82. F [ α β ; γ ] = 1 3 ( F α β ; γ + F β γ ; α + F γ α ; β ) = 1 3 ( F α β , γ + F β γ , α + F γ α , β ) = 0. F_{[\alpha\beta;\gamma]}=\frac{1}{3}\left(F_{\alpha\beta;\gamma}+F_{\beta% \gamma;\alpha}+F_{\gamma\alpha;\beta}\right)=\frac{1}{3}\left(F_{\alpha\beta,% \gamma}+F_{\beta\gamma,\alpha}+F_{\gamma\alpha,\beta}\right)=0.\!
  83. F α β = A α ; β - A β ; α = A α , β - A β , α F_{\alpha\beta}=A_{\alpha;\beta}-A_{\beta;\alpha}=A_{\alpha,\beta}-A_{\beta,% \alpha}\!
  84. det ( g ) = 1 24 ε α β γ δ ε κ λ μ ν g α κ g β λ g γ μ g δ ν \det(g)=\frac{1}{24}\varepsilon^{\alpha\beta\gamma\delta}\varepsilon^{\kappa% \lambda\mu\nu}g_{\alpha\kappa}g_{\beta\lambda}g_{\gamma\mu}g_{\delta\nu}\,
  85. g α κ = 1 6 ε α β γ δ ε κ λ μ ν g β λ g γ μ g δ ν / det ( g ) . g^{\alpha\kappa}=\frac{1}{6}\varepsilon^{\alpha\beta\gamma\delta}\varepsilon^{% \kappa\lambda\mu\nu}g_{\beta\lambda}g_{\gamma\mu}g_{\delta\nu}/\det(g)\,.

Einstein_ring.html

  1. θ E = 4 G M c 2 d L S d L d S , \theta_{E}=\sqrt{\frac{4GM}{c^{2}}\;\frac{d_{LS}}{d_{L}d_{S}}},
  2. G G
  3. M M
  4. c c
  5. d L d_{L}
  6. d S d_{S}
  7. d L S d_{LS}
  8. d L S d S - d L d_{LS}\neq d_{S}-d_{L}
  9. θ E \theta_{E}

Einstein–Cartan_theory.html

  1. M \mathcal{L}_{\mathrm{M}}
  2. G \mathcal{L}_{\mathrm{G}}
  3. G = 1 2 κ R | g | \mathcal{L}_{\mathrm{G}}=\frac{1}{2\kappa}R\sqrt{|g|}
  4. S = ( G + M ) d 4 x , S=\int\left(\mathcal{L}_{\mathrm{G}}+\mathcal{L}_{\mathrm{M}}\right)\,d^{4}x,
  5. g g
  6. κ \kappa
  7. 8 π G / c 4 8\pi G/c^{4}
  8. S S
  9. δ S = 0. \delta S=0.
  10. g a b g^{ab}
  11. δ G δ g a b - 1 2 P a b = 0 \frac{\delta\mathcal{L}_{\mathrm{G}}}{\delta g^{ab}}-\frac{1}{2}P_{ab}=0
  12. R a b - 1 2 R g a b = κ P a b R_{ab}-\frac{1}{2}Rg_{ab}=\kappa P_{ab}
  13. R a b R_{ab}
  14. P a b P_{ab}
  15. P a b P_{ab}
  16. T a b c {T^{ab}}_{c}
  17. δ G δ T a b c - 1 2 σ a b c = 0 \frac{\delta\mathcal{L}_{\mathrm{G}}}{\delta{T^{ab}}_{c}}-\frac{1}{2}{\sigma_{% ab}}^{c}=0
  18. T a b c + g a c T b d d - g b c T a d d = κ σ a b c {T_{ab}}^{c}+{g_{a}}^{c}{T_{bd}}^{d}-{g_{b}}^{c}{T_{ad}}^{d}=\kappa{\sigma_{ab% }}^{c}
  19. σ a b c {\sigma_{ab}}^{c}

Eisenstein's_criterion.html

  1. Q = a n x n + a n - 1 x n - 1 + + a 1 x + a 0 Q=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0}
  2. p p
  3. p p
  4. i n i≠n
  5. p p
  6. Q Q
  7. Q Q
  8. p p
  9. Q Q
  10. Q Q
  11. Q Q
  12. p p
  13. p p
  14. 15 15
  15. 10 10
  16. p = 5 p=5
  17. 5 5
  18. 3 3
  19. 25 25
  20. 10 10
  21. Q Q
  22. 𝐐 \mathbf{Q}
  23. 𝐙 \mathbf{Z}
  24. Q Q
  25. Q Q
  26. a a
  27. x + a x+a
  28. x x
  29. x x
  30. H H
  31. x + 3 x+3
  32. x x
  33. H H
  34. 7 7
  35. 𝐐 x x \mathbf{Q}xx
  36. H H
  37. x x
  38. R x x Rxx
  39. R R
  40. x x
  41. p = 2 p=2
  42. 𝐙 x x \mathbf{Z}xx
  43. p p
  44. x 1 x−1
  45. 1 1
  46. p > 2 p>2
  47. x p - 1 x - 1 = x p - 1 + x p - 2 + + x + 1. \frac{x^{p}-1}{x-1}=x^{p-1}+x^{p-2}+\cdots+x+1.
  48. H H
  49. 1 1
  50. p p
  51. x + 1 x+1
  52. x x
  53. ( x + 1 ) p - 1 x = x p - 1 + ( p p - 1 ) x p - 2 + + ( p 2 ) x + ( p 1 ) , \frac{(x+1)^{p}-1}{x}=x^{p-1}+{\left({{p}\atop{p-1}}\right)}x^{p-2}+\cdots+{% \left({{p}\atop{2}}\right)}x+{\left({{p}\atop{1}}\right)},
  54. p p
  55. p p
  56. p p
  57. p p
  58. ( x + 1 ) p - 1 x x p + 1 p - 1 x = x p x = x p - 1 ( mod p ) , \frac{(x+1)^{p}-1}{x}\equiv\frac{x^{p}+1^{p}-1}{x}=\frac{x^{p}}{x}=x^{p-1}\;\;% (\mathop{{\rm mod}}p),
  59. p p
  60. p p
  61. 1 1
  62. x + 1 x+1
  63. x x
  64. ( x a ) < s u p > n + p F ( x ) (x−a)<sup>n+pF(x)
  65. a a
  66. 0
  67. F ( x ) F(x)
  68. F ( a ) F(a)
  69. p p
  70. p F ( a ) pF(a)
  71. p p
  72. F ( x ) F(x)
  73. n n
  74. F ( x ) F(x)
  75. n n
  76. 𝐙 x x \mathbf{Z}xx
  77. F ( x ) F(x)
  78. x x
  79. 1 1
  80. m m
  81. ε m εm
  82. ε ε
  83. m m
  84. F ( x ) F(x)
  85. ( x μ + a 1 x μ - 1 + + a μ ) ( x ν + b 1 x ν - 1 + + b ν ) \left(x^{\mu}+a_{1}x^{\mu-1}+\cdots+a_{\mu}\right)\left(x^{\nu}+b_{1}x^{\nu-1}% +\cdots+b_{\nu}\right)
  86. μ , ν 1 μ,ν≥1
  87. μ + ν = d e g ( F ( x ) ) μ+ν=deg(F(x))
  88. a a
  89. b b
  90. F ( x ) = 0 F(x)=0
  91. Q Q
  92. p p
  93. 𝐐 x x \mathbf{Q}xx
  94. Q Q
  95. 𝐙 x x \mathbf{Z}xx
  96. Q = G H Q=GH
  97. G , H G,H
  98. Q Q
  99. Q / c Q/c
  100. c c
  101. Q Q
  102. c c
  103. Q Q
  104. Q = G H Q=GH
  105. p p
  106. ( 𝐙 / p 𝐙 ) x x (\mathbf{Z}/p\mathbf{Z})xx
  107. Q Q
  108. a 𝐙 / p 𝐙 a∈\mathbf{Z}/p\mathbf{Z}
  109. p p
  110. G G
  111. H H
  112. ( 𝐙 / p 𝐙 ) x x (\mathbf{Z}/p\mathbf{Z})xx
  113. G G
  114. H H
  115. p p
  116. Q Q
  117. p p
  118. p p
  119. p p
  120. ( n , 0 ) (n,0)
  121. 1 / n −1/n
  122. Q Q
  123. p p
  124. 1 / n 1/n
  125. Q Q
  126. p p
  127. p p
  128. p p
  129. p p
  130. p p
  131. 𝐐 \mathbf{Q}
  132. Q Q
  133. Q Q
  134. 7 −7
  135. 7 7
  136. 7 7
  137. 0 m o d 7 0mod7
  138. p p
  139. p p
  140. p p
  141. p p
  142. D D
  143. Q = i = 0 n a i x i Q=\sum_{i=0}^{n}a_{i}x^{i}
  144. D x x Dxx
  145. D D
  146. 𝐩 \mathbf{p}
  147. D D
  148. a < s u b > i 𝐩 a<sub>i∈\mathbf{p}

Ejection_fraction.html

  1. 70 / 120 {70}/{120}
  2. d ( v o l u m e ) d ( p r e s s u r e ) \frac{d(volume)}{d(pressure)}
  3. E f ( % ) = S V E D V × 100 E_{f}(\%)=\frac{SV}{EDV}\times 100
  4. S V = E D V - E S V SV=EDV-ESV

Ekman_number.html

  1. Ek = ν 2 D 2 Ω sin φ \mathrm{Ek}=\frac{\nu}{2D^{2}\Omega\sin\varphi}
  2. Ek = ν Ω L 2 . \mathrm{Ek}=\frac{\nu}{\Omega L^{2}}.
  3. Ek = ν 2 Ω L 2 = Ro Re . \mathrm{Ek}=\sqrt{\frac{\nu}{2\Omega L^{2}}}=\sqrt{\frac{\mathrm{Ro}}{\mathrm{% Re}}}.

Elastic_modulus.html

  1. λ = def stress strain \lambda\ \stackrel{\,\text{def}}{=}\ \frac{\,\text{stress}}{\,\text{strain}}
  2. μ \mu\,

Electric_field_gradient.html

  1. V i j = 2 V x i x j . V_{ij}=\frac{\partial^{2}V}{\partial x_{i}\partial x_{j}}.
  2. φ i j = V i j - 1 3 δ i j 2 V , \varphi_{ij}=V_{ij}-\frac{1}{3}\delta_{ij}\nabla^{2}V,
  3. η = V x x - V y y V z z . \eta=\frac{V_{xx}-V_{yy}}{V_{zz}}.
  4. | V z z | | V y y | | V x x | |V_{zz}|\geq|V_{yy}|\geq|V_{xx}|
  5. V z z + V y y + V x x = 0 V_{zz}+V_{yy}+V_{xx}=0
  6. 0 η 1 0\leq\eta\leq 1

Electrical_efficiency.html

  1. Efficiency = Useful power output Total power input \mathrm{Efficiency}=\frac{\mathrm{Useful\ power\ output}}{\mathrm{Total\ power% \ input}}

Electro-osmosis.html

  1. 𝐔 = 0 \nabla\cdot\mathbf{U}=0
  2. ρ D 𝐔 D t = - p + μ 2 𝐔 + ρ e ( ψ + ϕ ) \rho\frac{D\mathbf{U}}{Dt}=-\nabla p+\mu\nabla^{2}\mathbf{U}+\rho_{e}\nabla% \left(\psi+\phi\right)
  3. 𝐔 \mathbf{U}
  4. ρ ρ
  5. D / D t D/Dt
  6. μ μ
  7. Φ Φ
  8. ψ ψ
  9. 2 ϕ = 0 \nabla^{2}\phi=0
  10. 2 ψ = - ρ e ϵ ϵ 0 \nabla^{2}\psi=\frac{-\rho_{e}}{\epsilon\epsilon_{0}}
  11. ε ε
  12. 2 ψ = k 2 ψ \nabla^{2}\psi=k^{2}\psi
  13. 1 / k 1/k
  14. ρ e = - ϵ ϵ 0 k 2 ψ \rho_{e}=-\epsilon\epsilon_{0}k^{2}\psi\,
  15. K + K^{+}

Electrodynamic_suspension.html

  1. = - N d Φ B d t \mathcal{E}=-N{{d\Phi_{B}}\over dt}
  2. K = R + i ω L K=R+i\omega L\,
  3. = I K \mathcal{E}=IK

Electromigration.html

  1. F r e s = F e - F p = q Z * E = q Z * j ρ F_{res}=F_{e}-F_{p}=q\cdot Z^{*}\cdot E=q\cdot Z^{*}\cdot j\cdot\rho
  2. N t + J = 0 \frac{\partial N}{\partial t}+\nabla\cdot\vec{J}=0
  3. N ( x , t ) N(\vec{x},t)
  4. x = ( x , y , z ) \vec{x}=(x,y,z)
  5. t t
  6. J J
  7. J J
  8. J = J c + J T + J σ + J N \vec{J}=\vec{J}_{c}+\vec{J}_{T}+\vec{J}_{\sigma}+\vec{J}_{N}
  9. J c = N e Z D ρ k T j \vec{J}_{c}=\frac{NeZD\rho}{kT}\vec{j}
  10. e e
  11. e Z eZ
  12. ρ \rho
  13. j \vec{j}
  14. k k
  15. T T
  16. D ( x , t ) D(\vec{x},t)
  17. J T = - N D Q k T 2 T \vec{J}_{T}=-\frac{NDQ}{kT^{2}}\nabla T
  18. Q Q
  19. J σ = N D Ω k T H \vec{J}_{\sigma}=\frac{ND\Omega}{kT}\nabla H
  20. Ω = 1 / N 0 \Omega=1/N_{0}
  21. N 0 N_{0}
  22. H = ( σ 11 + σ 22 + σ 33 ) / 3 H=(\sigma_{11}+\sigma_{22}+\sigma_{33})/3
  23. σ 11 , σ 22 , σ 33 \sigma_{11},\sigma_{22},\sigma_{33}
  24. J N = - D N \vec{J}_{N}=-D\nabla N
  25. D D
  26. D = D 0 exp ( Ω H - E A k T ) D=D_{0}\exp(\frac{\Omega H-E_{A}}{kT})
  27. E A E_{A}
  28. MTTF = ( A / ( J n ) ) e E a k T \,\text{MTTF}=(A/(J^{n}))e^{\frac{E_{a}}{kT}}
  29. A A
  30. J J
  31. E a E_{a}
  32. k k
  33. T T
  34. n n

Electron_cyclotron_resonance.html

  1. ω c e = e B m \omega_{ce}=\frac{eB}{m}
  2. e e
  3. m m
  4. ω c e = q B γ m 0 \omega_{ce}=\frac{qB}{\gamma\cdot m_{0}}
  5. γ = 1 1 - ( v / c ) 2 \gamma=\frac{1}{\sqrt{1-(v/c)^{2}}}
  6. m * \begin{Vmatrix}m^{*}\end{Vmatrix}
  7. ω c e > 1 / τ ω c e > k B T \begin{matrix}\omega_{ce}>1/\tau\\ \hbar\omega_{ce}>k_{B}T\\ \end{matrix}
  8. τ \tau
  9. k B k_{B}
  10. T T

Electron_ionization.html

  1. M + e - M + + 2 e - , M+e^{-}\to M^{+\bullet}+2e^{-},

Electron_wave-packet_interference.html

  1. ψ ( x , y ) \psi\left(x,y\right)
  2. | ψ ( x , y ) | 2 |\psi\left(x,y\right)|^{2}

Electrophile.html

  1. ω = χ 2 2 η \omega=\frac{\chi^{2}}{2\eta}\,
  2. χ \chi\,
  3. η \eta\,
  4. P = V 2 R P=\frac{V^{2}}{R}\,
  5. R R\,
  6. V V\,

Electrophoresis.html

  1. F t o t = 0 = F e l + F f + F r e t F_{tot}=0=F_{el}+F_{f}+F_{ret}
  2. μ e = v E . \mu_{e}={v\over E}.
  3. μ e = ε r ε 0 ζ η \mu_{e}=\frac{\varepsilon_{r}\varepsilon_{0}\zeta}{\eta}
  4. a κ 1 a\kappa\gg 1
  5. D u 1 Du\ll 1
  6. a κ < 1 a\kappa<\!\,1
  7. μ e = 2 ε r ε 0 ζ 3 η \mu_{e}=\frac{2\varepsilon_{r}\varepsilon_{0}\zeta}{3\eta}

Electrostatics.html

  1. q q
  2. Q Q
  3. q q
  4. Q Q
  5. r r
  6. F = 1 4 π ε 0 q Q r 2 = k e q Q r 2 , F=\frac{1}{4\pi\varepsilon_{0}}\frac{qQ}{r^{2}}=k_{e}\frac{qQ}{r^{2}}\,,
  7. ε 0 = 10 - 9 36 π C 2 N - 1 m - 2 8.854 187 817 × 10 - 12 C 2 N - 1 m - 2 . \varepsilon_{0}={10^{-9}\over 36\pi}\;\;\mathrm{C^{2}\ N^{-1}\ m^{-2}}\approx 8% .854\ 187\ 817\times 10^{-12}\;\;\mathrm{C^{2}\ N^{-1}\ m^{-2}}.
  8. k e 1 4 π ε 0 8.987 551 787 × 10 9 N m 2 C - 2 . k_{e}\approx\frac{1}{4\pi\varepsilon_{0}}\approx 8.987\ 551\ 787\times 10^{9}% \;\;\mathrm{N\ m^{2}\ C}^{-2}.
  9. e 1.602 176 565 × 10 - 19 C . e\approx 1.602\ 176\ 565\times 10^{-19}\;\;\mathrm{C}.
  10. E \vec{E}
  11. F = q E . \vec{F}=q\vec{E}.\,
  12. N N
  13. Q i Q_{i}
  14. r i \vec{r}_{i}
  15. r \vec{r}
  16. E ( r ) = 1 4 π ε 0 i = 1 N ^ i Q i R i 2 , \vec{E}(\vec{r})=\frac{1}{4\pi\varepsilon_{0}}\sum_{i=1}^{N}\frac{\widehat{% \mathcal{R}}_{i}Q_{i}}{\left\|\mathcal{\vec{}}R_{i}\right\|^{2}},
  17. i = r - r i , \vec{\mathcal{R}}_{i}=\vec{r}-\vec{r}_{i},
  18. r i \vec{r}_{i}
  19. r \vec{r}
  20. ^ i = i / i \widehat{\mathcal{R}}_{i}=\vec{\mathcal{R}}_{i}/\left\|\vec{\mathcal{R}}_{i}\right\|
  21. E = k e Q / 2 , E=k_{e}Q/\mathcal{R}^{2},
  22. ρ ( r ) \rho(\vec{r})
  23. E ( r ) = 1 4 π ε 0 r - r r - r 3 ρ ( r ) d 3 r \vec{E}(\vec{r})=\frac{1}{4\pi\varepsilon_{0}}\iiint\frac{\vec{r}-\vec{r}\,^{% \prime}}{\left\|\vec{r}-\vec{r}\,^{\prime}\right\|^{3}}\rho(\vec{r}\,^{\prime}% )\operatorname{d}^{3}r\,^{\prime}
  24. S E d A = 1 ε 0 Q e n c l o s e d = V ρ ε 0 d 3 r , \oint_{S}\vec{E}\cdot\mathrm{d}\vec{A}=\frac{1}{\varepsilon_{0}}\,Q_{enclosed}% =\int_{V}{\rho\over\varepsilon_{0}}\cdot\operatorname{d}^{3}r,
  25. d 3 r = d x d y d z \operatorname{d}^{3}r=\mathrm{d}x\ \mathrm{d}y\ \mathrm{d}z
  26. ρ d 3 r \rho\mathrm{d}^{3}r
  27. σ d A \sigma\mathrm{d}A
  28. λ d \lambda\mathrm{d}\ell
  29. E = ρ ε 0 . \vec{\nabla}\cdot\vec{E}={\rho\over\varepsilon_{0}}.
  30. \vec{\nabla}\cdot
  31. 2 ϕ = - ρ ε 0 . {\nabla}^{2}\phi=-{\rho\over\varepsilon_{0}}.
  32. 2 ϕ = 0 , {\nabla}^{2}\phi=0,
  33. × E = 0. \vec{\nabla}\times\vec{E}=0.
  34. B t = 0. {\partial\vec{B}\over\partial t}=0.
  35. ϕ \phi
  36. E E
  37. E = - ϕ . \vec{E}=-\vec{\nabla}\phi.
  38. a a
  39. b b
  40. - a b E d = ϕ ( b ) - ϕ ( a ) . -\int_{a}^{b}{\vec{E}\cdot\mathrm{d}\vec{\ell}}=\phi(\vec{b})-\phi(\vec{a}).
  41. U E single U_{\mathrm{E}}^{\,\text{single}}
  42. q n E d q_{n}\vec{E}\cdot\mathrm{d}\vec{\ell}
  43. N N
  44. Q n Q_{n}
  45. r i \vec{r}_{i}
  46. U E single = q ϕ ( r ) = q 4 π ε 0 i = 1 N Q i 𝒾 U_{\mathrm{E}}^{\,\text{single}}=q\phi(\vec{r})=\frac{q}{4\pi\varepsilon_{0}}% \sum_{i=1}^{N}\frac{Q_{i}}{\left\|\mathcal{\vec{R}_{i}}\right\|}
  47. 𝒾 = r - r i \vec{\mathcal{R_{i}}}=\vec{r}-\vec{r}_{i}
  48. Q i Q_{i}
  49. q q
  50. r \vec{r}
  51. ϕ ( r ) \phi(\vec{r})
  52. r \vec{r}
  53. k e Q 1 Q 2 / r k_{e}Q_{1}Q_{2}/r
  54. U E total = 1 4 π ε 0 j = 1 N Q j i = 1 j - 1 Q i r i j = 1 2 i = 1 N Q i ϕ i , U_{\mathrm{E}}^{\,\text{total}}=\frac{1}{4\pi\varepsilon_{0}}\sum_{j=1}^{N}Q_{% j}\sum_{i=1}^{j-1}\frac{Q_{i}}{r_{ij}}=\frac{1}{2}\sum_{i=1}^{N}Q_{i}\phi_{i},
  55. ϕ i = 1 4 π ε 0 j = 1 ( j i ) N Q j r i j . \phi_{i}=\frac{1}{4\pi\varepsilon_{0}}\sum_{j=1(j\neq i)}^{N}\frac{Q_{j}}{r_{% ij}}.
  56. ϕ i \phi_{i}
  57. r i \vec{r}_{i}
  58. Q i Q_{i}
  59. ( ) ( ) ρ d 3 r \sum(\cdots)\rightarrow\int(\cdots)\rho\mathrm{d}^{3}r
  60. U E total = 1 2 ρ ( r ) ϕ ( r ) d 3 r = ε 0 2 | 𝐄 | 2 d 3 r U_{\mathrm{E}}^{\,\text{total}}=\frac{1}{2}\int\rho(\vec{r})\phi(\vec{r})% \operatorname{d}^{3}r=\frac{\varepsilon_{0}}{2}\int\left|{\mathbf{E}}\right|^{% 2}\operatorname{d}^{3}r
  61. 1 2 ρ ϕ \frac{1}{2}\rho\phi
  62. ε 0 2 E 2 \frac{\varepsilon_{0}}{2}E^{2}
  63. P = ε 0 2 E 2 P=\frac{\varepsilon_{0}}{2}E^{2}

