wpmath0000012_0

(ε,_δ)-definition_of_limit.html

  1. ε , δ \varepsilon,\delta
  2. ε , δ \varepsilon,\delta
  3. lim x c f ( x ) = L \lim_{x\to c}f(x)=L\,
  4. ( ε , δ ) (\varepsilon,\delta)
  5. f : D f:D\rightarrow\mathbb{R}
  6. D D\subseteq\mathbb{R}
  7. c c
  8. D D
  9. L L
  10. f f
  11. L L
  12. c c
  13. ε > 0 \varepsilon>0
  14. δ > 0 \delta>0
  15. x x
  16. D D
  17. 0 < | x - c | < δ 0<|x-c|<\delta
  18. | f ( x ) - L | < ε |f(x)-L|<\varepsilon
  19. lim x c f ( x ) = L ( ε > 0 ) ( δ > 0 ) ( x D ) ( 0 < | x - c | < δ | f ( x ) - L | < ε ) \lim_{x\to c}f(x)=L\iff(\forall\varepsilon>0)(\exists\ \delta>0)(\forall x\in D% )(0<|x-c|<\delta\ \Rightarrow\ |f(x)-L|<\varepsilon)
  20. lim x 5 ( 3 x - 3 ) = 12. \lim_{x\to 5}(3x-3)=12.
  21. x x
  22. δ \delta
  23. c c
  24. f ( x ) f(x)
  25. ε \varepsilon
  26. L L
  27. x x
  28. δ \delta
  29. 3 x - 3 3x-3
  30. ε \varepsilon
  31. δ \delta
  32. ε \varepsilon
  33. 0 < | x - 5 | < δ | ( 3 x - 3 ) - 12 | < ε . 0<|x-5|<\delta\ \Rightarrow\ |(3x-3)-12|<\varepsilon.
  34. | x - 5 | < ε / 3 , |x-5|<\varepsilon/3,
  35. δ = ε / 3. \delta=\varepsilon/3.
  36. x x
  37. f ( x ) f(x)
  38. y = 3 x - 3. y=3x-3.
  39. lim x c f ( x ) = f ( c ) . \lim_{x\to c}f(x)=f(c).
  40. f ( x ) f(x)
  41. x x
  42. f ( x + e ) f(x+e)

14_(number).html

  1. x 1 , x 2 , , x n x_{1},x_{2},\dots,x_{n}
  2. i = 1 n x i x i + 1 + x i + 2 < n 2 \sum_{i=1}^{n}\frac{x_{i}}{x_{i+1}+x_{i+2}}<\frac{n}{2}
  3. x n + 1 = x 1 , x n + 2 = x 2 x_{n+1}=x_{1},x_{n+2}=x_{2}

1_33_honeycomb.html

  1. E ~ 7 {\tilde{E}}_{7}
  2. E ~ 7 {\tilde{E}}_{7}
  3. F ~ 4 {\tilde{F}}_{4}
  4. E ~ 7 {\tilde{E}}_{7}
  5. F ~ 4 {\tilde{F}}_{4}
  6. E ~ 7 {\tilde{E}}_{7}
  7. A ~ 7 {\tilde{A}}_{7}
  8. E ~ 7 {\tilde{E}}_{7}
  9. A ~ 7 {\tilde{A}}_{7}
  10. A 7 A_{7}

1_52_honeycomb.html

  1. E ~ 8 {\tilde{E}}_{8}

2',3'-Cyclic-nucleotide_3'-phosphodiesterase.html

  1. \rightleftharpoons

2-group.html

  1. π 1 ( X , x ) = π 1 ( Π 2 ( X , x ) ) . \pi_{1}(X,x)=\pi_{1}(\Pi_{2}(X,x)).\!
  2. π 2 ( X , x ) = π 2 ( Π 2 ( X , x ) ) . \pi_{2}(X,x)=\pi_{2}(\Pi_{2}(X,x)).\!

2008_Primera_División_de_Chile_season.html

  1. P d = 0.4 P 2007 19 20 + 0.6 ( P A p - 2008 + P C l - 2008 19 18 ) , P_{d}=0.4\cdot\;P_{2007}\cdot\;{\frac{19}{20}}+0.6\cdot\left(\;P_{Ap-2008}+\;P% _{Cl-2008}\cdot\;{\frac{19}{18}}\right),
  2. P d = P A p - 2008 + P C l - 2008 19 18 . P_{d}=P_{Ap-2008}+P_{Cl-2008}\cdot\;{\frac{19}{18}}.

2008–09_Olympique_Lyonnais_season.html

  1. \leftarrow
  2. \rightarrow

25::8.html

  1. 25 / 8 = 3.125 25/8=3.125

2_22_honeycomb.html

  1. E ~ 6 {\tilde{E}}_{6}
  2. E ~ 6 {\tilde{E}}_{6}
  3. E ~ 6 {\tilde{E}}_{6}
  4. F ~ 4 {\tilde{F}}_{4}
  5. E ~ 6 {\tilde{E}}_{6}
  6. F ~ 4 {\tilde{F}}_{4}

2_51_honeycomb.html

  1. E ~ 8 {\tilde{E}}_{8}

3_31_honeycomb.html

  1. E ~ 7 {\tilde{E}}_{7}
  2. E ~ 7 {\tilde{E}}_{7}
  3. A ~ 7 {\tilde{A}}_{7}
  4. E ~ 7 {\tilde{E}}_{7}
  5. A ~ 7 {\tilde{A}}_{7}
  6. A 7 A_{7}

4-Hydroxy-3-methylbut-2-enyl_diphosphate_reductase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

4_21_polytope.html

  1. ( ± 2 , ± 2 , 0 , 0 , 0 , 0 , 0 , 0 ) (\pm 2,\pm 2,0,0,0,0,0,0)\,
  2. ( ± 1 , ± 1 , ± 1 , ± 1 , ± 1 , ± 1 , ± 1 , ± 1 ) (\pm 1,\pm 1,\pm 1,\pm 1,\pm 1,\pm 1,\pm 1,\pm 1)\,
  3. ( 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 ) (1,1,1,1,1,1,1,1)
  4. ( 2 , 2 , 0 , 0 , 0 , 0 , 0 , 0 ) (2,2,0,0,0,0,0,0)\,
  5. ( 1 , 1 , 1 , 1 , 1 , 1 , - 1 , - 1 ) (1,1,1,1,1,1,-1,-1)
  6. ( 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 ) (1,1,1,1,1,1,1,1)
  7. ( 2 , - 2 , 0 , 0 , 0 , 0 , 0 , 0 ) (2,-2,0,0,0,0,0,0)\,
  8. ( 1 , 1 , 1 , 1 , - 1 , - 1 , - 1 , - 1 ) (1,1,1,1,-1,-1,-1,-1)
  9. 1 + 56 + 126 + 56 + 1 = 240 1+56+126+56+1=240

A_value.html

  1. Δ S = R l n σ \Delta S=Rln\sigma

ABA_problem.html

  1. P 1 P_{1}
  2. P 1 P_{1}
  3. P 2 P_{2}
  4. P 2 P_{2}
  5. P 1 P_{1}
  6. P 1 P_{1}

Abbe_error.html

  1. ϵ = h sin θ \epsilon=h\sin\theta
  2. ϵ \epsilon
  3. h h
  4. θ \theta

Abelian_sandpile_model.html

  1. z ( x , y ) 𝐙 z(x,y)\in\mathbf{Z}
  2. x , y z ( x , y ) < \sum_{x,y}z(x,y)<\infty
  3. ( x , y ) (x,y)
  4. z ( x , y ) 4 z(x,y)\geq 4
  5. z ( x , y ) z ( x , y ) - 4 z(x,y)\rightarrow z(x,y)-4
  6. z ( x ± 1 , y ) z ( x ± 1 , y ) + 1 z(x\pm 1,y)\rightarrow z(x\pm 1,y)+1
  7. z ( x , y ± 1 ) z ( x , y ± 1 ) + 1. z(x,y\pm 1)\rightarrow z(x,y\pm 1)+1.
  8. v v
  9. z ( v ) deg ( v ) z(v)\geq\mathrm{deg}(v)
  10. z ( v ) z ( v ) - deg ( v ) z(v)\rightarrow z(v)-\mathrm{deg}(v)
  11. u v u\sim v
  12. z ( u ) z ( u ) + 1. z(u)\rightarrow z(u)+1.
  13. Δ \Delta
  14. Δ \Delta
  15. Δ \Delta^{\prime}
  16. 𝐱 \mathbf{x}
  17. z z
  18. v v
  19. 𝐱 ( v ) \mathbf{x}(v)
  20. z - 𝐱 Δ z-\mathbf{x}\Delta^{\prime}
  21. z z
  22. z z^{\circ}
  23. z z
  24. z * w ( z + w ) z*w\to(z+w)^{\circ}
  25. 𝐙 n - 1 / 𝐙 n - 1 Δ \mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta^{\prime}
  26. n n
  27. 𝐙 n - 1 Δ \mathbf{Z}^{n-1}\Delta^{\prime}
  28. Δ \Delta^{\prime}
  29. Δ \Delta^{\prime}
  30. v v
  31. z ( v ) deg + ( v ) z(v)\geq\mathrm{deg}^{+}(v)
  32. z ( v ) z ( v ) - deg + ( v ) + deg ( v , v ) z(v)\rightarrow z(v)-\mathrm{deg}^{+}(v)+\mathrm{deg}(v,v)
  33. u v u\neq v
  34. z ( u ) z ( u ) + deg ( v , u ) z(u)\rightarrow z(u)+\mathrm{deg}(v,u)
  35. deg ( v , u ) \mathrm{deg}(v,u)
  36. v v
  37. u u
  38. s s
  39. s s
  40. 𝐙 n - 1 / 𝐙 n - 1 Δ \mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta^{\prime}
  41. Δ \Delta^{\prime}
  42. 𝐮 \mathbf{u}
  43. 𝐮 ( v ) \mathbf{u}(v)
  44. v v
  45. z z
  46. 𝐧 \mathbf{n}
  47. z - 𝐧 Δ z-\mathbf{n}\Delta^{\prime}
  48. 𝐮 ( v ) 𝐧 ( v ) \mathbf{u}(v)\leq\mathbf{n}(v)
  49. v v

Absolute_income_hypothesis.html

  1. C t = λ Y t C_{t}=\lambda Y_{t}
  2. C t C_{t}
  3. λ \lambda
  4. 0 < λ < 1 0<\lambda<1
  5. Y t Y_{t}

Absolute_molar_mass.html

  1. V h V_{h}

Academic_grading_in_China.html

  1. G P A = { 4 - 3 × ( 100 - x ) 2 / 1600 60 x 100 0 0 x < 60 GPA=\begin{cases}4-3\times(100-x)^{2}/1600&60\leqslant x\leqslant 100\\ 0&0\leqslant x<60\end{cases}
  2. x x

Accommodation_index.html

  1. A = 1 N - k - 1 i = k N ( i s i i - i s i i - 1 ) ( i s i i + i s i i - 1 ) A=\frac{1}{N-k-1}\displaystyle\sum_{i=k}^{N}\frac{(isi_{i}-isi_{i-1})}{(isi_{i% }+isi_{i-1})}

Acetylenic.html

  1. \equiv

Acid–base_imbalance.html

  1. HCO 3 - + H + H 2 CO 3 CO 2 + H 2 O \rm HCO_{3}^{-}+H^{+}\leftrightarrow H_{2}CO_{3}\leftrightarrow CO_{2}+H_{2}O

Ackermann_ordinal.html

  1. ϕ < m t p l > Ω 2 ( 0 ) \phi_{<}mtpl>{{\Omega^{2}}}(0)
  2. θ ( Ω 2 ) \theta(\Omega^{2})
  3. ψ ( Ω Ω 2 ) \psi(\Omega^{\Omega^{2}})

Acoplanarity.html

  1. φ = π - ( ϕ 2 - ϕ 1 ) \varphi=\pi-(\phi_{2}-\phi_{1})
  2. ϕ i \phi_{i}
  3. A = 4 min ( i | p o u t i | i | p i | ) 2 A=4\min{\left(\frac{\sum_{i}|p_{out}^{i}|}{\sum_{i}|p_{i}|}\right)^{2}}
  4. p i p_{i}
  5. p o u t i p_{out}^{i}

Acoustic_microscopy.html

  1. R = ( z 2 - z 1 ) ( z 2 + z 1 ) R=\frac{\left(z_{2}-z_{1}\right)}{\left(z_{2}+z_{1}\right)}

Acoustic_streaming.html

  1. α = 2 η ω 2 / ( 3 ρ c 3 ) \alpha=2\eta\omega^{2}/(3\rho c^{3})
  2. α - 1 \alpha^{-1}
  3. α - 1 \alpha^{-1}
  4. δ = [ η / ( ρ ω ) ] 1 / 2 \delta=[\eta/(\rho\omega)]^{1/2}
  5. u = v + u ¯ {u}=v+\overline{u}
  6. v v
  7. ρ ¯ t u ¯ i + ρ ¯ u ¯ j j u ¯ i = - p ¯ i + η j j 2 u ¯ i - j ( ρ v i v j ¯ ) . \overline{\rho}{\partial_{t}\overline{u}_{i}}+\overline{\rho}\overline{u}_{j}{% \partial_{j}\overline{u}_{i}}=-{\partial\overline{p}_{i}}+\eta{\partial^{2}_{% jj}\overline{u}_{i}}-{\partial_{j}}(\overline{\rho v_{i}v_{j}}).
  8. f i = - ( ρ v i v j ¯ ) / x j f_{i}=-{\partial}(\overline{\rho v_{i}v_{j}})/{\partial x_{j}}
  9. - ρ v i v j ¯ -\overline{\rho v_{i}v_{j}}
  10. ϵ cos ( ω t ) \epsilon\cos(\omega t)
  11. ϵ 2 cos 2 ( ω t ) ¯ = ϵ 2 / 2 \scriptstyle\overline{\epsilon^{2}\cos^{2}(\omega t)}=\epsilon^{2}/2
  12. U - 3 / ( 4 ω ) × v 0 d v 0 / d x , U\sim-{3}/{(4\omega)}\times v_{0}dv_{0}/dx,
  13. v 0 v_{0}
  14. x x
  15. ϵ = δ r / a \epsilon=\delta r/a
  16. r = ϵ a sin ( ω t ) r=\epsilon a\sin(\omega t)
  17. ϵ = δ x / a \epsilon^{\prime}=\delta x/a
  18. x = ϵ a sin ( ω t / ϕ ) x=\epsilon^{\prime}a\sin(\omega t/\phi)
  19. ϕ \phi
  20. U ϵ ϵ a ω sin ϕ \displaystyle U\sim\epsilon\epsilon^{\prime}a\omega\sin\phi
  21. U α P / ( π μ c ) U\sim\alpha P/(\pi\mu c)
  22. P P
  23. μ \mu
  24. c c
  25. P P

Acyclic_model.html

  1. 𝒦 \mathcal{K}
  2. 𝒞 ( R ) \mathcal{C}(R)
  3. R R
  4. F , V : 𝒦 𝒞 ( R ) F,V:\mathcal{K}\to\mathcal{C}(R)
  5. F i = V i = 0 F_{i}=V_{i}=0
  6. i < 0 i<0
  7. k 𝒦 \mathcal{M}_{k}\subseteq\mathcal{K}
  8. k 0 k\geq 0
  9. F k F_{k}
  10. k \mathcal{M}_{k}
  11. F F
  12. V V
  13. k k
  14. ( k + 1 ) (k+1)
  15. H k ( V ( M ) ) = 0 H_{k}(V(M))=0
  16. k > 0 k>0
  17. M k k + 1 M\in\mathcal{M}_{k}\cup\mathcal{M}_{k+1}
  18. φ : H 0 ( F ) H 0 ( V ) \varphi:H_{0}(F)\to H_{0}(V)
  19. f : F V f:F\to V
  20. φ , ψ : H 0 ( F ) H 0 ( V ) \varphi,\psi:H_{0}(F)\to H_{0}(V)
  21. f , g : F V f,g:F\to V
  22. φ M = ψ M \varphi^{M}=\psi^{M}
  23. M 0 M\in\mathcal{M}_{0}
  24. f f
  25. g g
  26. f f
  27. K K
  28. L L
  29. K 0 L 0 K_{0}\to L_{0}
  30. K L K\to L
  31. 𝒞 ( R ) 𝒦 \mathcal{C}(R)^{\mathcal{K}}
  32. V V
  33. 𝒦 \mathcal{K}
  34. F F
  35. V V
  36. V V
  37. 𝒜 \mathcal{A}
  38. 𝒞 ( R ) \mathcal{C}(R)
  39. 𝒞 ( R ) 𝒦 \mathcal{C}(R)^{\mathcal{K}}
  40. Γ \Gamma
  41. 𝒜 \mathcal{A}
  42. Γ \Gamma
  43. C C
  44. Γ \Gamma
  45. C C
  46. K K
  47. L L
  48. K Γ K\in\Gamma
  49. L Γ L\in\Gamma
  50. Γ \Gamma
  51. D D
  52. Γ \Gamma
  53. D D
  54. Γ \Gamma
  55. Σ \Sigma
  56. Γ \Gamma
  57. Σ \Sigma
  58. Σ - 1 C \Sigma^{-1}C
  59. Σ \Sigma
  60. G G
  61. C C
  62. ϵ : G I d \epsilon:G\to Id
  63. C C
  64. K K
  65. G G
  66. n n
  67. K n G m + 1 K n G m K n \cdots K_{n}G^{m+1}\to K_{n}G^{m}\to\cdots\to K_{n}
  68. Γ \Gamma
  69. ( - 1 ) i K n G i ϵ G m - i : K n G m + 1 K n G m \sum(-1)^{i}K_{n}G^{i}\epsilon G^{m-i}:K_{n}G^{m+1}\to K_{n}G^{m}
  70. L L
  71. G G
  72. L H 0 ( L ) 0 L\to H_{0}(L)\to 0
  73. Γ \Gamma
  74. Γ \Gamma
  75. Σ \Sigma
  76. K K
  77. G G
  78. L L
  79. G G
  80. f 0 : H 0 ( K ) H 0 ( L ) f_{0}:H_{0}(K)\to H_{0}(L)
  81. Σ - 1 ( C ) \Sigma^{-1}(C)
  82. f : K L f:K\to L
  83. Σ - 1 ( C ) \Sigma^{-1}(C)
  84. L L
  85. G G
  86. K K
  87. G G
  88. f 0 f_{0}
  89. f f
  90. X X
  91. C C
  92. X X
  93. K K
  94. L L
  95. E : X X E:X\to X
  96. X X
  97. n 0 Hom ( Δ n , X ) Δ n \sum_{n\geq 0}\sum_{\textrm{Hom}(\Delta_{n},X)}\Delta_{n}
  98. Δ n \Delta_{n}
  99. n n
  100. X X
  101. n n
  102. Δ n X \Delta_{n}\to X
  103. G G
  104. G ( C ) = C E G(C)=CE
  105. E X X EX\to X
  106. G G
  107. K K
  108. L L
  109. G G
  110. G G
  111. L L
  112. Γ \Gamma
  113. H 0 ( K ) H 0 ( L ) H_{0}(K)\simeq H_{0}(L)
  114. X X

Additive_model.html

  1. { y i , x i 1 , , x i p } i = 1 n \{y_{i},\,x_{i1},\ldots,x_{ip}\}_{i=1}^{n}
  2. { x i 1 , , x i p } i = 1 n \{x_{i1},\ldots,x_{ip}\}_{i=1}^{n}
  3. y i y_{i}
  4. E [ y i | x i 1 , , x i p ] = β 0 + j = 1 p f j ( x i j ) E[y_{i}|x_{i1},\ldots,x_{ip}]=\beta_{0}+\sum_{j=1}^{p}f_{j}(x_{ij})
  5. Y = β 0 + j = 1 p f j ( X j ) + ε Y=\beta_{0}+\sum_{j=1}^{p}f_{j}(X_{j})+\varepsilon
  6. E [ ϵ ] = 0 E[\epsilon]=0
  7. V a r ( ϵ ) = σ 2 Var(\epsilon)=\sigma^{2}
  8. E [ f j ( X j ) ] = 0 E[f_{j}(X_{j})]=0
  9. f j ( x i j ) f_{j}(x_{ij})
  10. f j ( x i j ) f_{j}(x_{ij})

Additive_smoothing.html

  1. θ ^ i = x i + α N + α d ( i = 1 , , d ) , \hat{\theta}_{i}=\frac{x_{i}+\alpha}{N+\alpha d}\qquad(i=1,\ldots,d),
  2. θ ^ i = x i + μ i α d N + α d ( i = 1 , , d ) , \hat{\theta}_{i}=\frac{x_{i}+\mu_{i}\alpha d}{N+\alpha d}\qquad(i=1,\ldots,d),

Additively_indecomposable_ordinal.html

  1. β , γ < α \beta,\gamma<\alpha
  2. β + γ < α . \beta+\gamma<\alpha.
  3. . \mathbb{H}.
  4. β < α \beta<\alpha
  5. β + α = α . \beta+\alpha=\alpha.
  6. 1 1\in\mathbb{H}
  7. 0 + 0 < 1. 0+0<1.
  8. 1 1
  9. . \mathbb{H}.
  10. ω \omega\in\mathbb{H}
  11. . \mathbb{H}.
  12. \mathbb{H}
  13. \mathbb{H}
  14. f ( α ) = ω α . f_{\mathbb{H}}(\alpha)=\omega^{\alpha}.
  15. f ( α ) f_{\mathbb{H}}^{\prime}(\alpha)
  16. ϵ α . \epsilon_{\alpha}.
  17. f f_{\mathbb{H}}
  18. ϵ 0 = ω ω ω \epsilon_{0}=\omega^{\omega^{\omega^{\cdot^{\cdot^{\cdot}}}}}
  19. ω , ω ω , ω ω ω , \omega,\omega^{\omega}\!,\omega^{\omega^{\omega}}\!\!,\ldots
  20. ω ω α \omega^{\omega^{\alpha}}\,

Adult_Literacy_Index.html

  1. A L R - 0 100 - 0 \frac{ALR-0}{100-0}

Advance–decline_line.html

  1. A D l i n e = t o d a y s a d v a n c i n g s t o c k s - t o d a y s d e c l i n i n g s t o c k s + y e s t e r d a y s A D l i n e v a l u e ADline=today^{\prime}s\ advancing\ stocks-today^{\prime}s\ declining\ stocks+% yesterday^{\prime}s\ AD\ line\ value
  2. A D v o l u m e l i n e = A d v a n c e d V o l u m e - D e c l i n e d V o l u m e + y e s t e r d a y s A D v o l u m e l i n e AD\ volume\ line=Advanced\ Volume-Declined\ Volume+yesterday^{\prime}s\ AD\ % volume\ line

Aerodynamic_potential-flow_code.html

  1. V = 0 \nabla\cdot V=0
  2. × V = 0 \nabla\times V=0
  3. t = 0 \frac{\partial}{\partial t}=0
  4. 2 ϕ = 0 \nabla^{2}\phi=0
  5. ( 1 - M 2 ) ϕ x x + ϕ y y + ϕ z z = 0 (1-M_{\infty}^{2})\phi_{xx}+\phi_{yy}+\phi_{zz}=0
  6. V ( 𝐅 ) d V = S 𝐅 𝐧 d S \iiint\limits_{V}\left(\nabla\cdot\mathbf{F}\right)dV=\iint\limits_{S}\mathbf{% F}\cdot\mathbf{n}\,dS
  7. ϕ \phi
  8. R = | P - Q | R=|P-Q|
  9. U p = - 1 4 π V ( 2 𝐔 R ) d V Q U_{p}=-\frac{1}{4\pi}\iiint\limits_{V}\left(\frac{\nabla^{2}\cdot\mathbf{U}}{R% }\right)dV_{Q}
  10. - 1 4 π S ( 𝐧 𝐔 R ) d S Q -\frac{1}{4\pi}\iint\limits_{S}\left(\frac{\mathbf{n}\cdot\nabla\mathbf{U}}{R}% \right)dS_{Q}
  11. + 1 4 π S ( 𝐔𝐧 1 R ) d S Q +\frac{1}{4\pi}\iint\limits_{S}\left(\mathbf{U}\mathbf{n}\cdot\nabla\frac{1}{R% }\right)dS_{Q}
  12. 2 ϕ = 0 \nabla^{2}\phi=0
  13. ϕ p = - 1 4 π S ( 𝐧 ϕ U - ϕ L R - 𝐧 ( ϕ U - ϕ L ) 1 R ) d S Q \phi_{p}=-\frac{1}{4\pi}\iint\limits_{S}\left(\mathbf{n}\frac{\nabla\phi_{U}-% \nabla\phi_{L}}{R}-\mathbf{n}\left(\phi_{U}-\phi_{L}\right)\nabla\frac{1}{R}% \right)dS_{Q}
  14. σ = 𝐧 ( ϕ U - ϕ L ) \sigma=\nabla\mathbf{n}(\nabla\phi_{U}-\nabla\phi_{L})
  15. μ = ϕ U - ϕ L \mu=\phi_{U}-\phi_{L}
  16. ϕ p = - 1 4 π S ( σ R - μ 𝐧 1 R ) d S \phi_{p}=-\frac{1}{4\pi}\iint\limits_{S}\left(\frac{\sigma}{R}-\mu\cdot\mathbf% {n}\cdot\nabla\frac{1}{R}\right)dS
  17. ϕ L = 0 \phi_{L}=0
  18. μ = ϕ U - ϕ L \mu=\phi_{U}-\phi_{L}
  19. μ = ϕ U \mu=\phi_{U}
  20. ϕ U = - V 𝐧 \phi_{U}=-V_{\infty}\cdot\mathbf{n}
  21. μ P = 1 4 π S ( V 𝐧 R ) d S U + 1 4 π S ( μ 𝐧 1 R ) d S \mu_{P}=\frac{1}{4\pi}\iint\limits_{S}\left(\frac{V_{\infty}\cdot\mathbf{n}}{R% }\right)dS_{U}+\frac{1}{4\pi}\iint\limits_{S}\left(\mu\cdot\mathbf{n}\cdot% \nabla\frac{1}{R}\right)dS
  22. d S U dS_{U}
  23. d S dS
  24. λ \lambda
  25. λ \lambda
  26. σ Q = i = 1 n λ i s i ( Q ) = 0 \sigma_{Q}=\sum_{i=1}^{n}\lambda_{i}s_{i}(Q)=0
  27. μ Q = i = 1 n λ i m i ( Q ) \mu_{Q}=\sum_{i=1}^{n}\lambda_{i}m_{i}(Q)
  28. s i = l n ( r ) s_{i}=ln(r)
  29. m i = m_{i}=
  30. λ \lambda
  31. C p = p - p q = p - p 1 2 ρ V 2 = p - p γ 2 p M 2 C_{p}=\frac{p-p_{\infty}}{q_{\infty}}=\frac{p-p_{\infty}}{\frac{1}{2}\rho_{% \infty}V_{\infty}^{2}}=\frac{p-p_{\infty}}{\frac{\gamma}{2}p_{\infty}M_{\infty% }^{2}}
  32. C p = 2 γ M 2 ( ( 1 + γ - 1 2 M 2 [ 1 - | V | 2 | V | 2 ] ) γ γ - 1 - 1 ) C_{p}=\frac{2}{\gamma M_{\infty}^{2}}\left(\left(1+\frac{\gamma-1}{2}M_{\infty% }^{2}\left[\frac{1-|\vec{V}|^{2}}{|\vec{V_{\infty}}|^{2}}\right]\right)^{\frac% {\gamma}{\gamma-1}}-1\right)
  33. C p = 1 - | V | 2 | V | 2 C_{p}=1-\frac{|\vec{V}|^{2}}{|\vec{V_{\infty}}|^{2}}
  34. C p = 1 - | V | 2 + M 2 u 2 C_{p}=1-|\vec{V}|^{2}+M_{\infty}^{2}u^{2}
  35. C p = - ( 2 u + v 2 + w 2 ) C_{p}=-(2u+v^{2}+w^{2})
  36. C p = - 2 u C_{p}=-2u
  37. C p = 1 - | V | 2 C_{p}=1-|\vec{V}|^{2}

Affine_focal_set.html

  1. U n U\subset\mathbb{R}^{n}
  2. u = ( u 1 , , u n ) {u}=(u_{1},\ldots,u_{n})
  3. X : U n + 1 {X}:U\to\mathbb{R}^{n+1}
  4. A {A}
  5. A : U T X ( U ) n + 1 . {A}:U\to T_{{X}(U)}\mathbb{R}^{n+1}.\,
  6. u 0 U {u}_{0}\in U
  7. X ( u 0 ) {X}({u}_{0})
  8. t X ( u 0 ) + t A ( u 0 ) . t\mapsto{X}({u}_{0})+t{A}({u}_{0}).
  9. u U {u}\in U
  10. u + d u {u}+d{u}
  11. d u d{u}
  12. X ( u ) {X}({u})
  13. X ( u + d u ) {X}({u}+d{u})
  14. d u d{u}
  15. X ( u ) + t A ( u ) = X ( u + d u ) + t A ( u + d u ) . {X}({u})+t{A}({u})={X}({u}+d{u})+t{A}({u}+d{u}).
  16. S v = D v A Sv=D_{v}{A}
  17. X ( u ) + t A ( u ) = X ( u + d u ) + t A ( u + d u ) {X}({u})+t{A}({u})={X}({u}+d{u})+t{A}({u}+d{u})
  18. d u d{u}
  19. 0 k [ n / 2 ] 0\leq k\leq[n/2]
  20. [ - ] [-]
  21. x 2 + a = 0 x^{2}+a=0
  22. Δ : n + 1 × M . \Delta:\mathbb{R}^{n+1}\times M\to\mathbb{R}.\,
  23. x {x}
  24. x {x}
  25. A {A}
  26. x - p = Z ( x , p ) + Δ ( x , p ) A ( p ) {x}-p=Z({x},p)+\Delta({x},p){A}(p)
  27. Δ : { x } × M \Delta:\{{x}\}\times M\to\mathbb{R}
  28. D X ( x - p ) = D X ( Z + Δ A ) , D_{X}({x}-p)=D_{X}(Z+\Delta{A}),
  29. - X = X Z + h ( X , Z ) A + d X Δ A - Δ S X , -X=\nabla_{X}Z+h(X,Z){A}+d_{X}\Delta{A}-\Delta SX,
  30. ( X Z + ( I - Δ S ) X ) + ( h ( X , Z ) + d X Δ ) A = 0 , (\nabla_{X}Z+(I-\Delta S)X)+(h(X,Z)+d_{X}\Delta){A}=0,
  31. X Z = ( Δ S - I ) X \nabla_{X}Z=(\Delta S-I)X
  32. h ( X , Z ) = - d X Δ h(X,Z)=-d_{X}\Delta
  33. h ( - , Z ) = d Δ h(-,Z)=d\Delta
  34. d Δ d\Delta
  35. d X Δ = 0 d_{X}\Delta=0
  36. h ( - , Z ) = d Δ h(-,Z)=d\Delta
  37. d Δ d\Delta
  38. h ( - , Z ) h(-,Z)
  39. Δ : { x } × M \Delta:\{{x}\}\times M\to\mathbb{R}
  40. ( X , Y ) d Y ( d X Δ ) (X,Y)\mapsto d_{Y}(d_{X}\Delta)
  41. h ( X , Z ) = - d X Δ h(X,Z)=-d_{X}\Delta
  42. d Y ( d X Δ ) = - d Y ( h ( X , Z ) ) . d_{Y}(d_{X}\Delta)=-d_{Y}(h(X,Z)).
  43. ( X , Y ) - d Y ( h ( X , Z ) ) = - ( Y h ) ( X , Z ) - h ( Y X , Z ) - h ( X , Y Z ) (X,Y)\mapsto-d_{Y}(h(X,Z))=-(\nabla_{Y}h)(X,Z)-h(\nabla_{Y}X,Z)-h(X,\nabla_{Y}Z)
  44. ( X , Y ) - h ( X , Y Z ) (X,Y)\mapsto-h(X,\nabla_{Y}Z)
  45. X Z = ( Δ S - I ) X \nabla_{X}Z=(\Delta S-I)X
  46. ( X , Y ) h ( X , ( I - Δ S ) Y ) (X,Y)\mapsto h(X,(I-\Delta S)Y)
  47. det ( I - Δ S ) = 0 \det(I-\Delta S)=0
  48. Y ker ( I - Δ S ) Y\in\ker(I-\Delta S)
  49. x = p + t A {x}=p+t{A}
  50. { p + t A ( p ) : p M , det ( I - t S ) = 0 } . \{p+t{A}(p):p\in M,\det(I-tS)=0\}\ .
  51. D X ( p + t A ) = ( I - t S ) X + d X t A . D_{X}(p+t{A})=(I-tS)X+d_{X}t{A}.
  52. D X ( p + t A ) = 0 D_{X}(p+t{A})=0
  53. A 3 A_{3}
  54. A 4 A_{4}
  55. D 4 + D_{4}^{+}
  56. D 4 - D_{4}^{-}
  57. A k A_{k}
  58. D k D_{k}

Affine_geometry_of_curves.html

  1. SL ( n , ) n . \mbox{SL}~{}(n,\mathbb{R})\ltimes\mathbb{R}^{n}.
  2. det [ 𝐱 ˙ , 𝐱 ¨ , , 𝐱 ( n ) ] = ± 1. \det\begin{bmatrix}\dot{\mathbf{x}},&\ddot{\mathbf{x}},&\dots,&{\mathbf{x}}^{(% n)}\end{bmatrix}=\pm 1.
  3. t [ 𝐱 ( t ) , 𝐱 ˙ ( t ) , , 𝐱 ( n ) ( t ) ] t\mapsto[\mathbf{x}(t),\dot{\mathbf{x}}(t),\dots,\mathbf{x}^{(n)}(t)]
  4. 𝐱 , 𝐱 ˙ , , 𝐱 ( n ) \mathbf{x},\dot{\mathbf{x}},\dots,\mathbf{x}^{(n)}
  5. 𝐱 ˙ , , 𝐱 ( n ) \dot{\mathbf{x}},\dots,\mathbf{x}^{(n)}
  6. det [ 𝐱 ˙ , 𝐱 ¨ , , 𝐱 ( n ) ] = ± 1. \det\begin{bmatrix}\dot{\mathbf{x}},&\ddot{\mathbf{x}},&\dots,&{\mathbf{x}}^{(% n)}\end{bmatrix}=\pm 1.
  7. 𝐱 ( n + 1 ) = k 1 𝐱 ˙ + + k n - 1 𝐱 ( n - 1 ) . \mathbf{x}^{(n+1)}=k_{1}\dot{\mathbf{x}}+\cdots+k_{n-1}\mathbf{x}^{(n-1)}.
  8. 0 = det [ 𝐱 ˙ , 𝐱 ¨ , , 𝐱 ( n ) ] ˙ = det [ 𝐱 ˙ , 𝐱 ¨ , , 𝐱 ( n + 1 ) ] 0=\det\begin{bmatrix}\dot{\mathbf{x}},&\ddot{\mathbf{x}},&\dots,&{\mathbf{x}}^% {(n)}\end{bmatrix}\dot{}\,=\det\begin{bmatrix}\dot{\mathbf{x}},&\ddot{\mathbf{% x}},&\dots,&{\mathbf{x}}^{(n+1)}\end{bmatrix}
  9. A = [ 𝐱 ˙ , 𝐱 ¨ , , 𝐱 ( n ) ] A=\begin{bmatrix}\dot{\mathbf{x}},&\ddot{\mathbf{x}},&\dots,&{\mathbf{x}}^{(n)% }\end{bmatrix}
  10. A ˙ = [ 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 k 1 k 2 k 3 k 4 k n - 1 0 ] A = C A . \dot{A}=\begin{bmatrix}0&1&0&0&\cdots&0&0\\ 0&0&1&0&\cdots&0&0\\ \vdots&\vdots&\vdots&\cdots&\cdots&\vdots&\vdots\\ 0&0&0&0&\cdots&1&0\\ 0&0&0&0&\cdots&0&1\\ k_{1}&k_{2}&k_{3}&k_{4}&\cdots&k_{n-1}&0\end{bmatrix}A=CA.

Affine_Grassmannian.html

  1. k - Alg k-\mathrm{Alg}
  2. Set \mathrm{Set}
  3. X : k - Alg Set X:k-\mathrm{Alg}\to\mathrm{Set}
  4. 𝒦 = k ( ( t ) ) \mathcal{K}=k((t))
  5. 𝒪 = k [ [ t ] ] \mathcal{O}=k[[t]]
  6. 𝒪 \mathcal{O}
  7. G ( 𝒦 ) / G ( 𝒪 ) G(\mathcal{K})/G(\mathcal{O})

Affine_Grassmannian_(manifold).html

  1. p ( U ) p(U)^{\perp}
  2. Graff k ( V ) E ( n ) E ( k ) × O ( n - k ) \mathrm{Graff}_{k}(V)\simeq\frac{E(n)}{E(k)\times O(n-k)}
  3. dim [ Graff k ( V ) ] = ( n - k ) ( k + 1 ) . \dim\left[\mathrm{Graff}_{k}(V)\right]=(n-k)(k+1)\,.
  4. a 11 x 1 + + a 1 n x n = a 1 , n + 1 a n - k , 1 x 1 + + a n - k , n x n = a n - k , n + 1 . \begin{aligned}\displaystyle a_{11}x_{1}+\cdots+a_{1n}x_{n}&\displaystyle=a_{1% ,n+1}\\ &\displaystyle\vdots&\\ \displaystyle a_{n-k,1}x_{1}+\cdots+a_{n-k,n}x_{n}&\displaystyle=a_{n-k,n+1}.% \end{aligned}
  5. a 11 x 1 + + a 1 n x n = a 1 , n + 1 x n + 1 a n - k , 1 x 1 + + a n - k , n x n = a n - k , n + 1 x n + 1 . \begin{aligned}\displaystyle a_{11}x_{1}+\cdots+a_{1n}x_{n}&\displaystyle=a_{1% ,n+1}x_{n+1}\\ &\displaystyle\vdots&\\ \displaystyle a_{n-k,1}x_{1}+\cdots+a_{n-k,n}x_{n}&\displaystyle=a_{n-k,n+1}x_% {n+1}.\end{aligned}

Agmon's_inequality.html

  1. L L^{\infty}
  2. H s H^{s}
  3. u H 2 ( Ω ) H 0 1 ( Ω ) u\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega)
  4. Ω 3 \Omega\subset\mathbb{R}^{3}
  5. C C
  6. u L ( Ω ) C u H 1 ( Ω ) 1 / 2 u H 2 ( Ω ) 1 / 2 , \displaystyle\|u\|_{L^{\infty}(\Omega)}\leq C\|u\|_{H^{1}(\Omega)}^{1/2}\|u\|_% {H^{2}(\Omega)}^{1/2},
  7. u L ( Ω ) C u L 2 ( Ω ) 1 / 4 u H 2 ( Ω ) 3 / 4 . \displaystyle\|u\|_{L^{\infty}(\Omega)}\leq C\|u\|_{L^{2}(\Omega)}^{1/4}\|u\|_% {H^{2}(\Omega)}^{3/4}.
  8. u H 2 ( Ω ) H 0 1 ( Ω ) u\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega)
  9. Ω 2 \Omega\subset\mathbb{R}^{2}
  10. C C
  11. u L ( Ω ) C u L 2 ( Ω ) 1 / 2 u H 2 ( Ω ) 1 / 2 . \displaystyle\|u\|_{L^{\infty}(\Omega)}\leq C\|u\|_{L^{2}(\Omega)}^{1/2}\|u\|_% {H^{2}(\Omega)}^{1/2}.
  12. n n
  13. s 1 s_{1}
  14. s 2 s_{2}
  15. s 1 < n 2 < s 2 s_{1}<\tfrac{n}{2}<s_{2}
  16. 0 < θ < 1 0<\theta<1
  17. n 2 = θ s 1 + ( 1 - θ ) s 2 \tfrac{n}{2}=\theta s_{1}+(1-\theta)s_{2}
  18. u H s 2 ( Ω ) u\in H^{s_{2}}(\Omega)
  19. u L ( Ω ) C u H s 1 ( Ω ) θ u H s 2 ( Ω ) 1 - θ \displaystyle\|u\|_{L^{\infty}(\Omega)}\leq C\|u\|_{H^{s_{1}}(\Omega)}^{\theta% }\|u\|_{H^{s_{2}}(\Omega)}^{1-\theta}

Ahlswede–Daykin_inequality.html

  1. f 1 , f 2 , f 3 , f 4 f_{1},f_{2},f_{3},f_{4}
  2. f 1 ( x ) f 2 ( y ) f 3 ( x y ) f 4 ( x y ) f_{1}(x)f_{2}(y)\leq f_{3}(x\vee y)f_{4}(x\wedge y)
  3. f 1 ( X ) f 2 ( Y ) f 3 ( X Y ) f 4 ( X Y ) f_{1}(X)f_{2}(Y)\leq f_{3}(X\vee Y)f_{4}(X\wedge Y)
  4. f ( X ) = x X f ( x ) f(X)=\sum_{x\in X}f(x)
  5. X Y = { x y x X , y Y } X\vee Y=\{x\vee y\mid x\in X,y\in Y\}
  6. X Y = { x y x X , y Y } . X\wedge Y=\{x\wedge y\mid x\in X,y\in Y\}.