Electrowetting.html

  1. γ w s \gamma_{ws}\,
  2. γ w s 0 \gamma_{ws}^{0}\,
  3. γ s \gamma_{s}\,
  4. γ w \gamma_{w}\,
  5. θ \theta
  6. C C
  7. V V
  8. γ w s = γ w s 0 - C V 2 2 \gamma_{ws}=\gamma_{ws}^{0}-\frac{CV^{2}}{2}\,
  9. γ w s \gamma_{ws}
  10. γ w s = γ s - γ w cos ( θ ) \gamma_{ws}=\gamma_{s}-\gamma_{w}\cos(\theta)\,
  11. cos θ = ( γ s - γ w s 0 + C V 2 2 γ w ) \cos\theta=\left(\frac{\gamma_{s}-\gamma_{ws}^{0}+\frac{CV^{2}}{2}}{\gamma_{w}% }\right)\,

Elias_M._Stein.html

  1. H 1 H^{1}
  2. B M O BMO

Elliptic_Curve_Digital_Signature_Algorithm.html

  1. 2 80 2^{80}
  2. 4 t 4t
  3. t t
  4. ( CURVE , G , n ) (\textrm{CURVE},G,n)
  5. G G
  6. n n
  7. G G
  8. n × G = O n\times G=O
  9. d A d_{A}
  10. [ 1 , n - 1 ] [1,n-1]
  11. Q A = d A × G Q_{A}=d_{A}\times G
  12. × \times
  13. m m
  14. e = HASH ( m ) e=\textrm{HASH}(m)
  15. z z
  16. L n L_{n}
  17. e e
  18. L n L_{n}
  19. n n
  20. k k
  21. [ 1 , n - 1 ] [1,n-1]
  22. ( x 1 , y 1 ) = k × G (x_{1},y_{1})=k\times G
  23. r = x 1 mod n r=x_{1}\,\bmod\,n
  24. r = 0 r=0
  25. s = k - 1 ( z + r d A ) mod n s=k^{-1}(z+rd_{A})\,\bmod\,n
  26. s = 0 s=0
  27. ( r , s ) (r,s)
  28. s s
  29. z z
  30. HASH ( m ) \textrm{HASH}(m)
  31. z z
  32. n n
  33. k k
  34. d A d_{A}
  35. ( r , s ) (r,s)
  36. ( r , s ) (r,s^{\prime})
  37. k k
  38. m m
  39. m m^{\prime}
  40. z z
  41. z z^{\prime}
  42. s - s = k - 1 ( z - z ) s-s^{\prime}=k^{-1}(z-z^{\prime})
  43. n n
  44. k = z - z s - s k=\frac{z-z^{\prime}}{s-s^{\prime}}
  45. s = k - 1 ( z + r d A ) s=k^{-1}(z+rd_{A})
  46. d A = s k - z r d_{A}=\frac{sk-z}{r}
  47. k k
  48. k k
  49. k k
  50. Q A Q_{A}
  51. Q A Q_{A}
  52. Q A Q_{A}
  53. O O
  54. Q A Q_{A}
  55. n × Q A = O n\times Q_{A}=O
  56. r r
  57. s s
  58. [ 1 , n - 1 ] [1,n-1]
  59. e = HASH ( m ) e=\textrm{HASH}(m)
  60. z z
  61. L n L_{n}
  62. e e
  63. w = s - 1 mod n w=s^{-1}\,\bmod\,n
  64. u 1 = z w mod n u_{1}=zw\,\bmod\,n
  65. u 2 = r w mod n u_{2}=rw\,\bmod\,n
  66. ( x 1 , y 1 ) = u 1 × G + u 2 × Q A (x_{1},y_{1})=u_{1}\times G+u_{2}\times Q_{A}
  67. r x 1 ( mod n ) r\equiv x_{1}\;\;(\mathop{{\rm mod}}n)
  68. u 1 × G + u 2 × Q A u_{1}\times G+u_{2}\times Q_{A}
  69. C C
  70. C = u 1 × G + u 2 × Q A C=u_{1}\times G+u_{2}\times Q_{A}
  71. Q A = d A × G Q_{A}=d_{A}\times G
  72. C = u 1 × G + u 2 d A × G C=u_{1}\times G+u_{2}d_{A}\times G
  73. C = ( u 1 + u 2 d A ) × G C=(u_{1}+u_{2}d_{A})\times G
  74. u 1 u_{1}
  75. u 2 u_{2}
  76. C = ( z s - 1 + r d A s - 1 ) × G C=(zs^{-1}+rd_{A}s^{-1})\times G
  77. s - 1 s^{-1}
  78. C = ( z + r d A ) s - 1 × G C=(z+rd_{A})s^{-1}\times G
  79. s s
  80. C = ( z + r d A ) ( z + r d A ) - 1 ( k - 1 ) - 1 × G C=(z+rd_{A})(z+rd_{A})^{-1}(k^{-1})^{-1}\times G
  81. C = k × G C=k\times G
  82. r r
  83. k k
  84. d A d_{A}

Elliptic_operator.html

  1. Ω \Omega
  2. L u = | α | m a α ( x ) α u Lu=\sum_{|\alpha|\leq m}a_{\alpha}(x)\partial^{\alpha}u\,
  3. α = ( α 1 , , α n ) \alpha=(\alpha_{1},...,\alpha_{n})
  4. α u = α 1 α n u \partial^{\alpha}u=\partial^{\alpha_{1}}\cdots\partial^{\alpha_{n}}u
  5. Ω \Omega
  6. ξ \xi
  7. | α | = m a α ( x ) ξ α 0 , \sum_{|\alpha|=m}a_{\alpha}(x)\xi^{\alpha}\neq 0,\,
  8. ξ α = ξ 1 α 1 ξ n α n \xi^{\alpha}=\xi_{1}^{\alpha_{1}}\cdots\xi_{n}^{\alpha_{n}}
  9. ( - 1 ) k | α | = 2 k a α ( x ) ξ α > C | ξ | 2 k , (-1)^{k}\sum_{|\alpha|=2k}a_{\alpha}(x)\xi^{\alpha}>C|\xi|^{2k},\,
  10. L ( u ) = F ( x , u , ( α u ) | α | 2 k ) L(u)=F(x,u,(\partial^{\alpha}u)_{|\alpha|\leq 2k})\,
  11. - Δ u = - i = 1 d i 2 u -\Delta u=-\sum_{i=1}^{d}\partial_{i}^{2}u\,
  12. - Δ Φ = 4 π ρ . -\Delta\Phi=4\pi\rho.\,
  13. L u = - i ( a i j ( x ) j u ) + b j ( x ) j u + c u Lu=-\partial_{i}(a^{ij}(x)\partial_{j}u)+b^{j}(x)\partial_{j}u+cu\,
  14. L ( u ) = - i = 1 d i ( | u | p - 2 i u ) . L(u)=-\sum_{i=1}^{d}\partial_{i}(|\nabla u|^{p-2}\partial_{i}u).\,
  15. τ i j = B ( k , l = 1 3 ( l u k ) 2 ) - 1 3 1 2 ( j u i + i u j ) \tau_{ij}=B\left(\sum_{k,l=1}^{3}(\partial_{l}u_{k})^{2}\right)^{-\frac{1}{3}}% \cdot\frac{1}{2}(\partial_{j}u_{i}+\partial_{i}u_{j})\,
  16. j = 1 3 j τ i j + ρ g i - i p = Q , \sum_{j=1}^{3}\partial_{j}\tau_{ij}+\rho g_{i}-\partial_{i}p=Q,\,
  17. f f
  18. f f
  19. D D
  20. σ ξ ( D ) \sigma_{\xi}(D)
  21. ξ \xi
  22. \nabla
  23. ξ \xi
  24. D D
  25. σ ξ ( D ) \sigma_{\xi}(D)
  26. ξ \xi
  27. D D
  28. c > 0 c>0
  29. ( [ σ ξ ( D ) ] ( v ) , v ) c v 2 ([\sigma_{\xi}(D)](v),v)\geq c\|v\|^{2}
  30. ξ = 1 \|\xi\|=1
  31. v v
  32. ( , ) (\cdot,\cdot)
  33. ξ \xi
  34. v v
  35. D D
  36. D D
  37. ξ \xi

Embree–Trefethen_constant.html

  1. x n + 1 = x n ± β x n - 1 x_{n+1}=x_{n}\pm\beta x_{n-1}\,
  2. σ ( β ) = lim n ( | x n | 1 / n ) \sigma(\beta)=\lim_{n\to\infty}(|x_{n}|^{1/n})\,

Endergonic_reaction.html

  1. Δ G > 0 \Delta G^{\circ}>0
  2. K = e - Δ G R T K=e^{-\frac{\Delta G^{\circ}}{RT}}
  3. K < 1 K<1\,
  4. X + Y X Y X+Y\longrightarrow XY
  5. X + 𝐴𝑇𝑃 𝑋𝑃 + 𝐴𝐷𝑃 X+\mathit{ATP}\longrightarrow\mathit{XP}+\mathit{ADP}
  6. 𝑋𝑃 + Y 𝑋𝑌 + P i \mathit{XP}+Y\longrightarrow\mathit{XY}+P_{i}

Endogenous_growth_theory.html

  1. Y = A K Y=AK\,
  2. A A\,
  3. K K\,
  4. A > 0 A>0\,
  5. y = A k y=Ak\,
  6. k k
  7. y y
  8. f ( k ) k = A \frac{f(k)}{k}=A\,
  9. k k\,
  10. γ K = k ˙ / k = s . f ( k ) / k - ( n + δ ) , \gamma_{K}=\dot{k}/k=s.f(k)/k-(n+\delta)\ ,
  11. A A\,
  12. γ K = s A - ( n + δ ) , \gamma_{K}=sA-(n+\delta)\ ,
  13. x = 0 x=0\,
  14. s A sA\,
  15. γ K \gamma_{K}\,
  16. s A > sA>\,
  17. γ K > 0 \gamma_{K}>0\,
  18. γ K \gamma_{K}\,
  19. K K\,
  20. K K\,
  21. γ K * = s A - ( n + δ ) , \gamma_{K}^{*}=sA-(n+\delta)\ ,
  22. y = A K y=AK\,
  23. γ K \gamma_{K}\,
  24. γ K * \gamma_{K}^{*}\,
  25. c = ( 1 - s ) y c=(1-s)y\,
  26. c c\,
  27. γ K * \gamma_{K}^{*}\,
  28. γ * = s A - ( n + δ ) , \gamma^{*}=sA-(n+\delta)\ ,
  29. y = A K y=AK\,
  30. γ * \gamma^{*}\,

Energy_development.html

  1. E = E=
  2. m m
  3. c 2 \cdot c^{2}

Enneadecagon.html

  1. R = t 2 csc 180 19 R=\frac{t}{2}\csc\frac{180}{19}
  2. 19 4 t 2 cot π 19 28.4652 t 2 . \frac{19}{4}t^{2}\cot\frac{\pi}{19}\simeq 28.4652\,t^{2}.
  3. \scriptstyle\angle{}
  4. \scriptstyle\angle{}

Envelope_detector.html

  1. x ( t ) x(t)
  2. x ( t ) = R ( t ) cos ( ω t + ϕ ( t ) ) x(t)=R(t)\cos(\omega t+\phi(t))\,
  3. ω \omega
  4. x ( t ) = ( C + m ( t ) ) cos ( ω t ) x(t)=(C+m(t))\cos(\omega t)\,
  5. x ( t ) x(t)

Epicyclic_gearing.html

  1. Ns ω s + Np ω p - ( Ns + Np ) ω c \displaystyle\,\text{N}\text{s}\omega\text{s}+\,\text{N}\text{p}\omega\text{p}% -(\,\text{N}\text{s}+\,\text{N}\text{p})\omega\text{c}
  2. ω a , ω s , ω p , ω c {\omega\text{a}},{\omega\text{s}},{\omega\text{p}},{\omega\text{c}}
  3. Na , Ns , Np {\,\text{N}\text{a}},{\,\text{N}\text{s}},{\,\text{N}\text{p}}
  4. Ns ω s + Na ω a = ( Ns + Na ) ω c \,\text{N}\text{s}\omega\text{s}+\,\text{N}\text{a}\omega\text{a}=(\,\text{N}% \text{s}+\,\text{N}\text{a})\omega\text{c}
  5. - Na Ns = ω s - ω c ω a - ω c -\frac{\,\text{N}\text{a}}{\,\text{N}\text{s}}=\frac{\omega\text{s}-\omega% \text{c}}{\omega\text{a}-\omega\text{c}}
  6. ω a ω c \omega\text{a}\neq\omega\text{c}
  7. N a = N s + 2 N p N\text{a}=N\text{s}+2N\text{p}
  8. n ω s + ( 2 + n ) ω a - 2 ( 1 + n ) ω c = 0 n\omega\text{s}+(2+n)\omega\text{a}-2(1+n)\omega\text{c}=0
  9. n = N s / N p n=N\text{s}/N\text{p}
  10. s {}_{s}
  11. p {}_{p}
  12. a {}_{a}
  13. p {}_{p}
  14. a {}_{a}
  15. - N s / N p -N\text{s}/N\text{p}
  16. N p / N a N\text{p}/N\text{a}
  17. - N s / N a -N\text{s}/N\text{a}
  18. a {}_{a}
  19. s {}_{s}
  20. a {}_{a}
  21. s {}_{s}
  22. s {}_{s}
  23. a {}_{a}
  24. a {}_{a}
  25. R = ω s ω a = - N a N s . R=\frac{\omega_{s}}{\omega_{a}}=-\frac{N_{a}}{N_{s}}.
  26. R = ω s - ω c ω a - ω c . R=\frac{\omega_{s}-\omega_{c}}{\omega_{a}-\omega_{c}}.
  27. ω s ω a = R , so ω s ω a = - N a N s . \frac{\omega_{s}}{\omega_{a}}=R,\quad\mbox{so}~{}\quad\frac{\omega_{s}}{\omega% _{a}}=-\frac{N_{a}}{N_{s}}.
  28. ω s - ω c - ω c = R , or ω s ω c = 1 - R , so ω s ω c = 1 + N a N s . \frac{\omega_{s}-\omega_{c}}{-\omega_{c}}=R,\quad\mbox{or}~{}\quad\frac{\omega% _{s}}{\omega_{c}}=1-R,\quad\mbox{so}~{}\quad\frac{\omega_{s}}{\omega_{c}}=1+% \frac{N_{a}}{N_{s}}.
  29. - ω c ω a - ω c = R , or ω a ω c = 1 - 1 R , so ω a ω c = 1 + N s N a . \frac{-\omega_{c}}{\omega_{a}-\omega_{c}}=R,\quad\mbox{or}~{}\quad\frac{\omega% _{a}}{\omega_{c}}=1-\frac{1}{R},\quad\mbox{so}~{}\quad\frac{\omega_{a}}{\omega% _{c}}=1+\frac{N_{s}}{N_{a}}.
  30. ω s - ω c ω a - ω c = - 1 , \frac{\omega_{s}-\omega_{c}}{\omega_{a}-\omega_{c}}=-1,
  31. ω c = 1 2 ( ω s + ω a ) . \omega_{c}=\frac{1}{2}(\omega_{s}+\omega_{a}).
  32. ( R - 1 ) ω c = R ω a - ω s (R-1)\omega\text{c}=R\omega\text{a}-\omega\text{s}
  33. R = N a / N s R=N\text{a}/N\text{s}
  34. ω a = ω s ( 1 / R ) \omega\text{a}=\omega\text{s}(1/R)
  35. ω a = ω c ( R - 1 ) / R \omega\text{a}=\omega\text{c}(R-1)/R
  36. ω s = - ω c ( R - 1 ) \omega\text{s}=-\omega\text{c}(R-1)

Equation_of_time.html

  1. 1 / 3 {1}/{3}
  2. 5 / 9 {5}/{9}
  3. EOT = GHA - GMHA \mbox{EOT}~{}=\mbox{GHA}~{}-\mbox{GMHA}~{}
  4. π π
  5. ° °
  6. π π
  7. ° °
  8. EOT = GAST - α - UT + Offset \mbox{EOT = GAST}~{}-\alpha-\mbox{UT}~{}+\mbox{Offset}~{}
  9. EOT = α M - α \mbox{EOT}~{}=\alpha_{M}-\alpha
  10. α α
  11. E = GMST - α - UT + Offset \mbox{E = GMST}~{}-\alpha-\mbox{UT}~{}+\mbox{Offset}~{}
  12. Δ Δ
  13. Δ Δ
  14. Δ t = Λ - α \Delta t=\Lambda-\alpha
  15. Λ Λ
  16. Λ Λ
  17. Δ Δ
  18. Δ t = M + λ p - α \Delta t=M+\lambda_{p}-\alpha
  19. π π
  20. π π
  21. Λ Λ
  22. λ λ
  23. α α
  24. ν ν
  25. λ λ
  26. ε ε
  27. α α
  28. M = E - e sin E M=E-e\sin E
  29. ν ν
  30. ν = 2 tan - 1 [ 1 + e 1 - e tan 1 2 E ] \nu=2\tan^{-1}\left[\sqrt{\frac{1+e}{1-e}}\tan\tfrac{1}{2}E\right]
  31. ν ν
  32. ν ν
  33. π π
  34. π π
  35. π π
  36. π π
  37. ν ν
  38. ν ν
  39. ν ν
  40. ν M + 2 e sin M + ( 5 / 4 ) e 2 sin 2 M \nu\sim M+2e\sin M+(5/4)e^{2}\sin 2M
  41. ν ν
  42. ν ν
  43. λ = ν + λ p \lambda=\nu+\lambda_{p}
  44. λ λ
  45. ν ν
  46. λ λ
  47. λ λ
  48. λ λ
  49. α α
  50. α = tan - 1 [ cos ε tan λ ] \alpha=\tan^{-1}[\cos\varepsilon\,\tan\lambda]
  51. α α
  52. λ λ
  53. λ λ
  54. π π
  55. α α
  56. ε ε
  57. λ λ
  58. π π
  59. λ λ
  60. λ λ
  61. λ λ
  62. α α
  63. λ λ
  64. α α
  65. ν ν
  66. α = λ - sin - 1 [ y sin ( α + λ ) ] \alpha=\lambda-\sin^{-1}[y\sin(\alpha+\lambda)]
  67. ε ε
  68. ε ε
  69. α α
  70. λ λ
  71. α α
  72. λ λ
  73. λ λ
  74. λ λ
  75. Δ Δ
  76. λ λ
  77. α α
  78. λ λ
  79. λ λ
  80. α α
  81. λ λ
  82. Δ Δ
  83. Δ t Δt
  84. Δ t e y = - 2 e sin M + y sin ( 2 M + 2 λ p ) = [ - 7.659 sin M + 9.863 sin ( 2 M + 3.5932 ) ] min \Delta t_{ey}=-2e\sin M+y\sin(2M+2\lambda_{p})=[-7.659\sin M+9.863\sin(2M+3.59% 32)]\mbox{min}~{}
  85. Λ Λ
  86. ε ε
  87. ° °
  88. λ λ
  89. ° °
  90. Δ Δ
  91. λ λ
  92. π π
  93. Δ t e 2 y 2 = Δ t e y - ( 5 / 4 ) e 2 sin 2 M + e y sin M cos ( 2 M + 2 λ p ) - ( 1 / 2 ) y 2 sin ( 4 M + 4 λ p ) \Delta t_{e^{2}y^{2}}=\Delta t_{ey}-(5/4)e^{2}\sin 2M+ey\sin M\cos(2M+2\lambda% _{p})-(1/2)y^{2}\sin(4M+4\lambda_{p})
  94. ε ε
  95. λ λ
  96. Δ Δ
  97. α α
  98. Δ Δ
  99. M = 2 π t Y n day = M D + 2 π t Y D day = 6.24004077 + 0.01720197 D M=\frac{2\pi}{t_{Y}}n\mbox{ day}~{}=M_{D}+\frac{2\pi}{t_{Y}}D\mbox{ day}~{}=6.% 24004077+0.01720197D
  100. π π
  101. π π
  102. π π
  103. π π
  104. π π
  105. π π
  106. π π
  107. Δ Δ
  108. Δ Δ
  109. Δ Δ
  110. Δ Δ
  111. Δ Δ
  112. arctan \arctan
  113. arctan η x = arctan x + π round η - arctan x π . \arctan_{\eta}x=\arctan x+\pi\cdot\rm{round}\frac{\eta-\arctan x}{\pi}.
  114. η \eta
  115. round \rm{round}
  116. Δ t ( M ) = M + λ p - arctan M + λ p ( cos ε tan λ ) . \Delta t(M)=M+\lambda_{p}-\arctan_{M+\lambda p}\left(\cos\varepsilon\tan% \lambda\right).
  117. M + λ p M+\lambda_{p}
  118. Δ t \Delta t
  119. Δ Δ
  120. e = 1.6709 10 - 2 - 4.193 10 - 5 ( D 36525 ) - 1.26 10 - 7 ( D 36525 ) 2 e=1.6709\cdot 10^{-2}-4.193\cdot 10^{-5}\left(\frac{D}{36525}\right)-1.26\cdot 1% 0^{-7}\left(\frac{D}{36525}\right)^{2}
  121. ε = [ 23.4393 - 0.013 ( D 36525 ) - 2 10 - 7 ( D 36525 ) 2 + 5 10 - 7 ( D 36525 ) 3 ] deg \varepsilon=\left[23.4393-0.013\left(\frac{D}{36525}\right)-2\cdot 10^{-7}% \left(\frac{D}{36525}\right)^{2}+5\cdot 10^{-7}\left(\frac{D}{36525}\right)^{3% }\right]\mbox{ deg}~{}
  122. λ p = [ 282.93807 + 1.7195 ( D 36525 ) + 3.025 10 - 4 ( D 36525 ) 2 ] deg \lambda_{p}=\left[282.93807+1.7195\left(\frac{D}{36525}\right)+3.025\cdot 10^{% -4}\left(\frac{D}{36525}\right)^{2}\right]\mbox{ deg}~{}
  123. λ λ
  124. ε ε
  125. Δ t Δt
  126. Δ Δ
  127. Δ t Δt
  128. W = 360 / 365.24 W=360/365.24
  129. A = W × ( D + 10 ) A=W\times(D+10)
  130. B = A + ( 360 / π ) × 0.0167 × sin ( W × ( D - 2 ) ) B=A+(360/\pi)\times 0.0167\times\sin(W\times(D-2))
  131. B = A + 1.914 × sin ( W × ( D - 2 ) ) B=A+1.914\times\sin(W\times(D-2))
  132. C = ( A - arctan ( tan ( B ) / cos ( 23.44 ) ) ) / 180 C=(A-\arctan(\tan(B)/\cos(23.44)))/180
  133. 1 {}^{−1}
  134. EoT = 720 × ( C - nint ( C ) ) \,\text{EoT}=720\times(C-\,\text{nint}(C))
  135. Declination = - arcsin ( sin ( 23.44 ) × cos ( B ) ) \,\text{Declination}=-\arcsin(\sin(23.44)\times\cos(B))

Equation_solving.html

  1. 2 ¯ \overline{2}
  2. 2 ¯ \overline{2}
  3. 8 x + 7 = 4 x + 35 or 4 x + 9 3 x + 4 = 2 , 8x+7=4x+35\quad\,\text{or}\quad\frac{4x+9}{3x+4}=2\,,
  4. 2 x 5 - 5 x 4 - x 3 - 7 x 2 + 2 x + 3 = 0 2x^{5}-5x^{4}-x^{3}-7x^{2}+2x+3=0\,
  5. tan x + cot x = 2 \tan x+\cot x=2
  6. tan 2 x - 2 tan x + 1 tan x = 0 , \frac{\tan^{2}x-2\tan x+1}{\tan x}=0,
  7. ( tan x - 1 ) 2 tan x = 0. \frac{(\tan x-1)^{2}}{\tan x}=0.
  8. x = π 4 + k π , k = , - 2 , - 1 , 0 , 1 , 2 , . x=\tfrac{\pi}{4}+k\pi,k=\cdots,-2,-1,0,1,2,\ldots.

Equicontinuity.html

  1. d Y ( f ( y ) , f ( x ) ) < ϵ d_{Y}(f(y),f(x))<\epsilon\,
  2. | f j ( x ) - f j ( z ) | < ϵ / 3 |f_{j}(x)-f_{j}(z)|<\epsilon/3\,
  3. | f j ( z ) - f k ( z ) | < ϵ / 3 |f_{j}(z)-f_{k}(z)|<\epsilon/3\,
  4. sup X | f j - f k | < ϵ \sup_{X}|f_{j}-f_{k}|<\epsilon
  5. | f j ( x ) - f k ( x ) | | f j ( x ) - f j ( z ) | + | f j ( z ) - f k ( z ) | + | f k ( z ) - f k ( x ) | < ϵ |f_{j}(x)-f_{k}(x)|\leq|f_{j}(x)-f_{j}(z)|+|f_{j}(z)-f_{k}(z)|+|f_{k}(z)-f_{k}% (x)|<\epsilon
  6. sup { T : T Γ } < \sup\{\|T\|:T\in\Gamma\}<\infty

Equinumerosity.html

  1. A B A\approx B\,
  2. A B A\sim B
  3. | A | = | B | . |A|=|B|.
  4. 0 \beth_{0}
  5. 1 \beth_{1}
  6. 2 \beth_{2}
  7. 0 \beth_{0}
  8. 0 \aleph_{0}
  9. 1 \beth_{1}
  10. 𝔠 \mathfrak{c}

Equipartition_theorem.html

  1. 1 / 2 {1}/{2}
  2. 1 / 2 {1}/{2}
  3. H kin = 1 2 m | 𝐯 | 2 = 1 2 m ( v x 2 + v y 2 + v z 2 ) , H_{\,\text{kin}}=\tfrac{1}{2}m|\mathbf{v}|^{2}=\tfrac{1}{2}m\left(v_{x}^{2}+v_% {y}^{2}+v_{z}^{2}\right),
  4. 1 / 2 {1}/{2}
  5. v rms = v 2 = 3 k B T m = 3 R T M , v_{\,\text{rms}}=\sqrt{\langle v^{2}\rangle}=\sqrt{\frac{3k_{B}T}{m}}=\sqrt{% \frac{3RT}{M}},
  6. H rot = 1 2 ( I 1 ω 1 2 + I 2 ω 2 2 + I 3 ω 3 2 ) , H_{\mathrm{rot}}=\tfrac{1}{2}(I_{1}\omega_{1}^{2}+I_{2}\omega_{2}^{2}+I_{3}% \omega_{3}^{2}),
  7. H pot = 1 2 a q 2 , H_{\,\text{pot}}=\tfrac{1}{2}aq^{2},\,
  8. H kin = 1 2 m v 2 = p 2 2 m , H_{\,\text{kin}}=\frac{1}{2}mv^{2}=\frac{p^{2}}{2m},
  9. H = H kin + H pot = p 2 2 m + 1 2 a q 2 . H=H_{\,\text{kin}}+H_{\,\text{pot}}=\frac{p^{2}}{2m}+\frac{1}{2}aq^{2}.
  10. H = H kin + H pot = 1 2 k B T + 1 2 k B T = k B T , \langle H\rangle=\langle H_{\,\text{kin}}\rangle+\langle H_{\,\text{pot}}% \rangle=\tfrac{1}{2}k_{B}T+\tfrac{1}{2}k_{B}T=k_{B}T,
  11. \left\langle\ldots\right\rangle
  12. H grav = m b g z H^{\mathrm{grav}}=m_{\rm b}gz\,
  13. x m H x n = δ m n k B T . \!\Bigl\langle x_{m}\frac{\partial H}{\partial x_{n}}\Bigr\rangle=\delta_{mn}k% _{B}T.
  14. \left\langle\ldots\right\rangle
  15. x n H x n = k B T for all n \Bigl\langle x_{n}\frac{\partial H}{\partial x_{n}}\Bigr\rangle=k_{B}T\quad% \mbox{for all }~{}n
  16. x m H x n = 0 for all m n . \Bigl\langle x_{m}\frac{\partial H}{\partial x_{n}}\Bigr\rangle=0\quad\mbox{% for all }~{}m\neq n.
  17. k B T = x n H x n = 2 a n x n 2 , k_{B}T=\Bigl\langle x_{n}\frac{\partial H}{\partial x_{n}}\Bigr\rangle=2% \langle a_{n}x_{n}^{2}\rangle,
  18. H \langle H\rangle
  19. p k H p k = q k H q k = k B T . \Bigl\langle p_{k}\frac{\partial H}{\partial p_{k}}\Bigr\rangle=\Bigl\langle q% _{k}\frac{\partial H}{\partial q_{k}}\Bigr\rangle=k_{\rm B}T.
  20. p k d q k d t = - q k d p k d t = k B T . \Bigl\langle p_{k}\frac{dq_{k}}{dt}\Bigr\rangle=-\Bigl\langle q_{k}\frac{dp_{k% }}{dt}\Bigr\rangle=k_{\rm B}T.
  21. q j H p k = p j H q k = 0 for all j , k \Bigl\langle q_{j}\frac{\partial H}{\partial p_{k}}\Bigr\rangle=\Bigl\langle p% _{j}\frac{\partial H}{\partial q_{k}}\Bigr\rangle=0\quad\mbox{ for all }~{}\,j,k
  22. q j H q k = p j H p k = 0 for all j k . \Bigl\langle q_{j}\frac{\partial H}{\partial q_{k}}\Bigr\rangle=\Bigl\langle p% _{j}\frac{\partial H}{\partial p_{k}}\Bigr\rangle=0\quad\mbox{ for all }~{}\,j% \neq k.
  23. k q k H q k = k p k H p k = k p k d q k d t = - k q k d p k d t , \Bigl\langle\sum_{k}q_{k}\frac{\partial H}{\partial q_{k}}\Bigr\rangle=\Bigl% \langle\sum_{k}p_{k}\frac{\partial H}{\partial p_{k}}\Bigr\rangle=\Bigl\langle% \sum_{k}p_{k}\frac{dq_{k}}{dt}\Bigr\rangle=-\Bigl\langle\sum_{k}q_{k}\frac{dp_% {k}}{dt}\Bigr\rangle,
  24. H kin = 1 2 m p x 2 + p y 2 + p z 2 = 1 2 ( p x H kin p x + p y H kin p y + p z H kin p z ) = 3 2 k B T \begin{aligned}\displaystyle\langle H^{\mathrm{kin}}\rangle&\displaystyle=% \frac{1}{2m}\langle p_{x}^{2}+p_{y}^{2}+p_{z}^{2}\rangle\\ &\displaystyle=\frac{1}{2}\biggl(\Bigl\langle p_{x}\frac{\partial H^{\mathrm{% kin}}}{\partial p_{x}}\Bigr\rangle+\Bigl\langle p_{y}\frac{\partial H^{\mathrm% {kin}}}{\partial p_{y}}\Bigr\rangle+\Bigl\langle p_{z}\frac{\partial H^{% \mathrm{kin}}}{\partial p_{z}}\Bigr\rangle\biggr)=\frac{3}{2}k_{B}T\end{aligned}
  25. 𝐪 𝐅 = q x d p x d t + q y d p y d t + q z d p z d t = - q x H q x - q y H q y - q z H q z = - 3 k B T , \begin{aligned}\displaystyle\langle\mathbf{q}\cdot\mathbf{F}\rangle&% \displaystyle=\Bigl\langle q_{x}\frac{dp_{x}}{dt}\Bigr\rangle+\Bigl\langle q_{% y}\frac{dp_{y}}{dt}\Bigr\rangle+\Bigl\langle q_{z}\frac{dp_{z}}{dt}\Bigr% \rangle\\ &\displaystyle=-\Bigl\langle q_{x}\frac{\partial H}{\partial q_{x}}\Bigr% \rangle-\Bigl\langle q_{y}\frac{\partial H}{\partial q_{y}}\Bigr\rangle-\Bigl% \langle q_{z}\frac{\partial H}{\partial q_{z}}\Bigr\rangle=-3k_{B}T,\end{aligned}
  26. 3 N k B T = - k = 1 N 𝐪 k 𝐅 k . 3Nk_{B}T=-\biggl\langle\sum_{k=1}^{N}\mathbf{q}_{k}\cdot\mathbf{F}_{k}\biggr\rangle.
  27. - k = 1 N 𝐪 k 𝐅 k = P surface 𝐪 𝐝𝐒 , -\biggl\langle\sum_{k=1}^{N}\mathbf{q}_{k}\cdot\mathbf{F}_{k}\biggr\rangle=P% \oint_{\mathrm{surface}}\mathbf{q}\cdot\mathbf{dS},
  28. s y m b o l 𝐪 = q x q x + q y q y + q z q z = 3 , symbol\nabla\cdot\mathbf{q}=\frac{\partial q_{x}}{\partial q_{x}}+\frac{% \partial q_{y}}{\partial q_{y}}+\frac{\partial q_{z}}{\partial q_{z}}=3,
  29. P surface 𝐪 𝐝𝐒 = P volume ( s y m b o l 𝐪 ) d V = 3 P V , P\oint_{\mathrm{surface}}\mathbf{q}\cdot\mathbf{dS}=P\int_{\mathrm{volume}}% \left(symbol\nabla\cdot\mathbf{q}\right)\,dV=3PV,
  30. 3 N k B T = - k = 1 N 𝐪 k 𝐅 k = 3 P V , 3Nk_{B}T=-\biggl\langle\sum_{k=1}^{N}\mathbf{q}_{k}\cdot\mathbf{F}_{k}\biggr% \rangle=3PV,
  31. P V = N k B T = n R T , PV=Nk_{B}T=nRT,\,
  32. H = | 𝐩 1 | 2 2 m 1 + | 𝐩 2 | 2 2 m 2 + 1 2 a q 2 , H=\frac{\left|\mathbf{p}_{1}\right|^{2}}{2m_{1}}+\frac{\left|\mathbf{p}_{2}% \right|^{2}}{2m_{2}}+\frac{1}{2}aq^{2},
  33. 1 / 2 {1}/{2}
  34. 1 / 2 {1}/{2}
  35. 1 / 2 {1}/{2}
  36. H kin c p = c p x 2 + p y 2 + p z 2 . H_{\mathrm{kin}}\approx cp=c\sqrt{p_{x}^{2}+p_{y}^{2}+p_{z}^{2}}.
  37. p x H kin p x = c p x 2 p x 2 + p y 2 + p z 2 p_{x}\frac{\partial H_{\mathrm{kin}}}{\partial p_{x}}=c\frac{p_{x}^{2}}{\sqrt{% p_{x}^{2}+p_{y}^{2}+p_{z}^{2}}}
  38. H kin = c p x 2 + p y 2 + p z 2 p x 2 + p y 2 + p z 2 = p x H kin p x + p y H kin p y + p z H kin p z = 3 k B T \begin{aligned}\displaystyle\langle H_{\mathrm{kin}}\rangle&\displaystyle=% \biggl\langle c\frac{p_{x}^{2}+p_{y}^{2}+p_{z}^{2}}{\sqrt{p_{x}^{2}+p_{y}^{2}+% p_{z}^{2}}}\biggr\rangle\\ &\displaystyle=\Bigl\langle p_{x}\frac{\partial H^{\mathrm{kin}}}{\partial p_{% x}}\Bigr\rangle+\Bigl\langle p_{y}\frac{\partial H^{\mathrm{kin}}}{\partial p_% {y}}\Bigr\rangle+\Bigl\langle p_{z}\frac{\partial H^{\mathrm{kin}}}{\partial p% _{z}}\Bigr\rangle\\ &\displaystyle=3k_{B}T\end{aligned}
  39. h pot = 0 4 π r 2 ρ U ( r ) g ( r ) d r . \langle h_{\mathrm{pot}}\rangle=\int_{0}^{\infty}4\pi r^{2}\rho U(r)g(r)\,dr.
  40. H p o t = 1 2 N h pot \langle H_{pot}\rangle=\tfrac{1}{2}N\langle h_{\mathrm{pot}}\rangle
  41. 1 / 2 {1}/{2}
  42. H = H kin + H pot = 3 2 N k B T + 2 π N ρ 0 r 2 U ( r ) g ( r ) d r . H=\langle H_{\mathrm{kin}}\rangle+\langle H_{\mathrm{pot}}\rangle=\frac{3}{2}% Nk_{B}T+2\pi N\rho\int_{0}^{\infty}r^{2}U(r)g(r)\,dr.
  43. 3 N k B T = 3 P V + 2 π N ρ 0 r 3 U ( r ) g ( r ) d r . 3Nk_{\rm B}T=3PV+2\pi N\rho\int_{0}^{\infty}r^{3}U^{\prime}(r)g(r)\,dr.
  44. H pot = C q s , H_{\mathrm{pot}}=Cq^{s},\,
  45. k B T = q H pot q = q s C q s - 1 = s C q s = s H pot . k_{\rm B}T=\Bigl\langle q\frac{\partial H_{\mathrm{pot}}}{\partial q}\Bigr% \rangle=\langle q\cdot sCq^{s-1}\rangle=\langle sCq^{s}\rangle=s\langle H_{% \mathrm{pot}}\rangle.
  46. H pot = n = 2 C n q n H_{\mathrm{pot}}=\sum_{n=2}^{\infty}C_{n}q^{n}
  47. k B T = q H pot q = n = 2 q n C n q n - 1 = n = 2 n C n q n . k_{B}T=\Bigl\langle q\frac{\partial H_{\mathrm{pot}}}{\partial q}\Bigr\rangle=% \sum_{n=2}^{\infty}\langle q\cdot nC_{n}q^{n-1}\rangle=\sum_{n=2}^{\infty}nC_{% n}\langle q^{n}\rangle.
  48. H pot = 1 2 k B T - n = 3 ( n - 2 2 ) C n q n \langle H_{\mathrm{pot}}\rangle=\frac{1}{2}k_{\rm B}T-\sum_{n=3}^{\infty}\left% (\frac{n-2}{2}\right)C_{n}\langle q^{n}\rangle
  49. d 𝐯 d t = 1 m 𝐅 = - 𝐯 τ + 1 m 𝐅 rnd , \frac{d\mathbf{v}}{dt}=\frac{1}{m}\mathbf{F}=-\frac{\mathbf{v}}{\tau}+\frac{1}% {m}\mathbf{F}_{\mathrm{rnd}},
  50. 𝐫 d 𝐯 d t + 1 τ 𝐫 𝐯 = 0 \Bigl\langle\mathbf{r}\cdot\frac{d\mathbf{v}}{dt}\Bigr\rangle+\frac{1}{\tau}% \langle\mathbf{r}\cdot\mathbf{v}\rangle=0
  51. d d t ( 𝐫 𝐫 ) = d d t ( r 2 ) = 2 ( 𝐫 𝐯 ) \frac{d}{dt}\left(\mathbf{r}\cdot\mathbf{r}\right)=\frac{d}{dt}\left(r^{2}% \right)=2\left(\mathbf{r}\cdot\mathbf{v}\right)
  52. d d t ( 𝐫 𝐯 ) = v 2 + 𝐫 d 𝐯 d t , \frac{d}{dt}\left(\mathbf{r}\cdot\mathbf{v}\right)=v^{2}+\mathbf{r}\cdot\frac{% d\mathbf{v}}{dt},
  53. d 2 d t 2 r 2 + 1 τ d d t r 2 = 2 v 2 = 6 m k B T , \frac{d^{2}}{dt^{2}}\langle r^{2}\rangle+\frac{1}{\tau}\frac{d}{dt}\langle r^{% 2}\rangle=2\langle v^{2}\rangle=\frac{6}{m}k_{\rm B}T,
  54. H kin = p 2 2 m = 1 2 m v 2 = 3 2 k B T . \langle H_{\mathrm{kin}}\rangle=\Bigl\langle\frac{p^{2}}{2m}\Bigr\rangle=% \langle\tfrac{1}{2}mv^{2}\rangle=\tfrac{3}{2}k_{\rm B}T.
  55. r 2 \langle r^{2}\rangle
  56. r 2 = 6 k B T τ 2 m ( e - t / τ - 1 + t τ ) . \langle r^{2}\rangle=\frac{6k_{\rm B}T\tau^{2}}{m}\left(e^{-t/\tau}-1+\frac{t}% {\tau}\right).
  57. r 2 6 k B T τ m t = 6 k B T t γ . \langle r^{2}\rangle\approx\frac{6k_{B}T\tau}{m}t=\frac{6k_{B}Tt}{\gamma}.
  58. H grav = - 0 R 4 π r 2 G r M ( r ) ρ ( r ) d r , H_{\mathrm{grav}}=-\int_{0}^{R}\frac{4\pi r^{2}G}{r}M(r)\,\rho(r)\,dr,
  59. H grav = - 3 G M 2 5 R , H_{\mathrm{grav}}=-\frac{3GM^{2}}{5R},
  60. H grav = H grav N = - 3 G M 2 5 R N , \langle H_{\mathrm{grav}}\rangle=\frac{H_{\mathrm{grav}}}{N}=-\frac{3GM^{2}}{5% RN},
  61. r H grav r = - H grav = k B T = 3 G M 2 5 R N . \Bigl\langle r\frac{\partial H_{\mathrm{grav}}}{\partial r}\Bigr\rangle=% \langle-H_{\mathrm{grav}}\rangle=k_{B}T=\frac{3GM^{2}}{5RN}.
  62. 3 G M 2 5 R > 3 N k B T . \frac{3GM^{2}}{5R}>3Nk_{B}T.
  63. M = 4 3 π R 3 ρ M=\frac{4}{3}\pi R^{3}\rho
  64. M J 2 = ( 5 k B T G m p ) 3 ( 3 4 π ρ ) . M_{\rm J}^{2}=\left(\frac{5k_{B}T}{Gm_{p}}\right)^{3}\left(\frac{3}{4\pi\rho}% \right).
  65. × 10 16 \times 10^{−}16
  66. f ( v ) = 4 π ( m 2 π k B T ) 3 / 2 v 2 exp ( - m v 2 2 k B T ) f(v)=4\pi\left(\frac{m}{2\pi k_{\rm B}T}\right)^{3/2}\!\!v^{2}\exp\Bigl(\frac{% -mv^{2}}{2k_{\rm B}T}\Bigr)
  67. v x 2 + v y 2 + v z 2 \sqrt{v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}
  68. 𝐯 = ( v x , v y , v z ) . \mathbf{v}=(v_{x},v_{y},v_{z}).
  69. H kin = 1 2 m v 2 = 0 1 2 m v 2 f ( v ) d v = 3 2 k B T , \langle H_{\mathrm{kin}}\rangle=\langle\tfrac{1}{2}mv^{2}\rangle=\int_{0}^{% \infty}\tfrac{1}{2}mv^{2}\ f(v)\ dv=\tfrac{3}{2}k_{\rm B}T,
  70. Z x = - d x e - β A x 2 = π β A , Z_{x}=\int_{-\infty}^{\infty}dx\ e^{-\beta Ax^{2}}=\sqrt{\frac{\pi}{\beta A}},
  71. H x = - log Z x β = 1 2 β = 1 2 k B T \langle H_{x}\rangle=-\frac{\partial\log Z_{x}}{\partial\beta}=\frac{1}{2\beta% }=\frac{1}{2}k_{\rm B}T
  72. d Γ = i d q i d p i d\Gamma=\prod_{i}dq_{i}\,dp_{i}\,
  73. Γ ( E , Δ E ) = H [ E , E + Δ E ] d Γ . \Gamma(E,\Delta E)=\int_{H\in\left[E,E+\Delta E\right]}d\Gamma.
  74. H [ E , E + Δ E ] d Γ = Δ E E H < E d Γ , \int_{H\in\left[E,E+\Delta E\right]}\ldots d\Gamma=\Delta E\frac{\partial}{% \partial E}\int_{H<E}\ldots d\Gamma,
  75. Γ = Δ E Σ E = Δ E ρ ( E ) , \Gamma=\Delta E\ \frac{\partial\Sigma}{\partial E}=\Delta E\ \rho(E),
  76. 1 T = S E = k B log Σ E = k B 1 Σ Σ E . \frac{1}{T}=\frac{\partial S}{\partial E}=k_{\rm B}\frac{\partial\log\Sigma}{% \partial E}=k_{\rm B}\frac{1}{\Sigma}\,\frac{\partial\Sigma}{\partial E}.
  77. 𝒩 \mathcal{N}
  78. 𝒩 e - β H ( p , q ) d Γ = 1 , \mathcal{N}\int e^{-\beta H(p,q)}d\Gamma=1,
  79. 𝒩 [ e - β H ( p , q ) x k ] x k = a x k = b d Γ k + 𝒩 e - β H ( p , q ) x k β H x k d Γ = 1 , \mathcal{N}\int\left[e^{-\beta H(p,q)}x_{k}\right]_{x_{k}=a}^{x_{k}=b}d\Gamma_% {k}+\mathcal{N}\int e^{-\beta H(p,q)}x_{k}\beta\frac{\partial H}{\partial x_{k% }}d\Gamma=1,
  80. 𝒩 e - β H ( p , q ) x k H x k d Γ = x k H x k = 1 β = k B T . \mathcal{N}\int e^{-\beta H(p,q)}x_{k}\frac{\partial H}{\partial x_{k}}\,d% \Gamma=\Bigl\langle x_{k}\frac{\partial H}{\partial x_{k}}\Bigr\rangle=\frac{1% }{\beta}=k_{B}T.
  81. \langle\ldots\rangle
  82. x m H x n = 1 Γ H [ E , E + Δ E ] x m H x n d Γ = Δ E Γ E H < E x m H x n d Γ = 1 ρ E H < E x m ( H - E ) x n d Γ , \begin{aligned}\displaystyle\Bigl\langle x_{m}\frac{\partial H}{\partial x_{n}% }\Bigr\rangle&\displaystyle=\frac{1}{\Gamma}\,\int_{H\in\left[E,E+\Delta E% \right]}x_{m}\frac{\partial H}{\partial x_{n}}\,d\Gamma\\ &\displaystyle=\frac{\Delta E}{\Gamma}\,\frac{\partial}{\partial E}\int_{H<E}x% _{m}\frac{\partial H}{\partial x_{n}}\,d\Gamma\\ &\displaystyle=\frac{1}{\rho}\,\frac{\partial}{\partial E}\int_{H<E}x_{m}\frac% {\partial\left(H-E\right)}{\partial x_{n}}\,d\Gamma,\end{aligned}
  83. H < E x m ( H - E ) x n d Γ = H < E x n ( x m ( H - E ) ) d Γ - H < E δ m n ( H - E ) d Γ = δ m n H < E ( E - H ) d Γ , \begin{aligned}\displaystyle\int_{H<E}x_{m}\frac{\partial(H-E)}{\partial x_{n}% }\,d\Gamma&\displaystyle=\int_{H<E}\frac{\partial}{\partial x_{n}}\bigl(x_{m}(% H-E)\bigr)\,d\Gamma-\int_{H<E}\delta_{mn}(H-E)d\Gamma\\ &\displaystyle=\delta_{mn}\int_{H<E}(E-H)\,d\Gamma,\end{aligned}
  84. x m H x n = δ m n 1 ρ E H < E ( E - H ) d Γ = δ m n 1 ρ H < E d Γ = δ m n Σ ρ . \Bigl\langle x_{m}\frac{\partial H}{\partial x_{n}}\Bigr\rangle=\delta_{mn}% \frac{1}{\rho}\,\frac{\partial}{\partial E}\int_{H<E}\left(E-H\right)\,d\Gamma% =\delta_{mn}\frac{1}{\rho}\,\int_{H<E}\,d\Gamma=\delta_{mn}\frac{\Sigma}{\rho}.
  85. ρ = Σ E \rho=\frac{\partial\Sigma}{\partial E}
  86. x m H x n = δ m n ( 1 Σ Σ E ) - 1 = δ m n ( log Σ E ) - 1 = δ m n k B T . \Bigl\langle x_{m}\frac{\partial H}{\partial x_{n}}\Bigr\rangle=\delta_{mn}% \Bigl(\frac{1}{\Sigma}\frac{\partial\Sigma}{\partial E}\Bigr)^{-1}=\delta_{mn}% \Bigl(\frac{\partial\log\Sigma}{\partial E}\Bigr)^{-1}=\delta_{mn}k_{B}T.
  87. x m H x n = δ m n k B T , \!\Bigl\langle x_{m}\frac{\partial H}{\partial x_{n}}\Bigr\rangle=\delta_{mn}k% _{B}T,
  88. P ( E n ) = e - n β h ν Z , P(E_{n})=\frac{e^{-n\beta h\nu}}{Z},
  89. Z = n = 0 e - n β h ν = 1 1 - e - β h ν . Z=\sum_{n=0}^{\infty}e^{-n\beta h\nu}=\frac{1}{1-e^{-\beta h\nu}}.
  90. H = n = 0 E n P ( E n ) = 1 Z n = 0 n h ν e - n β h ν = - 1 Z Z β = - log Z β . \langle H\rangle=\sum_{n=0}^{\infty}E_{n}P(E_{n})=\frac{1}{Z}\sum_{n=0}^{% \infty}nh\nu\ e^{-n\beta h\nu}=-\frac{1}{Z}\frac{\partial Z}{\partial\beta}=-% \frac{\partial\log Z}{\partial\beta}.
  91. H = h ν e - β h ν 1 - e - β h ν . \langle H\rangle=h\nu\frac{e^{-\beta h\nu}}{1-e^{-\beta h\nu}}.