Air_mass_(solar_energy).html

  1. L L
  2. z z
  3. A M = L L o 1 cos z AM=\frac{L}{L_{\mathrm{o}}}\approx\frac{1}{\cos\,z}\,
  4. L o L_{\mathrm{o}}
  5. z z
  6. z z
  7. A M = 1 cos z + 0.50572 ( 96.07995 - z ) - 1.6364 AM=\frac{1}{\cos\,z+0.50572\,(96.07995-z)^{-1.6364}}\,
  8. z z
  9. A M = ( r cos z ) 2 + 2 r + 1 - r cos z AM=\sqrt{(r\cos z)^{2}+2r+1}\;-\;r\cos z\,
  10. R E R_{\mathrm{E}}
  11. y atm y_{\mathrm{atm}}
  12. r = R E / y atm r=R_{\mathrm{E}}/y_{\mathrm{atm}}
  13. z z
  14. \infty
  15. z z
  16. z z
  17. z z
  18. z z
  19. z z
  20. z z
  21. I = 1.1 × I o × 0.7 ( A M ) ( 0.678 ) I=1.1\times I_{\mathrm{o}}\times 0.7^{(AM)^{(0.678)}}\,
  22. I o I_{\mathrm{o}}
  23. z z
  24. z z
  25. I = 1.1 × I o × 0.56 ( A M 0.715 ) I=1.1\times I_{\mathrm{o}}\times 0.56^{(AM^{0.715})}\,
  26. I = 1.1 × I o × 0.76 ( A M 0.618 ) I=1.1\times I_{\mathrm{o}}\times 0.76^{(AM^{0.618})}\,
  27. I = 1.1 × I o × [ ( 1 - h / 7.1 ) 0.7 ( A M ) 0.678 ) + h / 7.1 ] I=1.1\times I_{\mathrm{o}}\times[(1-h/7.1)0.7^{(AM)^{0.678})}+h/7.1]\,
  28. h h
  29. A M AM
  30. A M = ( r + c ) 2 cos 2 z + ( 2 r + 1 + c ) ( 1 - c ) - ( r + c ) cos z AM=\sqrt{(r+c)^{2}\cos^{2}z+(2r+1+c)(1-c)}\;-\;(r+c)\cos z\,
  31. r = R E / y atm r=R_{\mathrm{E}}/y_{\mathrm{atm}}
  32. c = h / y atm c=h/y_{\mathrm{atm}}

Airborne_particulate_radioactivity_monitoring.html

  1. d C ˙ F F d t = ε k F m ϕ Q ( t ) - λ C ˙ F F {{d\dot{C}_{FF}}\over{dt}}\,\,\,=\,\,\,\varepsilon\,k\,F_{m}\,\phi\,Q(t)\,\,-% \,\,\lambda\,\dot{C}_{FF}
  2. C ˙ F F ( t ) = ε k F m ϕ exp ( - λ t ) 0 t Q ( τ ) exp ( λ τ ) d τ + C ˙ 0 exp ( - λ t ) \dot{C}_{FF}(t)\,\,=\,\,\varepsilon\,k\,F_{m}\,\phi\,\,\exp\left({-\lambda\,t}% \right)\,\,\int_{0}^{t}{Q(\tau)\,\,\exp(\lambda\tau)\,d\tau\,\,\,+\,\,\,\dot{C% }_{0}\exp\left({-\lambda\,t}\right)}
  3. C ˙ R W ( t ) = ε k F m ϕ exp ( - λ t ) L [ 0 v t t - ( < m t p l > x v ) t Q ( τ ) exp ( λ τ ) d τ d x + v t L 0 t Q ( τ ) exp ( λ τ ) d τ d x ] \dot{C}_{RW}(t)\,\,\,=\,\,\,{{\varepsilon\,k\,F_{m}\,\phi\,\,\exp\left({-% \lambda\,t}\right)}\over L}\left[{\int_{0}^{v\,t}{\int_{t\,-\,\left(<mtpl>{{x% \over v}}\right)}^{t}{Q(\tau)\exp(\lambda\,\tau)\,d\tau\,dx\,\,\,+\,\,\,\int_{% v\,t}^{L}{\int_{0}^{t}{Q(\tau)\exp(\lambda\,\tau)\,d\tau\,dx}}}}}\right]
  4. C ˙ \dot{C}
  5. C ˙ F F ( t ) = ε k F m ϕ Q 0 1 - exp ( - λ t ) λ \dot{C}_{FF}(t)=\varepsilon\,k\,F_{m}\,\phi\,Q_{0}{{1\,\,\,-\,\,\exp(-\lambda% \,t)}\over\lambda}
  6. C ˙ F F ( t ) = ε k F m ϕ Q 0 t \dot{C}_{FF}(t)=\varepsilon\,k\,F_{m}\,\phi\,Q_{0}\,t
  7. C ˙ R W ( t ) = ε k F m ϕ Q 0 λ 2 v L [ λ t - 1 + exp ( - λ t ) ] + ε k F m ϕ Q 0 λ ( 1 - v t L ) [ 1 - exp ( - λ t ) ] \dot{C}_{RW}(t)\,\,\,=\,\,\,{{\varepsilon\,k\,F_{m}\,\phi\,Q_{0}}\over{\lambda% ^{2}}}{v\over L}\left[{\lambda\,t\,\,-\,\,1\,\,+\,\,\exp(-\lambda\,t)}\right]% \,\,\,+\,\,\,{{\varepsilon\,k\,F_{m}\,\phi\,Q_{0}}\over\lambda}\left({1-{{v\,t% }\over L}}\right)\left[{1\,\,-\,\,\exp(-\lambda\,t)}\right]
  8. C ˙ R W ( t ) = ε k F m ϕ Q 0 ( t - v t 2 2 L ) \dot{C}_{RW}(t)\,\,\,=\,\,\,\varepsilon\,k\,F_{m}\,\phi\,Q_{0}\left({t\,\,-\,% \,{{v\,t^{2}}\over{2L}}}\right)
  9. C ˙ R W ( t ) = ε k F m ϕ Q 0 { 1 λ - v λ 2 L [ 1 - exp ( - λ L v ) ] } \dot{C}_{RW}(t)\,\,\,=\,\,\,\varepsilon\,k\,F_{m}\,\phi\,Q_{0}\,\,\left\{{{1% \over\lambda}\,\,\,-\,\,\,{v\over{\lambda^{2}L}}\left[{1\,\,\,-\,\,\,\exp\left% ({-\lambda{L\over v}}\right)}\right]\,}\right\}
  10. C ˙ R W ( t ) = ε k F m ϕ Q 0 L 2 v \dot{C}_{RW}(t)\,\,\,=\,\,\,\varepsilon\,k\,F_{m}\,\phi\,Q_{0}\,{L\over{2\,v\,}}
  11. L C W = 16 R 3 π T C W = 2 R v L C W v T R W = L v L_{CW}=\,\,{{16\,R}\over{3\,\pi}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,T_{CW}\,\,=\,% \,{{2R}\over v}\,\,\,\,\neq\,\,\,{{L_{CW}}\over v}\,\,\,\,\,\,\,\,\,\,\,\,T_{% RW}=\,\,{L\over v}
  12. μ \mu
  13. μ \mu
  14. C ˙ R W = ε k F m ϕ Q 0 T 2 Q ^ 0 = 2 v C ˙ R W ε k F m ϕ L \dot{C}_{RW}\,\,\,=\,\,\,\varepsilon\,k\,F_{m}\,\phi\,Q_{0}{T\over{2\,}}\,\,\,% \,\,\,\,\,\,\,\,\,\Rightarrow\,\,\,\,\,\,\hat{Q}_{0}\,\,=\,\,{{2\,v\,\dot{C}_{% RW}}\over{\varepsilon\,k\,F_{m}\,\phi\,L}}
  15. C ˙ C W = ε k F m ϕ Q 0 8 R 3 π v Q ^ 0 = 3 π v C ˙ C W 8 R ε k F m ϕ \dot{C}_{CW}\,\,=\,\,\,\varepsilon\,k\,F_{m}\,\phi\,Q_{0}{{8R}\over{3\,\pi\,v}% }\,\,\,\,\,\,\Rightarrow\,\,\,\,\,\,\hat{Q}_{0}\,\,=\,\,{{3\,\pi\,v\,\dot{C}_{% CW}}\over{8R\,\varepsilon\,k\,F_{m}\,\phi}}
  16. η \eta
  17. R s t a c k ( η ) = 0 η Q ( τ ) F s t a c k ( τ ) d τ R_{stack}\left(\eta\right)\,\,\,=\,\,\,\int_{0}^{\eta}{Q(\tau)\,F_{stack}(\tau% )\,d\tau}
  18. R s t a c k ( η ) = F s t a c k [ C ˙ ( η ) + λ 0 η C ˙ ( τ ) d τ ] ε k F m ϕ R_{stack}\left(\eta\right)\,\,\,=\,\,\,{{F_{stack}\left[{\dot{C}\left(\eta% \right)\,\,\,+\,\,\,\lambda\,\int_{0}^{\eta}{\dot{C}\left(\tau\right)\,\,d\tau% }}\right]}\over{\varepsilon\,k\,F_{m}\,\phi}}

Airwatt.html

  1. P = 0.117354 F S P=0.117354\cdot F\cdot S
  2. 1 m 3 s 1 N m 2 = 1 N m s = 1 J s = 1 W 1~{}\frac{\mathrm{m}^{3}}{\mathrm{s}}\cdot 1~{}\frac{\mathrm{N}}{\mathrm{m}^{2% }}=1~{}\frac{\mathrm{N}\cdot\mathrm{m}}{\mathrm{s}}=1~{}\frac{\mathrm{J}}{% \mathrm{s}}=1~{}\mathrm{W}
  3. P = a i r f l o w [ C F M ] s u c t i o n [ i n c h e s o f w a t e r ] 8.5 P=\frac{airflow[CFM]\cdot suction[inches\;of\;water]}{8.5}
  4. 1 W = 1 kPa L s 1~{}\mathrm{W}=1~{}\frac{\mathrm{kPa}\cdot\mathrm{L}}{\mathrm{s}}

Airy_wave_theory.html

  1. η ( x , t ) = a cos ( k x - ω t ) \eta(x,t)\,=\,a\,\cos\,\left(kx\,-\,\omega t\right)
  2. k = 2 π λ , k\,=\,\frac{2\pi}{\lambda},\,
  3. ω = 2 π T = 2 π f . \omega\,=\,\frac{2\pi}{T}\,=\,2\pi\,f.\,
  4. c p = ω k = λ T . c_{p}\,=\,\frac{\omega}{k}\,=\,\frac{\lambda}{T}.
  5. H = 2 a and a = 1 2 H , H\,=\,2\,a\qquad\,\text{and}\qquad a\,=\,\frac{1}{2}\,H,
  6. u x = Φ x and u z = Φ z . u_{x}\,=\,\frac{\partial\Phi}{\partial x}\quad\,\text{and}\quad u_{z}\,=\,% \frac{\partial\Phi}{\partial z}.
  7. ( 1 ) 2 Φ x 2 + 2 Φ z 2 = 0. (1)\qquad\frac{\partial^{2}\Phi}{\partial x^{2}}\,+\,\frac{\partial^{2}\Phi}{% \partial z^{2}}\,=\,0.
  8. ( 2 ) Φ z = 0 at z = - h . (2)\qquad\frac{\partial\Phi}{\partial z}\,=\,0\quad\,\text{ at }z\,=\,-h.
  9. ( 3 ) η t = Φ z at z = η ( x , t ) . (3)\qquad\frac{\partial\eta}{\partial t}\,=\,\frac{\partial\Phi}{\partial z}% \quad\,\text{ at }z\,=\,\eta(x,t).
  10. ( 4 ) Φ t + g η = 0 at z = η ( x , t ) . (4)\qquad\frac{\partial\Phi}{\partial t}\,+\,g\,\eta\,=\,0\quad\,\text{ at }z% \,=\,\eta(x,t).
  11. η = a cos ( k x - ω t ) . \eta\,=\,a\,\cos\,(kx\,-\,\omega t).
  12. Φ = ω k a cosh ( k ( z + h ) ) sinh ( k h ) sin ( k x - ω t ) , \Phi\,=\,\frac{\omega}{k}\,a\,\frac{\cosh\,\bigl(k\,(z+h)\bigr)}{\sinh\,(k\,h)% }\,\sin\,(kx\,-\,\omega t),
  13. ω 2 = g k tanh ( k h ) , \omega^{2}\,=\,g\,k\,\tanh\,(kh),
  14. 1 / 20 {1}/{20}
  15. η ( s y m b o l x , t ) \eta(symbol{x},t)\,
  16. a cos θ ( s y m b o l x , t ) a\,\cos\,\theta(symbol{x},t)\,
  17. θ ( s y m b o l x , t ) \theta(symbol{x},t)\,
  18. s y m b o l k \cdotsymbol x - ω t symbol{k}\cdotsymbol{x}\,-\,\omega\,t\,
  19. ω \omega\,
  20. ( ω - s y m b o l k \cdotsymbol U ) 2 = ( Ω ( k ) ) 2 with k = | s y m b o l k | \left(\omega\,-\,symbol{k}\cdotsymbol{U}\right)^{2}\,=\,\bigl(\Omega(k)\bigr)^% {2}\quad\,\text{ with }\quad k\,=\,|symbol{k}|\,
  21. σ \sigma\,
  22. σ 2 = ( Ω ( k ) ) 2 with σ = ω - s y m b o l k \cdotsymbol U \quad\sigma^{2}\,=\,\bigl(\Omega(k)\bigr)^{2}\quad\,\text{ with }\quad\sigma\,% =\,\omega\,-\,symbol{k}\cdotsymbol{U}\,
  23. s y m b o l e k symbol{e}_{k}\,
  24. s y m b o l k k \frac{symbol{k}}{k}\,
  25. Ω ( k ) \Omega(k)\,
  26. Ω ( k ) = g k \Omega(k)\,=\,\sqrt{g\,k}
  27. Ω ( k ) = k g h \Omega(k)\,=\,k\,\sqrt{g\,h}\,
  28. Ω ( k ) = g k tanh ( k h ) \Omega(k)\,=\,\sqrt{g\,k\,\tanh\,(k\,h)}\,
  29. c p = Ω ( k ) k c_{p}=\frac{\Omega(k)}{k}\,
  30. g k = g σ \sqrt{\frac{g}{k}}\,=\,\frac{g}{\sigma}\,
  31. g h \sqrt{gh}
  32. g k tanh ( k h ) \sqrt{\frac{g}{k}\,\tanh\,(k\,h)\,}
  33. c g = Ω k c_{g}=\frac{\partial\Omega}{\partial k}
  34. 1 2 g k = 1 2 g σ \frac{1}{2}\,\sqrt{\frac{g}{k}}\,=\,\frac{1}{2}\,\frac{g}{\sigma}\,
  35. g h \sqrt{gh}\,
  36. 1 2 c p ( 1 + k h 1 - tanh 2 ( k h ) tanh ( k h ) ) \frac{1}{2}\,c_{p}\,\left(1\,+\,k\,h\,\frac{1\,-\,\tanh^{2}\,(k\,h)}{\tanh\,(k% \,h)}\right)
  37. c g c p \frac{c_{g}}{c_{p}}\,
  38. 1 2 \frac{1}{2}\,
  39. 1 1\,
  40. 1 2 ( 1 + k h 1 - tanh 2 ( k h ) tanh ( k h ) ) \frac{1}{2}\,\left(1\,+\,k\,h\,\frac{1\,-\,\tanh^{2}\,(k\,h)}{\tanh\,(k\,h)}\right)
  41. s y m b o l u x ( s y m b o l x , z , t ) symbol{u}_{x}(symbol{x},z,t)\,
  42. s y m b o l e k σ a e k z cos θ symbol{e}_{k}\,\sigma\,a\;\,\text{e}^{\displaystyle k\,z}\,\cos\,\theta\,
  43. s y m b o l e k g h a cos θ symbol{e}_{k}\,\sqrt{\frac{g}{h}}\,a\,\cos\,\theta\,
  44. s y m b o l e k σ a cosh ( k ( z + h ) ) sinh ( k h ) cos θ symbol{e}_{k}\,\sigma\,a\,\frac{\cosh\,\bigl(k\,(z+h)\bigr)}{\sinh\,(k\,h)}\,% \cos\,\theta\,
  45. u z ( s y m b o l x , z , t ) u_{z}(symbol{x},z,t)\,
  46. σ a e k z sin θ \sigma\,a\;\,\text{e}^{\displaystyle k\,z}\,\sin\,\theta\,
  47. σ a z + h h sin θ \sigma\,a\,\frac{z\,+\,h}{h}\,\sin\,\theta\,
  48. σ a sinh ( k ( z + h ) ) sinh ( k h ) sin θ \sigma\,a\,\frac{\sinh\,\bigl(k\,(z+h)\bigr)}{\sinh\,(k\,h)}\,\sin\,\theta\,
  49. s y m b o l ξ x ( s y m b o l x , z , t ) symbol{\xi}_{x}(symbol{x},z,t)\,
  50. - s y m b o l e k a e k z sin θ -symbol{e}_{k}\,a\;\,\text{e}^{\displaystyle k\,z}\,\sin\,\theta\,
  51. - s y m b o l e k 1 k h a sin θ -symbol{e}_{k}\,\frac{1}{k\,h}\,a\,\sin\,\theta\,
  52. - s y m b o l e k a cosh ( k ( z + h ) ) sinh ( k h ) sin θ -symbol{e}_{k}\,a\,\frac{\cosh\,\bigl(k\,(z+h)\bigr)}{\sinh\,(k\,h)}\,\sin\,\theta\,
  53. ξ z ( s y m b o l x , z , t ) \xi_{z}(symbol{x},z,t)\,
  54. a e k z cos θ a\;\,\text{e}^{\displaystyle k\,z}\,\cos\,\theta\,
  55. a z + h h cos θ a\,\frac{z\,+\,h}{h}\,\cos\,\theta\,
  56. a sinh ( k ( z + h ) ) sinh ( k h ) cos θ a\,\frac{\sinh\,\bigl(k\,(z+h)\bigr)}{\sinh\,(k\,h)}\,\cos\,\theta\,
  57. p ( s y m b o l x , z , t ) p(symbol{x},z,t)\,
  58. ρ g a e k z cos θ \rho\,g\,a\;\,\text{e}^{\displaystyle k\,z}\,\cos\,\theta\,
  59. ρ g a cos θ \rho\,g\,a\,\cos\,\theta\,
  60. ρ g a cosh ( k ( z + h ) ) cosh ( k h ) cos θ \rho\,g\,a\,\frac{\cosh\,\bigl(k\,(z+h)\bigr)}{\cosh\,(k\,h)}\,\cos\,\theta\,
  61. g σ / ρ 4 \scriptstyle\sqrt[4]{g\sigma/\rho}
  62. 1 λ σ / ( ρ g ) \scriptstyle\frac{1}{\lambda}\sqrt{\sigma/(\rho g)}
  63. Ω 2 ( k ) = ( g + γ ρ k 2 ) k tanh ( k h ) , \Omega^{2}(k)\,=\,\left(g\,+\,\frac{\gamma}{\rho}\,k^{2}\right)\,k\;\tanh\,(k% \,h),
  64. g ~ = g + γ ρ k 2 . \tilde{g}\,=\,g\,+\,\frac{\gamma}{\rho}\,k^{2}.
  65. Ω 2 ( k ) = | k | ( ρ - ρ ρ + ρ g + γ ρ + ρ k 2 ) , \Omega^{2}(k)\,=\,|k|\,\left(\frac{\rho-\rho^{\prime}}{\rho+\rho^{\prime}}g\,+% \,\frac{\gamma}{\rho+\rho^{\prime}}\,k^{2}\right),
  66. Ω 2 ( k ) = g k ( ρ - ρ ) ρ coth ( k h ) + ρ coth ( k h ) , \Omega^{2}(k)=\frac{g\,k(\rho-\rho^{\prime})}{\rho\,\coth(kh)+\rho^{\prime}\,% \coth(kh^{\prime})},
  67. E E\,
  68. E = 1 2 ρ g a 2 E\,=\,\frac{1}{2}\,\rho\,g\,a^{2}\,
  69. S x x S_{xx}\,
  70. S x x = ( 2 c g c p - 1 2 ) E S_{xx}\,=\,\left(2\,\frac{c_{g}}{c_{p}}\,-\,\frac{1}{2}\right)\,E\,
  71. 𝒜 \mathcal{A}\,
  72. 𝒜 = E σ = E ω - k U \mathcal{A}\,=\,\frac{E}{\sigma}\,=\,\frac{E}{\omega\,-\,k\,U}\,
  73. M M\,
  74. M = E c p = k E σ M\,=\,\frac{E}{c_{p}}\,=\,k\,\frac{E}{\sigma}\,
  75. U ~ \tilde{U}\,
  76. U ~ = U + M ρ h = U + E ρ h c p \tilde{U}\,=\,U\,+\,\frac{M}{\rho\,h}\,=\,U\,+\,\frac{E}{\rho\,h\,c_{p}}\,
  77. u ¯ S \bar{u}_{S}\,
  78. u ¯ S = 1 2 σ k a 2 cosh 2 k ( z + h ) sinh 2 ( k h ) \bar{u}_{S}\,=\,\frac{1}{2}\,\sigma\,k\,a^{2}\,\frac{\cosh\,2\,k\,(z+h)}{\sinh% ^{2}\,(k\,h)}\,
  79. E t + x ( ( U + c g ) E ) + S x x U x = 0 \frac{\partial E}{\partial t}\,+\,\frac{\partial}{\partial x}\Bigl((U\,+\,c_{g% })\,E\Bigr)\,+\,S_{xx}\,\frac{\partial U}{\partial x}\,=\,0\,
  80. 𝒜 t + x ( ( U + c g ) 𝒜 ) = 0 \frac{\partial\mathcal{A}}{\partial t}\,+\,\frac{\partial}{\partial x}\Bigl((U% \,+\,c_{g})\,\mathcal{A}\Bigr)\,=\,0\,
  81. k t + ω x = 0 \frac{\partial k}{\partial t}\,+\,\frac{\partial\omega}{\partial x}\,=\,0\,
  82. ω = Ω ( k ) + k U \omega\,=\,\Omega(k)\,+\,k\,U\,
  83. t ( ρ h ) + x ( ρ h U ~ ) = 0 \frac{\partial}{\partial t}\Bigl(\rho\,h\Bigr)\,+\,\frac{\partial}{\partial x}% \Bigl(\rho\,h\,\tilde{U}\Bigr)\,=\,0\,
  84. t ( ρ h U ~ ) + x ( ρ h U ~ 2 + 1 2 ρ g h 2 + S x x ) = ρ g h d x \frac{\partial}{\partial t}\Bigl(\rho\,h\,\tilde{U}\Bigr)\,+\,\frac{\partial}{% \partial x}\left(\rho\,h\,\tilde{U}^{2}\,+\,\frac{1}{2}\,\rho\,g\,h^{2}\,+\,S_% {xx}\right)\,=\,\rho\,g\,h\,\frac{\partial d}{\partial x}\,
  85. U ~ \tilde{U}
  86. E pot = - h η ρ g z d z ¯ - - h 0 ρ g z d z = 1 2 ρ g η 2 ¯ = 1 4 ρ g a 2 , E\text{pot}\,=\,\overline{\int_{-h}^{\eta}\rho\,g\,z\;\,\text{d}z}\,-\,\int_{-% h}^{0}\rho\,g\,z\;\,\text{d}z\,=\,\overline{\frac{1}{2}\,\rho\,g\,\eta^{2}}\,=% \,\frac{1}{4}\,\rho\,g\,a^{2},
  87. E kin = - h 0 1 2 ρ [ | s y m b o l U + s y m b o l u x | 2 + u z 2 ] d z ¯ - - h 0 1 2 ρ | s y m b o l U | 2 d z = 1 4 ρ σ 2 k tanh ( k h ) a 2 , E\text{kin}\,=\,\overline{\int_{-h}^{0}\frac{1}{2}\,\rho\,\left[\,\left|symbol% {U}\,+\,symbol{u}_{x}\right|^{2}\,+\,u_{z}^{2}\,\right]\;\,\text{d}z}\,-\,\int% _{-h}^{0}\frac{1}{2}\,\rho\,\left|symbol{U}\right|^{2}\;\,\text{d}z\,=\,\frac{% 1}{4}\,\rho\,\frac{\sigma^{2}}{k\,\tanh\,(k\,h)}\,a^{2},
  88. E kin = 1 4 ρ g a 2 . E\text{kin}\,=\,\frac{1}{4}\,\rho\,g\,a^{2}.
  89. E = E pot + E kin = 1 2 ρ g a 2 . E\,=\,E\text{pot}\,+\,E\text{kin}\,=\,\frac{1}{2}\,\rho\,g\,a^{2}.
  90. E pot = E kin = 1 4 ( ρ g + γ k 2 ) a 2 , so E = E pot + E kin = 1 2 ( ρ g + γ k 2 ) a 2 , E\text{pot}\,=\,E\text{kin}\,=\,\frac{1}{4}\,\left(\rho\,g\,+\,\gamma\,k^{2}% \right)\,a^{2},\qquad\,\text{so}\qquad E\,=\,E\text{pot}\,+\,E\text{kin}\,=\,% \frac{1}{2}\,\left(\rho\,g\,+\,\gamma\,k^{2}\right)\,a^{2},
  91. 𝒜 = E / σ : \mathcal{A}=E/\sigma:
  92. 𝒜 t + [ ( s y m b o l U + s y m b o l c g ) 𝒜 ] = 0 , \frac{\partial\mathcal{A}}{\partial t}\,+\,\nabla\cdot\left[\left(symbol{U}+% symbol{c}_{g}\right)\,\mathcal{A}\right]\,=\,0,
  93. ( s y m b o l U + s y m b o l c g ) 𝒜 \left(symbol{U}+symbol{c}_{g}\right)\,\mathcal{A}
  94. s y m b o l c g = c g s y m b o l e k symbol{c}_{g}=c_{g}\,symbol{e}_{k}
  95. E t + [ ( s y m b o l U + s y m b o l c g ) E ] + 𝕊 : ( \nablasymbol U ) = 0 , \frac{\partial E}{\partial t}\,+\,\nabla\cdot\left[\left(symbol{U}+symbol{c}_{% g}\right)\,E\right]\,+\,\mathbb{S}:\left(\nablasymbol{U}\right)\,=\,0,
  96. ( s y m b o l U + s y m b o l c g ) E \left(symbol{U}+symbol{c}_{g}\right)\,E
  97. 𝕊 \mathbb{S}
  98. \nablasymbol U \nablasymbol{U}
  99. 𝕊 : ( \nablasymbol U ) \mathbb{S}:(\nablasymbol{U})
  100. \nablasymbol U = 𝟢 , \nablasymbol{U}=\mathsf{0},
  101. E E
  102. 𝕊 \mathbb{S}
  103. \nablasymbol U \nablasymbol{U}
  104. 𝕊 = ( S x x S x y S y x S y y ) = 𝕀 ( c g c p - 1 2 ) E + 1 k 2 ( k x k x k x k y k y k x k y k y ) c g c p E , 𝕀 = ( 1 0 0 1 ) and s y m b o l U = ( U x x U y x U x y U y y ) , \begin{aligned}\displaystyle\mathbb{S}&\displaystyle=\,\begin{pmatrix}S_{xx}&S% _{xy}\\ S_{yx}&S_{yy}\end{pmatrix}\,=\,\mathbb{I}\,\left(\frac{c_{g}}{c_{p}}-\frac{1}{% 2}\right)\,E\,+\,\frac{1}{k^{2}}\,\begin{pmatrix}k_{x}\,k_{x}&k_{x}\,k_{y}\\ k_{y}\,k_{x}&k_{y}\,k_{y}\end{pmatrix}\,\frac{c_{g}}{c_{p}}\,E,\\ \displaystyle\mathbb{I}&\displaystyle=\,\begin{pmatrix}1&0\\ 0&1\end{pmatrix}\quad\,\text{and}\\ \displaystyle\nabla symbol{U}&\displaystyle=\,\begin{pmatrix}\displaystyle% \frac{\partial U_{x}}{\partial x}&\displaystyle\frac{\partial U_{y}}{\partial x% }\\ \displaystyle\frac{\partial U_{x}}{\partial y}&\displaystyle\frac{\partial U_{% y}}{\partial y}\end{pmatrix},\end{aligned}
  105. k x k_{x}
  106. k y k_{y}
  107. s y m b o l k symbol{k}
  108. U x U_{x}
  109. U y U_{y}
  110. s y m b o l U symbol{U}
  111. s y m b o l M symbol{M}
  112. s y m b o l M = - h η ρ ( s y m b o l U + s y m b o l u x ) d z ¯ - - h 0 ρ s y m b o l U d z = E c p s y m b o l e k , symbol{M}\,=\,\overline{\int_{-h}^{\eta}\rho\,\left(symbol{U}+symbol{u}_{x}% \right)\;\,\text{d}z}\,-\,\int_{-h}^{0}\rho\,symbol{U}\;\,\text{d}z\,=\,\frac{% E}{c_{p}}\,symbol{e}_{k},
  113. s y m b o l U ~ \tilde{symbol{U}}
  114. s y m b o l U ~ = s y m b o l U + s y m b o l M ρ h . \tilde{symbol{U}}\,=\,symbol{U}\,+\,\frac{symbol{M}}{\rho\,h}.
  115. s y m b o l U symbol{U}
  116. s y m b o l U ~ \tilde{symbol{U}}
  117. t ( ρ h ) + ( ρ h s y m b o l U ~ ) = 0 , \frac{\partial}{\partial t}\left(\rho\,h\,\right)\,+\,\nabla\cdot\left(\rho\,h% \,\tilde{symbol{U}}\right)\,=\,0,
  118. t ( ρ h s y m b o l U ~ ) + ( ρ h s y m b o l U ~ s y m b o l U ~ + 1 2 ρ g h 2 𝕀 + 𝕊 ) = ρ g h d , \frac{\partial}{\partial t}\left(\rho\,h\,\tilde{symbol{U}}\right)\,+\,\nabla% \cdot\left(\rho\,h\,\tilde{symbol{U}}\otimes\tilde{symbol{U}}\,+\,\frac{1}{2}% \,\rho\,g\,h^{2}\,\mathbb{I}\,+\,\mathbb{S}\right)\,=\,\rho\,g\,h\,\nabla d,
  119. 𝕊 \mathbb{S}
  120. 𝕀 \mathbb{I}
  121. \otimes
  122. s y m b o l U ~ s y m b o l U ~ = ( U ~ x U ~ x U ~ x U ~ y U ~ y U ~ x U ~ y U ~ y ) . \tilde{symbol{U}}\otimes\tilde{symbol{U}}\,=\,\begin{pmatrix}\tilde{U}_{x}\,% \tilde{U}_{x}&\tilde{U}_{x}\,\tilde{U}_{y}\\ \tilde{U}_{y}\,\tilde{U}_{x}&\tilde{U}_{y}\,\tilde{U}_{y}\end{pmatrix}.
  123. t ( E σ ) + [ ( s y m b o l U + s y m b o l c g ) E σ ] = 0. \frac{\partial}{\partial t}\left(\frac{E}{\sigma}\,\right)+\,\nabla\cdot\left[% \left(symbol{U}+symbol{c}_{g}\right)\,\frac{E}{\sigma}\right]\,=\,0.
  124. s y m b o l k t + ω = s y m b o l 0 , \frac{\partial symbol{k}}{\partial t}\,+\,\nabla\omega\,=\,symbol{0},
  125. ( s y m b o l U = s y m b o l 0 ) , (symbol{U}=symbol{0}),
  126. s y m b o l u ¯ S \bar{symbol{u}}_{S}
  127. s y m b o l u ¯ S = 1 2 σ k a 2 cosh 2 k ( z + h ) sinh 2 ( k h ) s y m b o l e k , \bar{symbol{u}}_{S}\,=\,\frac{1}{2}\,\sigma\,k\,a^{2}\,\frac{\cosh\,2\,k\,(z+h% )}{\sinh^{2}\,(k\,h)}\,symbol{e}_{k},
  128. ρ s y m b o l u ¯ S \rho\,\bar{symbol{u}}_{S}
  129. s y m b o l M symbol{M}

Aizerman's_conjecture.html

  1. d x d t = P x + q f ( e ) , e = r * x x R n , \frac{dx}{dt}=Px+qf(e),\quad e=r^{*}x\quad x\in R^{n},
  2. k 1 < f ( e ) < k 2. k1<f(e)<k2.

Akbulut_cork.html

  1. W int K ( M int A ) × [ 0 , 1 ] , W\setminus\operatorname{int}\,K\cong\left(M\setminus\operatorname{int}\,A% \right)\times\left[0,1\right],

Alan_Baker_(mathematician).html

  1. α 1 , , α n \alpha_{1},...,\alpha_{n}
  2. β 1 , . . , β n \beta_{1},..,\beta_{n}
  3. { 1 , β 1 , , β n } \{1,\beta_{1},...,\beta_{n}\}
  4. α 1 β 1 α 2 β 2 α n β n \alpha_{1}^{\beta_{1}}\alpha_{2}^{\beta_{2}}\cdots\alpha_{n}^{\beta_{n}}

Albrecht_v._Herald_Co..html

  1. n n
  2. a a
  3. n = N ( P n , a ) n=N(P_{n},a)
  4. a = A ( P a , n ) a=A(P_{a},n)
  5. P n = C n - A n ( P a - C a ) 1 - δ ϵ n P_{n}=\frac{C_{n}-\frac{\partial A}{\partial n}(P_{a}-\frac{\partial C}{% \partial a})}{1-\frac{\delta}{\epsilon_{n}}}
  6. C n C_{n}
  7. C a \frac{\partial C}{\partial a}
  8. ϵ n = - N P n P n n \epsilon_{n}=-\frac{\partial N}{\partial P_{n}}\frac{P_{n}}{n}
  9. δ = 1 - N a A n \delta=1-\frac{\partial N}{\partial a}\frac{\partial A}{\partial n}
  10. - A n ( P a - C a ) -\frac{\partial A}{\partial n}(P_{a}-\frac{\partial C}{\partial a})

Algebra.html

  1. a x 2 + b x + c = 0 ax^{2}+bx+c=0
  2. a , b , c a,b,c
  3. a a
  4. x + 2 = 5 x+2=5
  5. x x
  6. x = 3 x=3
  7. E = m c 2 E=mc^{2}
  8. E E
  9. m m
  10. c c
  11. a x 2 + b x + c = 0 , ax^{2}+bx+c=0,
  12. a , b , c a,b,c
  13. a a
  14. 0
  15. x x

Algebra_tile.html

  1. 6 - 3 = ? 6-3=?
  2. 6 - 3 = 3 6-3=3
  3. - 4 - ( - 2 ) = ? -4-(-2)=?
  4. - 4 + 2 -4+2
  5. - ( - 2 ) = 2 -(-2)=2
  6. 5 - 8 5-8
  7. 5 x - 3 5x-3
  8. 4 x + 5 - 2 x - 3 4x+5-2x-3
  9. 2 x + 2 2x+2
  10. 3 ( x + 1 ) = 3 x + 3 3(x+1)=3x+3
  11. 3 x + 3 3x+3
  12. x - 6 = 2 x-6=2
  13. x = 8 x=8
  14. x + 7 = 10 x+7=10
  15. x = 3 x=3
  16. x + 1 x+1
  17. x + 2 x+2
  18. ( x + 3 ) ( x + 3 ) (x+3)(x+3)

Algebraic_Riccati_equation.html

  1. A T X + X A - X B R - 1 B T X + Q = 0 A^{T}X+XA-XBR^{-1}B^{T}X+Q=0\,
  2. X = A T X A - ( A T X B ) ( R + B T X B ) - 1 ( B T X A ) + Q . X=A^{T}XA-(A^{T}XB)(R+B^{T}XB)^{-1}(B^{T}XA)+Q.\,
  3. t = 1 T ( y t T Q y t + u t T R u t ) \sum_{t=1}^{T}(y_{t}^{T}Qy_{t}+u_{t}^{T}Ru_{t})
  4. y t = A y t - 1 + B u t , y_{t}=Ay_{t-1}+Bu_{t},
  5. u t * = - ( B T X t B + R ) - 1 ( B T X t A ) y t - 1 , u_{t}^{*}=-(B^{T}X_{t}B+R)^{-1}(B^{T}X_{t}A)y_{t-1},
  6. X T = Q X_{T}=Q
  7. X t - 1 = Q + A T X t A - A T X t B ( B T X t B + R ) - 1 B T X t A , X_{t-1}=Q+A^{T}X_{t}A-A^{T}X_{t}B(B^{T}X_{t}B+R)^{-1}B^{T}X_{t}A,\,
  8. K = R - 1 B T X K=R^{-1}B^{T}X
  9. A - B K = A - B R - 1 B T X A-BK=A-BR^{-1}B^{T}X
  10. K = ( R + B T X B ) - 1 B T X A K=(R+B^{T}XB)^{-1}B^{T}XA
  11. A - B K = A - B ( R + B T X B ) - 1 B T X A A-BK=A-B(R+B^{T}XB)^{-1}B^{T}XA
  12. Z = ( A - B R - 1 B T - Q - A T ) Z=\begin{pmatrix}A&-BR^{-1}B^{T}\\ -Q&-A^{T}\end{pmatrix}
  13. Z \scriptstyle Z
  14. 2 n × n \scriptstyle 2n\times n
  15. ( U 1 U 2 ) \begin{pmatrix}U_{1}\\ U_{2}\end{pmatrix}
  16. X = U 2 U 1 - 1 X=U_{2}U_{1}^{-1}
  17. A - B R - 1 B T X \scriptstyle A-BR^{-1}B^{T}X
  18. Z \scriptstyle Z
  19. A A
  20. Z = ( A + B R - 1 B T ( A - 1 ) T Q - B R - 1 B T ( A - 1 ) T - ( A - 1 ) T Q ( A - 1 ) T ) Z=\begin{pmatrix}A+BR^{-1}B^{T}(A^{-1})^{T}Q&-BR^{-1}B^{T}(A^{-1})^{T}\\ -(A^{-1})^{T}Q&(A^{-1})^{T}\end{pmatrix}
  21. Z \scriptstyle Z
  22. 2 n × n \scriptstyle 2n\times n
  23. ( U 1 U 2 ) \begin{pmatrix}U_{1}\\ U_{2}\end{pmatrix}
  24. X = U 2 U 1 - 1 X=U_{2}U_{1}^{-1}
  25. A - B ( R + B T X B ) - 1 B T X A \scriptstyle A-B(R+B^{T}XB)^{-1}B^{T}XA
  26. Z \scriptstyle Z