Equity_theory.html

  1. individual’s outcomes individual’s own inputs = relational partner’s outcomes relational partner’s inputs \frac{\,\text{individual's outcomes}}{\,\text{individual's own inputs}}=\frac{% \,\text{relational partner's outcomes}}{\,\text{relational partner's inputs}}

Equivalence_of_categories.html

  1. C C
  2. c c
  3. 1 c 1_{c}
  4. D D
  5. d 1 d_{1}
  6. d 2 d_{2}
  7. 1 d 1 1_{d_{1}}
  8. 1 d 2 1_{d_{2}}
  9. α : d 1 d 2 \alpha\colon d_{1}\to d_{2}
  10. β : d 2 d 1 \beta\colon d_{2}\to d_{1}
  11. C C
  12. D D
  13. F F
  14. c c
  15. d 1 d_{1}
  16. G G
  17. D D
  18. c c
  19. 1 c 1_{c}
  20. C C
  21. E E
  22. C C
  23. c c
  24. 1 c , f : c c 1_{c},f\colon c\to c
  25. 1 c 1_{c}
  26. c c
  27. f f = 1 f\circ f=1
  28. C C
  29. 1 c 1_{c}
  30. 𝐈 C \mathbf{I}_{C}
  31. f f
  32. 𝐈 C \mathbf{I}_{C}
  33. C C
  34. D = Mat ( ) D=\mathrm{Mat}(\mathbb{R})
  35. C C
  36. D D
  37. G : D C G\colon D\to C
  38. A n A_{n}
  39. D D
  40. n \mathbb{R}^{n}
  41. D D
  42. G G
  43. F F
  44. X X
  45. X X
  46. B B
  47. B B

Equivalent_dose.html

  1. H T = R W R D T , R H_{T}=\sum_{R}W_{R}\cdot D_{T,R}

Equivariant_map.html

  1. g g\cdot
  2. z z
  3. g z g\cdot z

Ernst_Abbe.html

  1. d = λ 2 N A d=\frac{\lambda}{2NA}

Errett_Bishop.html

  1. n \mathbb{C}^{n}

Errors_and_residuals.html

  1. X 1 , , X n N ( μ , σ 2 ) X_{1},\dots,X_{n}\sim N(\mu,\sigma^{2})\,
  2. X ¯ = X 1 + + X n n \overline{X}={X_{1}+\cdots+X_{n}\over n}
  3. X ¯ N ( μ , σ 2 / n ) . \overline{X}\sim N(\mu,\sigma^{2}/n).
  4. e i = X i - μ , e_{i}=X_{i}-\mu,\,
  5. r i = X i - X ¯ . r_{i}=X_{i}-\overline{X}.
  6. 1 σ 2 i = 1 n e i 2 χ n 2 . \frac{1}{\sigma^{2}}\sum_{i=1}^{n}e_{i}^{2}\sim\chi^{2}_{n}.
  7. 1 σ 2 i = 1 n r i 2 χ n - 1 2 . \frac{1}{\sigma^{2}}\sum_{i=1}^{n}r_{i}^{2}\sim\chi^{2}_{n-1}.
  8. X ¯ n - μ S n / n . {\overline{X}_{n}-\mu\over S_{n}/\sqrt{n}}.