Algebraic_statistics.html

  1. p i = Pr ( X = i ) , i = 0 , 1 , 2 p_{i}=\mathrm{Pr}(X=i),\quad i=0,1,2
  2. i = 0 2 p i = 1 and 0 p i 1. \sum_{i=0}^{2}p_{i}=1\quad\mbox{and}~{}\quad 0\leq p_{i}\leq 1.
  3. p i = Pr ( X = i ) = ( 2 i ) q i ( 1 - q ) 2 - i p_{i}=\mathrm{Pr}(X=i)={2\choose i}q^{i}(1-q)^{2-i}
  4. 4 p 0 p 2 - p 1 2 = 0. 4p_{0}p_{2}-p_{1}^{2}=0.
  5. i = 0 2 p i = 1 and 0 p i 1 , \sum_{i=0}^{2}p_{i}=1\quad\mbox{and}~{}\quad 0\leq p_{i}\leq 1,

Algorithmic_inference.html

  1. μ \mu
  2. σ 2 \sigma^{2}
  3. { X 1 , , X m } \{X_{1},\ldots,X_{m}\}
  4. S μ = i = 1 m X i S_{\mu}=\sum_{i=1}^{m}X_{i}
  5. S σ 2 = i = 1 m ( X i - X ¯ ) 2 , where X ¯ = S μ m S_{\sigma^{2}}=\sum_{i=1}^{m}(X_{i}-\overline{X})^{2},\,\text{ where }% \overline{X}=\frac{S_{\mu}}{m}
  6. T = S μ - m μ S σ 2 m - 1 m = X ¯ - μ S σ 2 / ( m ( m - 1 ) ) T=\frac{S_{\mu}-m\mu}{\sqrt{S_{\sigma^{2}}}}\sqrt{\frac{m-1}{m}}=\frac{% \overline{X}-\mu}{\sqrt{S_{\sigma^{2}}/(m(m-1))}}
  7. f T ( t ) = Γ ( m / 2 ) Γ ( ( m - 1 ) / 2 ) 1 π ( m - 1 ) ( 1 + t 2 m - 1 ) m / 2 . f_{T}(t)=\frac{\Gamma(m/2)}{\Gamma((m-1)/2)}\frac{1}{\sqrt{\pi(m-1)}}\left(1+% \frac{t^{2}}{m-1}\right)^{m/2}.
  8. μ \mu
  9. μ \mu
  10. 𝐱 = { 7.14 , 6.3 , 3.9 , 6.46 , 0.2 , 2.94 , 4.14 , 4.69 , 6.02 , 1.58 } \mathbf{x}=\{7.14,6.3,3.9,6.46,0.2,2.94,4.14,4.69,6.02,1.58\}
  11. s μ = 43.37 s_{\mu}=43.37
  12. s σ 2 = 46.07 s_{\sigma^{2}}=46.07
  13. μ \mu
  14. ( Z , g s y m b o l θ ) (Z,g_{symbol\theta})
  15. g s y m b o l θ g_{symbol\theta}
  16. s y m b o l θ symbol\theta
  17. 𝚯 \mathbf{\Theta}
  18. [ 0 , 1 ] [0,1]
  19. F X ( x ) = ( 1 - k x a ) I [ k , ) ( x ) , F_{X}(x)=\left(1-\frac{k}{x}^{a}\right)I_{[k,\infty)}(x),
  20. ( U , g ( a , k ) ) (U,g_{(a,k)})
  21. g ( a , k ) ( u ) = k ( 1 - u ) - 1 a , g_{(a,k)}(u)=k(1-u)^{-\frac{1}{a}},
  22. g ( a , k ) ( u ) = k u - 1 / a . g_{(a,k)}(u)=ku^{-1/a}.
  23. s = h ( x 1 , , x m ) = h ( g s y m b o l θ ( z 1 ) , , g s y m b o l θ ( z m ) ) s=h(x_{1},\ldots,x_{m})=h(g_{symbol\theta}(z_{1}),\ldots,g_{symbol\theta}(z_{m% }))
  24. s = ρ ( s y m b o l θ ; z 1 , , z m ) s=\rho(symbol\theta;z_{1},\ldots,z_{m})
  25. s 1 = i = 1 m log x i s_{1}=\sum_{i=1}^{m}\log x_{i}
  26. s 2 = min i = 1 , , m { x i } s_{2}=\min_{i=1,\ldots,m}\{x_{i}\}
  27. g ( a , k ) g_{(a,k)}
  28. s 1 = m log k + 1 / a i = 1 m log u i s_{1}=m\log k+1/a\sum_{i=1}^{m}\log u_{i}
  29. s 2 = min i = 1 , , m { k u i - 1 a } , s_{2}=\min_{i=1,\ldots,m}\{ku_{i}^{-\frac{1}{a}}\},
  30. ( a , k ) (a,k)
  31. a = log u i - m log min { u i } s 1 - m log s 2 . a=\frac{\sum\log u_{i}-m\log\min\{u_{i}\}}{s_{1}-m\log s_{2}}.
  32. k = e a s 1 - log u i m a k=\mathrm{e}^{\frac{as_{1}-\sum\log u_{i}}{ma}}
  33. s 1 s_{1}
  34. s 2 s_{2}
  35. u 1 , , u m u_{1},\ldots,u_{m}
  36. X = ( Z , g s y m b o l θ ) \mathcal{M}_{X}=(Z,g_{symbol\theta})
  37. s y m b o l θ symbol\theta
  38. 𝚯 \mathbf{\Theta}
  39. F M ( μ ) F_{M}(\mu)
  40. s M = i = 1 m x i s_{M}=\sum_{i=1}^{m}x_{i}
  41. Σ 2 \Sigma^{2}
  42. σ 2 \sigma^{2}
  43. F M ( μ ) = Φ ( m μ - s M σ m ) , F_{M}(\mu)=\Phi\left(\frac{m\mu-s_{M}}{\sigma\sqrt{m}}\right),
  44. Φ \Phi
  45. δ / 2 \delta/2
  46. 1 - δ / 2 1-\delta/2
  47. ( x , y ) (x,y)
  48. t = ( 9 , 13 , > 13 , 18 , 12 , 23 , 31 , 34 , > 45 , 48 , > 161 ) , t=(9,13,>13,18,12,23,31,34,>45,48,>161),\,

Almost_Mathieu_operator.html

  1. [ H ω λ , α u ] ( n ) = u ( n + 1 ) + u ( n - 1 ) + 2 λ cos ( 2 π ( ω + n α ) ) u ( n ) , [H^{\lambda,\alpha}_{\omega}u](n)=u(n+1)+u(n-1)+2\lambda\cos(2\pi(\omega+n% \alpha))u(n),\,
  2. 2 ( ) \ell^{2}(\mathbb{Z})
  3. α , ω 𝕋 , λ > 0 \alpha,\omega\in\mathbb{T},\lambda>0
  4. λ = 1 \lambda=1
  5. α \alpha
  6. H ω λ , α H^{\lambda,\alpha}_{\omega}
  7. α \alpha
  8. ω ω + α \omega\mapsto\omega+\alpha
  9. H ω λ , α H^{\lambda,\alpha}_{\omega}
  10. ω \omega
  11. ω \omega
  12. 0 < λ < 1 0<\lambda<1
  13. H ω λ , α H^{\lambda,\alpha}_{\omega}
  14. λ = 1 \lambda=1
  15. H ω λ , α H^{\lambda,\alpha}_{\omega}
  16. λ > 1 \lambda>1
  17. H ω λ , α H^{\lambda,\alpha}_{\omega}
  18. λ 1 \lambda\geq 1
  19. γ ( E ) \gamma(E)
  20. γ ( E ) max { 0 , log ( λ ) } . \gamma(E)\geq\max\{0,\log(\lambda)\}.\,
  21. E E
  22. α \alpha
  23. λ > 0 \lambda>0
  24. L e b ( σ ( H ω λ , α ) ) = | 4 - 4 λ | Leb(\sigma(H^{\lambda,\alpha}_{\omega}))=|4-4\lambda|\,
  25. λ > 0 \lambda>0
  26. λ = 1 \lambda=1
  27. λ 1 \lambda\neq 1
  28. λ = 1 \lambda=1

Alpha_beta_filter.html

  1. (1) 𝐱 ^ k 𝐱 ^ k - 1 + Δ T 𝐯 ^ k - 1 \,\text{(1)}\quad\hat{\,\textbf{x}}_{k}\leftarrow\hat{\,\textbf{x}}_{k-1}+% \Delta\textrm{T}\ \,\textbf{ }\hat{\,\textbf{v}}_{k-1}
  2. (2) 𝐯 ^ k 𝐯 ^ k - 1 \,\text{(2)}\quad\hat{\,\textbf{v}}_{k}\leftarrow\hat{\,\textbf{v}}_{k-1}
  3. (3) 𝐫 ^ k 𝐱 k - 𝐱 ^ k {\,\textbf{(3)}}\quad\hat{\,\textbf{r}}_{k}\leftarrow\,\textbf{x}_{k}-\hat{\,% \textbf{x}}_{k}
  4. (4) 𝐱 ^ k 𝐱 ^ k + ( α ) 𝐫 ^ k \,\textbf{(4)}\quad\hat{\,\textbf{x}}_{k}\leftarrow\hat{\,\textbf{x}}_{k}+(% \alpha)\ \hat{\,\textbf{r}}_{k}
  5. (5) 𝐯 ^ k 𝐯 ^ k + ( β / [ Δ T ] ) 𝐫 ^ k \,\textbf{(5)}\quad\hat{\,\textbf{v}}_{k}\leftarrow\hat{\,\textbf{v}}_{k}+(% \beta/[\Delta\textrm{T}])\ \hat{\,\textbf{r}}_{k}
  6. 0 < α < 1 \quad 0<\alpha<1
  7. 0 < β 2 \quad 0<\beta\leq 2
  8. 0 < 4 - 2 α - β \quad 0<4-2\alpha-\beta
  9. 0 < β < 1 0<\beta<1
  10. T T
  11. σ w 2 \sigma_{w}^{2}
  12. σ v 2 \sigma_{v}^{2}
  13. λ = σ w T 2 σ v \lambda=\frac{\sigma_{w}T^{2}}{\sigma_{v}}
  14. r = 4 + λ - 8 λ + λ 2 4 r=\frac{4+\lambda-\sqrt{8\lambda+\lambda^{2}}}{4}
  15. α = 1 - r 2 \alpha=1-r^{2}
  16. β = 2 ( 2 - α ) - 4 1 - α \beta=2\left(2-\alpha\right)-4\sqrt{1-\alpha}
  17. 𝐱 ^ k 𝐱 ^ k + ( α ) 𝐫 k \hat{\,\textbf{x}}_{k}\leftarrow\hat{\,\textbf{x}}_{k}+(\alpha)\ \,\textbf{r}_% {k}
  18. 𝐯 ^ k 𝐯 ^ k + ( β / [ Δ T ] ) 𝐫 k \hat{\,\textbf{v}}_{k}\leftarrow\hat{\,\textbf{v}}_{k}+(\beta/[\Delta\textrm{T% }])\ \,\textbf{r}_{k}
  19. 𝐚 ^ k 𝐚 ^ k + ( γ 2 / [ Δ T ] 2 ) 𝐫 k \hat{\,\textbf{a}}_{k}\leftarrow\hat{\,\textbf{a}}_{k}+(\gamma\ 2/[\Delta% \textrm{T}]^{\textrm{2}})\ \,\textbf{r}_{k}
  20. λ = σ w T 2 σ v b = λ 2 - 3 c = λ 2 + 3 d = - 1 p = c - b 2 3 q = 2 b 3 27 - b c 3 + d v = q 2 + 4 p 3 27 z = - q + v 2 3 s = z - p 3 z - b 3 α = 1 - s 2 β = 2 ( 1 - s ) 2 γ = β 2 2 α \begin{aligned}\displaystyle\lambda&\displaystyle=\frac{\sigma_{w}T^{2}}{% \sigma_{v}}\\ \displaystyle b&\displaystyle=\frac{\lambda}{2}-3\\ \displaystyle c&\displaystyle=\frac{\lambda}{2}+3\\ \displaystyle d&\displaystyle=-1\\ \displaystyle p&\displaystyle=c-\frac{b^{2}}{3}\\ \displaystyle q&\displaystyle=\frac{2b^{3}}{27}-\frac{bc}{3}+d\\ \displaystyle v&\displaystyle=\sqrt{q^{2}+\frac{4p^{3}}{27}}\\ \displaystyle z&\displaystyle=-\sqrt[3]{q+\frac{v}{2}}\\ \displaystyle s&\displaystyle=z-\frac{p}{3z}-\frac{b}{3}\\ \displaystyle\alpha&\displaystyle=1-s^{2}\\ \displaystyle\beta&\displaystyle=2(1-s)^{2}\\ \displaystyle\gamma&\displaystyle=\frac{\beta^{2}}{2\alpha}\end{aligned}
  21. 𝐱 ^ k 𝐱 ^ k + ( α ) 𝐫 ^ k \hat{\,\textbf{x}}_{k}\leftarrow\hat{\,\textbf{x}}_{k}+(\alpha)\ \hat{\,% \textbf{r}}_{k}
  22. λ = σ w T 2 σ v α = - λ 2 + λ 4 + 16 λ 2 8 \begin{aligned}\displaystyle\lambda&\displaystyle=\frac{\sigma_{w}T^{2}}{% \sigma_{v}}\\ \displaystyle\alpha&\displaystyle=\frac{-\lambda^{2}+\sqrt{\lambda^{4}+16% \lambda^{2}}}{8}\end{aligned}

Alpha–beta_transformation.html

  1. α β γ \alpha\beta\gamma
  2. α β γ \alpha\beta\gamma
  3. α β γ \alpha\beta\gamma
  4. i α β γ ( t ) = T i a b c ( t ) = 2 3 [ 1 - 1 2 - 1 2 0 3 2 - 3 2 1 2 1 2 1 2 ] [ i a ( t ) i b ( t ) i c ( t ) ] i_{\alpha\beta\gamma}(t)=Ti_{abc}(t)=\frac{2}{3}\begin{bmatrix}1&-\frac{1}{2}&% -\frac{1}{2}\\ 0&\frac{\sqrt{3}}{2}&-\frac{\sqrt{3}}{2}\\ \frac{1}{2}&\frac{1}{2}&\frac{1}{2}\\ \end{bmatrix}\begin{bmatrix}i_{a}(t)\\ i_{b}(t)\\ i_{c}(t)\end{bmatrix}
  5. i a b c ( t ) i_{abc}(t)
  6. i α β γ ( t ) i_{\alpha\beta\gamma}(t)
  7. T T
  8. i a b c ( t ) = T - 1 i α β γ ( t ) = [ 1 0 1 - 1 2 3 2 1 - 1 2 - 3 2 1 ] [ i α ( t ) i β ( t ) i γ ( t ) ] . i_{abc}(t)=T^{-1}i_{\alpha\beta\gamma}(t)=\begin{bmatrix}1&0&1\\ -\frac{1}{2}&\frac{\sqrt{3}}{2}&1\\ -\frac{1}{2}&-\frac{\sqrt{3}}{2}&1\end{bmatrix}\begin{bmatrix}i_{\alpha}(t)\\ i_{\beta}(t)\\ i_{\gamma}(t)\end{bmatrix}.
  9. i a ( t ) = 2 I cos θ ( t ) , i b ( t ) = 2 I cos ( θ ( t ) - 2 3 π ) , i c ( t ) = 2 I cos ( θ ( t ) + 2 3 π ) , \begin{aligned}\displaystyle i_{a}(t)=&\displaystyle\sqrt{2}I\cos\theta(t),\\ \displaystyle i_{b}(t)=&\displaystyle\sqrt{2}I\cos\left(\theta(t)-\frac{2}{3}% \pi\right),\\ \displaystyle i_{c}(t)=&\displaystyle\sqrt{2}I\cos\left(\theta(t)+\frac{2}{3}% \pi\right),\end{aligned}
  10. I I
  11. i a ( t ) i_{a}(t)
  12. i b ( t ) i_{b}(t)
  13. i c ( t ) i_{c}(t)
  14. θ ( t ) \theta(t)
  15. ω t \omega t
  16. T T
  17. i α = 2 I cos θ ( t ) , i β = 2 I sin θ ( t ) , i γ = 0 , \begin{aligned}\displaystyle i_{\alpha}=&\displaystyle\sqrt{2}I\cos\theta(t),% \\ \displaystyle i_{\beta}=&\displaystyle\sqrt{2}I\sin\theta(t),\\ \displaystyle i_{\gamma}=&\displaystyle 0,\end{aligned}
  18. α β γ \alpha\beta\gamma
  19. T T
  20. i α β γ ( t ) = T i a b c ( t ) = 2 3 [ 1 - 1 2 - 1 2 0 3 2 - 3 2 1 2 1 2 1 2 ] [ i a ( t ) i b ( t ) i c ( t ) ] , i_{\alpha\beta\gamma}(t)=Ti_{abc}(t)=\sqrt{\frac{2}{3}}\begin{bmatrix}1&-\frac% {1}{2}&-\frac{1}{2}\\ 0&\frac{\sqrt{3}}{2}&-\frac{\sqrt{3}}{2}\\ \frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\ \end{bmatrix}\begin{bmatrix}i_{a}(t)\\ i_{b}(t)\\ i_{c}(t)\end{bmatrix},
  21. i α = 3 I cos θ ( t ) , i β = 3 I sin θ ( t ) , i γ = 0. \begin{aligned}\displaystyle i_{\alpha}=&\displaystyle\sqrt{3}I\cos\theta(t),% \\ \displaystyle i_{\beta}=&\displaystyle\sqrt{3}I\sin\theta(t),\\ \displaystyle i_{\gamma}=&\displaystyle 0.\end{aligned}
  22. i a b c ( t ) = 2 3 [ 1 0 2 2 - 1 2 3 2 2 2 - 1 2 - 3 2 2 2 ] [ i α ( t ) i β ( t ) i γ ( t ) ] . i_{abc}(t)=\sqrt{\frac{2}{3}}\begin{bmatrix}1&0&\frac{\sqrt{2}}{2}\\ -\frac{1}{2}&\frac{\sqrt{3}}{2}&\frac{\sqrt{2}}{2}\\ -\frac{1}{2}&-\frac{\sqrt{3}}{2}&\frac{\sqrt{2}}{2}\\ \end{bmatrix}\begin{bmatrix}i_{\alpha}(t)\\ i_{\beta}(t)\\ i_{\gamma}(t)\end{bmatrix}.
  23. i a ( t ) + i b ( t ) + i c ( t ) = 0 i_{a}(t)+i_{b}(t)+i_{c}(t)=0
  24. i γ ( t ) = 0 i_{\gamma}(t)=0
  25. i α β ( t ) = 2 3 [ 1 - 1 2 - 1 2 0 3 2 - 3 2 ] [ i a ( t ) i b ( t ) i c ( t ) ] i_{\alpha\beta}(t)=\frac{2}{3}\begin{bmatrix}1&-\frac{1}{2}&-\frac{1}{2}\\ 0&\frac{\sqrt{3}}{2}&-\frac{\sqrt{3}}{2}\end{bmatrix}\begin{bmatrix}i_{a}(t)\\ i_{b}(t)\\ i_{c}(t)\end{bmatrix}
  26. i a b c ( t ) = 3 2 [ 2 3 0 - 1 3 3 3 - 1 3 - 3 3 ] [ i α ( t ) i β ( t ) ] . i_{abc}(t)=\frac{3}{2}\begin{bmatrix}\frac{2}{3}&0\\ -\frac{1}{3}&\frac{\sqrt{3}}{3}\\ -\frac{1}{3}&-\frac{\sqrt{3}}{3}\end{bmatrix}\begin{bmatrix}i_{\alpha}(t)\\ i_{\beta}(t)\end{bmatrix}.
  27. α β γ \alpha\beta\gamma
  28. α β γ \alpha\beta\gamma
  29. δ \delta
  30. α \alpha
  31. β \beta
  32. α \alpha
  33. I α β γ I_{\alpha\beta\gamma}
  34. ω \omega
  35. γ \gamma
  36. d q o dqo
  37. d q o dqo
  38. α β γ \alpha\beta\gamma
  39. α β γ \alpha\beta\gamma

Alternated_hypercubic_honeycomb.html

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

Alternating_polynomial.html

  1. f ( x 1 , , x n ) f(x_{1},\dots,x_{n})
  2. f ( x 1 , , x j , , x i , , x n ) = - f ( x 1 , , x i , , x j , , x n ) . f(x_{1},\dots,x_{j},\dots,x_{i},\dots,x_{n})=-f(x_{1},\dots,x_{i},\dots,x_{j},% \dots,x_{n}).
  3. f ( x σ ( 1 ) , , x σ ( n ) ) = sgn ( σ ) f ( x 1 , , x n ) . f\left(x_{\sigma(1)},\dots,x_{\sigma(n)}\right)=\mathrm{sgn}(\sigma)f(x_{1},% \dots,x_{n}).
  4. f ( x 1 , , x n , y 1 , , y t ) f(x_{1},\dots,x_{n},y_{1},\dots,y_{t})
  5. x 1 , , x n x_{1},\dots,x_{n}
  6. x i x_{i}
  7. y j y_{j}
  8. x 1 , , x n x_{1},\dots,x_{n}
  9. 𝐙 2 \mathbf{Z}_{2}
  10. v n = 1 i < j n ( x j - x i ) . v_{n}=\prod_{1\leq i<j\leq n}(x_{j}-x_{i}).
  11. a = v n s a=v_{n}\cdot s
  12. s s
  13. v n v_{n}
  14. ( x j - x i ) (x_{j}-x_{i})
  15. x i = x j x_{i}=x_{j}
  16. f ( x 1 , , x i , , x j , , x n ) = f ( x 1 , , x j , , x i , , x n ) = - f ( x 1 , , x i , , x j , , x n ) , f(x_{1},\dots,x_{i},\dots,x_{j},\dots,x_{n})=f(x_{1},\dots,x_{j},\dots,x_{i},% \dots,x_{n})=-f(x_{1},\dots,x_{i},\dots,x_{j},\dots,x_{n}),
  17. ( x j - x i ) (x_{j}-x_{i})
  18. v n v_{n}
  19. v n v_{n}
  20. Λ n [ v n ] \Lambda_{n}[v_{n}]
  21. Λ n [ v n ] / v n 2 - Δ \Lambda_{n}[v_{n}]/\langle v_{n}^{2}-\Delta\rangle
  22. Δ = v n 2 \Delta=v_{n}^{2}
  23. R [ e 1 , , e n , v n ] / v n 2 - Δ . R[e_{1},\dots,e_{n},v_{n}]/\langle v_{n}^{2}-\Delta\rangle.
  24. W n W_{n}
  25. n > 2 n>2
  26. x n x_{n}
  27. n = 3 n=3
  28. x 1 x_{1}
  29. x 2 x_{2}
  30. ( x 2 - x 1 ) (x_{2}-x_{1})
  31. ( x 1 - x 2 ) = - ( x 2 - x 1 ) (x_{1}-x_{2})=-(x_{2}-x_{1})
  32. ( x 3 - x 1 ) (x_{3}-x_{1})
  33. ( x 3 - x 2 ) (x_{3}-x_{2})

Amalgam_(chemistry).html

  1. 2 H 3 N - Hg - H Δ T 2 NH 3 + H 2 + 2 Hg \mathrm{2\ H_{3}N{-}Hg{-}H\ \xrightarrow{\Delta T}\ 2\ NH_{3}+H_{2}+2\ Hg}

Ambient_calculus.html

  1. i n m . P in\;m.P
  2. m m
  3. P P
  4. o u t m . P out\;m.P
  5. m m
  6. o p e n m . P open\;m.P
  7. m m
  8. c o p y m . copy\;m.
  9. m m

Ambient_ionization.html

  1. H e * + [ ( H 2 O ) n H ] [ ( H 2 O ) n - 1 H ] + + O H + e - He^{*}+\left[(H_{2}O\right)_{n}H]\to\left[(H_{2}O\right)_{n-1}H]^{+}+OH^{% \centerdot}+e^{-}
  2. [ ( H 2 O ) n H ] + + S S H + + n H 2 O \left[(H_{2}O\right)_{n}H]^{+}+S\to SH^{+}+nH_{2}O

American_Standard_(Adams).html

  1. 2 4 {}^{4}_{2}

Ammann–Beenker_tiling.html

  1. 1 + 2 1+\sqrt{2}
  2. R R r R ; r R R\to RrR;r\to R
  3. r r
  4. R R
  5. 1 + 2 1+\sqrt{2}
  6. 1 - 2 1-\sqrt{2}
  7. 1 + 2 1+\sqrt{2}
  8. 2 2 2\sqrt{2}
  9. 1 + 2 1+\sqrt{2}
  10. 2 - 1 \sqrt{2}-1
  11. A = [ 0 0 0 - 1 1 0 0 0 0 - 1 0 0 0 0 - 1 0 ] . A=\begin{bmatrix}0&0&0&-1\\ 1&0&0&0\\ 0&-1&0&0\\ 0&0&-1&0\end{bmatrix}.
  12. B = [ - 1 / 2 0 - 1 / 2 2 / 2 1 / 2 2 / 2 - 1 / 2 0 - 1 / 2 0 - 1 / 2 - 2 / 2 - 1 / 2 2 / 2 1 / 2 0 ] B=\begin{bmatrix}-1/2&0&-1/2&\sqrt{2}/2\\ 1/2&\sqrt{2}/2&-1/2&0\\ -1/2&0&-1/2&-\sqrt{2}/2\\ -1/2&\sqrt{2}/2&1/2&0\end{bmatrix}
  13. B A B - 1 = [ 2 / 2 2 / 2 0 0 - 2 / 2 2 / 2 0 0 0 0 - 2 / 2 2 / 2 0 0 - 2 / 2 - 2 / 2 ] . BAB^{-1}=\begin{bmatrix}\sqrt{2}/2&\sqrt{2}/2&0&0\\ -\sqrt{2}/2&\sqrt{2}/2&0&0\\ 0&0&-\sqrt{2}/2&\sqrt{2}/2\\ 0&0&-\sqrt{2}/2&-\sqrt{2}/2\end{bmatrix}.

Ammonia_volatilization_from_urea.html

  1. ( N H 2 ) 2 C O + H 2 O u r e a s e N H 3 + H 2 N C O O H 2 N H 3 ( g a s ) + C O 2 ( g a s ) (NH_{2})_{2}CO+H_{2}O\stackrel{urease}{\rightarrow}NH_{3}+H_{2}NCOOH% \rightarrow 2NH_{3(gas)}+CO_{2(gas)}
  2. N H 3 ( g a s ) + H 2 O N H 4 + + O H - NH_{3(gas)}+H_{2}O\rightarrow NH^{+}_{4}+OH^{-}

Amott_test.html

  1. I w I_{w}
  2. I o I_{o}
  3. A I = I w - I o AI=I_{w}-I_{o}
  4. I w = S s p w - S c w S o r - S c w I_{w}=\frac{S_{spw}-S_{cw}}{S_{or}-S_{cw}}
  5. S s p w S_{spw}
  6. S c w S_{cw}
  7. S o r S_{or}
  8. I o = S o r - S s p o S o r - S c w I_{o}=\frac{S_{or}-S_{spo}}{S_{or}-S_{cw}}
  9. S s p o S_{spo}
  10. S c w S_{cw}
  11. S o r S_{or}

Ampère's_force_law.html

  1. F m L = 2 k A I 1 I 2 r \frac{F_{m}}{L}=2k_{A}\frac{I_{1}I_{2}}{r}
  2. k A = def μ 0 4 π k_{A}\ \overset{\underset{\mathrm{def}}{}}{=}\ \frac{\mu_{0}}{4\pi}
  3. μ 0 = def 4 π × 10 - 7 \mu_{0}\ \overset{\underset{\mathrm{def}}{}}{=}\ 4\pi\times 10^{-7}
  4. 2 × 10 - 7 \displaystyle 2\times 10^{-7}
  5. F 12 = μ 0 4 π L 1 L 2 I 1 d 1 × ( I 2 d 2 × 𝐫 ^ 21 ) | r | 2 \vec{F}_{12}=\frac{\mu_{0}}{4\pi}\int_{L_{1}}\int_{L_{2}}\frac{I_{1}d\vec{\ell% }_{1}\ \mathbf{\times}\ (I_{2}d\vec{\ell}_{2}\ \mathbf{\times}\ \hat{\mathbf{r% }}_{21})}{|r|^{2}}
  6. F 12 \vec{F}_{12}
  7. d 1 d\vec{\ell}_{1}
  8. d 2 d\vec{\ell}_{2}
  9. 𝐫 ^ 21 \hat{\mathbf{r}}_{21}
  10. d n d\vec{\ell}_{n}
  11. d 1 d\vec{\ell}_{1}
  12. d F x = k I I d s d s cos ( x d s ) cos ( r d s ) - cos ( r x ) cos ( d s d s ) r 2 . dF_{x}=kII^{\prime}ds^{\prime}\int ds\frac{\cos(xds)\cos(rds^{\prime})-\cos(rx% )\cos(dsds^{\prime})}{r^{2}}.
  13. cos ( x d s ) cos ( r d s ) r 2 = - cos ( r x ) ( cos ϵ - 3 cos ϕ cos ϕ ) r 2 \frac{\cos(xds)\cos(rds^{\prime})}{r^{2}}=-\cos(rx)\frac{(\cos\epsilon-3\cos% \phi\cos\phi^{\prime})}{r^{2}}
  14. cos ( r x ) cos ( d s d s ) r 2 = cos ( r x ) cos ϵ r 2 \frac{\cos(rx)\cos(dsds^{\prime})}{r^{2}}=\frac{\cos(rx)\cos\epsilon}{r^{2}}
  15. d F x = k I I d s d s cos ( r x ) 2 cos ϵ - 3 cos ϕ cos ϕ r 2 dF_{x}=kII^{\prime}ds^{\prime}\int ds^{\prime}\cos(rx)\frac{2\cos\epsilon-3% \cos\phi\cos\phi^{\prime}}{r^{2}}
  16. r s = cos ϕ , r s = - cos ϕ \frac{\partial r}{\partial s}=\cos\phi,\frac{\partial r}{\partial s^{\prime}}=% -\cos\phi^{\prime}
  17. 2 r s s = - cos ϵ + cos ϕ cos ϕ r \frac{\partial^{2}r}{\partial s\partial s^{\prime}}=\frac{-\cos\epsilon+\cos% \phi\cos\phi^{\prime}}{r}
  18. d 2 F = k I I d s d s r 2 ( r s r s - 2 r 2 r s s ) d^{2}F=\frac{kII^{\prime}dsds^{\prime}}{r^{2}}\left(\frac{\partial r}{\partial s% }\frac{\partial r}{\partial s^{\prime}}-2r\frac{\partial^{2}r}{\partial s% \partial s^{\prime}}\right)
  19. d 2 F x = k I I d s d s [ ( 1 r 2 ( r s r s - 2 r 2 r s s ) + r 2 Q s s ) cos ( r x ) + Q s cos ( x d s ) - Q s cos ( x d s ) ] d^{2}F_{x}=kII^{\prime}dsds^{\prime}\left[\left(\frac{1}{r^{2}}\left(\frac{% \partial r}{\partial s}\frac{\partial r}{\partial s^{\prime}}-2r\frac{\partial% ^{2}r}{\partial s\partial s^{\prime}}\right)+r\frac{\partial^{2}Q}{\partial s% \partial s^{\prime}}\right)\cos(rx)+\frac{\partial Q}{\partial s^{\prime}}\cos% (xds)-\frac{\partial Q}{\partial s}\cos(xds^{\prime})\right]
  20. Q = - ( 1 + k ) 2 r Q=-\frac{(1+k)}{2r}
  21. 𝐝 𝟐 𝐅 = - k I I 2 r 2 [ ( 3 - k ) 𝐫 𝟏 ^ ( 𝐝𝐬𝐝𝐬 ) - 3 ( 1 - k ) 𝐫 𝟏 ^ ( 𝐫 𝟏 ^ 𝐝𝐬 ) ( 𝐫 𝟏 ^ 𝐝𝐬 ) - ( 1 + k ) 𝐝𝐬 ( 𝐫 𝟏 ^ 𝐝𝐬 ) - ( 1 + k ) 𝐝 𝐬 ( 𝐫 𝟏 ^ 𝐝𝐬 ) ] \mathbf{d^{2}F}=-\frac{kII^{\prime}}{2r^{2}}\left[(3-k)\hat{\mathbf{r_{1}}}(% \mathbf{dsds^{\prime}})-3(1-k)\hat{\mathbf{r_{1}}}(\mathbf{\hat{r_{1}}ds})(% \mathbf{\hat{r_{1}}ds^{\prime}})-(1+k)\mathbf{ds}(\mathbf{\hat{r_{1}}ds^{% \prime}})-(1+k)\mathbf{d^{\prime}s}(\mathbf{\hat{r_{1}}ds})\right]
  22. 𝐝 𝟐 𝐅 = - k I I r 3 [ 2 𝐫 ( 𝐝𝐬𝐝𝐬 ) - 3 𝐫 ( 𝐫𝐝𝐬 ) ( 𝐫𝐝𝐬 ) ] \mathbf{d^{2}F}=-\frac{kII^{\prime}}{r^{3}}\left[2\mathbf{r}(\mathbf{dsds^{% \prime}})-3\mathbf{r}(\mathbf{rds})(\mathbf{rds^{\prime}})\right]
  23. 𝐝 𝟐 𝐅 = - k I I r 3 [ 𝐫 ( 𝐝𝐬𝐝𝐬 ) - 𝐝𝐬 ( 𝐫𝐝𝐬 ) - 𝐝𝐬 ( 𝐫𝐝𝐬 ) ] \mathbf{d^{2}F}=-\frac{kII^{\prime}}{r^{3}}\left[\mathbf{r}(\mathbf{dsds^{% \prime}})-\mathbf{ds(rds^{\prime})}-\mathbf{ds^{\prime}(rds)}\right]
  24. 𝐝 𝟐 𝐅 = k I I r 3 [ ( 𝐝𝐬 × 𝐝𝐬 × 𝐫 ) + 𝐝𝐬 ( 𝐫𝐝𝐬 ) ] \mathbf{d^{2}F}=\frac{kII^{\prime}}{r^{3}}\left[\left(\mathbf{ds}\times\mathbf% {ds^{\prime}}\times\mathbf{r}\right)+\mathbf{ds^{\prime}(rds)}\right]
  25. 𝐅 = k I I 𝐝𝐬 × ( 𝐝𝐬 × 𝐫 ) | r | 3 \mathbf{F}=kII^{\prime}\int\int\frac{\mathbf{ds}\times(\mathbf{ds^{\prime}}% \times\mathbf{r})}{|r|^{3}}
  26. F 12 = μ 0 4 π L 1 L 2 I 1 d 1 × ( I 2 d 2 × 𝐫 ^ 21 ) | r | 2 \vec{F}_{12}=\frac{\mu_{0}}{4\pi}\int_{L_{1}}\int_{L_{2}}\frac{I_{1}d\vec{\ell% }_{1}\ \mathbf{\times}\ (I_{2}d\vec{\ell}_{2}\ \mathbf{\times}\ \hat{\mathbf{r% }}_{21})}{|r|^{2}}
  27. x 1 , x 2 x_{1},x_{2}
  28. ( x 1 , D , 0 ) (x_{1},D,0)
  29. ( x 2 , 0 , 0 ) (x_{2},0,0)
  30. d 1 = ( d x 1 , 0 , 0 ) d\vec{\ell}_{1}=(dx_{1},0,0)
  31. d 2 = ( d x 2 , 0 , 0 ) d\vec{\ell}_{2}=(dx_{2},0,0)
  32. 𝐫 ^ 21 = 1 ( x 1 - x 2 ) 2 + D 2 ( x 1 - x 2 , D , 0 ) \hat{\mathbf{r}}_{21}=\frac{1}{\sqrt{(x_{1}-x_{2})^{2}+D^{2}}}(x_{1}-x_{2},D,0)
  33. | r | = ( x 1 - x 2 ) 2 + D 2 |r|=\sqrt{(x_{1}-x_{2})^{2}+D^{2}}
  34. F 12 = μ 0 I 1 I 2 4 π L 1 L 2 ( d x 1 , 0 , 0 ) × [ ( d x 2 , 0 , 0 ) × ( x 1 - x 2 , D , 0 ) ] | ( x 1 - x 2 ) 2 + D 2 | 3 / 2 \vec{F}_{12}=\frac{\mu_{0}I_{1}I_{2}}{4\pi}\int_{L_{1}}\int_{L_{2}}\frac{(dx_{% 1},0,0)\ \mathbf{\times}\ \left[(dx_{2},0,0)\ \mathbf{\times}\ (x_{1}-x_{2},D,% 0)\right]}{|(x_{1}-x_{2})^{2}+D^{2}|^{3/2}}
  35. F 12 = μ 0 I 1 I 2 4 π L 1 L 2 d x 1 d x 2 ( 0 , - D , 0 ) | ( x 1 - x 2 ) 2 + D 2 | 3 / 2 \vec{F}_{12}=\frac{\mu_{0}I_{1}I_{2}}{4\pi}\int_{L_{1}}\int_{L_{2}}dx_{1}dx_{2% }\frac{(0,-D,0)}{|(x_{1}-x_{2})^{2}+D^{2}|^{3/2}}
  36. x 2 x_{2}
  37. - -\infty
  38. + +\infty
  39. F 12 = μ 0 I 1 I 2 4 π 2 D ( 0 , - 1 , 0 ) L 1 d x 1 \vec{F}_{12}=\frac{\mu_{0}I_{1}I_{2}}{4\pi}\frac{2}{D}(0,-1,0)\int_{L_{1}}dx_{1}
  40. L 1 L_{1}
  41. F 12 = μ 0 I 1 I 2 4 π 2 D ( 0 , - 1 , 0 ) L 1 \vec{F}_{12}=\frac{\mu_{0}I_{1}I_{2}}{4\pi}\frac{2}{D}(0,-1,0)L_{1}
  42. F 12 L 1 = μ 0 I 1 I 2 2 π D ( 0 , - 1 , 0 ) \frac{\vec{F}_{12}}{L_{1}}=\frac{\mu_{0}I_{1}I_{2}}{2\pi D}(0,-1,0)
  43. F m L \frac{F_{m}}{L}

Amplitude_amplification.html

  1. \mathcal{H}
  2. B := { | k } k = 0 N - 1 B:=\{|k\rangle\}_{k=0}^{N-1}
  3. P : P:\mathcal{H}\to\mathcal{H}
  4. P P
  5. χ : { 0 , 1 } \chi:\mathbb{Z}\to\{0,1\}
  6. B op := { | ω k } k = 0 N - 1 B_{\,\text{op}}:=\{|\omega_{k}\rangle\}_{k=0}^{N-1}
  7. P := χ ( k ) = 1 | ω k ω k | P:=\sum_{\chi(k)=1}|\omega_{k}\rangle\langle\omega_{k}|
  8. P P
  9. \mathcal{H}
  10. 1 \mathcal{H}_{1}
  11. 0 \mathcal{H}_{0}
  12. 1 \displaystyle\mathcal{H}_{1}
  13. | ψ |\psi\rangle\in\mathcal{H}
  14. | ψ = sin ( θ ) | ψ 1 + cos ( θ ) | ψ 0 |\psi\rangle=\sin(\theta)|\psi_{1}\rangle+\cos(\theta)|\psi_{0}\rangle
  15. θ = arcsin ( | P | ψ | ) [ 0 , π / 2 ] \theta=\arcsin\left(\left|P|\psi\rangle\right|\right)\in[0,\pi/2]
  16. | ψ 1 |\psi_{1}\rangle
  17. | ψ 0 |\psi_{0}\rangle
  18. | ψ |\psi\rangle
  19. 1 \mathcal{H}_{1}
  20. 0 \mathcal{H}_{0}
  21. ψ \mathcal{H}_{\psi}
  22. | ψ 0 |\psi_{0}\rangle
  23. | ψ 1 |\psi_{1}\rangle
  24. sin 2 ( θ ) \sin^{2}(\theta)
  25. Q ( ψ , P ) := - S ψ S P Q(\psi,P):=-S_{\psi}S_{P}\,\!
  26. S ψ = I - 2 | ψ ψ | and S P = I - 2 P . \begin{aligned}\displaystyle S_{\psi}&\displaystyle=I-2|\psi\rangle\langle\psi% |\quad\,\text{and}\\ \displaystyle S_{P}&\displaystyle=I-2P.\end{aligned}
  27. S P S_{P}
  28. S ψ S_{\psi}
  29. | ψ |\psi\rangle
  30. ψ \mathcal{H}_{\psi}
  31. Q | ψ 0 = - S ψ | ψ 0 = ( 2 cos 2 ( θ ) - 1 ) | ψ 0 + 2 sin ( θ ) cos ( θ ) | ψ 1 Q|\psi_{0}\rangle=-S_{\psi}|\psi_{0}\rangle=(2\cos^{2}(\theta)-1)|\psi_{0}% \rangle+2\sin(\theta)\cos(\theta)|\psi_{1}\rangle
  32. Q | ψ 1 = S ψ | ψ 1 = - 2 sin ( θ ) cos ( θ ) | ψ 0 + ( 1 - 2 sin 2 ( θ ) ) | ψ 1 Q|\psi_{1}\rangle=S_{\psi}|\psi_{1}\rangle=-2\sin(\theta)\cos(\theta)|\psi_{0}% \rangle+(1-2\sin^{2}(\theta))|\psi_{1}\rangle
  33. ψ \mathcal{H}_{\psi}
  34. Q Q
  35. 2 θ 2\theta\,\!
  36. Q = ( cos ( 2 θ ) sin ( 2 θ ) - sin ( 2 θ ) cos ( 2 θ ) ) Q=\begin{pmatrix}\cos(2\theta)&\sin(2\theta)\\ -\sin(2\theta)&\cos(2\theta)\end{pmatrix}
  37. Q Q
  38. n n
  39. | ψ |\psi\rangle
  40. Q n | ψ = cos ( ( 2 n + 1 ) θ ) | ψ 0 + sin ( ( 2 n + 1 ) θ ) | ψ 1 Q^{n}|\psi\rangle=\cos((2n+1)\theta)|\psi_{0}\rangle+\sin((2n+1)\theta)|\psi_{% 1}\rangle
  41. n n
  42. sin 2 ( ( 2 n + 1 ) θ ) \sin^{2}((2n+1)\theta)\,\!
  43. n = π 4 θ n=\left\lfloor\frac{\pi}{4\theta}\right\rfloor
  44. χ \chi
  45. B op = B B_{\,\text{op}}=B
  46. | ψ = 1 N k = 0 N - 1 | k |\psi\rangle=\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}|k\rangle
  47. sin ( θ ) = | P | ψ | = G / N \sin(\theta)=|P|\psi\rangle|=\sqrt{G/N}
  48. sin ( θ ) 1 \sin(\theta)\ll 1
  49. n = π 4 θ π 4 sin ( θ ) = π 4 N G = O ( N ) . n=\left\lfloor\frac{\pi}{4\theta}\right\rfloor\approx\left\lfloor\frac{\pi}{4% \sin(\theta)}\right\rfloor=\left\lfloor\frac{\pi}{4}\sqrt{\frac{N}{G}}\right% \rfloor=O(\sqrt{N}).
  50. S P S_{P}
  51. O ( N ) O(\sqrt{N})

Anaerobic_corrosion.html

  1. F e F e 2 + + 2 e - Fe\;\rightarrow\;Fe^{2+}\;+\;2\,e^{-}
  2. 2 H + + 2 e - H 2 2\,H^{+}\;+\;2\,e^{-}\rightarrow\;H_{2}\;
  3. 2 H 2 O + 2 e - H 2 + 2 O H - 2\,H_{2}O\;+\;2\,e^{-}\;\rightarrow\;H_{2}\;+\;2\,OH^{-}
  4. F e + 2 H 2 O F e ( O H ) 2 + H 2 Fe\;+\;2\,H_{2}O\;\rightarrow\;Fe(OH)_{2}\;+\;H_{2}\;

Ancestral_relation.html

  1. ( P x x R y ) P y (Px\land xRy)\rightarrow Py
  2. 𝟕𝟔 : a R * b F [ x ( a R x F x ) x y ( F x x R y F y ) F b ] \mathbf{76:}\ \vdash aR^{*}b\leftrightarrow\forall F[\forall x(aRx\to Fx)\land% \forall x\forall y(Fx\land xRy\to Fy)\to Fb]
  3. 𝟗𝟖 : ( a R * b b R * c ) a R * c \mathbf{98:}\ \vdash(aR^{*}b\land bR^{*}c)\rightarrow aR^{*}c
  4. 𝟏𝟏𝟓 : I ( R ) x y z [ ( x R y x R z ) y = z ] \mathbf{115:}\ \vdash I(R)\leftrightarrow\forall x\forall y\forall z[(xRy\land xRz% )\rightarrow y=z]
  5. 𝟏𝟑𝟑 : ( I ( R ) a R * b a R * c ) ( b R * c b = c c R * b ) \mathbf{133:}\ \vdash(I(R)\land aR^{*}b\land aR^{*}c)\rightarrow(bR^{*}c\lor b% =c\lor cR^{*}b)
  6. R * R^{*}
  7. R + R^{+}
  8. R R
  9. R * R^{*}
  10. R * R^{*}
  11. R R
  12. R * R^{*}
  13. R + R^{+}
  14. a R * b aR^{*}b
  15. F x Fx
  16. a R + x aR^{+}x
  17. x ( a R x F x ) \forall x(aRx\to Fx)
  18. x y ( F x x R y F y ) \forall x\forall y(Fx\land xRy\to Fy)
  19. F b Fb
  20. a R + b aR^{+}b
  21. F F

AND-OR-Invert.html

  1. F = ( A B ) ( C D ) ¯ F=\overline{(A\wedge B)\vee(C\wedge D)}
  2. F = A ( B C ) ¯ F=\overline{A\vee(B\wedge C)}

Andreas_von_Ettingshausen.html

  1. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}

Andronov–Pontryagin_criterion.html

  1. x ˙ = v ( x ) , \dot{x}=v(x),

Anger_function.html

  1. 𝐉 ν ( z ) = 1 π 0 π cos ( ν θ - z sin θ ) d θ \mathbf{J}_{\nu}(z)=\frac{1}{\pi}\int_{0}^{\pi}\cos(\nu\theta-z\sin\theta)\,d\theta
  2. 𝐄 ν ( z ) = 1 π 0 π sin ( ν θ - z sin θ ) d θ \mathbf{E}_{\nu}(z)=\frac{1}{\pi}\int_{0}^{\pi}\sin(\nu\theta-z\sin\theta)\,d\theta
  3. sin ( π ν ) 𝐉 ν ( z ) = cos ( π ν ) 𝐄 ν ( z ) - 𝐄 - ν ( z ) \sin(\pi\nu)\mathbf{J}_{\nu}(z)=\cos(\pi\nu)\mathbf{E}_{\nu}(z)-\mathbf{E}_{-% \nu}(z)
  4. - sin ( π ν ) 𝐄 ν ( z ) = cos ( π ν ) 𝐉 ν ( z ) - 𝐉 - ν ( z ) -\sin(\pi\nu)\mathbf{E}_{\nu}(z)=\cos(\pi\nu)\mathbf{J}_{\nu}(z)-\mathbf{J}_{-% \nu}(z)
  5. z 2 y ′′ + z y + ( z 2 - ν 2 ) y = 0 z^{2}y^{\prime\prime}+zy^{\prime}+(z^{2}-\nu^{2})y=0
  6. z 2 y ′′ + z y + ( z 2 - ν 2 ) y = ( z - ν ) sin ( π z ) / π z^{2}y^{\prime\prime}+zy^{\prime}+(z^{2}-\nu^{2})y=(z-\nu)\sin(\pi z)/\pi
  7. z 2 y ′′ + z y + ( z 2 - ν 2 ) y = - ( ( z + ν ) + ( z - ν ) cos ( π z ) ) / π . z^{2}y^{\prime\prime}+zy^{\prime}+(z^{2}-\nu^{2})y=-((z+\nu)+(z-\nu)\cos(\pi z% ))/\pi.

Annual_effective_discount_rate.html

  1. 100 - 95 100 = 5.00 % \frac{100-95}{100}=5.00\%
  2. 100 - 95 95 = 5.26 % \frac{100-95}{95}=5.26\%
  3. d = i 1 + i d=\frac{i}{1+i}
  4. i = d 1 - d i=\frac{d}{1-d}
  5. d \,d
  6. i \,i
  7. d = 1 + i 1 + i - 1 1 + i = 1 - v d=\frac{1+i}{1+i}-\frac{1}{1+i}\ =1-v
  8. v v
  9. v = 1 - d v=1-d
  10. d = i v \,d=iv
  11. i d = i - d id=i-d
  12. p \,p
  13. p \,p
  14. 1 - d = ( 1 - d ( p ) p ) p 1-d=\left(1-\frac{d^{(p)}}{p}\right)^{p}
  15. d ( p ) \,d^{(p)}
  16. p \,p
  17. 1 - d = exp ( - d ( ) ) 1-d=\exp(-d^{(\infty)})
  18. d ( ) = δ \,d^{(\infty)}=\delta
  19. d ( p ) \,d^{(p)}

Annular_velocity.html

  1. A V = 1029.4 ( P O b p m ) I D 2 - O D 2 AV=\frac{1029.4(PO_{bpm})}{ID^{2}-OD^{2}}\,
  2. A V = 24.5 ( P O g p m ) I D 2 - O D 2 AV=\frac{24.5(PO_{gpm})}{ID^{2}-OD^{2}}\,

Anonymous_veto_network.html

  1. G \scriptstyle G
  2. g \scriptstyle g
  3. q \scriptstyle q
  4. n \scriptstyle n
  5. i \scriptstyle i
  6. x i R q \scriptstyle x_{i}\,\in_{R}\,\mathbb{Z}_{q}
  7. g x i \scriptstyle g^{x_{i}}
  8. x i \scriptstyle x_{i}
  9. g y i = j < i g x j / j > i g x j g^{y_{i}}=\prod_{j<i}g^{x_{j}}/\prod_{j>i}g^{x_{j}}
  10. i \scriptstyle i
  11. g c i y i \scriptstyle g^{c_{i}y_{i}}
  12. c i \scriptstyle c_{i}
  13. c i = x i \scriptstyle c_{i}\;=\;x_{i}
  14. g c i y i \scriptstyle\prod g^{c_{i}y_{i}}
  15. g c i y i = 1 \scriptstyle\prod g^{c_{i}y_{i}}\;=\;1
  16. g c i y i 1 \scriptstyle\prod g^{c_{i}y_{i}}\;\neq\;1
  17. x i y i = 0 \scriptstyle\sum{x_{i}\cdot y_{i}}\;=\;0
  18. x 1 y 1 + x 1 y 2 + x 3 y 3 = x 1 ( - x 2 - x 3 ) + x 2 ( x 1 - x 3 ) + x 3 ( x 1 + x 2 ) = 0 \scriptstyle x_{1}\cdot y_{1}\,+\,x_{1}\cdot y_{2}\,+\,x_{3}\cdot y_{3}\;=\;x_% {1}\cdot(-x_{2}\,-\,x_{3})\,+\,x_{2}\cdot(x_{1}\,-\,x_{3})\,+\,x_{3}\cdot(x_{1% }\,+\,x_{2})\;=\;0

Ant_on_a_rubber_rope.html

  1. x x
  2. x = 0 x=0
  3. x = c x=c
  4. c > 0 c>0
  5. t = 0 t=0
  6. x = 0 x=0
  7. v > 0 v>0
  8. t = 0 t=0
  9. α > 0 \alpha>0
  10. v < α v<\alpha
  11. α / 2 v \alpha/2v
  12. α / v \alpha/v
  13. P 1 P_{1}
  14. P 1 P_{1}
  15. P 1 P_{1}
  16. P 2 P_{2}
  17. P 1 P_{1}
  18. P 1 P_{1}
  19. P 2 P_{2}
  20. P 3 P_{3}
  21. P 2 P_{2}
  22. P 2 P_{2}
  23. P 1 P_{1}
  24. P 3 P_{3}
  25. P 1 P_{1}
  26. P 2 P_{2}
  27. P 3 P_{3}
  28. P 1 P_{1}
  29. 1 / 200000 t h 1/200000th
  30. P 2 P_{2}
  31. 1 / 100000 t h 1/100000th
  32. P 1 P_{1}
  33. P 3 P_{3}
  34. 3 / 200000 t h s 3/200000ths
  35. 1 / 200000 t h 1/200000th
  36. x = c x=c\,\!
  37. x = c + v x=c+v\,\!
  38. t = 0 t=0\,\!
  39. x = c + v x=c+v\,\!
  40. x = c + 2 v x=c+2v\,\!
  41. t = 1 t=1\,\!
  42. θ ( t ) \theta(t)\,\!
  43. θ ( 0 ) = 0 \theta(0)=0\,\!
  44. α \alpha\,\!
  45. α c + v \frac{\alpha}{c+v}\,\!
  46. c + v c+v\,\!
  47. θ ( t ) \theta(t)\,\!
  48. θ ( 1 ) = α c + v \theta(1)=\frac{\alpha}{c+v}\,\!
  49. α \alpha\,\!
  50. α c + 2 v \frac{\alpha}{c+2v}\,\!
  51. c + 2 v c+2v\,\!
  52. θ ( 2 ) = α c + v + α c + 2 v \theta(2)=\frac{\alpha}{c+v}+\frac{\alpha}{c+2v}\,\!
  53. n n\in\mathbb{N}\,\!
  54. θ ( n ) = α c + v + α c + 2 v + + α c + n v \theta(n)=\frac{\alpha}{c+v}+\frac{\alpha}{c+2v}+\cdots+\frac{\alpha}{c+nv}\,\!
  55. i i\in\mathbb{N}\,\!
  56. α c + i v α i c + i v = ( α c + v ) ( 1 i ) \frac{\alpha}{c+iv}\geqslant\frac{\alpha}{ic+iv}=\left(\frac{\alpha}{c+v}% \right)\left(\frac{1}{i}\right)\,\!
  57. θ ( n ) ( α c + v ) ( 1 + 1 2 + + 1 n ) \theta(n)\geqslant\left(\frac{\alpha}{c+v}\right)\left(1+\frac{1}{2}+\cdots+% \frac{1}{n}\right)\,\!
  58. ( 1 + 1 2 + + 1 n ) \left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right)\,\!
  59. N N\in\mathbb{N}\,\!
  60. 1 + 1 2 + + 1 N c + v α 1+\frac{1}{2}+\cdots+\frac{1}{N}\geqslant\frac{c+v}{\alpha}\,\!
  61. θ ( N ) 1 \theta(N)\geqslant 1\,\!
  62. t > 0 t>0\,\!
  63. α \alpha\,\!
  64. v v\,\!
  65. t t\,\!
  66. x = c + v t x=c+vt\,\!
  67. t t\,\!
  68. x = X x=X\,\!
  69. v X c + v t \frac{vX}{c+vt}\,\!
  70. t t\,\!
  71. y ( t ) y(t)\,\!
  72. t t\,\!
  73. y ( t ) y^{\prime}(t)\,\!
  74. y ( t ) = α + v y ( t ) c + v t y^{\prime}(t)=\alpha+\frac{v\,y(t)}{c+vt}\,\!
  75. ψ \psi\,\!
  76. ψ = 0 \psi=0\,\!
  77. ψ = 1 \psi=1\,\!
  78. ψ \psi\,\!
  79. t 0 t\geqslant 0\,\!
  80. x = X x=X\,\!
  81. ψ = X c + v t \psi=\frac{X}{c+vt}\,\!
  82. α \alpha\,\!
  83. x x\,\!
  84. α c + v t \frac{\alpha}{c+vt}\,\!
  85. ψ \psi\,\!
  86. ψ \psi\,\!
  87. t t\,\!
  88. ϕ ( t ) \phi(t)\,\!
  89. ψ \psi\,\!
  90. t t\,\!
  91. ϕ ( t ) \phi^{\prime}(t)\,\!
  92. ϕ ( t ) = α c + v t \phi^{\prime}(t)=\frac{\alpha}{c+vt}
  93. ϕ ( t ) = α c + v t d t = α v ln ( c + v t ) + κ \therefore\phi(t)=\int{\frac{\alpha}{c+vt}\,dt}=\frac{\alpha}{v}\ln(c+vt)+\kappa
  94. κ \kappa\,\!
  95. ϕ ( 0 ) = 0 \phi(0)=0\,\!
  96. κ = - α v ln c \kappa=-\frac{\alpha}{v}\ln{c}\,\!
  97. ϕ ( t ) = α v ln ( c + v t c ) \phi(t)=\frac{\alpha}{v}\ln{\left(\frac{c+vt}{c}\right)}\,\!
  98. ψ = 1 \psi=1\,\!
  99. t = T t=T\,\!
  100. ϕ ( T ) = 1 \phi(T)=1\,\!
  101. α v ln ( c + v T c ) = 1 \frac{\alpha}{v}\ln{\left(\frac{c+vT}{c}\right)}=1\,\!
  102. T = c v ( e v / α - 1 ) \therefore T=\frac{c}{v}\left(e^{v/\alpha}-1\right)\,\!
  103. T T\,\!
  104. c c\,\!
  105. v v\,\!
  106. α \alpha\,\!
  107. v > 0 v>0\,\!
  108. α > 0 \alpha>0\,\!
  109. c = 1 km c=1\,\mathrm{km}\,\!
  110. v = 1 km / s v=1\,\mathrm{km}/\mathrm{s}\,\!
  111. α = 1 cm / s \alpha=1\,\mathrm{cm}/\mathrm{s}\,\!
  112. T = ( e 100 , 000 - 1 ) s 2.8 × 10 43 , 429 s T=(e^{100,000}-1)\,\mathrm{s}\,\!\approx 2.8\times 10^{43,429}\,\mathrm{s}\,\!

Antiplane_shear.html

  1. s y m b o l ε = [ 0 0 ϵ 13 0 0 ϵ 23 ϵ 13 ϵ 23 0 ] symbol{\varepsilon}=\begin{bmatrix}0&0&\epsilon_{13}\\ 0&0&\epsilon_{23}\\ \epsilon_{13}&\epsilon_{23}&0\end{bmatrix}
  2. 12 12\,
  3. 3 3\,
  4. u 1 = u 2 = 0 ; u 3 = u ^ 3 ( x 1 , x 2 ) u_{1}=u_{2}=0~{};~{}~{}u_{3}=\hat{u}_{3}(x_{1},x_{2})
  5. u i , i = 1 , 2 , 3 u_{i},~{}i=1,2,3
  6. x 1 , x 2 , x 3 x_{1},x_{2},x_{3}\,
  7. s y m b o l σ [ σ 11 σ 12 σ 13 σ 12 σ 22 σ 23 σ 13 σ 23 σ 33 ] = [ 0 0 μ u 3 x 1 0 0 μ u 3 x 2 μ u 3 x 1 μ u 3 x 2 0 ] symbol{\sigma}\equiv\begin{bmatrix}\sigma_{11}&\sigma_{12}&\sigma_{13}\\ \sigma_{12}&\sigma_{22}&\sigma_{23}\\ \sigma_{13}&\sigma_{23}&\sigma_{33}\end{bmatrix}=\begin{bmatrix}0&0&\mu~{}% \cfrac{\partial u_{3}}{\partial x_{1}}\\ 0&0&\mu~{}\cfrac{\partial u_{3}}{\partial x_{2}}\\ \mu~{}\cfrac{\partial u_{3}}{\partial x_{1}}&\mu~{}\cfrac{\partial u_{3}}{% \partial x_{2}}&0\end{bmatrix}
  8. μ \mu\,
  9. μ 2 u 3 + b 3 ( x 1 , x 2 ) = 0 \mu~{}\nabla^{2}u_{3}+b_{3}(x_{1},x_{2})=0
  10. b 3 b_{3}
  11. x 3 x_{3}
  12. 2 u 3 = 2 u 3 x 1 2 + 2 u 3 x 2 2 \nabla^{2}u_{3}=\cfrac{\partial^{2}u_{3}}{\partial x_{1}^{2}}+\cfrac{\partial^% {2}u_{3}}{\partial x_{2}^{2}}

Antiresonance.html

  1. g g
  2. F F
  3. x ¨ 1 + 2 γ 1 x ˙ 1 - 2 g ω 1 x 2 + ω 1 2 x 1 = 2 F cos ω t x ¨ 2 + 2 γ 2 x ˙ 2 - 2 g ω 2 x 1 + ω 2 2 x 2 = 0 \begin{array}[]{lcl}\ddot{x}_{1}+2\gamma_{1}\dot{x}_{1}-2g\omega_{1}x_{2}+% \omega_{1}^{2}x_{1}&=&2F\cos\omega t\\ \ddot{x}_{2}+2\gamma_{2}\dot{x}_{2}-2g\omega_{2}x_{1}+\omega_{2}^{2}x_{2}&=&0% \end{array}
  4. ω i \omega_{i}
  5. γ i \gamma_{i}
  6. α 1 = ω 1 x 1 + i p 1 / m 1 \alpha_{1}=\omega_{1}x_{1}+ip_{1}/m_{1}
  7. α 2 = ω 2 x 2 + i p 2 / m 1 \alpha_{2}=\omega_{2}x_{2}+ip_{2}/m_{1}
  8. α ˙ 1 = i ω 1 α 1 - γ 1 ( α 1 - α 1 * ) - i g ω 1 ω 2 ( α 2 + α 2 * ) + i F ( e i ω t + e - i ω t ) α ˙ 2 = i ω 2 α 2 - γ 2 ( α 2 - α 2 * ) - i g ω 2 ω 1 ( α 1 + α 1 * ) \begin{array}[]{lcl}\dot{\alpha}_{1}&=&i\omega_{1}\alpha_{1}-\gamma_{1}(\alpha% _{1}-\alpha_{1}^{*})-ig\tfrac{\omega_{1}}{\omega_{2}}(\alpha_{2}+\alpha_{2}^{*% })+iF(e^{i\omega t}+e^{-i\omega t})\\ \dot{\alpha}_{2}&=&i\omega_{2}\alpha_{2}-\gamma_{2}(\alpha_{2}-\alpha_{2}^{*})% -ig\tfrac{\omega_{2}}{\omega_{1}}(\alpha_{1}+\alpha_{1}^{*})\end{array}
  9. α i α i e - i ω t \alpha_{i}\rightarrow\alpha_{i}e^{-i\omega t}
  10. α ˙ 1 = i Δ 1 α 1 - γ 1 ( α 1 - α 1 * e 2 i ω t ) - i g ω 1 ω 2 ( α 2 + α 2 * e 2 i ω t ) + i F ( 1 + e 2 i ω t ) α ˙ 2 = i Δ 2 α 2 - γ 2 ( α 2 - α 2 * e 2 i ω t ) - i g ω 2 ω 1 ( α 1 + α 1 * e 2 i ω t ) \begin{array}[]{lcl}\dot{\alpha}_{1}&=&i\Delta_{1}\alpha_{1}-\gamma_{1}(\alpha% _{1}-\alpha_{1}^{*}e^{2i\omega t})-ig\tfrac{\omega_{1}}{\omega_{2}}(\alpha_{2}% +\alpha_{2}^{*}e^{2i\omega t})+iF(1+e^{2i\omega t})\\ \dot{\alpha}_{2}&=&i\Delta_{2}\alpha_{2}-\gamma_{2}(\alpha_{2}-\alpha_{2}^{*}e% ^{2i\omega t})-ig\tfrac{\omega_{2}}{\omega_{1}}(\alpha_{1}+\alpha_{1}^{*}e^{2i% \omega t})\end{array}
  11. Δ i = ω - ω i \Delta_{i}=\omega-\omega_{i}
  12. e 2 i ω t e^{2i\omega t}
  13. ω + ω i ω - ω i \omega+\omega_{i}\gg\omega-\omega_{i}
  14. α ˙ 1 = i ( Δ 1 + i γ 1 ) α 1 - i g ω 1 ω 2 α 2 + i F α ˙ 2 = i ( Δ 2 + i γ 2 ) α 2 - i g ω 2 ω 1 α 1 \begin{array}[]{lcl}\dot{\alpha}_{1}&=&i(\Delta_{1}+i\gamma_{1})\alpha_{1}-ig% \tfrac{\omega_{1}}{\omega_{2}}\alpha_{2}+iF\\ \dot{\alpha}_{2}&=&i(\Delta_{2}+i\gamma_{2})\alpha_{2}-ig\tfrac{\omega_{2}}{% \omega_{1}}\alpha_{1}\end{array}
  15. α i ( t ) = α i ( 0 ) e i Δ t \alpha_{i}(t)=\alpha_{i}(0)e^{i\Delta t}
  16. α \alpha
  17. Δ \Delta
  18. α ˙ 1 = α ˙ 2 = 0 \dot{\alpha}_{1}=\dot{\alpha}_{2}=0
  19. α 1 , s s = - F ( Δ 2 + i γ 2 ) ( Δ 1 + i γ 1 ) ( Δ 2 + i γ 2 ) - g 2 α 2 , s s = ω 2 ω 1 - F g ( Δ 1 + i γ 1 ) ( Δ 2 + i γ 2 ) - g 2 \begin{array}[]{lcl}\alpha_{1,ss}&=&\dfrac{-F(\Delta_{2}+i\gamma_{2})}{(\Delta% _{1}+i\gamma_{1})(\Delta_{2}+i\gamma_{2})-g^{2}}\\ \alpha_{2,ss}&=&\dfrac{\omega_{2}}{\omega_{1}}\dfrac{-Fg}{(\Delta_{1}+i\gamma_% {1})(\Delta_{2}+i\gamma_{2})-g^{2}}\end{array}
  20. F F

Antoine_equation.html

  1. log 10 p = A - B C + T . \log_{10}p=A-\frac{B}{C+T}.
  2. T T
  3. A A
  4. B B
  5. C C
  6. C C
  7. log 10 p = A - B T \log_{10}p=A-\frac{B}{T}
  8. T = B A - log 10 p - C T=\frac{B}{A-\log_{10}\,p}-C
  9. P = 10 ( 8.20417 - 1642.89 78.32 + 230.300 ) = 760.0 mmHg P=10^{(8{.}20417-\frac{1642{.}89}{78{.}32+230{.}300})}=760{.}0\ \mathrm{mmHg}
  10. P = 10 ( 7.68117 - 1332.04 78.32 + 199.200 ) = 761.0 mmHg P=10^{(7{.}68117-\frac{1332{.}04}{78{.}32+199{.}200})}=761{.}0\ \mathrm{mmHg}
  11. A Pa = A mmHg + log 10 101325 760 = A mmHg + 2.124903. A_{\mathrm{Pa}}=A_{\mathrm{mmHg}}+\log_{10}\frac{101325}{760}=A_{\mathrm{mmHg}% }+2.124903.
  12. log 10 ( P ) = 10.3291 - 1642.89 351.47 - 42.85 = 5.005727378 = log 10 ( 101328 Pa ) . \log_{10}(P)=10{.}3291-\frac{1642{.}89}{351{.}47-42{.}85}=5{.}005727378=\log_{% 10}(101328\ \mathrm{Pa}).
  13. ln P = 23.7836 - 3782.89 351.47 - 42.85 = 11.52616367 = ln ( 101332 Pa ) . \ln P=23{.}7836-\frac{3782{.}89}{351{.}47-42{.}85}=11{.}52616367=\ln(101332\,% \mathrm{Pa}).
  14. P = exp ( A + B C + T + D T + E T 2 + F ln ( T ) ) P=\exp{\left(A+\frac{B}{C+T}+D\cdot T+E\cdot T^{2}+F\cdot\ln\left(T\right)% \right)}
  15. P = exp ( A + B C + T + D ln ( T ) + E T F ) . P=\exp\left(A+\frac{B}{C+T}+D\cdot\ln\left(T\right)+E\cdot T^{F}\right).

Apostolico–Giancarlo_algorithm.html

  1. P P
  2. T T
  3. P P
  4. T T
  5. P P
  6. T T
  7. T T
  8. P P
  9. T T
  10. n n
  11. P P
  12. O ( n m ) O(nm)
  13. m m
  14. T T
  15. T T
  16. P P

Apparent_viscosity.html

  1. η = τ γ ˙ \eta=\frac{\tau}{\dot{\gamma}}
  2. τ x y \tau_{xy}
  3. τ x y = k ( d u d y ) n \tau_{xy}=k\left(\frac{du}{dy}\right)^{n}
  4. τ x y \tau_{xy}
  5. τ y x = k | d u d y | n - 1 d u d y = η d u d y \tau_{yx}=k\left|\frac{du}{dy}\right|^{n-1}\frac{du}{dy}=\eta\frac{du}{dy}
  6. η = k | d u d y | n - 1 \eta=k\left|\frac{du}{dy}\right|^{n-1}

Appell_series.html

  1. ( q ) n = q ( q + 1 ) ( q + n - 1 ) = Γ ( q + n ) Γ ( q ) , (q)_{n}=q\,(q+1)\cdots(q+n-1)=\frac{\Gamma(q+n)}{\Gamma(q)}~{},
  2. q q
  3. q = 0 , - 1 , - 2 , q=0,-1,-2,\ldots
  4. ( a - b 1 - b 2 ) F 1 ( a , b 1 , b 2 , c ; x , y ) - a F 1 ( a + 1 , b 1 , b 2 , c ; x , y ) + b 1 F 1 ( a , b 1 + 1 , b 2 , c ; x , y ) + b 2 F 1 ( a , b 1 , b 2 + 1 , c ; x , y ) = 0 , (a-b_{1}-b_{2})F_{1}(a,b_{1},b_{2},c;x,y)-a\,F_{1}(a+1,b_{1},b_{2},c;x,y)+b_{1% }F_{1}(a,b_{1}+1,b_{2},c;x,y)+b_{2}F_{1}(a,b_{1},b_{2}+1,c;x,y)=0~{},
  5. c F 1 ( a , b 1 , b 2 , c ; x , y ) - ( c - a ) F 1 ( a , b 1 , b 2 , c + 1 ; x , y ) - a F 1 ( a + 1 , b 1 , b 2 , c + 1 ; x , y ) = 0 , c\,F_{1}(a,b_{1},b_{2},c;x,y)-(c-a)F_{1}(a,b_{1},b_{2},c+1;x,y)-a\,F_{1}(a+1,b% _{1},b_{2},c+1;x,y)=0~{},
  6. c F 1 ( a , b 1 , b 2 , c ; x , y ) + c ( x - 1 ) F 1 ( a , b 1 + 1 , b 2 , c ; x , y ) - ( c - a ) x F 1 ( a , b 1 + 1 , b 2 , c + 1 ; x , y ) = 0 , c\,F_{1}(a,b_{1},b_{2},c;x,y)+c(x-1)F_{1}(a,b_{1}+1,b_{2},c;x,y)-(c-a)x\,F_{1}% (a,b_{1}+1,b_{2},c+1;x,y)=0~{},
  7. c F 1 ( a , b 1 , b 2 , c ; x , y ) + c ( y - 1 ) F 1 ( a , b 1 , b 2 + 1 , c ; x , y ) - ( c - a ) y F 1 ( a , b 1 , b 2 + 1 , c + 1 ; x , y ) = 0 . c\,F_{1}(a,b_{1},b_{2},c;x,y)+c(y-1)F_{1}(a,b_{1},b_{2}+1,c;x,y)-(c-a)y\,F_{1}% (a,b_{1},b_{2}+1,c+1;x,y)=0~{}.
  8. c F 3 ( a 1 , a 2 , b 1 , b 2 , c ; x , y ) + ( a 1 + a 2 - c ) F 3 ( a 1 , a 2 , b 1 , b 2 , c + 1 ; x , y ) - a 1 F 3 ( a 1 + 1 , a 2 , b 1 , b 2 , c + 1 ; x , y ) - a 2 F 3 ( a 1 , a 2 + 1 , b 1 , b 2 , c + 1 ; x , y ) = 0 , c\,F_{3}(a_{1},a_{2},b_{1},b_{2},c;x,y)+(a_{1}+a_{2}-c)F_{3}(a_{1},a_{2},b_{1}% ,b_{2},c+1;x,y)-a_{1}F_{3}(a_{1}+1,a_{2},b_{1},b_{2},c+1;x,y)-a_{2}F_{3}(a_{1}% ,a_{2}+1,b_{1},b_{2},c+1;x,y)=0~{},
  9. c F 3 ( a 1 , a 2 , b 1 , b 2 , c ; x , y ) - c F 3 ( a 1 + 1 , a 2 , b 1 , b 2 , c ; x , y ) + b 1 x F 3 ( a 1 + 1 , a 2 , b 1 + 1 , b 2 , c + 1 ; x , y ) = 0 , c\,F_{3}(a_{1},a_{2},b_{1},b_{2},c;x,y)-c\,F_{3}(a_{1}+1,a_{2},b_{1},b_{2},c;x% ,y)+b_{1}x\,F_{3}(a_{1}+1,a_{2},b_{1}+1,b_{2},c+1;x,y)=0~{},
  10. c F 3 ( a 1 , a 2 , b 1 , b 2 , c ; x , y ) - c F 3 ( a 1 , a 2 + 1 , b 1 , b 2 , c ; x , y ) + b 2 y F 3 ( a 1 , a 2 + 1 , b 1 , b 2 + 1 , c + 1 ; x , y ) = 0 , c\,F_{3}(a_{1},a_{2},b_{1},b_{2},c;x,y)-c\,F_{3}(a_{1},a_{2}+1,b_{1},b_{2},c;x% ,y)+b_{2}y\,F_{3}(a_{1},a_{2}+1,b_{1},b_{2}+1,c+1;x,y)=0~{},
  11. c F 3 ( a 1 , a 2 , b 1 , b 2 , c ; x , y ) - c F 3 ( a 1 , a 2 , b 1 + 1 , b 2 , c ; x , y ) + a 1 x F 3 ( a 1 + 1 , a 2 , b 1 + 1 , b 2 , c + 1 ; x , y ) = 0 , c\,F_{3}(a_{1},a_{2},b_{1},b_{2},c;x,y)-c\,F_{3}(a_{1},a_{2},b_{1}+1,b_{2},c;x% ,y)+a_{1}x\,F_{3}(a_{1}+1,a_{2},b_{1}+1,b_{2},c+1;x,y)=0~{},
  12. c F 3 ( a 1 , a 2 , b 1 , b 2 , c ; x , y ) - c F 3 ( a 1 , a 2 , b 1 , b 2 + 1 , c ; x , y ) + a 2 y F 3 ( a 1 , a 2 + 1 , b 1 , b 2 + 1 , c + 1 ; x , y ) = 0 . c\,F_{3}(a_{1},a_{2},b_{1},b_{2},c;x,y)-c\,F_{3}(a_{1},a_{2},b_{1},b_{2}+1,c;x% ,y)+a_{2}y\,F_{3}(a_{1},a_{2}+1,b_{1},b_{2}+1,c+1;x,y)=0~{}.
  13. x F 1 ( a , b 1 , b 2 , c ; x , y ) = a b 1 c F 1 ( a + 1 , b 1 + 1 , b 2 , c + 1 ; x , y ) , \frac{\partial}{\partial x}F_{1}(a,b_{1},b_{2},c;x,y)=\frac{ab_{1}}{c}F_{1}(a+% 1,b_{1}+1,b_{2},c+1;x,y)~{},
  14. y F 1 ( a , b 1 , b 2 , c ; x , y ) = a b 2 c F 1 ( a + 1 , b 1 , b 2 + 1 , c + 1 ; x , y ) . \frac{\partial}{\partial y}F_{1}(a,b_{1},b_{2},c;x,y)=\frac{ab_{2}}{c}F_{1}(a+% 1,b_{1},b_{2}+1,c+1;x,y)~{}.
  15. ( x ( 1 - x ) 2 x 2 + y ( 1 - x ) 2 x y + [ c - ( a + b 1 + 1 ) x ] x - b 1 y y - a b 1 ) F 1 ( x , y ) = 0 , \left(x(1-x)\frac{\partial^{2}}{\partial x^{2}}+y(1-x)\frac{\partial^{2}}{% \partial x\partial y}+[c-(a+b_{1}+1)x]\frac{\partial}{\partial x}-b_{1}y\frac{% \partial}{\partial y}-ab_{1}\right)F_{1}(x,y)=0~{},
  16. ( y ( 1 - y ) 2 y 2 + x ( 1 - y ) 2 x y + [ c - ( a + b 2 + 1 ) y ] y - b 2 x x - a b 2 ) F 1 ( x , y ) = 0 . \left(y(1-y)\frac{\partial^{2}}{\partial y^{2}}+x(1-y)\frac{\partial^{2}}{% \partial x\partial y}+[c-(a+b_{2}+1)y]\frac{\partial}{\partial y}-b_{2}x\frac{% \partial}{\partial x}-ab_{2}\right)F_{1}(x,y)=0~{}.
  17. x F 3 ( a 1 , a 2 , b 1 , b 2 , c ; x , y ) = a 1 b 1 c F 3 ( a 1 + 1 , a 2 , b 1 + 1 , b 2 , c + 1 ; x , y ) , \frac{\partial}{\partial x}F_{3}(a_{1},a_{2},b_{1},b_{2},c;x,y)=\frac{a_{1}b_{% 1}}{c}F_{3}(a_{1}+1,a_{2},b_{1}+1,b_{2},c+1;x,y)~{},
  18. y F 3 ( a 1 , a 2 , b 1 , b 2 , c ; x , y ) = a 2 b 2 c F 3 ( a 1 , a 2 + 1 , b 1 , b 2 + 1 , c + 1 ; x , y ) . \frac{\partial}{\partial y}F_{3}(a_{1},a_{2},b_{1},b_{2},c;x,y)=\frac{a_{2}b_{% 2}}{c}F_{3}(a_{1},a_{2}+1,b_{1},b_{2}+1,c+1;x,y)~{}.
  19. ( x ( 1 - x ) 2 x 2 + y 2 x y + [ c - ( a 1 + b 1 + 1 ) x ] x - a 1 b 1 ) F 3 ( x , y ) = 0 , \left(x(1-x)\frac{\partial^{2}}{\partial x^{2}}+y\frac{\partial^{2}}{\partial x% \partial y}+[c-(a_{1}+b_{1}+1)x]\frac{\partial}{\partial x}-a_{1}b_{1}\right)F% _{3}(x,y)=0~{},
  20. ( y ( 1 - y ) 2 y 2 + x 2 x y + [ c - ( a 2 + b 2 + 1 ) y ] y - a 2 b 2 ) F 3 ( x , y ) = 0 . \left(y(1-y)\frac{\partial^{2}}{\partial y^{2}}+x\frac{\partial^{2}}{\partial x% \partial y}+[c-(a_{2}+b_{2}+1)y]\frac{\partial}{\partial y}-a_{2}b_{2}\right)F% _{3}(x,y)=0~{}.
  21. F 1 ( a , b 1 , b 2 , c ; x , y ) = Γ ( c ) Γ ( a ) Γ ( c - a ) 0 1 t a - 1 ( 1 - t ) c - a - 1 ( 1 - x t ) - b 1 ( 1 - y t ) - b 2 d t , \real c > \real a > 0 . F_{1}(a,b_{1},b_{2},c;x,y)=\frac{\Gamma(c)}{\Gamma(a)\Gamma(c-a)}\int_{0}^{1}t% ^{a-1}(1-t)^{c-a-1}(1-xt)^{-b_{1}}(1-yt)^{-b_{2}}\,\mathrm{d}t,\quad\real\,c>% \real\,a>0~{}.
  22. F ( ϕ , k ) = 0 ϕ d θ 1 - k 2 sin 2 θ = sin ( ϕ ) F 1 ( 1 2 , 1 2 , 1 2 , 3 2 ; sin 2 ϕ , k 2 sin 2 ϕ ) , | \real ϕ | < π 2 , F(\phi,k)=\int_{0}^{\phi}\frac{\mathrm{d}\theta}{\sqrt{1-k^{2}\sin^{2}\theta}}% =\sin(\phi)\,F_{1}(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{3}{2};\sin^{2% }\phi,k^{2}\sin^{2}\phi),\quad|\real\,\phi|<\frac{\pi}{2}~{},
  23. E ( ϕ , k ) = 0 ϕ 1 - k 2 sin 2 θ d θ = sin ( ϕ ) F 1 ( 1 2 , 1 2 , - 1 2 , 3 2 ; sin 2 ϕ , k 2 sin 2 ϕ ) , | \real ϕ | < π 2 , E(\phi,k)=\int_{0}^{\phi}\sqrt{1-k^{2}\sin^{2}\theta}\,\mathrm{d}\theta=\sin(% \phi)\,F_{1}(\tfrac{1}{2},\tfrac{1}{2},-\tfrac{1}{2},\tfrac{3}{2};\sin^{2}\phi% ,k^{2}\sin^{2}\phi),\quad|\real\,\phi|<\frac{\pi}{2}~{},
  24. Π ( n , k ) = 0 π / 2 d θ ( 1 - n sin 2 θ ) 1 - k 2 sin 2 θ = π 2 F 1 ( 1 2 , 1 , 1 2 , 1 ; n , k 2 ) . \Pi(n,k)=\int_{0}^{\pi/2}\frac{\mathrm{d}\theta}{(1-n\sin^{2}\theta)\sqrt{1-k^% {2}\sin^{2}\theta}}=\frac{\pi}{2}\,F_{1}(\tfrac{1}{2},1,\tfrac{1}{2},1;n,k^{2}% )~{}.
  25. ( y - x ) F 1 ( a , b 1 + 1 , b 2 + 1 , c , x , y ) = y F 1 ( a , b 1 , b 2 + 1 , c , x , y ) - x F 1 ( a , b 1 + 1 , b 2 , c , x , y ) (y-x)F_{1}(a,b_{1}+1,b_{2}+1,c,x,y)=y\,F_{1}(a,b_{1},b_{2}+1,c,x,y)-x\,F_{1}(a% ,b_{1}+1,b_{2},c,x,y)

Applications_of_capacitors.html

  1. f = 1 2 π L C . f=\frac{1}{2\pi\sqrt{LC}}.