Euler_angles.html

  1. α α
  2. β β
  3. γ γ
  4. φ φ
  5. θ θ
  6. ψ ψ
  7. z z
  8. α α
  9. x x
  10. β β
  11. z z
  12. γ γ
  13. ( z - x - z , x - y - x , y - z - y , z - y - z , x - z - x , y - x - y ) (z-x-z,x-y-x,y-z-y,z-y-z,x-z-x,y-x-y)
  14. ( x - y - z , y - z - x , z - x - y , x - z - y , z - y - x , y - x - z ) (x-y-z,y-z-x,z-x-y,x-z-y,z-y-x,y-x-z)
  15. φ \varphi
  16. θ \theta
  17. ψ \psi
  18. φ \varphi
  19. θ \theta
  20. ψ \psi
  21. [ π , π ] [−π, π]
  22. [ 0 , π ] [0, π]
  23. [ π / 2 , π / 2 ] [−π/2, π/2]
  24. cos ( β ) = Z 3 . \cos(\beta)=Z_{3}.
  25. sin 2 x = 1 - cos 2 x , \sin^{2}x=1-\cos^{2}x,
  26. sin ( β ) = 1 - Z 3 2 . \sin(\beta)=\sqrt{1-Z_{3}^{2}}.
  27. Z 2 Z_{2}
  28. cos ( α ) sin ( β ) = Z 2 , \cos(\alpha)\cdot\sin(\beta)=Z_{2},
  29. cos ( α ) = Z 2 / 1 - Z 3 2 . \cos(\alpha)=Z_{2}/\sqrt{1-Z_{3}^{2}}.
  30. Y 3 Y_{3}
  31. π / 2 - β \pi/2-\beta
  32. cos ( π / 2 - β ) = sin ( β ) \cos(\pi/2-\beta)=\sin(\beta)
  33. sin ( β ) cos ( γ ) = Y 3 , \sin(\beta)\cdot\cos(\gamma)=Y_{3},
  34. cos ( γ ) = Y 3 / 1 - Z 3 2 , \cos(\gamma)=Y_{3}/\sqrt{1-Z_{3}^{2}},
  35. α = arccos ( Z 2 / 1 - Z 3 2 ) , \alpha=\arccos(Z_{2}/\sqrt{1-Z_{3}^{2}}),
  36. β = arccos ( Z 3 ) , \beta=\arccos(Z_{3}),
  37. γ = arccos ( Y 3 / 1 - Z 3 2 ) . \gamma=\arccos(Y_{3}/\sqrt{1-Z_{3}^{2}}).
  38. α = atan2 ( Z 1 , Z 2 ) , \alpha=\operatorname{atan2}(Z_{1},Z_{2}),
  39. γ = atan2 ( X 3 , Y 3 ) . \gamma=\operatorname{atan2}(X_{3},Y_{3}).
  40. R = X ( α ) Y ( β ) Z ( γ ) R=X(\alpha)Y(\beta)Z(\gamma)
  41. R = Z ( γ ) Y ( β ) X ( α ) R=Z(\gamma)Y(\beta)X(\alpha)
  42. R = Z ( γ ) Y ( β ) X ( α ) R=Z(\gamma)Y(\beta)X(\alpha)
  43. R = X ( α ) Y ( β ) Z ( γ ) R=X(\alpha)Y(\beta)Z(\gamma)
  44. R = X ( α ) Y ( β ) Z ( γ ) R=X(\alpha)Y(\beta)Z(\gamma)
  45. R = Z ( γ ) Y ( β ) X ( α ) R=Z(\gamma)Y(\beta)X(\alpha)
  46. R = X ( α ) Y ( β ) Z ( γ ) R=X(\alpha)Y(\beta)Z(\gamma)
  47. R = Z 1 X 2 Z 3 R=Z_{1}X_{2}Z_{3}
  48. Z 1 X 2 Z 3 Z_{1}X_{2}Z_{3}
  49. X 1 Z 2 X 3 = [ c 2 - c 3 s 2 s 2 s 3 c 1 s 2 c 1 c 2 c 3 - s 1 s 3 - c 3 s 1 - c 1 c 2 s 3 s 1 s 2 c 1 s 3 + c 2 c 3 s 1 c 1 c 3 - c 2 s 1 s 3 ] X_{1}Z_{2}X_{3}=\begin{bmatrix}c_{2}&-c_{3}s_{2}&s_{2}s_{3}\\ c_{1}s_{2}&c_{1}c_{2}c_{3}-s_{1}s_{3}&-c_{3}s_{1}-c_{1}c_{2}s_{3}\\ s_{1}s_{2}&c_{1}s_{3}+c_{2}c_{3}s_{1}&c_{1}c_{3}-c_{2}s_{1}s_{3}\end{bmatrix}
  50. X 1 Z 2 Y 3 = [ c 2 c 3 - s 2 c 2 s 3 s 1 s 3 + c 1 c 3 s 2 c 1 c 2 c 1 s 2 s 3 - c 3 s 1 c 3 s 1 s 2 - c 1 s 3 c 2 s 1 c 1 c 3 + s 1 s 2 s 3 ] X_{1}Z_{2}Y_{3}=\begin{bmatrix}c_{2}c_{3}&-s_{2}&c_{2}s_{3}\\ s_{1}s_{3}+c_{1}c_{3}s_{2}&c_{1}c_{2}&c_{1}s_{2}s_{3}-c_{3}s_{1}\\ c_{3}s_{1}s_{2}-c_{1}s_{3}&c_{2}s_{1}&c_{1}c_{3}+s_{1}s_{2}s_{3}\end{bmatrix}
  51. X 1 Y 2 X 3 = [ c 2 s 2 s 3 c 3 s 2 s 1 s 2 c 1 c 3 - c 2 s 1 s 3 - c 1 s 3 - c 2 c 3 s 1 - c 1 s 2 c 3 s 1 + c 1 c 2 s 3 c 1 c 2 c 3 - s 1 s 3 ] X_{1}Y_{2}X_{3}=\begin{bmatrix}c_{2}&s_{2}s_{3}&c_{3}s_{2}\\ s_{1}s_{2}&c_{1}c_{3}-c_{2}s_{1}s_{3}&-c_{1}s_{3}-c_{2}c_{3}s_{1}\\ -c_{1}s_{2}&c_{3}s_{1}+c_{1}c_{2}s_{3}&c_{1}c_{2}c_{3}-s_{1}s_{3}\end{bmatrix}
  52. X 1 Y 2 Z 3 = [ c 2 c 3 - c 2 s 3 s 2 c 1 s 3 + c 3 s 1 s 2 c 1 c 3 - s 1 s 2 s 3 - c 2 s 1 s 1 s 3 - c 1 c 3 s 2 c 3 s 1 + c 1 s 2 s 3 c 1 c 2 ] X_{1}Y_{2}Z_{3}=\begin{bmatrix}c_{2}c_{3}&-c_{2}s_{3}&s_{2}\\ c_{1}s_{3}+c_{3}s_{1}s_{2}&c_{1}c_{3}-s_{1}s_{2}s_{3}&-c_{2}s_{1}\\ s_{1}s_{3}-c_{1}c_{3}s_{2}&c_{3}s_{1}+c_{1}s_{2}s_{3}&c_{1}c_{2}\end{bmatrix}
  53. Y 1 X 2 Y 3 = [ c 1 c 3 - c 2 s 1 s 3 s 1 s 2 c 1 s 3 + c 2 c 3 s 1 s 2 s 3 c 2 - c 3 s 2 - c 3 s 1 - c 1 c 2 s 3 c 1 s 2 c 1 c 2 c 3 - s 1 s 3 ] Y_{1}X_{2}Y_{3}=\begin{bmatrix}c_{1}c_{3}-c_{2}s_{1}s_{3}&s_{1}s_{2}&c_{1}s_{3% }+c_{2}c_{3}s_{1}\\ s_{2}s_{3}&c_{2}&-c_{3}s_{2}\\ -c_{3}s_{1}-c_{1}c_{2}s_{3}&c_{1}s_{2}&c_{1}c_{2}c_{3}-s_{1}s_{3}\end{bmatrix}
  54. Y 1 X 2 Z 3 = [ c 1 c 3 + s 1 s 2 s 3 c 3 s 1 s 2 - c 1 s 3 c 2 s 1 c 2 s 3 c 2 c 3 - s 2 c 1 s 2 s 3 - c 3 s 1 c 1 c 3 s 2 + s 1 s 3 c 1 c 2 ] Y_{1}X_{2}Z_{3}=\begin{bmatrix}c_{1}c_{3}+s_{1}s_{2}s_{3}&c_{3}s_{1}s_{2}-c_{1% }s_{3}&c_{2}s_{1}\\ c_{2}s_{3}&c_{2}c_{3}&-s_{2}\\ c_{1}s_{2}s_{3}-c_{3}s_{1}&c_{1}c_{3}s_{2}+s_{1}s_{3}&c_{1}c_{2}\end{bmatrix}
  55. Y 1 Z 2 Y 3 = [ c 1 c 2 c 3 - s 1 s 3 - c 1 s 2 c 3 s 1 + c 1 c 2 s 3 c 3 s 2 c 2 s 2 s 3 - c 1 s 3 - c 2 c 3 s 1 s 1 s 2 c 1 c 3 - c 2 s 1 s 3 ] Y_{1}Z_{2}Y_{3}=\begin{bmatrix}c_{1}c_{2}c_{3}-s_{1}s_{3}&-c_{1}s_{2}&c_{3}s_{% 1}+c_{1}c_{2}s_{3}\\ c_{3}s_{2}&c_{2}&s_{2}s_{3}\\ -c_{1}s_{3}-c_{2}c_{3}s_{1}&s_{1}s_{2}&c_{1}c_{3}-c_{2}s_{1}s_{3}\end{bmatrix}
  56. Y 1 Z 2 X 3 = [ c 1 c 2 s 1 s 3 - c 1 c 3 s 2 c 3 s 1 + c 1 s 2 s 3 s 2 c 2 c 3 - c 2 s 3 - c 2 s 1 c 1 s 3 + c 3 s 1 s 2 c 1 c 3 - s 1 s 2 s 3 ] Y_{1}Z_{2}X_{3}=\begin{bmatrix}c_{1}c_{2}&s_{1}s_{3}-c_{1}c_{3}s_{2}&c_{3}s_{1% }+c_{1}s_{2}s_{3}\\ s_{2}&c_{2}c_{3}&-c_{2}s_{3}\\ -c_{2}s_{1}&c_{1}s_{3}+c_{3}s_{1}s_{2}&c_{1}c_{3}-s_{1}s_{2}s_{3}\end{bmatrix}
  57. Z 1 Y 2 Z 3 = [ c 1 c 2 c 3 - s 1 s 3 - c 3 s 1 - c 1 c 2 s 3 c 1 s 2 c 1 s 3 + c 2 c 3 s 1 c 1 c 3 - c 2 s 1 s 3 s 1 s 2 - c 3 s 2 s 2 s 3 c 2 ] Z_{1}Y_{2}Z_{3}=\begin{bmatrix}c_{1}c_{2}c_{3}-s_{1}s_{3}&-c_{3}s_{1}-c_{1}c_{% 2}s_{3}&c_{1}s_{2}\\ c_{1}s_{3}+c_{2}c_{3}s_{1}&c_{1}c_{3}-c_{2}s_{1}s_{3}&s_{1}s_{2}\\ -c_{3}s_{2}&s_{2}s_{3}&c_{2}\end{bmatrix}
  58. Z 1 Y 2 X 3 = [ c 1 c 2 c 1 s 2 s 3 - c 3 s 1 s 1 s 3 + c 1 c 3 s 2 c 2 s 1 c 1 c 3 + s 1 s 2 s 3 c 3 s 1 s 2 - c 1 s 3 - s 2 c 2 s 3 c 2 c 3 ] Z_{1}Y_{2}X_{3}=\begin{bmatrix}c_{1}c_{2}&c_{1}s_{2}s_{3}-c_{3}s_{1}&s_{1}s_{3% }+c_{1}c_{3}s_{2}\\ c_{2}s_{1}&c_{1}c_{3}+s_{1}s_{2}s_{3}&c_{3}s_{1}s_{2}-c_{1}s_{3}\\ -s_{2}&c_{2}s_{3}&c_{2}c_{3}\end{bmatrix}
  59. Z 1 X 2 Z 3 = [ c 1 c 3 - c 2 s 1 s 3 - c 1 s 3 - c 2 c 3 s 1 s 1 s 2 c 3 s 1 + c 1 c 2 s 3 c 1 c 2 c 3 - s 1 s 3 - c 1 s 2 s 2 s 3 c 3 s 2 c 2 ] Z_{1}X_{2}Z_{3}=\begin{bmatrix}c_{1}c_{3}-c_{2}s_{1}s_{3}&-c_{1}s_{3}-c_{2}c_{% 3}s_{1}&s_{1}s_{2}\\ c_{3}s_{1}+c_{1}c_{2}s_{3}&c_{1}c_{2}c_{3}-s_{1}s_{3}&-c_{1}s_{2}\\ s_{2}s_{3}&c_{3}s_{2}&c_{2}\end{bmatrix}
  60. Z 1 X 2 Y 3 = [ c 1 c 3 - s 1 s 2 s 3 - c 2 s 1 c 1 s 3 + c 3 s 1 s 2 c 3 s 1 + c 1 s 2 s 3 c 1 c 2 s 1 s 3 - c 1 c 3 s 2 - c 2 s 3 s 2 c 2 c 3 ] Z_{1}X_{2}Y_{3}=\begin{bmatrix}c_{1}c_{3}-s_{1}s_{2}s_{3}&-c_{2}s_{1}&c_{1}s_{% 3}+c_{3}s_{1}s_{2}\\ c_{3}s_{1}+c_{1}s_{2}s_{3}&c_{1}c_{2}&s_{1}s_{3}-c_{1}c_{3}s_{2}\\ -c_{2}s_{3}&s_{2}&c_{2}c_{3}\end{bmatrix}
  61. \R = [ cos ( θ / 2 ) - I u sin ( θ / 2 ) ] {\R}=[\cos(\theta/2)-Iu\sin(\theta/2)]
  62. θ = \theta=
  63. ( u ) = (u)=
  64. ( I ) = (I)=
  65. 3 \mathbb{R}^{3}
  66. D D
  67. N r o t = ( D 2 ) = D ( D - 1 ) / 2 N_{rot}={\left({{D}\atop{2}}\right)}=D(D-1)/2
  68. D = 2 , 3 , 4 D=2,3,4
  69. N r o t = 1 , 3 , 6 N_{rot}=1,3,6

Euler_equations_(fluid_dynamics).html

  1. D D t {D\over Dt}
  2. \cdot
  3. 𝐮 \mathbf{u}\cdot\nabla
  4. 𝐠 \mathbf{g}
  5. { u t + u u = - w + g u = 0 \left\{\begin{aligned}\displaystyle{\partial u\over\partial t}+u\cdot\nabla u=% -\nabla w+{g}\\ \displaystyle\nabla\cdot u=0\end{aligned}\right.
  6. w ( p ρ 0 ) = 1 ρ 0 p \nabla w\equiv\nabla\left(\frac{p}{\rho_{0}}\right)=\frac{1}{\rho_{0}}\nabla p
  7. ρ t = 0 \frac{\partial\rho}{\partial t}=0
  8. { u j t + i = 1 N u j ( u i + w e ^ i ) r i = 0 i = 1 N u i r i = 0 \left\{\begin{aligned}\displaystyle{\partial u_{j}\over\partial t}+\sum_{i=1}^% {N}u_{j}{\partial(u_{i}+w\hat{e}_{i})\over\partial r_{i}}=0\\ \displaystyle\sum_{i=1}^{N}{\partial u_{i}\over\partial r_{i}}=0\end{aligned}\right.
  9. { t u j + i ( u i u j + w δ i j ) = 0 i u i = 0 , \left\{\begin{aligned}\displaystyle\partial_{t}u_{j}+\partial_{i}(u_{i}u_{j}+w% \delta_{ij})=0\\ \displaystyle\partial_{i}u_{i}=0,\end{aligned}\right.
  10. t t \partial_{t}\equiv\frac{\partial}{\partial t}
  11. i r i \partial_{i}\equiv\frac{\partial}{\partial r_{i}}
  12. t ( 1 2 u 2 ) + ( u 2 u + w I ) = 0 {\partial\over\partial t}\left(\frac{1}{2}u^{2}\right)+\nabla\cdot(u^{2}u+wI)=0
  13. u t + u u x = 0 {\partial u\over\partial t}+u{\partial u\over\partial x}=0
  14. u * u u 0 , u^{*}\equiv\frac{u}{u_{0}},
  15. r * r r 0 , r^{*}\equiv\frac{r}{r_{0}},
  16. t * u 0 r 0 t , t^{*}\equiv\frac{u_{0}}{r_{0}}t,
  17. p * w u 0 2 , p^{*}\equiv\frac{w}{u_{0}^{2}},
  18. * r 0 \nabla^{*}\equiv r_{0}\nabla
  19. 𝐠 ^ 𝐠 g , \hat{\mathbf{g}}\equiv\frac{\mathbf{g}}{g},
  20. y t + F = 0 , \frac{\partial y}{\partial t}+\nabla\cdot F={0},
  21. y j t + f i j r i = 0 i , \frac{\partial y_{j}}{\partial t}+\frac{\partial f_{ij}}{\partial r_{i}}=0_{i},
  22. p = ρ w p=\rho w
  23. ρ t + u ρ + ρ u = 0 {\partial\rho\over\partial t}+u\cdot\nabla\rho+\rho\nabla\cdot u=0
  24. y = ( ρ ρ u 0 ) ; F = ( ρ u ρ u u + p I u ) . {y}=\begin{pmatrix}\rho\\ \rho u\\ 0\end{pmatrix};\qquad{F}=\begin{pmatrix}\rho u\\ \rho u\otimes u+pI\\ u\end{pmatrix}.
  25. t ( ρ ρ u 0 ) + ( ρ u ρ u u + p I u ) = ( 0 ρ g 0 ) \frac{\partial}{\partial t}\begin{pmatrix}\rho\\ \rho u\\ 0\end{pmatrix}+\nabla\cdot\begin{pmatrix}\rho u\\ \rho u\otimes u+pI\\ u\end{pmatrix}=\begin{pmatrix}0\\ \rho g\\ 0\end{pmatrix}
  26. y = ( ρ j 0 ) ; F = ( j 1 ρ j j + p I j ρ ) . {y}=\begin{pmatrix}\rho\\ j\\ 0\end{pmatrix};\qquad{F}=\begin{pmatrix}j\\ \frac{1}{\rho}\,j\otimes j+pI\\ \frac{j}{\rho}\end{pmatrix}.
  27. { ρ t + u ρ + ρ u = 0 𝐮 t + 𝐮 𝐮 + p ρ = 𝐠 e t + u e + p ρ u = 0 \left\{\begin{aligned}\displaystyle{\partial\rho\over\partial t}+u\cdot\nabla% \rho+\rho\nabla\cdot u=0\\ \displaystyle\frac{\partial\mathbf{u}}{\partial t}+\mathbf{u}\cdot\nabla% \mathbf{u}+\frac{\nabla p}{\rho}=\mathbf{g}\\ \displaystyle{\partial e\over\partial t}+u\cdot\nabla e+\frac{p}{\rho}\nabla% \cdot u=0\end{aligned}\right.
  28. D e D t = 0 {De\over Dt}=0
  29. ρ e t + ( ρ e 𝐮 ) = 0 {\partial\rho e\over\partial t}+\nabla\cdot(\rho e\mathbf{u})=0
  30. h = e + p ρ h=e+\frac{p}{\rho}
  31. D e D t = D h D t - 1 ρ ( D p D t - p ρ D ρ D t ) {De\over Dt}={Dh\over Dt}-\frac{1}{\rho}\left({Dp\over Dt}-\frac{p}{\rho}{D% \rho\over Dt}\right)
  32. D e D t = D h D t + 1 ρ ( p 𝐮 - D p D t ) {De\over Dt}={Dh\over Dt}+\frac{1}{\rho}\left(p\nabla\cdot\mathbf{u}-{Dp\over Dt% }\right)
  33. D h D t = 1 ρ D p D t {Dh\over Dt}=\frac{1}{\rho}{Dp\over Dt}
  34. ρ s t + ( ρ s 𝐮 ) = 0 {\partial\rho s\over\partial t}+\nabla\cdot(\rho s\mathbf{u})=0
  35. e = e ( v , s ) e=e(v,s)
  36. y = ( ρ j E t ) ; F = ( j 1 ρ j j + p I ( E t + p ) j ρ ) . {y}=\begin{pmatrix}\rho\\ j\\ E^{t}\end{pmatrix};\qquad{F}=\begin{pmatrix}j\\ \frac{1}{\rho}j\otimes j+pI\\ (E^{t}+p)\frac{j}{\rho}\end{pmatrix}.
  37. t ( ρ s ) + ( ρ s 𝐮 ) = 0 {\partial\over\partial t}(\rho s)+\nabla\cdot(\rho s\mathbf{u})=0
  38. H t t + ( H t 𝐮 ) = 𝐮 𝐟 - p t \frac{\partial H^{t}}{\partial t}+\nabla\cdot(H^{t}\mathbf{u})=\mathbf{u}\cdot% \mathbf{f}-\frac{\partial p}{\partial t}
  39. y t + A i y r i = 0. \frac{\partial y}{\partial t}+A_{i}\frac{\partial y}{\partial r_{i}}={0}.
  40. A i ( y ) = f i ( y ) y . A_{i}(y)=\frac{\partial f_{i}(y)}{\partial y}.
  41. 𝐏 = [ 𝐩 1 , 𝐩 2 , , 𝐩 n ] \mathbf{P}=[\mathbf{p}_{1},\mathbf{p}_{2},...,\mathbf{p}_{n}]
  42. 𝐰 = 𝐏 - 1 𝐲 , \mathbf{w}=\mathbf{P}^{-1}\mathbf{y},
  43. w i t + λ j w i r j = 0 i \frac{\partial w_{i}}{\partial t}+\lambda_{j}\frac{\partial w_{i}}{\partial r_% {j}}=0_{i}
  44. w i ( x , t ) = w i ( x - λ i t , 0 ) w_{i}(x,t)=w_{i}(x-\lambda_{i}t,0)
  45. 𝐲 = 𝐏𝐰 , \mathbf{y}=\mathbf{P}\mathbf{w},
  46. 𝐲 ( x , t ) = i = 1 m w i ( x - λ i t , 0 ) 𝐩 i , \mathbf{y}(x,t)=\sum_{i=1}^{m}w_{i}(x-\lambda_{i}t,0)\mathbf{p}_{i},
  47. { v t + u v x - v u x = 0 u t + u u x - e v v v v x - e v s v s x = 0 s t + u s t = 0 \left\{\begin{aligned}\displaystyle{\partial v\over\partial t}+u{\partial v% \over\partial x}-v{\partial u\over\partial x}=0\\ \displaystyle{\partial u\over\partial t}+u{\partial u\over\partial x}-e_{vv}v{% \partial v\over\partial x}-e_{vs}v{\partial s\over\partial x}=0\\ \displaystyle{\partial s\over\partial t}+u{\partial s\over\partial t}=0\end{% aligned}\right.
  48. y = ( v u s ) {y}=\begin{pmatrix}v\\ u\\ s\end{pmatrix}
  49. A = ( u - v 0 - e v v v u - e v s v 0 0 u ) . {A}=\begin{pmatrix}u&-v&0\\ -e_{vv}v&u&-e_{vs}v\\ 0&0&u\end{pmatrix}.
  50. det ( 𝐀 ( 𝐲 ) - λ ( 𝐲 ) 𝐈 ) ) = 0 \det(\mathbf{A}(\mathbf{y})-\lambda(\mathbf{y})\mathbf{I}))=0
  51. det [ u - λ - v 0 - e v v v u - e v s v 0 0 u - λ ] = 0 \det\begin{bmatrix}u-\lambda&-v&0\\ -e_{vv}v&u&-e_{vs}v\\ 0&0&u-\lambda\end{bmatrix}=0
  52. ( u - λ ) det [ u - λ - v - e v v v u - λ ] = 0 (u-\lambda)\det\begin{bmatrix}u-\lambda&-v\\ -e_{vv}v&u-\lambda\end{bmatrix}=0
  53. ( u - λ ) ( ( u - λ ) 2 - e v v v 2 ) = 0 (u-\lambda)\left((u-\lambda)^{2}-e_{vv}v^{2}\right)=0
  54. a ( v , s ) v e v v a(v,s)\equiv v\sqrt{e_{vv}}
  55. a ( ρ , p ) p ρ a(\rho,p)\equiv\sqrt{\partial p\over\partial\rho}
  56. ( e v v e v s e v s e s s ) \begin{pmatrix}e_{vv}&e_{vs}\\ e_{vs}&e_{ss}\end{pmatrix}
  57. { e v v > 0 e v v e s s - e v s 2 > 0 \left\{\begin{aligned}\displaystyle e_{vv}>0\\ \displaystyle e_{vv}e_{ss}-e_{vs}^{2}>0\end{aligned}\right.
  58. ( u - λ ) ( ( u - λ ) 2 - a 2 ) = 0 (u-\lambda)\left((u-\lambda)^{2}-a^{2}\right)=0
  59. λ 1 ( v , u , s ) = u - a ( v , s ) λ 2 ( u ) = u , λ 3 ( v , u , s ) = u + a ( v , s ) \lambda_{1}(v,u,s)=u-a(v,s)\quad\lambda_{2}(u)=u,\quad\lambda_{3}(v,u,s)=u+a(v% ,s)
  60. ( a - v 0 - e v v v a - e v s v 0 0 a ) ( v 1 u 1 s 1 ) = 0 \begin{pmatrix}a&-v&0\\ -e_{vv}v&a&-e_{vs}v\\ 0&0&a\end{pmatrix}\begin{pmatrix}v_{1}\\ u_{1}\\ s_{1}\end{pmatrix}=0
  61. ( a - v - a 2 / v a ) ( v 1 u 1 ) = 0 \begin{pmatrix}a&-v\\ -a^{2}/v&a\end{pmatrix}\begin{pmatrix}v_{1}\\ u_{1}\end{pmatrix}=0
  62. 𝐩 1 = ( v a 0 ) \mathbf{p}_{1}=\begin{pmatrix}v\\ a\\ 0\end{pmatrix}
  63. 𝐩 2 = ( e v s 0 - ( a v ) 2 ) , 𝐩 3 = ( v - a 0 ) \mathbf{p}_{2}=\begin{pmatrix}e_{vs}\\ 0\\ -\left(\frac{a}{v}\right)^{2}\end{pmatrix},\qquad\mathbf{p}_{3}=\begin{pmatrix% }v\\ -a\\ 0\end{pmatrix}
  64. 𝐏 ( v , u , s ) = ( 𝐩 1 , 𝐩 2 , 𝐩 3 ) = ( v e v s v a 0 - a 0 - ( a v ) 2 0 ) \mathbf{P}(v,u,s)=(\mathbf{p}_{1},\mathbf{p}_{2},\mathbf{p}_{3})=\begin{% pmatrix}v&e_{vs}&v\\ a&0&-a\\ 0&-\left(\frac{a}{v}\right)^{2}&0\end{pmatrix}
  65. a s ( p ρ ) s a_{s}\equiv\sqrt{\left({\partial p\over\partial\rho}\right)_{s}}
  66. a s ( ρ , p ) ( p ρ ) s a_{s}(\rho,p)\equiv\sqrt{\left({\partial p\over\partial\rho}\right)_{s}}
  67. K s ( ρ , p ) 1 ρ ( p ρ ) s K_{s}(\rho,p)\equiv\frac{1}{\rho}\left({\partial p\over\partial\rho}\right)_{s}
  68. a s K s ρ a_{s}\equiv\sqrt{\frac{K_{s}}{\rho}}
  69. a s ( T ) = γ T m a_{s}(T)=\sqrt{\gamma\frac{T}{m}}
  70. h = c p T = γ γ - 1 T m h=c_{p}T=\frac{\gamma}{\gamma-1}\frac{T}{m}
  71. a s ( h ) = ( γ - 1 ) h a_{s}(h)=\sqrt{(\gamma-1)h}
  72. 𝐯 × ( × 𝐅 ) = F ( 𝐯 𝐅 ) - 𝐯 𝐅 , \mathbf{v\ \times}\left(\mathbf{\nabla\times F}\right)=\nabla_{F}\left(\mathbf% {v\cdot F}\right)-\mathbf{v\cdot\nabla}\mathbf{F}\ ,
  73. 𝐮 𝐮 = 1 2 ( u 2 ) + ( × 𝐮 ) × 𝐮 \mathbf{u}\cdot\nabla\mathbf{u}=\frac{1}{2}\nabla(u^{2})+(\nabla\times\mathbf{% u})\times\mathbf{u}
  74. 𝐮 t + 1 2 ( u 2 ) + ( × 𝐮 ) × 𝐮 + p ρ = 𝐠 \frac{\partial\mathbf{u}}{\partial t}+\frac{1}{2}\nabla(u^{2})+(\nabla\times% \mathbf{u})\times\mathbf{u}+\frac{\nabla p}{\rho}=\mathbf{g}
  75. ( p ρ ) = p ρ - p ρ 2 ρ \nabla\left(\frac{p}{\rho}\right)=\frac{\nabla p}{\rho}-\frac{p}{\rho^{2}}\nabla\rho
  76. ( 1 2 u 2 + p ρ ) - 𝐠 = p ρ 2 ρ + 𝐮 × ( × 𝐮 ) - u t \nabla\left(\frac{1}{2}u^{2}+\frac{p}{\rho}\right)-\mathbf{g}=\frac{p}{\rho^{2% }}\nabla\rho+\mathbf{u}\times(\nabla\times\mathbf{u})-\frac{\partial u}{% \partial t}
  77. ( 1 2 u 2 + ϕ + p ρ ) = p ρ 2 ρ + 𝐮 × ( × 𝐮 ) - u t \nabla\left(\frac{1}{2}u^{2}+\phi+\frac{p}{\rho}\right)=\frac{p}{\rho^{2}}% \nabla\rho+\mathbf{u}\times(\nabla\times\mathbf{u})-\frac{\partial u}{\partial t}
  78. ( 1 2 u 2 + ϕ + p ρ ) = p ρ 2 ρ + 𝐮 × ( × 𝐮 ) \nabla\left(\frac{1}{2}u^{2}+\phi+\frac{p}{\rho}\right)=\frac{p}{\rho^{2}}% \nabla\rho+\mathbf{u}\times(\nabla\times\mathbf{u})
  79. 𝐮 ( 1 2 u 2 + ϕ + p ρ ) = p ρ 2 𝐮 ρ \mathbf{u}\cdot\nabla\left(\frac{1}{2}u^{2}+\phi+\frac{p}{\rho}\right)=\frac{p% }{\rho^{2}}\mathbf{u}\cdot\nabla\rho
  80. 𝐮 ρ = 0 \mathbf{u}\cdot\nabla\rho=0
  81. 𝐮 ( 1 2 u 2 + ϕ + p ρ ) = 0 \mathbf{u}\cdot\nabla\left(\frac{1}{2}u^{2}+\phi+\frac{p}{\rho}\right)=0
  82. b l 1 2 u 2 + ϕ + p ρ b_{l}\equiv\frac{1}{2}u^{2}+\phi+\frac{p}{\rho}
  83. 𝐮 b l = 0 \mathbf{u}\cdot\nabla b_{l}=0
  84. 𝐣 = 0 \nabla\cdot\mathbf{j}=0
  85. ( ( e + 1 2 u 2 + p ρ ) 𝐣 ) = 𝐣 𝐠 \nabla\cdot\left(\left(e+\frac{1}{2}u^{2}+\frac{p}{\rho}\right)\mathbf{j}% \right)=\mathbf{j}\cdot\mathbf{g}
  86. ( ( e + 1 2 u 2 + p ρ ) 𝐣 ) = ρ 𝐮 ( e + 1 2 u 2 + p ρ ) \nabla\cdot\left(\left(e+\frac{1}{2}u^{2}+\frac{p}{\rho}\right)\mathbf{j}% \right)=\rho\mathbf{u}\cdot\nabla\left(e+\frac{1}{2}u^{2}+\frac{p}{\rho}\right)
  87. 𝐣 𝐠 = - ρ 𝐮 ϕ \mathbf{j}\cdot\mathbf{g}=-\rho\mathbf{u}\cdot\nabla\phi
  88. 𝐮 ( e + p ρ + 1 2 u 2 + ϕ ) = 0 \mathbf{u}\cdot\nabla\left(e+\frac{p}{\rho}+\frac{1}{2}u^{2}+\phi\right)=0
  89. h t e + p ρ + 1 2 u 2 h^{t}\equiv e+\frac{p}{\rho}+\frac{1}{2}u^{2}
  90. b g h t + ϕ = b l + e b_{g}\equiv h^{t}+\phi=b_{l}+e
  91. 𝐮 b g = 0 \mathbf{u}\cdot\nabla b_{g}=0
  92. 𝐮 h t 0 \mathbf{u}\cdot\nabla h^{t}\sim 0
  93. v p = - T s + h v\nabla p=-T\nabla s+\nabla h
  94. D 𝐮 D t = T s - h \frac{D\mathbf{u}}{Dt}=T\nabla\,s-\nabla\,h
  95. 𝐮 t + 1 2 ( u 2 ) + ( × 𝐮 ) × 𝐮 + p ρ = 𝐠 \frac{\partial\mathbf{u}}{\partial t}+\frac{1}{2}\nabla(u^{2})+(\nabla\times% \mathbf{u})\times\mathbf{u}+\frac{\nabla p}{\rho}=\mathbf{g}
  96. h t = h + 1 2 u 2 h^{t}=h+\frac{1}{2}u^{2}
  97. 𝐮 t + ( × 𝐮 ) × 𝐮 - T s + h t = 𝐠 \frac{\partial\mathbf{u}}{\partial t}+(\nabla\times\mathbf{u})\times\mathbf{u}% -T\nabla s+\nabla h^{t}=\mathbf{g}
  98. { 𝐮 × × 𝐮 + T s - h t = 𝐠 𝐮 s = 0 𝐮 h t = 0 \left\{\begin{aligned}\displaystyle\mathbf{u}\times\nabla\times\mathbf{u}+T% \nabla s-\nabla h^{t}=\mathbf{g}\\ \displaystyle\mathbf{u}\cdot\nabla s=0\\ \displaystyle\mathbf{u}\cdot\nabla h^{t}=0\end{aligned}\right.
  99. T s = ( T s ) T\nabla s=\nabla(Ts)
  100. g t h t + T s g^{t}\equiv h^{t}+Ts
  101. { 𝐮 × × 𝐮 - g t = 𝐠 𝐮 g t = 0 \left\{\begin{aligned}\displaystyle\mathbf{u}\times\nabla\times\mathbf{u}-% \nabla g^{t}=\mathbf{g}\\ \displaystyle\mathbf{u}\cdot\nabla g^{t}=0\end{aligned}\right.
  102. 𝐅 = 𝟎 \nabla\cdot\mathbf{F}=\mathbf{0}
  103. V m 𝐅 d V = 𝟎 \int_{V_{m}}\nabla\cdot\mathbf{F}dV=\mathbf{0}
  104. V m 𝐅 d s = 𝟎 \oint_{\partial V_{m}}\mathbf{F}ds=\mathbf{0}
  105. x m x m + 1 𝐅 ( x ) d x = 𝟎 \int_{x_{m}}^{x_{m+1}}\mathbf{F}(x^{\prime})dx^{\prime}=\mathbf{0}
  106. Δ 𝐅 = 𝟎 \Delta\mathbf{F}=\mathbf{0}
  107. 𝐅 m + 1 - 𝐅 m = 𝟎 \mathbf{F}_{m+1}-\mathbf{F}_{m}=\mathbf{0}
  108. 𝐅 m = 𝐅 ( x m ) \mathbf{F}_{m}=\mathbf{F}(x_{m})
  109. 𝐅 - 𝐅 0 = 𝟎 \mathbf{F}-\mathbf{F}_{0}=\mathbf{0}
  110. y t + 𝐅 = 𝟎 {\partial y\over\partial t}+\nabla\cdot\mathbf{F}=\mathbf{0}
  111. d x d t Δ u = Δ 𝐅 \frac{dx}{dt}\,\Delta u=\Delta\mathbf{F}
  112. 𝐲 = ( 1 / v j E t ) \mathbf{y}=\begin{pmatrix}1/v\\ j\\ E^{t}\end{pmatrix}
  113. 𝐅 = ( j v j 2 + p v j ( E t + p ) ) \mathbf{F}=\begin{pmatrix}j\\ vj^{2}+p\\ vj(E^{t}+p)\end{pmatrix}
  114. { d x d t Δ ( 1 / v ) = Δ j d x d t Δ j = Δ ( v j 2 + p ) d x d t Δ E t = Δ ( j v ( E t + p ) ) \left\{\begin{aligned}\displaystyle\frac{dx}{dt}\Delta(1/v)=\Delta j\\ \displaystyle\frac{dx}{dt}\Delta j=\Delta(vj^{2}+p)\\ \displaystyle\frac{dx}{dt}\Delta E^{t}=\Delta(jv(E^{t}+p))\end{aligned}\right.
  115. { Δ j = 0 Δ ( v j 2 + p ) = 0 Δ ( j ( E t ρ + p ρ ) ) = 0 \left\{\begin{aligned}\displaystyle\Delta j=0\\ \displaystyle\Delta(vj^{2}+p)=0\\ \displaystyle\Delta(j(\frac{E^{t}}{\rho}+\frac{p}{\rho}))=0\end{aligned}\right.
  116. { Δ j = 0 Δ ( v j 2 + p ) = 0 Δ h t = 0 \left\{\begin{aligned}\displaystyle\Delta j=0\\ \displaystyle\Delta(vj^{2}+p)=0\\ \displaystyle\Delta h^{t}=0\end{aligned}\right.
  117. { Δ j = 0 Δ ( u 2 v + p ) = 0 Δ ( e + 1 2 u 2 + p v ) = 0 \left\{\begin{aligned}\displaystyle\Delta j=0\\ \displaystyle\Delta(\frac{u^{2}}{v}+p)=0\\ \displaystyle\Delta\left(e+\frac{1}{2}u^{2}+pv\right)=0\end{aligned}\right.
  118. Δ p Δ v = - u 0 2 v 0 \frac{\Delta p}{\Delta v}=-\frac{u_{0}^{2}}{v_{0}}
  119. { ρ u = ρ 0 u 0 ρ u 2 + p = ρ 0 u 0 2 + p 0 e + 1 2 u 2 + p ρ = e 0 + 1 2 u 0 2 + p 0 ρ 0 \left\{\begin{aligned}\displaystyle\rho u=\rho_{0}u_{0}\\ \displaystyle\rho u^{2}+p=\rho_{0}u_{0}^{2}+p_{0}\\ \displaystyle e+\frac{1}{2}u^{2}+\frac{p}{\rho}=e_{0}+\frac{1}{2}u_{0}^{2}+% \frac{p_{0}}{\rho_{0}}\end{aligned}\right.
  120. { u 2 ( v , p ) = u 0 2 + ( p - p 0 ) ( v 0 + v ) e ( v , p ) = e 0 + 1 2 ( p + p 0 ) ( v 0 - v ) \left\{\begin{aligned}\displaystyle u^{2}(v,p)=u_{0}^{2}+(p-p_{0})(v_{0}+v)\\ \displaystyle e(v,p)=e_{0}+\frac{1}{2}(p+p_{0})(v_{0}-v)\end{aligned}\right.
  121. e = e ( v , p ) e=e(v,p)
  122. 𝔥 ( v , s ) e ( v , s ) - e 0 + 1 2 ( p ( v , s ) + p 0 ) ( v - v 0 ) \mathfrak{h}(v,s)\equiv e(v,s)-e_{0}+\frac{1}{2}(p(v,s)+p_{0})(v-v_{0})
  123. 𝐲 t + 𝐅 = 𝐬 \frac{\partial\mathbf{y}}{\partial t}+\nabla\cdot\mathbf{F}=\mathbf{s}
  124. d d t V m 𝐲 d V + V m 𝐅 n ^ d s = 𝐒 \frac{d}{dt}\int_{V_{m}}\mathbf{y}dV+\oint_{\partial V_{m}}\mathbf{F}\cdot\hat% {n}ds=\mathbf{S}
  125. V m 𝐲 ( 𝐫 , t n + 1 ) d V - V m 𝐲 ( 𝐫 , t n ) d V + t n t n + 1 V m 𝐅 n ^ d s d t = 𝟎 \int_{V_{m}}\mathbf{y}(\mathbf{r},t_{n+1})dV-\int_{V_{m}}\mathbf{y}(\mathbf{r}% ,t_{n})dV+\int_{t_{n}}^{t_{n+1}}\oint_{\partial V_{m}}\mathbf{F}\cdot\hat{n}% dsdt=\mathbf{0}
  126. 𝐲 m , n 1 V m V m 𝐲 ( 𝐫 , t n ) d V \mathbf{y}_{m,n}\equiv\frac{1}{V_{m}}\int_{V_{m}}\mathbf{y}(\mathbf{r},t_{n})dV
  127. 𝐲 m , n + 1 = 𝐲 m , n - 1 V m t n t n + 1 V m 𝐅 n ^ d s d t \mathbf{y}_{m,n+1}=\mathbf{y}_{m,n}-\frac{1}{V_{m}}\int_{t_{n}}^{t_{n+1}}\oint% _{\partial V_{m}}\mathbf{F}\cdot\hat{n}dsdt
  128. { 𝐮 m , n = 𝐣 m , n ρ m , n e m , n = E m , n t ρ m , n - 1 2 u m , n 2 \left\{\begin{aligned}\displaystyle\mathbf{u}_{m,n}=\frac{\mathbf{j}_{m,n}}{% \rho_{m,n}}\\ \displaystyle e_{m,n}=\frac{E^{t}_{m,n}}{\rho_{m,n}}-\frac{1}{2}u^{2}_{m,n}\\ \end{aligned}\right.
  129. e ( v , s ) = e 0 e ( γ - 1 ) m ( s - s 0 ) ( v 0 v ) γ - 1 e(v,s)=e_{0}e^{(\gamma-1)m(s-s_{0})}\left({v_{0}\over v}\right)^{\gamma-1}
  130. p ( v , e ) - e v = ( γ - 1 ) e v p(v,e)\equiv-{\partial e\over\partial v}=(\gamma-1)\frac{e}{v}
  131. e ( v , p ) = p v γ - 1 e(v,p)=\frac{pv}{\gamma-1}
  132. y t + A y x = 0. \frac{\partial y}{\partial t}+A\frac{\partial y}{\partial x}={0}.
  133. y = ( v u p ) {y}=\begin{pmatrix}v\\ u\\ p\end{pmatrix}
  134. A = ( u - v 0 0 u v 0 γ p u ) . {A}=\begin{pmatrix}u&-v&0\\ 0&u&v\\ 0&\gamma p&u\end{pmatrix}.
  135. s y m b o l u s y m b o l u = - 1 ρ p , symbol{u}\cdot\nabla symbol{u}=-\frac{1}{\rho}\nabla p,
  136. s y m b o l u s y m b o l u \displaystyle symbol{u}\cdot\nabla symbol{u}
  137. { u u s = - 1 ρ p s , u 2 R = - 1 ρ p n ( / n \equivsymbol e n ) , 0 = - 1 ρ p b ( / b \equivsymbol e b ) . \begin{cases}\displaystyle u\frac{\partial u}{\partial s}=-\frac{1}{\rho}\frac% {\partial p}{\partial s},\\ \displaystyle{u^{2}\over R}=-\frac{1}{\rho}\frac{\partial p}{\partial n}&({% \partial/\partial n}\equivsymbol{e}_{n}\cdot\nabla),\\ \displaystyle 0=-\frac{1}{\rho}\frac{\partial p}{\partial b}&({\partial/% \partial b}\equivsymbol{e}_{b}\cdot\nabla).\end{cases}
  138. s ( u 2 2 + d p ρ ) = 0. \frac{\partial}{\partial s}\left(\frac{u^{2}}{2}+\int\frac{\mathrm{d}p}{\rho}% \right)=0.
  139. p r = ρ u 2 r ( > 0 ) , \frac{\partial p}{\partial r}=\rho\frac{u^{2}}{r}~{}(>0),
  140. / r = - / n . {\partial/\partial r}=-{\partial/\partial n}.
  141. y = ( ρ ρ u 1 ρ u 2 ρ u 3 0 ) ; F = ( ρ u 1 ρ u 2 ρ u 3 ρ u 1 2 + p ρ u 1 u 2 ρ u 1 u 3 ρ u 1 u 2 ρ u 2 2 + p ρ u 2 u 3 ρ u 3 u 1 ρ u 3 u 2 ρ u 3 2 + p u 1 u 2 u 3 ) . {y}=\begin{pmatrix}\rho\\ \rho u_{1}\\ \rho u_{2}\\ \rho u_{3}\\ 0\end{pmatrix};\quad{F}=\begin{pmatrix}\rho u_{1}&\rho u_{2}&\rho u_{3}\\ \rho u_{1}^{2}+p&\rho u_{1}u_{2}&\rho u_{1}u_{3}\\ \rho u_{1}u_{2}&\rho u_{2}^{2}+p&\rho u_{2}u_{3}\\ \rho u_{3}u_{1}&\rho u_{3}u_{2}&\rho u_{3}^{2}+p\\ u_{1}&u_{2}&u_{3}\end{pmatrix}.
  142. y = ( j 1 j 2 j 3 ) ; F = ( j 1 j 2 j 3 j 1 2 ρ + p j 1 j 2 ρ j 1 j 3 ρ j 1 j 2 ρ j 2 2 ρ + p j 2 j 3 ρ j 3 j 1 ρ j 3 j 2 ρ j 3 2 ρ + p ( E t + p ) j 1 ρ ( E t + p ) j 2 ρ ( E t + p ) j 3 ρ ) . {y}=\begin{pmatrix}j_{1}\\ j_{2}\\ j_{3}\end{pmatrix};\quad{F}=\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ \frac{j_{1}^{2}}{\rho}+p&\frac{j_{1}j_{2}}{\rho}&\frac{j_{1}j_{3}}{\rho}\\ \frac{j_{1}j_{2}}{\rho}&\frac{j_{2}^{2}}{\rho}+p&\frac{j_{2}j_{3}}{\rho}\\ \frac{j_{3}j_{1}}{\rho}&\frac{j_{3}j_{2}}{\rho}&\frac{j_{3}^{2}}{\rho}+p\\ (E^{t}+p)\frac{j_{1}}{\rho}&(E^{t}+p)\frac{j_{2}}{\rho}&(E^{t}+p)\frac{j_{3}}{% \rho}\end{pmatrix}.