Approximately_finite-dimensional_C*-algebra.html

  1. k M n k , \oplus_{k}M_{n_{k}},
  2. Φ ( a ) = a I r , \Phi(a)=a\otimes I_{r},
  3. Φ : 1 s M n k 1 t M m l \Phi:\oplus_{1}^{s}M_{n_{k}}\rightarrow\oplus_{1}^{t}M_{m_{l}}
  4. k r l k n k = m l . \sum_{k}r_{lk}n_{k}=m_{l}.\;
  5. A = lim A i α i A i + 1 , A=\underrightarrow{\lim}\cdots\rightarrow A_{i}\,\stackrel{\alpha_{i}}{% \rightarrow}A_{i+1}\rightarrow\cdots,
  6. A = n A n ¯ . A=\overline{\cup_{n}A_{n}}.
  7. 1 2 4 8 1\rightrightarrows 2\rightrightarrows 4\rightrightarrows 8\rightrightarrows\dots
  8. M ( A ) = lim M n ( A ) M n + 1 ( A ) . M_{\infty}(A)=\underrightarrow{\lim}\cdots\rightarrow M_{n}(A)\rightarrow M_{n% +1}(A)\rightarrow\cdots.
  9. [ p ] + [ q ] = [ p q ] [p]+[q]=[p\oplus q]
  10. A = k = 1 m M n k , A=\oplus_{k=1}^{m}M_{n_{k}},
  11. ( K 0 ( A ) , K 0 ( A ) + ) = ( m , + m ) . (K_{0}(A),K_{0}(A)^{+})=(\mathbb{Z}^{m},\mathbb{Z}_{+}^{m}).
  12. \mapsto
  13. Γ ( A ) = { [ p ] | p * = p 2 = p A } . \Gamma(A)=\{[p]\,|\,p^{*}=p^{2}=p\in A\}.
  14. Γ ( A ) = { x K 0 ( A ) | 0 x [ 1 A ] } . \Gamma(A)=\{x\in K_{0}(A)\,|\,0\leq x\leq[1_{A}]\}.
  15. ( G , G + ) = lim ( n k , + n k ) , (G,G^{+})=\underrightarrow{\lim}(\mathbb{Z}^{n_{k}},\mathbb{Z}^{n_{k}}_{+}),
  16. ( G , G + ) = lim ( H k , H k + ) , where ( H , H k + ) = ( n k , + n k ) . (G,G^{+})=\underrightarrow{\lim}(H_{k},H_{k}^{+}),\quad\mbox{where}~{}\quad(H,% H_{k}^{+})=(\mathbb{Z}^{n_{k}},\mathbb{Z}^{n_{k}}_{+}).
  17. Γ ( H 1 ) = { v H 1 + | ϕ 1 ( v ) Γ ( G ) } , \Gamma(H_{1})=\{v\in H_{1}^{+}|\phi_{1}(v)\in\Gamma(G)\},
  18. A = lim A k , A=\underrightarrow{\lim}A_{k},
  19. lim ( G k , G k + ) , \underrightarrow{\lim}(G_{k},G_{k}^{+}),
  20. k ϕ k [ 0 , u k ] = [ 0 , u G ] . \cup_{k}\phi_{k}[0,u_{k}]=[0,u_{G}].
  21. Ell ( A ) = def ( ( K 0 ( A ) , K 0 ( A ) + , Γ ( A ) ) , K 1 ( A ) , T + ( A ) , ρ A ) , \mbox{Ell}~{}(A)\;\stackrel{\mbox{def}~{}}{=}\;(\;(K_{0}(A),K_{0}(A)^{+},% \Gamma(A)),K_{1}(A),T^{+}(A),\rho_{A}\;),

Apsidal_precession.html

  1. α \alpha
  2. ϵ = 24 π 3 α 2 T 2 c 2 ( 1 - e 2 ) \epsilon=24\pi^{3}\frac{\alpha^{2}}{T^{2}c^{2}(1-e^{2})}
  3. ϵ \epsilon

Arakelov_theory.html

  1. CH ^ p ( X ) \widehat{\mathrm{CH}}_{p}(X)
  2. ch ^ ( f * ( [ E ] ) ) = f * ( ch ^ ( E ) Td ^ R ( T X / Y ) ) \hat{\mathrm{ch}}(f_{*}([E]))=f_{*}(\hat{\mathrm{ch}}(E)\widehat{\mathrm{Td}}^% {R}(T_{X/Y}))
  3. ch ^ \hat{\mathrm{ch}}
  4. Td ^ \hat{\mathrm{Td}}
  5. Td ^ R ( E ) \hat{\mathrm{Td}}^{R}(E)
  6. Td ^ ( E ) ( 1 - ϵ ( R ( E ) ) ) \hat{\mathrm{Td}}(E)(1-\epsilon(R(E)))
  7. m > 0 m odd X m m ! [ 2 ζ ( - m ) + ζ ( - m ) ( 1 1 + 1 2 + + 1 m ) ] . \sum_{{m>0\atop m\,\text{ odd}}}\frac{X^{m}}{m!}\left[2\zeta^{\prime}(-m)+% \zeta(-m)\left({1\over 1}+{1\over 2}+\cdots+{1\over m}\right)\right].

Archimedean_circle.html

  1. ρ = 1 2 r ( 1 - r ) , \rho=\frac{1}{2}r\left(1-r\right),

Archimedes'_quadruplets.html

  1. r 1 r 2 r . \frac{r_{1}\cdot r_{2}}{r}.
  2. ( H E ) 2 = ( r 1 ) 2 + ( r 2 ) 2 . \left(HE\right)^{2}=\left(r_{1}\right)^{2}+\left(r_{2}\right)^{2}.
  3. ( H J i ) 2 = ( H E ) 2 + x 2 = ( r 1 ) 2 + ( r 2 ) 2 + x 2 \left(HJ_{i}\right)^{2}=\left(HE\right)^{2}+x^{2}=\left(r_{1}\right)^{2}+\left% (r_{2}\right)^{2}+x^{2}
  4. H J i = H L i - x = r - x = r 1 + r 2 - x HJ_{i}=HL_{i}-x=r-x=r_{1}+r_{2}-x~{}
  5. ( r 1 ) 2 + ( r 2 ) 2 + x 2 = ( r 1 + r 2 - x ) 2 \left(r_{1}\right)^{2}+\left(r_{2}\right)^{2}+x^{2}=\left(r_{1}+r_{2}-x\right)% ^{2}
  6. 2 r 1 r 2 - 2 x ( r 1 + r 2 ) = 0 2r_{1}r_{2}-2x\left(r_{1}+r_{2}\right)=0
  7. x = r 1 r 2 r 1 + r 2 = r 1 r 2 r x=\frac{r_{1}\cdot r_{2}}{r_{1}+r_{2}}=\frac{r_{1}\cdot r_{2}}{r}

Archimedes'_twin_circles.html

  1. A A
  2. B B
  3. C C
  4. B B
  5. A A
  6. C C
  7. H H
  8. A C AC
  9. B B
  10. B H BH
  11. B H BH
  12. A B AB
  13. B C BC
  14. d = a b a + b d=\frac{ab}{a+b}
  15. s s
  16. 1 - s 1-s
  17. d = s ( 1 - s ) d=s(1-s)

Area_theorem_(conformal_mapping).html

  1. f f
  2. 𝔻 { 0 } \mathbb{D}\setminus\{0\}
  3. f ( z ) = 1 z + n = 0 a n z n , z 𝔻 { 0 } , f(z)=\frac{1}{z}+\sum_{n=0}^{\infty}a_{n}z^{n},\qquad z\in\mathbb{D}\setminus% \{0\},
  4. a n a_{n}
  5. n = 0 n | a n | 2 1. \sum_{n=0}^{\infty}n|a_{n}|^{2}\leq 1.
  6. f f
  7. r ( 0 , 1 ) r\in(0,1)
  8. γ r ( θ ) := f ( r e - i θ ) , θ [ 0 , 2 π ] . \gamma_{r}(\theta):=f(r\,e^{-i\theta}),\qquad\theta\in[0,2\pi].
  9. γ r \gamma_{r}
  10. D r D_{r}
  11. γ [ 0 , 2 π ] \mathbb{C}\setminus\gamma[0,2\pi]
  12. D r D_{r}
  13. D D
  14. γ \gamma
  15. area ( D ) = γ x d y = - γ y d x , \mathrm{area}(D)=\int_{\gamma}x\,dy=-\int_{\gamma}y\,dx\,,
  16. γ \gamma
  17. D D
  18. γ r \gamma_{r}
  19. D r D_{r}
  20. γ r \gamma_{r}
  21. γ r \gamma_{r}
  22. area ( D r ) = 0 2 π ( f ( r e - i θ ) ) ( - i r e - i θ f ( r e - i θ ) ) d θ = - 0 2 π ( f ( r e - i θ ) ) ( - i r e - i θ f ( r e - i θ ) ) d θ . \mathrm{area}(D_{r})=\int_{0}^{2\pi}\Re\bigl(f(re^{-i\theta})\bigr)\,\Im\bigl(% -i\,r\,e^{-i\theta}\,f^{\prime}(re^{-i\theta})\bigr)\,d\theta=-\int_{0}^{2\pi}% \Im\bigl(f(re^{-i\theta})\bigr)\,\Re\bigl(-i\,r\,e^{-i\theta}\,f^{\prime}(re^{% -i\theta})\bigr)d\theta.
  23. D r D_{r}
  24. area ( D r ) = - 1 2 0 2 π f ( r e - i θ ) r e - i θ f ( r e - i θ ) ¯ d θ , \mathrm{area}(D_{r})=-\frac{1}{2}\,\Re\int_{0}^{2\pi}f(r\,e^{-i\theta})\,% \overline{r\,e^{-i\theta}\,f^{\prime}(r\,e^{-i\theta})}\,d\theta,
  25. z ¯ \overline{z}
  26. a - 1 = 1 a_{-1}=1
  27. f f
  28. area ( D r ) = - 1 2 0 2 π n = - 1 m = - 1 m r n + m a n a m ¯ e i ( m - n ) θ d θ . \mathrm{area}(D_{r})=-\frac{1}{2}\,\Re\int_{0}^{2\pi}\sum_{n=-1}^{\infty}\sum_% {m=-1}^{\infty}m\,r^{n+m}\,a_{n}\,\overline{a_{m}}\,e^{i\,(m-n)\,\theta}\,d% \theta\,.
  29. 0 2 π n = - 1 m = - 1 m r n + m | a n | | a m | d θ < , \int_{0}^{2\pi}\sum_{n=-1}^{\infty}\sum_{m=-1}^{\infty}m\,r^{n+m}\,|a_{n}|\,|a% _{m}|\,d\theta<\infty\,,
  30. 0 2 π e i ( m - n ) θ d θ \int_{0}^{2\pi}e^{i\,(m-n)\,\theta}\,d\theta
  31. 2 π 2\pi
  32. n = m n=m
  33. area ( D r ) = - π n = - 1 n r 2 n | a n | 2 . \mathrm{area}(D_{r})=-\pi\sum_{n=-1}^{\infty}n\,r^{2n}\,|a_{n}|^{2}.
  34. D r D_{r}
  35. a - 1 = 1 a_{-1}=1
  36. r 1 r\to 1
  37. γ r \gamma_{r}
  38. D r D_{r}
  39. r r^{\prime}
  40. r < r < 1 r<r^{\prime}<1
  41. z 0 = f ( r ) z_{0}=f(r^{\prime})
  42. s > 0 s>0
  43. γ s \gamma_{s}
  44. z 0 z_{0}
  45. 1 1
  46. γ t \gamma_{t}
  47. z 0 z_{0}
  48. t r t\neq r^{\prime}
  49. f f
  50. z 0 z_{0}
  51. γ r \gamma_{r}
  52. z 0 z_{0}
  53. 1 1
  54. z 0 D r z_{0}\in D_{r}
  55. γ r \gamma_{r}
  56. D r D_{r}

Arithmetic_combinatorics.html

  1. A + A := { x + y : x , y A } , A+A:=\{x+y:x,y\in A\},
  2. A - A := { x - y : x , y A } , A-A:=\{x-y:x,y\in A\},
  3. A A := { x y : x , y A } A\cdot A:=\{xy:x,y\in A\}

Arithmetic_derivative.html

  1. p = 1 p^{\prime}\;=\;1\!
  2. p p\!
  3. ( a b ) = a b + a b (ab)^{\prime}\;=\;a^{\prime}b\,+\,ab^{\prime}\!
  4. a , b a\textrm{,}\,b\;\in\;\mathbb{N}
  5. 1 1^{\prime}
  6. 0
  7. 0 0^{\prime}
  8. x = p 1 e 1 p k e k , x=p_{1}^{e_{1}}\cdots p_{k}^{e_{k}}\textrm{,}
  9. p 1 , , p k p_{1},\,\dots,\,p_{k}
  10. e 1 , , e k e_{1},\,\dots,\,e_{k}
  11. x = i = 1 k e i p 1 e 1 p i - 1 e i - 1 p i e i - 1 p i + 1 e i + 1 p k e k = i = 1 k e i x p i . x^{\prime}=\sum_{i=1}^{k}e_{i}p_{1}^{e_{1}}\cdots p_{i-1}^{e_{i-1}}p_{i}^{e_{i% }-1}p_{i+1}^{e_{i+1}}\cdots p_{k}^{e_{k}}=\sum_{i=1}^{k}e_{i}\frac{x}{p_{i}}.
  12. ( p a ) = a p a - 1 , (p^{a})^{\prime}=ap^{a-1}\textrm{,}\!
  13. p p
  14. a a
  15. 81 = ( 3 4 ) \displaystyle 81^{\prime}=(3^{4})^{\prime}
  16. ( - x ) = - ( x ) (-x)^{\prime}\;=\;-(x^{\prime})
  17. ( a b ) = a b - b a b 2 . \left(\frac{a}{b}\right)^{\prime}=\frac{a^{\prime}b-b^{\prime}a}{b^{2}}\ .
  18. e i e_{i}
  19. n n \frac{n^{\prime}}{n}
  20. n x n n = T 0 x + O ( log x log log x ) \sum_{n\leq x}\frac{n^{\prime}}{n}=T_{0}x+O(\log x\log\log x)
  21. n x n = ( 1 / 2 ) T 0 x 2 + O ( x 1 + δ ) \sum_{n\leq x}n^{\prime}=(1/2)T_{0}x^{2}+O(x^{1+\delta})
  22. T 0 = p 1 p ( p - 1 ) . T_{0}=\sum_{p}\frac{1}{p(p-1)}.
  23. n n log k n k n^{\prime}\leq\frac{n\log_{k}n}{k}
  24. n s n s - 1 s n^{\prime}\geq sn^{\frac{s-1}{s}}
  25. n = 2 m n=2^{m}

Arithmetic_dynamics.html

  1. p p
  2. 𝐂 \mathbf{C}
  3. p p
  4. S S
  5. F : S S F:S→S
  6. S S
  7. F F
  8. n n
  9. F ( n ) = F F F . F^{(n)}=F\circ F\circ\cdots\circ F.
  10. P S P∈S
  11. n > 1 n>1
  12. k 1 k≥1
  13. P P
  14. O F ( P ) = { P , F ( P ) , F ( 2 ) ( P ) , F ( 3 ) ( P ) , } . O_{F}(P)=\left\{P,F(P),F^{(2)}(P),F^{(3)}(P),\cdots\right\}.
  15. P P
  16. O < s u b > F ( P ) O<sub>F(P)

Arrival_theorem.html

  1. 𝐧 = ( n 1 , n 2 , , n m ) \scriptstyle{\mathbf{n}=(n_{1},n_{2},\ldots,n_{m})}
  2. π ( 𝐧 ) \scriptstyle{\pi(\mathbf{n})}
  3. 𝐧 \scriptstyle{\mathbf{n}}
  4. 𝐧 \scriptstyle{\mathbf{n}}
  5. π ( 𝐧 ) \scriptstyle{\pi(\mathbf{n})}
  6. 𝐧 = ( n 1 , n 2 , , n m ) \scriptstyle{\mathbf{n}=(n_{1},n_{2},\ldots,n_{m})}
  7. α i ( 𝐧 - 𝐞 i ) \scriptstyle{\alpha_{i}(\mathbf{n}-\mathbf{e}_{i})}
  8. 𝐧 - 𝐞 i = ( n 1 , n 2 , , n i - 1 , , n m ) . \mathbf{n}-\mathbf{e}_{i}=(n_{1},n_{2},\ldots,n_{i}-1,\ldots,n_{m}).\,
  9. α i ( 𝐧 - 𝐞 i ) \scriptstyle{\alpha_{i}(\mathbf{n}-\mathbf{e}_{i})}
  10. 𝐧 - 𝐞 i \scriptstyle{\mathbf{n}-\mathbf{e}_{i}}

Arrowhead_matrix.html

  1. A = [ * * * * * * * 0 0 0 * 0 * 0 0 * 0 0 * 0 * 0 0 0 * ] . A=\begin{bmatrix}\,\!*&*&*&*&*\\ \,\!*&*&0&0&0\\ \,\!*&0&*&0&0\\ \,\!*&0&0&*&0\\ \,\!*&0&0&0&*\end{bmatrix}.
  2. P T A P P^{T}AP
  3. A = [ D z z T α ] , A=\left[\begin{array}[]{cc}D&z\\ z^{T}&\alpha\end{array}\right],
  4. D = diag ( d 1 , d 2 , , d n - 1 ) D=\mathop{\mathrm{diag}}(d_{1},d_{2},\ldots,d_{n-1})
  5. z = [ ζ 1 ζ 2 ζ n - 1 ] T z=\begin{bmatrix}\zeta_{1}&\zeta_{2}&\cdots&\zeta_{n-1}\end{bmatrix}^{T}
  6. α \alpha
  7. A = V Λ V T A=V\Lambda V^{T}
  8. Λ = diag ( λ 1 , λ 2 , , λ n ) \Lambda=\mathop{\mathrm{diag}}(\lambda_{1},\lambda_{2},\ldots,\lambda_{n})
  9. V = [ v 1 v n ] V=\begin{bmatrix}v_{1}&\cdots&v_{n}\end{bmatrix}
  10. ζ i = 0 \zeta_{i}=0
  11. ( d i , e i ) (d_{i},e_{i})
  12. e i e_{i}
  13. ζ i 0 \zeta_{i}\neq 0
  14. d i d_{i}
  15. d 1 d 2 d n - 1 d_{1}\geq d_{2}\geq\cdots\geq d_{n-1}
  16. λ i \lambda_{i}
  17. λ 1 d 1 λ 2 d 2 λ n - 1 d n - 1 λ n \lambda_{1}\geq d_{1}\geq\lambda_{2}\geq d_{2}\geq\cdots\geq\lambda_{n-1}\geq d% _{n-1}\geq\lambda_{n}
  18. d i = d j d_{i}=d_{j}
  19. i j i\neq j
  20. d i d_{i}
  21. ζ j \zeta_{j}
  22. ( i , j ) (i,j)
  23. ζ i 0 \zeta_{i}\neq 0
  24. d i d j d_{i}\neq d_{j}
  25. i j i\neq j
  26. f ( λ ) = α - λ - i = 1 n - 1 ζ i 2 d i - λ α - λ - z T ( D - λ I ) - 1 z = 0 f(\lambda)=\alpha-\lambda-\sum_{i=1}^{n-1}\frac{\zeta_{i}^{2}}{d_{i}-\lambda}% \equiv\alpha-\lambda-z^{T}(D-\lambda I)^{-1}z=0
  27. v i = x i x i 2 , x i = [ ( D - λ i I ) - 1 z - 1 ] , i = 1 , , n . v_{i}=\frac{x_{i}}{\|x_{i}\|_{2}},\quad x_{i}=\begin{bmatrix}\left(D-\lambda_{% i}I\right)^{-1}z\\ -1\end{bmatrix},\quad i=1,\ldots,n.
  28. d i = 0 d_{i}=0
  29. A - 1 = [ D 1 - 1 w 1 0 0 w 1 T b w 2 T 1 / ζ i 0 w 2 D 2 - 1 0 0 1 / ζ i 0 0 ] A^{-1}=\begin{bmatrix}D_{1}^{-1}&w_{1}&0&0\\ w_{1}^{T}&b&w_{2}^{T}&1/\zeta_{i}\\ 0&w_{2}&D_{2}^{-1}&0\\ 0&1/\zeta_{i}&0&0\end{bmatrix}
  30. D 1 = diag ( d 1 , d 2 , , d i - 1 ) , D 2 = diag ( d i + 1 , d i + 2 , , d n - 1 ) , z 1 = [ ζ 1 ζ 2 ζ i - 1 ] T , z 2 = [ ζ i + 1 ζ i + 2 ζ n - 1 ] T , w 1 = - D 1 - 1 z 1 1 ζ i , w 2 = - D 2 - 1 z 2 1 ζ i , b = 1 ζ i 2 ( - a + z 1 T D 1 - 1 z 1 + z 2 T D 2 - 1 z 2 ) . \begin{aligned}\displaystyle D_{1}&\displaystyle=\mathop{\mathrm{diag}}(d_{1},% d_{2},\ldots,d_{i-1}),\\ \displaystyle D_{2}&\displaystyle=\mathop{\mathrm{diag}}(d_{i+1},d_{i+2},% \ldots,d_{n-1}),\\ \displaystyle z_{1}&\displaystyle=\begin{bmatrix}\zeta_{1}&\zeta_{2}&\cdots&% \zeta_{i-1}\end{bmatrix}^{T},\\ \displaystyle z_{2}&\displaystyle=\begin{bmatrix}\zeta_{i+1}&\zeta_{i+2}&% \cdots&\zeta_{n-1}\end{bmatrix}^{T},\\ \displaystyle w_{1}&\displaystyle=-D_{1}^{-1}z_{1}\frac{1}{\zeta_{i}},\\ \displaystyle w_{2}&\displaystyle=-D_{2}^{-1}z_{2}\frac{1}{\zeta_{i}},\\ \displaystyle b&\displaystyle=\frac{1}{\zeta_{i}^{2}}\left(-a+z_{1}^{T}D_{1}^{% -1}z_{1}+z_{2}^{T}D_{2}^{-1}z_{2}\right).\end{aligned}
  31. d i 0 d_{i}\neq 0
  32. A - 1 = [ D - 1 0 ] + ρ u u T , A^{-1}=\begin{bmatrix}D^{-1}&\\ &0\end{bmatrix}+\rho uu^{T},
  33. u = [ D - 1 z - 1 ] , ρ = 1 α - z T D - 1 z . u=\begin{bmatrix}D^{-1}z\\ -1\end{bmatrix},\quad\rho=\frac{1}{\alpha-z^{T}D^{-1}z}.

Ars_Conjectandi.html

  1. b a \begin{smallmatrix}\frac{b}{a}\end{smallmatrix}
  2. E = p 0 a 0 + p 1 a 1 + p 2 a 2 + + p n a n p 0 + p 1 + + p n . E=\frac{p_{0}a_{0}+p_{1}a_{1}+p_{2}a_{2}+\cdots+p_{n}a_{n}}{p_{0}+p_{1}+\cdots% +p_{n}}.
  3. ( n r ) \begin{smallmatrix}{\left({{n}\atop{r}}\right)}\end{smallmatrix}
  4. P = i = 0 e - d ( e d + i ) ( a b ) a + v ( b - a b ) e - d - i . P=\sum_{i=0}^{e-d}{\left({{e}\atop{d+i}}\right)}\left(\frac{a}{b}\right)^{a+v}% \left(\frac{b-a}{b}\right)^{e-d-i}.

Arthur–Selberg_trace_formula.html

  1. R ( f ) ( ϕ ) ( x ) = G f ( y ) ϕ ( x y ) d y = Γ \ G γ Γ f ( x - 1 γ y ) ϕ ( y ) d y . \displaystyle R(f)(\phi)(x)=\int_{G}f(y)\phi(xy)\,dy=\int_{\Gamma\backslash G}% \sum_{\gamma\in\Gamma}f(x^{-1}\gamma y)\phi(y)\,dy.
  2. K f ( x , y ) = γ Γ f ( x - 1 γ y ) . \displaystyle K_{f}(x,y)=\sum_{\gamma\in\Gamma}f(x^{-1}\gamma y).
  3. Tr ( R ( f ) ) = Γ \ G K f ( x , x ) d x . \displaystyle\operatorname{Tr}(R(f))=\int_{\Gamma\backslash G}K_{f}(x,x)\,dx.
  4. K f ( x , y ) = o O K o ( x , y ) K_{f}(x,y)=\sum_{o\in O}K_{o}(x,y)
  5. K o ( x , y ) = γ o f ( x - 1 γ y ) = δ Γ γ \ Γ f ( x - 1 δ - 1 γ δ y ) K_{o}(x,y)=\sum_{\gamma\in o}f(x^{-1}\gamma y)=\sum_{\delta\in\Gamma_{\gamma}% \backslash\Gamma}f(x^{-1}\delta^{-1}\gamma\delta y)
  6. Tr ( R ( f ) ) = π m ( π ) Tr ( R ( f ) | π ) \displaystyle\operatorname{Tr}(R(f))=\sum_{\pi}m(\pi)\operatorname{Tr}(R(f)|\pi)
  7. o O J o T = χ X J χ T . \sum_{o\in O}J_{o}^{T}=\sum_{\chi\in X}J_{\chi}^{T}.
  8. M | W 0 M | | W 0 G | γ ( M ( Q ) ) a M ( γ ) I M ( γ , f ) = M | W 0 M | | W 0 G | Π ( M ) a M ( π ) I M ( π , f ) d π \sum_{M}\frac{|W_{0}^{M}|}{|W_{0}^{G}|}\sum_{\gamma\in(M(Q))}a^{M}(\gamma)I_{M% }(\gamma,f)=\sum_{M}\frac{|W_{0}^{M}|}{|W_{0}^{G}|}\int_{\Pi(M)}a^{M}(\pi)I_{M% }(\pi,f)\,d\pi
  9. I M ( γ , f ) \displaystyle I_{M}(\gamma,f)
  10. M ( A , γ ) \ M ( A ) f ( x - 1 γ x ) d x \displaystyle\int_{M(A,\gamma)\backslash M(A)}f(x^{-1}\gamma x)\,dx
  11. I M ( π , f ) \displaystyle I_{M}(\pi,f)
  12. M ( A ) f ( x ) π ( x ) d x \displaystyle\int_{M(A)}f(x)\pi(x)\,dx
  13. n N ( A ) f ( x n y ) d n = 0 \int_{n\in N(A)}f(xny)\,dn=0

Artinian_ideal.html

  1. R = k [ x , y , z ] R=k[x,y,z]
  2. I = ( x 2 , y 5 , z 4 ) , J = ( x 3 , y 2 , z 6 , x 2 y z 4 , y z 3 ) I=(x^{2},y^{5},z^{4}),\;J=(x^{3},y^{2},z^{6},x^{2}yz^{4},yz^{3})
  3. K = ( x 3 , y 4 , x 2 z 7 ) \displaystyle{K=(x^{3},y^{4},x^{2}z^{7})}
  4. I \displaystyle{I}
  5. J \displaystyle{J}
  6. K \displaystyle{K}
  7. K \displaystyle{K}
  8. z \displaystyle{z}
  9. K \displaystyle{K}
  10. K ^ \displaystyle{\hat{K}}
  11. K \displaystyle{K}
  12. x 3 y 4 z 7 \displaystyle{x^{3}y^{4}z^{7}}
  13. x 4 , y 5 \displaystyle{x^{4},y^{5}}
  14. z 8 \displaystyle{z^{8}}
  15. K \displaystyle{K}
  16. K ^ = ( x 3 , y 4 , z 8 , x 2 z 7 ) \displaystyle{\hat{K}}=(x^{3},y^{4},z^{8},x^{2}z^{7})

ASC_CDL.html

  1. o u t = ( i × s + o ) p out=(i\times s+o)^{p}\,
  2. o u t out
  3. i i
  4. s s
  5. o o
  6. p p

Askey–Gasper_inequality.html

  1. β 0 , α + β 2 β≥0,α+β≥−2
  2. 1 x 1 −1≤x≤1
  3. k = 0 n P k ( α , β ) ( x ) P k ( β , α ) ( 1 ) 0 \sum_{k=0}^{n}\frac{P_{k}^{(\alpha,\beta)}(x)}{P_{k}^{(\beta,\alpha)}(1)}\geq 0
  4. P k ( α , β ) ( x ) P_{k}^{(\alpha,\beta)}(x)
  5. β = 0 β=0
  6. F 2 3 ( - n , n + α + 2 , 1 2 ( α + 1 ) ; 1 2 ( α + 3 ) , α + 1 ; t ) > 0 , 0 t < 1 , α > - 1. {}_{3}F_{2}\left(-n,n+\alpha+2,\tfrac{1}{2}(\alpha+1);\tfrac{1}{2}(\alpha+3),% \alpha+1;t\right)>0,\qquad 0\leq t<1,\quad\alpha>-1.
  7. α α
  8. ( α + 2 ) n n ! × F 2 3 ( - n , n + α + 2 , 1 2 ( α + 1 ) ; 1 2 ( α + 3 ) , α + 1 ; t ) = = ( 1 2 ) j ( α 2 + 1 ) n - j ( α 2 + 3 2 ) n - 2 j ( α + 1 ) n - 2 j j ! ( α 2 + 3 2 ) n - j ( α 2 + 1 2 ) n - 2 j ( n - 2 j ) ! × F 2 3 ( - n + 2 j , n - 2 j + α + 1 , 1 2 ( α + 1 ) ; 1 2 ( α + 2 ) , α + 1 ; t ) \begin{aligned}\displaystyle\frac{(\alpha+2)_{n}}{n!}&\displaystyle\times{}_{3% }F_{2}\left(-n,n+\alpha+2,\tfrac{1}{2}(\alpha+1);\tfrac{1}{2}(\alpha+3),\alpha% +1;t\right)=\\ &\displaystyle=\frac{\left(\tfrac{1}{2}\right)_{j}\left(\tfrac{\alpha}{2}+1% \right)_{n-j}\left(\tfrac{\alpha}{2}+\tfrac{3}{2}\right)_{n-2j}(\alpha+1)_{n-2% j}}{j!\left(\tfrac{\alpha}{2}+\tfrac{3}{2}\right)_{n-j}\left(\tfrac{\alpha}{2}% +\tfrac{1}{2}\right)_{n-2j}(n-2j)!}\times{}_{3}F_{2}\left(-n+2j,n-2j+\alpha+1,% \tfrac{1}{2}(\alpha+1);\tfrac{1}{2}(\alpha+2),\alpha+1;t\right)\end{aligned}

Askey–Wilson_polynomials.html

  1. p n ( x ; a , b , c , d | q ) = ( a b , a c , a d ; q ) n a 4 - n ϕ 3 [ q - n a b c d q n - 1 a e i θ a e - i θ a b a c a d ; q , q ] p_{n}(x;a,b,c,d|q)=(ab,ac,ad;q)_{n}a^{-n}\;_{4}\phi_{3}\left[\begin{matrix}q^{% -n}&abcdq^{n-1}&ae^{i\theta}&ae^{-i\theta}\\ ab&ac&ad\end{matrix};q,q\right]

Assembly_map.html

  1. h * h_{*}
  2. E E
  3. h * ( X ) π * ( X + E ) , h_{*}(X)\cong\pi_{*}(X_{+}\wedge E),
  4. X + := X { * } X_{+}:=X\coprod\{*\}
  5. X X + E X\mapsto X_{+}\wedge E
  6. h * h_{*}
  7. h * h_{*}
  8. h * h_{*}
  9. F F
  10. α : F % F \alpha\colon F^{\%}\to F
  11. F % F^{\%}
  12. F F
  13. F % ( * ) F ( * ) F^{\%}(*)\to F(*)
  14. h * := π * F % h_{*}:=\pi_{*}\circ F^{\%}
  15. h * π * F h_{*}\to\pi_{*}\circ F
  16. π * F \pi_{*}\circ F
  17. k * π * F k_{*}\to\pi_{*}\circ F
  18. k * k_{*}
  19. k * h * k_{*}\to h_{*}
  20. X X
  21. X X
  22. L ( X ) L(X)
  23. X X
  24. A ( X ) A(X)
  25. X X
  26. X X
  27. K K
  28. L L
  29. L L
  30. L % ( M ) L_{\%}(M)
  31. L % ( M ) L ( M ) L^{\%}(M)\to L(M)
  32. M M
  33. M M
  34. L % ( M ) L % ( M ) L ( M ) L_{\%}(M)\to L^{\%}(M)\to L(M)
  35. M M
  36. A A
  37. A % ( M ) A_{\%}(M)
  38. M M
  39. M M
  40. π 1 ( M ) \pi_{1}(M)

Associated_prime.html

  1. Ass R ( M ) \operatorname{Ass}_{R}(M)\,
  2. Ass R ( R / J ) \operatorname{Ass}_{R}(R/J)\,
  3. Ann R ( N ) = Ann R ( N ) \mathrm{Ann}_{R}(N)=\mathrm{Ann}_{R}(N^{\prime})\,
  4. Ann R ( N ) \mathrm{Ann}_{R}(N)\,
  5. Ann R ( N ) \mathrm{Ann}_{R}(N)\,
  6. Ann R ( m ) \mathrm{Ann}_{R}(m)\,
  7. R / P R/P
  8. Ass R ( M ) \operatorname{Ass}_{R}(M)
  9. M / N M/N
  10. Ass R ( R / I ) = { P } \operatorname{Ass}_{R}(R/I)=\{P\}
  11. Ass R ( M ) Ass R ( M ) \mathrm{Ass}_{R}(M^{\prime})\subseteq\mathrm{Ass}_{R}(M)\,
  12. Spec ( R ) \mathrm{Spec}(R)\,
  13. E ( R / 𝔭 ) E(R/\mathfrak{p})\,
  14. E ( - ) E(-)\,
  15. 𝔭 \mathfrak{p}\,
  16. Ass R ( R / J ) \mathrm{Ass}_{R}(R/J)\,
  17. Ass R ( R / J ) \mathrm{Ass}_{R}(R/J)\,
  18. Ass R ( R / J ) \mathrm{Ass}_{R}(R/J)\,
  19. 0 = M 0 M 1 M n - 1 M n = M 0=M_{0}\subset M_{1}\subset\cdots\subset M_{n-1}\subset M_{n}=M\,
  20. f : Spec ( S - 1 R ) Spec ( R ) f:\operatorname{Spec}(S^{-1}R)\to\operatorname{Spec}(R)
  21. Ass R ( S - 1 M ) = f ( Ass S - 1 R ( S - 1 M ) ) = Ass R ( M ) { P | P S = } \operatorname{Ass}_{R}(S^{-1}M)=f(\operatorname{Ass}_{S^{-1}R}(S^{-1}M))=% \operatorname{Ass}_{R}(M)\cap\{P|P\cap S=\emptyset\}
  22. Ass ( M ) Supp ( M ) \mathrm{Ass}(M)\subseteq\mathrm{Supp}(M)
  23. Supp ( M ) \mathrm{Supp}(M)
  24. Ass ( M ) \mathrm{Ass}(M)
  25. Ass ( M ) \mathrm{Ass}(M)
  26. Ass ( M ) \mathrm{Ass}(M)

Astroecology.html

  1. m b i o m a s s , X = m r e s o u r c e , X c r e s o u r c e , X c b i o m a s s , X m_{biomass,\,X}=m_{resource,\,X}\frac{c_{resource,\,X}}{c_{biomass,\,X}}
  2. M b i o m a s s ( t ) = M b i o m a s s ( 0 ) e - k t M_{biomass}(t)=M_{biomass}(0)e^{-kt}\,
  3. B I O T A = M b i o m a s s ( 0 ) k BIOTA=\frac{M_{biomass}(0)}{k}
  4. M b i o m a s s ( 0 ) M_{biomass}(0)

Asymptotology.html

  1. f ( x ) f(x)
  2. ϕ n ( x ) {\phi_{n}(x)}
  3. f ( x ) = n = 0 a n ϕ n ( x ) f(x)=\sum_{n=0}^{\infty}a_{n}\phi_{n}(x)
  4. x x
  5. 0
  6. S N ( x ) S_{N}(x)
  7. N N
  8. Δ N ( x ) = | f ( x ) - S N ( x ) | \Delta_{N}(x)=|f(x)-S_{N}(x)|
  9. N N
  10. x x
  11. x x
  12. N N
  13. Δ \Delta
  14. x x
  15. N N
  16. Δ \Delta
  17. E i ( y ) = - y e ζ ζ - 1 d ζ , y < 0 Ei(y)=\int_{-\infty}^{y}e^{\zeta}\zeta^{-1}d{\zeta},y<0
  18. E i ( y ) e y n = 1 ( n - 1 ) ! y - n Ei(y)\sim e^{y}\sum_{n=1}^{\infty}(n-1)!y^{-n}
  19. y y
  20. - -\infty
  21. f ( x ) = - e - y E i ( y ) f(x)=-e-yEi(y)
  22. y = - x - 1 y=-x-1
  23. Δ N ( x ) \Delta_{N}(x)
  24. f ( x ) f(x)
  25. x x
  26. x x
  27. f ( x ) f(x)
  28. Δ 1 \Delta_{1}
  29. Δ 2 \Delta_{2}
  30. Δ 3 \Delta_{3}
  31. Δ 4 \Delta_{4}
  32. Δ 5 \Delta_{5}
  33. Δ 6 \Delta_{6}
  34. Δ 7 \Delta_{7}
  35. x x
  36. N N
  37. N N
  38. x x