Euler_product.html

  1. a a
  2. n a ( n ) n - s \sum_{n}a(n)n^{-s}\,
  3. p P ( p , s ) \prod_{p}P(p,s)\,
  4. p p
  5. P ( p , s ) P(p,s)
  6. 1 + a ( p ) p - s + a ( p 2 ) p - 2 s + . 1+a(p)p^{-s}+a(p^{2})p^{-2s}+\cdots.
  7. a ( n ) a(n)
  8. a ( n ) a(n)
  9. a ( p k ) a(p^{k})
  10. n n
  11. p k p^{k}
  12. p p
  13. a ( n ) a(n)
  14. P ( p , s ) P(p,s)
  15. P ( p , s ) = 1 1 - a ( p ) p - s , P(p,s)=\frac{1}{1-a(p)p^{-s}},
  16. a ( n ) = 1 a(n)=1
  17. ζ ( s ) \zeta(s)
  18. p ( 1 - p - s ) - 1 = p ( n = 0 p - n s ) = n = 1 1 n s = ζ ( s ) \prod_{p}(1-p^{-s})^{-1}=\prod_{p}\Big(\sum_{n=0}^{\infty}p^{-ns}\Big)=\sum_{n% =1}^{\infty}\frac{1}{n^{s}}=\zeta(s)
  19. λ ( n ) = ( - 1 ) Ω ( n ) \lambda(n)=(-1)^{\Omega(n)}
  20. p ( 1 + p - s ) - 1 = n = 1 λ ( n ) n s = ζ ( 2 s ) ζ ( s ) \prod_{p}(1+p^{-s})^{-1}=\sum_{n=1}^{\infty}\frac{\lambda(n)}{n^{s}}=\frac{% \zeta(2s)}{\zeta(s)}
  21. μ ( n ) \mu(n)
  22. p ( 1 - p - s ) = n = 1 μ ( n ) n s = 1 ζ ( s ) \prod_{p}(1-p^{-s})=\sum_{n=1}^{\infty}\frac{\mu(n)}{n^{s}}=\frac{1}{\zeta(s)}
  23. p ( 1 + p - s ) = n = 1 | μ ( n ) | n s = ζ ( s ) ζ ( 2 s ) \prod_{p}(1+p^{-s})=\sum_{n=1}^{\infty}\frac{|\mu(n)|}{n^{s}}=\frac{\zeta(s)}{% \zeta(2s)}
  24. p ( 1 + p - s 1 - p - s ) = p ( p s + 1 p s - 1 ) = ζ ( s ) 2 ζ ( 2 s ) \prod_{p}\Big(\frac{1+p^{-s}}{1-p^{-s}}\Big)=\prod_{p}\Big(\frac{p^{s}+1}{p^{s% }-1}\Big)=\frac{\zeta(s)^{2}}{\zeta(2s)}
  25. ζ ( s ) \zeta(s)
  26. π s \pi^{s}
  27. ζ ( 2 ) = π 2 / 6 \zeta(2)=\pi^{2}/6
  28. ζ ( 4 ) = π 4 / 90 \zeta(4)=\pi^{4}/90
  29. ζ ( 8 ) = π 8 / 9450 \zeta(8)=\pi^{8}/9450
  30. p ( p 2 + 1 p 2 - 1 ) = 5 2 \prod_{p}\Big(\frac{p^{2}+1}{p^{2}-1}\Big)=\frac{5}{2}
  31. p ( p 4 + 1 p 4 - 1 ) = 7 6 \prod_{p}\Big(\frac{p^{4}+1}{p^{4}-1}\Big)=\frac{7}{6}
  32. p ( 1 + 2 p - s + 2 p - 2 s + ) = n = 1 2 ω ( n ) n - s = ζ ( s ) 2 ζ ( 2 s ) \prod_{p}(1+2p^{-s}+2p^{-2s}+\cdots)=\sum_{n=1}^{\infty}2^{\omega(n)}n^{-s}=% \frac{\zeta(s)^{2}}{\zeta(2s)}
  33. ω ( n ) \omega(n)
  34. 2 ω ( n ) 2^{\omega(n)}
  35. χ ( n ) \chi(n)
  36. N N
  37. χ \chi
  38. χ ( n ) \chi(n)
  39. χ ( n ) = 0 \chi(n)=0
  40. p ( 1 - χ ( p ) p - s ) - 1 = n = 1 χ ( n ) n - s \prod_{p}(1-\chi(p)p^{-s})^{-1}=\sum_{n=1}^{\infty}\chi(n)n^{-s}
  41. p ( x - p - s ) 1 Li s ( x ) \prod_{p}(x-p^{-s})\approx\frac{1}{\operatorname{Li}_{s}(x)}
  42. s > 1 s>1
  43. Li s ( x ) \operatorname{Li}_{s}(x)
  44. x = 1 x=1
  45. 1 / ζ ( s ) . 1/\zeta(s).
  46. π / 4 = n = 0 ( - 1 ) n 2 n + 1 = 1 - 1 3 + 1 5 - 1 7 + , \pi/4=\sum_{n=0}^{\infty}\,\frac{(-1)^{n}}{2n+1}=1-\frac{1}{3}+\frac{1}{5}-% \frac{1}{7}+\cdots,
  47. π / 4 = ( p 1 ( mod 4 ) p p - 1 ) ( p 3 ( mod 4 ) p p + 1 ) = 3 4 5 4 7 8 11 12 13 12 , \pi/4=\left(\prod_{p\equiv 1\;\;(\mathop{{\rm mod}}4)}\frac{p}{p-1}\right)% \cdot\left(\prod_{p\equiv 3\;\;(\mathop{{\rm mod}}4)}\frac{p}{p+1}\right)=% \frac{3}{4}\cdot\frac{5}{4}\cdot\frac{7}{8}\cdot\frac{11}{12}\cdot\frac{13}{12% }\cdots,
  48. p > 2 ( 1 - 1 ( p - 1 ) 2 ) = 0.660161... \prod_{p>2}\Big(1-\frac{1}{(p-1)^{2}}\Big)=0.660161...
  49. π 4 p = 1 mod 4 ( 1 - 1 p 2 ) 1 / 2 = 0.764223... \frac{\pi}{4}\prod_{p=1\,\,\text{mod}\,4}\Big(1-\frac{1}{p^{2}}\Big)^{1/2}=0.7% 64223...
  50. 1 2 p = 3 mod 4 ( 1 - 1 p 2 ) - 1 / 2 = 0.764223... \frac{1}{\sqrt{2}}\prod_{p=3\,\,\text{mod}\,4}\Big(1-\frac{1}{p^{2}}\Big)^{-1/% 2}=0.764223...
  51. p ( 1 + 1 ( p - 1 ) 2 ) = 2.826419... \prod_{p}\Big(1+\frac{1}{(p-1)^{2}}\Big)=2.826419...
  52. × ζ ( 2 ) 2 \times\zeta(2)^{2}
  53. p ( 1 - 1 ( p + 1 ) 2 ) = 0.775883... \prod_{p}\Big(1-\frac{1}{(p+1)^{2}}\Big)=0.775883...
  54. p ( 1 - 1 p ( p - 1 ) ) = 0.373955... \prod_{p}\Big(1-\frac{1}{p(p-1)}\Big)=0.373955...
  55. p ( 1 + 1 p ( p - 1 ) ) = 315 2 π 4 ζ ( 3 ) = 1.943596... \prod_{p}\Big(1+\frac{1}{p(p-1)}\Big)=\frac{315}{2\pi^{4}}\zeta(3)=1.943596...
  56. × ζ ( 2 ) \times\zeta(2)
  57. p ( 1 - 1 p ( p + 1 ) ) = 0.704442... \prod_{p}\Big(1-\frac{1}{p(p+1)}\Big)=0.704442...
  58. p ( 1 + 1 p 2 + p - 1 ) = 1.419562... \prod_{p}\Big(1+\frac{1}{p^{2}+p-1}\Big)=1.419562...
  59. 1 2 + 1 2 p ( 1 - 2 p 2 ) = 0.661317... \frac{1}{2}+\frac{1}{2}\prod_{p}\Big(1-\frac{2}{p^{2}}\Big)=0.661317...
  60. p ( 1 - 1 p 2 ( p + 1 ) ) = 0.881513... \prod_{p}\Big(1-\frac{1}{p^{2}(p+1)}\Big)=0.881513...
  61. p ( 1 + 1 p 2 ( p - 1 ) ) = 1.339784... \prod_{p}\Big(1+\frac{1}{p^{2}(p-1)}\Big)=1.339784...
  62. p > 2 ( 1 - p + 2 p 3 ) = 0.723648... \prod_{p>2}\Big(1-\frac{p+2}{p^{3}}\Big)=0.723648...
  63. p ( 1 - 2 p - 1 p 3 ) = 0.428249... \prod_{p}\Big(1-\frac{2p-1}{p^{3}}\Big)=0.428249...
  64. p ( 1 - 3 p - 2 p 3 ) = 0.286747... \prod_{p}\Big(1-\frac{3p-2}{p^{3}}\Big)=0.286747...
  65. p ( 1 - p p 3 - 1 ) = 0.575959... \prod_{p}\Big(1-\frac{p}{p^{3}-1}\Big)=0.575959...
  66. p ( 1 + 3 p 2 - 1 p ( p + 1 ) ( p 2 - 1 ) ) = 2.596536... \prod_{p}\Big(1+\frac{3p^{2}-1}{p(p+1)(p^{2}-1)}\Big)=2.596536...
  67. p ( 1 - 3 p 3 + 2 p 4 + 1 p 5 - 1 p 6 ) = 0.678234... \prod_{p}\Big(1-\frac{3}{p^{3}}+\frac{2}{p^{4}}+\frac{1}{p^{5}}-\frac{1}{p^{6}% }\Big)=0.678234...
  68. p ( 1 - 1 p ) 7 ( 1 + 7 p + 1 p 2 ) = 0.0013176... \prod_{p}\Big(1-\frac{1}{p}\Big)^{7}\Big(1+\frac{7p+1}{p^{2}}\Big)=0.0013176...

Eulerian_path.html

  1. O ( | E | log 3 | E | log log | E | ) O(|E|\log^{3}|E|\log\log|E|)
  2. e c ( K n ) = 2 ( n + 1 ) / 2 π 1 / 2 e - n 2 / 2 + 11 / 12 n ( n - 2 ) ( n + 1 ) / 2 ( 1 + O ( n - 1 / 2 + ϵ ) ) . ec(K_{n})=2^{(n+1)/2}\pi^{1/2}e^{-n^{2}/2+11/12}n^{(n-2)(n+1)/2}\bigl(1+O(n^{-% 1/2+\epsilon})\bigr).
  3. e c ( K n , n ) = ( n / 2 - 1 ) ! 2 n 2 n 2 - n + 1 / 2 π - n + 1 / 2 n n - 1 ( 1 + O ( n - 1 / 2 + ϵ ) ) . ec(K_{n,n})=(n/2-1)!^{2n}2^{n^{2}-n+1/2}\pi^{-n+1/2}n^{n-1}\bigl(1+O(n^{-1/2+% \epsilon})\bigr).

Even_and_odd_functions.html

  1. f ( x ) = f ( - x ) , f(x)=f(-x),\,
  2. f ( x ) - f ( - x ) = 0. f(x)-f(-x)=0.\,
  3. | |
  4. | |
  5. - f ( x ) = f ( - x ) , -f(x)=f(-x),\,
  6. f ( x ) + f ( - x ) = 0. f(x)+f(-x)=0.\,
  7. f ( x ) = f e ( x ) + f o ( x ) , f(x)=f\text{e}(x)+f\text{o}(x)\,,
  8. f e ( x ) = 1 2 [ f ( x ) + f ( - x ) ] f\text{e}(x)=\tfrac{1}{2}[f(x)+f(-x)]
  9. f o ( x ) = 1 2 [ f ( x ) - f ( - x ) ] f\text{o}(x)=\tfrac{1}{2}[f(x)-f(-x)]
  10. t t
  11. t t
  12. V out ( t ) = f ( V in ( t ) ) V\text{out}(t)=f(V\text{in}(t))
  13. f f
  14. 0 f , 2 f , 4 f , 6 f , 0f,2f,4f,6f,\dots
  15. 0 f 0f
  16. 1 f , 3 f , 5 f , 1f,3f,5f,\dots
  17. 1 f , 2 f , 3 f , 1f,2f,3f,\dots

Event-related_potential.html

  1. σ 2 \sigma^{2}
  2. k k
  3. t t
  4. k k
  5. x ( t , k ) = s ( t ) + n ( t , k ) x(t,k)=s(t)+n(t,k)
  6. s ( t ) s(t)
  7. n ( t , k ) n(t,k)
  8. N N
  9. x ¯ ( t ) = 1 N k = 1 N x ( t , k ) = s ( t ) + 1 N k = 1 N n ( t , k ) \bar{x}(t)=\frac{1}{N}\sum_{k=1}^{N}x(t,k)=s(t)+\frac{1}{N}\sum_{k=1}^{N}n(t,k)
  10. x ¯ ( t ) \bar{x}(t)
  11. E [ x ¯ ( t ) ] = s ( t ) \operatorname{E}[\bar{x}(t)]=s(t)
  12. Var [ x ¯ ( t ) ] = E [ ( x ¯ ( t ) - E [ x ¯ ( t ) ] ) 2 ] = 1 N 2 E [ ( k = 1 N n ( t , k ) ) 2 ] = 1 N 2 k = 1 N E [ n ( t , k ) 2 ] = σ 2 N \operatorname{Var}[\bar{x}(t)]=\operatorname{E}\left[\left(\bar{x}(t)-% \operatorname{E}[\bar{x}(t)]\right)^{2}\right]=\frac{1}{N^{2}}\operatorname{E}% \left[\left(\sum_{k=1}^{N}n(t,k)\right)^{2}\right]=\frac{1}{N^{2}}\sum_{k=1}^{% N}\operatorname{E}\left[n(t,k)^{2}\right]=\frac{\sigma^{2}}{N}
  13. N N
  14. 1 / N 1/{\sqrt{N}}

Exact_differential.html

  1. A ( x , y , z ) d x + B ( x , y , z ) d y + C ( x , y , z ) d z A(x,y,z)dx+B(x,y,z)dy+C(x,y,z)dz
  2. D 3 D\subset\mathbb{R}^{3}
  3. Q = Q ( x , y , z ) Q=Q(x,y,z)
  4. D D
  5. d Q ( Q x ) y , z d x + ( Q y ) z , x d y + ( Q z ) x , y d z , \;dQ\;\equiv\;\left(\frac{\partial Q}{\partial x}\right)_{{y,z}}dx\quad+\quad% \left(\frac{\partial Q}{\partial y}\right)_{{z,x}}dy\quad+\quad\left(\frac{% \partial Q}{\partial z}\right)_{{x,y}}dz,
  6. d Q = A d x + B d y + C d z dQ=Adx+Bdy+Cdz
  7. ( A , B , C ) (A,B,C)
  8. Q Q
  9. A ( x ) d x A(x)\,dx
  10. A A
  11. Q Q
  12. A A
  13. A A
  14. d Q = A ( x ) d x dQ=A(x)\,dx
  15. Q Q
  16. 2 Q x y = 2 Q y x \frac{\partial^{2}Q}{\partial x\partial y}=\frac{\partial^{2}Q}{\partial y% \partial x}
  17. A ( x , y ) d x + B ( x , y ) d y A(x,y)\,dx+B(x,y)\,dy
  18. ( A y ) x = ( B x ) y \left(\frac{\partial A}{\partial y}\right)_{x}=\left(\frac{\partial B}{% \partial x}\right)_{y}
  19. d Q = A ( x , y , z ) d x + B ( x , y , z ) d y + C ( x , y , z ) d z dQ=A(x,y,z)\,dx+B(x,y,z)\,dy+C(x,y,z)\,dz
  20. ( A y ) x , z = ( B x ) y , z \left(\frac{\partial A}{\partial y}\right)_{x,z}\!\!\!=\left(\frac{\partial B}% {\partial x}\right)_{y,z}
  21. ( A z ) x , y = ( C x ) y , z \left(\frac{\partial A}{\partial z}\right)_{x,y}\!\!\!=\left(\frac{\partial C}% {\partial x}\right)_{y,z}
  22. ( B z ) x , y = ( C y ) x , z \left(\frac{\partial B}{\partial z}\right)_{x,y}\!\!\!=\left(\frac{\partial C}% {\partial y}\right)_{x,z}
  23. i f d Q = Q ( f ) - Q ( i ) , \int_{i}^{f}dQ=Q(f)-Q(i),
  24. x x
  25. y y
  26. z z
  27. F ( x , y , z ) = constant F(x,y,z)=\,\text{constant}
  28. F ( x , y , z ) F(x,y,z)
  29. d x = ( x y ) x d y + ( x z ) y d z dx={\left(\frac{\partial x}{\partial y}\right)}_{x}\,dy+{\left(\frac{\partial x% }{\partial z}\right)}_{y}\,dz
  30. d z = ( z x ) y d x + ( z y ) x d y . dz={\left(\frac{\partial z}{\partial x}\right)}_{y}\,dx+{\left(\frac{\partial z% }{\partial y}\right)}_{x}\,dy.
  31. d z = ( z x ) y [ ( x y ) z d y + ( x z ) y d z ] + ( z y ) x d y , dz={\left(\frac{\partial z}{\partial x}\right)}_{y}\left[{\left(\frac{\partial x% }{\partial y}\right)}_{z}dy+{\left(\frac{\partial x}{\partial z}\right)}_{y}dz% \right]+{\left(\frac{\partial z}{\partial y}\right)}_{x}dy,
  32. d z = [ ( z x ) y ( x y ) z + ( z y ) x ] d y + ( z x ) y ( x z ) y d z , dz=\left[{\left(\frac{\partial z}{\partial x}\right)}_{y}{\left(\frac{\partial x% }{\partial y}\right)}_{z}+{\left(\frac{\partial z}{\partial y}\right)}_{x}% \right]dy+{\left(\frac{\partial z}{\partial x}\right)}_{y}{\left(\frac{% \partial x}{\partial z}\right)}_{y}dz,
  33. [ 1 - ( z x ) y ( x z ) y ] d z = [ ( z x ) y ( x y ) z + ( z y ) x ] d y . \left[1-{\left(\frac{\partial z}{\partial x}\right)}_{y}{\left(\frac{\partial x% }{\partial z}\right)}_{y}\right]dz=\left[{\left(\frac{\partial z}{\partial x}% \right)}_{y}{\left(\frac{\partial x}{\partial y}\right)}_{z}+{\left(\frac{% \partial z}{\partial y}\right)}_{x}\right]dy.
  34. y y
  35. z z
  36. d y dy
  37. d z dz
  38. ( z x ) y ( x z ) y = 1. {\left(\frac{\partial z}{\partial x}\right)}_{y}{\left(\frac{\partial x}{% \partial z}\right)}_{y}=1.
  39. ( z x ) y = 1 ( x z ) y . {\left(\frac{\partial z}{\partial x}\right)}_{y}=\frac{1}{{\left(\frac{% \partial x}{\partial z}\right)}_{y}}.
  40. x x
  41. y y
  42. z z
  43. ( z x ) y ( x y ) z = - ( z y ) x . {\left(\frac{\partial z}{\partial x}\right)}_{y}{\left(\frac{\partial x}{% \partial y}\right)}_{z}=-{\left(\frac{\partial z}{\partial y}\right)}_{x}.
  44. z y \tfrac{\partial z}{\partial y}
  45. ( x y ) z ( y z ) x ( z x ) y = - 1. {\left(\frac{\partial x}{\partial y}\right)}_{z}{\left(\frac{\partial y}{% \partial z}\right)}_{x}{\left(\frac{\partial z}{\partial x}\right)}_{y}=-1.
  46. x y \tfrac{\partial x}{\partial y}
  47. ( y x ) z = - ( z x ) y ( z y ) x . {\left(\frac{\partial y}{\partial x}\right)}_{z}=-\frac{{\left(\frac{\partial z% }{\partial x}\right)}_{y}}{{\left(\frac{\partial z}{\partial y}\right)}_{x}}.
  48. z , x , y , u z,x,y,u
  49. v v
  50. ( 1 ) d z = ( z x ) y d x + ( z y ) x d y = ( z u ) v d u + ( z v ) u d v (1)~{}~{}~{}~{}~{}dz=\left(\frac{\partial z}{\partial x}\right)_{y}dx+\left(% \frac{\partial z}{\partial y}\right)_{x}dy=\left(\frac{\partial z}{\partial u}% \right)_{v}du+\left(\frac{\partial z}{\partial v}\right)_{u}dv
  51. ( 2 ) d x = ( x u ) v d u + ( x v ) u d v (2)~{}~{}~{}~{}~{}dx=\left(\frac{\partial x}{\partial u}\right)_{v}du+\left(% \frac{\partial x}{\partial v}\right)_{u}dv
  52. ( 3 ) d y = ( y u ) v d u + ( y v ) u d v (3)~{}~{}~{}~{}~{}dy=\left(\frac{\partial y}{\partial u}\right)_{v}du+\left(% \frac{\partial y}{\partial v}\right)_{u}dv
  53. ( 4 ) d z = [ ( z x ) y ( x u ) v + ( z y ) x ( y u ) v ] d u (4)~{}~{}~{}~{}~{}dz=\left[\left(\frac{\partial z}{\partial x}\right)_{y}\left% (\frac{\partial x}{\partial u}\right)_{v}+\left(\frac{\partial z}{\partial y}% \right)_{x}\left(\frac{\partial y}{\partial u}\right)_{v}\right]du
  54. + [ ( z x ) y ( x v ) u + ( z y ) x ( y v ) u ] d v +\left[\left(\frac{\partial z}{\partial x}\right)_{y}\left(\frac{\partial x}{% \partial v}\right)_{u}+\left(\frac{\partial z}{\partial y}\right)_{x}\left(% \frac{\partial y}{\partial v}\right)_{u}\right]dv
  55. ( 5 ) ( z u ) v = ( z x ) y ( x u ) v + ( z y ) x ( y u ) v (5)~{}~{}~{}~{}~{}\left(\frac{\partial z}{\partial u}\right)_{v}=\left(\frac{% \partial z}{\partial x}\right)_{y}\left(\frac{\partial x}{\partial u}\right)_{% v}+\left(\frac{\partial z}{\partial y}\right)_{x}\left(\frac{\partial y}{% \partial u}\right)_{v}
  56. v = y v=y
  57. ( 6 ) ( z u ) y = ( z x ) y ( x u ) y (6)~{}~{}~{}~{}~{}\left(\frac{\partial z}{\partial u}\right)_{y}=\left(\frac{% \partial z}{\partial x}\right)_{y}\left(\frac{\partial x}{\partial u}\right)_{y}
  58. u = y u=y
  59. ( 7 ) ( z y ) v = ( z y ) x + ( z x ) y ( x y ) v (7)~{}~{}~{}~{}~{}\left(\frac{\partial z}{\partial y}\right)_{v}=\left(\frac{% \partial z}{\partial y}\right)_{x}+\left(\frac{\partial z}{\partial x}\right)_% {y}\left(\frac{\partial x}{\partial y}\right)_{v}
  60. u = y u=y
  61. v = z v=z
  62. ( 8 ) ( z y ) x = - ( z x ) y ( x y ) z (8)~{}~{}~{}~{}~{}\left(\frac{\partial z}{\partial y}\right)_{x}=-\left(\frac{% \partial z}{\partial x}\right)_{y}\left(\frac{\partial x}{\partial y}\right)_{z}
  63. a / b ) c = 1 / ( b / a ) c \partial a/\partial b)_{c}=1/(\partial b/\partial a)_{c}
  64. ( 9 ) ( z x ) y ( x y ) z ( y z ) x = - 1 (9)~{}~{}~{}~{}~{}\left(\frac{\partial z}{\partial x}\right)_{y}\left(\frac{% \partial x}{\partial y}\right)_{z}\left(\frac{\partial y}{\partial z}\right)_{% x}=-1