Atiyah–Segal_completion_theorem.html

  1. π : X × E G X \pi\colon X\times EG\to X
  2. π * : K G * ( X ) I ^ K G * ( X × E G ) \pi^{*}\colon K_{G}^{*}(X)\hat{{}_{I}}\to K_{G}^{*}(X\times EG)
  3. K * ( B G ) R ( G ) I ^ K^{*}(BG)\cong R(G)\hat{{}_{I}}

Atomic_mass.html

  1. 1 u = M u N A = 1 g / mol N A 1\ {\rm{u}}={M_{\rm{u}}\over N_{\rm A}}\ ={{1\ \rm{g/mol}}\over N_{\rm A}}
  2. M u M_{\rm u}
  3. N A N_{\rm A}

Atomic_nucleus.html

  1. R = r 0 A 1 / 3 R=r_{0}A^{1/3}\,

ATS_theorem.html

  1. S = a < k b φ ( k ) e 2 π i f ( k ) ( 1 ) S=\sum_{a<k\leq b}\varphi(k)e^{2\pi if(k)}\ \ \ (1)
  2. φ ( x ) \varphi(x)
  3. f ( x ) f(x)
  4. i 2 = - 1. i^{2}=-1.
  5. S S
  6. b - a b-a
  7. a a
  8. b , b,
  9. S S
  10. φ ( x ) \varphi(x)
  11. f ( x ) f(x)
  12. S S
  13. S 1 , S_{1},
  14. S 1 = α < k β Φ ( k ) e 2 π i F ( k ) , ( 2 ) S_{1}=\sum_{\alpha<k\leq\beta}\Phi(k)e^{2\pi iF(k)},\ \ \ (2)
  15. β - α \beta-\alpha
  16. b - a . b-a.
  17. S = S 1 + R , ( 3 ) S=S_{1}+R,\ \ \ (3)
  18. S , S,
  19. S 1 S_{1}
  20. R R
  21. φ ( x ) \varphi(x)
  22. f ( x ) , f(x),
  23. ζ ( s ) \zeta(s)
  24. φ ( x ) \varphi(x)
  25. f ( x ) f(x)
  26. φ ( x ) \varphi(x)
  27. f ( x ) , f(x),
  28. B > 0 , B + , B>0,B\to+\infty,
  29. B 0 , B\to 0,
  30. 1 A B 1 1\ll\frac{A}{B}\ll 1
  31. C 1 > 0 C_{1}>0
  32. C 2 > 0 , C_{2}>0,
  33. C 1 | A | B C 2 . C_{1}\leq\frac{|A|}{B}\leq C_{2}.
  34. α , \alpha,
  35. || α || ||\alpha||
  36. || α || = min ( { α } , 1 - { α } ) , ||\alpha||=\min(\{\alpha\},1-\{\alpha\}),
  37. { α } \{\alpha\}
  38. α . \alpha.
  39. φ ( x ) \varphi(x)
  40. f ′′′′ ( x ) f^{\prime\prime\prime\prime}(x)
  41. φ ′′ ( x ) \varphi^{\prime\prime}(x)
  42. H , H,
  43. U U
  44. V V
  45. H > 0 , 1 U V , 0 < b - a V H>0,\qquad 1\ll U\ll V,\qquad 0<b-a\leq V
  46. 1 U f ′′ ( x ) 1 U , φ ( x ) H , f ′′′ ( x ) 1 U V , φ ( x ) H V , f ′′′′ ( x ) 1 U V 2 , φ ′′ ( x ) H V 2 . \begin{array}[]{rc}\frac{1}{U}\ll f^{\prime\prime}(x)\ll\frac{1}{U}\ ,&\varphi% (x)\ll H,\\ \\ f^{\prime\prime\prime}(x)\ll\frac{1}{UV}\ ,&\varphi^{\prime}(x)\ll\frac{H}{V},% \\ \\ f^{\prime\prime\prime\prime}(x)\ll\frac{1}{UV^{2}}\ ,&\varphi^{\prime\prime}(x% )\ll\frac{H}{V^{2}}.\\ \\ \end{array}
  47. x μ x_{\mu}
  48. f ( x μ ) = μ , f^{\prime}(x_{\mu})=\mu,
  49. a < μ b φ ( μ ) e 2 π i f ( μ ) = f ( a ) μ f ( b ) C ( μ ) Z ( μ ) + R , \sum_{a<\mu\leq b}\varphi(\mu)e^{2\pi if(\mu)}=\sum_{f^{\prime}(a)\leq\mu\leq f% ^{\prime}(b)}C(\mu)Z(\mu)+R,
  50. R = O ( H U b - a + H T a + H T b + H log ( f ( b ) - f ( a ) + 2 ) ) ; R=O\left(\frac{HU}{b-a}+HT_{a}+HT_{b}+H\log\left(f^{\prime}(b)-f^{\prime}(a)+2% \right)\right);
  51. T j = { 0 , if f ( j ) is an integer ; min ( 1 || f ( j ) || , U ) , if || f ( j ) || 0 ; T_{j}=\begin{cases}0,&\,\text{if }f^{\prime}(j)\,\text{ is an integer};\\ \min\left(\frac{1}{||f^{\prime}(j)||},\sqrt{U}\right),&\,\text{if }||f^{\prime% }(j)||\neq 0;\\ \end{cases}
  52. j = a , b ; j=a,b;
  53. C ( μ ) = { 1 , if f ( a ) < μ < f ( b ) ; 1 2 , if μ = f ( a ) or μ = f ( b ) ; C(\mu)=\begin{cases}1,&\,\text{if }f^{\prime}(a)<\mu<f^{\prime}(b);\\ \frac{1}{2},&\,\text{if }\mu=f^{\prime}(a)\,\text{ or }\mu=f^{\prime}(b);\\ \end{cases}
  54. Z ( μ ) = 1 + i 2 φ ( x μ ) f ′′ ( x μ ) e 2 π i ( f ( x μ ) - μ x μ ) . Z(\mu)=\frac{1+i}{\sqrt{2}}\frac{\varphi(x_{\mu})}{\sqrt{f^{\prime\prime}(x_{% \mu})}}e^{2\pi i(f(x_{\mu})-\mu x_{\mu})}\ .
  55. f ( x ) f(x)
  56. a < x b , a<x\leq b,
  57. f ( x ) f^{\prime}(x)
  58. δ \delta
  59. 0 < δ < 1 0<\delta<1
  60. | f ( x ) | δ . |f^{\prime}(x)|\leq\delta.
  61. a < k b e 2 π i f ( k ) = a b e 2 π i f ( x ) d x + θ ( 3 + 2 δ 1 - δ ) , \sum_{a<k\leq b}e^{2\pi if(k)}=\int_{a}^{b}e^{2\pi if(x)}dx+\theta\left(3+% \frac{2\delta}{1-\delta}\right),
  62. | θ | 1. |\theta|\leq 1.
  63. a a
  64. b b
  65. a < k b e 2 π i f ( k ) = a b e 2 π i f ( x ) d x + 1 2 e 2 π i f ( b ) - 1 2 e 2 π i f ( a ) + θ 2 δ 1 - δ , \sum_{a<k\leq b}e^{2\pi if(k)}=\int_{a}^{b}e^{2\pi if(x)}dx+\frac{1}{2}e^{2\pi if% (b)}-\frac{1}{2}e^{2\pi if(a)}+\theta\frac{2\delta}{1-\delta},
  66. | θ | 1. |\theta|\leq 1.

Attribute_Hierarchy_Method.html

  1. [ 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] \begin{bmatrix}0&1&0&1&0&0&0&0&0\\ 0&0&1&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0\\ 0&0&0&0&1&0&1&0&1\\ 0&0&0&0&0&1&0&0&0\\ 0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0\end{bmatrix}
  2. R = ( A + I ) n R=(A+I)^{n}
  3. n = 1 , 2 , , m n=1,2,\cdots,m
  4. [ 1 1 1 1 1 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 ] \begin{bmatrix}1&1&1&1&1&1&1&1&1\\ 0&1&1&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0&0\\ 0&0&0&1&1&1&1&1&1\\ 0&0&0&0&1&1&0&0&0\\ 0&0&0&0&0&1&0&0&0\\ 0&0&0&0&0&0&1&1&0\\ 0&0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&0&0&1\end{bmatrix}
  5. [ 1 1 1 1 1 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 ] \begin{bmatrix}1&1&1&1&1&1&1&1&1\\ 0&1&1&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0&0\\ 0&0&0&1&1&1&1&1&1\\ 0&0&0&0&1&1&0&0&0\\ 0&0&0&0&0&1&0&0&0\\ 0&0&0&0&0&0&1&1&0\\ 0&0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&0&0&1\end{bmatrix}
  6. [ 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 0 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 0 0 0 1 1 0 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 0 0 1 1 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1 0 1 1 1 0 0 0 1 0 0 1 0 0 1 0 0 1 1 0 1 0 0 1 0 0 1 1 1 1 0 0 1 0 0 1 0 0 1 1 0 1 0 0 1 1 0 1 1 0 1 0 0 1 1 1 1 1 0 1 0 0 1 0 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 0 1 1 0 1 1 1 1 1 0 1 0 0 1 0 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 1 1 1 0 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 0 0 1 0 0 0 0 1 1 1 0 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 0 0 1 1 0 0 0 1 1 1 0 1 1 0 0 0 1 1 1 1 1 1 0 0 0 1 1 0 0 1 1 1 0 0 1 1 1 0 1 1 1 0 0 1 1 1 1 1 1 1 0 0 1 1 0 0 1 0 0 1 0 1 1 1 0 1 0 0 1 0 1 1 1 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0 1 1 1 0 1 1 0 1 0 1 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 1 0 1 1 1 0 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1 1 1 1 1 1 1 0 0 1 1 1 1 0 0 1 1 0 1 1 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] \begin{bmatrix}0&0&0&0&0&0&0&0&0\\ 1&0&0&0&0&0&0&0&0\\ 1&1&0&0&0&0&0&0&0\\ 1&1&1&0&0&0&0&0&0\\ 1&0&0&1&0&0&0&0&0\\ 1&1&0&1&0&0&0&0&0\\ 1&1&1&1&0&0&0&0&0\\ 1&0&0&1&1&0&0&0&0\\ 1&1&0&1&1&0&0&0&0\\ 1&1&1&1&1&0&0&0&0\\ 1&0&0&1&1&1&0&0&0\\ 1&1&0&1&1&1&0&0&0\\ 1&1&0&1&1&1&0&0&0\\ 1&0&0&1&0&0&1&0&0\\ 1&1&0&1&0&0&1&0&0\\ 1&1&1&1&0&0&1&0&0\\ 1&0&0&1&1&0&1&0&0\\ 1&1&0&1&1&0&1&0&0\\ 1&1&1&1&1&0&1&0&0\\ 1&0&0&1&1&1&1&0&0\\ 1&1&0&1&1&1&1&0&0\\ 1&1&1&1&1&1&1&0&0\\ 1&0&0&1&0&0&1&1&0\\ 1&1&0&1&0&0&1&1&0\\ 1&1&1&1&1&0&1&0&0\\ 1&0&0&1&1&0&1&1&0\\ 1&1&0&1&1&0&1&1&0\\ 1&1&1&1&1&0&1&1&0\\ 1&0&0&1&1&1&1&1&0\\ 1&1&0&1&1&1&1&1&0\\ 1&1&1&1&1&1&1&1&0\\ 1&0&0&1&0&0&0&0&1\\ 1&1&0&1&0&0&0&0&1\\ 1&1&1&1&0&0&0&0&1\\ 1&0&0&1&1&0&0&0&1\\ 1&1&0&1&1&0&0&0&1\\ 1&1&1&1&1&0&0&0&1\\ 1&0&0&1&1&1&0&0&1\\ 1&1&0&1&1&1&0&0&1\\ 1&1&1&1&1&1&0&0&1\\ 1&0&0&1&0&0&1&0&1\\ 1&1&0&1&0&0&1&0&1\\ 1&1&1&1&0&0&1&0&1\\ 1&0&0&1&1&0&1&0&1\\ 1&1&0&1&1&0&1&0&1\\ 1&1&1&1&1&0&1&0&1\\ 1&0&0&1&1&1&1&0&1\\ 1&1&0&1&1&1&1&0&1\\ 1&1&1&1&1&1&1&0&1\\ 1&0&0&1&0&0&1&1&1\\ 1&1&0&1&0&0&1&1&1\\ 1&1&1&1&0&0&1&1&1\\ 1&0&0&1&1&0&1&1&1\\ 1&1&0&1&1&0&1&1&1\\ 1&1&1&1&1&0&1&1&1\\ 1&0&0&1&1&1&1&1&1\\ 1&1&0&1&1&1&1&1&1\\ 1&1&1&1&1&1&1&1&1\end{bmatrix}
  7. 4 ( t + u ) + 3 = 19 4(t+u)+3=19
  8. If x + y a - b = 2 3 , then 9 x + 9 y 10 a - 10 b = ? \mbox{If }~{}\frac{x+y}{a-b}=\frac{2}{3}\mbox{ , then }~{}\frac{9x+9y}{10a-10b% }=?
  9. H C I i = 1 - j = 1 J g s j X i j ( 1 - X i g ) N c i HCI_{i}=1-\frac{\sum_{j=1}^{J}\sum_{g\in s_{j}}X_{i_{j}}(1-X_{i_{g}})}{N_{c_{i% }}}

Augustus_De_Morgan.html

  1. + +
  2. + +
  3. + +
  4. - -
  5. A A
  6. B B
  7. C C
  8. = =
  9. A = B A=B
  10. A A
  11. B B
  12. + +
  13. × \times
  14. a a
  15. b b
  16. a - b a-b
  17. a + b - 1 a+b\sqrt{-1}
  18. - 1 \sqrt{-1}
  19. a + b - 1 a+b\sqrt{-1}
  20. a a
  21. b b
  22. e a - 1 e^{a\sqrt{-1}}
  23. b + q - 1 b+q\sqrt{-1}
  24. a + b - 1 a+b\sqrt{-1}
  25. a + b - 1 + c - 1 - 1 a+b\sqrt{-1}+c\sqrt{-1}\,^{\sqrt{-1}}
  26. - 1 - 1 = e - 1 2 π \sqrt{-1}\,^{\sqrt{-1}}=e^{-\frac{1}{2}\pi}
  27. 0
  28. 1 1
  29. + +
  30. - -
  31. × \times
  32. ÷ \div
  33. ( ) ()
  34. + +
  35. × \times
  36. a = 0 + a a=0+a
  37. = + a =+a
  38. = a + 0 =a+0
  39. = a - 0 =a-0
  40. = 1 × a =1\times a
  41. = × a =\times a
  42. = a × 1 =a\times 1
  43. = a ÷ 1 =a\div 1
  44. = 0 + 1 × a =0+1\times a
  45. + ( + a ) = + a , +(+a)=+a,
  46. + ( - a ) = - a , +(-a)=-a,
  47. - ( + a ) = - a , -(+a)=-a,
  48. - ( - a ) = + a , -(-a)=+a,
  49. × ( × a ) = × a , \times(\times a)=\times a,
  50. × ( ÷ a ) = ÷ a , \times(\div a)=\div a,
  51. ÷ ( × a ) = ÷ a , \div(\times a)=\div a,
  52. ÷ ( ÷ a ) = × a \div(\div a)=\times a
  53. a + b = b + a , a+b=b+a,
  54. a × b = b × a a\times b=b\times a
  55. a ( b + c ) = a b + a c , a(b+c)=ab+ac,
  56. a ( b - c ) = a b - a c , a(b-c)=ab-ac,
  57. ( b + c ) ÷ a = ( b ÷ a ) + ( c ÷ a ) , (b+c)\div a=(b\div a)+(c\div a),
  58. ( b - c ) ÷ a = ( b ÷ a ) - ( c ÷ a ) (b-c)\div a=(b\div a)-(c\div a)
  59. a 0 = 1 , a^{0}=1,
  60. a 1 = a , a^{1}=a,
  61. ( a × b ) c = a c × b c , (a\times b)^{c}=a^{c}\times b^{c},
  62. a b × a c = a b + c , a^{b}\times a^{c}=a^{b+c},
  63. ( a b ) c = a b × c (a^{b})^{c}=a^{b\times c}
  64. ( a + b ) + c = a + ( b + c ) , ( a b ) c = a ( b c ) (a+b)+c=a+(b+c),(ab)c=a(bc)
  65. m n m^{n}
  66. n m n^{m}
  67. m m
  68. a a
  69. b b
  70. ( a + b - m ) (a+b-m)
  71. π = 3 1 8 \pi=3\tfrac{1}{8}
  72. π = 3 1 8 \pi=3\tfrac{1}{8}
  73. π \pi
  74. π = 3 1 8 \pi=3\tfrac{1}{8}
  75. x x
  76. x 2 x^{2}

Automatic_calculation_of_particle_interaction_or_decay.html

  1. p p n jets pp\rightarrow n\text{jets}
  2. p p
  3. n jets n\text{jets}

Automatic_semigroup.html

  1. S S
  2. A A
  3. S S
  4. A A
  5. L L
  6. A A
  7. S S
  8. L L
  9. a A { ε } a\in A\cup\{\varepsilon\}
  10. ( u , v ) (u,v)
  11. u a = v ua=v

Auxiliary_particle_filter.html

  1. p ( x t | z 1 : t ) i = 1 M ω t ( i ) δ ( x t - x t ( i ) ) . p(x_{t}|z_{1:t})\approx\sum_{i=1}^{M}\omega^{(i)}_{t}\delta\left(x_{t}-x^{(i)}% _{t}\right).
  2. k k
  3. t - 1 t-1
  4. t t
  5. μ t ( i ) \mu^{(i)}_{t}
  6. x t | x t - 1 x_{t}|x_{t-1}
  7. k ( i ) P ( i = k | z t ) ω t ( i ) p ( z t | μ t ( i ) ) k^{(i)}\sim P(i=k|z_{t})\propto\omega^{(i)}_{t}p(z_{t}|\mu^{(i)}_{t})
  8. i = 1 , 2 , , M i=1,2,\dots,M
  9. x t ( i ) p ( x | x t - 1 k ( i ) ) . x_{t}^{(i)}\sim p(x|x^{k^{(i)}}_{t-1}).
  10. μ t k ( i ) \mu_{t}^{k^{(i)}}
  11. ω t ( i ) p ( z t | x t ( i ) ) p ( z t | μ t k ( i ) ) . \omega_{t}^{(i)}\propto\frac{p(z_{t}|x^{(i)}_{t})}{p(z_{t}|\mu^{k^{(i)}}_{t})}.

Average_crossing_number.html

  1. 1 4 π S 2 n ( v ) d A \frac{1}{4\pi}\int_{S^{2}}n(v)\,dA
  2. f : S 1 3 . f:S^{1}\rightarrow\mathbb{R}^{3}.
  3. g : S 1 × S 1 S 2 g:S^{1}\times S^{1}\rightarrow S^{2}
  4. g ( s , t ) = f ( s ) - f ( t ) | f ( s ) - f ( t ) | . g(s,t)=\frac{f(s)-f(t)}{|f(s)-f(t)|}.
  5. 1 4 π T 2 | ( f ( s ) × f ( t ) ) ( f ( s ) - f ( t ) ) | | ( f ( s ) - f ( t ) ) | 3 d s d t . \frac{1}{4\pi}\int_{T^{2}}\frac{|(f^{\prime}(s)\times f^{\prime}(t))\cdot(f(s)% -f(t))|}{|(f(s)-f(t))|^{3}}\,ds\,dt.

Average_order_of_an_arithmetic_function.html

  1. n x f ( n ) n x g ( n ) \sum_{n\leq x}f(n)\sim\sum_{n\leq x}g(n)
  2. lim N 1 N n N f ( n ) = c \lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n\leq N}f(n)=c
  3. F ( n ) = d | n f ( d ) , F(n)=\sum_{d\mathop{|}n}f(d),
  4. n x F ( n ) = d x f ( d ) n x , d | n 1 = d x f ( d ) [ x / d ] = x d x f ( d ) d + O ( d x | f ( d ) | ) . ( 1 ) \sum_{n\leq x}F(n)=\sum_{d\leq x}f(d)\sum_{n\leq x,d|n}1=\sum_{d\leq x}f(d)[x/% d]=x\sum_{d\leq x}\frac{f(d)}{d}\,\text{ }+O(\sum_{d\leq x}|f(d)|).\qquad% \qquad(1)
  5. N N
  6. Q k := { n | n is not divisible by d k for any integer d 2 } Q_{k}:=\{n\in\mathbb{Z}|\;n\,\text{ is not divisible by }d^{k}\,\text{ for any% integer }d\geq 2\}
  7. 𝐍 \mathbf{N}
  8. Q k n - s = n δ ( n ) n - s = p ( 1 + p - s + + p - s ( k - 1 ) ) = p ( 1 - p - s k 1 - p - s ) = ζ ( s ) ζ ( s k ) \sum_{Q_{k}}n^{-s}=\sum_{n}\delta(n)n^{-s}=\prod_{p}(1+p^{-s}+\cdots+p^{-s(k-1% )})=\prod_{p}\left(\frac{1-p^{-sk}}{1-p^{-s}}\right)=\frac{\zeta(s)}{\zeta(sk)}
  9. 1 ζ ( k s ) = n μ ( n ) n - k s , \frac{1}{\zeta(ks)}=\sum_{n}\mu(n)n^{-ks},
  10. μ \mu
  11. 1 ζ ( k s ) = n f ( n ) n - s , \frac{1}{\zeta(ks)}=\sum_{n}f(n)n^{-s},
  12. f ( n ) = { μ ( d ) n = d k 0 otherwise , f(n)=\begin{cases}\;\;\,\mu(d)&n=d^{k}\\ \;\;\,0&\,\text{otherwise},\end{cases}
  13. ζ ( s ) ζ ( s k ) = n ( d | n f ( d ) ) n - s \frac{\zeta(s)}{\zeta(sk)}=\sum_{n}(\sum_{d|n}f(d))n^{-s}
  14. δ ( n ) = d | n f ( d ) n - s \delta(n)=\sum_{d|n}f(d)n^{-s}
  15. d x δ ( d ) = x d x ( f ( d ) / d ) + O ( x 1 / k ) \sum_{d\leq x}\delta(d)=x\sum_{d\leq x}(f(d)/d)+O(x^{1/k})
  16. n Q k , n x 1 = x ζ ( k ) + O ( x 1 / k ) \sum_{n\in Q_{k},n\leq x}1=\frac{x}{\zeta(k)}+O(x^{1/k})
  17. n ( f ( n ) / n ) = n f ( n k ) n - k = n μ ( n ) n - k = 1 ζ ( k ) \sum_{n}(f(n)/n)=\sum_{n}f(n^{k})n^{-k}=\sum_{n}\mu(n)n^{-k}=\frac{1}{\zeta(k)}
  18. ζ ( 2 ) - 1 = 6 π 2 \zeta(2)^{-1}=\frac{6}{\pi^{2}}
  19. φ ( n ) n \frac{\varphi(n)}{n}
  20. { ( r , s ) : max ( | r | , | s | ) = n } \{(r,s)\in\mathbb{N}:\max(|r|,|s|)=n\}
  21. lim N 1 N n N φ ( n ) n = 6 π 2 = 1 ζ ( 2 ) \lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n\leq N}\frac{\varphi(n)}{n}=\frac{6% }{\pi^{2}}=\frac{1}{\zeta(2)}
  22. 1 ζ ( 2 ) \frac{1}{\zeta(2)}
  23. 𝐍 \mathbf{N}
  24. k \mathbb{Z}^{k}
  25. 1 ζ ( k ) \frac{1}{\zeta(k)}
  26. 𝐍 \mathbf{N}
  27. d ( n ) d(n)
  28. σ α ( n ) = d | n d α \sigma_{\alpha}(n)=\sum_{d|n}d^{\alpha}
  29. n x σ α ( n ) = { n x σ α ( n ) = ζ ( α + 1 ) α + 1 x α + 1 + O ( x β ) if α is positive n x σ - 1 ( n ) = ζ ( 2 ) x + O ( l o g x ) if α = - 1 n x σ α ( n ) = ζ ( - α + 1 ) x + O ( x m a x ( 0 , 1 + α ) ) otherwise \sum_{n\leq x}\sigma_{\alpha}(n)=\begin{cases}\;\;\sum_{n\leq x}\sigma_{\alpha% }(n)=\frac{\zeta(\alpha+1)}{\alpha+1}x^{\alpha+1}+O(x^{\beta})\mbox{if }~{}% \alpha\mbox{ is positive}\\ \;\;\sum_{n\leq x}\sigma_{-1}(n)=\zeta(2)x+O(logx)\mbox{if }~{}\alpha=-1\\ \;\;\sum_{n\leq x}\sigma_{\alpha}(n)=\zeta(-\alpha+1)x+O(x^{max(0,1+\alpha)})% \mbox{otherwise }\end{cases}
  30. β = m a x ( 1 , α ) \beta=max(1,\alpha)
  31. n x d ( n ) = x log x + ( 2 γ - 1 ) x + o ( x ) \sum_{n\leq x}d(n)=x\log x+(2\gamma-1)x+o(x)
  32. γ \gamma
  33. n x log n = x log x - x + O ( log x ) , \sum_{n\leq x}\log n=x\log x-x+O(\log x),
  34. n x ( d ( n ) - ( log n + 2 γ ) ) = o ( x ) ( x ) , \sum_{n\leq x}(d(n)-(\log n+2\gamma))=o(x)\quad(x\rightarrow\infty),
  35. log n + 2 γ \log n+2\gamma
  36. d ( n ) d(n)
  37. log n \log n
  38. n 1 n\geq 1
  39. Ave n ( h ) = 1 q n f monic , deg ( f ) = n h ( f ) \,\text{Ave}_{n}(h)=\frac{1}{q^{n}}\sum_{f\,\text{ monic},\,\text{ deg}(f)=n}h% (f)
  40. Ave n ( h ) g ( n ) \,\text{Ave}_{n}(h)\sim g(n)
  41. lim n Ave n ( h ) = c \lim_{n\rightarrow\infty}\,\text{Ave}_{n}(h)=c
  42. D h ( s ) = f monic h ( f ) | f | - s D_{h}(s)=\sum_{f\,\text{ monic}}h(f)|f|^{-s}
  43. g A g\in A
  44. | g | = q d e g ( g ) |g|=q^{deg(g)}
  45. g 0 g\neq 0
  46. | g | = 0 |g|=0
  47. ζ A ( s ) = f monic | f | - s \zeta_{A}(s)=\sum_{f\,\text{ monic}}|f|^{-s}
  48. 𝐍 \mathbf{N}
  49. D h ( s ) = P ( n = 0 h ( P n ) | P | - s n ) D_{h}(s)=\prod_{P}(\sum_{n\mathop{=}0}^{\infty}h(P^{n})|P|^{-sn})
  50. ζ A ( s ) = P ( 1 - | P | - s ) - 1 \zeta_{A}(s)=\prod_{P}(1-|P|^{-s})^{-1}
  51. ζ A ( s ) \zeta_{A}(s)
  52. ζ A ( s ) = f ( | f | - s ) = n deg(f)=n q - s n = n ( q n - s n ) = ( 1 - q 1 - s ) - 1 \zeta_{A}(s)=\sum_{f}(|f|^{-s})=\sum_{n}\sum_{\,\text{deg(f)=n}}q^{-sn}=\sum_{% n}(q^{n-sn})=(1-q^{1-s})^{-1}
  53. ( f * g ) ( m ) = d m f ( m ) g ( m d ) = a b = m f ( a ) g ( b ) \begin{aligned}\displaystyle(f*g)(m)&\displaystyle=\sum_{d\,\mid\,m}f(m)g\left% (\frac{m}{d}\right)\\ &\displaystyle=\sum_{ab\,=\,m}f(a)g(b)\end{aligned}
  54. D h D g = D h * g D_{h}D_{g}=D_{h*g}
  55. δ ( f ) \delta(f)
  56. f f
  57. δ \delta
  58. δ \delta
  59. f δ ( f ) | f | s = P ( j = 0 k - 1 ( | P | - j s ) ) = P 1 - | P | - s k 1 - | P | - s = ζ A ( s ) ζ A ( s k ) = 1 - q 1 - k s 1 - q 1 - s = ζ A ( s ) ζ A ( k s ) \sum_{f}\frac{\delta(f)}{|f|^{s}}=\prod_{P}(\sum_{j\mathop{=}0}^{k-1}(|P|^{-js% }))=\prod_{P}\frac{1-|P|^{-sk}}{1-|P|^{-s}}=\frac{\zeta_{A}(s)}{\zeta_{A}(sk)}% =\frac{1-q^{1-ks}}{1-q^{1-s}}=\frac{\zeta_{A}(s)}{\zeta_{A}(ks)}
  60. b n b_{n}
  61. f δ ( f ) | f | s = n def f = n δ ( f ) | f | - s = n b n q - s n \sum_{f}\frac{\delta(f)}{|f|^{s}}=\sum_{n}\sum_{\,\text{def}f=n}\delta(f)|f|^{% -s}=\sum_{n}b_{n}q^{-sn}
  62. u = q - s u=q^{-s}
  63. 1 - q u k 1 - q u = n = 0 b n u n \frac{1-qu^{k}}{1-qu}=\sum_{n\mathop{=}0}^{\infty}b_{n}u^{n}
  64. u n u^{n}
  65. b n = { q n n k - 1 q n ( 1 - q 1 - k ) otherwise b_{n}=\begin{cases}\;\;\,q^{n}&n\leq k-1\\ \;\;\,q^{n}(1-q^{1-k})&\,\text{otherwise}\\ \end{cases}
  66. Ave n ( δ ) = 1 - q 1 - k = 1 ζ A ( k ) \,\text{Ave}_{n}(\delta)=1-q^{1-k}=\frac{1}{\zeta_{A}(k)}
  67. δ ( f ) \delta(f)
  68. σ k ( m ) = f | m , f monic | f | k \sigma_{k}(m)=\sum_{f|m\,\text{, f monic}}|f|^{k}
  69. Ave n ( σ k ) \,\text{Ave}_{n}(\sigma_{k})
  70. k 1 k\geq 1
  71. σ k ( m ) = h * 𝕀 ( m ) \sigma_{k}(m)=h*\mathbb{I}(m)
  72. h ( f ) = | f | k h(f)=|f|^{k}
  73. 𝕀 ( f ) = 1 f \;\mathbb{I}(f)=1\;\;\forall{f}
  74. m σ k ( m ) | m | - s = ζ A ( s ) m h ( m ) | m | - s \sum_{m}\sigma_{k}(m)|m|^{-s}=\zeta_{A}(s)\sum_{m}h(m)|m|^{-s}
  75. q - s = u q^{-s}=u
  76. LHS = n ( deg ( m ) = n σ k ( m ) ) u n \,\text{LHS}=\sum_{n}(\sum_{\,\text{deg}(m)=n}\sigma_{k}(m))u^{n}
  77. RHS = n q n ( 1 - s ) n ( deg ( m ) = n h ( m ) ) u n = n q n u n l q l q l k u l = n ( j = 0 n q n - j q j k + j ) = n ( q n ( 1 - q k ( n + 1 ) 1 - q k ) ) u n \begin{aligned}\displaystyle\,\text{RHS}&\displaystyle=\sum_{n}q^{n(1-s)}\sum_% {n}(\sum_{\,\text{deg}(m)=n}h(m))u^{n}\\ &\displaystyle=\sum_{n}q^{n}u^{n}\sum_{l}q^{l}q^{lk}u^{l}\\ &\displaystyle=\sum_{n}(\sum_{j\mathop{=}0}^{n}q^{n-j}q^{jk+j})\\ &\displaystyle=\sum_{n}(q^{n}(\frac{1-q^{k(n+1)}}{1-q^{k}}))u^{n}\end{aligned}
  78. Ave n σ k = 1 - q k ( n + 1 ) 1 - q k \,\text{Ave}_{n}\sigma_{k}=\frac{1-q^{k(n+1)}}{1-q^{k}}
  79. q n A v e n σ k = q n ( k + 1 ) ( 1 - q - k ( n + 1 ) 1 - q - k ) = q n ( k + 1 ) ( ζ ( k + 1 ) ζ ( k n + k + 1 ) ) q^{n}Ave_{n}\sigma_{k}=q^{n(k+1)}(\frac{1-q^{-k(n+1)}}{1-q^{-k}})=q^{n(k+1)}(% \frac{\zeta(k+1)}{\zeta(kn+k+1)})
  80. x = q n x=q^{n}
  81. deg ( m ) = n , m monic σ k ( m ) = x k + 1 ( ζ ( k + 1 ) ζ ( k n + k + 1 ) ) \sum_{\,\text{deg}(m)=n,m\,\text{ monic}}\sigma_{k}(m)=x^{k+1}(\frac{\zeta(k+1% )}{\zeta(kn+k+1)})
  82. n x σ k ( n ) = ζ ( k + 1 ) k + 1 x k + 1 + O ( x k ) \sum_{n\leq x}\sigma_{k}(n)=\frac{\zeta(k+1)}{k+1}x^{k+1}+O(x^{k})
  83. d ( f ) d(f)
  84. D ( n ) D(n)
  85. d ( f ) d(f)
  86. ζ A ( s ) 2 = ( h | h | - s ) ( g | g | - s ) = f ( h g = f 1 ) | f | - s = f d ( f ) | f | - s = D d ( s ) = n = 0 D ( n ) u n \zeta_{A}(s)^{2}=(\sum_{h}|h|^{-s})(\sum_{g}|g|^{-s})=\sum_{f}(\sum_{hg=f}1)|f% |^{-s}=\sum_{f}d(f)|f|^{-s}=D_{d}(s)=\sum_{n\mathop{=}0}^{\infty}D(n)u^{n}
  87. u = q - s u=q^{-s}
  88. D ( n ) = ( n + 1 ) q n D(n)=(n+1)q^{n}
  89. x = q n x=q^{n}
  90. D ( n ) = x l o g q ( x ) + x D(n)=xlog_{q}(x)+x
  91. k = 1 n d ( k ) = x l o g x + ( 2 γ - 1 ) x + O ( x ) \sum_{k\mathop{=}1}^{n}d(k)=xlogx+(2\gamma-1)x+O(\sqrt{x})
  92. γ \gamma
  93. ζ A ( s ) \zeta_{A}(s)
  94. Λ A ( f ) = { log | P | if f = | P | k for some prime monic P and integer k 1 , 0 otherwise. \Lambda_{A}(f)=\begin{cases}\log|P|&\mbox{if }~{}f=|P|^{k}\,\text{ for some % prime monic}P\,\text{ and integer }k\geq 1,\\ 0&\mbox{otherwise.}\end{cases}
  95. Λ A \Lambda_{A}
  96. m = i = 1 l P i e i m=\prod_{i\mathop{=}1}^{l}P_{i}^{e_{i}}
  97. f | m Λ A ( f ) = ( i 1 , , i l ) | 0 i j e j Λ A ( j = 1 l P j i j ) = j = 1 l i = 1 e i Λ A ( P j i ) = j = 1 l i = 1 e i log | P j | = j = 1 l e j log | P j | = j = 1 l log | P j | e j = log | ( i = 1 l P i e i ) | = log ( m ) \begin{aligned}\displaystyle\sum_{f|m}\Lambda_{A}(f)&\displaystyle=\sum_{(i_{1% },...,i_{l})|0\leq i_{j}\leq e_{j}}\Lambda_{A}(\prod_{j\mathop{=}1}^{l}P_{j}^{% i_{j}})=\sum_{j\mathop{=}1}^{l}\sum_{i\mathop{=}1}^{e_{i}}\Lambda_{A}(P_{j}^{i% })=\sum_{j\mathop{=}1}^{l}\sum_{i\mathop{=}1}^{e_{i}}\log|P_{j}|\\ &\displaystyle=\sum_{j\mathop{=}1}^{l}e_{j}\log|P_{j}|=\sum_{j\mathop{=}1}^{l}% \log|P_{j}|^{e_{j}}=\log|(\prod_{i\mathop{=}1}^{l}P_{i}^{e_{i}})|\\ &\displaystyle=\log(m)\end{aligned}
  98. 𝕀 * Λ A ( m ) = l o g | m | \mathbb{I}*\Lambda_{A}(m)=log|m|
  99. ζ A ( s ) D Λ A ( s ) = m l o g | m | | m | - s \zeta_{A}(s)D_{\Lambda_{A}}(s)=\sum_{m}log|m||m|^{-s}
  100. m | m | s = n deg m = n u n = n q n u n = n q n ( 1 - s ) \sum_{m}|m|^{s}=\sum_{n}\sum_{\,\text{deg}m=n}u^{n}=\sum_{n}q^{n}u^{n}=\sum_{n% }q^{n(1-s)}
  101. d d s m | m | s = - n l o g ( q n ) q n ( 1 - s ) = - n d e g ( f ) = n l o g ( q n ) q - n s = - f l o g | f | | f | - s \frac{d}{ds}\sum_{m}|m|^{s}=-\sum_{n}log(q^{n})q^{n(1-s)}=-\sum_{n}\sum_{deg(f% )=n}log(q^{n})q^{-ns}=-\sum_{f}log|f||f|^{-s}
  102. D Λ A ( s ) = - ζ A ( s ) ζ A ( s ) D_{\Lambda_{A}}(s)=\frac{-\zeta^{\prime}_{A}(s)}{\zeta_{A}(s)}
  103. m Λ A ( m ) | m | - s = n ( d e g ( m ) = n Λ A ( m ) q - s m ) = n ( d e g ( m ) = n Λ A ( m ) ) u n = - ζ A ( s ) ζ A ( s ) = q 1 - s l o g ( q ) 1 - q 1 - s = l o g ( q ) n = 1 q n u n \sum_{m}\Lambda_{A}(m)|m|^{-s}=\sum_{n}(\sum_{deg(m)=n}\Lambda_{A}(m)q^{-sm})=% \sum_{n}(\sum_{deg(m)=n}\Lambda_{A}(m))u^{n}=\frac{-\zeta^{\prime}_{A}(s)}{% \zeta_{A}(s)}=\frac{q^{1-s}log(q)}{1-q^{1-s}}=log(q)\sum_{n\mathop{=}1}^{% \infty}q^{n}u^{n}
  104. d e g ( m ) = n Λ A ( m ) = q n l o g ( q ) \sum_{deg(m)=n}\Lambda_{A}(m)=q^{n}log(q)
  105. q n q^{n}
  106. A v e n Λ A ( m ) = l o g ( q ) = 1 Ave_{n}\Lambda_{A}(m)=log(q)=1
  107. Φ \Phi
  108. ( A / f A ) * (A/fA)^{*}
  109. deg f = n , f monic Φ ( f ) = q 2 n ( 1 - q - 1 ) \sum_{\,\text{deg}f=n,f\,\text{monic}}\Phi(f)=q^{2n}(1-q^{-1})

Axiom_schema_of_predicative_separation.html

  1. x y z ( z y z x ϕ ( z ) ) \forall x\;\exists y\;\forall z\;(z\in y\leftrightarrow z\in x\wedge\phi(z))
  2. x y ψ ( x ) \exists x\in y\;\psi(x)
  3. x y ψ ( x ) \forall x\in y\;\psi(x)

A¹_homotopy_theory.html

  1. 0 , 11 0,11
  2. T T
  3. 𝐒𝐡𝐯 ( T ) \mathbf{Shv}(T)
  4. Δ Δ
  5. 𝐒𝐡𝐯 ( T ) \mathbf{Shv}(T)
  6. T T
  7. T T
  8. T T
  9. 𝐒𝐞𝐭 \mathbf{Set}
  10. f : 𝒳 𝒴 f:\mathcal{X}\to\mathcal{Y}
  11. f f
  12. x x
  13. T T
  14. x * f : x * 𝒳 x * 𝒴 x^{*}f:x^{*}\mathcal{X}\to x^{*}\mathcal{Y}
  15. f f
  16. f f
  17. s ( T ) \mathcal{H}_{s}(T)
  18. I I
  19. T T
  20. p t pt
  21. I I
  22. μ : I × I I μ:I×I→I
  23. p p
  24. I p t I→pt
  25. I I
  26. 𝒳 \mathcal{X}
  27. I I
  28. 𝒴 \mathcal{Y}
  29. Hom s ( T ) ( 𝒴 × I , 𝒳 ) Hom s ( T ) ( 𝒴 , 𝒳 ) \,\text{Hom}_{\mathcal{H}_{s}(T)}(\mathcal{Y}\times I,\mathcal{X})\to\,\text{% Hom}_{\mathcal{H}_{s}(T)}(\mathcal{Y},\mathcal{X})
  30. f : 𝒳 𝒴 f:\mathcal{X}\to\mathcal{Y}
  31. I I
  32. I I
  33. 𝒵 \mathcal{Z}
  34. Hom s ( T ) ( 𝒴 , 𝒵 ) Hom s ( T ) ( 𝒳 , 𝒵 ) \,\text{Hom}_{\mathcal{H}_{s}(T)}(\mathcal{Y},\mathcal{Z})\to\,\text{Hom}_{% \mathcal{H}_{s}(T)}(\mathcal{X},\mathcal{Z})
  35. ( T , I ) (T,I)
  36. I I
  37. ( T , I ) \mathcal{H}(T,I)
  38. 𝐀 < s u p > 1 \mathbf{A}<sup>1

Babenko–Beckner_inequality.html

  1. q , p = sup f L p ( n ) f q f p , where 1 < p 2 , and 1 p + 1 q = 1. \|\mathcal{F}\|_{q,p}=\sup_{f\in L^{p}(\mathbb{R}^{n})}\frac{\|\mathcal{F}f\|_% {q}}{\|f\|_{p}},\,\text{ where }1<p\leq 2,\,\text{ and }\frac{1}{p}+\frac{1}{q% }=1.
  2. q 2 q\geq 2
  3. q , p = ( p 1 / p / q 1 / q ) n / 2 . \|\mathcal{F}\|_{q,p}=\left(p^{1/p}/q^{1/q}\right)^{n/2}.
  4. f q ( p 1 / p / q 1 / q ) n / 2 f p . \|\mathcal{F}f\|_{q}\leq\left(p^{1/p}/q^{1/q}\right)^{n/2}\|f\|_{p}.
  5. g ( y ) e - 2 π i x y f ( x ) d x and f ( x ) e 2 π i x y g ( y ) d y , g(y)\approx\int_{\mathbb{R}}e^{-2\pi ixy}f(x)\,dx\,\text{ and }f(x)\approx\int% _{\mathbb{R}}e^{2\pi ixy}g(y)\,dy,
  6. ( | g ( y ) | q d y ) 1 / q ( p 1 / p / q 1 / q ) 1 / 2 ( | f ( x ) | p d x ) 1 / p \left(\int_{\mathbb{R}}|g(y)|^{q}\,dy\right)^{1/q}\leq\left(p^{1/p}/q^{1/q}% \right)^{1/2}\left(\int_{\mathbb{R}}|f(x)|^{p}\,dx\right)^{1/p}
  7. ( q | g ( y ) | q d y ) 1 / q ( p | f ( x ) | p d x ) 1 / p . \left(\sqrt{q}\int_{\mathbb{R}}|g(y)|^{q}\,dy\right)^{1/q}\leq\left(\sqrt{p}% \int_{\mathbb{R}}|f(x)|^{p}\,dx\right)^{1/p}.
  8. 1 < p 2 , 1 p + 1 q = 1 , and ω = 1 - p = i p - 1 . 1<p\leq 2,\quad\frac{1}{p}+\frac{1}{q}=1,\quad\,\text{and}\quad\omega=\sqrt{1-% p}=i\sqrt{p-1}.
  9. d ν ( x ) d\nu(x)
  10. 1 / 2 1/2
  11. x = ± 1. x=\pm 1.
  12. C : a + b x a + ω b x C:a+bx\rightarrow a+\omega bx\,
  13. L p ( d ν ) L^{p}(d\nu)
  14. L q ( d ν ) L^{q}(d\nu)
  15. [ | a + ω b x | q d ν ( x ) ] 1 / q [ | a + b x | p d ν ( x ) ] 1 / p , \left[\int|a+\omega bx|^{q}d\nu(x)\right]^{1/q}\leq\left[\int|a+bx|^{p}d\nu(x)% \right]^{1/p},
  16. [ | a + ω b | q + | a - ω b | q 2 ] 1 / q [ | a + b | p + | a - b | p 2 ] 1 / p \left[\frac{|a+\omega b|^{q}+|a-\omega b|^{q}}{2}\right]^{1/q}\leq\left[\frac{% |a+b|^{p}+|a-b|^{p}}{2}\right]^{1/p}
  17. d ν d\nu
  18. d ν n ( x ) d\nu_{n}(x)
  19. d ν ( n x ) d\nu(\sqrt{n}x)
  20. d ν n ( x ) d\nu_{n}(x)
  21. d ν n ( x ) d\nu_{n}(x)
  22. d μ ( x ) = 1 2 π e - x 2 / 2 d x d\mu(x)=\frac{1}{\sqrt{2\pi}}e^{-x^{2}/2}\,dx
  23. d ν n ( x ) d\nu_{n}(x)

Backhouse's_constant.html

  1. 1 + 1 2 + 1 5 + 1 5 + 1 4 + 1+\cfrac{1}{2+\cfrac{1}{5+\cfrac{1}{5+\cfrac{1}{4+\ddots}}}}
  2. P ( x ) = 1 + k = 1 p k x k = 1 + 2 x + 3 x 2 + 5 x 3 + 7 x 4 + P(x)=1+\sum_{k=1}^{\infty}p_{k}x^{k}=1+2x+3x^{2}+5x^{3}+7x^{4}+\cdots
  3. Q ( x ) = 1 P ( x ) = k = 0 q k x k . Q(x)=\frac{1}{P(x)}=\sum_{k=0}^{\infty}q_{k}x^{k}.
  4. lim k | q k + 1 q k | = 1.45607 \lim_{k\to\infty}\left|\frac{q_{k+1}}{q_{k}}\right|=1.45607\ldots

Backstepping.html

  1. { 𝐱 ˙ = f x ( 𝐱 ) + g x ( 𝐱 ) z 1 z ˙ 1 = f 1 ( 𝐱 , z 1 ) + g 1 ( 𝐱 , z 1 ) z 2 z ˙ 2 = f 2 ( 𝐱 , z 1 , z 2 ) + g 2 ( 𝐱 , z 1 , z 2 ) z 3 z ˙ i = f i ( 𝐱 , z 1 , z 2 , , z i - 1 , z i ) + g i ( 𝐱 , z 1 , z 2 , , z i - 1 , z i ) z i + 1 for 1 i < k - 1 z ˙ k - 1 = f k - 1 ( 𝐱 , z 1 , z 2 , , z k - 1 ) + g k - 1 ( 𝐱 , z 1 , z 2 , , z k - 1 ) z k z ˙ k = f k ( 𝐱 , z 1 , z 2 , , z k - 1 , z k ) + g k ( 𝐱 , z 1 , z 2 , , z k - 1 , z k ) u \begin{cases}\dot{\mathbf{x}}=f_{x}(\mathbf{x})+g_{x}(\mathbf{x})z_{1}\\ \dot{z}_{1}=f_{1}(\mathbf{x},z_{1})+g_{1}(\mathbf{x},z_{1})z_{2}\\ \dot{z}_{2}=f_{2}(\mathbf{x},z_{1},z_{2})+g_{2}(\mathbf{x},z_{1},z_{2})z_{3}\\ \vdots\\ \dot{z}_{i}=f_{i}(\mathbf{x},z_{1},z_{2},\ldots,z_{i-1},z_{i})+g_{i}(\mathbf{x% },z_{1},z_{2},\ldots,z_{i-1},z_{i})z_{i+1}\quad\,\text{ for }1\leq i<k-1\\ \vdots\\ \dot{z}_{k-1}=f_{k-1}(\mathbf{x},z_{1},z_{2},\ldots,z_{k-1})+g_{k-1}(\mathbf{x% },z_{1},z_{2},\ldots,z_{k-1})z_{k}\\ \dot{z}_{k}=f_{k}(\mathbf{x},z_{1},z_{2},\ldots,z_{k-1},z_{k})+g_{k}(\mathbf{x% },z_{1},z_{2},\dots,z_{k-1},z_{k})u\end{cases}
  2. 𝐱 n \mathbf{x}\in\mathbb{R}^{n}
  3. n 1 n\geq 1
  4. z 1 , z 2 , , z i , , z k - 1 , z k z_{1},z_{2},\ldots,z_{i},\ldots,z_{k-1},z_{k}
  5. u u
  6. f x , f 1 , f 2 , , f i , , f k - 1 , f k f_{x},f_{1},f_{2},\ldots,f_{i},\ldots,f_{k-1},f_{k}
  7. f i ( 0 , 0 , , 0 ) = 0 f_{i}(0,0,\dots,0)=0
  8. g 1 , g 2 , , g i , , g k - 1 , g k g_{1},g_{2},\ldots,g_{i},\ldots,g_{k-1},g_{k}
  9. g i ( 𝐱 , z 1 , , z k ) 0 g_{i}(\mathbf{x},z_{1},\ldots,z_{k})\neq 0
  10. 1 i k 1\leq i\leq k
  11. 𝐱 ˙ = f x ( 𝐱 ) + g x ( 𝐱 ) u x ( 𝐱 ) \dot{\mathbf{x}}=f_{x}(\mathbf{x})+g_{x}(\mathbf{x})u_{x}(\mathbf{x})
  12. 𝐱 = 𝟎 \mathbf{x}=\mathbf{0}\,
  13. u x ( 𝐱 ) u_{x}(\mathbf{x})
  14. u x ( 𝟎 ) = 0 u_{x}(\mathbf{0})=0
  15. V x V_{x}
  16. 𝐱 \mathbf{x}
  17. 𝐳 \,\textbf{z}
  18. 𝐱 \mathbf{x}
  19. u u
  20. z n z_{n}
  21. z n z_{n}
  22. z n - 1 z_{n-1}
  23. z i z_{i}
  24. z i + 1 z_{i+1}
  25. 𝐱 \mathbf{x}
  26. z 1 z_{1}
  27. z 2 z_{2}
  28. z 1 z_{1}
  29. 𝐱 \mathbf{x}
  30. 𝐱 \mathbf{x}
  31. u u
  32. 𝐱 ˙ = f x ( 𝐱 ) + g x ( 𝐱 ) u x ( 𝐱 ) \dot{\mathbf{x}}=f_{x}(\mathbf{x})+g_{x}(\mathbf{x})u_{x}(\mathbf{x})
  33. u x ( 𝐱 ) u_{x}(\mathbf{x})
  34. u x ( 𝟎 ) = 0 u_{x}(\mathbf{0})=0
  35. u x u_{x}
  36. V x V_{x}
  37. u 1 ( 𝐱 , z 1 ) u_{1}(\mathbf{x},z_{1})
  38. z ˙ 1 = f 1 ( 𝐱 , z 1 ) + g 1 ( 𝐱 , z 1 ) u 1 ( 𝐱 , z 1 ) \dot{z}_{1}=f_{1}(\mathbf{x},z_{1})+g_{1}(\mathbf{x},z_{1})u_{1}(\mathbf{x},z_% {1})
  39. z 1 z_{1}
  40. u x u_{x}
  41. V 1 ( 𝐱 , z 1 ) = V x ( 𝐱 ) + 1 2 ( z 1 - u x ( 𝐱 ) ) 2 V_{1}(\mathbf{x},z_{1})=V_{x}(\mathbf{x})+\frac{1}{2}(z_{1}-u_{x}(\mathbf{x}))% ^{2}
  42. u 1 u_{1}
  43. V ˙ 1 \dot{V}_{1}
  44. u 2 ( 𝐱 , z 1 , z 2 ) u_{2}(\mathbf{x},z_{1},z_{2})
  45. z ˙ 2 = f 2 ( 𝐱 , z 1 , z 2 ) + g 2 ( 𝐱 , z 1 , z 2 ) u 2 ( 𝐱 , z 1 , z 2 ) \dot{z}_{2}=f_{2}(\mathbf{x},z_{1},z_{2})+g_{2}(\mathbf{x},z_{1},z_{2})u_{2}(% \mathbf{x},z_{1},z_{2})
  46. z 2 z_{2}
  47. u 1 u_{1}
  48. V 2 ( 𝐱 , z 1 , z 2 ) = V 1 ( 𝐱 , z 1 ) + 1 2 ( z 2 - u 1 ( 𝐱 , z 1 ) ) 2 V_{2}(\mathbf{x},z_{1},z_{2})=V_{1}(\mathbf{x},z_{1})+\frac{1}{2}(z_{2}-u_{1}(% \mathbf{x},z_{1}))^{2}
  49. u 2 u_{2}
  50. V ˙ 2 \dot{V}_{2}
  51. u u
  52. u u
  53. z k z_{k}
  54. u k - 1 u_{k-1}
  55. u k - 1 u_{k-1}
  56. z k - 1 z_{k-1}
  57. u k - 2 u_{k-2}
  58. u k - 2 u_{k-2}
  59. z k - 2 z_{k-2}
  60. u k - 3 u_{k-3}
  61. u 2 u_{2}
  62. z 2 z_{2}
  63. u 1 u_{1}
  64. u 1 u_{1}
  65. z 1 z_{1}
  66. u x u_{x}
  67. u x u_{x}
  68. 𝐱 \mathbf{x}
  69. f i f_{i}
  70. 0 i k 0\leq i\leq k
  71. g i g_{i}
  72. 1 i k 1\leq i\leq k
  73. u x u_{x}
  74. u x ( 𝟎 ) = 0 u_{x}(\mathbf{0})=0
  75. 𝐱 = 𝟎 \mathbf{x}=\mathbf{0}\,
  76. z 1 = 0 z_{1}=0
  77. z 2 = 0 z_{2}=0
  78. z k - 1 = 0 z_{k-1}=0
  79. z k = 0 z_{k}=0
  80. { 𝐱 ˙ = f x ( 𝐱 ) + g x ( 𝐱 ) z 1 z ˙ 1 = u 1 \begin{cases}\dot{\mathbf{x}}=f_{x}(\mathbf{x})+g_{x}(\mathbf{x})z_{1}\\ \dot{z}_{1}=u_{1}\end{cases}
  81. ( 1 ) (1)\,
  82. 𝐱 n \mathbf{x}\in\mathbb{R}^{n}
  83. z 1 z_{1}
  84. 𝐱 \mathbf{x}
  85. u u
  86. z 1 z_{1}
  87. 𝐱 \mathbf{x}
  88. f x ( 𝟎 ) = 0 f_{x}(\mathbf{0})=0
  89. u 1 = 0 u_{1}=0
  90. 𝐱 = 𝟎 \mathbf{x}=\mathbf{0}\,
  91. z 1 = 0 z_{1}=0
  92. { 𝐱 ˙ = f x ( 𝟎 𝐱 ) + ( g x ( 𝟎 𝐱 ) ) ( 0 z 1 ) = 0 + ( g x ( 𝟎 ) ) ( 0 ) = 𝟎 (i.e., 𝐱 = 𝟎 is stationary) z ˙ 1 = 0 u 1 (i.e., z 1 = 0 is stationary) \begin{cases}\dot{\mathbf{x}}=f_{x}(\underbrace{\mathbf{0}}_{\mathbf{x}})+(g_{% x}(\underbrace{\mathbf{0}}_{\mathbf{x}}))(\underbrace{0}_{z_{1}})=0+(g_{x}(% \mathbf{0}))(0)=\mathbf{0}&\quad\,\text{ (i.e., }\mathbf{x}=\mathbf{0}\,\text{% is stationary)}\\ \dot{z}_{1}=\overbrace{0}^{u_{1}}&\quad\,\text{ (i.e., }z_{1}=0\,\text{ is % stationary)}\end{cases}
  93. ( 𝐱 , z 1 ) = ( 𝟎 , 0 ) (\mathbf{x},z_{1})=(\mathbf{0},0)
  94. u 1 ( 𝐱 , z 1 ) u_{1}(\mathbf{x},z_{1})
  95. ( 𝐱 , z 1 ) (\mathbf{x},z_{1})
  96. ( 𝟎 , 0 ) (\mathbf{0},0)
  97. 𝐱 ˙ = F ( 𝐱 ) where F ( 𝐱 ) f x ( 𝐱 ) + g x ( 𝐱 ) u x ( 𝐱 ) \dot{\mathbf{x}}=F(\mathbf{x})\qquad\,\text{where}\qquad F(\mathbf{x})% \triangleq f_{x}(\mathbf{x})+g_{x}(\mathbf{x})u_{x}(\mathbf{x})
  98. u x ( 𝟎 ) = 0 u_{x}(\mathbf{0})=0
  99. V x ( 𝐱 ) > 0 V_{x}(\mathbf{x})>0
  100. V ˙ x = V x 𝐱 ( f x ( 𝐱 ) + g x ( 𝐱 ) u x ( 𝐱 ) ) - W ( 𝐱 ) \dot{V}_{x}=\frac{\partial V_{x}}{\partial\mathbf{x}}(f_{x}(\mathbf{x})+g_{x}(% \mathbf{x})u_{x}(\mathbf{x}))\leq-W(\mathbf{x})
  101. W ( 𝐱 ) W(\mathbf{x})
  102. 𝐱 \mathbf{x}
  103. V x V_{x}
  104. 𝐱 \mathbf{x}
  105. 𝐱 \mathbf{x}
  106. V x ( 𝐱 ) V_{x}(\mathbf{x})
  107. V x ( 𝐱 ( t ) ) V_{x}(\mathbf{x}(t))
  108. 𝐱 \mathbf{x}
  109. 𝐱 = 𝟎 \mathbf{x}=\mathbf{0}\,
  110. 𝐱 = 𝟎 \mathbf{x}=\mathbf{0}\,
  111. 𝐱 \mathbf{x}
  112. W ( 𝐱 ) W(\mathbf{x})
  113. W ( 𝐱 ) > 0 W(\mathbf{x})>0
  114. 𝐱 = 𝟎 \mathbf{x}=\mathbf{0}\,
  115. W ( 𝟎 ) = 0 W(\mathbf{0})=0
  116. V ˙ x - W ( 𝐱 ) \dot{V}_{x}\leq-W(\mathbf{x})
  117. V ˙ x \dot{V}_{x}
  118. 𝐱 = 𝟎 \mathbf{x}=\mathbf{0}\,
  119. u u
  120. ( 𝐱 , z 1 ) (\mathbf{x},z_{1})
  121. u u
  122. g x ( 𝐱 ) u x ( 𝐱 ) g_{x}(\mathbf{x})u_{x}(\mathbf{x})
  123. 𝐱 ˙ \dot{\mathbf{x}}
  124. ( 𝐱 , z 1 ) (\mathbf{x},z_{1})
  125. { 𝐱 ˙ = f x ( 𝐱 ) + g x ( 𝐱 ) z 1 + ( g x ( 𝐱 ) u x ( 𝐱 ) - g x ( 𝐱 ) u x ( 𝐱 ) ) 0 z ˙ 1 = u 1 \begin{cases}\dot{\mathbf{x}}=f_{x}(\mathbf{x})+g_{x}(\mathbf{x})z_{1}+% \mathord{\underbrace{\left(g_{x}(\mathbf{x})u_{x}(\mathbf{x})-g_{x}(\mathbf{x}% )u_{x}(\mathbf{x})\right)}_{0}}\\ \dot{z}_{1}=u_{1}\end{cases}
  126. { x ˙ = ( f x ( 𝐱 ) + g x ( 𝐱 ) u x ( 𝐱 ) ) F ( 𝐱 ) + g x ( 𝐱 ) ( z 1 - u x ( 𝐱 ) ) z 1 error tracking u x z ˙ 1 = u 1 \begin{cases}\dot{x}=\mathord{\underbrace{\left(f_{x}(\mathbf{x})+g_{x}(% \mathbf{x})u_{x}(\mathbf{x})\right)}_{F(\mathbf{x})}}+g_{x}(\mathbf{x})% \underbrace{\left(z_{1}-u_{x}(\mathbf{x})\right)}_{z_{1}\,\text{ error % tracking }u_{x}}\\ \dot{z}_{1}=u_{1}\end{cases}
  127. 𝐱 ˙ = F ( 𝐱 ) \dot{\mathbf{x}}=F(\mathbf{x})
  128. ( 𝐱 , z 1 ) (\mathbf{x},z_{1})
  129. ( 𝐱 , e 1 ) (\mathbf{x},e_{1})
  130. e 1 z 1 - u x ( 𝐱 ) e_{1}\triangleq z_{1}-u_{x}(\mathbf{x})
  131. { 𝐱 ˙ = ( f x ( 𝐱 ) + g x ( 𝐱 ) u x ( 𝐱 ) ) + g x ( 𝐱 ) e 1 e ˙ 1 = u 1 - u ˙ x \begin{cases}\dot{\mathbf{x}}=(f_{x}(\mathbf{x})+g_{x}(\mathbf{x})u_{x}(% \mathbf{x}))+g_{x}(\mathbf{x})e_{1}\\ \dot{e}_{1}=u_{1}-\dot{u}_{x}\end{cases}
  132. v 1 u 1 - u ˙ x v_{1}\triangleq u_{1}-\dot{u}_{x}
  133. u 1 = v 1 + u ˙ x u_{1}=v_{1}+\dot{u}_{x}
  134. { 𝐱 ˙ = ( f x ( 𝐱 ) + g x ( 𝐱 ) u x ( 𝐱 ) ) + g x ( 𝐱 ) e 1 e ˙ 1 = v 1 \begin{cases}\dot{\mathbf{x}}=(f_{x}(\mathbf{x})+g_{x}(\mathbf{x})u_{x}(% \mathbf{x}))+g_{x}(\mathbf{x})e_{1}\\ \dot{e}_{1}=v_{1}\end{cases}
  135. v 1 v_{1}
  136. e 1 = 0 e_{1}=0
  137. z 1 z_{1}
  138. u x u_{x}
  139. 𝐱 \mathbf{x}
  140. V x V_{x}
  141. V 1 ( 𝐱 , e 1 ) V x ( 𝐱 ) + 1 2 e 1 2 V_{1}(\mathbf{x},e_{1})\triangleq V_{x}(\mathbf{x})+\frac{1}{2}e_{1}^{2}
  142. V ˙ 1 = V ˙ x ( 𝐱 ) + 1 2 ( 2 e 1 e ˙ 1 ) = V ˙ x ( 𝐱 ) + e 1 e ˙ 1 = V ˙ x ( 𝐱 ) + e 1 v 1 e ˙ 1 = V x 𝐱 𝐱 ˙ (i.e., d 𝐱 d t ) V ˙ x (i.e., d V x d t ) + e 1 v 1 = V x 𝐱 ( ( f x ( 𝐱 ) + g x ( 𝐱 ) u x ( 𝐱 ) ) + g x ( 𝐱 ) e 1 ) 𝐱 ˙ V ˙ x + e 1 v 1 \dot{V}_{1}=\dot{V}_{x}(\mathbf{x})+\frac{1}{2}\left(2e_{1}\dot{e}_{1}\right)=% \dot{V}_{x}(\mathbf{x})+e_{1}\dot{e}_{1}=\dot{V}_{x}(\mathbf{x})+e_{1}% \overbrace{v_{1}}^{\dot{e}_{1}}=\overbrace{\frac{\partial V_{x}}{\partial% \mathbf{x}}\underbrace{\dot{\mathbf{x}}}_{\,\text{(i.e., }\frac{\operatorname{% d}\mathbf{x}}{\operatorname{d}t}\,\text{)}}}^{\dot{V}_{x}\,\text{ (i.e.,}\frac% {\operatorname{d}V_{x}}{\operatorname{d}t}\,\text{)}}+e_{1}v_{1}=\overbrace{% \frac{\partial V_{x}}{\partial\mathbf{x}}\underbrace{\left((f_{x}(\mathbf{x})+% g_{x}(\mathbf{x})u_{x}(\mathbf{x}))+g_{x}(\mathbf{x})e_{1}\right)}_{\dot{% \mathbf{x}}}}^{\dot{V}_{x}}+e_{1}v_{1}
  143. V x / 𝐱 \partial V_{x}/\partial\mathbf{x}
  144. V ˙ 1 = V x 𝐱 ( f x ( 𝐱 ) + g x ( 𝐱 ) u x ( 𝐱 ) ) - W ( 𝐱 ) + V x 𝐱 g x ( 𝐱 ) e 1 + e 1 v 1 - W ( 𝐱 ) + V x 𝐱 g x ( 𝐱 ) e 1 + e 1 v 1 \dot{V}_{1}=\overbrace{\frac{\partial V_{x}}{\partial\mathbf{x}}(f_{x}(\mathbf% {x})+g_{x}(\mathbf{x})u_{x}(\mathbf{x}))}^{{}\leq-W(\mathbf{x})}+\frac{% \partial V_{x}}{\partial\mathbf{x}}g_{x}(\mathbf{x})e_{1}+e_{1}v_{1}\leq-W(% \mathbf{x})+\frac{\partial V_{x}}{\partial\mathbf{x}}g_{x}(\mathbf{x})e_{1}+e_% {1}v_{1}
  145. V ˙ 1 - W ( 𝐱 ) < 0 \dot{V}_{1}\leq-W(\mathbf{x})<0
  146. v 1 = - V x 𝐱 g x ( 𝐱 ) - k 1 e 1 v_{1}=-\frac{\partial V_{x}}{\partial\mathbf{x}}g_{x}(\mathbf{x})-k_{1}e_{1}
  147. k 1 > 0 k_{1}>0
  148. V ˙ 1 = - W ( 𝐱 ) + V x 𝐱 g x ( 𝐱 ) e 1 + e 1 ( - V x 𝐱 g x ( 𝐱 ) - k 1 e 1 ) v 1 \dot{V}_{1}=-W(\mathbf{x})+\frac{\partial V_{x}}{\partial\mathbf{x}}g_{x}(% \mathbf{x})e_{1}+e_{1}\overbrace{\left(-\frac{\partial V_{x}}{\partial\mathbf{% x}}g_{x}(\mathbf{x})-k_{1}e_{1}\right)}^{v_{1}}
  149. e 1 e_{1}
  150. V ˙ 1 = - W ( 𝐱 ) + V x 𝐱 g x ( 𝐱 ) e 1 - e 1 V x 𝐱 g x ( 𝐱 ) 0 - k 1 e 1 2 = - W ( 𝐱 ) - k 1 e 1 2 - W ( 𝐱 ) < 0 \dot{V}_{1}=-W(\mathbf{x})+\mathord{\overbrace{\frac{\partial V_{x}}{\partial% \mathbf{x}}g_{x}(\mathbf{x})e_{1}-e_{1}\frac{\partial V_{x}}{\partial\mathbf{x% }}g_{x}(\mathbf{x})}^{0}}-k_{1}e_{1}^{2}=-W(\mathbf{x})-k_{1}e_{1}^{2}\leq-W(% \mathbf{x})<0
  151. V 1 V_{1}
  152. v 1 v_{1}
  153. u 1 u_{1}
  154. v 1 u 1 - u ˙ x v_{1}\triangleq u_{1}-\dot{u}_{x}
  155. V 1 ( 𝐱 , z 1 ) V x ( 𝐱 ) + 1 2 ( z 1 - u x ( 𝐱 ) ) 2 V_{1}(\mathbf{x},z_{1})\triangleq V_{x}(\mathbf{x})+\frac{1}{2}(z_{1}-u_{x}(% \mathbf{x}))^{2}
  156. ( 2 ) (2)\,
  157. v 1 v_{1}
  158. u 1 ( 𝐱 , z 1 ) = v 1 + u ˙ x By definition of v 1 = - V x 𝐱 g x ( 𝐱 ) - k 1 ( z 1 - u x ( 𝐱 ) e 1 ) v 1 + u x 𝐱 ( f x ( 𝐱 ) + g x ( 𝐱 ) z 1 𝐱 ˙ (i.e., d 𝐱 d t ) ) u ˙ x (i.e., d u x d t ) \underbrace{u_{1}(\mathbf{x},z_{1})=v_{1}+\dot{u}_{x}}_{\,\text{By definition % of }v_{1}}=\overbrace{-\frac{\partial V_{x}}{\partial\mathbf{x}}g_{x}(\mathbf{% x})-k_{1}(\underbrace{z_{1}-u_{x}(\mathbf{x})}_{e_{1}})}^{v_{1}}\,+\,% \overbrace{\frac{\partial u_{x}}{\partial\mathbf{x}}(\underbrace{f_{x}(\mathbf% {x})+g_{x}(\mathbf{x})z_{1}}_{\dot{\mathbf{x}}\,\text{ (i.e., }\frac{% \operatorname{d}\mathbf{x}}{\operatorname{d}t}\,\text{)}})}^{\dot{u}_{x}\,% \text{ (i.e., }\frac{\operatorname{d}u_{x}}{\operatorname{d}t}\,\text{)}}
  159. ( 3 ) (3)\,
  160. 𝐱 \mathbf{x}
  161. z 1 z_{1}
  162. f x f_{x}
  163. g x g_{x}
  164. u x u_{x}
  165. 𝐱 ˙ = F ( 𝐱 ) \dot{\mathbf{x}}=F(\mathbf{x})
  166. k 1 > 0 k_{1}>0
  167. ( 𝐱 , z 1 ) = ( 𝟎 , 0 ) (\mathbf{x},z_{1})=(\mathbf{0},0)
  168. u 1 u_{1}
  169. u x u_{x}
  170. u 1 u_{1}
  171. u ˙ x \dot{u}_{x}
  172. u ˙ x \dot{u}_{x}
  173. u 1 ( 𝐱 , z 1 ) u_{1}(\mathbf{x},z_{1})
  174. V 1 ( 𝐱 , z 1 ) V_{1}(\mathbf{x},z_{1})
  175. V ˙ 1 ( 𝐱 , z 1 ) - W ( 𝐱 ) < 0 \dot{V}_{1}(\mathbf{x},z_{1})\leq-W(\mathbf{x})<0
  176. { 𝐱 ˙ = f x ( 𝐱 ) + g x ( 𝐱 ) z 1 z ˙ 1 = z 2 z ˙ 2 = u 2 \begin{cases}\dot{\mathbf{x}}=f_{x}(\mathbf{x})+g_{x}(\mathbf{x})z_{1}\\ \dot{z}_{1}=z_{2}\\ \dot{z}_{2}=u_{2}\end{cases}
  177. ( 4 ) (4)\,
  178. 𝐱 n \mathbf{x}\in\mathbb{R}^{n}
  179. z 1 z_{1}
  180. z 2 z_{2}
  181. u 2 u_{2}
  182. u 1 u_{1}
  183. 𝐲 [ 𝐱 z 1 ] \mathbf{y}\triangleq\begin{bmatrix}\mathbf{x}\\ z_{1}\end{bmatrix}\,
  184. f y ( 𝐲 ) [ f x ( 𝐱 ) + g x ( 𝐱 ) z 1 0 ] f_{y}(\mathbf{y})\triangleq\begin{bmatrix}f_{x}(\mathbf{x})+g_{x}(\mathbf{x})z% _{1}\\ 0\end{bmatrix}\,
  185. g y ( 𝐲 ) [ 𝟎 1 ] , g_{y}(\mathbf{y})\triangleq\begin{bmatrix}\mathbf{0}\\ 1\end{bmatrix},\,
  186. { 𝐲 ˙ = f y ( 𝐲 ) + g y ( 𝐲 ) z 2 ( where this 𝐲 subsystem is stabilized by z 2 = u 1 ( 𝐱 , z 1 ) ) z ˙ 2 = u 2 . \begin{cases}\dot{\mathbf{y}}=f_{y}(\mathbf{y})+g_{y}(\mathbf{y})z_{2}&\quad\,% \text{( where this }\mathbf{y}\,\text{ subsystem is stabilized by }z_{2}=u_{1}% (\mathbf{x},z_{1})\,\text{ )}\\ \dot{z}_{2}=u_{2}.\end{cases}
  187. ( 5 ) (5)\,
  188. u y ( 𝐲 ) u 1 ( 𝐱 , z 1 ) u_{y}(\mathbf{y})\triangleq u_{1}(\mathbf{x},z_{1})
  189. z 2 z_{2}
  190. 𝐲 \mathbf{y}
  191. V 1 ( 𝐱 , z 1 ) V_{1}(\mathbf{x},z_{1})
  192. u 2 u_{2}
  193. u 1 u_{1}
  194. 𝐱 ˙ = f x ( 𝐱 ) + g x ( 𝐱 ) u x \dot{\mathbf{x}}=f_{x}(\mathbf{x})+g_{x}(\mathbf{x})u_{x}
  195. u x u_{x}
  196. 𝐱 = [ x 1 , x 2 , , x n ] T n \mathbf{x}=[x_{1},x_{2},\ldots,x_{n}]^{\,\text{T}}\in\mathbb{R}^{n}
  197. f x ( 𝐱 ) = 𝟎 f_{x}(\mathbf{x})=\mathbf{0}
  198. u x = 0 u_{x}=0
  199. 𝐱 = 𝟎 \mathbf{x}=\mathbf{0}\,
  200. u x ( 𝐱 ) u_{x}(\mathbf{x})
  201. V x ( 𝐱 ) V_{x}(\mathbf{x})
  202. 𝐱 \mathbf{x}
  203. u x u_{x}
  204. u x ( 𝐱 ) u_{x}(\mathbf{x})
  205. u x u_{x}
  206. u x u_{x}
  207. u 1 u_{1}
  208. 𝐱 \mathbf{x}
  209. { 𝐱 ˙ = f x ( 𝐱 ) + g x ( 𝐱 ) z 1 z ˙ 1 = u 1 \begin{cases}\dot{\mathbf{x}}=f_{x}(\mathbf{x})+g_{x}(\mathbf{x})z_{1}\\ \dot{z}_{1}=u_{1}\end{cases}
  210. z 1 z_{1}
  211. 𝐱 \mathbf{x}
  212. u 1 u_{1}
  213. u 1 ( 𝐱 , z 1 ) = - V x 𝐱 g x ( 𝐱 ) - k 1 ( z 1 - u x ( 𝐱 ) ) + u x 𝐱 ( f x ( 𝐱 ) + g x ( 𝐱 ) z 1 ) u_{1}(\mathbf{x},z_{1})=-\frac{\partial V_{x}}{\partial\mathbf{x}}g_{x}(% \mathbf{x})-k_{1}(z_{1}-u_{x}(\mathbf{x}))+\frac{\partial u_{x}}{\partial% \mathbf{x}}(f_{x}(\mathbf{x})+g_{x}(\mathbf{x})z_{1})
  214. k 1 > 0 k_{1}>0
  215. z 1 z_{1}
  216. 𝐱 \mathbf{x}
  217. z 1 = 0 z_{1}=0
  218. 𝐱 = 𝟎 \mathbf{x}=\mathbf{0}\,
  219. u 1 u_{1}
  220. V 1 ( 𝐱 , z 1 ) = V x ( 𝐱 ) + 1 2 ( z 1 - u x ( 𝐱 ) ) 2 V_{1}(\mathbf{x},z_{1})=V_{x}(\mathbf{x})+\frac{1}{2}(z_{1}-u_{x}(\mathbf{x}))% ^{2}
  221. u 1 u_{1}
  222. V 1 V_{1}
  223. u 1 u_{1}
  224. u 2 u_{2}
  225. 𝐱 \mathbf{x}
  226. { 𝐱 ˙ = f x ( 𝐱 ) + g x ( 𝐱 ) z 1 z ˙ 1 = z 2 z ˙ 2 = u 2 \begin{cases}\dot{\mathbf{x}}=f_{x}(\mathbf{x})+g_{x}(\mathbf{x})z_{1}\\ \dot{z}_{1}=z_{2}\\ \dot{z}_{2}=u_{2}\end{cases}
  227. { [ 𝐱 ˙ z ˙ 1 ] 𝐱 ˙ 1 = [ f x ( 𝐱 ) + g x ( 𝐱 ) z 1 0 ] f 1 ( 𝐱 1 ) + [ 𝟎 1 ] g 1 ( 𝐱 1 ) z 2 ( by Lyapunov function V 1 , subsystem stabilized by u 1 ( 𝐱 1 ) ) z ˙ 2 = u 2 \begin{cases}\overbrace{\begin{bmatrix}\dot{\mathbf{x}}\\ \dot{z}_{1}\end{bmatrix}}^{\triangleq\,\dot{\mathbf{x}}_{1}}=\overbrace{\begin% {bmatrix}f_{x}(\mathbf{x})+g_{x}(\mathbf{x})z_{1}\\ 0\end{bmatrix}}^{\triangleq\,f_{1}(\mathbf{x}_{1})}+\overbrace{\begin{bmatrix}% \mathbf{0}\\ 1\end{bmatrix}}^{\triangleq\,g_{1}(\mathbf{x}_{1})}z_{2}&\qquad\,\text{ ( by % Lyapunov function }V_{1},\,\text{ subsystem stabilized by }u_{1}(\,\textbf{x}_% {1})\,\text{ )}\\ \dot{z}_{2}=u_{2}\end{cases}
  228. 𝐱 1 \mathbf{x}_{1}
  229. f 1 f_{1}
  230. g 1 g_{1}
  231. { 𝐱 ˙ 1 = f 1 ( 𝐱 1 ) + g 1 ( 𝐱 1 ) z 2 ( by Lyapunov function V 1 , subsystem stabilized by u 1 ( 𝐱 1 ) ) z ˙ 2 = u 2 \begin{cases}\dot{\mathbf{x}}_{1}=f_{1}(\mathbf{x}_{1})+g_{1}(\mathbf{x}_{1})z% _{2}&\qquad\,\text{ ( by Lyapunov function }V_{1},\,\text{ subsystem % stabilized by }u_{1}(\,\textbf{x}_{1})\,\text{ )}\\ \dot{z}_{2}=u_{2}\end{cases}
  232. z 1 z_{1}
  233. z 2 z_{2}
  234. 𝐱 \mathbf{x}
  235. u 2 u_{2}
  236. u 2 ( 𝐱 , z 1 , z 2 ) = - V 1 𝐱 1 g 1 ( 𝐱 1 ) - k 2 ( z 2 - u 1 ( 𝐱 1 ) ) + u 1 𝐱 1 ( f 1 ( 𝐱 1 ) + g 1 ( 𝐱 1 ) z 2 ) u_{2}(\mathbf{x},z_{1},z_{2})=-\frac{\partial V_{1}}{\partial\mathbf{x}_{1}}g_% {1}(\mathbf{x}_{1})-k_{2}(z_{2}-u_{1}(\mathbf{x}_{1}))+\frac{\partial u_{1}}{% \partial\mathbf{x}_{1}}(f_{1}(\mathbf{x}_{1})+g_{1}(\mathbf{x}_{1})z_{2})
  237. k 2 > 0 k_{2}>0
  238. z 1 z_{1}
  239. z 2 z_{2}
  240. 𝐱 \mathbf{x}
  241. z 1 = 0 z_{1}=0
  242. z 2 = 0 z_{2}=0
  243. 𝐱 = 𝟎 \mathbf{x}=\mathbf{0}\,
  244. u 2 u_{2}
  245. V 2 ( 𝐱 , z 1 , z 2 ) = V 1 ( 𝐱 1 ) + 1 2 ( z 2 - u 1 ( 𝐱 1 ) ) 2 V_{2}(\mathbf{x},z_{1},z_{2})=V_{1}(\mathbf{x}_{1})+\frac{1}{2}(z_{2}-u_{1}(% \mathbf{x}_{1}))^{2}
  246. u 2 u_{2}
  247. V 2 V_{2}
  248. u 2 u_{2}
  249. u 3 u_{3}
  250. 𝐱 \mathbf{x}
  251. { 𝐱 ˙ = f x ( 𝐱 ) + g x ( 𝐱 ) z 1 z ˙ 1 = z 2 z ˙ 2 = z 3 z ˙ 3 = u 3 \begin{cases}\dot{\mathbf{x}}=f_{x}(\mathbf{x})+g_{x}(\mathbf{x})z_{1}\\ \dot{z}_{1}=z_{2}\\ \dot{z}_{2}=z_{3}\\ \dot{z}_{3}=u_{3}\end{cases}
  252. { [ 𝐱 ˙ z ˙ 1 z ˙ 2 ] 𝐱 ˙ 2 = [ f x ( 𝐱 ) + g x ( 𝐱 ) z 2 z 2 0 ] f 2 ( 𝐱 2 ) + [ 𝟎 0 1 ] g 2 ( 𝐱 2 ) z 3 ( by Lyapunov function V 2 , subsystem stabilized by u 2 ( 𝐱 2 ) ) z ˙ 3 = u 3 \begin{cases}\overbrace{\begin{bmatrix}\dot{\mathbf{x}}\\ \dot{z}_{1}\\ \dot{z}_{2}\end{bmatrix}}^{\triangleq\,\dot{\mathbf{x}}_{2}}=\overbrace{\begin% {bmatrix}f_{x}(\mathbf{x})+g_{x}(\mathbf{x})z_{2}\\ z_{2}\\ 0\end{bmatrix}}^{\triangleq\,f_{2}(\mathbf{x}_{2})}+\overbrace{\begin{bmatrix}% \mathbf{0}\\ 0\\ 1\end{bmatrix}}^{\triangleq\,g_{2}(\mathbf{x}_{2})}z_{3}&\qquad\,\text{ ( by % Lyapunov function }V_{2},\,\text{ subsystem stabilized by }u_{2}(\,\textbf{x}_% {2})\,\text{ )}\\ \dot{z}_{3}=u_{3}\end{cases}
  253. 𝐱 1 \mathbf{x}_{1}
  254. f 1 f_{1}
  255. g 1 g_{1}
  256. { [ 𝐱 ˙ 1 z ˙ 2 ] 𝐱 ˙ 2 = [ f 1 ( 𝐱 1 ) + g 1 ( 𝐱 1 ) z 2 0 ] f 2 ( 𝐱 2 ) + [ 𝟎 1 ] g 2 ( 𝐱 2 ) z 3 ( by Lyapunov function V 2 , subsystem stabilized by u 2 ( 𝐱 2 ) ) z ˙ 3 = u 3 \begin{cases}\overbrace{\begin{bmatrix}\dot{\mathbf{x}}_{1}\\ \dot{z}_{2}\end{bmatrix}}^{\dot{\mathbf{x}}_{2}}=\overbrace{\begin{bmatrix}f_{% 1}(\mathbf{x}_{1})+g_{1}(\mathbf{x}_{1})z_{2}\\ 0\end{bmatrix}}^{f_{2}(\mathbf{x}_{2})}+\overbrace{\begin{bmatrix}\mathbf{0}\\ 1\end{bmatrix}}^{g_{2}(\mathbf{x}_{2})}z_{3}&\qquad\,\text{ ( by Lyapunov % function }V_{2},\,\text{ subsystem stabilized by }u_{2}(\,\textbf{x}_{2})\,% \text{ )}\\ \dot{z}_{3}=u_{3}\end{cases}
  257. 𝐱 2 \mathbf{x}_{2}
  258. f 2 f_{2}
  259. g 2 g_{2}
  260. { 𝐱 ˙ 2 = f 2 ( 𝐱 2 ) + g 2 ( 𝐱 2 ) z 3 ( by Lyapunov function V 2 , subsystem stabilized by u 2 ( 𝐱 2 ) ) z ˙ 3 = u 3 \begin{cases}\dot{\mathbf{x}}_{2}=f_{2}(\mathbf{x}_{2})+g_{2}(\mathbf{x}_{2})z% _{3}&\qquad\,\text{ ( by Lyapunov function }V_{2},\,\text{ subsystem % stabilized by }u_{2}(\,\textbf{x}_{2})\,\text{ )}\\ \dot{z}_{3}=u_{3}\end{cases}
  261. z 1 z_{1}
  262. z 2 z_{2}
  263. z 3 z_{3}
  264. 𝐱 \mathbf{x}
  265. u 3 u_{3}
  266. u 3 ( 𝐱 , z 1 , z 2 , z 3 ) = - V 2 𝐱 2 g 2 ( 𝐱 2 ) - k 3 ( z 3 - u 2 ( 𝐱 2 ) ) + u 2 𝐱 2 ( f 2 ( 𝐱 2 ) + g 2 ( 𝐱 2 ) z 3 ) u_{3}(\mathbf{x},z_{1},z_{2},z_{3})=-\frac{\partial V_{2}}{\partial\mathbf{x}_% {2}}g_{2}(\mathbf{x}_{2})-k_{3}(z_{3}-u_{2}(\mathbf{x}_{2}))+\frac{\partial u_% {2}}{\partial\mathbf{x}_{2}}(f_{2}(\mathbf{x}_{2})+g_{2}(\mathbf{x}_{2})z_{3})
  267. k 3 > 0 k_{3}>0
  268. z 1 z_{1}
  269. z 2 z_{2}
  270. z 3 z_{3}
  271. 𝐱 \mathbf{x}
  272. z 1 = 0 z_{1}=0
  273. z 2 = 0 z_{2}=0
  274. z 3 = 0 z_{3}=0
  275. 𝐱 = 𝟎 \mathbf{x}=\mathbf{0}\,
  276. u 3 u_{3}
  277. V 3 ( 𝐱 , z 1 , z 2 , z 3 ) = V 2 ( 𝐱 2 ) + 1 2 ( z 3 - u 2 ( 𝐱 2 ) ) 2 V_{3}(\mathbf{x},z_{1},z_{2},z_{3})=V_{2}(\mathbf{x}_{2})+\frac{1}{2}(z_{3}-u_% {2}(\mathbf{x}_{2}))^{2}
  278. u 3 u_{3}
  279. V 3 V_{3}
  280. { 𝐱 ˙ = f x ( 𝐱 ) + g x ( 𝐱 ) z 1 ( by Lyapunov function V x , subsystem stabilized by u x ( 𝐱 ) ) z ˙ 1 = z 2 z ˙ 2 = z 3 z ˙ i = z i + 1 z ˙ k - 2 = z k - 1 z ˙ k - 1 = z k z ˙ k = u \begin{cases}\dot{\mathbf{x}}=f_{x}(\mathbf{x})+g_{x}(\mathbf{x})z_{1}&\qquad% \,\text{ ( by Lyapunov function }V_{x},\,\text{ subsystem stabilized by }u_{x}% (\,\textbf{x})\,\text{ )}\\ \dot{z}_{1}=z_{2}\\ \dot{z}_{2}=z_{3}\\ \vdots\\ \dot{z}_{i}=z_{i+1}\\ \vdots\\ \dot{z}_{k-2}=z_{k-1}\\ \dot{z}_{k-1}=z_{k}\\ \dot{z}_{k}=u\end{cases}
  281. { { { { { { { { 𝐱 ˙ = f x ( 𝐱 ) + g x ( 𝐱 ) z 1 ( by Lyapunov function V x , subsystem stabilized by u x ( 𝐱 ) ) z ˙ 1 = z 2 z ˙ 2 = z 3 z ˙ i = z i + 1 z ˙ k - 2 = z k - 1 z ˙ k - 1 = z k z ˙ k = u \begin{cases}\begin{cases}\begin{cases}\begin{cases}\begin{cases}\begin{cases}% \begin{cases}\begin{cases}\dot{\mathbf{x}}=f_{x}(\mathbf{x})+g_{x}(\mathbf{x})% z_{1}&\qquad\,\text{ ( by Lyapunov function }V_{x},\,\text{ subsystem % stabilized by }u_{x}(\,\textbf{x})\,\text{ )}\\ \dot{z}_{1}=z_{2}\end{cases}\\ \dot{z}_{2}=z_{3}\end{cases}\\ \vdots\end{cases}\\ \dot{z}_{i}=z_{i+1}\end{cases}\\ \vdots\end{cases}\\ \dot{z}_{k-2}=z_{k-1}\end{cases}\\ \dot{z}_{k-1}=z_{k}\end{cases}\\ \dot{z}_{k}=u\end{cases}
  282. ( 𝐱 , z 1 ) (\mathbf{x},z_{1})
  283. z 2 z_{2}
  284. 𝐱 \mathbf{x}
  285. u u
  286. i i
  287. { [ 𝐱 ˙ z ˙ 1 z ˙ 2 z ˙ i - 2 z ˙ i - 1 ] 𝐱 ˙ i - 1 = [ f i - 2 ( 𝐱 i - 2 ) + g i - 2 ( 𝐱 i - 1 ) z i - 2 0 ] f i - 1 ( 𝐱 i - 1 ) + [ 𝟎 1 ] g i - 1 ( 𝐱 i - 1 ) z i ( by Lyap. func. V i - 1 , subsystem stabilized by u i - 1 ( 𝐱 i - 1 ) ) z ˙ i = u i \begin{cases}\overbrace{\begin{bmatrix}\dot{\mathbf{x}}\\ \dot{z}_{1}\\ \dot{z}_{2}\\ \vdots\\ \dot{z}_{i-2}\\ \dot{z}_{i-1}\end{bmatrix}}^{\triangleq\,\dot{\mathbf{x}}_{i-1}}=\overbrace{% \begin{bmatrix}f_{i-2}(\mathbf{x}_{i-2})+g_{i-2}(\mathbf{x}_{i-1})z_{i-2}\\ 0\end{bmatrix}}^{\triangleq\,f_{i-1}(\mathbf{x}_{i-1})}+\overbrace{\begin{% bmatrix}\mathbf{0}\\ 1\end{bmatrix}}^{\triangleq\,g_{i-1}(\mathbf{x}_{i-1})}z_{i}&\quad\,\text{ ( % by Lyap. func. }V_{i-1},\,\text{ subsystem stabilized by }u_{i-1}(\,\textbf{x}% _{i-1})\,\text{ )}\\ \dot{z}_{i}=u_{i}\end{cases}
  288. u i ( 𝐱 , z 1 , z 2 , , z i 𝐱 i ) = - V i - 1 𝐱 i - 1 g i - 1 ( 𝐱 i - 1 ) - k i ( z i - u i - 1 ( 𝐱 i - 1 ) ) + u i - 1 𝐱 i - 1 ( f i - 1 ( 𝐱 i - 1 ) + g i - 1 ( 𝐱 i - 1 ) z i ) u_{i}(\overbrace{\mathbf{x},z_{1},z_{2},\dots,z_{i}}^{\triangleq\,\mathbf{x}_{% i}})=-\frac{\partial V_{i-1}}{\partial\mathbf{x}_{i-1}}g_{i-1}(\mathbf{x}_{i-1% })\,-\,k_{i}(z_{i}\,-\,u_{i-1}(\mathbf{x}_{i-1}))\,+\,\frac{\partial u_{i-1}}{% \partial\mathbf{x}_{i-1}}(f_{i-1}(\mathbf{x}_{i-1})\,+\,g_{i-1}(\mathbf{x}_{i-% 1})z_{i})
  289. k i > 0 k_{i}>0
  290. V i ( 𝐱 i ) = V i - 1 ( 𝐱 i - 1 ) + 1 2 ( z i - u i - 1 ( 𝐱 i - 1 ) ) 2 V_{i}(\mathbf{x}_{i})=V_{i-1}(\mathbf{x}_{i-1})+\frac{1}{2}(z_{i}-u_{i-1}(% \mathbf{x}_{i-1}))^{2}
  291. u ( 𝐱 , z 1 , z 2 , , z k ) = u k ( 𝐱 k ) u(\mathbf{x},z_{1},z_{2},\ldots,z_{k})=u_{k}(\mathbf{x}_{k})
  292. i = k i=k
  293. { 𝐱 ˙ = f x ( 𝐱 ) + g x ( 𝐱 ) z 1 z ˙ 1 = f 1 ( 𝐱 , z 1 ) + g 1 ( 𝐱 , z 1 ) u 1 \begin{cases}\dot{\mathbf{x}}=f_{x}(\mathbf{x})+g_{x}(\mathbf{x})z_{1}\\ \dot{z}_{1}=f_{1}(\mathbf{x},z_{1})+g_{1}(\mathbf{x},z_{1})u_{1}\end{cases}
  294. ( 6 ) (6)\,
  295. 𝐱 = [ x 1 , x 2 , , x n ] T n \mathbf{x}=[x_{1},x_{2},\ldots,x_{n}]^{\,\text{T}}\in\mathbb{R}^{n}
  296. z 1 z_{1}
  297. u 1 u_{1}
  298. 𝐱 \mathbf{x}
  299. z 1 z_{1}
  300. g 1 ( 𝐱 , z 1 ) 0 g_{1}(\mathbf{x},z_{1})\neq 0
  301. u 1 u_{1}
  302. u a 1 u_{a1}
  303. u 1 ( 𝐱 , z 1 ) = 1 g 1 ( 𝐱 , z 1 ) ( u a 1 - f 1 ( 𝐱 , z 1 ) ) u_{1}(\mathbf{x},z_{1})=\frac{1}{g_{1}(\mathbf{x},z_{1})}\left(u_{a1}-f_{1}(% \mathbf{x},z_{1})\right)
  304. g 1 0 g_{1}\neq 0
  305. { 𝐱 ˙ = f x ( 𝐱 ) + g x ( 𝐱 ) z 1 z ˙ 1 = f 1 ( 𝐱 , z 1 ) + g 1 ( 𝐱 , z 1 ) 1 g 1 ( 𝐱 , z 1 ) ( u a 1 - f 1 ( 𝐱 , z 1 ) ) u 1 ( 𝐱 , z 1 ) \begin{cases}\dot{\mathbf{x}}=f_{x}(\mathbf{x})+g_{x}(\mathbf{x})z_{1}\\ \dot{z}_{1}=f_{1}(\mathbf{x},z_{1})+g_{1}(\mathbf{x},z_{1})\overbrace{\frac{1}% {g_{1}(\mathbf{x},z_{1})}\left(u_{a1}-f_{1}(\mathbf{x},z_{1})\right)}^{u_{1}(% \mathbf{x},z_{1})}\end{cases}
  306. { 𝐱 ˙ = f x ( 𝐱 ) + g x ( 𝐱 ) z 1 z ˙ 1 = u a 1 \begin{cases}\dot{\mathbf{x}}=f_{x}(\mathbf{x})+g_{x}(\mathbf{x})z_{1}\\ \dot{z}_{1}=u_{a1}\end{cases}
  307. u a 1 u_{a1}
  308. 𝐱 \mathbf{x}
  309. u x ( 𝐱 ) u_{x}(\mathbf{x})
  310. V x ( 𝐱 ) V_{x}(\mathbf{x})
  311. u a 1 ( 𝐱 , z 1 ) = - V x 𝐱 g x ( 𝐱 ) - k 1 ( z 1 - u x ( 𝐱 ) ) + u x 𝐱 ( f x ( 𝐱 ) + g x ( 𝐱 ) z 1 ) u_{a1}(\mathbf{x},z_{1})=-\frac{\partial V_{x}}{\partial\mathbf{x}}g_{x}(% \mathbf{x})-k_{1}(z_{1}-u_{x}(\mathbf{x}))+\frac{\partial u_{x}}{\partial% \mathbf{x}}(f_{x}(\mathbf{x})+g_{x}(\mathbf{x})z_{1})
  312. k 1 > 0 k_{1}>0
  313. u 1 ( 𝐱 , z 1 ) = 1 g 1 ( 𝐱 , z 1 ) ( - V x 𝐱 g x ( 𝐱 ) - k 1 ( z 1 - u x ( 𝐱 ) ) + u x 𝐱 ( f x ( 𝐱 ) + g x ( 𝐱 ) z 1 ) u a 1 ( 𝐱 , z 1 ) - f 1 ( 𝐱 , z 1 ) ) u_{1}(\mathbf{x},z_{1})=\frac{1}{g_{1}(\mathbf{x},z_{1})}\left(\overbrace{-% \frac{\partial V_{x}}{\partial\mathbf{x}}g_{x}(\mathbf{x})-k_{1}(z_{1}-u_{x}(% \mathbf{x}))+\frac{\partial u_{x}}{\partial\mathbf{x}}(f_{x}(\mathbf{x})+g_{x}% (\mathbf{x})z_{1})}^{u_{a1}(\mathbf{x},z_{1})}\,-\,f_{1}(\mathbf{x},z_{1})\right)
  314. ( 7 ) (7)\,
  315. k 1 > 0 k_{1}>0
  316. V 1 ( 𝐱 , z 1 ) = V x ( 𝐱 ) + 1 2 ( z 1 - u x ( 𝐱 ) ) 2 V_{1}(\mathbf{x},z_{1})=V_{x}(\mathbf{x})+\frac{1}{2}(z_{1}-u_{x}(\mathbf{x}))% ^{2}
  317. ( 8 ) (8)\,
  318. { 𝐱 ˙ = f x ( 𝐱 ) + g x ( 𝐱 ) z 1 ( by Lyapunov function V x , subsystem stabilized by u x ( 𝐱 ) ) z ˙ 1 = f 1 ( 𝐱 , z 1 ) + g 1 ( 𝐱 , z 1 ) z 2 z ˙ 2 = f 2 ( 𝐱 , z 1 , z 2 ) + g 2 ( 𝐱 , z 1 , z 2 ) z 3 z ˙ i = f i ( 𝐱 , z 1 , z 2 , , z i ) + g i ( 𝐱 , z 1 , z 2 , , z i ) z i + 1 z ˙ k - 2 = f k - 2 ( 𝐱 , z 1 , z 2 , z k - 2 ) + g k - 2 ( 𝐱 , z 1 , z 2 , , z k - 2 ) z k - 1 z ˙ k - 1 = f k - 1 ( 𝐱 , z 1 , z 2 , z k - 2 , z k - 1 ) + g k - 1 ( 𝐱 , z 1 , z 2 , , z k - 2 , z k - 1 ) z k z ˙ k = f k ( 𝐱 , z 1 , z 2 , z k - 1 , z k ) + g k ( 𝐱 , z 1 , z 2 , , z k - 1 , z k ) u \begin{cases}\dot{\mathbf{x}}=f_{x}(\mathbf{x})+g_{x}(\mathbf{x})z_{1}&\qquad% \,\text{ ( by Lyapunov function }V_{x},\,\text{ subsystem stabilized by }u_{x}% (\,\textbf{x})\,\text{ )}\\ \dot{z}_{1}=f_{1}(\mathbf{x},z_{1})+g_{1}(\mathbf{x},z_{1})z_{2}\\ \dot{z}_{2}=f_{2}(\mathbf{x},z_{1},z_{2})+g_{2}(\mathbf{x},z_{1},z_{2})z_{3}\\ \vdots\\ \dot{z}_{i}=f_{i}(\mathbf{x},z_{1},z_{2},\ldots,z_{i})+g_{i}(\mathbf{x},z_{1},% z_{2},\ldots,z_{i})z_{i+1}\\ \vdots\\ \dot{z}_{k-2}=f_{k-2}(\mathbf{x},z_{1},z_{2},\ldots z_{k-2})+g_{k-2}(\mathbf{x% },z_{1},z_{2},\ldots,z_{k-2})z_{k-1}\\ \dot{z}_{k-1}=f_{k-1}(\mathbf{x},z_{1},z_{2},\ldots z_{k-2},z_{k-1})+g_{k-1}(% \mathbf{x},z_{1},z_{2},\ldots,z_{k-2},z_{k-1})z_{k}\\ \dot{z}_{k}=f_{k}(\mathbf{x},z_{1},z_{2},\ldots z_{k-1},z_{k})+g_{k}(\mathbf{x% },z_{1},z_{2},\ldots,z_{k-1},z_{k})u\end{cases}
  319. { { { { { { { { 𝐱 ˙ = f x ( 𝐱 ) + g x ( 𝐱 ) z 1 ( by Lyapunov function V x , subsystem stabilized by u x ( 𝐱 ) ) z ˙ 1 = f 1 ( 𝐱 , z 1 ) + g 1 ( 𝐱 , z 1 ) z 2 z ˙ 2 = f 2 ( 𝐱 , z 1 , z 2 ) + g 2 ( 𝐱 , z 1 , z 2 ) z 3 z ˙ i = f i ( 𝐱 , z 1 , z 2 , , z i ) + g i ( 𝐱 , z 1 , z 2 , , z i ) z i + 1 z ˙ k - 2 = f k - 2 ( 𝐱 , z 1 , z 2 , z k - 2 ) + g k - 2 ( 𝐱 , z 1 , z 2 , , z k - 2 ) z k - 1 z ˙ k - 1 = f k - 1 ( 𝐱 , z 1 , z 2 , z k - 2 , z k - 1 ) + g k - 1 ( 𝐱 , z 1 , z 2 , , z k - 2 , z k - 1 ) z k z ˙ k = f k ( 𝐱 , z 1 , z 2 , z k - 1 , z k ) + g k ( 𝐱 , z 1 , z 2 , , z k - 1 , z k ) u \begin{cases}\begin{cases}\begin{cases}\begin{cases}\begin{cases}\begin{cases}% \begin{cases}\begin{cases}\dot{\mathbf{x}}=f_{x}(\mathbf{x})+g_{x}(\mathbf{x})% z_{1}&\qquad\,\text{ ( by Lyapunov function }V_{x},\,\text{ subsystem % stabilized by }u_{x}(\,\textbf{x})\,\text{ )}\\ \dot{z}_{1}=f_{1}(\mathbf{x},z_{1})+g_{1}(\mathbf{x},z_{1})z_{2}\end{cases}\\ \dot{z}_{2}=f_{2}(\mathbf{x},z_{1},z_{2})+g_{2}(\mathbf{x},z_{1},z_{2})z_{3}% \end{cases}\\ \vdots\\ \end{cases}\\ \dot{z}_{i}=f_{i}(\mathbf{x},z_{1},z_{2},\ldots,z_{i})+g_{i}(\mathbf{x},z_{1},% z_{2},\ldots,z_{i})z_{i+1}\end{cases}\\ \vdots\end{cases}\\ \dot{z}_{k-2}=f_{k-2}(\mathbf{x},z_{1},z_{2},\ldots z_{k-2})+g_{k-2}(\mathbf{x% },z_{1},z_{2},\ldots,z_{k-2})z_{k-1}\end{cases}\\ \dot{z}_{k-1}=f_{k-1}(\mathbf{x},z_{1},z_{2},\ldots z_{k-2},z_{k-1})+g_{k-1}(% \mathbf{x},z_{1},z_{2},\ldots,z_{k-2},z_{k-1})z_{k}\end{cases}\\ \dot{z}_{k}=f_{k}(\mathbf{x},z_{1},z_{2},\ldots z_{k-1},z_{k})+g_{k}(\mathbf{x% },z_{1},z_{2},\ldots,z_{k-1},z_{k})u\end{cases}
  320. ( 𝐱 , z 1 ) (\mathbf{x},z_{1})
  321. z 2 z_{2}
  322. 𝐱 \mathbf{x}
  323. u u
  324. i i
  325. { [ 𝐱 ˙ z ˙ 1 z ˙ 2 z ˙ i - 2 z ˙ i - 1 ] 𝐱 ˙ i - 1 = [ f i - 2 ( 𝐱 i - 2 ) + g i - 2 ( 𝐱 i - 2 ) z i - 2 f i - 1 ( 𝐱 i ) ] f i - 1 ( 𝐱 i - 1 ) + [ 𝟎 g i - 1 ( 𝐱 i ) ] g i - 1 ( 𝐱 i - 1 ) z i ( by Lyap. func. V i - 1 , subsystem stabilized by u i - 1 ( 𝐱 i - 1 ) ) z ˙ i = f i ( 𝐱 i ) + g i ( 𝐱 i ) u i \begin{cases}\overbrace{\begin{bmatrix}\dot{\mathbf{x}}\\ \dot{z}_{1}\\ \dot{z}_{2}\\ \vdots\\ \dot{z}_{i-2}\\ \dot{z}_{i-1}\end{bmatrix}}^{\triangleq\,\dot{\mathbf{x}}_{i-1}}=\overbrace{% \begin{bmatrix}f_{i-2}(\mathbf{x}_{i-2})+g_{i-2}(\mathbf{x}_{i-2})z_{i-2}\\ f_{i-1}(\mathbf{x}_{i})\end{bmatrix}}^{\triangleq\,f_{i-1}(\mathbf{x}_{i-1})}+% \overbrace{\begin{bmatrix}\mathbf{0}\\ g_{i-1}(\mathbf{x}_{i})\end{bmatrix}}^{\triangleq\,g_{i-1}(\mathbf{x}_{i-1})}z% _{i}&\quad\,\text{ ( by Lyap. func. }V_{i-1},\,\text{ subsystem stabilized by % }u_{i-1}(\,\textbf{x}_{i-1})\,\text{ )}\\ \dot{z}_{i}=f_{i}(\mathbf{x}_{i})+g_{i}(\mathbf{x}_{i})u_{i}\end{cases}
  326. u i ( 𝐱 , z 1 , z 2 , , z i 𝐱 i ) = 1 g i ( 𝐱 i ) ( - V i - 1 𝐱 i - 1 g i - 1 ( 𝐱 i - 1 ) - k i ( z i - u i - 1 ( 𝐱 i - 1 ) ) + u i - 1 𝐱 i - 1 ( f i - 1 ( 𝐱 i - 1 ) + g i - 1 ( 𝐱 i - 1 ) z i ) Single-integrator stabilizing control u a i ( 𝐱 i ) - f i ( 𝐱 i - 1 ) ) u_{i}(\overbrace{\mathbf{x},z_{1},z_{2},\dots,z_{i}}^{\triangleq\,\mathbf{x}_{% i}})=\frac{1}{g_{i}(\mathbf{x}_{i})}\left(\overbrace{-\frac{\partial V_{i-1}}{% \partial\mathbf{x}_{i-1}}g_{i-1}(\mathbf{x}_{i-1})\,-\,k_{i}\left(z_{i}\,-\,u_% {i-1}(\mathbf{x}_{i-1})\right)\,+\,\frac{\partial u_{i-1}}{\partial\mathbf{x}_% {i-1}}(f_{i-1}(\mathbf{x}_{i-1})\,+\,g_{i-1}(\mathbf{x}_{i-1})z_{i})}^{\,\text% {Single-integrator stabilizing control }u_{a\;\!i}(\mathbf{x}_{i})}\,-\,f_{i}(% \mathbf{x}_{i-1})\right)
  327. k i > 0 k_{i}>0
  328. V i ( 𝐱 i ) = V i - 1 ( 𝐱 i - 1 ) + 1 2 ( z i - u i - 1 ( 𝐱 i - 1 ) ) 2 V_{i}(\mathbf{x}_{i})=V_{i-1}(\mathbf{x}_{i-1})+\frac{1}{2}(z_{i}-u_{i-1}(% \mathbf{x}_{i-1}))^{2}
  329. u ( 𝐱 , z 1 , z 2 , , z k ) = u k ( 𝐱 k ) u(\mathbf{x},z_{1},z_{2},\ldots,z_{k})=u_{k}(\mathbf{x}_{k})
  330. i = k i=k