Exergonic_reaction.html

  1. Δ G = G products - G reactants < 0. \Delta G=G_{\rm{products}}-G_{\rm{reactants}}<0.\,

Expectation–maximization_algorithm.html

  1. 𝐗 \mathbf{X}
  2. 𝐙 \mathbf{Z}
  3. s y m b o l θ symbol\theta
  4. L ( s y m b o l θ ; 𝐗 , 𝐙 ) = p ( 𝐗 , 𝐙 | s y m b o l θ ) L(symbol\theta;\mathbf{X},\mathbf{Z})=p(\mathbf{X},\mathbf{Z}|symbol\theta)
  5. L ( s y m b o l θ ; 𝐗 ) = p ( 𝐗 | s y m b o l θ ) = 𝐙 p ( 𝐗 , 𝐙 | s y m b o l θ ) L(symbol\theta;\mathbf{X})=p(\mathbf{X}|symbol\theta)=\sum_{\mathbf{Z}}p(% \mathbf{X},\mathbf{Z}|symbol\theta)
  6. 𝐙 \mathbf{Z}
  7. 𝐙 \mathbf{Z}
  8. 𝐗 \mathbf{X}
  9. s y m b o l θ ( t ) symbol\theta^{(t)}
  10. Q ( s y m b o l θ | s y m b o l θ ( t ) ) = E 𝐙 | 𝐗 , s y m b o l θ ( t ) [ log L ( s y m b o l θ ; 𝐗 , 𝐙 ) ] Q(symbol\theta|symbol\theta^{(t)})=\operatorname{E}_{\mathbf{Z}|\mathbf{X},% symbol\theta^{(t)}}\left[\log L(symbol\theta;\mathbf{X},\mathbf{Z})\right]\,
  11. s y m b o l θ ( t + 1 ) = arg max s y m b o l θ Q ( s y m b o l θ | s y m b o l θ ( t ) ) symbol\theta^{(t+1)}=\underset{symbol\theta}{\operatorname{arg\,max}}\ Q(% symbol\theta|symbol\theta^{(t)})\,
  12. 𝐗 \mathbf{X}
  13. 𝐙 \mathbf{Z}
  14. s y m b o l θ symbol\theta
  15. 𝐙 \mathbf{Z}
  16. 𝐙 \mathbf{Z}
  17. 𝐙 \mathbf{Z}
  18. 𝐙 \mathbf{Z}
  19. s y m b o l θ symbol\theta
  20. s y m b o l θ symbol\theta
  21. 𝐙 \mathbf{Z}
  22. s y m b o l θ symbol\theta
  23. 𝐙 \mathbf{Z}
  24. 𝐙 \mathbf{Z}
  25. s y m b o l θ symbol\theta
  26. 𝐙 \mathbf{Z}
  27. 𝐙 \mathbf{Z}
  28. 𝐙 \mathbf{Z}
  29. 𝐙 \mathbf{Z}
  30. 𝐙 \mathbf{Z}
  31. 𝐙 \mathbf{Z}
  32. Q ( s y m b o l θ | s y m b o l θ ( t ) ) Q(symbol\theta|symbol\theta^{(t)})
  33. log p ( 𝐗 | s y m b o l θ ) \log p(\mathbf{X}|symbol\theta)
  34. 𝐙 \mathbf{Z}
  35. p ( 𝐙 | 𝐗 , s y m b o l θ ) p(\mathbf{Z}|\mathbf{X},symbol\theta)
  36. log p ( 𝐗 | s y m b o l θ ) = log p ( 𝐗 , 𝐙 | s y m b o l θ ) - log p ( 𝐙 | 𝐗 , s y m b o l θ ) . \log p(\mathbf{X}|symbol\theta)=\log p(\mathbf{X},\mathbf{Z}|symbol\theta)-% \log p(\mathbf{Z}|\mathbf{X},symbol\theta)\,.
  37. 𝐙 \mathbf{Z}
  38. p ( 𝐙 | 𝐗 , s y m b o l θ ( t ) ) p(\mathbf{Z}|\mathbf{X},symbol\theta^{(t)})
  39. 𝐙 \mathbf{Z}
  40. log p ( 𝐗 | s y m b o l θ ) \displaystyle\log p(\mathbf{X}|symbol\theta)
  41. H ( s y m b o l θ | s y m b o l θ ( t ) ) H(symbol\theta|symbol\theta^{(t)})
  42. s y m b o l θ symbol\theta
  43. s y m b o l θ = s y m b o l θ ( t ) symbol\theta=symbol\theta^{(t)}
  44. log p ( 𝐗 | s y m b o l θ ( t ) ) = Q ( s y m b o l θ ( t ) | s y m b o l θ ( t ) ) + H ( s y m b o l θ ( t ) | s y m b o l θ ( t ) ) , \log p(\mathbf{X}|symbol\theta^{(t)})=Q(symbol\theta^{(t)}|symbol\theta^{(t)})% +H(symbol\theta^{(t)}|symbol\theta^{(t)})\,,
  45. log p ( 𝐗 | s y m b o l θ ) - log p ( 𝐗 | s y m b o l θ ( t ) ) = Q ( s y m b o l θ | s y m b o l θ ( t ) ) - Q ( s y m b o l θ ( t ) | s y m b o l θ ( t ) ) + H ( s y m b o l θ | s y m b o l θ ( t ) ) - H ( s y m b o l θ ( t ) | s y m b o l θ ( t ) ) , \log p(\mathbf{X}|symbol\theta)-\log p(\mathbf{X}|symbol\theta^{(t)})=Q(symbol% \theta|symbol\theta^{(t)})-Q(symbol\theta^{(t)}|symbol\theta^{(t)})+H(symbol% \theta|symbol\theta^{(t)})-H(symbol\theta^{(t)}|symbol\theta^{(t)})\,,
  46. H ( s y m b o l θ | s y m b o l θ ( t ) ) H ( s y m b o l θ ( t ) | s y m b o l θ ( t ) ) H(symbol\theta|symbol\theta^{(t)})\geq H(symbol\theta^{(t)}|symbol\theta^{(t)})
  47. log p ( 𝐗 | s y m b o l θ ) - log p ( 𝐗 | s y m b o l θ ( t ) ) Q ( s y m b o l θ | s y m b o l θ ( t ) ) - Q ( s y m b o l θ ( t ) | s y m b o l θ ( t ) ) . \log p(\mathbf{X}|symbol\theta)-\log p(\mathbf{X}|symbol\theta^{(t)})\geq Q(% symbol\theta|symbol\theta^{(t)})-Q(symbol\theta^{(t)}|symbol\theta^{(t)})\,.
  48. s y m b o l θ symbol\theta
  49. Q ( s y m b o l θ | s y m b o l θ ( t ) ) Q(symbol\theta|symbol\theta^{(t)})
  50. Q ( s y m b o l θ ( t ) | s y m b o l θ ( t ) ) Q(symbol\theta^{(t)}|symbol\theta^{(t)})
  51. log p ( 𝐗 | s y m b o l θ ) \log p(\mathbf{X}|symbol\theta)
  52. log p ( 𝐗 | s y m b o l θ ( t ) ) \log p(\mathbf{X}|symbol\theta^{(t)})
  53. F ( q , θ ) = E q [ log L ( θ ; x , Z ) ] + H ( q ) = - D KL ( q p Z | X ( | x ; θ ) ) + log L ( θ ; x ) F(q,\theta)=\operatorname{E}_{q}[\log L(\theta;x,Z)]+H(q)=-D_{\mathrm{KL}}\big% (q\big\|p_{Z|X}(\cdot|x;\theta)\big)+\log L(\theta;x)
  54. q ( t ) = arg max q F ( q , θ ( t ) ) q^{(t)}=\operatorname*{arg\,max}_{q}\ F(q,\theta^{(t)})
  55. θ ( t + 1 ) = arg max θ F ( q ( t ) , θ ) \theta^{(t+1)}=\operatorname*{arg\,max}_{\theta}\ F(q^{(t)},\theta)
  56. σ ^ v 2 = 1 N k = 1 N ( z k - x ^ k ) 2 \hat{\sigma}^{2}_{v}=\frac{1}{N}\sum_{k=1}^{N}{(z_{k}-\hat{x}_{k})}^{2}
  57. x ^ k \hat{x}_{k}
  58. z k {z_{k}}
  59. σ ^ w 2 = 1 N k = 1 N ( x ^ k + 1 - F ^ < ^ m t p l > x k ) 2 \hat{\sigma}^{2}_{w}=\frac{1}{N}\sum_{k=1}^{N}{(\hat{x}_{k+1}-\hat{F}\hat{<}% mtpl>{{x}}_{k})}^{2}
  60. x ^ k \hat{x}_{k}
  61. x ^ k + 1 \hat{x}_{k+1}
  62. F ^ = k = 1 N ( x ^ k + 1 - F ^ x ^ k ) k = 1 N x ^ k 2 \hat{F}=\frac{\sum_{k=1}^{N}(\hat{x}_{k+1}-\hat{F}\hat{x}_{k})}{\sum_{k=1}^{N}% \hat{x}_{k}^{2}}
  63. 𝐱 = ( 𝐱 1 , 𝐱 2 , , 𝐱 n ) \mathbf{x}=(\mathbf{x}_{1},\mathbf{x}_{2},\ldots,\mathbf{x}_{n})
  64. n n
  65. d d
  66. 𝐳 = ( z 1 , z 2 , , z n ) \mathbf{z}=(z_{1},z_{2},\ldots,z_{n})
  67. X i | ( Z i = 1 ) 𝒩 d ( s y m b o l μ 1 , Σ 1 ) X_{i}|(Z_{i}=1)\sim\mathcal{N}_{d}(symbol{\mu}_{1},\Sigma_{1})
  68. X i | ( Z i = 2 ) 𝒩 d ( s y m b o l μ 2 , Σ 2 ) X_{i}|(Z_{i}=2)\sim\mathcal{N}_{d}(symbol{\mu}_{2},\Sigma_{2})
  69. P ( Z i = 1 ) = τ 1 \operatorname{P}(Z_{i}=1)=\tau_{1}\,
  70. P ( Z i = 2 ) = τ 2 = 1 - τ 1 \operatorname{P}(Z_{i}=2)=\tau_{2}=1-\tau_{1}
  71. θ = ( s y m b o l τ , s y m b o l μ 1 , s y m b o l μ 2 , Σ 1 , Σ 2 ) \theta=\big(symbol{\tau},symbol{\mu}_{1},symbol{\mu}_{2},\Sigma_{1},\Sigma_{2}\big)
  72. L ( θ ; 𝐱 ) = i = 1 n j = 1 2 τ j f ( 𝐱 i ; s y m b o l μ j , Σ j ) L(\theta;\mathbf{x})=\prod_{i=1}^{n}\sum_{j=1}^{2}\tau_{j}\ f(\mathbf{x}_{i};% symbol{\mu}_{j},\Sigma_{j})
  73. L ( θ ; 𝐱 , 𝐳 ) = P ( 𝐱 , 𝐳 | θ ) = i = 1 n j = 1 2 𝕀 ( z i = j ) f ( 𝐱 i ; s y m b o l μ j , Σ j ) τ j L(\theta;\mathbf{x},\mathbf{z})=P(\mathbf{x},\mathbf{z}|\theta)=\prod_{i=1}^{n% }\sum_{j=1}^{2}\mathbb{I}(z_{i}=j)\ f(\mathbf{x}_{i};symbol{\mu}_{j},\Sigma_{j% })\tau_{j}
  74. L ( θ ; 𝐱 , 𝐳 ) = exp { i = 1 n j = 1 2 𝕀 ( z i = j ) [ log τ j - 1 2 log | Σ j | - 1 2 ( 𝐱 i - s y m b o l μ j ) Σ j - 1 ( 𝐱 i - s y m b o l μ j ) - d 2 log ( 2 π ) ] } . L(\theta;\mathbf{x},\mathbf{z})=\exp\left\{\sum_{i=1}^{n}\sum_{j=1}^{2}\mathbb% {I}(z_{i}=j)\big[\log\tau_{j}-\tfrac{1}{2}\log|\Sigma_{j}|-\tfrac{1}{2}(% \mathbf{x}_{i}-symbol{\mu}_{j})^{\top}\Sigma_{j}^{-1}(\mathbf{x}_{i}-symbol{% \mu}_{j})-\tfrac{d}{2}\log(2\pi)\big]\right\}.
  75. 𝕀 \mathbb{I}
  76. f f
  77. 𝕀 ( z i = j ) \mathbb{I}(z_{i}=j)
  78. T j , i ( t ) := P ( Z i = j | X i = 𝐱 i ; θ ( t ) ) = τ j ( t ) f ( 𝐱 i ; s y m b o l μ j ( t ) , Σ j ( t ) ) τ 1 ( t ) f ( 𝐱 i ; s y m b o l μ 1 ( t ) , Σ 1 ( t ) ) + τ 2 ( t ) f ( 𝐱 i ; s y m b o l μ 2 ( t ) , Σ 2 ( t ) ) T_{j,i}^{(t)}:=\operatorname{P}(Z_{i}=j|X_{i}=\mathbf{x}_{i};\theta^{(t)})=% \frac{\tau_{j}^{(t)}\ f(\mathbf{x}_{i};symbol{\mu}_{j}^{(t)},\Sigma_{j}^{(t)})% }{\tau_{1}^{(t)}\ f(\mathbf{x}_{i};symbol{\mu}_{1}^{(t)},\Sigma_{1}^{(t)})+% \tau_{2}^{(t)}\ f(\mathbf{x}_{i};symbol{\mu}_{2}^{(t)},\Sigma_{2}^{(t)})}
  79. Q ( θ | θ ( t ) ) \displaystyle Q(\theta|\theta^{(t)})
  80. s y m b o l τ ( t + 1 ) \displaystyle symbol{\tau}^{(t+1)}
  81. τ j ( t + 1 ) = i = 1 n T j , i ( t ) i = 1 n ( T 1 , i ( t ) + T 2 , i ( t ) ) = 1 n i = 1 n T j , i ( t ) \tau^{(t+1)}_{j}=\frac{\sum_{i=1}^{n}T_{j,i}^{(t)}}{\sum_{i=1}^{n}(T_{1,i}^{(t% )}+T_{2,i}^{(t)})}=\frac{1}{n}\sum_{i=1}^{n}T_{j,i}^{(t)}
  82. ( s y m b o l μ 1 ( t + 1 ) , Σ 1 ( t + 1 ) ) \displaystyle(symbol{\mu}_{1}^{(t+1)},\Sigma_{1}^{(t+1)})
  83. s y m b o l μ 1 ( t + 1 ) = i = 1 n T 1 , i ( t ) 𝐱 i i = 1 n T 1 , i ( t ) symbol{\mu}_{1}^{(t+1)}=\frac{\sum_{i=1}^{n}T_{1,i}^{(t)}\mathbf{x}_{i}}{\sum_% {i=1}^{n}T_{1,i}^{(t)}}
  84. Σ 1 ( t + 1 ) = i = 1 n T 1 , i ( t ) ( 𝐱 i - s y m b o l μ 1 ( t + 1 ) ) ( 𝐱 i - s y m b o l μ 1 ( t + 1 ) ) i = 1 n T 1 , i ( t ) \Sigma_{1}^{(t+1)}=\frac{\sum_{i=1}^{n}T_{1,i}^{(t)}(\mathbf{x}_{i}-symbol{\mu% }_{1}^{(t+1)})(\mathbf{x}_{i}-symbol{\mu}_{1}^{(t+1)})^{\top}}{\sum_{i=1}^{n}T% _{1,i}^{(t)}}
  85. s y m b o l μ 2 ( t + 1 ) = i = 1 n T 2 , i ( t ) 𝐱 i i = 1 n T 2 , i ( t ) symbol{\mu}_{2}^{(t+1)}=\frac{\sum_{i=1}^{n}T_{2,i}^{(t)}\mathbf{x}_{i}}{\sum_% {i=1}^{n}T_{2,i}^{(t)}}
  86. Σ 2 ( t + 1 ) = i = 1 n T 2 , i ( t ) ( 𝐱 i - s y m b o l μ 2 ( t + 1 ) ) ( 𝐱 i - s y m b o l μ 2 ( t + 1 ) ) i = 1 n T 2 , i ( t ) \Sigma_{2}^{(t+1)}=\frac{\sum_{i=1}^{n}T_{2,i}^{(t)}(\mathbf{x}_{i}-symbol{\mu% }_{2}^{(t+1)})(\mathbf{x}_{i}-symbol{\mu}_{2}^{(t+1)})^{\top}}{\sum_{i=1}^{n}T% _{2,i}^{(t)}}
  87. log L ( θ t ; 𝐱 , 𝐙 ) log L ( θ ( t - 1 ) ; 𝐱 , 𝐙 ) + ϵ \log L(\theta^{t};\mathbf{x},\mathbf{Z})\leq\log L(\theta^{(t-1)};\mathbf{x},% \mathbf{Z})+\epsilon
  88. ϵ \epsilon