Backward_differentiation_formula.html

  1. y = f ( t , y ) , y ( t 0 ) = y 0 . y^{\prime}=f(t,y),\quad y(t_{0})=y_{0}.
  2. k = 0 s a k y n + k = h β f ( t n + s , y n + s ) , \sum_{k=0}^{s}a_{k}y_{n+k}=h\beta f(t_{n+s},y_{n+s}),
  3. h h
  4. t n = t 0 + n h t_{n}=t_{0}+nh
  5. a k a_{k}
  6. β \beta
  7. s s
  8. y n + 1 - y n = h f ( t n + 1 , y n + 1 ) y_{n+1}-y_{n}=hf(t_{n+1},y_{n+1})
  9. y n + 2 - 4 3 y n + 1 + 1 3 y n = 2 3 h f ( t n + 2 , y n + 2 ) ; y_{n+2}-\tfrac{4}{3}y_{n+1}+\tfrac{1}{3}y_{n}=\tfrac{2}{3}hf(t_{n+2},y_{n+2});
  10. y n + 3 - 18 11 y n + 2 + 9 11 y n + 1 - 2 11 y n = 6 11 h f ( t n + 3 , y n + 3 ) y_{n+3}-\tfrac{18}{11}y_{n+2}+\tfrac{9}{11}y_{n+1}-\tfrac{2}{11}y_{n}=\tfrac{6% }{11}hf(t_{n+3},y_{n+3})
  11. y n + 4 - 48 25 y n + 3 + 36 25 y n + 2 - 16 25 y n + 1 + 3 25 y n = 12 25 h f ( t n + 4 , y n + 4 ) y_{n+4}-\tfrac{48}{25}y_{n+3}+\tfrac{36}{25}y_{n+2}-\tfrac{16}{25}y_{n+1}+% \tfrac{3}{25}y_{n}=\tfrac{12}{25}hf(t_{n+4},y_{n+4})
  12. y n + 5 - 300 137 y n + 4 + 300 137 y n + 3 - 200 137 y n + 2 + 75 137 y n + 1 - 12 137 y n = 60 137 h f ( t n + 5 , y n + 5 ) y_{n+5}-\tfrac{300}{137}y_{n+4}+\tfrac{300}{137}y_{n+3}-\tfrac{200}{137}y_{n+2% }+\tfrac{75}{137}y_{n+1}-\tfrac{12}{137}y_{n}=\tfrac{60}{137}hf(t_{n+5},y_{n+5})
  13. y n + 6 - 360 147 y n + 5 + 450 147 y n + 4 - 400 147 y n + 3 + 225 147 y n + 2 - 72 147 y n + 1 + 10 147 y n = 60 147 h f ( t n + 6 , y n + 6 ) . y_{n+6}-\tfrac{360}{147}y_{n+5}+\tfrac{450}{147}y_{n+4}-\tfrac{400}{147}y_{n+3% }+\tfrac{225}{147}y_{n+2}-\tfrac{72}{147}y_{n+1}+\tfrac{10}{147}y_{n}=\tfrac{6% 0}{147}hf(t_{n+6},y_{n+6}).

Bacterial_adhesion_in_aquatic_system.html

  1. V E D L = π ε 0 ε r a p { 2 ψ p ψ c l n [ 1 + e x p ( - κ h ) 1 - e x p ( - κ h ) ] + ( ψ p 2 + ψ c 2 ) l n [ 1 - e x p ( - 2 κ h ) ] } V_{EDL}=\pi\varepsilon_{0}\varepsilon_{r}a_{p}\bigg\{2\psi_{p}\psi_{c}ln\bigg[% \frac{1+exp(-\kappa h)}{1-exp(-\kappa h)}\bigg]+(\psi_{p}^{2}+\psi_{c}^{2})ln% \big[1-exp(-2\kappa h)\big]\bigg\}
  2. V V d W = - A a p 6 h [ 1 + 14 h λ ] - 1 V_{VdW}=-\frac{Aa_{p}}{6h}\bigg[1+\frac{14h}{\lambda}\bigg]^{-1}
  3. V T O T = π ε 0 ε r a p { 2 ψ p ψ c l n [ 1 + e x p ( - κ h ) 1 - e x p ( - κ h ) ] + ( ψ p 2 + ψ c 2 ) l n [ 1 - e x p ( - 2 κ h ) ] } - A a p 6 h [ 1 + 14 h λ ] - 1 V_{TOT}=\pi\varepsilon_{0}\varepsilon_{r}a_{p}\bigg\{2\psi_{p}\psi_{c}ln\bigg[% \frac{1+exp(-\kappa h)}{1-exp(-\kappa h)}\bigg]+(\psi_{p}^{2}+\psi_{c}^{2})ln% \big[1-exp(-2\kappa h)\big]\bigg\}-\frac{Aa_{p}}{6h}\bigg[1+\frac{14h}{\lambda% }\bigg]^{-1}

Baer–Specker_group.html

  1. P * S P^{*}\cong S
  2. P P
  3. \mathbb{Z}
  4. S S
  5. P P
  6. S S
  7. P P

Bailey_pair.html

  1. ( a ; q ) n (a;q)_{n}
  2. ( a ; q ) n = 0 j < n ( 1 - a q j ) = ( 1 - a ) ( 1 - a q ) ( 1 - a q n - 1 ) . (a;q)_{n}=\prod_{0\leq j<n}(1-aq^{j})=(1-a)(1-aq)\cdots(1-aq^{n-1}).
  3. β n = r = 0 n α r ( q ; q ) n - r ( a q ; q ) n + r \beta_{n}=\sum_{r=0}^{n}\frac{\alpha_{r}}{(q;q)_{n-r}(aq;q)_{n+r}}
  4. α n = ( 1 - a q 2 n ) j = 0 n ( a q ; q ) n + j - 1 ( - 1 ) n - j q ( n - j 2 ) β j ( q ; q ) n - j . \alpha_{n}=(1-aq^{2n})\sum_{j=0}^{n}\frac{(aq;q)_{n+j-1}(-1)^{n-j}q^{n-j% \choose 2}\beta_{j}}{(q;q)_{n-j}}.
  5. α n = ( ρ 1 ; q ) n ( ρ 2 ; q ) n ( a q / ρ 1 ρ 2 ) n α n ( a q / ρ 1 ; q ) n ( a q / ρ 2 ; q ) n \alpha^{\prime}_{n}=\frac{(\rho_{1};q)_{n}(\rho_{2};q)_{n}(aq/\rho_{1}\rho_{2}% )^{n}\alpha_{n}}{(aq/\rho_{1};q)_{n}(aq/\rho_{2};q)_{n}}
  6. β n = j 0 ( ρ 1 ; q ) j ( ρ 2 ; q ) j ( a q / ρ 1 ρ 2 ; q ) n - j ( a q / ρ 1 ρ 2 ) j β j ( q ; q ) n - j ( a q / ρ 1 ; q ) n ( a q / ρ 2 ; q ) n . \beta^{\prime}_{n}=\sum_{j\geq 0}\frac{(\rho_{1};q)_{j}(\rho_{2};q)_{j}(aq/% \rho_{1}\rho_{2};q)_{n-j}(aq/\rho_{1}\rho_{2})^{j}\beta_{j}}{(q;q)_{n-j}(aq/% \rho_{1};q)_{n}(aq/\rho_{2};q)_{n}}.
  7. α n = q n 2 + n j = - n n ( - 1 ) j q - j 2 , β n = ( - q ) n ( q 2 ; q 2 ) n . \alpha_{n}=q^{n^{2}+n}\sum_{j=-n}^{n}(-1)^{j}q^{-j^{2}},\quad\beta_{n}=\frac{(% -q)^{n}}{(q^{2};q^{2})_{n}}.

Balayage.html

  1. f ( x ) = D f ( y ) d ν x ( y ) . f(x)=\int_{\partial D}f(y)\,d\nu_{x}(y).

Banach–Mazur_compactum.html

  1. δ ( X , Y ) = log ( inf { T T - 1 : T GL ( X , Y ) } ) . \delta(X,Y)=\log\Bigl(\inf\{\|T\|\|T^{-1}\|:T\in\operatorname{GL}(X,Y)\}\Bigr).
  2. d ( X , Y ) := e δ ( X , Y ) = inf { T T - 1 : T GL ( X , Y ) } , d(X,Y):=\mathrm{e}^{\delta(X,Y)}=\inf\{\|T\|\|T^{-1}\|:T\in\operatorname{GL}(X% ,Y)\},
  3. d ( X , 2 n ) n , d(X,\ell^{n}_{2})\leq\sqrt{n},\,

Banach–Tarski_paradox.html

  1. A A
  2. B B
  3. A A
  4. B B
  5. i i
  6. 1 1
  7. k k
  8. A A
  9. B B
  10. A A
  11. B B
  12. A A
  13. E ( n ) E(n)
  14. n = 1 , 2 n=1,2
  15. n 3 n≥3
  16. G G
  17. X X
  18. X X
  19. n n
  20. G G
  21. X X
  22. X X
  23. E ( n ) E(n)
  24. G G
  25. A A
  26. B B
  27. X X
  28. G G
  29. G G
  30. A A
  31. B B
  32. G G
  33. X X
  34. A = i = 1 k A i , B = i = 1 k B i , A i A j = , B i B j = , 1 i , j k , g i ( A i ) = B i , 1 i k , g i G , A=\bigcup_{i=1}^{k}A_{i},\quad B=\bigcup_{i=1}^{k}B_{i},\quad A_{i}\cap A_{j}=% \emptyset,\quad B_{i}\cap B_{j}=\emptyset,\quad 1\leq i,j\leq k,\qquad g_{i}(A% _{i})=B_{i},\quad 1\leq i\leq k,\quad g_{i}\in G,
  35. A A
  36. B B
  37. G G
  38. k k
  39. E E
  40. A A
  41. B B
  42. A A
  43. E E
  44. B B
  45. E E
  46. G G
  47. E E
  48. A A
  49. B B
  50. B B
  51. A A
  52. A A
  53. B B
  54. X X
  55. G G
  56. G G
  57. B , C , D B,C,D
  58. E E
  59. B , C , D B,C,D
  60. B B
  61. C C
  62. D D
  63. F 2 F_{2}
  64. F 2 = { e } S ( a ) S ( a - 1 ) S ( b ) S ( b - 1 ) F_{2}=\{e\}\cup S(a)\cup S(a^{-1})\cup S(b)\cup S(b^{-1})
  65. F 2 = a S ( a - 1 ) S ( a ) , F_{2}=aS(a^{-1})\cup S(a),\,
  66. F 2 = b S ( b - 1 ) S ( b ) , F_{2}=bS(b^{-1})\cup S(b),\,
  67. a a - 1 b aa^{-1}b
  68. a S ( a - 1 ) aS(a^{-1})
  69. a a
  70. a - 1 a^{-1}
  71. b b
  72. a - 1 a^{-1}
  73. a a - 1 a - 1 aa^{-1}a^{-1}
  74. a - 1 a^{-1}
  75. a S ( a - 1 ) aS(a^{-1})
  76. b b
  77. b - 1 b^{-1}
  78. a - 1 a^{-1}
  79. F 2 F_{2}
  80. F 2 F_{2}
  81. θ = arccos ( 1 3 ) \theta=\arccos\left(\frac{1}{3}\right)
  82. θ \theta
  83. ω \omega
  84. ω = A k 1 B k 2 B n \omega=\ldots A^{k_{1}}B^{k_{2}}\ldots B^{n}
  85. ω \omega
  86. ( 1 , 0 , 0 ) (1,0,0)
  87. ( a 3 N , b 2 3 N , c 3 N ) \left(\frac{a}{3^{N}},\frac{b\sqrt{2}}{3^{N}},\frac{c}{3^{N}}\right)
  88. a , b , c , N a,b,c\in\mathbb{Z},N\in\mathbb{N}
  89. a , b a,b
  90. c c
  91. b 0 b\neq 0
  92. ρ \rho\in
  93. ρ e \rho\neq e
  94. A 1 = S ( a ) M M B A_{1}=S(a)M\cup M\cup B
  95. A 2 = S ( a - 1 ) M B A_{2}=S(a^{-1})M\setminus B
  96. A 3 = S ( b ) M \displaystyle A_{3}=S(b)M
  97. A 4 = S ( b - 1 ) M \displaystyle A_{4}=S(b^{-1})M
  98. S ( a ) M = { s ( x ) | s S ( a ) , x M } S(a)M=\{s(x)|s\in S(a),x\in M\}
  99. B = a - 1 M a - 2 M B=a^{-1}M\cup a^{-2}M\cup\dots
  100. a A 2 = A 2 A 3 A 4 aA_{2}=A_{2}\cup A_{3}\cup A_{4}
  101. b A 4 = A 1 A 2 A 4 bA_{4}=A_{1}\cup A_{2}\cup A_{4}
  102. 2 0 2^{\aleph_{0}}